4, k\\ge 2$ , corresponds to a set $\\mathcal {Q}$ of $q^k$ affine points in $\\operatorname{PG}(2k,q)$ whose set of determined directions $D$ is an $\\mathbb {F}_2$ -linear set of pseudoregulus type.","Consequently, every line meets $D$ in $0,1,3$ or $q-1$ points.","This paper is organised as follows.","In Section we give the necessary definitions and background, in section , we provide a proof of Theorem REF .","Finally, we use this result in Section to generalise the result of Barwick-Jackson [4]."],["Linear sets","Linear sets are a central object in finite geometry and have been studied intensively, mainly due to the connection with other objects such as semifield planes, blocking sets, and more recently, MRD codes.","(see e.g.","[9], [10], [14]).","Let $V$ be an $r$ -dimensional vector space over $\\mathbb {F}_{q^n}$ , let $\\Omega $ be the projective space $\\operatorname{PG}(V) = \\operatorname{PG}(r-1, q^n)$ .","A set $T$ is said to be an $\\mathbb {F}_q$ -linear set of $\\Omega $ of rank $t$ if it is defined by the non-zero vectors of an $\\mathbb {F}_q$ -vector subspace $U$ of $V$ of dimension $t$ , i.e.","$T = L_U = \\lbrace \\langle u \\rangle _{\\mathbb {F}_{q^n}}| u \\in U \\setminus \\lbrace 0\\rbrace \\rbrace .$ The points of $\\operatorname{PG}(r-1,q^n)$ correspond to 1-dimensional subspaces of $\\mathbb {F}_{q^n}^{r}$ , and hence to $n$ -dimensional subspaces of $\\mathbb {F}_q^{rn}$ .","In this way, the point set of $\\operatorname{PG}(r-1,q^n)$ corresponds to a set $\\mathcal {D}$ of $(n-1)$ -dimensional subspaces of $\\operatorname{PG}(rn-1,q)$ , which partitions the point set of $\\operatorname{PG}(rn-1,q)$ .","The set $\\mathcal {D}$ is called a Desarguesian spread, and we have a one-to-one correspondence between the points of $\\operatorname{PG}(r-1,q^n)$ and the elements of $\\mathcal {D}$ .","Using coordinates, we see that a point $P=(x_0,x_1,\\dots x_{r-1})_{q^n} \\in \\operatorname{PG}(r-1,q^n)$ corresponds to the set $\\lbrace (\\alpha x_0,\\alpha x_1, \\dots , \\alpha x_{r-1})_q| \\alpha \\in \\mathbb {F}_{q^n} \\rbrace $ in $\\operatorname{PG}(rn-1,q)$ .","Note that we have used $r$ coordinates from $\\mathbb {F}_{q^n}$ , defined up to $\\mathbb {F}_q$ -scalar multiple to define points of $\\operatorname{PG}(rn-1,q)$ , and the set $\\lbrace (\\alpha x_0,\\alpha x_1, \\dots , \\alpha x_{r-1})_q| \\alpha \\in \\mathbb {F}_{q^n} \\rbrace $ consists of $\\frac{q^n-1}{q-1}$ different points forming an $(n-1)$ -dimensional space.","Hence, we find that $\\mathcal {D}$ is given by the set of $(n-1)$ -spaces $\\lbrace (\\alpha x_0,\\alpha x_1, \\dots , \\alpha x_{r-1})_q| \\alpha \\in \\mathbb {F}_{q^n} \\rbrace \\; \\mathrm {for\\ all\\ }(x_0,x_1,\\ldots ,x_{r-1})\\in \\mathbb {F}_{q^n}^r.$ Note that these coordinates for points in $\\operatorname{PG}(rn-1,q)$ can be transformed into the usual coordinates consisting of $rn$ elements of $\\mathbb {F}_q$ by representing the elements of $\\mathbb {F}_{q^n}$ as the $n$ coordinates with respect to a fixed basis of $\\mathbb {F}_{q^n}$ over $\\mathbb {F}_q$ .","We also have a more geometric perspective on the notion of a linear set; namely, an $\\mathbb {F}_q$ -linear set is a set $T$ of points of $\\operatorname{PG}(r-1,q^n)$ for which there exists a subspace $\\pi $ in $\\operatorname{PG}(rn-1,q)$ such that the points of $T$ correspond to the elements of $\\mathcal {D}$ that have a non-empty intersection with $\\pi $ .","For more on this approach to linear sets, we refer to [10].","If the subspace $\\pi $ intersects each spread element in at most a point, then $\\pi $ is called scattered with respect to $\\mathcal {D}$ and the associated linear set is called a scattered linear set.","Note that if $\\pi $ is $(n-1)$ -dimensional and scattered, then the associated $\\mathbb {F}_q$ -linear set has rank $n$ and has exactly $\\frac{q^n-1}{q-1}$ points, and conversely.","In this paper, we will make use of the following bound on the rank of a scattered linear set.","Result 2.1 ([5]) The rank of a scattered $\\mathbb {F}_q$ -linear set in $\\operatorname{PG}(r-1,q^n)$ is at most $ rn/2$ .","A maximum scattered linear set is a scattered $\\mathbb {F}_q$ -linear set in $\\operatorname{PG}(r-1,q^n)$ with rank $rn/2$ .","In this article we work with maximum scattered linear sets to which a geometric structure, called pseudoregulus, can be associated.","For more information, we refer to [12] and [13].","Definition 2.2 Let $S$ be a scattered $\\mathbb {F}_q$ -linear set of $\\operatorname{PG}(2k-1,q^n)$ of rank $kn$ , where $n, k\\ge 2$ .","We say that $S$ is of pseudoregulus type if there exist $m=\\frac{q^{nk}-1}{q^n-1}$ pairwise disjoint lines of $\\operatorname{PG}(2k-1, q^n)$ , say $s_1, s_2, \\dots , s_m$ , such that $| S\\cap s_i|=\\frac{q^n-1}{q-1}\\quad \\forall i=1,\\dots ,m,$ there exist exactly two $(k-1)$ -dimensional subspaces $T_1$ and $T_2$ of $\\operatorname{PG}(2k-1,q^n)$ disjoint from $S$ such that $T_j\\cap s_i \\ne \\emptyset $ for each $i = 1,\\dots , m$ and $j = 1, 2$ .","The set of lines $s_i$ , $i=1,\\dots m$ is called the pseudoregulus of $\\operatorname{PG}(2k-1,q^n)$ associated with the linear set $S$ and we refer to $T_1$ and $T_2$ as transversal spaces to this pseudoregulus.","Since a maximum scattered linear set spans the whole space, we see that the transversal spaces are disjoint.","Throughout this paper we need the following result of [13] on pseudoreguli .","Applied to $\\mathbb {F}_2$ -linear sets, this gives us the following result.","Result 2.3 ([13]) Each $\\mathbb {F}_2$ -linear set of $\\operatorname{PG}(2k-1, q)$ , $q$ even, of pseudoregulus type, is of the form $L_{\\rho ,f}$ with $L_{\\rho ,f} = \\lbrace ( u, \\rho f(u))_q| u\\in U_0 \\rbrace ,$ with $\\rho \\in \\mathbb {F}_{q}^*$ , $U_0,U_\\infty $ the $k$ -dimensional vector spaces corresponding to the transversal spaces $T_0,T_\\infty $ and with $f:U_0\\rightarrow U_\\infty $ an invertible semilinear map with companion automorphism $\\sigma \\in Aut(\\mathbb {F}_q)$ , $Fix(\\sigma )=\\lbrace 0,1\\rbrace $ .","Note that in the previous result, $\\operatorname{PG}(2k-1,q)$ is identified with $\\operatorname{PG}(V)$ , $V= U_0 \\oplus U_\\infty $ and a point, corresponding to a vector $v=v_1+v_2\\in U_0 \\oplus U_\\infty $ , has coordinates $(v_1,v_2)_q$ ."],["The Barlotti-Cofman and André/Bruck-Bose constructions","In this paper, we will switch between three different representations of a projective plane $\\operatorname{PG}(2,q^k)$ , $q=2^h$ .","Using the André/Bruck-Bose correspondence, we can, on one hand, model this plane as a subset of points and $k$ -spaces in $\\operatorname{PG}(2k,q)$ , determined by a $(k-1)$ -spread at infinity.","On the other hand, we can see it as a subset of points and $hk$ -spaces of $\\operatorname{PG}(2hk,2)$ determined by a $(hk-1)$ -spread at infinity.","We can switch between the $\\operatorname{PG}(2k,q)$ -setting an the $\\operatorname{PG}(2hk,2)$ -setting by the Barlotti-Cofman correspondence, which is a natural generalization of the André/Bruck-Bose correspondence.","The Barlotti-Cofman representation of the projective space $\\operatorname{PG}(2k,2^h)$ in $\\operatorname{PG}(2hk,2)$ is defined as follows (see [2]).","Let $\\mathcal {S}^{\\prime }$ be a Desarguesian $(h-1)$ -spread in $\\operatorname{PG}(2hk-1, 2)$ .","Embed $\\operatorname{PG}(2hk-1, 2)$ as a hyperplane $\\tilde{H_\\infty }$ in $\\operatorname{PG}(2hk, 2)$ .","Consider the following incidence structure $\\mathcal {P}(\\mathcal {S}) = (\\mathcal {P},\\mathcal {L} )$ , where incidence is natural: The set $\\mathcal {P}$ of points consists of the $2^{2hk}$ affine points $P_i$ in $\\operatorname{PG}(2hk,2)$ (i.e.","the points not in $\\tilde{H_\\infty }$ ) together with elements of the $(h-1)$ -spread $\\mathcal {S}^{\\prime }$ in $\\tilde{H_\\infty }$ The set $\\mathcal {L}$ of lines consist of the following two sets of subspaces in $\\operatorname{PG}(2hk,2)$ .","The set of $h$ -spaces spanned by an element of $\\mathcal {S}^{\\prime }$ and an affine point of $\\operatorname{PG}(2hk,2)$ .","The set of $(2h-1)$ -spaces in $\\tilde{H_\\infty }$ spanned by two different elements of $\\mathcal {S}^{\\prime }$ .","This incidence structure $(\\mathcal {P},\\mathcal {L} )$ is isomorphic to $\\operatorname{PG}(2k,2^h)$ .","We use the notation $P_i$ for the affine point of $\\operatorname{PG}(2k,2^h)$ (i.e.","a point not contained in $H_\\infty $ ) which corresponds to the affine point $\\tilde{P_i}\\in \\operatorname{PG}(2hk,2)$ .","A point, say $R_i$ in $H_\\infty $ , corresponds to the element $\\mathcal {S}^{\\prime }(R_i)$ of the $(h-1)$ -spread $\\mathcal {S}^{\\prime }$ in $\\tilde{H_\\infty }$ .","As already mentioned above, during this paper we will work in the following three projective spaces: The $2k$ -dimensional projective space $\\Pi _q=\\operatorname{PG}(2k,q)$ , $q=2^h, h>2$ , with the $(2k-1)$ -space at infinity called $H_\\infty $ .","The projective plane $\\Pi _{q^k}=\\operatorname{PG}(2,q^k)$ , $q=2^h$ with line at infinity called $\\ell _\\infty $ .","Given a Desarguesian $(k-1)$ -spread $\\mathcal {S}$ in $H_\\infty $ in $\\Pi _q$ , the plane $\\Pi _{q^k}$ is obtained by the André-Bruck-Bose construction using $\\mathcal {S}$ .","The $2hk$ -dimensional projective space $\\Pi _2=\\operatorname{PG}(2hk,2)$ , with the $(2hk-1)$ -space $\\tilde{H_\\infty }$ at infinity.","Note that the Barlotti-Cofman representation of $\\Pi _q$ defines a Desarguesian $(h-1)$ -spread $\\mathcal {S}^{\\prime }$ in $\\tilde{H_\\infty }$ .","Moreover, if $\\mathcal {S}$ is the $(k-1)$ -spread in $H_\\infty $ in $\\Pi _q$ such that $\\Pi _{q^ k}$ is the corresponding projective plane, the André-Bruck-Bose representation of $\\Pi _{q^k}$ in $\\Pi _2$ gives rise to a Desarguesian $(hk-1)$ -spread $\\tilde{\\mathcal {S}}$ in $\\tilde{H_\\infty }$ , such that $\\mathcal {S}^{\\prime }$ is a subspread of $\\tilde{\\mathcal {S}}$ ."],["The proof of the main theorem","Consider $\\Pi _q=\\operatorname{PG}(2k,q)$ and the hyperplane $H_\\infty $ of $\\operatorname{PG}(2k,q)$ .","Recall that a point of $\\operatorname{PG}(2k,q)$ is called affine if it is not contained in $H_\\infty $ .","Likewise, a line is called affine if it is not contained in $H_\\infty $ .","Let $P_1,P_2$ be affine points, then the point $P_1P_2\\cap H_\\infty $ is the direction determined by the line $P_1P_2$ .","If $\\mathcal {Q}$ is a set of affine points, then the directions determined by $\\mathcal {Q}$ are all points of $H_\\infty $ that appear as the direction of a line $P_iP_j$ for some $P_i,P_j\\in \\mathcal {Q}$ .","From now on, we consider a set $\\mathcal {Q}$ satisfying the conditions of Theorem REF : $\\mathcal {Q}$ is a set of $q^k$ affine points in $\\operatorname{PG}(2k,q)$ , $q=2^h$ , $h\\ge 4$ , $k \\ge 2$ ; $D$ , the set of directions determined by $\\mathcal {Q}$ at the hyperplane at infinity $H_\\infty $ has size $q^k-1$ ; Every line has 0, 1, 3 or $q-1$ points in common with the point set $D$ ."],["The $(q-1)$ -secants to {{formula:824f1829-06a2-4e3c-9246-2c2a3663df00}} are disjoint","Definition 3.1 A 0-point in $H_\\infty $ is a point $P\\notin D$ such that $P$ is contained in at least one $(q-1)$ -secant to $D$ .","From Proposition REF , it will follow that a 0-point is contained in precisely one $(q-1)$ -secant to $D$ .","We first start with two lemmas.","Lemma 3.2 No three points of $\\mathcal {Q}$ are collinear.","Let $l$ be an affine line in $\\operatorname{PG}(2k,q)$ containing $3\\le t\\le q$ points of $\\mathcal {Q}$ , and let $P^{\\prime }=l\\cap H_\\infty $ .","A point $P_i\\in \\mathcal {Q}\\setminus l$ determines a plane $\\alpha _i=\\langle P_i,l\\rangle $ such that the line $l_i= \\alpha _i\\cap H_\\infty $ is a $(q-1)$ -secant: the lines through $P_i$ and a point of $l\\cap \\mathcal {Q}$ determine $t\\ge 3$ directions of $D$ on the line $l_i$ , different from the point $P^{\\prime } \\in D$ .","So $l$ contains more than three points of $D$ , showing that $l_i$ is a $(q-1)$ -secant.","Furthermore, the plane $\\alpha _i$ contains at most $q$ affine points of $\\mathcal {Q}$ , as every affine line in $\\alpha $ through a 0-point of $l_i$ contains at most one element of $\\mathcal {Q}$ .","This implies that each of the $q^k-t\\ge q^k-q$ points of $\\mathcal {Q}\\setminus l$ define a plane $\\alpha $ , with $\\alpha \\cap H_\\infty $ a $(q-1)$ -secant, and so that $\\alpha $ contains at most $q-t\\le q-3$ points of $\\mathcal {Q}\\setminus l$ .","This shows that the number of such planes $\\alpha _i$ through $l$ , and hence the number of $(q-1)$ -secants through $P^{\\prime }$ , is at least $\\frac{q^k-q}{q-3}$ .","This gives that there are at least $1+\\frac{q^k-q}{q-3}(q-2)>q^k-1$ points of $D$ , a contradiction.","Lemma 3.3 Let $\\gamma $ be a plane in $\\operatorname{PG}(2k,q)$ containing 4 points $P_1,P_2,P_3$ and $P_4$ of $\\mathcal {Q}$ , such that $P_1P_2 \\cap P_3P_4\\notin \\mathcal {Q}\\cup D$ .","Then $\\gamma $ meets $H_\\infty $ in a $(q-1)$ -secant to $D$ .","By Lemma REF , no three points of $P_1, P_2,P_3,P_4$ are collinear.","Since $P_1P_2 \\cap P_3P_4\\notin D$ , we see that $P_1P_2$ and $P_3P_4$ define two different directions in $H_\\infty $ .","The four points $P_1,P_2,P_3$ and $P_4$ determine at least 4 directions on the line $\\gamma \\cap H_\\infty $ .","The statement follows since a line contains $0,1,3$ or $q-1$ points of $D$ .","Proposition 3.4 Every two $(q-1)$ -secants to $D$ are disjoint.","Consider a point $P_0\\in \\mathcal {Q}$ .","Then, by Lemma REF , all points of $D$ are defined by the lines $P_0 P_i$ with $P_i\\in \\mathcal {Q}\\setminus \\lbrace P_0\\rbrace $ .","Let $P_i^{\\prime }$ denote the direction of the line $P_0 P_i$ , that is, the point $P_0P_i\\cap H_\\infty $ .","We see that a line through a point $P_i^{\\prime }\\in D$ contains 0 or 2 points of $\\mathcal {Q}$ .","Let $l_\\alpha $ and $l_\\beta $ be two lines, both containing $q-1$ points of $D$ , with $P^{\\prime }=l_\\alpha \\cap l_\\beta $ .","Let $\\alpha =\\langle P_0, l_\\alpha \\rangle $ and $\\beta =\\langle P_0, l_\\beta \\rangle $ and let $\\lbrace P_{1 \\alpha },P_{2 \\alpha }\\rbrace $ and $\\lbrace P_{1 \\beta },P_{2 \\beta }\\rbrace $ be the 0-points in $l_\\alpha $ and $l_\\beta $ .","Note that $P^{\\prime }$ may be amongst these points.","It follows from the argument above that there are precisely $q$ points in $\\alpha \\cap \\mathcal {Q}$ and that the affine points of $\\mathcal {Q}$ in $\\alpha $ together with the two points $P_{1 \\alpha },P_{2 \\alpha }$ form a hyperoval $H_\\alpha $ .","Similarly, we find a hyperoval $H_\\beta $ in $\\beta $ .","We first suppose that $P^{\\prime }\\in D$ .","This implies that there is a point $P\\ne P_0$ of $\\mathcal {Q}$ on the line $P_0 P^{\\prime }$ .","Note that $P_0$ and $P$ are contained in $H_\\alpha \\cap H_\\beta $ .","Consider a point $R\\in l_\\alpha $ , different from $P^{\\prime }, P_{1 \\alpha },P_{2 \\alpha }$ .","Then $R\\in D$ and through $R$ , there are $\\frac{q}{2}$ bisecants to $H_\\alpha \\ne l_\\alpha $ .","One of these bisecants contains $P$ and another one contains $P_0$ .","Since $q>8$ , there exists a bisecant to $H_\\alpha $ through $R$ which intersects the line $P_0 P$ in a point $R_0 \\notin \\lbrace P_0,P,P^{\\prime }\\rbrace $ .","Through $R_0$ , there are $\\frac{q}{2}-2$ bisecants $r_i$ to $H_\\beta $ , different from the lines $R_0 P$ , $R_0P_{1\\beta }$ and $R_0P_{2\\beta }$ .","Let $r_i\\cap l_\\beta =R_i, i=1,\\dots ,\\frac{q}{2}-2$ .","A plane $\\langle R, r_i \\rangle $ contains two lines, $r_i$ and $m=R R_0$ , both containing two points of $\\mathcal {Q}$ and $r_i\\cap m=R_0\\notin \\mathcal {Q}$ .","Hence, by Lemma REF we find that every line $R R_i$ is a $(q-1)$ -secant to $D$ .","So there are $\\frac{q}{2}-2$ $(q-1)$ -secants of the form $RR_i$ , and the total number of 0-points on these lines is $2(\\frac{q}{2}-2)=q-4$ .","Let $\\Omega $ be the set of these 0-points.","We call a $(\\le 3)$ -secant in $\\langle l_\\alpha , l_\\beta \\rangle $ a line with at most 3 points of $D$ .","A line through $P^{\\prime }$ in $\\langle l_\\alpha ,l_\\beta \\rangle $ intersects all lines $RR_i$ .","The $q-4$ points of $\\Omega $ lie on the $q-1$ lines through $P^{\\prime }$ different from $l_\\alpha $ and $l_\\beta $ .","Since every line $RR_i$ contains precisely two 0-points, we find that for $q>8$ there are at most 3 $(\\le 3)$ -secants through $P^{\\prime }$ : if there are at least four $(\\le 3)$ -secants through $P^{\\prime }$ in $\\langle l_\\alpha , l_\\beta \\rangle $ , then there are at least $\\frac{q}{2}-2-2$ 0-points of $\\Omega $ on each of these lines, as we supposed that $P^{\\prime }\\in D$ .","This implies that there would be at least $4(\\frac{q}{2}-4)>q-4$ 0-points in $\\Omega $ , which gives a contradiction for $q\\ge 16$ .","Now we distinguish different cases depending on the number of $(\\le 3)$ -secants through $P^{\\prime }$ .","In each of the cases we will show that there exists at least two $(\\le 3)$ -secants $l_1,l_2$ in $\\langle l_\\alpha ,l_\\beta \\rangle $ , and a point $X\\notin D$ not on these lines.","This leads to a contradiction since there are at least $q+1-7$ lines through $X$ , both intersecting $l_1$ and $l_2$ in a point not in $D$ , and not through $l_1\\cap l_2$ .","These lines contain at least 3 points not in $D$ so they have to be $(\\le 3)$ -secants.","But this implies that there are at least $1+(q-6)(q-3)=q^2-9q+19$ points in $\\langle l_\\alpha ,l_\\beta \\rangle $ , not contained in $D$ .","On the other hand, there are at most three $(\\le 3)$ -secants through $P^{\\prime }$ and the other lines through $P^{\\prime }$ contain two 0-points.","This implies that there are at most $3q+ 2(q-2)=5q-44$ .","Let $K$ be the $(hk-1)$ -dimensional subspace in $\\operatorname{PG}(2hk-1,2)$ defining the $\\mathbb {F}_2$ -linear set $D$ and let $\\mathcal {S}^{\\prime }$ be the $(h-1)$ -spread that corresponds to the point set of $\\operatorname{PG}(2k-1,q)$ .","For every $hk$ -space $Y$ through $K$ in $\\operatorname{PG}(2hk-1,2)$ , we find at least one element of $\\mathcal {S}^{\\prime }$ that intersects $Y$ in a line since $D$ is maximum scattered.","Every line $l$ , through a point of $K$ , such that $l$ lies in an element of $\\mathcal {S}^{\\prime }$ , defines a $hk$ -space through $K$ , and the number of $hk$ -spaces through $K$ is $2^{hk}-1$ .","This implies that there are on average $2^{h-1}-1>2$ lines contained in different spread elements of $\\mathcal {S}^{\\prime }$ in a $hk$ -space through $K$ in $\\operatorname{PG}(2hk-1,2)$ .","Take a $hk$ -space $Y$ through $K$ with at least two lines contained in spread elements, and let $S_1$ and $S_2$ be two elements of $\\mathcal {S}^{\\prime }$ that intersect $Y$ in the lines $y_1$ and $y_2$ respectively.","The $(2h-1)$ -space $\\langle S_1,S_2\\rangle $ intersects $K$ in at least a plane, as $y_1$ and $y_2$ span a 3-space.","But this implies that the line $l$ in $\\operatorname{PG}(2k-1,q)$ , corresponding with $\\langle S_1,S_2\\rangle $ contains at least 7 points of $D$ .","This implies that $l$ is a $(q-1)$ -secant of $D$ , and that $\\langle S_1,S_2\\rangle $ intersects $K$ in a $(h-1)$ -space $\\alpha $ as a $(h-1)$ -space contains $2^h-1=q-1$ points.","Consider now the $h$ -space $\\beta =Y\\cap \\langle S_1,S_2 \\rangle $ through $\\alpha $ .","Since all of the $2^h+1$ $(h-1)$ -spaces of $\\mathcal {S}^{\\prime }$ in $\\langle S_1,S_2 \\rangle $ intersect $\\beta $ in a point or a line, we find that there are precisely $2^{h-1}-1$ elements of $\\mathcal {S}^{\\prime }$ , meeting $\\beta $ , and so $Y$ , in a line.","Hence, this proves that a $hk$ -space $Y$ through $K$ , containing at least 2 lines $y_1,y_2$ in $S_1,S_2$ respectively, contains at least $2^{h-1}-1$ lines $y_i$ in different spread elements of $\\mathcal {S}^{\\prime }$ .","Now we prove, by contradiction, that $Y$ cannot contain more lines $y_i$ contained in a spread element.","Suppose $Y$ contains another line $y_0 \\subset S_0$ with $S_0 \\in \\mathcal {S}^{\\prime }$ , then $y_0 \\notin \\langle S_1, S_2 \\rangle $ .","Repeating the previous argument for $y_1$ and $y_2$ shows that there are two $(2h-1)$ -spaces $\\langle S_1, S_2 \\rangle $ and $\\langle S_0, S_1 \\rangle $ , both meeting $K$ in a $(h-1)$ -space and so, there are two $(q-1)$ -secants through $P_1\\in H_\\infty $ , the point corresponding to the spread element $S_1$ .","This gives a contradiction by Proposition REF .","Since the average number of lines contained in a spreadelement in a $hk$ -space through $K$ is $2^{h-1}-1>2$ , we find that every $hk$ -space through $K$ contains exactly $2^{h-1}-1$ lines contained in a spreadelement.","In particular, every line $y_i \\subset S_i$ , with $S_i \\in \\mathcal {S}^{\\prime }$ and $y_i$ through a point of $K$ , defines a $hk$ -space though $K$ , and so a $(q-1)$ -secant.","So we find that every point in $D$ is contained in at least one $(q-1)$ -secant.","As we already proved that two $(q-1)$ -secants are disjoint (see Lemma REF ), we find $\\frac{q^k-1}{q-1}$ pairwise disjoint $(q-1)$ -secants in $\\operatorname{PG}(2k-1,q)$ .","We will first show that the linear set is of pseudoregulus type when $k=2$ .","To prove this, we begin with a lemma.","Lemma 3.10 Assume that $k=2$ .","Let $l$ be a line in $H_\\infty $ through two 0-points, not on the same $(q-1)$ -secant, then $l$ contains no points of $D$ .","Let $l_1$ and $l_2$ be two $(q-1)$ -secants in $\\operatorname{PG}(3,q)$ and let $l$ be a line through a 0-point of $l_1$ and through a 0-point of $l_2$ .","Recall that $l_1$ and $l_2$ are disjoint by Lemma REF .","Every two points $(A,B)$ , $A\\in l_1$ , $B\\in l_2$ , define a third point in $D$ on the line $AB$ .","Hence we find, since $|D|=q^2-1$ , that every point $P\\in D\\setminus \\lbrace l_1,l_2\\rbrace $ is uniquely defined as a third point on a line, defined by two points $A$ and $B$ of $D$ in $l_1$ and $l_2$ respectively.","Now suppose that $l$ contains a point $X\\in D$ , then $X$ lies on a unique line $l^{\\prime }$ , intersecting $l_1$ and $l_2$ in precisely one point.","But then $l_1$ and $l_2$ lie in a plane spanned by $l$ and $l^{\\prime }$ , a contradiction since $l_1$ and $l_2$ are disjoint by Lemma REF .","Proposition 3.11 Assume that $k=2$ .","The $(q-1)$ -secants to $D$ in $\\operatorname{PG}(3,q)$ form a pseudoregulus.","By Lemma REF it is sufficient to prove that there exist 2 lines in $\\operatorname{PG}(3,q)$ that have a point in common with all $(q-1)$ -secants to $D$ .","Consider three $(q-1)$ -secants $l_1$ , $l_2$ and $l_3$ and let $P_i,Q_i\\in l_i$ , $i=1,2,3$ be the corresponding 0-points.","Let $l_0$ be the unique line through $P_1$ that intersects $l_2$ and $l_3$ both in a point, say $R_2=l_0\\cap l_2$ and $R_3=l_0\\cap l_3$ respectively.","By Corollary $\\ref {q-1en3disjunct}$ , $R_2$ and $R_3$ cannot both belong to $\\mathcal {Q}$ , so suppose $R_2$ is a 0-point of $l_2$ (w.l.o.g.","$R_2=P_2$ ).","We see that $l_0=P_1P_2$ is a line through two 0-points, so $R_3$ is also a 0-point by Corollary REF , w.l.o.g.","$R_3=P_3$ .","By the same argument, we see that $Q_1,Q_2$ and $Q_3$ are contained in a line, say $l_\\infty $ .","Now we want to show that every other $(q-1)$ -secant has a 0-point in common with both $l_0$ and $l_\\infty $ .","Consider an $(q-1)$ -secant $l_4$ , different from $l_1,l_2,l_3$ , with 0-points $P_4$ and $Q_4$ .","Consider now again the unique line $m$ through $P_4$ that intersects $l_1$ and $l_2$ in a point.","By the previous arguments $m$ has to contain a 0-point of $l_1$ and a 0-point of $l_2$ , so $m=l_0$ , $m=l_\\infty $ , $m=P_1Q_2$ or $m=Q_1P_2$ .","We will show that only the first two possibilities can occur, which then proves that every other 0-point lies on $l_0$ or $l_\\infty $ .","Suppose to the contrary that $m=P_1Q_2P_4$ (the case $m=Q_1P_2P_4$ is completely analogous).","Then the unique line through $Q_4$ , meeting $l_1$ and $l_2$ is the line $Q_1P_2$ .","Consider now the unique line $m^{\\prime }$ through $P_4$ meeting $l_2$ and $l_3$ in a point.","As we supposed that $m\\ne l_0$ and $m\\ne l_\\infty $ , we see that $P_4$ cannot lie on these lines, so $m^{\\prime }$ contains the points $P_4,P_2,Q_3$ or the points $P_4,Q_2,P_3$ .","In the former case both lines $l_0$ and $l_\\infty $ are contained in the plane spanned by $m^{\\prime }=P_4Q_3P_2$ and $m=P_1Q_2P_4$ .","This implies that the disjoint lines $l_1$ and $l_2$ are contained in this plane, a contradiction.","If $m^{\\prime }=P_4P_3Q_2$ , then $m$ and $m^{\\prime }$ both contain $P_4$ and $Q_2$ but intersect $l_0$ in different points, a contradiction.","We conclude that $P_4$ , and analoguously $P_4^{\\prime }$ , is contained in the line $l_0$ or $l_\\infty $ .","Using the previous proposition, we will prove that for all $k$ , the $\\mathbb {F}_2$ -linear set $D$ in $\\operatorname{PG}(2k-1,q)$ is of pseudoregulus type.","Theorem 3.12 The $(q-1)$ -secants to $D$ in $\\operatorname{PG}(2k-1,q)$ form a pseudoregulus.","By Lemma REF it is sufficient to prove that there exist two $(k-1)$ -spaces in $\\operatorname{PG}(2k-1,q)$ that both have a point in common with all $(q-1)$ -secants to $D$ .","Consider a $(q-1)$ -secant $l_0$ , and let $P_0$ and $P_0^{\\prime }$ be the 0-points on $l_0$ .","Let $l_i$ be a $(q-1)$ -secant, different from $l_0$ .","The lines $l_0$ and $l_i$ span a 3-space $\\gamma $ and since $D$ is a scattered $\\mathbb {F}_2$ -linear set, $\\gamma \\cap D$ is also a scattered $\\mathbb {F}_2$ -linear set.","Since $\\gamma $ contains $2(q-1)$ points of $D$ on the lines $l_i,l_j$ and $(q-1)^2$ points of $D$ defined in a unique way as a third point on the line $A_1 A_2$ , with $A_1 \\in l_0$ , $A_2\\in l_i$ , we have that $|D\\cap \\gamma |=q^2-1$ , and hence it is a maximum scattered linear set.","By Theorem REF , we find that $\\gamma \\cap D$ is of pseudoregulus type.","This means that it has transversal lines, say $m_i$ and $m_i^{\\prime }$ , where $P_0$ lies on $m_i$ and $P_0^{\\prime }$ lies on $m_i^{\\prime }$ .","This holds for every $(q-1)$ -secant $l_i$ .","Since there are exactly $\\frac{q^k-1}{q-1}$ $(q-1)$ -secants to $D$ , which are mutually disjoint, there are exactly $2\\frac{q^k-1}{q-1}$ 0-points.","We have proven that a 0-point lies on $\\frac{q^{k-1}-1}{q-1}$ lines full of 0-points (call such lines 0-lines) and on $\\frac{q^k-1}{q-1}$ lines containing exactly 1 other 0-point.","Let $A$ and $A^{\\prime }$ be the set of all points on the lines $m_i$ and $m_i^{\\prime }$ respectively.","Then we will to show that $A\\cup A^{\\prime }$ is the union of two disjoint $(k-1)$ -spaces.","Consider a line containing two 0-points $P_1,P_2$ , with $l_1$ and $l_2$ the $(q-1)$ -secants through $P_1,P_2$ .","Then, as seen before, the intersection of the 3-space spanned by $l_1$ and $l_2$ with $D$ is a linear set of pseudoregulus type, and hence the line $P_1P_2$ contains 2 or $q+1$ 0-points.","This shows that every line in $\\operatorname{PG}(2k-1,q)$ intersects $A \\cup A^{\\prime }$ in $0,1,2$ or $q+1$ points.","This in turn implies that a plane with three 0-lines only contains 0-points.","Consider now a point $P_3$ on a 0-line through $P_0$ , and consider a 0-line $m\\ne P_0P_3$ through $P_3$ .","If $m$ contains a point $P_4 \\ne P_3$ such that $P_4P_0$ is a 0-line through $P_0$ , then we see that the plane $\\langle P_0,m \\rangle $ only contains 0-points.","In the other case, $M$ contains at least two 0-points on 0-lines through $P_0^{\\prime }$ .","In this case, all the points in the plane $\\langle P_0^{\\prime }, m \\rangle $ are 0-points, and hence the line $P_1P_0^{\\prime }$ is a 0-line, a contradiction.","So we find that every 0-line through a 0-point of $A$ is contained in $A$ .","Since every point of $A$ lies on $\\frac{q^{k-1}-1}{q-1}$ 0-lines, and $A$ contains $\\frac{q^k-1}{q-1}$ 0-points, we find that every 2 points of $A$ are contained in a 0-line of $A$ .","The same argument works for the set $A^{\\prime }$ .","This shows that $A$ forms a subspace and likewise $A^{\\prime }$ forms a subspace.","Since $|A|=|A^{\\prime }|=\\frac{q^k-1}{q-1}$ , these subspaces are $(k-1)$ -dimensional."],["There exists a suitable Desarguesian $(k-1)$ -spread {{formula:5d299a62-d518-48c3-80a5-84f442bd5791}} in {{formula:32a97683-3610-493b-8c79-c869c3ed8c0b}}","Consider the scattered linear set $D\\subset H_\\infty $ of pseudoregulus type.","Let $T_0$ and $T_\\infty $ be the transversal $(k-1)$ -spaces to the pseudoregulus defined by $D$ found in Theorem REF .","Now we want to show that there exists a Desarguesian $(k-1)$ -spread $\\mathcal {S}$ in $\\operatorname{PG}(2k-1,q)$ such that $T_0,T_\\infty \\in \\mathcal {S}$ and such that every other $(k-1)$ -space of $\\mathcal {S}$ has precisely one point in common with $D$ .","Lemma 3.13 There exists a Desarguesian $(k-1)$ -spread $\\mathcal {S}$ in $\\operatorname{PG}(2k-1,q)$ , such that $T_0,T_\\infty \\in \\mathcal {S}$ and such that every other element of $\\mathcal {S}$ has precisely one point in common with $D$ .","We prove this lemma using the representation of Result REF .","By [13] we find that the linear sets $L_{\\rho ,f}$ and $L_{\\rho ^{\\prime },g}$ are equivalent if and only if $\\sigma _f=\\sigma _g^{\\pm 1}$ , where $\\sigma _f$ and $\\sigma _g$ are the automorphisms associated with $f$ and $g$ respectively.","Hence, up to equivalence, we may suppose that $\\rho =1$ and $f:\\mathbb {F}_{q^k} \\rightarrow \\mathbb {F}_{q^k}: t\\rightarrow t^{2^i}$ , $\\gcd (i,hk)=1$ .","Considering $U_0,U_\\infty $ as $\\mathbb {F}_{q^k}$ , it follows that $D$ is equivalent to the set of points $P_u$ with $P_u:=(u,u^{2^i})_q,u\\in \\mathbb {F}_{q^k}^*.$ The transversal spaces $T_0$ and $T_\\infty $ are the point sets $T_0=\\lbrace (u,0)|u\\in \\mathbb {F}_{q^k}^*\\rbrace $ and $T_\\infty =\\lbrace (0,u)|u\\in \\mathbb {F}_{q^k}^*\\rbrace $ .","Consider now the set $\\mathcal {S}_0$ of $(k-1)$ -spaces $T_u$ , $u\\in \\mathbb {F}_{q^k}^*$ with $T_u:=\\lbrace (\\alpha u, \\alpha u^{2^i})_{q}|\\alpha \\in \\mathbb {F}_{q^k}^*\\rbrace .$ We will show that the set $\\mathcal {S}=\\mathcal {S}_0\\cup \\lbrace T_0,T_\\infty \\rbrace $ is a $(k-1)$ -spread of $\\operatorname{PG}(2k-1,q)$ .","Suppose that $P=T_{u_1}\\cap T_{u_2}$ , for some $u_1, u_2\\notin \\lbrace 0, \\infty \\rbrace $ , then there exists elements $\\alpha _1,\\alpha _2 \\in \\mathbb {F}_{q^k}^*, \\mu \\in \\mathbb {F}_{q}^*$ such that $\\left\\lbrace \\begin{array}{ll}\\alpha _1 u_1&=\\mu \\alpha _2 u_2 \\\\\\alpha _1 u_1^{2^i}&=\\mu \\alpha _2 u_2^{2^i} \\end{array}\\right.$ With $\\mu \\in \\mathbb {F}_q^*$ .","This implies that $u_1^{2^i-1}=u_2^{2^i-1}$ or $\\left(\\frac{u_1}{u_2}\\right)^{2^i}=\\frac{u_1}{u_2}$ .","Hence $\\frac{u_1}{u_2}\\in \\mathbb {F}_{2^i}\\cap \\mathbb {F}_{2^{hk}}$ which is $\\mathbb {F}_2$ since $\\gcd (i,hk)=1$ .","Since $u_1,u_2\\in \\mathbb {F}_{q^k}^*$ , this implies that $u_1=u_2$ , and that $T_{u_1}=T_{u_2}$ .","In particular, we see that $T_u \\ne T_{u^{\\prime }}$ for $u\\ne u^{\\prime } \\in \\mathbb {F}_{q^k}^*$ .","Since $T_0$ and $T_\\infty $ are distinct from $T_u$ for all $u\\in \\mathbb {F}_{q^k}^*$ , we obtain that $|\\mathcal {S}|=q^k+1$ .","We will now show that $T_u\\cap T_0=\\emptyset $ for all $u\\in \\mathbb {F}_{q^k}^*$ .","If $P=T_u\\cap T_0$ , $u\\notin \\lbrace 0, \\infty \\rbrace $ for some $u\\in \\mathbb {F}_{q^k}^*$ then $P=(u^{\\prime },0)_q$ with $u^{\\prime }\\in \\mathbb {F}_{q^k}^*$ and $\\left\\lbrace \\begin{array}{ll}\\alpha u&=\\mu u^{\\prime } \\\\\\alpha u^{2^i}&=0 \\end{array}\\right.$ for some $\\mu \\in \\mathbb {F}_q^*$ and $\\alpha \\in \\mathbb {F}_{q^k}^*$ .","The second equality gives a contradiction since $u\\ne 0 \\ne \\alpha $ .","Hence $T_u \\cap T_0 = \\emptyset $ .","It follows from a similar argument that $T_u \\cap T_\\infty =\\emptyset $ .","This shows that $\\mathcal {S}$ is a spread which is Desarguesian as seen in Subsection REF .","Remark 3.14 In [13] a geometric construction of the Desarguesian spread, found in Lemma REF , using indicator sets, is given."],["The point set $\\mathcal {Q}$ defines a translation hyperoval in the André/Bruck-Bose plane {{formula:fe441269-42d1-4751-97d9-6854ca55cc19}}","The spread $\\mathcal {S}$ found in Lemma REF defines a projective plane $\\mathcal {P}(\\mathcal {S})=\\Pi _{q^k}\\cong \\operatorname{PG}(2,q^k)$ by the André/Bruck-Bose construction.","The transversal $(k-1)$ -spaces $T_0, T_\\infty \\in \\mathcal {S}$ to the pseudoregulus associated with $D$ correspond to points $P_0, P_\\infty $ contained in the line $\\ell _\\infty $ at infinity of $\\operatorname{PG}(2,q^k)$ .","Theorem 3.15 The set $\\mathcal {Q}$ , together with $T_0$ and $T_\\infty $ , defines a translation hyperoval in $\\Pi _{q^k}\\cong \\operatorname{PG}(2,q^k)$ .","Let $\\mathcal {A}$ be the set of points in $\\Pi _{q^k}$ corresponding to the point set $\\mathcal {Q}$ of $\\Pi _q$ .","Recall that $T_0$ corresponds to a point $P_0$ and $T_\\infty $ to a point $P_\\infty $ , contained in the line $\\ell _\\infty $ of $\\Pi _{q^k}$ .","We first show that every line in $\\operatorname{PG}(2,q^k)$ contains at most 2 points of the set $\\mathcal {H}=\\mathcal {A}\\cup P_0\\cup P_\\infty $ .","The line $\\ell _\\infty $ at infinity only contains the points $P_0$ and $P_\\infty $ .","Consider a line $l\\ne \\ell _\\infty $ through $P_0$ in $\\operatorname{PG}(2,q^k)$ .","This line corresponds to a $k$ -space through $T_0$ in $\\operatorname{PG}(2k,q)$ .","As $P_0\\in l\\cap \\mathcal {H}$ , we have to show that this $k$ -space contains at most one affine point of $\\mathcal {Q}$ .","If this space would contain 2 (or more) affine points $X_1,X_2\\in \\mathcal {Q}$ , then they would define a direction of $D$ at infinity in $T_0$ .","But this is impossible as $T_0$ has no points of $D$ , see Corollary REF .","This argument also works for the lines through $P_\\infty $ , different from $\\ell _\\infty $ .","Consider a line $l$ through a point $P_i$ , $i\\notin \\lbrace 0,\\infty \\rbrace $ at infinity.","This point $P_i$ corresponds to an element $T_i\\in \\mathcal {S}$ that intersects the pseudoregulus $D$ in a unique point $X_i$ .","The line $l$ corresponds to a $k$ -space $\\gamma $ in $\\operatorname{PG}(2k,q)$ through $T_i$ .","Suppose that $\\gamma $ contains at least 3 points from $\\mathcal {Q}$ , say $X,Y,Z$ .","By Lemma REF these points are not collinear, hence they determine at least two different points of $D$ which are contained in $T_i$ , a contradiction.","This proves that $\\gamma $ contains at most two points of $\\mathcal {Q}$ , which implies that the line $l$ contains at most two points of $\\mathcal {A}$ .","Since $\\mathcal {H}$ has size $q^k+2$ , it follows that $\\mathcal {H}$ is a hyperoval.","Finally consider the group $G$ of elations in $\\operatorname{PG}(2hk,2)$ with axis the hyperplane at infinity $\\tilde{H}_\\infty $ .","Since the points of $\\tilde{{Q}}$ form a subspace, we see that $G$ acts transitively on the points of $\\tilde{\\mathcal {Q}}$ .","Every element of $G$ induces an element of the group $G^{\\prime }$ of elations in $\\operatorname{PG}(2,q^k)$ with axis the line $P_0P_\\infty $ .","Hence, $G^{\\prime }$ acts transitively on the points of $\\mathcal {A}$ in $\\operatorname{PG}(2,q^k)$ .","This shows that $\\mathcal {H}$ is a translation hyperoval."],["Every translation hyperoval defines a linear set of pseudoregulus type","In this section, we show that the vice versa part of Theorem REF holds.","Proposition 3.16 Via the André/Bruck-Bose construction, the set of affine points of a translation hyperoval in $\\operatorname{PG}(2,q^k)$ , $q=2^h$ , where $h,k\\ge 2$ corresponds to a set $\\mathcal {Q}$ of $q^k$ affine points in $\\operatorname{PG}(2k,q)$ whose set of determined directions $D$ is an $\\mathbb {F}_2$ -linear set of pseudoregulus type.","Consider a translation hyperoval $H\\in \\operatorname{PG}(2,q^k)$ .","Without loss of generality we may suppose that $H=\\lbrace (1,t,t^{2^i})_{q^k} | t\\in \\mathbb {F}_{q^k}\\rbrace \\cup \\lbrace (0,1,0)_{q^k},(0,0,1)_{q^k}\\rbrace $ with $\\gcd (i,hk)=1$ .","The set of affine points of $H$ corresponds to the set of points $H^{\\prime }=\\lbrace (1,t,t^{2^i})_{q} \\in \\mathbb {F}_q \\oplus \\mathbb {F}_{q^k} \\oplus \\mathbb {F}_{q^k} | t\\in \\mathbb {F}_{q^k}\\rbrace $ in $\\operatorname{PG}(2k,q)$ (for more information about the use of these coordinates for $H$ and $H^{\\prime }$ , see [15]).","The determined directions in the hyperplane at infinity $H_\\infty : X_0=0$ have coordinates $(0,t_1-t_2,t_1^{2^i}-t_2^{2^i})_q$ where $t_1,t_2 \\in \\mathbb {F}_{q^k}$ .","So the set $D=\\lbrace (0,u,u^{2^i} ) _q| u\\in \\mathbb {F}_{q^k} \\rbrace $ is precisely the set of directions determined by the points of $H$ .","By Result REF we find that this set of directions $D$ is an $\\mathbb {F}_2$ -linear set of pseudoregulus type in the hyperplane $H_\\infty $ .","We will now show that every line in $\\operatorname{PG}(2k-1,q)$ intersects the points of the linear set $D$ in $0,1,3$ or $q-1$ points.","Proposition 3.17 Let $D$ be the set of points of an $\\mathbb {F}_2$ -linear set of pseudoregulus type in $\\operatorname{PG}(2k-1,q)$ , $q=2^h$ , $h>2$ , $k\\ge 2$ .","Then every line of $\\operatorname{PG}(2k-1,q)$ meets $D$ in $0,1,3$ or $q-1$ points.","We use the representation of Result REF for the points of $D$ .","Let $R_1=(u_1,f(u_1))_q$ and $R_2=(u_2,f(u_2))_q$ , $u_1,u_2 \\in U_0$ , be two points of $D$ not on the same line of the pseudoregulus, so the vectors $\\langle u_1 \\rangle $ and $\\langle u_2 \\rangle $ in $V(k,q)$ are not an $\\mathbb {F}_q$ -multiple (in short $\\langle u_1 \\rangle _q \\ne \\langle u_2 \\rangle _q$ ).","Recall that $f$ is a invertible semilinear map with automorphism $\\sigma \\in Aut(\\mathbb {F}_q)$ , $Fix(\\sigma )=\\lbrace 0,1\\rbrace $ .","A third point $R_3=(u_3,f(u_3))_q\\in D$ is contained in $R_1R_2$ if and only if there are $\\mu ,\\lambda \\in \\mathbb {F}_q$ such that $\\left\\lbrace \\begin{array}{ll}u_1+\\lambda u_2&=\\mu u_3 \\\\f(u_1)+\\lambda f(u_2) &= \\mu f(u_3)\\end{array}\\right.\\Leftrightarrow \\left\\lbrace \\begin{array}{ll}f(u_1)+\\lambda ^\\sigma f(u_2)&=\\mu ^\\sigma f(u_3) \\\\f(u_1)+\\lambda f(u_2) &= \\mu f(u_3)\\end{array} \\right.\\\\\\Leftrightarrow \\left\\lbrace \\begin{array}{ll}u_1+\\lambda u_2&=\\mu u_3 \\\\(\\lambda ^\\sigma - \\lambda ) f(u_2) &= f((\\mu -\\mu ^{\\sigma ^{-1}})u_3)\\end{array} \\right.\\Leftrightarrow \\left\\lbrace \\begin{array}{ll}u_1+\\lambda u_2&=\\mu u_3 \\\\(\\lambda ^\\sigma - \\lambda )^{\\sigma ^{-1}} u_2 &= (\\mu -\\mu ^{\\sigma ^{-1}})u_3\\end{array} \\right.$ As $R_2$ and $R_3$ lie on different $(q-1)$ -secants to $D$ , we have that $\\langle u_2 \\rangle _q \\ne \\langle u_3 \\rangle _q$ .","It follows that $\\lambda ^\\sigma - \\lambda =\\mu -\\mu ^{\\sigma ^{-1}}=0$ , so $\\lambda ,\\mu \\in Fix(\\sigma )=\\lbrace 0,1\\rbrace $ .","We find that there is only one solution of this system, such that $R_1\\ne R_3$ (i.e.","$\\langle u_1 \\rangle _q \\ne \\langle u_3 \\rangle _q$ ), namely when $\\lambda =\\mu =1$ .","Hence, given two points $R_1,R_2$ in $D$ , there is an unique point $R_3\\in D\\cap R_1R_2$ , different from $R_1$ and $R_2$ ."],["The generalisation of a characterisation of Barwick and Jackson","Using Theorem REF , we are now able to generalise the following result of Barwick-Jackson which concerns translation hyperovals in $\\operatorname{PG}(2,q^2)$ ([4]).","Result 4.1 [4] Consider $\\operatorname{PG}(4,q)$ , $q$ even, $q>2$ , with the hyperplane at infinity denoted by $\\Sigma _\\infty $ .","Let $\\mathcal {C}$ be a set of $q^2$ affine points, called $\\mathcal {C}$ -points and consider a set of planes called $\\mathcal {C}$ -planes which satisfies the following: (A1) Each $\\mathcal {C}$ -plane meets $\\mathcal {C}$ in a $q$ -arc.","(A2) Any two distinct $\\mathcal {C}$ -points lie in a unique $\\mathcal {C}$ -plane.","(A3) The affine points that are not in $\\mathcal {C}$ lie on exactly one $\\mathcal {C}$ -plane.","(A4) Every plane which meets $\\mathcal {C}$ in at least 3 points either meets $\\mathcal {C}$ in 4 points or is a $\\mathcal {C}$ -plane.","Then there exists a Desarguesian spread $\\mathcal {S}$ in $\\Sigma _\\infty $ such that in the Bruck-Bose plane $\\mathcal {P}(\\mathcal {S})\\cong \\operatorname{PG}(2,q^2)$ , the $\\mathcal {C}$ -points, together with 2 extra points on $\\ell _\\infty $ form a translation hyperoval in $\\operatorname{PG}(2,q^2)$ .","Remark 4.2 At two different points, the proofs of [4] are inherently linked to the fact that they are dealing with hyperovals in $\\operatorname{PG}(2,q^2)$ .","In [4] the authors show the existence of a design which is isomorphic to an affine plane, of which they later need to use the parallel classes.","In [4], they use the Klein correspondence to represent lines in $\\operatorname{PG}(3,q)$ in $\\operatorname{PG}(5,q)$ .","Both techniques cannot be extended in a straightforward way to $q^k$ , $k>2$ .","The following Proposition shows that a set of $\\mathcal {C}$ -planes as defined by Barwick and Jackson in [4] (using $\\operatorname{PG}(2k,q)$ instead of $\\operatorname{PG}(4,q)$ ) satisfies the conditions of Theorem REF .","Proposition 4.3 Consider $\\operatorname{PG}(2k,q)$ , $q$ even, $q>2$ , with the hyperplane at infinity denoted by $\\Sigma _\\infty $ .","Let $\\mathcal {C}$ be a set of $q^k$ affine points, called $\\mathcal {C}$ -points and consider a set of planes called $\\mathcal {C}$ -planes which satisfies the following: (A1) Each $\\mathcal {C}$ -plane meets $\\mathcal {C}$ in a $q$ -arc.","(A2) Any two distinct $\\mathcal {C}$ -points lie in a unique $\\mathcal {C}$ -plane.","(A3) The affine points that are not in $\\mathcal {C}$ lie on exactly one $\\mathcal {C}$ -plane.","(A4) Every plane which meets $\\mathcal {C}$ in at least 3 points either meets $\\mathcal {C}$ in 4 points or is a $\\mathcal {C}$ -plane.","Then $\\mathcal {C}$ determines a set of $q^k-1$ directions $D$ in $\\Sigma _\\infty $ such that every line of $\\Sigma _\\infty $ meets $D$ in $0,1,3$ or $q-1$ points.","As before, we call the points that are not contained in $\\Sigma _\\infty $ affine points.","Note that all $\\mathcal {C}$ -points are affine.","Since every two $\\mathcal {C}$ -points lie on a $\\mathcal {C}$ -plane which meets $\\mathcal {C}$ in a $q$ -arc, we have that no three $\\mathcal {C}$ -points are collinear.","Let $P_0$ be a $\\mathcal {C}$ -point and let $D_0$ be the set of points of the form $P_0P_i\\cap \\Sigma _\\infty $ , where $P_i\\ne P_0$ is a point of $\\mathcal {C}$ .","We first show that every line meets $D_0$ in $0,1,3$ or $q-1$ points.","Let $M$ be a line of $\\Sigma _\\infty $ containing 2 points of $D_0$ , say $R_1^{\\prime }=P_0R_1\\cap \\Sigma _\\infty $ , $R_2^{\\prime }=P_0R_2\\cap \\Sigma _\\infty $ , where $R_1,R_2\\in \\mathcal {C}$ .","Then $\\langle M,P_0\\rangle $ contains at least 3 points of $\\mathcal {C}$ , and hence, by (A4), either it is a $\\mathcal {C}$ -plane or it contains exactly 4 points of $\\mathcal {C}$ .","If $\\langle M,P_0\\rangle $ is a $\\mathcal {C}$ -plane, it contains $q$ points of $\\mathcal {C}$ forming a $q$ -arc, and hence, $M$ contains $q-1$ points of $D_0$ .","Now suppose that $\\langle M,P_0\\rangle $ contains exactly 4 $\\mathcal {C}$ -points, then $M$ contains 3 points of $D_0$ .","Now let $P_1\\ne P_0$ be a point of $\\mathcal {C}$ and let $D_1$ be the set of points of the form $P_1P_i\\cap \\Sigma _\\infty $ , where $P_i\\ne P_1$ is a point of $\\mathcal {C}$ .","We claim that $D_0=D_1$ .","Let $P_1^{\\prime }=P_0P_1\\cap \\Sigma _\\infty $ .","We see that $P_1^{\\prime }\\in D_0\\cap D_1$ .","Consider a point $P_2^{\\prime }\\ne P_1^{\\prime }$ in $D_0$ , then $P_0P_2\\cap \\Sigma _\\infty =P_2^{\\prime }$ for some $P_2\\in \\mathcal {C}$ .","Consider the plane $\\pi =\\langle P_0,P_1,P_2\\rangle $ .","Suppose first that $\\pi $ is not a $\\mathcal {C}$ -plane, then, by (A4), $\\pi $ contains exactly one extra point, say $P_3$ of $\\mathcal {C}$ .","The lines $P_0P_1$ and $P_2P_3$ lie in $\\pi $ and hence, meet in a point $Q$ .","By $(A2)$ , there is a $\\mathcal {C}$ -plane $\\mu $ through $P_0P_1$ , and likewise, there is a $\\mathcal {C}$ -plane $\\mu ^{\\prime }$ through $P_2P_3$ .","Since $\\pi $ is not a $\\mathcal {C}$ -plane, $\\mu $ and $\\mu ^{\\prime }$ are two distinct $\\mathcal {C}$ -planes through $Q$ .","By (A3) his implies that $Q$ is a point of $\\Sigma _\\infty $ .","Likewise, $P_0P_2\\cap P_1P_3$ and $P_0P_3\\cap P_1P_2$ are points of $\\Sigma _\\infty $ .","It follows that $D_0\\cap \\pi =D_1\\cap \\pi $ .","This argument shows that for all points $R\\ne P_1^{\\prime }\\in D_0$ such that $\\langle P_0,P_1,R\\rangle $ is not a $\\mathcal {C}$ -plane, we have that $R\\in D_1$ .","Now $P_0P_1$ lies on a unique $\\mathcal {C}$ -plane, say $\\nu $ .","Let $\\nu \\cap \\Sigma _\\infty =L$ , then we have shown that $\\langle P_0,P_1,R\\rangle $ is not a $\\mathcal {C}$ -plane as long as $R\\in \\Sigma _\\infty $ is not on $L$ .","We conclude that $D_0\\setminus L=D_1\\setminus L$ .","Now assume that $D_0\\ne D_1$ and let $X$ be a point in $D_1$ which is not contained in $D_0$ .","Then $X\\in L$ and $P_1X$ contains a point $Y\\ne P_1\\in \\mathcal {C}$ .","Consider a point $P_4^{\\prime }\\in D_1$ , not on $L$ , then $P_1P_4^{\\prime }$ contains a point $P_4\\ne P_1$ of $\\mathcal {C}$ .","Since $P_4^{\\prime }\\in D_1\\setminus L$ , $P_4^{\\prime }\\in D_0$ so the line $P_4^{\\prime }P_0$ contains a point $P_5\\ne P_1$ of $\\mathcal {C}$ .","The plane $\\langle P_1,P_4^{\\prime },X\\rangle $ is not a $\\mathcal {C}$ -plane since otherwise, the points $P_1$ and $Y$ of $\\mathcal {C}$ would lie in two different $\\mathcal {C}$ -planes.","This implies that $\\langle P_1,P_4,X\\rangle $ which contains the $\\mathcal {C}$ -points $P_1,P_4,Y$ contains exactly one extra point of $\\mathcal {C}$ , say $P_6$ .","Denote $P_1P_6\\cap \\Sigma _\\infty $ by $P^{\\prime }_6$ .","We see that there are exactly 3 points of $D_1$ on the line $P_4^{\\prime }X$ , namely $P_4^{\\prime },X$ and $P_6^{\\prime }$ .","Now $P^{\\prime }_6$ is a point of $D_1$ , not on $L$ , so $P^{\\prime }_6\\in D_0$ .","Hence, there is a point $S\\ne P_0\\in \\mathcal {C}$ on the line $P_0P^{\\prime }_6$ .","If $\\langle P_4^{\\prime },P^{\\prime }_6,P_0\\rangle $ is not a $\\mathcal {C}$ -plane, then, since it contains $P_0,P_5,S$ of $\\mathcal {C}$ it contains precisely 3 points of $D_0$ at infinity.","These are the points $P_4^{\\prime },P_6^{\\prime }$ and one other point, say $T$ , which needs to be different from $X$ by our assumption that $X\\notin D_0$ .","That implies that $T$ is not on $L$ , and hence, $T\\in D_1$ .","This is a contradiction since we have seen that the only points of $D_1$ on $P_4^{\\prime }X$ are $P_4^{\\prime },X$ and $P_6^{\\prime }$ .","Now if $\\langle P_4^{\\prime },P_6,P_0\\rangle $ is a $\\mathcal {C}$ -plane, we find $q-1$ points of $D_0$ on $P_4^{\\prime }X$ , all of them are not on $L$ .","Hence, we find $q-1$ points of $D_1$ on $P_4^{\\prime }X$ , not on $L$ .","This is again a contradiction since $P_4^{\\prime }X$ has only the points $P_4^{\\prime }$ and $P_6^{\\prime }$ of $D_1$ not on $L$ .","This proves our claim that $D_0=D_1$ .","Since $P_1$ was chosen arbitrarily, different from $P_0$ , and $D_0=D_1$ , we find that the set $D$ of directions determined by $\\mathcal {C}$ is precisely the set $D_0$ .","The statement now follows from the fact that a line meets $D_0$ in $0,1,3$ or $q-1$ points.","Proposition REF shows that the set $\\mathcal {C}$ satisfies the criteria of Theorem REF .","Hence, we find the following generalisation of Result REF .","Theorem 4.4 Consider $\\operatorname{PG}(2k,q)$ , $q$ even, $q>2$ , with the hyperplane at infinity denoted by $\\Sigma _\\infty $ .","Let $\\mathcal {C}$ be a set of $q^k$ affine points, called $\\mathcal {C}$ -points and consider a set of planes called $\\mathcal {C}$ -planes which satisfies the following: (A1) Each $\\mathcal {C}$ -plane meets $\\mathcal {C}$ in a $q$ -arc.","(A2) Any two distinct $\\mathcal {C}$ -points lie in a unique $\\mathcal {C}$ -plane.","(A3) The affine points that are not in $\\mathcal {C}$ lie on exactly one $\\mathcal {C}$ -plane.","(A4) Every plane which meets $\\mathcal {C}$ in at least 3 points either meets $\\mathcal {C}$ in 4 points or is a $\\mathcal {C}$ -plane.","Then there exists a Desarguesian spread $\\mathcal {S}$ in $\\Sigma _\\infty $ such that in the Bruck-Bose plane $\\mathcal {P}(\\mathcal {S})\\cong \\operatorname{PG}(2,q^k)$ , the $\\mathcal {C}$ -points, together with 2 extra points on $\\ell _\\infty $ form a translation hyperoval in $\\operatorname{PG}(2,q^k)$ ."]],"string":"[\n [\n \"Translation hyperovals and $\\\\mathbb{F}_2$-linear sets of pseudoregulus\\n type\"\n ],\n [\n \"Abstract In this paper, we study translation hyperovals in PG$(2,q^k)$.\",\n \"The main result of this paper characterises the point sets defined by translation hyperovals in the Andr\\\\'e/Bruck-Bose representation.\",\n \"We show that the affine point sets of translation hyperovals in PG$(2,q^k)$ are precisely those that have a scattered $\\\\mathbb{F}_2$-linear set of pseudoregulus type in PG$(2k-1,q)$ as set of directions.\",\n \"This correspondence is used to generalise the results of Barwick and Jackson who provided a characterisation for translation hyperovals in PG$(2,q^2)$.\"\n ],\n [\n \"Introduction\",\n \"Let $\\\\operatorname{PG}(n,q)$ denote the $n$ -dimensional projective space over the finite field $\\\\mathbb {F}_q$ with $q$ elements.\",\n \"A $k$ -arc in $\\\\operatorname{PG}(2,q)$ is a set of $k$ points such that no three of them are collinear.\",\n \"A hyperoval in $\\\\operatorname{PG}(2,q)$ is a $(q+2)$ -arc.\",\n \"Hyperovals only exist when $q$ is even.\",\n \"A translation hyperoval is a hyperoval $H$ such that there exists a bisecant $\\\\ell $ of $H$ such that the group of elations with axis $\\\\ell $ acts transitively on the points of $H$ not on $\\\\ell $ .\",\n \"It is well-known (see e.g.\",\n \"[7]) that every translation hyperoval in $\\\\operatorname{PG}(2,q)$ is $\\\\mathrm {PGL}$ -equivalent to a point set $\\\\lbrace (1,t,t^{2^i})|t\\\\in \\\\mathbb {F}_{q}\\\\rbrace \\\\cup \\\\lbrace (0,1,0),(0,0,1)\\\\rbrace ,$ where $q=2^h$ and $\\\\gcd (i,h)=1$ .\",\n \"In [4], Barwick and Jackson provided a chararacterisation of translation hyperovals in $\\\\operatorname{PG}(2,q^2)$ : they considered a set $\\\\mathcal {C}$ of points in $\\\\operatorname{PG}(4,q)$ , $q$ even, with certain combinatorial properties with respect to the planes of $\\\\operatorname{PG}(4,q)$ (see Section for details).\",\n \"They proved that the set $\\\\mathcal {C}^{\\\\prime }$ of directions determined by the points of $\\\\mathcal {C}$ has the property that every line intersects $\\\\mathcal {C}^{\\\\prime }$ in $0,1,3$ or $q-1$ points.\",\n \"They then used this to construct a Desarguesian line spread $\\\\mathcal {S}$ in $\\\\operatorname{PG}(3,q)$ , such that in the corresponding André/Bruck-Bose plane $\\\\mathcal {P}(\\\\mathcal {S})\\\\cong \\\\operatorname{PG}(2,q^2)$ , the points corresponding to $\\\\mathcal {C}$ form a translation hyperoval.\",\n \"This extended the work done in [3], where the same authors gave a similar characterisation of André/Bruck-Bose representation of conics for $q$ odd.\",\n \"In this paper, we will generalise the combinatorial characterisation provided by Barwick and Jackson for translation hyperovals.\",\n \"In order to do this, we prove our main theorem, linking translation hyperovals with $\\\\mathbb {F}_2$ -linear sets of pseudoregulus type: Theorem 1.1 Let $\\\\mathcal {Q}$ be a set of $q^k$ affine points in $\\\\operatorname{PG}(2k,q)$ , $q=2^h$ , $h\\\\ge 4$ , $k\\\\ge 2$ , determining a set $D$ of $q^k-1$ directions in the hyperplane at infinity $H_\\\\infty =\\\\operatorname{PG}(2k-1,q)$ .\",\n \"Suppose that every line has 0, 1, 3 or $q-1$ points in common with the point set $D$ .\",\n \"Then (1) $D$ is an $\\\\mathbb {F}_2$ -linear set of pseudoregulus type.\",\n \"(2) There exists a Desarguesian spread $\\\\mathcal {S}$ in $H_\\\\infty $ such that, in the Bruck-Bose plane $\\\\mathcal {P}(\\\\mathcal {S})\\\\cong \\\\operatorname{PG}(2,q^k)$ , with $H_\\\\infty $ corresponding to the line $l_\\\\infty $ , the points of $\\\\mathcal {Q}$ together with 2 extra points on $\\\\ell _\\\\infty $ , form a translation hyperoval in $\\\\operatorname{PG}(2,q^k)$ .\",\n \"Vice versa, via the André/Bruck-Bose construction, the set of affine points of a translation hyperoval in $\\\\operatorname{PG}(2,q^k)$ , $q> 4, k\\\\ge 2$ , corresponds to a set $\\\\mathcal {Q}$ of $q^k$ affine points in $\\\\operatorname{PG}(2k,q)$ whose set of determined directions $D$ is an $\\\\mathbb {F}_2$ -linear set of pseudoregulus type.\",\n \"Consequently, every line meets $D$ in $0,1,3$ or $q-1$ points.\",\n \"This paper is organised as follows.\",\n \"In Section we give the necessary definitions and background, in section , we provide a proof of Theorem REF .\",\n \"Finally, we use this result in Section to generalise the result of Barwick-Jackson [4].\"\n ],\n [\n \"Linear sets\",\n \"Linear sets are a central object in finite geometry and have been studied intensively, mainly due to the connection with other objects such as semifield planes, blocking sets, and more recently, MRD codes.\",\n \"(see e.g.\",\n \"[9], [10], [14]).\",\n \"Let $V$ be an $r$ -dimensional vector space over $\\\\mathbb {F}_{q^n}$ , let $\\\\Omega $ be the projective space $\\\\operatorname{PG}(V) = \\\\operatorname{PG}(r-1, q^n)$ .\",\n \"A set $T$ is said to be an $\\\\mathbb {F}_q$ -linear set of $\\\\Omega $ of rank $t$ if it is defined by the non-zero vectors of an $\\\\mathbb {F}_q$ -vector subspace $U$ of $V$ of dimension $t$ , i.e.\",\n \"$T = L_U = \\\\lbrace \\\\langle u \\\\rangle _{\\\\mathbb {F}_{q^n}}| u \\\\in U \\\\setminus \\\\lbrace 0\\\\rbrace \\\\rbrace .$ The points of $\\\\operatorname{PG}(r-1,q^n)$ correspond to 1-dimensional subspaces of $\\\\mathbb {F}_{q^n}^{r}$ , and hence to $n$ -dimensional subspaces of $\\\\mathbb {F}_q^{rn}$ .\",\n \"In this way, the point set of $\\\\operatorname{PG}(r-1,q^n)$ corresponds to a set $\\\\mathcal {D}$ of $(n-1)$ -dimensional subspaces of $\\\\operatorname{PG}(rn-1,q)$ , which partitions the point set of $\\\\operatorname{PG}(rn-1,q)$ .\",\n \"The set $\\\\mathcal {D}$ is called a Desarguesian spread, and we have a one-to-one correspondence between the points of $\\\\operatorname{PG}(r-1,q^n)$ and the elements of $\\\\mathcal {D}$ .\",\n \"Using coordinates, we see that a point $P=(x_0,x_1,\\\\dots x_{r-1})_{q^n} \\\\in \\\\operatorname{PG}(r-1,q^n)$ corresponds to the set $\\\\lbrace (\\\\alpha x_0,\\\\alpha x_1, \\\\dots , \\\\alpha x_{r-1})_q| \\\\alpha \\\\in \\\\mathbb {F}_{q^n} \\\\rbrace $ in $\\\\operatorname{PG}(rn-1,q)$ .\",\n \"Note that we have used $r$ coordinates from $\\\\mathbb {F}_{q^n}$ , defined up to $\\\\mathbb {F}_q$ -scalar multiple to define points of $\\\\operatorname{PG}(rn-1,q)$ , and the set $\\\\lbrace (\\\\alpha x_0,\\\\alpha x_1, \\\\dots , \\\\alpha x_{r-1})_q| \\\\alpha \\\\in \\\\mathbb {F}_{q^n} \\\\rbrace $ consists of $\\\\frac{q^n-1}{q-1}$ different points forming an $(n-1)$ -dimensional space.\",\n \"Hence, we find that $\\\\mathcal {D}$ is given by the set of $(n-1)$ -spaces $\\\\lbrace (\\\\alpha x_0,\\\\alpha x_1, \\\\dots , \\\\alpha x_{r-1})_q| \\\\alpha \\\\in \\\\mathbb {F}_{q^n} \\\\rbrace \\\\; \\\\mathrm {for\\\\ all\\\\ }(x_0,x_1,\\\\ldots ,x_{r-1})\\\\in \\\\mathbb {F}_{q^n}^r.$ Note that these coordinates for points in $\\\\operatorname{PG}(rn-1,q)$ can be transformed into the usual coordinates consisting of $rn$ elements of $\\\\mathbb {F}_q$ by representing the elements of $\\\\mathbb {F}_{q^n}$ as the $n$ coordinates with respect to a fixed basis of $\\\\mathbb {F}_{q^n}$ over $\\\\mathbb {F}_q$ .\",\n \"We also have a more geometric perspective on the notion of a linear set; namely, an $\\\\mathbb {F}_q$ -linear set is a set $T$ of points of $\\\\operatorname{PG}(r-1,q^n)$ for which there exists a subspace $\\\\pi $ in $\\\\operatorname{PG}(rn-1,q)$ such that the points of $T$ correspond to the elements of $\\\\mathcal {D}$ that have a non-empty intersection with $\\\\pi $ .\",\n \"For more on this approach to linear sets, we refer to [10].\",\n \"If the subspace $\\\\pi $ intersects each spread element in at most a point, then $\\\\pi $ is called scattered with respect to $\\\\mathcal {D}$ and the associated linear set is called a scattered linear set.\",\n \"Note that if $\\\\pi $ is $(n-1)$ -dimensional and scattered, then the associated $\\\\mathbb {F}_q$ -linear set has rank $n$ and has exactly $\\\\frac{q^n-1}{q-1}$ points, and conversely.\",\n \"In this paper, we will make use of the following bound on the rank of a scattered linear set.\",\n \"Result 2.1 ([5]) The rank of a scattered $\\\\mathbb {F}_q$ -linear set in $\\\\operatorname{PG}(r-1,q^n)$ is at most $ rn/2$ .\",\n \"A maximum scattered linear set is a scattered $\\\\mathbb {F}_q$ -linear set in $\\\\operatorname{PG}(r-1,q^n)$ with rank $rn/2$ .\",\n \"In this article we work with maximum scattered linear sets to which a geometric structure, called pseudoregulus, can be associated.\",\n \"For more information, we refer to [12] and [13].\",\n \"Definition 2.2 Let $S$ be a scattered $\\\\mathbb {F}_q$ -linear set of $\\\\operatorname{PG}(2k-1,q^n)$ of rank $kn$ , where $n, k\\\\ge 2$ .\",\n \"We say that $S$ is of pseudoregulus type if there exist $m=\\\\frac{q^{nk}-1}{q^n-1}$ pairwise disjoint lines of $\\\\operatorname{PG}(2k-1, q^n)$ , say $s_1, s_2, \\\\dots , s_m$ , such that $| S\\\\cap s_i|=\\\\frac{q^n-1}{q-1}\\\\quad \\\\forall i=1,\\\\dots ,m,$ there exist exactly two $(k-1)$ -dimensional subspaces $T_1$ and $T_2$ of $\\\\operatorname{PG}(2k-1,q^n)$ disjoint from $S$ such that $T_j\\\\cap s_i \\\\ne \\\\emptyset $ for each $i = 1,\\\\dots , m$ and $j = 1, 2$ .\",\n \"The set of lines $s_i$ , $i=1,\\\\dots m$ is called the pseudoregulus of $\\\\operatorname{PG}(2k-1,q^n)$ associated with the linear set $S$ and we refer to $T_1$ and $T_2$ as transversal spaces to this pseudoregulus.\",\n \"Since a maximum scattered linear set spans the whole space, we see that the transversal spaces are disjoint.\",\n \"Throughout this paper we need the following result of [13] on pseudoreguli .\",\n \"Applied to $\\\\mathbb {F}_2$ -linear sets, this gives us the following result.\",\n \"Result 2.3 ([13]) Each $\\\\mathbb {F}_2$ -linear set of $\\\\operatorname{PG}(2k-1, q)$ , $q$ even, of pseudoregulus type, is of the form $L_{\\\\rho ,f}$ with $L_{\\\\rho ,f} = \\\\lbrace ( u, \\\\rho f(u))_q| u\\\\in U_0 \\\\rbrace ,$ with $\\\\rho \\\\in \\\\mathbb {F}_{q}^*$ , $U_0,U_\\\\infty $ the $k$ -dimensional vector spaces corresponding to the transversal spaces $T_0,T_\\\\infty $ and with $f:U_0\\\\rightarrow U_\\\\infty $ an invertible semilinear map with companion automorphism $\\\\sigma \\\\in Aut(\\\\mathbb {F}_q)$ , $Fix(\\\\sigma )=\\\\lbrace 0,1\\\\rbrace $ .\",\n \"Note that in the previous result, $\\\\operatorname{PG}(2k-1,q)$ is identified with $\\\\operatorname{PG}(V)$ , $V= U_0 \\\\oplus U_\\\\infty $ and a point, corresponding to a vector $v=v_1+v_2\\\\in U_0 \\\\oplus U_\\\\infty $ , has coordinates $(v_1,v_2)_q$ .\"\n ],\n [\n \"The Barlotti-Cofman and André/Bruck-Bose constructions\",\n \"In this paper, we will switch between three different representations of a projective plane $\\\\operatorname{PG}(2,q^k)$ , $q=2^h$ .\",\n \"Using the André/Bruck-Bose correspondence, we can, on one hand, model this plane as a subset of points and $k$ -spaces in $\\\\operatorname{PG}(2k,q)$ , determined by a $(k-1)$ -spread at infinity.\",\n \"On the other hand, we can see it as a subset of points and $hk$ -spaces of $\\\\operatorname{PG}(2hk,2)$ determined by a $(hk-1)$ -spread at infinity.\",\n \"We can switch between the $\\\\operatorname{PG}(2k,q)$ -setting an the $\\\\operatorname{PG}(2hk,2)$ -setting by the Barlotti-Cofman correspondence, which is a natural generalization of the André/Bruck-Bose correspondence.\",\n \"The Barlotti-Cofman representation of the projective space $\\\\operatorname{PG}(2k,2^h)$ in $\\\\operatorname{PG}(2hk,2)$ is defined as follows (see [2]).\",\n \"Let $\\\\mathcal {S}^{\\\\prime }$ be a Desarguesian $(h-1)$ -spread in $\\\\operatorname{PG}(2hk-1, 2)$ .\",\n \"Embed $\\\\operatorname{PG}(2hk-1, 2)$ as a hyperplane $\\\\tilde{H_\\\\infty }$ in $\\\\operatorname{PG}(2hk, 2)$ .\",\n \"Consider the following incidence structure $\\\\mathcal {P}(\\\\mathcal {S}) = (\\\\mathcal {P},\\\\mathcal {L} )$ , where incidence is natural: The set $\\\\mathcal {P}$ of points consists of the $2^{2hk}$ affine points $P_i$ in $\\\\operatorname{PG}(2hk,2)$ (i.e.\",\n \"the points not in $\\\\tilde{H_\\\\infty }$ ) together with elements of the $(h-1)$ -spread $\\\\mathcal {S}^{\\\\prime }$ in $\\\\tilde{H_\\\\infty }$ The set $\\\\mathcal {L}$ of lines consist of the following two sets of subspaces in $\\\\operatorname{PG}(2hk,2)$ .\",\n \"The set of $h$ -spaces spanned by an element of $\\\\mathcal {S}^{\\\\prime }$ and an affine point of $\\\\operatorname{PG}(2hk,2)$ .\",\n \"The set of $(2h-1)$ -spaces in $\\\\tilde{H_\\\\infty }$ spanned by two different elements of $\\\\mathcal {S}^{\\\\prime }$ .\",\n \"This incidence structure $(\\\\mathcal {P},\\\\mathcal {L} )$ is isomorphic to $\\\\operatorname{PG}(2k,2^h)$ .\",\n \"We use the notation $P_i$ for the affine point of $\\\\operatorname{PG}(2k,2^h)$ (i.e.\",\n \"a point not contained in $H_\\\\infty $ ) which corresponds to the affine point $\\\\tilde{P_i}\\\\in \\\\operatorname{PG}(2hk,2)$ .\",\n \"A point, say $R_i$ in $H_\\\\infty $ , corresponds to the element $\\\\mathcal {S}^{\\\\prime }(R_i)$ of the $(h-1)$ -spread $\\\\mathcal {S}^{\\\\prime }$ in $\\\\tilde{H_\\\\infty }$ .\",\n \"As already mentioned above, during this paper we will work in the following three projective spaces: The $2k$ -dimensional projective space $\\\\Pi _q=\\\\operatorname{PG}(2k,q)$ , $q=2^h, h>2$ , with the $(2k-1)$ -space at infinity called $H_\\\\infty $ .\",\n \"The projective plane $\\\\Pi _{q^k}=\\\\operatorname{PG}(2,q^k)$ , $q=2^h$ with line at infinity called $\\\\ell _\\\\infty $ .\",\n \"Given a Desarguesian $(k-1)$ -spread $\\\\mathcal {S}$ in $H_\\\\infty $ in $\\\\Pi _q$ , the plane $\\\\Pi _{q^k}$ is obtained by the André-Bruck-Bose construction using $\\\\mathcal {S}$ .\",\n \"The $2hk$ -dimensional projective space $\\\\Pi _2=\\\\operatorname{PG}(2hk,2)$ , with the $(2hk-1)$ -space $\\\\tilde{H_\\\\infty }$ at infinity.\",\n \"Note that the Barlotti-Cofman representation of $\\\\Pi _q$ defines a Desarguesian $(h-1)$ -spread $\\\\mathcal {S}^{\\\\prime }$ in $\\\\tilde{H_\\\\infty }$ .\",\n \"Moreover, if $\\\\mathcal {S}$ is the $(k-1)$ -spread in $H_\\\\infty $ in $\\\\Pi _q$ such that $\\\\Pi _{q^ k}$ is the corresponding projective plane, the André-Bruck-Bose representation of $\\\\Pi _{q^k}$ in $\\\\Pi _2$ gives rise to a Desarguesian $(hk-1)$ -spread $\\\\tilde{\\\\mathcal {S}}$ in $\\\\tilde{H_\\\\infty }$ , such that $\\\\mathcal {S}^{\\\\prime }$ is a subspread of $\\\\tilde{\\\\mathcal {S}}$ .\"\n ],\n [\n \"The proof of the main theorem\",\n \"Consider $\\\\Pi _q=\\\\operatorname{PG}(2k,q)$ and the hyperplane $H_\\\\infty $ of $\\\\operatorname{PG}(2k,q)$ .\",\n \"Recall that a point of $\\\\operatorname{PG}(2k,q)$ is called affine if it is not contained in $H_\\\\infty $ .\",\n \"Likewise, a line is called affine if it is not contained in $H_\\\\infty $ .\",\n \"Let $P_1,P_2$ be affine points, then the point $P_1P_2\\\\cap H_\\\\infty $ is the direction determined by the line $P_1P_2$ .\",\n \"If $\\\\mathcal {Q}$ is a set of affine points, then the directions determined by $\\\\mathcal {Q}$ are all points of $H_\\\\infty $ that appear as the direction of a line $P_iP_j$ for some $P_i,P_j\\\\in \\\\mathcal {Q}$ .\",\n \"From now on, we consider a set $\\\\mathcal {Q}$ satisfying the conditions of Theorem REF : $\\\\mathcal {Q}$ is a set of $q^k$ affine points in $\\\\operatorname{PG}(2k,q)$ , $q=2^h$ , $h\\\\ge 4$ , $k \\\\ge 2$ ; $D$ , the set of directions determined by $\\\\mathcal {Q}$ at the hyperplane at infinity $H_\\\\infty $ has size $q^k-1$ ; Every line has 0, 1, 3 or $q-1$ points in common with the point set $D$ .\"\n ],\n [\n \"The $(q-1)$ -secants to {{formula:824f1829-06a2-4e3c-9246-2c2a3663df00}} are disjoint\",\n \"Definition 3.1 A 0-point in $H_\\\\infty $ is a point $P\\\\notin D$ such that $P$ is contained in at least one $(q-1)$ -secant to $D$ .\",\n \"From Proposition REF , it will follow that a 0-point is contained in precisely one $(q-1)$ -secant to $D$ .\",\n \"We first start with two lemmas.\",\n \"Lemma 3.2 No three points of $\\\\mathcal {Q}$ are collinear.\",\n \"Let $l$ be an affine line in $\\\\operatorname{PG}(2k,q)$ containing $3\\\\le t\\\\le q$ points of $\\\\mathcal {Q}$ , and let $P^{\\\\prime }=l\\\\cap H_\\\\infty $ .\",\n \"A point $P_i\\\\in \\\\mathcal {Q}\\\\setminus l$ determines a plane $\\\\alpha _i=\\\\langle P_i,l\\\\rangle $ such that the line $l_i= \\\\alpha _i\\\\cap H_\\\\infty $ is a $(q-1)$ -secant: the lines through $P_i$ and a point of $l\\\\cap \\\\mathcal {Q}$ determine $t\\\\ge 3$ directions of $D$ on the line $l_i$ , different from the point $P^{\\\\prime } \\\\in D$ .\",\n \"So $l$ contains more than three points of $D$ , showing that $l_i$ is a $(q-1)$ -secant.\",\n \"Furthermore, the plane $\\\\alpha _i$ contains at most $q$ affine points of $\\\\mathcal {Q}$ , as every affine line in $\\\\alpha $ through a 0-point of $l_i$ contains at most one element of $\\\\mathcal {Q}$ .\",\n \"This implies that each of the $q^k-t\\\\ge q^k-q$ points of $\\\\mathcal {Q}\\\\setminus l$ define a plane $\\\\alpha $ , with $\\\\alpha \\\\cap H_\\\\infty $ a $(q-1)$ -secant, and so that $\\\\alpha $ contains at most $q-t\\\\le q-3$ points of $\\\\mathcal {Q}\\\\setminus l$ .\",\n \"This shows that the number of such planes $\\\\alpha _i$ through $l$ , and hence the number of $(q-1)$ -secants through $P^{\\\\prime }$ , is at least $\\\\frac{q^k-q}{q-3}$ .\",\n \"This gives that there are at least $1+\\\\frac{q^k-q}{q-3}(q-2)>q^k-1$ points of $D$ , a contradiction.\",\n \"Lemma 3.3 Let $\\\\gamma $ be a plane in $\\\\operatorname{PG}(2k,q)$ containing 4 points $P_1,P_2,P_3$ and $P_4$ of $\\\\mathcal {Q}$ , such that $P_1P_2 \\\\cap P_3P_4\\\\notin \\\\mathcal {Q}\\\\cup D$ .\",\n \"Then $\\\\gamma $ meets $H_\\\\infty $ in a $(q-1)$ -secant to $D$ .\",\n \"By Lemma REF , no three points of $P_1, P_2,P_3,P_4$ are collinear.\",\n \"Since $P_1P_2 \\\\cap P_3P_4\\\\notin D$ , we see that $P_1P_2$ and $P_3P_4$ define two different directions in $H_\\\\infty $ .\",\n \"The four points $P_1,P_2,P_3$ and $P_4$ determine at least 4 directions on the line $\\\\gamma \\\\cap H_\\\\infty $ .\",\n \"The statement follows since a line contains $0,1,3$ or $q-1$ points of $D$ .\",\n \"Proposition 3.4 Every two $(q-1)$ -secants to $D$ are disjoint.\",\n \"Consider a point $P_0\\\\in \\\\mathcal {Q}$ .\",\n \"Then, by Lemma REF , all points of $D$ are defined by the lines $P_0 P_i$ with $P_i\\\\in \\\\mathcal {Q}\\\\setminus \\\\lbrace P_0\\\\rbrace $ .\",\n \"Let $P_i^{\\\\prime }$ denote the direction of the line $P_0 P_i$ , that is, the point $P_0P_i\\\\cap H_\\\\infty $ .\",\n \"We see that a line through a point $P_i^{\\\\prime }\\\\in D$ contains 0 or 2 points of $\\\\mathcal {Q}$ .\",\n \"Let $l_\\\\alpha $ and $l_\\\\beta $ be two lines, both containing $q-1$ points of $D$ , with $P^{\\\\prime }=l_\\\\alpha \\\\cap l_\\\\beta $ .\",\n \"Let $\\\\alpha =\\\\langle P_0, l_\\\\alpha \\\\rangle $ and $\\\\beta =\\\\langle P_0, l_\\\\beta \\\\rangle $ and let $\\\\lbrace P_{1 \\\\alpha },P_{2 \\\\alpha }\\\\rbrace $ and $\\\\lbrace P_{1 \\\\beta },P_{2 \\\\beta }\\\\rbrace $ be the 0-points in $l_\\\\alpha $ and $l_\\\\beta $ .\",\n \"Note that $P^{\\\\prime }$ may be amongst these points.\",\n \"It follows from the argument above that there are precisely $q$ points in $\\\\alpha \\\\cap \\\\mathcal {Q}$ and that the affine points of $\\\\mathcal {Q}$ in $\\\\alpha $ together with the two points $P_{1 \\\\alpha },P_{2 \\\\alpha }$ form a hyperoval $H_\\\\alpha $ .\",\n \"Similarly, we find a hyperoval $H_\\\\beta $ in $\\\\beta $ .\",\n \"We first suppose that $P^{\\\\prime }\\\\in D$ .\",\n \"This implies that there is a point $P\\\\ne P_0$ of $\\\\mathcal {Q}$ on the line $P_0 P^{\\\\prime }$ .\",\n \"Note that $P_0$ and $P$ are contained in $H_\\\\alpha \\\\cap H_\\\\beta $ .\",\n \"Consider a point $R\\\\in l_\\\\alpha $ , different from $P^{\\\\prime }, P_{1 \\\\alpha },P_{2 \\\\alpha }$ .\",\n \"Then $R\\\\in D$ and through $R$ , there are $\\\\frac{q}{2}$ bisecants to $H_\\\\alpha \\\\ne l_\\\\alpha $ .\",\n \"One of these bisecants contains $P$ and another one contains $P_0$ .\",\n \"Since $q>8$ , there exists a bisecant to $H_\\\\alpha $ through $R$ which intersects the line $P_0 P$ in a point $R_0 \\\\notin \\\\lbrace P_0,P,P^{\\\\prime }\\\\rbrace $ .\",\n \"Through $R_0$ , there are $\\\\frac{q}{2}-2$ bisecants $r_i$ to $H_\\\\beta $ , different from the lines $R_0 P$ , $R_0P_{1\\\\beta }$ and $R_0P_{2\\\\beta }$ .\",\n \"Let $r_i\\\\cap l_\\\\beta =R_i, i=1,\\\\dots ,\\\\frac{q}{2}-2$ .\",\n \"A plane $\\\\langle R, r_i \\\\rangle $ contains two lines, $r_i$ and $m=R R_0$ , both containing two points of $\\\\mathcal {Q}$ and $r_i\\\\cap m=R_0\\\\notin \\\\mathcal {Q}$ .\",\n \"Hence, by Lemma REF we find that every line $R R_i$ is a $(q-1)$ -secant to $D$ .\",\n \"So there are $\\\\frac{q}{2}-2$ $(q-1)$ -secants of the form $RR_i$ , and the total number of 0-points on these lines is $2(\\\\frac{q}{2}-2)=q-4$ .\",\n \"Let $\\\\Omega $ be the set of these 0-points.\",\n \"We call a $(\\\\le 3)$ -secant in $\\\\langle l_\\\\alpha , l_\\\\beta \\\\rangle $ a line with at most 3 points of $D$ .\",\n \"A line through $P^{\\\\prime }$ in $\\\\langle l_\\\\alpha ,l_\\\\beta \\\\rangle $ intersects all lines $RR_i$ .\",\n \"The $q-4$ points of $\\\\Omega $ lie on the $q-1$ lines through $P^{\\\\prime }$ different from $l_\\\\alpha $ and $l_\\\\beta $ .\",\n \"Since every line $RR_i$ contains precisely two 0-points, we find that for $q>8$ there are at most 3 $(\\\\le 3)$ -secants through $P^{\\\\prime }$ : if there are at least four $(\\\\le 3)$ -secants through $P^{\\\\prime }$ in $\\\\langle l_\\\\alpha , l_\\\\beta \\\\rangle $ , then there are at least $\\\\frac{q}{2}-2-2$ 0-points of $\\\\Omega $ on each of these lines, as we supposed that $P^{\\\\prime }\\\\in D$ .\",\n \"This implies that there would be at least $4(\\\\frac{q}{2}-4)>q-4$ 0-points in $\\\\Omega $ , which gives a contradiction for $q\\\\ge 16$ .\",\n \"Now we distinguish different cases depending on the number of $(\\\\le 3)$ -secants through $P^{\\\\prime }$ .\",\n \"In each of the cases we will show that there exists at least two $(\\\\le 3)$ -secants $l_1,l_2$ in $\\\\langle l_\\\\alpha ,l_\\\\beta \\\\rangle $ , and a point $X\\\\notin D$ not on these lines.\",\n \"This leads to a contradiction since there are at least $q+1-7$ lines through $X$ , both intersecting $l_1$ and $l_2$ in a point not in $D$ , and not through $l_1\\\\cap l_2$ .\",\n \"These lines contain at least 3 points not in $D$ so they have to be $(\\\\le 3)$ -secants.\",\n \"But this implies that there are at least $1+(q-6)(q-3)=q^2-9q+19$ points in $\\\\langle l_\\\\alpha ,l_\\\\beta \\\\rangle $ , not contained in $D$ .\",\n \"On the other hand, there are at most three $(\\\\le 3)$ -secants through $P^{\\\\prime }$ and the other lines through $P^{\\\\prime }$ contain two 0-points.\",\n \"This implies that there are at most $3q+ 2(q-2)=5q-44$ .\",\n \"Let $K$ be the $(hk-1)$ -dimensional subspace in $\\\\operatorname{PG}(2hk-1,2)$ defining the $\\\\mathbb {F}_2$ -linear set $D$ and let $\\\\mathcal {S}^{\\\\prime }$ be the $(h-1)$ -spread that corresponds to the point set of $\\\\operatorname{PG}(2k-1,q)$ .\",\n \"For every $hk$ -space $Y$ through $K$ in $\\\\operatorname{PG}(2hk-1,2)$ , we find at least one element of $\\\\mathcal {S}^{\\\\prime }$ that intersects $Y$ in a line since $D$ is maximum scattered.\",\n \"Every line $l$ , through a point of $K$ , such that $l$ lies in an element of $\\\\mathcal {S}^{\\\\prime }$ , defines a $hk$ -space through $K$ , and the number of $hk$ -spaces through $K$ is $2^{hk}-1$ .\",\n \"This implies that there are on average $2^{h-1}-1>2$ lines contained in different spread elements of $\\\\mathcal {S}^{\\\\prime }$ in a $hk$ -space through $K$ in $\\\\operatorname{PG}(2hk-1,2)$ .\",\n \"Take a $hk$ -space $Y$ through $K$ with at least two lines contained in spread elements, and let $S_1$ and $S_2$ be two elements of $\\\\mathcal {S}^{\\\\prime }$ that intersect $Y$ in the lines $y_1$ and $y_2$ respectively.\",\n \"The $(2h-1)$ -space $\\\\langle S_1,S_2\\\\rangle $ intersects $K$ in at least a plane, as $y_1$ and $y_2$ span a 3-space.\",\n \"But this implies that the line $l$ in $\\\\operatorname{PG}(2k-1,q)$ , corresponding with $\\\\langle S_1,S_2\\\\rangle $ contains at least 7 points of $D$ .\",\n \"This implies that $l$ is a $(q-1)$ -secant of $D$ , and that $\\\\langle S_1,S_2\\\\rangle $ intersects $K$ in a $(h-1)$ -space $\\\\alpha $ as a $(h-1)$ -space contains $2^h-1=q-1$ points.\",\n \"Consider now the $h$ -space $\\\\beta =Y\\\\cap \\\\langle S_1,S_2 \\\\rangle $ through $\\\\alpha $ .\",\n \"Since all of the $2^h+1$ $(h-1)$ -spaces of $\\\\mathcal {S}^{\\\\prime }$ in $\\\\langle S_1,S_2 \\\\rangle $ intersect $\\\\beta $ in a point or a line, we find that there are precisely $2^{h-1}-1$ elements of $\\\\mathcal {S}^{\\\\prime }$ , meeting $\\\\beta $ , and so $Y$ , in a line.\",\n \"Hence, this proves that a $hk$ -space $Y$ through $K$ , containing at least 2 lines $y_1,y_2$ in $S_1,S_2$ respectively, contains at least $2^{h-1}-1$ lines $y_i$ in different spread elements of $\\\\mathcal {S}^{\\\\prime }$ .\",\n \"Now we prove, by contradiction, that $Y$ cannot contain more lines $y_i$ contained in a spread element.\",\n \"Suppose $Y$ contains another line $y_0 \\\\subset S_0$ with $S_0 \\\\in \\\\mathcal {S}^{\\\\prime }$ , then $y_0 \\\\notin \\\\langle S_1, S_2 \\\\rangle $ .\",\n \"Repeating the previous argument for $y_1$ and $y_2$ shows that there are two $(2h-1)$ -spaces $\\\\langle S_1, S_2 \\\\rangle $ and $\\\\langle S_0, S_1 \\\\rangle $ , both meeting $K$ in a $(h-1)$ -space and so, there are two $(q-1)$ -secants through $P_1\\\\in H_\\\\infty $ , the point corresponding to the spread element $S_1$ .\",\n \"This gives a contradiction by Proposition REF .\",\n \"Since the average number of lines contained in a spreadelement in a $hk$ -space through $K$ is $2^{h-1}-1>2$ , we find that every $hk$ -space through $K$ contains exactly $2^{h-1}-1$ lines contained in a spreadelement.\",\n \"In particular, every line $y_i \\\\subset S_i$ , with $S_i \\\\in \\\\mathcal {S}^{\\\\prime }$ and $y_i$ through a point of $K$ , defines a $hk$ -space though $K$ , and so a $(q-1)$ -secant.\",\n \"So we find that every point in $D$ is contained in at least one $(q-1)$ -secant.\",\n \"As we already proved that two $(q-1)$ -secants are disjoint (see Lemma REF ), we find $\\\\frac{q^k-1}{q-1}$ pairwise disjoint $(q-1)$ -secants in $\\\\operatorname{PG}(2k-1,q)$ .\",\n \"We will first show that the linear set is of pseudoregulus type when $k=2$ .\",\n \"To prove this, we begin with a lemma.\",\n \"Lemma 3.10 Assume that $k=2$ .\",\n \"Let $l$ be a line in $H_\\\\infty $ through two 0-points, not on the same $(q-1)$ -secant, then $l$ contains no points of $D$ .\",\n \"Let $l_1$ and $l_2$ be two $(q-1)$ -secants in $\\\\operatorname{PG}(3,q)$ and let $l$ be a line through a 0-point of $l_1$ and through a 0-point of $l_2$ .\",\n \"Recall that $l_1$ and $l_2$ are disjoint by Lemma REF .\",\n \"Every two points $(A,B)$ , $A\\\\in l_1$ , $B\\\\in l_2$ , define a third point in $D$ on the line $AB$ .\",\n \"Hence we find, since $|D|=q^2-1$ , that every point $P\\\\in D\\\\setminus \\\\lbrace l_1,l_2\\\\rbrace $ is uniquely defined as a third point on a line, defined by two points $A$ and $B$ of $D$ in $l_1$ and $l_2$ respectively.\",\n \"Now suppose that $l$ contains a point $X\\\\in D$ , then $X$ lies on a unique line $l^{\\\\prime }$ , intersecting $l_1$ and $l_2$ in precisely one point.\",\n \"But then $l_1$ and $l_2$ lie in a plane spanned by $l$ and $l^{\\\\prime }$ , a contradiction since $l_1$ and $l_2$ are disjoint by Lemma REF .\",\n \"Proposition 3.11 Assume that $k=2$ .\",\n \"The $(q-1)$ -secants to $D$ in $\\\\operatorname{PG}(3,q)$ form a pseudoregulus.\",\n \"By Lemma REF it is sufficient to prove that there exist 2 lines in $\\\\operatorname{PG}(3,q)$ that have a point in common with all $(q-1)$ -secants to $D$ .\",\n \"Consider three $(q-1)$ -secants $l_1$ , $l_2$ and $l_3$ and let $P_i,Q_i\\\\in l_i$ , $i=1,2,3$ be the corresponding 0-points.\",\n \"Let $l_0$ be the unique line through $P_1$ that intersects $l_2$ and $l_3$ both in a point, say $R_2=l_0\\\\cap l_2$ and $R_3=l_0\\\\cap l_3$ respectively.\",\n \"By Corollary $\\\\ref {q-1en3disjunct}$ , $R_2$ and $R_3$ cannot both belong to $\\\\mathcal {Q}$ , so suppose $R_2$ is a 0-point of $l_2$ (w.l.o.g.\",\n \"$R_2=P_2$ ).\",\n \"We see that $l_0=P_1P_2$ is a line through two 0-points, so $R_3$ is also a 0-point by Corollary REF , w.l.o.g.\",\n \"$R_3=P_3$ .\",\n \"By the same argument, we see that $Q_1,Q_2$ and $Q_3$ are contained in a line, say $l_\\\\infty $ .\",\n \"Now we want to show that every other $(q-1)$ -secant has a 0-point in common with both $l_0$ and $l_\\\\infty $ .\",\n \"Consider an $(q-1)$ -secant $l_4$ , different from $l_1,l_2,l_3$ , with 0-points $P_4$ and $Q_4$ .\",\n \"Consider now again the unique line $m$ through $P_4$ that intersects $l_1$ and $l_2$ in a point.\",\n \"By the previous arguments $m$ has to contain a 0-point of $l_1$ and a 0-point of $l_2$ , so $m=l_0$ , $m=l_\\\\infty $ , $m=P_1Q_2$ or $m=Q_1P_2$ .\",\n \"We will show that only the first two possibilities can occur, which then proves that every other 0-point lies on $l_0$ or $l_\\\\infty $ .\",\n \"Suppose to the contrary that $m=P_1Q_2P_4$ (the case $m=Q_1P_2P_4$ is completely analogous).\",\n \"Then the unique line through $Q_4$ , meeting $l_1$ and $l_2$ is the line $Q_1P_2$ .\",\n \"Consider now the unique line $m^{\\\\prime }$ through $P_4$ meeting $l_2$ and $l_3$ in a point.\",\n \"As we supposed that $m\\\\ne l_0$ and $m\\\\ne l_\\\\infty $ , we see that $P_4$ cannot lie on these lines, so $m^{\\\\prime }$ contains the points $P_4,P_2,Q_3$ or the points $P_4,Q_2,P_3$ .\",\n \"In the former case both lines $l_0$ and $l_\\\\infty $ are contained in the plane spanned by $m^{\\\\prime }=P_4Q_3P_2$ and $m=P_1Q_2P_4$ .\",\n \"This implies that the disjoint lines $l_1$ and $l_2$ are contained in this plane, a contradiction.\",\n \"If $m^{\\\\prime }=P_4P_3Q_2$ , then $m$ and $m^{\\\\prime }$ both contain $P_4$ and $Q_2$ but intersect $l_0$ in different points, a contradiction.\",\n \"We conclude that $P_4$ , and analoguously $P_4^{\\\\prime }$ , is contained in the line $l_0$ or $l_\\\\infty $ .\",\n \"Using the previous proposition, we will prove that for all $k$ , the $\\\\mathbb {F}_2$ -linear set $D$ in $\\\\operatorname{PG}(2k-1,q)$ is of pseudoregulus type.\",\n \"Theorem 3.12 The $(q-1)$ -secants to $D$ in $\\\\operatorname{PG}(2k-1,q)$ form a pseudoregulus.\",\n \"By Lemma REF it is sufficient to prove that there exist two $(k-1)$ -spaces in $\\\\operatorname{PG}(2k-1,q)$ that both have a point in common with all $(q-1)$ -secants to $D$ .\",\n \"Consider a $(q-1)$ -secant $l_0$ , and let $P_0$ and $P_0^{\\\\prime }$ be the 0-points on $l_0$ .\",\n \"Let $l_i$ be a $(q-1)$ -secant, different from $l_0$ .\",\n \"The lines $l_0$ and $l_i$ span a 3-space $\\\\gamma $ and since $D$ is a scattered $\\\\mathbb {F}_2$ -linear set, $\\\\gamma \\\\cap D$ is also a scattered $\\\\mathbb {F}_2$ -linear set.\",\n \"Since $\\\\gamma $ contains $2(q-1)$ points of $D$ on the lines $l_i,l_j$ and $(q-1)^2$ points of $D$ defined in a unique way as a third point on the line $A_1 A_2$ , with $A_1 \\\\in l_0$ , $A_2\\\\in l_i$ , we have that $|D\\\\cap \\\\gamma |=q^2-1$ , and hence it is a maximum scattered linear set.\",\n \"By Theorem REF , we find that $\\\\gamma \\\\cap D$ is of pseudoregulus type.\",\n \"This means that it has transversal lines, say $m_i$ and $m_i^{\\\\prime }$ , where $P_0$ lies on $m_i$ and $P_0^{\\\\prime }$ lies on $m_i^{\\\\prime }$ .\",\n \"This holds for every $(q-1)$ -secant $l_i$ .\",\n \"Since there are exactly $\\\\frac{q^k-1}{q-1}$ $(q-1)$ -secants to $D$ , which are mutually disjoint, there are exactly $2\\\\frac{q^k-1}{q-1}$ 0-points.\",\n \"We have proven that a 0-point lies on $\\\\frac{q^{k-1}-1}{q-1}$ lines full of 0-points (call such lines 0-lines) and on $\\\\frac{q^k-1}{q-1}$ lines containing exactly 1 other 0-point.\",\n \"Let $A$ and $A^{\\\\prime }$ be the set of all points on the lines $m_i$ and $m_i^{\\\\prime }$ respectively.\",\n \"Then we will to show that $A\\\\cup A^{\\\\prime }$ is the union of two disjoint $(k-1)$ -spaces.\",\n \"Consider a line containing two 0-points $P_1,P_2$ , with $l_1$ and $l_2$ the $(q-1)$ -secants through $P_1,P_2$ .\",\n \"Then, as seen before, the intersection of the 3-space spanned by $l_1$ and $l_2$ with $D$ is a linear set of pseudoregulus type, and hence the line $P_1P_2$ contains 2 or $q+1$ 0-points.\",\n \"This shows that every line in $\\\\operatorname{PG}(2k-1,q)$ intersects $A \\\\cup A^{\\\\prime }$ in $0,1,2$ or $q+1$ points.\",\n \"This in turn implies that a plane with three 0-lines only contains 0-points.\",\n \"Consider now a point $P_3$ on a 0-line through $P_0$ , and consider a 0-line $m\\\\ne P_0P_3$ through $P_3$ .\",\n \"If $m$ contains a point $P_4 \\\\ne P_3$ such that $P_4P_0$ is a 0-line through $P_0$ , then we see that the plane $\\\\langle P_0,m \\\\rangle $ only contains 0-points.\",\n \"In the other case, $M$ contains at least two 0-points on 0-lines through $P_0^{\\\\prime }$ .\",\n \"In this case, all the points in the plane $\\\\langle P_0^{\\\\prime }, m \\\\rangle $ are 0-points, and hence the line $P_1P_0^{\\\\prime }$ is a 0-line, a contradiction.\",\n \"So we find that every 0-line through a 0-point of $A$ is contained in $A$ .\",\n \"Since every point of $A$ lies on $\\\\frac{q^{k-1}-1}{q-1}$ 0-lines, and $A$ contains $\\\\frac{q^k-1}{q-1}$ 0-points, we find that every 2 points of $A$ are contained in a 0-line of $A$ .\",\n \"The same argument works for the set $A^{\\\\prime }$ .\",\n \"This shows that $A$ forms a subspace and likewise $A^{\\\\prime }$ forms a subspace.\",\n \"Since $|A|=|A^{\\\\prime }|=\\\\frac{q^k-1}{q-1}$ , these subspaces are $(k-1)$ -dimensional.\"\n ],\n [\n \"There exists a suitable Desarguesian $(k-1)$ -spread {{formula:5d299a62-d518-48c3-80a5-84f442bd5791}} in {{formula:32a97683-3610-493b-8c79-c869c3ed8c0b}}\",\n \"Consider the scattered linear set $D\\\\subset H_\\\\infty $ of pseudoregulus type.\",\n \"Let $T_0$ and $T_\\\\infty $ be the transversal $(k-1)$ -spaces to the pseudoregulus defined by $D$ found in Theorem REF .\",\n \"Now we want to show that there exists a Desarguesian $(k-1)$ -spread $\\\\mathcal {S}$ in $\\\\operatorname{PG}(2k-1,q)$ such that $T_0,T_\\\\infty \\\\in \\\\mathcal {S}$ and such that every other $(k-1)$ -space of $\\\\mathcal {S}$ has precisely one point in common with $D$ .\",\n \"Lemma 3.13 There exists a Desarguesian $(k-1)$ -spread $\\\\mathcal {S}$ in $\\\\operatorname{PG}(2k-1,q)$ , such that $T_0,T_\\\\infty \\\\in \\\\mathcal {S}$ and such that every other element of $\\\\mathcal {S}$ has precisely one point in common with $D$ .\",\n \"We prove this lemma using the representation of Result REF .\",\n \"By [13] we find that the linear sets $L_{\\\\rho ,f}$ and $L_{\\\\rho ^{\\\\prime },g}$ are equivalent if and only if $\\\\sigma _f=\\\\sigma _g^{\\\\pm 1}$ , where $\\\\sigma _f$ and $\\\\sigma _g$ are the automorphisms associated with $f$ and $g$ respectively.\",\n \"Hence, up to equivalence, we may suppose that $\\\\rho =1$ and $f:\\\\mathbb {F}_{q^k} \\\\rightarrow \\\\mathbb {F}_{q^k}: t\\\\rightarrow t^{2^i}$ , $\\\\gcd (i,hk)=1$ .\",\n \"Considering $U_0,U_\\\\infty $ as $\\\\mathbb {F}_{q^k}$ , it follows that $D$ is equivalent to the set of points $P_u$ with $P_u:=(u,u^{2^i})_q,u\\\\in \\\\mathbb {F}_{q^k}^*.$ The transversal spaces $T_0$ and $T_\\\\infty $ are the point sets $T_0=\\\\lbrace (u,0)|u\\\\in \\\\mathbb {F}_{q^k}^*\\\\rbrace $ and $T_\\\\infty =\\\\lbrace (0,u)|u\\\\in \\\\mathbb {F}_{q^k}^*\\\\rbrace $ .\",\n \"Consider now the set $\\\\mathcal {S}_0$ of $(k-1)$ -spaces $T_u$ , $u\\\\in \\\\mathbb {F}_{q^k}^*$ with $T_u:=\\\\lbrace (\\\\alpha u, \\\\alpha u^{2^i})_{q}|\\\\alpha \\\\in \\\\mathbb {F}_{q^k}^*\\\\rbrace .$ We will show that the set $\\\\mathcal {S}=\\\\mathcal {S}_0\\\\cup \\\\lbrace T_0,T_\\\\infty \\\\rbrace $ is a $(k-1)$ -spread of $\\\\operatorname{PG}(2k-1,q)$ .\",\n \"Suppose that $P=T_{u_1}\\\\cap T_{u_2}$ , for some $u_1, u_2\\\\notin \\\\lbrace 0, \\\\infty \\\\rbrace $ , then there exists elements $\\\\alpha _1,\\\\alpha _2 \\\\in \\\\mathbb {F}_{q^k}^*, \\\\mu \\\\in \\\\mathbb {F}_{q}^*$ such that $\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\alpha _1 u_1&=\\\\mu \\\\alpha _2 u_2 \\\\\\\\\\\\alpha _1 u_1^{2^i}&=\\\\mu \\\\alpha _2 u_2^{2^i} \\\\end{array}\\\\right.$ With $\\\\mu \\\\in \\\\mathbb {F}_q^*$ .\",\n \"This implies that $u_1^{2^i-1}=u_2^{2^i-1}$ or $\\\\left(\\\\frac{u_1}{u_2}\\\\right)^{2^i}=\\\\frac{u_1}{u_2}$ .\",\n \"Hence $\\\\frac{u_1}{u_2}\\\\in \\\\mathbb {F}_{2^i}\\\\cap \\\\mathbb {F}_{2^{hk}}$ which is $\\\\mathbb {F}_2$ since $\\\\gcd (i,hk)=1$ .\",\n \"Since $u_1,u_2\\\\in \\\\mathbb {F}_{q^k}^*$ , this implies that $u_1=u_2$ , and that $T_{u_1}=T_{u_2}$ .\",\n \"In particular, we see that $T_u \\\\ne T_{u^{\\\\prime }}$ for $u\\\\ne u^{\\\\prime } \\\\in \\\\mathbb {F}_{q^k}^*$ .\",\n \"Since $T_0$ and $T_\\\\infty $ are distinct from $T_u$ for all $u\\\\in \\\\mathbb {F}_{q^k}^*$ , we obtain that $|\\\\mathcal {S}|=q^k+1$ .\",\n \"We will now show that $T_u\\\\cap T_0=\\\\emptyset $ for all $u\\\\in \\\\mathbb {F}_{q^k}^*$ .\",\n \"If $P=T_u\\\\cap T_0$ , $u\\\\notin \\\\lbrace 0, \\\\infty \\\\rbrace $ for some $u\\\\in \\\\mathbb {F}_{q^k}^*$ then $P=(u^{\\\\prime },0)_q$ with $u^{\\\\prime }\\\\in \\\\mathbb {F}_{q^k}^*$ and $\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\alpha u&=\\\\mu u^{\\\\prime } \\\\\\\\\\\\alpha u^{2^i}&=0 \\\\end{array}\\\\right.$ for some $\\\\mu \\\\in \\\\mathbb {F}_q^*$ and $\\\\alpha \\\\in \\\\mathbb {F}_{q^k}^*$ .\",\n \"The second equality gives a contradiction since $u\\\\ne 0 \\\\ne \\\\alpha $ .\",\n \"Hence $T_u \\\\cap T_0 = \\\\emptyset $ .\",\n \"It follows from a similar argument that $T_u \\\\cap T_\\\\infty =\\\\emptyset $ .\",\n \"This shows that $\\\\mathcal {S}$ is a spread which is Desarguesian as seen in Subsection REF .\",\n \"Remark 3.14 In [13] a geometric construction of the Desarguesian spread, found in Lemma REF , using indicator sets, is given.\"\n ],\n [\n \"The point set $\\\\mathcal {Q}$ defines a translation hyperoval in the André/Bruck-Bose plane {{formula:fe441269-42d1-4751-97d9-6854ca55cc19}}\",\n \"The spread $\\\\mathcal {S}$ found in Lemma REF defines a projective plane $\\\\mathcal {P}(\\\\mathcal {S})=\\\\Pi _{q^k}\\\\cong \\\\operatorname{PG}(2,q^k)$ by the André/Bruck-Bose construction.\",\n \"The transversal $(k-1)$ -spaces $T_0, T_\\\\infty \\\\in \\\\mathcal {S}$ to the pseudoregulus associated with $D$ correspond to points $P_0, P_\\\\infty $ contained in the line $\\\\ell _\\\\infty $ at infinity of $\\\\operatorname{PG}(2,q^k)$ .\",\n \"Theorem 3.15 The set $\\\\mathcal {Q}$ , together with $T_0$ and $T_\\\\infty $ , defines a translation hyperoval in $\\\\Pi _{q^k}\\\\cong \\\\operatorname{PG}(2,q^k)$ .\",\n \"Let $\\\\mathcal {A}$ be the set of points in $\\\\Pi _{q^k}$ corresponding to the point set $\\\\mathcal {Q}$ of $\\\\Pi _q$ .\",\n \"Recall that $T_0$ corresponds to a point $P_0$ and $T_\\\\infty $ to a point $P_\\\\infty $ , contained in the line $\\\\ell _\\\\infty $ of $\\\\Pi _{q^k}$ .\",\n \"We first show that every line in $\\\\operatorname{PG}(2,q^k)$ contains at most 2 points of the set $\\\\mathcal {H}=\\\\mathcal {A}\\\\cup P_0\\\\cup P_\\\\infty $ .\",\n \"The line $\\\\ell _\\\\infty $ at infinity only contains the points $P_0$ and $P_\\\\infty $ .\",\n \"Consider a line $l\\\\ne \\\\ell _\\\\infty $ through $P_0$ in $\\\\operatorname{PG}(2,q^k)$ .\",\n \"This line corresponds to a $k$ -space through $T_0$ in $\\\\operatorname{PG}(2k,q)$ .\",\n \"As $P_0\\\\in l\\\\cap \\\\mathcal {H}$ , we have to show that this $k$ -space contains at most one affine point of $\\\\mathcal {Q}$ .\",\n \"If this space would contain 2 (or more) affine points $X_1,X_2\\\\in \\\\mathcal {Q}$ , then they would define a direction of $D$ at infinity in $T_0$ .\",\n \"But this is impossible as $T_0$ has no points of $D$ , see Corollary REF .\",\n \"This argument also works for the lines through $P_\\\\infty $ , different from $\\\\ell _\\\\infty $ .\",\n \"Consider a line $l$ through a point $P_i$ , $i\\\\notin \\\\lbrace 0,\\\\infty \\\\rbrace $ at infinity.\",\n \"This point $P_i$ corresponds to an element $T_i\\\\in \\\\mathcal {S}$ that intersects the pseudoregulus $D$ in a unique point $X_i$ .\",\n \"The line $l$ corresponds to a $k$ -space $\\\\gamma $ in $\\\\operatorname{PG}(2k,q)$ through $T_i$ .\",\n \"Suppose that $\\\\gamma $ contains at least 3 points from $\\\\mathcal {Q}$ , say $X,Y,Z$ .\",\n \"By Lemma REF these points are not collinear, hence they determine at least two different points of $D$ which are contained in $T_i$ , a contradiction.\",\n \"This proves that $\\\\gamma $ contains at most two points of $\\\\mathcal {Q}$ , which implies that the line $l$ contains at most two points of $\\\\mathcal {A}$ .\",\n \"Since $\\\\mathcal {H}$ has size $q^k+2$ , it follows that $\\\\mathcal {H}$ is a hyperoval.\",\n \"Finally consider the group $G$ of elations in $\\\\operatorname{PG}(2hk,2)$ with axis the hyperplane at infinity $\\\\tilde{H}_\\\\infty $ .\",\n \"Since the points of $\\\\tilde{{Q}}$ form a subspace, we see that $G$ acts transitively on the points of $\\\\tilde{\\\\mathcal {Q}}$ .\",\n \"Every element of $G$ induces an element of the group $G^{\\\\prime }$ of elations in $\\\\operatorname{PG}(2,q^k)$ with axis the line $P_0P_\\\\infty $ .\",\n \"Hence, $G^{\\\\prime }$ acts transitively on the points of $\\\\mathcal {A}$ in $\\\\operatorname{PG}(2,q^k)$ .\",\n \"This shows that $\\\\mathcal {H}$ is a translation hyperoval.\"\n ],\n [\n \"Every translation hyperoval defines a linear set of pseudoregulus type\",\n \"In this section, we show that the vice versa part of Theorem REF holds.\",\n \"Proposition 3.16 Via the André/Bruck-Bose construction, the set of affine points of a translation hyperoval in $\\\\operatorname{PG}(2,q^k)$ , $q=2^h$ , where $h,k\\\\ge 2$ corresponds to a set $\\\\mathcal {Q}$ of $q^k$ affine points in $\\\\operatorname{PG}(2k,q)$ whose set of determined directions $D$ is an $\\\\mathbb {F}_2$ -linear set of pseudoregulus type.\",\n \"Consider a translation hyperoval $H\\\\in \\\\operatorname{PG}(2,q^k)$ .\",\n \"Without loss of generality we may suppose that $H=\\\\lbrace (1,t,t^{2^i})_{q^k} | t\\\\in \\\\mathbb {F}_{q^k}\\\\rbrace \\\\cup \\\\lbrace (0,1,0)_{q^k},(0,0,1)_{q^k}\\\\rbrace $ with $\\\\gcd (i,hk)=1$ .\",\n \"The set of affine points of $H$ corresponds to the set of points $H^{\\\\prime }=\\\\lbrace (1,t,t^{2^i})_{q} \\\\in \\\\mathbb {F}_q \\\\oplus \\\\mathbb {F}_{q^k} \\\\oplus \\\\mathbb {F}_{q^k} | t\\\\in \\\\mathbb {F}_{q^k}\\\\rbrace $ in $\\\\operatorname{PG}(2k,q)$ (for more information about the use of these coordinates for $H$ and $H^{\\\\prime }$ , see [15]).\",\n \"The determined directions in the hyperplane at infinity $H_\\\\infty : X_0=0$ have coordinates $(0,t_1-t_2,t_1^{2^i}-t_2^{2^i})_q$ where $t_1,t_2 \\\\in \\\\mathbb {F}_{q^k}$ .\",\n \"So the set $D=\\\\lbrace (0,u,u^{2^i} ) _q| u\\\\in \\\\mathbb {F}_{q^k} \\\\rbrace $ is precisely the set of directions determined by the points of $H$ .\",\n \"By Result REF we find that this set of directions $D$ is an $\\\\mathbb {F}_2$ -linear set of pseudoregulus type in the hyperplane $H_\\\\infty $ .\",\n \"We will now show that every line in $\\\\operatorname{PG}(2k-1,q)$ intersects the points of the linear set $D$ in $0,1,3$ or $q-1$ points.\",\n \"Proposition 3.17 Let $D$ be the set of points of an $\\\\mathbb {F}_2$ -linear set of pseudoregulus type in $\\\\operatorname{PG}(2k-1,q)$ , $q=2^h$ , $h>2$ , $k\\\\ge 2$ .\",\n \"Then every line of $\\\\operatorname{PG}(2k-1,q)$ meets $D$ in $0,1,3$ or $q-1$ points.\",\n \"We use the representation of Result REF for the points of $D$ .\",\n \"Let $R_1=(u_1,f(u_1))_q$ and $R_2=(u_2,f(u_2))_q$ , $u_1,u_2 \\\\in U_0$ , be two points of $D$ not on the same line of the pseudoregulus, so the vectors $\\\\langle u_1 \\\\rangle $ and $\\\\langle u_2 \\\\rangle $ in $V(k,q)$ are not an $\\\\mathbb {F}_q$ -multiple (in short $\\\\langle u_1 \\\\rangle _q \\\\ne \\\\langle u_2 \\\\rangle _q$ ).\",\n \"Recall that $f$ is a invertible semilinear map with automorphism $\\\\sigma \\\\in Aut(\\\\mathbb {F}_q)$ , $Fix(\\\\sigma )=\\\\lbrace 0,1\\\\rbrace $ .\",\n \"A third point $R_3=(u_3,f(u_3))_q\\\\in D$ is contained in $R_1R_2$ if and only if there are $\\\\mu ,\\\\lambda \\\\in \\\\mathbb {F}_q$ such that $\\\\left\\\\lbrace \\\\begin{array}{ll}u_1+\\\\lambda u_2&=\\\\mu u_3 \\\\\\\\f(u_1)+\\\\lambda f(u_2) &= \\\\mu f(u_3)\\\\end{array}\\\\right.\\\\Leftrightarrow \\\\left\\\\lbrace \\\\begin{array}{ll}f(u_1)+\\\\lambda ^\\\\sigma f(u_2)&=\\\\mu ^\\\\sigma f(u_3) \\\\\\\\f(u_1)+\\\\lambda f(u_2) &= \\\\mu f(u_3)\\\\end{array} \\\\right.\\\\\\\\\\\\Leftrightarrow \\\\left\\\\lbrace \\\\begin{array}{ll}u_1+\\\\lambda u_2&=\\\\mu u_3 \\\\\\\\(\\\\lambda ^\\\\sigma - \\\\lambda ) f(u_2) &= f((\\\\mu -\\\\mu ^{\\\\sigma ^{-1}})u_3)\\\\end{array} \\\\right.\\\\Leftrightarrow \\\\left\\\\lbrace \\\\begin{array}{ll}u_1+\\\\lambda u_2&=\\\\mu u_3 \\\\\\\\(\\\\lambda ^\\\\sigma - \\\\lambda )^{\\\\sigma ^{-1}} u_2 &= (\\\\mu -\\\\mu ^{\\\\sigma ^{-1}})u_3\\\\end{array} \\\\right.$ As $R_2$ and $R_3$ lie on different $(q-1)$ -secants to $D$ , we have that $\\\\langle u_2 \\\\rangle _q \\\\ne \\\\langle u_3 \\\\rangle _q$ .\",\n \"It follows that $\\\\lambda ^\\\\sigma - \\\\lambda =\\\\mu -\\\\mu ^{\\\\sigma ^{-1}}=0$ , so $\\\\lambda ,\\\\mu \\\\in Fix(\\\\sigma )=\\\\lbrace 0,1\\\\rbrace $ .\",\n \"We find that there is only one solution of this system, such that $R_1\\\\ne R_3$ (i.e.\",\n \"$\\\\langle u_1 \\\\rangle _q \\\\ne \\\\langle u_3 \\\\rangle _q$ ), namely when $\\\\lambda =\\\\mu =1$ .\",\n \"Hence, given two points $R_1,R_2$ in $D$ , there is an unique point $R_3\\\\in D\\\\cap R_1R_2$ , different from $R_1$ and $R_2$ .\"\n ],\n [\n \"The generalisation of a characterisation of Barwick and Jackson\",\n \"Using Theorem REF , we are now able to generalise the following result of Barwick-Jackson which concerns translation hyperovals in $\\\\operatorname{PG}(2,q^2)$ ([4]).\",\n \"Result 4.1 [4] Consider $\\\\operatorname{PG}(4,q)$ , $q$ even, $q>2$ , with the hyperplane at infinity denoted by $\\\\Sigma _\\\\infty $ .\",\n \"Let $\\\\mathcal {C}$ be a set of $q^2$ affine points, called $\\\\mathcal {C}$ -points and consider a set of planes called $\\\\mathcal {C}$ -planes which satisfies the following: (A1) Each $\\\\mathcal {C}$ -plane meets $\\\\mathcal {C}$ in a $q$ -arc.\",\n \"(A2) Any two distinct $\\\\mathcal {C}$ -points lie in a unique $\\\\mathcal {C}$ -plane.\",\n \"(A3) The affine points that are not in $\\\\mathcal {C}$ lie on exactly one $\\\\mathcal {C}$ -plane.\",\n \"(A4) Every plane which meets $\\\\mathcal {C}$ in at least 3 points either meets $\\\\mathcal {C}$ in 4 points or is a $\\\\mathcal {C}$ -plane.\",\n \"Then there exists a Desarguesian spread $\\\\mathcal {S}$ in $\\\\Sigma _\\\\infty $ such that in the Bruck-Bose plane $\\\\mathcal {P}(\\\\mathcal {S})\\\\cong \\\\operatorname{PG}(2,q^2)$ , the $\\\\mathcal {C}$ -points, together with 2 extra points on $\\\\ell _\\\\infty $ form a translation hyperoval in $\\\\operatorname{PG}(2,q^2)$ .\",\n \"Remark 4.2 At two different points, the proofs of [4] are inherently linked to the fact that they are dealing with hyperovals in $\\\\operatorname{PG}(2,q^2)$ .\",\n \"In [4] the authors show the existence of a design which is isomorphic to an affine plane, of which they later need to use the parallel classes.\",\n \"In [4], they use the Klein correspondence to represent lines in $\\\\operatorname{PG}(3,q)$ in $\\\\operatorname{PG}(5,q)$ .\",\n \"Both techniques cannot be extended in a straightforward way to $q^k$ , $k>2$ .\",\n \"The following Proposition shows that a set of $\\\\mathcal {C}$ -planes as defined by Barwick and Jackson in [4] (using $\\\\operatorname{PG}(2k,q)$ instead of $\\\\operatorname{PG}(4,q)$ ) satisfies the conditions of Theorem REF .\",\n \"Proposition 4.3 Consider $\\\\operatorname{PG}(2k,q)$ , $q$ even, $q>2$ , with the hyperplane at infinity denoted by $\\\\Sigma _\\\\infty $ .\",\n \"Let $\\\\mathcal {C}$ be a set of $q^k$ affine points, called $\\\\mathcal {C}$ -points and consider a set of planes called $\\\\mathcal {C}$ -planes which satisfies the following: (A1) Each $\\\\mathcal {C}$ -plane meets $\\\\mathcal {C}$ in a $q$ -arc.\",\n \"(A2) Any two distinct $\\\\mathcal {C}$ -points lie in a unique $\\\\mathcal {C}$ -plane.\",\n \"(A3) The affine points that are not in $\\\\mathcal {C}$ lie on exactly one $\\\\mathcal {C}$ -plane.\",\n \"(A4) Every plane which meets $\\\\mathcal {C}$ in at least 3 points either meets $\\\\mathcal {C}$ in 4 points or is a $\\\\mathcal {C}$ -plane.\",\n \"Then $\\\\mathcal {C}$ determines a set of $q^k-1$ directions $D$ in $\\\\Sigma _\\\\infty $ such that every line of $\\\\Sigma _\\\\infty $ meets $D$ in $0,1,3$ or $q-1$ points.\",\n \"As before, we call the points that are not contained in $\\\\Sigma _\\\\infty $ affine points.\",\n \"Note that all $\\\\mathcal {C}$ -points are affine.\",\n \"Since every two $\\\\mathcal {C}$ -points lie on a $\\\\mathcal {C}$ -plane which meets $\\\\mathcal {C}$ in a $q$ -arc, we have that no three $\\\\mathcal {C}$ -points are collinear.\",\n \"Let $P_0$ be a $\\\\mathcal {C}$ -point and let $D_0$ be the set of points of the form $P_0P_i\\\\cap \\\\Sigma _\\\\infty $ , where $P_i\\\\ne P_0$ is a point of $\\\\mathcal {C}$ .\",\n \"We first show that every line meets $D_0$ in $0,1,3$ or $q-1$ points.\",\n \"Let $M$ be a line of $\\\\Sigma _\\\\infty $ containing 2 points of $D_0$ , say $R_1^{\\\\prime }=P_0R_1\\\\cap \\\\Sigma _\\\\infty $ , $R_2^{\\\\prime }=P_0R_2\\\\cap \\\\Sigma _\\\\infty $ , where $R_1,R_2\\\\in \\\\mathcal {C}$ .\",\n \"Then $\\\\langle M,P_0\\\\rangle $ contains at least 3 points of $\\\\mathcal {C}$ , and hence, by (A4), either it is a $\\\\mathcal {C}$ -plane or it contains exactly 4 points of $\\\\mathcal {C}$ .\",\n \"If $\\\\langle M,P_0\\\\rangle $ is a $\\\\mathcal {C}$ -plane, it contains $q$ points of $\\\\mathcal {C}$ forming a $q$ -arc, and hence, $M$ contains $q-1$ points of $D_0$ .\",\n \"Now suppose that $\\\\langle M,P_0\\\\rangle $ contains exactly 4 $\\\\mathcal {C}$ -points, then $M$ contains 3 points of $D_0$ .\",\n \"Now let $P_1\\\\ne P_0$ be a point of $\\\\mathcal {C}$ and let $D_1$ be the set of points of the form $P_1P_i\\\\cap \\\\Sigma _\\\\infty $ , where $P_i\\\\ne P_1$ is a point of $\\\\mathcal {C}$ .\",\n \"We claim that $D_0=D_1$ .\",\n \"Let $P_1^{\\\\prime }=P_0P_1\\\\cap \\\\Sigma _\\\\infty $ .\",\n \"We see that $P_1^{\\\\prime }\\\\in D_0\\\\cap D_1$ .\",\n \"Consider a point $P_2^{\\\\prime }\\\\ne P_1^{\\\\prime }$ in $D_0$ , then $P_0P_2\\\\cap \\\\Sigma _\\\\infty =P_2^{\\\\prime }$ for some $P_2\\\\in \\\\mathcal {C}$ .\",\n \"Consider the plane $\\\\pi =\\\\langle P_0,P_1,P_2\\\\rangle $ .\",\n \"Suppose first that $\\\\pi $ is not a $\\\\mathcal {C}$ -plane, then, by (A4), $\\\\pi $ contains exactly one extra point, say $P_3$ of $\\\\mathcal {C}$ .\",\n \"The lines $P_0P_1$ and $P_2P_3$ lie in $\\\\pi $ and hence, meet in a point $Q$ .\",\n \"By $(A2)$ , there is a $\\\\mathcal {C}$ -plane $\\\\mu $ through $P_0P_1$ , and likewise, there is a $\\\\mathcal {C}$ -plane $\\\\mu ^{\\\\prime }$ through $P_2P_3$ .\",\n \"Since $\\\\pi $ is not a $\\\\mathcal {C}$ -plane, $\\\\mu $ and $\\\\mu ^{\\\\prime }$ are two distinct $\\\\mathcal {C}$ -planes through $Q$ .\",\n \"By (A3) his implies that $Q$ is a point of $\\\\Sigma _\\\\infty $ .\",\n \"Likewise, $P_0P_2\\\\cap P_1P_3$ and $P_0P_3\\\\cap P_1P_2$ are points of $\\\\Sigma _\\\\infty $ .\",\n \"It follows that $D_0\\\\cap \\\\pi =D_1\\\\cap \\\\pi $ .\",\n \"This argument shows that for all points $R\\\\ne P_1^{\\\\prime }\\\\in D_0$ such that $\\\\langle P_0,P_1,R\\\\rangle $ is not a $\\\\mathcal {C}$ -plane, we have that $R\\\\in D_1$ .\",\n \"Now $P_0P_1$ lies on a unique $\\\\mathcal {C}$ -plane, say $\\\\nu $ .\",\n \"Let $\\\\nu \\\\cap \\\\Sigma _\\\\infty =L$ , then we have shown that $\\\\langle P_0,P_1,R\\\\rangle $ is not a $\\\\mathcal {C}$ -plane as long as $R\\\\in \\\\Sigma _\\\\infty $ is not on $L$ .\",\n \"We conclude that $D_0\\\\setminus L=D_1\\\\setminus L$ .\",\n \"Now assume that $D_0\\\\ne D_1$ and let $X$ be a point in $D_1$ which is not contained in $D_0$ .\",\n \"Then $X\\\\in L$ and $P_1X$ contains a point $Y\\\\ne P_1\\\\in \\\\mathcal {C}$ .\",\n \"Consider a point $P_4^{\\\\prime }\\\\in D_1$ , not on $L$ , then $P_1P_4^{\\\\prime }$ contains a point $P_4\\\\ne P_1$ of $\\\\mathcal {C}$ .\",\n \"Since $P_4^{\\\\prime }\\\\in D_1\\\\setminus L$ , $P_4^{\\\\prime }\\\\in D_0$ so the line $P_4^{\\\\prime }P_0$ contains a point $P_5\\\\ne P_1$ of $\\\\mathcal {C}$ .\",\n \"The plane $\\\\langle P_1,P_4^{\\\\prime },X\\\\rangle $ is not a $\\\\mathcal {C}$ -plane since otherwise, the points $P_1$ and $Y$ of $\\\\mathcal {C}$ would lie in two different $\\\\mathcal {C}$ -planes.\",\n \"This implies that $\\\\langle P_1,P_4,X\\\\rangle $ which contains the $\\\\mathcal {C}$ -points $P_1,P_4,Y$ contains exactly one extra point of $\\\\mathcal {C}$ , say $P_6$ .\",\n \"Denote $P_1P_6\\\\cap \\\\Sigma _\\\\infty $ by $P^{\\\\prime }_6$ .\",\n \"We see that there are exactly 3 points of $D_1$ on the line $P_4^{\\\\prime }X$ , namely $P_4^{\\\\prime },X$ and $P_6^{\\\\prime }$ .\",\n \"Now $P^{\\\\prime }_6$ is a point of $D_1$ , not on $L$ , so $P^{\\\\prime }_6\\\\in D_0$ .\",\n \"Hence, there is a point $S\\\\ne P_0\\\\in \\\\mathcal {C}$ on the line $P_0P^{\\\\prime }_6$ .\",\n \"If $\\\\langle P_4^{\\\\prime },P^{\\\\prime }_6,P_0\\\\rangle $ is not a $\\\\mathcal {C}$ -plane, then, since it contains $P_0,P_5,S$ of $\\\\mathcal {C}$ it contains precisely 3 points of $D_0$ at infinity.\",\n \"These are the points $P_4^{\\\\prime },P_6^{\\\\prime }$ and one other point, say $T$ , which needs to be different from $X$ by our assumption that $X\\\\notin D_0$ .\",\n \"That implies that $T$ is not on $L$ , and hence, $T\\\\in D_1$ .\",\n \"This is a contradiction since we have seen that the only points of $D_1$ on $P_4^{\\\\prime }X$ are $P_4^{\\\\prime },X$ and $P_6^{\\\\prime }$ .\",\n \"Now if $\\\\langle P_4^{\\\\prime },P_6,P_0\\\\rangle $ is a $\\\\mathcal {C}$ -plane, we find $q-1$ points of $D_0$ on $P_4^{\\\\prime }X$ , all of them are not on $L$ .\",\n \"Hence, we find $q-1$ points of $D_1$ on $P_4^{\\\\prime }X$ , not on $L$ .\",\n \"This is again a contradiction since $P_4^{\\\\prime }X$ has only the points $P_4^{\\\\prime }$ and $P_6^{\\\\prime }$ of $D_1$ not on $L$ .\",\n \"This proves our claim that $D_0=D_1$ .\",\n \"Since $P_1$ was chosen arbitrarily, different from $P_0$ , and $D_0=D_1$ , we find that the set $D$ of directions determined by $\\\\mathcal {C}$ is precisely the set $D_0$ .\",\n \"The statement now follows from the fact that a line meets $D_0$ in $0,1,3$ or $q-1$ points.\",\n \"Proposition REF shows that the set $\\\\mathcal {C}$ satisfies the criteria of Theorem REF .\",\n \"Hence, we find the following generalisation of Result REF .\",\n \"Theorem 4.4 Consider $\\\\operatorname{PG}(2k,q)$ , $q$ even, $q>2$ , with the hyperplane at infinity denoted by $\\\\Sigma _\\\\infty $ .\",\n \"Let $\\\\mathcal {C}$ be a set of $q^k$ affine points, called $\\\\mathcal {C}$ -points and consider a set of planes called $\\\\mathcal {C}$ -planes which satisfies the following: (A1) Each $\\\\mathcal {C}$ -plane meets $\\\\mathcal {C}$ in a $q$ -arc.\",\n \"(A2) Any two distinct $\\\\mathcal {C}$ -points lie in a unique $\\\\mathcal {C}$ -plane.\",\n \"(A3) The affine points that are not in $\\\\mathcal {C}$ lie on exactly one $\\\\mathcal {C}$ -plane.\",\n \"(A4) Every plane which meets $\\\\mathcal {C}$ in at least 3 points either meets $\\\\mathcal {C}$ in 4 points or is a $\\\\mathcal {C}$ -plane.\",\n \"Then there exists a Desarguesian spread $\\\\mathcal {S}$ in $\\\\Sigma _\\\\infty $ such that in the Bruck-Bose plane $\\\\mathcal {P}(\\\\mathcal {S})\\\\cong \\\\operatorname{PG}(2,q^k)$ , the $\\\\mathcal {C}$ -points, together with 2 extra points on $\\\\ell _\\\\infty $ form a translation hyperoval in $\\\\operatorname{PG}(2,q^k)$ .\"\n ]\n]"},"id":{"kind":"string","value":"1906.04537"}}},{"rowIdx":2793,"cells":{"text":{"kind":"list like","value":[["Stochastic Neural Network with Kronecker Flow"],["Abstract Recent advances in variational inference enable the modelling of highly structured joint distributions, but are limited in their capacity to scale to the high-dimensional setting of stochastic neural networks.","This limitation motivates a need for scalable parameterizations of the noise generation process, in a manner that adequately captures the dependencies among the various parameters.","In this work, we address this need and present the Kronecker Flow, a generalization of the Kronecker product to invertible mappings designed for stochastic neural networks.","We apply our method to variational Bayesian neural networks on predictive tasks, PAC-Bayes generalization bound estimation, and approximate Thompson sampling in contextual bandits.","In all setups, our methods prove to be competitive with existing methods and better than the baselines."],["Introduction","Work done while Chin-Wei was an intern at Element AI and Ahmed at Facebook Research.","Stochastic neural networks (SNN) are a central tool in many subfields of machine learning, including (1) Bayesian deep learning [33], [6], [19], [14], (2) exploration in reinforcement learning [23], [39], [43], and (3) statistical learning theory such as PAC-Bayesian learning [34], [29], [12].","Perturbations of the network parameters induce a distribution over the model, and this intrinsic uncertainty is the subject of great interest to machine learning practitioners and theoreticians alike.","For example, deep Bayesian models are often used to adequately measure uncertainty, and determine whether the model itself is inherently familiar with the unseen data.","This is especially important in the context of autonomous vehicles, where decisions must be made to meet specific safety standards [35].","Conversely, the lack of confidence can be leveraged to efficiently guide exploration in reinforcement learning, via randomizing the approximate value function [2], [49] or maximizing intrinsic rewards [20].","Furthermore, a considerable proportion of statistical learning theory is devoted to understanding what implies generalization, or what constitutes an appropriate measure of complexity [3], [1], [38].","PAC-Bayesian learning theory [34] specifically explores the generalization property of a randomized prediction rule, and has been recently studied in the context of stochastic neural networks [12].","In this particular study, the working hypothesis is that good generalization can be guaranteed on the premise that stochastic gradient descent [45] finds a solution that obtains certain structural properties, such as flatness.","For computational reasons, considerable effort has been devoted to modelling uncertainty through the injection of independent noise to the network parameters [18], [6], [26].","However, noise independence largely restricts the expressivity of the noise distribution and thus the resulting uncertainty measures are ill-calibrated [36], [50].","Attempts have been made to correlate parameters of a neural network, including [32], [28], [40], for example, by adapting expressive non-linear invertible transformations developed in the variational inference literature [42], [25], [21], or via implicit methods [17].","However, these methods are limited due to their inability to scale well.","[32], for instance, resort to a specific multiplicative noise sampled from a lower dimensional space and have to use an auxiliary method to bound the entropy.","[28], on the other hand, give up on injecting noise on the entire set of parameters and model the distribution of the scale and shift parameter of the pre-activations.","In attempts to address some of the challenges articulated above and efficiently model the joint distribution of a network's parameters, we take inspiration from the Kronecker product, which we notice can be thought of as left-transforming a matrix via a linear map, and then right-transforming it using another linear map, thus providing us an efficient way to correlate the weight parameters.","We propose the Kronecker Flow, an invertible transformation-based method that generalizes the Kronecker product to its nonlinear counterparts.","Our contributions are as follows.","We extend the idea of Kronecker product to more general invertible mappings to induce non-linear dependencies, and apply this trick to parameterizing deep stochastic neural networks.","We apply our method to predictive tasks and show that our methods work better on larger architectures compared to existing methods.","We are the first to apply flow-based methods to tighten the PAC-Bayes bound.","We show that the KL divergence in the PAC-Bayes bound can be estimated with high probability, and demonstrate the generalization gap can be further reduced and explained by leveraging the structure in the parameter space.","Our methods prove to be competitive over other methods in approximate Thompson sampling in contextual bandit problems."],["Background","Stochastic neural networks with parameter perturbation normally follow the stochastic process: $\\Theta \\sim q_\\phi (\\Theta )$ , $y|x\\sim p(y|x, \\Theta )=f_\\Theta (x)$ , where $\\Theta $ is the parameters of the neural network $f$ , which outputs the prediction probability vector for classification or the predicted values for regression.","We let $D=\\lbrace (x_i,y_i):{i\\in [m]}\\rbrace $ be the training set of size $m$ We use the notation $[n]$ to compactly describe the set of integers $\\lbrace 1,2,\\dots ,n\\rbrace $ ., $H$ be the differential entropy $H[q]=- \\mathbb {E}_q[\\log q]$ , $\\beta >0$ be the coefficient controlling the amount of noise injected into the model and the degree of regularization, $l(y,\\bar{y})$ be the loss function and $\\hat{R}_{D}(\\Theta )=\\frac{1}{m}\\sum _{i=1}^{m} l(y_i, f_\\Theta (x_i))$ be the empirical risk."],["Variational Bayesian neural networks","Variational Bayesian neural networks are a type of stochastic neural network.","Bayesian inference updates our prior belief $p(\\Theta )$ over the model parameters according to the Bayes rule $p(\\Theta |D)\\propto p(D|\\Theta )p(\\Theta )$ , by incorporating information from the training set through the likelihood function $p(D|\\Theta )$ .","Variational inference is a computational realization of Bayesian inference, which casts inference as an optimization problem, where one maximizes the variational lower bound (also known as the evidence lower bound, or the ELBO) on the log marginal likelihood: $\\log p(D) \\ge \\mathbb {E}_{q_\\phi }[ \\log p(D|\\Theta ) + \\log p(\\Theta )] + H(q_\\phi (\\Theta )),$ where $q_\\phi $ is the variational approximate posterior and $p(D|\\Theta )$ can be decomposed into $\\prod _{i=1}^{m} p(y_i|x_i, \\Theta )$ due to conditional independence assumption.","The optimal $q$ is the true posterior, i.e.","$q^*(\\Theta )=\\frac{p(D|\\Theta )p(\\Theta )}{p(D)}$ .","In our case, we use $\\Theta $ to parameterize a neural network.","Prediction can be carried out via the (approximate) predictive posterior $p(y|x,D) & = \\mathbb {E}_{\\Theta \\sim p(\\Theta |D)} [p(y|x,\\Theta )] \\\\& \\approx \\mathbb {E}_{\\Theta \\sim q_\\phi (\\Theta )}[p(y|x,\\Theta )] \\\\& \\approx \\frac{1}{K}\\sum _{k=1}^K p(y|x,\\Theta _k)$ for $\\lbrace \\Theta _k\\rbrace _{k\\in [K]}$ drawn i.i.d.","from $q_\\phi (\\Theta )$ , where we use the variational distribution $q$ to approximate $p(\\Theta |D)$ and a Monte Carlo estimate to estimate the integral.","The prior distribution can be used to encode some form of inductive bias, such as one that is in favor of parameter values closer to some $\\Theta _0$ chosen a priori.","We choose the prior to be an isotropic Gaussian, centered at the random initialization $\\Theta _0$ , i.e., $p(\\Theta ) = {\\mathcal {N}}(\\Theta ;\\Theta _0, \\lambda {\\mathbf {I}})$ .","The entropy term ensures the variational posterior does not collapse to a point estimate.","Both of them can be thought of as some form of regularizer, so we attach a coefficient $\\beta $ in front of them as a hyperparameter Like the $\\lambda $ parameter in [53]."],["PAC-Bayes generalization bound","Another use case of stochastic neural networks is to understand generalization, via PAC-Bayes bounds.","The aim is to bound a divergence between the empirical risk, $\\hat{{\\mathcal {L}}}[q]=\\mathbb {E}_q[\\hat{R}_D(\\Theta )]$ , and the risk measured on the true distribution $\\mathcal {D}$ , ${\\mathcal {L}}[q]=\\mathbb {E}_q[\\mathbb {E}_{\\mathcal {D}}[l(y,f_\\Theta (x))]]$ .","While this quantity is unbounded in the general case, assuming a bounded loss function $l$ (e.g.",": zero-one loss), we can obtain a probabilistic bound that holds with probability $1-\\delta $ over the choice of $D$ , for $\\delta >0$ .","More specifically, with probability $1-\\delta $ , $\\Delta (\\hat{{\\mathcal {L}}}[q], {\\mathcal {L}}[q])\\le \\Omega (D_{\\mathrm {KL}}(q||p),m,\\delta )$ , where $\\Omega $ is a measure of complexity that scales proportionally with the Kullback–Leibler (KL) divergence and $\\Delta $ a measure of divergence (e.g.",": square distance or convex functions [16]).","For instance, [12] minimize the following bound originally due to [34] and then tightened by [29]: Theorem 1 Let $l$ be the zero-one loss.","For any $\\delta >0$ and data distribution ${\\mathcal {D}}$ , and any distribution $p$ on the space of $\\Theta $ , with probability at least $1-\\delta $ over the choice of a training set $D\\sim {\\mathcal {D}}^m$ , for all distributions $q$ on the space of $\\Theta $ , $D_{\\mathrm {KL}}(\\hat{{\\mathcal {L}}}[q]||{\\mathcal {L}}[q])\\le \\frac{D_{\\mathrm {KL}}(q||p)+\\log \\frac{m}{\\delta }}{m-1},$ where the KL on the LHS is between two Bernoulli distributions, defined by the probability of performing an error.","We refer to the above bound as the McAllester bound.","The KL divergence on the RHS of the bound, also known as the information gain, tells us to what extent the posterior $q$ is dependent on the training data.","The sharper and more confident $q$ is, and the farther away it is from the prior $p$ , the larger the KL will be, which in turn is reflected by the larger bound on the generalization gap.","This is consistent with traditional notion of bias-variance trade-off.","Alternatively, we consider the following bound due to [9]: Theorem 2 With the $l$ , $\\delta $ , $\\mathcal {D}$ , and $p$ as defined in Theorem REF , and with a fixed $\\beta >1/2$ , the following bound holds with probability over $1-\\delta $ : ${\\mathcal {L}}[q] &\\le \\frac{1}{1-\\frac{1}{2\\beta }} \\left(\\hat{{\\mathcal {L}}}[q] + \\frac{\\beta }{m} \\left( D_{\\mathrm {KL}}(q||p) + \\ln \\frac{1}{\\delta } \\right) \\right).$ We refer to this bound as the Catoni bound.","We notice the linear relationship (which is also noticed by [15]) between the empirical risk and the KL divergence.","This allows us to make use of the linearity of expectation to perform change of variable (see the next section).","We also note that the optimal $\\beta $ in Equation REF is always larger than 1, so the PAC-Bayes bound is actually more conservative than Bayesian inference in this sense."],["Normalizing flows","Minimization of Equation REF and REF requires (i) computing the gradient with respect to the parameter of the (PAC-)Bayesian posterior $\\phi $ , and (ii) computing the entropy of $q$ .","One approach to do this is via change of variable under an invertible mapping.","Let $\\mathbf {\\epsilon }\\sim q_0$ be a random variable in $\\mathbb {R}^d$ , and ${\\mathbf {g}}_\\phi :\\mathbb {R}^d\\rightarrow \\mathbb {R}^d$ be a bijection parameterized by $\\phi $ .","Let $\\Theta ={\\mathbf {g}}_\\phi (\\mathbf {\\epsilon })$ and $q_\\phi $ be its density.","Then we can rewrite the loss function as Since the weighting coefficient $\\beta $ can be absorbed into the loss function $l$ , we neglect it for simplicity now.","$\\mathbb {E}_{\\Theta }[\\hat{R}_{D}(\\Theta ) + \\log q_\\phi (\\Theta )]& =\\mathbb {E}_{\\mathbf {\\epsilon }}[\\hat{R}_{D}({\\mathbf {g}}_\\phi (\\mathbf {\\epsilon })) + \\log q_0(\\mathbf {\\epsilon }) \\\\& - \\log \\left|\\det \\frac{\\partial {\\mathbf {g}}_\\phi (\\mathbf {\\epsilon })}{\\partial \\mathbf {\\epsilon }}\\right|],$ where we apply the change of variable (see Appendix for the detailed derivation).","The log-determinant (logdet) term ensures that we obtain a valid probability density function after $g_\\phi $ is applied, which can be a sequence of invertible mappings itself, hence referred to as the normalizing flow [42].","This way, the random variable and the parameters are decoupled, so that we can differentiate the integrand to have an unbiased estimate of the gradient (fixing some $\\mathbf {\\epsilon }\\sim q_0$ ).","We let $q_0$ be the standard normal."],["Kronecker Flows","We consider maximizing the ELBO and minimizing the Catoni bound via normalizing flow-based SNNs.","Conventionally, mean-field approximation using factorized distributions (such as multivariate Gaussian with diagonal covariance) has been well explored in the variational inference (VI) literature [6].","We are interested in better capturing the structure in the parameter space as restricted VI methods are known to exhibit overconfidence [36], [50].","However, the parameters of a neural network are usually very high dimensional (on the order of millions), requiring a novel way to parameterize the joint distribution over the parameters.","In its general form, neural networks can be represented by a collection of tensors i.e.","$\\Theta =\\lbrace {\\mathbf {W}}_l:{l\\in [L]}\\rbrace $ .","While our method below can easily be generalized to high-dimensional tensors (such as for convolutional kernels), to simplify notation, we describe the matrix form."],["Linear Kronecker Flow","The matrix-variate normal ($\\mathcal {MN}$ ) distribution generalizes the multivariate normal distribution to matrix-valued random variables, such as weight matrices of a neural network [31].","Matrix normal is a multivariate normal distribution whose covariance matrix is a Kronecker product ($\\otimes $ ), which allows us to model the correlation among the parameters to some degree.","More concretely, assume ${\\mathbf {E}}_{ij}\\overset{\\textnormal {i.i.d.","}}{\\sim } {\\mathcal {N}}(0,1)$ is an $n\\times p$ random Gaussian matrix, and ${\\mathbf {A}}\\in \\mathbb {R}^{n\\times n}$ , ${\\mathbf {B}}\\in \\mathbb {R}^{p\\times p}$ and ${\\mathbf {M}}\\in \\mathbb {R}^{n\\times p}$ are real-valued matrices.","Then ${\\mathbf {M}}+ {\\mathbf {A}}{\\mathbf {E}}{\\mathbf {B}}$ has a matrix normal distribution, as $\\textnormal {vec}({\\mathbf {M}}+ {\\mathbf {A}}{\\mathbf {E}}{\\mathbf {B}})\\sim {\\mathcal {N}}(\\textnormal {vec}({\\mathbf {M}}), {\\mathbf {B}}^\\top {\\mathbf {B}}\\otimes {\\mathbf {A}}{\\mathbf {A}}^\\top ),$ where $\\textnormal {vec}$ is the vectorization of a matrix that concatenates all the columns.","This allows us to represent the covariance matrix in a more compact manner ($n^2p^2/2$ parameters versus $n^2/2+p^2/2$ parameters for Kronecker product)."],["Limitation of the Kronecker product.","The Kronecker product covariance matrix is not a strict generalization of diagonal covariance matrix.","To observe this, let ${\\mathbf {U}}=\\mathop {\\mathrm {diag}}\\nolimits ({\\mathbf {u}})$ , ${\\mathbf {V}}=\\mathop {\\mathrm {diag}}\\nolimits ({\\mathbf {v}})$ (this is the case of [31]), and ${\\mathbf {S}}=\\mathop {\\mathrm {diag}}\\nolimits ({\\mathbf {s}})$ , where ${\\mathbf {u}}\\in \\mathbb {R}^n_{>0}$ , ${\\mathbf {v}}\\in \\mathbb {R}^p_{>0}$ , and ${\\mathbf {s}}\\in \\mathbb {R}^{np}_{>0}$ .","Then ${\\mathbf {U}}\\otimes {\\mathbf {V}}$ is also a diagonal matrix of size $np\\times np$ .","Equating ${\\mathbf {U}}\\otimes {\\mathbf {V}}={\\mathbf {S}}$ to solve for ${\\mathbf {u}}$ and ${\\mathbf {v}}$ will result in $np$ nonlinear equations with $n+p$ variables, which can be over-determined for $n,p>2$ .","For example, let $n=2,p=3$ , and ${\\mathbf {s}}=[1,\\epsilon ,\\epsilon ,1,1,1]$ for some $\\epsilon >0$ .","Then the nonlinear system below does not have a solution: ${\\mathbf {U}}\\otimes {\\mathbf {V}}= {\\mathbf {S}}\\quad \\Longleftrightarrow & \\quad {\\mathbf {u}}_1{\\mathbf {v}}_1\\overset{(a)}{=}1 \\quad {\\mathbf {u}}_1{\\mathbf {v}}_2\\overset{(b)}{=}\\epsilon \\quad {\\mathbf {u}}_1{\\mathbf {v}}_3\\overset{(c)}{=}\\epsilon \\quad \\\\& {\\mathbf {u}}_2{\\mathbf {v}}_1\\overset{(d)}{=}1 \\quad {\\mathbf {u}}_2{\\mathbf {v}}_2\\overset{(e)}{=}1 \\quad {\\mathbf {u}}_2{\\mathbf {v}}_3\\overset{(f)}{=}1$ To see this, dividing $(a)$ by $(b)$ and dividing $(d)$ by $(e)$ yield ${\\mathbf {v}}_1={\\mathbf {v}}_2/\\epsilon $ and ${\\mathbf {v}}_1={\\mathbf {v}}_2$ , respectively, which doesn't have a solution if $\\epsilon \\ne 1$ .","This is because the Kronecker product is essentially parameter sharing, which can heavily restrict the matrix it can represent.","To remedy the above limitation, we can further decouple the reparameterization of the parameter matrix into two parts: (1) one that models the marginal variance and (2) one that models correlations.","Assume ${\\mathbf {S}}\\in \\mathbb {R}^{n\\times p}_{>0}$ is a positive-valued matrix, and let ${\\mathbf {W}}:={\\mathbf {M}}+ {\\mathbf {A}}({\\mathbf {E}}\\circ {\\mathbf {S}}){\\mathbf {B}}$ .","Then $\\textnormal {vec}({\\mathbf {W}})$ is a Gaussian distribution with the following property, which is useful in calculating the KL divergence: Property 1 Let ${\\mathbf {W}}$ be given as above, with $\\mu =\\mathbb {E}[\\textnormal {vec}({\\mathbf {W}})]$ and $\\Sigma =\\mathrm {Var}(\\textnormal {vec}({\\mathbf {W}}))$ .","Then $\\mu =\\textnormal {vec}({\\mathbf {M}})$ , and $\\Sigma =({\\mathbf {B}}^\\top \\otimes {\\mathbf {A}}) \\mathop {\\mathrm {diag}}\\nolimits (\\textnormal {vec}({\\mathbf {S}}^2)) ({\\mathbf {B}}\\otimes {\\mathbf {A}}^\\top )$ $\\det (\\Sigma )=\\det ({\\mathbf {A}})^{2p}\\det ({\\mathbf {B}})^{2n}\\prod _{ij}{\\mathbf {S}}_{ij}^2$ $\\textnormal {Tr}(\\Sigma )=\\sum _{ij}\\left({\\mathbf {A}}^2{\\mathbf {S}}^2{\\mathbf {B}}^2\\right)_{ij}$ See Appendix for the derivation and interpretation of the property.","Naive implemetations of this can be inefficient and numerically unstable, as the entropy term involves computing the log-determinant of ${\\mathbf {A}}$ and ${\\mathbf {B}}$ , requiring the standard automatic differentiation libraries to resort to singular value decomposition when the matrix is near-singular.","Thus, we choose to parameterize ${\\mathbf {A}}$ and ${\\mathbf {B}}$ as lower triangular matricesThis is achieved by masking.","with ones on the diagonal, leaving the uncertainty to be modeled by ${\\mathbf {S}}$ .","This means $\\det (\\Sigma )=\\prod _{ij}{\\mathbf {S}}_{ij}^2$ .","Figure: Random 3D Gaussian tensor"],["Simulation.","To validate the limited expressiveness of kronecker product, we randomly initialize a target density $p$ to be a multivariate Gaussian with mean zero, and covariance being the square of a random standard Gaussian matrix.","We choose the dimensionality $d$ of the Gaussian such that it can be decomposed into a product of integers, and parameterize $q$ using independent Gaussian (dubbed Diag), the Kronecker product with diagonal $A$ and $B$ (K-Diag), and the Kronecker product with elementwise scaling (K-Linear).","We minimize $D_{\\mathrm {KL}}(q||p)$ ; see Figure REF for the results.","We also conduct the same experiment with 3D tensors (instead of matrices).","We see that K-Diag consistently underperforms when compared to Diag, which indicates parameter sharing does restrict the family of distributions it can represent, and K-Linear is consistently better as it captures some correlation."],["Nonlinear Kronecker Flow","In this section, we generalize the Kronecker product to more general non-linear mappings.","In Appendix , we make a connection to non-decreasing triangle maps [52] that are general enough to model any probability distributions.","First, notice that left-multiplying ${\\mathbf {E}}$ by ${\\mathbf {A}}$ amounts to introducing linear correlation among the $n$ rows of ${\\mathbf {E}}$ , applied to each of the $p$ columns.","Likewise, right-multiplying ${\\mathbf {E}}$ by ${\\mathbf {B}}$ amounts to correlating column entries of each row of ${\\mathbf {E}}$ .","Inspired by this, we consider applying an invertible mapping to each row of the random weight matrix, and another invertible mapping to each column of the matrix.","We call this the Kronecker Flow To differentiate this from K-Linear from the previous section, we refer to using non-linear ${\\mathbf {g}}$ as K-Nonlinear..","Specifically, let ${\\mathbf {g}}_A:\\mathbb {R}^{n}\\rightarrow \\mathbb {R}^{n}$ and ${\\mathbf {g}}_B:\\mathbb {R}^{p}\\rightarrow \\mathbb {R}^{p}$ be invertible mappings.","We define the matrix-matrix function ${\\mathbf {G}}:\\mathbb {R}^{n\\times p}\\rightarrow \\mathbb {R}^{n\\times p}$ as ${\\mathbf {G}}_B({\\mathbf {G}}_A({\\mathbf {E}}^\\top )^\\top )$ , with the following batch-operations (for $i\\in [n]$ and $j\\in [p]$ ): ${\\mathbf {G}}_A({\\mathbf {E}}^\\top )_{j:} := {\\mathbf {g}}_A({\\mathbf {E}}_{:j}) \\qquad \\qquad {\\mathbf {G}}_B({\\mathbf {E}})_{i:} := {\\mathbf {g}}_B({\\mathbf {E}}_{i:}) $ It is easy to verify that ${\\mathbf {G}}$ is invertible.","Due to the partial dependency of ${\\mathbf {G}}_A$ and ${\\mathbf {G}}_B$ , the Jacobians of the vectorized forms (after proper permutation) are block-diagonal, so we have $\\det &\\frac{\\partial \\textnormal {vec}({\\mathbf {G}}({\\mathbf {E}}))}{\\partial \\textnormal {vec}({\\mathbf {E}})} \\\\&= \\prod _{j\\in [p]}\\det \\frac{\\partial {\\mathbf {g}}_A({\\mathbf {E}}_{:j})}{\\partial {\\mathbf {E}}_{:j}}\\cdot \\prod _{i\\in [n]}\\det \\frac{\\partial {\\mathbf {g}}_B({\\mathbf {G}}_A({\\mathbf {E}}^\\top )_{:i})}{\\partial {\\mathbf {G}}_A({\\mathbf {E}}^\\top )_{:i}}.$ In practice, we use the volume preserving version of RealNVP [11] and inverse autoregressive flow (IAF) [25] to parameterize ${\\mathbf {g}}_A$ and ${\\mathbf {g}}_B$ for our experiments We also experimented with Block NAF by [10], but did not include it in the manuscript: its performance was similar to IAF, but it is much slower to sample from (we adapted the open implementation from [10])..","The K-Linear from the previous section can be thought of as using a linear map as ${\\mathbf {g}}_A$ and ${\\mathbf {g}}_B$ ."],["Concentration of empirical KL with normalizing flows","In their study, [12] use independent Gaussian for $q$ to minimize the McAllester bound, so they can compute the KL between Gaussians analytically.","This is no longer feasible when we use more flexible families for $q$ , such as normalizing flows.","Moreover, a Monte Carlo estimate might result in underestimating the bound after inverting the KL between Bernoullis on the LHS of Equation REF (which is a concave function; see Appendix A of [41] for an illustration).","This necessitates a high probability bound on the concentration of the empirical estimate.","In Section REF , we have established $D_{\\mathrm {KL}}(q||p)$ can be written in the following form $\\leavevmode \\xbox {resizebox}{\\XMLaddatt {width}{427.0pt}\\mathbb {E}_{\\mathbf {\\epsilon }} \\left[\\log {\\mathcal {N}}(\\mathbf {\\epsilon }; \\mathbf {0},{\\mathbf {I}}) - \\log \\left|\\det \\frac{\\partial {\\mathbf {g}}_\\phi (\\mathbf {\\epsilon })}{\\partial \\mathbf {\\epsilon }}\\right|-\\log {\\mathcal {N}}({\\mathbf {g}}_\\phi (\\mathbf {\\epsilon }); \\mathbf {0}, {\\mathbf {I}}) \\right],}$ where both $q_0$ and $p$ are standard Gaussian (the mean and variance can be absorbed into the invertible mapping ${\\mathbf {g}}$ if this is not the case).","The first term in the KL can be computed analytically.","The second term usually can be almost surely bounded (e.g.","using Block neural autoregressive flows) so that we can use Hoeffding-type concentration or it can simply be made zero using e.g.","volume preserving flows.","The challenge now lies in the third term, which has a quadratic form $\\frac{1}{2}||{\\mathbf {g}}(\\mathbf {\\epsilon })||^2$ , neglecting the normalizing constant.","Now assume ${\\mathbf {g}}$ is a $L_0$ -Lipschitz The following flows are all Lipschitz (with Lipschitz activation functions): volume preserving version of [11], [25], [5], [4], [10], etc.. Let $g(\\mathbf {\\epsilon })=\\frac{1}{\\sqrt{2}}||{\\mathbf {g}}(\\mathbf {\\epsilon })||$ .","Then $g$ is $L_0/\\sqrt{2}$ -Lipschitz: $\\big |g(\\mathbf {\\epsilon }_1)-g(\\mathbf {\\epsilon }_2)\\big |= &\\frac{1}{\\sqrt{2}}\\big |||{\\mathbf {g}}(\\mathbf {\\epsilon }_1)|| - ||{\\mathbf {g}}(\\mathbf {\\epsilon }_2)||\\big | \\\\& \\le \\frac{1}{\\sqrt{2}}||{\\mathbf {g}}(\\mathbf {\\epsilon }_1) - {\\mathbf {g}}(\\mathbf {\\epsilon }_2)||\\le \\frac{L_0}{\\sqrt{2}}||\\mathbf {\\epsilon }_1-\\mathbf {\\epsilon }_2||.$ This is key in deriving a tail bound on $g^2$ , as Lipschitz functions of canonical Gaussian random variables are sub-Gaussians, meaning they have a tail that decays faster than a Gaussian random variable.","The following theorem provides a concentration bound for the empirical average of $g^2$ similar to that of a Chi-square random variable, as $g^2$ (square of a sub-Gaussian) is sub-exponential.","Theorem 3 Let $g$ be defined as above with a Lipschitz constant $L=L_0/\\sqrt{2}$ .","Let $\\bar{g}^2=\\frac{1}{K}\\sum _{k=1}^K g_k^2$ .","Then the following concentration bound holds ${\\mathbb {P}}(\\bar{g}^2-\\mathbb {E}[g^2]>\\epsilon )\\le \\exp \\left(-\\frac{K\\epsilon ^2}{2(4C^2+C\\epsilon )}\\right),$ where $C=\\left(6L^2+\\frac{L}{\\sqrt{\\log 2}}(\\sqrt{d}+||{\\mathbf {g}}^{-1}(\\mathbf {0})||)\\right)^2$ .","Note that in practice the empirical KL that we use is inversely scaled by the size of the training set $m$ (see Equation REF ), so the Lipschitz constant can be made small in practice to dominate the dimensionality."],["Experiments","We evaluate our proposed method in the context of two prediction tasks (Section REF ), PAC-Bayes bound minimization (Section REF ) and contextual bandit (Section REF ).","For the two prediction tasks, we use the MNIST handwritten digit dataset [30] and CIFAR-10 [27].","See Appendix for a detailed description.","Table: Test error with LeNet (%) on MNIST and the first 5 classes of CIFAR-10.","First 3 columns are from .","K-Diag on CIFAR-5 diverged, so we did not include the result.Table: Test error with modified version of VGG16 (%) on CIFAR10.First 4 columns are from .R means regular training and D means training with data augmentation.Table: PAC-Bayes bound estimation: We minimize the Pinsker bound (an upper bound on the McAllester bound) and the Catani bound using different flows, and estimate the McAllester bound at inference time using Newton's method.Table: Cumulative regret incurred by different algorithms on the bandit benchmarks described in .","Values reported are the mean over 3 independent trials with standard error of the mean, normalized with respect to the performance of the uniform policy."],["Classification","One benefit of Bayesian neural networks compared to the regular ones is that the trade-off between the prior and the likelihood is a form of regularization.","In this section, we evaluate the generalization performance of our method applied to Bayesian neural networks.","We consider two architectures: LeNet-5 [30] and a modified version VGG-16 [46] proposed by [53].","We first compare to the multiplicative normalizing flow (MNFG) proposed by [32], applying our method to LeNet-5 (see Table REF ).","Our Diag matches the performance of their FFG (fully factorized Gaussian).","K-Diag outperforms Diag in this case, perhaps due to the smaller number of parameters which makes it easier to optimize.","K-Nonlinear yields the best generalization error in this case.","On the CIFAR-5 experiment (we take the first 5 classes of CIFAR-10), our methods are on par with MNFG.","Second, we compare with the noisy K-FAC proposed by [53], applying our methods to the larger architecture VGG-16 (see Table REF ).","Noisy K-FAC applies an approximate natural gradient method.","Despite this advantage, our methods (K-Linear and K-Nonlinear) have simiar prediction accuracy in the regular setup.","We also include the results of data augmentation with horizontal flip and random crop where K-Nonlinear outperforms all the other methods."],["PAC Bayes bound minimization","For the PAC-Bayes bound estimation, we minimize Equation REF .","We follow the recipe of [12].","We upper bound the zero-one loss by cross-entropy divided by $\\log |{\\mathcal {Y}}|$ (where $|{\\mathcal {Y}}|$ is the number of classes) to make the upper bound tight.","We set the prior to be ${\\mathcal {N}}(\\Theta _0,\\lambda {\\mathbf {I}})$ , where $\\Theta _0$ is the initial value of the parameters, and apply a union bound to tune the prior variance $\\lambda $ .","We also tune the $\\beta $ coefficient as a parameter during training We are allowed to do so since we treat Equation REF as an optimization objective, rather than report it as a bound.","We report the McAllester bound, which holds for any $q$ , even if it depends on $\\beta $ ., and report the McAllester bound for comparison (since it is the tightest).","For more details, see [12] for reference.","We test with a multi-layer perceptron with 1 or 2 hidden layers with 600 neurons and LeNet-5, evaluated on the MNIST dataset (see Table REF ).","For further clarification, we follow the steps of [12] by minimizing the McAllester bound, using Pinsker's inequality to bound the inverse of the Bernoulli KL (which we call the Pinsker bound).","Since this bound has a square root in the complexity term, we can only use the Gaussian family with an analytic form of the KL.","The result we have is slightly looser than [12] since we have a 10-class problem and they deal with a binary version of MNIST.","We see that the bound can indeed be improved by capturing the correlation among the parameters.","We then compare to minimizing the Catoni bound, which is slightly looser since the linear relationship between the empirical risk and the KL term penalizes the latter more when the KL is larger.","However, by modelling the non-linear dependencies, K-Nonlinear clearly outperforms the other methods (even compared to the ones minimizing the Pinsker bound).","This indicates there exists a considerable amount of structure in the parameter space that may explain the gap between the test error and the generalization bound.","We also notice that, despite the linear relationship, the Catoni bound focuses more on the complexity term than the ELBO.","For example, the empirical risks of LeNet-5 in Table REF are much higher compared to the test loss of Table REF .","The reasons are: (1) the optimal $\\beta $ in Equation REF is larger than 1 (depending on the relative value of the KL), and (2) to properly upper bound the zero-one loss, we scale down the cross-entropy loss by $\\log |{\\mathcal {Y}}|$ during optimization.","This means a learning algorithm based on a tight PAC-Bayes risk bound cannot overfit by much; see [13] for a recent demonstration.","A smaller value of $\\beta $ could bring down the risk and empirical risk, resulting in a looser bound but usually better test performance.","This trade-off between test set performance and the tightness of the bound is a general issue for generalization bounds.","One reason for the interest in PAC-Bayes bounds is that their optimization leads to training algorithms with generalization guarantees.","The bounds, however, are considerably looser than held-out estimates.","A recent work by [44] shows PAC-Bayes bound can potentially be made tight; however, we could not reproduce the results and the best bound using Gaussian prior was around 40% in their setting.","Our work produces much tighter bounds by building flexible families of distributions on neural network weight matrices."],["Contextual bandit","Uncertainty modeling lies at the heart of the exploration-exploitation dilemma in sequential decision-making.","In order to maximize its collected cumulative rewards, an agent should trade off exploring different actions and gaining more knowledge about the reward estimate vs. exploiting the current estimate and allocating resources to the actions that are likely rewarding.","Thompson sampling (TS) [48] is one the popular approaches that deals with the latter trade-off by maintaining posterior distribution over reward models and randomizing actions on the basis of their probability of being optimal.","In this section, we investigate the effectiveness of our proposed method for performing an approximate Thompson sampling in the particular setting of contextual bandits.","In the latter setting, at each time $t = 1 \\ldots T$ , the agent sees a $d$ -dimensional context $X_t$ , selects one of the $k$ available actions, $a_t$ , and earns a reward $r_t$ generated by the environment.","The agent aims to minimize its cumulative regret defined as $R = \\mathbb {E}[\\sum _{t=1}^T r^{\\star }_t - r_t]$ where $r^\\star _t$ is the highest expected reward given the context $X_t$ and the expectation is over the randomness of both environment and the agent's choice of actions.","We compare different methods on a range of real-world bandit problems introduced by [43].","We train the models every 50 time steps for 200 iterations using a batch-size of 512.","We ran each experiment with 3 different random seeds and we report the means and standard errors of cumulative regret normalized with respect to the uniform baseline in the table REF .","We include the functional variational Bayesian neural networks (fBNN), recently introduced by [47] as a baseline, and we use their open sourced implementation of fBNN in the bandit setting.","From table REF , we see that across the 6 bandit problems, our proposed method (K-Linear and K-Nonlinear) provides competitive and consistent results.","They outperform other baselines in 4 problems out of 6."],["Conclusion","In this work, we present the Kronecker Flow, a flow-based method to induce complex distribution inspired by the Kronecker product.","Our methods scale to larger architectures such as VGG-16 since it takes advantage of the shape of the parameters.","We demonstrate our methods work better than vanilla Kronecker product with diagonal matrices on multiple setups, including classification and approximate Thompson sampling in contexual bandit, and prove to be competitive with existing methods in the Bayesian neural network literature.","We are also the first to apply flow-based methods to obtain a tighter numerical generalization bound.","Our work shows that the dependencies among network parameters constitute a non-negligible portion of the gap between risk and PAC-Bayes generalization bound."],["Acknowledgement","CWH would like to thank Kris Sankaran for pointing to the TIS inequality for Gaussian concentration, which is a key component in deriving the tail bound on Lipschitz flows.","Law of the unconscious statistician Let $(\\Omega , {\\mathcal {F}}, {\\mathbb {P}})$ be our probability space.","Let $\\mathbf {\\epsilon }\\in \\mathbb {R}^d$ be a random variable following the (Lebesgue) density $q_0(\\mathbf {\\epsilon })=\\frac{\\mathrm {d}\\mathbf {\\epsilon }_*{\\mathbb {P}}}{\\mathrm {d}\\mu }$ and $\\mathbf {\\epsilon }_*{\\mathbb {P}}$ being its pushforward measure, and write $\\Theta ={\\mathbf {g}}_\\phi (\\mathbf {\\epsilon })\\in \\mathbb {R}^d$ with $q_\\phi =\\frac{\\mathrm {d}({\\mathbf {g}}_\\phi \\circ \\mathbf {\\epsilon })_*{\\mathbb {P}}}{\\mathrm {d}\\mu }$ being its density and $({\\mathbf {g}}_\\phi \\circ \\mathbf {\\epsilon })_*{\\mathbb {P}}$ being its pushforward measure, and $A=\\hat{R}_D(\\Theta )+\\log q_\\phi (\\Theta )\\in \\mathbb {R}$ .","Then $\\mathbb {E}[A] &= \\int _{\\mathbb {R}^d} A \\,\\,\\mathrm {d}({\\mathbf {g}}_\\phi \\circ \\mathbf {\\epsilon })_*{\\mathbb {P}}&&= \\int _{\\mathbb {R}^d} \\left(\\hat{R}_D(\\Theta ) + \\log q_\\phi (\\Theta ) \\right) q_\\phi (\\Theta ) \\,\\mathrm {d}\\Theta \\\\&= \\int _{\\mathbb {R}^d} A\\circ \\Theta \\,\\,\\mathrm {d}\\mathbf {\\epsilon }_*{\\mathbb {P}}&&= \\int _{\\mathbb {R}^d} \\left(\\hat{R}_D({\\mathbf {g}}_\\phi (\\mathbf {\\epsilon })) + \\log q_\\phi ({\\mathbf {g}}_\\phi (\\mathbf {\\epsilon })) \\right) q_0(\\mathbf {\\epsilon }) \\,\\mathrm {d}\\mathbf {\\epsilon }\\\\&{}&&=\\int _{\\mathbb {R}^d} \\left(\\hat{R}_{D}({\\mathbf {g}}_\\phi (\\mathbf {\\epsilon })) + \\log q_0(\\mathbf {\\epsilon }) - \\log \\left|\\det \\frac{\\partial {\\mathbf {g}}_\\phi (\\mathbf {\\epsilon })}{\\partial \\mathbf {\\epsilon }}\\right|\\right)q_0(\\mathbf {\\epsilon })\\,\\mathrm {d}\\mathbf {\\epsilon }$ Derivation and interpretation of Property REF We first derive Property REF algebraically, and give an interpretation that can be genralized to higher dimensional tensor operation.","Recall that we have the following givens: Assume ${\\mathbf {E}}_{ij}\\overset{\\textnormal {i.i.d.","}}{\\sim } {\\mathcal {N}}(0,1)$ is a $n\\times p$ random Gaussian matrix.","Assume ${\\mathbf {A}}\\in \\mathbb {R}^{n\\times n}$ and ${\\mathbf {B}}\\in \\mathbb {R}^{p\\times p}$ .","${\\mathbf {S}}\\in \\mathbb {R}^{n\\times p}_{>0}$ .","${\\mathbf {M}}\\in \\mathbb {R}^{n\\times p}$ .","If we rescale ${\\mathbf {E}}$ elementwise by ${\\mathbf {S}}$ before inducing the column-wise and row-wise correlation, we have: (superscript is Hadamard power) $\\Sigma :=\\mathrm {Var}(\\textnormal {vec}({\\mathbf {M}}+ {\\mathbf {A}}({\\mathbf {E}}\\circ {\\mathbf {S}}){\\mathbf {B}}))&=\\mathrm {Var}( ({\\mathbf {B}}^\\top \\otimes {\\mathbf {A}})\\textnormal {vec}({\\mathbf {E}}\\circ {\\mathbf {S}}))\\\\&=({\\mathbf {B}}^\\top \\otimes {\\mathbf {A}}) \\mathop {\\mathrm {diag}}\\nolimits (\\textnormal {vec}({\\mathbf {S}}^2)) ({\\mathbf {B}}^\\top \\otimes {\\mathbf {A}})^\\top \\\\&=({\\mathbf {B}}^\\top \\otimes {\\mathbf {A}}) \\mathop {\\mathrm {diag}}\\nolimits (\\textnormal {vec}({\\mathbf {S}}^2)) ({\\mathbf {B}}\\otimes {\\mathbf {A}}^\\top )$ If ${\\mathbf {S}}$ is a matrix of ones, the RHS equals $({\\mathbf {B}}^\\top \\otimes {\\mathbf {A}}) ({\\mathbf {B}}\\otimes {\\mathbf {A}}^\\top )= ({\\mathbf {B}}^\\top {\\mathbf {B}})\\otimes ({\\mathbf {A}}{\\mathbf {A}}^\\top )$ , which is the covariance of the matrix normal.","Generally, ${\\mathbf {S}}$ might not be a matrix of ones.","But we can still compute the determinant and trace of the covariance matrix (useful in computing KL): $\\det (\\Sigma )=\\det ({\\mathbf {A}})^{2p}\\det ({\\mathbf {B}})^{2n}\\prod _{ij}{\\mathbf {S}}_{ij}^2$ $\\textnormal {Tr}(\\Sigma )&=\\sum _i \\Sigma _{ii} \\\\&=\\sum _i \\sum _{j} ({\\mathbf {B}}^\\top \\otimes {\\mathbf {A}})_{ij}\\textnormal {vec}({\\mathbf {S}}^2)_j ({\\mathbf {B}}\\otimes {\\mathbf {A}}^\\top )_{ji}\\\\&=\\sum _i \\sum _{j} ({\\mathbf {B}}^{2\\top }\\otimes {\\mathbf {A}}^2)_{ij}\\textnormal {vec}({\\mathbf {S}}^2)_j\\\\&=\\sum _i\\left( ({\\mathbf {B}}^{2\\top }\\otimes {\\mathbf {A}}^2)\\textnormal {vec}({\\mathbf {S}}^2) \\right)_i\\\\&=\\sum _{ij}\\left({\\mathbf {A}}^2{\\mathbf {S}}^2{\\mathbf {B}}^2\\right)_{ij}$ Interpretation of the determinant and trace.","The determinant measures the change in volume due to the linear map.","Since each operation (elementwise multiplication with ${\\mathbf {S}}$ , left-multiplication with ${\\mathbf {A}}$ , and right-multiplication with ${\\mathbf {B}}$ ) is an invertible map, the determinant of the composition is a product of determinants.","After elementwise multiplication with ${\\mathbf {S}}$ (hence $\\prod {\\mathbf {S}}_{ij}$ )The power 2 comes from the fact that we are looking at the determinant of the covariance.","Direct computation of the likelihood involves $\\frac{1}{2}\\log \\det (\\Sigma )$ , which is equivalent to the log-determinant of the invertible map., we apply the same linear map ${\\mathbf {A}}$ to the columns of $({\\mathbf {E}}\\circ {\\mathbf {S}})$ ; in vector form, this corresponds to left-multiplication with a block diagonal of $p$ ${\\mathbf {A}}$ 's, hence $\\det ({\\mathbf {A}})^{p}$ .","The same reasoning explains $\\det ({\\mathbf {B}})^n$ .","The trace of the covariance can be written as $\\textnormal {Tr}(\\Sigma )=\\sum _{ij}\\mathrm {Var}({\\mathbf {W}}_{ij})$ , i.e.","the sum of marginal variances.","Each of the ${\\mathbf {W}}_{ij}$ is a linear combination of the entries of ${\\mathbf {E}}$ , which have unit variance and are uncorrelated, so by the additive property of variance of sum of uncorrelated random variables and the quadratic scaling property of variance, $\\mathrm {Var}({\\mathbf {W}}_{ij})=A_{i:}^2{\\mathbf {S}}^2{\\mathbf {B}}_{:j}^2$ .","Connection to triangular maps.","Much of the recent work on normalizing flows has been dedicated to inverse autoregressive transformations [25], [21], [37], [10], [24], as they are general enough to induce any density function [22], [7], [52].","When such transformations are used for ${\\mathbf {g}}_A$ and ${\\mathbf {g}}_B$ , the overall transformation ${\\mathbf {G}}$ is also a triangle map, since ${\\mathbf {G}}({\\mathbf {E}})_{ij}$ depends on ${\\mathbf {E}}_{i^{\\prime }j^{\\prime }}$ for $i^{\\prime }\\le i$ and $j^{\\prime }\\le j$ .","Such a function has some “blind spots” similar to the ones discovered by [51].","One avenue for improvement is to design a transformation that increase the connectivity.","Another avenue for improvement is to condition each (row-wise or column-wise) transformation on a learnable embedding of the row/column, such that each row/column is transformed by a slightly different function than another We try this idea in the preliminary stage of the project, but find it harder to optimize.","This is potentially due to the extra parameters that have to be learned.. Tail bound of empirical KL We begin with some preliminaries and lemmas in Section REF , and prove the main result in Section REF .","Basic tail bounds and Bernstein inequality The tools developed in this section is to translate the coefficients (such as variance) of sub-Gaussian random variables and sub-exponential random variables.","We start with the definition of sub-Gaussians: Definition 1 We write $X\\sim \\textnormal {sub}{\\mathcal {N}}(L^2)$ if $X$ is a random variable satisfying ${\\mathbb {P}}(|X|>t)\\le 2\\exp \\left(-\\frac{t^2}{2L^2}\\right)$ We write $\\Gamma (\\cdot )$ as the Gamma function: $\\Gamma (z)=\\int _0^\\infty e^{-u}u^{z-1}\\mathrm {d}u$ .","Note that for positive integers $z$ , $\\Gamma (z)=(z-1)!$ .","The following lemma gives an upper-bound on the moments of a sub-Gaussian.","Lemma 4 For $X\\sim \\textnormal {sub}{\\mathcal {N}}(L^2)$ , for any integer $p\\ge 1$ , $\\mathbb {E}[|X|^p]\\le (2L^2)^{p/2}p\\Gamma (p/2)$ .","Since $|X|^p$ is non-negative, similar to Lemma REF , we have $\\mathbb {E}[|X|^p]&= \\int _{0}^\\infty {\\mathbb {P}}(|X|^p\\ge s)\\mathrm {d}s=\\int _0^\\infty {\\mathbb {P}}(|X|\\ge t)pt^{p-1} \\mathrm {d}t \\\\&\\le 2p\\int _0^\\infty e^{-{t^2}/{2L^2}}t^{p-1} \\mathrm {d}t=\\le p(2L^2)^{p/2}\\int _0^\\infty e^{-u}u^{p/2-1}\\mathrm {d}u = p(2L^2)^{p/2}\\Gamma (p/2)$ where we let $s=t^p$ and $u=t^2/2L^2$ .","The following definition is the main tool for translating the coefficients.","Definition 2 Let $X$ be a random variable.","For integer $k\\ge 1$ , define the $\\psi _k$ -Orlicz norm as $||X||_{\\psi _k}:=\\inf \\lbrace t>0:\\mathbb {E}[\\exp (|X|^k/t^k)]\\le 2\\rbrace $ i.e, the smallest constant $t>0$ for which the super-exponential moment of $X^k/t^k$ is bounded by 2.","The Orlicz norm is infinity if there's no finite $t$ for which $\\mathbb {E}[\\exp (|X|^k/t^k)]$ exists.","It is easy to verify that the Orlicz norm is indeed a norm.","We call $||\\cdot ||_{\\psi _2}$ the sub-Gaussian norm, and $||\\cdot ||_{\\psi _1}$ the sub-exponential norm.","Note that $||X^2||_{\\psi _1} = ||X||_{\\psi _2}^2$ .","The following lemma upper bounds the sub-Gaussian norm by its variance.","Lemma 5 If $X\\sim \\textnormal {sub}{\\mathcal {N}}(L^2)$ , $||X||_{\\psi _2}\\le 6L^2$ .","By power series expansion of the exponential function, $\\mathbb {E}[e^{cX^2}]=1+\\sum _{p=1}^\\infty \\frac{c^p\\mathbb {E}[X^{2p}]}{p!","}\\le 1+\\sum _{p=1}^\\infty \\frac{c^p}{p!}","2(2L^2)^pp!=1+2\\sum _{p=1}^\\infty (2cL^2)^p$ where we used Lemma REF for the inequality.","The RHS converges and is equal to 2 if $c=1/6L^2$ .","Thus, $||X||_{\\psi _2}\\le 6L^2$ .","The following lemma gives an upper bound on the moments of sub-exponential random variables.","Lemma 6 If for some $C>0$ , $\\mathbb {E}[\\exp (|X|/C)]\\le 2$ , then $\\mathbb {E}[|X|^p]\\le 2C^pp!$ .","By Markov's inequality, ${\\mathbb {P}}(|X|>t)\\le \\frac{\\mathbb {E}[\\exp (|X|/C)]}{\\exp (t/C)}\\le 2e^{-t/C}$ For $p\\in {\\mathbb {Z}}^+$ , since $|X|^p$ is non-negative, $\\mathbb {E}[|X|^p] &= \\int _{0}^\\infty {\\mathbb {P}}(|X|^p\\ge s)ds= \\int _{0}^\\infty {\\mathbb {P}}(|X|\\ge t)pt^{p-1}dt \\\\&\\le 2p\\int _{0}^\\infty e^{-t/C}t^{p-1}dt =2pC^p\\int _{0}^\\infty e^{-u}u^{p-1} du=2pC^p\\Gamma (p)=2C^pp!$ where we let $s=t^p$ and $u=t/C$ .","Finally, we derive a concentration bound for sub-exponential random variables.","Theorem 7 (Bernstein's inequality for sub-exponential random variables) Let $(X_i)_{i\\in [n]}$ be independent real-valued random variables satisfying $\\mathbb {E}[\\exp (|X|/C)]\\le 2$ for some $C>0$ , with mean $\\mu _X=\\mathbb {E}[X]$ , and let $\\bar{X}=\\frac{1}{n}\\sum _{i=1}^nX_i$ .","Then, for any $\\epsilon >0$ , the following concentration bound holds: ${\\mathbb {P}}(\\bar{X}-\\mu _X > \\epsilon ) \\le \\exp \\left(-\\frac{n\\epsilon ^2}{2(4C^2+C\\epsilon )}\\right)$ Let $\\nu =4nC^2$ and $c=C$ .","Then by Lemma REF , $\\sum _{i=1}^n\\mathbb {E}[X_i^2]\\le n\\cdot 4C^2=\\nu $ and for integers $p>2$ : $\\sum _{i=1}^n \\mathbb {E}[|X_i|^p] \\le 2nC^pp!","= \\nu C^{p-2}p!/2 = \\nu c^{p-2}p!/2$ .","Then by Corollary 2.11 of [8], we have ${\\mathbb {P}}(\\bar{X}-\\mu _X > \\epsilon )={\\mathbb {P}}\\left(\\sum _{i=1}^n(X_i-\\mu _X)>n\\epsilon \\right)\\le \\exp \\left(-\\frac{n\\epsilon ^2}{2(4C^2+C\\epsilon )}\\right)$ Proof of Theorem REF Since $\\bar{g}(\\mathbf {\\epsilon }):=g(\\mathbf {\\epsilon })-\\mathbb {E}[g(\\mathbf {\\epsilon })]$ is $L$ -Lipschitz, according to Theorem 5.5 and 5.6 of [8], $\\bar{g}\\sim \\textnormal {sub}{\\mathcal {N}}(L^2)$ .","And we have that $||g^2||_{\\psi _1}=||g||_{\\psi _2}^2=||\\bar{g}+\\mathbb {E}[g]||_{\\psi _2}^2\\le \\left(||\\bar{g}||_{\\psi _2} + \\frac{\\mathbb {E}[g]}{\\sqrt{\\log 2}}\\right)^2$ due to triangle inequality of the norm.","Now since $g$ is $L$ -Lipschitz, its expectation can be bounded by $\\mathbb {E}[g]=\\mathbb {E}\\left[\\frac{1}{\\sqrt{2}}||{\\mathbf {g}}(\\mathbf {\\epsilon })-\\mathbf {0}||\\right]\\le L\\mathbb {E}\\left[||\\mathbf {\\epsilon }-{\\mathbf {g}}^{-1}(\\mathbf {0})||\\right]\\le L(\\mathbb {E}\\left[||\\mathbf {\\epsilon }||\\right]+||{\\mathbf {g}}^{-1}(\\mathbf {0})||) $ Since $\\mathbf {\\epsilon }$ is standard-normally distributed, $||\\mathbf {\\epsilon }||$ follows the chi distribution with $d$ degrees of freedom, which has an expectation that can be upper-bounded using Gautschi's inequality (using Wendel's version of the upper bound): $\\mathbb {E}[||\\mathbf {\\epsilon }||]=\\sqrt{2}\\frac{\\Gamma ((d+1)/2)}{\\Gamma (d/2)}\\le \\sqrt{2}\\left(\\frac{d}{2}\\right)^{1/2}=\\sqrt{d}$ Combining the above and using Lemma REF , we have $||g^2||_{\\psi _1}\\le \\left(6L^2+\\frac{L}{\\sqrt{\\log 2}}(\\sqrt{d}+||{\\mathbf {g}}^{-1}(\\mathbf {0})||)\\right)^2$ Setting $C$ to be the RHS and applying Theorem REF yield the desired result.","Experimental Details For the predictive tasks (Section REF ), we use a cosine annealing schedule for the learning rate, scaling down to 0.01 of the initial learning rate, and pretrain a deterministic network for 10 epochs using the Adam optimizer with a learning rate of 0.001, to initialize the mean of the Gaussian $q_0$ , and train $q$ for 200 epochs.","LeNet-5 MNIST.","We use a linear annealing schedule of the $\\beta $ coefficient (from 0 back to 1) for 50,000 iterations.","We use the Adam optimizer with a learning rate of 0.0005.","The result we get for K-Linear uses polyak averaging with exponential decay coefficient 0.995.","We use the volume preserving version of RealNVP for the K-Nonlinear.","We use the standard Gaussian prior for $p$ .","LeNet-5 CIFAR-5 We use the same architecture as [32] (192 convolutional kernels and 1,000 hidden units for the fully connected layers).","We use a linear annealing schedule of the $\\beta $ coefficient (from 0 back to 1) for 20,000 iterations for Diag, and no annealing for K-Linear and K-Nonlinear.","We use the Adam optimizer with a learning rate of 0.0003, 0.0003, 0.0005 for Diag, K-Linear and K-Nonlinear, respectively.","We use the volume preserving version of RealNVP for K-Nonlinear.","We use the standard Gaussian prior for $p$ .","VGG-16 CIFAR-10 We use the modified version of VGG-16 proposed by [53].","We use a learning rate of 0.0005 for all experiments but K-Nonlinear in the regular setup (where we use 0.001).","We use the isotropic Gaussian prior with variance being 0.1, and set $\\beta $ to be [0.5, 0.1, 0.1, 0.5] in the regular setup and [0.5, 0.1, 0.1, 0.1] in the data augmented setup for Diag, K-Diag, K-Linear, and K-Nonlinear, respectively.","We use the volume preserving version of RealNVP for the K-Nonlinear.","PAC-Bayes MLP We follow the same steps as [12], except we did not discretize the prior variance after tuning.","In practice this does not affect the bound much.","We also did not initialize the mean of $q_0$ in our setup using SGD for our experiments.","We train the stochastic network for 300 epochs, with a learning rate of 0.002.","The bound holds with probability at least 0.965 over the choice of prior and the training set.","The $b$ and $c$ coefficients in [12] are set as 100 and 0.1.","We use the volume preserving version of IAF for the K-Nonlinear.","PAC-Bayes LeNet-5 The same setup as PAC-Bayes MLP, except with polyak averaging with coefficient 0.995.","We use the volume preserving version of IAF for the K-Nonlinear.","Bandit Benchmark All the models share the same architechture: one hidden layer with 50 units.","We use the volume preserving version of RealNVP for K-NonLinear.","We train models every 50 time steps for 200 training iterations using a batch-size of 512.","Table: Description of bandit problem: number of actions and number of contexts used for experiments.","Comparing to benchmark, we restrict ourself to 50000 contexts for Covertype instead of 150000 contexts.Table: Additional results: Cumulative regret incurred by different algorithms on the bandit benchmarks described in .","Values reported are the mean over 5 independent trials with standard error of the mean, normalized with respect to the performance of the uniform policy.","We use the same hyperparameters for different algorithms without any finetuning: learning rate = 0.0001 and 100 training epochs.Figure: Posterior predictive (with 20 samples) on rotated MNIST digit 3 and 5.","Top left: Diag; top right: K-Diag; bottom left: K-Linear; bottom right: K-Nonlinear."],["Law of the unconscious statistician","Let $(\\Omega , {\\mathcal {F}}, {\\mathbb {P}})$ be our probability space.","Let $\\mathbf {\\epsilon }\\in \\mathbb {R}^d$ be a random variable following the (Lebesgue) density $q_0(\\mathbf {\\epsilon })=\\frac{\\mathrm {d}\\mathbf {\\epsilon }_*{\\mathbb {P}}}{\\mathrm {d}\\mu }$ and $\\mathbf {\\epsilon }_*{\\mathbb {P}}$ being its pushforward measure, and write $\\Theta ={\\mathbf {g}}_\\phi (\\mathbf {\\epsilon })\\in \\mathbb {R}^d$ with $q_\\phi =\\frac{\\mathrm {d}({\\mathbf {g}}_\\phi \\circ \\mathbf {\\epsilon })_*{\\mathbb {P}}}{\\mathrm {d}\\mu }$ being its density and $({\\mathbf {g}}_\\phi \\circ \\mathbf {\\epsilon })_*{\\mathbb {P}}$ being its pushforward measure, and $A=\\hat{R}_D(\\Theta )+\\log q_\\phi (\\Theta )\\in \\mathbb {R}$ .","Then $\\mathbb {E}[A] &= \\int _{\\mathbb {R}^d} A \\,\\,\\mathrm {d}({\\mathbf {g}}_\\phi \\circ \\mathbf {\\epsilon })_*{\\mathbb {P}}&&= \\int _{\\mathbb {R}^d} \\left(\\hat{R}_D(\\Theta ) + \\log q_\\phi (\\Theta ) \\right) q_\\phi (\\Theta ) \\,\\mathrm {d}\\Theta \\\\&= \\int _{\\mathbb {R}^d} A\\circ \\Theta \\,\\,\\mathrm {d}\\mathbf {\\epsilon }_*{\\mathbb {P}}&&= \\int _{\\mathbb {R}^d} \\left(\\hat{R}_D({\\mathbf {g}}_\\phi (\\mathbf {\\epsilon })) + \\log q_\\phi ({\\mathbf {g}}_\\phi (\\mathbf {\\epsilon })) \\right) q_0(\\mathbf {\\epsilon }) \\,\\mathrm {d}\\mathbf {\\epsilon }\\\\&{}&&=\\int _{\\mathbb {R}^d} \\left(\\hat{R}_{D}({\\mathbf {g}}_\\phi (\\mathbf {\\epsilon })) + \\log q_0(\\mathbf {\\epsilon }) - \\log \\left|\\det \\frac{\\partial {\\mathbf {g}}_\\phi (\\mathbf {\\epsilon })}{\\partial \\mathbf {\\epsilon }}\\right|\\right)q_0(\\mathbf {\\epsilon })\\,\\mathrm {d}\\mathbf {\\epsilon }$"],["Derivation and interpretation of Property ","We first derive Property REF algebraically, and give an interpretation that can be genralized to higher dimensional tensor operation.","Recall that we have the following givens: Assume ${\\mathbf {E}}_{ij}\\overset{\\textnormal {i.i.d.","}}{\\sim } {\\mathcal {N}}(0,1)$ is a $n\\times p$ random Gaussian matrix.","Assume ${\\mathbf {A}}\\in \\mathbb {R}^{n\\times n}$ and ${\\mathbf {B}}\\in \\mathbb {R}^{p\\times p}$ .","${\\mathbf {S}}\\in \\mathbb {R}^{n\\times p}_{>0}$ .","${\\mathbf {M}}\\in \\mathbb {R}^{n\\times p}$ .","If we rescale ${\\mathbf {E}}$ elementwise by ${\\mathbf {S}}$ before inducing the column-wise and row-wise correlation, we have: (superscript is Hadamard power) $\\Sigma :=\\mathrm {Var}(\\textnormal {vec}({\\mathbf {M}}+ {\\mathbf {A}}({\\mathbf {E}}\\circ {\\mathbf {S}}){\\mathbf {B}}))&=\\mathrm {Var}( ({\\mathbf {B}}^\\top \\otimes {\\mathbf {A}})\\textnormal {vec}({\\mathbf {E}}\\circ {\\mathbf {S}}))\\\\&=({\\mathbf {B}}^\\top \\otimes {\\mathbf {A}}) \\mathop {\\mathrm {diag}}\\nolimits (\\textnormal {vec}({\\mathbf {S}}^2)) ({\\mathbf {B}}^\\top \\otimes {\\mathbf {A}})^\\top \\\\&=({\\mathbf {B}}^\\top \\otimes {\\mathbf {A}}) \\mathop {\\mathrm {diag}}\\nolimits (\\textnormal {vec}({\\mathbf {S}}^2)) ({\\mathbf {B}}\\otimes {\\mathbf {A}}^\\top )$ If ${\\mathbf {S}}$ is a matrix of ones, the RHS equals $({\\mathbf {B}}^\\top \\otimes {\\mathbf {A}}) ({\\mathbf {B}}\\otimes {\\mathbf {A}}^\\top )= ({\\mathbf {B}}^\\top {\\mathbf {B}})\\otimes ({\\mathbf {A}}{\\mathbf {A}}^\\top )$ , which is the covariance of the matrix normal.","Generally, ${\\mathbf {S}}$ might not be a matrix of ones.","But we can still compute the determinant and trace of the covariance matrix (useful in computing KL): $\\det (\\Sigma )=\\det ({\\mathbf {A}})^{2p}\\det ({\\mathbf {B}})^{2n}\\prod _{ij}{\\mathbf {S}}_{ij}^2$ $\\textnormal {Tr}(\\Sigma )&=\\sum _i \\Sigma _{ii} \\\\&=\\sum _i \\sum _{j} ({\\mathbf {B}}^\\top \\otimes {\\mathbf {A}})_{ij}\\textnormal {vec}({\\mathbf {S}}^2)_j ({\\mathbf {B}}\\otimes {\\mathbf {A}}^\\top )_{ji}\\\\&=\\sum _i \\sum _{j} ({\\mathbf {B}}^{2\\top }\\otimes {\\mathbf {A}}^2)_{ij}\\textnormal {vec}({\\mathbf {S}}^2)_j\\\\&=\\sum _i\\left( ({\\mathbf {B}}^{2\\top }\\otimes {\\mathbf {A}}^2)\\textnormal {vec}({\\mathbf {S}}^2) \\right)_i\\\\&=\\sum _{ij}\\left({\\mathbf {A}}^2{\\mathbf {S}}^2{\\mathbf {B}}^2\\right)_{ij}$"],["Interpretation of the determinant and trace.","The determinant measures the change in volume due to the linear map.","Since each operation (elementwise multiplication with ${\\mathbf {S}}$ , left-multiplication with ${\\mathbf {A}}$ , and right-multiplication with ${\\mathbf {B}}$ ) is an invertible map, the determinant of the composition is a product of determinants.","After elementwise multiplication with ${\\mathbf {S}}$ (hence $\\prod {\\mathbf {S}}_{ij}$ )The power 2 comes from the fact that we are looking at the determinant of the covariance.","Direct computation of the likelihood involves $\\frac{1}{2}\\log \\det (\\Sigma )$ , which is equivalent to the log-determinant of the invertible map., we apply the same linear map ${\\mathbf {A}}$ to the columns of $({\\mathbf {E}}\\circ {\\mathbf {S}})$ ; in vector form, this corresponds to left-multiplication with a block diagonal of $p$ ${\\mathbf {A}}$ 's, hence $\\det ({\\mathbf {A}})^{p}$ .","The same reasoning explains $\\det ({\\mathbf {B}})^n$ .","The trace of the covariance can be written as $\\textnormal {Tr}(\\Sigma )=\\sum _{ij}\\mathrm {Var}({\\mathbf {W}}_{ij})$ , i.e.","the sum of marginal variances.","Each of the ${\\mathbf {W}}_{ij}$ is a linear combination of the entries of ${\\mathbf {E}}$ , which have unit variance and are uncorrelated, so by the additive property of variance of sum of uncorrelated random variables and the quadratic scaling property of variance, $\\mathrm {Var}({\\mathbf {W}}_{ij})=A_{i:}^2{\\mathbf {S}}^2{\\mathbf {B}}_{:j}^2$ ."],["Connection to triangular maps.","Much of the recent work on normalizing flows has been dedicated to inverse autoregressive transformations [25], [21], [37], [10], [24], as they are general enough to induce any density function [22], [7], [52].","When such transformations are used for ${\\mathbf {g}}_A$ and ${\\mathbf {g}}_B$ , the overall transformation ${\\mathbf {G}}$ is also a triangle map, since ${\\mathbf {G}}({\\mathbf {E}})_{ij}$ depends on ${\\mathbf {E}}_{i^{\\prime }j^{\\prime }}$ for $i^{\\prime }\\le i$ and $j^{\\prime }\\le j$ .","Such a function has some “blind spots” similar to the ones discovered by [51].","One avenue for improvement is to design a transformation that increase the connectivity.","Another avenue for improvement is to condition each (row-wise or column-wise) transformation on a learnable embedding of the row/column, such that each row/column is transformed by a slightly different function than another We try this idea in the preliminary stage of the project, but find it harder to optimize.","This is potentially due to the extra parameters that have to be learned.."],["Tail bound of empirical KL","We begin with some preliminaries and lemmas in Section REF , and prove the main result in Section REF ."],["Basic tail bounds and Bernstein inequality","The tools developed in this section is to translate the coefficients (such as variance) of sub-Gaussian random variables and sub-exponential random variables.","We start with the definition of sub-Gaussians: Definition 1 We write $X\\sim \\textnormal {sub}{\\mathcal {N}}(L^2)$ if $X$ is a random variable satisfying ${\\mathbb {P}}(|X|>t)\\le 2\\exp \\left(-\\frac{t^2}{2L^2}\\right)$ We write $\\Gamma (\\cdot )$ as the Gamma function: $\\Gamma (z)=\\int _0^\\infty e^{-u}u^{z-1}\\mathrm {d}u$ .","Note that for positive integers $z$ , $\\Gamma (z)=(z-1)!$ .","The following lemma gives an upper-bound on the moments of a sub-Gaussian.","Lemma 4 For $X\\sim \\textnormal {sub}{\\mathcal {N}}(L^2)$ , for any integer $p\\ge 1$ , $\\mathbb {E}[|X|^p]\\le (2L^2)^{p/2}p\\Gamma (p/2)$ .","Since $|X|^p$ is non-negative, similar to Lemma REF , we have $\\mathbb {E}[|X|^p]&= \\int _{0}^\\infty {\\mathbb {P}}(|X|^p\\ge s)\\mathrm {d}s=\\int _0^\\infty {\\mathbb {P}}(|X|\\ge t)pt^{p-1} \\mathrm {d}t \\\\&\\le 2p\\int _0^\\infty e^{-{t^2}/{2L^2}}t^{p-1} \\mathrm {d}t=\\le p(2L^2)^{p/2}\\int _0^\\infty e^{-u}u^{p/2-1}\\mathrm {d}u = p(2L^2)^{p/2}\\Gamma (p/2)$ where we let $s=t^p$ and $u=t^2/2L^2$ .","The following definition is the main tool for translating the coefficients.","Definition 2 Let $X$ be a random variable.","For integer $k\\ge 1$ , define the $\\psi _k$ -Orlicz norm as $||X||_{\\psi _k}:=\\inf \\lbrace t>0:\\mathbb {E}[\\exp (|X|^k/t^k)]\\le 2\\rbrace $ i.e, the smallest constant $t>0$ for which the super-exponential moment of $X^k/t^k$ is bounded by 2.","The Orlicz norm is infinity if there's no finite $t$ for which $\\mathbb {E}[\\exp (|X|^k/t^k)]$ exists.","It is easy to verify that the Orlicz norm is indeed a norm.","We call $||\\cdot ||_{\\psi _2}$ the sub-Gaussian norm, and $||\\cdot ||_{\\psi _1}$ the sub-exponential norm.","Note that $||X^2||_{\\psi _1} = ||X||_{\\psi _2}^2$ .","The following lemma upper bounds the sub-Gaussian norm by its variance.","Lemma 5 If $X\\sim \\textnormal {sub}{\\mathcal {N}}(L^2)$ , $||X||_{\\psi _2}\\le 6L^2$ .","By power series expansion of the exponential function, $\\mathbb {E}[e^{cX^2}]=1+\\sum _{p=1}^\\infty \\frac{c^p\\mathbb {E}[X^{2p}]}{p!","}\\le 1+\\sum _{p=1}^\\infty \\frac{c^p}{p!}","2(2L^2)^pp!=1+2\\sum _{p=1}^\\infty (2cL^2)^p$ where we used Lemma REF for the inequality.","The RHS converges and is equal to 2 if $c=1/6L^2$ .","Thus, $||X||_{\\psi _2}\\le 6L^2$ .","The following lemma gives an upper bound on the moments of sub-exponential random variables.","Lemma 6 If for some $C>0$ , $\\mathbb {E}[\\exp (|X|/C)]\\le 2$ , then $\\mathbb {E}[|X|^p]\\le 2C^pp!$ .","By Markov's inequality, ${\\mathbb {P}}(|X|>t)\\le \\frac{\\mathbb {E}[\\exp (|X|/C)]}{\\exp (t/C)}\\le 2e^{-t/C}$ For $p\\in {\\mathbb {Z}}^+$ , since $|X|^p$ is non-negative, $\\mathbb {E}[|X|^p] &= \\int _{0}^\\infty {\\mathbb {P}}(|X|^p\\ge s)ds= \\int _{0}^\\infty {\\mathbb {P}}(|X|\\ge t)pt^{p-1}dt \\\\&\\le 2p\\int _{0}^\\infty e^{-t/C}t^{p-1}dt =2pC^p\\int _{0}^\\infty e^{-u}u^{p-1} du=2pC^p\\Gamma (p)=2C^pp!$ where we let $s=t^p$ and $u=t/C$ .","Finally, we derive a concentration bound for sub-exponential random variables.","Theorem 7 (Bernstein's inequality for sub-exponential random variables) Let $(X_i)_{i\\in [n]}$ be independent real-valued random variables satisfying $\\mathbb {E}[\\exp (|X|/C)]\\le 2$ for some $C>0$ , with mean $\\mu _X=\\mathbb {E}[X]$ , and let $\\bar{X}=\\frac{1}{n}\\sum _{i=1}^nX_i$ .","Then, for any $\\epsilon >0$ , the following concentration bound holds: ${\\mathbb {P}}(\\bar{X}-\\mu _X > \\epsilon ) \\le \\exp \\left(-\\frac{n\\epsilon ^2}{2(4C^2+C\\epsilon )}\\right)$ Let $\\nu =4nC^2$ and $c=C$ .","Then by Lemma REF , $\\sum _{i=1}^n\\mathbb {E}[X_i^2]\\le n\\cdot 4C^2=\\nu $ and for integers $p>2$ : $\\sum _{i=1}^n \\mathbb {E}[|X_i|^p] \\le 2nC^pp!","= \\nu C^{p-2}p!/2 = \\nu c^{p-2}p!/2$ .","Then by Corollary 2.11 of [8], we have ${\\mathbb {P}}(\\bar{X}-\\mu _X > \\epsilon )={\\mathbb {P}}\\left(\\sum _{i=1}^n(X_i-\\mu _X)>n\\epsilon \\right)\\le \\exp \\left(-\\frac{n\\epsilon ^2}{2(4C^2+C\\epsilon )}\\right)$"],["Proof of Theorem ","Since $\\bar{g}(\\mathbf {\\epsilon }):=g(\\mathbf {\\epsilon })-\\mathbb {E}[g(\\mathbf {\\epsilon })]$ is $L$ -Lipschitz, according to Theorem 5.5 and 5.6 of [8], $\\bar{g}\\sim \\textnormal {sub}{\\mathcal {N}}(L^2)$ .","And we have that $||g^2||_{\\psi _1}=||g||_{\\psi _2}^2=||\\bar{g}+\\mathbb {E}[g]||_{\\psi _2}^2\\le \\left(||\\bar{g}||_{\\psi _2} + \\frac{\\mathbb {E}[g]}{\\sqrt{\\log 2}}\\right)^2$ due to triangle inequality of the norm.","Now since $g$ is $L$ -Lipschitz, its expectation can be bounded by $\\mathbb {E}[g]=\\mathbb {E}\\left[\\frac{1}{\\sqrt{2}}||{\\mathbf {g}}(\\mathbf {\\epsilon })-\\mathbf {0}||\\right]\\le L\\mathbb {E}\\left[||\\mathbf {\\epsilon }-{\\mathbf {g}}^{-1}(\\mathbf {0})||\\right]\\le L(\\mathbb {E}\\left[||\\mathbf {\\epsilon }||\\right]+||{\\mathbf {g}}^{-1}(\\mathbf {0})||) $ Since $\\mathbf {\\epsilon }$ is standard-normally distributed, $||\\mathbf {\\epsilon }||$ follows the chi distribution with $d$ degrees of freedom, which has an expectation that can be upper-bounded using Gautschi's inequality (using Wendel's version of the upper bound): $\\mathbb {E}[||\\mathbf {\\epsilon }||]=\\sqrt{2}\\frac{\\Gamma ((d+1)/2)}{\\Gamma (d/2)}\\le \\sqrt{2}\\left(\\frac{d}{2}\\right)^{1/2}=\\sqrt{d}$ Combining the above and using Lemma REF , we have $||g^2||_{\\psi _1}\\le \\left(6L^2+\\frac{L}{\\sqrt{\\log 2}}(\\sqrt{d}+||{\\mathbf {g}}^{-1}(\\mathbf {0})||)\\right)^2$ Setting $C$ to be the RHS and applying Theorem REF yield the desired result."],["Experimental Details","For the predictive tasks (Section REF ), we use a cosine annealing schedule for the learning rate, scaling down to 0.01 of the initial learning rate, and pretrain a deterministic network for 10 epochs using the Adam optimizer with a learning rate of 0.001, to initialize the mean of the Gaussian $q_0$ , and train $q$ for 200 epochs."],["LeNet-5 MNIST.","We use a linear annealing schedule of the $\\beta $ coefficient (from 0 back to 1) for 50,000 iterations.","We use the Adam optimizer with a learning rate of 0.0005.","The result we get for K-Linear uses polyak averaging with exponential decay coefficient 0.995.","We use the volume preserving version of RealNVP for the K-Nonlinear.","We use the standard Gaussian prior for $p$ ."],["LeNet-5 CIFAR-5","We use the same architecture as [32] (192 convolutional kernels and 1,000 hidden units for the fully connected layers).","We use a linear annealing schedule of the $\\beta $ coefficient (from 0 back to 1) for 20,000 iterations for Diag, and no annealing for K-Linear and K-Nonlinear.","We use the Adam optimizer with a learning rate of 0.0003, 0.0003, 0.0005 for Diag, K-Linear and K-Nonlinear, respectively.","We use the volume preserving version of RealNVP for K-Nonlinear.","We use the standard Gaussian prior for $p$ ."],["VGG-16 CIFAR-10","We use the modified version of VGG-16 proposed by [53].","We use a learning rate of 0.0005 for all experiments but K-Nonlinear in the regular setup (where we use 0.001).","We use the isotropic Gaussian prior with variance being 0.1, and set $\\beta $ to be [0.5, 0.1, 0.1, 0.5] in the regular setup and [0.5, 0.1, 0.1, 0.1] in the data augmented setup for Diag, K-Diag, K-Linear, and K-Nonlinear, respectively.","We use the volume preserving version of RealNVP for the K-Nonlinear."],["PAC-Bayes MLP","We follow the same steps as [12], except we did not discretize the prior variance after tuning.","In practice this does not affect the bound much.","We also did not initialize the mean of $q_0$ in our setup using SGD for our experiments.","We train the stochastic network for 300 epochs, with a learning rate of 0.002.","The bound holds with probability at least 0.965 over the choice of prior and the training set.","The $b$ and $c$ coefficients in [12] are set as 100 and 0.1.","We use the volume preserving version of IAF for the K-Nonlinear."],["PAC-Bayes LeNet-5","The same setup as PAC-Bayes MLP, except with polyak averaging with coefficient 0.995.","We use the volume preserving version of IAF for the K-Nonlinear."],["Bandit Benchmark","All the models share the same architechture: one hidden layer with 50 units.","We use the volume preserving version of RealNVP for K-NonLinear.","We train models every 50 time steps for 200 training iterations using a batch-size of 512.","Table: Description of bandit problem: number of actions and number of contexts used for experiments.","Comparing to benchmark, we restrict ourself to 50000 contexts for Covertype instead of 150000 contexts.Table: Additional results: Cumulative regret incurred by different algorithms on the bandit benchmarks described in .","Values reported are the mean over 5 independent trials with standard error of the mean, normalized with respect to the performance of the uniform policy.","We use the same hyperparameters for different algorithms without any finetuning: learning rate = 0.0001 and 100 training epochs."]],"string":"[\n [\n \"Stochastic Neural Network with Kronecker Flow\"\n ],\n [\n \"Abstract Recent advances in variational inference enable the modelling of highly structured joint distributions, but are limited in their capacity to scale to the high-dimensional setting of stochastic neural networks.\",\n \"This limitation motivates a need for scalable parameterizations of the noise generation process, in a manner that adequately captures the dependencies among the various parameters.\",\n \"In this work, we address this need and present the Kronecker Flow, a generalization of the Kronecker product to invertible mappings designed for stochastic neural networks.\",\n \"We apply our method to variational Bayesian neural networks on predictive tasks, PAC-Bayes generalization bound estimation, and approximate Thompson sampling in contextual bandits.\",\n \"In all setups, our methods prove to be competitive with existing methods and better than the baselines.\"\n ],\n [\n \"Introduction\",\n \"Work done while Chin-Wei was an intern at Element AI and Ahmed at Facebook Research.\",\n \"Stochastic neural networks (SNN) are a central tool in many subfields of machine learning, including (1) Bayesian deep learning [33], [6], [19], [14], (2) exploration in reinforcement learning [23], [39], [43], and (3) statistical learning theory such as PAC-Bayesian learning [34], [29], [12].\",\n \"Perturbations of the network parameters induce a distribution over the model, and this intrinsic uncertainty is the subject of great interest to machine learning practitioners and theoreticians alike.\",\n \"For example, deep Bayesian models are often used to adequately measure uncertainty, and determine whether the model itself is inherently familiar with the unseen data.\",\n \"This is especially important in the context of autonomous vehicles, where decisions must be made to meet specific safety standards [35].\",\n \"Conversely, the lack of confidence can be leveraged to efficiently guide exploration in reinforcement learning, via randomizing the approximate value function [2], [49] or maximizing intrinsic rewards [20].\",\n \"Furthermore, a considerable proportion of statistical learning theory is devoted to understanding what implies generalization, or what constitutes an appropriate measure of complexity [3], [1], [38].\",\n \"PAC-Bayesian learning theory [34] specifically explores the generalization property of a randomized prediction rule, and has been recently studied in the context of stochastic neural networks [12].\",\n \"In this particular study, the working hypothesis is that good generalization can be guaranteed on the premise that stochastic gradient descent [45] finds a solution that obtains certain structural properties, such as flatness.\",\n \"For computational reasons, considerable effort has been devoted to modelling uncertainty through the injection of independent noise to the network parameters [18], [6], [26].\",\n \"However, noise independence largely restricts the expressivity of the noise distribution and thus the resulting uncertainty measures are ill-calibrated [36], [50].\",\n \"Attempts have been made to correlate parameters of a neural network, including [32], [28], [40], for example, by adapting expressive non-linear invertible transformations developed in the variational inference literature [42], [25], [21], or via implicit methods [17].\",\n \"However, these methods are limited due to their inability to scale well.\",\n \"[32], for instance, resort to a specific multiplicative noise sampled from a lower dimensional space and have to use an auxiliary method to bound the entropy.\",\n \"[28], on the other hand, give up on injecting noise on the entire set of parameters and model the distribution of the scale and shift parameter of the pre-activations.\",\n \"In attempts to address some of the challenges articulated above and efficiently model the joint distribution of a network's parameters, we take inspiration from the Kronecker product, which we notice can be thought of as left-transforming a matrix via a linear map, and then right-transforming it using another linear map, thus providing us an efficient way to correlate the weight parameters.\",\n \"We propose the Kronecker Flow, an invertible transformation-based method that generalizes the Kronecker product to its nonlinear counterparts.\",\n \"Our contributions are as follows.\",\n \"We extend the idea of Kronecker product to more general invertible mappings to induce non-linear dependencies, and apply this trick to parameterizing deep stochastic neural networks.\",\n \"We apply our method to predictive tasks and show that our methods work better on larger architectures compared to existing methods.\",\n \"We are the first to apply flow-based methods to tighten the PAC-Bayes bound.\",\n \"We show that the KL divergence in the PAC-Bayes bound can be estimated with high probability, and demonstrate the generalization gap can be further reduced and explained by leveraging the structure in the parameter space.\",\n \"Our methods prove to be competitive over other methods in approximate Thompson sampling in contextual bandit problems.\"\n ],\n [\n \"Background\",\n \"Stochastic neural networks with parameter perturbation normally follow the stochastic process: $\\\\Theta \\\\sim q_\\\\phi (\\\\Theta )$ , $y|x\\\\sim p(y|x, \\\\Theta )=f_\\\\Theta (x)$ , where $\\\\Theta $ is the parameters of the neural network $f$ , which outputs the prediction probability vector for classification or the predicted values for regression.\",\n \"We let $D=\\\\lbrace (x_i,y_i):{i\\\\in [m]}\\\\rbrace $ be the training set of size $m$ We use the notation $[n]$ to compactly describe the set of integers $\\\\lbrace 1,2,\\\\dots ,n\\\\rbrace $ ., $H$ be the differential entropy $H[q]=- \\\\mathbb {E}_q[\\\\log q]$ , $\\\\beta >0$ be the coefficient controlling the amount of noise injected into the model and the degree of regularization, $l(y,\\\\bar{y})$ be the loss function and $\\\\hat{R}_{D}(\\\\Theta )=\\\\frac{1}{m}\\\\sum _{i=1}^{m} l(y_i, f_\\\\Theta (x_i))$ be the empirical risk.\"\n ],\n [\n \"Variational Bayesian neural networks\",\n \"Variational Bayesian neural networks are a type of stochastic neural network.\",\n \"Bayesian inference updates our prior belief $p(\\\\Theta )$ over the model parameters according to the Bayes rule $p(\\\\Theta |D)\\\\propto p(D|\\\\Theta )p(\\\\Theta )$ , by incorporating information from the training set through the likelihood function $p(D|\\\\Theta )$ .\",\n \"Variational inference is a computational realization of Bayesian inference, which casts inference as an optimization problem, where one maximizes the variational lower bound (also known as the evidence lower bound, or the ELBO) on the log marginal likelihood: $\\\\log p(D) \\\\ge \\\\mathbb {E}_{q_\\\\phi }[ \\\\log p(D|\\\\Theta ) + \\\\log p(\\\\Theta )] + H(q_\\\\phi (\\\\Theta )),$ where $q_\\\\phi $ is the variational approximate posterior and $p(D|\\\\Theta )$ can be decomposed into $\\\\prod _{i=1}^{m} p(y_i|x_i, \\\\Theta )$ due to conditional independence assumption.\",\n \"The optimal $q$ is the true posterior, i.e.\",\n \"$q^*(\\\\Theta )=\\\\frac{p(D|\\\\Theta )p(\\\\Theta )}{p(D)}$ .\",\n \"In our case, we use $\\\\Theta $ to parameterize a neural network.\",\n \"Prediction can be carried out via the (approximate) predictive posterior $p(y|x,D) & = \\\\mathbb {E}_{\\\\Theta \\\\sim p(\\\\Theta |D)} [p(y|x,\\\\Theta )] \\\\\\\\& \\\\approx \\\\mathbb {E}_{\\\\Theta \\\\sim q_\\\\phi (\\\\Theta )}[p(y|x,\\\\Theta )] \\\\\\\\& \\\\approx \\\\frac{1}{K}\\\\sum _{k=1}^K p(y|x,\\\\Theta _k)$ for $\\\\lbrace \\\\Theta _k\\\\rbrace _{k\\\\in [K]}$ drawn i.i.d.\",\n \"from $q_\\\\phi (\\\\Theta )$ , where we use the variational distribution $q$ to approximate $p(\\\\Theta |D)$ and a Monte Carlo estimate to estimate the integral.\",\n \"The prior distribution can be used to encode some form of inductive bias, such as one that is in favor of parameter values closer to some $\\\\Theta _0$ chosen a priori.\",\n \"We choose the prior to be an isotropic Gaussian, centered at the random initialization $\\\\Theta _0$ , i.e., $p(\\\\Theta ) = {\\\\mathcal {N}}(\\\\Theta ;\\\\Theta _0, \\\\lambda {\\\\mathbf {I}})$ .\",\n \"The entropy term ensures the variational posterior does not collapse to a point estimate.\",\n \"Both of them can be thought of as some form of regularizer, so we attach a coefficient $\\\\beta $ in front of them as a hyperparameter Like the $\\\\lambda $ parameter in [53].\"\n ],\n [\n \"PAC-Bayes generalization bound\",\n \"Another use case of stochastic neural networks is to understand generalization, via PAC-Bayes bounds.\",\n \"The aim is to bound a divergence between the empirical risk, $\\\\hat{{\\\\mathcal {L}}}[q]=\\\\mathbb {E}_q[\\\\hat{R}_D(\\\\Theta )]$ , and the risk measured on the true distribution $\\\\mathcal {D}$ , ${\\\\mathcal {L}}[q]=\\\\mathbb {E}_q[\\\\mathbb {E}_{\\\\mathcal {D}}[l(y,f_\\\\Theta (x))]]$ .\",\n \"While this quantity is unbounded in the general case, assuming a bounded loss function $l$ (e.g.\",\n \": zero-one loss), we can obtain a probabilistic bound that holds with probability $1-\\\\delta $ over the choice of $D$ , for $\\\\delta >0$ .\",\n \"More specifically, with probability $1-\\\\delta $ , $\\\\Delta (\\\\hat{{\\\\mathcal {L}}}[q], {\\\\mathcal {L}}[q])\\\\le \\\\Omega (D_{\\\\mathrm {KL}}(q||p),m,\\\\delta )$ , where $\\\\Omega $ is a measure of complexity that scales proportionally with the Kullback–Leibler (KL) divergence and $\\\\Delta $ a measure of divergence (e.g.\",\n \": square distance or convex functions [16]).\",\n \"For instance, [12] minimize the following bound originally due to [34] and then tightened by [29]: Theorem 1 Let $l$ be the zero-one loss.\",\n \"For any $\\\\delta >0$ and data distribution ${\\\\mathcal {D}}$ , and any distribution $p$ on the space of $\\\\Theta $ , with probability at least $1-\\\\delta $ over the choice of a training set $D\\\\sim {\\\\mathcal {D}}^m$ , for all distributions $q$ on the space of $\\\\Theta $ , $D_{\\\\mathrm {KL}}(\\\\hat{{\\\\mathcal {L}}}[q]||{\\\\mathcal {L}}[q])\\\\le \\\\frac{D_{\\\\mathrm {KL}}(q||p)+\\\\log \\\\frac{m}{\\\\delta }}{m-1},$ where the KL on the LHS is between two Bernoulli distributions, defined by the probability of performing an error.\",\n \"We refer to the above bound as the McAllester bound.\",\n \"The KL divergence on the RHS of the bound, also known as the information gain, tells us to what extent the posterior $q$ is dependent on the training data.\",\n \"The sharper and more confident $q$ is, and the farther away it is from the prior $p$ , the larger the KL will be, which in turn is reflected by the larger bound on the generalization gap.\",\n \"This is consistent with traditional notion of bias-variance trade-off.\",\n \"Alternatively, we consider the following bound due to [9]: Theorem 2 With the $l$ , $\\\\delta $ , $\\\\mathcal {D}$ , and $p$ as defined in Theorem REF , and with a fixed $\\\\beta >1/2$ , the following bound holds with probability over $1-\\\\delta $ : ${\\\\mathcal {L}}[q] &\\\\le \\\\frac{1}{1-\\\\frac{1}{2\\\\beta }} \\\\left(\\\\hat{{\\\\mathcal {L}}}[q] + \\\\frac{\\\\beta }{m} \\\\left( D_{\\\\mathrm {KL}}(q||p) + \\\\ln \\\\frac{1}{\\\\delta } \\\\right) \\\\right).$ We refer to this bound as the Catoni bound.\",\n \"We notice the linear relationship (which is also noticed by [15]) between the empirical risk and the KL divergence.\",\n \"This allows us to make use of the linearity of expectation to perform change of variable (see the next section).\",\n \"We also note that the optimal $\\\\beta $ in Equation REF is always larger than 1, so the PAC-Bayes bound is actually more conservative than Bayesian inference in this sense.\"\n ],\n [\n \"Normalizing flows\",\n \"Minimization of Equation REF and REF requires (i) computing the gradient with respect to the parameter of the (PAC-)Bayesian posterior $\\\\phi $ , and (ii) computing the entropy of $q$ .\",\n \"One approach to do this is via change of variable under an invertible mapping.\",\n \"Let $\\\\mathbf {\\\\epsilon }\\\\sim q_0$ be a random variable in $\\\\mathbb {R}^d$ , and ${\\\\mathbf {g}}_\\\\phi :\\\\mathbb {R}^d\\\\rightarrow \\\\mathbb {R}^d$ be a bijection parameterized by $\\\\phi $ .\",\n \"Let $\\\\Theta ={\\\\mathbf {g}}_\\\\phi (\\\\mathbf {\\\\epsilon })$ and $q_\\\\phi $ be its density.\",\n \"Then we can rewrite the loss function as Since the weighting coefficient $\\\\beta $ can be absorbed into the loss function $l$ , we neglect it for simplicity now.\",\n \"$\\\\mathbb {E}_{\\\\Theta }[\\\\hat{R}_{D}(\\\\Theta ) + \\\\log q_\\\\phi (\\\\Theta )]& =\\\\mathbb {E}_{\\\\mathbf {\\\\epsilon }}[\\\\hat{R}_{D}({\\\\mathbf {g}}_\\\\phi (\\\\mathbf {\\\\epsilon })) + \\\\log q_0(\\\\mathbf {\\\\epsilon }) \\\\\\\\& - \\\\log \\\\left|\\\\det \\\\frac{\\\\partial {\\\\mathbf {g}}_\\\\phi (\\\\mathbf {\\\\epsilon })}{\\\\partial \\\\mathbf {\\\\epsilon }}\\\\right|],$ where we apply the change of variable (see Appendix for the detailed derivation).\",\n \"The log-determinant (logdet) term ensures that we obtain a valid probability density function after $g_\\\\phi $ is applied, which can be a sequence of invertible mappings itself, hence referred to as the normalizing flow [42].\",\n \"This way, the random variable and the parameters are decoupled, so that we can differentiate the integrand to have an unbiased estimate of the gradient (fixing some $\\\\mathbf {\\\\epsilon }\\\\sim q_0$ ).\",\n \"We let $q_0$ be the standard normal.\"\n ],\n [\n \"Kronecker Flows\",\n \"We consider maximizing the ELBO and minimizing the Catoni bound via normalizing flow-based SNNs.\",\n \"Conventionally, mean-field approximation using factorized distributions (such as multivariate Gaussian with diagonal covariance) has been well explored in the variational inference (VI) literature [6].\",\n \"We are interested in better capturing the structure in the parameter space as restricted VI methods are known to exhibit overconfidence [36], [50].\",\n \"However, the parameters of a neural network are usually very high dimensional (on the order of millions), requiring a novel way to parameterize the joint distribution over the parameters.\",\n \"In its general form, neural networks can be represented by a collection of tensors i.e.\",\n \"$\\\\Theta =\\\\lbrace {\\\\mathbf {W}}_l:{l\\\\in [L]}\\\\rbrace $ .\",\n \"While our method below can easily be generalized to high-dimensional tensors (such as for convolutional kernels), to simplify notation, we describe the matrix form.\"\n ],\n [\n \"Linear Kronecker Flow\",\n \"The matrix-variate normal ($\\\\mathcal {MN}$ ) distribution generalizes the multivariate normal distribution to matrix-valued random variables, such as weight matrices of a neural network [31].\",\n \"Matrix normal is a multivariate normal distribution whose covariance matrix is a Kronecker product ($\\\\otimes $ ), which allows us to model the correlation among the parameters to some degree.\",\n \"More concretely, assume ${\\\\mathbf {E}}_{ij}\\\\overset{\\\\textnormal {i.i.d.\",\n \"}}{\\\\sim } {\\\\mathcal {N}}(0,1)$ is an $n\\\\times p$ random Gaussian matrix, and ${\\\\mathbf {A}}\\\\in \\\\mathbb {R}^{n\\\\times n}$ , ${\\\\mathbf {B}}\\\\in \\\\mathbb {R}^{p\\\\times p}$ and ${\\\\mathbf {M}}\\\\in \\\\mathbb {R}^{n\\\\times p}$ are real-valued matrices.\",\n \"Then ${\\\\mathbf {M}}+ {\\\\mathbf {A}}{\\\\mathbf {E}}{\\\\mathbf {B}}$ has a matrix normal distribution, as $\\\\textnormal {vec}({\\\\mathbf {M}}+ {\\\\mathbf {A}}{\\\\mathbf {E}}{\\\\mathbf {B}})\\\\sim {\\\\mathcal {N}}(\\\\textnormal {vec}({\\\\mathbf {M}}), {\\\\mathbf {B}}^\\\\top {\\\\mathbf {B}}\\\\otimes {\\\\mathbf {A}}{\\\\mathbf {A}}^\\\\top ),$ where $\\\\textnormal {vec}$ is the vectorization of a matrix that concatenates all the columns.\",\n \"This allows us to represent the covariance matrix in a more compact manner ($n^2p^2/2$ parameters versus $n^2/2+p^2/2$ parameters for Kronecker product).\"\n ],\n [\n \"Limitation of the Kronecker product.\",\n \"The Kronecker product covariance matrix is not a strict generalization of diagonal covariance matrix.\",\n \"To observe this, let ${\\\\mathbf {U}}=\\\\mathop {\\\\mathrm {diag}}\\\\nolimits ({\\\\mathbf {u}})$ , ${\\\\mathbf {V}}=\\\\mathop {\\\\mathrm {diag}}\\\\nolimits ({\\\\mathbf {v}})$ (this is the case of [31]), and ${\\\\mathbf {S}}=\\\\mathop {\\\\mathrm {diag}}\\\\nolimits ({\\\\mathbf {s}})$ , where ${\\\\mathbf {u}}\\\\in \\\\mathbb {R}^n_{>0}$ , ${\\\\mathbf {v}}\\\\in \\\\mathbb {R}^p_{>0}$ , and ${\\\\mathbf {s}}\\\\in \\\\mathbb {R}^{np}_{>0}$ .\",\n \"Then ${\\\\mathbf {U}}\\\\otimes {\\\\mathbf {V}}$ is also a diagonal matrix of size $np\\\\times np$ .\",\n \"Equating ${\\\\mathbf {U}}\\\\otimes {\\\\mathbf {V}}={\\\\mathbf {S}}$ to solve for ${\\\\mathbf {u}}$ and ${\\\\mathbf {v}}$ will result in $np$ nonlinear equations with $n+p$ variables, which can be over-determined for $n,p>2$ .\",\n \"For example, let $n=2,p=3$ , and ${\\\\mathbf {s}}=[1,\\\\epsilon ,\\\\epsilon ,1,1,1]$ for some $\\\\epsilon >0$ .\",\n \"Then the nonlinear system below does not have a solution: ${\\\\mathbf {U}}\\\\otimes {\\\\mathbf {V}}= {\\\\mathbf {S}}\\\\quad \\\\Longleftrightarrow & \\\\quad {\\\\mathbf {u}}_1{\\\\mathbf {v}}_1\\\\overset{(a)}{=}1 \\\\quad {\\\\mathbf {u}}_1{\\\\mathbf {v}}_2\\\\overset{(b)}{=}\\\\epsilon \\\\quad {\\\\mathbf {u}}_1{\\\\mathbf {v}}_3\\\\overset{(c)}{=}\\\\epsilon \\\\quad \\\\\\\\& {\\\\mathbf {u}}_2{\\\\mathbf {v}}_1\\\\overset{(d)}{=}1 \\\\quad {\\\\mathbf {u}}_2{\\\\mathbf {v}}_2\\\\overset{(e)}{=}1 \\\\quad {\\\\mathbf {u}}_2{\\\\mathbf {v}}_3\\\\overset{(f)}{=}1$ To see this, dividing $(a)$ by $(b)$ and dividing $(d)$ by $(e)$ yield ${\\\\mathbf {v}}_1={\\\\mathbf {v}}_2/\\\\epsilon $ and ${\\\\mathbf {v}}_1={\\\\mathbf {v}}_2$ , respectively, which doesn't have a solution if $\\\\epsilon \\\\ne 1$ .\",\n \"This is because the Kronecker product is essentially parameter sharing, which can heavily restrict the matrix it can represent.\",\n \"To remedy the above limitation, we can further decouple the reparameterization of the parameter matrix into two parts: (1) one that models the marginal variance and (2) one that models correlations.\",\n \"Assume ${\\\\mathbf {S}}\\\\in \\\\mathbb {R}^{n\\\\times p}_{>0}$ is a positive-valued matrix, and let ${\\\\mathbf {W}}:={\\\\mathbf {M}}+ {\\\\mathbf {A}}({\\\\mathbf {E}}\\\\circ {\\\\mathbf {S}}){\\\\mathbf {B}}$ .\",\n \"Then $\\\\textnormal {vec}({\\\\mathbf {W}})$ is a Gaussian distribution with the following property, which is useful in calculating the KL divergence: Property 1 Let ${\\\\mathbf {W}}$ be given as above, with $\\\\mu =\\\\mathbb {E}[\\\\textnormal {vec}({\\\\mathbf {W}})]$ and $\\\\Sigma =\\\\mathrm {Var}(\\\\textnormal {vec}({\\\\mathbf {W}}))$ .\",\n \"Then $\\\\mu =\\\\textnormal {vec}({\\\\mathbf {M}})$ , and $\\\\Sigma =({\\\\mathbf {B}}^\\\\top \\\\otimes {\\\\mathbf {A}}) \\\\mathop {\\\\mathrm {diag}}\\\\nolimits (\\\\textnormal {vec}({\\\\mathbf {S}}^2)) ({\\\\mathbf {B}}\\\\otimes {\\\\mathbf {A}}^\\\\top )$ $\\\\det (\\\\Sigma )=\\\\det ({\\\\mathbf {A}})^{2p}\\\\det ({\\\\mathbf {B}})^{2n}\\\\prod _{ij}{\\\\mathbf {S}}_{ij}^2$ $\\\\textnormal {Tr}(\\\\Sigma )=\\\\sum _{ij}\\\\left({\\\\mathbf {A}}^2{\\\\mathbf {S}}^2{\\\\mathbf {B}}^2\\\\right)_{ij}$ See Appendix for the derivation and interpretation of the property.\",\n \"Naive implemetations of this can be inefficient and numerically unstable, as the entropy term involves computing the log-determinant of ${\\\\mathbf {A}}$ and ${\\\\mathbf {B}}$ , requiring the standard automatic differentiation libraries to resort to singular value decomposition when the matrix is near-singular.\",\n \"Thus, we choose to parameterize ${\\\\mathbf {A}}$ and ${\\\\mathbf {B}}$ as lower triangular matricesThis is achieved by masking.\",\n \"with ones on the diagonal, leaving the uncertainty to be modeled by ${\\\\mathbf {S}}$ .\",\n \"This means $\\\\det (\\\\Sigma )=\\\\prod _{ij}{\\\\mathbf {S}}_{ij}^2$ .\",\n \"Figure: Random 3D Gaussian tensor\"\n ],\n [\n \"Simulation.\",\n \"To validate the limited expressiveness of kronecker product, we randomly initialize a target density $p$ to be a multivariate Gaussian with mean zero, and covariance being the square of a random standard Gaussian matrix.\",\n \"We choose the dimensionality $d$ of the Gaussian such that it can be decomposed into a product of integers, and parameterize $q$ using independent Gaussian (dubbed Diag), the Kronecker product with diagonal $A$ and $B$ (K-Diag), and the Kronecker product with elementwise scaling (K-Linear).\",\n \"We minimize $D_{\\\\mathrm {KL}}(q||p)$ ; see Figure REF for the results.\",\n \"We also conduct the same experiment with 3D tensors (instead of matrices).\",\n \"We see that K-Diag consistently underperforms when compared to Diag, which indicates parameter sharing does restrict the family of distributions it can represent, and K-Linear is consistently better as it captures some correlation.\"\n ],\n [\n \"Nonlinear Kronecker Flow\",\n \"In this section, we generalize the Kronecker product to more general non-linear mappings.\",\n \"In Appendix , we make a connection to non-decreasing triangle maps [52] that are general enough to model any probability distributions.\",\n \"First, notice that left-multiplying ${\\\\mathbf {E}}$ by ${\\\\mathbf {A}}$ amounts to introducing linear correlation among the $n$ rows of ${\\\\mathbf {E}}$ , applied to each of the $p$ columns.\",\n \"Likewise, right-multiplying ${\\\\mathbf {E}}$ by ${\\\\mathbf {B}}$ amounts to correlating column entries of each row of ${\\\\mathbf {E}}$ .\",\n \"Inspired by this, we consider applying an invertible mapping to each row of the random weight matrix, and another invertible mapping to each column of the matrix.\",\n \"We call this the Kronecker Flow To differentiate this from K-Linear from the previous section, we refer to using non-linear ${\\\\mathbf {g}}$ as K-Nonlinear..\",\n \"Specifically, let ${\\\\mathbf {g}}_A:\\\\mathbb {R}^{n}\\\\rightarrow \\\\mathbb {R}^{n}$ and ${\\\\mathbf {g}}_B:\\\\mathbb {R}^{p}\\\\rightarrow \\\\mathbb {R}^{p}$ be invertible mappings.\",\n \"We define the matrix-matrix function ${\\\\mathbf {G}}:\\\\mathbb {R}^{n\\\\times p}\\\\rightarrow \\\\mathbb {R}^{n\\\\times p}$ as ${\\\\mathbf {G}}_B({\\\\mathbf {G}}_A({\\\\mathbf {E}}^\\\\top )^\\\\top )$ , with the following batch-operations (for $i\\\\in [n]$ and $j\\\\in [p]$ ): ${\\\\mathbf {G}}_A({\\\\mathbf {E}}^\\\\top )_{j:} := {\\\\mathbf {g}}_A({\\\\mathbf {E}}_{:j}) \\\\qquad \\\\qquad {\\\\mathbf {G}}_B({\\\\mathbf {E}})_{i:} := {\\\\mathbf {g}}_B({\\\\mathbf {E}}_{i:}) $ It is easy to verify that ${\\\\mathbf {G}}$ is invertible.\",\n \"Due to the partial dependency of ${\\\\mathbf {G}}_A$ and ${\\\\mathbf {G}}_B$ , the Jacobians of the vectorized forms (after proper permutation) are block-diagonal, so we have $\\\\det &\\\\frac{\\\\partial \\\\textnormal {vec}({\\\\mathbf {G}}({\\\\mathbf {E}}))}{\\\\partial \\\\textnormal {vec}({\\\\mathbf {E}})} \\\\\\\\&= \\\\prod _{j\\\\in [p]}\\\\det \\\\frac{\\\\partial {\\\\mathbf {g}}_A({\\\\mathbf {E}}_{:j})}{\\\\partial {\\\\mathbf {E}}_{:j}}\\\\cdot \\\\prod _{i\\\\in [n]}\\\\det \\\\frac{\\\\partial {\\\\mathbf {g}}_B({\\\\mathbf {G}}_A({\\\\mathbf {E}}^\\\\top )_{:i})}{\\\\partial {\\\\mathbf {G}}_A({\\\\mathbf {E}}^\\\\top )_{:i}}.$ In practice, we use the volume preserving version of RealNVP [11] and inverse autoregressive flow (IAF) [25] to parameterize ${\\\\mathbf {g}}_A$ and ${\\\\mathbf {g}}_B$ for our experiments We also experimented with Block NAF by [10], but did not include it in the manuscript: its performance was similar to IAF, but it is much slower to sample from (we adapted the open implementation from [10])..\",\n \"The K-Linear from the previous section can be thought of as using a linear map as ${\\\\mathbf {g}}_A$ and ${\\\\mathbf {g}}_B$ .\"\n ],\n [\n \"Concentration of empirical KL with normalizing flows\",\n \"In their study, [12] use independent Gaussian for $q$ to minimize the McAllester bound, so they can compute the KL between Gaussians analytically.\",\n \"This is no longer feasible when we use more flexible families for $q$ , such as normalizing flows.\",\n \"Moreover, a Monte Carlo estimate might result in underestimating the bound after inverting the KL between Bernoullis on the LHS of Equation REF (which is a concave function; see Appendix A of [41] for an illustration).\",\n \"This necessitates a high probability bound on the concentration of the empirical estimate.\",\n \"In Section REF , we have established $D_{\\\\mathrm {KL}}(q||p)$ can be written in the following form $\\\\leavevmode \\\\xbox {resizebox}{\\\\XMLaddatt {width}{427.0pt}\\\\mathbb {E}_{\\\\mathbf {\\\\epsilon }} \\\\left[\\\\log {\\\\mathcal {N}}(\\\\mathbf {\\\\epsilon }; \\\\mathbf {0},{\\\\mathbf {I}}) - \\\\log \\\\left|\\\\det \\\\frac{\\\\partial {\\\\mathbf {g}}_\\\\phi (\\\\mathbf {\\\\epsilon })}{\\\\partial \\\\mathbf {\\\\epsilon }}\\\\right|-\\\\log {\\\\mathcal {N}}({\\\\mathbf {g}}_\\\\phi (\\\\mathbf {\\\\epsilon }); \\\\mathbf {0}, {\\\\mathbf {I}}) \\\\right],}$ where both $q_0$ and $p$ are standard Gaussian (the mean and variance can be absorbed into the invertible mapping ${\\\\mathbf {g}}$ if this is not the case).\",\n \"The first term in the KL can be computed analytically.\",\n \"The second term usually can be almost surely bounded (e.g.\",\n \"using Block neural autoregressive flows) so that we can use Hoeffding-type concentration or it can simply be made zero using e.g.\",\n \"volume preserving flows.\",\n \"The challenge now lies in the third term, which has a quadratic form $\\\\frac{1}{2}||{\\\\mathbf {g}}(\\\\mathbf {\\\\epsilon })||^2$ , neglecting the normalizing constant.\",\n \"Now assume ${\\\\mathbf {g}}$ is a $L_0$ -Lipschitz The following flows are all Lipschitz (with Lipschitz activation functions): volume preserving version of [11], [25], [5], [4], [10], etc.. Let $g(\\\\mathbf {\\\\epsilon })=\\\\frac{1}{\\\\sqrt{2}}||{\\\\mathbf {g}}(\\\\mathbf {\\\\epsilon })||$ .\",\n \"Then $g$ is $L_0/\\\\sqrt{2}$ -Lipschitz: $\\\\big |g(\\\\mathbf {\\\\epsilon }_1)-g(\\\\mathbf {\\\\epsilon }_2)\\\\big |= &\\\\frac{1}{\\\\sqrt{2}}\\\\big |||{\\\\mathbf {g}}(\\\\mathbf {\\\\epsilon }_1)|| - ||{\\\\mathbf {g}}(\\\\mathbf {\\\\epsilon }_2)||\\\\big | \\\\\\\\& \\\\le \\\\frac{1}{\\\\sqrt{2}}||{\\\\mathbf {g}}(\\\\mathbf {\\\\epsilon }_1) - {\\\\mathbf {g}}(\\\\mathbf {\\\\epsilon }_2)||\\\\le \\\\frac{L_0}{\\\\sqrt{2}}||\\\\mathbf {\\\\epsilon }_1-\\\\mathbf {\\\\epsilon }_2||.$ This is key in deriving a tail bound on $g^2$ , as Lipschitz functions of canonical Gaussian random variables are sub-Gaussians, meaning they have a tail that decays faster than a Gaussian random variable.\",\n \"The following theorem provides a concentration bound for the empirical average of $g^2$ similar to that of a Chi-square random variable, as $g^2$ (square of a sub-Gaussian) is sub-exponential.\",\n \"Theorem 3 Let $g$ be defined as above with a Lipschitz constant $L=L_0/\\\\sqrt{2}$ .\",\n \"Let $\\\\bar{g}^2=\\\\frac{1}{K}\\\\sum _{k=1}^K g_k^2$ .\",\n \"Then the following concentration bound holds ${\\\\mathbb {P}}(\\\\bar{g}^2-\\\\mathbb {E}[g^2]>\\\\epsilon )\\\\le \\\\exp \\\\left(-\\\\frac{K\\\\epsilon ^2}{2(4C^2+C\\\\epsilon )}\\\\right),$ where $C=\\\\left(6L^2+\\\\frac{L}{\\\\sqrt{\\\\log 2}}(\\\\sqrt{d}+||{\\\\mathbf {g}}^{-1}(\\\\mathbf {0})||)\\\\right)^2$ .\",\n \"Note that in practice the empirical KL that we use is inversely scaled by the size of the training set $m$ (see Equation REF ), so the Lipschitz constant can be made small in practice to dominate the dimensionality.\"\n ],\n [\n \"Experiments\",\n \"We evaluate our proposed method in the context of two prediction tasks (Section REF ), PAC-Bayes bound minimization (Section REF ) and contextual bandit (Section REF ).\",\n \"For the two prediction tasks, we use the MNIST handwritten digit dataset [30] and CIFAR-10 [27].\",\n \"See Appendix for a detailed description.\",\n \"Table: Test error with LeNet (%) on MNIST and the first 5 classes of CIFAR-10.\",\n \"First 3 columns are from .\",\n \"K-Diag on CIFAR-5 diverged, so we did not include the result.Table: Test error with modified version of VGG16 (%) on CIFAR10.First 4 columns are from .R means regular training and D means training with data augmentation.Table: PAC-Bayes bound estimation: We minimize the Pinsker bound (an upper bound on the McAllester bound) and the Catani bound using different flows, and estimate the McAllester bound at inference time using Newton's method.Table: Cumulative regret incurred by different algorithms on the bandit benchmarks described in .\",\n \"Values reported are the mean over 3 independent trials with standard error of the mean, normalized with respect to the performance of the uniform policy.\"\n ],\n [\n \"Classification\",\n \"One benefit of Bayesian neural networks compared to the regular ones is that the trade-off between the prior and the likelihood is a form of regularization.\",\n \"In this section, we evaluate the generalization performance of our method applied to Bayesian neural networks.\",\n \"We consider two architectures: LeNet-5 [30] and a modified version VGG-16 [46] proposed by [53].\",\n \"We first compare to the multiplicative normalizing flow (MNFG) proposed by [32], applying our method to LeNet-5 (see Table REF ).\",\n \"Our Diag matches the performance of their FFG (fully factorized Gaussian).\",\n \"K-Diag outperforms Diag in this case, perhaps due to the smaller number of parameters which makes it easier to optimize.\",\n \"K-Nonlinear yields the best generalization error in this case.\",\n \"On the CIFAR-5 experiment (we take the first 5 classes of CIFAR-10), our methods are on par with MNFG.\",\n \"Second, we compare with the noisy K-FAC proposed by [53], applying our methods to the larger architecture VGG-16 (see Table REF ).\",\n \"Noisy K-FAC applies an approximate natural gradient method.\",\n \"Despite this advantage, our methods (K-Linear and K-Nonlinear) have simiar prediction accuracy in the regular setup.\",\n \"We also include the results of data augmentation with horizontal flip and random crop where K-Nonlinear outperforms all the other methods.\"\n ],\n [\n \"PAC Bayes bound minimization\",\n \"For the PAC-Bayes bound estimation, we minimize Equation REF .\",\n \"We follow the recipe of [12].\",\n \"We upper bound the zero-one loss by cross-entropy divided by $\\\\log |{\\\\mathcal {Y}}|$ (where $|{\\\\mathcal {Y}}|$ is the number of classes) to make the upper bound tight.\",\n \"We set the prior to be ${\\\\mathcal {N}}(\\\\Theta _0,\\\\lambda {\\\\mathbf {I}})$ , where $\\\\Theta _0$ is the initial value of the parameters, and apply a union bound to tune the prior variance $\\\\lambda $ .\",\n \"We also tune the $\\\\beta $ coefficient as a parameter during training We are allowed to do so since we treat Equation REF as an optimization objective, rather than report it as a bound.\",\n \"We report the McAllester bound, which holds for any $q$ , even if it depends on $\\\\beta $ ., and report the McAllester bound for comparison (since it is the tightest).\",\n \"For more details, see [12] for reference.\",\n \"We test with a multi-layer perceptron with 1 or 2 hidden layers with 600 neurons and LeNet-5, evaluated on the MNIST dataset (see Table REF ).\",\n \"For further clarification, we follow the steps of [12] by minimizing the McAllester bound, using Pinsker's inequality to bound the inverse of the Bernoulli KL (which we call the Pinsker bound).\",\n \"Since this bound has a square root in the complexity term, we can only use the Gaussian family with an analytic form of the KL.\",\n \"The result we have is slightly looser than [12] since we have a 10-class problem and they deal with a binary version of MNIST.\",\n \"We see that the bound can indeed be improved by capturing the correlation among the parameters.\",\n \"We then compare to minimizing the Catoni bound, which is slightly looser since the linear relationship between the empirical risk and the KL term penalizes the latter more when the KL is larger.\",\n \"However, by modelling the non-linear dependencies, K-Nonlinear clearly outperforms the other methods (even compared to the ones minimizing the Pinsker bound).\",\n \"This indicates there exists a considerable amount of structure in the parameter space that may explain the gap between the test error and the generalization bound.\",\n \"We also notice that, despite the linear relationship, the Catoni bound focuses more on the complexity term than the ELBO.\",\n \"For example, the empirical risks of LeNet-5 in Table REF are much higher compared to the test loss of Table REF .\",\n \"The reasons are: (1) the optimal $\\\\beta $ in Equation REF is larger than 1 (depending on the relative value of the KL), and (2) to properly upper bound the zero-one loss, we scale down the cross-entropy loss by $\\\\log |{\\\\mathcal {Y}}|$ during optimization.\",\n \"This means a learning algorithm based on a tight PAC-Bayes risk bound cannot overfit by much; see [13] for a recent demonstration.\",\n \"A smaller value of $\\\\beta $ could bring down the risk and empirical risk, resulting in a looser bound but usually better test performance.\",\n \"This trade-off between test set performance and the tightness of the bound is a general issue for generalization bounds.\",\n \"One reason for the interest in PAC-Bayes bounds is that their optimization leads to training algorithms with generalization guarantees.\",\n \"The bounds, however, are considerably looser than held-out estimates.\",\n \"A recent work by [44] shows PAC-Bayes bound can potentially be made tight; however, we could not reproduce the results and the best bound using Gaussian prior was around 40% in their setting.\",\n \"Our work produces much tighter bounds by building flexible families of distributions on neural network weight matrices.\"\n ],\n [\n \"Contextual bandit\",\n \"Uncertainty modeling lies at the heart of the exploration-exploitation dilemma in sequential decision-making.\",\n \"In order to maximize its collected cumulative rewards, an agent should trade off exploring different actions and gaining more knowledge about the reward estimate vs. exploiting the current estimate and allocating resources to the actions that are likely rewarding.\",\n \"Thompson sampling (TS) [48] is one the popular approaches that deals with the latter trade-off by maintaining posterior distribution over reward models and randomizing actions on the basis of their probability of being optimal.\",\n \"In this section, we investigate the effectiveness of our proposed method for performing an approximate Thompson sampling in the particular setting of contextual bandits.\",\n \"In the latter setting, at each time $t = 1 \\\\ldots T$ , the agent sees a $d$ -dimensional context $X_t$ , selects one of the $k$ available actions, $a_t$ , and earns a reward $r_t$ generated by the environment.\",\n \"The agent aims to minimize its cumulative regret defined as $R = \\\\mathbb {E}[\\\\sum _{t=1}^T r^{\\\\star }_t - r_t]$ where $r^\\\\star _t$ is the highest expected reward given the context $X_t$ and the expectation is over the randomness of both environment and the agent's choice of actions.\",\n \"We compare different methods on a range of real-world bandit problems introduced by [43].\",\n \"We train the models every 50 time steps for 200 iterations using a batch-size of 512.\",\n \"We ran each experiment with 3 different random seeds and we report the means and standard errors of cumulative regret normalized with respect to the uniform baseline in the table REF .\",\n \"We include the functional variational Bayesian neural networks (fBNN), recently introduced by [47] as a baseline, and we use their open sourced implementation of fBNN in the bandit setting.\",\n \"From table REF , we see that across the 6 bandit problems, our proposed method (K-Linear and K-Nonlinear) provides competitive and consistent results.\",\n \"They outperform other baselines in 4 problems out of 6.\"\n ],\n [\n \"Conclusion\",\n \"In this work, we present the Kronecker Flow, a flow-based method to induce complex distribution inspired by the Kronecker product.\",\n \"Our methods scale to larger architectures such as VGG-16 since it takes advantage of the shape of the parameters.\",\n \"We demonstrate our methods work better than vanilla Kronecker product with diagonal matrices on multiple setups, including classification and approximate Thompson sampling in contexual bandit, and prove to be competitive with existing methods in the Bayesian neural network literature.\",\n \"We are also the first to apply flow-based methods to obtain a tighter numerical generalization bound.\",\n \"Our work shows that the dependencies among network parameters constitute a non-negligible portion of the gap between risk and PAC-Bayes generalization bound.\"\n ],\n [\n \"Acknowledgement\",\n \"CWH would like to thank Kris Sankaran for pointing to the TIS inequality for Gaussian concentration, which is a key component in deriving the tail bound on Lipschitz flows.\",\n \"Law of the unconscious statistician Let $(\\\\Omega , {\\\\mathcal {F}}, {\\\\mathbb {P}})$ be our probability space.\",\n \"Let $\\\\mathbf {\\\\epsilon }\\\\in \\\\mathbb {R}^d$ be a random variable following the (Lebesgue) density $q_0(\\\\mathbf {\\\\epsilon })=\\\\frac{\\\\mathrm {d}\\\\mathbf {\\\\epsilon }_*{\\\\mathbb {P}}}{\\\\mathrm {d}\\\\mu }$ and $\\\\mathbf {\\\\epsilon }_*{\\\\mathbb {P}}$ being its pushforward measure, and write $\\\\Theta ={\\\\mathbf {g}}_\\\\phi (\\\\mathbf {\\\\epsilon })\\\\in \\\\mathbb {R}^d$ with $q_\\\\phi =\\\\frac{\\\\mathrm {d}({\\\\mathbf {g}}_\\\\phi \\\\circ \\\\mathbf {\\\\epsilon })_*{\\\\mathbb {P}}}{\\\\mathrm {d}\\\\mu }$ being its density and $({\\\\mathbf {g}}_\\\\phi \\\\circ \\\\mathbf {\\\\epsilon })_*{\\\\mathbb {P}}$ being its pushforward measure, and $A=\\\\hat{R}_D(\\\\Theta )+\\\\log q_\\\\phi (\\\\Theta )\\\\in \\\\mathbb {R}$ .\",\n \"Then $\\\\mathbb {E}[A] &= \\\\int _{\\\\mathbb {R}^d} A \\\\,\\\\,\\\\mathrm {d}({\\\\mathbf {g}}_\\\\phi \\\\circ \\\\mathbf {\\\\epsilon })_*{\\\\mathbb {P}}&&= \\\\int _{\\\\mathbb {R}^d} \\\\left(\\\\hat{R}_D(\\\\Theta ) + \\\\log q_\\\\phi (\\\\Theta ) \\\\right) q_\\\\phi (\\\\Theta ) \\\\,\\\\mathrm {d}\\\\Theta \\\\\\\\&= \\\\int _{\\\\mathbb {R}^d} A\\\\circ \\\\Theta \\\\,\\\\,\\\\mathrm {d}\\\\mathbf {\\\\epsilon }_*{\\\\mathbb {P}}&&= \\\\int _{\\\\mathbb {R}^d} \\\\left(\\\\hat{R}_D({\\\\mathbf {g}}_\\\\phi (\\\\mathbf {\\\\epsilon })) + \\\\log q_\\\\phi ({\\\\mathbf {g}}_\\\\phi (\\\\mathbf {\\\\epsilon })) \\\\right) q_0(\\\\mathbf {\\\\epsilon }) \\\\,\\\\mathrm {d}\\\\mathbf {\\\\epsilon }\\\\\\\\&{}&&=\\\\int _{\\\\mathbb {R}^d} \\\\left(\\\\hat{R}_{D}({\\\\mathbf {g}}_\\\\phi (\\\\mathbf {\\\\epsilon })) + \\\\log q_0(\\\\mathbf {\\\\epsilon }) - \\\\log \\\\left|\\\\det \\\\frac{\\\\partial {\\\\mathbf {g}}_\\\\phi (\\\\mathbf {\\\\epsilon })}{\\\\partial \\\\mathbf {\\\\epsilon }}\\\\right|\\\\right)q_0(\\\\mathbf {\\\\epsilon })\\\\,\\\\mathrm {d}\\\\mathbf {\\\\epsilon }$ Derivation and interpretation of Property REF We first derive Property REF algebraically, and give an interpretation that can be genralized to higher dimensional tensor operation.\",\n \"Recall that we have the following givens: Assume ${\\\\mathbf {E}}_{ij}\\\\overset{\\\\textnormal {i.i.d.\",\n \"}}{\\\\sim } {\\\\mathcal {N}}(0,1)$ is a $n\\\\times p$ random Gaussian matrix.\",\n \"Assume ${\\\\mathbf {A}}\\\\in \\\\mathbb {R}^{n\\\\times n}$ and ${\\\\mathbf {B}}\\\\in \\\\mathbb {R}^{p\\\\times p}$ .\",\n \"${\\\\mathbf {S}}\\\\in \\\\mathbb {R}^{n\\\\times p}_{>0}$ .\",\n \"${\\\\mathbf {M}}\\\\in \\\\mathbb {R}^{n\\\\times p}$ .\",\n \"If we rescale ${\\\\mathbf {E}}$ elementwise by ${\\\\mathbf {S}}$ before inducing the column-wise and row-wise correlation, we have: (superscript is Hadamard power) $\\\\Sigma :=\\\\mathrm {Var}(\\\\textnormal {vec}({\\\\mathbf {M}}+ {\\\\mathbf {A}}({\\\\mathbf {E}}\\\\circ {\\\\mathbf {S}}){\\\\mathbf {B}}))&=\\\\mathrm {Var}( ({\\\\mathbf {B}}^\\\\top \\\\otimes {\\\\mathbf {A}})\\\\textnormal {vec}({\\\\mathbf {E}}\\\\circ {\\\\mathbf {S}}))\\\\\\\\&=({\\\\mathbf {B}}^\\\\top \\\\otimes {\\\\mathbf {A}}) \\\\mathop {\\\\mathrm {diag}}\\\\nolimits (\\\\textnormal {vec}({\\\\mathbf {S}}^2)) ({\\\\mathbf {B}}^\\\\top \\\\otimes {\\\\mathbf {A}})^\\\\top \\\\\\\\&=({\\\\mathbf {B}}^\\\\top \\\\otimes {\\\\mathbf {A}}) \\\\mathop {\\\\mathrm {diag}}\\\\nolimits (\\\\textnormal {vec}({\\\\mathbf {S}}^2)) ({\\\\mathbf {B}}\\\\otimes {\\\\mathbf {A}}^\\\\top )$ If ${\\\\mathbf {S}}$ is a matrix of ones, the RHS equals $({\\\\mathbf {B}}^\\\\top \\\\otimes {\\\\mathbf {A}}) ({\\\\mathbf {B}}\\\\otimes {\\\\mathbf {A}}^\\\\top )= ({\\\\mathbf {B}}^\\\\top {\\\\mathbf {B}})\\\\otimes ({\\\\mathbf {A}}{\\\\mathbf {A}}^\\\\top )$ , which is the covariance of the matrix normal.\",\n \"Generally, ${\\\\mathbf {S}}$ might not be a matrix of ones.\",\n \"But we can still compute the determinant and trace of the covariance matrix (useful in computing KL): $\\\\det (\\\\Sigma )=\\\\det ({\\\\mathbf {A}})^{2p}\\\\det ({\\\\mathbf {B}})^{2n}\\\\prod _{ij}{\\\\mathbf {S}}_{ij}^2$ $\\\\textnormal {Tr}(\\\\Sigma )&=\\\\sum _i \\\\Sigma _{ii} \\\\\\\\&=\\\\sum _i \\\\sum _{j} ({\\\\mathbf {B}}^\\\\top \\\\otimes {\\\\mathbf {A}})_{ij}\\\\textnormal {vec}({\\\\mathbf {S}}^2)_j ({\\\\mathbf {B}}\\\\otimes {\\\\mathbf {A}}^\\\\top )_{ji}\\\\\\\\&=\\\\sum _i \\\\sum _{j} ({\\\\mathbf {B}}^{2\\\\top }\\\\otimes {\\\\mathbf {A}}^2)_{ij}\\\\textnormal {vec}({\\\\mathbf {S}}^2)_j\\\\\\\\&=\\\\sum _i\\\\left( ({\\\\mathbf {B}}^{2\\\\top }\\\\otimes {\\\\mathbf {A}}^2)\\\\textnormal {vec}({\\\\mathbf {S}}^2) \\\\right)_i\\\\\\\\&=\\\\sum _{ij}\\\\left({\\\\mathbf {A}}^2{\\\\mathbf {S}}^2{\\\\mathbf {B}}^2\\\\right)_{ij}$ Interpretation of the determinant and trace.\",\n \"The determinant measures the change in volume due to the linear map.\",\n \"Since each operation (elementwise multiplication with ${\\\\mathbf {S}}$ , left-multiplication with ${\\\\mathbf {A}}$ , and right-multiplication with ${\\\\mathbf {B}}$ ) is an invertible map, the determinant of the composition is a product of determinants.\",\n \"After elementwise multiplication with ${\\\\mathbf {S}}$ (hence $\\\\prod {\\\\mathbf {S}}_{ij}$ )The power 2 comes from the fact that we are looking at the determinant of the covariance.\",\n \"Direct computation of the likelihood involves $\\\\frac{1}{2}\\\\log \\\\det (\\\\Sigma )$ , which is equivalent to the log-determinant of the invertible map., we apply the same linear map ${\\\\mathbf {A}}$ to the columns of $({\\\\mathbf {E}}\\\\circ {\\\\mathbf {S}})$ ; in vector form, this corresponds to left-multiplication with a block diagonal of $p$ ${\\\\mathbf {A}}$ 's, hence $\\\\det ({\\\\mathbf {A}})^{p}$ .\",\n \"The same reasoning explains $\\\\det ({\\\\mathbf {B}})^n$ .\",\n \"The trace of the covariance can be written as $\\\\textnormal {Tr}(\\\\Sigma )=\\\\sum _{ij}\\\\mathrm {Var}({\\\\mathbf {W}}_{ij})$ , i.e.\",\n \"the sum of marginal variances.\",\n \"Each of the ${\\\\mathbf {W}}_{ij}$ is a linear combination of the entries of ${\\\\mathbf {E}}$ , which have unit variance and are uncorrelated, so by the additive property of variance of sum of uncorrelated random variables and the quadratic scaling property of variance, $\\\\mathrm {Var}({\\\\mathbf {W}}_{ij})=A_{i:}^2{\\\\mathbf {S}}^2{\\\\mathbf {B}}_{:j}^2$ .\",\n \"Connection to triangular maps.\",\n \"Much of the recent work on normalizing flows has been dedicated to inverse autoregressive transformations [25], [21], [37], [10], [24], as they are general enough to induce any density function [22], [7], [52].\",\n \"When such transformations are used for ${\\\\mathbf {g}}_A$ and ${\\\\mathbf {g}}_B$ , the overall transformation ${\\\\mathbf {G}}$ is also a triangle map, since ${\\\\mathbf {G}}({\\\\mathbf {E}})_{ij}$ depends on ${\\\\mathbf {E}}_{i^{\\\\prime }j^{\\\\prime }}$ for $i^{\\\\prime }\\\\le i$ and $j^{\\\\prime }\\\\le j$ .\",\n \"Such a function has some “blind spots” similar to the ones discovered by [51].\",\n \"One avenue for improvement is to design a transformation that increase the connectivity.\",\n \"Another avenue for improvement is to condition each (row-wise or column-wise) transformation on a learnable embedding of the row/column, such that each row/column is transformed by a slightly different function than another We try this idea in the preliminary stage of the project, but find it harder to optimize.\",\n \"This is potentially due to the extra parameters that have to be learned.. Tail bound of empirical KL We begin with some preliminaries and lemmas in Section REF , and prove the main result in Section REF .\",\n \"Basic tail bounds and Bernstein inequality The tools developed in this section is to translate the coefficients (such as variance) of sub-Gaussian random variables and sub-exponential random variables.\",\n \"We start with the definition of sub-Gaussians: Definition 1 We write $X\\\\sim \\\\textnormal {sub}{\\\\mathcal {N}}(L^2)$ if $X$ is a random variable satisfying ${\\\\mathbb {P}}(|X|>t)\\\\le 2\\\\exp \\\\left(-\\\\frac{t^2}{2L^2}\\\\right)$ We write $\\\\Gamma (\\\\cdot )$ as the Gamma function: $\\\\Gamma (z)=\\\\int _0^\\\\infty e^{-u}u^{z-1}\\\\mathrm {d}u$ .\",\n \"Note that for positive integers $z$ , $\\\\Gamma (z)=(z-1)!$ .\",\n \"The following lemma gives an upper-bound on the moments of a sub-Gaussian.\",\n \"Lemma 4 For $X\\\\sim \\\\textnormal {sub}{\\\\mathcal {N}}(L^2)$ , for any integer $p\\\\ge 1$ , $\\\\mathbb {E}[|X|^p]\\\\le (2L^2)^{p/2}p\\\\Gamma (p/2)$ .\",\n \"Since $|X|^p$ is non-negative, similar to Lemma REF , we have $\\\\mathbb {E}[|X|^p]&= \\\\int _{0}^\\\\infty {\\\\mathbb {P}}(|X|^p\\\\ge s)\\\\mathrm {d}s=\\\\int _0^\\\\infty {\\\\mathbb {P}}(|X|\\\\ge t)pt^{p-1} \\\\mathrm {d}t \\\\\\\\&\\\\le 2p\\\\int _0^\\\\infty e^{-{t^2}/{2L^2}}t^{p-1} \\\\mathrm {d}t=\\\\le p(2L^2)^{p/2}\\\\int _0^\\\\infty e^{-u}u^{p/2-1}\\\\mathrm {d}u = p(2L^2)^{p/2}\\\\Gamma (p/2)$ where we let $s=t^p$ and $u=t^2/2L^2$ .\",\n \"The following definition is the main tool for translating the coefficients.\",\n \"Definition 2 Let $X$ be a random variable.\",\n \"For integer $k\\\\ge 1$ , define the $\\\\psi _k$ -Orlicz norm as $||X||_{\\\\psi _k}:=\\\\inf \\\\lbrace t>0:\\\\mathbb {E}[\\\\exp (|X|^k/t^k)]\\\\le 2\\\\rbrace $ i.e, the smallest constant $t>0$ for which the super-exponential moment of $X^k/t^k$ is bounded by 2.\",\n \"The Orlicz norm is infinity if there's no finite $t$ for which $\\\\mathbb {E}[\\\\exp (|X|^k/t^k)]$ exists.\",\n \"It is easy to verify that the Orlicz norm is indeed a norm.\",\n \"We call $||\\\\cdot ||_{\\\\psi _2}$ the sub-Gaussian norm, and $||\\\\cdot ||_{\\\\psi _1}$ the sub-exponential norm.\",\n \"Note that $||X^2||_{\\\\psi _1} = ||X||_{\\\\psi _2}^2$ .\",\n \"The following lemma upper bounds the sub-Gaussian norm by its variance.\",\n \"Lemma 5 If $X\\\\sim \\\\textnormal {sub}{\\\\mathcal {N}}(L^2)$ , $||X||_{\\\\psi _2}\\\\le 6L^2$ .\",\n \"By power series expansion of the exponential function, $\\\\mathbb {E}[e^{cX^2}]=1+\\\\sum _{p=1}^\\\\infty \\\\frac{c^p\\\\mathbb {E}[X^{2p}]}{p!\",\n \"}\\\\le 1+\\\\sum _{p=1}^\\\\infty \\\\frac{c^p}{p!}\",\n \"2(2L^2)^pp!=1+2\\\\sum _{p=1}^\\\\infty (2cL^2)^p$ where we used Lemma REF for the inequality.\",\n \"The RHS converges and is equal to 2 if $c=1/6L^2$ .\",\n \"Thus, $||X||_{\\\\psi _2}\\\\le 6L^2$ .\",\n \"The following lemma gives an upper bound on the moments of sub-exponential random variables.\",\n \"Lemma 6 If for some $C>0$ , $\\\\mathbb {E}[\\\\exp (|X|/C)]\\\\le 2$ , then $\\\\mathbb {E}[|X|^p]\\\\le 2C^pp!$ .\",\n \"By Markov's inequality, ${\\\\mathbb {P}}(|X|>t)\\\\le \\\\frac{\\\\mathbb {E}[\\\\exp (|X|/C)]}{\\\\exp (t/C)}\\\\le 2e^{-t/C}$ For $p\\\\in {\\\\mathbb {Z}}^+$ , since $|X|^p$ is non-negative, $\\\\mathbb {E}[|X|^p] &= \\\\int _{0}^\\\\infty {\\\\mathbb {P}}(|X|^p\\\\ge s)ds= \\\\int _{0}^\\\\infty {\\\\mathbb {P}}(|X|\\\\ge t)pt^{p-1}dt \\\\\\\\&\\\\le 2p\\\\int _{0}^\\\\infty e^{-t/C}t^{p-1}dt =2pC^p\\\\int _{0}^\\\\infty e^{-u}u^{p-1} du=2pC^p\\\\Gamma (p)=2C^pp!$ where we let $s=t^p$ and $u=t/C$ .\",\n \"Finally, we derive a concentration bound for sub-exponential random variables.\",\n \"Theorem 7 (Bernstein's inequality for sub-exponential random variables) Let $(X_i)_{i\\\\in [n]}$ be independent real-valued random variables satisfying $\\\\mathbb {E}[\\\\exp (|X|/C)]\\\\le 2$ for some $C>0$ , with mean $\\\\mu _X=\\\\mathbb {E}[X]$ , and let $\\\\bar{X}=\\\\frac{1}{n}\\\\sum _{i=1}^nX_i$ .\",\n \"Then, for any $\\\\epsilon >0$ , the following concentration bound holds: ${\\\\mathbb {P}}(\\\\bar{X}-\\\\mu _X > \\\\epsilon ) \\\\le \\\\exp \\\\left(-\\\\frac{n\\\\epsilon ^2}{2(4C^2+C\\\\epsilon )}\\\\right)$ Let $\\\\nu =4nC^2$ and $c=C$ .\",\n \"Then by Lemma REF , $\\\\sum _{i=1}^n\\\\mathbb {E}[X_i^2]\\\\le n\\\\cdot 4C^2=\\\\nu $ and for integers $p>2$ : $\\\\sum _{i=1}^n \\\\mathbb {E}[|X_i|^p] \\\\le 2nC^pp!\",\n \"= \\\\nu C^{p-2}p!/2 = \\\\nu c^{p-2}p!/2$ .\",\n \"Then by Corollary 2.11 of [8], we have ${\\\\mathbb {P}}(\\\\bar{X}-\\\\mu _X > \\\\epsilon )={\\\\mathbb {P}}\\\\left(\\\\sum _{i=1}^n(X_i-\\\\mu _X)>n\\\\epsilon \\\\right)\\\\le \\\\exp \\\\left(-\\\\frac{n\\\\epsilon ^2}{2(4C^2+C\\\\epsilon )}\\\\right)$ Proof of Theorem REF Since $\\\\bar{g}(\\\\mathbf {\\\\epsilon }):=g(\\\\mathbf {\\\\epsilon })-\\\\mathbb {E}[g(\\\\mathbf {\\\\epsilon })]$ is $L$ -Lipschitz, according to Theorem 5.5 and 5.6 of [8], $\\\\bar{g}\\\\sim \\\\textnormal {sub}{\\\\mathcal {N}}(L^2)$ .\",\n \"And we have that $||g^2||_{\\\\psi _1}=||g||_{\\\\psi _2}^2=||\\\\bar{g}+\\\\mathbb {E}[g]||_{\\\\psi _2}^2\\\\le \\\\left(||\\\\bar{g}||_{\\\\psi _2} + \\\\frac{\\\\mathbb {E}[g]}{\\\\sqrt{\\\\log 2}}\\\\right)^2$ due to triangle inequality of the norm.\",\n \"Now since $g$ is $L$ -Lipschitz, its expectation can be bounded by $\\\\mathbb {E}[g]=\\\\mathbb {E}\\\\left[\\\\frac{1}{\\\\sqrt{2}}||{\\\\mathbf {g}}(\\\\mathbf {\\\\epsilon })-\\\\mathbf {0}||\\\\right]\\\\le L\\\\mathbb {E}\\\\left[||\\\\mathbf {\\\\epsilon }-{\\\\mathbf {g}}^{-1}(\\\\mathbf {0})||\\\\right]\\\\le L(\\\\mathbb {E}\\\\left[||\\\\mathbf {\\\\epsilon }||\\\\right]+||{\\\\mathbf {g}}^{-1}(\\\\mathbf {0})||) $ Since $\\\\mathbf {\\\\epsilon }$ is standard-normally distributed, $||\\\\mathbf {\\\\epsilon }||$ follows the chi distribution with $d$ degrees of freedom, which has an expectation that can be upper-bounded using Gautschi's inequality (using Wendel's version of the upper bound): $\\\\mathbb {E}[||\\\\mathbf {\\\\epsilon }||]=\\\\sqrt{2}\\\\frac{\\\\Gamma ((d+1)/2)}{\\\\Gamma (d/2)}\\\\le \\\\sqrt{2}\\\\left(\\\\frac{d}{2}\\\\right)^{1/2}=\\\\sqrt{d}$ Combining the above and using Lemma REF , we have $||g^2||_{\\\\psi _1}\\\\le \\\\left(6L^2+\\\\frac{L}{\\\\sqrt{\\\\log 2}}(\\\\sqrt{d}+||{\\\\mathbf {g}}^{-1}(\\\\mathbf {0})||)\\\\right)^2$ Setting $C$ to be the RHS and applying Theorem REF yield the desired result.\",\n \"Experimental Details For the predictive tasks (Section REF ), we use a cosine annealing schedule for the learning rate, scaling down to 0.01 of the initial learning rate, and pretrain a deterministic network for 10 epochs using the Adam optimizer with a learning rate of 0.001, to initialize the mean of the Gaussian $q_0$ , and train $q$ for 200 epochs.\",\n \"LeNet-5 MNIST.\",\n \"We use a linear annealing schedule of the $\\\\beta $ coefficient (from 0 back to 1) for 50,000 iterations.\",\n \"We use the Adam optimizer with a learning rate of 0.0005.\",\n \"The result we get for K-Linear uses polyak averaging with exponential decay coefficient 0.995.\",\n \"We use the volume preserving version of RealNVP for the K-Nonlinear.\",\n \"We use the standard Gaussian prior for $p$ .\",\n \"LeNet-5 CIFAR-5 We use the same architecture as [32] (192 convolutional kernels and 1,000 hidden units for the fully connected layers).\",\n \"We use a linear annealing schedule of the $\\\\beta $ coefficient (from 0 back to 1) for 20,000 iterations for Diag, and no annealing for K-Linear and K-Nonlinear.\",\n \"We use the Adam optimizer with a learning rate of 0.0003, 0.0003, 0.0005 for Diag, K-Linear and K-Nonlinear, respectively.\",\n \"We use the volume preserving version of RealNVP for K-Nonlinear.\",\n \"We use the standard Gaussian prior for $p$ .\",\n \"VGG-16 CIFAR-10 We use the modified version of VGG-16 proposed by [53].\",\n \"We use a learning rate of 0.0005 for all experiments but K-Nonlinear in the regular setup (where we use 0.001).\",\n \"We use the isotropic Gaussian prior with variance being 0.1, and set $\\\\beta $ to be [0.5, 0.1, 0.1, 0.5] in the regular setup and [0.5, 0.1, 0.1, 0.1] in the data augmented setup for Diag, K-Diag, K-Linear, and K-Nonlinear, respectively.\",\n \"We use the volume preserving version of RealNVP for the K-Nonlinear.\",\n \"PAC-Bayes MLP We follow the same steps as [12], except we did not discretize the prior variance after tuning.\",\n \"In practice this does not affect the bound much.\",\n \"We also did not initialize the mean of $q_0$ in our setup using SGD for our experiments.\",\n \"We train the stochastic network for 300 epochs, with a learning rate of 0.002.\",\n \"The bound holds with probability at least 0.965 over the choice of prior and the training set.\",\n \"The $b$ and $c$ coefficients in [12] are set as 100 and 0.1.\",\n \"We use the volume preserving version of IAF for the K-Nonlinear.\",\n \"PAC-Bayes LeNet-5 The same setup as PAC-Bayes MLP, except with polyak averaging with coefficient 0.995.\",\n \"We use the volume preserving version of IAF for the K-Nonlinear.\",\n \"Bandit Benchmark All the models share the same architechture: one hidden layer with 50 units.\",\n \"We use the volume preserving version of RealNVP for K-NonLinear.\",\n \"We train models every 50 time steps for 200 training iterations using a batch-size of 512.\",\n \"Table: Description of bandit problem: number of actions and number of contexts used for experiments.\",\n \"Comparing to benchmark, we restrict ourself to 50000 contexts for Covertype instead of 150000 contexts.Table: Additional results: Cumulative regret incurred by different algorithms on the bandit benchmarks described in .\",\n \"Values reported are the mean over 5 independent trials with standard error of the mean, normalized with respect to the performance of the uniform policy.\",\n \"We use the same hyperparameters for different algorithms without any finetuning: learning rate = 0.0001 and 100 training epochs.Figure: Posterior predictive (with 20 samples) on rotated MNIST digit 3 and 5.\",\n \"Top left: Diag; top right: K-Diag; bottom left: K-Linear; bottom right: K-Nonlinear.\"\n ],\n [\n \"Law of the unconscious statistician\",\n \"Let $(\\\\Omega , {\\\\mathcal {F}}, {\\\\mathbb {P}})$ be our probability space.\",\n \"Let $\\\\mathbf {\\\\epsilon }\\\\in \\\\mathbb {R}^d$ be a random variable following the (Lebesgue) density $q_0(\\\\mathbf {\\\\epsilon })=\\\\frac{\\\\mathrm {d}\\\\mathbf {\\\\epsilon }_*{\\\\mathbb {P}}}{\\\\mathrm {d}\\\\mu }$ and $\\\\mathbf {\\\\epsilon }_*{\\\\mathbb {P}}$ being its pushforward measure, and write $\\\\Theta ={\\\\mathbf {g}}_\\\\phi (\\\\mathbf {\\\\epsilon })\\\\in \\\\mathbb {R}^d$ with $q_\\\\phi =\\\\frac{\\\\mathrm {d}({\\\\mathbf {g}}_\\\\phi \\\\circ \\\\mathbf {\\\\epsilon })_*{\\\\mathbb {P}}}{\\\\mathrm {d}\\\\mu }$ being its density and $({\\\\mathbf {g}}_\\\\phi \\\\circ \\\\mathbf {\\\\epsilon })_*{\\\\mathbb {P}}$ being its pushforward measure, and $A=\\\\hat{R}_D(\\\\Theta )+\\\\log q_\\\\phi (\\\\Theta )\\\\in \\\\mathbb {R}$ .\",\n \"Then $\\\\mathbb {E}[A] &= \\\\int _{\\\\mathbb {R}^d} A \\\\,\\\\,\\\\mathrm {d}({\\\\mathbf {g}}_\\\\phi \\\\circ \\\\mathbf {\\\\epsilon })_*{\\\\mathbb {P}}&&= \\\\int _{\\\\mathbb {R}^d} \\\\left(\\\\hat{R}_D(\\\\Theta ) + \\\\log q_\\\\phi (\\\\Theta ) \\\\right) q_\\\\phi (\\\\Theta ) \\\\,\\\\mathrm {d}\\\\Theta \\\\\\\\&= \\\\int _{\\\\mathbb {R}^d} A\\\\circ \\\\Theta \\\\,\\\\,\\\\mathrm {d}\\\\mathbf {\\\\epsilon }_*{\\\\mathbb {P}}&&= \\\\int _{\\\\mathbb {R}^d} \\\\left(\\\\hat{R}_D({\\\\mathbf {g}}_\\\\phi (\\\\mathbf {\\\\epsilon })) + \\\\log q_\\\\phi ({\\\\mathbf {g}}_\\\\phi (\\\\mathbf {\\\\epsilon })) \\\\right) q_0(\\\\mathbf {\\\\epsilon }) \\\\,\\\\mathrm {d}\\\\mathbf {\\\\epsilon }\\\\\\\\&{}&&=\\\\int _{\\\\mathbb {R}^d} \\\\left(\\\\hat{R}_{D}({\\\\mathbf {g}}_\\\\phi (\\\\mathbf {\\\\epsilon })) + \\\\log q_0(\\\\mathbf {\\\\epsilon }) - \\\\log \\\\left|\\\\det \\\\frac{\\\\partial {\\\\mathbf {g}}_\\\\phi (\\\\mathbf {\\\\epsilon })}{\\\\partial \\\\mathbf {\\\\epsilon }}\\\\right|\\\\right)q_0(\\\\mathbf {\\\\epsilon })\\\\,\\\\mathrm {d}\\\\mathbf {\\\\epsilon }$\"\n ],\n [\n \"Derivation and interpretation of Property \",\n \"We first derive Property REF algebraically, and give an interpretation that can be genralized to higher dimensional tensor operation.\",\n \"Recall that we have the following givens: Assume ${\\\\mathbf {E}}_{ij}\\\\overset{\\\\textnormal {i.i.d.\",\n \"}}{\\\\sim } {\\\\mathcal {N}}(0,1)$ is a $n\\\\times p$ random Gaussian matrix.\",\n \"Assume ${\\\\mathbf {A}}\\\\in \\\\mathbb {R}^{n\\\\times n}$ and ${\\\\mathbf {B}}\\\\in \\\\mathbb {R}^{p\\\\times p}$ .\",\n \"${\\\\mathbf {S}}\\\\in \\\\mathbb {R}^{n\\\\times p}_{>0}$ .\",\n \"${\\\\mathbf {M}}\\\\in \\\\mathbb {R}^{n\\\\times p}$ .\",\n \"If we rescale ${\\\\mathbf {E}}$ elementwise by ${\\\\mathbf {S}}$ before inducing the column-wise and row-wise correlation, we have: (superscript is Hadamard power) $\\\\Sigma :=\\\\mathrm {Var}(\\\\textnormal {vec}({\\\\mathbf {M}}+ {\\\\mathbf {A}}({\\\\mathbf {E}}\\\\circ {\\\\mathbf {S}}){\\\\mathbf {B}}))&=\\\\mathrm {Var}( ({\\\\mathbf {B}}^\\\\top \\\\otimes {\\\\mathbf {A}})\\\\textnormal {vec}({\\\\mathbf {E}}\\\\circ {\\\\mathbf {S}}))\\\\\\\\&=({\\\\mathbf {B}}^\\\\top \\\\otimes {\\\\mathbf {A}}) \\\\mathop {\\\\mathrm {diag}}\\\\nolimits (\\\\textnormal {vec}({\\\\mathbf {S}}^2)) ({\\\\mathbf {B}}^\\\\top \\\\otimes {\\\\mathbf {A}})^\\\\top \\\\\\\\&=({\\\\mathbf {B}}^\\\\top \\\\otimes {\\\\mathbf {A}}) \\\\mathop {\\\\mathrm {diag}}\\\\nolimits (\\\\textnormal {vec}({\\\\mathbf {S}}^2)) ({\\\\mathbf {B}}\\\\otimes {\\\\mathbf {A}}^\\\\top )$ If ${\\\\mathbf {S}}$ is a matrix of ones, the RHS equals $({\\\\mathbf {B}}^\\\\top \\\\otimes {\\\\mathbf {A}}) ({\\\\mathbf {B}}\\\\otimes {\\\\mathbf {A}}^\\\\top )= ({\\\\mathbf {B}}^\\\\top {\\\\mathbf {B}})\\\\otimes ({\\\\mathbf {A}}{\\\\mathbf {A}}^\\\\top )$ , which is the covariance of the matrix normal.\",\n \"Generally, ${\\\\mathbf {S}}$ might not be a matrix of ones.\",\n \"But we can still compute the determinant and trace of the covariance matrix (useful in computing KL): $\\\\det (\\\\Sigma )=\\\\det ({\\\\mathbf {A}})^{2p}\\\\det ({\\\\mathbf {B}})^{2n}\\\\prod _{ij}{\\\\mathbf {S}}_{ij}^2$ $\\\\textnormal {Tr}(\\\\Sigma )&=\\\\sum _i \\\\Sigma _{ii} \\\\\\\\&=\\\\sum _i \\\\sum _{j} ({\\\\mathbf {B}}^\\\\top \\\\otimes {\\\\mathbf {A}})_{ij}\\\\textnormal {vec}({\\\\mathbf {S}}^2)_j ({\\\\mathbf {B}}\\\\otimes {\\\\mathbf {A}}^\\\\top )_{ji}\\\\\\\\&=\\\\sum _i \\\\sum _{j} ({\\\\mathbf {B}}^{2\\\\top }\\\\otimes {\\\\mathbf {A}}^2)_{ij}\\\\textnormal {vec}({\\\\mathbf {S}}^2)_j\\\\\\\\&=\\\\sum _i\\\\left( ({\\\\mathbf {B}}^{2\\\\top }\\\\otimes {\\\\mathbf {A}}^2)\\\\textnormal {vec}({\\\\mathbf {S}}^2) \\\\right)_i\\\\\\\\&=\\\\sum _{ij}\\\\left({\\\\mathbf {A}}^2{\\\\mathbf {S}}^2{\\\\mathbf {B}}^2\\\\right)_{ij}$\"\n ],\n [\n \"Interpretation of the determinant and trace.\",\n \"The determinant measures the change in volume due to the linear map.\",\n \"Since each operation (elementwise multiplication with ${\\\\mathbf {S}}$ , left-multiplication with ${\\\\mathbf {A}}$ , and right-multiplication with ${\\\\mathbf {B}}$ ) is an invertible map, the determinant of the composition is a product of determinants.\",\n \"After elementwise multiplication with ${\\\\mathbf {S}}$ (hence $\\\\prod {\\\\mathbf {S}}_{ij}$ )The power 2 comes from the fact that we are looking at the determinant of the covariance.\",\n \"Direct computation of the likelihood involves $\\\\frac{1}{2}\\\\log \\\\det (\\\\Sigma )$ , which is equivalent to the log-determinant of the invertible map., we apply the same linear map ${\\\\mathbf {A}}$ to the columns of $({\\\\mathbf {E}}\\\\circ {\\\\mathbf {S}})$ ; in vector form, this corresponds to left-multiplication with a block diagonal of $p$ ${\\\\mathbf {A}}$ 's, hence $\\\\det ({\\\\mathbf {A}})^{p}$ .\",\n \"The same reasoning explains $\\\\det ({\\\\mathbf {B}})^n$ .\",\n \"The trace of the covariance can be written as $\\\\textnormal {Tr}(\\\\Sigma )=\\\\sum _{ij}\\\\mathrm {Var}({\\\\mathbf {W}}_{ij})$ , i.e.\",\n \"the sum of marginal variances.\",\n \"Each of the ${\\\\mathbf {W}}_{ij}$ is a linear combination of the entries of ${\\\\mathbf {E}}$ , which have unit variance and are uncorrelated, so by the additive property of variance of sum of uncorrelated random variables and the quadratic scaling property of variance, $\\\\mathrm {Var}({\\\\mathbf {W}}_{ij})=A_{i:}^2{\\\\mathbf {S}}^2{\\\\mathbf {B}}_{:j}^2$ .\"\n ],\n [\n \"Connection to triangular maps.\",\n \"Much of the recent work on normalizing flows has been dedicated to inverse autoregressive transformations [25], [21], [37], [10], [24], as they are general enough to induce any density function [22], [7], [52].\",\n \"When such transformations are used for ${\\\\mathbf {g}}_A$ and ${\\\\mathbf {g}}_B$ , the overall transformation ${\\\\mathbf {G}}$ is also a triangle map, since ${\\\\mathbf {G}}({\\\\mathbf {E}})_{ij}$ depends on ${\\\\mathbf {E}}_{i^{\\\\prime }j^{\\\\prime }}$ for $i^{\\\\prime }\\\\le i$ and $j^{\\\\prime }\\\\le j$ .\",\n \"Such a function has some “blind spots” similar to the ones discovered by [51].\",\n \"One avenue for improvement is to design a transformation that increase the connectivity.\",\n \"Another avenue for improvement is to condition each (row-wise or column-wise) transformation on a learnable embedding of the row/column, such that each row/column is transformed by a slightly different function than another We try this idea in the preliminary stage of the project, but find it harder to optimize.\",\n \"This is potentially due to the extra parameters that have to be learned..\"\n ],\n [\n \"Tail bound of empirical KL\",\n \"We begin with some preliminaries and lemmas in Section REF , and prove the main result in Section REF .\"\n ],\n [\n \"Basic tail bounds and Bernstein inequality\",\n \"The tools developed in this section is to translate the coefficients (such as variance) of sub-Gaussian random variables and sub-exponential random variables.\",\n \"We start with the definition of sub-Gaussians: Definition 1 We write $X\\\\sim \\\\textnormal {sub}{\\\\mathcal {N}}(L^2)$ if $X$ is a random variable satisfying ${\\\\mathbb {P}}(|X|>t)\\\\le 2\\\\exp \\\\left(-\\\\frac{t^2}{2L^2}\\\\right)$ We write $\\\\Gamma (\\\\cdot )$ as the Gamma function: $\\\\Gamma (z)=\\\\int _0^\\\\infty e^{-u}u^{z-1}\\\\mathrm {d}u$ .\",\n \"Note that for positive integers $z$ , $\\\\Gamma (z)=(z-1)!$ .\",\n \"The following lemma gives an upper-bound on the moments of a sub-Gaussian.\",\n \"Lemma 4 For $X\\\\sim \\\\textnormal {sub}{\\\\mathcal {N}}(L^2)$ , for any integer $p\\\\ge 1$ , $\\\\mathbb {E}[|X|^p]\\\\le (2L^2)^{p/2}p\\\\Gamma (p/2)$ .\",\n \"Since $|X|^p$ is non-negative, similar to Lemma REF , we have $\\\\mathbb {E}[|X|^p]&= \\\\int _{0}^\\\\infty {\\\\mathbb {P}}(|X|^p\\\\ge s)\\\\mathrm {d}s=\\\\int _0^\\\\infty {\\\\mathbb {P}}(|X|\\\\ge t)pt^{p-1} \\\\mathrm {d}t \\\\\\\\&\\\\le 2p\\\\int _0^\\\\infty e^{-{t^2}/{2L^2}}t^{p-1} \\\\mathrm {d}t=\\\\le p(2L^2)^{p/2}\\\\int _0^\\\\infty e^{-u}u^{p/2-1}\\\\mathrm {d}u = p(2L^2)^{p/2}\\\\Gamma (p/2)$ where we let $s=t^p$ and $u=t^2/2L^2$ .\",\n \"The following definition is the main tool for translating the coefficients.\",\n \"Definition 2 Let $X$ be a random variable.\",\n \"For integer $k\\\\ge 1$ , define the $\\\\psi _k$ -Orlicz norm as $||X||_{\\\\psi _k}:=\\\\inf \\\\lbrace t>0:\\\\mathbb {E}[\\\\exp (|X|^k/t^k)]\\\\le 2\\\\rbrace $ i.e, the smallest constant $t>0$ for which the super-exponential moment of $X^k/t^k$ is bounded by 2.\",\n \"The Orlicz norm is infinity if there's no finite $t$ for which $\\\\mathbb {E}[\\\\exp (|X|^k/t^k)]$ exists.\",\n \"It is easy to verify that the Orlicz norm is indeed a norm.\",\n \"We call $||\\\\cdot ||_{\\\\psi _2}$ the sub-Gaussian norm, and $||\\\\cdot ||_{\\\\psi _1}$ the sub-exponential norm.\",\n \"Note that $||X^2||_{\\\\psi _1} = ||X||_{\\\\psi _2}^2$ .\",\n \"The following lemma upper bounds the sub-Gaussian norm by its variance.\",\n \"Lemma 5 If $X\\\\sim \\\\textnormal {sub}{\\\\mathcal {N}}(L^2)$ , $||X||_{\\\\psi _2}\\\\le 6L^2$ .\",\n \"By power series expansion of the exponential function, $\\\\mathbb {E}[e^{cX^2}]=1+\\\\sum _{p=1}^\\\\infty \\\\frac{c^p\\\\mathbb {E}[X^{2p}]}{p!\",\n \"}\\\\le 1+\\\\sum _{p=1}^\\\\infty \\\\frac{c^p}{p!}\",\n \"2(2L^2)^pp!=1+2\\\\sum _{p=1}^\\\\infty (2cL^2)^p$ where we used Lemma REF for the inequality.\",\n \"The RHS converges and is equal to 2 if $c=1/6L^2$ .\",\n \"Thus, $||X||_{\\\\psi _2}\\\\le 6L^2$ .\",\n \"The following lemma gives an upper bound on the moments of sub-exponential random variables.\",\n \"Lemma 6 If for some $C>0$ , $\\\\mathbb {E}[\\\\exp (|X|/C)]\\\\le 2$ , then $\\\\mathbb {E}[|X|^p]\\\\le 2C^pp!$ .\",\n \"By Markov's inequality, ${\\\\mathbb {P}}(|X|>t)\\\\le \\\\frac{\\\\mathbb {E}[\\\\exp (|X|/C)]}{\\\\exp (t/C)}\\\\le 2e^{-t/C}$ For $p\\\\in {\\\\mathbb {Z}}^+$ , since $|X|^p$ is non-negative, $\\\\mathbb {E}[|X|^p] &= \\\\int _{0}^\\\\infty {\\\\mathbb {P}}(|X|^p\\\\ge s)ds= \\\\int _{0}^\\\\infty {\\\\mathbb {P}}(|X|\\\\ge t)pt^{p-1}dt \\\\\\\\&\\\\le 2p\\\\int _{0}^\\\\infty e^{-t/C}t^{p-1}dt =2pC^p\\\\int _{0}^\\\\infty e^{-u}u^{p-1} du=2pC^p\\\\Gamma (p)=2C^pp!$ where we let $s=t^p$ and $u=t/C$ .\",\n \"Finally, we derive a concentration bound for sub-exponential random variables.\",\n \"Theorem 7 (Bernstein's inequality for sub-exponential random variables) Let $(X_i)_{i\\\\in [n]}$ be independent real-valued random variables satisfying $\\\\mathbb {E}[\\\\exp (|X|/C)]\\\\le 2$ for some $C>0$ , with mean $\\\\mu _X=\\\\mathbb {E}[X]$ , and let $\\\\bar{X}=\\\\frac{1}{n}\\\\sum _{i=1}^nX_i$ .\",\n \"Then, for any $\\\\epsilon >0$ , the following concentration bound holds: ${\\\\mathbb {P}}(\\\\bar{X}-\\\\mu _X > \\\\epsilon ) \\\\le \\\\exp \\\\left(-\\\\frac{n\\\\epsilon ^2}{2(4C^2+C\\\\epsilon )}\\\\right)$ Let $\\\\nu =4nC^2$ and $c=C$ .\",\n \"Then by Lemma REF , $\\\\sum _{i=1}^n\\\\mathbb {E}[X_i^2]\\\\le n\\\\cdot 4C^2=\\\\nu $ and for integers $p>2$ : $\\\\sum _{i=1}^n \\\\mathbb {E}[|X_i|^p] \\\\le 2nC^pp!\",\n \"= \\\\nu C^{p-2}p!/2 = \\\\nu c^{p-2}p!/2$ .\",\n \"Then by Corollary 2.11 of [8], we have ${\\\\mathbb {P}}(\\\\bar{X}-\\\\mu _X > \\\\epsilon )={\\\\mathbb {P}}\\\\left(\\\\sum _{i=1}^n(X_i-\\\\mu _X)>n\\\\epsilon \\\\right)\\\\le \\\\exp \\\\left(-\\\\frac{n\\\\epsilon ^2}{2(4C^2+C\\\\epsilon )}\\\\right)$\"\n ],\n [\n \"Proof of Theorem \",\n \"Since $\\\\bar{g}(\\\\mathbf {\\\\epsilon }):=g(\\\\mathbf {\\\\epsilon })-\\\\mathbb {E}[g(\\\\mathbf {\\\\epsilon })]$ is $L$ -Lipschitz, according to Theorem 5.5 and 5.6 of [8], $\\\\bar{g}\\\\sim \\\\textnormal {sub}{\\\\mathcal {N}}(L^2)$ .\",\n \"And we have that $||g^2||_{\\\\psi _1}=||g||_{\\\\psi _2}^2=||\\\\bar{g}+\\\\mathbb {E}[g]||_{\\\\psi _2}^2\\\\le \\\\left(||\\\\bar{g}||_{\\\\psi _2} + \\\\frac{\\\\mathbb {E}[g]}{\\\\sqrt{\\\\log 2}}\\\\right)^2$ due to triangle inequality of the norm.\",\n \"Now since $g$ is $L$ -Lipschitz, its expectation can be bounded by $\\\\mathbb {E}[g]=\\\\mathbb {E}\\\\left[\\\\frac{1}{\\\\sqrt{2}}||{\\\\mathbf {g}}(\\\\mathbf {\\\\epsilon })-\\\\mathbf {0}||\\\\right]\\\\le L\\\\mathbb {E}\\\\left[||\\\\mathbf {\\\\epsilon }-{\\\\mathbf {g}}^{-1}(\\\\mathbf {0})||\\\\right]\\\\le L(\\\\mathbb {E}\\\\left[||\\\\mathbf {\\\\epsilon }||\\\\right]+||{\\\\mathbf {g}}^{-1}(\\\\mathbf {0})||) $ Since $\\\\mathbf {\\\\epsilon }$ is standard-normally distributed, $||\\\\mathbf {\\\\epsilon }||$ follows the chi distribution with $d$ degrees of freedom, which has an expectation that can be upper-bounded using Gautschi's inequality (using Wendel's version of the upper bound): $\\\\mathbb {E}[||\\\\mathbf {\\\\epsilon }||]=\\\\sqrt{2}\\\\frac{\\\\Gamma ((d+1)/2)}{\\\\Gamma (d/2)}\\\\le \\\\sqrt{2}\\\\left(\\\\frac{d}{2}\\\\right)^{1/2}=\\\\sqrt{d}$ Combining the above and using Lemma REF , we have $||g^2||_{\\\\psi _1}\\\\le \\\\left(6L^2+\\\\frac{L}{\\\\sqrt{\\\\log 2}}(\\\\sqrt{d}+||{\\\\mathbf {g}}^{-1}(\\\\mathbf {0})||)\\\\right)^2$ Setting $C$ to be the RHS and applying Theorem REF yield the desired result.\"\n ],\n [\n \"Experimental Details\",\n \"For the predictive tasks (Section REF ), we use a cosine annealing schedule for the learning rate, scaling down to 0.01 of the initial learning rate, and pretrain a deterministic network for 10 epochs using the Adam optimizer with a learning rate of 0.001, to initialize the mean of the Gaussian $q_0$ , and train $q$ for 200 epochs.\"\n ],\n [\n \"LeNet-5 MNIST.\",\n \"We use a linear annealing schedule of the $\\\\beta $ coefficient (from 0 back to 1) for 50,000 iterations.\",\n \"We use the Adam optimizer with a learning rate of 0.0005.\",\n \"The result we get for K-Linear uses polyak averaging with exponential decay coefficient 0.995.\",\n \"We use the volume preserving version of RealNVP for the K-Nonlinear.\",\n \"We use the standard Gaussian prior for $p$ .\"\n ],\n [\n \"LeNet-5 CIFAR-5\",\n \"We use the same architecture as [32] (192 convolutional kernels and 1,000 hidden units for the fully connected layers).\",\n \"We use a linear annealing schedule of the $\\\\beta $ coefficient (from 0 back to 1) for 20,000 iterations for Diag, and no annealing for K-Linear and K-Nonlinear.\",\n \"We use the Adam optimizer with a learning rate of 0.0003, 0.0003, 0.0005 for Diag, K-Linear and K-Nonlinear, respectively.\",\n \"We use the volume preserving version of RealNVP for K-Nonlinear.\",\n \"We use the standard Gaussian prior for $p$ .\"\n ],\n [\n \"VGG-16 CIFAR-10\",\n \"We use the modified version of VGG-16 proposed by [53].\",\n \"We use a learning rate of 0.0005 for all experiments but K-Nonlinear in the regular setup (where we use 0.001).\",\n \"We use the isotropic Gaussian prior with variance being 0.1, and set $\\\\beta $ to be [0.5, 0.1, 0.1, 0.5] in the regular setup and [0.5, 0.1, 0.1, 0.1] in the data augmented setup for Diag, K-Diag, K-Linear, and K-Nonlinear, respectively.\",\n \"We use the volume preserving version of RealNVP for the K-Nonlinear.\"\n ],\n [\n \"PAC-Bayes MLP\",\n \"We follow the same steps as [12], except we did not discretize the prior variance after tuning.\",\n \"In practice this does not affect the bound much.\",\n \"We also did not initialize the mean of $q_0$ in our setup using SGD for our experiments.\",\n \"We train the stochastic network for 300 epochs, with a learning rate of 0.002.\",\n \"The bound holds with probability at least 0.965 over the choice of prior and the training set.\",\n \"The $b$ and $c$ coefficients in [12] are set as 100 and 0.1.\",\n \"We use the volume preserving version of IAF for the K-Nonlinear.\"\n ],\n [\n \"PAC-Bayes LeNet-5\",\n \"The same setup as PAC-Bayes MLP, except with polyak averaging with coefficient 0.995.\",\n \"We use the volume preserving version of IAF for the K-Nonlinear.\"\n ],\n [\n \"Bandit Benchmark\",\n \"All the models share the same architechture: one hidden layer with 50 units.\",\n \"We use the volume preserving version of RealNVP for K-NonLinear.\",\n \"We train models every 50 time steps for 200 training iterations using a batch-size of 512.\",\n \"Table: Description of bandit problem: number of actions and number of contexts used for experiments.\",\n \"Comparing to benchmark, we restrict ourself to 50000 contexts for Covertype instead of 150000 contexts.Table: Additional results: Cumulative regret incurred by different algorithms on the bandit benchmarks described in .\",\n \"Values reported are the mean over 5 independent trials with standard error of the mean, normalized with respect to the performance of the uniform policy.\",\n \"We use the same hyperparameters for different algorithms without any finetuning: learning rate = 0.0001 and 100 training epochs.\"\n ]\n]"},"id":{"kind":"string","value":"1906.04282"}}},{"rowIdx":2794,"cells":{"text":{"kind":"list like","value":[["Leptogenesis after superconformal subcritical hybrid inflation"],["Abstract We consider an extended version of superconformal subcritical hybrid inflation model by introducing three right-handed neutrinos that have the Majorana mass terms.","In the model one of the right-handed sneutrinos plays a role of the inflaton field, and it decays to reheat the universe after inflation.","The vacuum expectation value for the waterfall field gives an unconventional pattern of the light neutrino mass matrix, and the neutrino Yukawa couplings that determine the reheating temperature are not constrained by the neutrino oscillation data.","Consequently thermal leptogenesis or sneutrino leptogenesis is realized."],["Introduction","Inflation paradigm is strongly supported by the observations of the cosmic microwave background radiation (CMB).","Slow-roll scalar field in the early universe is a promising candidate for inflation, and many types of inflation models have been proposed so far.","In a theoretical point of view, it would be tempting to ask what the underlying physics or symmetry of the inflaton field is.","Supersymmetry (SUSY) might be one of the answers.","It protects the flatness of the inflaton direction, which is suitable for inflation.","Recently supersymmetric D-term hybrid inflation has been revisited in various point of view.","Under shift symmetric Kähler potential [1], subcritical hybrid inflation was found, where inflation continues for subcritical point value of the inflaton field [2], [3].","On the other hand, it was shown in Refs.","[4], [5] that Starobinsky model [6] emerges in the framework of superconformal supergravity [7], [8], [9], [10].","It turned out in the following study that this framework has another new regime of inflation.","It was shown that a general class of superconformal $\\alpha $ -attractor model [11], [12] appears in the subcritical regime of inflation, which we call superconformal subcritical hybrid inflation [13].","In addition, the energy scale of inflation should coincide with the grand unification scale to be consistent with the Planck observation data, which is the feature found in the subcritical hybrid inflation [2], [3].","Namely, the superconformal subcritical hybrid inflation has both features of the superconformal $\\alpha $ -attractor models and the subcritical hybrid inflation.","The shift symmetry and superconformality are crucial for them.","In this paper we will study the thermal history after the end of superconformal subcritical hybrid inflation.","(See Refs.","[14], [15] that study the phenomenology of Pati-Salam version of subcritical hybrid inflation.","Recently Ref.","[16] comprehensively studies the D-term hybrid inflation, including reheating, leptogenesis, and the SUSY breaking mechanism.)","For the purpose, we introduce three right-handed neutrinos that interact with the minimal supersymmetric standard model (MSSM) sector.","In fermionic sector, the mass matrix for the light neutrinos is given by the seesaw mechanism [17], but it has an unconventional structure.","In bosonic sector, on the other hand, it will be shown that one of sneutrinos can play a role of the inflaton field.","In addition, baryon asymmetry that is sufficient amount to explain the observed value is generated via leptogenesis [18].","This paper is organized as follows.","In the next section the model that we consider is described.","Then Sec.","shows the conditions required for the superconformal subcritical hybrid inflation in this model.","Mass matrices of the heavy and light (s)neutrinos, including parametrization of the neutrino Yukawa couplings, are given in Sec.",", then we discuss the reheating and leptogenesis after inflation in Sec. .","Sec.","is dedicated to conclusion."],["The model","We consider a model described by the superpotential $W=W_{\\rm MSSM}+W_{\\rm neu}\\,,$ where $W_{\\rm MSSM}$ is the superpotential of the MSSM sector and $W_{\\rm neu}=\\frac{1}{2}M_{ij}N^c_iN^c_j+y_{\\nu \\,ij}N^c_i L_jH_u+\\lambda _i N_i^c S_+ S_-\\,.$ Here $N_i^c$ , $L_i=(\\nu _{Li},l_{Li})^T$ , and $H_u=(H_u^+,H_u^0)^T$ are the chiral superfields of the right-handed neutrinos, left-handed leptons, and up-type Higgs, respectively, and $L_jH_u=\\nu _{Lj}H_u^0-l_{Lj}H_u^+$ .","$S_{\\pm }$ are the local U(1) fields with charge $\\pm q$ $(q>0)$ , one of which plays a role of the waterfall field.1We write the superpartners with tilde for the MSSM fields and right-handed neutrinos.","For $S_\\pm $ , the same symbols are used for scalar fields while fermionic parts are expressed with tilde.","In the current and the next sections, we adopt the unit in which the reduced Planck mass $M_{pl}\\simeq 2.4\\times 10^{18}\\,{\\rm GeV}$ is taken to be unity unless otherwise mentioned.","Indices are summed over $i,j=1$ – 3.","(We will use the same contraction in the following discussion unless otherwise mentioned.)","$M_{ij}$ terms are explicit superconformal breaking terms that are added by phenomenological purpose.","While we do not argue their origin, $M_{ij} \\ll 1$ are expected.","The Kähler potential, on the other hand, is given by ${\\cal K}=-3\\log \\Omega ^{-2}\\,,$ where $\\Omega ^{-2}=1-\\frac{1}{3}\\left(\\sum _{\\rm MSSM} |I_{\\rm MSSM}|^2+\\sum _i |N_i^c|^2+|S_+|^2+|S_-|^2\\right)-\\sum _i\\frac{\\chi _i}{6}\\left(N_i^{c\\,2}+\\bar{N}_i^{c\\,2}\\right)\\,.$ Here $I_{\\rm MSSM}$ are chiral superfields in the MSSM sector.","The last term is superconformal breaking term that is considered in Refs.","[4], [10].","With the superpotential and Kähler potential, the scalar potential is given by $V_{\\rm tot}=V_F+V_D,$ where $V_F$ and $V_D$ are F- and D-terms, respectively, and given by [4] $&V_F=\\Omega ^4\\left[\\delta ^{\\alpha \\bar{\\alpha }}W_\\alpha W_{\\bar{\\alpha }}+\\frac{1}{\\Delta }|\\delta ^{\\alpha \\bar{\\alpha }}W_\\alpha \\Phi _{\\bar{\\alpha }}-3W|^2\\right]\\,,\\\\&V_D=\\frac{1}{2}g^2\\left[q\\Omega ^2(|S_+|^2-|S_-|^2)-\\xi \\right]^2\\,.$ Here $\\Phi \\equiv -3\\Omega ^{-2}$ and $\\Delta \\equiv \\Phi -\\delta ^{\\alpha \\bar{\\alpha }}\\Phi _\\alpha \\Phi _{\\bar{\\alpha }}$ have been additionally introduced.","Subscript in $W$ and $\\Phi $ stands for the field derivative, e.g., $W_\\alpha \\equiv \\partial W/\\partial z^\\alpha $ where $z^\\alpha $ is a chiral superfield.","In the D-term, we have introduced the Fayet-Iliopoulos (FI) term $\\xi \\,(>0)$ associated with the U(1).2The origin of the FI term in canonical superconformal supergravity model [10] is discussed in Ref. [4].","See also Ref.","[16] for recent development.","Due to the FI term, $S_+$ has a vacuum expectation value (VEV) at the global minimum, which is obtained as $\\langle S_+\\rangle =\\sqrt{\\xi /q(1+\\tilde{\\xi })}$ with $\\tilde{\\xi }\\equiv \\xi /3q$ .","As in Ref.","[13], we take $\\chi _i \\le 0$ without loss of generality.","In the present model, we impose the following condition: $\\chi _3\\simeq -1,~~ \\chi _{1},\\,\\chi _2\\simeq 0\\,.$ This distinguishes $N^c_3$ from $N^c_{1}$ and $N^c_{2}$ .","$\\phi \\equiv \\sqrt{2}{\\rm Re}\\,\\tilde{N}^c_3 $ has an approximate shift symmetry that is explicitly broken by $\\lambda _3\\,(\\ll 1)$ .","Then, $\\phi $ is expected to be the inflaton as studied in Ref. [13].","In the inflation model, $ s \\equiv \\sqrt{2}|S_+|$ plays a role of the waterfall field.","$N^c_1$ and $N^c_2$ , on the other hand, have no such symmetry.","Instead there is a freedom to choose any basis for $N^c_1$ and $N^c_2$ with a redefinition of $M_{ij}$ and $\\lambda _i$ due to $\\chi _{1,2}\\simeq 0$ .","For simple notation, hereafter we omit subscripts of $\\chi _3$ and $\\lambda _3$ and introduce $m_\\phi $ , which will be identified as the inflaton mass in Sec.","REF (with another assumption in Sec.","REF ), $\\chi \\equiv \\chi _3, &~~ \\lambda \\equiv \\lambda _3\\,, \\\\m_\\phi \\equiv &\\, \\lambda \\langle S_+ \\rangle \\,.$ It is sometimes convenient to use $\\delta \\chi $ ($0<\\delta \\chi <1$ ) defined by $\\delta \\chi /(1+\\chi ) = -qg^2\\xi /3\\lambda ^2$ .","$0<\\delta \\chi <1$ guarantees $\\Omega (\\phi ,s)^2>0$ and $\\phi _{c,0}^2>0$ , which will be defined in Eqs.","(REF ) and (REF ), respectively.","Then inflation that is consistent with the observations of the CMB is realized in the parameter space [13], $\\lambda \\simeq (0.5\\, {\\rm }{\\rm }\\,1)\\times 10^{-3}\\,, &~~\\xi ^{1/2}\\simeq (3\\, {\\rm }{\\rm }\\,1)\\times 10^{16}\\,{\\rm GeV}\\,,\\nonumber \\\\m_{\\phi } \\simeq (1\\,{\\rm }{\\rm }\\,2)\\times &10^{13}\\,{\\rm GeV}\\,,$ for $q=g=1$ , $\\delta \\chi =0.9$ and the number of $e$ -folds $N_e=55$ – 60, which we take in the later numerical study.3We will estimate the number of $e$ -folds in Sec.","REF to confirm this.","In the parameter space, e.g., $\\chi \\simeq -1.16$ for $\\lambda =10^{-3}$ , $\\xi ^{1/2}=10^{16}\\,{\\rm GeV}$ , and $\\delta \\chi =0.9$ .","The mass terms in the superpotential, however, have a possibility to alter the inflationary path.","In the next section, we will derive the conditions in order not to affect the inflationary dynamics."],["Inflation","We define several variables that are used in the following analysis.","During inflation, the other fields except for the inflaton and waterfall field are irrelevant.","Thus it is convenient to define following potentials, $& V(\\phi ,s)\\equiv V_{\\rm tot}|_{\\sqrt{2}{\\rm Re}\\,\\tilde{N}_3^c=\\phi ,\\,\\sqrt{2}|S_+|=s,\\,{\\rm the\\,others}=0}\\,,\\\\& \\Omega (\\phi ,s)\\equiv \\Omega |_{\\sqrt{2}{\\rm Re}\\,\\tilde{N}_3^c=\\phi ,\\,\\sqrt{2}|S_+|=s,\\,{\\rm the\\,others}=0}\\,.$ Then the critical point value $\\phi _c$ is defined as a field value below which the waterfall field becomes tachyonic.","It receives ${\\cal O}(M_{ij}^2)$ corrections as $\\phi _c^2=\\phi _{c,0}^2\\left[1-\\frac{2\\Delta M^2(\\phi _{c,0})}{3\\lambda ^2}\\right]+{\\cal O}(M_{i3}^4)\\,,$ where $\\phi _{c,0}^2&=\\frac{6qg^2\\xi }{3\\lambda ^2+(1+\\chi )qg^2\\xi }\\,,\\\\\\Delta M^2(\\phi ) &= \\sum _{i=1}^3|M_{i3}|^2-\\frac{(1-2\\chi )^2}{24}\\frac{\\phi ^2}{1+\\phi ^2\\chi (1+\\chi )/6}|M_{33}|^2\\,.$ For example, $\\phi _{c,0}\\simeq 18$ or in terms of canonically-normalized field $\\hat{\\phi }_{c,0}\\simeq 11$ defined in Eq.","(REF ) for $\\lambda =10^{-3}$ , $\\xi ^{1/2}=10^{16}\\,{\\rm GeV}$ and $\\delta \\chi =0.9$ .","This perturbative expansion is valid when4We have checked that ${\\cal O}(M^4_{i3})$ term is irrelevant when Eq.","() is satisfied, thus we ignore it in the following discussion.","$|M_{13}|^2+|M_{23}|^2 &\\ll 3\\lambda ^2/2\\sim (10^{15}\\,{\\rm GeV})^2\\,, \\\\|M_{33}|^2 &\\ll 2\\lambda ^4/qg^2\\xi \\sim (10^{14}\\,{\\rm GeV})^2\\,.$ It will be checked in this section that the above conditions are satisfied in this inflation model.","Finally it is useful to define $\\Psi \\equiv \\frac{\\Omega (\\phi ,0)\\phi }{\\Omega (\\phi _{c,0},0)\\phi _{c,0}}=\\frac{\\Omega (\\phi ,0)\\phi }{\\sqrt{2qg^2\\xi /\\lambda ^2}}\\,,$ when potential is expressed in terms of canonically-normalized inflaton field."],["Pre-critical regime","Let us begin with the regime where the inflaton is approaching down to the critical point value.","Since the waterfall field is stabilized at the origin in this regime, the relevant Lagrangian is given as ${\\cal L}_{\\rm pre}=\\frac{f(\\phi ,0)}{2}(\\partial _\\mu \\phi )^2-V_{\\rm pre}(\\phi )\\,,$ where5It is noted that the term proportional to $\\Delta M^2$ in Eq.","(REF ) is equivalent to $\\Omega ^4(\\phi ,0)\\phi ^2\\Delta M^2(\\phi )/2$ .","$f(\\phi ,s)&=\\Omega ^2(\\phi ,s)\\left[1+\\Omega ^2(\\phi ,s)\\frac{(1+\\chi )^2}{6}\\phi ^2\\right]\\,, \\\\V_{\\rm pre}(\\phi )&=\\frac{1}{2}g^2\\xi ^2\\left[1+2\\Psi ^2 \\frac{\\Omega ^2(\\phi ,0)\\Delta M^2(\\phi )}{\\lambda ^2\\xi /q}\\right]\\,.$ It is seen that $\\Delta M^2$ term gives a gradient to the inflaton field, which should not invade the slow-roll conditions.","To see the impact of $\\Delta M^2$ term, it is instructive to change dynamical variable $\\phi $ to canonically-normalized field $\\hat{\\phi }$ .","Since $\\chi \\simeq -1$ , it is a good approximation that $f(\\phi ,0)\\simeq \\Omega ^2(\\phi ,0)$ .","Then $d\\phi /d\\hat{\\phi }=f(\\phi )^{-1/2}\\simeq \\Omega ^{-1}(\\phi ,0)$ can be solved easily to obtain, $&\\phi \\simeq \\beta ^{-1/2}\\sinh \\beta ^{1/2}\\hat{\\phi }\\,, \\\\&\\Psi \\simeq \\delta \\chi ^{-1/2}\\tanh \\beta ^{1/2}\\hat{\\phi }\\,, \\\\&\\Omega ^{-1}(\\phi ,0) \\simeq \\cosh \\beta ^{1/2}\\hat{\\phi }\\,,$ where $\\beta =-(1+\\chi )/6=\\lambda ^2\\delta \\chi /2qg^2\\xi $ .","Then the potential in terms of $\\hat{\\phi }$ is given as $V_{\\rm pre} &\\simeq \\frac{1}{2}g^2\\xi ^2\\biggr [1+\\frac{\\tanh ^2\\beta ^{1/2}\\hat{\\phi }}{\\cosh ^2\\beta ^{1/2}\\hat{\\phi } }\\frac{2q}{\\lambda ^2\\xi \\delta \\chi }\\biggr \\lbrace \\sum _{i=1}^3|M_{i3}|^2- \\frac{3}{8\\beta }|M_{33}|^2\\tanh ^{2}\\beta ^{1/2}\\hat{\\phi }\\biggl \\rbrace \\biggl ]\\,.$ On the other hand, it was shown in Ref.","[4] that there is one-loop corrections to the tree-level potential.","In terms of the canonically-normalized field, it is given by $V_{1l}\\simeq \\frac{1}{2}g^2\\xi ^2\\times \\frac{q^2g^2}{8\\pi ^2}\\log \\left[\\delta \\chi ^{-1}\\tanh ^2\\beta ^{1/2}\\hat{\\phi }\\right]\\,.$ Therefore, in order not to affect the inflationary trajectory, it is sufficient that the terms proportional to $|M_{i3}|^2$ are subdominant compared to the one-loop potential.","In the parameter space given in Eq.","(REF ), $\\sqrt{\\beta }\\hat{\\phi _c}\\simeq \\mathrm {arcsinh} \\sqrt{\\beta }\\phi _c \\sim {\\cal O}(1)$ .","Then, the conditions are given as $|M_{13}|^2+|M_{23}|^2 &\\lesssim \\frac{qg^2\\delta \\chi \\lambda ^2\\xi }{16\\pi ^2}\\lesssim (1\\times 10^{12}\\,{\\rm GeV})^2\\,,\\\\|M_{33}|^2 &\\lesssim \\frac{\\delta \\chi ^2\\lambda ^4}{12\\pi ^2}\\lesssim (2\\times 10^{11}\\,{\\rm GeV})^2\\,.$ It is easy to check that the slow-roll conditions are satisfied under the constraints.","Since the constraints are more stringent than Eqs.","(REF ) and (), it has been confirmed that the perturbative expansion to obtain Eq.","(REF ) is valid."],["Subcritical regime","In the previous subsection, we have seen that the slow-roll conditions are satisfied before reaching to the critical point value.","After the inflaton field becomes subcritical point value, the tachyonic growth of the waterfall field occurs.","It is expected that the inflation continues in the subcritical regime when $M_{i3}\\rightarrow 0$ .","In this subsection, we will derive the conditions under which the inflaton and waterfall field dynamics are not affected with non-zero $M_{i3}$ .","As seen in the previous section, the perturbative expression for $\\phi _c$ is valid under the conditions given in Eqs.","(REF ) and ().","Then, the tachyonic growth of the waterfall field is not affected by the additional gradient in the inflaton direction due to $|M_{i3}|^2$ terms since $\\phi _c\\simeq \\phi _{c,0}$ .","Consequently, the dynamics of the waterfall field around the critical point is the same as one discussed in Ref. [13].","The tachyonic growth is qualitatively the same as the subcritical hybrid inflation [2], [3].","Namely, due to the tachyonic growth, the waterfall field relaxes to the local minimum value $s_{\\rm min}$ just after a few Hubble-unit time.","(See, for example, Figure 1. of Ref. [2].)","In the present model, $s_{\\rm min}$ is found to be $s_{\\rm min}^2&=\\frac{2\\xi }{q(1+\\tilde{\\xi })}\\frac{\\Omega ^{-2}(\\phi ,0)}{1+\\tilde{\\xi }\\Psi ^2/(1+\\tilde{\\xi })}\\left[1-\\Psi ^2\\left\\lbrace 1+\\frac{2\\Omega ^2(\\phi ,0)\\Delta M^2(\\phi )}{3\\lambda ^2} \\right\\rbrace \\right]\\,.$ Then the potential in the subcritical regime of the inflaton field is effectively given by $V(\\phi ,s_{\\rm min})$ and the dynamics reduces to single field inflation that is described by the Lagrangian, ${\\cal L}=\\frac{f(\\phi ,s_{\\rm min})}{2}(\\partial _\\mu \\phi )^2- V_{\\rm inf}(\\phi )\\,,$ where $V_{\\rm inf}(\\phi )&=g^2\\xi ^2\\frac{(1+\\tilde{\\xi })\\Psi ^2}{1+2\\tilde{\\xi }\\Psi ^2}\\biggl [1-\\frac{\\Psi ^2}{2(1+\\tilde{\\xi })}+\\frac{\\lbrace 1+\\tilde{\\xi }\\Psi ^2/(1+\\tilde{\\xi })\\rbrace ^2}{(1+2\\tilde{\\xi }\\Psi ^2)/(1+\\tilde{\\xi })}\\frac{\\Omega ^2(\\phi ,0)\\Delta M^2(\\phi )}{\\lambda ^2\\xi /q}\\biggr ]\\nonumber \\\\ &+{\\cal O}(M_{i3}^4) \\,.$ Recall that $\\tilde{\\xi }\\,,\\xi \\ll 1$ in the parameters in Eq.","(REF ).","Then it is clear that the additional term proportional to $\\Delta M^2$ is the same as in Eq.","(REF ).","Note that $s_{\\rm min}$ is negligible in $\\Omega (\\phi ,s_{\\rm min})$ .","Then, the canonically-normalized inflaton field is given by Eqs.","(REF )–().","Consequently, $V_{\\rm inf}$ is given by $V_{\\rm inf} &\\simeq g^2\\xi ^2 \\delta \\chi ^{-1} \\tanh ^2\\beta ^{1/2} \\hat{\\phi }\\biggr [1-\\frac{\\delta \\chi ^{-1}}{2} \\tanh ^2\\beta ^{1/2}\\hat{\\phi } \\nonumber \\\\&+\\frac{\\cosh ^{-2}\\beta ^{1/2}\\hat{\\phi }}{\\lambda ^2\\xi /q}\\biggr \\lbrace \\sum _{i=1}^3|M_{i3}|^2-\\frac{3}{8\\beta }|M_{33}|^2\\tanh ^{2}\\beta ^{1/2}\\hat{\\phi }\\biggl \\rbrace \\biggl ]\\,.$ Therefore, if Eqs.","(REF ) and () are satisfied, then the dynamics in the subcritical regime reduces to one in Ref. [13].","In addition, it is clear that $m_\\phi $ defined in Eq.","(REF ) is indeed the inflaton mass since $V_{\\rm inf}\\simeq \\frac{1}{2}m_\\phi ^2 \\phi ^2$ around $\\phi \\,(\\simeq \\hat{\\phi })\\sim 0$ .","To summarize the present and previous subsections, the inflaton-waterfall field dynamics is unchanged when $|M_{i3}|^2=|M_{3i}|^2 \\lesssim (2\\times 10^{11}\\,{\\rm GeV})^2\\,,$ for $i=1$ –3 are satisfied."],["Stability of inflationary trajectory","It was pointed out in Ref.","[19] that $\\tilde{L}_iH_u$ may become tachyonic in sneutrino inflation.","In order to find out the stability condition, let us derive the mass matrix in $\\tilde{L}_i$ and $H_u$ basis.","From $V_{\\rm tot}$ , it is obtained by ${\\cal L}_{\\rm mass}^{\\tilde{L}H_u}=&-\\frac{\\Omega (\\phi ,s)^4 \\phi ^2}{2} |y_{\\nu 3i} \\tilde{L}_i|^2-\\frac{\\Omega (\\phi ,s)^4 \\phi ^2}{2} |y_{\\nu 3i}|^2|H_u|^2\\nonumber \\\\&+\\left[\\frac{M_{33}^* \\Omega (\\phi ,s)^6\\phi ^3}{4\\sqrt{2}}y_{\\nu 3i} \\tilde{L}_iH_u+{\\rm h.c.}\\right]\\,,$ On the other hand, the kinetic terms of $\\tilde{L}_i$ and $H_u$ are given by ${\\cal L}_{\\rm kin}^{\\tilde{L}H_u}=\\Omega (\\phi ,s)^2\\left[|\\partial _\\mu \\tilde{L}_i|^2+|\\partial _\\mu H_u|^2\\right]\\,.$ Therefore, using canonically-normalized fields, $\\hat{\\tilde{L}}_i\\equiv \\Omega (\\phi ,s)\\tilde{L}_i$ and $\\hat{H}_u\\equiv \\Omega (\\phi ,s)H_u$ , the mass terms are rewritten as ${\\cal L}_{\\rm mass}^{\\tilde{L}H_u}=&-\\frac{\\Omega (\\phi ,s)^2 \\phi ^2}{2} |y_{\\nu 3i} \\hat{\\tilde{L}}_i|^2-\\frac{\\Omega (\\phi ,s)^2 \\phi ^2}{2} |y_{\\nu 3i}|^2|\\hat{H}_u|^2\\nonumber \\\\&+\\left[\\frac{M_{33}^* \\Omega (\\phi ,s)^4\\phi ^3}{4\\sqrt{2}}y_{\\nu 3i} \\hat{\\tilde{L}}_i\\hat{H}_u+{\\rm h.c.}\\right]\\nonumber \\\\=&-\\frac{\\Omega (\\phi ,s)^2 \\phi ^2}{2}(y_\\nu y_\\nu ^\\dagger )_{33}|\\hat{\\tilde{L}}_3^\\prime |^2-\\frac{\\Omega (\\phi ,s)^2 \\phi ^2}{2}(y_\\nu y_\\nu ^\\dagger )_{33} |\\hat{H}_u|^2\\nonumber \\\\&+\\left[\\frac{M_{33}^* \\Omega (\\phi ,s)^4\\phi ^3}{4\\sqrt{2}}(y_\\nu y_\\nu ^\\dagger )_{33}^{1/2}\\hat{\\tilde{L}}_3^\\prime \\hat{H}_u+{\\rm h.c.}\\right]\\,,$ where we have defined $(y_\\nu y_\\nu ^\\dagger )_{33}^{1/2}\\hat{\\tilde{L}}_3^\\prime \\equiv y_{\\nu 3i}\\hat{\\tilde{L}}_i$ in the second line following Ref. [19].","Then, the stability condition is given by $|M_{33}|<2\\sqrt{2}(y_\\nu y_\\nu ^\\dagger )_{33}^{1/2}\\frac{\\Omega (\\phi ,s)^{-2}}{\\phi }\\,.$ Using Eqs.","(REF ) and (), it turns out that $|M_{33}|&<4\\sqrt{2}(y_\\nu y_\\nu ^\\dagger )_{33}^{1/2}\\sqrt{\\beta }\\nonumber \\\\&\\simeq 1.4 \\times 10^{14}\\,{\\rm GeV}\\left(\\frac{(y_\\nu y_\\nu ^\\dagger )_{33}}{10^{-7}}\\frac{\\beta }{10^{-3}}\\right)^{1/2}\\,.$ This upper bound is weaker than (REF ) in most of the parameter space, which will be seen later."],["Neutrino mass","In this section, we derive the mass matrices for the heavy and light neutrinos.","Around the global minimum, $\\Omega \\simeq 1$ since $\\xi \\ll 1$ .","Consequently, all the fields are canonical.","Thus, the mass terms are derived similarly in global SUSY model."],["Mass matrix","The superpotential (REF ) gives the Majorana masses for the light neutrinos.","To see how the masses are generated, we write down the mass terms for fermionic part of $N_i^c$ , $\\nu _{Li}$ and $\\tilde{S}_-$ , ${\\cal L}_{\\nu }^{\\rm mass}=-\\frac{1}{2}(\\bar{\\psi }{\\cal M}P_L\\psi +{\\rm h.c.})\\,,$ where $\\psi &=(N^c_1,N^c_2,N^c_3,\\tilde{S}_-,\\nu _{L1},\\nu _{L2},\\nu _{L3})^T\\,,\\\\{\\cal M}&=\\left(\\begin{array}{cc}\\tilde{M} & \\tilde{m}_\\nu \\\\\\tilde{m}_\\nu ^T & {\\bf 0}\\end{array}\\right)\\,.$ $\\tilde{M}$ and $\\tilde{m}_\\nu $ are $4\\times 4$ and $4\\times 3$ matrices, respectively, and given by $\\tilde{M}&=\\left(\\begin{array}{cccc}&&&\\lambda _1\\langle S_+\\rangle \\\\&\\mbox{\\smash{\\Large M}}&&\\lambda _2\\langle S_+\\rangle \\\\&&& m_\\phi \\\\\\lambda _1\\langle S_+\\rangle &\\lambda _2\\langle S_+\\rangle &m_\\phi & 0\\end{array}\\right)\\,, \\\\\\tilde{m}_\\nu &=\\left(\\begin{array}{ccc}&& \\\\&\\mbox{\\smash{\\Large m_\\nu }}&\\\\&& \\\\0&0&0\\end{array}\\right)\\,.$ Here $m_{\\nu \\,ij}=y_{\\nu \\,ij}\\langle H_u^0\\rangle $ with $\\langle H_u^0\\rangle $ being the VEV of the up-type neutral Higgs.","Then mass matrix $M_{\\nu }$ for the light neutrinos are obtained by the seesaw mechanism [17], $M_{\\nu } =-\\tilde{m}_\\nu ^T \\tilde{M}^{-1}\\tilde{m}_\\nu \\,.$ An important consequence of the mass matrix is that the one of three light neutrinos is massless.","This is because the rank of $M_\\nu $ is two.","Using this mass matrix, it is possible to constrain the parameters by the observed neutrino masses.","In the later discussion we assume $\\lambda _1\\,,~\\lambda _2\\,\\ll \\lambda \\,.$ Here, recall that there is a freedom to choose a basis for $N^c_1$ and $N^c_2$ .","Then, $M_{12}$ can be rotated away.","As a result, $M_{\\nu }$ is given in the following simple expression, $M_{\\nu \\, ij} = -\\langle H^0_u\\rangle ^2\\sum _{k=1}^2\\frac{y_{\\nu ki}y_{\\nu kj}}{M_{kk}}+{\\cal O}\\left(\\frac{\\lambda _{1,2}}{\\lambda }\\frac{m_{\\nu \\, il}m_{\\nu \\, jm}}{M_{kk}}\\right)\\,.$ Here in the second term of right-hand side, $k=1$ or 2 and $i$ , $j$ , $l$ , $m$ can be 1–3, and we have ignored $M_{i3}$ based on the discussion of the previous section.","The leading order term, on the other hand, is independent of $\\lambda _{1,2}$ , $M_{i3}$ , $m_\\phi $ and $y_{\\nu 3i}$ ($i=1$ –3).","Therefore, they are not constrained by the neutrino oscillation data.","This fact is important in the estimation of the reheating temperature, which we will see later.","To ensure the non-zero $\\lambda _{1,2}$ does not affect our later analysis at percent level, we implicitly assume $\\lambda _{1,2}<0.01\\lambda $ .","Before further discussing the light neutrino mass matrix, let us note that the mass matrix $\\tilde{M}$ corresponds to the mass matrix in the superpotential around the global minimum, $W_{\\rm neu}=\\frac{1}{2} (N^c_1,N^c_2,N^c_3,S_-)\\tilde{M}\\left(\\begin{array}{c}N^c_1 \\\\N^c_2 \\\\N^c_3 \\\\S_-\\end{array}\\right)+\\cdots \\,.$ Recall that we have the requirement (REF ) for successful inflation.","Therefore, $\\tilde{M}$ should be almost block-diagonal as $\\tilde{M}&\\simeq \\left(\\begin{array}{ccc:c}M_1&&&0 \\\\&M_2&&0\\\\&& M_3& m_\\phi \\\\0 &0&m_\\phi & 0\\end{array}\\right)\\,,$ where $M_i>0$ .","Here we have left $M_3$ for later discussion.","In the following analysis we use Eq.","(REF ) for $\\tilde{M}$ .","$\\tilde{M}$ is further diagonalized by a unitary matrix $U_{\\tilde{M}}$ as, $D_{\\tilde{M}}\\simeq {\\rm diag}(M_1,M_2,m_\\phi , m_\\phi )\\simeq U^T_{\\tilde{M}} \\tilde{M} U_{\\tilde{M}}\\,,$ where $U_{\\tilde{M}}=\\left(\\begin{array}{cc:c}1 & & \\\\& 1 & \\\\& &{\\scriptsize \\frac{1}{\\sqrt{2}}\\bigl (\\begin{array}{cc}1 & i\\\\1 & -i\\end{array}\\bigr )}\\end{array}\\right)\\,.$ Here we have omitted ${\\cal O}(M_3/m_\\phi )$ corrections since they are irrelevant in the later analysis.","Non-zero $\\lambda _{1,2}$ corrections enter as ${\\cal O}(\\lambda _{1,2}/\\lambda )$ and ${\\cal O}((m_\\phi /M_{1,2})\\lambda _{1,2}/\\lambda )$ for $m_\\phi \\gtrsim M_{1,2}$ and $m_\\phi \\lesssim M_{1,2}$ , respectively.","The corrections due to non-zero $\\lambda _{1,2}$ are similar in the scalar sector since the mass matrix in $(\\tilde{N}_1,\\tilde{N}_2,\\tilde{N}_3,S_-)$ basis is given by, $M_{\\rm scalar}^2=\\tilde{M}^\\dagger \\tilde{M}\\,.$ Therefore the assumption (REF ) assures that the dynamics of the inflaton and the waterfall field discussed in the previous section, as well as subsequent reheating and leptogenesis, is not affected.","If it is not satisfied, then the inflationary trajectory would be altered so that we need further analysis to find the paramter space that is consistent with the CMB observations.","Accordingly, the subsequent reheating and leptogenesis scenario change.","We leave the detailed analysis as an interesting future work."],["Parametrization of neutrino Yukawa couplings","Now let us discuss $M_\\nu $ .","It can be diagonalized by a unitary matrix $U_\\nu $ as $D_{M_\\nu }={\\rm diag}(m_1,m_2,m_3)=U_\\nu ^T M_\\nu U_\\nu \\,.$ As noted above, one of the three light neutrino masses is zero.","We follow the standard convention that $m_3>m_2>m_1(=0)$ for the normal hierarchy (NH) case and $m_2>m_1>m_3(=0)$ for the inverted hierarchy (IH) case and use the values given in Ref.","[20], which are listed in Table REF .","Before discussing the parametrization of the neutrino Yukawa couplings, it is instructive to count the number of parameters.","The situation is the same as one discussed Refs.","[19], [21] since the mass matrix of the light neutrinos (REF ) is similar.","Since one neutrino is massless, there are 7 parameters in low energy, i.e., 2 neutrino masses $+$ 3 real mixing angles $+$ 2 phases.","On the other hand, $M_\\nu $ includes $y_{\\nu ki}$ and $M_k$ where $k=1,2$ and $i=1,2,3$ , which means 12 real parameters (neutrino Yukawa couplings) $+$ 2 real parameters (right-handed neutrino masses).","However, 3 phases can be absorbed by lepton doublets and 2 real parameters are unphysical since $M_\\nu $ is unchanged by the rescalings $y_{\\nu ki}\\rightarrow \\gamma _k y_{\\nu ki}$ and $M_k \\rightarrow \\gamma ^2_kM_k$ with $\\gamma _k$ being real constants.","Therefore, we have 9 independent parameters in $M_\\nu $ to determine 7 parameters in the light neutrino sector.","As it will be seen below, however, the parametrization of the Yukawa couplings is different, especially for $y_{\\nu 3i}$ that are important parameters for the estimation of the reheating temperature.","Let us discuss the NH case first.","We define $4\\times 3$ matrix $R$ in the similar manner in Refs.","[22], [23], $iR=D_{\\tilde{M}}^{-1/2}U_{\\tilde{M}}^T\\tilde{m}_\\nu U_\\nu D_{M_\\nu }^{-1/2}\\,,$ which satisfies $R^TR={\\rm diag}(0,1,1)\\,.$ Here $D_{\\tilde{M}}^{-1/2}$ and $D_{M_\\nu }^{-1/2}$ are matrices that satisfy $(D_{\\tilde{M}}^{-1/2})^2=D_{\\tilde{M}}^{-1}$ and $(D_{M_\\nu }^{-1/2})^2={\\rm diag}(0,m_2^{-1},m_3^{-1})$ , respectively.","It is found that $R$ is more restrictive than Eq.","(REF ).","Namely, $r^Tr=rr^T={\\bf 1}\\,,\\\\R_{42}/R_{32}=R_{43}/R_{33}=i\\,, \\\\R_{l1}=0~~~~(l=14)\\,,$ where $r\\equiv \\left(\\begin{array}{cc}R_{12} & R_{13} \\\\R_{22} & R_{23}\\end{array}\\right)\\,.$ Using the relations, the neutrino Yukawa couplings can be expressed in terms of $R$ .","For later discussion, it is useful to give following quantities: $(y_\\nu y_\\nu ^\\dagger )_{ii}&=M_i\\sum _{j=2}^3 |R_{ij}|^2 m_j/\\langle H_u^0 \\rangle ^2~~~~~~~~~(i=1,2)\\,,\\\\(y_\\nu y_\\nu ^\\dagger )_{33}&=2m_\\phi \\sum _{j=2}^3|R_{3j}|^2 m_j/\\langle H_u^0 \\rangle ^2\\,,\\\\{\\rm Im}\\left[(y_\\nu y_\\nu ^\\dagger )_{21}^2\\right]\\frac{M_1}{M_2}&=-\\frac{M_1^2}{\\langle H_u^0\\rangle ^4}{\\rm Im}\\left[\\sum _{j=2}^3R_{1j}^2m_j^2\\right]\\,, \\\\\\sum _{i=1}^2{\\rm Im}\\left[(y_\\nu y_\\nu ^\\dagger )_{i3}^2\\right]\\frac{M_3}{M_i}&=-\\frac{2M_3 m_\\phi }{\\langle H_u^0\\rangle ^4}{\\rm Im}\\left[\\sum _{j=2}^3R_{3j}^2m_j^2\\right]\\,.$ It is seen that $(y_\\nu y_\\nu ^\\dagger )_{11}$ and $(y_\\nu y_\\nu ^\\dagger )_{22}$ are constrained by the neutrino oscillation data, meanwhile $(y_\\nu y_\\nu ^\\dagger )_{33}$ is basically a free parameter since $R_{3j}$ is not constrained.","This is consistent with the fact that $M_\\nu $ is independent of $y_{\\nu 3i}$ .","The discussion is quite similar in the IH case.","The definition of $R$ is the same form as in Eq.","(REF ), but satisfies $R^TR={\\rm diag}(1,1,0)$ with $(D_{M_\\nu }^{-1/2})^2={\\rm diag}(m_1^{-1},m_2^{-1},0)$ .","Then, defining $r$ as $r\\equiv \\left(\\begin{array}{cc}R_{11} & R_{12} \\\\R_{21} & R_{22}\\end{array}\\right)\\,,$ we get $r^Tr=rr^T={\\bf 1}\\,,\\\\R_{41}/R_{31}=R_{42}/R_{32}=i\\,, \\\\R_{l3}=0~~~~(l=14)\\,,$ and the neutrino Yukawa couplings are given by, $(y_\\nu y_\\nu ^\\dagger )_{ii}&=M_i\\sum _{j=1}^2 |R_{ij}|^2 m_j/\\langle H_u^0 \\rangle ^2~~~~~~~~~(i=1,2)\\,,\\\\(y_\\nu y_\\nu ^\\dagger )_{33}&=2m_\\phi \\sum _{j=1}^2|R_{3j}|^2 m_j/\\langle H_u^0 \\rangle ^2\\,,\\\\{\\rm Im}\\left[(y_\\nu y_\\nu ^\\dagger )_{21}^2\\right]\\frac{M_1}{M_2}&=-\\frac{M_1^2}{\\langle H_u^0\\rangle ^4}{\\rm Im}\\left[\\sum _{j=1}^2R_{1j}^2m_j^2\\right]\\,, \\\\\\sum _{i=1}^{2}{\\rm Im}\\left[(y_\\nu y_\\nu ^\\dagger )_{i3}^2\\right]\\frac{M_3}{M_i}&=-\\frac{2M_3 m_\\phi }{\\langle H_u^0\\rangle ^4}{\\rm Im}\\left[\\sum _{j=1}^2R_{3j}^2m_j^2\\right]\\,.$"],["Post inflationary regime","After the end of inflation, the inflaton oscillates around the global minimum and decays eventually.","Due to the decay the universe is reheated and thermal plasma is created.","In this section, we estimate the reheating temperature and discuss how the lepton number asymmetry is generated.","As in the previous section, we take $\\Omega \\simeq 1$ .","After the universe is reheated, gravitinos are produced in various ways.","We discuss the gravitino problem at the end of this section."],["Reheating","The reheating temperature $T_R$ due to the inflaton decay is estimated by, $T_R\\simeq (90/\\pi ^2g_*(T_R))^{1/4}\\sqrt{\\Gamma _\\phi M_{pl}}\\,,$ where $g_*(T)$ is the effective degree of freedom of radiation fields at temperature $T$ and $\\Gamma _\\phi $ is the decay rate of the inflaton.","This expression is valid when the neutrino Yukawa couplings that are responsible for the decay is sufficiently small to satisfy $T_R\\lesssim m_\\phi $ [24], [25], which is the situation we focus on.6Of course, it is possible to consider a higher reheating temperature than the inflaton mass.","Such a case is discussed in Ref. [19].","We will comment on the impact of such high reheating temperature on leptogenesis in the next subsection.","The inflaton decays as $\\phi \\rightarrow L\\tilde{H}_u$ , $\\bar{L}\\bar{\\tilde{H}}_u$ , $\\tilde{L}H_u$ , $\\tilde{L}^*H_u^*$ .7In general, the inflaton decays to gravitino pair or gravitino and right-handed neutrino.","We will discuss those processes in Sec.","REF .","Here flavor indices and SU(2) doublet components are summed implicitly.","Then the decay rates for the modes are given by $&\\Gamma _{\\phi \\rightarrow L\\tilde{H}_u}=\\Gamma _{\\phi \\rightarrow \\bar{L}\\bar{\\tilde{H}}_u}=\\frac{(y_{\\nu }y_\\nu ^\\dagger )_{33}}{16\\pi }m_\\phi \\,, \\\\& \\Gamma _{\\phi \\rightarrow \\tilde{L}H_u}=\\Gamma _{\\phi \\rightarrow \\tilde{L}^*H_u^*}=\\frac{(y_{\\nu }y_\\nu ^\\dagger )_{33}}{16\\pi }\\frac{M_3^2}{m_\\phi }\\,.$ Since $\\Gamma _{\\phi \\rightarrow \\tilde{L}H_u}$ (=$\\Gamma _{\\phi \\rightarrow \\tilde{L}^*H_u^*}$ ) is suppressed by $(M_3/m_\\phi )^2$ , the total decay rate is given by $\\Gamma _\\phi \\simeq \\Gamma _{\\phi \\rightarrow L\\tilde{H}_u}+\\Gamma _{\\phi \\rightarrow \\bar{L}\\bar{\\tilde{H}}_u}=\\frac{(y_{\\nu }y_\\nu ^\\dagger )_{33}}{8\\pi }m_\\phi \\,.$ Then the reheating temperature is estimated as $T_R\\simeq 1.4\\times 10^{10}\\,{\\rm GeV}\\left(\\frac{m_\\phi }{10^{13}\\,{\\rm GeV}}\\right)^{1/2}\\left(\\frac{(y_{\\nu }y_\\nu ^\\dagger )_{33}}{10^{-9}}\\right)^{1/2}\\left(\\frac{g_{*}(T_R)}{228.75}\\right)^{-1/4}\\,.$ Recall that $(y_\\nu y_\\nu ^\\dagger )_{33}$ is not constrained by the neutrino observations.","As a consequence, it is possible to consider a wide range of values for the reheating temperature, which is suitable for leptogenesis.","To end this subsection, we derive the number of $e$ -folds before the end of inflation.","In this model, the inflaton oscillates after the end of inflation and eventually decays to reheat the universe.","Therefore, it is given by $N_e\\simeq &\\, 55+\\log \\left(\\frac{L}{{\\rm Gpc}}\\right)+\\frac{1}{3}\\log \\left(\\frac{T_R}{10^{10}\\,{\\rm GeV}}\\right)\\nonumber \\\\&+\\log \\left(\\frac{H_e}{10^{13}\\,{\\rm GeV}}\\right)-\\frac{2}{3}\\log \\left(\\frac{H_{\\rm end}}{10^{13}\\,{\\rm GeV}}\\right)\\,,$ where $L$ is the present cosmological scale, and $H_e$ and $H_{\\rm end}$ are the Hubble scale corresponding to $N_e$ and at the end of inflation, respectively.","It is now clear that the parameters given in Eq.","(REF ) is consistent with $N_e=55$ – 60."],["Leptogenesis","Now we discuss the lepton number asymmetry.","The lepton number is generated via leptogenesis [18] (see, for example, Refs.","[26], [27] for review).","In the following numerical study, we discuss following representative cases: $({\\rm I}).~& M_1,M_2 < m_\\phi \\,,\\\\({\\rm II}).~& M_1,M_2 > m_\\phi \\,.$ In both cases, $M_3$ should satisfy Eq.","(REF ).","Let us consider case (I) first.","For simplicity, we consider the Majorana masses are further hierarchical, i.e., $M_1\\ll M_2$ .8$M_3$ is irrelevant for leptogenesis if Eq.","(REF ) is satisfied.","Similar situation has been studied intensively in the literature [28], [29].","Even though the lepton number is generated by the inflaton decay, it is possibly washed out when the reheating temperature is comparable or higher than $M_{1}$ .9It would be possible that $\\tilde{N}_i$ ($i=1,2$ ) have initial amplitude of the Hubble parameter during inflation $H_{\\rm inf}\\sim g\\xi /\\sqrt{6}M_{pl}$ .","Then $\\tilde{N}_i$ start to oscillate when $H\\sim M_i$ , and eventually decays.","However, the effect of coherent oscillation of $\\tilde{N}_i$ is negligible since the energy density ratio of $\\tilde{N}_i$ to radiation at the decay is estimated as less than $\\xi ^2/18M_{pl}^4\\sim 10^{-9}$ .","To see this more explicitly, it is convenient to introduce the effective neutrino mass [30] and equilibrium neutrino mass [31], $&\\tilde{m}_1=\\frac{(m_\\nu m_\\nu ^\\dagger )_{11}}{M_1}\\,,\\\\&m_*=\\frac{4\\pi ^2\\sqrt{g_*(M_1)} \\langle H_u^0\\rangle ^2 }{3\\sqrt{10} M_{pl}}\\simeq 3.9 \\times 10^{-4}\\,{\\rm eV}\\left(\\frac{\\langle H_u^0\\rangle }{v/2}\\right)^2\\,,$ where $v\\simeq 246.7~{\\rm GeV}$ .","If $\\tilde{m}_1/m_*$ is larger than unity, then it is the strong washout regime and the lepton number generated at the reheating is washed out.","$\\tilde{m}_1$ is estimated by using Eqs.","(REF ) and (REF ), $\\tilde{m}_1=\\left\\lbrace \\begin{array}{ll}\\sum _{j=2}^3 |R_{1j}|^2 m_j &({\\rm NH}) \\\\\\sum _{j=1}^2 |R_{1j}|^2 m_j &({\\rm IH})\\end{array}\\right.\\,.$ Using the neutrino mass data and Eqs.","(REF ) and (REF ), it is straightforward to find that $\\tilde{m}_1$ has a lower bound, $\\tilde{m}_1 \\ge \\left\\lbrace \\begin{array}{ll}m_2 \\simeq 8.6 \\times 10^{-3}\\,{\\rm eV} &({\\rm NH}) \\\\m_1 \\simeq 4.9\\times 10^{-2}\\,{\\rm eV} &({\\rm IH})\\end{array}\\right.\\,.$ Therefore, it is the strong washout regime in either case.","Figure: Allowed region for M 1 M_1 as function of effective neutrinomass m ˜ 1 \\tilde{m}_1 defined in Eq.","() for the normalhierarchy (NH) and inverted hierarchy (IH) cases.","Lower bound onM 1 M_1 is obtained from η B max ≥η B obs \\eta _B^{\\rm max}\\ge \\eta _B^{\\rm obs}while upper bound is ().","Lower bound onm ˜ 1 \\tilde{m}_1 is given by Eq.","().Although the primordial lepton number is washed out, the lepton number is regenerated by the decay of the lightest right-handed (s)neutrino, i.e., $N_1$ and $\\tilde{N}_1$ in the present case.","Then the lepton number, strictly speaking lepton number minus baryon number, is converted to baryon number via the sphaleron effect.","This scenario works if $T_R\\gtrsim M_1$ [28], [29], which is always possible as confirmed in the previous subsection.","Then the resultant baryon number becomes independent of $T_R$ .","In our study, we adopt the analytic expressions in Ref.","[29] for the calculation of the baryon number.","Note that although the results there are given in non-supersymmetric model, the results in supersymmetric model do not change much both quantitatively and qualitatively [32], [33], [27].","In our study we adopt the discussion given in Ref. [27].","Then the baryon number is determined by $\\eta _B\\equiv \\frac{n_B}{n_\\gamma }=\\frac{3\\sqrt{2}}{4}\\frac{a_{\\rm sph}}{f}\\epsilon _1 \\kappa _f\\simeq 2.7\\times 10^{-10} \\left(\\frac{\\epsilon _1}{10^{-6}}\\right)\\left(\\frac{\\kappa _f}{2\\times 10^{-2}}\\right)\\,,$ where $n_B$ and $n_\\gamma $ are number densities of baryon and photon at present, respectively, $a_{\\rm sph}=28/79$ , $f=2387/86$ , and a factor of $\\sqrt{2}$ counts the supersymmetric effect.","The efficiency factor $\\kappa _f$ is given by [29] $\\kappa _f=(2\\pm 1)\\times 10^{-2}\\left(\\frac{0.01~{\\rm eV}}{\\tilde{m}_1}\\right)^{1.1\\pm 0.1}\\,.$ Finally, referring Ref.","[34], the asymmetric parameter $\\epsilon _1$ in our model is given by $\\epsilon _1=-\\frac{3}{16\\pi }\\frac{{\\rm Im}\\,\\left[(y_\\nu y_\\nu ^\\dagger )_{21}^2\\right] }{(y_\\nu y_\\nu ^\\dagger )_{11}}\\frac{M_1}{M_2}=\\frac{3}{16\\pi }\\frac{M_1}{\\langle H^0_u \\rangle ^2}m_{\\rm eff}\\,,$ where we have used Eqs.","(REF ), (), (REF ), and () to obtain $m_{\\rm eff} =\\left\\lbrace \\begin{array}{ll}\\frac{{\\rm Im}\\sum _{j=2}^3R_{1j}^2m_j^2}{\\sum _{j=2}^3|R_{1j}|^2m_j}&({\\rm NH}) \\\\[4mm]\\frac{{\\rm Im}\\sum _{j=1}^2R_{1j}^2m_j^2}{\\sum _{j=1}^2|R_{1j}|^2m_j}&({\\rm IH})\\end{array}\\right.\\,.$ It turns out that the maximum value of $m_{\\rm eff}$ , denoted as $m_{\\rm eff}^{\\rm max}$ , is $m_{\\rm eff}^{\\rm max} =\\left\\lbrace \\begin{array}{ll}m_3 - m_2 \\simeq 4.2\\times 10^{-2}\\,{\\rm eV}&({\\rm NH}) \\\\m_2 - m_1\\simeq 7.4 \\times 10^{-4}\\,{\\rm eV}&({\\rm IH})\\end{array}\\right.\\,.$ Therefore, parametrizing $m_{\\rm eff}$ as $m_{\\rm eff}=m_{\\rm eff}^{\\rm max} \\sin \\delta $ , the asymmetric parameter is given by $\\epsilon _1\\simeq \\left\\lbrace \\begin{array}{ll}8.2 \\times 10^{-7}&({\\rm NH}) \\\\1.5\\times 10^{-8}&({\\rm IH})\\end{array}\\right\\rbrace \\times \\left(\\frac{M_1}{10^{10}\\,{\\rm GeV}}\\right)\\left(\\frac{\\langle H^0_u \\rangle }{v/2}\\right)^{-2}\\left(\\frac{\\sin \\delta }{0.5}\\right)\\,.$ Using Eqs.","(REF ), (REF ) and (REF ), the lower limit on $M_1$ to explain the present baryon number is determined from $\\eta _B^{\\rm max} \\ge \\eta _B^{\\rm obs}$ where $\\sin \\delta =1$ and $\\eta _B^{\\rm obs}$ is given by [35] $\\eta _B^{\\rm obs}=(6.12\\pm 0.03)\\times 10^{-10}\\,.$ In Fig.","REF , allowed regions are depicted for the NH and IH cases.","Here we consider so-called high-scale SUSY and take $\\langle H^0_u \\rangle =v/2$ to get 125 GeV Higgs mass [36], [37].","In the plot upper bound on $M_1$ is given by (REF ), i.e., $M_110^{-5}\\,{\\rm eV}$ because $\\tilde{m}_3\\ll m_*$ is no longer valid.","In the plot, contours of $T_R$ are depicted.","It is found that the leptogenesis is successful in a wide parameter space for both the NH and IH cases.","Lower bound on $M_3$ behaves similarly to region C in Fig.","1 of Ref.","[42] by reading $M_1$ and $\\tilde{m}_1$ as $M_3$ and $\\tilde{m}_3$ , respectively, i.e., the lower bound is proportional to $1/\\sqrt{\\tilde{m}_3}$ .","Quantitatively, the lower bound in our model is relaxed by roughly a factor of 4 compared to the result in the reference.","This can be understood as follows; first, the decay rate of the inflaton in our model is different from one in the literature (see Eq.","(REF )), leading to a $1/\\sqrt{2}$ suppression in $T_R/m_\\phi $ ; as another consequence, the expression (REF ) (with $m_{\\rm eff}^{\\prime }=m_{\\rm eff}^{\\prime {\\rm max}}$ ) is enhanced by a factor of two compared to $|\\epsilon _1^{\\rm max}|$ in the reference; finally $\\tan \\beta =\\infty $ is taken in the work meanwhile $\\tan \\beta =1$ in our model.","Thus, in total, a factor of 4 enhancement is obtained in $\\eta _B$ .","In the case where $\\tilde{m}_3$ gets much larger than $m_*$ , the situation reduces to case (I).","Namely, the reheating temperature is so high that both $\\tilde{N}_3$ and $N_3$ are thermalized and thermal leptogenesis takes place.","Resultant allowed region is the same as the NH case of Fig.","REF , by replacing $M_1$ and $\\tilde{m}_1$ by $M_3$ and $\\tilde{m}_3$ , respectively, but there is no lower bound on $\\tilde{m}_3$ meanwhile there is the upper bound on $M_3$ .","In the intermediate case, $\\tilde{m}_3\\sim m_*$ , on the other hand, the Boltzmann equations should be solved numerically to get the lepton number, which is already done in Ref. [42].","The result corresponds to region B in Fig.","1 of the reference.","Strictly speaking, the effective dissipation rate should be used instead of the decay rate of the inflaton [25] in the Boltzmann equations.","As shown in the reference, the reheating process is so efficient when the dissipation rate is taken into account that the reheating temperature can exceed the mass of the inflaton mass and consequently $N_3$ and $\\tilde{N}_3$ are easily thermalized.","Once they are thermalized, the thermal leptogenesis takes place, where the resultant baryon number becomes independent of the reheating temperature.","Eventually the situation reduces to the case (I).","Such qualitative behavior can be confirmed by numerical study, which is left for the future work.","Crucial difference from sneutrino leptogenesis [38], [39], [40], [41], [42] is that although $M_3$ and $\\tilde{m}_3$ , i.e., $(y_\\nu y_\\nu ^\\dagger )_{33}$ , are important parameters to determine baryon number, they are sequestered from other physical quantities, such as the heavy right-handed (s)neutrino masses or the light neutrino mass matrix.","Therefore, there is no consequence in other low energy experiments.","This is a feature of case (II)."],["Gravitino problem","In the framework of supergravity, a fair amount of gravitino $\\psi _\\mu $ can be produced in various ways in the thermal history of the universe.","Since the interactions of gravitino with the MSSM particles are Planck-suppressed, gravitino is long-lived and its decay can spoil the successful big-bang nucleosynthesis if it is unstable.","Although this problem can be avoided when gravitino is enough heavy to have the lifetime much shorter than 1 sec, gravitino decay produces the lightest superparticle (LSP).","Then the LSP produced by the decay may overclose the universe if the R-parity is conserved.","There are three types of production mechanism of gravitino in the model we consider; (i) the inflaton decay; (ii) thermal scattering from the thermal bath [51], [52], [53], [54]; (iii) decay of superparticles in the thermal bath [55], [56].","In general, process (i) includes gravitino pair production.","The decay width of the mode, however, depends on the inflaton VEV [57], [58], [59], [60], [61], [62].","In our case, therefore, this process can be ignored since the inflaton does not have a VEV.","On the other hand, the inflaton can decay to gravitino and right-handed neutrino.","The decay width is given by $\\Gamma _{\\phi \\rightarrow \\psi _{\\mu }N_3}=\\frac{\\beta _f^{3}m_\\phi ^5}{48\\pi M_{pl}^2m_{3/2}^2}\\left[1-\\frac{(m_f+m_{3/2})^2}{m_\\phi ^2}\\right]\\,,$ where $\\beta _f^{2}=1-2(m_f^2+m_{3/2}^2)/m_\\phi ^2+(m_f^2-m_{3/2}^2)^2/m_\\phi ^4\\,.$ Here $m_f$ and $m_{3/2}$ are masses of $N_3$ and $\\psi _\\mu $ , respectively.","The mass difference between $\\phi $ and $N_3$ is expected to be given by the soft SUSY breaking mass scale for scalar superpartners, $|m_\\phi -m_f|\\sim \\tilde{m}$ .","Let us assume this decay happens by taking $m_\\phi =m_f+\\tilde{m}$ where $\\tilde{m}=k m_{3/2}$ ($k>1$ ).","In the limit $km_{3/2}/m_\\phi \\ll 1$ , we obtain $\\Gamma _{\\phi \\rightarrow \\psi _{\\mu }N_3}\\simeq \\frac{Cm_\\phi m_{3/2}^2}{3\\pi M^2_{pl}}\\,,$ where $C=(k-1)(k^2-1)^{3/2}$ .","Then the branching fraction of this mode is ${\\rm Br}_{\\phi \\rightarrow \\psi _{\\mu }N_3} =\\frac{\\Gamma _{\\phi \\rightarrow \\psi _{\\mu }N_3}}{\\Gamma _{\\phi }}\\simeq 4.5\\times 10^{-16}C\\left(\\frac{m_{3/2}}{10^{5}\\,{\\rm GeV}}\\right)^2\\left(\\frac{10^{-11}}{(y_\\nu y_\\nu ^\\dagger )_{33}}\\right)\\,.$ Therefore, it is sure that the inflaton decay reheats the universe in wide range of gravitino mass region.","On the other hand, the resultant gravitino abundance produced by the decay is estimated as $\\Omega _{3/2}^{\\rm inf}h^2\\sim 1.3\\times 10^{-6} C\\left(\\frac{m_{3/2}}{10^{5}\\,{\\rm GeV}}\\right)^3\\left(\\frac{10^{13}\\,{\\rm GeV}}{m_{\\phi }}\\right)^{1/2}\\left(\\frac{10^{-11}}{(y_\\nu y_\\nu ^\\dagger )_{33}}\\right)^{1/2}\\,,$ where $h$ is the scale factor of Hubble expansion rate.","Gravitino abundance via process (ii) is most effective at high temperature, thus it is proportional to $T_R$ , meanwhile in process (iii) gravitino is dominantly produced when the temperature is around the mass of decaying particle.","Adopting the expression given in Ref.","[63], the abundances via processes (ii) and (iii) are given by $&\\Omega _{3/2}^{\\rm TH}h^2\\sim 4\\left(\\frac{T_R}{10^{9}\\,{\\rm GeV}}\\right)\\left(\\frac{m_{3/2}}{10^{5}\\,{\\rm GeV}}\\right)\\,,\\\\&\\Omega _{3/2}^{\\rm FI}h^2\\sim 5\\times 10^{-4}k^3\\left(\\frac{m_{3/2}}{10^5\\,{\\rm GeV}}\\right)^2\\,.$ Here the contribution of the longitudinal mode of gravitino is suppressed in $\\Omega _{3/2}^{\\rm TH}$ by considering gluino is lighter than gravitino.","In $\\Omega _{3/2}^{\\rm FI}$ , we have assumed that all scalar leptons and quarks in the MSSM (whose mass scale is $\\tilde{m}$ ) are heavier than gravitino and that they are thermalized.","And as in the discussion of the inflaton decay, $\\tilde{m}=km_{3/2}$ has been taken.11We have checked that the contribution from $\\tilde{N}_1$ is negligible even if $\\tilde{N}_1$ is thermalized, i.e., in case (I).","Then the relic abundance of the LSP due to gravitino decay is estimated as $\\Omega _{\\rm LSP}^{\\rm nonth}h^2=\\frac{m_{\\rm LSP}}{m_{3/2}}\\Omega _{3/2}h^2\\,,$ where $m_{\\rm LSP}$ is the LSP mass and $\\Omega _{3/2}=\\Omega _{3/2}^{\\rm inf}+\\Omega _{3/2}^{\\rm TH}+\\Omega _{3/2}^{\\rm FI}$ is the total gravitino abundance.","$\\Omega _{\\rm LSP}^{\\rm nonth}h^2$ should not exceed the observed dark matter abundance $\\Omega _{\\rm DM}h^2\\simeq 0.12$ [35], which gives a constraint on gravitino mass.","Figure: Relic density of the LSP as function of gravitino mass.“inf” (dotted green), “TH” (dot-dashed red), and “FI”(dashed blue) are the contributions from process (i), (ii), and(iii), respectively.","Ω DM h 2 ≃0.12\\Omega _{\\rm DM}h^2\\simeq 0.12 (solid brown) isindicated as a reference.","Right region from vertical line (dottedviolet) indicates T 3/2 >m LSP T_{3/2}>m_{\\rm LSP}, and shaded region isexcluded.","m LSP =1 TeV m_{\\rm LSP}=1\\,{\\rm TeV}, m φ =10 13 GeV m_\\phi =10^{13}\\,{\\rm GeV},(y ν y ν † ) 33 =10 -11 (y_\\nu y_\\nu ^\\dagger )_{33}=10^{-11}, and k=m ˜/m 3/2 =2k=\\tilde{m}/m_{3/2}=2(left), 10 (right) are taken.Fig.","REF shows the resultant $\\Omega _{\\rm LSP}^{\\rm nonth}$ from the each contribution.","Here $m_{\\rm LSP}=1\\,{\\rm TeV}$ is taken by considering 1 TeV Higgsino or Wino dark matter with a mass of 2.7 – 3 TeV [64].","$m_\\phi =10^{13}\\,{\\rm GeV}$ , $(y_\\nu y_\\nu ^\\dagger )_{33}=10^{-11}$ , and $k=2$ (left), 10 (right) are taken to determine the contributions from processes (i) and (iii).","It is seen that in lower gravitino mass region, the dominant contribution to $\\Omega _{\\rm LSP}^{\\rm nonth}$ is from process (ii).","In order for the contribution not to exceed the dark matter abundance, $T_R\\lesssim 10^9\\,{\\rm GeV}$ is required, which is well-known result.","This gives a stringent constraint on the parameter space for successful leptogenesis shown in Figs.","REF and REF if gravitino is not heavy enough.","Namely, $\\tilde{m}_1\\sim 10^{-2}\\,{\\rm eV}$ is the allowed region in case (I) meanwhile $10^{-13}\\,{\\rm eV}\\lesssim \\tilde{m}_3\\lesssim 10^{-11}\\,{\\rm eV}$ is allowed in case (II).","On the other hand, the contributions from processes (i) and (iii) depend on the parameters, especially $k=\\tilde{m}/m_{3/2}$ (and $m_{3/2}$ ).","In order for $\\Omega _{\\rm LSP}^{\\rm nonth}$ not to exceed the dark matter abundance, the upper bound on gravitino mass is obtained depending on the mass spectrum of squarks and sleptons, e.g., $m_{3/2}\\lesssim 10^8$ ($10^{6}$ ) GeV for $\\tilde{m}\\sim m_{3/2}$ $(10m_{3/2})$ .","Such gravitino mass is preferred in minimal or mini split supersymmetry [65], [66], pure gravity mediation [67], [68], and spread supersymmetry [63], [69].","On the other hand, there is also an allowed region in higher gravitino mass region.","This is because gravitino decays before thermal freeze-out of the LSP in that region.","The allowed region can be estimated by imposing gravitino decay temperature $T_{3/2}$ larger than the LSP mass.","The gravitino decay temperature is defined by $T_{3/2}\\simeq (90/\\pi ^2g_*(T_{3/2}))^{1/4}\\sqrt{\\Gamma _{3/2} M_{pl}}\\,,$ where $\\Gamma _{3/2}$ is the decay rate of gravitino.","Then $m_{3/2}\\gtrsim 3\\times 10^{8}\\,{\\rm GeV}$ is obtained from $T_{3/2}>m_{\\rm LSP}$ for $m_{\\rm LSP}=1\\,{\\rm TeV}$ (see e.g., Ref. [70]).","Such high gravitino mass can be considered high-scale SUSY [71], intermediate scale supersymmetry [72], and unified inflation model [16].","However, gravitino cannot be too heavy because ${\\rm Br}_{\\phi \\rightarrow \\psi _{\\mu }N_3}$ should be less than unity for the reheating.","Taking ${\\rm Br}_{\\phi \\rightarrow \\psi _{\\mu }N_3} \\lesssim 0.1$ , for example, upper bound on gravitino mass is obtained as $m_{3/2}\\lesssim 5\\times 10^{11}\\,{\\rm GeV}$ ($2\\times 10^{10}\\,{\\rm GeV}$ ) for $k=2$ (10).","Another option is the R-parity violation.","Under the R-parity violation, the LSP decays to the standard-model particles.","Then, the LSP does not contribute to the matter abundance of the universe so that there is no constraint on $m_{3/2}$ ."],["Conclusion","Superconformal subcritical hybrid inflation is one of attractive inflation models that are consistent with the observed cosmological parameters by the Planck satellite.","In this paper we have studied the cosmology of an extended version of the model.","In the model three right-handed neutrinos are introduced.","The superpotential consists of one in the supersymmetric seesaw model and the interaction terms of the right-handed neutrinos with the additional matter fields, one of which plays the role of the waterfall field.","In the Kähler potential, on the other hand, it is possible for the sneutrinos to have shift symmetry by introducing explicit superconformal breaking terms of ${\\cal O}(1)$ .","Due to the shift symmetry, one of the sneutrinos becomes the inflaton field similarly in superconformal subcritical hybrid inflation.","Although the mass terms of the sneutrinos can affect the trajectory of the inflaton, it has turned out that the effect is restrictive and viable inflation is realized.","After inflation, the inflaton field decays to Higgses and sleptons or Higgsinos and leptons to reheat the universe.","Light neutrino masses are given by the seesaw mechanism.","However, the mass matrix is different from the conventional one.","It turns out that one of the neutrinos is massless.","Assuming that suppressed couplings of the other right-handed neutrinos to the waterfall field, it has been found that the neutrino Yukawa couplings that couples the inflaton to the MSSM sector are not constrained by the neutrino oscillation data.","Consequently, the reheating temperature is a free parameter, which is suitable for leptogenesis.","We have considered two representative cases; (I) the other right-handed (s)neutrinos are lighter than the inflaton; (II) the other right-handed (s)neutrinos are heavier than the inflaton.","In case (I), thermal leptogenesis is possible if the reheating temperature is larger than $\\sim 10^{9}\\,{\\rm GeV}$ .","It has been found leptogenesis is successful in a wide range of parameter space in the normal hierarchy case while the parameter space for leptogenesis is relatively limited in the inverted hierarchy case.","In case (II), on the other hand, sneutrino leptogenesis takes place if the reheating temperature is larger than $\\sim 10^{8}\\,{\\rm GeV}$ .","It has turned out that in both the normal and inverted hierarchy cases successful leptogenesis is realized in wide range of parameter space."],["Acknowledgments","We are grateful to Wilfried Buchmüller, Pasquale Di Bari, Oleg Lebedev, and Fuminobu Takahashi for valuable discussions.","This work was supported by JSPS KAKENHI Grant Numbers JP17H05402, JP17K14278, JP17H02875 and JP18H05542 (KI)."]],"string":"[\n [\n \"Leptogenesis after superconformal subcritical hybrid inflation\"\n ],\n [\n \"Abstract We consider an extended version of superconformal subcritical hybrid inflation model by introducing three right-handed neutrinos that have the Majorana mass terms.\",\n \"In the model one of the right-handed sneutrinos plays a role of the inflaton field, and it decays to reheat the universe after inflation.\",\n \"The vacuum expectation value for the waterfall field gives an unconventional pattern of the light neutrino mass matrix, and the neutrino Yukawa couplings that determine the reheating temperature are not constrained by the neutrino oscillation data.\",\n \"Consequently thermal leptogenesis or sneutrino leptogenesis is realized.\"\n ],\n [\n \"Introduction\",\n \"Inflation paradigm is strongly supported by the observations of the cosmic microwave background radiation (CMB).\",\n \"Slow-roll scalar field in the early universe is a promising candidate for inflation, and many types of inflation models have been proposed so far.\",\n \"In a theoretical point of view, it would be tempting to ask what the underlying physics or symmetry of the inflaton field is.\",\n \"Supersymmetry (SUSY) might be one of the answers.\",\n \"It protects the flatness of the inflaton direction, which is suitable for inflation.\",\n \"Recently supersymmetric D-term hybrid inflation has been revisited in various point of view.\",\n \"Under shift symmetric Kähler potential [1], subcritical hybrid inflation was found, where inflation continues for subcritical point value of the inflaton field [2], [3].\",\n \"On the other hand, it was shown in Refs.\",\n \"[4], [5] that Starobinsky model [6] emerges in the framework of superconformal supergravity [7], [8], [9], [10].\",\n \"It turned out in the following study that this framework has another new regime of inflation.\",\n \"It was shown that a general class of superconformal $\\\\alpha $ -attractor model [11], [12] appears in the subcritical regime of inflation, which we call superconformal subcritical hybrid inflation [13].\",\n \"In addition, the energy scale of inflation should coincide with the grand unification scale to be consistent with the Planck observation data, which is the feature found in the subcritical hybrid inflation [2], [3].\",\n \"Namely, the superconformal subcritical hybrid inflation has both features of the superconformal $\\\\alpha $ -attractor models and the subcritical hybrid inflation.\",\n \"The shift symmetry and superconformality are crucial for them.\",\n \"In this paper we will study the thermal history after the end of superconformal subcritical hybrid inflation.\",\n \"(See Refs.\",\n \"[14], [15] that study the phenomenology of Pati-Salam version of subcritical hybrid inflation.\",\n \"Recently Ref.\",\n \"[16] comprehensively studies the D-term hybrid inflation, including reheating, leptogenesis, and the SUSY breaking mechanism.)\",\n \"For the purpose, we introduce three right-handed neutrinos that interact with the minimal supersymmetric standard model (MSSM) sector.\",\n \"In fermionic sector, the mass matrix for the light neutrinos is given by the seesaw mechanism [17], but it has an unconventional structure.\",\n \"In bosonic sector, on the other hand, it will be shown that one of sneutrinos can play a role of the inflaton field.\",\n \"In addition, baryon asymmetry that is sufficient amount to explain the observed value is generated via leptogenesis [18].\",\n \"This paper is organized as follows.\",\n \"In the next section the model that we consider is described.\",\n \"Then Sec.\",\n \"shows the conditions required for the superconformal subcritical hybrid inflation in this model.\",\n \"Mass matrices of the heavy and light (s)neutrinos, including parametrization of the neutrino Yukawa couplings, are given in Sec.\",\n \", then we discuss the reheating and leptogenesis after inflation in Sec. .\",\n \"Sec.\",\n \"is dedicated to conclusion.\"\n ],\n [\n \"The model\",\n \"We consider a model described by the superpotential $W=W_{\\\\rm MSSM}+W_{\\\\rm neu}\\\\,,$ where $W_{\\\\rm MSSM}$ is the superpotential of the MSSM sector and $W_{\\\\rm neu}=\\\\frac{1}{2}M_{ij}N^c_iN^c_j+y_{\\\\nu \\\\,ij}N^c_i L_jH_u+\\\\lambda _i N_i^c S_+ S_-\\\\,.$ Here $N_i^c$ , $L_i=(\\\\nu _{Li},l_{Li})^T$ , and $H_u=(H_u^+,H_u^0)^T$ are the chiral superfields of the right-handed neutrinos, left-handed leptons, and up-type Higgs, respectively, and $L_jH_u=\\\\nu _{Lj}H_u^0-l_{Lj}H_u^+$ .\",\n \"$S_{\\\\pm }$ are the local U(1) fields with charge $\\\\pm q$ $(q>0)$ , one of which plays a role of the waterfall field.1We write the superpartners with tilde for the MSSM fields and right-handed neutrinos.\",\n \"For $S_\\\\pm $ , the same symbols are used for scalar fields while fermionic parts are expressed with tilde.\",\n \"In the current and the next sections, we adopt the unit in which the reduced Planck mass $M_{pl}\\\\simeq 2.4\\\\times 10^{18}\\\\,{\\\\rm GeV}$ is taken to be unity unless otherwise mentioned.\",\n \"Indices are summed over $i,j=1$ – 3.\",\n \"(We will use the same contraction in the following discussion unless otherwise mentioned.)\",\n \"$M_{ij}$ terms are explicit superconformal breaking terms that are added by phenomenological purpose.\",\n \"While we do not argue their origin, $M_{ij} \\\\ll 1$ are expected.\",\n \"The Kähler potential, on the other hand, is given by ${\\\\cal K}=-3\\\\log \\\\Omega ^{-2}\\\\,,$ where $\\\\Omega ^{-2}=1-\\\\frac{1}{3}\\\\left(\\\\sum _{\\\\rm MSSM} |I_{\\\\rm MSSM}|^2+\\\\sum _i |N_i^c|^2+|S_+|^2+|S_-|^2\\\\right)-\\\\sum _i\\\\frac{\\\\chi _i}{6}\\\\left(N_i^{c\\\\,2}+\\\\bar{N}_i^{c\\\\,2}\\\\right)\\\\,.$ Here $I_{\\\\rm MSSM}$ are chiral superfields in the MSSM sector.\",\n \"The last term is superconformal breaking term that is considered in Refs.\",\n \"[4], [10].\",\n \"With the superpotential and Kähler potential, the scalar potential is given by $V_{\\\\rm tot}=V_F+V_D,$ where $V_F$ and $V_D$ are F- and D-terms, respectively, and given by [4] $&V_F=\\\\Omega ^4\\\\left[\\\\delta ^{\\\\alpha \\\\bar{\\\\alpha }}W_\\\\alpha W_{\\\\bar{\\\\alpha }}+\\\\frac{1}{\\\\Delta }|\\\\delta ^{\\\\alpha \\\\bar{\\\\alpha }}W_\\\\alpha \\\\Phi _{\\\\bar{\\\\alpha }}-3W|^2\\\\right]\\\\,,\\\\\\\\&V_D=\\\\frac{1}{2}g^2\\\\left[q\\\\Omega ^2(|S_+|^2-|S_-|^2)-\\\\xi \\\\right]^2\\\\,.$ Here $\\\\Phi \\\\equiv -3\\\\Omega ^{-2}$ and $\\\\Delta \\\\equiv \\\\Phi -\\\\delta ^{\\\\alpha \\\\bar{\\\\alpha }}\\\\Phi _\\\\alpha \\\\Phi _{\\\\bar{\\\\alpha }}$ have been additionally introduced.\",\n \"Subscript in $W$ and $\\\\Phi $ stands for the field derivative, e.g., $W_\\\\alpha \\\\equiv \\\\partial W/\\\\partial z^\\\\alpha $ where $z^\\\\alpha $ is a chiral superfield.\",\n \"In the D-term, we have introduced the Fayet-Iliopoulos (FI) term $\\\\xi \\\\,(>0)$ associated with the U(1).2The origin of the FI term in canonical superconformal supergravity model [10] is discussed in Ref. [4].\",\n \"See also Ref.\",\n \"[16] for recent development.\",\n \"Due to the FI term, $S_+$ has a vacuum expectation value (VEV) at the global minimum, which is obtained as $\\\\langle S_+\\\\rangle =\\\\sqrt{\\\\xi /q(1+\\\\tilde{\\\\xi })}$ with $\\\\tilde{\\\\xi }\\\\equiv \\\\xi /3q$ .\",\n \"As in Ref.\",\n \"[13], we take $\\\\chi _i \\\\le 0$ without loss of generality.\",\n \"In the present model, we impose the following condition: $\\\\chi _3\\\\simeq -1,~~ \\\\chi _{1},\\\\,\\\\chi _2\\\\simeq 0\\\\,.$ This distinguishes $N^c_3$ from $N^c_{1}$ and $N^c_{2}$ .\",\n \"$\\\\phi \\\\equiv \\\\sqrt{2}{\\\\rm Re}\\\\,\\\\tilde{N}^c_3 $ has an approximate shift symmetry that is explicitly broken by $\\\\lambda _3\\\\,(\\\\ll 1)$ .\",\n \"Then, $\\\\phi $ is expected to be the inflaton as studied in Ref. [13].\",\n \"In the inflation model, $ s \\\\equiv \\\\sqrt{2}|S_+|$ plays a role of the waterfall field.\",\n \"$N^c_1$ and $N^c_2$ , on the other hand, have no such symmetry.\",\n \"Instead there is a freedom to choose any basis for $N^c_1$ and $N^c_2$ with a redefinition of $M_{ij}$ and $\\\\lambda _i$ due to $\\\\chi _{1,2}\\\\simeq 0$ .\",\n \"For simple notation, hereafter we omit subscripts of $\\\\chi _3$ and $\\\\lambda _3$ and introduce $m_\\\\phi $ , which will be identified as the inflaton mass in Sec.\",\n \"REF (with another assumption in Sec.\",\n \"REF ), $\\\\chi \\\\equiv \\\\chi _3, &~~ \\\\lambda \\\\equiv \\\\lambda _3\\\\,, \\\\\\\\m_\\\\phi \\\\equiv &\\\\, \\\\lambda \\\\langle S_+ \\\\rangle \\\\,.$ It is sometimes convenient to use $\\\\delta \\\\chi $ ($0<\\\\delta \\\\chi <1$ ) defined by $\\\\delta \\\\chi /(1+\\\\chi ) = -qg^2\\\\xi /3\\\\lambda ^2$ .\",\n \"$0<\\\\delta \\\\chi <1$ guarantees $\\\\Omega (\\\\phi ,s)^2>0$ and $\\\\phi _{c,0}^2>0$ , which will be defined in Eqs.\",\n \"(REF ) and (REF ), respectively.\",\n \"Then inflation that is consistent with the observations of the CMB is realized in the parameter space [13], $\\\\lambda \\\\simeq (0.5\\\\, {\\\\rm }{\\\\rm }\\\\,1)\\\\times 10^{-3}\\\\,, &~~\\\\xi ^{1/2}\\\\simeq (3\\\\, {\\\\rm }{\\\\rm }\\\\,1)\\\\times 10^{16}\\\\,{\\\\rm GeV}\\\\,,\\\\nonumber \\\\\\\\m_{\\\\phi } \\\\simeq (1\\\\,{\\\\rm }{\\\\rm }\\\\,2)\\\\times &10^{13}\\\\,{\\\\rm GeV}\\\\,,$ for $q=g=1$ , $\\\\delta \\\\chi =0.9$ and the number of $e$ -folds $N_e=55$ – 60, which we take in the later numerical study.3We will estimate the number of $e$ -folds in Sec.\",\n \"REF to confirm this.\",\n \"In the parameter space, e.g., $\\\\chi \\\\simeq -1.16$ for $\\\\lambda =10^{-3}$ , $\\\\xi ^{1/2}=10^{16}\\\\,{\\\\rm GeV}$ , and $\\\\delta \\\\chi =0.9$ .\",\n \"The mass terms in the superpotential, however, have a possibility to alter the inflationary path.\",\n \"In the next section, we will derive the conditions in order not to affect the inflationary dynamics.\"\n ],\n [\n \"Inflation\",\n \"We define several variables that are used in the following analysis.\",\n \"During inflation, the other fields except for the inflaton and waterfall field are irrelevant.\",\n \"Thus it is convenient to define following potentials, $& V(\\\\phi ,s)\\\\equiv V_{\\\\rm tot}|_{\\\\sqrt{2}{\\\\rm Re}\\\\,\\\\tilde{N}_3^c=\\\\phi ,\\\\,\\\\sqrt{2}|S_+|=s,\\\\,{\\\\rm the\\\\,others}=0}\\\\,,\\\\\\\\& \\\\Omega (\\\\phi ,s)\\\\equiv \\\\Omega |_{\\\\sqrt{2}{\\\\rm Re}\\\\,\\\\tilde{N}_3^c=\\\\phi ,\\\\,\\\\sqrt{2}|S_+|=s,\\\\,{\\\\rm the\\\\,others}=0}\\\\,.$ Then the critical point value $\\\\phi _c$ is defined as a field value below which the waterfall field becomes tachyonic.\",\n \"It receives ${\\\\cal O}(M_{ij}^2)$ corrections as $\\\\phi _c^2=\\\\phi _{c,0}^2\\\\left[1-\\\\frac{2\\\\Delta M^2(\\\\phi _{c,0})}{3\\\\lambda ^2}\\\\right]+{\\\\cal O}(M_{i3}^4)\\\\,,$ where $\\\\phi _{c,0}^2&=\\\\frac{6qg^2\\\\xi }{3\\\\lambda ^2+(1+\\\\chi )qg^2\\\\xi }\\\\,,\\\\\\\\\\\\Delta M^2(\\\\phi ) &= \\\\sum _{i=1}^3|M_{i3}|^2-\\\\frac{(1-2\\\\chi )^2}{24}\\\\frac{\\\\phi ^2}{1+\\\\phi ^2\\\\chi (1+\\\\chi )/6}|M_{33}|^2\\\\,.$ For example, $\\\\phi _{c,0}\\\\simeq 18$ or in terms of canonically-normalized field $\\\\hat{\\\\phi }_{c,0}\\\\simeq 11$ defined in Eq.\",\n \"(REF ) for $\\\\lambda =10^{-3}$ , $\\\\xi ^{1/2}=10^{16}\\\\,{\\\\rm GeV}$ and $\\\\delta \\\\chi =0.9$ .\",\n \"This perturbative expansion is valid when4We have checked that ${\\\\cal O}(M^4_{i3})$ term is irrelevant when Eq.\",\n \"() is satisfied, thus we ignore it in the following discussion.\",\n \"$|M_{13}|^2+|M_{23}|^2 &\\\\ll 3\\\\lambda ^2/2\\\\sim (10^{15}\\\\,{\\\\rm GeV})^2\\\\,, \\\\\\\\|M_{33}|^2 &\\\\ll 2\\\\lambda ^4/qg^2\\\\xi \\\\sim (10^{14}\\\\,{\\\\rm GeV})^2\\\\,.$ It will be checked in this section that the above conditions are satisfied in this inflation model.\",\n \"Finally it is useful to define $\\\\Psi \\\\equiv \\\\frac{\\\\Omega (\\\\phi ,0)\\\\phi }{\\\\Omega (\\\\phi _{c,0},0)\\\\phi _{c,0}}=\\\\frac{\\\\Omega (\\\\phi ,0)\\\\phi }{\\\\sqrt{2qg^2\\\\xi /\\\\lambda ^2}}\\\\,,$ when potential is expressed in terms of canonically-normalized inflaton field.\"\n ],\n [\n \"Pre-critical regime\",\n \"Let us begin with the regime where the inflaton is approaching down to the critical point value.\",\n \"Since the waterfall field is stabilized at the origin in this regime, the relevant Lagrangian is given as ${\\\\cal L}_{\\\\rm pre}=\\\\frac{f(\\\\phi ,0)}{2}(\\\\partial _\\\\mu \\\\phi )^2-V_{\\\\rm pre}(\\\\phi )\\\\,,$ where5It is noted that the term proportional to $\\\\Delta M^2$ in Eq.\",\n \"(REF ) is equivalent to $\\\\Omega ^4(\\\\phi ,0)\\\\phi ^2\\\\Delta M^2(\\\\phi )/2$ .\",\n \"$f(\\\\phi ,s)&=\\\\Omega ^2(\\\\phi ,s)\\\\left[1+\\\\Omega ^2(\\\\phi ,s)\\\\frac{(1+\\\\chi )^2}{6}\\\\phi ^2\\\\right]\\\\,, \\\\\\\\V_{\\\\rm pre}(\\\\phi )&=\\\\frac{1}{2}g^2\\\\xi ^2\\\\left[1+2\\\\Psi ^2 \\\\frac{\\\\Omega ^2(\\\\phi ,0)\\\\Delta M^2(\\\\phi )}{\\\\lambda ^2\\\\xi /q}\\\\right]\\\\,.$ It is seen that $\\\\Delta M^2$ term gives a gradient to the inflaton field, which should not invade the slow-roll conditions.\",\n \"To see the impact of $\\\\Delta M^2$ term, it is instructive to change dynamical variable $\\\\phi $ to canonically-normalized field $\\\\hat{\\\\phi }$ .\",\n \"Since $\\\\chi \\\\simeq -1$ , it is a good approximation that $f(\\\\phi ,0)\\\\simeq \\\\Omega ^2(\\\\phi ,0)$ .\",\n \"Then $d\\\\phi /d\\\\hat{\\\\phi }=f(\\\\phi )^{-1/2}\\\\simeq \\\\Omega ^{-1}(\\\\phi ,0)$ can be solved easily to obtain, $&\\\\phi \\\\simeq \\\\beta ^{-1/2}\\\\sinh \\\\beta ^{1/2}\\\\hat{\\\\phi }\\\\,, \\\\\\\\&\\\\Psi \\\\simeq \\\\delta \\\\chi ^{-1/2}\\\\tanh \\\\beta ^{1/2}\\\\hat{\\\\phi }\\\\,, \\\\\\\\&\\\\Omega ^{-1}(\\\\phi ,0) \\\\simeq \\\\cosh \\\\beta ^{1/2}\\\\hat{\\\\phi }\\\\,,$ where $\\\\beta =-(1+\\\\chi )/6=\\\\lambda ^2\\\\delta \\\\chi /2qg^2\\\\xi $ .\",\n \"Then the potential in terms of $\\\\hat{\\\\phi }$ is given as $V_{\\\\rm pre} &\\\\simeq \\\\frac{1}{2}g^2\\\\xi ^2\\\\biggr [1+\\\\frac{\\\\tanh ^2\\\\beta ^{1/2}\\\\hat{\\\\phi }}{\\\\cosh ^2\\\\beta ^{1/2}\\\\hat{\\\\phi } }\\\\frac{2q}{\\\\lambda ^2\\\\xi \\\\delta \\\\chi }\\\\biggr \\\\lbrace \\\\sum _{i=1}^3|M_{i3}|^2- \\\\frac{3}{8\\\\beta }|M_{33}|^2\\\\tanh ^{2}\\\\beta ^{1/2}\\\\hat{\\\\phi }\\\\biggl \\\\rbrace \\\\biggl ]\\\\,.$ On the other hand, it was shown in Ref.\",\n \"[4] that there is one-loop corrections to the tree-level potential.\",\n \"In terms of the canonically-normalized field, it is given by $V_{1l}\\\\simeq \\\\frac{1}{2}g^2\\\\xi ^2\\\\times \\\\frac{q^2g^2}{8\\\\pi ^2}\\\\log \\\\left[\\\\delta \\\\chi ^{-1}\\\\tanh ^2\\\\beta ^{1/2}\\\\hat{\\\\phi }\\\\right]\\\\,.$ Therefore, in order not to affect the inflationary trajectory, it is sufficient that the terms proportional to $|M_{i3}|^2$ are subdominant compared to the one-loop potential.\",\n \"In the parameter space given in Eq.\",\n \"(REF ), $\\\\sqrt{\\\\beta }\\\\hat{\\\\phi _c}\\\\simeq \\\\mathrm {arcsinh} \\\\sqrt{\\\\beta }\\\\phi _c \\\\sim {\\\\cal O}(1)$ .\",\n \"Then, the conditions are given as $|M_{13}|^2+|M_{23}|^2 &\\\\lesssim \\\\frac{qg^2\\\\delta \\\\chi \\\\lambda ^2\\\\xi }{16\\\\pi ^2}\\\\lesssim (1\\\\times 10^{12}\\\\,{\\\\rm GeV})^2\\\\,,\\\\\\\\|M_{33}|^2 &\\\\lesssim \\\\frac{\\\\delta \\\\chi ^2\\\\lambda ^4}{12\\\\pi ^2}\\\\lesssim (2\\\\times 10^{11}\\\\,{\\\\rm GeV})^2\\\\,.$ It is easy to check that the slow-roll conditions are satisfied under the constraints.\",\n \"Since the constraints are more stringent than Eqs.\",\n \"(REF ) and (), it has been confirmed that the perturbative expansion to obtain Eq.\",\n \"(REF ) is valid.\"\n ],\n [\n \"Subcritical regime\",\n \"In the previous subsection, we have seen that the slow-roll conditions are satisfied before reaching to the critical point value.\",\n \"After the inflaton field becomes subcritical point value, the tachyonic growth of the waterfall field occurs.\",\n \"It is expected that the inflation continues in the subcritical regime when $M_{i3}\\\\rightarrow 0$ .\",\n \"In this subsection, we will derive the conditions under which the inflaton and waterfall field dynamics are not affected with non-zero $M_{i3}$ .\",\n \"As seen in the previous section, the perturbative expression for $\\\\phi _c$ is valid under the conditions given in Eqs.\",\n \"(REF ) and ().\",\n \"Then, the tachyonic growth of the waterfall field is not affected by the additional gradient in the inflaton direction due to $|M_{i3}|^2$ terms since $\\\\phi _c\\\\simeq \\\\phi _{c,0}$ .\",\n \"Consequently, the dynamics of the waterfall field around the critical point is the same as one discussed in Ref. [13].\",\n \"The tachyonic growth is qualitatively the same as the subcritical hybrid inflation [2], [3].\",\n \"Namely, due to the tachyonic growth, the waterfall field relaxes to the local minimum value $s_{\\\\rm min}$ just after a few Hubble-unit time.\",\n \"(See, for example, Figure 1. of Ref. [2].)\",\n \"In the present model, $s_{\\\\rm min}$ is found to be $s_{\\\\rm min}^2&=\\\\frac{2\\\\xi }{q(1+\\\\tilde{\\\\xi })}\\\\frac{\\\\Omega ^{-2}(\\\\phi ,0)}{1+\\\\tilde{\\\\xi }\\\\Psi ^2/(1+\\\\tilde{\\\\xi })}\\\\left[1-\\\\Psi ^2\\\\left\\\\lbrace 1+\\\\frac{2\\\\Omega ^2(\\\\phi ,0)\\\\Delta M^2(\\\\phi )}{3\\\\lambda ^2} \\\\right\\\\rbrace \\\\right]\\\\,.$ Then the potential in the subcritical regime of the inflaton field is effectively given by $V(\\\\phi ,s_{\\\\rm min})$ and the dynamics reduces to single field inflation that is described by the Lagrangian, ${\\\\cal L}=\\\\frac{f(\\\\phi ,s_{\\\\rm min})}{2}(\\\\partial _\\\\mu \\\\phi )^2- V_{\\\\rm inf}(\\\\phi )\\\\,,$ where $V_{\\\\rm inf}(\\\\phi )&=g^2\\\\xi ^2\\\\frac{(1+\\\\tilde{\\\\xi })\\\\Psi ^2}{1+2\\\\tilde{\\\\xi }\\\\Psi ^2}\\\\biggl [1-\\\\frac{\\\\Psi ^2}{2(1+\\\\tilde{\\\\xi })}+\\\\frac{\\\\lbrace 1+\\\\tilde{\\\\xi }\\\\Psi ^2/(1+\\\\tilde{\\\\xi })\\\\rbrace ^2}{(1+2\\\\tilde{\\\\xi }\\\\Psi ^2)/(1+\\\\tilde{\\\\xi })}\\\\frac{\\\\Omega ^2(\\\\phi ,0)\\\\Delta M^2(\\\\phi )}{\\\\lambda ^2\\\\xi /q}\\\\biggr ]\\\\nonumber \\\\\\\\ &+{\\\\cal O}(M_{i3}^4) \\\\,.$ Recall that $\\\\tilde{\\\\xi }\\\\,,\\\\xi \\\\ll 1$ in the parameters in Eq.\",\n \"(REF ).\",\n \"Then it is clear that the additional term proportional to $\\\\Delta M^2$ is the same as in Eq.\",\n \"(REF ).\",\n \"Note that $s_{\\\\rm min}$ is negligible in $\\\\Omega (\\\\phi ,s_{\\\\rm min})$ .\",\n \"Then, the canonically-normalized inflaton field is given by Eqs.\",\n \"(REF )–().\",\n \"Consequently, $V_{\\\\rm inf}$ is given by $V_{\\\\rm inf} &\\\\simeq g^2\\\\xi ^2 \\\\delta \\\\chi ^{-1} \\\\tanh ^2\\\\beta ^{1/2} \\\\hat{\\\\phi }\\\\biggr [1-\\\\frac{\\\\delta \\\\chi ^{-1}}{2} \\\\tanh ^2\\\\beta ^{1/2}\\\\hat{\\\\phi } \\\\nonumber \\\\\\\\&+\\\\frac{\\\\cosh ^{-2}\\\\beta ^{1/2}\\\\hat{\\\\phi }}{\\\\lambda ^2\\\\xi /q}\\\\biggr \\\\lbrace \\\\sum _{i=1}^3|M_{i3}|^2-\\\\frac{3}{8\\\\beta }|M_{33}|^2\\\\tanh ^{2}\\\\beta ^{1/2}\\\\hat{\\\\phi }\\\\biggl \\\\rbrace \\\\biggl ]\\\\,.$ Therefore, if Eqs.\",\n \"(REF ) and () are satisfied, then the dynamics in the subcritical regime reduces to one in Ref. [13].\",\n \"In addition, it is clear that $m_\\\\phi $ defined in Eq.\",\n \"(REF ) is indeed the inflaton mass since $V_{\\\\rm inf}\\\\simeq \\\\frac{1}{2}m_\\\\phi ^2 \\\\phi ^2$ around $\\\\phi \\\\,(\\\\simeq \\\\hat{\\\\phi })\\\\sim 0$ .\",\n \"To summarize the present and previous subsections, the inflaton-waterfall field dynamics is unchanged when $|M_{i3}|^2=|M_{3i}|^2 \\\\lesssim (2\\\\times 10^{11}\\\\,{\\\\rm GeV})^2\\\\,,$ for $i=1$ –3 are satisfied.\"\n ],\n [\n \"Stability of inflationary trajectory\",\n \"It was pointed out in Ref.\",\n \"[19] that $\\\\tilde{L}_iH_u$ may become tachyonic in sneutrino inflation.\",\n \"In order to find out the stability condition, let us derive the mass matrix in $\\\\tilde{L}_i$ and $H_u$ basis.\",\n \"From $V_{\\\\rm tot}$ , it is obtained by ${\\\\cal L}_{\\\\rm mass}^{\\\\tilde{L}H_u}=&-\\\\frac{\\\\Omega (\\\\phi ,s)^4 \\\\phi ^2}{2} |y_{\\\\nu 3i} \\\\tilde{L}_i|^2-\\\\frac{\\\\Omega (\\\\phi ,s)^4 \\\\phi ^2}{2} |y_{\\\\nu 3i}|^2|H_u|^2\\\\nonumber \\\\\\\\&+\\\\left[\\\\frac{M_{33}^* \\\\Omega (\\\\phi ,s)^6\\\\phi ^3}{4\\\\sqrt{2}}y_{\\\\nu 3i} \\\\tilde{L}_iH_u+{\\\\rm h.c.}\\\\right]\\\\,,$ On the other hand, the kinetic terms of $\\\\tilde{L}_i$ and $H_u$ are given by ${\\\\cal L}_{\\\\rm kin}^{\\\\tilde{L}H_u}=\\\\Omega (\\\\phi ,s)^2\\\\left[|\\\\partial _\\\\mu \\\\tilde{L}_i|^2+|\\\\partial _\\\\mu H_u|^2\\\\right]\\\\,.$ Therefore, using canonically-normalized fields, $\\\\hat{\\\\tilde{L}}_i\\\\equiv \\\\Omega (\\\\phi ,s)\\\\tilde{L}_i$ and $\\\\hat{H}_u\\\\equiv \\\\Omega (\\\\phi ,s)H_u$ , the mass terms are rewritten as ${\\\\cal L}_{\\\\rm mass}^{\\\\tilde{L}H_u}=&-\\\\frac{\\\\Omega (\\\\phi ,s)^2 \\\\phi ^2}{2} |y_{\\\\nu 3i} \\\\hat{\\\\tilde{L}}_i|^2-\\\\frac{\\\\Omega (\\\\phi ,s)^2 \\\\phi ^2}{2} |y_{\\\\nu 3i}|^2|\\\\hat{H}_u|^2\\\\nonumber \\\\\\\\&+\\\\left[\\\\frac{M_{33}^* \\\\Omega (\\\\phi ,s)^4\\\\phi ^3}{4\\\\sqrt{2}}y_{\\\\nu 3i} \\\\hat{\\\\tilde{L}}_i\\\\hat{H}_u+{\\\\rm h.c.}\\\\right]\\\\nonumber \\\\\\\\=&-\\\\frac{\\\\Omega (\\\\phi ,s)^2 \\\\phi ^2}{2}(y_\\\\nu y_\\\\nu ^\\\\dagger )_{33}|\\\\hat{\\\\tilde{L}}_3^\\\\prime |^2-\\\\frac{\\\\Omega (\\\\phi ,s)^2 \\\\phi ^2}{2}(y_\\\\nu y_\\\\nu ^\\\\dagger )_{33} |\\\\hat{H}_u|^2\\\\nonumber \\\\\\\\&+\\\\left[\\\\frac{M_{33}^* \\\\Omega (\\\\phi ,s)^4\\\\phi ^3}{4\\\\sqrt{2}}(y_\\\\nu y_\\\\nu ^\\\\dagger )_{33}^{1/2}\\\\hat{\\\\tilde{L}}_3^\\\\prime \\\\hat{H}_u+{\\\\rm h.c.}\\\\right]\\\\,,$ where we have defined $(y_\\\\nu y_\\\\nu ^\\\\dagger )_{33}^{1/2}\\\\hat{\\\\tilde{L}}_3^\\\\prime \\\\equiv y_{\\\\nu 3i}\\\\hat{\\\\tilde{L}}_i$ in the second line following Ref. [19].\",\n \"Then, the stability condition is given by $|M_{33}|<2\\\\sqrt{2}(y_\\\\nu y_\\\\nu ^\\\\dagger )_{33}^{1/2}\\\\frac{\\\\Omega (\\\\phi ,s)^{-2}}{\\\\phi }\\\\,.$ Using Eqs.\",\n \"(REF ) and (), it turns out that $|M_{33}|&<4\\\\sqrt{2}(y_\\\\nu y_\\\\nu ^\\\\dagger )_{33}^{1/2}\\\\sqrt{\\\\beta }\\\\nonumber \\\\\\\\&\\\\simeq 1.4 \\\\times 10^{14}\\\\,{\\\\rm GeV}\\\\left(\\\\frac{(y_\\\\nu y_\\\\nu ^\\\\dagger )_{33}}{10^{-7}}\\\\frac{\\\\beta }{10^{-3}}\\\\right)^{1/2}\\\\,.$ This upper bound is weaker than (REF ) in most of the parameter space, which will be seen later.\"\n ],\n [\n \"Neutrino mass\",\n \"In this section, we derive the mass matrices for the heavy and light neutrinos.\",\n \"Around the global minimum, $\\\\Omega \\\\simeq 1$ since $\\\\xi \\\\ll 1$ .\",\n \"Consequently, all the fields are canonical.\",\n \"Thus, the mass terms are derived similarly in global SUSY model.\"\n ],\n [\n \"Mass matrix\",\n \"The superpotential (REF ) gives the Majorana masses for the light neutrinos.\",\n \"To see how the masses are generated, we write down the mass terms for fermionic part of $N_i^c$ , $\\\\nu _{Li}$ and $\\\\tilde{S}_-$ , ${\\\\cal L}_{\\\\nu }^{\\\\rm mass}=-\\\\frac{1}{2}(\\\\bar{\\\\psi }{\\\\cal M}P_L\\\\psi +{\\\\rm h.c.})\\\\,,$ where $\\\\psi &=(N^c_1,N^c_2,N^c_3,\\\\tilde{S}_-,\\\\nu _{L1},\\\\nu _{L2},\\\\nu _{L3})^T\\\\,,\\\\\\\\{\\\\cal M}&=\\\\left(\\\\begin{array}{cc}\\\\tilde{M} & \\\\tilde{m}_\\\\nu \\\\\\\\\\\\tilde{m}_\\\\nu ^T & {\\\\bf 0}\\\\end{array}\\\\right)\\\\,.$ $\\\\tilde{M}$ and $\\\\tilde{m}_\\\\nu $ are $4\\\\times 4$ and $4\\\\times 3$ matrices, respectively, and given by $\\\\tilde{M}&=\\\\left(\\\\begin{array}{cccc}&&&\\\\lambda _1\\\\langle S_+\\\\rangle \\\\\\\\&\\\\mbox{\\\\smash{\\\\Large M}}&&\\\\lambda _2\\\\langle S_+\\\\rangle \\\\\\\\&&& m_\\\\phi \\\\\\\\\\\\lambda _1\\\\langle S_+\\\\rangle &\\\\lambda _2\\\\langle S_+\\\\rangle &m_\\\\phi & 0\\\\end{array}\\\\right)\\\\,, \\\\\\\\\\\\tilde{m}_\\\\nu &=\\\\left(\\\\begin{array}{ccc}&& \\\\\\\\&\\\\mbox{\\\\smash{\\\\Large m_\\\\nu }}&\\\\\\\\&& \\\\\\\\0&0&0\\\\end{array}\\\\right)\\\\,.$ Here $m_{\\\\nu \\\\,ij}=y_{\\\\nu \\\\,ij}\\\\langle H_u^0\\\\rangle $ with $\\\\langle H_u^0\\\\rangle $ being the VEV of the up-type neutral Higgs.\",\n \"Then mass matrix $M_{\\\\nu }$ for the light neutrinos are obtained by the seesaw mechanism [17], $M_{\\\\nu } =-\\\\tilde{m}_\\\\nu ^T \\\\tilde{M}^{-1}\\\\tilde{m}_\\\\nu \\\\,.$ An important consequence of the mass matrix is that the one of three light neutrinos is massless.\",\n \"This is because the rank of $M_\\\\nu $ is two.\",\n \"Using this mass matrix, it is possible to constrain the parameters by the observed neutrino masses.\",\n \"In the later discussion we assume $\\\\lambda _1\\\\,,~\\\\lambda _2\\\\,\\\\ll \\\\lambda \\\\,.$ Here, recall that there is a freedom to choose a basis for $N^c_1$ and $N^c_2$ .\",\n \"Then, $M_{12}$ can be rotated away.\",\n \"As a result, $M_{\\\\nu }$ is given in the following simple expression, $M_{\\\\nu \\\\, ij} = -\\\\langle H^0_u\\\\rangle ^2\\\\sum _{k=1}^2\\\\frac{y_{\\\\nu ki}y_{\\\\nu kj}}{M_{kk}}+{\\\\cal O}\\\\left(\\\\frac{\\\\lambda _{1,2}}{\\\\lambda }\\\\frac{m_{\\\\nu \\\\, il}m_{\\\\nu \\\\, jm}}{M_{kk}}\\\\right)\\\\,.$ Here in the second term of right-hand side, $k=1$ or 2 and $i$ , $j$ , $l$ , $m$ can be 1–3, and we have ignored $M_{i3}$ based on the discussion of the previous section.\",\n \"The leading order term, on the other hand, is independent of $\\\\lambda _{1,2}$ , $M_{i3}$ , $m_\\\\phi $ and $y_{\\\\nu 3i}$ ($i=1$ –3).\",\n \"Therefore, they are not constrained by the neutrino oscillation data.\",\n \"This fact is important in the estimation of the reheating temperature, which we will see later.\",\n \"To ensure the non-zero $\\\\lambda _{1,2}$ does not affect our later analysis at percent level, we implicitly assume $\\\\lambda _{1,2}<0.01\\\\lambda $ .\",\n \"Before further discussing the light neutrino mass matrix, let us note that the mass matrix $\\\\tilde{M}$ corresponds to the mass matrix in the superpotential around the global minimum, $W_{\\\\rm neu}=\\\\frac{1}{2} (N^c_1,N^c_2,N^c_3,S_-)\\\\tilde{M}\\\\left(\\\\begin{array}{c}N^c_1 \\\\\\\\N^c_2 \\\\\\\\N^c_3 \\\\\\\\S_-\\\\end{array}\\\\right)+\\\\cdots \\\\,.$ Recall that we have the requirement (REF ) for successful inflation.\",\n \"Therefore, $\\\\tilde{M}$ should be almost block-diagonal as $\\\\tilde{M}&\\\\simeq \\\\left(\\\\begin{array}{ccc:c}M_1&&&0 \\\\\\\\&M_2&&0\\\\\\\\&& M_3& m_\\\\phi \\\\\\\\0 &0&m_\\\\phi & 0\\\\end{array}\\\\right)\\\\,,$ where $M_i>0$ .\",\n \"Here we have left $M_3$ for later discussion.\",\n \"In the following analysis we use Eq.\",\n \"(REF ) for $\\\\tilde{M}$ .\",\n \"$\\\\tilde{M}$ is further diagonalized by a unitary matrix $U_{\\\\tilde{M}}$ as, $D_{\\\\tilde{M}}\\\\simeq {\\\\rm diag}(M_1,M_2,m_\\\\phi , m_\\\\phi )\\\\simeq U^T_{\\\\tilde{M}} \\\\tilde{M} U_{\\\\tilde{M}}\\\\,,$ where $U_{\\\\tilde{M}}=\\\\left(\\\\begin{array}{cc:c}1 & & \\\\\\\\& 1 & \\\\\\\\& &{\\\\scriptsize \\\\frac{1}{\\\\sqrt{2}}\\\\bigl (\\\\begin{array}{cc}1 & i\\\\\\\\1 & -i\\\\end{array}\\\\bigr )}\\\\end{array}\\\\right)\\\\,.$ Here we have omitted ${\\\\cal O}(M_3/m_\\\\phi )$ corrections since they are irrelevant in the later analysis.\",\n \"Non-zero $\\\\lambda _{1,2}$ corrections enter as ${\\\\cal O}(\\\\lambda _{1,2}/\\\\lambda )$ and ${\\\\cal O}((m_\\\\phi /M_{1,2})\\\\lambda _{1,2}/\\\\lambda )$ for $m_\\\\phi \\\\gtrsim M_{1,2}$ and $m_\\\\phi \\\\lesssim M_{1,2}$ , respectively.\",\n \"The corrections due to non-zero $\\\\lambda _{1,2}$ are similar in the scalar sector since the mass matrix in $(\\\\tilde{N}_1,\\\\tilde{N}_2,\\\\tilde{N}_3,S_-)$ basis is given by, $M_{\\\\rm scalar}^2=\\\\tilde{M}^\\\\dagger \\\\tilde{M}\\\\,.$ Therefore the assumption (REF ) assures that the dynamics of the inflaton and the waterfall field discussed in the previous section, as well as subsequent reheating and leptogenesis, is not affected.\",\n \"If it is not satisfied, then the inflationary trajectory would be altered so that we need further analysis to find the paramter space that is consistent with the CMB observations.\",\n \"Accordingly, the subsequent reheating and leptogenesis scenario change.\",\n \"We leave the detailed analysis as an interesting future work.\"\n ],\n [\n \"Parametrization of neutrino Yukawa couplings\",\n \"Now let us discuss $M_\\\\nu $ .\",\n \"It can be diagonalized by a unitary matrix $U_\\\\nu $ as $D_{M_\\\\nu }={\\\\rm diag}(m_1,m_2,m_3)=U_\\\\nu ^T M_\\\\nu U_\\\\nu \\\\,.$ As noted above, one of the three light neutrino masses is zero.\",\n \"We follow the standard convention that $m_3>m_2>m_1(=0)$ for the normal hierarchy (NH) case and $m_2>m_1>m_3(=0)$ for the inverted hierarchy (IH) case and use the values given in Ref.\",\n \"[20], which are listed in Table REF .\",\n \"Before discussing the parametrization of the neutrino Yukawa couplings, it is instructive to count the number of parameters.\",\n \"The situation is the same as one discussed Refs.\",\n \"[19], [21] since the mass matrix of the light neutrinos (REF ) is similar.\",\n \"Since one neutrino is massless, there are 7 parameters in low energy, i.e., 2 neutrino masses $+$ 3 real mixing angles $+$ 2 phases.\",\n \"On the other hand, $M_\\\\nu $ includes $y_{\\\\nu ki}$ and $M_k$ where $k=1,2$ and $i=1,2,3$ , which means 12 real parameters (neutrino Yukawa couplings) $+$ 2 real parameters (right-handed neutrino masses).\",\n \"However, 3 phases can be absorbed by lepton doublets and 2 real parameters are unphysical since $M_\\\\nu $ is unchanged by the rescalings $y_{\\\\nu ki}\\\\rightarrow \\\\gamma _k y_{\\\\nu ki}$ and $M_k \\\\rightarrow \\\\gamma ^2_kM_k$ with $\\\\gamma _k$ being real constants.\",\n \"Therefore, we have 9 independent parameters in $M_\\\\nu $ to determine 7 parameters in the light neutrino sector.\",\n \"As it will be seen below, however, the parametrization of the Yukawa couplings is different, especially for $y_{\\\\nu 3i}$ that are important parameters for the estimation of the reheating temperature.\",\n \"Let us discuss the NH case first.\",\n \"We define $4\\\\times 3$ matrix $R$ in the similar manner in Refs.\",\n \"[22], [23], $iR=D_{\\\\tilde{M}}^{-1/2}U_{\\\\tilde{M}}^T\\\\tilde{m}_\\\\nu U_\\\\nu D_{M_\\\\nu }^{-1/2}\\\\,,$ which satisfies $R^TR={\\\\rm diag}(0,1,1)\\\\,.$ Here $D_{\\\\tilde{M}}^{-1/2}$ and $D_{M_\\\\nu }^{-1/2}$ are matrices that satisfy $(D_{\\\\tilde{M}}^{-1/2})^2=D_{\\\\tilde{M}}^{-1}$ and $(D_{M_\\\\nu }^{-1/2})^2={\\\\rm diag}(0,m_2^{-1},m_3^{-1})$ , respectively.\",\n \"It is found that $R$ is more restrictive than Eq.\",\n \"(REF ).\",\n \"Namely, $r^Tr=rr^T={\\\\bf 1}\\\\,,\\\\\\\\R_{42}/R_{32}=R_{43}/R_{33}=i\\\\,, \\\\\\\\R_{l1}=0~~~~(l=14)\\\\,,$ where $r\\\\equiv \\\\left(\\\\begin{array}{cc}R_{12} & R_{13} \\\\\\\\R_{22} & R_{23}\\\\end{array}\\\\right)\\\\,.$ Using the relations, the neutrino Yukawa couplings can be expressed in terms of $R$ .\",\n \"For later discussion, it is useful to give following quantities: $(y_\\\\nu y_\\\\nu ^\\\\dagger )_{ii}&=M_i\\\\sum _{j=2}^3 |R_{ij}|^2 m_j/\\\\langle H_u^0 \\\\rangle ^2~~~~~~~~~(i=1,2)\\\\,,\\\\\\\\(y_\\\\nu y_\\\\nu ^\\\\dagger )_{33}&=2m_\\\\phi \\\\sum _{j=2}^3|R_{3j}|^2 m_j/\\\\langle H_u^0 \\\\rangle ^2\\\\,,\\\\\\\\{\\\\rm Im}\\\\left[(y_\\\\nu y_\\\\nu ^\\\\dagger )_{21}^2\\\\right]\\\\frac{M_1}{M_2}&=-\\\\frac{M_1^2}{\\\\langle H_u^0\\\\rangle ^4}{\\\\rm Im}\\\\left[\\\\sum _{j=2}^3R_{1j}^2m_j^2\\\\right]\\\\,, \\\\\\\\\\\\sum _{i=1}^2{\\\\rm Im}\\\\left[(y_\\\\nu y_\\\\nu ^\\\\dagger )_{i3}^2\\\\right]\\\\frac{M_3}{M_i}&=-\\\\frac{2M_3 m_\\\\phi }{\\\\langle H_u^0\\\\rangle ^4}{\\\\rm Im}\\\\left[\\\\sum _{j=2}^3R_{3j}^2m_j^2\\\\right]\\\\,.$ It is seen that $(y_\\\\nu y_\\\\nu ^\\\\dagger )_{11}$ and $(y_\\\\nu y_\\\\nu ^\\\\dagger )_{22}$ are constrained by the neutrino oscillation data, meanwhile $(y_\\\\nu y_\\\\nu ^\\\\dagger )_{33}$ is basically a free parameter since $R_{3j}$ is not constrained.\",\n \"This is consistent with the fact that $M_\\\\nu $ is independent of $y_{\\\\nu 3i}$ .\",\n \"The discussion is quite similar in the IH case.\",\n \"The definition of $R$ is the same form as in Eq.\",\n \"(REF ), but satisfies $R^TR={\\\\rm diag}(1,1,0)$ with $(D_{M_\\\\nu }^{-1/2})^2={\\\\rm diag}(m_1^{-1},m_2^{-1},0)$ .\",\n \"Then, defining $r$ as $r\\\\equiv \\\\left(\\\\begin{array}{cc}R_{11} & R_{12} \\\\\\\\R_{21} & R_{22}\\\\end{array}\\\\right)\\\\,,$ we get $r^Tr=rr^T={\\\\bf 1}\\\\,,\\\\\\\\R_{41}/R_{31}=R_{42}/R_{32}=i\\\\,, \\\\\\\\R_{l3}=0~~~~(l=14)\\\\,,$ and the neutrino Yukawa couplings are given by, $(y_\\\\nu y_\\\\nu ^\\\\dagger )_{ii}&=M_i\\\\sum _{j=1}^2 |R_{ij}|^2 m_j/\\\\langle H_u^0 \\\\rangle ^2~~~~~~~~~(i=1,2)\\\\,,\\\\\\\\(y_\\\\nu y_\\\\nu ^\\\\dagger )_{33}&=2m_\\\\phi \\\\sum _{j=1}^2|R_{3j}|^2 m_j/\\\\langle H_u^0 \\\\rangle ^2\\\\,,\\\\\\\\{\\\\rm Im}\\\\left[(y_\\\\nu y_\\\\nu ^\\\\dagger )_{21}^2\\\\right]\\\\frac{M_1}{M_2}&=-\\\\frac{M_1^2}{\\\\langle H_u^0\\\\rangle ^4}{\\\\rm Im}\\\\left[\\\\sum _{j=1}^2R_{1j}^2m_j^2\\\\right]\\\\,, \\\\\\\\\\\\sum _{i=1}^{2}{\\\\rm Im}\\\\left[(y_\\\\nu y_\\\\nu ^\\\\dagger )_{i3}^2\\\\right]\\\\frac{M_3}{M_i}&=-\\\\frac{2M_3 m_\\\\phi }{\\\\langle H_u^0\\\\rangle ^4}{\\\\rm Im}\\\\left[\\\\sum _{j=1}^2R_{3j}^2m_j^2\\\\right]\\\\,.$\"\n ],\n [\n \"Post inflationary regime\",\n \"After the end of inflation, the inflaton oscillates around the global minimum and decays eventually.\",\n \"Due to the decay the universe is reheated and thermal plasma is created.\",\n \"In this section, we estimate the reheating temperature and discuss how the lepton number asymmetry is generated.\",\n \"As in the previous section, we take $\\\\Omega \\\\simeq 1$ .\",\n \"After the universe is reheated, gravitinos are produced in various ways.\",\n \"We discuss the gravitino problem at the end of this section.\"\n ],\n [\n \"Reheating\",\n \"The reheating temperature $T_R$ due to the inflaton decay is estimated by, $T_R\\\\simeq (90/\\\\pi ^2g_*(T_R))^{1/4}\\\\sqrt{\\\\Gamma _\\\\phi M_{pl}}\\\\,,$ where $g_*(T)$ is the effective degree of freedom of radiation fields at temperature $T$ and $\\\\Gamma _\\\\phi $ is the decay rate of the inflaton.\",\n \"This expression is valid when the neutrino Yukawa couplings that are responsible for the decay is sufficiently small to satisfy $T_R\\\\lesssim m_\\\\phi $ [24], [25], which is the situation we focus on.6Of course, it is possible to consider a higher reheating temperature than the inflaton mass.\",\n \"Such a case is discussed in Ref. [19].\",\n \"We will comment on the impact of such high reheating temperature on leptogenesis in the next subsection.\",\n \"The inflaton decays as $\\\\phi \\\\rightarrow L\\\\tilde{H}_u$ , $\\\\bar{L}\\\\bar{\\\\tilde{H}}_u$ , $\\\\tilde{L}H_u$ , $\\\\tilde{L}^*H_u^*$ .7In general, the inflaton decays to gravitino pair or gravitino and right-handed neutrino.\",\n \"We will discuss those processes in Sec.\",\n \"REF .\",\n \"Here flavor indices and SU(2) doublet components are summed implicitly.\",\n \"Then the decay rates for the modes are given by $&\\\\Gamma _{\\\\phi \\\\rightarrow L\\\\tilde{H}_u}=\\\\Gamma _{\\\\phi \\\\rightarrow \\\\bar{L}\\\\bar{\\\\tilde{H}}_u}=\\\\frac{(y_{\\\\nu }y_\\\\nu ^\\\\dagger )_{33}}{16\\\\pi }m_\\\\phi \\\\,, \\\\\\\\& \\\\Gamma _{\\\\phi \\\\rightarrow \\\\tilde{L}H_u}=\\\\Gamma _{\\\\phi \\\\rightarrow \\\\tilde{L}^*H_u^*}=\\\\frac{(y_{\\\\nu }y_\\\\nu ^\\\\dagger )_{33}}{16\\\\pi }\\\\frac{M_3^2}{m_\\\\phi }\\\\,.$ Since $\\\\Gamma _{\\\\phi \\\\rightarrow \\\\tilde{L}H_u}$ (=$\\\\Gamma _{\\\\phi \\\\rightarrow \\\\tilde{L}^*H_u^*}$ ) is suppressed by $(M_3/m_\\\\phi )^2$ , the total decay rate is given by $\\\\Gamma _\\\\phi \\\\simeq \\\\Gamma _{\\\\phi \\\\rightarrow L\\\\tilde{H}_u}+\\\\Gamma _{\\\\phi \\\\rightarrow \\\\bar{L}\\\\bar{\\\\tilde{H}}_u}=\\\\frac{(y_{\\\\nu }y_\\\\nu ^\\\\dagger )_{33}}{8\\\\pi }m_\\\\phi \\\\,.$ Then the reheating temperature is estimated as $T_R\\\\simeq 1.4\\\\times 10^{10}\\\\,{\\\\rm GeV}\\\\left(\\\\frac{m_\\\\phi }{10^{13}\\\\,{\\\\rm GeV}}\\\\right)^{1/2}\\\\left(\\\\frac{(y_{\\\\nu }y_\\\\nu ^\\\\dagger )_{33}}{10^{-9}}\\\\right)^{1/2}\\\\left(\\\\frac{g_{*}(T_R)}{228.75}\\\\right)^{-1/4}\\\\,.$ Recall that $(y_\\\\nu y_\\\\nu ^\\\\dagger )_{33}$ is not constrained by the neutrino observations.\",\n \"As a consequence, it is possible to consider a wide range of values for the reheating temperature, which is suitable for leptogenesis.\",\n \"To end this subsection, we derive the number of $e$ -folds before the end of inflation.\",\n \"In this model, the inflaton oscillates after the end of inflation and eventually decays to reheat the universe.\",\n \"Therefore, it is given by $N_e\\\\simeq &\\\\, 55+\\\\log \\\\left(\\\\frac{L}{{\\\\rm Gpc}}\\\\right)+\\\\frac{1}{3}\\\\log \\\\left(\\\\frac{T_R}{10^{10}\\\\,{\\\\rm GeV}}\\\\right)\\\\nonumber \\\\\\\\&+\\\\log \\\\left(\\\\frac{H_e}{10^{13}\\\\,{\\\\rm GeV}}\\\\right)-\\\\frac{2}{3}\\\\log \\\\left(\\\\frac{H_{\\\\rm end}}{10^{13}\\\\,{\\\\rm GeV}}\\\\right)\\\\,,$ where $L$ is the present cosmological scale, and $H_e$ and $H_{\\\\rm end}$ are the Hubble scale corresponding to $N_e$ and at the end of inflation, respectively.\",\n \"It is now clear that the parameters given in Eq.\",\n \"(REF ) is consistent with $N_e=55$ – 60.\"\n ],\n [\n \"Leptogenesis\",\n \"Now we discuss the lepton number asymmetry.\",\n \"The lepton number is generated via leptogenesis [18] (see, for example, Refs.\",\n \"[26], [27] for review).\",\n \"In the following numerical study, we discuss following representative cases: $({\\\\rm I}).~& M_1,M_2 < m_\\\\phi \\\\,,\\\\\\\\({\\\\rm II}).~& M_1,M_2 > m_\\\\phi \\\\,.$ In both cases, $M_3$ should satisfy Eq.\",\n \"(REF ).\",\n \"Let us consider case (I) first.\",\n \"For simplicity, we consider the Majorana masses are further hierarchical, i.e., $M_1\\\\ll M_2$ .8$M_3$ is irrelevant for leptogenesis if Eq.\",\n \"(REF ) is satisfied.\",\n \"Similar situation has been studied intensively in the literature [28], [29].\",\n \"Even though the lepton number is generated by the inflaton decay, it is possibly washed out when the reheating temperature is comparable or higher than $M_{1}$ .9It would be possible that $\\\\tilde{N}_i$ ($i=1,2$ ) have initial amplitude of the Hubble parameter during inflation $H_{\\\\rm inf}\\\\sim g\\\\xi /\\\\sqrt{6}M_{pl}$ .\",\n \"Then $\\\\tilde{N}_i$ start to oscillate when $H\\\\sim M_i$ , and eventually decays.\",\n \"However, the effect of coherent oscillation of $\\\\tilde{N}_i$ is negligible since the energy density ratio of $\\\\tilde{N}_i$ to radiation at the decay is estimated as less than $\\\\xi ^2/18M_{pl}^4\\\\sim 10^{-9}$ .\",\n \"To see this more explicitly, it is convenient to introduce the effective neutrino mass [30] and equilibrium neutrino mass [31], $&\\\\tilde{m}_1=\\\\frac{(m_\\\\nu m_\\\\nu ^\\\\dagger )_{11}}{M_1}\\\\,,\\\\\\\\&m_*=\\\\frac{4\\\\pi ^2\\\\sqrt{g_*(M_1)} \\\\langle H_u^0\\\\rangle ^2 }{3\\\\sqrt{10} M_{pl}}\\\\simeq 3.9 \\\\times 10^{-4}\\\\,{\\\\rm eV}\\\\left(\\\\frac{\\\\langle H_u^0\\\\rangle }{v/2}\\\\right)^2\\\\,,$ where $v\\\\simeq 246.7~{\\\\rm GeV}$ .\",\n \"If $\\\\tilde{m}_1/m_*$ is larger than unity, then it is the strong washout regime and the lepton number generated at the reheating is washed out.\",\n \"$\\\\tilde{m}_1$ is estimated by using Eqs.\",\n \"(REF ) and (REF ), $\\\\tilde{m}_1=\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\sum _{j=2}^3 |R_{1j}|^2 m_j &({\\\\rm NH}) \\\\\\\\\\\\sum _{j=1}^2 |R_{1j}|^2 m_j &({\\\\rm IH})\\\\end{array}\\\\right.\\\\,.$ Using the neutrino mass data and Eqs.\",\n \"(REF ) and (REF ), it is straightforward to find that $\\\\tilde{m}_1$ has a lower bound, $\\\\tilde{m}_1 \\\\ge \\\\left\\\\lbrace \\\\begin{array}{ll}m_2 \\\\simeq 8.6 \\\\times 10^{-3}\\\\,{\\\\rm eV} &({\\\\rm NH}) \\\\\\\\m_1 \\\\simeq 4.9\\\\times 10^{-2}\\\\,{\\\\rm eV} &({\\\\rm IH})\\\\end{array}\\\\right.\\\\,.$ Therefore, it is the strong washout regime in either case.\",\n \"Figure: Allowed region for M 1 M_1 as function of effective neutrinomass m ˜ 1 \\\\tilde{m}_1 defined in Eq.\",\n \"() for the normalhierarchy (NH) and inverted hierarchy (IH) cases.\",\n \"Lower bound onM 1 M_1 is obtained from η B max ≥η B obs \\\\eta _B^{\\\\rm max}\\\\ge \\\\eta _B^{\\\\rm obs}while upper bound is ().\",\n \"Lower bound onm ˜ 1 \\\\tilde{m}_1 is given by Eq.\",\n \"().Although the primordial lepton number is washed out, the lepton number is regenerated by the decay of the lightest right-handed (s)neutrino, i.e., $N_1$ and $\\\\tilde{N}_1$ in the present case.\",\n \"Then the lepton number, strictly speaking lepton number minus baryon number, is converted to baryon number via the sphaleron effect.\",\n \"This scenario works if $T_R\\\\gtrsim M_1$ [28], [29], which is always possible as confirmed in the previous subsection.\",\n \"Then the resultant baryon number becomes independent of $T_R$ .\",\n \"In our study, we adopt the analytic expressions in Ref.\",\n \"[29] for the calculation of the baryon number.\",\n \"Note that although the results there are given in non-supersymmetric model, the results in supersymmetric model do not change much both quantitatively and qualitatively [32], [33], [27].\",\n \"In our study we adopt the discussion given in Ref. [27].\",\n \"Then the baryon number is determined by $\\\\eta _B\\\\equiv \\\\frac{n_B}{n_\\\\gamma }=\\\\frac{3\\\\sqrt{2}}{4}\\\\frac{a_{\\\\rm sph}}{f}\\\\epsilon _1 \\\\kappa _f\\\\simeq 2.7\\\\times 10^{-10} \\\\left(\\\\frac{\\\\epsilon _1}{10^{-6}}\\\\right)\\\\left(\\\\frac{\\\\kappa _f}{2\\\\times 10^{-2}}\\\\right)\\\\,,$ where $n_B$ and $n_\\\\gamma $ are number densities of baryon and photon at present, respectively, $a_{\\\\rm sph}=28/79$ , $f=2387/86$ , and a factor of $\\\\sqrt{2}$ counts the supersymmetric effect.\",\n \"The efficiency factor $\\\\kappa _f$ is given by [29] $\\\\kappa _f=(2\\\\pm 1)\\\\times 10^{-2}\\\\left(\\\\frac{0.01~{\\\\rm eV}}{\\\\tilde{m}_1}\\\\right)^{1.1\\\\pm 0.1}\\\\,.$ Finally, referring Ref.\",\n \"[34], the asymmetric parameter $\\\\epsilon _1$ in our model is given by $\\\\epsilon _1=-\\\\frac{3}{16\\\\pi }\\\\frac{{\\\\rm Im}\\\\,\\\\left[(y_\\\\nu y_\\\\nu ^\\\\dagger )_{21}^2\\\\right] }{(y_\\\\nu y_\\\\nu ^\\\\dagger )_{11}}\\\\frac{M_1}{M_2}=\\\\frac{3}{16\\\\pi }\\\\frac{M_1}{\\\\langle H^0_u \\\\rangle ^2}m_{\\\\rm eff}\\\\,,$ where we have used Eqs.\",\n \"(REF ), (), (REF ), and () to obtain $m_{\\\\rm eff} =\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\frac{{\\\\rm Im}\\\\sum _{j=2}^3R_{1j}^2m_j^2}{\\\\sum _{j=2}^3|R_{1j}|^2m_j}&({\\\\rm NH}) \\\\\\\\[4mm]\\\\frac{{\\\\rm Im}\\\\sum _{j=1}^2R_{1j}^2m_j^2}{\\\\sum _{j=1}^2|R_{1j}|^2m_j}&({\\\\rm IH})\\\\end{array}\\\\right.\\\\,.$ It turns out that the maximum value of $m_{\\\\rm eff}$ , denoted as $m_{\\\\rm eff}^{\\\\rm max}$ , is $m_{\\\\rm eff}^{\\\\rm max} =\\\\left\\\\lbrace \\\\begin{array}{ll}m_3 - m_2 \\\\simeq 4.2\\\\times 10^{-2}\\\\,{\\\\rm eV}&({\\\\rm NH}) \\\\\\\\m_2 - m_1\\\\simeq 7.4 \\\\times 10^{-4}\\\\,{\\\\rm eV}&({\\\\rm IH})\\\\end{array}\\\\right.\\\\,.$ Therefore, parametrizing $m_{\\\\rm eff}$ as $m_{\\\\rm eff}=m_{\\\\rm eff}^{\\\\rm max} \\\\sin \\\\delta $ , the asymmetric parameter is given by $\\\\epsilon _1\\\\simeq \\\\left\\\\lbrace \\\\begin{array}{ll}8.2 \\\\times 10^{-7}&({\\\\rm NH}) \\\\\\\\1.5\\\\times 10^{-8}&({\\\\rm IH})\\\\end{array}\\\\right\\\\rbrace \\\\times \\\\left(\\\\frac{M_1}{10^{10}\\\\,{\\\\rm GeV}}\\\\right)\\\\left(\\\\frac{\\\\langle H^0_u \\\\rangle }{v/2}\\\\right)^{-2}\\\\left(\\\\frac{\\\\sin \\\\delta }{0.5}\\\\right)\\\\,.$ Using Eqs.\",\n \"(REF ), (REF ) and (REF ), the lower limit on $M_1$ to explain the present baryon number is determined from $\\\\eta _B^{\\\\rm max} \\\\ge \\\\eta _B^{\\\\rm obs}$ where $\\\\sin \\\\delta =1$ and $\\\\eta _B^{\\\\rm obs}$ is given by [35] $\\\\eta _B^{\\\\rm obs}=(6.12\\\\pm 0.03)\\\\times 10^{-10}\\\\,.$ In Fig.\",\n \"REF , allowed regions are depicted for the NH and IH cases.\",\n \"Here we consider so-called high-scale SUSY and take $\\\\langle H^0_u \\\\rangle =v/2$ to get 125 GeV Higgs mass [36], [37].\",\n \"In the plot upper bound on $M_1$ is given by (REF ), i.e., $M_110^{-5}\\\\,{\\\\rm eV}$ because $\\\\tilde{m}_3\\\\ll m_*$ is no longer valid.\",\n \"In the plot, contours of $T_R$ are depicted.\",\n \"It is found that the leptogenesis is successful in a wide parameter space for both the NH and IH cases.\",\n \"Lower bound on $M_3$ behaves similarly to region C in Fig.\",\n \"1 of Ref.\",\n \"[42] by reading $M_1$ and $\\\\tilde{m}_1$ as $M_3$ and $\\\\tilde{m}_3$ , respectively, i.e., the lower bound is proportional to $1/\\\\sqrt{\\\\tilde{m}_3}$ .\",\n \"Quantitatively, the lower bound in our model is relaxed by roughly a factor of 4 compared to the result in the reference.\",\n \"This can be understood as follows; first, the decay rate of the inflaton in our model is different from one in the literature (see Eq.\",\n \"(REF )), leading to a $1/\\\\sqrt{2}$ suppression in $T_R/m_\\\\phi $ ; as another consequence, the expression (REF ) (with $m_{\\\\rm eff}^{\\\\prime }=m_{\\\\rm eff}^{\\\\prime {\\\\rm max}}$ ) is enhanced by a factor of two compared to $|\\\\epsilon _1^{\\\\rm max}|$ in the reference; finally $\\\\tan \\\\beta =\\\\infty $ is taken in the work meanwhile $\\\\tan \\\\beta =1$ in our model.\",\n \"Thus, in total, a factor of 4 enhancement is obtained in $\\\\eta _B$ .\",\n \"In the case where $\\\\tilde{m}_3$ gets much larger than $m_*$ , the situation reduces to case (I).\",\n \"Namely, the reheating temperature is so high that both $\\\\tilde{N}_3$ and $N_3$ are thermalized and thermal leptogenesis takes place.\",\n \"Resultant allowed region is the same as the NH case of Fig.\",\n \"REF , by replacing $M_1$ and $\\\\tilde{m}_1$ by $M_3$ and $\\\\tilde{m}_3$ , respectively, but there is no lower bound on $\\\\tilde{m}_3$ meanwhile there is the upper bound on $M_3$ .\",\n \"In the intermediate case, $\\\\tilde{m}_3\\\\sim m_*$ , on the other hand, the Boltzmann equations should be solved numerically to get the lepton number, which is already done in Ref. [42].\",\n \"The result corresponds to region B in Fig.\",\n \"1 of the reference.\",\n \"Strictly speaking, the effective dissipation rate should be used instead of the decay rate of the inflaton [25] in the Boltzmann equations.\",\n \"As shown in the reference, the reheating process is so efficient when the dissipation rate is taken into account that the reheating temperature can exceed the mass of the inflaton mass and consequently $N_3$ and $\\\\tilde{N}_3$ are easily thermalized.\",\n \"Once they are thermalized, the thermal leptogenesis takes place, where the resultant baryon number becomes independent of the reheating temperature.\",\n \"Eventually the situation reduces to the case (I).\",\n \"Such qualitative behavior can be confirmed by numerical study, which is left for the future work.\",\n \"Crucial difference from sneutrino leptogenesis [38], [39], [40], [41], [42] is that although $M_3$ and $\\\\tilde{m}_3$ , i.e., $(y_\\\\nu y_\\\\nu ^\\\\dagger )_{33}$ , are important parameters to determine baryon number, they are sequestered from other physical quantities, such as the heavy right-handed (s)neutrino masses or the light neutrino mass matrix.\",\n \"Therefore, there is no consequence in other low energy experiments.\",\n \"This is a feature of case (II).\"\n ],\n [\n \"Gravitino problem\",\n \"In the framework of supergravity, a fair amount of gravitino $\\\\psi _\\\\mu $ can be produced in various ways in the thermal history of the universe.\",\n \"Since the interactions of gravitino with the MSSM particles are Planck-suppressed, gravitino is long-lived and its decay can spoil the successful big-bang nucleosynthesis if it is unstable.\",\n \"Although this problem can be avoided when gravitino is enough heavy to have the lifetime much shorter than 1 sec, gravitino decay produces the lightest superparticle (LSP).\",\n \"Then the LSP produced by the decay may overclose the universe if the R-parity is conserved.\",\n \"There are three types of production mechanism of gravitino in the model we consider; (i) the inflaton decay; (ii) thermal scattering from the thermal bath [51], [52], [53], [54]; (iii) decay of superparticles in the thermal bath [55], [56].\",\n \"In general, process (i) includes gravitino pair production.\",\n \"The decay width of the mode, however, depends on the inflaton VEV [57], [58], [59], [60], [61], [62].\",\n \"In our case, therefore, this process can be ignored since the inflaton does not have a VEV.\",\n \"On the other hand, the inflaton can decay to gravitino and right-handed neutrino.\",\n \"The decay width is given by $\\\\Gamma _{\\\\phi \\\\rightarrow \\\\psi _{\\\\mu }N_3}=\\\\frac{\\\\beta _f^{3}m_\\\\phi ^5}{48\\\\pi M_{pl}^2m_{3/2}^2}\\\\left[1-\\\\frac{(m_f+m_{3/2})^2}{m_\\\\phi ^2}\\\\right]\\\\,,$ where $\\\\beta _f^{2}=1-2(m_f^2+m_{3/2}^2)/m_\\\\phi ^2+(m_f^2-m_{3/2}^2)^2/m_\\\\phi ^4\\\\,.$ Here $m_f$ and $m_{3/2}$ are masses of $N_3$ and $\\\\psi _\\\\mu $ , respectively.\",\n \"The mass difference between $\\\\phi $ and $N_3$ is expected to be given by the soft SUSY breaking mass scale for scalar superpartners, $|m_\\\\phi -m_f|\\\\sim \\\\tilde{m}$ .\",\n \"Let us assume this decay happens by taking $m_\\\\phi =m_f+\\\\tilde{m}$ where $\\\\tilde{m}=k m_{3/2}$ ($k>1$ ).\",\n \"In the limit $km_{3/2}/m_\\\\phi \\\\ll 1$ , we obtain $\\\\Gamma _{\\\\phi \\\\rightarrow \\\\psi _{\\\\mu }N_3}\\\\simeq \\\\frac{Cm_\\\\phi m_{3/2}^2}{3\\\\pi M^2_{pl}}\\\\,,$ where $C=(k-1)(k^2-1)^{3/2}$ .\",\n \"Then the branching fraction of this mode is ${\\\\rm Br}_{\\\\phi \\\\rightarrow \\\\psi _{\\\\mu }N_3} =\\\\frac{\\\\Gamma _{\\\\phi \\\\rightarrow \\\\psi _{\\\\mu }N_3}}{\\\\Gamma _{\\\\phi }}\\\\simeq 4.5\\\\times 10^{-16}C\\\\left(\\\\frac{m_{3/2}}{10^{5}\\\\,{\\\\rm GeV}}\\\\right)^2\\\\left(\\\\frac{10^{-11}}{(y_\\\\nu y_\\\\nu ^\\\\dagger )_{33}}\\\\right)\\\\,.$ Therefore, it is sure that the inflaton decay reheats the universe in wide range of gravitino mass region.\",\n \"On the other hand, the resultant gravitino abundance produced by the decay is estimated as $\\\\Omega _{3/2}^{\\\\rm inf}h^2\\\\sim 1.3\\\\times 10^{-6} C\\\\left(\\\\frac{m_{3/2}}{10^{5}\\\\,{\\\\rm GeV}}\\\\right)^3\\\\left(\\\\frac{10^{13}\\\\,{\\\\rm GeV}}{m_{\\\\phi }}\\\\right)^{1/2}\\\\left(\\\\frac{10^{-11}}{(y_\\\\nu y_\\\\nu ^\\\\dagger )_{33}}\\\\right)^{1/2}\\\\,,$ where $h$ is the scale factor of Hubble expansion rate.\",\n \"Gravitino abundance via process (ii) is most effective at high temperature, thus it is proportional to $T_R$ , meanwhile in process (iii) gravitino is dominantly produced when the temperature is around the mass of decaying particle.\",\n \"Adopting the expression given in Ref.\",\n \"[63], the abundances via processes (ii) and (iii) are given by $&\\\\Omega _{3/2}^{\\\\rm TH}h^2\\\\sim 4\\\\left(\\\\frac{T_R}{10^{9}\\\\,{\\\\rm GeV}}\\\\right)\\\\left(\\\\frac{m_{3/2}}{10^{5}\\\\,{\\\\rm GeV}}\\\\right)\\\\,,\\\\\\\\&\\\\Omega _{3/2}^{\\\\rm FI}h^2\\\\sim 5\\\\times 10^{-4}k^3\\\\left(\\\\frac{m_{3/2}}{10^5\\\\,{\\\\rm GeV}}\\\\right)^2\\\\,.$ Here the contribution of the longitudinal mode of gravitino is suppressed in $\\\\Omega _{3/2}^{\\\\rm TH}$ by considering gluino is lighter than gravitino.\",\n \"In $\\\\Omega _{3/2}^{\\\\rm FI}$ , we have assumed that all scalar leptons and quarks in the MSSM (whose mass scale is $\\\\tilde{m}$ ) are heavier than gravitino and that they are thermalized.\",\n \"And as in the discussion of the inflaton decay, $\\\\tilde{m}=km_{3/2}$ has been taken.11We have checked that the contribution from $\\\\tilde{N}_1$ is negligible even if $\\\\tilde{N}_1$ is thermalized, i.e., in case (I).\",\n \"Then the relic abundance of the LSP due to gravitino decay is estimated as $\\\\Omega _{\\\\rm LSP}^{\\\\rm nonth}h^2=\\\\frac{m_{\\\\rm LSP}}{m_{3/2}}\\\\Omega _{3/2}h^2\\\\,,$ where $m_{\\\\rm LSP}$ is the LSP mass and $\\\\Omega _{3/2}=\\\\Omega _{3/2}^{\\\\rm inf}+\\\\Omega _{3/2}^{\\\\rm TH}+\\\\Omega _{3/2}^{\\\\rm FI}$ is the total gravitino abundance.\",\n \"$\\\\Omega _{\\\\rm LSP}^{\\\\rm nonth}h^2$ should not exceed the observed dark matter abundance $\\\\Omega _{\\\\rm DM}h^2\\\\simeq 0.12$ [35], which gives a constraint on gravitino mass.\",\n \"Figure: Relic density of the LSP as function of gravitino mass.“inf” (dotted green), “TH” (dot-dashed red), and “FI”(dashed blue) are the contributions from process (i), (ii), and(iii), respectively.\",\n \"Ω DM h 2 ≃0.12\\\\Omega _{\\\\rm DM}h^2\\\\simeq 0.12 (solid brown) isindicated as a reference.\",\n \"Right region from vertical line (dottedviolet) indicates T 3/2 >m LSP T_{3/2}>m_{\\\\rm LSP}, and shaded region isexcluded.\",\n \"m LSP =1 TeV m_{\\\\rm LSP}=1\\\\,{\\\\rm TeV}, m φ =10 13 GeV m_\\\\phi =10^{13}\\\\,{\\\\rm GeV},(y ν y ν † ) 33 =10 -11 (y_\\\\nu y_\\\\nu ^\\\\dagger )_{33}=10^{-11}, and k=m ˜/m 3/2 =2k=\\\\tilde{m}/m_{3/2}=2(left), 10 (right) are taken.Fig.\",\n \"REF shows the resultant $\\\\Omega _{\\\\rm LSP}^{\\\\rm nonth}$ from the each contribution.\",\n \"Here $m_{\\\\rm LSP}=1\\\\,{\\\\rm TeV}$ is taken by considering 1 TeV Higgsino or Wino dark matter with a mass of 2.7 – 3 TeV [64].\",\n \"$m_\\\\phi =10^{13}\\\\,{\\\\rm GeV}$ , $(y_\\\\nu y_\\\\nu ^\\\\dagger )_{33}=10^{-11}$ , and $k=2$ (left), 10 (right) are taken to determine the contributions from processes (i) and (iii).\",\n \"It is seen that in lower gravitino mass region, the dominant contribution to $\\\\Omega _{\\\\rm LSP}^{\\\\rm nonth}$ is from process (ii).\",\n \"In order for the contribution not to exceed the dark matter abundance, $T_R\\\\lesssim 10^9\\\\,{\\\\rm GeV}$ is required, which is well-known result.\",\n \"This gives a stringent constraint on the parameter space for successful leptogenesis shown in Figs.\",\n \"REF and REF if gravitino is not heavy enough.\",\n \"Namely, $\\\\tilde{m}_1\\\\sim 10^{-2}\\\\,{\\\\rm eV}$ is the allowed region in case (I) meanwhile $10^{-13}\\\\,{\\\\rm eV}\\\\lesssim \\\\tilde{m}_3\\\\lesssim 10^{-11}\\\\,{\\\\rm eV}$ is allowed in case (II).\",\n \"On the other hand, the contributions from processes (i) and (iii) depend on the parameters, especially $k=\\\\tilde{m}/m_{3/2}$ (and $m_{3/2}$ ).\",\n \"In order for $\\\\Omega _{\\\\rm LSP}^{\\\\rm nonth}$ not to exceed the dark matter abundance, the upper bound on gravitino mass is obtained depending on the mass spectrum of squarks and sleptons, e.g., $m_{3/2}\\\\lesssim 10^8$ ($10^{6}$ ) GeV for $\\\\tilde{m}\\\\sim m_{3/2}$ $(10m_{3/2})$ .\",\n \"Such gravitino mass is preferred in minimal or mini split supersymmetry [65], [66], pure gravity mediation [67], [68], and spread supersymmetry [63], [69].\",\n \"On the other hand, there is also an allowed region in higher gravitino mass region.\",\n \"This is because gravitino decays before thermal freeze-out of the LSP in that region.\",\n \"The allowed region can be estimated by imposing gravitino decay temperature $T_{3/2}$ larger than the LSP mass.\",\n \"The gravitino decay temperature is defined by $T_{3/2}\\\\simeq (90/\\\\pi ^2g_*(T_{3/2}))^{1/4}\\\\sqrt{\\\\Gamma _{3/2} M_{pl}}\\\\,,$ where $\\\\Gamma _{3/2}$ is the decay rate of gravitino.\",\n \"Then $m_{3/2}\\\\gtrsim 3\\\\times 10^{8}\\\\,{\\\\rm GeV}$ is obtained from $T_{3/2}>m_{\\\\rm LSP}$ for $m_{\\\\rm LSP}=1\\\\,{\\\\rm TeV}$ (see e.g., Ref. [70]).\",\n \"Such high gravitino mass can be considered high-scale SUSY [71], intermediate scale supersymmetry [72], and unified inflation model [16].\",\n \"However, gravitino cannot be too heavy because ${\\\\rm Br}_{\\\\phi \\\\rightarrow \\\\psi _{\\\\mu }N_3}$ should be less than unity for the reheating.\",\n \"Taking ${\\\\rm Br}_{\\\\phi \\\\rightarrow \\\\psi _{\\\\mu }N_3} \\\\lesssim 0.1$ , for example, upper bound on gravitino mass is obtained as $m_{3/2}\\\\lesssim 5\\\\times 10^{11}\\\\,{\\\\rm GeV}$ ($2\\\\times 10^{10}\\\\,{\\\\rm GeV}$ ) for $k=2$ (10).\",\n \"Another option is the R-parity violation.\",\n \"Under the R-parity violation, the LSP decays to the standard-model particles.\",\n \"Then, the LSP does not contribute to the matter abundance of the universe so that there is no constraint on $m_{3/2}$ .\"\n ],\n [\n \"Conclusion\",\n \"Superconformal subcritical hybrid inflation is one of attractive inflation models that are consistent with the observed cosmological parameters by the Planck satellite.\",\n \"In this paper we have studied the cosmology of an extended version of the model.\",\n \"In the model three right-handed neutrinos are introduced.\",\n \"The superpotential consists of one in the supersymmetric seesaw model and the interaction terms of the right-handed neutrinos with the additional matter fields, one of which plays the role of the waterfall field.\",\n \"In the Kähler potential, on the other hand, it is possible for the sneutrinos to have shift symmetry by introducing explicit superconformal breaking terms of ${\\\\cal O}(1)$ .\",\n \"Due to the shift symmetry, one of the sneutrinos becomes the inflaton field similarly in superconformal subcritical hybrid inflation.\",\n \"Although the mass terms of the sneutrinos can affect the trajectory of the inflaton, it has turned out that the effect is restrictive and viable inflation is realized.\",\n \"After inflation, the inflaton field decays to Higgses and sleptons or Higgsinos and leptons to reheat the universe.\",\n \"Light neutrino masses are given by the seesaw mechanism.\",\n \"However, the mass matrix is different from the conventional one.\",\n \"It turns out that one of the neutrinos is massless.\",\n \"Assuming that suppressed couplings of the other right-handed neutrinos to the waterfall field, it has been found that the neutrino Yukawa couplings that couples the inflaton to the MSSM sector are not constrained by the neutrino oscillation data.\",\n \"Consequently, the reheating temperature is a free parameter, which is suitable for leptogenesis.\",\n \"We have considered two representative cases; (I) the other right-handed (s)neutrinos are lighter than the inflaton; (II) the other right-handed (s)neutrinos are heavier than the inflaton.\",\n \"In case (I), thermal leptogenesis is possible if the reheating temperature is larger than $\\\\sim 10^{9}\\\\,{\\\\rm GeV}$ .\",\n \"It has been found leptogenesis is successful in a wide range of parameter space in the normal hierarchy case while the parameter space for leptogenesis is relatively limited in the inverted hierarchy case.\",\n \"In case (II), on the other hand, sneutrino leptogenesis takes place if the reheating temperature is larger than $\\\\sim 10^{8}\\\\,{\\\\rm GeV}$ .\",\n \"It has turned out that in both the normal and inverted hierarchy cases successful leptogenesis is realized in wide range of parameter space.\"\n ],\n [\n \"Acknowledgments\",\n \"We are grateful to Wilfried Buchmüller, Pasquale Di Bari, Oleg Lebedev, and Fuminobu Takahashi for valuable discussions.\",\n \"This work was supported by JSPS KAKENHI Grant Numbers JP17H05402, JP17K14278, JP17H02875 and JP18H05542 (KI).\"\n ]\n]"},"id":{"kind":"string","value":"1906.04530"}}},{"rowIdx":2795,"cells":{"text":{"kind":"list like","value":[["Retrospective Motion Correction of MR Images using Prior-Assisted Deep\n Learning"],["Abstract In MRI, motion artefacts are among the most common types of artefacts.","They can degrade images and render them unusable for accurate diagnosis.","Traditional methods, such as prospective or retrospective motion correction, have been proposed to avoid or alleviate motion artefacts.","Recently, several other methods based on deep learning approaches have been proposed to solve this problem.","This work proposes to enhance the performance of existing deep learning models by the inclusion of additional information present as image priors.","The proposed approach has shown promising results and will be further investigated for clinical validity."],["Introduction","One of the most common causes of artefacts in MR imaging is patient motion , , .","Physiological motion, such as cardiac or respiratory movement, can be controlled by gating or by specific sequence design .","The focus of this work is on correcting for less predictable voluntary or involuntary movement of the patient, often called, bulk motion , which e.g.","in Parkinsonism and leads to ghosting and blurring .","If the level of artefacts is too high, the scan must be repeated.","Even a moderate level may lead to an incomplete or inaccurate diagnosis [1].","The most common strategies for handling motion artefacts can be divided into three main groups: motion prevention, artefact reduction, and motion correction.","Motion prevention spans from training the subject, foam restraints, feed and wrap (for babies), sedation till breath-hold.","Artefact reduction can be achieved with faster imaging, triggering and gating, phase reordering, and similar techniques.","Finally, there are motion correction techniques such as retrospective motion correction (RMC) and prospective motion correction (PMC) .","In practice, RMC modifies the MR image data during the reconstruction process, while PMC performs a real-time adaptive update of the data acquisition with the possibility of using different tracking modalities .","Recently, an increasing number of attempts using deep learning techniques have shown that it is possible to remove or reduce motion artefacts , , , , , , .","It has also been observed that supplying additional prior knowledge, such as image priors, can help improve the performance of deep learning based reconstructions , .","This work aims to remove motion artefacts from corrupted brain images by improving existing deep learning models, by exploiting available non-corrupted image priors."],["Data Preparation","T1, T2, and PD images of 100 subjects (for each - training, testing, and validation) from the publicly available IXI DatasetDataset available at: https://brain-development.org/ixi-dataset/ were used in this study.","T2-weighted images were artificially corrupted with motion using a modified version of the RandomMotion transformation of TorchIO (v0.17.45).","The initial phase of the experiments were performed with 10 simulated movements with a rotation in the range of -1.75 to +1.75 degrees without any translation.","In this modified version of the RandomMotion function, in-plane motion corruption was randomly performed in X or Y direction."],["Image Priors","Supplying additional images as prior knowledge along with the corrupted image can help improve the performance of deep learning models , .","In this research, experiments were performed using two different types of image priors - similar slices from different subjects of the same MRI contrast and different MRI contrasts of the same subject."],["Similar Slices:","During the motion correction, ten similar (same slice position) slices of the same MRI contrast were randomly chosen from different subjects and supplied as prior along with the motion corrupted image.","The motivation behind this type of prior is that while performing motion correction on certain image, images of the same contrast but of different subjects, which are not corrupted by motion, can readily be available.","In these experiments, only T2-weighted images from the IXI Dataset were used."],["Different Contrasts of the Same Subject:","Multiple contrasts of the same subject are usually acquired during clinical routine.","If one of those various contrasts is corrupted by motion, then that image can be corrected by using the other contrasts of the same subject as prior.","All three available contrasts of the IXI Dataset were co-registered against the T2-weighted images.","T2-weighted images were artificially corrupted; T1 and PD images were used as priors during the correction process."],["Network Architectures","UNet and a modified version of the ResNet were used as the baselines for this work.","The baseline networks were modified such that they can receive priors.","Two different methods of supplying priors were tested."],["Multi-Channel Network:","Each motion corrupted image and the priors were concatenated on the channel dimension and were supplied to the network as multi-channel input.","The baselines were receiving only one channel image as input, where for the multi-channel approach, the models received $1+n\\_{prior}$ channel images as input."],["Dual-Branch Network:","For this approach, modified versions of the baselines were created by adding an additional branch to the baselines for the priors.","The main branch was supplied with the motion corrupted image and the priors were supplied to the auxiliary branch.","For the UNet, the auxiliary branch was identical to the contraction path and the latent space of the UNet except for the number of input channels.","The skip-connections were supplied only from the main branch of the network, and no skip-connections were supplied from the auxiliary branch.","For the ResNet, the auxiliary branch was identical to the downsampling blocks of the ResNet up to the residual blocks, with the only difference being the number of input channels.","For both network models, the main branch and the auxiliary branch generated two different latent space representations.","These latent representations were combined and forwarded for generating the final output.","Two different methods were considered for combining the latent spaces - by simple addition or by concatenation and convolution with a kernel size of one - to generate the final combined latent space.","This combined latent space was forwarded to the expansion path of the UNet.","In the case of ResNet, this latent representation was forwarded to the residual blocks for further operation."],["Results and Discussions","Figure REF shows the performance of the different methods based on the SSIM values and Figure REF shows a representative example result.","Between the two different types of priors, it was observed that supplying ten similar slices of the same contrast but of different subjects did not improve the motion correction.","Nonetheless, supplying different contrasts of the same subject improved the motion correction significantly for most of the tests carried out.","ResNet's performance improved for both types of prior supply methods - multi-channel and dual-branch.","For UNet however, only the multi-channel approach has shown significant improvement.","Figure: Plots showing the performance of the various methods, based on SSIMFigure: One example slice to show the motion correction performance of the various methods"],["Conclusion and Future Work","The initial experiments presented here have shown promising results for supplying other contrasts as prior for deep learning based motion correction.","For ResNet, the multi-channel and the dual-network approach both have shown promising results.","However, for UNet only the multi-channel approach has shown improvements over the baseline.","The reason for the failure could be attributed to the lack of skip connections from the auxiliary branch, but the skip connections will be making it similar to the multi-channel approach.","Further investigations will be performed to determine the cause of the failure, and also other strategies for supplying the priors to the UNet.","Further experiments will be performed to check the performance of these approaches when the corrupted volumes and the prior volumes are not co-registered.","Supplying ten similar slices as prior did not show any improvement for any of the networks.","Further investigations will be performed to determine the reasons and possible solutions.","Furthermore, the strategies will be validated with clinical data.","It will be assessed whether pathologies that can only be seen in a corrupted contrast, are preserved after performing the correction."],["Broader Impact","The current stage of the work has been carried out on simulated motion artefacts only and using a publicly available dataset and has been carried out in compliance with ethical standards.","To the best of the authors' knowledge, there are not evident direct or indirect effects on societal aspects.","It is well known that low-quality images invalidate the clinical diagnosis.","In addition, the repetition of scans involves the use of energy, money and human resources.","Therefore, researchers and clinicians working in the field of MRI may benefit from this work.","As this research is in its preliminary stage, it has still to be investigated whether the usage of priors exclude pathologies from the corrected data.","This work was partially conducted within the context of the International Graduate School MEMoRIAL at Otto von Guericke University (OVGU) Magdeburg, Germany, kindly supported by the European Structural and Investment Funds (ESF) under the programme \"Sachsen-Anhalt WISSENSCHAFT Internationalisierung\" (project no.","ZS/2016/08/80646).","This work was also partially conducted within the context of the Initial Training Network program, HiMR, funded by the FP7 Marie Curie Actions of the European Commission, grant number FP7-PEOPLE-2012-ITN-316716, and supported by the NIH grant number 1R01-DA021146, and by the federal state of Saxony-Anhalt under grant number 'I 88'."]],"string":"[\n [\n \"Retrospective Motion Correction of MR Images using Prior-Assisted Deep\\n Learning\"\n ],\n [\n \"Abstract In MRI, motion artefacts are among the most common types of artefacts.\",\n \"They can degrade images and render them unusable for accurate diagnosis.\",\n \"Traditional methods, such as prospective or retrospective motion correction, have been proposed to avoid or alleviate motion artefacts.\",\n \"Recently, several other methods based on deep learning approaches have been proposed to solve this problem.\",\n \"This work proposes to enhance the performance of existing deep learning models by the inclusion of additional information present as image priors.\",\n \"The proposed approach has shown promising results and will be further investigated for clinical validity.\"\n ],\n [\n \"Introduction\",\n \"One of the most common causes of artefacts in MR imaging is patient motion , , .\",\n \"Physiological motion, such as cardiac or respiratory movement, can be controlled by gating or by specific sequence design .\",\n \"The focus of this work is on correcting for less predictable voluntary or involuntary movement of the patient, often called, bulk motion , which e.g.\",\n \"in Parkinsonism and leads to ghosting and blurring .\",\n \"If the level of artefacts is too high, the scan must be repeated.\",\n \"Even a moderate level may lead to an incomplete or inaccurate diagnosis [1].\",\n \"The most common strategies for handling motion artefacts can be divided into three main groups: motion prevention, artefact reduction, and motion correction.\",\n \"Motion prevention spans from training the subject, foam restraints, feed and wrap (for babies), sedation till breath-hold.\",\n \"Artefact reduction can be achieved with faster imaging, triggering and gating, phase reordering, and similar techniques.\",\n \"Finally, there are motion correction techniques such as retrospective motion correction (RMC) and prospective motion correction (PMC) .\",\n \"In practice, RMC modifies the MR image data during the reconstruction process, while PMC performs a real-time adaptive update of the data acquisition with the possibility of using different tracking modalities .\",\n \"Recently, an increasing number of attempts using deep learning techniques have shown that it is possible to remove or reduce motion artefacts , , , , , , .\",\n \"It has also been observed that supplying additional prior knowledge, such as image priors, can help improve the performance of deep learning based reconstructions , .\",\n \"This work aims to remove motion artefacts from corrupted brain images by improving existing deep learning models, by exploiting available non-corrupted image priors.\"\n ],\n [\n \"Data Preparation\",\n \"T1, T2, and PD images of 100 subjects (for each - training, testing, and validation) from the publicly available IXI DatasetDataset available at: https://brain-development.org/ixi-dataset/ were used in this study.\",\n \"T2-weighted images were artificially corrupted with motion using a modified version of the RandomMotion transformation of TorchIO (v0.17.45).\",\n \"The initial phase of the experiments were performed with 10 simulated movements with a rotation in the range of -1.75 to +1.75 degrees without any translation.\",\n \"In this modified version of the RandomMotion function, in-plane motion corruption was randomly performed in X or Y direction.\"\n ],\n [\n \"Image Priors\",\n \"Supplying additional images as prior knowledge along with the corrupted image can help improve the performance of deep learning models , .\",\n \"In this research, experiments were performed using two different types of image priors - similar slices from different subjects of the same MRI contrast and different MRI contrasts of the same subject.\"\n ],\n [\n \"Similar Slices:\",\n \"During the motion correction, ten similar (same slice position) slices of the same MRI contrast were randomly chosen from different subjects and supplied as prior along with the motion corrupted image.\",\n \"The motivation behind this type of prior is that while performing motion correction on certain image, images of the same contrast but of different subjects, which are not corrupted by motion, can readily be available.\",\n \"In these experiments, only T2-weighted images from the IXI Dataset were used.\"\n ],\n [\n \"Different Contrasts of the Same Subject:\",\n \"Multiple contrasts of the same subject are usually acquired during clinical routine.\",\n \"If one of those various contrasts is corrupted by motion, then that image can be corrected by using the other contrasts of the same subject as prior.\",\n \"All three available contrasts of the IXI Dataset were co-registered against the T2-weighted images.\",\n \"T2-weighted images were artificially corrupted; T1 and PD images were used as priors during the correction process.\"\n ],\n [\n \"Network Architectures\",\n \"UNet and a modified version of the ResNet were used as the baselines for this work.\",\n \"The baseline networks were modified such that they can receive priors.\",\n \"Two different methods of supplying priors were tested.\"\n ],\n [\n \"Multi-Channel Network:\",\n \"Each motion corrupted image and the priors were concatenated on the channel dimension and were supplied to the network as multi-channel input.\",\n \"The baselines were receiving only one channel image as input, where for the multi-channel approach, the models received $1+n\\\\_{prior}$ channel images as input.\"\n ],\n [\n \"Dual-Branch Network:\",\n \"For this approach, modified versions of the baselines were created by adding an additional branch to the baselines for the priors.\",\n \"The main branch was supplied with the motion corrupted image and the priors were supplied to the auxiliary branch.\",\n \"For the UNet, the auxiliary branch was identical to the contraction path and the latent space of the UNet except for the number of input channels.\",\n \"The skip-connections were supplied only from the main branch of the network, and no skip-connections were supplied from the auxiliary branch.\",\n \"For the ResNet, the auxiliary branch was identical to the downsampling blocks of the ResNet up to the residual blocks, with the only difference being the number of input channels.\",\n \"For both network models, the main branch and the auxiliary branch generated two different latent space representations.\",\n \"These latent representations were combined and forwarded for generating the final output.\",\n \"Two different methods were considered for combining the latent spaces - by simple addition or by concatenation and convolution with a kernel size of one - to generate the final combined latent space.\",\n \"This combined latent space was forwarded to the expansion path of the UNet.\",\n \"In the case of ResNet, this latent representation was forwarded to the residual blocks for further operation.\"\n ],\n [\n \"Results and Discussions\",\n \"Figure REF shows the performance of the different methods based on the SSIM values and Figure REF shows a representative example result.\",\n \"Between the two different types of priors, it was observed that supplying ten similar slices of the same contrast but of different subjects did not improve the motion correction.\",\n \"Nonetheless, supplying different contrasts of the same subject improved the motion correction significantly for most of the tests carried out.\",\n \"ResNet's performance improved for both types of prior supply methods - multi-channel and dual-branch.\",\n \"For UNet however, only the multi-channel approach has shown significant improvement.\",\n \"Figure: Plots showing the performance of the various methods, based on SSIMFigure: One example slice to show the motion correction performance of the various methods\"\n ],\n [\n \"Conclusion and Future Work\",\n \"The initial experiments presented here have shown promising results for supplying other contrasts as prior for deep learning based motion correction.\",\n \"For ResNet, the multi-channel and the dual-network approach both have shown promising results.\",\n \"However, for UNet only the multi-channel approach has shown improvements over the baseline.\",\n \"The reason for the failure could be attributed to the lack of skip connections from the auxiliary branch, but the skip connections will be making it similar to the multi-channel approach.\",\n \"Further investigations will be performed to determine the cause of the failure, and also other strategies for supplying the priors to the UNet.\",\n \"Further experiments will be performed to check the performance of these approaches when the corrupted volumes and the prior volumes are not co-registered.\",\n \"Supplying ten similar slices as prior did not show any improvement for any of the networks.\",\n \"Further investigations will be performed to determine the reasons and possible solutions.\",\n \"Furthermore, the strategies will be validated with clinical data.\",\n \"It will be assessed whether pathologies that can only be seen in a corrupted contrast, are preserved after performing the correction.\"\n ],\n [\n \"Broader Impact\",\n \"The current stage of the work has been carried out on simulated motion artefacts only and using a publicly available dataset and has been carried out in compliance with ethical standards.\",\n \"To the best of the authors' knowledge, there are not evident direct or indirect effects on societal aspects.\",\n \"It is well known that low-quality images invalidate the clinical diagnosis.\",\n \"In addition, the repetition of scans involves the use of energy, money and human resources.\",\n \"Therefore, researchers and clinicians working in the field of MRI may benefit from this work.\",\n \"As this research is in its preliminary stage, it has still to be investigated whether the usage of priors exclude pathologies from the corrected data.\",\n \"This work was partially conducted within the context of the International Graduate School MEMoRIAL at Otto von Guericke University (OVGU) Magdeburg, Germany, kindly supported by the European Structural and Investment Funds (ESF) under the programme \\\"Sachsen-Anhalt WISSENSCHAFT Internationalisierung\\\" (project no.\",\n \"ZS/2016/08/80646).\",\n \"This work was also partially conducted within the context of the Initial Training Network program, HiMR, funded by the FP7 Marie Curie Actions of the European Commission, grant number FP7-PEOPLE-2012-ITN-316716, and supported by the NIH grant number 1R01-DA021146, and by the federal state of Saxony-Anhalt under grant number 'I 88'.\"\n ]\n]"},"id":{"kind":"string","value":"2011.14134"}}},{"rowIdx":2796,"cells":{"text":{"kind":"list like","value":[["Non-uniform dependence on initial data for the 2D viscous shallow water\n equations"],["Abstract The failure of uniform dependence on the data is an interesting property of classical solution for a hyperbolic system.","In this paper, we consider the solution map of the Cauchy problem to the 2D viscous shallow water equations which is a hyperbolic-parabolic system.","We prove that the solution map of this problem is not uniformly continuous in Sobolev spaces $H^s\\times H^{s}$ for $s>2$."],["Introduction","The 2D viscous shallow water equations are given by the following system, which have systematically introduced in [2], [3] $\\left\\lbrace \\begin{array}{ll}\\partial _t\\rho +\\rho u)=0,\\\\\\partial _t(\\rho u)+\\rho u\\otimes u)-\\mu \\rho \\nabla u)+\\rho \\nabla \\rho =0,\\\\\\rho |_{t=0}=\\rho _0,\\;u|_{t=0}=u_0.\\end{array}\\right.$ where $\\rho (x,t)$ is the height of fluid surface, $u(x,t)$ is the horizontal velocity field and $\\mu >0$ is the viscous coefficient.","We suppose that the initial data $\\rho _0(x)$ is a small perturbation of some positive constant $\\overline{\\rho }_0$ .","Classical solutions and well-posedness of the initial (boundary) value problem for the shallow water equations (REF ) have been studied extensively.","By using Lagrangian coordinates and Hölder space estimates, Bui [1] obtained the local existence and uniqueness of classical solutions to the Cauchy-Dirichlet problem for (REF ) with initial data in $C^{2+\\alpha }$ .","With the help of the energy method of Matsumura and Nishida [33], Kloeden [26] and Sundbye [34] independently showed the global existence and uniqueness of classical solutions to the Cauchy-Dirichlet problem for (REF ).","Subsequently, Sundbye [35] also proved the existence and uniqueness of classical solutions to the Cauchy problem for (REF ) using the method of [33].","Due to the strong nonlinearity of system (REF ), the problem of existence of solutions for large initial data is difficult.","By applying the Littlewood-Paley decomposition theory for Sobolev spaces to obtain a losing energy estimate in $H^s$ for any $s > 2$ , Wang–Xu [36] obtained local solutions for any initial data and global solutions to (REF ) for small initial data $(u_0, \\rho _0-\\bar{\\rho }_0) \\in H^s\\times H^s$ with $s>2$ .","Liu–Yin [30], [31], [32] improved the result of [36] in the Sobolev spaces with low regularity and inhomogeneous Besov spaces.","Chen–Miao–Zhang [6] obtained the local well-posedness of system (REF ) for general initial data in critical $L^p$ type Besov spaces with $1\\le p<4$ by developing a new method which relies on the smoothing properties of the heat equations.","Recently, Li–Hong–Zhu [28] proved that the system (REF ) is ill-posed in the critical Besov spaces with $p > 4$ .","The continuous dependence is particularly important when PDEs are used to model phenomena in the natural world since measurements are always associated with errors.","One of the first results of this type was proved by Kato [27] who showed that the solution operator for the (inviscid) Burgers equation is not Hölder continuous in the $H^s(\\mathbb {T})$ -norm $(s > 3/2)$ for any Hölder exponent.","After the phenomenon of non-uniform continuity for some dispersive equations was studied by Kenig et al.","[24], many results with regard to the non-uniform dependence on the initial data have been obtained for other nonlinear PDEs including the Euler equations [18], the Camassa-Holm equation [19], [20], [29], the Benjamin-Ono equation [25], the compressible gas dynamics [21], [23], the Hunter-Saxton equation [22] and so on.","Nevertheless we notice that almost the above system mentioned is hyperbolic.","As stated in [23], the exhibition of nonuniform behavior in a hyperbolic system related to the incompressible system indicates that the nonuniform dependence is hyperbolic in nature.","From the PDE's point of view, classical solution is uniform dependence on the data for a parabolic system.","Naturally, one may wonder if uniform dependence on the data of solution for a coupled system which is hyperbolic-parabolic can persist or not.","In this paper, we focus on the 2D viscous shallow water equations $\\left\\lbrace \\begin{array}{ll}\\partial _t\\rho +\\rho u)=0,\\\\\\partial _t(\\rho u)+\\rho u\\otimes u)-\\mu 2\\rho \\mathbb {D}u)+\\rho \\nabla \\rho =0,\\\\\\rho |_{t=0}=\\rho _0,\\;u|_{t=0}=u_0.\\end{array}\\right.$ where the strain tensor $\\mathbb {D}u =\\frac{1}{2}(\\nabla u+\\nabla ^{\\mathsf {T}} u)$ is the symmetric part of the velocity gradient.","For the sake of convenience, we take $\\bar{\\rho }_0 = 1$ and denote $\\varrho =\\rho -1$ , we can reformulate the system (REF ) equivalently as follows $\\left\\lbrace \\begin{array}{ll}\\partial _t\\varrho +u̥+u\\cdot \\nabla \\varrho =-\\varrho u̥,\\\\\\partial _tu-{\\mu } \\Delta u-{\\mu } \\nabla u̥+u\\cdot \\nabla u+\\nabla \\varrho =2\\nabla (\\ln (1+\\varrho ))\\cdot \\mathbb {D}u,\\\\\\varrho |_{t=0}=\\varrho _0,\\;u|_{t=0}=u_0.\\end{array}\\right.$ Roughly speaking, the system (REF ) can be regarded as a special case of compressible Navier-Stokes equations (see Refs.","[6], [7], [8], [9], [10], [11], [12], [14], [15], [16], [17] and the references therein).","We emphasize that the equations (REF ) form a quasi-linear hyperbolic-parabolic system and possess strong nonlinear terms.","Although the system (REF ) is partially parabolic, owing to the first equation of (REF ) which is of hyperbolic type, this creates the possibility of the non-uniformity for the solution map.","In this paper, we prove the failure of uniform dependence on the data for (REF ).","To the best of our knowledge, it allows us to give a first kind of answer to the problem of the non-uniform dependence on the data for the 2D viscous shallow water equations.","Our main result of this paper is stated: Theorem 1.1 (Nonuniform dependence on initial data) Let $s>2$ .","The data-to-solution map $(u_0,\\varrho _0)\\mapsto \\big (u(t),\\varrho (t)\\big )$ of the Cauchy problem (REF ) is not uniformly continuous from any bounded subset in $H^{s}\\times H^{s}$ into $\\mathcal {C}([0,T];H^{s}\\times H^{s})$ , namely, there exists two sequences of solutions $\\big (u_{1,n}(t),\\varrho _{1,n}(t)\\big )$ and $\\big (u_{2,n}(t),\\varrho _{2,n}(t)\\big )$ such that (C.1) $\\Vert u_{1,n},\\varrho _{1,n}\\Vert _{H^{s}}+\\Vert u_{2,n},\\varrho _{2,n}\\Vert _{H^{s}}\\lesssim 1;$ (C.2) $\\lim \\limits _{n\\rightarrow \\infty }\\big (\\Vert {u}_{2,n}(0)-u_{1,n}(0)\\Vert _{H^{s}}+\\Vert \\varrho _{2,n}(0)-\\varrho _{1,n}(0)\\Vert _{H^{s}}\\big )=0;$ (C.3) $\\liminf \\limits _{n\\rightarrow \\infty }\\big (\\Vert u_{2,n}(t)-u_{1,n}(t)\\Vert _{H^{s}}+\\Vert \\varrho _{2,n}(t)-\\varrho _{1,n}(t)\\Vert _{H^{s}}\\big )\\gtrsim t$ for any $t\\in [0,T_0]$ with small time $T_0$ .","Remark 1.1 It should be mentioned that most non-uniformity results for solutions to other nonlinear PDEs were obtained by the Himonas-Misiołek construction in [18].","The method we used in proving Theorem REF is motivated by [29] and is completely different from that [18].","Remark 1.2 We note that the local well-posedness result holds for $s>1$ .","However, the restriction on $s>2$ is essential in the present paper.","Outline of the proof to Theorem REF There are two key points in proving Theorem REF : On one hand, we construct one sequence of initial data $(u_{1,n}(0),\\varrho _{1,n}(0))=(0,f_n)$ , which leads to the solutions $(u_{1,n}(t),\\varrho _{1,n}(t))$ to (REF ) and $(u^{\\rm {ap}}_{1,n},\\varrho ^{\\rm {ap}}_{1,n})$ to the linearized system (REF ) respectively.","Based on the special choice of $f_n$ , we can prove the distance between the real solution $(u_{1,n}(t),\\varrho _{1,n}(t))$ and the approximate solution $(u^{\\rm {ap}}_{1,n},\\varrho ^{\\rm {ap}}_{1,n})$ will tends to zero in $H^s$ as $n\\rightarrow \\infty $ .","See Propositions REF .","On the other hand, we construct another sequence of initial data $(u_{2,n}(0),\\varrho _{2,n}(0))=(g_n,f_n)$ , which leads to the solutions $(u_{2,n}(t),\\varrho _{2,n}(t))$ to (REF ) and $(u^{\\rm {ap}}_{2,n},\\varrho ^{\\rm {ap}}_{2,n})$ to the linearized system (REF ) respectively.","Next, we aim to show that the mentioned solution $(u^{\\rm {ap}}_{2,n},\\varrho ^{\\rm {ap}}_{2,n})$ cannot approximate the real solution $(u_{2,n}(t),\\varrho _{2,n}(t))$ .","For more details, see Propositions REF .","Combining the precious steps, we can conclude that the distance of two solution maps at the initial time is converging to zero, while at any later time it is bounded below by a positive constant, which means the solution maps are not uniformly continuous.","The structure of the paper In Section we recall some notations and known results which will be used in the sequel.","Also, we investigate the spectrum properties of the linearized system corresponding to (REF ), which will play a crucial role in the construction of approximate solutions.","In Section we prove Theorem REF based on the well-posedness result and some key error estimates.","In Section we present some details in the computations."],["Notations","Firstly, we introduce some notations which shall be used in this paper.","The notation $A\\lesssim B$ (resp., $A \\gtrsim B$ ) means that there exists a harmless positive constant $c$ such that $A \\le cB$ (resp., $A \\ge cB$ ).","$A\\approx B$ means $A\\lesssim B$ and $A\\gtrsim B$ .","Given a Banach space $X$ , we denote its norm by $\\Vert \\cdot \\Vert _{X}$ .","We shall use the simplified notation $\\Vert f,\\cdots ,g\\Vert _X=\\Vert f\\Vert _X+\\cdots +\\Vert g\\Vert _X$ if there is no confusion.","We shall denote by $\\langle f,g\\rangle $ the $L^2$ inner product of $f$ and $g$ .","For all $f\\in \\mathcal {S}^{\\prime }$ , the Fourier transform $\\mathcal {F}f$ (also denoted by $\\widehat{f}$ ) is defined by $\\mathcal {F}f(\\xi )=\\widehat{f}(\\xi )=\\int _{\\mathbb {R}^2}e^{-ix\\xi }f(x)\\mathrm {d}x \\quad \\text{for any}\\; \\xi \\in \\mathbb {R}^2.$ We denote $\\Lambda =(-\\Delta )^{\\frac{1}{2}}$ .","For $s\\in \\mathbb {R}$ , the operator $\\Lambda ^s$ is defined by $\\widehat{\\Lambda ^s f}(\\xi )=|\\xi |^{s}\\widehat{f}(\\xi ).$ The homogeneous and nonhomogeneous Sobolev space are defined by $&\\Vert f\\Vert ^2_{\\dot{H}^s}=\\Vert \\Lambda ^sf\\Vert ^2_{L^2}=\\int _{\\mathbb {R}^2}|\\xi |^{2s}|\\hat{f}(\\xi )|^2\\mathrm {d}\\xi ,\\\\&\\Vert f\\Vert ^2_{H^s}=\\int _{\\mathbb {R}^2}(1+|\\xi |^2)^s|\\hat{f}(\\xi )|^2\\mathrm {d}\\xi .$ Then, for $s>0$ , we have $\\Vert f\\Vert ^2_{H^s}\\approx \\Vert f\\Vert ^2_{L^2}+\\Vert f\\Vert ^2_{\\dot{H}^s}.$"],["Useful Tools","Next, we review some useful tools involving the commutator and product estimates.","Lemma 2.1 (See [13]) Let $s>1$ and $\\nabla f,g\\in H^s(\\mathbb {R}^2)$ .","Then we have $\\Vert [\\Lambda ^s,f]\\cdot \\nabla g\\Vert _{L^2}\\le C\\Vert \\nabla f\\Vert _{H^s}\\Vert g\\Vert _{H^s}.$ Lemma 2.2 (See [27]) Let $s>0$ and $f,g\\in {\\rm {Lip}}\\cap H^s(\\mathbb {R}^2)$ and $g\\in L^\\infty \\cap H^{s-1}(\\mathbb {R}^2)$ .","Then we have $\\Vert [\\Lambda ^s,f]g\\Vert _{L^2}\\le C\\big (\\Vert \\nabla f\\Vert _{L^\\infty }\\Vert g\\Vert _{H^{s-1}}+\\Vert f\\Vert _{H^s}\\Vert g\\Vert _{L^\\infty }\\big ).$ Lemma 2.3 (See [4]) Let $s>2$ and $f\\in H^{s-2}(\\mathbb {R}^2),g\\in H^{s-1}(\\mathbb {R}^2)$ .","Then we have $\\Vert fg\\Vert _{H^{s-2}}\\le C\\Vert f\\Vert _{H^{s-2}}\\Vert g\\Vert _{H^{s-1}}.$ Lemma 2.4 (See [4]) Let $s>0$ and $f,g\\in L^\\infty \\cap H^s(\\mathbb {R}^2)$ .","Then we have $\\Vert fg\\Vert _{H^s}\\le C\\big (\\Vert f\\Vert _{L^\\infty }\\Vert g\\Vert _{H^s}+\\Vert f\\Vert _{H^s}\\Vert g\\Vert _{L^\\infty }\\big ).$ Moreover, for $s>1$ , we have the algebra estimate $\\Vert fg\\Vert _{H^s}\\le C\\Vert f\\Vert _{H^s}\\Vert g\\Vert _{H^s}.$ Lemma 2.5 (See [4]) Let $s>0$ and $f$ be a smooth function such that $f(0)=0.$ If $u \\in H^{s}\\left(\\mathbb {R}^{2}\\right) \\cap L^{\\infty }\\left(\\mathbb {R}^{2}\\right)$ then there exists a function $C$ depending only on $s$ and $f$ such that $\\Vert f(u)\\Vert _{H^{s}} \\le C\\big (\\Vert u\\Vert _{L^{\\infty }}\\big )\\Vert u\\Vert _{H^{s}}.$ Lemma 2.6 (See [4]) Let $s>1$ and $f$ be a smooth function such that $f^{\\prime }(0)=0 .$ If $u, v \\in H^{s}\\left(\\mathbb {R}^{2}\\right) \\cap L^{\\infty }\\left(\\mathbb {R}^{2}\\right)$ , then there exists a function $C$ depending only on $s$ and $f$ such that $\\Vert f(u)-f(v)\\Vert _{H^{s}} \\le C\\big (\\Vert u\\Vert _{L^{\\infty }},\\Vert v\\Vert _{L^{\\infty }}\\big )\\Vert u-v\\Vert _{H^{s}}\\big (\\Vert u\\Vert _{H^{s}}+\\Vert v\\Vert _{H^{s}}\\big ).$"],["The Linearized System","In this subsection, borrowing the idea from [7], we investigate the spectrum properties of the linearized system: $\\left\\lbrace \\begin{array}{ll}\\partial _t\\varrho +u̥=0,\\\\\\partial _tu-{\\mu } \\Delta u-{\\mu } \\nabla u̥+\\nabla \\varrho =0,\\\\\\varrho |_{t=0}=\\varrho _0,\\;u|_{t=0}=u_0.\\end{array}\\right.$ Denote $d=\\Lambda ^{-1}u̥\\quad \\text{and}\\quad c=\\Lambda ^{-1}\\mathrm {curl} u,$ then we deduce from (REF ) $\\left\\lbrace \\begin{array}{ll}\\partial _t\\varrho +\\Lambda d=0,\\\\\\partial _td-2{\\mu } \\Delta d-\\Lambda \\varrho =0,\\\\\\partial _tc-{\\mu } \\Delta c=0\\end{array}\\right.$ Let $\\mathbf {A}=\\left(\\begin{array}{cc}0 & -|\\xi | \\\\|\\xi | & -2\\mu |\\xi |^2 \\\\\\end{array}\\right),$ by taking the Fourier transform of $(\\ref {l1})_1$ and $(\\ref {l1})_2$ , we find that $\\partial _t\\left(\\begin{array}{c}\\widehat{\\varrho }\\\\\\widehat{d} \\\\\\end{array}\\right)=\\mathbf {A}\\left(\\begin{array}{c}\\widehat{\\varrho }\\\\\\widehat{d} \\\\\\end{array}\\right).$ Straightforward calculations give the eigenvalues of the matric $\\mathbf {A}$ as follows $\\lambda _{\\pm }=-\\mu |\\xi |^2\\pm \\sqrt{\\mu ^2|\\xi |^4-|\\xi |^2}$ and $\\mathbf {T}^{-1}\\mathbf {A}\\mathbf {T}=\\left(\\begin{array}{cc}\\lambda _{+} & 0 \\\\0 & \\lambda _{-} \\\\\\end{array}\\right),$ where the matrixes $\\mathbf {T}$ and $\\mathbf {T}^{-1}$ are given by $\\mathbf {T}=\\left(\\begin{array}{cc}\\lambda _{-} & \\lambda _{+} \\\\-|\\xi | & -|\\xi | \\\\\\end{array}\\right)\\quad \\text{and}\\quad \\mathbf {T}^{-1}=\\left(\\begin{array}{cc}-\\frac{1}{\\lambda _{+}-\\lambda _{-}} & -\\frac{\\lambda _{+}}{|\\xi |(\\lambda _{+}-\\lambda _{-})} \\\\\\frac{1}{\\lambda _{+}-\\lambda _{-}} & \\frac{\\lambda _{-}}{|\\xi |(\\lambda _{+}-\\lambda _{-})} \\\\\\end{array}\\right).$ Applying $\\mathbf {T}^{-1}$ to (REF ) and using Duhamel's principle, we get $\\mathbf {T}^{-1}\\left(\\begin{array}{c}\\widehat{\\varrho }\\\\\\widehat{d} \\\\\\end{array}\\right)=\\left(\\begin{array}{cc}e^{\\lambda _+t}&0\\\\0&e^{\\lambda _-t}\\\\\\end{array}\\right)\\mathbf {T}^{-1}\\left(\\begin{array}{c}\\widehat{\\varrho }_0\\\\\\widehat{d}_0 \\\\\\end{array}\\right),$ which implies that $&\\widehat{\\varrho }(t,\\xi )=\\frac{\\lambda _+e^{\\lambda _-t}-\\lambda _-e^{\\lambda _+t}}{\\lambda _+-\\lambda _-}\\widehat{\\varrho }_0-\\frac{e^{\\lambda _+t}-e^{\\lambda _-t}}{\\lambda _+-\\lambda _-}|\\xi |\\widehat{d}_0,\\\\&\\widehat{d}(t,\\xi )=\\frac{e^{\\lambda _+t}-e^{\\lambda _-t}}{\\lambda _+-\\lambda _-}|\\xi |\\widehat{\\varrho }_0+\\frac{\\lambda _+e^{\\lambda _+t}-\\lambda _-e^{\\lambda _-t}}{\\lambda _+-\\lambda _-}\\widehat{d}_0.$ From $(\\ref {l1})_3$ , we also have $&\\widehat{c}(t,\\xi )=e^{-\\mu t|\\xi |^2}\\widehat{c}_0.$ Due to the vector identity $\\Delta u=\\nabla u̥+\\nabla ^{\\bot }\\mathrm {curl}u=\\Lambda (\\nabla d+\\nabla ^{\\bot } c),$ which implies $u=-\\Lambda ^{-1}\\nabla d-\\Lambda ^{-1}\\nabla ^{\\bot } c,$ then we deduce from () and (REF ) $\\widehat{u}(t,\\xi )&=-\\frac{e^{\\lambda _+t}-e^{\\lambda _-t}}{\\lambda _+-\\lambda _-}\\mathcal {F}(\\nabla \\varrho _0)-\\frac{\\lambda _+e^{\\lambda _+t}-\\lambda _-e^{\\lambda _-t}}{\\lambda _+-\\lambda _-}\\mathcal {F}\\big (\\Lambda ^{-2}\\nabla u̥_0\\big )\\nonumber \\\\&\\quad -e^{-\\mu |\\xi |^2t }\\mathcal {F}(\\Lambda ^{-2}\\nabla ^{\\bot }\\mathrm {curl} u_0).$"],["Proof of Theorem ","This section is devoted to proving Theorem REF .","We begin with the well-posedness result for (REF ).","For the details of the proof, we refer to [30], [31], [36]."],["Global Well-posedness ","Lemma 3.1 Assume that $s>2$ .","For any initial data $(u_0,\\varrho _0)$ which belongs to $B_R=\\big \\lbrace (\\psi ,\\phi )\\in H^s\\times H^s: \\Vert \\psi \\Vert _{H^s}+\\Vert \\phi \\Vert _{H^s}\\le c\\big \\rbrace \\quad \\text{for small}\\;c>0,$ then System (REF ) has a unique global solution $\\big (u(t),\\varrho (t)\\big )\\in \\mathcal {C}([0,+\\infty );H^s\\times H^s)$ .","Moreover, we have the solution size estimate $\\Vert u(t)\\Vert _{H^{\\sigma }}+\\Vert \\varrho (t)\\Vert _{H^{\\sigma }}\\le C\\Vert u_0,\\varrho _0\\Vert _{H^{\\sigma }}\\quad \\text{for all}\\; \\sigma \\ge s.$"],["Approximate Solutions","Let $\\widehat{\\phi }\\in \\mathcal {C}^\\infty _0(\\mathbb {R})$ be an even, real-valued and non-negative function on $\\mathbb {R}$ and satisfy ()= 1,if $|\\xi |\\le \\frac{1}{4}$ , 0,if $|\\xi |\\ge \\frac{1}{2}$ .","Let $(u^{\\rm {ap}}_{i,n},\\varrho ^{\\rm {ap}}_{i,n})$ ($i=1,2$ ) be the solution of the following equation $\\left\\lbrace \\begin{array}{ll}\\partial _t\\varrho ^{\\rm {ap}}_{i,n}+u̥^{\\rm {ap}}_{i,n}=0,\\\\\\partial _tu^{\\rm {ap}}_{i,n}-\\mu \\Delta u^{\\rm {ap}}_{i,n}-\\mu \\nabla u̥^{\\rm {ap}}_{i,n}+\\nabla \\varrho ^{\\rm {ap}}_{i,n}=0,\\\\\\end{array}\\right.$ supplemented with the following initial condition, respectively, $(u^{\\rm {ap}}_{1,n},\\varrho ^{\\rm {ap}}_{1,n})|_{t=0}=(0,f_n)\\quad \\text{and}\\quad (u^{\\rm {ap}}_{2,n},\\varrho ^{\\rm {ap}}_{2,n})|_{t=0}=(g_n,f_n),$ where $f_n=2^{-ns}\\phi (x_1)\\sin (2^nx_1)\\phi (x_2)\\quad \\text{and}\\quad g_n=2^{-n}\\nabla \\big (\\phi (x_1)\\phi (x_2)\\big ),\\quad n\\gg 1.$ The following Lemma will play an important role in the proof of Theorem REF .","Lemma 3.2 Let $\\sigma \\in \\mathbb {R}$ .","Assume that $(u^{\\rm {ap}}_{i,n},\\varrho ^{\\rm {ap}}_{i,n})$ with $i=1,2$ solves System (REF )–(REF ).","Then there exists a constant $C=C(\\sigma )$ such that the following statement holds $&\\Vert \\varrho ^{\\rm {ap}}_{1,n},u^{\\rm {ap}}_{1,n}\\Vert ^2_{H^\\sigma }+\\mu \\int ^t_0\\Vert \\nabla u^{\\rm {ap}}_{1,n},u̥^{\\rm {ap}}_{1,n}\\Vert ^2_{H^\\sigma }\\mathrm {d}\\tau \\le C2^{2n(\\sigma -s)},\\\\&\\Vert \\varrho ^{\\rm {ap}}_{2,n},u^{\\rm {ap}}_{2,n}\\Vert ^2_{H^\\sigma }+\\mu \\int ^t_0\\Vert \\nabla u^{\\rm {ap}}_{2,n},u̥^{\\rm {ap}}_{2,n}\\Vert ^2_{H^\\sigma }\\mathrm {d}\\tau \\le C2^{2n \\max \\lbrace \\sigma -s,-1\\rbrace }.$ Proof.","Standard energy method directly gives us that $\\frac{\\mathrm {d}}{\\mathrm {d}t}\\Vert \\varrho ^{\\rm {ap}}_{i,n},u^{\\rm {ap}}_{i,n}\\Vert ^2_{H^\\sigma }+\\mu \\Vert \\nabla u^{\\rm {ap}}_{i,n}\\Vert ^2_{H^\\sigma }+\\mu \\Vert u̥^{\\rm {ap}}_{i,n}\\Vert ^2_{H^\\sigma }=0,$ which implies $&\\Vert \\varrho ^{\\rm {ap}}_{1,n},u^{\\rm {ap}}_{1,n}\\Vert ^2_{H^\\sigma }+\\mu \\int ^t_0\\Vert \\nabla u^{\\rm {ap}}_{1,n},u̥^{\\rm {ap}}_{1,n}\\Vert ^2_{H^\\sigma }\\mathrm {d}\\tau =\\Vert f_n\\Vert ^2_{H^\\sigma },\\\\&\\Vert \\varrho ^{\\rm {ap}}_{2,n},u^{\\rm {ap}}_{2,n}\\Vert ^2_{H^\\sigma }+\\mu \\int ^t_0\\Vert \\nabla u^{\\rm {ap}}_{2,n},u̥^{\\rm {ap}}_{2,n}\\Vert ^2_{H^\\sigma }\\mathrm {d}\\tau =\\Vert f_n,g_n\\Vert ^2_{H^\\sigma }.$ Obviously, we have $&\\Vert g_n\\Vert _{H^\\sigma }\\lesssim 2^{-n}\\Vert \\phi (x_1)\\Vert _{L^2(\\mathbb {R})}\\Vert \\phi (x_2)\\Vert _{L^2(\\mathbb {R})}\\lesssim 2^{-n},\\nonumber \\\\&\\Vert f_n\\Vert _{H^\\sigma }\\lesssim 2^{-ns}\\Vert \\widetilde{f_n}(x_1,x_2)\\Vert _{H^\\sigma (\\mathbb {R}^2)},$ where $\\widetilde{f_n}(x_1,x_2)=\\phi (x_1)\\sin (2^nx_1)\\phi (x_2)$ .","Easy computations give that $\\mathcal {F}\\big (\\widetilde{f_n}\\big )(\\xi _1,\\xi _2)=\\frac{i}{2}\\Big [\\hat{\\phi }(\\xi _1+2^n)-\\hat{\\phi }(\\xi _1-2^n)\\Big ]\\hat{\\phi }(\\xi _2),$ which implies $\\mathrm {supp} \\ \\mathcal {F}\\big (\\widetilde{f_n}\\big )\\subset \\mathcal {C}_n:=\\Big \\lbrace \\xi \\in \\mathbb {R}^2: \\ 2^n-1\\le |\\xi |\\le 2^n+1\\Big \\rbrace ,$ By the definition of Sobolev space, we get $\\Vert \\widetilde{f_n}(x_1,x_2)\\Vert _{H^\\sigma (\\mathbb {R}^2)}&\\lesssim \\int _{\\mathbb {R}^2}(1+|\\xi |^2)^{\\sigma /2}|\\mathcal {F}\\big (\\widetilde{f_n}\\big )|^2\\mathrm {d}\\xi \\nonumber \\\\&\\lesssim ~2^{n\\sigma }\\int _{\\mathcal {C}_n }|\\mathcal {F}\\big (\\widetilde{f_n}\\big )|^2\\mathrm {d}\\xi \\nonumber \\\\&\\lesssim 2^{n\\sigma }.$ Inserting (REF ) into (REF ) enables us to finish the proof of Lemma REF .","Before proceeding on, we give two data-to-solution maps for the Cauchy problem (REF ) $&\\big (u_{1,n}(0)=0,\\varrho _{1,n}(0)=f_n\\big )\\mapsto \\big (u_{1,n},\\varrho _{1,n}\\big ),\\\\&\\big ({u}_{2,n}(0)=g_n,{\\varrho }_{2,n}(0)=f_n\\big )\\mapsto \\big ({u}_{2,n},{\\varrho }_{2,n}\\big ),$ where $f_{n}$ and $g_n$ are defined in Section REF .","Let $\\varepsilon _s=\\frac{1}{2}(s-2)$ and $s^{\\prime }=s-\\varepsilon _s>2$ .","For the initial data with $H^{s^{\\prime }}$ norm, we have $\\Vert u_{1,n}(0),u_{2,n}(0)\\Vert _{H^{s^{\\prime }}}+\\Vert \\varrho _{1,n}(0),\\varrho _{2,n}(0)\\Vert _{H^{s^{\\prime }}}\\le 2^{-n\\min \\lbrace \\varepsilon _s,1\\rbrace },$ which tends to 0 when $n$ tends to infinity.","Therefore, by Lemma REF , we have the solutions size estimate which will be used implicitly in the sequel $&\\Vert u_{1,n},\\varrho _{1,n}\\Vert _{H^\\gamma }+\\Vert u_{2,n},\\varrho _{2,n}\\Vert _{H^{\\gamma }}\\lesssim 2^{-n\\min \\lbrace (s-\\gamma ),1\\rbrace }\\quad \\text{for all}\\; \\gamma \\ge s^{\\prime }.$ In particular, this implies that the condition $(\\bf {C.1})$ in Theorem REF holds."],["Error Estimates","Letting $\\varrho _{1,n}^{\\rm {er}}=\\varrho _{1,n}-\\varrho _{1,n}^{\\rm {ap}}$ and $u_{1,n}^{\\rm {er}}=u_{1,n}-u_{1,n}^{\\rm {ap}}$ , we can find that $(\\varrho _{1,n}^{\\rm {er}},u_{1,n}^{\\rm {er}})$ satisfies $\\left\\lbrace \\begin{array}{ll}\\partial _t\\varrho _{1,n}^{\\rm {er}}+u̥_{1,n}^{\\rm {er}}+u_{1,n}\\cdot \\nabla \\varrho _{1,n}^{\\rm {er}}=\\mathbf {F_1},\\\\\\partial _tu_{1,n}^{\\rm {er}}-{\\mu } \\Delta u_{1,n}^{\\rm {er}}-{\\mu } \\nabla u̥_{1,n}^{\\rm {er}}+\\nabla \\varrho _{1,n}^{\\rm {er}}+u_{1,n}\\cdot \\nabla u_{1,n}^{\\rm {er}}=\\mathbf {F_2}+\\mathbf {F_3},\\\\\\varrho _{1,n}^{\\rm {er}}|_{t=0}=0,\\; u_{1,n}^{\\rm {er}}|_{t=0}=0, \\end{array}\\right.$ where $&\\mathbf {F_1}:=-u_{1,n}^{\\rm {er}}\\cdot \\nabla \\varrho ^{\\rm {ap}}_{1,n}-\\varrho _{1,n}u̥_{1,n}^{\\rm {er}}-\\varrho _{1,n}^{\\rm {er}}u̥_{1,n}^{\\rm {ap}}-\\varrho ^{\\rm {ap}}_{1,n}u_{1,n}^{\\rm {ap}}),\\\\&\\mathbf {F_2}:=-u_{1,n}^{\\rm {er}}\\cdot \\nabla u^{\\rm {ap}}_{1,n}- u^{\\rm {ap}}_{1,n}\\cdot \\nabla u^{\\rm {ap}}_{1,n},\\\\&\\mathbf {F_3}:=2\\nabla (\\ln (1+\\varrho _{1,n}))\\cdot \\mathbb {D}u_{1,n}.$ The following proposition implies that the error of approximate solution and actual solution will tends to zero in $H^s$ as $n\\rightarrow \\infty $ .","Proposition 3.1 Under the assumptions of Theorem REF , then we have for $t\\le 1$ $\\Vert u_{1,n}-u_{1,n}^{\\rm {ap}}\\Vert _{H^s}+\\Vert \\varrho _{1,n}-\\varrho _{1,n}^{\\rm {ap}}\\Vert _{H^s}\\le C2^{-\\frac{n}{2}\\min \\lbrace \\varepsilon _s,1\\rbrace }.$ Proof.","Taking the $L^2$ inner product of $(\\ref {u1})_1$ and $(\\ref {u1})_2$ with $(\\varrho _{1,n}^{\\rm {er}},u_{1,n}^{\\rm {er}})$ yields $&\\frac{1}{2}\\frac{\\mathrm {d}}{\\mathrm {d}t}\\Vert \\varrho _{1,n}^{\\rm {er}},u_{1,n}^{\\rm {er}}\\Vert ^2_{L^2}+\\mu \\Vert \\nabla u_{1,n}^{\\rm {er}},u̥_{1,n}^{\\rm {er}}\\Vert ^2_{L^{2}}\\nonumber \\\\=&~\\frac{1}{2} \\big + \\big <\\mathbf {F_1}, \\varrho _{1,n}^{\\rm {er}}\\big >+\\big <\\mathbf {F_2}+\\mathbf {F_3}, u_{1,n}^{\\rm {er}}\\big >\\nonumber \\\\\\lesssim &~ \\Vert u_{1,n}\\Vert _{H^{s}}\\Vert \\varrho _{1,n}^{\\rm {er}},u_{1,n}^{\\rm {er}}\\Vert ^2_{L^{2}}+\\Vert \\mathbf {F_1}\\Vert _{L^{2}}\\Vert \\varrho _{1,n}^{\\rm {er}}\\Vert _{L^2}+\\Vert \\mathbf {F_2},\\mathbf {F_3},u_{1,n}^{\\rm {er}}\\Vert ^2_{L^{2}}.$ Applying the operators $\\Lambda ^{s-1}\\varrho _{1,n}^{\\rm {er}}\\Lambda ^{s-1}$ and $\\Lambda ^{s-1}u_{1,n}^{\\rm {er}}\\Lambda ^{s-1}$ to $(\\ref {u1})_1$ and $(\\ref {u1})_2$ , respectively, then integrating the resulting over $\\mathbb {R}^2$ , we obtain $&\\frac{1}{2}\\frac{\\mathrm {d}}{\\mathrm {d}t}\\Vert \\varrho _{1,n}^{\\rm {er}},u_{1,n}^{\\rm {er}}\\Vert ^2_{{\\dot{H}}^{s-1}}+\\mu \\Vert \\nabla u_{1,n}^{\\rm {er}},u̥_{1,n}^{\\rm {er}}\\Vert ^2_{{\\dot{H}}^{s-1}}\\nonumber \\\\=&~\\frac{1}{2} \\big \\nonumber \\\\&-\\big <[\\Lambda ^{s-1},u_{1,n}]\\cdot \\nabla \\varrho _{1,n}^{\\rm {er}},\\Lambda ^{s-1}\\varrho _{1,n}^{\\rm {er}}\\big >-\\big <[\\Lambda ^{s-1},u_{1,n}]\\cdot \\nabla u_{1,n}^{\\rm {er}},\\Lambda ^{s-1}u_{1,n}^{\\rm {er}}\\big >\\nonumber \\\\&+ \\big <\\Lambda ^{s-1}\\mathbf {F_1}, \\Lambda ^{s-1}\\varrho _{1,n}^{\\rm {er}}\\big >+\\big < \\Lambda ^{s-2}\\mathbf {F_2}, \\Lambda ^{s}u_{1,n}^{\\rm {er}}\\big >+\\big < \\Lambda ^{s-2}\\mathbf {F_3}, \\Lambda ^{s}u_{1,n}^{\\rm {er}}\\big >\\nonumber \\\\\\lesssim &~ (1+\\Vert u_{1,n}\\Vert _{H^{s}})\\Vert \\varrho _{1,n}^{\\rm {er}},u_{1,n}^{\\rm {er}}\\Vert ^2_{{{H}}^{s-1}}+\\Vert \\mathbf {F_1}\\Vert _{{\\dot{H}}^{s-1}}\\Vert \\varrho _{1,n}^{\\rm {er}}\\Vert _{{\\dot{H}}^{s-1}}\\nonumber \\\\&+\\Vert \\mathbf {F_2},\\mathbf {F_3}\\Vert ^2_{{\\dot{H}}^{s-2}}+\\varepsilon \\Vert \\nabla u_{1,n}^{\\rm {er}}\\Vert ^2_{{\\dot{H}}^{s-1}},$ where we have used the commutator estimate from Lemma REF .","Combing (REF ) and (REF ), we get $\\frac{\\mathrm {d}}{\\mathrm {d}t}\\Vert \\varrho _{1,n}^{\\rm {er}},u_{1,n}^{\\rm {er}}\\Vert ^2_{{{H}}^{s-1}}&+\\mu \\Vert \\nabla u_{1,n}^{\\rm {er}},u̥_{1,n}^{\\rm {er}}\\Vert ^2_{{{H}}^{s-1}}\\lesssim \\Vert \\varrho _{1,n}^{\\rm {er}},u_{1,n}^{\\rm {er}}\\Vert ^2_{{{H}}^{s-1}}\\nonumber \\\\&\\quad +\\Vert \\mathbf {F_1}\\Vert _{{{H}}^{s-1}}\\Vert \\varrho _{1,n}^{\\rm {er}}\\Vert _{{{H}}^{s-1}}+\\Vert \\mathbf {F_2},\\mathbf {F_3}\\Vert ^2_{{{H}}^{s-2}}+\\varepsilon \\Vert \\nabla u_{1,n}^{\\rm {er}}\\Vert ^2_{{\\dot{H}}^{s-1}}.$ Estimate of $\\mathbf {F_1}$ .","Notice that $H^{s-1}(\\mathbb {R}^2)$ with $s>2$ is a Banach algebra, we have $\\Vert u_{1,n}^{\\rm {er}}\\cdot \\nabla \\varrho ^{\\rm {ap}}_{1,n}\\Vert _{H^{s-1}}&\\lesssim \\Vert u_{1,n}^{\\rm {er}}\\Vert _{H^{s-1}}\\Vert \\varrho ^{\\rm {ap}}_{1,n}\\Vert _{H^s}\\lesssim \\Vert u_{1,n}^{\\rm {er}}\\Vert _{H^{s-1}},\\\\\\Vert \\varrho _{1,n}u̥_{1,n}^{\\rm {er}}\\Vert _{H^{s-1}}&\\lesssim \\Vert \\varrho _{1,n}\\Vert _{H^{s-1}}\\Vert u̥_{1,n}^{\\rm {er}}\\Vert _{H^{s-1}}\\lesssim \\Vert u̥_{1,n}^{\\rm {er}}\\Vert _{H^{s-1}},\\\\\\Vert \\varrho _{1,n}^{\\rm {er}}u̥_{1,n}^{\\rm {ap}}\\Vert _{H^{s-1}}&\\lesssim \\Vert \\varrho _{1,n}^{\\rm {er}}\\Vert _{H^{s-1}}\\Vert u^{\\rm {ap}}_{1,n}\\Vert _{H^s}\\lesssim \\Vert \\varrho _{1,n}^{\\rm {er}}\\Vert _{H^{s-1}},\\\\\\Vert \\varrho ^{\\rm {ap}}_{1,n}u_{1,n}^{\\rm {ap}})\\Vert _{H^{s-1}}&\\lesssim \\Vert \\varrho ^{\\rm {ap}}_{1,n}u_{1,n}^{\\rm {ap}}\\Vert _{H^{s}}\\lesssim \\Vert \\varrho ^{\\rm {ap}}_{1,n}\\Vert _{L^\\infty }\\Vert u_{1,n}^{\\rm {ap}}\\Vert _{H^{s}}+\\Vert \\varrho ^{\\rm {ap}}_{1,n}\\Vert _{H^{s}}\\Vert u_{1,n}^{\\rm {ap}}\\Vert _{L^\\infty }\\\\&\\lesssim \\Vert \\varrho ^{\\rm {ap}}_{1,n}\\Vert _{H^{s-1-\\varepsilon _s}}\\Vert u_{1,n}^{\\rm {ap}}\\Vert _{H^{s}}+\\Vert \\varrho ^{\\rm {ap}}_{1,n}\\Vert _{H^{s}}\\Vert u_{1,n}^{\\rm {ap}}\\Vert _{H^{s-1-\\varepsilon _s}}\\lesssim 2^{-(1+\\varepsilon _s)n},$ which implies $\\Vert \\mathbf {F_1}\\Vert _{H^{s-1}}&\\lesssim \\Vert u_{1,n}^{\\rm {er}},\\varrho _{1,n}^{\\rm {er}}\\Vert _{H^{s-1}}+\\Vert u̥_{1,n}^{\\rm {er}}\\Vert _{H^{s-1}}+2^{-(1+\\varepsilon _s)n}.$ Estimate of $\\mathbf {F_2}$ .","Using Lemma REF yields $\\Vert u_{1,n}^{\\rm {er}}\\cdot \\nabla u^{\\rm {ap}}_{1,n}\\Vert _{H^{s-2}}&\\lesssim \\Vert u_{1,n}^{\\rm {er}}\\Vert _{H^{s-1}}\\Vert u^{\\rm {ap}}_{1,n}\\Vert _{H^{s-1}}\\lesssim \\Vert u_{1,n}^{\\rm {er}}\\Vert _{H^{s-1}},\\\\\\Vert u^{\\rm {ap}}_{1,n}\\cdot \\nabla u^{\\rm {ap}}_{1,n}\\Vert _{H^{s-2}}&\\lesssim \\Vert u^{\\rm {ap}}_{1,n}\\Vert ^2_{H^{s-1}}\\lesssim 2^{-2n},$ which implies $\\Vert \\mathbf {F_2}\\Vert _{H^{s-2}}&\\lesssim \\Vert u_{1,n}^{\\rm {er}}\\Vert _{H^{s-1}}+2^{-2n}.$ Estimate of $\\mathbf {F_3}$ .","We can decompose the term $\\nabla (\\ln (1+\\varrho _{1,n}))\\cdot \\mathbb {D}u_{1,n}$ as $\\nabla (\\ln (1+\\varrho _{1,n}))\\cdot \\mathbb {D}u_{1,n}&=\\nabla (\\ln (1+\\varrho _{1,n}))\\cdot \\mathbb {D}u_{1,n}^{\\rm {er}}+\\nabla (\\ln (1+\\varrho _{1,n})-\\ln (1+\\varrho ^{\\rm {ap}}_{1,n}))\\cdot \\mathbb {D}u^{\\rm {ap}}_{1,n}\\\\&\\quad +\\nabla (\\ln (1+\\varrho ^{\\rm {ap}}_{1,n}))\\cdot \\mathbb {D}u^{\\rm {ap}}_{1,n}\\\\&=\\mathbf {F_{3,1}}+\\mathbf {F_{3,2}}+\\mathbf {F_{3,3}}.$ For the first two terms, by Lemma REF and Lemmas REF –REF , we have $&\\Vert \\mathbf {F_{3,1}}\\Vert _{H^{s-2}}\\lesssim \\Vert \\mathbb {D}u_{1,n}^{\\rm {er}}\\Vert _{H^{s-2}}\\Vert \\varrho _{1,n}\\Vert _{H^s}\\lesssim \\Vert u_{1,n}^{\\rm {er}}\\Vert _{H^{s-1}},\\\\&\\Vert \\mathbf {F_{3,2}}\\Vert _{H^{s-2}}\\lesssim \\Vert \\varrho _{1,n}^{\\rm {er}}\\Vert _{H^{s-1}}\\Vert u^{\\rm {ap}}_{1,n}\\Vert _{H^s}\\lesssim \\Vert \\varrho _{1,n}^{\\rm {er}}\\Vert _{H^{s-1}}.$ For the third term, by Lemmas REF and REF , we have $\\Vert \\mathbf {F_{3,3}}\\Vert _{H^{s-2}}&\\lesssim \\Vert \\nabla (\\ln (1+\\varrho ^{\\rm {ap}}_{1,n}))\\Vert _{L^\\infty }\\Vert u_{1,n}^{\\rm {ap}}\\Vert _{H^{s-1}}+\\Vert \\nabla (\\ln (1+\\varrho ^{\\rm {ap}}_{1,n}))\\Vert _{H^{s-2}}\\Vert \\nabla u_{1,n}^{\\rm {ap}}\\Vert _{L^\\infty }\\\\&\\lesssim \\Vert \\varrho ^{\\rm {ap}}_{1,n}\\Vert _{H^{s-\\varepsilon _s}}\\Vert u_{1,n}^{\\rm {ap}}\\Vert _{H^{s-1}}+\\Vert \\varrho ^{\\rm {ap}}_{1,n}\\Vert _{H^{s-1}}\\Vert u_{1,n}^{\\rm {ap}}\\Vert _{H^{s-\\varepsilon _s}}\\lesssim 2^{-(1+\\varepsilon _s)n},$ which implies $\\Vert \\mathbf {F_3}\\Vert _{H^{s-2}}&\\lesssim \\Vert u_{1,n}^{\\rm {er}},\\varrho _{1,n}^{\\rm {er}}\\Vert _{H^{s-1}}+2^{-(1+\\varepsilon _s)n}.$ Putting the above estimates (REF )–(REF ) together with (REF ), we can obtain $\\frac{\\mathrm {d}}{\\mathrm {d}t}\\Vert \\varrho _{1,n}^{\\rm {er}},u_{1,n}^{\\rm {er}}\\Vert ^2_{{{H}}^{s-1}}+\\mu \\Vert \\nabla u_{1,n}^{\\rm {er}},u̥_{1,n}^{\\rm {er}}\\Vert ^2_{{{H}}^{s-1}}\\lesssim &~ \\Vert \\varrho _{1,n}^{\\rm {er}},u_{1,n}^{\\rm {er}}\\Vert ^2_{H^{s-1}}+2^{-2n(1+\\min \\lbrace \\varepsilon _s,1\\rbrace )}\\\\&+\\varepsilon \\Vert \\nabla u_{1,n}^{\\rm {er}},u̥_{1,n}^{\\rm {er}}\\Vert ^2_{{{H}}^{s-1}}.$ Absorbing the $\\varepsilon $ -term and using Gronwall's inequality yield $\\Vert \\varrho _{1,n}^{\\rm {er}},u_{1,n}^{\\rm {er}}\\Vert _{{{H}}^{s-1}}\\le C2^{-n(1+\\min \\lbrace \\varepsilon _s,1\\rbrace )}.$ An interpolation argument leads to $\\Vert \\varrho _{1,n}^{\\rm {er}},u_{1,n}^{\\rm {er}}\\Vert _{{{H}}^{s}}\\lesssim \\Vert \\varrho _{1,n}^{\\rm {er}},u_{1,n}^{\\rm {er}}\\Vert _{{{H}}^{s-1}}^{\\frac{1}{2}}\\Vert \\varrho _{1,n}^{\\rm {er}},u_{1,n}^{\\rm {er}}\\Vert _{{{H}}^{s+1}}^{\\frac{1}{2}}\\lesssim 2^{-\\frac{n}{2}\\min \\lbrace \\varepsilon _s,1\\rbrace }.$ Thus, we have finished the proof of Proposition REF .","Denoting $V^{\\rm {ap}}_{n}=-u_{2,n}^{\\rm {ap}}\\cdot \\nabla \\varrho _{2,n}^{\\rm {ap}}$ and introducing the errors $\\varrho _{2,n}^{\\rm {er}}=\\varrho _{2,n}-\\varrho _{2,n}^{\\rm {ap}}-tV^{\\rm {ap}}_{n}\\quad \\text{and}\\quad u_{2,n}^{\\rm {er}}=u_{2,n}-u_{2,n}^{\\rm {ap}},$ we can find that $(\\varrho _{2,n}^{\\rm {er}},u_{2,n}^{\\rm {er}})$ satisfies $\\left\\lbrace \\begin{array}{ll}\\partial _t\\varrho _{2,n}^{\\rm {er}}+u̥_{2,n}^{\\rm {er}}+u_{2,n}\\cdot \\nabla \\varrho _{2,n}^{\\rm {er}}=\\mathbf {G_1}-t\\mathbf {G_4}-t\\partial _tV^{\\rm {ap}}_{n},\\\\\\partial _tu_{2,n}^{\\rm {er}}-{\\mu } \\Delta u_{2,n}^{\\rm {er}}-{\\mu } \\nabla u̥_{2,n}^{\\rm {er}}+\\nabla \\varrho _{2,n}^{\\rm {er}}+u_{2,n}\\cdot \\nabla u_{2,n}^{\\rm {er}}=\\mathbf {G_2}+\\mathbf {G_3}-t\\nabla V^{\\rm {ap}}_{n}, \\\\\\varrho _{2,n}^{\\rm {er}}|_{t=0}=0,\\; u_{2,n}^{\\rm {er}}|_{t=0}=0, \\end{array}\\right.$ where $&\\mathbf {G_1}:=-u_{2,n}^{\\rm {er}}\\cdot \\nabla \\varrho ^{\\rm {ap}}_{2,n}-\\varrho _{2,n}u̥_{2,n}^{\\rm {er}}-\\varrho _{2,n}^{\\rm {er}}u̥_{2,n}^{\\rm {ap}}-\\varrho ^{\\rm {ap}}_{2,n}u̥_{2,n}^{\\rm {ap}},\\\\&\\mathbf {G_2}:=-u_{2,n}^{\\rm {er}}\\cdot \\nabla u^{\\rm {ap}}_{2,n}- u^{\\rm {ap}}_{2,n}\\cdot \\nabla u^{\\rm {ap}}_{2,n},\\\\&\\mathbf {G_3}:=2\\nabla (\\ln (1+\\varrho _{2,n}))\\cdot \\mathbb {D}u_{2,n},\\\\&\\mathbf {G_4}:=u_{2,n}^{\\rm {er}}\\cdot \\nabla V^{\\rm {ap}}_{n}+V^{\\rm {ap}}_{n} u_{2,n}^{\\rm {ap}}).$ Proposition 3.2 Under the assumptions of Theorem REF , then we have for $t\\le 1$ $\\Vert u_{2,n}-u_{2,n}^{\\rm {ap}}\\Vert _{H^s}+\\Vert \\varrho _{2,n}-\\varrho _{2,n}^{\\rm {ap}}-tV^{\\rm {ap}}_{n}\\Vert _{H^s}\\le C2^{-n\\min \\lbrace \\varepsilon _s,1\\rbrace }+Ct^{\\frac{3}{2}}.$ Proof.","Following the procedure in (REF ) and (REF ), we obtain $&\\frac{\\mathrm {d}}{\\mathrm {d}t}\\Vert \\varrho _{2,n}^{\\rm {er}},u_{2,n}^{\\rm {er}}\\Vert ^2_{{{H}}^{s}}+\\mu \\Vert \\nabla u_{2,n}^{\\rm {er}},u̥_{2,n}^{\\rm {er}}\\Vert ^2_{{{H}}^{s}}\\nonumber \\\\\\lesssim &~ (1+\\Vert u_{2,n}\\Vert _{H^{s}})\\Vert \\varrho _{2,n}^{\\rm {er}},u_{2,n}^{\\rm {er}}\\Vert ^2_{{{H}}^{s}}\\nonumber \\\\&+\\big (\\Vert \\mathbf {G_1}\\Vert _{{{H}}^{s}}+\\Vert t\\partial _tV^{\\rm {ap}}_{n}\\Vert _{{{H}}^{s}}+t\\Vert \\mathbf {G_4}\\Vert _{{{H}}^{s}}\\big )\\Vert \\varrho _{2,n}^{\\rm {er}}\\Vert _{{{H}}^{s}}\\nonumber \\\\&+\\Vert \\mathbf {G_2},\\mathbf {G_3},t\\nabla V^{\\rm {ap}}_{n}\\Vert ^2_{{{H}}^{s-1}}\\nonumber \\\\\\lesssim &~ \\Vert \\varrho _{2,n}^{\\rm {er}},u_{2,n}^{\\rm {er}}\\Vert ^2_{{{H}}^{s}}+\\Vert \\mathbf {G_1}\\Vert _{{{H}}^{s}}\\Vert \\varrho _{2,n}^{\\rm {er}}\\Vert _{{{H}}^{s}}+\\Vert \\mathbf {G_2},\\mathbf {G_3}\\Vert ^2_{{{H}}^{s-1}}+\\Vert t\\partial _tV^{\\rm {ap}}_{n}\\Vert ^2_{{{H}}^{s}}\\nonumber \\\\&+t^2\\big (\\Vert \\mathbf {G_4}\\Vert ^2_{{{H}}^{s}}+\\Vert V^{\\rm {ap}}_{n}\\Vert ^2_{{{H}}^{s}}\\big ).$ We should mentioned that the commutator estimate from Lemma REF was used here.","It is not hard to deduce that for $k\\in \\lbrace -1,0,1\\rbrace $ $\\Vert V^{\\rm {ap}}_{n}\\Vert _{H^{s+k}}&\\lesssim \\Vert u_{2,n}^{\\rm {ap}}\\Vert _{L^\\infty }\\Vert \\nabla \\varrho _{2,n}^{\\rm {ap}}\\Vert _{H^{s+k}}+\\Vert u_{2,n}^{\\rm {ap}}\\Vert _{H^{s+k}}\\Vert \\nabla \\varrho _{2,n}^{\\rm {ap}}\\Vert _{L^\\infty }\\\\&\\lesssim \\Vert u_{2,n}^{\\rm {ap}}\\Vert _{H^{s-1}}\\Vert \\nabla \\varrho _{2,n}^{\\rm {ap}}\\Vert _{H^{s+k}}+\\Vert u_{2,n}^{\\rm {ap}}\\Vert _{H^{s+k}}\\Vert \\nabla \\varrho _{2,n}^{\\rm {ap}}\\Vert _{H^{s-1}}\\\\&\\lesssim 2^{kn}.$ Estimate of $\\mathbf {G_1}$ .","Using Lemma REF yields $\\Vert u_{2,n}^{\\rm {er}}\\cdot \\nabla \\varrho ^{\\rm {ap}}_{2,n}\\Vert _{H^{s}}&\\lesssim \\Vert u_{2,n}^{\\rm {er}}\\Vert _{L^\\infty }\\Vert \\varrho ^{\\rm {ap}}_{2,n}\\Vert _{H^{s+1}}+\\Vert u_{2,n}^{\\rm {er}}\\Vert _{H^s}\\Vert \\nabla \\varrho ^{\\rm {ap}}_{2,n}\\Vert _{L^\\infty }\\\\&\\lesssim 2^n\\Vert u_{2,n}^{\\rm {er}}\\Vert _{H^{s-1}}+\\Vert u_{2,n}^{\\rm {er}}\\Vert _{H^{s}},\\\\\\Vert \\varrho _{2,n}u̥_{2,n}^{\\rm {er}}\\Vert _{H^{s}}&\\lesssim \\Vert u̥_{2,n}^{\\rm {er}}\\Vert _{H^{s}},\\\\\\Vert \\varrho _{2,n}^{\\rm {er}}u̥_{2,n}^{\\rm {ap}}\\Vert _{H^{s}}&\\lesssim 2^n\\Vert \\varrho _{2,n}^{\\rm {er}}\\Vert _{H^{s-1}}+\\Vert \\varrho _{2,n}^{\\rm {er}}\\Vert _{H^{s}},\\\\\\Vert \\varrho ^{\\rm {ap}}_{2,n}u̥_{2,n}^{\\rm {ap}}\\Vert _{H^{s}}&\\lesssim \\Vert \\varrho _{2,n}^{\\rm {ap}}\\Vert _{H^{s-1}}\\Vert u̥^{\\rm {ap}}_{2,n}\\Vert _{H^{s}}+\\Vert \\varrho _{2,n}^{\\rm {ap}}\\Vert _{H^s}\\Vert u^{\\rm {ap}}_{2,n}\\Vert _{H^{s-\\varepsilon _s}}\\\\&\\lesssim 2^{-n}\\Vert u̥^{\\rm {ap}}_{2,n}\\Vert _{H^{s}}+2^{-n\\min \\lbrace \\varepsilon _s,1\\rbrace },$ which implies that $\\Vert \\mathbf {G_{1}}\\Vert _{H^{s}}&\\lesssim 2^n\\Vert u_{2,n}^{\\rm {er}},\\varrho _{2,n}^{\\rm {er}}\\Vert _{H^{s-1}}+\\Vert u̥_{2,n}^{\\rm {er}}\\Vert _{H^{s}}+\\Vert u_{2,n}^{\\rm {er}},\\varrho _{2,n}^{\\rm {er}}\\Vert _{H^{s}}\\nonumber \\\\&\\quad +2^{-n}\\Vert u̥^{\\rm {ap}}_{2,n}\\Vert _{H^{s}}+2^{-n\\min \\lbrace \\varepsilon _s,1\\rbrace }.$ Estimate of $\\mathbf {G_2}$ .","Notice that $H^{s-1}(\\mathbb {R}^2)$ with $s>2$ is a Banach algebra, we have $\\Vert u_{2,n}^{\\rm {er}}\\cdot \\nabla u^{\\rm {ap}}_{2,n}\\Vert _{H^{s-1}}&\\lesssim \\Vert u_{2,n}^{\\rm {er}}\\Vert _{H^{s-1}}\\Vert u_{2,n}^{\\rm {ap}}\\Vert _{H^{s}}\\lesssim \\Vert u_{2,n}^{\\rm {er}}\\Vert _{H^{s}},\\\\\\Vert u^{\\rm {ap}}_{2,n}\\cdot \\nabla u^{\\rm {ap}}_{2,n}\\Vert _{H^{s-1}}&\\lesssim \\Vert u^{\\rm {ap}}_{2,n}\\Vert _{H^{s-1}}\\Vert u_{2,n}^{\\rm {ap}}\\Vert _{H^{s}}\\lesssim 2^{-n},$ which implies that $\\Vert \\mathbf {G_{2}}\\Vert _{H^{s-1}}&\\lesssim \\Vert u_{2,n}^{\\rm {er}}\\Vert _{H^{s}}+2^{-n}.$ Estimate of $\\mathbf {G_3}$ .","For the term $\\nabla (\\ln (1+\\varrho _{2,n}))\\cdot \\mathbb {D}u_{2,n}$ , we can decompose it as $\\nabla (\\ln (1+\\varrho _{2,n}))\\cdot \\mathbb {D}u_{2,n}&=\\nabla (\\ln (1+\\varrho _{2,n}))\\cdot \\mathbb {D}u_{2,n}^{\\rm {er}}\\\\&\\quad +\\nabla (\\ln (1+\\varrho _{2,n})-\\ln (1+\\varrho ^{\\rm {ap}}_{2,n}+tV^{\\rm {ap}}_{n}))\\cdot \\mathbb {D}u^{\\rm {ap}}_{2,n}\\\\&\\quad +\\nabla (\\ln (1+\\varrho ^{\\rm {ap}}_{2,n}+tV^{\\rm {ap}}_{n}))\\cdot \\mathbb {D}u^{\\rm {ap}}_{2,n}\\\\&=\\mathbf {G_{3,1}}+\\mathbf {G_{3,2}}+\\mathbf {G_{3,3}}.$ By Lemmas REF and REF , it is easy to deduce that $&\\Vert \\mathbf {G_{3,1}}\\Vert _{H^{s-1}}\\lesssim \\Vert u_{2,n}^{\\rm {er}}\\Vert _{H^{s}},\\\\&\\Vert \\mathbf {G_{3,2}}\\Vert _{H^{s-1}}\\lesssim \\Vert \\varrho _{2,n}^{\\rm {er}}\\Vert _{H^{s}},\\\\&\\Vert \\mathbf {G_{3,3}}\\Vert _{H^{s-1}}\\lesssim 2^{-n\\min \\lbrace \\varepsilon _s,1\\rbrace },$ which gives that $\\Vert \\mathbf {G_{3}}\\Vert _{H^{s-1}}&\\lesssim \\Vert u_{2,n}^{\\rm {er}},\\varrho _{2,n}^{\\rm {er}}\\Vert _{H^{s}}+2^{-n\\min \\lbrace \\varepsilon _s,1\\rbrace }.$ Estimate of $\\mathbf {G_4}$ .","By Lemma REF again, we have $\\Vert u_{2,n}^{\\rm {er}}\\cdot \\nabla V^{\\rm {ap}}_{n}\\Vert _{H^{s}}&\\lesssim \\Vert u_{2,n}^{\\rm {er}}\\Vert _{L^\\infty }\\Vert V^{\\rm {ap}}_{n}\\Vert _{H^{s+1}}+\\Vert u_{2,n}^{\\rm {er}}\\Vert _{H^s}\\Vert \\nabla V^{\\rm {ap}}_{n}\\Vert _{L^\\infty }\\lesssim 2^n\\Vert u_{2,n}^{\\rm {er}}\\Vert _{H^{s-1}}+\\Vert u_{2,n}^{\\rm {er}}\\Vert _{H^{s}},\\\\\\Vert V^{\\rm {ap}}_{2,n}u_{2,n}^{\\rm {ap}})\\Vert _{H^{s}}&\\lesssim \\Vert V^{\\rm {ap}}_{2,n}u̥_{2,n}^{\\rm {ap}}\\Vert _{H^{s}}+ \\Vert u_{2,n}^{\\rm {ap}}\\cdot \\nabla V^{\\rm {ap}}_{2,n}\\Vert _{H^{s}}\\lesssim 1$ and $\\Vert u_{2,n}^{\\rm {er}}\\cdot \\nabla V^{\\rm {ap}}_{n}\\Vert _{H^{s-1}}&\\lesssim \\Vert u_{2,n}^{\\rm {er}}\\Vert _{L^\\infty }\\Vert V^{\\rm {ap}}_{n}\\Vert _{H^{s}}+\\Vert u_{2,n}^{\\rm {er}}\\Vert _{H^{s-1}}\\Vert \\nabla V^{\\rm {ap}}_{n}\\Vert _{L^\\infty }\\lesssim \\Vert u_{2,n}^{\\rm {er}}\\Vert _{H^{s-1}},\\\\\\Vert V^{\\rm {ap}}_{2,n}u_{2,n}^{\\rm {ap}})\\Vert _{H^{s-1}}&\\lesssim \\Vert V^{\\rm {ap}}_{2,n}u_{2,n}^{\\rm {ap}}\\Vert _{H^{s}}\\lesssim 2^{-n},$ from which, we obtain $\\Vert \\mathbf {G_{4}}\\Vert _{H^{s}}&\\lesssim 2^n\\Vert u_{2,n}^{\\rm {er}}\\Vert _{H^{s-1}}+\\Vert u_{2,n}^{\\rm {er}}\\Vert _{H^{s}}+1\\quad \\text{and}\\quad \\Vert \\mathbf {G_{4}}\\Vert _{H^{s-1}}\\lesssim \\Vert u_{2,n}^{\\rm {er}}\\Vert _{H^{s-1}}+2^{-n}.$ Next, we need to deal with the involved term $\\Vert t\\partial _tV^{\\rm {ap}}_{n}\\Vert _{{{H}}^{s}}$ .","Estimate of $\\Vert t\\partial _tV^{\\rm {ap}}_{n}\\Vert _{{{H}}^{s}}$ .","Direct calculation gives that $&-\\partial _tV_{n}^{\\rm {ap}}=\\partial _tu_{2,n}^{\\rm {ap}}\\cdot \\nabla \\varrho _{2,n}^{\\rm {ap}}+u_{2,n}^{\\rm {ap}}\\cdot \\nabla \\partial _t\\varrho _{2,n}^{\\rm {ap}},$ using the following estimates whose proof are relegated to A.1–A.2 in Section $&\\Vert t\\partial _tu_{2,n}^{\\rm {ap}}\\Vert _{H^{s+k}}\\le C2^{(k-1)n}+Ct2^{-n},\\\\&\\Vert \\partial _t\\varrho _{2,n}^{\\rm {ap}}\\Vert _{H^{s+k}}\\le C2^{kn},$ then we can estimate $\\Vert t\\partial _tV^{\\rm {ap}}_{n}\\Vert _{{{H}}^{s}}$ as $\\Vert t\\partial _tV^{\\rm {ap}}_{n}\\Vert _{H^{s+k}}&\\lesssim \\Vert t\\partial _tu_{2,n}^{\\rm {ap}}\\Vert _{L^\\infty }\\Vert \\nabla \\varrho _{2,n}^{\\rm {ap}}\\Vert _{H^{s+k}}+\\Vert t\\partial _tu_{2,n}^{\\rm {ap}}\\Vert _{H^{s+k}}\\Vert \\nabla \\varrho _{2,n}^{\\rm {ap}}\\Vert _{L^\\infty }\\\\&\\quad +t\\Vert u_{2,n}^{\\rm {ap}}\\Vert _{L^\\infty }\\Vert \\nabla \\partial _t\\varrho _{2,n}^{\\rm {ap}}\\Vert _{H^{s+k}}+t\\Vert u_{2,n}^{\\rm {ap}}\\Vert _{H^{s+k}}\\Vert \\nabla \\partial _t\\varrho _{2,n}^{\\rm {ap}}\\Vert _{L^\\infty }\\\\&\\lesssim \\Vert t\\partial _tu_{2,n}^{\\rm {ap}}\\Vert _{H^{s-1}}\\Vert \\varrho _{2,n}^{\\rm {ap}}\\Vert _{H^{s+k+1}}+\\Vert t\\partial _tu_{2,n}^{\\rm {ap}}\\Vert _{H^{s+k}}\\Vert \\varrho _{2,n}^{\\rm {ap}}\\Vert _{H^{s}}\\\\&\\quad +t\\Vert u_{2,n}^{\\rm {ap}}\\Vert _{H^{s-1}}\\Vert \\partial _t\\varrho _{2,n}^{\\rm {ap}}\\Vert _{H^{s+k+1}}+t\\Vert u_{2,n}^{\\rm {ap}}\\Vert _{H^{s+k}}\\Vert \\partial _t\\varrho _{2,n}^{\\rm {ap}}\\Vert _{H^{s}}\\\\&\\lesssim t2^{kn}+2^{(k-1)n}.$ Putting the above estimates into (REF ) yields $\\frac{\\mathrm {d}}{\\mathrm {d}t}\\Vert \\varrho _{2,n}^{\\rm {er}},u_{2,n}^{\\rm {er}}\\Vert ^2_{{{H}}^{s}}\\lesssim &~\\Vert \\varrho _{2,n}^{\\rm {er}},u_{2,n}^{\\rm {er}}\\Vert ^2_{{{H}}^{s}}+2^{2n}\\Vert \\varrho _{2,n}^{\\rm {er}},u_{2,n}^{\\rm {er}}\\Vert ^2_{H^{s-1}}+2^{-2n}\\Vert u̥_{2,n}^{\\rm {ap}}\\Vert ^2_{H^{s}}\\nonumber \\\\&+2^{-2n\\min \\lbrace \\varepsilon _s,1\\rbrace }+t^2.$ Finally, to close the above, we have to estimate $\\Vert \\varrho _{2,n}^{\\rm {er}},u_{2,n}^{\\rm {er}}\\Vert ^2_{H^{s-1}}$ .","Following the procedure in (REF ) and (REF ) once again, we obtain $&\\frac{\\mathrm {d}}{\\mathrm {d}t}\\Vert \\varrho _{2,n}^{\\rm {er}},u_{2,n}^{\\rm {er}}\\Vert ^2_{{{H}}^{s-1}}+\\mu \\Vert \\nabla u_{2,n}^{\\rm {er}},u̥_{2,n}^{\\rm {er}}\\Vert ^2_{{{H}}^{s-1}}\\nonumber \\\\\\lesssim &~ (1+\\Vert u_{2,n}\\Vert _{H^{s}})\\Vert \\varrho _{2,n}^{\\rm {er}},u_{2,n}^{\\rm {er}}\\Vert ^2_{{{H}}^{s-1}}\\nonumber \\\\&+\\big (\\Vert \\mathbf {G_1}\\Vert _{{{H}}^{s-1}}+\\Vert t\\partial _tV^{\\rm {ap}}_{n}\\Vert _{{{H}}^{s-1}}\\big )\\Vert \\varrho _{2,n}^{\\rm {er}}\\Vert _{H^{s-1}}+\\Vert \\mathbf {G_2},\\mathbf {G_3}\\Vert ^2_{{{H}}^{s-2}}\\nonumber \\\\&+t^2\\big (\\Vert \\mathbf {G_4}\\Vert ^2_{H^{s-1}}+\\Vert V^{\\rm {ap}}_{n}\\Vert ^2_{{{H}}^{s-1}}\\big )\\nonumber \\\\\\lesssim &~\\Vert \\varrho _{2,n}^{\\rm {er}},u_{2,n}^{\\rm {er}}\\Vert ^2_{{{H}}^{s-1}}+\\Vert \\mathbf {G_1}\\Vert _{{{H}}^{s-1}}\\Vert \\varrho _{2,n}^{\\rm {er}}\\Vert _{H^{s-1}}+\\Vert \\mathbf {G_2},\\mathbf {G_3}\\Vert ^2_{{{H}}^{s-2}}+\\Vert t\\partial _tV^{\\rm {ap}}_{n}\\Vert ^2_{{{H}}^{s-1}}\\nonumber \\\\&+t^2\\big (\\Vert \\mathbf {G_4}\\Vert ^2_{H^{s-1}}+\\Vert V^{\\rm {ap}}_{n}\\Vert ^2_{{{H}}^{s-1}}\\big ).$ It should be mentioned that the first three terms of $\\mathbf {G_1}$ can be done as that of $\\mathbf {F_1}$ .","The only difference is the forth term of $\\mathbf {G_1}$ .","In fact, we estimate it as $\\Vert \\varrho ^{\\rm {ap}}_{2,n}u̥_{2,n}^{\\rm {ap}}\\Vert _{{{H}}^{s-1}}\\lesssim \\Vert \\varrho ^{\\rm {ap}}_{2,n}\\Vert _{{{H}}^{s-1}}\\Vert u̥_{2,n}^{\\rm {ap}}\\Vert _{{{H}}^{s-1}}\\lesssim 2^{-n}\\Vert u̥_{2,n}^{\\rm {ap}}\\Vert _{{{H}}^{s-1}}.$ Then we have $\\Vert \\mathbf {G_1}\\Vert _{H^{s-1}}&\\lesssim \\Vert u_{1,n}^{\\rm {er}},\\varrho _{1,n}^{\\rm {er}}\\Vert _{H^{s-1}}+\\Vert u̥_{1,n}^{\\rm {er}}\\Vert _{H^{s-1}}+2^{-n}\\Vert u̥_{2,n}^{\\rm {ap}}\\Vert _{{{H}}^{s-1}}.$ The estimates of $\\mathbf {G_2}$ and $\\mathbf {G_3}$ in $H^{s-2}$ can be done as that of $\\mathbf {F_2}$ and $\\mathbf {F_3}$ , respectively.","We have $\\Vert \\mathbf {G_2},\\mathbf {G_3}\\Vert _{H^{s-2}}&\\lesssim \\Vert u_{1,n}^{\\rm {er}},\\varrho _{1,n}^{\\rm {er}}\\Vert _{H^{s-1}}+2^{-(1+\\varepsilon _s)n}+2^{-2n}.$ Combining the above estimates with (REF ), we can obtain $&\\frac{\\mathrm {d}}{\\mathrm {d}t}\\Vert \\varrho _{2,n}^{\\rm {er}},u_{2,n}^{\\rm {er}}\\Vert ^2_{{{H}}^{s-1}}\\lesssim \\Vert \\varrho _{2,n}^{\\rm {er}},u_{2,n}^{\\rm {er}}\\Vert ^2_{{{H}}^{s-1}}+2^{-2n}\\Vert u̥_{2,n}^{\\rm {ap}}\\Vert ^2_{{{H}}^{s-1}}+2^{-2n(1+\\min \\lbrace \\varepsilon _s,1\\rbrace )}+2^{-2n}t^2,$ which follows from Gronwall's inequality and Lemma REF that $\\Vert \\varrho _{2,n}^{\\rm {er}},u_{2,n}^{\\rm {er}}\\Vert ^2_{{{H}}^{s-1}}\\lesssim 2^{-2n(1+\\min \\lbrace \\varepsilon _s,1\\rbrace )}+2^{-2n}t^3.$ Plugging (REF ) into (REF ) yields $\\frac{\\mathrm {d}}{\\mathrm {d}t}\\Vert \\varrho _{2,n}^{\\rm {er}},u_{2,n}^{\\rm {er}}\\Vert ^2_{{{H}}^{s}}\\lesssim &~\\Vert \\varrho _{2,n}^{\\rm {er}},u_{2,n}^{\\rm {er}}\\Vert ^2_{{{H}}^{s}}+2^{-2n}\\Vert u̥_{2,n}^{\\rm {ap}}\\Vert ^2_{H^{s}}+2^{-2n\\min \\lbrace \\varepsilon _s,1\\rbrace }+t^2.$ Using Gronwall's inequality and Lemma REF yields $\\Vert \\varrho _{2,n}^{\\rm {er}},u_{2,n}^{\\rm {er}}\\Vert ^2_{{{H}}^{s}}\\lesssim &~2^{-2n\\min \\lbrace \\varepsilon _s,1\\rbrace }+t^3.$ Thus, we have finished the proof of Proposition REF ."],["Non-uniform Continuous Dependence","With Propositions REF –REF in hand, we can prove Theorem REF .","Behavior at time $t=0$ .","Obviously, we have $\\Vert {u}_{2,n}(0)-u_{1,n}(0)\\Vert _{H^{s}}+\\Vert \\varrho _{2,n}(0)-\\varrho _{1,n}(0)\\Vert _{H^{s}}=\\Vert g_n\\Vert _{H^{s}}\\le C2^{-n},$ which means that the condition $(\\bf {C.2})$ in Theorem REF holds.","Behavior at time $t>0$ .","Notice that $&u_{2,n}-u_{1,n}=u^{\\rm {er}}_{2,n}-u^{\\rm {er}}_{1,n}+u^{\\rm {ap}}_{2,n}-u^{\\rm {ap}}_{1,n},\\nonumber \\\\&\\varrho _{2,n}-\\varrho _{1,n}=t{V}_n^{\\rm {ap}}+\\varrho ^{\\rm {er}}_{2,n}-\\varrho ^{\\rm {er}}_{1,n}+\\varrho ^{\\rm {ap}}_{2,n}-\\varrho ^{\\rm {ap}}_{1,n},$ by the triangle inequality and Propositions REF and REF , we deduce that for $t\\le 1$ $&\\Vert \\varrho _{2,n}(t)-\\varrho _{1,n}(t)\\Vert _{H^{s}}+\\Vert u_{2,n}(t)-u_{1,n}(t)\\Vert _{H^{s}}\\nonumber \\\\\\ge &~ t\\Vert V^{\\rm {ap}}_{n}\\Vert _{H^s}-\\big (\\Vert u^{\\rm {ap}}_{2,n}-u^{\\rm {ap}}_{1,n}\\Vert _{H^s}+\\Vert \\varrho ^{\\rm {ap}}_{2,n}-\\varrho ^{\\rm {ap}}_{1,n}\\Vert _{H^s}\\big )-\\big (\\Vert u^{\\rm {er}}_{1,n},\\varrho ^{\\rm {er}}_{1,n}\\Vert _{H^s}+\\Vert u^{\\rm {er}}_{2,n},\\varrho ^{\\rm {er}}_{2,n}\\Vert _{H^s}\\big )\\nonumber \\\\\\gtrsim &~ t\\Vert V^{\\rm {ap}}_{n}\\Vert _{H^s}-2^{-n}-2^{-\\frac{n}{2}\\min \\lbrace \\varepsilon _s,1\\rbrace }-2^{-n\\min \\lbrace \\varepsilon _s,1\\rbrace }-t^{\\frac{3}{2}},$ where we have used $\\Vert u^{\\rm {ap}}_{2,n}-u^{\\rm {ap}}_{1,n}\\Vert ^2_{H^s}+\\Vert \\varrho ^{\\rm {ap}}_{2,n}-\\varrho ^{\\rm {ap}}_{1,n}\\Vert ^2_{H^s}\\lesssim \\Vert g_n\\Vert ^2_{H^s}\\lesssim 2^{-2n}.$ Notice that $-V^{\\rm {ap}}_{n}=u_{2,n}^{\\rm {ap}}\\cdot \\nabla \\varrho _{2,n}^{\\rm {ap}}$ , we decompose it as $-V^{\\rm {ap}}_{n}=(u_{2,n}^{\\rm {ap}})_1\\partial _1 \\varrho _{2,n}^{\\rm {ap}}+(u_{2,n}^{\\rm {ap}})_2\\partial _2 \\varrho _{2,n}^{\\rm {ap}}=(g_n)_1\\partial _1f_n+E,$ where $E:=(u_{2,n}^{\\rm {ap}}-g_n)_1\\partial _1\\varrho _{2,n}^{\\rm {ap}}+(g_n)_1\\partial _1(\\varrho _{2,n}^{\\rm {ap}}-f_n)+(u_{2,n}^{\\rm {ap}})_2\\partial _2\\varrho _{2,n}^{\\rm {ap}}.$ Then we have for $t\\le 1$ $\\Vert E\\Vert _{H^{s}}\\le Ct+C2^{-n}.$ For more details of proof, see A.3 in Section .","Furthermore, (REF ) reduces to $\\Vert \\varrho _{2,n}(t)-\\varrho _{1,n}(t)\\Vert _{H^{s}}+\\Vert u_{2,n}(t)-u_{1,n}(t)\\Vert _{H^{s}}\\gtrsim t\\Vert (g_n)_1\\partial _1f_n\\Vert _{H^s}-2^{-\\frac{n}{2}\\min \\lbrace \\varepsilon _s,1\\rbrace }-t^{\\frac{3}{2}},$ combining the following estimate whose proof is postponed to A.4 in Section $\\liminf _{n\\rightarrow \\infty }\\Vert (g_n)_1\\partial _1f_n\\Vert _{H^s}\\gtrsim \\Vert \\phi ^{\\prime }\\phi \\Vert ^2_{L^2}\\Vert \\phi ^2\\Vert _{L^2},$ we get from (REF ) that $\\liminf _{n\\rightarrow \\infty }\\big (\\Vert \\varrho _{2,n}(t)-\\varrho _{1,n}(t)\\Vert _{H^{s}}+\\Vert u_{2,n}(t)-u_{1,n}(t)\\Vert _{H^{s}}\\big )\\gtrsim t\\quad \\text{for} \\ t \\ \\text{small enough},$ which is nothing but the condition $(\\bf {C.3})$ in Theorem REF .","Thus, we complete the proof of Theorem REF ."],["Appendix","For the sake of convenience, here we present more details in the computations.","A.1 Proof of (REF ).","When $|\\xi |\\rightarrow \\infty $ , we have $\\lambda _-(\\xi )\\sim -2\\mu |\\xi |^2 \\quad \\text{and}\\quad \\lambda _+(\\xi )\\sim -\\frac{1}{2\\mu }.$ From (REF ), then $\\mathcal {F}\\big (u_{2,n}^{\\rm {ap}}\\big )(t,\\xi )&=-\\frac{e^{\\lambda _+t}-e^{\\lambda _-t}}{\\lambda _+-\\lambda _-}\\mathcal {F}\\big (\\nabla f_n\\big )-\\frac{\\lambda _+e^{\\lambda _+t}-\\lambda _-e^{\\lambda _-t}}{\\lambda _+-\\lambda _-}\\mathcal {F}\\big (\\Lambda ^{-2}\\nabla g̥_n\\big )$ which gives $&\\mathcal {F}\\big (t\\partial _tu_{2,n}^{\\rm {ap}}\\big )=-\\frac{t\\lambda _+e^{\\lambda _+t}-t\\lambda _-e^{\\lambda _-t}}{\\lambda _+-\\lambda _-}\\mathcal {F}\\big (\\nabla f_n\\big )-t\\frac{\\lambda _{+}^2e^{\\lambda _+t}-\\lambda _{-}^2e^{\\lambda _-t}}{\\lambda _+-\\lambda _-}\\mathcal {F}\\big (\\Lambda ^{-2}\\nabla g̥_n\\big ).$ Using the simple facts $&|t\\lambda _{\\pm }e^{\\lambda _{\\pm }t}|\\le 1\\quad \\text{if }\\;\\xi \\in \\mathcal {C}_n \\\\&\\Big |\\frac{\\lambda _{+}^2e^{\\lambda _+t}-\\lambda _{-}^2e^{\\lambda _-t}}{\\lambda _+-\\lambda _-}\\Big |\\lesssim |\\lambda _++\\lambda _-|+t|\\lambda _-|^2\\lesssim 1\\quad \\text{if }\\;\\xi \\in \\mathcal {B}(0,1),$ then we have $\\big |\\mathcal {F}\\big (t\\partial _tu_{2,n}^{\\rm {ap}}\\big )\\big |&\\le \\Big |\\frac{t\\lambda _+e^{\\lambda _+t}-t\\lambda _-e^{\\lambda _-t}}{\\lambda _+-\\lambda _-}\\mathcal {F}\\big (\\nabla f_n\\big )\\Big |+t\\Big |\\frac{\\lambda _{+}^2e^{\\lambda _+t}-\\lambda _{-}^2e^{\\lambda _-t}}{\\lambda _+-\\lambda _-}\\mathcal {F}\\big (\\Lambda ^{-2}\\nabla g̥_n\\big )\\Big |\\\\&\\le \\frac{2}{|\\lambda _+-\\lambda _-|}\\big |\\mathcal {F}\\big (\\nabla f_n\\big )\\big |+t\\big |\\mathcal {F}\\big (\\Lambda ^{-2}\\nabla g̥_n\\big )\\big |\\\\&\\approx \\big |\\mathcal {F}\\big (\\Lambda ^{-2}\\nabla f_n\\big )\\big |+t\\big |\\mathcal {F}\\big (\\Lambda ^{-2}\\nabla g̥_n\\big )\\big |,$ which in turn gives $\\Vert t\\partial _tu_{2,n}^{\\rm {ap}}\\Vert _{H^{s+k}}&=\\Big [\\int _{\\mathbb {R}^2}(1+|\\xi |^2)^{\\frac{s+k}{2}}\\big |\\mathcal {F}\\big (t\\partial _tu_{2,n}^{\\rm {ap}}\\big )\\big |^2\\mathrm {d}\\xi \\Big ]^{\\frac{1}{2}}\\\\&\\lesssim \\Big [\\int _{\\mathcal {C}_n }(1+|\\xi |^2)^{\\frac{s+k}{2}}\\big |\\mathcal {F}\\big (\\Lambda ^{-2}\\nabla f_n\\big )\\big |^2\\mathrm {d}\\xi \\Big ]^{\\frac{1}{2}}+t\\Big [\\int _{\\mathcal {B}(0,1)}\\big |\\mathcal {F}\\big (\\Lambda ^{-2}\\nabla g̥_n\\big )\\big |^2\\mathrm {d}\\xi \\Big ]^{\\frac{1}{2}}\\\\&\\lesssim \\Vert f_n\\Vert _{H^{s+k-1}}+t\\Vert g_n\\Vert _{L^{2}}.$ A.2 Proof of ().","From (REF ), then $\\mathcal {F}\\big (\\partial _t\\varrho _{2,n}^{\\rm {ap}}\\big )&=-\\frac{\\lambda _+\\lambda _-(e^{\\lambda _+t}-e^{\\lambda _-t})}{\\lambda _+-\\lambda _-}\\mathcal {F}\\big (f_n\\big )-\\frac{\\lambda _{+}e^{\\lambda _+t}-\\lambda _{-}e^{\\lambda _-t}}{\\lambda _+-\\lambda _-}\\mathcal {F}\\big ( g̥_n\\big )\\\\&=-\\frac{e^{\\lambda _+t}-e^{\\lambda _-t}}{\\sqrt{\\mu ^2-|\\xi |^{-2}}}\\mathcal {F}\\big (f_n\\big )-\\frac{\\lambda _{+}e^{\\lambda _+t}-\\lambda _{-}e^{\\lambda _-t}}{\\lambda _+-\\lambda _-}\\mathcal {F}\\big ( g̥_n\\big ),$ similarly, we have $\\Vert \\partial _t\\varrho _{2,n}^{\\rm {ap}}\\Vert _{H^{s+k}}&=\\Big [\\int _{\\mathbb {R}^2}(1+|\\xi |^2)^{\\frac{s+k}{2}}\\big |\\mathcal {F}\\big (\\partial _t\\varrho _{2,n}^{\\rm {ap}}\\big )\\big |^2\\mathrm {d}\\xi \\Big ]^{\\frac{1}{2}}\\\\&\\lesssim \\Big [\\int _{\\mathcal {C}_n }(1+|\\xi |^2)^{\\frac{s+k}{2}}\\big |\\widehat{f_n}(\\xi )\\big |^2\\mathrm {d}\\xi \\Big ]^{\\frac{1}{2}}+\\Big [\\int _{\\mathcal {B}(0,1)}(1+|\\xi |^2)^{\\frac{s+k}{2}}\\big |\\mathcal {F}\\big ( g̥_n\\big )\\big |^2\\mathrm {d}\\xi \\Big ]^{\\frac{1}{2}}\\\\&\\lesssim \\Vert f_n\\Vert _{H^{s+k}}+\\Vert g_n\\Vert _{L^{2}}.$ A.3 Proof of (REF ).","Notice that $&\\widehat{\\varrho ^{\\rm {ap}}_{2,n}}-\\widehat{f_n}=\\frac{\\lambda _+(e^{\\lambda _-t}-e^{\\lambda _+t})}{\\lambda _+-\\lambda _-}\\widehat{f_n}+(e^{\\lambda _+t}-1)\\widehat{f_n}-\\frac{e^{\\lambda _+t}-e^{\\lambda _-t}}{\\lambda _+-\\lambda _-}\\mathcal {F}(g̥_n),\\\\&\\widehat{u^{\\rm {ap}}_{2,n}}-\\widehat{g_n}=-\\frac{e^{\\lambda _+t}-e^{\\lambda _-t}}{\\lambda _+-\\lambda _-}\\widehat{\\nabla f_n}+\\Big (e^{\\lambda _+t}-1+\\frac{\\lambda _-(e^{\\lambda _+t}-e^{\\lambda _-t})}{\\lambda _+-\\lambda _-}\\Big )\\widehat{g_n},$ then by (REF ) $\\Vert \\varrho ^{\\rm {ap}}_{2,n}-f_n\\Vert _{H^{s+1}}&\\lesssim t\\Big [\\int _{\\mathcal {C}_n }(1+|\\xi |^2)^{\\frac{s+1}{2}}|\\widehat{f_n}|^2\\mathrm {d}\\xi +\\int _{\\mathcal {B}(0,1)}(1+|\\xi |^2)^{\\frac{s+1}{2}}\\big |\\mathcal {F}\\big (g̥_n\\big )\\big |^2\\mathrm {d}\\xi \\Big ]^{\\frac{1}{2}}\\\\&\\lesssim t\\big (\\Vert f_n\\Vert _{H^{s+1}}+\\Vert g_n\\Vert _{L^{2}}\\big )\\lesssim t2^{n}$ and $\\Vert u^{\\rm {ap}}_{2,n}-g_n\\Vert _{H^{s+k}}&\\lesssim \\Big [\\int _{\\mathcal {C}_n }(1+|\\xi |^2)^{\\frac{s+k}{2}}\\big |\\widehat{\\Lambda ^{-1}f_n}\\big |^2\\mathrm {d}\\xi +t^2\\int _{\\mathcal {B}(0,1)}(1+|\\xi |^2)^{\\frac{s+k}{2}}\\big |\\mathcal {F}\\big ( g_n\\big )\\big |^2\\mathrm {d}\\xi \\Big ]^{\\frac{1}{2}}\\\\&\\lesssim \\Vert f_n\\Vert _{H^{s+k-1}}+t\\Vert g_n\\Vert _{L^{2}}\\lesssim 2^{(k-1)n}+t2^{-n}.$ By Lemma REF , we deduce $&\\Vert (u_{2,n}^{\\rm {ap}}-g_n)_1\\partial _1\\varrho _{2,n}^{\\rm {ap}}\\Vert _{H^{s}}\\lesssim \\Vert u_{2,n}^{\\rm {ap}}-g_n\\Vert _{H^{s-1}}\\Vert \\varrho _{2,n}^{\\rm {ap}}\\Vert _{H^{s+1}}+\\Vert u_{2,n}^{\\rm {ap}}-g_n\\Vert _{H^{s}}\\Vert \\varrho _{2,n}^{\\rm {ap}}\\Vert _{H^{s}}\\lesssim t+2^{-n},\\\\&\\Vert (g_n)_1\\partial _1(\\varrho _{2,n}^{\\rm {ap}}-f_n)\\Vert _{H^{s}}\\lesssim \\Vert g_n\\Vert _{H^{s}}\\Vert \\varrho _{2,n}^{\\rm {ap}}-f_n\\Vert _{H^{s+1}}\\lesssim t,\\\\&\\Vert (u_{2,n}^{\\rm {ap}})_2\\partial _2\\varrho _{2,n}^{\\rm {ap}}\\Vert _{H^{s}}\\lesssim \\Vert u_{2,n}^{\\rm {ap}}\\Vert _{H^{s-1}}\\Vert \\partial _2\\varrho _{2,n}^{\\rm {ap}}\\Vert _{H^{s}}+\\Vert u_{2,n}^{\\rm {ap}}\\Vert _{H^{s}}\\Vert \\partial _2\\varrho _{2,n}^{\\rm {ap}}\\Vert _{H^{s-1}}\\lesssim 2^{-n},$ which implies (REF ).","A.4 Proof of (REF ).","Notice that the support condition of $\\widehat{\\phi }$ , we deduce that $2^{-n}\\Vert \\partial _1\\big (\\phi (x_1)\\phi (x_2)\\big )\\partial _1f_n\\Vert _{H^s}\\gtrsim &~\\Vert \\phi ^{\\prime }(x_1)\\phi (x_1)\\phi ^2(x_2)\\cos (2^nx_1)\\Vert _{L^2}\\nonumber \\\\&-2^{-n}\\Vert (\\phi ^{\\prime }(x_1)\\phi (x_2))^2\\sin (2^nx_1)\\Vert _{L^2}\\nonumber \\\\\\gtrsim &~\\Vert \\phi ^2(x_2)\\Vert _{L^2}\\Vert \\phi ^{\\prime }(x_1)\\phi (x_1)\\cos (2^nx_1)\\Vert _{L^2}\\nonumber \\\\&-2^{-n}\\Vert (\\phi ^{\\prime }(x_1)\\phi (x_2))^2\\Vert _{L^2}.$ Using the simple formula $2\\cos ^2(2^nx_1)=1-\\cos (2^{n+1}x_1),$ we have $\\Vert \\phi ^{\\prime }(x_1)\\phi (x_1)\\cos (2^nx_1)\\Vert ^2_{L^2}=\\frac{1}{2}\\Vert \\phi ^{\\prime }(x_1)\\phi (x_1)\\Vert ^2_{L^2}-\\frac{1}{2}\\int _{\\mathbb {R}}|\\phi ^{\\prime }(x_1)\\phi (x_1)|^2\\cos (2^{n+1}x_1)\\mathrm {d}x_1,$ which follows from Riemann-Lebesgue's Theorem that $\\lim _{n\\rightarrow \\infty }\\Vert \\phi ^{\\prime }(x_1)\\phi (x_1)\\cos (2^nx_1)\\Vert ^2_{L^2}=\\frac{1}{2}\\Vert \\phi ^{\\prime }(x_1)\\phi (x_1)\\Vert ^2_{L^2}.$ Combining (REF ) and (REF ) yields the desired (REF )."],["Acknowledgments","J. Li is supported by the National Natural Science Foundation of China (Grant No.11801090).","Y. Yu is supported by the Natural Science Foundation of Anhui Province (No.1908085QA05).","W. Zhu is partially supported by the National Natural Science Foundation of China (Grant No.11901092) and Natural Science Foundation of Guangdong Province (No.2017A030310634)."]],"string":"[\n [\n \"Non-uniform dependence on initial data for the 2D viscous shallow water\\n equations\"\n ],\n [\n \"Abstract The failure of uniform dependence on the data is an interesting property of classical solution for a hyperbolic system.\",\n \"In this paper, we consider the solution map of the Cauchy problem to the 2D viscous shallow water equations which is a hyperbolic-parabolic system.\",\n \"We prove that the solution map of this problem is not uniformly continuous in Sobolev spaces $H^s\\\\times H^{s}$ for $s>2$.\"\n ],\n [\n \"Introduction\",\n \"The 2D viscous shallow water equations are given by the following system, which have systematically introduced in [2], [3] $\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\partial _t\\\\rho +\\\\rho u)=0,\\\\\\\\\\\\partial _t(\\\\rho u)+\\\\rho u\\\\otimes u)-\\\\mu \\\\rho \\\\nabla u)+\\\\rho \\\\nabla \\\\rho =0,\\\\\\\\\\\\rho |_{t=0}=\\\\rho _0,\\\\;u|_{t=0}=u_0.\\\\end{array}\\\\right.$ where $\\\\rho (x,t)$ is the height of fluid surface, $u(x,t)$ is the horizontal velocity field and $\\\\mu >0$ is the viscous coefficient.\",\n \"We suppose that the initial data $\\\\rho _0(x)$ is a small perturbation of some positive constant $\\\\overline{\\\\rho }_0$ .\",\n \"Classical solutions and well-posedness of the initial (boundary) value problem for the shallow water equations (REF ) have been studied extensively.\",\n \"By using Lagrangian coordinates and Hölder space estimates, Bui [1] obtained the local existence and uniqueness of classical solutions to the Cauchy-Dirichlet problem for (REF ) with initial data in $C^{2+\\\\alpha }$ .\",\n \"With the help of the energy method of Matsumura and Nishida [33], Kloeden [26] and Sundbye [34] independently showed the global existence and uniqueness of classical solutions to the Cauchy-Dirichlet problem for (REF ).\",\n \"Subsequently, Sundbye [35] also proved the existence and uniqueness of classical solutions to the Cauchy problem for (REF ) using the method of [33].\",\n \"Due to the strong nonlinearity of system (REF ), the problem of existence of solutions for large initial data is difficult.\",\n \"By applying the Littlewood-Paley decomposition theory for Sobolev spaces to obtain a losing energy estimate in $H^s$ for any $s > 2$ , Wang–Xu [36] obtained local solutions for any initial data and global solutions to (REF ) for small initial data $(u_0, \\\\rho _0-\\\\bar{\\\\rho }_0) \\\\in H^s\\\\times H^s$ with $s>2$ .\",\n \"Liu–Yin [30], [31], [32] improved the result of [36] in the Sobolev spaces with low regularity and inhomogeneous Besov spaces.\",\n \"Chen–Miao–Zhang [6] obtained the local well-posedness of system (REF ) for general initial data in critical $L^p$ type Besov spaces with $1\\\\le p<4$ by developing a new method which relies on the smoothing properties of the heat equations.\",\n \"Recently, Li–Hong–Zhu [28] proved that the system (REF ) is ill-posed in the critical Besov spaces with $p > 4$ .\",\n \"The continuous dependence is particularly important when PDEs are used to model phenomena in the natural world since measurements are always associated with errors.\",\n \"One of the first results of this type was proved by Kato [27] who showed that the solution operator for the (inviscid) Burgers equation is not Hölder continuous in the $H^s(\\\\mathbb {T})$ -norm $(s > 3/2)$ for any Hölder exponent.\",\n \"After the phenomenon of non-uniform continuity for some dispersive equations was studied by Kenig et al.\",\n \"[24], many results with regard to the non-uniform dependence on the initial data have been obtained for other nonlinear PDEs including the Euler equations [18], the Camassa-Holm equation [19], [20], [29], the Benjamin-Ono equation [25], the compressible gas dynamics [21], [23], the Hunter-Saxton equation [22] and so on.\",\n \"Nevertheless we notice that almost the above system mentioned is hyperbolic.\",\n \"As stated in [23], the exhibition of nonuniform behavior in a hyperbolic system related to the incompressible system indicates that the nonuniform dependence is hyperbolic in nature.\",\n \"From the PDE's point of view, classical solution is uniform dependence on the data for a parabolic system.\",\n \"Naturally, one may wonder if uniform dependence on the data of solution for a coupled system which is hyperbolic-parabolic can persist or not.\",\n \"In this paper, we focus on the 2D viscous shallow water equations $\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\partial _t\\\\rho +\\\\rho u)=0,\\\\\\\\\\\\partial _t(\\\\rho u)+\\\\rho u\\\\otimes u)-\\\\mu 2\\\\rho \\\\mathbb {D}u)+\\\\rho \\\\nabla \\\\rho =0,\\\\\\\\\\\\rho |_{t=0}=\\\\rho _0,\\\\;u|_{t=0}=u_0.\\\\end{array}\\\\right.$ where the strain tensor $\\\\mathbb {D}u =\\\\frac{1}{2}(\\\\nabla u+\\\\nabla ^{\\\\mathsf {T}} u)$ is the symmetric part of the velocity gradient.\",\n \"For the sake of convenience, we take $\\\\bar{\\\\rho }_0 = 1$ and denote $\\\\varrho =\\\\rho -1$ , we can reformulate the system (REF ) equivalently as follows $\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\partial _t\\\\varrho +u̥+u\\\\cdot \\\\nabla \\\\varrho =-\\\\varrho u̥,\\\\\\\\\\\\partial _tu-{\\\\mu } \\\\Delta u-{\\\\mu } \\\\nabla u̥+u\\\\cdot \\\\nabla u+\\\\nabla \\\\varrho =2\\\\nabla (\\\\ln (1+\\\\varrho ))\\\\cdot \\\\mathbb {D}u,\\\\\\\\\\\\varrho |_{t=0}=\\\\varrho _0,\\\\;u|_{t=0}=u_0.\\\\end{array}\\\\right.$ Roughly speaking, the system (REF ) can be regarded as a special case of compressible Navier-Stokes equations (see Refs.\",\n \"[6], [7], [8], [9], [10], [11], [12], [14], [15], [16], [17] and the references therein).\",\n \"We emphasize that the equations (REF ) form a quasi-linear hyperbolic-parabolic system and possess strong nonlinear terms.\",\n \"Although the system (REF ) is partially parabolic, owing to the first equation of (REF ) which is of hyperbolic type, this creates the possibility of the non-uniformity for the solution map.\",\n \"In this paper, we prove the failure of uniform dependence on the data for (REF ).\",\n \"To the best of our knowledge, it allows us to give a first kind of answer to the problem of the non-uniform dependence on the data for the 2D viscous shallow water equations.\",\n \"Our main result of this paper is stated: Theorem 1.1 (Nonuniform dependence on initial data) Let $s>2$ .\",\n \"The data-to-solution map $(u_0,\\\\varrho _0)\\\\mapsto \\\\big (u(t),\\\\varrho (t)\\\\big )$ of the Cauchy problem (REF ) is not uniformly continuous from any bounded subset in $H^{s}\\\\times H^{s}$ into $\\\\mathcal {C}([0,T];H^{s}\\\\times H^{s})$ , namely, there exists two sequences of solutions $\\\\big (u_{1,n}(t),\\\\varrho _{1,n}(t)\\\\big )$ and $\\\\big (u_{2,n}(t),\\\\varrho _{2,n}(t)\\\\big )$ such that (C.1) $\\\\Vert u_{1,n},\\\\varrho _{1,n}\\\\Vert _{H^{s}}+\\\\Vert u_{2,n},\\\\varrho _{2,n}\\\\Vert _{H^{s}}\\\\lesssim 1;$ (C.2) $\\\\lim \\\\limits _{n\\\\rightarrow \\\\infty }\\\\big (\\\\Vert {u}_{2,n}(0)-u_{1,n}(0)\\\\Vert _{H^{s}}+\\\\Vert \\\\varrho _{2,n}(0)-\\\\varrho _{1,n}(0)\\\\Vert _{H^{s}}\\\\big )=0;$ (C.3) $\\\\liminf \\\\limits _{n\\\\rightarrow \\\\infty }\\\\big (\\\\Vert u_{2,n}(t)-u_{1,n}(t)\\\\Vert _{H^{s}}+\\\\Vert \\\\varrho _{2,n}(t)-\\\\varrho _{1,n}(t)\\\\Vert _{H^{s}}\\\\big )\\\\gtrsim t$ for any $t\\\\in [0,T_0]$ with small time $T_0$ .\",\n \"Remark 1.1 It should be mentioned that most non-uniformity results for solutions to other nonlinear PDEs were obtained by the Himonas-Misiołek construction in [18].\",\n \"The method we used in proving Theorem REF is motivated by [29] and is completely different from that [18].\",\n \"Remark 1.2 We note that the local well-posedness result holds for $s>1$ .\",\n \"However, the restriction on $s>2$ is essential in the present paper.\",\n \"Outline of the proof to Theorem REF There are two key points in proving Theorem REF : On one hand, we construct one sequence of initial data $(u_{1,n}(0),\\\\varrho _{1,n}(0))=(0,f_n)$ , which leads to the solutions $(u_{1,n}(t),\\\\varrho _{1,n}(t))$ to (REF ) and $(u^{\\\\rm {ap}}_{1,n},\\\\varrho ^{\\\\rm {ap}}_{1,n})$ to the linearized system (REF ) respectively.\",\n \"Based on the special choice of $f_n$ , we can prove the distance between the real solution $(u_{1,n}(t),\\\\varrho _{1,n}(t))$ and the approximate solution $(u^{\\\\rm {ap}}_{1,n},\\\\varrho ^{\\\\rm {ap}}_{1,n})$ will tends to zero in $H^s$ as $n\\\\rightarrow \\\\infty $ .\",\n \"See Propositions REF .\",\n \"On the other hand, we construct another sequence of initial data $(u_{2,n}(0),\\\\varrho _{2,n}(0))=(g_n,f_n)$ , which leads to the solutions $(u_{2,n}(t),\\\\varrho _{2,n}(t))$ to (REF ) and $(u^{\\\\rm {ap}}_{2,n},\\\\varrho ^{\\\\rm {ap}}_{2,n})$ to the linearized system (REF ) respectively.\",\n \"Next, we aim to show that the mentioned solution $(u^{\\\\rm {ap}}_{2,n},\\\\varrho ^{\\\\rm {ap}}_{2,n})$ cannot approximate the real solution $(u_{2,n}(t),\\\\varrho _{2,n}(t))$ .\",\n \"For more details, see Propositions REF .\",\n \"Combining the precious steps, we can conclude that the distance of two solution maps at the initial time is converging to zero, while at any later time it is bounded below by a positive constant, which means the solution maps are not uniformly continuous.\",\n \"The structure of the paper In Section we recall some notations and known results which will be used in the sequel.\",\n \"Also, we investigate the spectrum properties of the linearized system corresponding to (REF ), which will play a crucial role in the construction of approximate solutions.\",\n \"In Section we prove Theorem REF based on the well-posedness result and some key error estimates.\",\n \"In Section we present some details in the computations.\"\n ],\n [\n \"Notations\",\n \"Firstly, we introduce some notations which shall be used in this paper.\",\n \"The notation $A\\\\lesssim B$ (resp., $A \\\\gtrsim B$ ) means that there exists a harmless positive constant $c$ such that $A \\\\le cB$ (resp., $A \\\\ge cB$ ).\",\n \"$A\\\\approx B$ means $A\\\\lesssim B$ and $A\\\\gtrsim B$ .\",\n \"Given a Banach space $X$ , we denote its norm by $\\\\Vert \\\\cdot \\\\Vert _{X}$ .\",\n \"We shall use the simplified notation $\\\\Vert f,\\\\cdots ,g\\\\Vert _X=\\\\Vert f\\\\Vert _X+\\\\cdots +\\\\Vert g\\\\Vert _X$ if there is no confusion.\",\n \"We shall denote by $\\\\langle f,g\\\\rangle $ the $L^2$ inner product of $f$ and $g$ .\",\n \"For all $f\\\\in \\\\mathcal {S}^{\\\\prime }$ , the Fourier transform $\\\\mathcal {F}f$ (also denoted by $\\\\widehat{f}$ ) is defined by $\\\\mathcal {F}f(\\\\xi )=\\\\widehat{f}(\\\\xi )=\\\\int _{\\\\mathbb {R}^2}e^{-ix\\\\xi }f(x)\\\\mathrm {d}x \\\\quad \\\\text{for any}\\\\; \\\\xi \\\\in \\\\mathbb {R}^2.$ We denote $\\\\Lambda =(-\\\\Delta )^{\\\\frac{1}{2}}$ .\",\n \"For $s\\\\in \\\\mathbb {R}$ , the operator $\\\\Lambda ^s$ is defined by $\\\\widehat{\\\\Lambda ^s f}(\\\\xi )=|\\\\xi |^{s}\\\\widehat{f}(\\\\xi ).$ The homogeneous and nonhomogeneous Sobolev space are defined by $&\\\\Vert f\\\\Vert ^2_{\\\\dot{H}^s}=\\\\Vert \\\\Lambda ^sf\\\\Vert ^2_{L^2}=\\\\int _{\\\\mathbb {R}^2}|\\\\xi |^{2s}|\\\\hat{f}(\\\\xi )|^2\\\\mathrm {d}\\\\xi ,\\\\\\\\&\\\\Vert f\\\\Vert ^2_{H^s}=\\\\int _{\\\\mathbb {R}^2}(1+|\\\\xi |^2)^s|\\\\hat{f}(\\\\xi )|^2\\\\mathrm {d}\\\\xi .$ Then, for $s>0$ , we have $\\\\Vert f\\\\Vert ^2_{H^s}\\\\approx \\\\Vert f\\\\Vert ^2_{L^2}+\\\\Vert f\\\\Vert ^2_{\\\\dot{H}^s}.$\"\n ],\n [\n \"Useful Tools\",\n \"Next, we review some useful tools involving the commutator and product estimates.\",\n \"Lemma 2.1 (See [13]) Let $s>1$ and $\\\\nabla f,g\\\\in H^s(\\\\mathbb {R}^2)$ .\",\n \"Then we have $\\\\Vert [\\\\Lambda ^s,f]\\\\cdot \\\\nabla g\\\\Vert _{L^2}\\\\le C\\\\Vert \\\\nabla f\\\\Vert _{H^s}\\\\Vert g\\\\Vert _{H^s}.$ Lemma 2.2 (See [27]) Let $s>0$ and $f,g\\\\in {\\\\rm {Lip}}\\\\cap H^s(\\\\mathbb {R}^2)$ and $g\\\\in L^\\\\infty \\\\cap H^{s-1}(\\\\mathbb {R}^2)$ .\",\n \"Then we have $\\\\Vert [\\\\Lambda ^s,f]g\\\\Vert _{L^2}\\\\le C\\\\big (\\\\Vert \\\\nabla f\\\\Vert _{L^\\\\infty }\\\\Vert g\\\\Vert _{H^{s-1}}+\\\\Vert f\\\\Vert _{H^s}\\\\Vert g\\\\Vert _{L^\\\\infty }\\\\big ).$ Lemma 2.3 (See [4]) Let $s>2$ and $f\\\\in H^{s-2}(\\\\mathbb {R}^2),g\\\\in H^{s-1}(\\\\mathbb {R}^2)$ .\",\n \"Then we have $\\\\Vert fg\\\\Vert _{H^{s-2}}\\\\le C\\\\Vert f\\\\Vert _{H^{s-2}}\\\\Vert g\\\\Vert _{H^{s-1}}.$ Lemma 2.4 (See [4]) Let $s>0$ and $f,g\\\\in L^\\\\infty \\\\cap H^s(\\\\mathbb {R}^2)$ .\",\n \"Then we have $\\\\Vert fg\\\\Vert _{H^s}\\\\le C\\\\big (\\\\Vert f\\\\Vert _{L^\\\\infty }\\\\Vert g\\\\Vert _{H^s}+\\\\Vert f\\\\Vert _{H^s}\\\\Vert g\\\\Vert _{L^\\\\infty }\\\\big ).$ Moreover, for $s>1$ , we have the algebra estimate $\\\\Vert fg\\\\Vert _{H^s}\\\\le C\\\\Vert f\\\\Vert _{H^s}\\\\Vert g\\\\Vert _{H^s}.$ Lemma 2.5 (See [4]) Let $s>0$ and $f$ be a smooth function such that $f(0)=0.$ If $u \\\\in H^{s}\\\\left(\\\\mathbb {R}^{2}\\\\right) \\\\cap L^{\\\\infty }\\\\left(\\\\mathbb {R}^{2}\\\\right)$ then there exists a function $C$ depending only on $s$ and $f$ such that $\\\\Vert f(u)\\\\Vert _{H^{s}} \\\\le C\\\\big (\\\\Vert u\\\\Vert _{L^{\\\\infty }}\\\\big )\\\\Vert u\\\\Vert _{H^{s}}.$ Lemma 2.6 (See [4]) Let $s>1$ and $f$ be a smooth function such that $f^{\\\\prime }(0)=0 .$ If $u, v \\\\in H^{s}\\\\left(\\\\mathbb {R}^{2}\\\\right) \\\\cap L^{\\\\infty }\\\\left(\\\\mathbb {R}^{2}\\\\right)$ , then there exists a function $C$ depending only on $s$ and $f$ such that $\\\\Vert f(u)-f(v)\\\\Vert _{H^{s}} \\\\le C\\\\big (\\\\Vert u\\\\Vert _{L^{\\\\infty }},\\\\Vert v\\\\Vert _{L^{\\\\infty }}\\\\big )\\\\Vert u-v\\\\Vert _{H^{s}}\\\\big (\\\\Vert u\\\\Vert _{H^{s}}+\\\\Vert v\\\\Vert _{H^{s}}\\\\big ).$\"\n ],\n [\n \"The Linearized System\",\n \"In this subsection, borrowing the idea from [7], we investigate the spectrum properties of the linearized system: $\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\partial _t\\\\varrho +u̥=0,\\\\\\\\\\\\partial _tu-{\\\\mu } \\\\Delta u-{\\\\mu } \\\\nabla u̥+\\\\nabla \\\\varrho =0,\\\\\\\\\\\\varrho |_{t=0}=\\\\varrho _0,\\\\;u|_{t=0}=u_0.\\\\end{array}\\\\right.$ Denote $d=\\\\Lambda ^{-1}u̥\\\\quad \\\\text{and}\\\\quad c=\\\\Lambda ^{-1}\\\\mathrm {curl} u,$ then we deduce from (REF ) $\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\partial _t\\\\varrho +\\\\Lambda d=0,\\\\\\\\\\\\partial _td-2{\\\\mu } \\\\Delta d-\\\\Lambda \\\\varrho =0,\\\\\\\\\\\\partial _tc-{\\\\mu } \\\\Delta c=0\\\\end{array}\\\\right.$ Let $\\\\mathbf {A}=\\\\left(\\\\begin{array}{cc}0 & -|\\\\xi | \\\\\\\\|\\\\xi | & -2\\\\mu |\\\\xi |^2 \\\\\\\\\\\\end{array}\\\\right),$ by taking the Fourier transform of $(\\\\ref {l1})_1$ and $(\\\\ref {l1})_2$ , we find that $\\\\partial _t\\\\left(\\\\begin{array}{c}\\\\widehat{\\\\varrho }\\\\\\\\\\\\widehat{d} \\\\\\\\\\\\end{array}\\\\right)=\\\\mathbf {A}\\\\left(\\\\begin{array}{c}\\\\widehat{\\\\varrho }\\\\\\\\\\\\widehat{d} \\\\\\\\\\\\end{array}\\\\right).$ Straightforward calculations give the eigenvalues of the matric $\\\\mathbf {A}$ as follows $\\\\lambda _{\\\\pm }=-\\\\mu |\\\\xi |^2\\\\pm \\\\sqrt{\\\\mu ^2|\\\\xi |^4-|\\\\xi |^2}$ and $\\\\mathbf {T}^{-1}\\\\mathbf {A}\\\\mathbf {T}=\\\\left(\\\\begin{array}{cc}\\\\lambda _{+} & 0 \\\\\\\\0 & \\\\lambda _{-} \\\\\\\\\\\\end{array}\\\\right),$ where the matrixes $\\\\mathbf {T}$ and $\\\\mathbf {T}^{-1}$ are given by $\\\\mathbf {T}=\\\\left(\\\\begin{array}{cc}\\\\lambda _{-} & \\\\lambda _{+} \\\\\\\\-|\\\\xi | & -|\\\\xi | \\\\\\\\\\\\end{array}\\\\right)\\\\quad \\\\text{and}\\\\quad \\\\mathbf {T}^{-1}=\\\\left(\\\\begin{array}{cc}-\\\\frac{1}{\\\\lambda _{+}-\\\\lambda _{-}} & -\\\\frac{\\\\lambda _{+}}{|\\\\xi |(\\\\lambda _{+}-\\\\lambda _{-})} \\\\\\\\\\\\frac{1}{\\\\lambda _{+}-\\\\lambda _{-}} & \\\\frac{\\\\lambda _{-}}{|\\\\xi |(\\\\lambda _{+}-\\\\lambda _{-})} \\\\\\\\\\\\end{array}\\\\right).$ Applying $\\\\mathbf {T}^{-1}$ to (REF ) and using Duhamel's principle, we get $\\\\mathbf {T}^{-1}\\\\left(\\\\begin{array}{c}\\\\widehat{\\\\varrho }\\\\\\\\\\\\widehat{d} \\\\\\\\\\\\end{array}\\\\right)=\\\\left(\\\\begin{array}{cc}e^{\\\\lambda _+t}&0\\\\\\\\0&e^{\\\\lambda _-t}\\\\\\\\\\\\end{array}\\\\right)\\\\mathbf {T}^{-1}\\\\left(\\\\begin{array}{c}\\\\widehat{\\\\varrho }_0\\\\\\\\\\\\widehat{d}_0 \\\\\\\\\\\\end{array}\\\\right),$ which implies that $&\\\\widehat{\\\\varrho }(t,\\\\xi )=\\\\frac{\\\\lambda _+e^{\\\\lambda _-t}-\\\\lambda _-e^{\\\\lambda _+t}}{\\\\lambda _+-\\\\lambda _-}\\\\widehat{\\\\varrho }_0-\\\\frac{e^{\\\\lambda _+t}-e^{\\\\lambda _-t}}{\\\\lambda _+-\\\\lambda _-}|\\\\xi |\\\\widehat{d}_0,\\\\\\\\&\\\\widehat{d}(t,\\\\xi )=\\\\frac{e^{\\\\lambda _+t}-e^{\\\\lambda _-t}}{\\\\lambda _+-\\\\lambda _-}|\\\\xi |\\\\widehat{\\\\varrho }_0+\\\\frac{\\\\lambda _+e^{\\\\lambda _+t}-\\\\lambda _-e^{\\\\lambda _-t}}{\\\\lambda _+-\\\\lambda _-}\\\\widehat{d}_0.$ From $(\\\\ref {l1})_3$ , we also have $&\\\\widehat{c}(t,\\\\xi )=e^{-\\\\mu t|\\\\xi |^2}\\\\widehat{c}_0.$ Due to the vector identity $\\\\Delta u=\\\\nabla u̥+\\\\nabla ^{\\\\bot }\\\\mathrm {curl}u=\\\\Lambda (\\\\nabla d+\\\\nabla ^{\\\\bot } c),$ which implies $u=-\\\\Lambda ^{-1}\\\\nabla d-\\\\Lambda ^{-1}\\\\nabla ^{\\\\bot } c,$ then we deduce from () and (REF ) $\\\\widehat{u}(t,\\\\xi )&=-\\\\frac{e^{\\\\lambda _+t}-e^{\\\\lambda _-t}}{\\\\lambda _+-\\\\lambda _-}\\\\mathcal {F}(\\\\nabla \\\\varrho _0)-\\\\frac{\\\\lambda _+e^{\\\\lambda _+t}-\\\\lambda _-e^{\\\\lambda _-t}}{\\\\lambda _+-\\\\lambda _-}\\\\mathcal {F}\\\\big (\\\\Lambda ^{-2}\\\\nabla u̥_0\\\\big )\\\\nonumber \\\\\\\\&\\\\quad -e^{-\\\\mu |\\\\xi |^2t }\\\\mathcal {F}(\\\\Lambda ^{-2}\\\\nabla ^{\\\\bot }\\\\mathrm {curl} u_0).$\"\n ],\n [\n \"Proof of Theorem \",\n \"This section is devoted to proving Theorem REF .\",\n \"We begin with the well-posedness result for (REF ).\",\n \"For the details of the proof, we refer to [30], [31], [36].\"\n ],\n [\n \"Global Well-posedness \",\n \"Lemma 3.1 Assume that $s>2$ .\",\n \"For any initial data $(u_0,\\\\varrho _0)$ which belongs to $B_R=\\\\big \\\\lbrace (\\\\psi ,\\\\phi )\\\\in H^s\\\\times H^s: \\\\Vert \\\\psi \\\\Vert _{H^s}+\\\\Vert \\\\phi \\\\Vert _{H^s}\\\\le c\\\\big \\\\rbrace \\\\quad \\\\text{for small}\\\\;c>0,$ then System (REF ) has a unique global solution $\\\\big (u(t),\\\\varrho (t)\\\\big )\\\\in \\\\mathcal {C}([0,+\\\\infty );H^s\\\\times H^s)$ .\",\n \"Moreover, we have the solution size estimate $\\\\Vert u(t)\\\\Vert _{H^{\\\\sigma }}+\\\\Vert \\\\varrho (t)\\\\Vert _{H^{\\\\sigma }}\\\\le C\\\\Vert u_0,\\\\varrho _0\\\\Vert _{H^{\\\\sigma }}\\\\quad \\\\text{for all}\\\\; \\\\sigma \\\\ge s.$\"\n ],\n [\n \"Approximate Solutions\",\n \"Let $\\\\widehat{\\\\phi }\\\\in \\\\mathcal {C}^\\\\infty _0(\\\\mathbb {R})$ be an even, real-valued and non-negative function on $\\\\mathbb {R}$ and satisfy ()= 1,if $|\\\\xi |\\\\le \\\\frac{1}{4}$ , 0,if $|\\\\xi |\\\\ge \\\\frac{1}{2}$ .\",\n \"Let $(u^{\\\\rm {ap}}_{i,n},\\\\varrho ^{\\\\rm {ap}}_{i,n})$ ($i=1,2$ ) be the solution of the following equation $\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\partial _t\\\\varrho ^{\\\\rm {ap}}_{i,n}+u̥^{\\\\rm {ap}}_{i,n}=0,\\\\\\\\\\\\partial _tu^{\\\\rm {ap}}_{i,n}-\\\\mu \\\\Delta u^{\\\\rm {ap}}_{i,n}-\\\\mu \\\\nabla u̥^{\\\\rm {ap}}_{i,n}+\\\\nabla \\\\varrho ^{\\\\rm {ap}}_{i,n}=0,\\\\\\\\\\\\end{array}\\\\right.$ supplemented with the following initial condition, respectively, $(u^{\\\\rm {ap}}_{1,n},\\\\varrho ^{\\\\rm {ap}}_{1,n})|_{t=0}=(0,f_n)\\\\quad \\\\text{and}\\\\quad (u^{\\\\rm {ap}}_{2,n},\\\\varrho ^{\\\\rm {ap}}_{2,n})|_{t=0}=(g_n,f_n),$ where $f_n=2^{-ns}\\\\phi (x_1)\\\\sin (2^nx_1)\\\\phi (x_2)\\\\quad \\\\text{and}\\\\quad g_n=2^{-n}\\\\nabla \\\\big (\\\\phi (x_1)\\\\phi (x_2)\\\\big ),\\\\quad n\\\\gg 1.$ The following Lemma will play an important role in the proof of Theorem REF .\",\n \"Lemma 3.2 Let $\\\\sigma \\\\in \\\\mathbb {R}$ .\",\n \"Assume that $(u^{\\\\rm {ap}}_{i,n},\\\\varrho ^{\\\\rm {ap}}_{i,n})$ with $i=1,2$ solves System (REF )–(REF ).\",\n \"Then there exists a constant $C=C(\\\\sigma )$ such that the following statement holds $&\\\\Vert \\\\varrho ^{\\\\rm {ap}}_{1,n},u^{\\\\rm {ap}}_{1,n}\\\\Vert ^2_{H^\\\\sigma }+\\\\mu \\\\int ^t_0\\\\Vert \\\\nabla u^{\\\\rm {ap}}_{1,n},u̥^{\\\\rm {ap}}_{1,n}\\\\Vert ^2_{H^\\\\sigma }\\\\mathrm {d}\\\\tau \\\\le C2^{2n(\\\\sigma -s)},\\\\\\\\&\\\\Vert \\\\varrho ^{\\\\rm {ap}}_{2,n},u^{\\\\rm {ap}}_{2,n}\\\\Vert ^2_{H^\\\\sigma }+\\\\mu \\\\int ^t_0\\\\Vert \\\\nabla u^{\\\\rm {ap}}_{2,n},u̥^{\\\\rm {ap}}_{2,n}\\\\Vert ^2_{H^\\\\sigma }\\\\mathrm {d}\\\\tau \\\\le C2^{2n \\\\max \\\\lbrace \\\\sigma -s,-1\\\\rbrace }.$ Proof.\",\n \"Standard energy method directly gives us that $\\\\frac{\\\\mathrm {d}}{\\\\mathrm {d}t}\\\\Vert \\\\varrho ^{\\\\rm {ap}}_{i,n},u^{\\\\rm {ap}}_{i,n}\\\\Vert ^2_{H^\\\\sigma }+\\\\mu \\\\Vert \\\\nabla u^{\\\\rm {ap}}_{i,n}\\\\Vert ^2_{H^\\\\sigma }+\\\\mu \\\\Vert u̥^{\\\\rm {ap}}_{i,n}\\\\Vert ^2_{H^\\\\sigma }=0,$ which implies $&\\\\Vert \\\\varrho ^{\\\\rm {ap}}_{1,n},u^{\\\\rm {ap}}_{1,n}\\\\Vert ^2_{H^\\\\sigma }+\\\\mu \\\\int ^t_0\\\\Vert \\\\nabla u^{\\\\rm {ap}}_{1,n},u̥^{\\\\rm {ap}}_{1,n}\\\\Vert ^2_{H^\\\\sigma }\\\\mathrm {d}\\\\tau =\\\\Vert f_n\\\\Vert ^2_{H^\\\\sigma },\\\\\\\\&\\\\Vert \\\\varrho ^{\\\\rm {ap}}_{2,n},u^{\\\\rm {ap}}_{2,n}\\\\Vert ^2_{H^\\\\sigma }+\\\\mu \\\\int ^t_0\\\\Vert \\\\nabla u^{\\\\rm {ap}}_{2,n},u̥^{\\\\rm {ap}}_{2,n}\\\\Vert ^2_{H^\\\\sigma }\\\\mathrm {d}\\\\tau =\\\\Vert f_n,g_n\\\\Vert ^2_{H^\\\\sigma }.$ Obviously, we have $&\\\\Vert g_n\\\\Vert _{H^\\\\sigma }\\\\lesssim 2^{-n}\\\\Vert \\\\phi (x_1)\\\\Vert _{L^2(\\\\mathbb {R})}\\\\Vert \\\\phi (x_2)\\\\Vert _{L^2(\\\\mathbb {R})}\\\\lesssim 2^{-n},\\\\nonumber \\\\\\\\&\\\\Vert f_n\\\\Vert _{H^\\\\sigma }\\\\lesssim 2^{-ns}\\\\Vert \\\\widetilde{f_n}(x_1,x_2)\\\\Vert _{H^\\\\sigma (\\\\mathbb {R}^2)},$ where $\\\\widetilde{f_n}(x_1,x_2)=\\\\phi (x_1)\\\\sin (2^nx_1)\\\\phi (x_2)$ .\",\n \"Easy computations give that $\\\\mathcal {F}\\\\big (\\\\widetilde{f_n}\\\\big )(\\\\xi _1,\\\\xi _2)=\\\\frac{i}{2}\\\\Big [\\\\hat{\\\\phi }(\\\\xi _1+2^n)-\\\\hat{\\\\phi }(\\\\xi _1-2^n)\\\\Big ]\\\\hat{\\\\phi }(\\\\xi _2),$ which implies $\\\\mathrm {supp} \\\\ \\\\mathcal {F}\\\\big (\\\\widetilde{f_n}\\\\big )\\\\subset \\\\mathcal {C}_n:=\\\\Big \\\\lbrace \\\\xi \\\\in \\\\mathbb {R}^2: \\\\ 2^n-1\\\\le |\\\\xi |\\\\le 2^n+1\\\\Big \\\\rbrace ,$ By the definition of Sobolev space, we get $\\\\Vert \\\\widetilde{f_n}(x_1,x_2)\\\\Vert _{H^\\\\sigma (\\\\mathbb {R}^2)}&\\\\lesssim \\\\int _{\\\\mathbb {R}^2}(1+|\\\\xi |^2)^{\\\\sigma /2}|\\\\mathcal {F}\\\\big (\\\\widetilde{f_n}\\\\big )|^2\\\\mathrm {d}\\\\xi \\\\nonumber \\\\\\\\&\\\\lesssim ~2^{n\\\\sigma }\\\\int _{\\\\mathcal {C}_n }|\\\\mathcal {F}\\\\big (\\\\widetilde{f_n}\\\\big )|^2\\\\mathrm {d}\\\\xi \\\\nonumber \\\\\\\\&\\\\lesssim 2^{n\\\\sigma }.$ Inserting (REF ) into (REF ) enables us to finish the proof of Lemma REF .\",\n \"Before proceeding on, we give two data-to-solution maps for the Cauchy problem (REF ) $&\\\\big (u_{1,n}(0)=0,\\\\varrho _{1,n}(0)=f_n\\\\big )\\\\mapsto \\\\big (u_{1,n},\\\\varrho _{1,n}\\\\big ),\\\\\\\\&\\\\big ({u}_{2,n}(0)=g_n,{\\\\varrho }_{2,n}(0)=f_n\\\\big )\\\\mapsto \\\\big ({u}_{2,n},{\\\\varrho }_{2,n}\\\\big ),$ where $f_{n}$ and $g_n$ are defined in Section REF .\",\n \"Let $\\\\varepsilon _s=\\\\frac{1}{2}(s-2)$ and $s^{\\\\prime }=s-\\\\varepsilon _s>2$ .\",\n \"For the initial data with $H^{s^{\\\\prime }}$ norm, we have $\\\\Vert u_{1,n}(0),u_{2,n}(0)\\\\Vert _{H^{s^{\\\\prime }}}+\\\\Vert \\\\varrho _{1,n}(0),\\\\varrho _{2,n}(0)\\\\Vert _{H^{s^{\\\\prime }}}\\\\le 2^{-n\\\\min \\\\lbrace \\\\varepsilon _s,1\\\\rbrace },$ which tends to 0 when $n$ tends to infinity.\",\n \"Therefore, by Lemma REF , we have the solutions size estimate which will be used implicitly in the sequel $&\\\\Vert u_{1,n},\\\\varrho _{1,n}\\\\Vert _{H^\\\\gamma }+\\\\Vert u_{2,n},\\\\varrho _{2,n}\\\\Vert _{H^{\\\\gamma }}\\\\lesssim 2^{-n\\\\min \\\\lbrace (s-\\\\gamma ),1\\\\rbrace }\\\\quad \\\\text{for all}\\\\; \\\\gamma \\\\ge s^{\\\\prime }.$ In particular, this implies that the condition $(\\\\bf {C.1})$ in Theorem REF holds.\"\n ],\n [\n \"Error Estimates\",\n \"Letting $\\\\varrho _{1,n}^{\\\\rm {er}}=\\\\varrho _{1,n}-\\\\varrho _{1,n}^{\\\\rm {ap}}$ and $u_{1,n}^{\\\\rm {er}}=u_{1,n}-u_{1,n}^{\\\\rm {ap}}$ , we can find that $(\\\\varrho _{1,n}^{\\\\rm {er}},u_{1,n}^{\\\\rm {er}})$ satisfies $\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\partial _t\\\\varrho _{1,n}^{\\\\rm {er}}+u̥_{1,n}^{\\\\rm {er}}+u_{1,n}\\\\cdot \\\\nabla \\\\varrho _{1,n}^{\\\\rm {er}}=\\\\mathbf {F_1},\\\\\\\\\\\\partial _tu_{1,n}^{\\\\rm {er}}-{\\\\mu } \\\\Delta u_{1,n}^{\\\\rm {er}}-{\\\\mu } \\\\nabla u̥_{1,n}^{\\\\rm {er}}+\\\\nabla \\\\varrho _{1,n}^{\\\\rm {er}}+u_{1,n}\\\\cdot \\\\nabla u_{1,n}^{\\\\rm {er}}=\\\\mathbf {F_2}+\\\\mathbf {F_3},\\\\\\\\\\\\varrho _{1,n}^{\\\\rm {er}}|_{t=0}=0,\\\\; u_{1,n}^{\\\\rm {er}}|_{t=0}=0, \\\\end{array}\\\\right.$ where $&\\\\mathbf {F_1}:=-u_{1,n}^{\\\\rm {er}}\\\\cdot \\\\nabla \\\\varrho ^{\\\\rm {ap}}_{1,n}-\\\\varrho _{1,n}u̥_{1,n}^{\\\\rm {er}}-\\\\varrho _{1,n}^{\\\\rm {er}}u̥_{1,n}^{\\\\rm {ap}}-\\\\varrho ^{\\\\rm {ap}}_{1,n}u_{1,n}^{\\\\rm {ap}}),\\\\\\\\&\\\\mathbf {F_2}:=-u_{1,n}^{\\\\rm {er}}\\\\cdot \\\\nabla u^{\\\\rm {ap}}_{1,n}- u^{\\\\rm {ap}}_{1,n}\\\\cdot \\\\nabla u^{\\\\rm {ap}}_{1,n},\\\\\\\\&\\\\mathbf {F_3}:=2\\\\nabla (\\\\ln (1+\\\\varrho _{1,n}))\\\\cdot \\\\mathbb {D}u_{1,n}.$ The following proposition implies that the error of approximate solution and actual solution will tends to zero in $H^s$ as $n\\\\rightarrow \\\\infty $ .\",\n \"Proposition 3.1 Under the assumptions of Theorem REF , then we have for $t\\\\le 1$ $\\\\Vert u_{1,n}-u_{1,n}^{\\\\rm {ap}}\\\\Vert _{H^s}+\\\\Vert \\\\varrho _{1,n}-\\\\varrho _{1,n}^{\\\\rm {ap}}\\\\Vert _{H^s}\\\\le C2^{-\\\\frac{n}{2}\\\\min \\\\lbrace \\\\varepsilon _s,1\\\\rbrace }.$ Proof.\",\n \"Taking the $L^2$ inner product of $(\\\\ref {u1})_1$ and $(\\\\ref {u1})_2$ with $(\\\\varrho _{1,n}^{\\\\rm {er}},u_{1,n}^{\\\\rm {er}})$ yields $&\\\\frac{1}{2}\\\\frac{\\\\mathrm {d}}{\\\\mathrm {d}t}\\\\Vert \\\\varrho _{1,n}^{\\\\rm {er}},u_{1,n}^{\\\\rm {er}}\\\\Vert ^2_{L^2}+\\\\mu \\\\Vert \\\\nabla u_{1,n}^{\\\\rm {er}},u̥_{1,n}^{\\\\rm {er}}\\\\Vert ^2_{L^{2}}\\\\nonumber \\\\\\\\=&~\\\\frac{1}{2} \\\\big + \\\\big <\\\\mathbf {F_1}, \\\\varrho _{1,n}^{\\\\rm {er}}\\\\big >+\\\\big <\\\\mathbf {F_2}+\\\\mathbf {F_3}, u_{1,n}^{\\\\rm {er}}\\\\big >\\\\nonumber \\\\\\\\\\\\lesssim &~ \\\\Vert u_{1,n}\\\\Vert _{H^{s}}\\\\Vert \\\\varrho _{1,n}^{\\\\rm {er}},u_{1,n}^{\\\\rm {er}}\\\\Vert ^2_{L^{2}}+\\\\Vert \\\\mathbf {F_1}\\\\Vert _{L^{2}}\\\\Vert \\\\varrho _{1,n}^{\\\\rm {er}}\\\\Vert _{L^2}+\\\\Vert \\\\mathbf {F_2},\\\\mathbf {F_3},u_{1,n}^{\\\\rm {er}}\\\\Vert ^2_{L^{2}}.$ Applying the operators $\\\\Lambda ^{s-1}\\\\varrho _{1,n}^{\\\\rm {er}}\\\\Lambda ^{s-1}$ and $\\\\Lambda ^{s-1}u_{1,n}^{\\\\rm {er}}\\\\Lambda ^{s-1}$ to $(\\\\ref {u1})_1$ and $(\\\\ref {u1})_2$ , respectively, then integrating the resulting over $\\\\mathbb {R}^2$ , we obtain $&\\\\frac{1}{2}\\\\frac{\\\\mathrm {d}}{\\\\mathrm {d}t}\\\\Vert \\\\varrho _{1,n}^{\\\\rm {er}},u_{1,n}^{\\\\rm {er}}\\\\Vert ^2_{{\\\\dot{H}}^{s-1}}+\\\\mu \\\\Vert \\\\nabla u_{1,n}^{\\\\rm {er}},u̥_{1,n}^{\\\\rm {er}}\\\\Vert ^2_{{\\\\dot{H}}^{s-1}}\\\\nonumber \\\\\\\\=&~\\\\frac{1}{2} \\\\big \\\\nonumber \\\\\\\\&-\\\\big <[\\\\Lambda ^{s-1},u_{1,n}]\\\\cdot \\\\nabla \\\\varrho _{1,n}^{\\\\rm {er}},\\\\Lambda ^{s-1}\\\\varrho _{1,n}^{\\\\rm {er}}\\\\big >-\\\\big <[\\\\Lambda ^{s-1},u_{1,n}]\\\\cdot \\\\nabla u_{1,n}^{\\\\rm {er}},\\\\Lambda ^{s-1}u_{1,n}^{\\\\rm {er}}\\\\big >\\\\nonumber \\\\\\\\&+ \\\\big <\\\\Lambda ^{s-1}\\\\mathbf {F_1}, \\\\Lambda ^{s-1}\\\\varrho _{1,n}^{\\\\rm {er}}\\\\big >+\\\\big < \\\\Lambda ^{s-2}\\\\mathbf {F_2}, \\\\Lambda ^{s}u_{1,n}^{\\\\rm {er}}\\\\big >+\\\\big < \\\\Lambda ^{s-2}\\\\mathbf {F_3}, \\\\Lambda ^{s}u_{1,n}^{\\\\rm {er}}\\\\big >\\\\nonumber \\\\\\\\\\\\lesssim &~ (1+\\\\Vert u_{1,n}\\\\Vert _{H^{s}})\\\\Vert \\\\varrho _{1,n}^{\\\\rm {er}},u_{1,n}^{\\\\rm {er}}\\\\Vert ^2_{{{H}}^{s-1}}+\\\\Vert \\\\mathbf {F_1}\\\\Vert _{{\\\\dot{H}}^{s-1}}\\\\Vert \\\\varrho _{1,n}^{\\\\rm {er}}\\\\Vert _{{\\\\dot{H}}^{s-1}}\\\\nonumber \\\\\\\\&+\\\\Vert \\\\mathbf {F_2},\\\\mathbf {F_3}\\\\Vert ^2_{{\\\\dot{H}}^{s-2}}+\\\\varepsilon \\\\Vert \\\\nabla u_{1,n}^{\\\\rm {er}}\\\\Vert ^2_{{\\\\dot{H}}^{s-1}},$ where we have used the commutator estimate from Lemma REF .\",\n \"Combing (REF ) and (REF ), we get $\\\\frac{\\\\mathrm {d}}{\\\\mathrm {d}t}\\\\Vert \\\\varrho _{1,n}^{\\\\rm {er}},u_{1,n}^{\\\\rm {er}}\\\\Vert ^2_{{{H}}^{s-1}}&+\\\\mu \\\\Vert \\\\nabla u_{1,n}^{\\\\rm {er}},u̥_{1,n}^{\\\\rm {er}}\\\\Vert ^2_{{{H}}^{s-1}}\\\\lesssim \\\\Vert \\\\varrho _{1,n}^{\\\\rm {er}},u_{1,n}^{\\\\rm {er}}\\\\Vert ^2_{{{H}}^{s-1}}\\\\nonumber \\\\\\\\&\\\\quad +\\\\Vert \\\\mathbf {F_1}\\\\Vert _{{{H}}^{s-1}}\\\\Vert \\\\varrho _{1,n}^{\\\\rm {er}}\\\\Vert _{{{H}}^{s-1}}+\\\\Vert \\\\mathbf {F_2},\\\\mathbf {F_3}\\\\Vert ^2_{{{H}}^{s-2}}+\\\\varepsilon \\\\Vert \\\\nabla u_{1,n}^{\\\\rm {er}}\\\\Vert ^2_{{\\\\dot{H}}^{s-1}}.$ Estimate of $\\\\mathbf {F_1}$ .\",\n \"Notice that $H^{s-1}(\\\\mathbb {R}^2)$ with $s>2$ is a Banach algebra, we have $\\\\Vert u_{1,n}^{\\\\rm {er}}\\\\cdot \\\\nabla \\\\varrho ^{\\\\rm {ap}}_{1,n}\\\\Vert _{H^{s-1}}&\\\\lesssim \\\\Vert u_{1,n}^{\\\\rm {er}}\\\\Vert _{H^{s-1}}\\\\Vert \\\\varrho ^{\\\\rm {ap}}_{1,n}\\\\Vert _{H^s}\\\\lesssim \\\\Vert u_{1,n}^{\\\\rm {er}}\\\\Vert _{H^{s-1}},\\\\\\\\\\\\Vert \\\\varrho _{1,n}u̥_{1,n}^{\\\\rm {er}}\\\\Vert _{H^{s-1}}&\\\\lesssim \\\\Vert \\\\varrho _{1,n}\\\\Vert _{H^{s-1}}\\\\Vert u̥_{1,n}^{\\\\rm {er}}\\\\Vert _{H^{s-1}}\\\\lesssim \\\\Vert u̥_{1,n}^{\\\\rm {er}}\\\\Vert _{H^{s-1}},\\\\\\\\\\\\Vert \\\\varrho _{1,n}^{\\\\rm {er}}u̥_{1,n}^{\\\\rm {ap}}\\\\Vert _{H^{s-1}}&\\\\lesssim \\\\Vert \\\\varrho _{1,n}^{\\\\rm {er}}\\\\Vert _{H^{s-1}}\\\\Vert u^{\\\\rm {ap}}_{1,n}\\\\Vert _{H^s}\\\\lesssim \\\\Vert \\\\varrho _{1,n}^{\\\\rm {er}}\\\\Vert _{H^{s-1}},\\\\\\\\\\\\Vert \\\\varrho ^{\\\\rm {ap}}_{1,n}u_{1,n}^{\\\\rm {ap}})\\\\Vert _{H^{s-1}}&\\\\lesssim \\\\Vert \\\\varrho ^{\\\\rm {ap}}_{1,n}u_{1,n}^{\\\\rm {ap}}\\\\Vert _{H^{s}}\\\\lesssim \\\\Vert \\\\varrho ^{\\\\rm {ap}}_{1,n}\\\\Vert _{L^\\\\infty }\\\\Vert u_{1,n}^{\\\\rm {ap}}\\\\Vert _{H^{s}}+\\\\Vert \\\\varrho ^{\\\\rm {ap}}_{1,n}\\\\Vert _{H^{s}}\\\\Vert u_{1,n}^{\\\\rm {ap}}\\\\Vert _{L^\\\\infty }\\\\\\\\&\\\\lesssim \\\\Vert \\\\varrho ^{\\\\rm {ap}}_{1,n}\\\\Vert _{H^{s-1-\\\\varepsilon _s}}\\\\Vert u_{1,n}^{\\\\rm {ap}}\\\\Vert _{H^{s}}+\\\\Vert \\\\varrho ^{\\\\rm {ap}}_{1,n}\\\\Vert _{H^{s}}\\\\Vert u_{1,n}^{\\\\rm {ap}}\\\\Vert _{H^{s-1-\\\\varepsilon _s}}\\\\lesssim 2^{-(1+\\\\varepsilon _s)n},$ which implies $\\\\Vert \\\\mathbf {F_1}\\\\Vert _{H^{s-1}}&\\\\lesssim \\\\Vert u_{1,n}^{\\\\rm {er}},\\\\varrho _{1,n}^{\\\\rm {er}}\\\\Vert _{H^{s-1}}+\\\\Vert u̥_{1,n}^{\\\\rm {er}}\\\\Vert _{H^{s-1}}+2^{-(1+\\\\varepsilon _s)n}.$ Estimate of $\\\\mathbf {F_2}$ .\",\n \"Using Lemma REF yields $\\\\Vert u_{1,n}^{\\\\rm {er}}\\\\cdot \\\\nabla u^{\\\\rm {ap}}_{1,n}\\\\Vert _{H^{s-2}}&\\\\lesssim \\\\Vert u_{1,n}^{\\\\rm {er}}\\\\Vert _{H^{s-1}}\\\\Vert u^{\\\\rm {ap}}_{1,n}\\\\Vert _{H^{s-1}}\\\\lesssim \\\\Vert u_{1,n}^{\\\\rm {er}}\\\\Vert _{H^{s-1}},\\\\\\\\\\\\Vert u^{\\\\rm {ap}}_{1,n}\\\\cdot \\\\nabla u^{\\\\rm {ap}}_{1,n}\\\\Vert _{H^{s-2}}&\\\\lesssim \\\\Vert u^{\\\\rm {ap}}_{1,n}\\\\Vert ^2_{H^{s-1}}\\\\lesssim 2^{-2n},$ which implies $\\\\Vert \\\\mathbf {F_2}\\\\Vert _{H^{s-2}}&\\\\lesssim \\\\Vert u_{1,n}^{\\\\rm {er}}\\\\Vert _{H^{s-1}}+2^{-2n}.$ Estimate of $\\\\mathbf {F_3}$ .\",\n \"We can decompose the term $\\\\nabla (\\\\ln (1+\\\\varrho _{1,n}))\\\\cdot \\\\mathbb {D}u_{1,n}$ as $\\\\nabla (\\\\ln (1+\\\\varrho _{1,n}))\\\\cdot \\\\mathbb {D}u_{1,n}&=\\\\nabla (\\\\ln (1+\\\\varrho _{1,n}))\\\\cdot \\\\mathbb {D}u_{1,n}^{\\\\rm {er}}+\\\\nabla (\\\\ln (1+\\\\varrho _{1,n})-\\\\ln (1+\\\\varrho ^{\\\\rm {ap}}_{1,n}))\\\\cdot \\\\mathbb {D}u^{\\\\rm {ap}}_{1,n}\\\\\\\\&\\\\quad +\\\\nabla (\\\\ln (1+\\\\varrho ^{\\\\rm {ap}}_{1,n}))\\\\cdot \\\\mathbb {D}u^{\\\\rm {ap}}_{1,n}\\\\\\\\&=\\\\mathbf {F_{3,1}}+\\\\mathbf {F_{3,2}}+\\\\mathbf {F_{3,3}}.$ For the first two terms, by Lemma REF and Lemmas REF –REF , we have $&\\\\Vert \\\\mathbf {F_{3,1}}\\\\Vert _{H^{s-2}}\\\\lesssim \\\\Vert \\\\mathbb {D}u_{1,n}^{\\\\rm {er}}\\\\Vert _{H^{s-2}}\\\\Vert \\\\varrho _{1,n}\\\\Vert _{H^s}\\\\lesssim \\\\Vert u_{1,n}^{\\\\rm {er}}\\\\Vert _{H^{s-1}},\\\\\\\\&\\\\Vert \\\\mathbf {F_{3,2}}\\\\Vert _{H^{s-2}}\\\\lesssim \\\\Vert \\\\varrho _{1,n}^{\\\\rm {er}}\\\\Vert _{H^{s-1}}\\\\Vert u^{\\\\rm {ap}}_{1,n}\\\\Vert _{H^s}\\\\lesssim \\\\Vert \\\\varrho _{1,n}^{\\\\rm {er}}\\\\Vert _{H^{s-1}}.$ For the third term, by Lemmas REF and REF , we have $\\\\Vert \\\\mathbf {F_{3,3}}\\\\Vert _{H^{s-2}}&\\\\lesssim \\\\Vert \\\\nabla (\\\\ln (1+\\\\varrho ^{\\\\rm {ap}}_{1,n}))\\\\Vert _{L^\\\\infty }\\\\Vert u_{1,n}^{\\\\rm {ap}}\\\\Vert _{H^{s-1}}+\\\\Vert \\\\nabla (\\\\ln (1+\\\\varrho ^{\\\\rm {ap}}_{1,n}))\\\\Vert _{H^{s-2}}\\\\Vert \\\\nabla u_{1,n}^{\\\\rm {ap}}\\\\Vert _{L^\\\\infty }\\\\\\\\&\\\\lesssim \\\\Vert \\\\varrho ^{\\\\rm {ap}}_{1,n}\\\\Vert _{H^{s-\\\\varepsilon _s}}\\\\Vert u_{1,n}^{\\\\rm {ap}}\\\\Vert _{H^{s-1}}+\\\\Vert \\\\varrho ^{\\\\rm {ap}}_{1,n}\\\\Vert _{H^{s-1}}\\\\Vert u_{1,n}^{\\\\rm {ap}}\\\\Vert _{H^{s-\\\\varepsilon _s}}\\\\lesssim 2^{-(1+\\\\varepsilon _s)n},$ which implies $\\\\Vert \\\\mathbf {F_3}\\\\Vert _{H^{s-2}}&\\\\lesssim \\\\Vert u_{1,n}^{\\\\rm {er}},\\\\varrho _{1,n}^{\\\\rm {er}}\\\\Vert _{H^{s-1}}+2^{-(1+\\\\varepsilon _s)n}.$ Putting the above estimates (REF )–(REF ) together with (REF ), we can obtain $\\\\frac{\\\\mathrm {d}}{\\\\mathrm {d}t}\\\\Vert \\\\varrho _{1,n}^{\\\\rm {er}},u_{1,n}^{\\\\rm {er}}\\\\Vert ^2_{{{H}}^{s-1}}+\\\\mu \\\\Vert \\\\nabla u_{1,n}^{\\\\rm {er}},u̥_{1,n}^{\\\\rm {er}}\\\\Vert ^2_{{{H}}^{s-1}}\\\\lesssim &~ \\\\Vert \\\\varrho _{1,n}^{\\\\rm {er}},u_{1,n}^{\\\\rm {er}}\\\\Vert ^2_{H^{s-1}}+2^{-2n(1+\\\\min \\\\lbrace \\\\varepsilon _s,1\\\\rbrace )}\\\\\\\\&+\\\\varepsilon \\\\Vert \\\\nabla u_{1,n}^{\\\\rm {er}},u̥_{1,n}^{\\\\rm {er}}\\\\Vert ^2_{{{H}}^{s-1}}.$ Absorbing the $\\\\varepsilon $ -term and using Gronwall's inequality yield $\\\\Vert \\\\varrho _{1,n}^{\\\\rm {er}},u_{1,n}^{\\\\rm {er}}\\\\Vert _{{{H}}^{s-1}}\\\\le C2^{-n(1+\\\\min \\\\lbrace \\\\varepsilon _s,1\\\\rbrace )}.$ An interpolation argument leads to $\\\\Vert \\\\varrho _{1,n}^{\\\\rm {er}},u_{1,n}^{\\\\rm {er}}\\\\Vert _{{{H}}^{s}}\\\\lesssim \\\\Vert \\\\varrho _{1,n}^{\\\\rm {er}},u_{1,n}^{\\\\rm {er}}\\\\Vert _{{{H}}^{s-1}}^{\\\\frac{1}{2}}\\\\Vert \\\\varrho _{1,n}^{\\\\rm {er}},u_{1,n}^{\\\\rm {er}}\\\\Vert _{{{H}}^{s+1}}^{\\\\frac{1}{2}}\\\\lesssim 2^{-\\\\frac{n}{2}\\\\min \\\\lbrace \\\\varepsilon _s,1\\\\rbrace }.$ Thus, we have finished the proof of Proposition REF .\",\n \"Denoting $V^{\\\\rm {ap}}_{n}=-u_{2,n}^{\\\\rm {ap}}\\\\cdot \\\\nabla \\\\varrho _{2,n}^{\\\\rm {ap}}$ and introducing the errors $\\\\varrho _{2,n}^{\\\\rm {er}}=\\\\varrho _{2,n}-\\\\varrho _{2,n}^{\\\\rm {ap}}-tV^{\\\\rm {ap}}_{n}\\\\quad \\\\text{and}\\\\quad u_{2,n}^{\\\\rm {er}}=u_{2,n}-u_{2,n}^{\\\\rm {ap}},$ we can find that $(\\\\varrho _{2,n}^{\\\\rm {er}},u_{2,n}^{\\\\rm {er}})$ satisfies $\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\partial _t\\\\varrho _{2,n}^{\\\\rm {er}}+u̥_{2,n}^{\\\\rm {er}}+u_{2,n}\\\\cdot \\\\nabla \\\\varrho _{2,n}^{\\\\rm {er}}=\\\\mathbf {G_1}-t\\\\mathbf {G_4}-t\\\\partial _tV^{\\\\rm {ap}}_{n},\\\\\\\\\\\\partial _tu_{2,n}^{\\\\rm {er}}-{\\\\mu } \\\\Delta u_{2,n}^{\\\\rm {er}}-{\\\\mu } \\\\nabla u̥_{2,n}^{\\\\rm {er}}+\\\\nabla \\\\varrho _{2,n}^{\\\\rm {er}}+u_{2,n}\\\\cdot \\\\nabla u_{2,n}^{\\\\rm {er}}=\\\\mathbf {G_2}+\\\\mathbf {G_3}-t\\\\nabla V^{\\\\rm {ap}}_{n}, \\\\\\\\\\\\varrho _{2,n}^{\\\\rm {er}}|_{t=0}=0,\\\\; u_{2,n}^{\\\\rm {er}}|_{t=0}=0, \\\\end{array}\\\\right.$ where $&\\\\mathbf {G_1}:=-u_{2,n}^{\\\\rm {er}}\\\\cdot \\\\nabla \\\\varrho ^{\\\\rm {ap}}_{2,n}-\\\\varrho _{2,n}u̥_{2,n}^{\\\\rm {er}}-\\\\varrho _{2,n}^{\\\\rm {er}}u̥_{2,n}^{\\\\rm {ap}}-\\\\varrho ^{\\\\rm {ap}}_{2,n}u̥_{2,n}^{\\\\rm {ap}},\\\\\\\\&\\\\mathbf {G_2}:=-u_{2,n}^{\\\\rm {er}}\\\\cdot \\\\nabla u^{\\\\rm {ap}}_{2,n}- u^{\\\\rm {ap}}_{2,n}\\\\cdot \\\\nabla u^{\\\\rm {ap}}_{2,n},\\\\\\\\&\\\\mathbf {G_3}:=2\\\\nabla (\\\\ln (1+\\\\varrho _{2,n}))\\\\cdot \\\\mathbb {D}u_{2,n},\\\\\\\\&\\\\mathbf {G_4}:=u_{2,n}^{\\\\rm {er}}\\\\cdot \\\\nabla V^{\\\\rm {ap}}_{n}+V^{\\\\rm {ap}}_{n} u_{2,n}^{\\\\rm {ap}}).$ Proposition 3.2 Under the assumptions of Theorem REF , then we have for $t\\\\le 1$ $\\\\Vert u_{2,n}-u_{2,n}^{\\\\rm {ap}}\\\\Vert _{H^s}+\\\\Vert \\\\varrho _{2,n}-\\\\varrho _{2,n}^{\\\\rm {ap}}-tV^{\\\\rm {ap}}_{n}\\\\Vert _{H^s}\\\\le C2^{-n\\\\min \\\\lbrace \\\\varepsilon _s,1\\\\rbrace }+Ct^{\\\\frac{3}{2}}.$ Proof.\",\n \"Following the procedure in (REF ) and (REF ), we obtain $&\\\\frac{\\\\mathrm {d}}{\\\\mathrm {d}t}\\\\Vert \\\\varrho _{2,n}^{\\\\rm {er}},u_{2,n}^{\\\\rm {er}}\\\\Vert ^2_{{{H}}^{s}}+\\\\mu \\\\Vert \\\\nabla u_{2,n}^{\\\\rm {er}},u̥_{2,n}^{\\\\rm {er}}\\\\Vert ^2_{{{H}}^{s}}\\\\nonumber \\\\\\\\\\\\lesssim &~ (1+\\\\Vert u_{2,n}\\\\Vert _{H^{s}})\\\\Vert \\\\varrho _{2,n}^{\\\\rm {er}},u_{2,n}^{\\\\rm {er}}\\\\Vert ^2_{{{H}}^{s}}\\\\nonumber \\\\\\\\&+\\\\big (\\\\Vert \\\\mathbf {G_1}\\\\Vert _{{{H}}^{s}}+\\\\Vert t\\\\partial _tV^{\\\\rm {ap}}_{n}\\\\Vert _{{{H}}^{s}}+t\\\\Vert \\\\mathbf {G_4}\\\\Vert _{{{H}}^{s}}\\\\big )\\\\Vert \\\\varrho _{2,n}^{\\\\rm {er}}\\\\Vert _{{{H}}^{s}}\\\\nonumber \\\\\\\\&+\\\\Vert \\\\mathbf {G_2},\\\\mathbf {G_3},t\\\\nabla V^{\\\\rm {ap}}_{n}\\\\Vert ^2_{{{H}}^{s-1}}\\\\nonumber \\\\\\\\\\\\lesssim &~ \\\\Vert \\\\varrho _{2,n}^{\\\\rm {er}},u_{2,n}^{\\\\rm {er}}\\\\Vert ^2_{{{H}}^{s}}+\\\\Vert \\\\mathbf {G_1}\\\\Vert _{{{H}}^{s}}\\\\Vert \\\\varrho _{2,n}^{\\\\rm {er}}\\\\Vert _{{{H}}^{s}}+\\\\Vert \\\\mathbf {G_2},\\\\mathbf {G_3}\\\\Vert ^2_{{{H}}^{s-1}}+\\\\Vert t\\\\partial _tV^{\\\\rm {ap}}_{n}\\\\Vert ^2_{{{H}}^{s}}\\\\nonumber \\\\\\\\&+t^2\\\\big (\\\\Vert \\\\mathbf {G_4}\\\\Vert ^2_{{{H}}^{s}}+\\\\Vert V^{\\\\rm {ap}}_{n}\\\\Vert ^2_{{{H}}^{s}}\\\\big ).$ We should mentioned that the commutator estimate from Lemma REF was used here.\",\n \"It is not hard to deduce that for $k\\\\in \\\\lbrace -1,0,1\\\\rbrace $ $\\\\Vert V^{\\\\rm {ap}}_{n}\\\\Vert _{H^{s+k}}&\\\\lesssim \\\\Vert u_{2,n}^{\\\\rm {ap}}\\\\Vert _{L^\\\\infty }\\\\Vert \\\\nabla \\\\varrho _{2,n}^{\\\\rm {ap}}\\\\Vert _{H^{s+k}}+\\\\Vert u_{2,n}^{\\\\rm {ap}}\\\\Vert _{H^{s+k}}\\\\Vert \\\\nabla \\\\varrho _{2,n}^{\\\\rm {ap}}\\\\Vert _{L^\\\\infty }\\\\\\\\&\\\\lesssim \\\\Vert u_{2,n}^{\\\\rm {ap}}\\\\Vert _{H^{s-1}}\\\\Vert \\\\nabla \\\\varrho _{2,n}^{\\\\rm {ap}}\\\\Vert _{H^{s+k}}+\\\\Vert u_{2,n}^{\\\\rm {ap}}\\\\Vert _{H^{s+k}}\\\\Vert \\\\nabla \\\\varrho _{2,n}^{\\\\rm {ap}}\\\\Vert _{H^{s-1}}\\\\\\\\&\\\\lesssim 2^{kn}.$ Estimate of $\\\\mathbf {G_1}$ .\",\n \"Using Lemma REF yields $\\\\Vert u_{2,n}^{\\\\rm {er}}\\\\cdot \\\\nabla \\\\varrho ^{\\\\rm {ap}}_{2,n}\\\\Vert _{H^{s}}&\\\\lesssim \\\\Vert u_{2,n}^{\\\\rm {er}}\\\\Vert _{L^\\\\infty }\\\\Vert \\\\varrho ^{\\\\rm {ap}}_{2,n}\\\\Vert _{H^{s+1}}+\\\\Vert u_{2,n}^{\\\\rm {er}}\\\\Vert _{H^s}\\\\Vert \\\\nabla \\\\varrho ^{\\\\rm {ap}}_{2,n}\\\\Vert _{L^\\\\infty }\\\\\\\\&\\\\lesssim 2^n\\\\Vert u_{2,n}^{\\\\rm {er}}\\\\Vert _{H^{s-1}}+\\\\Vert u_{2,n}^{\\\\rm {er}}\\\\Vert _{H^{s}},\\\\\\\\\\\\Vert \\\\varrho _{2,n}u̥_{2,n}^{\\\\rm {er}}\\\\Vert _{H^{s}}&\\\\lesssim \\\\Vert u̥_{2,n}^{\\\\rm {er}}\\\\Vert _{H^{s}},\\\\\\\\\\\\Vert \\\\varrho _{2,n}^{\\\\rm {er}}u̥_{2,n}^{\\\\rm {ap}}\\\\Vert _{H^{s}}&\\\\lesssim 2^n\\\\Vert \\\\varrho _{2,n}^{\\\\rm {er}}\\\\Vert _{H^{s-1}}+\\\\Vert \\\\varrho _{2,n}^{\\\\rm {er}}\\\\Vert _{H^{s}},\\\\\\\\\\\\Vert \\\\varrho ^{\\\\rm {ap}}_{2,n}u̥_{2,n}^{\\\\rm {ap}}\\\\Vert _{H^{s}}&\\\\lesssim \\\\Vert \\\\varrho _{2,n}^{\\\\rm {ap}}\\\\Vert _{H^{s-1}}\\\\Vert u̥^{\\\\rm {ap}}_{2,n}\\\\Vert _{H^{s}}+\\\\Vert \\\\varrho _{2,n}^{\\\\rm {ap}}\\\\Vert _{H^s}\\\\Vert u^{\\\\rm {ap}}_{2,n}\\\\Vert _{H^{s-\\\\varepsilon _s}}\\\\\\\\&\\\\lesssim 2^{-n}\\\\Vert u̥^{\\\\rm {ap}}_{2,n}\\\\Vert _{H^{s}}+2^{-n\\\\min \\\\lbrace \\\\varepsilon _s,1\\\\rbrace },$ which implies that $\\\\Vert \\\\mathbf {G_{1}}\\\\Vert _{H^{s}}&\\\\lesssim 2^n\\\\Vert u_{2,n}^{\\\\rm {er}},\\\\varrho _{2,n}^{\\\\rm {er}}\\\\Vert _{H^{s-1}}+\\\\Vert u̥_{2,n}^{\\\\rm {er}}\\\\Vert _{H^{s}}+\\\\Vert u_{2,n}^{\\\\rm {er}},\\\\varrho _{2,n}^{\\\\rm {er}}\\\\Vert _{H^{s}}\\\\nonumber \\\\\\\\&\\\\quad +2^{-n}\\\\Vert u̥^{\\\\rm {ap}}_{2,n}\\\\Vert _{H^{s}}+2^{-n\\\\min \\\\lbrace \\\\varepsilon _s,1\\\\rbrace }.$ Estimate of $\\\\mathbf {G_2}$ .\",\n \"Notice that $H^{s-1}(\\\\mathbb {R}^2)$ with $s>2$ is a Banach algebra, we have $\\\\Vert u_{2,n}^{\\\\rm {er}}\\\\cdot \\\\nabla u^{\\\\rm {ap}}_{2,n}\\\\Vert _{H^{s-1}}&\\\\lesssim \\\\Vert u_{2,n}^{\\\\rm {er}}\\\\Vert _{H^{s-1}}\\\\Vert u_{2,n}^{\\\\rm {ap}}\\\\Vert _{H^{s}}\\\\lesssim \\\\Vert u_{2,n}^{\\\\rm {er}}\\\\Vert _{H^{s}},\\\\\\\\\\\\Vert u^{\\\\rm {ap}}_{2,n}\\\\cdot \\\\nabla u^{\\\\rm {ap}}_{2,n}\\\\Vert _{H^{s-1}}&\\\\lesssim \\\\Vert u^{\\\\rm {ap}}_{2,n}\\\\Vert _{H^{s-1}}\\\\Vert u_{2,n}^{\\\\rm {ap}}\\\\Vert _{H^{s}}\\\\lesssim 2^{-n},$ which implies that $\\\\Vert \\\\mathbf {G_{2}}\\\\Vert _{H^{s-1}}&\\\\lesssim \\\\Vert u_{2,n}^{\\\\rm {er}}\\\\Vert _{H^{s}}+2^{-n}.$ Estimate of $\\\\mathbf {G_3}$ .\",\n \"For the term $\\\\nabla (\\\\ln (1+\\\\varrho _{2,n}))\\\\cdot \\\\mathbb {D}u_{2,n}$ , we can decompose it as $\\\\nabla (\\\\ln (1+\\\\varrho _{2,n}))\\\\cdot \\\\mathbb {D}u_{2,n}&=\\\\nabla (\\\\ln (1+\\\\varrho _{2,n}))\\\\cdot \\\\mathbb {D}u_{2,n}^{\\\\rm {er}}\\\\\\\\&\\\\quad +\\\\nabla (\\\\ln (1+\\\\varrho _{2,n})-\\\\ln (1+\\\\varrho ^{\\\\rm {ap}}_{2,n}+tV^{\\\\rm {ap}}_{n}))\\\\cdot \\\\mathbb {D}u^{\\\\rm {ap}}_{2,n}\\\\\\\\&\\\\quad +\\\\nabla (\\\\ln (1+\\\\varrho ^{\\\\rm {ap}}_{2,n}+tV^{\\\\rm {ap}}_{n}))\\\\cdot \\\\mathbb {D}u^{\\\\rm {ap}}_{2,n}\\\\\\\\&=\\\\mathbf {G_{3,1}}+\\\\mathbf {G_{3,2}}+\\\\mathbf {G_{3,3}}.$ By Lemmas REF and REF , it is easy to deduce that $&\\\\Vert \\\\mathbf {G_{3,1}}\\\\Vert _{H^{s-1}}\\\\lesssim \\\\Vert u_{2,n}^{\\\\rm {er}}\\\\Vert _{H^{s}},\\\\\\\\&\\\\Vert \\\\mathbf {G_{3,2}}\\\\Vert _{H^{s-1}}\\\\lesssim \\\\Vert \\\\varrho _{2,n}^{\\\\rm {er}}\\\\Vert _{H^{s}},\\\\\\\\&\\\\Vert \\\\mathbf {G_{3,3}}\\\\Vert _{H^{s-1}}\\\\lesssim 2^{-n\\\\min \\\\lbrace \\\\varepsilon _s,1\\\\rbrace },$ which gives that $\\\\Vert \\\\mathbf {G_{3}}\\\\Vert _{H^{s-1}}&\\\\lesssim \\\\Vert u_{2,n}^{\\\\rm {er}},\\\\varrho _{2,n}^{\\\\rm {er}}\\\\Vert _{H^{s}}+2^{-n\\\\min \\\\lbrace \\\\varepsilon _s,1\\\\rbrace }.$ Estimate of $\\\\mathbf {G_4}$ .\",\n \"By Lemma REF again, we have $\\\\Vert u_{2,n}^{\\\\rm {er}}\\\\cdot \\\\nabla V^{\\\\rm {ap}}_{n}\\\\Vert _{H^{s}}&\\\\lesssim \\\\Vert u_{2,n}^{\\\\rm {er}}\\\\Vert _{L^\\\\infty }\\\\Vert V^{\\\\rm {ap}}_{n}\\\\Vert _{H^{s+1}}+\\\\Vert u_{2,n}^{\\\\rm {er}}\\\\Vert _{H^s}\\\\Vert \\\\nabla V^{\\\\rm {ap}}_{n}\\\\Vert _{L^\\\\infty }\\\\lesssim 2^n\\\\Vert u_{2,n}^{\\\\rm {er}}\\\\Vert _{H^{s-1}}+\\\\Vert u_{2,n}^{\\\\rm {er}}\\\\Vert _{H^{s}},\\\\\\\\\\\\Vert V^{\\\\rm {ap}}_{2,n}u_{2,n}^{\\\\rm {ap}})\\\\Vert _{H^{s}}&\\\\lesssim \\\\Vert V^{\\\\rm {ap}}_{2,n}u̥_{2,n}^{\\\\rm {ap}}\\\\Vert _{H^{s}}+ \\\\Vert u_{2,n}^{\\\\rm {ap}}\\\\cdot \\\\nabla V^{\\\\rm {ap}}_{2,n}\\\\Vert _{H^{s}}\\\\lesssim 1$ and $\\\\Vert u_{2,n}^{\\\\rm {er}}\\\\cdot \\\\nabla V^{\\\\rm {ap}}_{n}\\\\Vert _{H^{s-1}}&\\\\lesssim \\\\Vert u_{2,n}^{\\\\rm {er}}\\\\Vert _{L^\\\\infty }\\\\Vert V^{\\\\rm {ap}}_{n}\\\\Vert _{H^{s}}+\\\\Vert u_{2,n}^{\\\\rm {er}}\\\\Vert _{H^{s-1}}\\\\Vert \\\\nabla V^{\\\\rm {ap}}_{n}\\\\Vert _{L^\\\\infty }\\\\lesssim \\\\Vert u_{2,n}^{\\\\rm {er}}\\\\Vert _{H^{s-1}},\\\\\\\\\\\\Vert V^{\\\\rm {ap}}_{2,n}u_{2,n}^{\\\\rm {ap}})\\\\Vert _{H^{s-1}}&\\\\lesssim \\\\Vert V^{\\\\rm {ap}}_{2,n}u_{2,n}^{\\\\rm {ap}}\\\\Vert _{H^{s}}\\\\lesssim 2^{-n},$ from which, we obtain $\\\\Vert \\\\mathbf {G_{4}}\\\\Vert _{H^{s}}&\\\\lesssim 2^n\\\\Vert u_{2,n}^{\\\\rm {er}}\\\\Vert _{H^{s-1}}+\\\\Vert u_{2,n}^{\\\\rm {er}}\\\\Vert _{H^{s}}+1\\\\quad \\\\text{and}\\\\quad \\\\Vert \\\\mathbf {G_{4}}\\\\Vert _{H^{s-1}}\\\\lesssim \\\\Vert u_{2,n}^{\\\\rm {er}}\\\\Vert _{H^{s-1}}+2^{-n}.$ Next, we need to deal with the involved term $\\\\Vert t\\\\partial _tV^{\\\\rm {ap}}_{n}\\\\Vert _{{{H}}^{s}}$ .\",\n \"Estimate of $\\\\Vert t\\\\partial _tV^{\\\\rm {ap}}_{n}\\\\Vert _{{{H}}^{s}}$ .\",\n \"Direct calculation gives that $&-\\\\partial _tV_{n}^{\\\\rm {ap}}=\\\\partial _tu_{2,n}^{\\\\rm {ap}}\\\\cdot \\\\nabla \\\\varrho _{2,n}^{\\\\rm {ap}}+u_{2,n}^{\\\\rm {ap}}\\\\cdot \\\\nabla \\\\partial _t\\\\varrho _{2,n}^{\\\\rm {ap}},$ using the following estimates whose proof are relegated to A.1–A.2 in Section $&\\\\Vert t\\\\partial _tu_{2,n}^{\\\\rm {ap}}\\\\Vert _{H^{s+k}}\\\\le C2^{(k-1)n}+Ct2^{-n},\\\\\\\\&\\\\Vert \\\\partial _t\\\\varrho _{2,n}^{\\\\rm {ap}}\\\\Vert _{H^{s+k}}\\\\le C2^{kn},$ then we can estimate $\\\\Vert t\\\\partial _tV^{\\\\rm {ap}}_{n}\\\\Vert _{{{H}}^{s}}$ as $\\\\Vert t\\\\partial _tV^{\\\\rm {ap}}_{n}\\\\Vert _{H^{s+k}}&\\\\lesssim \\\\Vert t\\\\partial _tu_{2,n}^{\\\\rm {ap}}\\\\Vert _{L^\\\\infty }\\\\Vert \\\\nabla \\\\varrho _{2,n}^{\\\\rm {ap}}\\\\Vert _{H^{s+k}}+\\\\Vert t\\\\partial _tu_{2,n}^{\\\\rm {ap}}\\\\Vert _{H^{s+k}}\\\\Vert \\\\nabla \\\\varrho _{2,n}^{\\\\rm {ap}}\\\\Vert _{L^\\\\infty }\\\\\\\\&\\\\quad +t\\\\Vert u_{2,n}^{\\\\rm {ap}}\\\\Vert _{L^\\\\infty }\\\\Vert \\\\nabla \\\\partial _t\\\\varrho _{2,n}^{\\\\rm {ap}}\\\\Vert _{H^{s+k}}+t\\\\Vert u_{2,n}^{\\\\rm {ap}}\\\\Vert _{H^{s+k}}\\\\Vert \\\\nabla \\\\partial _t\\\\varrho _{2,n}^{\\\\rm {ap}}\\\\Vert _{L^\\\\infty }\\\\\\\\&\\\\lesssim \\\\Vert t\\\\partial _tu_{2,n}^{\\\\rm {ap}}\\\\Vert _{H^{s-1}}\\\\Vert \\\\varrho _{2,n}^{\\\\rm {ap}}\\\\Vert _{H^{s+k+1}}+\\\\Vert t\\\\partial _tu_{2,n}^{\\\\rm {ap}}\\\\Vert _{H^{s+k}}\\\\Vert \\\\varrho _{2,n}^{\\\\rm {ap}}\\\\Vert _{H^{s}}\\\\\\\\&\\\\quad +t\\\\Vert u_{2,n}^{\\\\rm {ap}}\\\\Vert _{H^{s-1}}\\\\Vert \\\\partial _t\\\\varrho _{2,n}^{\\\\rm {ap}}\\\\Vert _{H^{s+k+1}}+t\\\\Vert u_{2,n}^{\\\\rm {ap}}\\\\Vert _{H^{s+k}}\\\\Vert \\\\partial _t\\\\varrho _{2,n}^{\\\\rm {ap}}\\\\Vert _{H^{s}}\\\\\\\\&\\\\lesssim t2^{kn}+2^{(k-1)n}.$ Putting the above estimates into (REF ) yields $\\\\frac{\\\\mathrm {d}}{\\\\mathrm {d}t}\\\\Vert \\\\varrho _{2,n}^{\\\\rm {er}},u_{2,n}^{\\\\rm {er}}\\\\Vert ^2_{{{H}}^{s}}\\\\lesssim &~\\\\Vert \\\\varrho _{2,n}^{\\\\rm {er}},u_{2,n}^{\\\\rm {er}}\\\\Vert ^2_{{{H}}^{s}}+2^{2n}\\\\Vert \\\\varrho _{2,n}^{\\\\rm {er}},u_{2,n}^{\\\\rm {er}}\\\\Vert ^2_{H^{s-1}}+2^{-2n}\\\\Vert u̥_{2,n}^{\\\\rm {ap}}\\\\Vert ^2_{H^{s}}\\\\nonumber \\\\\\\\&+2^{-2n\\\\min \\\\lbrace \\\\varepsilon _s,1\\\\rbrace }+t^2.$ Finally, to close the above, we have to estimate $\\\\Vert \\\\varrho _{2,n}^{\\\\rm {er}},u_{2,n}^{\\\\rm {er}}\\\\Vert ^2_{H^{s-1}}$ .\",\n \"Following the procedure in (REF ) and (REF ) once again, we obtain $&\\\\frac{\\\\mathrm {d}}{\\\\mathrm {d}t}\\\\Vert \\\\varrho _{2,n}^{\\\\rm {er}},u_{2,n}^{\\\\rm {er}}\\\\Vert ^2_{{{H}}^{s-1}}+\\\\mu \\\\Vert \\\\nabla u_{2,n}^{\\\\rm {er}},u̥_{2,n}^{\\\\rm {er}}\\\\Vert ^2_{{{H}}^{s-1}}\\\\nonumber \\\\\\\\\\\\lesssim &~ (1+\\\\Vert u_{2,n}\\\\Vert _{H^{s}})\\\\Vert \\\\varrho _{2,n}^{\\\\rm {er}},u_{2,n}^{\\\\rm {er}}\\\\Vert ^2_{{{H}}^{s-1}}\\\\nonumber \\\\\\\\&+\\\\big (\\\\Vert \\\\mathbf {G_1}\\\\Vert _{{{H}}^{s-1}}+\\\\Vert t\\\\partial _tV^{\\\\rm {ap}}_{n}\\\\Vert _{{{H}}^{s-1}}\\\\big )\\\\Vert \\\\varrho _{2,n}^{\\\\rm {er}}\\\\Vert _{H^{s-1}}+\\\\Vert \\\\mathbf {G_2},\\\\mathbf {G_3}\\\\Vert ^2_{{{H}}^{s-2}}\\\\nonumber \\\\\\\\&+t^2\\\\big (\\\\Vert \\\\mathbf {G_4}\\\\Vert ^2_{H^{s-1}}+\\\\Vert V^{\\\\rm {ap}}_{n}\\\\Vert ^2_{{{H}}^{s-1}}\\\\big )\\\\nonumber \\\\\\\\\\\\lesssim &~\\\\Vert \\\\varrho _{2,n}^{\\\\rm {er}},u_{2,n}^{\\\\rm {er}}\\\\Vert ^2_{{{H}}^{s-1}}+\\\\Vert \\\\mathbf {G_1}\\\\Vert _{{{H}}^{s-1}}\\\\Vert \\\\varrho _{2,n}^{\\\\rm {er}}\\\\Vert _{H^{s-1}}+\\\\Vert \\\\mathbf {G_2},\\\\mathbf {G_3}\\\\Vert ^2_{{{H}}^{s-2}}+\\\\Vert t\\\\partial _tV^{\\\\rm {ap}}_{n}\\\\Vert ^2_{{{H}}^{s-1}}\\\\nonumber \\\\\\\\&+t^2\\\\big (\\\\Vert \\\\mathbf {G_4}\\\\Vert ^2_{H^{s-1}}+\\\\Vert V^{\\\\rm {ap}}_{n}\\\\Vert ^2_{{{H}}^{s-1}}\\\\big ).$ It should be mentioned that the first three terms of $\\\\mathbf {G_1}$ can be done as that of $\\\\mathbf {F_1}$ .\",\n \"The only difference is the forth term of $\\\\mathbf {G_1}$ .\",\n \"In fact, we estimate it as $\\\\Vert \\\\varrho ^{\\\\rm {ap}}_{2,n}u̥_{2,n}^{\\\\rm {ap}}\\\\Vert _{{{H}}^{s-1}}\\\\lesssim \\\\Vert \\\\varrho ^{\\\\rm {ap}}_{2,n}\\\\Vert _{{{H}}^{s-1}}\\\\Vert u̥_{2,n}^{\\\\rm {ap}}\\\\Vert _{{{H}}^{s-1}}\\\\lesssim 2^{-n}\\\\Vert u̥_{2,n}^{\\\\rm {ap}}\\\\Vert _{{{H}}^{s-1}}.$ Then we have $\\\\Vert \\\\mathbf {G_1}\\\\Vert _{H^{s-1}}&\\\\lesssim \\\\Vert u_{1,n}^{\\\\rm {er}},\\\\varrho _{1,n}^{\\\\rm {er}}\\\\Vert _{H^{s-1}}+\\\\Vert u̥_{1,n}^{\\\\rm {er}}\\\\Vert _{H^{s-1}}+2^{-n}\\\\Vert u̥_{2,n}^{\\\\rm {ap}}\\\\Vert _{{{H}}^{s-1}}.$ The estimates of $\\\\mathbf {G_2}$ and $\\\\mathbf {G_3}$ in $H^{s-2}$ can be done as that of $\\\\mathbf {F_2}$ and $\\\\mathbf {F_3}$ , respectively.\",\n \"We have $\\\\Vert \\\\mathbf {G_2},\\\\mathbf {G_3}\\\\Vert _{H^{s-2}}&\\\\lesssim \\\\Vert u_{1,n}^{\\\\rm {er}},\\\\varrho _{1,n}^{\\\\rm {er}}\\\\Vert _{H^{s-1}}+2^{-(1+\\\\varepsilon _s)n}+2^{-2n}.$ Combining the above estimates with (REF ), we can obtain $&\\\\frac{\\\\mathrm {d}}{\\\\mathrm {d}t}\\\\Vert \\\\varrho _{2,n}^{\\\\rm {er}},u_{2,n}^{\\\\rm {er}}\\\\Vert ^2_{{{H}}^{s-1}}\\\\lesssim \\\\Vert \\\\varrho _{2,n}^{\\\\rm {er}},u_{2,n}^{\\\\rm {er}}\\\\Vert ^2_{{{H}}^{s-1}}+2^{-2n}\\\\Vert u̥_{2,n}^{\\\\rm {ap}}\\\\Vert ^2_{{{H}}^{s-1}}+2^{-2n(1+\\\\min \\\\lbrace \\\\varepsilon _s,1\\\\rbrace )}+2^{-2n}t^2,$ which follows from Gronwall's inequality and Lemma REF that $\\\\Vert \\\\varrho _{2,n}^{\\\\rm {er}},u_{2,n}^{\\\\rm {er}}\\\\Vert ^2_{{{H}}^{s-1}}\\\\lesssim 2^{-2n(1+\\\\min \\\\lbrace \\\\varepsilon _s,1\\\\rbrace )}+2^{-2n}t^3.$ Plugging (REF ) into (REF ) yields $\\\\frac{\\\\mathrm {d}}{\\\\mathrm {d}t}\\\\Vert \\\\varrho _{2,n}^{\\\\rm {er}},u_{2,n}^{\\\\rm {er}}\\\\Vert ^2_{{{H}}^{s}}\\\\lesssim &~\\\\Vert \\\\varrho _{2,n}^{\\\\rm {er}},u_{2,n}^{\\\\rm {er}}\\\\Vert ^2_{{{H}}^{s}}+2^{-2n}\\\\Vert u̥_{2,n}^{\\\\rm {ap}}\\\\Vert ^2_{H^{s}}+2^{-2n\\\\min \\\\lbrace \\\\varepsilon _s,1\\\\rbrace }+t^2.$ Using Gronwall's inequality and Lemma REF yields $\\\\Vert \\\\varrho _{2,n}^{\\\\rm {er}},u_{2,n}^{\\\\rm {er}}\\\\Vert ^2_{{{H}}^{s}}\\\\lesssim &~2^{-2n\\\\min \\\\lbrace \\\\varepsilon _s,1\\\\rbrace }+t^3.$ Thus, we have finished the proof of Proposition REF .\"\n ],\n [\n \"Non-uniform Continuous Dependence\",\n \"With Propositions REF –REF in hand, we can prove Theorem REF .\",\n \"Behavior at time $t=0$ .\",\n \"Obviously, we have $\\\\Vert {u}_{2,n}(0)-u_{1,n}(0)\\\\Vert _{H^{s}}+\\\\Vert \\\\varrho _{2,n}(0)-\\\\varrho _{1,n}(0)\\\\Vert _{H^{s}}=\\\\Vert g_n\\\\Vert _{H^{s}}\\\\le C2^{-n},$ which means that the condition $(\\\\bf {C.2})$ in Theorem REF holds.\",\n \"Behavior at time $t>0$ .\",\n \"Notice that $&u_{2,n}-u_{1,n}=u^{\\\\rm {er}}_{2,n}-u^{\\\\rm {er}}_{1,n}+u^{\\\\rm {ap}}_{2,n}-u^{\\\\rm {ap}}_{1,n},\\\\nonumber \\\\\\\\&\\\\varrho _{2,n}-\\\\varrho _{1,n}=t{V}_n^{\\\\rm {ap}}+\\\\varrho ^{\\\\rm {er}}_{2,n}-\\\\varrho ^{\\\\rm {er}}_{1,n}+\\\\varrho ^{\\\\rm {ap}}_{2,n}-\\\\varrho ^{\\\\rm {ap}}_{1,n},$ by the triangle inequality and Propositions REF and REF , we deduce that for $t\\\\le 1$ $&\\\\Vert \\\\varrho _{2,n}(t)-\\\\varrho _{1,n}(t)\\\\Vert _{H^{s}}+\\\\Vert u_{2,n}(t)-u_{1,n}(t)\\\\Vert _{H^{s}}\\\\nonumber \\\\\\\\\\\\ge &~ t\\\\Vert V^{\\\\rm {ap}}_{n}\\\\Vert _{H^s}-\\\\big (\\\\Vert u^{\\\\rm {ap}}_{2,n}-u^{\\\\rm {ap}}_{1,n}\\\\Vert _{H^s}+\\\\Vert \\\\varrho ^{\\\\rm {ap}}_{2,n}-\\\\varrho ^{\\\\rm {ap}}_{1,n}\\\\Vert _{H^s}\\\\big )-\\\\big (\\\\Vert u^{\\\\rm {er}}_{1,n},\\\\varrho ^{\\\\rm {er}}_{1,n}\\\\Vert _{H^s}+\\\\Vert u^{\\\\rm {er}}_{2,n},\\\\varrho ^{\\\\rm {er}}_{2,n}\\\\Vert _{H^s}\\\\big )\\\\nonumber \\\\\\\\\\\\gtrsim &~ t\\\\Vert V^{\\\\rm {ap}}_{n}\\\\Vert _{H^s}-2^{-n}-2^{-\\\\frac{n}{2}\\\\min \\\\lbrace \\\\varepsilon _s,1\\\\rbrace }-2^{-n\\\\min \\\\lbrace \\\\varepsilon _s,1\\\\rbrace }-t^{\\\\frac{3}{2}},$ where we have used $\\\\Vert u^{\\\\rm {ap}}_{2,n}-u^{\\\\rm {ap}}_{1,n}\\\\Vert ^2_{H^s}+\\\\Vert \\\\varrho ^{\\\\rm {ap}}_{2,n}-\\\\varrho ^{\\\\rm {ap}}_{1,n}\\\\Vert ^2_{H^s}\\\\lesssim \\\\Vert g_n\\\\Vert ^2_{H^s}\\\\lesssim 2^{-2n}.$ Notice that $-V^{\\\\rm {ap}}_{n}=u_{2,n}^{\\\\rm {ap}}\\\\cdot \\\\nabla \\\\varrho _{2,n}^{\\\\rm {ap}}$ , we decompose it as $-V^{\\\\rm {ap}}_{n}=(u_{2,n}^{\\\\rm {ap}})_1\\\\partial _1 \\\\varrho _{2,n}^{\\\\rm {ap}}+(u_{2,n}^{\\\\rm {ap}})_2\\\\partial _2 \\\\varrho _{2,n}^{\\\\rm {ap}}=(g_n)_1\\\\partial _1f_n+E,$ where $E:=(u_{2,n}^{\\\\rm {ap}}-g_n)_1\\\\partial _1\\\\varrho _{2,n}^{\\\\rm {ap}}+(g_n)_1\\\\partial _1(\\\\varrho _{2,n}^{\\\\rm {ap}}-f_n)+(u_{2,n}^{\\\\rm {ap}})_2\\\\partial _2\\\\varrho _{2,n}^{\\\\rm {ap}}.$ Then we have for $t\\\\le 1$ $\\\\Vert E\\\\Vert _{H^{s}}\\\\le Ct+C2^{-n}.$ For more details of proof, see A.3 in Section .\",\n \"Furthermore, (REF ) reduces to $\\\\Vert \\\\varrho _{2,n}(t)-\\\\varrho _{1,n}(t)\\\\Vert _{H^{s}}+\\\\Vert u_{2,n}(t)-u_{1,n}(t)\\\\Vert _{H^{s}}\\\\gtrsim t\\\\Vert (g_n)_1\\\\partial _1f_n\\\\Vert _{H^s}-2^{-\\\\frac{n}{2}\\\\min \\\\lbrace \\\\varepsilon _s,1\\\\rbrace }-t^{\\\\frac{3}{2}},$ combining the following estimate whose proof is postponed to A.4 in Section $\\\\liminf _{n\\\\rightarrow \\\\infty }\\\\Vert (g_n)_1\\\\partial _1f_n\\\\Vert _{H^s}\\\\gtrsim \\\\Vert \\\\phi ^{\\\\prime }\\\\phi \\\\Vert ^2_{L^2}\\\\Vert \\\\phi ^2\\\\Vert _{L^2},$ we get from (REF ) that $\\\\liminf _{n\\\\rightarrow \\\\infty }\\\\big (\\\\Vert \\\\varrho _{2,n}(t)-\\\\varrho _{1,n}(t)\\\\Vert _{H^{s}}+\\\\Vert u_{2,n}(t)-u_{1,n}(t)\\\\Vert _{H^{s}}\\\\big )\\\\gtrsim t\\\\quad \\\\text{for} \\\\ t \\\\ \\\\text{small enough},$ which is nothing but the condition $(\\\\bf {C.3})$ in Theorem REF .\",\n \"Thus, we complete the proof of Theorem REF .\"\n ],\n [\n \"Appendix\",\n \"For the sake of convenience, here we present more details in the computations.\",\n \"A.1 Proof of (REF ).\",\n \"When $|\\\\xi |\\\\rightarrow \\\\infty $ , we have $\\\\lambda _-(\\\\xi )\\\\sim -2\\\\mu |\\\\xi |^2 \\\\quad \\\\text{and}\\\\quad \\\\lambda _+(\\\\xi )\\\\sim -\\\\frac{1}{2\\\\mu }.$ From (REF ), then $\\\\mathcal {F}\\\\big (u_{2,n}^{\\\\rm {ap}}\\\\big )(t,\\\\xi )&=-\\\\frac{e^{\\\\lambda _+t}-e^{\\\\lambda _-t}}{\\\\lambda _+-\\\\lambda _-}\\\\mathcal {F}\\\\big (\\\\nabla f_n\\\\big )-\\\\frac{\\\\lambda _+e^{\\\\lambda _+t}-\\\\lambda _-e^{\\\\lambda _-t}}{\\\\lambda _+-\\\\lambda _-}\\\\mathcal {F}\\\\big (\\\\Lambda ^{-2}\\\\nabla g̥_n\\\\big )$ which gives $&\\\\mathcal {F}\\\\big (t\\\\partial _tu_{2,n}^{\\\\rm {ap}}\\\\big )=-\\\\frac{t\\\\lambda _+e^{\\\\lambda _+t}-t\\\\lambda _-e^{\\\\lambda _-t}}{\\\\lambda _+-\\\\lambda _-}\\\\mathcal {F}\\\\big (\\\\nabla f_n\\\\big )-t\\\\frac{\\\\lambda _{+}^2e^{\\\\lambda _+t}-\\\\lambda _{-}^2e^{\\\\lambda _-t}}{\\\\lambda _+-\\\\lambda _-}\\\\mathcal {F}\\\\big (\\\\Lambda ^{-2}\\\\nabla g̥_n\\\\big ).$ Using the simple facts $&|t\\\\lambda _{\\\\pm }e^{\\\\lambda _{\\\\pm }t}|\\\\le 1\\\\quad \\\\text{if }\\\\;\\\\xi \\\\in \\\\mathcal {C}_n \\\\\\\\&\\\\Big |\\\\frac{\\\\lambda _{+}^2e^{\\\\lambda _+t}-\\\\lambda _{-}^2e^{\\\\lambda _-t}}{\\\\lambda _+-\\\\lambda _-}\\\\Big |\\\\lesssim |\\\\lambda _++\\\\lambda _-|+t|\\\\lambda _-|^2\\\\lesssim 1\\\\quad \\\\text{if }\\\\;\\\\xi \\\\in \\\\mathcal {B}(0,1),$ then we have $\\\\big |\\\\mathcal {F}\\\\big (t\\\\partial _tu_{2,n}^{\\\\rm {ap}}\\\\big )\\\\big |&\\\\le \\\\Big |\\\\frac{t\\\\lambda _+e^{\\\\lambda _+t}-t\\\\lambda _-e^{\\\\lambda _-t}}{\\\\lambda _+-\\\\lambda _-}\\\\mathcal {F}\\\\big (\\\\nabla f_n\\\\big )\\\\Big |+t\\\\Big |\\\\frac{\\\\lambda _{+}^2e^{\\\\lambda _+t}-\\\\lambda _{-}^2e^{\\\\lambda _-t}}{\\\\lambda _+-\\\\lambda _-}\\\\mathcal {F}\\\\big (\\\\Lambda ^{-2}\\\\nabla g̥_n\\\\big )\\\\Big |\\\\\\\\&\\\\le \\\\frac{2}{|\\\\lambda _+-\\\\lambda _-|}\\\\big |\\\\mathcal {F}\\\\big (\\\\nabla f_n\\\\big )\\\\big |+t\\\\big |\\\\mathcal {F}\\\\big (\\\\Lambda ^{-2}\\\\nabla g̥_n\\\\big )\\\\big |\\\\\\\\&\\\\approx \\\\big |\\\\mathcal {F}\\\\big (\\\\Lambda ^{-2}\\\\nabla f_n\\\\big )\\\\big |+t\\\\big |\\\\mathcal {F}\\\\big (\\\\Lambda ^{-2}\\\\nabla g̥_n\\\\big )\\\\big |,$ which in turn gives $\\\\Vert t\\\\partial _tu_{2,n}^{\\\\rm {ap}}\\\\Vert _{H^{s+k}}&=\\\\Big [\\\\int _{\\\\mathbb {R}^2}(1+|\\\\xi |^2)^{\\\\frac{s+k}{2}}\\\\big |\\\\mathcal {F}\\\\big (t\\\\partial _tu_{2,n}^{\\\\rm {ap}}\\\\big )\\\\big |^2\\\\mathrm {d}\\\\xi \\\\Big ]^{\\\\frac{1}{2}}\\\\\\\\&\\\\lesssim \\\\Big [\\\\int _{\\\\mathcal {C}_n }(1+|\\\\xi |^2)^{\\\\frac{s+k}{2}}\\\\big |\\\\mathcal {F}\\\\big (\\\\Lambda ^{-2}\\\\nabla f_n\\\\big )\\\\big |^2\\\\mathrm {d}\\\\xi \\\\Big ]^{\\\\frac{1}{2}}+t\\\\Big [\\\\int _{\\\\mathcal {B}(0,1)}\\\\big |\\\\mathcal {F}\\\\big (\\\\Lambda ^{-2}\\\\nabla g̥_n\\\\big )\\\\big |^2\\\\mathrm {d}\\\\xi \\\\Big ]^{\\\\frac{1}{2}}\\\\\\\\&\\\\lesssim \\\\Vert f_n\\\\Vert _{H^{s+k-1}}+t\\\\Vert g_n\\\\Vert _{L^{2}}.$ A.2 Proof of ().\",\n \"From (REF ), then $\\\\mathcal {F}\\\\big (\\\\partial _t\\\\varrho _{2,n}^{\\\\rm {ap}}\\\\big )&=-\\\\frac{\\\\lambda _+\\\\lambda _-(e^{\\\\lambda _+t}-e^{\\\\lambda _-t})}{\\\\lambda _+-\\\\lambda _-}\\\\mathcal {F}\\\\big (f_n\\\\big )-\\\\frac{\\\\lambda _{+}e^{\\\\lambda _+t}-\\\\lambda _{-}e^{\\\\lambda _-t}}{\\\\lambda _+-\\\\lambda _-}\\\\mathcal {F}\\\\big ( g̥_n\\\\big )\\\\\\\\&=-\\\\frac{e^{\\\\lambda _+t}-e^{\\\\lambda _-t}}{\\\\sqrt{\\\\mu ^2-|\\\\xi |^{-2}}}\\\\mathcal {F}\\\\big (f_n\\\\big )-\\\\frac{\\\\lambda _{+}e^{\\\\lambda _+t}-\\\\lambda _{-}e^{\\\\lambda _-t}}{\\\\lambda _+-\\\\lambda _-}\\\\mathcal {F}\\\\big ( g̥_n\\\\big ),$ similarly, we have $\\\\Vert \\\\partial _t\\\\varrho _{2,n}^{\\\\rm {ap}}\\\\Vert _{H^{s+k}}&=\\\\Big [\\\\int _{\\\\mathbb {R}^2}(1+|\\\\xi |^2)^{\\\\frac{s+k}{2}}\\\\big |\\\\mathcal {F}\\\\big (\\\\partial _t\\\\varrho _{2,n}^{\\\\rm {ap}}\\\\big )\\\\big |^2\\\\mathrm {d}\\\\xi \\\\Big ]^{\\\\frac{1}{2}}\\\\\\\\&\\\\lesssim \\\\Big [\\\\int _{\\\\mathcal {C}_n }(1+|\\\\xi |^2)^{\\\\frac{s+k}{2}}\\\\big |\\\\widehat{f_n}(\\\\xi )\\\\big |^2\\\\mathrm {d}\\\\xi \\\\Big ]^{\\\\frac{1}{2}}+\\\\Big [\\\\int _{\\\\mathcal {B}(0,1)}(1+|\\\\xi |^2)^{\\\\frac{s+k}{2}}\\\\big |\\\\mathcal {F}\\\\big ( g̥_n\\\\big )\\\\big |^2\\\\mathrm {d}\\\\xi \\\\Big ]^{\\\\frac{1}{2}}\\\\\\\\&\\\\lesssim \\\\Vert f_n\\\\Vert _{H^{s+k}}+\\\\Vert g_n\\\\Vert _{L^{2}}.$ A.3 Proof of (REF ).\",\n \"Notice that $&\\\\widehat{\\\\varrho ^{\\\\rm {ap}}_{2,n}}-\\\\widehat{f_n}=\\\\frac{\\\\lambda _+(e^{\\\\lambda _-t}-e^{\\\\lambda _+t})}{\\\\lambda _+-\\\\lambda _-}\\\\widehat{f_n}+(e^{\\\\lambda _+t}-1)\\\\widehat{f_n}-\\\\frac{e^{\\\\lambda _+t}-e^{\\\\lambda _-t}}{\\\\lambda _+-\\\\lambda _-}\\\\mathcal {F}(g̥_n),\\\\\\\\&\\\\widehat{u^{\\\\rm {ap}}_{2,n}}-\\\\widehat{g_n}=-\\\\frac{e^{\\\\lambda _+t}-e^{\\\\lambda _-t}}{\\\\lambda _+-\\\\lambda _-}\\\\widehat{\\\\nabla f_n}+\\\\Big (e^{\\\\lambda _+t}-1+\\\\frac{\\\\lambda _-(e^{\\\\lambda _+t}-e^{\\\\lambda _-t})}{\\\\lambda _+-\\\\lambda _-}\\\\Big )\\\\widehat{g_n},$ then by (REF ) $\\\\Vert \\\\varrho ^{\\\\rm {ap}}_{2,n}-f_n\\\\Vert _{H^{s+1}}&\\\\lesssim t\\\\Big [\\\\int _{\\\\mathcal {C}_n }(1+|\\\\xi |^2)^{\\\\frac{s+1}{2}}|\\\\widehat{f_n}|^2\\\\mathrm {d}\\\\xi +\\\\int _{\\\\mathcal {B}(0,1)}(1+|\\\\xi |^2)^{\\\\frac{s+1}{2}}\\\\big |\\\\mathcal {F}\\\\big (g̥_n\\\\big )\\\\big |^2\\\\mathrm {d}\\\\xi \\\\Big ]^{\\\\frac{1}{2}}\\\\\\\\&\\\\lesssim t\\\\big (\\\\Vert f_n\\\\Vert _{H^{s+1}}+\\\\Vert g_n\\\\Vert _{L^{2}}\\\\big )\\\\lesssim t2^{n}$ and $\\\\Vert u^{\\\\rm {ap}}_{2,n}-g_n\\\\Vert _{H^{s+k}}&\\\\lesssim \\\\Big [\\\\int _{\\\\mathcal {C}_n }(1+|\\\\xi |^2)^{\\\\frac{s+k}{2}}\\\\big |\\\\widehat{\\\\Lambda ^{-1}f_n}\\\\big |^2\\\\mathrm {d}\\\\xi +t^2\\\\int _{\\\\mathcal {B}(0,1)}(1+|\\\\xi |^2)^{\\\\frac{s+k}{2}}\\\\big |\\\\mathcal {F}\\\\big ( g_n\\\\big )\\\\big |^2\\\\mathrm {d}\\\\xi \\\\Big ]^{\\\\frac{1}{2}}\\\\\\\\&\\\\lesssim \\\\Vert f_n\\\\Vert _{H^{s+k-1}}+t\\\\Vert g_n\\\\Vert _{L^{2}}\\\\lesssim 2^{(k-1)n}+t2^{-n}.$ By Lemma REF , we deduce $&\\\\Vert (u_{2,n}^{\\\\rm {ap}}-g_n)_1\\\\partial _1\\\\varrho _{2,n}^{\\\\rm {ap}}\\\\Vert _{H^{s}}\\\\lesssim \\\\Vert u_{2,n}^{\\\\rm {ap}}-g_n\\\\Vert _{H^{s-1}}\\\\Vert \\\\varrho _{2,n}^{\\\\rm {ap}}\\\\Vert _{H^{s+1}}+\\\\Vert u_{2,n}^{\\\\rm {ap}}-g_n\\\\Vert _{H^{s}}\\\\Vert \\\\varrho _{2,n}^{\\\\rm {ap}}\\\\Vert _{H^{s}}\\\\lesssim t+2^{-n},\\\\\\\\&\\\\Vert (g_n)_1\\\\partial _1(\\\\varrho _{2,n}^{\\\\rm {ap}}-f_n)\\\\Vert _{H^{s}}\\\\lesssim \\\\Vert g_n\\\\Vert _{H^{s}}\\\\Vert \\\\varrho _{2,n}^{\\\\rm {ap}}-f_n\\\\Vert _{H^{s+1}}\\\\lesssim t,\\\\\\\\&\\\\Vert (u_{2,n}^{\\\\rm {ap}})_2\\\\partial _2\\\\varrho _{2,n}^{\\\\rm {ap}}\\\\Vert _{H^{s}}\\\\lesssim \\\\Vert u_{2,n}^{\\\\rm {ap}}\\\\Vert _{H^{s-1}}\\\\Vert \\\\partial _2\\\\varrho _{2,n}^{\\\\rm {ap}}\\\\Vert _{H^{s}}+\\\\Vert u_{2,n}^{\\\\rm {ap}}\\\\Vert _{H^{s}}\\\\Vert \\\\partial _2\\\\varrho _{2,n}^{\\\\rm {ap}}\\\\Vert _{H^{s-1}}\\\\lesssim 2^{-n},$ which implies (REF ).\",\n \"A.4 Proof of (REF ).\",\n \"Notice that the support condition of $\\\\widehat{\\\\phi }$ , we deduce that $2^{-n}\\\\Vert \\\\partial _1\\\\big (\\\\phi (x_1)\\\\phi (x_2)\\\\big )\\\\partial _1f_n\\\\Vert _{H^s}\\\\gtrsim &~\\\\Vert \\\\phi ^{\\\\prime }(x_1)\\\\phi (x_1)\\\\phi ^2(x_2)\\\\cos (2^nx_1)\\\\Vert _{L^2}\\\\nonumber \\\\\\\\&-2^{-n}\\\\Vert (\\\\phi ^{\\\\prime }(x_1)\\\\phi (x_2))^2\\\\sin (2^nx_1)\\\\Vert _{L^2}\\\\nonumber \\\\\\\\\\\\gtrsim &~\\\\Vert \\\\phi ^2(x_2)\\\\Vert _{L^2}\\\\Vert \\\\phi ^{\\\\prime }(x_1)\\\\phi (x_1)\\\\cos (2^nx_1)\\\\Vert _{L^2}\\\\nonumber \\\\\\\\&-2^{-n}\\\\Vert (\\\\phi ^{\\\\prime }(x_1)\\\\phi (x_2))^2\\\\Vert _{L^2}.$ Using the simple formula $2\\\\cos ^2(2^nx_1)=1-\\\\cos (2^{n+1}x_1),$ we have $\\\\Vert \\\\phi ^{\\\\prime }(x_1)\\\\phi (x_1)\\\\cos (2^nx_1)\\\\Vert ^2_{L^2}=\\\\frac{1}{2}\\\\Vert \\\\phi ^{\\\\prime }(x_1)\\\\phi (x_1)\\\\Vert ^2_{L^2}-\\\\frac{1}{2}\\\\int _{\\\\mathbb {R}}|\\\\phi ^{\\\\prime }(x_1)\\\\phi (x_1)|^2\\\\cos (2^{n+1}x_1)\\\\mathrm {d}x_1,$ which follows from Riemann-Lebesgue's Theorem that $\\\\lim _{n\\\\rightarrow \\\\infty }\\\\Vert \\\\phi ^{\\\\prime }(x_1)\\\\phi (x_1)\\\\cos (2^nx_1)\\\\Vert ^2_{L^2}=\\\\frac{1}{2}\\\\Vert \\\\phi ^{\\\\prime }(x_1)\\\\phi (x_1)\\\\Vert ^2_{L^2}.$ Combining (REF ) and (REF ) yields the desired (REF ).\"\n ],\n [\n \"Acknowledgments\",\n \"J. Li is supported by the National Natural Science Foundation of China (Grant No.11801090).\",\n \"Y. Yu is supported by the Natural Science Foundation of Anhui Province (No.1908085QA05).\",\n \"W. Zhu is partially supported by the National Natural Science Foundation of China (Grant No.11901092) and Natural Science Foundation of Guangdong Province (No.2017A030310634).\"\n ]\n]"},"id":{"kind":"string","value":"2011.14125"}}},{"rowIdx":2797,"cells":{"text":{"kind":"list like","value":[["i3DMM: Deep Implicit 3D Morphable Model of Human Heads"],["Abstract We present the first deep implicit 3D morphable model (i3DMM) of full heads.","Unlike earlier morphable face models it not only captures identity-specific geometry, texture, and expressions of the frontal face, but also models the entire head, including hair.","We collect a new dataset consisting of 64 people with different expressions and hairstyles to train i3DMM.","Our approach has the following favorable properties: (i) It is the first full head morphable model that includes hair.","(ii) In contrast to mesh-based models it can be trained on merely rigidly aligned scans, without requiring difficult non-rigid registration.","(iii) We design a novel architecture to decouple the shape model into an implicit reference shape and a deformation of this reference shape.","With that, dense correspondences between shapes can be learned implicitly.","(iv) This architecture allows us to semantically disentangle the geometry and color components, as color is learned in the reference space.","Geometry is further disentangled as identity, expressions, and hairstyle, while color is disentangled as identity and hairstyle components.","We show the merits of i3DMM using ablation studies, comparisons to state-of-the-art models, and applications such as semantic head editing and texture transfer.","We will make our model publicly available."],["Introduction","3D morphable models (3DMMs) are parametric models of geometry and appearance of human faces, with widespread use in applications such as image and video editing, face recognition and cognitive science [19].","These models are trained using 3D scans of humans, e.g., laser scans [4], depth sensor-based scans [9], or photometric multi-view scans [28].","As 3DMMs usually learn a deformation space of a fixed template mesh, the training shapes commonly need to be brought into dense surface correspondence with each other.","Computing such correspondence for the face region alone is already hard and often requires a challenging non-convex optimizaton problem [19]; computing dense correspondence for the rest of the head and the hair is close to impossible.","Therefore, as well as due to the lack of large 3D scan datasets with hair, most 3DMMs only capture the face region.","While some recent approaches aim to model the complete head [15], [28], [41], [40], they do not capture the hair region, and the appearance in the head region (if captured) is very limited.","In this work, we present i3DMM, the first implicit 3D morphable model which captures the full head region, including hair deformations and appearance.","We capture a new dataset of full photogrammetric head scans of 64 people for training.","Each subject performs several expressions and natural hair is captured.","Since such scans are noisy, especially in the hair region, our training algorithm uses an adaptive sampling strategy based on the quality of reconstructions in different regions of the head, allowing us to effectively handle noise without smoothing out details.","In contrast to existing 3DMMs, which use a mesh-representation with a fixed template, we implicitly represent surfaces using signed distance functions.","This allows us to capture large deformations easily, which is particularly convenient for the hair region.","The implicit representation also allows us to avoid computing dense correspondence between training scans.","Our method is inspired by recent works on deep implicit surface modeling [36], [31], [32], which demonstrate the advantages of using an implicit representation compared to voxel grids, point clouds or meshes.","Our main technical innovation compared to these works is that we use a novel neural network architecture which decouples the learning process, separating it into learning a reference shape, learning geometry deformations with respect to this reference, and learning a color network.","This allows us to automatically compute dense correspondences between any two shapes with only sparse supervision on salient face landmarks.","This decoupling is also essential for disentangling the learned space into semantically meaningful parametric components, an important feature of 3DMMs.","Most classical models disentangle identity geometry, expression geometry, and appearance of the face.","This is sufficient for several applications in face editing [55], [53], [50], but does not allow the explicit control of head parts other than the facial region.","Our model allows for unprecedented control over the different semantic modes of full heads.","To this end, we learn to disentangle the geometry and color, where geometry is further separated into identity, expression, and hairstyle components, and color is further separated into identity and hairstyle components.","In summary, our main contributions are: A method for learning full head 3D morphable models directly from rigidly aligned real-world noisy 3D scans without dense ground truth correspondences.","A novel network architecture which can compute dense correspondences between 3D shapes represented using implicit functions.","This network is trained only with sparse supervision.","A training method for disentangling the color and geometry components, and the identity, expression, and hairstyle components of the geometry; and the identity and hairstyle components of the color.","We compare our model to several state-of-the-art 3DMMs by fitting to 3D scans.","We also demonstrate the quality of our learned dense correspondences by showing texture transfer between scans, and show that i3DMM benefits applications such as semantic head editing, and one-shot computation of segmentation masks and landmarks on 3D scans."],["Related Work","Most approaches for face and head modeling are mesh-based.","We summarize them here and refer the reader to Egger [19] for a more comprehensive report.","As our model is based on implicit representations, we also discuss methods which learn implicit shape and color.","Face and Head Morphable Models.","The work of Blanz and Vetter [5] showed the possibility of representing faces using a 3D Morphable Model (3DMM).","The model is learned by transforming the shape and texture of examples into low-dimensional vector representations using principal components analysis (PCA).","Further improvements were proposed [22], [38], [8], [6] by using better registration and higher quality scans.","Multilinear models have also been proposed to model identity-dependent expressions [10], [20].","Li [29] presented FLAME, a head model that combines a linear shape space with an articulated jaw, neck and eyeballs, pose-dependent corrective expression blendshapes, and global expressions.","The model is learned from a large 3D dataset, including data from D3DFACS [12].","Ranjan [42] learned a head model using an autoencoder based on spectral graph convolutions [17].","The LYHM head model [16], [14] uses a hierarchical parts-based template morphing framework to process the head shape, and uses optical flow to refine the texture.","The model is built using $1{,}200$ different identities.","Ploumpis [40], [39] combined the LYHM and LSFM [7] models as the Universal Head model (UHM), with a focus on face geometry.","Note that all existing full head models only model the cranium geometry without hair.","Implicit Representation.","Implicit representations have a long history in computer vision for inferring 3D shape [27], temporally evolving shape [13], [26] and textured shape [49].","Park [36] presented DeepSDF to represent a class of shapes using signed distance fields with an autodecoder.","Chen [11] and Mescheder [31] learned a generative model of a class of shapes by classifying a point in space as inside or outside a shape.","Recent approaches [56], [18], [21] have proposed to represent shapes using a collection of local implicit patches for higher quality.","Oechsle [35] extended OccupancyNets [31] to represent texture along with the geometry using monocular inputs.","Niemeyer [34] presented an approach for handling time-varying shapes by learning to predict motion vectors for each point in 3D space using OccupancyNets.","PIFu [45], [46] allows for 3D reconstruction of humans from monocular inputs.","Pixel-aligned implicit functions estimate a continuous field that determines whether a pixel is inside or outside the surface of the human subject, as well as the color on the surface.","Several recent methods [48], [32], [51] have presented ways to achieve higher-quality implicit representations by using periodic functions as activations.","Saito [44] presented an approach for 3D hair modeling.","Their technique learns a manifold for 3D hairstyles represented as implicit surfaces using a volumetric variational autoencoder."],["Method","In this section, we describe how we obtain our deep implicit 3D morphable model (i3DMM).","Our overall processing pipeline is illustrated in Fig.","REF .","In the following we will describe our data acquisition, the training data preparation, and the neural network architecture.","Figure: Overview of the captured expressions (red box: test set; blue box: hairstyles) for each subject.Figure: Multi-view scanning system with 135 cameras around the person (green), and 2 cameras (blue) on top.","Right: front view image, and reconstruction."],["Data Acquisition","For training, we have scanned 64 subjects (46 male, 18 female; 22 Caucasian, 9 Asian, 25 Indian Sub-continent, 3 Hispanic, 3 Middle Eastern, 2 undisclosed; with an age ranging from 19 to 69 with average 26).","For each subject we have recorded 10 facial expressions (a subset of which are chosen from paGAN [33]), including neutral expressions with 3-4 different “hairstyles” (open hair, tied hair for subjects with long hair, and two different caps), as shown in Fig.","REF .","Our data was acquired with the Treedyshttps://www.treedys.com/ multi-view scanning system, that comprises of 137 calibrated cameras, see Fig.","REF .","The total capture time per person amounts to about 20 minutes.","Photogrammetry-based 3D reconstruction on the multi-view data was applied to obtain the final textured meshes (see Fig.","REF ), where each mesh comprises of around $50{,}000$ vertices."],["Training Data","Given our recorded textured mesh data, we apply several preprocessing steps to make the data is suitable for head model learning.","This includes a semi-automated landmark annotation, cropping of the head area to discard the shoulders and parts of the upper body, rigidly aligning the meshes, and closing the hole at the neck to make the mesh watertight.","In the following we elaborate on these steps.","Landmark Annotation.","First, we apply the automated face landmark detector from dlib [24] on the front-facing image obtained from the multi-view camera acquisition setup, which produces a total of 66 face landmarks.","We select a subset of 8 landmarks, i.e.","the four corner points of the eyes, the tip of the nose, the two corner points of the mouth, and the chin, and then transfer these 2D landmarks to the 3D mesh.","We manually correct the inaccurately detected landmarks, and additionally annotate 8 landmarks for the corners of ears (top, bottom, left, and right) directly in 3D.","Head Cropping.","In order to remove the shoulders and parts of the upper body, we crop the head meshes based on the 3D landmarks.","To this end, we first compute the vector $\\mathbf {v}$ from the centroid of the four eye cornerpoints to the tip of the nose.","Then, for $\\mathbf {p}$ being the chin landmark in 3D space, we define a virtual plane that goes through the point $\\mathbf {p} + \\frac{1}{2} \\mathbf {v}$ and has the normal $\\mathbf {v}$ .","We only keep the side of the plane that contains the facial landmarks.","Rigid Alignment.","Based on the 3D facial landmarks, we perform a rigid alignment of all head scans in order to ensure that they are oriented consistently in 3D space.","We use a numerically stable implementation [3] of the transformation synchronization method [2], which solves the generalized Procrustes problem in an initialization-free and unbiased way.","To this end, we first rigidly align all pairs of heads based on Singular Value Decomposition (SVD), i.e.","we solve all pairwise Procrustes problems, and subsequently use transformation synchronization to establish cycle-consistency in the set of pairwise transformations.","Hole Closing.","Since our full head model is based on a signed distance representation, we require that the meshes are watertight.","After rigid alignment, we assume that the longitudinal axis of the head aligns with the y-axis.","With that, we use a flat patch to close the hole at the neck by extruding the respective boundary vertices to coincide with the smallest y-coordinate.","Finally, we scale all meshes with the same value such that all of them fit in the unit cube."],["Training","We learn a vector-valued function $f_{\\theta }(\\mathbf {x},\\mathbf {z})$ , where $\\theta $ are the neural network weights, $\\mathbf {x} \\in \\mathbb {R}^3$ is the query point, and $\\mathbf {z}$ is a code vector that encodes the head instance.","The output $f_{\\theta }(\\mathbf {x},\\mathbf {z}) = ({s}, \\mathbf {c})$ includes a scalar ${s}$ that represents the signed distance to the surface, as well as the color $\\mathbf {c} \\in \\mathbb {R}^3$ at the closest surface point from $\\mathbf {x}$ .","The shape boundary is represented as the zero level-set of the signed distance function (SDF), while the interior parts of the shapes have a negative signed distance value, and the exterior parts a positive value.","We use an autodecoder network architecture [36], where the weights of the network $\\theta $ and the input latent codes $\\mathbf {z}$ for all shapes are learned jointly.","Mesh Sampling.","We require $(\\mathbf {x},{s}, \\mathbf {c})$ -triplets (query point, signed distance value, color) for training.","We use a combination of two strategies for sampling these triplets.","First, we sample points on the mesh surface based on uniform random sampling.","However, in order to also account for higher accuracy in high-detail facial regions, i.e.","the eyes, nose and mouth, we additionally sample more points in these areas.","We take the center of each eye, the tip of the nose, and the center of the mouth, and place a sphere around them that covers the respective region of interest.","Then, we sample points on the mesh surface that lie within each sphere.","Eventually we mix the uniformly sampled points with the landmark-based sampled points in a 1 to 3 ratio.","After sampling, the points are perturbed by a uniform 3D Gaussian with standard deviation $0.005$ times the length of the bounding box, such that not only surface points but also points in the interior and exterior are used, see [36].","The color value for each sample is obtained by finding the closest point on the mesh surface, and then looking up its color in the texture map.","Latent Codes and Disentanglement.","As mentioned earlier, latent codes for each object (head scan) are also learned during training.","Existing approaches use a single object latent code to describe the shape.","In contrast, we design several separate latent spaces for our objects in order to learn a semantically disentangled model.","We use two separate latent vector spaces for geometry and color, $\\mathbf {z}_\\text{geo}$ and $\\mathbf {z}_\\text{col}$ , respectively.","The geometry space includes three code vectors for identity, expression, and hairstyle, and the color space includes two code vectors for identity and hairstyle.","Thus, $\\mathbf {z}_\\text{geo} = (\\mathbf {z}_\\text{geoId}, \\mathbf {z}_\\text{geoEx}, \\mathbf {z}_\\text{geoH})$ and $\\mathbf {z}_\\text{col} = (\\mathbf {z}_\\text{colId}, \\mathbf {z}_\\text{colH})$ .","During training, the number of different identity code vectors is equal to the number of training identities, 58.","The number of different expression vectors is fixed to 10 (cf.","Fig.","REF for the training expressions) and hairstyle to 4 (short, long, cap1 or cap2) for geometry, and to 3 (nocap, cap1 or cap2) for color.","For all scans of the $i$ -th subject we use the same latent variables for the geometry identity and color identity, $\\mathbf {z}_\\text{geoId}^i$ and $\\mathbf {z}_\\text{colId}^i$ .","Similarly, for each expression and hairstyle, the same variables $\\mathbf {z}_\\text{geoEx}, \\mathbf {z}_\\text{geoH}$ and $\\mathbf {z}_\\text{colH}$ are used across all identities.","By doing so, we are able to learn disentangled latent variables without imposing any explicit constraints.","At test time, we can control each latent space individually, leading to semantically meaningful editing.","Network Architecture.","Figure: Overview of our network architecture.","We learn weights of three network components, a Shape Deformation component, a Reference Shape component, and a Color component.","Moreover, the latent codes for each object are also optimized for.","The input of the network is a 3D query point, and the output is a signed distance value along with the corresponding color.Our network comprises three components, the Reference Shape Network (RefNet), the Shape Deformation Network (DeformNet) and the Color Network (ColorNet).","All networks are composed of fully-connected layers with a ReLU non-linearity after every layer, except the output layer.","The inputs to all our networks are encoded using sinusoidal positional encoding [32].","Our Reference Shape Network encodes a single reference shape, such that all individual head shapes can be obtained by deforming this shape.","The output at a query point $\\mathbf {x}$ is the signed distance value $f_\\theta ^r(\\mathbf {x}) \\in \\mathbb {R}$ .","Note that for the reference shape there is no latent code input since only a single reference shape is learned.","This can be seen analogously to the mean (or neutral) shape used in classical 3DMMs [5].","We use 3 fully connected layers for this network, where each hidden layer has dimensionality 512.","The Shape Deformation Network deforms the reference shape to represent the shape of an individual head.","The network takes the geometry latent code $\\mathbf {z}^i_\\text{geo}$ for an object $i$ , and a query point $\\mathbf {x}$ as input, and produces the output, $f_\\theta ^s(\\mathbf {x}, \\mathbf {z}^i_\\text{geo}) = \\delta \\in \\mathbb {R}^3\\,,$ which is a displacement vector that deforms the query point to the reference shape.","Thus, the signed distance value at any point $\\mathbf {x}$ of the object $i$ with geometry latent $\\mathbf {z}^i_\\text{geo}$ is ${s}(\\mathbf {x}, \\mathbf {z}^i_\\text{geo}) = f_\\theta ^r(\\mathbf {x} + \\delta )\\,,$ where ${s}(\\cdot )$ is the scalar signed distance value, and $\\delta $ is the deformation from Eq.","(REF ).","This formulation allows us to compute dense correspondences between any input head scan and the reference head shape.","The separation between a reference shape and deformations with respect to this reference shape is also common in classic morphable models [19].","However, these models require dense correspondences before training, which is not necessary in our approach.","We use 8 fully connected layers for this network, where each hidden layer has dimensionality 1024.","The introduction of the reference shape makes it possible to disentangle the geometry and color components.","The Color Network learns the color of the query point in the reference space.","Given a query point $\\mathbf {x}$ , deformation $\\delta $ from Eq.","(REF ), and color latent vector $\\mathbf {z}^i_\\text{col}$ for the object $i$ , the output is represented as $f_\\theta ^c(\\mathbf {x}+\\delta , \\mathbf {z}^i_\\text{col}) \\in \\mathbb {R}^3$ , which is the color at point $\\mathbf {x}$ .","Note that without this separation of a reference space, the ColorNet would also have to take into account information about object geometry, thus not being able to disentangle shape and color.","Note that the latent code for ColorNet, for a given identity and hairstyle, does not change with expressions.","DeformNet finds the right colors and geometry for different expressions by achieving dense correspondences.","We use 9 fully connected layers for this network, where each hidden layer has dimensionality 1024.","Loss Functions.","We define the loss function to train our network as $\\mathcal {L}_\\theta (\\mathbf {x}, \\mathbf {z_\\text{geo}}, \\mathbf {z_\\text{col}}) & = \\sum _{i=1}^K(\\mathcal {L}_\\theta ^\\text{geo}(\\mathbf {x}, \\mathbf {z}^i_\\text{geo}) +\\mathcal {L}_\\theta ^\\text{def}(\\mathbf {x}, \\mathbf {z}^i_\\text{geo}) \\nonumber \\\\ &+\\mathcal {L}_\\theta ^\\text{col}(\\mathbf {x}, \\mathbf {z}^i_\\text{col}) + \\mathcal {L}^\\text{reg}(\\mathbf {z}^i_\\text{geo}, \\mathbf {z}^i_\\text{col}) ) \\nonumber \\\\ &+ \\sum _{i\\ne j}\\mathcal {L}^\\text{lm}_\\theta (\\mathbf {z}^i_\\text{geo},\\mathbf {z}^j_\\text{geo})\\,,$ where, $i,\\,j = \\lbrace 1,\\dots ,K\\rbrace $ , $K$ is the number of scans in the batch, $\\mathbf {z}_\\text{geo} = \\lbrace \\mathbf {z}^1_\\text{geo},\\dots ,\\mathbf {z}^K_\\text{geo}\\rbrace $ , and $\\mathbf {z}_\\text{col} = \\lbrace \\mathbf {z}^1_\\text{col},\\dots ,\\mathbf {z}^K_\\text{col}\\rbrace $ .","The latent vectors of head $i$ are represented as $\\mathbf {z}_\\text{geo}^i$ and $\\mathbf {z}_\\text{col}^i$ .","Here, $\\mathcal {L}_\\theta ^\\text{geo}(\\cdot )$ enforces good geometry reconstructions, $\\mathcal {L}_\\theta ^\\text{def}(\\cdot )$ regularizes the deformation field, $\\mathcal {L}_\\theta ^\\text{col}(\\cdot )$ is used to train the ColorNet, $\\mathcal {L}^\\text{reg}(\\cdot )$ is a regularizer on the latent vectors, and $\\mathcal {L}^\\text{lm}_\\theta (\\cdot )$ is a sparse pairwise landmark supervision loss.","For the geometry term that we impose upon the signed distance values, we use the $\\ell _1$ -loss $\\mathcal {L}_\\theta ^\\text{geo}(\\mathbf {x}, \\mathbf {z}^i_\\text{geo}) = w_g \\big \\Vert \\text{cl}(s(\\mathbf {x}, \\mathbf {z}^i_\\text{geo}), t) - \\text{cl}(s_\\text{gt}(\\mathbf {x}), t) \\big \\Vert _1\\,{,}$ where $s_\\text{gt}(\\mathbf {x})$ is the ground truth signed distance value at $\\mathbf {x}$ , and $s(.",")$ is from Eq.","(REF ).","These values are (symmetrically) clamped at $t$ =$0.1$ , for which we define $\\text{cl}(x, t) := \\min (t, \\max (x, -t))$ .","We use a similar $\\ell _1$ -loss for the color component, i.e.","$\\mathcal {L}_\\theta ^\\text{col}(\\mathbf {x}, \\mathbf {z}^i_\\text{col}) = w_c \\big \\Vert f_\\theta ^c(\\mathbf {x}+\\delta , \\mathbf {z}^i_\\text{col}) - c_\\text{gt}(\\mathbf {x}) \\big \\Vert _1\\,{,}$ where $c_\\text{gt} (\\mathbf {x})$ is the ground truth color at $\\mathbf {x}$ .","We also enforce that the 16 landmarks $\\lbrace \\mathbf {x}^{\\ell }_i\\rbrace $ , as described in Sec.","REF , of each shape $i$ , deform to the same points in the reference space using the pairwise loss $ \\mathcal {L}^\\text{lm}_\\theta (\\mathbf {z}^i_\\text{geo},\\mathbf {z}^j_\\text{geo}) = w_{lm} \\sum _{\\ell =1}^{16} \\big \\Vert (\\mathbf {x}^\\ell _i + \\delta ^\\ell _i) - (\\mathbf {x}^\\ell _j + \\delta ^\\ell _j) \\big \\Vert _2\\,,$ where, $\\delta ^\\ell _i = f_\\theta ^s(\\mathbf {x}^\\ell _i, \\mathbf {z}^i_\\text{geo})$ as in Eq.","(REF ).","For scans with the ears covered by hair, we do not have any ear annotations.","We would like to compress the ear region in the reference shape to a single point in the reconstruction for these shapes.","Thus, we additionally optimize for one point for each ear.","We enforce pairwise constraints between the learnable ear points and the annotated ear points for the other shapes in the batch using Eq.","(REF ).","Further, to ensure regularized deformations to the reference shape, we impose a loss on the amount of deformation.","We use $\\mathcal {L}_\\theta ^\\text{def}(\\mathbf {x}, \\mathbf {z}^i_\\text{geo}) = w_s || f_\\theta ^s(\\mathbf {x}, \\mathbf {z}^i_\\text{geo}) ||_2$ .","Finally, we use an $\\ell _2$ -regularizer on the latent vectors assuming a Gaussian prior distribution, $\\mathcal {L}^\\text{reg}(\\mathbf {z}^i_\\text{geo}, \\mathbf {z}^i_\\text{col}) =w_r ( ||\\mathbf {z}^i_\\text{geo}||_2 + ||\\mathbf {z}^i_\\text{col}||_2)$ .","Optimization.","Given $N$ batches with $K$ objects per batch, we optimize for the network weights and the latent vectors by solving the optimization problem $\\operatornamewithlimits{argmin}_{\\theta , \\lbrace \\mathbf {z}^b_\\text{geo}, \\mathbf {z}^b_\\text{col}\\rbrace _{b=1}^N}~~ \\sum _{b=1}^N\\sum _{\\mathbf {x}} \\mathcal {L}_\\theta (\\mathbf {x}, \\mathbf {z}^b_\\text{geo}, \\mathbf {z}^b_\\text{col})\\,,$ where, the inner sum takes into account all sampled points, as explained above, and we abuse the notation $\\mathbf {z}^b_\\text{geo}, \\mathbf {z}^b_\\text{col}$ to now represent latent codes of all the scans in batch $b$ ."],["Experiments","In this section, we present an experimental evaluation of our head model.","We demonstrate that the model can be used for reconstructing scan data.","We present an ablation study to carefully analyze our design choices, and compare i3DMM to state-of-the-art head models.","We also show dense correspondence results and applications of our model.","Before we present these results, we provide additional information on the neural network training and testing.","Training Details.","We train our networks using PyTorch [37], where we use the Adam [25] solver with mini-batches of size 64.","We train for 1000 epochs with a learning rate of $0.0005$ , which decays by a factor of 2 every 250 epochs.","We initialize RefNet by pretraining it using only one mouth-open (top row, third from left in Fig.","REF ) training scan.","Our network takes about 2 days to train on 2 NVIDIA RTX8000 GPUs.","Test Data.","We collect a separate test set consisting of scans of 6 identities, which are not part of the training data.","We capture each identity in 5 novel expressions that are not part of the training expressions, see Fig.","REF .","Each scan is preprocessed analogously to the training scans."],["Reconstruction","For a given test scan, we fit our learned model to it.","This is done by optimizing for the latent vector that can best reproduce the scan, i.e.","by finding the latent variables that minimize the problem $\\operatornamewithlimits{argmin}_{\\mathbf {z}_\\text{geo}^i, \\mathbf {z}_\\text{col}^i}~~ \\sum _{\\mathbf {x}} &\\, (\\mathcal {L}_\\theta ^\\text{geo}(\\mathbf {x}, \\mathbf {z}^i_\\text{geo}) +\\mathcal {L}_\\theta ^\\text{def}(\\mathbf {x}, \\mathbf {z}^i_\\text{geo}) \\nonumber \\\\ + &\\mathcal {L}_\\theta ^\\text{col}(\\mathbf {x}, \\mathbf {z}^i_\\text{col}) + \\mathcal {L}^\\text{reg}(\\mathbf {z}^i_\\text{geo}, \\mathbf {z}^i_\\text{col}))\\,,$ where, $\\mathbf {z}_\\text{geo}^i, \\mathbf {z}_\\text{col}^i$ is the latent code for test scan $i$ .","This equation is similar to Eq.","(REF ), with the difference that the network weights are fixed here and the pairwise landmark supervision loss in Eq.","(REF ) is not enforced.","We use Adam [25] with a step size of $0.0005$ to solve this problem.","We show several reconstruction results on the test data in Fig.","REF .","We can generalize to unseen identities and expressions, even though our training data only consists of 58 people with 10 expressions.","Note that while we can generally preserve the detailed face region of the scans, we also smooth the scans in the noisy hair area."],["Correspondences","As mentioned in Sec.","REF , due to the particular design of our network, where the reference shape and the deformations are separated, our approach also establishes dense correspondences between shapes, with extremely sparse landmark supervision.","We demonstrate these correspondences in Fig.","REF , where the color is transferred from one scan to the other.","The correspondences are also used in the applications of segmentation and landmark transfer in Sec.","REF .","Our model can reliably find correspondences across different subjects, expressions and hairstyles, including long and short hair.","Please refer to the supplementary material for more evaluations.","Figure: Reconstruction quality of i3DMM on test data.","The top part shows texture and geometry of ground truth scans.","The middle part shows texture and geometry of i3DMM fits.","The last part shows texture transfer from the scan on right to the scan on left of the image.","In column 3, noise in ground truth does not transfer to the i3DMM fit, showing the robustness of our method."],["Ablative Analysis","We design several experiments to evaluate the most important design choices for our approach: landmark-based sampling described in Sec.","REF , sparse pairwise landmarks loss in Eq.","(REF ), and jointly training ColorNet, DeformNet, and RefNet.","We evaluate the qualitative impact of these choices by excluding them one at a time while training our model, see Fig.","REF .","Without landmarks-based sampling, the model focuses more on the noisy hair region and ignores the details in the face region (first column in Fig.","REF ).","Without landmark supervision, the network creates small (fake) ear regions in the hair for the long-haired scans (second column in Fig.","REF ).","It also leads to poor texture transfer around the ear.","Finally, for evaluating the importance of joint training of the color and geometry networks, we first train RefNet and DeformNet.","After convergence, we train ColorNet.","The correspondences in this case are learned only from geometry reconstructions, which leads to artifacts as shown in the third column of Fig.","REF .","Figure: Ablation study.","Top and middle row show reconstructions of two different test scans.","Last row shows texture transfer from the scans in the top row to those in the middle row."],["Comparisons to Existing Models","We compare our model with the full head FLAME [29], face only FLAME, and Basel Face Model (BFM) [38] which only models the frontal face.","We fit each model to the test set by optimizing for their model parameters.","Please refer to the supplemental for more details on the fitting.","We show qualitative results in Fig.","REF .","FLAME only models the cranium without hair, thus fitting to the full head region results in incorrect head shapes.","We also evaluate FLAME by only using the the face region for fitting.","This leads to higher quality results.","We obtain higher quality geometry both for the face and head regions.","In addition, we can also reconstruct the color of the head, while FLAME can only model the geometry.","We combine the BFM [38] identity geometry and appearance models with the expression model used in Tewari [52], and use this to fit to our test scans.","The expression model is a combination of two blendshape models [1], [9].","Since BFM also includes color, we jointly optimize to minimize the geometry as well as color alignment errors.","As can be seen in Fig.","REF (middle), this model can fit to the expression as well as color of the scans.","However, it is limited to only the face region, while i3DMM can reconstruct the full head.","In addition, our reconstructions are more personalized, with higher quality nose geometry and face colors.","We show the quantitative evaluations of the different models in terms of the symmetric Chamfer distance and F-score metrics in Table REF .","Chamfer distance is the mean distance of points from one shape to their closest points in another.","We compute the symmetric Chamfer distance as the mean of Chamfer distance from the ground truth to the fits and the Chamfer distance in the opposite direction.","F-score is (100x) the harmonic mean of these two distances after applying a threshold.","We use a similar metric for color.","First, the mean error in color at points in the ground truth and the color at their nearest points in the fits is computed, along with the mean error in the opposite direction Finally, the color error is reported as their average.","Please refer to the supplemental for details on the metrics used.","We sample $150{,}000$ points for computing the metrics.","We achieve similar geometric quality as FLAME and BFM in the face region, but significantly outperform FLAME for the full head reconstructions, see Table REF .","Moreover, in terms of color, our reconstructions are more realistic than BFM and also outperform it quantitatively.","Table: Quantitative comparison of head models.","Symmetric Chamfer distance is reported here.","F-score is computed with a threshold of 0.010.01."],["Application: Semantic Head Editing","Due to the semantic disentanglement in our model, we can selectively change semantic components of a head.","In Fig.","REF , we edit one feature of a test scan while keeping the other features fixed.","To obtain the edited latent codes, we first find the principle components of variation in each latent-subspace by running PCA on the training latent spaces.","We move along the principal components to pick the latent codes to edit the identity, and we pick others from the trained latent codes.","Unlike existing models, we can also edit hairstyles and add caps while keeping other components fixed.","We also model the mouth interior.","These features allow for semantic head editing applications much beyond the capabilities of the existing methods."],["Application: One-shot Annotation Transfer","As i3DMM can predict dense correspondences among head scans, it also enables us to transfer annotations across different reconstructions.","As our model has low reconstruction errors, it can also be used to annotate real-world 3D scans.","We show this application in Fig.","REF .","We first transfer the annotations from one manually annotated scan to the reference shape.","We can then transfer them to different i3DMM fits, and also to the real-world scans using nearest neighbors from the fits to the scans.","As i3DMM can be used to even annotate the hair region, it can be very useful to curate scan datasets.","Please refer to the supplemental for more results.","Figure: Application: One-shot annotations on real-world scans.","Coarsely drawn annotations (top row: semantic segmentation, bottom row: landmarks, not used during training) on one scan (left) can be automatically transferred to other real-world head scans (right)."],["Discussion and Future Work","Although i3DMM disentangles and models an unprecedented amount of variety in head features, and enables reconstructions with very high quality (Fig.","REF ), it still has some limitations.","Although i3DMM models varying hairstyle shapes, it is only a coarse approximation of the physical nature of hair comprising of individual hair strands.","Further, we model hairstyles that cover the ear by trying to collapse the ear to a single point (Eq.","(REF )).","Ideally, the layered geometry of ears behind the hair should be correctly represented.","Finally, there is room for extending i3DMM for more fine-grained control over hairstyles.","One challenge is that our scans are noisy in the hair region.","This is due to the photometric multi-view reconstruction techniques used, which struggle with hair due to their complex physical interaction with light."],["Conclusion","We have presented the first unified 3D morphable model of geometry and appearance of full heads, which includes different identities, expressions and hairstyles.","The head geometry and color is represented using implicit function parameterized with neural networks, learned from 610 multi-view scans.","By explicitly separating the network into reference shape, shape deformation and color components, our model can disentangle the different semantic components.","It also allows us to obtain dense correspondences across different scans represented using our model, enabling several applications.","We will make our code publicly available to the community."],["Acknowledgements","This work has been supported by the ERC Consolidator Grant 4DReply (770784), and by the BMBF-funded Munich Center for Machine Learning.","We thank Garvita Tiwari, Navami Kairanda, and Neng Qian for their support in capturing the dataset.","We thank all the subjects in the dataset for lending their data for research.","Figure: NO_CAPTIONSupplementary Model Visualization Our implicit 3D morphable model models the identity, expression, and hairstyle components of the geometry; as well as the identity and hairstyle components of the color of the head with independent parameteric controls.","We visualize these components in Fig.","REF .","We perform PCA on the identity geometry and color spaces in order to compute the principal components.","The expression component is visualized by moving along the directions of the training expressions, since they are semantically well defined.","As we model hairstyles that include caps, we show the joint space of geometry and color for hairstyles.","Note that the hairstyle geometry can only take four discrete values – short, long, cap1, or cap2.","Any variation within these categories is modelled by the identity-geometry component.","Similarly, hairstyle color can only take the values – nocap, cap1, or cap2.","The color of hair without any cap is determined by the identity-color component.","Experiments Here, we provide more details on the evaluations in the main paper, and include further evaluations.","Sampling i3DMM One of the important features of a 3D Morphable Model is the ability to randomly sample shapes in the parametric space.","This has been used for generating synthetic data for training CNNs [23], [43], [47].","We achieve sampling in i3DMM by performing principal component analysis (PCA) over the training latent codes for color-identity, geometry-identity, and expressions.","We weigh the singular values using a Gaussian random variable $\\mathcal {N}(0,0.1)$ for color-identity, and geometry-identity, and $\\mathcal {N}(0,0.25)$ for expressions.","Since the latent codes for hair shapes and colors can take very limited numbers of values and are very well defined semantically, we sample these from their training values.","We show several results in the supplemental video.","As can be seen, our model is biased towards generating male heads.","This is likely due to our gender-biased training dataset with 46 males and 18 females.","While this does not lead to any clear loss of quality when fitting to female test scans, see Fig.","6 in the main paper, it might lead to biased quality of results in other problems, for eg., if random samples from the model was used for training another network.","Comparisons As mentioned in Sec.","4.4 of the main paper, we compare our model to two existing models, BFM [38] and FLAME [29].","We show more qualitative model fitting results in Fig.","REF .","Next, we provide more details on the fitting algorithm used.","Fitting: We paint the face region of each model's template mesh to create a mask as shown in Fig.","REF .","We use these masked regions of the models for fitting the models to head scans.","To initialize, we mark 8 landmarks (eye corners, nose, lip corners, and chin) on the template mesh and rigidly align the template to each ground truth scan using Procrustes algorithm.","We allow for translation, rotation, and scaling.","We use the rigid alignment as initialization and optimize for the parameters of each model using a modified iterative closest point (ICP) algorithm which also updates the model parameters.","The fitting algorithm maintains the initial scale and translation but optimizes for rotation.","In each optimization step, we first compute the correspondences as the closest points from the masked region in the model, shown in Fig.","REF , to the scan data.","We compute the loss as shown in Eq.","(REF ) and update the model parameters along with Euler angles for global rotation.","We run the following optimization program up to convergence to fit the models to our scans: $ \\operatornamewithlimits{argmin}_{\\theta ,K,\\alpha ,\\beta ,\\gamma }& \\,\\, \\sum _{i=1}^N \\,\\,(\\big \\Vert sR(\\alpha ,\\beta ,\\gamma )x_i(\\theta ) + t - x_i \\big \\Vert _2 \\nonumber \\\\ &\\quad \\quad + \\big \\Vert Kc_i(\\theta ) - c_i \\big \\Vert _2 ) \\nonumber \\\\ &+ w_l \\sum _{j=1}^L \\big \\Vert sR(\\alpha ,\\beta ,\\gamma )l_j(\\theta ) + t - l_j\\big \\Vert _2 \\,,$ where, $x_i(\\theta ) \\in \\mathbb {R}^3$ is a vertex $i$ in the masked region of the model (containing $N$ vertices), $x_i$ is the point on the scan data corresponding to $x_i(\\theta )$ ; $R(\\alpha ,\\beta ,\\gamma ) \\in \\mathbb {R}^{3 \\times 3}$ is the global rotation matrix computed using the Euler angles, $\\alpha ,\\beta ,\\text{and } \\gamma $ ($\\in \\mathbb {R}$ ); $s \\in \\mathbb {R} $ , $t \\in \\mathbb {R}^3$ are the global scale and translation computed during initialization; $c_i(\\theta ) \\in \\mathbb {R}^3$ , $c_i \\in \\mathbb {R}^3$ are the colors at the vertices $x_i(\\theta )$ and $x_i$ respectively; $l_j \\in \\mathbb {R}^3$ and $l_j(\\theta ) \\in \\mathbb {R}^3$ are the $L(=8)$ ground truth and model landmarks respectively, as described earlier; and $K \\in R^{3\\times 3}$ is a diagonal matrix.","We set $w_l=0.1$ during the fitting process.","Note that the color loss is only enforced for BFM, as FLAME does not model colors.","Further, as the color intensities of BFM and our scans differ, we globally scale the color values using channel-specific scalars arranged as a diagonal matrix $K$ which we optimize for along with the model parameters.","Evaluation details: We describe the evaluation metrics in Sec.","4.4 of the main paper.","Here, we present details about the masks used to evaluate these metrics.","Face region: We manually paint face masks on the ground truth scans to obtain the ground truth masks.","We exclude the mouth interior of the ground truth scans.","We copy this mask to the i3DMM fits.","We do that by annotating a vertex in i3DMM reconstruction if the nearest point from that vertex on the ground truth scan is in the masked region.","We show the face masks used to fit BFM and FLAME to ground truth scans in Fig.","REF .","We obtain the symmetric metrics presented in Table.","1 of the main paper for the face region in the following way.","In one direction, we compute the errors from masked region of ground truth to the closest points on the (unmasked) models fit to the scan.","In the other direction, we compute the errors from the masked region of the models to the (unmasked) ground truth scan.","We compute errors between the masked regions of one mesh to unmasked regions of other mesh to avoid large error metrics due to annotation mistakes during manual mask painting.","Full Head: We only fit to FLAME full head model as BFM does not model the entire head.","We remove the neck region from FLAME as shown in Fig.","REF as the ground truth head scans do not have neck regions.","We also remove the vertices used to close the neck from ground truth as FLAME has a hole in the mesh at the neck.","We compute the metrics as we do for the face region between these two full head meshes.","We report the full head metrics for our model in the entire head region, including the closed hole at the neck mesh.","Ablative Analysis In Fig.","REF , we show additional qualitative results for the ablative analysis.","We also show the quantitative results for full head i3DMM fit in comparison to i3DMM variants.","We compare the four models that evaluate our design choices in Table REF .","The error metrics are computed for the face region using manually annotated face masks as described in Sec.","REF .","We only evaluate the face region, as the ground truth for hair is noisy, and small quantitative differences are not very indicative of degradation in quality.","Although the geometric reconstruction accuracy is marginally better without the landmark supervision loss, as compared to i3DMM, the color reconstruction accuracy of i3DMM is higher.","Also, as mentioned in the main paper, Sec.","4.3, texture transfer results around the ear regions with landmark supervision loss are worse compared to i3DMM.","Table: Quantitative results for ablation study.","The columns, from left to right, show results obtained with uniform sampling for SDF instead of landmark-based sampling, without sparse pairwise landmark supervision loss, independently training for representing geometry and color, final model (i3DMM), and ground truth.","Correspondence Evaluation We quantitatively evaluate the correspondences predicted by i3DMM by using the FLAME and BFM fits as ground truth correspondences.","To this end, we first find the closest points from the vertices of the (masked) model fits to the i3DMM reconstructions for different scans.","We will call these correspondences ground truth annotations here.","We use a KD tree algorithm for efficiency.","The masked face region contains 26370 vertices for BFM, and 1873 vertices for FLAME.","We also transfer the annotations for one i3DMM reconstruction, to all the other reconstructions.","This process is same as that described in annotation transfer application (see Sec.","4.5 of main paper).","We compute the correspondence error as the average of error between the transferred and the ground truth annotations.","Note that we transfer annotations from one i3DMM fit to every other i3DMM fit.","Therefore, we compute a symmetric error metric.","The resulting distribution of error is shown in Fig.","REF (evaluating with BFM as ground truth is plotted in red, while FLAME is plotted in blue).","The mean and median of errors for BFM is $5.08$ mm and $3.02$ mm respectively.","The mean and median of errors for FLAME is $2.36$ mm and $1.83$ mm respectively.","Note that, this error does not only capture the error in i3DMM's correspondence predictions but also the error in registrations of FLAME and BFM fits, see Table.","1 in the main paper.","Figure: Distribution of errors in correspondences predicted by i3DMM computed using FLAME fits (face) as ground truth (blue), and using BFM fits (face) as ground truth (red).","Applications Full Head Completion Figure: Completing face scans (left) with different hair styles (short hairstyle (middle-left), long hairstyle (middle-right), and cap (right)) using i3DMM as prior.We use the i3DMM prior to complete face scans with different hairstyles as shown in Fig.","REF .","To obtain the face meshes from our head meshes, we delete all the vertices that are outside a sphere around the the tip of the nose.","We learn the latent vector for the given test scan using the SDF samples of the face mesh, as described in Sec.","4.1 of the main paper.","Additionally, we semantically control the hair style of the completed scan by adding a regularizer that enforces the learned $\\mathbf {z}_\\text{geoH}$ and $\\mathbf {z}_\\text{colH}$ to be close to the hairstyle latent vectors learned during training.","It can be inferred from the results in Fig.","REF that our model learns a good prior distribution, generating plausible heads for the given faces.","i3DMM offers user-guided control for head completion and can be used to turn existing face-only 3DMMs into full head 3DMMs.","Further, our method can also be used as a prior distribution for applications such as monocular 3D reconstruction [54].","Figure: Additional annotation transfer results.","Top: segmentation transfer front view.","Middle: segmentation transfer side view.","Bottom: landmark transfer.","Left column shows i3DMM reconstructions with manual annotations.","Right part shows annotations transferred to head scans using i3DMM.","Annotation Transfer We show more results for annotation transfer described in Sec.","4.5 of the main paper, in Fig.","REF .","Figure: Additional comparison results between i3DMM (full head) fits, BFM (face) fit, and FLAME (full head and face) fits.Figure: Additional ablation results.","From left to right, i3DMM without landmark-based sampling, i3DMM without landmark supervision, i3DMM with independent color and geometry training, i3DMM, and ground truth results are shown.","Visualization Details The output of the i3DMM is a signed distance field.","Generally, marching cubes algorithm is used to reconstruct the surface from a SDF.","However, based on the resolution used, marching cubes algorithm introduces unpleasant surface artifacts.","To avoid these artifacts, we used a sphere tracer to render our results.","We used the Blinn-Phong reflection model to shade our geometry results.","We apply a gamma correction with $\\gamma =0.65$ for the color renders.","We use Redner [30] for rendering meshes."],["Model Visualization","Our implicit 3D morphable model models the identity, expression, and hairstyle components of the geometry; as well as the identity and hairstyle components of the color of the head with independent parameteric controls.","We visualize these components in Fig.","REF .","We perform PCA on the identity geometry and color spaces in order to compute the principal components.","The expression component is visualized by moving along the directions of the training expressions, since they are semantically well defined.","As we model hairstyles that include caps, we show the joint space of geometry and color for hairstyles.","Note that the hairstyle geometry can only take four discrete values – short, long, cap1, or cap2.","Any variation within these categories is modelled by the identity-geometry component.","Similarly, hairstyle color can only take the values – nocap, cap1, or cap2.","The color of hair without any cap is determined by the identity-color component."],["Experiments","Here, we provide more details on the evaluations in the main paper, and include further evaluations."],["Sampling i3DMM","One of the important features of a 3D Morphable Model is the ability to randomly sample shapes in the parametric space.","This has been used for generating synthetic data for training CNNs [23], [43], [47].","We achieve sampling in i3DMM by performing principal component analysis (PCA) over the training latent codes for color-identity, geometry-identity, and expressions.","We weigh the singular values using a Gaussian random variable $\\mathcal {N}(0,0.1)$ for color-identity, and geometry-identity, and $\\mathcal {N}(0,0.25)$ for expressions.","Since the latent codes for hair shapes and colors can take very limited numbers of values and are very well defined semantically, we sample these from their training values.","We show several results in the supplemental video.","As can be seen, our model is biased towards generating male heads.","This is likely due to our gender-biased training dataset with 46 males and 18 females.","While this does not lead to any clear loss of quality when fitting to female test scans, see Fig.","6 in the main paper, it might lead to biased quality of results in other problems, for eg., if random samples from the model was used for training another network."],["Comparisons","As mentioned in Sec.","4.4 of the main paper, we compare our model to two existing models, BFM [38] and FLAME [29].","We show more qualitative model fitting results in Fig.","REF .","Next, we provide more details on the fitting algorithm used.","Fitting: We paint the face region of each model's template mesh to create a mask as shown in Fig.","REF .","We use these masked regions of the models for fitting the models to head scans.","To initialize, we mark 8 landmarks (eye corners, nose, lip corners, and chin) on the template mesh and rigidly align the template to each ground truth scan using Procrustes algorithm.","We allow for translation, rotation, and scaling.","We use the rigid alignment as initialization and optimize for the parameters of each model using a modified iterative closest point (ICP) algorithm which also updates the model parameters.","The fitting algorithm maintains the initial scale and translation but optimizes for rotation.","In each optimization step, we first compute the correspondences as the closest points from the masked region in the model, shown in Fig.","REF , to the scan data.","We compute the loss as shown in Eq.","(REF ) and update the model parameters along with Euler angles for global rotation.","We run the following optimization program up to convergence to fit the models to our scans: $ \\operatornamewithlimits{argmin}_{\\theta ,K,\\alpha ,\\beta ,\\gamma }& \\,\\, \\sum _{i=1}^N \\,\\,(\\big \\Vert sR(\\alpha ,\\beta ,\\gamma )x_i(\\theta ) + t - x_i \\big \\Vert _2 \\nonumber \\\\ &\\quad \\quad + \\big \\Vert Kc_i(\\theta ) - c_i \\big \\Vert _2 ) \\nonumber \\\\ &+ w_l \\sum _{j=1}^L \\big \\Vert sR(\\alpha ,\\beta ,\\gamma )l_j(\\theta ) + t - l_j\\big \\Vert _2 \\,,$ where, $x_i(\\theta ) \\in \\mathbb {R}^3$ is a vertex $i$ in the masked region of the model (containing $N$ vertices), $x_i$ is the point on the scan data corresponding to $x_i(\\theta )$ ; $R(\\alpha ,\\beta ,\\gamma ) \\in \\mathbb {R}^{3 \\times 3}$ is the global rotation matrix computed using the Euler angles, $\\alpha ,\\beta ,\\text{and } \\gamma $ ($\\in \\mathbb {R}$ ); $s \\in \\mathbb {R} $ , $t \\in \\mathbb {R}^3$ are the global scale and translation computed during initialization; $c_i(\\theta ) \\in \\mathbb {R}^3$ , $c_i \\in \\mathbb {R}^3$ are the colors at the vertices $x_i(\\theta )$ and $x_i$ respectively; $l_j \\in \\mathbb {R}^3$ and $l_j(\\theta ) \\in \\mathbb {R}^3$ are the $L(=8)$ ground truth and model landmarks respectively, as described earlier; and $K \\in R^{3\\times 3}$ is a diagonal matrix.","We set $w_l=0.1$ during the fitting process.","Note that the color loss is only enforced for BFM, as FLAME does not model colors.","Further, as the color intensities of BFM and our scans differ, we globally scale the color values using channel-specific scalars arranged as a diagonal matrix $K$ which we optimize for along with the model parameters.","Evaluation details: We describe the evaluation metrics in Sec.","4.4 of the main paper.","Here, we present details about the masks used to evaluate these metrics.","Face region: We manually paint face masks on the ground truth scans to obtain the ground truth masks.","We exclude the mouth interior of the ground truth scans.","We copy this mask to the i3DMM fits.","We do that by annotating a vertex in i3DMM reconstruction if the nearest point from that vertex on the ground truth scan is in the masked region.","We show the face masks used to fit BFM and FLAME to ground truth scans in Fig.","REF .","We obtain the symmetric metrics presented in Table.","1 of the main paper for the face region in the following way.","In one direction, we compute the errors from masked region of ground truth to the closest points on the (unmasked) models fit to the scan.","In the other direction, we compute the errors from the masked region of the models to the (unmasked) ground truth scan.","We compute errors between the masked regions of one mesh to unmasked regions of other mesh to avoid large error metrics due to annotation mistakes during manual mask painting.","Full Head: We only fit to FLAME full head model as BFM does not model the entire head.","We remove the neck region from FLAME as shown in Fig.","REF as the ground truth head scans do not have neck regions.","We also remove the vertices used to close the neck from ground truth as FLAME has a hole in the mesh at the neck.","We compute the metrics as we do for the face region between these two full head meshes.","We report the full head metrics for our model in the entire head region, including the closed hole at the neck mesh."],["Ablative Analysis","In Fig.","REF , we show additional qualitative results for the ablative analysis.","We also show the quantitative results for full head i3DMM fit in comparison to i3DMM variants.","We compare the four models that evaluate our design choices in Table REF .","The error metrics are computed for the face region using manually annotated face masks as described in Sec.","REF .","We only evaluate the face region, as the ground truth for hair is noisy, and small quantitative differences are not very indicative of degradation in quality.","Although the geometric reconstruction accuracy is marginally better without the landmark supervision loss, as compared to i3DMM, the color reconstruction accuracy of i3DMM is higher.","Also, as mentioned in the main paper, Sec.","4.3, texture transfer results around the ear regions with landmark supervision loss are worse compared to i3DMM.","Table: Quantitative results for ablation study.","The columns, from left to right, show results obtained with uniform sampling for SDF instead of landmark-based sampling, without sparse pairwise landmark supervision loss, independently training for representing geometry and color, final model (i3DMM), and ground truth."],["Correspondence Evaluation","We quantitatively evaluate the correspondences predicted by i3DMM by using the FLAME and BFM fits as ground truth correspondences.","To this end, we first find the closest points from the vertices of the (masked) model fits to the i3DMM reconstructions for different scans.","We will call these correspondences ground truth annotations here.","We use a KD tree algorithm for efficiency.","The masked face region contains 26370 vertices for BFM, and 1873 vertices for FLAME.","We also transfer the annotations for one i3DMM reconstruction, to all the other reconstructions.","This process is same as that described in annotation transfer application (see Sec.","4.5 of main paper).","We compute the correspondence error as the average of error between the transferred and the ground truth annotations.","Note that we transfer annotations from one i3DMM fit to every other i3DMM fit.","Therefore, we compute a symmetric error metric.","The resulting distribution of error is shown in Fig.","REF (evaluating with BFM as ground truth is plotted in red, while FLAME is plotted in blue).","The mean and median of errors for BFM is $5.08$ mm and $3.02$ mm respectively.","The mean and median of errors for FLAME is $2.36$ mm and $1.83$ mm respectively.","Note that, this error does not only capture the error in i3DMM's correspondence predictions but also the error in registrations of FLAME and BFM fits, see Table.","1 in the main paper.","Figure: Distribution of errors in correspondences predicted by i3DMM computed using FLAME fits (face) as ground truth (blue), and using BFM fits (face) as ground truth (red)."],["Full Head Completion","We use the i3DMM prior to complete face scans with different hairstyles as shown in Fig.","REF .","To obtain the face meshes from our head meshes, we delete all the vertices that are outside a sphere around the the tip of the nose.","We learn the latent vector for the given test scan using the SDF samples of the face mesh, as described in Sec.","4.1 of the main paper.","Additionally, we semantically control the hair style of the completed scan by adding a regularizer that enforces the learned $\\mathbf {z}_\\text{geoH}$ and $\\mathbf {z}_\\text{colH}$ to be close to the hairstyle latent vectors learned during training.","It can be inferred from the results in Fig.","REF that our model learns a good prior distribution, generating plausible heads for the given faces.","i3DMM offers user-guided control for head completion and can be used to turn existing face-only 3DMMs into full head 3DMMs.","Further, our method can also be used as a prior distribution for applications such as monocular 3D reconstruction [54].","Figure: Additional annotation transfer results.","Top: segmentation transfer front view.","Middle: segmentation transfer side view.","Bottom: landmark transfer.","Left column shows i3DMM reconstructions with manual annotations.","Right part shows annotations transferred to head scans using i3DMM."],["Annotation Transfer","We show more results for annotation transfer described in Sec.","4.5 of the main paper, in Fig.","REF .","Figure: Additional comparison results between i3DMM (full head) fits, BFM (face) fit, and FLAME (full head and face) fits.Figure: Additional ablation results.","From left to right, i3DMM without landmark-based sampling, i3DMM without landmark supervision, i3DMM with independent color and geometry training, i3DMM, and ground truth results are shown."],["Visualization Details","The output of the i3DMM is a signed distance field.","Generally, marching cubes algorithm is used to reconstruct the surface from a SDF.","However, based on the resolution used, marching cubes algorithm introduces unpleasant surface artifacts.","To avoid these artifacts, we used a sphere tracer to render our results.","We used the Blinn-Phong reflection model to shade our geometry results.","We apply a gamma correction with $\\gamma =0.65$ for the color renders.","We use Redner [30] for rendering meshes."]],"string":"[\n [\n \"i3DMM: Deep Implicit 3D Morphable Model of Human Heads\"\n ],\n [\n \"Abstract We present the first deep implicit 3D morphable model (i3DMM) of full heads.\",\n \"Unlike earlier morphable face models it not only captures identity-specific geometry, texture, and expressions of the frontal face, but also models the entire head, including hair.\",\n \"We collect a new dataset consisting of 64 people with different expressions and hairstyles to train i3DMM.\",\n \"Our approach has the following favorable properties: (i) It is the first full head morphable model that includes hair.\",\n \"(ii) In contrast to mesh-based models it can be trained on merely rigidly aligned scans, without requiring difficult non-rigid registration.\",\n \"(iii) We design a novel architecture to decouple the shape model into an implicit reference shape and a deformation of this reference shape.\",\n \"With that, dense correspondences between shapes can be learned implicitly.\",\n \"(iv) This architecture allows us to semantically disentangle the geometry and color components, as color is learned in the reference space.\",\n \"Geometry is further disentangled as identity, expressions, and hairstyle, while color is disentangled as identity and hairstyle components.\",\n \"We show the merits of i3DMM using ablation studies, comparisons to state-of-the-art models, and applications such as semantic head editing and texture transfer.\",\n \"We will make our model publicly available.\"\n ],\n [\n \"Introduction\",\n \"3D morphable models (3DMMs) are parametric models of geometry and appearance of human faces, with widespread use in applications such as image and video editing, face recognition and cognitive science [19].\",\n \"These models are trained using 3D scans of humans, e.g., laser scans [4], depth sensor-based scans [9], or photometric multi-view scans [28].\",\n \"As 3DMMs usually learn a deformation space of a fixed template mesh, the training shapes commonly need to be brought into dense surface correspondence with each other.\",\n \"Computing such correspondence for the face region alone is already hard and often requires a challenging non-convex optimizaton problem [19]; computing dense correspondence for the rest of the head and the hair is close to impossible.\",\n \"Therefore, as well as due to the lack of large 3D scan datasets with hair, most 3DMMs only capture the face region.\",\n \"While some recent approaches aim to model the complete head [15], [28], [41], [40], they do not capture the hair region, and the appearance in the head region (if captured) is very limited.\",\n \"In this work, we present i3DMM, the first implicit 3D morphable model which captures the full head region, including hair deformations and appearance.\",\n \"We capture a new dataset of full photogrammetric head scans of 64 people for training.\",\n \"Each subject performs several expressions and natural hair is captured.\",\n \"Since such scans are noisy, especially in the hair region, our training algorithm uses an adaptive sampling strategy based on the quality of reconstructions in different regions of the head, allowing us to effectively handle noise without smoothing out details.\",\n \"In contrast to existing 3DMMs, which use a mesh-representation with a fixed template, we implicitly represent surfaces using signed distance functions.\",\n \"This allows us to capture large deformations easily, which is particularly convenient for the hair region.\",\n \"The implicit representation also allows us to avoid computing dense correspondence between training scans.\",\n \"Our method is inspired by recent works on deep implicit surface modeling [36], [31], [32], which demonstrate the advantages of using an implicit representation compared to voxel grids, point clouds or meshes.\",\n \"Our main technical innovation compared to these works is that we use a novel neural network architecture which decouples the learning process, separating it into learning a reference shape, learning geometry deformations with respect to this reference, and learning a color network.\",\n \"This allows us to automatically compute dense correspondences between any two shapes with only sparse supervision on salient face landmarks.\",\n \"This decoupling is also essential for disentangling the learned space into semantically meaningful parametric components, an important feature of 3DMMs.\",\n \"Most classical models disentangle identity geometry, expression geometry, and appearance of the face.\",\n \"This is sufficient for several applications in face editing [55], [53], [50], but does not allow the explicit control of head parts other than the facial region.\",\n \"Our model allows for unprecedented control over the different semantic modes of full heads.\",\n \"To this end, we learn to disentangle the geometry and color, where geometry is further separated into identity, expression, and hairstyle components, and color is further separated into identity and hairstyle components.\",\n \"In summary, our main contributions are: A method for learning full head 3D morphable models directly from rigidly aligned real-world noisy 3D scans without dense ground truth correspondences.\",\n \"A novel network architecture which can compute dense correspondences between 3D shapes represented using implicit functions.\",\n \"This network is trained only with sparse supervision.\",\n \"A training method for disentangling the color and geometry components, and the identity, expression, and hairstyle components of the geometry; and the identity and hairstyle components of the color.\",\n \"We compare our model to several state-of-the-art 3DMMs by fitting to 3D scans.\",\n \"We also demonstrate the quality of our learned dense correspondences by showing texture transfer between scans, and show that i3DMM benefits applications such as semantic head editing, and one-shot computation of segmentation masks and landmarks on 3D scans.\"\n ],\n [\n \"Related Work\",\n \"Most approaches for face and head modeling are mesh-based.\",\n \"We summarize them here and refer the reader to Egger [19] for a more comprehensive report.\",\n \"As our model is based on implicit representations, we also discuss methods which learn implicit shape and color.\",\n \"Face and Head Morphable Models.\",\n \"The work of Blanz and Vetter [5] showed the possibility of representing faces using a 3D Morphable Model (3DMM).\",\n \"The model is learned by transforming the shape and texture of examples into low-dimensional vector representations using principal components analysis (PCA).\",\n \"Further improvements were proposed [22], [38], [8], [6] by using better registration and higher quality scans.\",\n \"Multilinear models have also been proposed to model identity-dependent expressions [10], [20].\",\n \"Li [29] presented FLAME, a head model that combines a linear shape space with an articulated jaw, neck and eyeballs, pose-dependent corrective expression blendshapes, and global expressions.\",\n \"The model is learned from a large 3D dataset, including data from D3DFACS [12].\",\n \"Ranjan [42] learned a head model using an autoencoder based on spectral graph convolutions [17].\",\n \"The LYHM head model [16], [14] uses a hierarchical parts-based template morphing framework to process the head shape, and uses optical flow to refine the texture.\",\n \"The model is built using $1{,}200$ different identities.\",\n \"Ploumpis [40], [39] combined the LYHM and LSFM [7] models as the Universal Head model (UHM), with a focus on face geometry.\",\n \"Note that all existing full head models only model the cranium geometry without hair.\",\n \"Implicit Representation.\",\n \"Implicit representations have a long history in computer vision for inferring 3D shape [27], temporally evolving shape [13], [26] and textured shape [49].\",\n \"Park [36] presented DeepSDF to represent a class of shapes using signed distance fields with an autodecoder.\",\n \"Chen [11] and Mescheder [31] learned a generative model of a class of shapes by classifying a point in space as inside or outside a shape.\",\n \"Recent approaches [56], [18], [21] have proposed to represent shapes using a collection of local implicit patches for higher quality.\",\n \"Oechsle [35] extended OccupancyNets [31] to represent texture along with the geometry using monocular inputs.\",\n \"Niemeyer [34] presented an approach for handling time-varying shapes by learning to predict motion vectors for each point in 3D space using OccupancyNets.\",\n \"PIFu [45], [46] allows for 3D reconstruction of humans from monocular inputs.\",\n \"Pixel-aligned implicit functions estimate a continuous field that determines whether a pixel is inside or outside the surface of the human subject, as well as the color on the surface.\",\n \"Several recent methods [48], [32], [51] have presented ways to achieve higher-quality implicit representations by using periodic functions as activations.\",\n \"Saito [44] presented an approach for 3D hair modeling.\",\n \"Their technique learns a manifold for 3D hairstyles represented as implicit surfaces using a volumetric variational autoencoder.\"\n ],\n [\n \"Method\",\n \"In this section, we describe how we obtain our deep implicit 3D morphable model (i3DMM).\",\n \"Our overall processing pipeline is illustrated in Fig.\",\n \"REF .\",\n \"In the following we will describe our data acquisition, the training data preparation, and the neural network architecture.\",\n \"Figure: Overview of the captured expressions (red box: test set; blue box: hairstyles) for each subject.Figure: Multi-view scanning system with 135 cameras around the person (green), and 2 cameras (blue) on top.\",\n \"Right: front view image, and reconstruction.\"\n ],\n [\n \"Data Acquisition\",\n \"For training, we have scanned 64 subjects (46 male, 18 female; 22 Caucasian, 9 Asian, 25 Indian Sub-continent, 3 Hispanic, 3 Middle Eastern, 2 undisclosed; with an age ranging from 19 to 69 with average 26).\",\n \"For each subject we have recorded 10 facial expressions (a subset of which are chosen from paGAN [33]), including neutral expressions with 3-4 different “hairstyles” (open hair, tied hair for subjects with long hair, and two different caps), as shown in Fig.\",\n \"REF .\",\n \"Our data was acquired with the Treedyshttps://www.treedys.com/ multi-view scanning system, that comprises of 137 calibrated cameras, see Fig.\",\n \"REF .\",\n \"The total capture time per person amounts to about 20 minutes.\",\n \"Photogrammetry-based 3D reconstruction on the multi-view data was applied to obtain the final textured meshes (see Fig.\",\n \"REF ), where each mesh comprises of around $50{,}000$ vertices.\"\n ],\n [\n \"Training Data\",\n \"Given our recorded textured mesh data, we apply several preprocessing steps to make the data is suitable for head model learning.\",\n \"This includes a semi-automated landmark annotation, cropping of the head area to discard the shoulders and parts of the upper body, rigidly aligning the meshes, and closing the hole at the neck to make the mesh watertight.\",\n \"In the following we elaborate on these steps.\",\n \"Landmark Annotation.\",\n \"First, we apply the automated face landmark detector from dlib [24] on the front-facing image obtained from the multi-view camera acquisition setup, which produces a total of 66 face landmarks.\",\n \"We select a subset of 8 landmarks, i.e.\",\n \"the four corner points of the eyes, the tip of the nose, the two corner points of the mouth, and the chin, and then transfer these 2D landmarks to the 3D mesh.\",\n \"We manually correct the inaccurately detected landmarks, and additionally annotate 8 landmarks for the corners of ears (top, bottom, left, and right) directly in 3D.\",\n \"Head Cropping.\",\n \"In order to remove the shoulders and parts of the upper body, we crop the head meshes based on the 3D landmarks.\",\n \"To this end, we first compute the vector $\\\\mathbf {v}$ from the centroid of the four eye cornerpoints to the tip of the nose.\",\n \"Then, for $\\\\mathbf {p}$ being the chin landmark in 3D space, we define a virtual plane that goes through the point $\\\\mathbf {p} + \\\\frac{1}{2} \\\\mathbf {v}$ and has the normal $\\\\mathbf {v}$ .\",\n \"We only keep the side of the plane that contains the facial landmarks.\",\n \"Rigid Alignment.\",\n \"Based on the 3D facial landmarks, we perform a rigid alignment of all head scans in order to ensure that they are oriented consistently in 3D space.\",\n \"We use a numerically stable implementation [3] of the transformation synchronization method [2], which solves the generalized Procrustes problem in an initialization-free and unbiased way.\",\n \"To this end, we first rigidly align all pairs of heads based on Singular Value Decomposition (SVD), i.e.\",\n \"we solve all pairwise Procrustes problems, and subsequently use transformation synchronization to establish cycle-consistency in the set of pairwise transformations.\",\n \"Hole Closing.\",\n \"Since our full head model is based on a signed distance representation, we require that the meshes are watertight.\",\n \"After rigid alignment, we assume that the longitudinal axis of the head aligns with the y-axis.\",\n \"With that, we use a flat patch to close the hole at the neck by extruding the respective boundary vertices to coincide with the smallest y-coordinate.\",\n \"Finally, we scale all meshes with the same value such that all of them fit in the unit cube.\"\n ],\n [\n \"Training\",\n \"We learn a vector-valued function $f_{\\\\theta }(\\\\mathbf {x},\\\\mathbf {z})$ , where $\\\\theta $ are the neural network weights, $\\\\mathbf {x} \\\\in \\\\mathbb {R}^3$ is the query point, and $\\\\mathbf {z}$ is a code vector that encodes the head instance.\",\n \"The output $f_{\\\\theta }(\\\\mathbf {x},\\\\mathbf {z}) = ({s}, \\\\mathbf {c})$ includes a scalar ${s}$ that represents the signed distance to the surface, as well as the color $\\\\mathbf {c} \\\\in \\\\mathbb {R}^3$ at the closest surface point from $\\\\mathbf {x}$ .\",\n \"The shape boundary is represented as the zero level-set of the signed distance function (SDF), while the interior parts of the shapes have a negative signed distance value, and the exterior parts a positive value.\",\n \"We use an autodecoder network architecture [36], where the weights of the network $\\\\theta $ and the input latent codes $\\\\mathbf {z}$ for all shapes are learned jointly.\",\n \"Mesh Sampling.\",\n \"We require $(\\\\mathbf {x},{s}, \\\\mathbf {c})$ -triplets (query point, signed distance value, color) for training.\",\n \"We use a combination of two strategies for sampling these triplets.\",\n \"First, we sample points on the mesh surface based on uniform random sampling.\",\n \"However, in order to also account for higher accuracy in high-detail facial regions, i.e.\",\n \"the eyes, nose and mouth, we additionally sample more points in these areas.\",\n \"We take the center of each eye, the tip of the nose, and the center of the mouth, and place a sphere around them that covers the respective region of interest.\",\n \"Then, we sample points on the mesh surface that lie within each sphere.\",\n \"Eventually we mix the uniformly sampled points with the landmark-based sampled points in a 1 to 3 ratio.\",\n \"After sampling, the points are perturbed by a uniform 3D Gaussian with standard deviation $0.005$ times the length of the bounding box, such that not only surface points but also points in the interior and exterior are used, see [36].\",\n \"The color value for each sample is obtained by finding the closest point on the mesh surface, and then looking up its color in the texture map.\",\n \"Latent Codes and Disentanglement.\",\n \"As mentioned earlier, latent codes for each object (head scan) are also learned during training.\",\n \"Existing approaches use a single object latent code to describe the shape.\",\n \"In contrast, we design several separate latent spaces for our objects in order to learn a semantically disentangled model.\",\n \"We use two separate latent vector spaces for geometry and color, $\\\\mathbf {z}_\\\\text{geo}$ and $\\\\mathbf {z}_\\\\text{col}$ , respectively.\",\n \"The geometry space includes three code vectors for identity, expression, and hairstyle, and the color space includes two code vectors for identity and hairstyle.\",\n \"Thus, $\\\\mathbf {z}_\\\\text{geo} = (\\\\mathbf {z}_\\\\text{geoId}, \\\\mathbf {z}_\\\\text{geoEx}, \\\\mathbf {z}_\\\\text{geoH})$ and $\\\\mathbf {z}_\\\\text{col} = (\\\\mathbf {z}_\\\\text{colId}, \\\\mathbf {z}_\\\\text{colH})$ .\",\n \"During training, the number of different identity code vectors is equal to the number of training identities, 58.\",\n \"The number of different expression vectors is fixed to 10 (cf.\",\n \"Fig.\",\n \"REF for the training expressions) and hairstyle to 4 (short, long, cap1 or cap2) for geometry, and to 3 (nocap, cap1 or cap2) for color.\",\n \"For all scans of the $i$ -th subject we use the same latent variables for the geometry identity and color identity, $\\\\mathbf {z}_\\\\text{geoId}^i$ and $\\\\mathbf {z}_\\\\text{colId}^i$ .\",\n \"Similarly, for each expression and hairstyle, the same variables $\\\\mathbf {z}_\\\\text{geoEx}, \\\\mathbf {z}_\\\\text{geoH}$ and $\\\\mathbf {z}_\\\\text{colH}$ are used across all identities.\",\n \"By doing so, we are able to learn disentangled latent variables without imposing any explicit constraints.\",\n \"At test time, we can control each latent space individually, leading to semantically meaningful editing.\",\n \"Network Architecture.\",\n \"Figure: Overview of our network architecture.\",\n \"We learn weights of three network components, a Shape Deformation component, a Reference Shape component, and a Color component.\",\n \"Moreover, the latent codes for each object are also optimized for.\",\n \"The input of the network is a 3D query point, and the output is a signed distance value along with the corresponding color.Our network comprises three components, the Reference Shape Network (RefNet), the Shape Deformation Network (DeformNet) and the Color Network (ColorNet).\",\n \"All networks are composed of fully-connected layers with a ReLU non-linearity after every layer, except the output layer.\",\n \"The inputs to all our networks are encoded using sinusoidal positional encoding [32].\",\n \"Our Reference Shape Network encodes a single reference shape, such that all individual head shapes can be obtained by deforming this shape.\",\n \"The output at a query point $\\\\mathbf {x}$ is the signed distance value $f_\\\\theta ^r(\\\\mathbf {x}) \\\\in \\\\mathbb {R}$ .\",\n \"Note that for the reference shape there is no latent code input since only a single reference shape is learned.\",\n \"This can be seen analogously to the mean (or neutral) shape used in classical 3DMMs [5].\",\n \"We use 3 fully connected layers for this network, where each hidden layer has dimensionality 512.\",\n \"The Shape Deformation Network deforms the reference shape to represent the shape of an individual head.\",\n \"The network takes the geometry latent code $\\\\mathbf {z}^i_\\\\text{geo}$ for an object $i$ , and a query point $\\\\mathbf {x}$ as input, and produces the output, $f_\\\\theta ^s(\\\\mathbf {x}, \\\\mathbf {z}^i_\\\\text{geo}) = \\\\delta \\\\in \\\\mathbb {R}^3\\\\,,$ which is a displacement vector that deforms the query point to the reference shape.\",\n \"Thus, the signed distance value at any point $\\\\mathbf {x}$ of the object $i$ with geometry latent $\\\\mathbf {z}^i_\\\\text{geo}$ is ${s}(\\\\mathbf {x}, \\\\mathbf {z}^i_\\\\text{geo}) = f_\\\\theta ^r(\\\\mathbf {x} + \\\\delta )\\\\,,$ where ${s}(\\\\cdot )$ is the scalar signed distance value, and $\\\\delta $ is the deformation from Eq.\",\n \"(REF ).\",\n \"This formulation allows us to compute dense correspondences between any input head scan and the reference head shape.\",\n \"The separation between a reference shape and deformations with respect to this reference shape is also common in classic morphable models [19].\",\n \"However, these models require dense correspondences before training, which is not necessary in our approach.\",\n \"We use 8 fully connected layers for this network, where each hidden layer has dimensionality 1024.\",\n \"The introduction of the reference shape makes it possible to disentangle the geometry and color components.\",\n \"The Color Network learns the color of the query point in the reference space.\",\n \"Given a query point $\\\\mathbf {x}$ , deformation $\\\\delta $ from Eq.\",\n \"(REF ), and color latent vector $\\\\mathbf {z}^i_\\\\text{col}$ for the object $i$ , the output is represented as $f_\\\\theta ^c(\\\\mathbf {x}+\\\\delta , \\\\mathbf {z}^i_\\\\text{col}) \\\\in \\\\mathbb {R}^3$ , which is the color at point $\\\\mathbf {x}$ .\",\n \"Note that without this separation of a reference space, the ColorNet would also have to take into account information about object geometry, thus not being able to disentangle shape and color.\",\n \"Note that the latent code for ColorNet, for a given identity and hairstyle, does not change with expressions.\",\n \"DeformNet finds the right colors and geometry for different expressions by achieving dense correspondences.\",\n \"We use 9 fully connected layers for this network, where each hidden layer has dimensionality 1024.\",\n \"Loss Functions.\",\n \"We define the loss function to train our network as $\\\\mathcal {L}_\\\\theta (\\\\mathbf {x}, \\\\mathbf {z_\\\\text{geo}}, \\\\mathbf {z_\\\\text{col}}) & = \\\\sum _{i=1}^K(\\\\mathcal {L}_\\\\theta ^\\\\text{geo}(\\\\mathbf {x}, \\\\mathbf {z}^i_\\\\text{geo}) +\\\\mathcal {L}_\\\\theta ^\\\\text{def}(\\\\mathbf {x}, \\\\mathbf {z}^i_\\\\text{geo}) \\\\nonumber \\\\\\\\ &+\\\\mathcal {L}_\\\\theta ^\\\\text{col}(\\\\mathbf {x}, \\\\mathbf {z}^i_\\\\text{col}) + \\\\mathcal {L}^\\\\text{reg}(\\\\mathbf {z}^i_\\\\text{geo}, \\\\mathbf {z}^i_\\\\text{col}) ) \\\\nonumber \\\\\\\\ &+ \\\\sum _{i\\\\ne j}\\\\mathcal {L}^\\\\text{lm}_\\\\theta (\\\\mathbf {z}^i_\\\\text{geo},\\\\mathbf {z}^j_\\\\text{geo})\\\\,,$ where, $i,\\\\,j = \\\\lbrace 1,\\\\dots ,K\\\\rbrace $ , $K$ is the number of scans in the batch, $\\\\mathbf {z}_\\\\text{geo} = \\\\lbrace \\\\mathbf {z}^1_\\\\text{geo},\\\\dots ,\\\\mathbf {z}^K_\\\\text{geo}\\\\rbrace $ , and $\\\\mathbf {z}_\\\\text{col} = \\\\lbrace \\\\mathbf {z}^1_\\\\text{col},\\\\dots ,\\\\mathbf {z}^K_\\\\text{col}\\\\rbrace $ .\",\n \"The latent vectors of head $i$ are represented as $\\\\mathbf {z}_\\\\text{geo}^i$ and $\\\\mathbf {z}_\\\\text{col}^i$ .\",\n \"Here, $\\\\mathcal {L}_\\\\theta ^\\\\text{geo}(\\\\cdot )$ enforces good geometry reconstructions, $\\\\mathcal {L}_\\\\theta ^\\\\text{def}(\\\\cdot )$ regularizes the deformation field, $\\\\mathcal {L}_\\\\theta ^\\\\text{col}(\\\\cdot )$ is used to train the ColorNet, $\\\\mathcal {L}^\\\\text{reg}(\\\\cdot )$ is a regularizer on the latent vectors, and $\\\\mathcal {L}^\\\\text{lm}_\\\\theta (\\\\cdot )$ is a sparse pairwise landmark supervision loss.\",\n \"For the geometry term that we impose upon the signed distance values, we use the $\\\\ell _1$ -loss $\\\\mathcal {L}_\\\\theta ^\\\\text{geo}(\\\\mathbf {x}, \\\\mathbf {z}^i_\\\\text{geo}) = w_g \\\\big \\\\Vert \\\\text{cl}(s(\\\\mathbf {x}, \\\\mathbf {z}^i_\\\\text{geo}), t) - \\\\text{cl}(s_\\\\text{gt}(\\\\mathbf {x}), t) \\\\big \\\\Vert _1\\\\,{,}$ where $s_\\\\text{gt}(\\\\mathbf {x})$ is the ground truth signed distance value at $\\\\mathbf {x}$ , and $s(.\",\n \")$ is from Eq.\",\n \"(REF ).\",\n \"These values are (symmetrically) clamped at $t$ =$0.1$ , for which we define $\\\\text{cl}(x, t) := \\\\min (t, \\\\max (x, -t))$ .\",\n \"We use a similar $\\\\ell _1$ -loss for the color component, i.e.\",\n \"$\\\\mathcal {L}_\\\\theta ^\\\\text{col}(\\\\mathbf {x}, \\\\mathbf {z}^i_\\\\text{col}) = w_c \\\\big \\\\Vert f_\\\\theta ^c(\\\\mathbf {x}+\\\\delta , \\\\mathbf {z}^i_\\\\text{col}) - c_\\\\text{gt}(\\\\mathbf {x}) \\\\big \\\\Vert _1\\\\,{,}$ where $c_\\\\text{gt} (\\\\mathbf {x})$ is the ground truth color at $\\\\mathbf {x}$ .\",\n \"We also enforce that the 16 landmarks $\\\\lbrace \\\\mathbf {x}^{\\\\ell }_i\\\\rbrace $ , as described in Sec.\",\n \"REF , of each shape $i$ , deform to the same points in the reference space using the pairwise loss $ \\\\mathcal {L}^\\\\text{lm}_\\\\theta (\\\\mathbf {z}^i_\\\\text{geo},\\\\mathbf {z}^j_\\\\text{geo}) = w_{lm} \\\\sum _{\\\\ell =1}^{16} \\\\big \\\\Vert (\\\\mathbf {x}^\\\\ell _i + \\\\delta ^\\\\ell _i) - (\\\\mathbf {x}^\\\\ell _j + \\\\delta ^\\\\ell _j) \\\\big \\\\Vert _2\\\\,,$ where, $\\\\delta ^\\\\ell _i = f_\\\\theta ^s(\\\\mathbf {x}^\\\\ell _i, \\\\mathbf {z}^i_\\\\text{geo})$ as in Eq.\",\n \"(REF ).\",\n \"For scans with the ears covered by hair, we do not have any ear annotations.\",\n \"We would like to compress the ear region in the reference shape to a single point in the reconstruction for these shapes.\",\n \"Thus, we additionally optimize for one point for each ear.\",\n \"We enforce pairwise constraints between the learnable ear points and the annotated ear points for the other shapes in the batch using Eq.\",\n \"(REF ).\",\n \"Further, to ensure regularized deformations to the reference shape, we impose a loss on the amount of deformation.\",\n \"We use $\\\\mathcal {L}_\\\\theta ^\\\\text{def}(\\\\mathbf {x}, \\\\mathbf {z}^i_\\\\text{geo}) = w_s || f_\\\\theta ^s(\\\\mathbf {x}, \\\\mathbf {z}^i_\\\\text{geo}) ||_2$ .\",\n \"Finally, we use an $\\\\ell _2$ -regularizer on the latent vectors assuming a Gaussian prior distribution, $\\\\mathcal {L}^\\\\text{reg}(\\\\mathbf {z}^i_\\\\text{geo}, \\\\mathbf {z}^i_\\\\text{col}) =w_r ( ||\\\\mathbf {z}^i_\\\\text{geo}||_2 + ||\\\\mathbf {z}^i_\\\\text{col}||_2)$ .\",\n \"Optimization.\",\n \"Given $N$ batches with $K$ objects per batch, we optimize for the network weights and the latent vectors by solving the optimization problem $\\\\operatornamewithlimits{argmin}_{\\\\theta , \\\\lbrace \\\\mathbf {z}^b_\\\\text{geo}, \\\\mathbf {z}^b_\\\\text{col}\\\\rbrace _{b=1}^N}~~ \\\\sum _{b=1}^N\\\\sum _{\\\\mathbf {x}} \\\\mathcal {L}_\\\\theta (\\\\mathbf {x}, \\\\mathbf {z}^b_\\\\text{geo}, \\\\mathbf {z}^b_\\\\text{col})\\\\,,$ where, the inner sum takes into account all sampled points, as explained above, and we abuse the notation $\\\\mathbf {z}^b_\\\\text{geo}, \\\\mathbf {z}^b_\\\\text{col}$ to now represent latent codes of all the scans in batch $b$ .\"\n ],\n [\n \"Experiments\",\n \"In this section, we present an experimental evaluation of our head model.\",\n \"We demonstrate that the model can be used for reconstructing scan data.\",\n \"We present an ablation study to carefully analyze our design choices, and compare i3DMM to state-of-the-art head models.\",\n \"We also show dense correspondence results and applications of our model.\",\n \"Before we present these results, we provide additional information on the neural network training and testing.\",\n \"Training Details.\",\n \"We train our networks using PyTorch [37], where we use the Adam [25] solver with mini-batches of size 64.\",\n \"We train for 1000 epochs with a learning rate of $0.0005$ , which decays by a factor of 2 every 250 epochs.\",\n \"We initialize RefNet by pretraining it using only one mouth-open (top row, third from left in Fig.\",\n \"REF ) training scan.\",\n \"Our network takes about 2 days to train on 2 NVIDIA RTX8000 GPUs.\",\n \"Test Data.\",\n \"We collect a separate test set consisting of scans of 6 identities, which are not part of the training data.\",\n \"We capture each identity in 5 novel expressions that are not part of the training expressions, see Fig.\",\n \"REF .\",\n \"Each scan is preprocessed analogously to the training scans.\"\n ],\n [\n \"Reconstruction\",\n \"For a given test scan, we fit our learned model to it.\",\n \"This is done by optimizing for the latent vector that can best reproduce the scan, i.e.\",\n \"by finding the latent variables that minimize the problem $\\\\operatornamewithlimits{argmin}_{\\\\mathbf {z}_\\\\text{geo}^i, \\\\mathbf {z}_\\\\text{col}^i}~~ \\\\sum _{\\\\mathbf {x}} &\\\\, (\\\\mathcal {L}_\\\\theta ^\\\\text{geo}(\\\\mathbf {x}, \\\\mathbf {z}^i_\\\\text{geo}) +\\\\mathcal {L}_\\\\theta ^\\\\text{def}(\\\\mathbf {x}, \\\\mathbf {z}^i_\\\\text{geo}) \\\\nonumber \\\\\\\\ + &\\\\mathcal {L}_\\\\theta ^\\\\text{col}(\\\\mathbf {x}, \\\\mathbf {z}^i_\\\\text{col}) + \\\\mathcal {L}^\\\\text{reg}(\\\\mathbf {z}^i_\\\\text{geo}, \\\\mathbf {z}^i_\\\\text{col}))\\\\,,$ where, $\\\\mathbf {z}_\\\\text{geo}^i, \\\\mathbf {z}_\\\\text{col}^i$ is the latent code for test scan $i$ .\",\n \"This equation is similar to Eq.\",\n \"(REF ), with the difference that the network weights are fixed here and the pairwise landmark supervision loss in Eq.\",\n \"(REF ) is not enforced.\",\n \"We use Adam [25] with a step size of $0.0005$ to solve this problem.\",\n \"We show several reconstruction results on the test data in Fig.\",\n \"REF .\",\n \"We can generalize to unseen identities and expressions, even though our training data only consists of 58 people with 10 expressions.\",\n \"Note that while we can generally preserve the detailed face region of the scans, we also smooth the scans in the noisy hair area.\"\n ],\n [\n \"Correspondences\",\n \"As mentioned in Sec.\",\n \"REF , due to the particular design of our network, where the reference shape and the deformations are separated, our approach also establishes dense correspondences between shapes, with extremely sparse landmark supervision.\",\n \"We demonstrate these correspondences in Fig.\",\n \"REF , where the color is transferred from one scan to the other.\",\n \"The correspondences are also used in the applications of segmentation and landmark transfer in Sec.\",\n \"REF .\",\n \"Our model can reliably find correspondences across different subjects, expressions and hairstyles, including long and short hair.\",\n \"Please refer to the supplementary material for more evaluations.\",\n \"Figure: Reconstruction quality of i3DMM on test data.\",\n \"The top part shows texture and geometry of ground truth scans.\",\n \"The middle part shows texture and geometry of i3DMM fits.\",\n \"The last part shows texture transfer from the scan on right to the scan on left of the image.\",\n \"In column 3, noise in ground truth does not transfer to the i3DMM fit, showing the robustness of our method.\"\n ],\n [\n \"Ablative Analysis\",\n \"We design several experiments to evaluate the most important design choices for our approach: landmark-based sampling described in Sec.\",\n \"REF , sparse pairwise landmarks loss in Eq.\",\n \"(REF ), and jointly training ColorNet, DeformNet, and RefNet.\",\n \"We evaluate the qualitative impact of these choices by excluding them one at a time while training our model, see Fig.\",\n \"REF .\",\n \"Without landmarks-based sampling, the model focuses more on the noisy hair region and ignores the details in the face region (first column in Fig.\",\n \"REF ).\",\n \"Without landmark supervision, the network creates small (fake) ear regions in the hair for the long-haired scans (second column in Fig.\",\n \"REF ).\",\n \"It also leads to poor texture transfer around the ear.\",\n \"Finally, for evaluating the importance of joint training of the color and geometry networks, we first train RefNet and DeformNet.\",\n \"After convergence, we train ColorNet.\",\n \"The correspondences in this case are learned only from geometry reconstructions, which leads to artifacts as shown in the third column of Fig.\",\n \"REF .\",\n \"Figure: Ablation study.\",\n \"Top and middle row show reconstructions of two different test scans.\",\n \"Last row shows texture transfer from the scans in the top row to those in the middle row.\"\n ],\n [\n \"Comparisons to Existing Models\",\n \"We compare our model with the full head FLAME [29], face only FLAME, and Basel Face Model (BFM) [38] which only models the frontal face.\",\n \"We fit each model to the test set by optimizing for their model parameters.\",\n \"Please refer to the supplemental for more details on the fitting.\",\n \"We show qualitative results in Fig.\",\n \"REF .\",\n \"FLAME only models the cranium without hair, thus fitting to the full head region results in incorrect head shapes.\",\n \"We also evaluate FLAME by only using the the face region for fitting.\",\n \"This leads to higher quality results.\",\n \"We obtain higher quality geometry both for the face and head regions.\",\n \"In addition, we can also reconstruct the color of the head, while FLAME can only model the geometry.\",\n \"We combine the BFM [38] identity geometry and appearance models with the expression model used in Tewari [52], and use this to fit to our test scans.\",\n \"The expression model is a combination of two blendshape models [1], [9].\",\n \"Since BFM also includes color, we jointly optimize to minimize the geometry as well as color alignment errors.\",\n \"As can be seen in Fig.\",\n \"REF (middle), this model can fit to the expression as well as color of the scans.\",\n \"However, it is limited to only the face region, while i3DMM can reconstruct the full head.\",\n \"In addition, our reconstructions are more personalized, with higher quality nose geometry and face colors.\",\n \"We show the quantitative evaluations of the different models in terms of the symmetric Chamfer distance and F-score metrics in Table REF .\",\n \"Chamfer distance is the mean distance of points from one shape to their closest points in another.\",\n \"We compute the symmetric Chamfer distance as the mean of Chamfer distance from the ground truth to the fits and the Chamfer distance in the opposite direction.\",\n \"F-score is (100x) the harmonic mean of these two distances after applying a threshold.\",\n \"We use a similar metric for color.\",\n \"First, the mean error in color at points in the ground truth and the color at their nearest points in the fits is computed, along with the mean error in the opposite direction Finally, the color error is reported as their average.\",\n \"Please refer to the supplemental for details on the metrics used.\",\n \"We sample $150{,}000$ points for computing the metrics.\",\n \"We achieve similar geometric quality as FLAME and BFM in the face region, but significantly outperform FLAME for the full head reconstructions, see Table REF .\",\n \"Moreover, in terms of color, our reconstructions are more realistic than BFM and also outperform it quantitatively.\",\n \"Table: Quantitative comparison of head models.\",\n \"Symmetric Chamfer distance is reported here.\",\n \"F-score is computed with a threshold of 0.010.01.\"\n ],\n [\n \"Application: Semantic Head Editing\",\n \"Due to the semantic disentanglement in our model, we can selectively change semantic components of a head.\",\n \"In Fig.\",\n \"REF , we edit one feature of a test scan while keeping the other features fixed.\",\n \"To obtain the edited latent codes, we first find the principle components of variation in each latent-subspace by running PCA on the training latent spaces.\",\n \"We move along the principal components to pick the latent codes to edit the identity, and we pick others from the trained latent codes.\",\n \"Unlike existing models, we can also edit hairstyles and add caps while keeping other components fixed.\",\n \"We also model the mouth interior.\",\n \"These features allow for semantic head editing applications much beyond the capabilities of the existing methods.\"\n ],\n [\n \"Application: One-shot Annotation Transfer\",\n \"As i3DMM can predict dense correspondences among head scans, it also enables us to transfer annotations across different reconstructions.\",\n \"As our model has low reconstruction errors, it can also be used to annotate real-world 3D scans.\",\n \"We show this application in Fig.\",\n \"REF .\",\n \"We first transfer the annotations from one manually annotated scan to the reference shape.\",\n \"We can then transfer them to different i3DMM fits, and also to the real-world scans using nearest neighbors from the fits to the scans.\",\n \"As i3DMM can be used to even annotate the hair region, it can be very useful to curate scan datasets.\",\n \"Please refer to the supplemental for more results.\",\n \"Figure: Application: One-shot annotations on real-world scans.\",\n \"Coarsely drawn annotations (top row: semantic segmentation, bottom row: landmarks, not used during training) on one scan (left) can be automatically transferred to other real-world head scans (right).\"\n ],\n [\n \"Discussion and Future Work\",\n \"Although i3DMM disentangles and models an unprecedented amount of variety in head features, and enables reconstructions with very high quality (Fig.\",\n \"REF ), it still has some limitations.\",\n \"Although i3DMM models varying hairstyle shapes, it is only a coarse approximation of the physical nature of hair comprising of individual hair strands.\",\n \"Further, we model hairstyles that cover the ear by trying to collapse the ear to a single point (Eq.\",\n \"(REF )).\",\n \"Ideally, the layered geometry of ears behind the hair should be correctly represented.\",\n \"Finally, there is room for extending i3DMM for more fine-grained control over hairstyles.\",\n \"One challenge is that our scans are noisy in the hair region.\",\n \"This is due to the photometric multi-view reconstruction techniques used, which struggle with hair due to their complex physical interaction with light.\"\n ],\n [\n \"Conclusion\",\n \"We have presented the first unified 3D morphable model of geometry and appearance of full heads, which includes different identities, expressions and hairstyles.\",\n \"The head geometry and color is represented using implicit function parameterized with neural networks, learned from 610 multi-view scans.\",\n \"By explicitly separating the network into reference shape, shape deformation and color components, our model can disentangle the different semantic components.\",\n \"It also allows us to obtain dense correspondences across different scans represented using our model, enabling several applications.\",\n \"We will make our code publicly available to the community.\"\n ],\n [\n \"Acknowledgements\",\n \"This work has been supported by the ERC Consolidator Grant 4DReply (770784), and by the BMBF-funded Munich Center for Machine Learning.\",\n \"We thank Garvita Tiwari, Navami Kairanda, and Neng Qian for their support in capturing the dataset.\",\n \"We thank all the subjects in the dataset for lending their data for research.\",\n \"Figure: NO_CAPTIONSupplementary Model Visualization Our implicit 3D morphable model models the identity, expression, and hairstyle components of the geometry; as well as the identity and hairstyle components of the color of the head with independent parameteric controls.\",\n \"We visualize these components in Fig.\",\n \"REF .\",\n \"We perform PCA on the identity geometry and color spaces in order to compute the principal components.\",\n \"The expression component is visualized by moving along the directions of the training expressions, since they are semantically well defined.\",\n \"As we model hairstyles that include caps, we show the joint space of geometry and color for hairstyles.\",\n \"Note that the hairstyle geometry can only take four discrete values – short, long, cap1, or cap2.\",\n \"Any variation within these categories is modelled by the identity-geometry component.\",\n \"Similarly, hairstyle color can only take the values – nocap, cap1, or cap2.\",\n \"The color of hair without any cap is determined by the identity-color component.\",\n \"Experiments Here, we provide more details on the evaluations in the main paper, and include further evaluations.\",\n \"Sampling i3DMM One of the important features of a 3D Morphable Model is the ability to randomly sample shapes in the parametric space.\",\n \"This has been used for generating synthetic data for training CNNs [23], [43], [47].\",\n \"We achieve sampling in i3DMM by performing principal component analysis (PCA) over the training latent codes for color-identity, geometry-identity, and expressions.\",\n \"We weigh the singular values using a Gaussian random variable $\\\\mathcal {N}(0,0.1)$ for color-identity, and geometry-identity, and $\\\\mathcal {N}(0,0.25)$ for expressions.\",\n \"Since the latent codes for hair shapes and colors can take very limited numbers of values and are very well defined semantically, we sample these from their training values.\",\n \"We show several results in the supplemental video.\",\n \"As can be seen, our model is biased towards generating male heads.\",\n \"This is likely due to our gender-biased training dataset with 46 males and 18 females.\",\n \"While this does not lead to any clear loss of quality when fitting to female test scans, see Fig.\",\n \"6 in the main paper, it might lead to biased quality of results in other problems, for eg., if random samples from the model was used for training another network.\",\n \"Comparisons As mentioned in Sec.\",\n \"4.4 of the main paper, we compare our model to two existing models, BFM [38] and FLAME [29].\",\n \"We show more qualitative model fitting results in Fig.\",\n \"REF .\",\n \"Next, we provide more details on the fitting algorithm used.\",\n \"Fitting: We paint the face region of each model's template mesh to create a mask as shown in Fig.\",\n \"REF .\",\n \"We use these masked regions of the models for fitting the models to head scans.\",\n \"To initialize, we mark 8 landmarks (eye corners, nose, lip corners, and chin) on the template mesh and rigidly align the template to each ground truth scan using Procrustes algorithm.\",\n \"We allow for translation, rotation, and scaling.\",\n \"We use the rigid alignment as initialization and optimize for the parameters of each model using a modified iterative closest point (ICP) algorithm which also updates the model parameters.\",\n \"The fitting algorithm maintains the initial scale and translation but optimizes for rotation.\",\n \"In each optimization step, we first compute the correspondences as the closest points from the masked region in the model, shown in Fig.\",\n \"REF , to the scan data.\",\n \"We compute the loss as shown in Eq.\",\n \"(REF ) and update the model parameters along with Euler angles for global rotation.\",\n \"We run the following optimization program up to convergence to fit the models to our scans: $ \\\\operatornamewithlimits{argmin}_{\\\\theta ,K,\\\\alpha ,\\\\beta ,\\\\gamma }& \\\\,\\\\, \\\\sum _{i=1}^N \\\\,\\\\,(\\\\big \\\\Vert sR(\\\\alpha ,\\\\beta ,\\\\gamma )x_i(\\\\theta ) + t - x_i \\\\big \\\\Vert _2 \\\\nonumber \\\\\\\\ &\\\\quad \\\\quad + \\\\big \\\\Vert Kc_i(\\\\theta ) - c_i \\\\big \\\\Vert _2 ) \\\\nonumber \\\\\\\\ &+ w_l \\\\sum _{j=1}^L \\\\big \\\\Vert sR(\\\\alpha ,\\\\beta ,\\\\gamma )l_j(\\\\theta ) + t - l_j\\\\big \\\\Vert _2 \\\\,,$ where, $x_i(\\\\theta ) \\\\in \\\\mathbb {R}^3$ is a vertex $i$ in the masked region of the model (containing $N$ vertices), $x_i$ is the point on the scan data corresponding to $x_i(\\\\theta )$ ; $R(\\\\alpha ,\\\\beta ,\\\\gamma ) \\\\in \\\\mathbb {R}^{3 \\\\times 3}$ is the global rotation matrix computed using the Euler angles, $\\\\alpha ,\\\\beta ,\\\\text{and } \\\\gamma $ ($\\\\in \\\\mathbb {R}$ ); $s \\\\in \\\\mathbb {R} $ , $t \\\\in \\\\mathbb {R}^3$ are the global scale and translation computed during initialization; $c_i(\\\\theta ) \\\\in \\\\mathbb {R}^3$ , $c_i \\\\in \\\\mathbb {R}^3$ are the colors at the vertices $x_i(\\\\theta )$ and $x_i$ respectively; $l_j \\\\in \\\\mathbb {R}^3$ and $l_j(\\\\theta ) \\\\in \\\\mathbb {R}^3$ are the $L(=8)$ ground truth and model landmarks respectively, as described earlier; and $K \\\\in R^{3\\\\times 3}$ is a diagonal matrix.\",\n \"We set $w_l=0.1$ during the fitting process.\",\n \"Note that the color loss is only enforced for BFM, as FLAME does not model colors.\",\n \"Further, as the color intensities of BFM and our scans differ, we globally scale the color values using channel-specific scalars arranged as a diagonal matrix $K$ which we optimize for along with the model parameters.\",\n \"Evaluation details: We describe the evaluation metrics in Sec.\",\n \"4.4 of the main paper.\",\n \"Here, we present details about the masks used to evaluate these metrics.\",\n \"Face region: We manually paint face masks on the ground truth scans to obtain the ground truth masks.\",\n \"We exclude the mouth interior of the ground truth scans.\",\n \"We copy this mask to the i3DMM fits.\",\n \"We do that by annotating a vertex in i3DMM reconstruction if the nearest point from that vertex on the ground truth scan is in the masked region.\",\n \"We show the face masks used to fit BFM and FLAME to ground truth scans in Fig.\",\n \"REF .\",\n \"We obtain the symmetric metrics presented in Table.\",\n \"1 of the main paper for the face region in the following way.\",\n \"In one direction, we compute the errors from masked region of ground truth to the closest points on the (unmasked) models fit to the scan.\",\n \"In the other direction, we compute the errors from the masked region of the models to the (unmasked) ground truth scan.\",\n \"We compute errors between the masked regions of one mesh to unmasked regions of other mesh to avoid large error metrics due to annotation mistakes during manual mask painting.\",\n \"Full Head: We only fit to FLAME full head model as BFM does not model the entire head.\",\n \"We remove the neck region from FLAME as shown in Fig.\",\n \"REF as the ground truth head scans do not have neck regions.\",\n \"We also remove the vertices used to close the neck from ground truth as FLAME has a hole in the mesh at the neck.\",\n \"We compute the metrics as we do for the face region between these two full head meshes.\",\n \"We report the full head metrics for our model in the entire head region, including the closed hole at the neck mesh.\",\n \"Ablative Analysis In Fig.\",\n \"REF , we show additional qualitative results for the ablative analysis.\",\n \"We also show the quantitative results for full head i3DMM fit in comparison to i3DMM variants.\",\n \"We compare the four models that evaluate our design choices in Table REF .\",\n \"The error metrics are computed for the face region using manually annotated face masks as described in Sec.\",\n \"REF .\",\n \"We only evaluate the face region, as the ground truth for hair is noisy, and small quantitative differences are not very indicative of degradation in quality.\",\n \"Although the geometric reconstruction accuracy is marginally better without the landmark supervision loss, as compared to i3DMM, the color reconstruction accuracy of i3DMM is higher.\",\n \"Also, as mentioned in the main paper, Sec.\",\n \"4.3, texture transfer results around the ear regions with landmark supervision loss are worse compared to i3DMM.\",\n \"Table: Quantitative results for ablation study.\",\n \"The columns, from left to right, show results obtained with uniform sampling for SDF instead of landmark-based sampling, without sparse pairwise landmark supervision loss, independently training for representing geometry and color, final model (i3DMM), and ground truth.\",\n \"Correspondence Evaluation We quantitatively evaluate the correspondences predicted by i3DMM by using the FLAME and BFM fits as ground truth correspondences.\",\n \"To this end, we first find the closest points from the vertices of the (masked) model fits to the i3DMM reconstructions for different scans.\",\n \"We will call these correspondences ground truth annotations here.\",\n \"We use a KD tree algorithm for efficiency.\",\n \"The masked face region contains 26370 vertices for BFM, and 1873 vertices for FLAME.\",\n \"We also transfer the annotations for one i3DMM reconstruction, to all the other reconstructions.\",\n \"This process is same as that described in annotation transfer application (see Sec.\",\n \"4.5 of main paper).\",\n \"We compute the correspondence error as the average of error between the transferred and the ground truth annotations.\",\n \"Note that we transfer annotations from one i3DMM fit to every other i3DMM fit.\",\n \"Therefore, we compute a symmetric error metric.\",\n \"The resulting distribution of error is shown in Fig.\",\n \"REF (evaluating with BFM as ground truth is plotted in red, while FLAME is plotted in blue).\",\n \"The mean and median of errors for BFM is $5.08$ mm and $3.02$ mm respectively.\",\n \"The mean and median of errors for FLAME is $2.36$ mm and $1.83$ mm respectively.\",\n \"Note that, this error does not only capture the error in i3DMM's correspondence predictions but also the error in registrations of FLAME and BFM fits, see Table.\",\n \"1 in the main paper.\",\n \"Figure: Distribution of errors in correspondences predicted by i3DMM computed using FLAME fits (face) as ground truth (blue), and using BFM fits (face) as ground truth (red).\",\n \"Applications Full Head Completion Figure: Completing face scans (left) with different hair styles (short hairstyle (middle-left), long hairstyle (middle-right), and cap (right)) using i3DMM as prior.We use the i3DMM prior to complete face scans with different hairstyles as shown in Fig.\",\n \"REF .\",\n \"To obtain the face meshes from our head meshes, we delete all the vertices that are outside a sphere around the the tip of the nose.\",\n \"We learn the latent vector for the given test scan using the SDF samples of the face mesh, as described in Sec.\",\n \"4.1 of the main paper.\",\n \"Additionally, we semantically control the hair style of the completed scan by adding a regularizer that enforces the learned $\\\\mathbf {z}_\\\\text{geoH}$ and $\\\\mathbf {z}_\\\\text{colH}$ to be close to the hairstyle latent vectors learned during training.\",\n \"It can be inferred from the results in Fig.\",\n \"REF that our model learns a good prior distribution, generating plausible heads for the given faces.\",\n \"i3DMM offers user-guided control for head completion and can be used to turn existing face-only 3DMMs into full head 3DMMs.\",\n \"Further, our method can also be used as a prior distribution for applications such as monocular 3D reconstruction [54].\",\n \"Figure: Additional annotation transfer results.\",\n \"Top: segmentation transfer front view.\",\n \"Middle: segmentation transfer side view.\",\n \"Bottom: landmark transfer.\",\n \"Left column shows i3DMM reconstructions with manual annotations.\",\n \"Right part shows annotations transferred to head scans using i3DMM.\",\n \"Annotation Transfer We show more results for annotation transfer described in Sec.\",\n \"4.5 of the main paper, in Fig.\",\n \"REF .\",\n \"Figure: Additional comparison results between i3DMM (full head) fits, BFM (face) fit, and FLAME (full head and face) fits.Figure: Additional ablation results.\",\n \"From left to right, i3DMM without landmark-based sampling, i3DMM without landmark supervision, i3DMM with independent color and geometry training, i3DMM, and ground truth results are shown.\",\n \"Visualization Details The output of the i3DMM is a signed distance field.\",\n \"Generally, marching cubes algorithm is used to reconstruct the surface from a SDF.\",\n \"However, based on the resolution used, marching cubes algorithm introduces unpleasant surface artifacts.\",\n \"To avoid these artifacts, we used a sphere tracer to render our results.\",\n \"We used the Blinn-Phong reflection model to shade our geometry results.\",\n \"We apply a gamma correction with $\\\\gamma =0.65$ for the color renders.\",\n \"We use Redner [30] for rendering meshes.\"\n ],\n [\n \"Model Visualization\",\n \"Our implicit 3D morphable model models the identity, expression, and hairstyle components of the geometry; as well as the identity and hairstyle components of the color of the head with independent parameteric controls.\",\n \"We visualize these components in Fig.\",\n \"REF .\",\n \"We perform PCA on the identity geometry and color spaces in order to compute the principal components.\",\n \"The expression component is visualized by moving along the directions of the training expressions, since they are semantically well defined.\",\n \"As we model hairstyles that include caps, we show the joint space of geometry and color for hairstyles.\",\n \"Note that the hairstyle geometry can only take four discrete values – short, long, cap1, or cap2.\",\n \"Any variation within these categories is modelled by the identity-geometry component.\",\n \"Similarly, hairstyle color can only take the values – nocap, cap1, or cap2.\",\n \"The color of hair without any cap is determined by the identity-color component.\"\n ],\n [\n \"Experiments\",\n \"Here, we provide more details on the evaluations in the main paper, and include further evaluations.\"\n ],\n [\n \"Sampling i3DMM\",\n \"One of the important features of a 3D Morphable Model is the ability to randomly sample shapes in the parametric space.\",\n \"This has been used for generating synthetic data for training CNNs [23], [43], [47].\",\n \"We achieve sampling in i3DMM by performing principal component analysis (PCA) over the training latent codes for color-identity, geometry-identity, and expressions.\",\n \"We weigh the singular values using a Gaussian random variable $\\\\mathcal {N}(0,0.1)$ for color-identity, and geometry-identity, and $\\\\mathcal {N}(0,0.25)$ for expressions.\",\n \"Since the latent codes for hair shapes and colors can take very limited numbers of values and are very well defined semantically, we sample these from their training values.\",\n \"We show several results in the supplemental video.\",\n \"As can be seen, our model is biased towards generating male heads.\",\n \"This is likely due to our gender-biased training dataset with 46 males and 18 females.\",\n \"While this does not lead to any clear loss of quality when fitting to female test scans, see Fig.\",\n \"6 in the main paper, it might lead to biased quality of results in other problems, for eg., if random samples from the model was used for training another network.\"\n ],\n [\n \"Comparisons\",\n \"As mentioned in Sec.\",\n \"4.4 of the main paper, we compare our model to two existing models, BFM [38] and FLAME [29].\",\n \"We show more qualitative model fitting results in Fig.\",\n \"REF .\",\n \"Next, we provide more details on the fitting algorithm used.\",\n \"Fitting: We paint the face region of each model's template mesh to create a mask as shown in Fig.\",\n \"REF .\",\n \"We use these masked regions of the models for fitting the models to head scans.\",\n \"To initialize, we mark 8 landmarks (eye corners, nose, lip corners, and chin) on the template mesh and rigidly align the template to each ground truth scan using Procrustes algorithm.\",\n \"We allow for translation, rotation, and scaling.\",\n \"We use the rigid alignment as initialization and optimize for the parameters of each model using a modified iterative closest point (ICP) algorithm which also updates the model parameters.\",\n \"The fitting algorithm maintains the initial scale and translation but optimizes for rotation.\",\n \"In each optimization step, we first compute the correspondences as the closest points from the masked region in the model, shown in Fig.\",\n \"REF , to the scan data.\",\n \"We compute the loss as shown in Eq.\",\n \"(REF ) and update the model parameters along with Euler angles for global rotation.\",\n \"We run the following optimization program up to convergence to fit the models to our scans: $ \\\\operatornamewithlimits{argmin}_{\\\\theta ,K,\\\\alpha ,\\\\beta ,\\\\gamma }& \\\\,\\\\, \\\\sum _{i=1}^N \\\\,\\\\,(\\\\big \\\\Vert sR(\\\\alpha ,\\\\beta ,\\\\gamma )x_i(\\\\theta ) + t - x_i \\\\big \\\\Vert _2 \\\\nonumber \\\\\\\\ &\\\\quad \\\\quad + \\\\big \\\\Vert Kc_i(\\\\theta ) - c_i \\\\big \\\\Vert _2 ) \\\\nonumber \\\\\\\\ &+ w_l \\\\sum _{j=1}^L \\\\big \\\\Vert sR(\\\\alpha ,\\\\beta ,\\\\gamma )l_j(\\\\theta ) + t - l_j\\\\big \\\\Vert _2 \\\\,,$ where, $x_i(\\\\theta ) \\\\in \\\\mathbb {R}^3$ is a vertex $i$ in the masked region of the model (containing $N$ vertices), $x_i$ is the point on the scan data corresponding to $x_i(\\\\theta )$ ; $R(\\\\alpha ,\\\\beta ,\\\\gamma ) \\\\in \\\\mathbb {R}^{3 \\\\times 3}$ is the global rotation matrix computed using the Euler angles, $\\\\alpha ,\\\\beta ,\\\\text{and } \\\\gamma $ ($\\\\in \\\\mathbb {R}$ ); $s \\\\in \\\\mathbb {R} $ , $t \\\\in \\\\mathbb {R}^3$ are the global scale and translation computed during initialization; $c_i(\\\\theta ) \\\\in \\\\mathbb {R}^3$ , $c_i \\\\in \\\\mathbb {R}^3$ are the colors at the vertices $x_i(\\\\theta )$ and $x_i$ respectively; $l_j \\\\in \\\\mathbb {R}^3$ and $l_j(\\\\theta ) \\\\in \\\\mathbb {R}^3$ are the $L(=8)$ ground truth and model landmarks respectively, as described earlier; and $K \\\\in R^{3\\\\times 3}$ is a diagonal matrix.\",\n \"We set $w_l=0.1$ during the fitting process.\",\n \"Note that the color loss is only enforced for BFM, as FLAME does not model colors.\",\n \"Further, as the color intensities of BFM and our scans differ, we globally scale the color values using channel-specific scalars arranged as a diagonal matrix $K$ which we optimize for along with the model parameters.\",\n \"Evaluation details: We describe the evaluation metrics in Sec.\",\n \"4.4 of the main paper.\",\n \"Here, we present details about the masks used to evaluate these metrics.\",\n \"Face region: We manually paint face masks on the ground truth scans to obtain the ground truth masks.\",\n \"We exclude the mouth interior of the ground truth scans.\",\n \"We copy this mask to the i3DMM fits.\",\n \"We do that by annotating a vertex in i3DMM reconstruction if the nearest point from that vertex on the ground truth scan is in the masked region.\",\n \"We show the face masks used to fit BFM and FLAME to ground truth scans in Fig.\",\n \"REF .\",\n \"We obtain the symmetric metrics presented in Table.\",\n \"1 of the main paper for the face region in the following way.\",\n \"In one direction, we compute the errors from masked region of ground truth to the closest points on the (unmasked) models fit to the scan.\",\n \"In the other direction, we compute the errors from the masked region of the models to the (unmasked) ground truth scan.\",\n \"We compute errors between the masked regions of one mesh to unmasked regions of other mesh to avoid large error metrics due to annotation mistakes during manual mask painting.\",\n \"Full Head: We only fit to FLAME full head model as BFM does not model the entire head.\",\n \"We remove the neck region from FLAME as shown in Fig.\",\n \"REF as the ground truth head scans do not have neck regions.\",\n \"We also remove the vertices used to close the neck from ground truth as FLAME has a hole in the mesh at the neck.\",\n \"We compute the metrics as we do for the face region between these two full head meshes.\",\n \"We report the full head metrics for our model in the entire head region, including the closed hole at the neck mesh.\"\n ],\n [\n \"Ablative Analysis\",\n \"In Fig.\",\n \"REF , we show additional qualitative results for the ablative analysis.\",\n \"We also show the quantitative results for full head i3DMM fit in comparison to i3DMM variants.\",\n \"We compare the four models that evaluate our design choices in Table REF .\",\n \"The error metrics are computed for the face region using manually annotated face masks as described in Sec.\",\n \"REF .\",\n \"We only evaluate the face region, as the ground truth for hair is noisy, and small quantitative differences are not very indicative of degradation in quality.\",\n \"Although the geometric reconstruction accuracy is marginally better without the landmark supervision loss, as compared to i3DMM, the color reconstruction accuracy of i3DMM is higher.\",\n \"Also, as mentioned in the main paper, Sec.\",\n \"4.3, texture transfer results around the ear regions with landmark supervision loss are worse compared to i3DMM.\",\n \"Table: Quantitative results for ablation study.\",\n \"The columns, from left to right, show results obtained with uniform sampling for SDF instead of landmark-based sampling, without sparse pairwise landmark supervision loss, independently training for representing geometry and color, final model (i3DMM), and ground truth.\"\n ],\n [\n \"Correspondence Evaluation\",\n \"We quantitatively evaluate the correspondences predicted by i3DMM by using the FLAME and BFM fits as ground truth correspondences.\",\n \"To this end, we first find the closest points from the vertices of the (masked) model fits to the i3DMM reconstructions for different scans.\",\n \"We will call these correspondences ground truth annotations here.\",\n \"We use a KD tree algorithm for efficiency.\",\n \"The masked face region contains 26370 vertices for BFM, and 1873 vertices for FLAME.\",\n \"We also transfer the annotations for one i3DMM reconstruction, to all the other reconstructions.\",\n \"This process is same as that described in annotation transfer application (see Sec.\",\n \"4.5 of main paper).\",\n \"We compute the correspondence error as the average of error between the transferred and the ground truth annotations.\",\n \"Note that we transfer annotations from one i3DMM fit to every other i3DMM fit.\",\n \"Therefore, we compute a symmetric error metric.\",\n \"The resulting distribution of error is shown in Fig.\",\n \"REF (evaluating with BFM as ground truth is plotted in red, while FLAME is plotted in blue).\",\n \"The mean and median of errors for BFM is $5.08$ mm and $3.02$ mm respectively.\",\n \"The mean and median of errors for FLAME is $2.36$ mm and $1.83$ mm respectively.\",\n \"Note that, this error does not only capture the error in i3DMM's correspondence predictions but also the error in registrations of FLAME and BFM fits, see Table.\",\n \"1 in the main paper.\",\n \"Figure: Distribution of errors in correspondences predicted by i3DMM computed using FLAME fits (face) as ground truth (blue), and using BFM fits (face) as ground truth (red).\"\n ],\n [\n \"Full Head Completion\",\n \"We use the i3DMM prior to complete face scans with different hairstyles as shown in Fig.\",\n \"REF .\",\n \"To obtain the face meshes from our head meshes, we delete all the vertices that are outside a sphere around the the tip of the nose.\",\n \"We learn the latent vector for the given test scan using the SDF samples of the face mesh, as described in Sec.\",\n \"4.1 of the main paper.\",\n \"Additionally, we semantically control the hair style of the completed scan by adding a regularizer that enforces the learned $\\\\mathbf {z}_\\\\text{geoH}$ and $\\\\mathbf {z}_\\\\text{colH}$ to be close to the hairstyle latent vectors learned during training.\",\n \"It can be inferred from the results in Fig.\",\n \"REF that our model learns a good prior distribution, generating plausible heads for the given faces.\",\n \"i3DMM offers user-guided control for head completion and can be used to turn existing face-only 3DMMs into full head 3DMMs.\",\n \"Further, our method can also be used as a prior distribution for applications such as monocular 3D reconstruction [54].\",\n \"Figure: Additional annotation transfer results.\",\n \"Top: segmentation transfer front view.\",\n \"Middle: segmentation transfer side view.\",\n \"Bottom: landmark transfer.\",\n \"Left column shows i3DMM reconstructions with manual annotations.\",\n \"Right part shows annotations transferred to head scans using i3DMM.\"\n ],\n [\n \"Annotation Transfer\",\n \"We show more results for annotation transfer described in Sec.\",\n \"4.5 of the main paper, in Fig.\",\n \"REF .\",\n \"Figure: Additional comparison results between i3DMM (full head) fits, BFM (face) fit, and FLAME (full head and face) fits.Figure: Additional ablation results.\",\n \"From left to right, i3DMM without landmark-based sampling, i3DMM without landmark supervision, i3DMM with independent color and geometry training, i3DMM, and ground truth results are shown.\"\n ],\n [\n \"Visualization Details\",\n \"The output of the i3DMM is a signed distance field.\",\n \"Generally, marching cubes algorithm is used to reconstruct the surface from a SDF.\",\n \"However, based on the resolution used, marching cubes algorithm introduces unpleasant surface artifacts.\",\n \"To avoid these artifacts, we used a sphere tracer to render our results.\",\n \"We used the Blinn-Phong reflection model to shade our geometry results.\",\n \"We apply a gamma correction with $\\\\gamma =0.65$ for the color renders.\",\n \"We use Redner [30] for rendering meshes.\"\n ]\n]"},"id":{"kind":"string","value":"2011.14143"}}},{"rowIdx":2798,"cells":{"text":{"kind":"list like","value":[["EdgeBERT: Sentence-Level Energy Optimizations for Latency-Aware\n Multi-Task NLP Inference"],["Abstract Transformer-based language models such as BERT provide significant accuracy improvement for a multitude of natural language processing (NLP) tasks.","However, their hefty computational and memory demands make them challenging to deploy to resource-constrained edge platforms with strict latency requirements.","We present EdgeBERT, an in-depth algorithm-hardware co-design for latency-aware energy optimization for multi-task NLP.","EdgeBERT employs entropy-based early exit predication in order to perform dynamic voltage-frequency scaling (DVFS), at a sentence granularity, for minimal energy consumption while adhering to a prescribed target latency.","Computation and memory footprint overheads are further alleviated by employing a calibrated combination of adaptive attention span, selective network pruning, and floating-point quantization.","Furthermore, in order to maximize the synergistic benefits of these algorithms in always-on and intermediate edge computing settings, we specialize a 12nm scalable hardware accelerator system, integrating a fast-switching low-dropout voltage regulator (LDO), an all-digital phase-locked loop (ADPLL), as well as, high-density embedded non-volatile memories (eNVMs) wherein the sparse floating-point bit encodings of the shared multi-task parameters are carefully stored.","Altogether, latency-aware multi-task NLP inference acceleration on the EdgeBERT hardware system generates up to 7x, 2.5x, and 53x lower energy compared to the conventional inference without early stopping, the latency-unbounded early exit approach, and CUDA adaptations on an Nvidia Jetson Tegra X2 mobile GPU, respectively."],["Introduction","Transformer-based networks trained with large multi-domain datasets have unlocked a series of breakthroughs in natural language learning and representation.","A major catalyst of this success is the Bidirectional Encoder Representations from Transformers technique, or BERT [16], which substantially advanced nuance and context understanding.","Its pre-training strategy, which consists of learning intentionally hidden sections of text, have proven beneficial for several downstream natural language processing (NLP) tasks.","BERT has sparked leading-edge performance in NLP leaderboards [78], [58], and it is now applied at a global scale in web search engines [52] with marked improvements in the quality of query results.","Figure: (a) Conventional BERT inference, (b) Conventional latency-unbounded BERT inference with early exit.","(c) Proposed latency-bounded inference.","The entropy result from the first layer is used to auto-adjust the accelerator supply voltage and clock frequency for energy-optimal operation while meeting an application end-to-end latency target.Advances in NLP models are also fueling the growth of intelligent virtual assistants, which leverage NLP to implement interactive voice interfaces.","Currently, these applications are offloaded from the edge device to the cloud.","However, they are naturally better suited to deployment on edge devices, where personal data can be kept private and the round trip latency to the cloud is removed.","However, the impressive performance of BERT comes with a heavy compute and memory cost, which makes on-device inference prohibitive.","Most significantly, the BERT base model consumes a staggering 432 MB of memory in native 32-bit floating-point (FP32).","Therefore, the goal of deploying BERT on edge/mobile devices is challenging and requires tight co-design of the BERT model optimizations with dedicated hardware acceleration and memory system design.","The constraints on mobile can be quite different to the datacenter scenario, where BERT has been mainly deployed to date.","Firstly, since we are dealing with user input, we need to meet real time throughput requirements to prevent a noticeable lag to the user.","Secondly, energy consumption is a critical concern on mobile devices, both for the model inference and also the associated data movement cost.","A number of prior works have been proposed to reduce BERT storage and computation overheads [21].","In fact, most of the compression techniques (weight pruning [49], distillation [63], quantization [89], [68]) originally proposed for convolutional and recurrent neural networks (CNNs, RNNs) have been independently applied to Transformer-based DNNs.","In this work, we present EdgeBERT, a principled latency-driven approach to accelerate NLP workloads with minimal energy consumption via early exit prediction, dynamic voltage-frequency scaling (DFVS), and non-volatile memory bitmask encoding of the shared word embeddings.","In conventional BERT inference (Fig.","REF a), the final classification result is generated by the last Transformer layer.","Early exit mechanisms [87], [90], [73], [65] (Fig.","REF (b)) have been proposed to reduce the average energy and latency.","The early exit entropy, which is a probabilistic measure of the classification confidence, is evaluated at the output of each computed Transformer layer and the inference exits when the entropy value falls below a pre-defined threshold.","While this approach can appreciably reduce computation and energy costs, the achieved latency can vary drastically from one input sentence to another, potentially violating the strict real time latency constraint of the application.","In contrast, EdgeBERT uses this upper-bound latency and the target entropy as optimization constraints, and then dynamically auto-adjusts the accelerator supply voltage and clock frequency to minimize energy consumption (Fig.","REF (c)), while meeting the real time throughput requirement.","Since energy scales quadratically with $V_{DD}$ and linearly with the number of computation cycles, our DVFS algorithm finds the lowest possible frequency/voltage, while also minimizing the total number of FLOPs via adaptive attention span predication.","While the benefits of early exit and attention predications can be reaped on commodity GPUs, we unlock additional energy savings by co-designing the hardware datapaths.","Specifically, we exploit these algorithmic optimizations in the EdgeBERT accelerator system, which integrates a fast-switching low-dropout (LDO) voltage regulator and an all-digital phase-locked loop (ADPLL) for DVFS adjustments.","The EdgeBERT accelerator uses bit-mask encoding for compressed sparse computations, while optimizing key operations (entropy assessment, layer normalization, softmax and attention masking) for numerical stability and energy efficiency.","Furthermore, edge/IoT devices operate intermittently which motivates powering down as much as possible.","The model's weights, typically stored in on-chip SRAMs, either have to be reloaded from DRAM each wake up cycle or the on-chip SRAMs storing the weights must be kept on, wasting leakage power [39].","Embedded non-volatile memories (eNVMs), which have shown considerable progress in recent years, offer great promise, if used judiciously, to eliminate the power penalty associated with intermittent operation.","For this purpose, we perform monte-carlo fault injection simulations to identify robust and viable eNVM structures for storing the shared NLP multi-task parameters with bitmask encoding.","Our resulting eNVM configuration significantly alleviates the energy and latency costs associated with multi-task intermediate computing by as much as 66,000$\\times $ and 50$\\times $ , respectively.","Figure: Comparison between (a) BERT, and (b) ALBERT base models.","ALBERT uses a smaller embedding size and its Transformer encoder layers share the same parameters.Altogether, EdgeBERT generates on average up to 7$\\times $ , and 2.5$\\times $ per-sentence energy savings compared to the conventional BERT inference, and latency-unaware early exit approaches, respectively.","This paper therefore makes the following contributions: We propose EdgeBERT, a novel algorithm-hardware co-design approach to enable latency-bound NLP workloads on resource-constrained embedded devices.","Recognizing that BERT word embeddings are shared across NLP tasks, we significantly alleviate off-chip communication costs by identifying viable and robust multi-level eNVM structures for storing the multi-task word embeddings.","Leveraging the insights from this broad analysis, we propose and design a 12nm accelerator that integrates a fast-switching LDO, an ADPLL, and a compressed sparse hardware accelerator that efficiently computes the DVFS, entropy, and adaptive attention span predication algorithms and other key Transformer operations using specialized datapaths.","We evaluate the energy consumption of latency-bound inference on four NLP tasks, and find that the EdgeBERT hardware accelerator system generates up to 7$\\times $ , 2.5$\\times $ , and 53$\\times $ lower energy compared to the unoptimized baseline inference without early exit, the conventional latency-unaware early exit approach, and CUDA adaptations on an Nvidia Jetson Tegra X2 mobile GPU respectively.","The General Language Understanding Evaluation (GLUE) benchmark is the most widely used tool to evaluate NLP performance.","It consists of nine English sentence understanding tasks covering three categories: Single-Sentence, Similarity and Paraphrase, and Inference [78].","This collection of datasets is specifically designed to favor models that can adapt to a variety of NLP tasks.","To validate the robustness and generalization performance of the EdgeBERT methodology, we conduct our evaluation on the four GLUE tasks with the largest corpora, which cover all three GLUE categories: SST-2 (Single-Sentence), QQP (Similarity and Paraphrase), and QNLI and MNLI (Inference)."],["Variations of BERT","Since the advent of BERT with 110M parameters, a number of variants were proposed to alleviate its memory consumption or to further improve its prediction metrics.","RoBERTa [44] generalizes better on several GLUE tasks by training on significantly more data, and for a longer amount of time, but remains as computationally intensive as BERT.","DistilBERT [63] and MobileBERT [70] leverage knowledge distillation to reduce BERT size by 1.7$\\times $ and 4.3$\\times $ , respectively, with iso-accuracy.","SqueezeBERT [29] substitutes several operations in the Transformer encoder with 1D grouped convolutions achieving 4$\\times $ speedup while being 2$\\times $ smaller.","Q8BERT [89] employs a symmetric linear quantization scheme for quantizing both weights and activations into 8-bit integers.","In contrast, in this work we leverage the higher dynamic range of floating-point encodings for greater quantization resilience.","ALBERT [37] yields the smallest footprint to date for a compressed BERT variant with only 12M parameters, with competitive accuracy on the GLUE benchmarks.","Fig.","REF summarizes the key differences between the ALBERT model and the base BERT model.","While each of BERT's twelve encoder layers have a unique set of weights, ALBERT's encoder layers instead share and reuse the same parameters – resulting in significant compression.","The encoder block in both models has the same architecture as the legacy Transformer network [75], but with twelve parallel self-attention heads.","Moreover, ALBERT employs a smaller embedding size (128 vs. 768) thanks to factorization in the embedding layer.","In this work, we adopt the ALBERT variant as an efficient baseline.","This work further pursues strategies to reduce latency and storage requirements to suit embedded hardware platform constraints."],["Alleviating Transformer memory and computation costs","An accelerator's energy consumption can be abstracted as: $Energy\\propto \\alpha C V_{DD}^2 N_{cycles}$ where $\\alpha $ , $C$ , $V_{DD}$ and $N_{cycles}$ are the switching activity factor, the effective wire and device capacitance, the supply voltage, and the required number of clock cycles to complete the inference, respectively.","While the DVFS algorithm (Sec.","REF ) lowers the energy quadratically by bringing $V_{DD}$ down to the lowest optimal voltage, in this section, we explore avenues to further reduce the energy by minimizing $\\alpha $ , $C$ , and $N_{cycles}$ .","For this purpose, we carefully incorporate into the multi-task ALBERT inference: 1) adaptive attention span predication and early exit which reduce $N_{cycles}$ ; 2) network pruning, which ultimately reduces $\\alpha $ ; and 3) floating-point quantization helping decrease $C$ , altogether with minimal accuracy degradation.","While briefly describing these optimizations individually in this section, we provide a reasoned methodology for applying them to the ALBERT model, as shown in Fig.","REF ."],["Entropy-based Early Exit","The motivation behind early exit (EE) is to match linguistically complex sentences with larger (or deeper) models and simple sentences with smaller (or shallower) models [87], [13].","This is typically done by adding a lightweight classifier at the output of the Transformer layer so that a given input can exit inference earlier or later in the stack, depending on its structural and contextual complexity.","The classifier computes and compares the entropy of an output distribution with a preset “confidence\" threshold, $E_{T}$ , in order to assess whether the prediction should exit or continue inference in the next Transformer encoder layer.","The entropy metric quantifies the amount of uncertainty in the data.","Smaller entropy values at a Transformer layer output implies greater confidence in the correctness of the classification result.","The entropy H on sample x is estimated as: $\\leavevmode \\xbox {resizebox}{\\XMLaddatt {width}{0.0pt}H(x) = - \\sum p(x)\\log p(x)= \\ln (\\sum \\limits _{k=1}^{n} e^{x_k}) -\\frac{\\sum \\limits _{k=1}^{n} x_k e^{x_k}}{\\sum \\limits _{k=1}^{n} e^{x_k}}}$ The early exit condition is met when $H(x)$ $<$ $E_{T}$ .","Therefore, the larger $E_{T}$ becomes, the earlier the sample will exit (i.e.","$N_{cycles}$ becomes smaller) with potentially lower accuracy.","In this work, we modify the conventional EE inference approach by predicting the early exit layer from the output of the first Transformer layer in order to run the rest of the network computation in an energy-optimal and latency-bounded manner (Sec.",")."],["Adaptive Attention Span","The attention mechanism [8] is a powerful technique that allows neural networks to emphasize the most relevant tokens of information when making predictions.","The base ALBERT model contains up to twelve parallel attention heads – each learning their own saliency weights on the full length of the encoder input.","However, depending on the complexity of the task, many heads can be redundant and can be safely removed without impacting accuracy [51].","Furthermore, the cost of computing the attention mechanism scales quadratically with the sequence length.","Therefore, there is potentially a meaningful amount of computations and energy to be saved in optimizing the inspection reach of every attention head.","In the quest to avoid needless attention computations in ALBERT, a learnable parameter z is introduced in the datapath of each self-attention head in order to find its own optimal attention span [69].","The parameter z is mapped to a masking function with a [0, 1] output range, as shown in Fig.","REF .","The masked span is then applied on the attention weights in order to re-modulate their saliencies.","The optimal span is automatically learned during the fine-tuning process by adding back the average loss from the reduced span to the training cross-entropy loss.","The maximum sentence length for fine-tuning the GLUE tasks is 128.","As a result, shorter sentences are typically zero-padded to 128 during the tokenization pre-processing.","Table REF shows the final attention span learned by each self-attention head when fine-tuning with the adaptive attention span technique.","Strikingly, the twelve parallel self-attention heads in ALBERT do not need to inspect their inputs at maximum span.","In fact, more than half of the attention heads, 8 for MNLI and QQP and 7 for SST-2 and QNLI, can be completely turned off with minimal accuracy loss.","This amounts to a 1.22$\\times $ and 1.18$\\times $ reduction, respectively, in the total number of FLOPS (which linearly correlates with $N_{cycles}$ ) required for single-batch inference.","Table: Learned spans of every attention head in ALBERT.Baseline Acc: MNLI=85.16, QQP=90.76, SST-2=92.20, QNLI=89.48The twelve attention spans, learned during fine-tuning, are written to registers in the EdgeBERT accelerator in the form of a 128-wide vector – in order to predicate on the inference computation of the multi-head attention.","Notably, all the computations inside any of the twelve attention head units can be effectively skipped in case its associated attention span mask is 100% null.","The EdgeBERT accelerator takes advantage of this observation in a proactive manner during inference in the custom hardware (Sec.","REF )."],["Network Pruning","The EdgeBERT hardware accelerator (Sec. )","executes sparse computations and saves energy by gating MACs whenever input operands are null.","Therefore, the extent to which we can prune the ALBERT model, without appreciable accuracy loss, determines the overall accelerator energy efficiency.","In this work, we consider both movement pruning [64] and the well-known magnitude pruning [25] methods.","Movement pruning is a first-order pruning technique that is applied during model fine-tuning which eliminates weights that are dynamically shrinking towards 0 (i.e., according to the movement of the values).","In some cases, magnitude pruning may be a sub-optimal method to use during transfer learning, as pre-trained weights closer to zero may have a high chance of being eliminated regardless of the fine-tuning requirement.","We observe that movement pruning particularly outperforms magnitude-based pruning in high sparsity regimes, as each individual remaining weight becomes more important to learn the task at hand.","Therefore, choosing between the two pruning techniques would depend on the per-task tolerance to increasing sparsity levels.","We note that magnitude pruning is always applied to the ALBERT embedding layer in order to enforce uniformity in the data during multi-domain on-chip acceleration – as using movement pruning on the embedding layer would make its weights unique for each NLP domain, thereby forgoing opportunities for data reuse in hardware."],["Floating-Point Quantization","DNN algorithmic resilience allows for parameters to be represented in lower bit precision without accuracy loss.","Fixed-point or integer quantization techniques, commonly adopted in CNN models, suffer from limited range and may be inadequate for NLP models, whose weights can be more than an order of magnitude larger [72].","This phenomenon is owed to layer normalization [7], which is commonly adopted in NLP models and has invariance properties that do not reparameterize the network – unlike batch normalization [30], which produces a weight normalization side effect in CNNs.","In this work, we employ floating-point based quantization, which provides 2$\\times $ higher dynamic range compared to integer datatypes [32].","Both weights and activations are quantized across ALBERT layers to 8-bit precision.","We also performed a search on the optimal exponent bit width to satisfy the dynamic range requirements of the ALBERT model.","Setting the floating-point exponent space to 4 bits within the 8-bit word size, with the exponent being scaled at a per-layer granularity, provided the best accuracy performance across NLP tasks.","In contrast to task-specific encoder weights, word embedding parameters are deliberately fixed during fine-tuning and reused across different NLP tasks.","We seek to avoid the energy and latency costs of reloading the word embeddings from off-chip memory for different tasks by storing these shared parameters in embedded non-volatile memories (eNVMs).","eNVM storage also enables energy-efficient intermittent computing because the embedding weights will be retained if and when the system-on-chip powers off between inferences.","However, despite their compelling storage density and read characteristics, eNVMs exhibit two main drawbacks: potentially high write cost (in terms of energy and latency) and decreased reliability, particularly in multi-level cell (MLC) configurations [15].","Fortunately, the word embeddings are acting as read-only parameters on-chip, which makes them highly suitable for eNVM storage, but previous work highlights the need to study the impacts of faulty, highly-dense ReRAM storage on DNN task accuracy [56].","On the other hand, encoder weights need to be updated when switching across different NLP tasks.","To prevent the energy and latency degradation that would follow from updating the encoder weight values in eNVMs, we map the natural partition of shared and task-specific parameters to eNVMs and SRAMs, respectively [17]."],["eNVM Modeling Methodology","This work specifically considers dense, energy-efficient Resistive RAM (ReRAM) arrays [10], [43] as an on-chip storage solution for shared embedding parameters.","We selected ReRAMs for their relative maturity and demonstrated read characteristics.","However, we note that there is a larger design space of opportunities to be explored with other emerging MLC-capable NVM technologies such as PCM [14], but is beyond the scope of this work.","We evaluate the robustness of storing the 8-bit quantized word embeddings in eNVM storage.","In order to quantify the trade-offs between storage density and task accuracy, we use cell characteristics of 28nm ReRAM programmed with varying number of bits per cell [15], and evaluate 100 fault injection trials per storage configuration to identify robust eNVM storage solutions.","We leverage and extend Ares [59], which is an existing open-source fault injection framework for quantifying the resilience of DNNs.","After pruning, we store non-zero compressed embedding weights using a bitmask-style sparse encoding.","Previous work demonstrates that DNN weight bitmask values are vulnerable to MLC faults, so the bitmask is protectively stored in lower-risk SLC devices, while we experiment with MLC storage for the non-zero data values [56]."],["Optimal eNVM Configuration","Table REF uncovers exceptional resilience to storing word embeddings in MLC ReRAM.","Across many fault injection trials, we observe that MLC2 (ReRAM programmed at 2 bits-per-cell) does not degrade accuracy across multiple tasks, while MLC3 exhibits potentially catastrophic degradation in minimum accuracy and an appreciable decline in average accuracy for the QNLI task, highlighted in bold.","Based on this observation, the EdgeBERT accelerator system leverages MLC2 ReRAMs for word embedding storage (Sec.).","[!t] linenosize= $E_T$ := target entropy input sentence $i = 0$ to $n$ encoder layer l = 1 to 12 $z_l = f(x;\\theta | VDD_{nom}, Freq_{max})$ $entropy(z_l) < E_T$ exit inference Conventional early exit inference"],["EdgeBERT's Latency-Aware Inference","The conventional BERT inference (Algorithm REF ) with early exit (EE) can significantly reduce BERT inference latency.","To further reduce the energy consumption for NLP inference, a latency-aware inference scheme leveraging the EE predictor and dynamic voltage and frequency scaling (DVFS) is proposed to minimize end-to-end per-sentence energy consumption while satisfying the real-time latency target.","[!t] linenosize= $T$ := per-sentence latency target, $E_T$ := entropy target $N_{cycles} := \\text{number of clock cycles to compute the Transformer encoder}$ input sentence $i = 1$ to $n$ encoder layer l = 1 $z_l = f(x;\\theta | VDD_{nom}, Freq_{max})$ $entropy(z_l) < E_T$ exit inference $L_{predict} = LUT(entropy(z_1), E_T)$ $VDD_{opt}, Freq_{opt} = DVFS(L_{predict},T)$ encoder layer $l = 2$ to $L_{predict}$ $z_l = f(x;\\theta | VDD_{opt}, Freq_{opt})$ $entropy(z_l) < E_T$ exit inference exit inference EdgeBERT latency-aware inference.","Computations exit at the predicted exit layer or earlier."],["Methodology","DVFS is a widely used technique to dynamically scale down the voltage and frequency for less computationally intensive workloads.","In the past, DVFS has been widely deployed in commercial CPUs [74], [28] and GPUs [50].","However, these schemes typically adjust the voltage and frequency at a coarse granularity at workload-level.","In the era of AI, DVFS has started to be explored for DNN accelerators [38].","For example, a recent state-of-the-art AI chip has reported per-layer DVFS to save energy [3].","In this work, we explore a fine-grained sentence-level DVFS to reduce the energy consumption for NLP inference while meeting the latency target.","The proposed early exit -based latency-aware inference methodology is illustrated in Algorithm .","The inference of a sentence starts at nominal voltage and maximum frequency, and the entropy value is calculated at the output of the first Transformer encoder layer.","The entropy result is then sent to a trained classifier (EE predictor) to predict which following encoder layer should early exit (e.g.","early exit at encoder layer 6 before the final encoder layer 12).","Based on the predicted early exit layer, the voltage and frequency is scaled down to proper energy-optimal setting for the rest of encoder layers (e.g.","encoder layer 2 to 6) while meeting the latency target for each sentence.","This scheme produces a quadratic reduction in the accelerator power consumption.","In our work, the EE predictor is a ReLU-activated five-layer perceptron neural network with 64 cells in each of the hidden layers.","It takes the entropy of encoder layer 1 as input and forecasts the early exit Transformer layer which has an entropy below the desired threshold.","The neural network architecture of the EE predictor was empirically searched with the goal of minimizing the difference between the predicted and the true entropy-based exit layer.","For this purpose, we constructed parallel training and test datasets containing the entropy values at the output of the 12 Transformer layers during evaluation on the GLUE benchmarks.","The EE predictor is distilled as a lookup table (LUT) leading to negligible one-time (per-sentence) computational overhead.","Furthermore, implementing the EE predictor as a LUT simplifies its hardware operation.","As the neural network based LUT is error-prone, it may predict a higher exit layer than necessary.","Therefore, during the inference, the entropy is checked after each encoder layer for early stopping until the predicted layer.","If the computed entropy becomes lower than the exit threshold before the predicted encoder layer, the inference will terminate at that early exit condition point.","In case the inference reaches the predicted layer, termination occurs even if the entropy at that layer is still higher than the exit threshold in order to not violate timing constraints.","When assessing the impacts of using entropy prediction instead of traditional EE methods, we set a fixed accuracy degradation threshold of 1%, 2%, or 5% (relative to the inference accuracy of the full ALBERT model) and increased the entropy threshold until the accuracy dropped to the desired threshold.","This allowed us to compare energy savings between entropy prediction and conventional EE for a fixed accuracy target.","For the same accuracy threshold, the entropy threshold for entropy prediction was lower than the entropy threshold for conventional EE, leading to a slightly later average exit layer during inference.","However, entropy prediction allows for DVFS since the maximum exit layer is known after the first layer, whereas with the conventional EE appproach, the maximum exit layer is always the final encoder layer.","EdgeBERT latency-aware inference therefore achieves greater energy savings than the conventional EE approach by facilitating DVFS (Sec.","REF ).","Figure: EdgeBERT training and evaluation procedure."],["On-chip DVFS system","To realize fast per-sentence DVFS, the on-chip DVFS system is developed and integrated within EdgeBERT.","The DVFS system includes a DVFS controller, an on-chip synthesizable linear voltage regulator (LDO), and an all-digital PLL (ADPLL).","Compared with the conventional workload-level DVFS [74], the proposed scheme adjusts voltage and frequency at a finer-grained sentence-level granularity.","Based on the predicted early exit layer from the EE predictor, the required run cycles, $N_{cycles}$ , for the rest of the encoder layers before early exit can be known.","And, knowing the frontend elapsed time $T_{elapsed}$ up to the EE predictor within the per-sentence latency target $T$ , the optimal running frequency can be calculated as follows: $Freq_{opt} = N_{cycles} / (T - T_{elapsed}) $ Meanwhile, the corresponding energy-optimal supply voltage, $VDD_{opt}$ , is selected by the DVFS controller to achieve the lowest operational voltage value at $Freq_{opt}$ .","In the EdgeBERT accelerator system, this is done via indexing the look-up table containing the ADPLL frequency/voltage sweep coordinates.","The DVFS is performed for each real-time sentence inference due to its fast response time; the implementation details are shown in Sec.","REF ."],["Algorithmic Synergy","In order to quantify the different tradeoffs, and evaluate the synergistic impact on the model accuracy from the memory and latency optimizations, the eNVM modeling, and the EE predictor, we implemented the training and evaluation procedures illustrated in Fig.","REF on the base of HuggingFace’s Transformers infrastructure [85]."],["Training and Evaluation Procedure","The training methodology consists of two phases.","In the first phase, the model is pruned during fine-tuning: magnitude pruning is applied to the embedding layer and either movement or magnitude pruning is applied to the Transformer encoder layer.","An additional loss term comes from knowledge distillation using the base ALBERT model fine-tuned on the target task as a teacher.","The embeddings and the encoder layer are subject to separate pruning schedules.","At the same time, the attention heads learn their optimal spans.","In the second training phase, we freeze the model's parameters prior to fine-tuning the early exit highway off-ramps.","At evaluation time, 8-bit floating-point quantization is applied on all the weights and activations.","The quantized embedding weights are modeled according to a 2-bit per cell multi-level (MLC2) ReRAM NVM configuration.","The learned attention span mask is element-wise multiplied with the attention weights to re-modulate their saliencies.","Entropy prediction is then deployed along with early exit during inference according to Algorithm ."],["Impact on Model Accuracy, Computation, and Storage","Using the multi-step procedure illustrated in Fig.","REF , we amalgamate into ALBERT the various memory and latency reduction techniques at training and evaluation times.","Table REF summarizes the generated benefits of the synergistic inference with the following main observations: EdgeBERT latency-aware inference provides comparable average exit layer for the same accuracy threshold as the conventional EE approach, while allowing the DVFS algorithm to reduce the frequency and voltage in accordance with the predicted exit layer.","The EdgeBERT approach requires a lower entropy threshold than the conventional EE approach for the same accuracy target; this demonstrates that the we must predict conservatively due to the classification error introduced by the neural network-based entropy predictor.","Across the four corpora, a uniform 40% density in the embedding layer is achieved, establishing a compact memory baseline of 1.73MB to be stored in eNVMs."],["Required Computations in ALBERT","The Transformer encoder is the backbone of ALBERT/BERT, consuming more than 95% of inference computations.","Fig.","REF summarizes the computations required in this unit.","Assuming a sentence length of 128, the transformer encoder requires 1.9GFLOPs to compute matrix multiplications, layer normalizations, element-wise operations (add, mult.","), and softmax.","The attention span mask learned during fine-tuning is element-wise multiplied with the softmax output.","Notably, all the computations inside any of the twelve attention blackhead units can be effectively skipped in case its associated attention span mask is 100% null.","The EdgeBERT accelerator reaps this benefit by enforcing adaptive attention span masking during fine-tuning."],["The EdgeBERT Accelerator System","In order to maximize the benefits of the latency and memory reduction techniques during latency-aware inference, we designed a scalable accelerator system that exploits these algorithms for compute and energy efficiency blackwith the following key highlights: blackSpecialized datapath support for (i) early exit assessment, (ii) softmax and attention span masking, and (iii) layer normalization.","We notably reformulate their mathematical definitions in order to avoid numerical instability, and where possible, hardware components with long cyclic behaviors such as divisions.","blackNon-volatile and high density storage of the shared multi-task parameters substantially improves the accelerator's energy and area efficiency (Sec.","REF ).","On-demand DVFS aided by the integration of a fast-locking ADPLL and a fast-switching LDO regulator.","blackCompressed sparse execution via bitmask encoding.","The EdgeBERT hardware accelerator, illustrated in Fig.","REF , consists of a processing unit (PU), a special function unit (SFU), a LDO and ADPLL for latency-bounded DVFS.","The communication between the PU and SFU occurs via a custom-built bi-directional streaming channel.","An AXI splitter arbitrates the CPU-controlled flow of instructions and data bound for the PU and SFU AXI-slave partitions.","The multi-task embedding pruned weights and corresponding bitmask are stored in a 2MB ReRAM NVM buffer in order to avoid reloading them when powered on.","Specifically, the bitmask embedding values are stored in a single-level cell (SLC) ReRAM configuration while the nonzero embedding parameters are kept in a 2-bit per cell (MLC2) ReRAM structure, according to the learnings from the NVM studies (Sec.",")."],["Processing Unit","The processing unit (PU) is designed to execute matrix-matrix multiplications in linear layers and attention heads of ALBERT.","In the PU datapath in Fig.","REF , $n$ defines the number of parallel floating-point vector MACs (VMAC) and the vector size of each VMAC.","So, there are $n^2$ MAC units in total.","The PU datapath takes two $n*n$ matrices as input and computes $n*n*n$ MAC operations in $n$ clock cycles.","We use 8-bit floating point as the input and weight data type as no accuracy degradation was observed, and 32-bit fixed-point during accumulation.","The PU accumulator sums activation matrices and quantizes the final matrix back to 8-bit floating-point.","To exploit sparsity in both input and weight matrices, we (1) adopt bit-mask encoding and decoding for compressing and decompressing the sparse matrix, and (2) implement skipping logic in the datapath.","Bit-masks are binary tags to indicate zero and non-zero entries of a matrix so that only non-zero entries are stored in the decoder SRAM scratchpads.","For every cycle during decoding, a size $n$ vector is fetched and decoded.","The decoder first reads a $n$ -bit mask from the single-banked mask buffer to figure out what bank in the $n$ -banked input can be neglected, and inserts zero values back to the original zero entries.","The encoder also takes a similar approach.","It creates a bit mask vector and removes zero entries from the data vector before sending the compressed mask and data vector to one of the PU decoder blocks.","To save energy, the PU datapath skips the computation of a VMAC product-sum if one of the operand vectors contains only zero values.","Although the cycle-behavior of the datapath is not affected by the sparsity of inputs due to the fixed scheduling of data accesses and computations, skipping VMAC operations saves up to 1.65$\\times $ in energy consumption (Sec.","REF )."],["Special Function Unit","The special function unit (SFU) contains specialized datapaths that compute the EE assessment, DVFS control, element-wise addition, layer normalization, and softmax, all of which get invoked during the latency-aware EdgeBERT inference.","The SFU also integrates a 32KB auxiliary buffer to house the EE and DVFS LUTs, the layer normalization parameters, and the multi-head attention span masks learned during the fine-tuning process.","blackAll the computations in the SFU are in 16-bit fixed-point format."],["Computing the Multi-Head Attention","While the linear layers for the attention query, key and value tensors are computed in the PU, the proceeding softmax operation is optimized in the SFU softmax unit.","First, prior to computing an attention head, the SFU controller inspects its associated attention span mask in the auxiliary buffer.","In case the attention span mask for an attention head is null, the SFU controller proactively cancels and skips entirely the sequence of computations required for that head, and directly writes zero in the corresponding logical memory for its context vector stored in one of the PU decoder blocks.","[!t] attention matrix $A$ , and mask $A_M$ of size ($T*T$ ) masked softmax output matrix $A_O$ $T := \\text{number of tokens}$ ; $n := \\text{tile size}$ ; $i = 0$ to $T-1$ // Step 1: compute max value $max = -\\infty $ $j = 0$ to $T-1$ $vec <= load(A_{[i][n*j:n*j+n-1]})$ $max < max(vec)$ $max = max(vec)$ // Step 2: compute log-exponential-sum $sum_{exp} = 0$ $j = 0$ to $T-1$ $vec <= load(A_{[i][n*j:n*j+n-1]})$ $sum_{exp} += sum(exp(vec - max))$ $logsum_{exp} = ln(sum_{exp})$ // Step 3: Get softmax and modulate with attn span mask $j = 0$ to $T-1$ $vec<= load(A_{[i][n*j:n*j+n-1]})$ $mask <= load(A_{M[i][n*j:n*j+n-1]})$ $vec = exp(vec - max - logsum_{exp})$ $vec = vec * mask$ $store(vec) => A_{O[i][n*j:n*j+n-1]}$ Computing Softmax and Attention Span Masking In case the attention span mask for a head contains non-zero elements, the softmax unit takes advantage of the LogSumExp [19] and Max [48] tricks to vectorize the computation of the softmax function $SM()$ as: $ \\leavevmode \\xbox {resizebox}{\\XMLaddatt {width}{0.0pt}SM(A_{k})= exp[A_{k} - MAX_{k}(A)- ln (\\sum ^{K}_{k=1}exp(A_{k}-MAX_{k}(A)))]}$ By doing so, the hardware prevents numerical instability stemming from exponential overflow, and avoids the computationally intensive division operation from the original softmax function.","Upon completing the softmax operation, the softmax unit then performs element-wise multiplication between the resulting attention scores and the attention span mask as described in Algorithm REF ."],["Performing Early Exit Assessment","The EE assessment unit computes the numerically-stable version of the entropy function from equation REF as follows: $\\leavevmode \\xbox {resizebox}{\\XMLaddatt {width}{0.0pt}H(x_k) = \\ln (\\sum \\limits _{k=1}^{n} e^{x_k - MAX_{k}(x) }) - MAX_{k}(x) -\\frac{\\sum \\limits _{k=1}^{n} x_k e^{x_k - MAX_{k}(x)}}{\\sum \\limits _{k=1}^{n} e^{x_k - MAX_{k}(x) }}}$ The EE assessment unit then compares the result with the register value for the entropy threshold.","If the EE condition is met, the unit then triggers the accelerator's interrupt (IRQ).","Otherwise, the SFU controller initiates the computation of the next Transformer encoder.","In the case of latency-aware inference in intermittent mode, the EE assessment unit also indexes the EE predictor LUT stored in the auxiliary buffer in order to acquire the predicted exit layer value, which is then passed on to the DVFS controller."],["DVFS System","During each sentence inference, the DVFS FSM algorithm keeps track of the EE predictor result and manages the operating voltage and frequency accordingly.","Based on the predicted early exit layer, the DVFS controller indexes the logical memory for the $V/F$ LUT table in the auxiliary buffer and extracts the lowest corresponding supply voltage value, $VDD_{opt}$ .","At the same time, the DVFS controller simultaneously updates the ADPLL and LDO configuration registers with settings for $Freq_{opt}$ and $VDD_{opt}$ , respectively.","Table: Performance specs of LDO and ADPLLFigure: Spice simulations of LDO dynamic voltage adjustments.","The LDO stabilizes voltage transitions within 100ns.The synthesizable LDO is implemented using standard power header cells [9], and evenly distributed across the EdgeBERT accelerator.","The LDO is able to scale the accelerator voltage from 0.5V to 0.8V with a 25mV step.","With careful power header selection and layout resistance optimization, the LDO can achieve nearly linear scaled power efficiency and a fast response time of 3.8ns/50mV.","The ADPLL is also implemented using all-synthesizable approach with the PLL architecture from the FASoC open-source SoC design framework [4].","Following a frequency update request, the all-digital PLL can relock the frequency in a fast speed with low power consumption.","The 12nm performance specs of the LDO and ADPLL are shown in Table REF .","Fig.","REF show the spice-level simulation of the DVFS for a consecutive sequence of sentence inference.","For each sentence, the entropy is calculated after the computation of Encoder 1 and sent to the EE predictor to forecast the early exit layer.","Based on the predicted early exit encoder and latency requirement for the sentence, the DVFS controller select the lowest voltage level and proper frequency to meet the latency requirement $T_{target}$ .","Therefore, the remaining encoder stages will compute at a lower voltage level to save energy.","For example, the sentence 1 of Fig.","REF , the early exit layer is predicted as 8.","Therefore, the rest Encoders (i.e encoder 2-8) in sentence 1 are computed under a lower voltage 0.7V.","After the inference of the first sentence, the voltage level ramps back to nominal 0.8V for the computation of layer 1 in the following sentence.","As on-chip integrated LDO is used, the transition and settling time is optimized to be within 100ns, which is negligible considering the 50ms latency target.","The computation of the next sentence starts once the voltage transition is settled.","During idle times, EdgeBERT stays at standby 0.50V to save leakage energy.","Figure: Average latency (top row) and energy (Bottom row) per sentence as the PU MAC vector size scales at max frequency (1GHz) and nominal voltage (0.8V), highlighting impact of adaptive attention span (AAS), and sparsity in weights and activations (Sparse) on the EdgeBERT accelerator and TX2 mGPU.","MAC size of 16 yields the most energy efficient design.Figure: Average DVFS-driven supply voltage (top row) and clock frequency (middle row), as well as, generated energy expenditures (bottom row) of the EdgeBERT accelerator system with n=16n=16 during latency-aware inference (LAI), and latency-aware inference further improved with adaptive attention span and sparse execution (LAS+AAS+Sparse).","Different latency targets of 50ms (T=50), 75ms (T=75), and 100ms (T=100) are used for LAI executions.","Results are compared with the baseline 12-layer inference (Base) and the conventional early exit inference (EE)."],["Design and Verification Methodology","The EdgeBERT accelerator is designed in synthesizable SystemC with the aid of hardware components from the MatchLib [34] and HLSLibs [27] open-source libraries.","Verilog RTL is auto-generated by the Catapult high-level synthesis (HLS) tool [1] using a commercial 12nm process node.","HLS constraints are uniformly set with the goal to achieve maximum throughput on the pipelined design.","During the bottom-up HLS phase, the decoder and auxiliary buffers are mapped to synthesized memories from a foundry memory compiler, while the rest of the registers are mapped to D-latches.","The energy, performance, and area results are reported on the post-HLS Verilog netlists by the Catapult tool at the 0.8V/25c/typical corner.","The 28nm ReRAM cells are characterized in NVSIM [18] and its read latency, energy, and area are back-annotated into the accelerator results after scaling to a 12nm F$^2$ cell definition in order to match the process node used in the rest of the system.","To quantify the benefits of non-volatility (Sec.","REF ), we quantify the alternative cost of loading embeddings from off-chip using DRAMsim3 [40] to extract cycle-accurate LPDDR4 DRAM energy and latency metrics.","GPU results are obtained from CUDA implementations on an Nvidia TX2 mobible GPU (mGPU), whose small form-factor SoC targets embedded edge/IoT applications [2].","We start by measuring the energy-performance trade-offs of the EdgeBERT accelerator by scaling its PU MAC vector size.","Simultaneously, we further quantify the benefit of bitmask encoding and the predicating logic of the adaptive attention span mechanism by using the attained optimization results (i.e.","embedding and encoder sparsity percentage, and attention span) reported in Table REF in which the accuracy drop was at 1%-pt of the baseline.","Adaptive adaptive span is also applied to the mGPU platform in order to quantify and compare the extent of these benefits.","Fig.","REF shows that the per-sentence processing latency decreases by roughly 3.5$\\times $ as the vector size doubles.","Across the four tasks, the energy-optimal accelerator design is obtained with a MAC vector size, $n$ , of 16.","This is because the increase in the datapath power consumption with $n=32$ starts to subdue throughput gains.","The predication/skipping mechanism of adaptive attention span reduces the accelerator processing time and energy consumption by up to 1.2$\\times $ and 1.1$\\times $ , respectively.","Compressed sparse execution in the PU datapath amounts to an additional 1.4–1.7$\\times $ energy savings with QQP receiving the benefit the most.","The EdgeBERT accelerator starts to outperform the mGPU processing time with $n=16$ .","This energy-optimal design generates up 53$\\times $ lower energy compared to the mGPU when all the optimizations are factored in.","Fig.","REF breaks down the latency, energy, area and power contributions inside the placed-and-routed, energy-optimal ($n$ =16) EdgeBERT accelerator system which occupies 1.4mm$^2$ while consuming an average power of 86mW."],["DVFS-based Latency-Aware Inference","Fig.","REF shows the DVFS-controlled supply voltage and clock frequency, and the energy savings of the latency-aware inference (LAI) on the energy-optimal accelerator design (i.e.","with MAC vector size $n=16$ ) using latency targets between 50ms and 100ms (common latency thresholds for real-time human perception [57])).","The results show that EdgeBERT optimized LAI achieves up to 7$\\times $ , and 2.5$\\times $ per-inference energy savings compared to the conventional inference (Base), and latency-unbounded early exit (EE) approaches, respectively, as seen in the SST-2 case.","As AAS further cuts the number of computation cycles, we observe further relaxation of the supply voltage and clock frequency.","At some latency targets (e.g., 75ms and 100ms in QQP and SST-2), further energy savings are not possible as V/F scaling bottoms out.","blackTo underscore the different contributions to energy savings, at 75ms latency target for example in the case of MNLI, early exit prediction, adaptive attention span, DVFS, sparse execution, and eNVMs account for 21%, 12%, 23%, 39%, and 5%, respectively, of the total accelerator energy reduction.","For stricter latency targets (e.g.","$<$ 20ms), the proposed DFVS-based scheme can be used by scaling up to even higher MAC vector sizes (i.e.","$n \\ge 32$ )."],["Benefits of NVM Embeddings Storage","BERT word embeddings are a natural fit for non-volatile storage, given that in EdgeBERT, we freeze them during fine-tuning and reuses them during inference By virtue of this scheme, we have established a compact 1.73MB baseline wherein the bitmask of the word embeddings is stored in a SLC ReRAM while the nonzero parameters are stored in a 2-bit per cell (MLC2) ReRAM buffer.","Fig.","REF illustrates the immense gains of leveraging this eNVM configuration during single-batch inference after SoC power-on.","In EdgeBERT, ALBERT embeddings would only need to be read from the integrated ReRAM buffers due to being statically pre-loaded.","The conventional operation dictates reading the embedding weights from off-chip DRAM, then writing them to dedicated on-chip volatile SRAM memories so they can be reused for future token identifications.","The EdgeBERT approach enforces a latency and energy advantage that is, respectively, 50$\\times $ and 66,000$\\times $ greater than the overhead costs in the conventional operation.","The non-volatility of this embedded storage means that these benefits can further scale with the frequency of power cycles.","Figure: (a) Breakdown of latency and energy consumption in PU and SFU datapaths, and (b) 12nm physical layout, and area and power (@ 0.8V/1GHz) breakdown of the energy-optimal EdgeBERT accelerator (MAC size=16).Figure: Costs of reading all embedding weights after system power-on.","Storing embeddings in ReRAMs gives EdgeBERT significant energy and latency advantages compared to the conventional approach requiring DRAM read followed by SRAM write/read."],["Related Work","Over the last decade, there has been extensive research on the design of high-performance and energy-efficient DNN hardware accelerators [24], [60], [11], [82], [83], [12], [47], [84], [5], [6], [26], [42], [31], [33], [54], [53], [62], [61], [66], [36], [35], [41], [45], [22], [86], [67], [71].","As these accelerators are increasingly deployed at all computing scales, there is additional interest in the hardware community to automatically generate designs [77], [76], [80], [81].","However, most of these works focus on CNN and RNN [20] computations, and not as much scrutiny has been given to accelerating Transformer-based networks with self-attention mechanisms.","Recent work in accelerating Transformer-based NLP includes $A^{3}$ [23], which proposed a hardware architecture that reduces the number of computations in attention mechanisms via approximate and iterative candidate search.","blackHowever, the $A^{3}$ scheme fetches the full and uncompressed data from DRAM before dynamically reducing computations in the hardware.","In contrast, EdgeBERT learns the optimal attention search radius during the finetuning process and then leverages its very sparse mask to avoid unnecessary matrix multiplications.","Therefore, our approach substantially eliminates DRAM accesses as the computation and memory optimizations are pre-learned before hardware acceleration.","GOBO [88] blackfocuses on BERT quantization only via 3-bit clustering on the majority of BERT weights while storing the outlier weights and activations in full FP32 precision.","blackAlthough this scheme significantly reduces DRAM accesses, it requires a mixed-precision computational datapath and a non-uniform memory storage.","In contrast, EdgeBERT adopts uniform 8-bit data storage in SRAM and eNVMs memories.","Lu et al.","[46] proposes a dense systolic array accelerator for the Transformer's multi-head attention and feed-forward layers and optimizes Transformers' computations via matrix partitioning schemes.","The EdgeBERT accelerator executes compressed sparse inference for higher energy efficiency.","OPTIMUS [55] looks to holistically accelerate Transformers with compressed sparse matrix multiplications and by skipping redundant decoding computations.","FlexASR [71] accelerates attention-based RNNs in a specialized attention datapath and only saves energy by gating the MAC when decoder RNN inputs are null.","SpAtten [79] accelerates Transformer-based models via progressive cascade token and attention head pruning.","The importance of each attention head is determined during the computation via a top-k ranking system.","In contrast, EdgeBERT opts to learn the important attention heads during the fine-tuning process by activating adaptive attention spanning.","The optimized and sparse attention spans are then used by the EdgeBERT accelerator to predicate the NLP computation.","blackFinally, all the aforementioned NLP accelerators stores the embedding weights in traditional volatile SRAM memories.","By contrast, this work recognizes that embedding weights do not change across NLP tasks.","Therefore, EdgeBERT statically stores the word embeddings in high density eNVMs, generating substantial energy and latency benefits (Sec.","REF ).","Fig.","REF qualitatively contrasts some of the prior work with EdgeBERT.","Figure: blackComparison of EdgeBERT with prior work accelerating Transformer-based NLP models."],["Conclusion","As newer Transformer-based pre-trained models continue to generate impressive breakthroughs in language modeling, they characteristically exhibit complexities that levy hefty latency, memory, and energy taxes on resource-constrained edge platforms.","EdgeBERT provides an in-depth and principled latency-driven methodology to alleviate these computational challenges in both the algorithm and hardware architecture layers.","EdgeBERT adopts first-layer early exit prediction in order to perform dynamic voltage-frequency scaling (DVFS), at a sentence granularity, for minimal energy consumption while adhering to a prescribed target latency.","Latency and memory footprint overheads are further alleviated by employing a balanced combination of adaptive attention span, selective network pruning, floating-point quantization.","We further exploit and optimize the structure of eNVMs in order to store the shared multi-task parameters, granting EdgeBERT significant performance and energy savings from system power-on.","Sentence-level, latency-aware inference on the EdgeBERT accelerator notably consumes 7$\\times $ and 2.5$\\times $ lower energy than the conventional full-model inference, and the latency-unbounded early exit approach, respectively."],["Acknowledgement","This work was supported in part by the Center for Applications Driving Architectures (ADA), one of six centers of JUMP, a Semiconductor Research Corporation (SRC) program co-sponsored by DARPA; DARPA’s DSSoC program; NSF Awards 1704834 and 1718160; Intel Corp.; and Arm Inc."]],"string":"[\n [\n \"EdgeBERT: Sentence-Level Energy Optimizations for Latency-Aware\\n Multi-Task NLP Inference\"\n ],\n [\n \"Abstract Transformer-based language models such as BERT provide significant accuracy improvement for a multitude of natural language processing (NLP) tasks.\",\n \"However, their hefty computational and memory demands make them challenging to deploy to resource-constrained edge platforms with strict latency requirements.\",\n \"We present EdgeBERT, an in-depth algorithm-hardware co-design for latency-aware energy optimization for multi-task NLP.\",\n \"EdgeBERT employs entropy-based early exit predication in order to perform dynamic voltage-frequency scaling (DVFS), at a sentence granularity, for minimal energy consumption while adhering to a prescribed target latency.\",\n \"Computation and memory footprint overheads are further alleviated by employing a calibrated combination of adaptive attention span, selective network pruning, and floating-point quantization.\",\n \"Furthermore, in order to maximize the synergistic benefits of these algorithms in always-on and intermediate edge computing settings, we specialize a 12nm scalable hardware accelerator system, integrating a fast-switching low-dropout voltage regulator (LDO), an all-digital phase-locked loop (ADPLL), as well as, high-density embedded non-volatile memories (eNVMs) wherein the sparse floating-point bit encodings of the shared multi-task parameters are carefully stored.\",\n \"Altogether, latency-aware multi-task NLP inference acceleration on the EdgeBERT hardware system generates up to 7x, 2.5x, and 53x lower energy compared to the conventional inference without early stopping, the latency-unbounded early exit approach, and CUDA adaptations on an Nvidia Jetson Tegra X2 mobile GPU, respectively.\"\n ],\n [\n \"Introduction\",\n \"Transformer-based networks trained with large multi-domain datasets have unlocked a series of breakthroughs in natural language learning and representation.\",\n \"A major catalyst of this success is the Bidirectional Encoder Representations from Transformers technique, or BERT [16], which substantially advanced nuance and context understanding.\",\n \"Its pre-training strategy, which consists of learning intentionally hidden sections of text, have proven beneficial for several downstream natural language processing (NLP) tasks.\",\n \"BERT has sparked leading-edge performance in NLP leaderboards [78], [58], and it is now applied at a global scale in web search engines [52] with marked improvements in the quality of query results.\",\n \"Figure: (a) Conventional BERT inference, (b) Conventional latency-unbounded BERT inference with early exit.\",\n \"(c) Proposed latency-bounded inference.\",\n \"The entropy result from the first layer is used to auto-adjust the accelerator supply voltage and clock frequency for energy-optimal operation while meeting an application end-to-end latency target.Advances in NLP models are also fueling the growth of intelligent virtual assistants, which leverage NLP to implement interactive voice interfaces.\",\n \"Currently, these applications are offloaded from the edge device to the cloud.\",\n \"However, they are naturally better suited to deployment on edge devices, where personal data can be kept private and the round trip latency to the cloud is removed.\",\n \"However, the impressive performance of BERT comes with a heavy compute and memory cost, which makes on-device inference prohibitive.\",\n \"Most significantly, the BERT base model consumes a staggering 432 MB of memory in native 32-bit floating-point (FP32).\",\n \"Therefore, the goal of deploying BERT on edge/mobile devices is challenging and requires tight co-design of the BERT model optimizations with dedicated hardware acceleration and memory system design.\",\n \"The constraints on mobile can be quite different to the datacenter scenario, where BERT has been mainly deployed to date.\",\n \"Firstly, since we are dealing with user input, we need to meet real time throughput requirements to prevent a noticeable lag to the user.\",\n \"Secondly, energy consumption is a critical concern on mobile devices, both for the model inference and also the associated data movement cost.\",\n \"A number of prior works have been proposed to reduce BERT storage and computation overheads [21].\",\n \"In fact, most of the compression techniques (weight pruning [49], distillation [63], quantization [89], [68]) originally proposed for convolutional and recurrent neural networks (CNNs, RNNs) have been independently applied to Transformer-based DNNs.\",\n \"In this work, we present EdgeBERT, a principled latency-driven approach to accelerate NLP workloads with minimal energy consumption via early exit prediction, dynamic voltage-frequency scaling (DFVS), and non-volatile memory bitmask encoding of the shared word embeddings.\",\n \"In conventional BERT inference (Fig.\",\n \"REF a), the final classification result is generated by the last Transformer layer.\",\n \"Early exit mechanisms [87], [90], [73], [65] (Fig.\",\n \"REF (b)) have been proposed to reduce the average energy and latency.\",\n \"The early exit entropy, which is a probabilistic measure of the classification confidence, is evaluated at the output of each computed Transformer layer and the inference exits when the entropy value falls below a pre-defined threshold.\",\n \"While this approach can appreciably reduce computation and energy costs, the achieved latency can vary drastically from one input sentence to another, potentially violating the strict real time latency constraint of the application.\",\n \"In contrast, EdgeBERT uses this upper-bound latency and the target entropy as optimization constraints, and then dynamically auto-adjusts the accelerator supply voltage and clock frequency to minimize energy consumption (Fig.\",\n \"REF (c)), while meeting the real time throughput requirement.\",\n \"Since energy scales quadratically with $V_{DD}$ and linearly with the number of computation cycles, our DVFS algorithm finds the lowest possible frequency/voltage, while also minimizing the total number of FLOPs via adaptive attention span predication.\",\n \"While the benefits of early exit and attention predications can be reaped on commodity GPUs, we unlock additional energy savings by co-designing the hardware datapaths.\",\n \"Specifically, we exploit these algorithmic optimizations in the EdgeBERT accelerator system, which integrates a fast-switching low-dropout (LDO) voltage regulator and an all-digital phase-locked loop (ADPLL) for DVFS adjustments.\",\n \"The EdgeBERT accelerator uses bit-mask encoding for compressed sparse computations, while optimizing key operations (entropy assessment, layer normalization, softmax and attention masking) for numerical stability and energy efficiency.\",\n \"Furthermore, edge/IoT devices operate intermittently which motivates powering down as much as possible.\",\n \"The model's weights, typically stored in on-chip SRAMs, either have to be reloaded from DRAM each wake up cycle or the on-chip SRAMs storing the weights must be kept on, wasting leakage power [39].\",\n \"Embedded non-volatile memories (eNVMs), which have shown considerable progress in recent years, offer great promise, if used judiciously, to eliminate the power penalty associated with intermittent operation.\",\n \"For this purpose, we perform monte-carlo fault injection simulations to identify robust and viable eNVM structures for storing the shared NLP multi-task parameters with bitmask encoding.\",\n \"Our resulting eNVM configuration significantly alleviates the energy and latency costs associated with multi-task intermediate computing by as much as 66,000$\\\\times $ and 50$\\\\times $ , respectively.\",\n \"Figure: Comparison between (a) BERT, and (b) ALBERT base models.\",\n \"ALBERT uses a smaller embedding size and its Transformer encoder layers share the same parameters.Altogether, EdgeBERT generates on average up to 7$\\\\times $ , and 2.5$\\\\times $ per-sentence energy savings compared to the conventional BERT inference, and latency-unaware early exit approaches, respectively.\",\n \"This paper therefore makes the following contributions: We propose EdgeBERT, a novel algorithm-hardware co-design approach to enable latency-bound NLP workloads on resource-constrained embedded devices.\",\n \"Recognizing that BERT word embeddings are shared across NLP tasks, we significantly alleviate off-chip communication costs by identifying viable and robust multi-level eNVM structures for storing the multi-task word embeddings.\",\n \"Leveraging the insights from this broad analysis, we propose and design a 12nm accelerator that integrates a fast-switching LDO, an ADPLL, and a compressed sparse hardware accelerator that efficiently computes the DVFS, entropy, and adaptive attention span predication algorithms and other key Transformer operations using specialized datapaths.\",\n \"We evaluate the energy consumption of latency-bound inference on four NLP tasks, and find that the EdgeBERT hardware accelerator system generates up to 7$\\\\times $ , 2.5$\\\\times $ , and 53$\\\\times $ lower energy compared to the unoptimized baseline inference without early exit, the conventional latency-unaware early exit approach, and CUDA adaptations on an Nvidia Jetson Tegra X2 mobile GPU respectively.\",\n \"The General Language Understanding Evaluation (GLUE) benchmark is the most widely used tool to evaluate NLP performance.\",\n \"It consists of nine English sentence understanding tasks covering three categories: Single-Sentence, Similarity and Paraphrase, and Inference [78].\",\n \"This collection of datasets is specifically designed to favor models that can adapt to a variety of NLP tasks.\",\n \"To validate the robustness and generalization performance of the EdgeBERT methodology, we conduct our evaluation on the four GLUE tasks with the largest corpora, which cover all three GLUE categories: SST-2 (Single-Sentence), QQP (Similarity and Paraphrase), and QNLI and MNLI (Inference).\"\n ],\n [\n \"Variations of BERT\",\n \"Since the advent of BERT with 110M parameters, a number of variants were proposed to alleviate its memory consumption or to further improve its prediction metrics.\",\n \"RoBERTa [44] generalizes better on several GLUE tasks by training on significantly more data, and for a longer amount of time, but remains as computationally intensive as BERT.\",\n \"DistilBERT [63] and MobileBERT [70] leverage knowledge distillation to reduce BERT size by 1.7$\\\\times $ and 4.3$\\\\times $ , respectively, with iso-accuracy.\",\n \"SqueezeBERT [29] substitutes several operations in the Transformer encoder with 1D grouped convolutions achieving 4$\\\\times $ speedup while being 2$\\\\times $ smaller.\",\n \"Q8BERT [89] employs a symmetric linear quantization scheme for quantizing both weights and activations into 8-bit integers.\",\n \"In contrast, in this work we leverage the higher dynamic range of floating-point encodings for greater quantization resilience.\",\n \"ALBERT [37] yields the smallest footprint to date for a compressed BERT variant with only 12M parameters, with competitive accuracy on the GLUE benchmarks.\",\n \"Fig.\",\n \"REF summarizes the key differences between the ALBERT model and the base BERT model.\",\n \"While each of BERT's twelve encoder layers have a unique set of weights, ALBERT's encoder layers instead share and reuse the same parameters – resulting in significant compression.\",\n \"The encoder block in both models has the same architecture as the legacy Transformer network [75], but with twelve parallel self-attention heads.\",\n \"Moreover, ALBERT employs a smaller embedding size (128 vs. 768) thanks to factorization in the embedding layer.\",\n \"In this work, we adopt the ALBERT variant as an efficient baseline.\",\n \"This work further pursues strategies to reduce latency and storage requirements to suit embedded hardware platform constraints.\"\n ],\n [\n \"Alleviating Transformer memory and computation costs\",\n \"An accelerator's energy consumption can be abstracted as: $Energy\\\\propto \\\\alpha C V_{DD}^2 N_{cycles}$ where $\\\\alpha $ , $C$ , $V_{DD}$ and $N_{cycles}$ are the switching activity factor, the effective wire and device capacitance, the supply voltage, and the required number of clock cycles to complete the inference, respectively.\",\n \"While the DVFS algorithm (Sec.\",\n \"REF ) lowers the energy quadratically by bringing $V_{DD}$ down to the lowest optimal voltage, in this section, we explore avenues to further reduce the energy by minimizing $\\\\alpha $ , $C$ , and $N_{cycles}$ .\",\n \"For this purpose, we carefully incorporate into the multi-task ALBERT inference: 1) adaptive attention span predication and early exit which reduce $N_{cycles}$ ; 2) network pruning, which ultimately reduces $\\\\alpha $ ; and 3) floating-point quantization helping decrease $C$ , altogether with minimal accuracy degradation.\",\n \"While briefly describing these optimizations individually in this section, we provide a reasoned methodology for applying them to the ALBERT model, as shown in Fig.\",\n \"REF .\"\n ],\n [\n \"Entropy-based Early Exit\",\n \"The motivation behind early exit (EE) is to match linguistically complex sentences with larger (or deeper) models and simple sentences with smaller (or shallower) models [87], [13].\",\n \"This is typically done by adding a lightweight classifier at the output of the Transformer layer so that a given input can exit inference earlier or later in the stack, depending on its structural and contextual complexity.\",\n \"The classifier computes and compares the entropy of an output distribution with a preset “confidence\\\" threshold, $E_{T}$ , in order to assess whether the prediction should exit or continue inference in the next Transformer encoder layer.\",\n \"The entropy metric quantifies the amount of uncertainty in the data.\",\n \"Smaller entropy values at a Transformer layer output implies greater confidence in the correctness of the classification result.\",\n \"The entropy H on sample x is estimated as: $\\\\leavevmode \\\\xbox {resizebox}{\\\\XMLaddatt {width}{0.0pt}H(x) = - \\\\sum p(x)\\\\log p(x)= \\\\ln (\\\\sum \\\\limits _{k=1}^{n} e^{x_k}) -\\\\frac{\\\\sum \\\\limits _{k=1}^{n} x_k e^{x_k}}{\\\\sum \\\\limits _{k=1}^{n} e^{x_k}}}$ The early exit condition is met when $H(x)$ $<$ $E_{T}$ .\",\n \"Therefore, the larger $E_{T}$ becomes, the earlier the sample will exit (i.e.\",\n \"$N_{cycles}$ becomes smaller) with potentially lower accuracy.\",\n \"In this work, we modify the conventional EE inference approach by predicting the early exit layer from the output of the first Transformer layer in order to run the rest of the network computation in an energy-optimal and latency-bounded manner (Sec.\",\n \").\"\n ],\n [\n \"Adaptive Attention Span\",\n \"The attention mechanism [8] is a powerful technique that allows neural networks to emphasize the most relevant tokens of information when making predictions.\",\n \"The base ALBERT model contains up to twelve parallel attention heads – each learning their own saliency weights on the full length of the encoder input.\",\n \"However, depending on the complexity of the task, many heads can be redundant and can be safely removed without impacting accuracy [51].\",\n \"Furthermore, the cost of computing the attention mechanism scales quadratically with the sequence length.\",\n \"Therefore, there is potentially a meaningful amount of computations and energy to be saved in optimizing the inspection reach of every attention head.\",\n \"In the quest to avoid needless attention computations in ALBERT, a learnable parameter z is introduced in the datapath of each self-attention head in order to find its own optimal attention span [69].\",\n \"The parameter z is mapped to a masking function with a [0, 1] output range, as shown in Fig.\",\n \"REF .\",\n \"The masked span is then applied on the attention weights in order to re-modulate their saliencies.\",\n \"The optimal span is automatically learned during the fine-tuning process by adding back the average loss from the reduced span to the training cross-entropy loss.\",\n \"The maximum sentence length for fine-tuning the GLUE tasks is 128.\",\n \"As a result, shorter sentences are typically zero-padded to 128 during the tokenization pre-processing.\",\n \"Table REF shows the final attention span learned by each self-attention head when fine-tuning with the adaptive attention span technique.\",\n \"Strikingly, the twelve parallel self-attention heads in ALBERT do not need to inspect their inputs at maximum span.\",\n \"In fact, more than half of the attention heads, 8 for MNLI and QQP and 7 for SST-2 and QNLI, can be completely turned off with minimal accuracy loss.\",\n \"This amounts to a 1.22$\\\\times $ and 1.18$\\\\times $ reduction, respectively, in the total number of FLOPS (which linearly correlates with $N_{cycles}$ ) required for single-batch inference.\",\n \"Table: Learned spans of every attention head in ALBERT.Baseline Acc: MNLI=85.16, QQP=90.76, SST-2=92.20, QNLI=89.48The twelve attention spans, learned during fine-tuning, are written to registers in the EdgeBERT accelerator in the form of a 128-wide vector – in order to predicate on the inference computation of the multi-head attention.\",\n \"Notably, all the computations inside any of the twelve attention head units can be effectively skipped in case its associated attention span mask is 100% null.\",\n \"The EdgeBERT accelerator takes advantage of this observation in a proactive manner during inference in the custom hardware (Sec.\",\n \"REF ).\"\n ],\n [\n \"Network Pruning\",\n \"The EdgeBERT hardware accelerator (Sec. )\",\n \"executes sparse computations and saves energy by gating MACs whenever input operands are null.\",\n \"Therefore, the extent to which we can prune the ALBERT model, without appreciable accuracy loss, determines the overall accelerator energy efficiency.\",\n \"In this work, we consider both movement pruning [64] and the well-known magnitude pruning [25] methods.\",\n \"Movement pruning is a first-order pruning technique that is applied during model fine-tuning which eliminates weights that are dynamically shrinking towards 0 (i.e., according to the movement of the values).\",\n \"In some cases, magnitude pruning may be a sub-optimal method to use during transfer learning, as pre-trained weights closer to zero may have a high chance of being eliminated regardless of the fine-tuning requirement.\",\n \"We observe that movement pruning particularly outperforms magnitude-based pruning in high sparsity regimes, as each individual remaining weight becomes more important to learn the task at hand.\",\n \"Therefore, choosing between the two pruning techniques would depend on the per-task tolerance to increasing sparsity levels.\",\n \"We note that magnitude pruning is always applied to the ALBERT embedding layer in order to enforce uniformity in the data during multi-domain on-chip acceleration – as using movement pruning on the embedding layer would make its weights unique for each NLP domain, thereby forgoing opportunities for data reuse in hardware.\"\n ],\n [\n \"Floating-Point Quantization\",\n \"DNN algorithmic resilience allows for parameters to be represented in lower bit precision without accuracy loss.\",\n \"Fixed-point or integer quantization techniques, commonly adopted in CNN models, suffer from limited range and may be inadequate for NLP models, whose weights can be more than an order of magnitude larger [72].\",\n \"This phenomenon is owed to layer normalization [7], which is commonly adopted in NLP models and has invariance properties that do not reparameterize the network – unlike batch normalization [30], which produces a weight normalization side effect in CNNs.\",\n \"In this work, we employ floating-point based quantization, which provides 2$\\\\times $ higher dynamic range compared to integer datatypes [32].\",\n \"Both weights and activations are quantized across ALBERT layers to 8-bit precision.\",\n \"We also performed a search on the optimal exponent bit width to satisfy the dynamic range requirements of the ALBERT model.\",\n \"Setting the floating-point exponent space to 4 bits within the 8-bit word size, with the exponent being scaled at a per-layer granularity, provided the best accuracy performance across NLP tasks.\",\n \"In contrast to task-specific encoder weights, word embedding parameters are deliberately fixed during fine-tuning and reused across different NLP tasks.\",\n \"We seek to avoid the energy and latency costs of reloading the word embeddings from off-chip memory for different tasks by storing these shared parameters in embedded non-volatile memories (eNVMs).\",\n \"eNVM storage also enables energy-efficient intermittent computing because the embedding weights will be retained if and when the system-on-chip powers off between inferences.\",\n \"However, despite their compelling storage density and read characteristics, eNVMs exhibit two main drawbacks: potentially high write cost (in terms of energy and latency) and decreased reliability, particularly in multi-level cell (MLC) configurations [15].\",\n \"Fortunately, the word embeddings are acting as read-only parameters on-chip, which makes them highly suitable for eNVM storage, but previous work highlights the need to study the impacts of faulty, highly-dense ReRAM storage on DNN task accuracy [56].\",\n \"On the other hand, encoder weights need to be updated when switching across different NLP tasks.\",\n \"To prevent the energy and latency degradation that would follow from updating the encoder weight values in eNVMs, we map the natural partition of shared and task-specific parameters to eNVMs and SRAMs, respectively [17].\"\n ],\n [\n \"eNVM Modeling Methodology\",\n \"This work specifically considers dense, energy-efficient Resistive RAM (ReRAM) arrays [10], [43] as an on-chip storage solution for shared embedding parameters.\",\n \"We selected ReRAMs for their relative maturity and demonstrated read characteristics.\",\n \"However, we note that there is a larger design space of opportunities to be explored with other emerging MLC-capable NVM technologies such as PCM [14], but is beyond the scope of this work.\",\n \"We evaluate the robustness of storing the 8-bit quantized word embeddings in eNVM storage.\",\n \"In order to quantify the trade-offs between storage density and task accuracy, we use cell characteristics of 28nm ReRAM programmed with varying number of bits per cell [15], and evaluate 100 fault injection trials per storage configuration to identify robust eNVM storage solutions.\",\n \"We leverage and extend Ares [59], which is an existing open-source fault injection framework for quantifying the resilience of DNNs.\",\n \"After pruning, we store non-zero compressed embedding weights using a bitmask-style sparse encoding.\",\n \"Previous work demonstrates that DNN weight bitmask values are vulnerable to MLC faults, so the bitmask is protectively stored in lower-risk SLC devices, while we experiment with MLC storage for the non-zero data values [56].\"\n ],\n [\n \"Optimal eNVM Configuration\",\n \"Table REF uncovers exceptional resilience to storing word embeddings in MLC ReRAM.\",\n \"Across many fault injection trials, we observe that MLC2 (ReRAM programmed at 2 bits-per-cell) does not degrade accuracy across multiple tasks, while MLC3 exhibits potentially catastrophic degradation in minimum accuracy and an appreciable decline in average accuracy for the QNLI task, highlighted in bold.\",\n \"Based on this observation, the EdgeBERT accelerator system leverages MLC2 ReRAMs for word embedding storage (Sec.).\",\n \"[!t] linenosize= $E_T$ := target entropy input sentence $i = 0$ to $n$ encoder layer l = 1 to 12 $z_l = f(x;\\\\theta | VDD_{nom}, Freq_{max})$ $entropy(z_l) < E_T$ exit inference Conventional early exit inference\"\n ],\n [\n \"EdgeBERT's Latency-Aware Inference\",\n \"The conventional BERT inference (Algorithm REF ) with early exit (EE) can significantly reduce BERT inference latency.\",\n \"To further reduce the energy consumption for NLP inference, a latency-aware inference scheme leveraging the EE predictor and dynamic voltage and frequency scaling (DVFS) is proposed to minimize end-to-end per-sentence energy consumption while satisfying the real-time latency target.\",\n \"[!t] linenosize= $T$ := per-sentence latency target, $E_T$ := entropy target $N_{cycles} := \\\\text{number of clock cycles to compute the Transformer encoder}$ input sentence $i = 1$ to $n$ encoder layer l = 1 $z_l = f(x;\\\\theta | VDD_{nom}, Freq_{max})$ $entropy(z_l) < E_T$ exit inference $L_{predict} = LUT(entropy(z_1), E_T)$ $VDD_{opt}, Freq_{opt} = DVFS(L_{predict},T)$ encoder layer $l = 2$ to $L_{predict}$ $z_l = f(x;\\\\theta | VDD_{opt}, Freq_{opt})$ $entropy(z_l) < E_T$ exit inference exit inference EdgeBERT latency-aware inference.\",\n \"Computations exit at the predicted exit layer or earlier.\"\n ],\n [\n \"Methodology\",\n \"DVFS is a widely used technique to dynamically scale down the voltage and frequency for less computationally intensive workloads.\",\n \"In the past, DVFS has been widely deployed in commercial CPUs [74], [28] and GPUs [50].\",\n \"However, these schemes typically adjust the voltage and frequency at a coarse granularity at workload-level.\",\n \"In the era of AI, DVFS has started to be explored for DNN accelerators [38].\",\n \"For example, a recent state-of-the-art AI chip has reported per-layer DVFS to save energy [3].\",\n \"In this work, we explore a fine-grained sentence-level DVFS to reduce the energy consumption for NLP inference while meeting the latency target.\",\n \"The proposed early exit -based latency-aware inference methodology is illustrated in Algorithm .\",\n \"The inference of a sentence starts at nominal voltage and maximum frequency, and the entropy value is calculated at the output of the first Transformer encoder layer.\",\n \"The entropy result is then sent to a trained classifier (EE predictor) to predict which following encoder layer should early exit (e.g.\",\n \"early exit at encoder layer 6 before the final encoder layer 12).\",\n \"Based on the predicted early exit layer, the voltage and frequency is scaled down to proper energy-optimal setting for the rest of encoder layers (e.g.\",\n \"encoder layer 2 to 6) while meeting the latency target for each sentence.\",\n \"This scheme produces a quadratic reduction in the accelerator power consumption.\",\n \"In our work, the EE predictor is a ReLU-activated five-layer perceptron neural network with 64 cells in each of the hidden layers.\",\n \"It takes the entropy of encoder layer 1 as input and forecasts the early exit Transformer layer which has an entropy below the desired threshold.\",\n \"The neural network architecture of the EE predictor was empirically searched with the goal of minimizing the difference between the predicted and the true entropy-based exit layer.\",\n \"For this purpose, we constructed parallel training and test datasets containing the entropy values at the output of the 12 Transformer layers during evaluation on the GLUE benchmarks.\",\n \"The EE predictor is distilled as a lookup table (LUT) leading to negligible one-time (per-sentence) computational overhead.\",\n \"Furthermore, implementing the EE predictor as a LUT simplifies its hardware operation.\",\n \"As the neural network based LUT is error-prone, it may predict a higher exit layer than necessary.\",\n \"Therefore, during the inference, the entropy is checked after each encoder layer for early stopping until the predicted layer.\",\n \"If the computed entropy becomes lower than the exit threshold before the predicted encoder layer, the inference will terminate at that early exit condition point.\",\n \"In case the inference reaches the predicted layer, termination occurs even if the entropy at that layer is still higher than the exit threshold in order to not violate timing constraints.\",\n \"When assessing the impacts of using entropy prediction instead of traditional EE methods, we set a fixed accuracy degradation threshold of 1%, 2%, or 5% (relative to the inference accuracy of the full ALBERT model) and increased the entropy threshold until the accuracy dropped to the desired threshold.\",\n \"This allowed us to compare energy savings between entropy prediction and conventional EE for a fixed accuracy target.\",\n \"For the same accuracy threshold, the entropy threshold for entropy prediction was lower than the entropy threshold for conventional EE, leading to a slightly later average exit layer during inference.\",\n \"However, entropy prediction allows for DVFS since the maximum exit layer is known after the first layer, whereas with the conventional EE appproach, the maximum exit layer is always the final encoder layer.\",\n \"EdgeBERT latency-aware inference therefore achieves greater energy savings than the conventional EE approach by facilitating DVFS (Sec.\",\n \"REF ).\",\n \"Figure: EdgeBERT training and evaluation procedure.\"\n ],\n [\n \"On-chip DVFS system\",\n \"To realize fast per-sentence DVFS, the on-chip DVFS system is developed and integrated within EdgeBERT.\",\n \"The DVFS system includes a DVFS controller, an on-chip synthesizable linear voltage regulator (LDO), and an all-digital PLL (ADPLL).\",\n \"Compared with the conventional workload-level DVFS [74], the proposed scheme adjusts voltage and frequency at a finer-grained sentence-level granularity.\",\n \"Based on the predicted early exit layer from the EE predictor, the required run cycles, $N_{cycles}$ , for the rest of the encoder layers before early exit can be known.\",\n \"And, knowing the frontend elapsed time $T_{elapsed}$ up to the EE predictor within the per-sentence latency target $T$ , the optimal running frequency can be calculated as follows: $Freq_{opt} = N_{cycles} / (T - T_{elapsed}) $ Meanwhile, the corresponding energy-optimal supply voltage, $VDD_{opt}$ , is selected by the DVFS controller to achieve the lowest operational voltage value at $Freq_{opt}$ .\",\n \"In the EdgeBERT accelerator system, this is done via indexing the look-up table containing the ADPLL frequency/voltage sweep coordinates.\",\n \"The DVFS is performed for each real-time sentence inference due to its fast response time; the implementation details are shown in Sec.\",\n \"REF .\"\n ],\n [\n \"Algorithmic Synergy\",\n \"In order to quantify the different tradeoffs, and evaluate the synergistic impact on the model accuracy from the memory and latency optimizations, the eNVM modeling, and the EE predictor, we implemented the training and evaluation procedures illustrated in Fig.\",\n \"REF on the base of HuggingFace’s Transformers infrastructure [85].\"\n ],\n [\n \"Training and Evaluation Procedure\",\n \"The training methodology consists of two phases.\",\n \"In the first phase, the model is pruned during fine-tuning: magnitude pruning is applied to the embedding layer and either movement or magnitude pruning is applied to the Transformer encoder layer.\",\n \"An additional loss term comes from knowledge distillation using the base ALBERT model fine-tuned on the target task as a teacher.\",\n \"The embeddings and the encoder layer are subject to separate pruning schedules.\",\n \"At the same time, the attention heads learn their optimal spans.\",\n \"In the second training phase, we freeze the model's parameters prior to fine-tuning the early exit highway off-ramps.\",\n \"At evaluation time, 8-bit floating-point quantization is applied on all the weights and activations.\",\n \"The quantized embedding weights are modeled according to a 2-bit per cell multi-level (MLC2) ReRAM NVM configuration.\",\n \"The learned attention span mask is element-wise multiplied with the attention weights to re-modulate their saliencies.\",\n \"Entropy prediction is then deployed along with early exit during inference according to Algorithm .\"\n ],\n [\n \"Impact on Model Accuracy, Computation, and Storage\",\n \"Using the multi-step procedure illustrated in Fig.\",\n \"REF , we amalgamate into ALBERT the various memory and latency reduction techniques at training and evaluation times.\",\n \"Table REF summarizes the generated benefits of the synergistic inference with the following main observations: EdgeBERT latency-aware inference provides comparable average exit layer for the same accuracy threshold as the conventional EE approach, while allowing the DVFS algorithm to reduce the frequency and voltage in accordance with the predicted exit layer.\",\n \"The EdgeBERT approach requires a lower entropy threshold than the conventional EE approach for the same accuracy target; this demonstrates that the we must predict conservatively due to the classification error introduced by the neural network-based entropy predictor.\",\n \"Across the four corpora, a uniform 40% density in the embedding layer is achieved, establishing a compact memory baseline of 1.73MB to be stored in eNVMs.\"\n ],\n [\n \"Required Computations in ALBERT\",\n \"The Transformer encoder is the backbone of ALBERT/BERT, consuming more than 95% of inference computations.\",\n \"Fig.\",\n \"REF summarizes the computations required in this unit.\",\n \"Assuming a sentence length of 128, the transformer encoder requires 1.9GFLOPs to compute matrix multiplications, layer normalizations, element-wise operations (add, mult.\",\n \"), and softmax.\",\n \"The attention span mask learned during fine-tuning is element-wise multiplied with the softmax output.\",\n \"Notably, all the computations inside any of the twelve attention blackhead units can be effectively skipped in case its associated attention span mask is 100% null.\",\n \"The EdgeBERT accelerator reaps this benefit by enforcing adaptive attention span masking during fine-tuning.\"\n ],\n [\n \"The EdgeBERT Accelerator System\",\n \"In order to maximize the benefits of the latency and memory reduction techniques during latency-aware inference, we designed a scalable accelerator system that exploits these algorithms for compute and energy efficiency blackwith the following key highlights: blackSpecialized datapath support for (i) early exit assessment, (ii) softmax and attention span masking, and (iii) layer normalization.\",\n \"We notably reformulate their mathematical definitions in order to avoid numerical instability, and where possible, hardware components with long cyclic behaviors such as divisions.\",\n \"blackNon-volatile and high density storage of the shared multi-task parameters substantially improves the accelerator's energy and area efficiency (Sec.\",\n \"REF ).\",\n \"On-demand DVFS aided by the integration of a fast-locking ADPLL and a fast-switching LDO regulator.\",\n \"blackCompressed sparse execution via bitmask encoding.\",\n \"The EdgeBERT hardware accelerator, illustrated in Fig.\",\n \"REF , consists of a processing unit (PU), a special function unit (SFU), a LDO and ADPLL for latency-bounded DVFS.\",\n \"The communication between the PU and SFU occurs via a custom-built bi-directional streaming channel.\",\n \"An AXI splitter arbitrates the CPU-controlled flow of instructions and data bound for the PU and SFU AXI-slave partitions.\",\n \"The multi-task embedding pruned weights and corresponding bitmask are stored in a 2MB ReRAM NVM buffer in order to avoid reloading them when powered on.\",\n \"Specifically, the bitmask embedding values are stored in a single-level cell (SLC) ReRAM configuration while the nonzero embedding parameters are kept in a 2-bit per cell (MLC2) ReRAM structure, according to the learnings from the NVM studies (Sec.\",\n \").\"\n ],\n [\n \"Processing Unit\",\n \"The processing unit (PU) is designed to execute matrix-matrix multiplications in linear layers and attention heads of ALBERT.\",\n \"In the PU datapath in Fig.\",\n \"REF , $n$ defines the number of parallel floating-point vector MACs (VMAC) and the vector size of each VMAC.\",\n \"So, there are $n^2$ MAC units in total.\",\n \"The PU datapath takes two $n*n$ matrices as input and computes $n*n*n$ MAC operations in $n$ clock cycles.\",\n \"We use 8-bit floating point as the input and weight data type as no accuracy degradation was observed, and 32-bit fixed-point during accumulation.\",\n \"The PU accumulator sums activation matrices and quantizes the final matrix back to 8-bit floating-point.\",\n \"To exploit sparsity in both input and weight matrices, we (1) adopt bit-mask encoding and decoding for compressing and decompressing the sparse matrix, and (2) implement skipping logic in the datapath.\",\n \"Bit-masks are binary tags to indicate zero and non-zero entries of a matrix so that only non-zero entries are stored in the decoder SRAM scratchpads.\",\n \"For every cycle during decoding, a size $n$ vector is fetched and decoded.\",\n \"The decoder first reads a $n$ -bit mask from the single-banked mask buffer to figure out what bank in the $n$ -banked input can be neglected, and inserts zero values back to the original zero entries.\",\n \"The encoder also takes a similar approach.\",\n \"It creates a bit mask vector and removes zero entries from the data vector before sending the compressed mask and data vector to one of the PU decoder blocks.\",\n \"To save energy, the PU datapath skips the computation of a VMAC product-sum if one of the operand vectors contains only zero values.\",\n \"Although the cycle-behavior of the datapath is not affected by the sparsity of inputs due to the fixed scheduling of data accesses and computations, skipping VMAC operations saves up to 1.65$\\\\times $ in energy consumption (Sec.\",\n \"REF ).\"\n ],\n [\n \"Special Function Unit\",\n \"The special function unit (SFU) contains specialized datapaths that compute the EE assessment, DVFS control, element-wise addition, layer normalization, and softmax, all of which get invoked during the latency-aware EdgeBERT inference.\",\n \"The SFU also integrates a 32KB auxiliary buffer to house the EE and DVFS LUTs, the layer normalization parameters, and the multi-head attention span masks learned during the fine-tuning process.\",\n \"blackAll the computations in the SFU are in 16-bit fixed-point format.\"\n ],\n [\n \"Computing the Multi-Head Attention\",\n \"While the linear layers for the attention query, key and value tensors are computed in the PU, the proceeding softmax operation is optimized in the SFU softmax unit.\",\n \"First, prior to computing an attention head, the SFU controller inspects its associated attention span mask in the auxiliary buffer.\",\n \"In case the attention span mask for an attention head is null, the SFU controller proactively cancels and skips entirely the sequence of computations required for that head, and directly writes zero in the corresponding logical memory for its context vector stored in one of the PU decoder blocks.\",\n \"[!t] attention matrix $A$ , and mask $A_M$ of size ($T*T$ ) masked softmax output matrix $A_O$ $T := \\\\text{number of tokens}$ ; $n := \\\\text{tile size}$ ; $i = 0$ to $T-1$ // Step 1: compute max value $max = -\\\\infty $ $j = 0$ to $T-1$ $vec <= load(A_{[i][n*j:n*j+n-1]})$ $max < max(vec)$ $max = max(vec)$ // Step 2: compute log-exponential-sum $sum_{exp} = 0$ $j = 0$ to $T-1$ $vec <= load(A_{[i][n*j:n*j+n-1]})$ $sum_{exp} += sum(exp(vec - max))$ $logsum_{exp} = ln(sum_{exp})$ // Step 3: Get softmax and modulate with attn span mask $j = 0$ to $T-1$ $vec<= load(A_{[i][n*j:n*j+n-1]})$ $mask <= load(A_{M[i][n*j:n*j+n-1]})$ $vec = exp(vec - max - logsum_{exp})$ $vec = vec * mask$ $store(vec) => A_{O[i][n*j:n*j+n-1]}$ Computing Softmax and Attention Span Masking In case the attention span mask for a head contains non-zero elements, the softmax unit takes advantage of the LogSumExp [19] and Max [48] tricks to vectorize the computation of the softmax function $SM()$ as: $ \\\\leavevmode \\\\xbox {resizebox}{\\\\XMLaddatt {width}{0.0pt}SM(A_{k})= exp[A_{k} - MAX_{k}(A)- ln (\\\\sum ^{K}_{k=1}exp(A_{k}-MAX_{k}(A)))]}$ By doing so, the hardware prevents numerical instability stemming from exponential overflow, and avoids the computationally intensive division operation from the original softmax function.\",\n \"Upon completing the softmax operation, the softmax unit then performs element-wise multiplication between the resulting attention scores and the attention span mask as described in Algorithm REF .\"\n ],\n [\n \"Performing Early Exit Assessment\",\n \"The EE assessment unit computes the numerically-stable version of the entropy function from equation REF as follows: $\\\\leavevmode \\\\xbox {resizebox}{\\\\XMLaddatt {width}{0.0pt}H(x_k) = \\\\ln (\\\\sum \\\\limits _{k=1}^{n} e^{x_k - MAX_{k}(x) }) - MAX_{k}(x) -\\\\frac{\\\\sum \\\\limits _{k=1}^{n} x_k e^{x_k - MAX_{k}(x)}}{\\\\sum \\\\limits _{k=1}^{n} e^{x_k - MAX_{k}(x) }}}$ The EE assessment unit then compares the result with the register value for the entropy threshold.\",\n \"If the EE condition is met, the unit then triggers the accelerator's interrupt (IRQ).\",\n \"Otherwise, the SFU controller initiates the computation of the next Transformer encoder.\",\n \"In the case of latency-aware inference in intermittent mode, the EE assessment unit also indexes the EE predictor LUT stored in the auxiliary buffer in order to acquire the predicted exit layer value, which is then passed on to the DVFS controller.\"\n ],\n [\n \"DVFS System\",\n \"During each sentence inference, the DVFS FSM algorithm keeps track of the EE predictor result and manages the operating voltage and frequency accordingly.\",\n \"Based on the predicted early exit layer, the DVFS controller indexes the logical memory for the $V/F$ LUT table in the auxiliary buffer and extracts the lowest corresponding supply voltage value, $VDD_{opt}$ .\",\n \"At the same time, the DVFS controller simultaneously updates the ADPLL and LDO configuration registers with settings for $Freq_{opt}$ and $VDD_{opt}$ , respectively.\",\n \"Table: Performance specs of LDO and ADPLLFigure: Spice simulations of LDO dynamic voltage adjustments.\",\n \"The LDO stabilizes voltage transitions within 100ns.The synthesizable LDO is implemented using standard power header cells [9], and evenly distributed across the EdgeBERT accelerator.\",\n \"The LDO is able to scale the accelerator voltage from 0.5V to 0.8V with a 25mV step.\",\n \"With careful power header selection and layout resistance optimization, the LDO can achieve nearly linear scaled power efficiency and a fast response time of 3.8ns/50mV.\",\n \"The ADPLL is also implemented using all-synthesizable approach with the PLL architecture from the FASoC open-source SoC design framework [4].\",\n \"Following a frequency update request, the all-digital PLL can relock the frequency in a fast speed with low power consumption.\",\n \"The 12nm performance specs of the LDO and ADPLL are shown in Table REF .\",\n \"Fig.\",\n \"REF show the spice-level simulation of the DVFS for a consecutive sequence of sentence inference.\",\n \"For each sentence, the entropy is calculated after the computation of Encoder 1 and sent to the EE predictor to forecast the early exit layer.\",\n \"Based on the predicted early exit encoder and latency requirement for the sentence, the DVFS controller select the lowest voltage level and proper frequency to meet the latency requirement $T_{target}$ .\",\n \"Therefore, the remaining encoder stages will compute at a lower voltage level to save energy.\",\n \"For example, the sentence 1 of Fig.\",\n \"REF , the early exit layer is predicted as 8.\",\n \"Therefore, the rest Encoders (i.e encoder 2-8) in sentence 1 are computed under a lower voltage 0.7V.\",\n \"After the inference of the first sentence, the voltage level ramps back to nominal 0.8V for the computation of layer 1 in the following sentence.\",\n \"As on-chip integrated LDO is used, the transition and settling time is optimized to be within 100ns, which is negligible considering the 50ms latency target.\",\n \"The computation of the next sentence starts once the voltage transition is settled.\",\n \"During idle times, EdgeBERT stays at standby 0.50V to save leakage energy.\",\n \"Figure: Average latency (top row) and energy (Bottom row) per sentence as the PU MAC vector size scales at max frequency (1GHz) and nominal voltage (0.8V), highlighting impact of adaptive attention span (AAS), and sparsity in weights and activations (Sparse) on the EdgeBERT accelerator and TX2 mGPU.\",\n \"MAC size of 16 yields the most energy efficient design.Figure: Average DVFS-driven supply voltage (top row) and clock frequency (middle row), as well as, generated energy expenditures (bottom row) of the EdgeBERT accelerator system with n=16n=16 during latency-aware inference (LAI), and latency-aware inference further improved with adaptive attention span and sparse execution (LAS+AAS+Sparse).\",\n \"Different latency targets of 50ms (T=50), 75ms (T=75), and 100ms (T=100) are used for LAI executions.\",\n \"Results are compared with the baseline 12-layer inference (Base) and the conventional early exit inference (EE).\"\n ],\n [\n \"Design and Verification Methodology\",\n \"The EdgeBERT accelerator is designed in synthesizable SystemC with the aid of hardware components from the MatchLib [34] and HLSLibs [27] open-source libraries.\",\n \"Verilog RTL is auto-generated by the Catapult high-level synthesis (HLS) tool [1] using a commercial 12nm process node.\",\n \"HLS constraints are uniformly set with the goal to achieve maximum throughput on the pipelined design.\",\n \"During the bottom-up HLS phase, the decoder and auxiliary buffers are mapped to synthesized memories from a foundry memory compiler, while the rest of the registers are mapped to D-latches.\",\n \"The energy, performance, and area results are reported on the post-HLS Verilog netlists by the Catapult tool at the 0.8V/25c/typical corner.\",\n \"The 28nm ReRAM cells are characterized in NVSIM [18] and its read latency, energy, and area are back-annotated into the accelerator results after scaling to a 12nm F$^2$ cell definition in order to match the process node used in the rest of the system.\",\n \"To quantify the benefits of non-volatility (Sec.\",\n \"REF ), we quantify the alternative cost of loading embeddings from off-chip using DRAMsim3 [40] to extract cycle-accurate LPDDR4 DRAM energy and latency metrics.\",\n \"GPU results are obtained from CUDA implementations on an Nvidia TX2 mobible GPU (mGPU), whose small form-factor SoC targets embedded edge/IoT applications [2].\",\n \"We start by measuring the energy-performance trade-offs of the EdgeBERT accelerator by scaling its PU MAC vector size.\",\n \"Simultaneously, we further quantify the benefit of bitmask encoding and the predicating logic of the adaptive attention span mechanism by using the attained optimization results (i.e.\",\n \"embedding and encoder sparsity percentage, and attention span) reported in Table REF in which the accuracy drop was at 1%-pt of the baseline.\",\n \"Adaptive adaptive span is also applied to the mGPU platform in order to quantify and compare the extent of these benefits.\",\n \"Fig.\",\n \"REF shows that the per-sentence processing latency decreases by roughly 3.5$\\\\times $ as the vector size doubles.\",\n \"Across the four tasks, the energy-optimal accelerator design is obtained with a MAC vector size, $n$ , of 16.\",\n \"This is because the increase in the datapath power consumption with $n=32$ starts to subdue throughput gains.\",\n \"The predication/skipping mechanism of adaptive attention span reduces the accelerator processing time and energy consumption by up to 1.2$\\\\times $ and 1.1$\\\\times $ , respectively.\",\n \"Compressed sparse execution in the PU datapath amounts to an additional 1.4–1.7$\\\\times $ energy savings with QQP receiving the benefit the most.\",\n \"The EdgeBERT accelerator starts to outperform the mGPU processing time with $n=16$ .\",\n \"This energy-optimal design generates up 53$\\\\times $ lower energy compared to the mGPU when all the optimizations are factored in.\",\n \"Fig.\",\n \"REF breaks down the latency, energy, area and power contributions inside the placed-and-routed, energy-optimal ($n$ =16) EdgeBERT accelerator system which occupies 1.4mm$^2$ while consuming an average power of 86mW.\"\n ],\n [\n \"DVFS-based Latency-Aware Inference\",\n \"Fig.\",\n \"REF shows the DVFS-controlled supply voltage and clock frequency, and the energy savings of the latency-aware inference (LAI) on the energy-optimal accelerator design (i.e.\",\n \"with MAC vector size $n=16$ ) using latency targets between 50ms and 100ms (common latency thresholds for real-time human perception [57])).\",\n \"The results show that EdgeBERT optimized LAI achieves up to 7$\\\\times $ , and 2.5$\\\\times $ per-inference energy savings compared to the conventional inference (Base), and latency-unbounded early exit (EE) approaches, respectively, as seen in the SST-2 case.\",\n \"As AAS further cuts the number of computation cycles, we observe further relaxation of the supply voltage and clock frequency.\",\n \"At some latency targets (e.g., 75ms and 100ms in QQP and SST-2), further energy savings are not possible as V/F scaling bottoms out.\",\n \"blackTo underscore the different contributions to energy savings, at 75ms latency target for example in the case of MNLI, early exit prediction, adaptive attention span, DVFS, sparse execution, and eNVMs account for 21%, 12%, 23%, 39%, and 5%, respectively, of the total accelerator energy reduction.\",\n \"For stricter latency targets (e.g.\",\n \"$<$ 20ms), the proposed DFVS-based scheme can be used by scaling up to even higher MAC vector sizes (i.e.\",\n \"$n \\\\ge 32$ ).\"\n ],\n [\n \"Benefits of NVM Embeddings Storage\",\n \"BERT word embeddings are a natural fit for non-volatile storage, given that in EdgeBERT, we freeze them during fine-tuning and reuses them during inference By virtue of this scheme, we have established a compact 1.73MB baseline wherein the bitmask of the word embeddings is stored in a SLC ReRAM while the nonzero parameters are stored in a 2-bit per cell (MLC2) ReRAM buffer.\",\n \"Fig.\",\n \"REF illustrates the immense gains of leveraging this eNVM configuration during single-batch inference after SoC power-on.\",\n \"In EdgeBERT, ALBERT embeddings would only need to be read from the integrated ReRAM buffers due to being statically pre-loaded.\",\n \"The conventional operation dictates reading the embedding weights from off-chip DRAM, then writing them to dedicated on-chip volatile SRAM memories so they can be reused for future token identifications.\",\n \"The EdgeBERT approach enforces a latency and energy advantage that is, respectively, 50$\\\\times $ and 66,000$\\\\times $ greater than the overhead costs in the conventional operation.\",\n \"The non-volatility of this embedded storage means that these benefits can further scale with the frequency of power cycles.\",\n \"Figure: (a) Breakdown of latency and energy consumption in PU and SFU datapaths, and (b) 12nm physical layout, and area and power (@ 0.8V/1GHz) breakdown of the energy-optimal EdgeBERT accelerator (MAC size=16).Figure: Costs of reading all embedding weights after system power-on.\",\n \"Storing embeddings in ReRAMs gives EdgeBERT significant energy and latency advantages compared to the conventional approach requiring DRAM read followed by SRAM write/read.\"\n ],\n [\n \"Related Work\",\n \"Over the last decade, there has been extensive research on the design of high-performance and energy-efficient DNN hardware accelerators [24], [60], [11], [82], [83], [12], [47], [84], [5], [6], [26], [42], [31], [33], [54], [53], [62], [61], [66], [36], [35], [41], [45], [22], [86], [67], [71].\",\n \"As these accelerators are increasingly deployed at all computing scales, there is additional interest in the hardware community to automatically generate designs [77], [76], [80], [81].\",\n \"However, most of these works focus on CNN and RNN [20] computations, and not as much scrutiny has been given to accelerating Transformer-based networks with self-attention mechanisms.\",\n \"Recent work in accelerating Transformer-based NLP includes $A^{3}$ [23], which proposed a hardware architecture that reduces the number of computations in attention mechanisms via approximate and iterative candidate search.\",\n \"blackHowever, the $A^{3}$ scheme fetches the full and uncompressed data from DRAM before dynamically reducing computations in the hardware.\",\n \"In contrast, EdgeBERT learns the optimal attention search radius during the finetuning process and then leverages its very sparse mask to avoid unnecessary matrix multiplications.\",\n \"Therefore, our approach substantially eliminates DRAM accesses as the computation and memory optimizations are pre-learned before hardware acceleration.\",\n \"GOBO [88] blackfocuses on BERT quantization only via 3-bit clustering on the majority of BERT weights while storing the outlier weights and activations in full FP32 precision.\",\n \"blackAlthough this scheme significantly reduces DRAM accesses, it requires a mixed-precision computational datapath and a non-uniform memory storage.\",\n \"In contrast, EdgeBERT adopts uniform 8-bit data storage in SRAM and eNVMs memories.\",\n \"Lu et al.\",\n \"[46] proposes a dense systolic array accelerator for the Transformer's multi-head attention and feed-forward layers and optimizes Transformers' computations via matrix partitioning schemes.\",\n \"The EdgeBERT accelerator executes compressed sparse inference for higher energy efficiency.\",\n \"OPTIMUS [55] looks to holistically accelerate Transformers with compressed sparse matrix multiplications and by skipping redundant decoding computations.\",\n \"FlexASR [71] accelerates attention-based RNNs in a specialized attention datapath and only saves energy by gating the MAC when decoder RNN inputs are null.\",\n \"SpAtten [79] accelerates Transformer-based models via progressive cascade token and attention head pruning.\",\n \"The importance of each attention head is determined during the computation via a top-k ranking system.\",\n \"In contrast, EdgeBERT opts to learn the important attention heads during the fine-tuning process by activating adaptive attention spanning.\",\n \"The optimized and sparse attention spans are then used by the EdgeBERT accelerator to predicate the NLP computation.\",\n \"blackFinally, all the aforementioned NLP accelerators stores the embedding weights in traditional volatile SRAM memories.\",\n \"By contrast, this work recognizes that embedding weights do not change across NLP tasks.\",\n \"Therefore, EdgeBERT statically stores the word embeddings in high density eNVMs, generating substantial energy and latency benefits (Sec.\",\n \"REF ).\",\n \"Fig.\",\n \"REF qualitatively contrasts some of the prior work with EdgeBERT.\",\n \"Figure: blackComparison of EdgeBERT with prior work accelerating Transformer-based NLP models.\"\n ],\n [\n \"Conclusion\",\n \"As newer Transformer-based pre-trained models continue to generate impressive breakthroughs in language modeling, they characteristically exhibit complexities that levy hefty latency, memory, and energy taxes on resource-constrained edge platforms.\",\n \"EdgeBERT provides an in-depth and principled latency-driven methodology to alleviate these computational challenges in both the algorithm and hardware architecture layers.\",\n \"EdgeBERT adopts first-layer early exit prediction in order to perform dynamic voltage-frequency scaling (DVFS), at a sentence granularity, for minimal energy consumption while adhering to a prescribed target latency.\",\n \"Latency and memory footprint overheads are further alleviated by employing a balanced combination of adaptive attention span, selective network pruning, floating-point quantization.\",\n \"We further exploit and optimize the structure of eNVMs in order to store the shared multi-task parameters, granting EdgeBERT significant performance and energy savings from system power-on.\",\n \"Sentence-level, latency-aware inference on the EdgeBERT accelerator notably consumes 7$\\\\times $ and 2.5$\\\\times $ lower energy than the conventional full-model inference, and the latency-unbounded early exit approach, respectively.\"\n ],\n [\n \"Acknowledgement\",\n \"This work was supported in part by the Center for Applications Driving Architectures (ADA), one of six centers of JUMP, a Semiconductor Research Corporation (SRC) program co-sponsored by DARPA; DARPA’s DSSoC program; NSF Awards 1704834 and 1718160; Intel Corp.; and Arm Inc.\"\n ]\n]"},"id":{"kind":"string","value":"2011.14203"}}},{"rowIdx":2799,"cells":{"text":{"kind":"list like","value":[["Learning from Incomplete Features by Simultaneous Training of Neural\n Networks and Sparse Coding"],["Abstract In this paper, the problem of training a classifier on a dataset with incomplete features is addressed.","We assume that different subsets of features (random or structured) are available at each data instance.","This situation typically occurs in the applications when not all the features are collected for every data sample.","A new supervised learning method is developed to train a general classifier, such as a logistic regression or a deep neural network, using only a subset of features per sample, while assuming sparse representations of data vectors on an unknown dictionary.","Sufficient conditions are identified, such that, if it is possible to train a classifier on incomplete observations so that their reconstructions are well separated by a hyperplane, then the same classifier also correctly separates the original (unobserved) data samples.","Extensive simulation results on synthetic and well-known datasets are presented that validate our theoretical findings and demonstrate the effectiveness of the proposed method compared to traditional data imputation approaches and one state-of-the-art algorithm."],["Introduction","Learning methods from limited or imperfect data has attracted great attention in the literature recently.","Datasets with limited, weak, noisy labels or incomplete features represent an important and still open problem.","In this paper, we address the problem of training a classifier on a dataset with incomplete features, which arises in many machine learning applications where sometimes the measurements are incomplete, noisy or affected by artifacts.","Examples of this situation include: recommendation systems built upon the information gathered by different users where not all the users have fully completed their forms; medical datasets where typically not all tests can be performed on every patient; or a self-driving vehicle or robot where objects in the view field can be partially occluded.","Handling correctly the incomplete-features problem is a classical challenge in machine learning.","Skipping missing features by setting them to zero values damages the classification accuracy [1].","Most previous studies addressed this problem by using an imputation approach, which consists of performing data completion followed by training the classifier with those reconstructions (referred here as the sequential method).","However, this strategy cannot ensure the statistical consistence of the classifier, as data completion is usually fully unsupervised or label information is partially or inefficiently exploited.","In this work, a new supervised learning method is developed to train a general classifier, such as a logistic regression or a deep neural network, using only a subset of features per sample, while assuming sparse representations of data vectors on an unknown dictionary.","The proposed method simultaneously learns the classifier, the dictionary and the corresponding sparse representation of each input data sample.","In this way, we combine the approximation power and simplicity of sparse coding with the extraordinary ability of neural networks (NNs) to model complex decision functions (classifiers) with the goal to successfully train a classifier based on incomplete features.","We analyze the limitations of the sequential approach (section REF ), i.e.","imputation followed by training, and introduce the simultaneous classification and coding approach in section .","Our method consists of incorporating a sparse data representation model into a single cost function that is optimized for training the classifier and, at the same time, finding the best representation of the observed data.","A learning algorithm is presented in section REF to train a classifier on incomplete features and sufficient conditions under which such a classifier performs as good as the ideal classifier, i.e.","the one that can be obtained from complete observations, is identified (section REF ).","Extensive experimental results are presented in section , using synthetic and well known benchmark datasets that validate our theoretical findings and illustrate the effectiveness of the proposed method.","Practical sequential methods based on statistical imputation, such as computing the “mean”, “regression” and “multiple” imputation techniques are common practice [2].","Remarkably, it was shown that imputing with a constant, e.g.","the mean, is Bayes-risk consistent only when missing features are not informative [3].","More elaborated completion methods were also explored, such as $K$ -nearest neighbor estimators, multilayer or recurrent NNs and others, see [4] and references therein.","However, sequential methods do not fully exploit label information.","Data labels can provide valuable information about missing features that could potentially improve the classifier learning process.","Recent advances on probabilistic generative models have allowed for a formulation of supervised learning with incomplete features as a statistical inference problem arriving at algorithms that significantly outperformed sequential methods.","In the seminal work [5], a framework for maximum likelihood density estimation based on mixtures models was proposed and successfully applied to small incomplete-features problems.","In particular, a Gaussian Mixture Model (GMM) was fitted to incomplete-data through an Expectation-Maximization (EM) algorithm.","Building upon this generative model strategy, some approaches have considered integrating out the missing values based on a simple logistic regression function [6], [7].","Other versions of this approach proposed an explicit simultaneous learning of the model and the decision function [8], [9].","While probabilistic generative models provided a nice and elegant approach to the incomplete-features problem showing good results on small datasets, they are not suitable for many modern machine learning applications because: (1) despite some acceleration techniques were explored, e.g.","[10], [11], those algorithms are computationally expensive becoming prohibitive for moderate to large datasets; (2) GMM is impractical for modeling high-dimensional datasets because the number of parameters to achieve good approximations becomes unmanageable; and (3) they do not consider complex classification functions as the ones provided by deep NN architectures.","Recently, some approaches based on the low-rank property of the features data matrix were investigated and algorithms for data completion were proposed incorporating the label information [12], [13], [14].","Since the rank estimation of a matrix is a computationally expensive task, usually based on the Singular Value Decomposition (SVD), the obtained algorithms are prohibitive to solve modern machine learning problems with large datasets.","Additionally, as in the case of the probabilistic generative models, none of these methods considered complex classification functions.","To overcome this drawback, more recently, a framework based on various NN architectures such as autoencoders, multilayer perceptrons and Radial Basis Function Networks (RBFNs), was proposed for handling missing input data by setting a probabilistic model, e.g.","a GMM, for every missing feature, which is trained together with the NN weights [15].","This method combined the great capability of NNs to approximate complex decision functions with the nice formulation of the GMM to model missing data.","However, it inherited the drawbacks of GMMs, i.e.","they are not well suited to higher-dimensional datasets.","On the other hand, during the last few years in the signal processing community, there has been a rapid development of theory and algorithms for sparse coding approximations which, by exploiting the redundancy of natural signals, are able to provide simple and accurate models of complex data distributions, see [16], [17], [18], [19], [20] and references therein.","Sparse coding is nearly ubiquitous in Nature, for example, it is found in the way that neurons encode sensory information [21], [22].","Sparse representations of data showed to be useful also in classification problems.","In [23], a Linear Discriminant Analysis (LDA) classifier was trained on corrupted data providing a robust classification method.","In [24], algorithms for learning discriminative sparse models, instead of purely reconstructive ones, were proposed based on simple linear and bilinear classifiers.","Similar methods were also studied by either using class-specific dictionaries [25], [26] or using a single one for all classes [27].","However, these proposed methods neither were applied to the incomplete-features problem nor considered deep NN classifiers."],["Problem formulation","We assume a supervised learning scenario with vector samples and labels $\\lbrace {\\mathbf {x}}_i, y_i\\rbrace $ , $i=1,2,\\dots , I$ , ${\\mathbf {x}}_i \\in {R}^N$ and $y_i \\in \\lbrace 0,1,\\dots , C-1\\rbrace $ ($C$ classes).","However, we are constrained to observe only subsets of features and their labels: $\\lbrace {\\mathbf {x}}^o_i, y_i\\rbrace $ , ${\\mathbf {x}}^o_i \\in {R}^{M_i}$ with $M_i < N$ .","Unobserved (missing) features are denoted by ${\\mathbf {x}}^m_i \\in {R}^{N-M_i}$ .","We consider arbitrary patterns of missing features, which are allowed to be different for each data instance $i$ .","The set of indices of missing features at sample $i$ is denoted by ${\\cal {M}}_i$ , i.e.","${\\mathbf {x}}^m_i = {\\mathbf {x}}_i({\\cal {M}}_i)$ and ${\\mathbf {x}}^o_i = {\\mathbf {x}}_i\\left({\\overline{\\mathcal {M}}}_i\\right)$ .","We define the set of all $K$ -sparse vectors $\\Sigma _K^P = \\lbrace {\\mathbf {s}}\\in {R}^P \\text{ s.t. }","\\Vert {\\mathbf {s}}\\Vert _0 \\le K\\rbrace $ (containing at most $K$ non-zero entries) and assume that data vectors ${\\mathbf {x}}_i$ admit $K$ -sparse representations over an unknown dictionary $\\mathbf {D}\\in {R}^{N\\times P}$ ($P \\ge N$ ): ${\\mathbf {x}}_i = \\mathbf {D}{\\mathbf {s}}_i, \\mbox{ with } {\\mathbf {s}}_i \\in \\Sigma _K^{P}.$ The columns of a dictionary are called “atoms” because every data vector can be written as a linear combination of at most $K$ elementary components.","Sometimes dictionaries are orthogonal such as the ones derived from the Discrete Cosine or Wavelet [19] transforms.","However, overcomplete ($P \\ge N$ ) nonorthogonal dictionaries have demonstrated to play an important role in image processing tasks such as denoising, inpainting, etc [17], [28].","By partitioning $\\mathbf {D}$ according to the pattern of missing features at sample $i$ , we obtain $\\mathbf {D}^o_i =\\mathbf {D}\\left({\\overline{\\mathcal {M}}}_i,:\\right) \\in {R}^{M_i\\times P}$ and $\\mathbf {D}^m_i =\\mathbf {D}({\\cal {M}}_i,:)\\in {R}^{{(N-M_i)}\\times P}$ , which according to equation (REF ) implies: ${\\mathbf {x}}^o_i = \\mathbf {D}^o_i {\\mathbf {s}}_i, \\mbox{ and } {\\mathbf {x}}^m_i = \\mathbf {D}^m_i {\\mathbf {s}}_i.$ Let us assume that a perfect classifier, e.g.","a logistic regression or deep NN, that assigns probability $p_{\\Theta }(\\hat{y} | {\\mathbf {x}})$ to predicted label $\\hat{y}$ given data ${\\mathbf {x}}$ can be trained on the complete dataset $\\lbrace {\\mathbf {x}}_i, y_i\\rbrace $ , such that, in a two-classes scenario ($C=2$ ), $p_{\\Theta }(\\hat{y} = y_i | {\\mathbf {x}}_i) > p_{\\Theta }(\\hat{y} \\ne y_i | {\\mathbf {x}}_i)$ , $\\forall i =1,2,\\dots , I$ , where $\\Theta $ is the set of trained parameters.","Our goal is to develop a method to obtain an estimate $\\hat{\\Theta }$ of parameters using only the incomplete dataset $\\lbrace {\\mathbf {x}}^o_i, y_i\\rbrace $ and to identify conditions under which such a classifier is compatible with the ideal one."],["Why training after imputation is difficult?","If the $K$ -sparse representations of the observations ${\\mathbf {x}}^o_i$ were unique, then ${\\mathbf {x}}_i$ can be perfectly reconstructed from the incomplete observations and the classifier can be successfully trained using these reconstructions.","In the particular case where the dictionary is known in advance, there exist conditions on the sampling patterns based on the coherence, spark or RIP (Restricted Isometry Property) of matrix $\\mathbf {D}^o_i$ that can guarantee uniqueness [20].","However, these conditions are difficult to meet in practice and determining RIP/Spark properties are NP-hard in general [29].","Moreover, in the general case where the dictionary $\\mathbf {D}$ is unknown and needs to be learned from data, it is even more difficult to obtain well separated reconstructions which certainly leads to suboptimal or wrong classifiers.","Next, we provide some intuition about the limitation of the sequential approach through a toy example.","Let us consider the classification of hand-written digit images belonging to two classes: “3s” and “8s” and assume that they admit 2-sparse representations over a dictionary.","Fig.","REF (a-b) shows the representations of two example vectors ${\\mathbf {x}}_i$ and ${\\mathbf {x}}_j$ belonging to classes “3” and “8”, respectively.","If only the right halves of the images are observed and no label information is provided, we are clearly faced with a problem because our observed samples from two different classes are identical, i.e.","${\\mathbf {x}}^o_i = {\\mathbf {x}}^o_j$ .","It is obvious that at least two possible 2-sparse representations for the observed data exist as illustrated in Fig.","REF (c).","When the sparse solution is not unique, we may end up reconstructing wrong vectors that could not be even well separated as illustrated in Fig.","REF (d-e).","In general, sequential methods using only the information of observed features are prone to fail because the non-uniqueness of solutions can make the training of a good classifier an impossible task.","However, we could solve this problem by incorporating the labelling information from the very beginning as it is proposed in the following section."],["Simultaneous learning and coding approach","We propose to train the classifier and find the proper representation, not only as sparse as possible but also providing the best separation of classes.","We want to combine the training of the classifier together with the learning of a dictionary and optimal sparse representations such that the reconstructed data vectors are compatible with observations and well separated.","To do that we propose to minimize the following global cost function: $J(\\Theta , \\mathbf {D}, {\\mathbf {s}}_i) = \\nonumber \\\\\\textstyle \\underbrace{\\frac{1}{I}\\sum _{i=1}^I\\big \\lbrace J_0(\\Theta , \\hat{{\\mathbf {x}}}_i, y_i) + \\lambda _{1} J_1(\\mathbf {D},{\\mathbf {s}}_i)\\big \\rbrace }_{F(\\Theta , \\mathbf {D}, {\\mathbf {s}}_i)} +\\underbrace{\\frac{1}{I}\\sum _{i=1}^I\\big \\lbrace \\lambda _{2}J_2({\\mathbf {s}}_i)\\big \\rbrace }_{G({\\mathbf {s}}_i)},$ with respect to $\\Theta $ , $\\mathbf {D}$ and ${\\mathbf {s}}_i$ ($i=1,2,\\dots , I$ ), where $\\Theta $ contains the classifier parameters, i.e.","the vector of weights in a deep NN classifier architecture; $\\mathbf {D}\\in {R}^{N\\times P}$ ($P \\ge N$ ) is a dictionary and ${\\mathbf {s}}_i \\in \\Sigma _K^{P}$ are the representation coefficients such that the reconstructed data vectors are $\\hat{{\\mathbf {x}}}_i = \\mathbf {D}{\\mathbf {s}}_i$ .","$J_0(\\Theta , \\hat{{\\mathbf {x}}}_i, y_i)$ is a measure of the classification error for the reconstructed sample vector $\\hat{{\\mathbf {x}}}_i$ .","Typically, we use the crossentropy measure, i.e.","$J_0(\\Theta , \\hat{{\\mathbf {x}}}_i, y_i) = -\\log [p_{\\Theta }(y_i | \\hat{{\\mathbf {x}}}_i)]$ , where $p_{\\Theta }(y_i | \\hat{{\\mathbf {x}}}_i)$ is the probability assigned by the classifier to sample $\\hat{{\\mathbf {x}}}_i$ as belonging to class $y_i$ .","$J_1(\\mathbf {D},{\\mathbf {s}}_i)$ is a measure of the approximation error of the reconstruction when it is restricted to observed features, which is defined as follows: $J_1(\\mathbf {D},{\\mathbf {s}}_i)=\\frac{M_i}{N}\\Vert {\\mathbf {m}}_i \\odot ({\\mathbf {x}}_i - \\mathbf {D}{\\mathbf {s}}_i)\\Vert ^2$ , where $\\odot $ stands for the entry-wise product, ${\\mathbf {m}}_i \\in {R}^N$ is the observation mask for sample $i$ , i.e.","$m_i(n) = 0$ (1) if data entry ${\\mathbf {x}}_i(n)$ is missing (available); and $J_2({\\mathbf {s}}_i) = \\frac{1}{N}\\Vert {\\mathbf {s}}_i\\Vert _1$ is proportional to the $\\ell _1$ -norm whose minimization promotes the sparsity of the representation since $\\ell _1$ -norm is a convenient proxy for $\\ell _0$ -norm [30].","Finally, the hyper-parameters $\\lambda _1$ and $\\lambda _2$ allow us to give more or less importance to the representation accuracy and its sparsity, with respect to the classification error.","Intuitively, minimizing equation (REF ) favors solutions that not only have sparse representations compatible with observed features, but also providing reconstructions that are best separated in the given classes.","Figure: Toy example: (a) 4 out PP dictionary elements 𝐝 i \\mathbf {d}_i (atoms).","(b) Digits “3” and “8” can be represented by combining only two atoms in the dictionary (2-sparse representations).","(c) A left-half occluded digit “3” or “8” admits more than one 2-sparse representation (sum of 𝐝 1 o \\mathbf {d}^o_1 and 𝐝 2 o \\mathbf {d}^o_2, or 𝐝 3 o \\mathbf {d}^o_3 and 𝐝 4 o \\mathbf {d}^o_4).","(d) Linearly separable samples from two classes (A and B) having two features: 𝐱=[x 1 ,x 2 ]∈R 2 \\mathbf {x} = [x_1, x_2]\\in {R}^2 where incomplete observations are taken by observing only one feature.","Note that 𝐱 A \\mathbf {x}_A and 𝐱 B \\mathbf {x}_B belong to different classes but their observations are identical.","(e) Without using label information, the sequential method could lead to wrong reconstructions of data vectors, i.e.","𝐱 ^ A ≠𝐱 A \\hat{\\mathbf {x}}_A\\ne \\mathbf {x}_A and 𝐱 ^ B ≠𝐱 B \\hat{\\mathbf {x}}_B\\ne \\mathbf {x}_B making the set of reconstructed vectors not linearly separable."],["A sparsity-promoting sub-gradient optimization algorithm","To minimize the cost function in equation (REF ) we propose to alternate between the optimization over ${\\mathbf {s}}_i$ ($i=1,2,\\dots , I$ ) and $\\lbrace \\Theta ,\\mathbf {D}\\rbrace $ using the training dataset (incomplete).","For fixed $\\lbrace \\Theta ,\\mathbf {D}\\rbrace $ , the optimization with respect to ${\\mathbf {s}}_i$ is a non-smooth separable minimization sub-problem, which was extensively studied in the literature [31], [32].","In this sub-problem, the objective function is written as the sum of $F(\\Theta , \\mathbf {D}, {\\mathbf {s}}_i)$ and a non-smooth separable function $G({\\mathbf {s}}_i)$ , for which highly specialized, efficient and provable convergent solvers, namely the Coordinate Gradient Descent (CGD), already exists.","However, the following key differences in our setting makes it not suitable for the CGD approach: first, our function $F(\\Theta , \\mathbf {D}, {\\mathbf {s}}_i)$ involves evaluation of a multi-layer NN classifier, which can be non-smooth due to involved activation functions like ReLU or others; second, and more importantly, the computation of its second derivatives (Hessian) becomes prohibitive.","Therefore, we choose a simpler and standard first order (stochastic sub-gradient based) search of local minima with back-propagation.","We take the strategy similar to the heuristics used in [33].","To update ${\\mathbf {s}}_i$ , we need to subtract $\\sigma _{{\\mathbf {s}}} \\frac{\\partial J}{\\partial {\\mathbf {s}}_i}(j)$ from each coordinate $j$ provided that we do not cross zero in the process in order to avoid escaping from a region where $G({\\mathbf {s}}_i)$ is differentiable.","In such a case, we let the new value of ${\\mathbf {s}}_i(j)$ be exactly zero.","More specifically, we define $\\Delta _i(j)= -\\sigma _{{\\mathbf {s}}} \\frac{\\partial J}{\\partial {\\mathbf {s}}_i}(j)$ and, if ${\\mathbf {s}}_i(j)[{\\mathbf {s}}_i(j)+\\Delta _i(j)] < 0$ (zero crossing condition), we re-define $\\Delta _i(j)=-{\\mathbf {s}}_i(j)$ ; finally we update ${\\mathbf {s}}_i \\leftarrow {\\mathbf {s}}_i + \\Delta _i$ .","It is noted that, once a coefficient ${\\mathbf {s}}_i(j)$ reaches zero at a coordinate $j$ , it becomes fixed, in other words, sparsity of solution ${\\mathbf {s}}_i$ is monotonically increasing with iterations.","When ${\\mathbf {s}}_i$ is fixed, our problem is reduced to minimize $F(\\Theta , \\mathbf {D}, {\\mathbf {s}}_i)$ with respect to $\\Theta $ and $\\mathbf {D}$ , which is easily done by standard first order (stochastic gradient based) search of local minima.","The algorithm proposed for the training phase is presented as Algorithm REF .","In addition, for the testing phase, if the test dataset is incomplete, we need to find first the sparsest representation for the given observations, compute the reconstructions $\\hat{{\\mathbf {x}}}_i = \\mathbf {D}{\\mathbf {s}}_i$ and then apply the previously learned classifier to them as presented in Supp.","material, Algorithm REF .",": Simultaneous classification and coding [1] $\\lbrace {\\mathbf {x}}^o_i, y_i\\rbrace $ , $i=1,2,\\dots , I$ , hyper-parameters $\\lambda _1$ and $\\lambda _2$ , $N_{iter}$ and update rates $\\sigma _{\\Theta }$ , $\\sigma _{\\mathbf {D}}$ and $\\sigma _{{\\mathbf {s}}}$ Weights $\\Theta $ and reconstructions $\\hat{{\\mathbf {x}}}_i= \\mathbf {D}{\\mathbf {s}}_i, \\forall i$ Randomly initialize $\\Theta , \\mathbf {D}, {\\mathbf {s}}_i, \\forall i$ $n\\le N_{iter}$ Fix ${\\mathbf {s}}_i$ , update $\\Theta $ and $\\mathbf {D}$ : $\\Theta = \\Theta - \\sigma _{\\Theta } \\frac{\\partial J}{\\partial \\Theta }$ $\\mathbf {D}= \\mathbf {D}- \\sigma _{\\mathbf {D}} \\frac{\\partial J}{\\partial \\mathbf {D}}$ Normalize columns of matrix $\\mathbf {D}$ Fix $\\Theta $ and $\\mathbf {D}$ , update ${\\mathbf {s}}_i$ , $\\forall i$ : $\\Delta _i= -\\sigma _{{\\mathbf {s}}} \\frac{\\partial J}{\\partial {\\mathbf {s}}_i}$ , $\\forall i$ ${\\mathbf {s}}_i(j)[{\\mathbf {s}}_i(j)+\\Delta _i(j)] < 0$ $\\Delta _i(j)=-{\\mathbf {s}}_i(j), \\forall (i, j)$ ; ${\\mathbf {s}}_i = {\\mathbf {s}}_i + \\Delta _i$ , $\\forall i$ return $\\Theta , \\mathbf {D}, {\\mathbf {s}}_i, \\hat{{\\mathbf {x}}}_i = \\mathbf {D}{\\mathbf {s}}_i, \\forall i$"],["Theoretical analysis","Here, we investigate about conditions under which a perfect classifier of the complete data can be obtained from incomplete data samples."],["Logistic regression","Let us first consider a logistic regression classifier [34] where the set of parameters $\\Theta = \\lbrace {\\mathbf {w}}, b\\rbrace $ are a vector ${\\mathbf {w}}\\in {R}^N$ and a scalar $b$ (bias).","A perfect classifier exists if there is a hyperplane that separates both classes, i.e., for each data vector ${\\mathbf {x}}_i$ : $f({\\mathbf {x}}_i) = \\langle {\\mathbf {w}}, {\\mathbf {x}}_i \\rangle + b > 0$ if $y_i=1$ , and $f({\\mathbf {x}}_i) \\le 0$ if $y_i=0$ .","We consider data samples admitting a $K$ -sparse representations ${\\mathbf {x}}_i = \\mathbf {D}{\\mathbf {s}}_i$ with dictionary $\\mathbf {D}\\in {R}^{N\\times P}$ having unit-norm columns.","We also assume an arbitrary pattern of missing features $\\mathcal {M}_i$ such that, data samples and dictionary are partitioned as $\\lbrace {\\mathbf {x}}^m_i,{\\mathbf {x}}^o_i \\rbrace $ and $\\lbrace \\mathbf {D}^m_i,\\mathbf {D}^o_i \\rbrace $ , respectively.","The following lemma identifies a sufficient condition under which, if we are able to train a classifier on incomplete observations such that the reconstructed data points are well separated by a hyperplane, then the same classifier correctly separates the original (unobserved) data vectors.","Lemma 2.1 (Sufficient condition type I) Suppose that we have obtained an alternative dictionary $\\mathbf {D}^{\\prime } \\ne \\mathbf {D}\\in {R}^{N\\times P}$ such that, for the incomplete observations ${\\mathbf {x}}^o_i \\in {R}^{M_i}$ , the $K$ -sparse representation solutions are non-unique, i.e.","$\\exists {\\mathbf {s}}_i, {\\mathbf {s}}_i^{\\prime } \\in \\Sigma _K^P$ such that ${\\mathbf {x}}^o_i = \\mathbf {D}^o_i{\\mathbf {s}}_i = \\mathbf {D}^{\\prime o}_i{\\mathbf {s}}_i^{\\prime }$ , where ${\\mathbf {s}}_i \\in {R}^P$ are the vectors of coefficients of the true data and ${\\mathbf {s}}_i^{\\prime }$ provides reconstructions $\\hat{{\\mathbf {x}}}_i = \\mathbf {D}^{\\prime }{\\mathbf {s}}_i^{\\prime }$ .","If a perfect classifier $\\lbrace {\\mathbf {w}}, b\\rbrace $ of the reconstructions $\\hat{{\\mathbf {x}}}_i$ exists s.t.","$|f(\\hat{{\\mathbf {x}}}_i)| > \\epsilon _i > 0$ and $\\epsilon _i > |\\langle {\\mathbf {w}}^m_i , {\\mathbf {e}}^m_i\\rangle |$ with ${\\mathbf {e}}^m_i = {\\mathbf {x}}^m_i - \\hat{{\\mathbf {x}}}^m_i$ , then the full data vectors ${\\mathbf {x}}_i$ are also perfectly separated with this classifier, in other words: $f({\\mathbf {x}}_i) = \\langle {\\mathbf {w}}_i, {\\mathbf {x}}_{i} \\rangle + b > 0$ ($\\le 0$ ) if $y_i =1$ ($y_i=0$ ).","By using the missing/observed partition and omitting the sample index $i$ , we can write: $f({\\mathbf {x}}) = \\langle {\\mathbf {w}}, {\\mathbf {x}}\\rangle + b = \\langle {\\mathbf {w}}^{o}, {\\mathbf {x}}^o \\rangle + \\langle {\\mathbf {w}}^{m}, {\\mathbf {x}}^m \\rangle + b$ .","If we add and subtract the term $\\langle {\\mathbf {w}}^{m}, \\hat{{\\mathbf {x}}}^m \\rangle $ on the right left hand, arrange terms and use the fact that $\\hat{{\\mathbf {x}}}^o = \\mathbf {D}^{\\prime o}{\\mathbf {s}}^{\\prime } = {\\mathbf {x}}^o$ , we get: $f({\\mathbf {x}}) = f(\\hat{{\\mathbf {x}}}) + \\langle {\\mathbf {w}}^{m}, {\\mathbf {e}}^m \\rangle .$ Since we assumed that $f(\\hat{{\\mathbf {x}}}) > \\epsilon >0$ (for $y_i = 1$ ) and $|\\langle {\\mathbf {w}}^{m}, {\\mathbf {e}}^m \\rangle | < \\epsilon $ , it implies that $f({\\mathbf {x}}) > 0$ .","Basically, condition (REF ) means requiring that reconstruction vector $\\hat{{\\mathbf {x}}}$ has a distance $\\epsilon $ to the separating hyperplane larger than the absolute dot product between ${\\mathbf {w}}$ and the residual $ {\\mathbf {e}}= {\\mathbf {x}}- \\hat{{\\mathbf {x}}}$ , which of course is true when the reconstruction is accurate, i.e.","${\\mathbf {x}}\\approx \\hat{{\\mathbf {x}}}$ .","However, in practice, reconstructions are not accurate so we are interested in conditions under which Lemma REF can still holds.","Below, we derive a more restrictive but useful sufficient condition: Proposition 2.1 (Sufficient condition type II) Under the same hypothesis of Lemma REF , the following condition is enough to guarantee a proper classifier trained on incomplete data: $\\epsilon > |\\langle {\\mathbf {w}}^{m}, {\\mathbf {x}}^m\\rangle | + |\\langle {\\mathbf {w}}^{m}, \\hat{{\\mathbf {x}}}^m \\rangle |.$ By using the fact that $|\\langle {\\mathbf {w}}^{m}, {\\mathbf {e}}^m \\rangle | = |\\langle {\\mathbf {w}}^{m}, {\\mathbf {x}}^m - \\hat{{\\mathbf {x}}}^m \\rangle | \\le |\\langle {\\mathbf {w}}^{m}, {\\mathbf {x}}^m \\rangle | + |\\langle {\\mathbf {w}}^{m}, \\hat{{\\mathbf {x}}}^m \\rangle |$ , and applying Lemma REF the proof is completed.","We highlight that, in our experiments, we were able to verify that Sufficient Condition type II is met in practice (see section , Fig.","REF ).","In Supp.","material section REF , we derive an additional sufficient condition based on the Restricted Isometry Property (RIP) of the dictionary $\\mathbf {D}$ and sparsity level $K$ , showing that sufficient condition (REF ) is easier to hold for datasets admitting highly sparse representations on dictionaries as close to orthogonal ones as possible."],["Multilayer-perceptron","Lemma REF can be straightforwardly generalized to multilayer-perceptron NNs where, if a softmax function is used at the output of the last layer then, as before, the prediction is based on the sign of the linear function: $\\small f({\\mathbf {x}}) = \\langle {\\mathbf {w}}, {\\mathbf {x}}^{(L)}\\rangle + b, \\mbox{ with } {\\mathbf {x}}^{(l)} = h\\left( \\mathbf {W}^T_l {\\mathbf {x}}^{(l-1)} + \\mathbf {B}_l\\right),$ $l=1,2,\\dots ,L$ , where $L+1$ is the total number of layers, $N_l$ is the number of neurons in layer $l$ , ${\\mathbf {w}}\\in {R}^{N_{L+1}}$ contains the weights in the last layer, $h(\\cdot )$ is an activation function, e.g.","ReLU, $\\mathbf {W}_l\\in {R}^{N_{l-1}\\times N_l}$ and $\\mathbf {B}_l \\in {R}^N_l$ contain the weights and biases associated to neurons at layer $l$ ; and ${\\mathbf {x}}^{(0)} = {\\mathbf {x}}$ is the input data vector.","In this case, the first layer matrix $\\mathbf {W}_1 \\in {R}^{N\\times N_1}$ can be partitioned into submatrices $\\mathbf {W}^o_{1i} \\in {R}^{M \\times N_1}$ and $\\mathbf {W}^m_{1i} \\in {R}^{(N-M) \\times N_1}$ according to the observed and missing input features, respectively.","Proposition 2.2 () Under the same conditions of Lemma REF , if a NN-classifier $\\lbrace {\\mathbf {W}_l,\\mathbf {B}_l} (l=1,2,\\dots L),{\\mathbf {w}}, b, h=ReLU\\rbrace $ of the reconstruction $\\hat{{\\mathbf {x}}}_i$ exists such that $\\epsilon _i > A \\max _j |\\langle \\mathbf {W}^m_{1i}(:,j) , {\\mathbf {e}}^m_i\\rangle | ,$ where $A = \\Vert {\\mathbf {w}}\\Vert \\prod _{l=2}^L \\Vert \\mathbf {W}_l \\Vert _2$ and ${\\mathbf {e}}^m_i = {\\mathbf {x}}^m_i - \\hat{{\\mathbf {x}}}^m_i$ , then the full data vector ${\\mathbf {x}}_i$ is also perfectly separated, in other words: $f({\\mathbf {x}}_i) > 0$ ($< 0$ ) if $y_i =1$ ($y_i=0$ ).","In the proof of Lemma REF , we were interested in finding a bound of the output error when the input ${\\mathbf {x}}$ of a classifier is perturbed, i.e.","we found conditions such that $|f({\\mathbf {x}}) - f({\\mathbf {x}}+ \\mathbf {\\delta })| < \\epsilon $ .","By generalizing the classifier to the case of a multilayer perceptron we can derive the proof as follows: Given a perturbation $\\mathbf {\\delta }^{(l-1)}\\in {R}^{N_l}$ at the input of layer $l-1$ , i.e.","$\\hat{{\\mathbf {x}}}^{(l-1)} = {\\mathbf {x}}^{(l-1)} + \\mathbf {\\delta }^{(l-1)}$ , it is propagated to the output of layer $l$ .","By writing the error at the output we obtain: $\\scriptstyle \\mathbf {\\delta }^{(l)} = h\\left( \\mathbf {W}_l^T {\\mathbf {x}}^{(l-1)} + \\mathbf {B}_l + \\mathbf {W}_l^T \\mathbf {\\delta }^{(l-1)} \\right) - h\\left( \\mathbf {W}_l^T {\\mathbf {x}}^{(l-1)} + \\mathbf {B}_l \\right),$ and, by using the sub-additivity of ReLU function $h(\\cdot )$ , i.e.","$h(a+b) \\le h(a) + h(b)$ , we derive the following entry-wise inequality: $\\small \\mathbf {\\delta }^{(l)} \\le h \\left( \\mathbf {W}_l^T \\mathbf {\\delta }^{(l-1)} \\right),$ and, by considering the property of ReLU activation function $\\Vert h({\\mathbf {x}})\\Vert \\le \\Vert {\\mathbf {x}}\\Vert $ , it turns out: $\\small \\Vert \\mathbf {\\delta }^{(l)}\\Vert \\le \\Vert \\mathbf {W}_l^T \\mathbf {\\delta }^{(l-1)} \\Vert ,$ Since the last layer of the NN is a linear classifier as in the case of Lemma REF , we can ask that $ \\langle {\\mathbf {w}}, \\mathbf {\\delta }^{(L)}\\rangle < \\epsilon $ .","Thus, by recursively using equation (REF ), we write $ \\langle {\\mathbf {w}}, \\mathbf {\\delta }^{(L)}\\rangle \\le \\Vert {\\mathbf {w}}\\Vert \\Vert \\mathbf {\\delta }^{(L)}\\Vert \\le \\Vert {\\mathbf {w}}\\Vert \\Vert \\mathbf {W}_L\\Vert _2 \\Vert \\mathbf {W}_{l-1}\\Vert _2 \\cdots \\Vert \\mathbf {W}_2\\Vert _2 \\Vert \\mathbf {\\delta }^{(1)} \\Vert .$ By defining $A = \\Vert {\\mathbf {w}}\\Vert \\prod _{l=2}^L\\Vert \\mathbf {W}_l\\Vert _2$ , evaluating equation (REF ) with $l=1$ and taking into account that perturbation at the input of first layer is $\\mathbf {\\delta }^{(0)} = {\\mathbf {e}}$ with ${\\mathbf {e}}^o = \\mathbf {0}$ , we arrive at: $\\small \\langle {\\mathbf {w}}, \\mathbf {\\delta }^{(L)}\\rangle \\le A \\Vert \\mathbf {W}_1^{mT} {\\mathbf {e}}^m\\Vert \\le A \\max _j |\\langle \\mathbf {W}^m_{1}(:,j) , {\\mathbf {e}}^m \\rangle | < \\epsilon ,$ which completes the proof.","It is interesting to note that $A=1$ is attained when unit-norm filters (columns of $\\mathbf {W}_l$ ) are orthogonal, which can be imposed by using orthogonality regularization [35]."],["Experimental results","We implemented all the algorithms in Pytorch 1.0.0 on a single GPU.","Implementation details are reported in Supplemental material, sections REF and REF .","The code is available at https://github.com/ccaiafa/SimultCodClass.","Synthetic datasets: We synthetically generated $I=11,000$ ($10,000$ training $+$ $1,000$ test) $K$ -sparse data vectors ${\\mathbf {x}}_i \\in {R}^{100}$ using a dictionary $\\mathbf {D}\\in {R}^{100 \\times 200}$ obtained from a Gaussian distribution with normalized atoms, i.e.","$\\Vert \\mathbf {D}(:,j)\\Vert =1, \\forall j$ .","A random hyperplane $\\lbrace {\\mathbf {w}}, b\\rbrace $ with ${\\mathbf {w}}\\in {R}^N$ , $b\\in {R}$ was randomly chosen dividing data vectors into two classes according to the sign of the expression $\\langle {\\mathbf {w}}, {\\mathbf {x}}_i \\rangle +b$ , which defined the label $y_i$ .","We also controlled the degree of separation between classes by discarding all data vectors with distances to the hyperplane lower than a pre-specified threshold, i.e.","$|\\langle {\\mathbf {w}}, {\\mathbf {x}}_i \\rangle +b| < d$ .","We used $n=10$ repetitions of each experiment with different masks and input data in order to compute statistics.","We applied our simultaneous method (Simult.)","with hyperparameters $\\lambda _1$ and $\\lambda _2$ in the cost function (REF ) tuned via cross-validation to train a logistic regression classifier on incomplete datasets with randomly distributed missing features.","Then, we computed the classification accuracy on the complete test dataset and compared the results against the following standard sequential methods: Sequential Sparsity based (Seq.","Sp.","): reconstructions are obtained by finding the sparsest representation compatible with the observations solving a LASSO problem.","We used Algorithm REF as shown in the Supp.","material; Zero Fill (ZF): missing features are filled with zeros, which is equivalent to ignore unknown values; Mean Unsupervised (MU): missing features are filled with the mean computed on the available values; Mean Supervised (MS): as in the previous case but the mean is computed on the same class vectors only; K-Nearest Neighbor (KNN): as in the previous case but the mean is computed on the K-nearest neighbors of the same class vectors only.","To compare the performance of classifiers, we computed the mean accuracy $\\pm $ standard error of the mean (s.e.m.","), with $n=10$ , on complete test datasets using all the methods for two levels of separation between classes ($d=0.0, 0.2$ ), two levels of sparsity ($K=4,32$ ) and missing features in the training dataset ranging from $25\\%$ to $95\\%$ as shown in Fig.","REF .","Our results show that the simultaneous algorithm clearly outperforms all the sequential methods.","A t-test was performed to evaluate the statistical significance with $p < 0.05$ of the difference between our algorithm and MS.","It is interesting to note that, when classes has some degree of separation ($d=0.2$ ), using the simple MS method, can give good results but not better than our algorithm.","Figure: Experimental results on synthetic dataset with random masks using our algorithm (red) and compared to various sequential methods.","Test accuracy (mean ±\\pm s.e.m with n=10n=10) is shown as a function of the percentage of missing features for separation of classes d=0.0,0.2d=0.0, 0.2 and levels of sparsity K=4,32K=4,32.","Statistical significance for the difference between Simult.","and MS is shown (p<0.05p < 0.05).In the second experiment, we generated $I =10,000$ $K$ -sparse data vectors ${\\mathbf {x}}_i \\in {R}^{100}$ using $\\mathbf {D}\\in {R}^{100 \\times 100}$ and we evaluated the sufficient condition of equation (REF ) on $n=10$ repetitions of the experiment with $95\\%$ missing features and separation $d=0.0$ .","Fig.","REF clearly shows that the sufficient condition is mostly met in practice, especially for highly sparse representations of input data (small $K$ ).","This means that in practice it is not necessary to accurately reconstruct the input vectors, it is enough to capture the intrinsic characteristics of the classes such that the distances of reconstructions to the separating hyperplane satisfy the sufficient condition (REF ).","Figure: Verification of the sufficient condition () for various levels of sparsity KK: 2D-histogram of ϵ\\epsilon versus g=|〈𝐰 m ,𝐱 m 〉|+|〈𝐰 m ,𝐱 ^ m 〉|g = |\\langle {\\mathbf {w}}^{m}, {\\mathbf {x}}^m\\rangle | + |\\langle {\\mathbf {w}}^{m}, \\hat{{\\mathbf {x}}}^m \\rangle |.","Mean + s.e.m (n=10n=10) percentage of correctly classified data samples are shown for ϵ>g\\epsilon > g and ϵ \\epsilon _i >0$ and $\\epsilon _i > 2 \\Vert {\\mathbf {w}}^m_i\\Vert _1\\sqrt{\\frac{K}{1-\\delta ^i_K}},$ then the full data vector ${\\mathbf {x}}_i$ is also perfectly separated with this classifier, in other words: $f({\\mathbf {x}}_i) = \\langle {\\mathbf {w}}_i, {\\mathbf {x}}_{i} \\rangle + b > 0$ ($\\le 0$ ) if $y_i =1$ ($y_i=0$ ).","Let us prove that the sufficient condition (REF ) is met under the hypothesis of Theorem REF .","Taking into account that ${\\mathbf {x}}_i^m = \\mathbf {D}_i^m {\\mathbf {s}}_i$ , we can write $\\small | \\langle {\\mathbf {w}}^{m}, \\mathbf {D}_i^m{\\mathbf {s}}_i \\rangle | & = \\Big |\\sum _{j=1}^{N-M_i} {\\mathbf {w}}^m(j) \\sum _{n=1}^N\\mathbf {D}_i^m(j,n) {\\mathbf {s}}_i(n) \\Big | \\nonumber \\\\& \\le \\sum _{j=1}^{N-M_i} |{\\mathbf {w}}^m(j)| \\sum _{n=1}^N |\\mathbf {D}_i^m(j,n)| |{\\mathbf {s}}_i(n)|.$ Since we assumed normalized vectors $\\Vert {\\mathbf {x}}_i\\Vert \\le 1$ , by applying the left-hand side of the RIP we obtain: $\\Vert {\\mathbf {s}}_i\\Vert \\le 1/\\sqrt{1-\\delta ^i_K}$ , and, taking into account that $\\Vert {\\mathbf {s}}_i\\Vert _1 \\le \\sqrt{K}\\Vert {\\mathbf {s}}_i\\Vert $ and $ |\\mathbf {D}_i^m(j,n)| \\le 1$ (columns of $\\mathbf {D}$ are unit-norm), we obtain: $\\small | \\langle {\\mathbf {w}}^{m}, \\mathbf {D}_i^m{\\mathbf {s}}_i \\rangle | \\le \\sqrt{\\frac{K}{1-\\delta ^i_K}} \\sum _{j=1}^{N-M_i} |{\\mathbf {w}}^m(j)| = \\sqrt{\\frac{K}{1-\\delta ^i_K}}\\Vert {\\mathbf {w}}^m_i\\Vert _1.$ Similarly, using $\\hat{{\\mathbf {x}}_i}^m = \\mathbf {D}_i^{\\prime m} {\\mathbf {s}}_i^{\\prime }$ , we can obtain that $| \\langle {\\mathbf {w}}^{m}, \\mathbf {D}_i^{\\prime m}{\\mathbf {s}}_i^{\\prime } \\rangle | \\le \\sqrt{\\frac{K}{1-\\delta ^i_K}}\\Vert {\\mathbf {w}}^m_i\\Vert _1.$ Putting equations (REF ) and (REF ) together with equation (REF ) complete the proof of the sufficient condition (REF )."],["Implementation","We implemented all the algorithms in Pytorch 1.0.0 on a single GPU.","The code is available athttps://github.com/ccaiafa/SimultCodClass.. Initializations of dictionary $\\mathbf {D}$ and coefficients ${\\mathbf {s}}_i$ were made at random.","However, we think some improvements in convergence could be achieved by using some dedicated dictionaries such as the case of Wavelet or Cosine Transform matrices.","To update NN weights ($\\Theta $ ), we used standard Stochastic Gradient Descent (SGD) with learning rate $\\sigma _{\\Theta }$ and momentum $m$ , while for updating dictionary $\\mathbf {D}$ and vector coefficients ${\\mathbf {s}}_i$ , we used fixed update rate $\\sigma = \\sigma _{\\mathbf {D}} = \\sigma _{{\\mathbf {s}}}$ .","It is noted that we used different update rates for training and testing stages.","In Table REF , we report the settings used for experiments for MNIST and CIFAR10 datasets, which includes Number of iterations $N_{iter}$ , batch size $bs$ , learning rate $\\sigma _{\\Theta }$ , momentum $m$ , update rate $\\sigma $ (training and test)."],["Hyperparameter tuning","In Table REF we present the results of the grid search for hyper-parameter tuning on MNIST and CIFAR10 datasets.","We fit our model to the training dataset for a range of values of parameters $\\lambda _1$ and $\\lambda _2$ and apply it to a validation data set.","Figure REF shows the validation accuracy obtained with different classifiers and levels of missing entries for MNIST dataset.","Table: Experimental settings for MNIST and CIFAR10 datasets: Number of iterations N iter N_{iter}, batch size bsbs, learning rate σ Θ \\sigma _{\\Theta }, momentum mm, update rate σ\\sigma (training and test)"],["Additional visual results","To visually evaluate our results, additional randomly selected examples of original (complete) images of the test dataset in MNIST and Fashion, together with their given incomplete observations and obtained reconstructions, are shown in Figure REF and Figure REF Table: Hyper-parameter tuning: crossvalidated hyperparameters λ 1 \\lambda _1 and λ 2 \\lambda _2 obtained for MNIST and CIFAR10 datasets with the classifiers used in our experiments.Figure: Test accuracy in the grid search for hyper-parameter tuning in MNIST dataset: λ 1 \\lambda _1 and λ 2 \\lambda _2 were tuned by cross-validation for various levels of missing entries: 25%, 50% and 75%.Figure: Reconstructions of incomplete test MNIST dataset images by applying our simultaneous classification and coding algorithm with the CNN4 architecture.Figure: Reconstructions of incomplete test Fashion dataset images by applying our simultaneous classification and coding algorithm with the CNN4 architecture with Batch Normalization (BN)."]],"string":"[\n [\n \"Learning from Incomplete Features by Simultaneous Training of Neural\\n Networks and Sparse Coding\"\n ],\n [\n \"Abstract In this paper, the problem of training a classifier on a dataset with incomplete features is addressed.\",\n \"We assume that different subsets of features (random or structured) are available at each data instance.\",\n \"This situation typically occurs in the applications when not all the features are collected for every data sample.\",\n \"A new supervised learning method is developed to train a general classifier, such as a logistic regression or a deep neural network, using only a subset of features per sample, while assuming sparse representations of data vectors on an unknown dictionary.\",\n \"Sufficient conditions are identified, such that, if it is possible to train a classifier on incomplete observations so that their reconstructions are well separated by a hyperplane, then the same classifier also correctly separates the original (unobserved) data samples.\",\n \"Extensive simulation results on synthetic and well-known datasets are presented that validate our theoretical findings and demonstrate the effectiveness of the proposed method compared to traditional data imputation approaches and one state-of-the-art algorithm.\"\n ],\n [\n \"Introduction\",\n \"Learning methods from limited or imperfect data has attracted great attention in the literature recently.\",\n \"Datasets with limited, weak, noisy labels or incomplete features represent an important and still open problem.\",\n \"In this paper, we address the problem of training a classifier on a dataset with incomplete features, which arises in many machine learning applications where sometimes the measurements are incomplete, noisy or affected by artifacts.\",\n \"Examples of this situation include: recommendation systems built upon the information gathered by different users where not all the users have fully completed their forms; medical datasets where typically not all tests can be performed on every patient; or a self-driving vehicle or robot where objects in the view field can be partially occluded.\",\n \"Handling correctly the incomplete-features problem is a classical challenge in machine learning.\",\n \"Skipping missing features by setting them to zero values damages the classification accuracy [1].\",\n \"Most previous studies addressed this problem by using an imputation approach, which consists of performing data completion followed by training the classifier with those reconstructions (referred here as the sequential method).\",\n \"However, this strategy cannot ensure the statistical consistence of the classifier, as data completion is usually fully unsupervised or label information is partially or inefficiently exploited.\",\n \"In this work, a new supervised learning method is developed to train a general classifier, such as a logistic regression or a deep neural network, using only a subset of features per sample, while assuming sparse representations of data vectors on an unknown dictionary.\",\n \"The proposed method simultaneously learns the classifier, the dictionary and the corresponding sparse representation of each input data sample.\",\n \"In this way, we combine the approximation power and simplicity of sparse coding with the extraordinary ability of neural networks (NNs) to model complex decision functions (classifiers) with the goal to successfully train a classifier based on incomplete features.\",\n \"We analyze the limitations of the sequential approach (section REF ), i.e.\",\n \"imputation followed by training, and introduce the simultaneous classification and coding approach in section .\",\n \"Our method consists of incorporating a sparse data representation model into a single cost function that is optimized for training the classifier and, at the same time, finding the best representation of the observed data.\",\n \"A learning algorithm is presented in section REF to train a classifier on incomplete features and sufficient conditions under which such a classifier performs as good as the ideal classifier, i.e.\",\n \"the one that can be obtained from complete observations, is identified (section REF ).\",\n \"Extensive experimental results are presented in section , using synthetic and well known benchmark datasets that validate our theoretical findings and illustrate the effectiveness of the proposed method.\",\n \"Practical sequential methods based on statistical imputation, such as computing the “mean”, “regression” and “multiple” imputation techniques are common practice [2].\",\n \"Remarkably, it was shown that imputing with a constant, e.g.\",\n \"the mean, is Bayes-risk consistent only when missing features are not informative [3].\",\n \"More elaborated completion methods were also explored, such as $K$ -nearest neighbor estimators, multilayer or recurrent NNs and others, see [4] and references therein.\",\n \"However, sequential methods do not fully exploit label information.\",\n \"Data labels can provide valuable information about missing features that could potentially improve the classifier learning process.\",\n \"Recent advances on probabilistic generative models have allowed for a formulation of supervised learning with incomplete features as a statistical inference problem arriving at algorithms that significantly outperformed sequential methods.\",\n \"In the seminal work [5], a framework for maximum likelihood density estimation based on mixtures models was proposed and successfully applied to small incomplete-features problems.\",\n \"In particular, a Gaussian Mixture Model (GMM) was fitted to incomplete-data through an Expectation-Maximization (EM) algorithm.\",\n \"Building upon this generative model strategy, some approaches have considered integrating out the missing values based on a simple logistic regression function [6], [7].\",\n \"Other versions of this approach proposed an explicit simultaneous learning of the model and the decision function [8], [9].\",\n \"While probabilistic generative models provided a nice and elegant approach to the incomplete-features problem showing good results on small datasets, they are not suitable for many modern machine learning applications because: (1) despite some acceleration techniques were explored, e.g.\",\n \"[10], [11], those algorithms are computationally expensive becoming prohibitive for moderate to large datasets; (2) GMM is impractical for modeling high-dimensional datasets because the number of parameters to achieve good approximations becomes unmanageable; and (3) they do not consider complex classification functions as the ones provided by deep NN architectures.\",\n \"Recently, some approaches based on the low-rank property of the features data matrix were investigated and algorithms for data completion were proposed incorporating the label information [12], [13], [14].\",\n \"Since the rank estimation of a matrix is a computationally expensive task, usually based on the Singular Value Decomposition (SVD), the obtained algorithms are prohibitive to solve modern machine learning problems with large datasets.\",\n \"Additionally, as in the case of the probabilistic generative models, none of these methods considered complex classification functions.\",\n \"To overcome this drawback, more recently, a framework based on various NN architectures such as autoencoders, multilayer perceptrons and Radial Basis Function Networks (RBFNs), was proposed for handling missing input data by setting a probabilistic model, e.g.\",\n \"a GMM, for every missing feature, which is trained together with the NN weights [15].\",\n \"This method combined the great capability of NNs to approximate complex decision functions with the nice formulation of the GMM to model missing data.\",\n \"However, it inherited the drawbacks of GMMs, i.e.\",\n \"they are not well suited to higher-dimensional datasets.\",\n \"On the other hand, during the last few years in the signal processing community, there has been a rapid development of theory and algorithms for sparse coding approximations which, by exploiting the redundancy of natural signals, are able to provide simple and accurate models of complex data distributions, see [16], [17], [18], [19], [20] and references therein.\",\n \"Sparse coding is nearly ubiquitous in Nature, for example, it is found in the way that neurons encode sensory information [21], [22].\",\n \"Sparse representations of data showed to be useful also in classification problems.\",\n \"In [23], a Linear Discriminant Analysis (LDA) classifier was trained on corrupted data providing a robust classification method.\",\n \"In [24], algorithms for learning discriminative sparse models, instead of purely reconstructive ones, were proposed based on simple linear and bilinear classifiers.\",\n \"Similar methods were also studied by either using class-specific dictionaries [25], [26] or using a single one for all classes [27].\",\n \"However, these proposed methods neither were applied to the incomplete-features problem nor considered deep NN classifiers.\"\n ],\n [\n \"Problem formulation\",\n \"We assume a supervised learning scenario with vector samples and labels $\\\\lbrace {\\\\mathbf {x}}_i, y_i\\\\rbrace $ , $i=1,2,\\\\dots , I$ , ${\\\\mathbf {x}}_i \\\\in {R}^N$ and $y_i \\\\in \\\\lbrace 0,1,\\\\dots , C-1\\\\rbrace $ ($C$ classes).\",\n \"However, we are constrained to observe only subsets of features and their labels: $\\\\lbrace {\\\\mathbf {x}}^o_i, y_i\\\\rbrace $ , ${\\\\mathbf {x}}^o_i \\\\in {R}^{M_i}$ with $M_i < N$ .\",\n \"Unobserved (missing) features are denoted by ${\\\\mathbf {x}}^m_i \\\\in {R}^{N-M_i}$ .\",\n \"We consider arbitrary patterns of missing features, which are allowed to be different for each data instance $i$ .\",\n \"The set of indices of missing features at sample $i$ is denoted by ${\\\\cal {M}}_i$ , i.e.\",\n \"${\\\\mathbf {x}}^m_i = {\\\\mathbf {x}}_i({\\\\cal {M}}_i)$ and ${\\\\mathbf {x}}^o_i = {\\\\mathbf {x}}_i\\\\left({\\\\overline{\\\\mathcal {M}}}_i\\\\right)$ .\",\n \"We define the set of all $K$ -sparse vectors $\\\\Sigma _K^P = \\\\lbrace {\\\\mathbf {s}}\\\\in {R}^P \\\\text{ s.t. }\",\n \"\\\\Vert {\\\\mathbf {s}}\\\\Vert _0 \\\\le K\\\\rbrace $ (containing at most $K$ non-zero entries) and assume that data vectors ${\\\\mathbf {x}}_i$ admit $K$ -sparse representations over an unknown dictionary $\\\\mathbf {D}\\\\in {R}^{N\\\\times P}$ ($P \\\\ge N$ ): ${\\\\mathbf {x}}_i = \\\\mathbf {D}{\\\\mathbf {s}}_i, \\\\mbox{ with } {\\\\mathbf {s}}_i \\\\in \\\\Sigma _K^{P}.$ The columns of a dictionary are called “atoms” because every data vector can be written as a linear combination of at most $K$ elementary components.\",\n \"Sometimes dictionaries are orthogonal such as the ones derived from the Discrete Cosine or Wavelet [19] transforms.\",\n \"However, overcomplete ($P \\\\ge N$ ) nonorthogonal dictionaries have demonstrated to play an important role in image processing tasks such as denoising, inpainting, etc [17], [28].\",\n \"By partitioning $\\\\mathbf {D}$ according to the pattern of missing features at sample $i$ , we obtain $\\\\mathbf {D}^o_i =\\\\mathbf {D}\\\\left({\\\\overline{\\\\mathcal {M}}}_i,:\\\\right) \\\\in {R}^{M_i\\\\times P}$ and $\\\\mathbf {D}^m_i =\\\\mathbf {D}({\\\\cal {M}}_i,:)\\\\in {R}^{{(N-M_i)}\\\\times P}$ , which according to equation (REF ) implies: ${\\\\mathbf {x}}^o_i = \\\\mathbf {D}^o_i {\\\\mathbf {s}}_i, \\\\mbox{ and } {\\\\mathbf {x}}^m_i = \\\\mathbf {D}^m_i {\\\\mathbf {s}}_i.$ Let us assume that a perfect classifier, e.g.\",\n \"a logistic regression or deep NN, that assigns probability $p_{\\\\Theta }(\\\\hat{y} | {\\\\mathbf {x}})$ to predicted label $\\\\hat{y}$ given data ${\\\\mathbf {x}}$ can be trained on the complete dataset $\\\\lbrace {\\\\mathbf {x}}_i, y_i\\\\rbrace $ , such that, in a two-classes scenario ($C=2$ ), $p_{\\\\Theta }(\\\\hat{y} = y_i | {\\\\mathbf {x}}_i) > p_{\\\\Theta }(\\\\hat{y} \\\\ne y_i | {\\\\mathbf {x}}_i)$ , $\\\\forall i =1,2,\\\\dots , I$ , where $\\\\Theta $ is the set of trained parameters.\",\n \"Our goal is to develop a method to obtain an estimate $\\\\hat{\\\\Theta }$ of parameters using only the incomplete dataset $\\\\lbrace {\\\\mathbf {x}}^o_i, y_i\\\\rbrace $ and to identify conditions under which such a classifier is compatible with the ideal one.\"\n ],\n [\n \"Why training after imputation is difficult?\",\n \"If the $K$ -sparse representations of the observations ${\\\\mathbf {x}}^o_i$ were unique, then ${\\\\mathbf {x}}_i$ can be perfectly reconstructed from the incomplete observations and the classifier can be successfully trained using these reconstructions.\",\n \"In the particular case where the dictionary is known in advance, there exist conditions on the sampling patterns based on the coherence, spark or RIP (Restricted Isometry Property) of matrix $\\\\mathbf {D}^o_i$ that can guarantee uniqueness [20].\",\n \"However, these conditions are difficult to meet in practice and determining RIP/Spark properties are NP-hard in general [29].\",\n \"Moreover, in the general case where the dictionary $\\\\mathbf {D}$ is unknown and needs to be learned from data, it is even more difficult to obtain well separated reconstructions which certainly leads to suboptimal or wrong classifiers.\",\n \"Next, we provide some intuition about the limitation of the sequential approach through a toy example.\",\n \"Let us consider the classification of hand-written digit images belonging to two classes: “3s” and “8s” and assume that they admit 2-sparse representations over a dictionary.\",\n \"Fig.\",\n \"REF (a-b) shows the representations of two example vectors ${\\\\mathbf {x}}_i$ and ${\\\\mathbf {x}}_j$ belonging to classes “3” and “8”, respectively.\",\n \"If only the right halves of the images are observed and no label information is provided, we are clearly faced with a problem because our observed samples from two different classes are identical, i.e.\",\n \"${\\\\mathbf {x}}^o_i = {\\\\mathbf {x}}^o_j$ .\",\n \"It is obvious that at least two possible 2-sparse representations for the observed data exist as illustrated in Fig.\",\n \"REF (c).\",\n \"When the sparse solution is not unique, we may end up reconstructing wrong vectors that could not be even well separated as illustrated in Fig.\",\n \"REF (d-e).\",\n \"In general, sequential methods using only the information of observed features are prone to fail because the non-uniqueness of solutions can make the training of a good classifier an impossible task.\",\n \"However, we could solve this problem by incorporating the labelling information from the very beginning as it is proposed in the following section.\"\n ],\n [\n \"Simultaneous learning and coding approach\",\n \"We propose to train the classifier and find the proper representation, not only as sparse as possible but also providing the best separation of classes.\",\n \"We want to combine the training of the classifier together with the learning of a dictionary and optimal sparse representations such that the reconstructed data vectors are compatible with observations and well separated.\",\n \"To do that we propose to minimize the following global cost function: $J(\\\\Theta , \\\\mathbf {D}, {\\\\mathbf {s}}_i) = \\\\nonumber \\\\\\\\\\\\textstyle \\\\underbrace{\\\\frac{1}{I}\\\\sum _{i=1}^I\\\\big \\\\lbrace J_0(\\\\Theta , \\\\hat{{\\\\mathbf {x}}}_i, y_i) + \\\\lambda _{1} J_1(\\\\mathbf {D},{\\\\mathbf {s}}_i)\\\\big \\\\rbrace }_{F(\\\\Theta , \\\\mathbf {D}, {\\\\mathbf {s}}_i)} +\\\\underbrace{\\\\frac{1}{I}\\\\sum _{i=1}^I\\\\big \\\\lbrace \\\\lambda _{2}J_2({\\\\mathbf {s}}_i)\\\\big \\\\rbrace }_{G({\\\\mathbf {s}}_i)},$ with respect to $\\\\Theta $ , $\\\\mathbf {D}$ and ${\\\\mathbf {s}}_i$ ($i=1,2,\\\\dots , I$ ), where $\\\\Theta $ contains the classifier parameters, i.e.\",\n \"the vector of weights in a deep NN classifier architecture; $\\\\mathbf {D}\\\\in {R}^{N\\\\times P}$ ($P \\\\ge N$ ) is a dictionary and ${\\\\mathbf {s}}_i \\\\in \\\\Sigma _K^{P}$ are the representation coefficients such that the reconstructed data vectors are $\\\\hat{{\\\\mathbf {x}}}_i = \\\\mathbf {D}{\\\\mathbf {s}}_i$ .\",\n \"$J_0(\\\\Theta , \\\\hat{{\\\\mathbf {x}}}_i, y_i)$ is a measure of the classification error for the reconstructed sample vector $\\\\hat{{\\\\mathbf {x}}}_i$ .\",\n \"Typically, we use the crossentropy measure, i.e.\",\n \"$J_0(\\\\Theta , \\\\hat{{\\\\mathbf {x}}}_i, y_i) = -\\\\log [p_{\\\\Theta }(y_i | \\\\hat{{\\\\mathbf {x}}}_i)]$ , where $p_{\\\\Theta }(y_i | \\\\hat{{\\\\mathbf {x}}}_i)$ is the probability assigned by the classifier to sample $\\\\hat{{\\\\mathbf {x}}}_i$ as belonging to class $y_i$ .\",\n \"$J_1(\\\\mathbf {D},{\\\\mathbf {s}}_i)$ is a measure of the approximation error of the reconstruction when it is restricted to observed features, which is defined as follows: $J_1(\\\\mathbf {D},{\\\\mathbf {s}}_i)=\\\\frac{M_i}{N}\\\\Vert {\\\\mathbf {m}}_i \\\\odot ({\\\\mathbf {x}}_i - \\\\mathbf {D}{\\\\mathbf {s}}_i)\\\\Vert ^2$ , where $\\\\odot $ stands for the entry-wise product, ${\\\\mathbf {m}}_i \\\\in {R}^N$ is the observation mask for sample $i$ , i.e.\",\n \"$m_i(n) = 0$ (1) if data entry ${\\\\mathbf {x}}_i(n)$ is missing (available); and $J_2({\\\\mathbf {s}}_i) = \\\\frac{1}{N}\\\\Vert {\\\\mathbf {s}}_i\\\\Vert _1$ is proportional to the $\\\\ell _1$ -norm whose minimization promotes the sparsity of the representation since $\\\\ell _1$ -norm is a convenient proxy for $\\\\ell _0$ -norm [30].\",\n \"Finally, the hyper-parameters $\\\\lambda _1$ and $\\\\lambda _2$ allow us to give more or less importance to the representation accuracy and its sparsity, with respect to the classification error.\",\n \"Intuitively, minimizing equation (REF ) favors solutions that not only have sparse representations compatible with observed features, but also providing reconstructions that are best separated in the given classes.\",\n \"Figure: Toy example: (a) 4 out PP dictionary elements 𝐝 i \\\\mathbf {d}_i (atoms).\",\n \"(b) Digits “3” and “8” can be represented by combining only two atoms in the dictionary (2-sparse representations).\",\n \"(c) A left-half occluded digit “3” or “8” admits more than one 2-sparse representation (sum of 𝐝 1 o \\\\mathbf {d}^o_1 and 𝐝 2 o \\\\mathbf {d}^o_2, or 𝐝 3 o \\\\mathbf {d}^o_3 and 𝐝 4 o \\\\mathbf {d}^o_4).\",\n \"(d) Linearly separable samples from two classes (A and B) having two features: 𝐱=[x 1 ,x 2 ]∈R 2 \\\\mathbf {x} = [x_1, x_2]\\\\in {R}^2 where incomplete observations are taken by observing only one feature.\",\n \"Note that 𝐱 A \\\\mathbf {x}_A and 𝐱 B \\\\mathbf {x}_B belong to different classes but their observations are identical.\",\n \"(e) Without using label information, the sequential method could lead to wrong reconstructions of data vectors, i.e.\",\n \"𝐱 ^ A ≠𝐱 A \\\\hat{\\\\mathbf {x}}_A\\\\ne \\\\mathbf {x}_A and 𝐱 ^ B ≠𝐱 B \\\\hat{\\\\mathbf {x}}_B\\\\ne \\\\mathbf {x}_B making the set of reconstructed vectors not linearly separable.\"\n ],\n [\n \"A sparsity-promoting sub-gradient optimization algorithm\",\n \"To minimize the cost function in equation (REF ) we propose to alternate between the optimization over ${\\\\mathbf {s}}_i$ ($i=1,2,\\\\dots , I$ ) and $\\\\lbrace \\\\Theta ,\\\\mathbf {D}\\\\rbrace $ using the training dataset (incomplete).\",\n \"For fixed $\\\\lbrace \\\\Theta ,\\\\mathbf {D}\\\\rbrace $ , the optimization with respect to ${\\\\mathbf {s}}_i$ is a non-smooth separable minimization sub-problem, which was extensively studied in the literature [31], [32].\",\n \"In this sub-problem, the objective function is written as the sum of $F(\\\\Theta , \\\\mathbf {D}, {\\\\mathbf {s}}_i)$ and a non-smooth separable function $G({\\\\mathbf {s}}_i)$ , for which highly specialized, efficient and provable convergent solvers, namely the Coordinate Gradient Descent (CGD), already exists.\",\n \"However, the following key differences in our setting makes it not suitable for the CGD approach: first, our function $F(\\\\Theta , \\\\mathbf {D}, {\\\\mathbf {s}}_i)$ involves evaluation of a multi-layer NN classifier, which can be non-smooth due to involved activation functions like ReLU or others; second, and more importantly, the computation of its second derivatives (Hessian) becomes prohibitive.\",\n \"Therefore, we choose a simpler and standard first order (stochastic sub-gradient based) search of local minima with back-propagation.\",\n \"We take the strategy similar to the heuristics used in [33].\",\n \"To update ${\\\\mathbf {s}}_i$ , we need to subtract $\\\\sigma _{{\\\\mathbf {s}}} \\\\frac{\\\\partial J}{\\\\partial {\\\\mathbf {s}}_i}(j)$ from each coordinate $j$ provided that we do not cross zero in the process in order to avoid escaping from a region where $G({\\\\mathbf {s}}_i)$ is differentiable.\",\n \"In such a case, we let the new value of ${\\\\mathbf {s}}_i(j)$ be exactly zero.\",\n \"More specifically, we define $\\\\Delta _i(j)= -\\\\sigma _{{\\\\mathbf {s}}} \\\\frac{\\\\partial J}{\\\\partial {\\\\mathbf {s}}_i}(j)$ and, if ${\\\\mathbf {s}}_i(j)[{\\\\mathbf {s}}_i(j)+\\\\Delta _i(j)] < 0$ (zero crossing condition), we re-define $\\\\Delta _i(j)=-{\\\\mathbf {s}}_i(j)$ ; finally we update ${\\\\mathbf {s}}_i \\\\leftarrow {\\\\mathbf {s}}_i + \\\\Delta _i$ .\",\n \"It is noted that, once a coefficient ${\\\\mathbf {s}}_i(j)$ reaches zero at a coordinate $j$ , it becomes fixed, in other words, sparsity of solution ${\\\\mathbf {s}}_i$ is monotonically increasing with iterations.\",\n \"When ${\\\\mathbf {s}}_i$ is fixed, our problem is reduced to minimize $F(\\\\Theta , \\\\mathbf {D}, {\\\\mathbf {s}}_i)$ with respect to $\\\\Theta $ and $\\\\mathbf {D}$ , which is easily done by standard first order (stochastic gradient based) search of local minima.\",\n \"The algorithm proposed for the training phase is presented as Algorithm REF .\",\n \"In addition, for the testing phase, if the test dataset is incomplete, we need to find first the sparsest representation for the given observations, compute the reconstructions $\\\\hat{{\\\\mathbf {x}}}_i = \\\\mathbf {D}{\\\\mathbf {s}}_i$ and then apply the previously learned classifier to them as presented in Supp.\",\n \"material, Algorithm REF .\",\n \": Simultaneous classification and coding [1] $\\\\lbrace {\\\\mathbf {x}}^o_i, y_i\\\\rbrace $ , $i=1,2,\\\\dots , I$ , hyper-parameters $\\\\lambda _1$ and $\\\\lambda _2$ , $N_{iter}$ and update rates $\\\\sigma _{\\\\Theta }$ , $\\\\sigma _{\\\\mathbf {D}}$ and $\\\\sigma _{{\\\\mathbf {s}}}$ Weights $\\\\Theta $ and reconstructions $\\\\hat{{\\\\mathbf {x}}}_i= \\\\mathbf {D}{\\\\mathbf {s}}_i, \\\\forall i$ Randomly initialize $\\\\Theta , \\\\mathbf {D}, {\\\\mathbf {s}}_i, \\\\forall i$ $n\\\\le N_{iter}$ Fix ${\\\\mathbf {s}}_i$ , update $\\\\Theta $ and $\\\\mathbf {D}$ : $\\\\Theta = \\\\Theta - \\\\sigma _{\\\\Theta } \\\\frac{\\\\partial J}{\\\\partial \\\\Theta }$ $\\\\mathbf {D}= \\\\mathbf {D}- \\\\sigma _{\\\\mathbf {D}} \\\\frac{\\\\partial J}{\\\\partial \\\\mathbf {D}}$ Normalize columns of matrix $\\\\mathbf {D}$ Fix $\\\\Theta $ and $\\\\mathbf {D}$ , update ${\\\\mathbf {s}}_i$ , $\\\\forall i$ : $\\\\Delta _i= -\\\\sigma _{{\\\\mathbf {s}}} \\\\frac{\\\\partial J}{\\\\partial {\\\\mathbf {s}}_i}$ , $\\\\forall i$ ${\\\\mathbf {s}}_i(j)[{\\\\mathbf {s}}_i(j)+\\\\Delta _i(j)] < 0$ $\\\\Delta _i(j)=-{\\\\mathbf {s}}_i(j), \\\\forall (i, j)$ ; ${\\\\mathbf {s}}_i = {\\\\mathbf {s}}_i + \\\\Delta _i$ , $\\\\forall i$ return $\\\\Theta , \\\\mathbf {D}, {\\\\mathbf {s}}_i, \\\\hat{{\\\\mathbf {x}}}_i = \\\\mathbf {D}{\\\\mathbf {s}}_i, \\\\forall i$\"\n ],\n [\n \"Theoretical analysis\",\n \"Here, we investigate about conditions under which a perfect classifier of the complete data can be obtained from incomplete data samples.\"\n ],\n [\n \"Logistic regression\",\n \"Let us first consider a logistic regression classifier [34] where the set of parameters $\\\\Theta = \\\\lbrace {\\\\mathbf {w}}, b\\\\rbrace $ are a vector ${\\\\mathbf {w}}\\\\in {R}^N$ and a scalar $b$ (bias).\",\n \"A perfect classifier exists if there is a hyperplane that separates both classes, i.e., for each data vector ${\\\\mathbf {x}}_i$ : $f({\\\\mathbf {x}}_i) = \\\\langle {\\\\mathbf {w}}, {\\\\mathbf {x}}_i \\\\rangle + b > 0$ if $y_i=1$ , and $f({\\\\mathbf {x}}_i) \\\\le 0$ if $y_i=0$ .\",\n \"We consider data samples admitting a $K$ -sparse representations ${\\\\mathbf {x}}_i = \\\\mathbf {D}{\\\\mathbf {s}}_i$ with dictionary $\\\\mathbf {D}\\\\in {R}^{N\\\\times P}$ having unit-norm columns.\",\n \"We also assume an arbitrary pattern of missing features $\\\\mathcal {M}_i$ such that, data samples and dictionary are partitioned as $\\\\lbrace {\\\\mathbf {x}}^m_i,{\\\\mathbf {x}}^o_i \\\\rbrace $ and $\\\\lbrace \\\\mathbf {D}^m_i,\\\\mathbf {D}^o_i \\\\rbrace $ , respectively.\",\n \"The following lemma identifies a sufficient condition under which, if we are able to train a classifier on incomplete observations such that the reconstructed data points are well separated by a hyperplane, then the same classifier correctly separates the original (unobserved) data vectors.\",\n \"Lemma 2.1 (Sufficient condition type I) Suppose that we have obtained an alternative dictionary $\\\\mathbf {D}^{\\\\prime } \\\\ne \\\\mathbf {D}\\\\in {R}^{N\\\\times P}$ such that, for the incomplete observations ${\\\\mathbf {x}}^o_i \\\\in {R}^{M_i}$ , the $K$ -sparse representation solutions are non-unique, i.e.\",\n \"$\\\\exists {\\\\mathbf {s}}_i, {\\\\mathbf {s}}_i^{\\\\prime } \\\\in \\\\Sigma _K^P$ such that ${\\\\mathbf {x}}^o_i = \\\\mathbf {D}^o_i{\\\\mathbf {s}}_i = \\\\mathbf {D}^{\\\\prime o}_i{\\\\mathbf {s}}_i^{\\\\prime }$ , where ${\\\\mathbf {s}}_i \\\\in {R}^P$ are the vectors of coefficients of the true data and ${\\\\mathbf {s}}_i^{\\\\prime }$ provides reconstructions $\\\\hat{{\\\\mathbf {x}}}_i = \\\\mathbf {D}^{\\\\prime }{\\\\mathbf {s}}_i^{\\\\prime }$ .\",\n \"If a perfect classifier $\\\\lbrace {\\\\mathbf {w}}, b\\\\rbrace $ of the reconstructions $\\\\hat{{\\\\mathbf {x}}}_i$ exists s.t.\",\n \"$|f(\\\\hat{{\\\\mathbf {x}}}_i)| > \\\\epsilon _i > 0$ and $\\\\epsilon _i > |\\\\langle {\\\\mathbf {w}}^m_i , {\\\\mathbf {e}}^m_i\\\\rangle |$ with ${\\\\mathbf {e}}^m_i = {\\\\mathbf {x}}^m_i - \\\\hat{{\\\\mathbf {x}}}^m_i$ , then the full data vectors ${\\\\mathbf {x}}_i$ are also perfectly separated with this classifier, in other words: $f({\\\\mathbf {x}}_i) = \\\\langle {\\\\mathbf {w}}_i, {\\\\mathbf {x}}_{i} \\\\rangle + b > 0$ ($\\\\le 0$ ) if $y_i =1$ ($y_i=0$ ).\",\n \"By using the missing/observed partition and omitting the sample index $i$ , we can write: $f({\\\\mathbf {x}}) = \\\\langle {\\\\mathbf {w}}, {\\\\mathbf {x}}\\\\rangle + b = \\\\langle {\\\\mathbf {w}}^{o}, {\\\\mathbf {x}}^o \\\\rangle + \\\\langle {\\\\mathbf {w}}^{m}, {\\\\mathbf {x}}^m \\\\rangle + b$ .\",\n \"If we add and subtract the term $\\\\langle {\\\\mathbf {w}}^{m}, \\\\hat{{\\\\mathbf {x}}}^m \\\\rangle $ on the right left hand, arrange terms and use the fact that $\\\\hat{{\\\\mathbf {x}}}^o = \\\\mathbf {D}^{\\\\prime o}{\\\\mathbf {s}}^{\\\\prime } = {\\\\mathbf {x}}^o$ , we get: $f({\\\\mathbf {x}}) = f(\\\\hat{{\\\\mathbf {x}}}) + \\\\langle {\\\\mathbf {w}}^{m}, {\\\\mathbf {e}}^m \\\\rangle .$ Since we assumed that $f(\\\\hat{{\\\\mathbf {x}}}) > \\\\epsilon >0$ (for $y_i = 1$ ) and $|\\\\langle {\\\\mathbf {w}}^{m}, {\\\\mathbf {e}}^m \\\\rangle | < \\\\epsilon $ , it implies that $f({\\\\mathbf {x}}) > 0$ .\",\n \"Basically, condition (REF ) means requiring that reconstruction vector $\\\\hat{{\\\\mathbf {x}}}$ has a distance $\\\\epsilon $ to the separating hyperplane larger than the absolute dot product between ${\\\\mathbf {w}}$ and the residual $ {\\\\mathbf {e}}= {\\\\mathbf {x}}- \\\\hat{{\\\\mathbf {x}}}$ , which of course is true when the reconstruction is accurate, i.e.\",\n \"${\\\\mathbf {x}}\\\\approx \\\\hat{{\\\\mathbf {x}}}$ .\",\n \"However, in practice, reconstructions are not accurate so we are interested in conditions under which Lemma REF can still holds.\",\n \"Below, we derive a more restrictive but useful sufficient condition: Proposition 2.1 (Sufficient condition type II) Under the same hypothesis of Lemma REF , the following condition is enough to guarantee a proper classifier trained on incomplete data: $\\\\epsilon > |\\\\langle {\\\\mathbf {w}}^{m}, {\\\\mathbf {x}}^m\\\\rangle | + |\\\\langle {\\\\mathbf {w}}^{m}, \\\\hat{{\\\\mathbf {x}}}^m \\\\rangle |.$ By using the fact that $|\\\\langle {\\\\mathbf {w}}^{m}, {\\\\mathbf {e}}^m \\\\rangle | = |\\\\langle {\\\\mathbf {w}}^{m}, {\\\\mathbf {x}}^m - \\\\hat{{\\\\mathbf {x}}}^m \\\\rangle | \\\\le |\\\\langle {\\\\mathbf {w}}^{m}, {\\\\mathbf {x}}^m \\\\rangle | + |\\\\langle {\\\\mathbf {w}}^{m}, \\\\hat{{\\\\mathbf {x}}}^m \\\\rangle |$ , and applying Lemma REF the proof is completed.\",\n \"We highlight that, in our experiments, we were able to verify that Sufficient Condition type II is met in practice (see section , Fig.\",\n \"REF ).\",\n \"In Supp.\",\n \"material section REF , we derive an additional sufficient condition based on the Restricted Isometry Property (RIP) of the dictionary $\\\\mathbf {D}$ and sparsity level $K$ , showing that sufficient condition (REF ) is easier to hold for datasets admitting highly sparse representations on dictionaries as close to orthogonal ones as possible.\"\n ],\n [\n \"Multilayer-perceptron\",\n \"Lemma REF can be straightforwardly generalized to multilayer-perceptron NNs where, if a softmax function is used at the output of the last layer then, as before, the prediction is based on the sign of the linear function: $\\\\small f({\\\\mathbf {x}}) = \\\\langle {\\\\mathbf {w}}, {\\\\mathbf {x}}^{(L)}\\\\rangle + b, \\\\mbox{ with } {\\\\mathbf {x}}^{(l)} = h\\\\left( \\\\mathbf {W}^T_l {\\\\mathbf {x}}^{(l-1)} + \\\\mathbf {B}_l\\\\right),$ $l=1,2,\\\\dots ,L$ , where $L+1$ is the total number of layers, $N_l$ is the number of neurons in layer $l$ , ${\\\\mathbf {w}}\\\\in {R}^{N_{L+1}}$ contains the weights in the last layer, $h(\\\\cdot )$ is an activation function, e.g.\",\n \"ReLU, $\\\\mathbf {W}_l\\\\in {R}^{N_{l-1}\\\\times N_l}$ and $\\\\mathbf {B}_l \\\\in {R}^N_l$ contain the weights and biases associated to neurons at layer $l$ ; and ${\\\\mathbf {x}}^{(0)} = {\\\\mathbf {x}}$ is the input data vector.\",\n \"In this case, the first layer matrix $\\\\mathbf {W}_1 \\\\in {R}^{N\\\\times N_1}$ can be partitioned into submatrices $\\\\mathbf {W}^o_{1i} \\\\in {R}^{M \\\\times N_1}$ and $\\\\mathbf {W}^m_{1i} \\\\in {R}^{(N-M) \\\\times N_1}$ according to the observed and missing input features, respectively.\",\n \"Proposition 2.2 () Under the same conditions of Lemma REF , if a NN-classifier $\\\\lbrace {\\\\mathbf {W}_l,\\\\mathbf {B}_l} (l=1,2,\\\\dots L),{\\\\mathbf {w}}, b, h=ReLU\\\\rbrace $ of the reconstruction $\\\\hat{{\\\\mathbf {x}}}_i$ exists such that $\\\\epsilon _i > A \\\\max _j |\\\\langle \\\\mathbf {W}^m_{1i}(:,j) , {\\\\mathbf {e}}^m_i\\\\rangle | ,$ where $A = \\\\Vert {\\\\mathbf {w}}\\\\Vert \\\\prod _{l=2}^L \\\\Vert \\\\mathbf {W}_l \\\\Vert _2$ and ${\\\\mathbf {e}}^m_i = {\\\\mathbf {x}}^m_i - \\\\hat{{\\\\mathbf {x}}}^m_i$ , then the full data vector ${\\\\mathbf {x}}_i$ is also perfectly separated, in other words: $f({\\\\mathbf {x}}_i) > 0$ ($< 0$ ) if $y_i =1$ ($y_i=0$ ).\",\n \"In the proof of Lemma REF , we were interested in finding a bound of the output error when the input ${\\\\mathbf {x}}$ of a classifier is perturbed, i.e.\",\n \"we found conditions such that $|f({\\\\mathbf {x}}) - f({\\\\mathbf {x}}+ \\\\mathbf {\\\\delta })| < \\\\epsilon $ .\",\n \"By generalizing the classifier to the case of a multilayer perceptron we can derive the proof as follows: Given a perturbation $\\\\mathbf {\\\\delta }^{(l-1)}\\\\in {R}^{N_l}$ at the input of layer $l-1$ , i.e.\",\n \"$\\\\hat{{\\\\mathbf {x}}}^{(l-1)} = {\\\\mathbf {x}}^{(l-1)} + \\\\mathbf {\\\\delta }^{(l-1)}$ , it is propagated to the output of layer $l$ .\",\n \"By writing the error at the output we obtain: $\\\\scriptstyle \\\\mathbf {\\\\delta }^{(l)} = h\\\\left( \\\\mathbf {W}_l^T {\\\\mathbf {x}}^{(l-1)} + \\\\mathbf {B}_l + \\\\mathbf {W}_l^T \\\\mathbf {\\\\delta }^{(l-1)} \\\\right) - h\\\\left( \\\\mathbf {W}_l^T {\\\\mathbf {x}}^{(l-1)} + \\\\mathbf {B}_l \\\\right),$ and, by using the sub-additivity of ReLU function $h(\\\\cdot )$ , i.e.\",\n \"$h(a+b) \\\\le h(a) + h(b)$ , we derive the following entry-wise inequality: $\\\\small \\\\mathbf {\\\\delta }^{(l)} \\\\le h \\\\left( \\\\mathbf {W}_l^T \\\\mathbf {\\\\delta }^{(l-1)} \\\\right),$ and, by considering the property of ReLU activation function $\\\\Vert h({\\\\mathbf {x}})\\\\Vert \\\\le \\\\Vert {\\\\mathbf {x}}\\\\Vert $ , it turns out: $\\\\small \\\\Vert \\\\mathbf {\\\\delta }^{(l)}\\\\Vert \\\\le \\\\Vert \\\\mathbf {W}_l^T \\\\mathbf {\\\\delta }^{(l-1)} \\\\Vert ,$ Since the last layer of the NN is a linear classifier as in the case of Lemma REF , we can ask that $ \\\\langle {\\\\mathbf {w}}, \\\\mathbf {\\\\delta }^{(L)}\\\\rangle < \\\\epsilon $ .\",\n \"Thus, by recursively using equation (REF ), we write $ \\\\langle {\\\\mathbf {w}}, \\\\mathbf {\\\\delta }^{(L)}\\\\rangle \\\\le \\\\Vert {\\\\mathbf {w}}\\\\Vert \\\\Vert \\\\mathbf {\\\\delta }^{(L)}\\\\Vert \\\\le \\\\Vert {\\\\mathbf {w}}\\\\Vert \\\\Vert \\\\mathbf {W}_L\\\\Vert _2 \\\\Vert \\\\mathbf {W}_{l-1}\\\\Vert _2 \\\\cdots \\\\Vert \\\\mathbf {W}_2\\\\Vert _2 \\\\Vert \\\\mathbf {\\\\delta }^{(1)} \\\\Vert .$ By defining $A = \\\\Vert {\\\\mathbf {w}}\\\\Vert \\\\prod _{l=2}^L\\\\Vert \\\\mathbf {W}_l\\\\Vert _2$ , evaluating equation (REF ) with $l=1$ and taking into account that perturbation at the input of first layer is $\\\\mathbf {\\\\delta }^{(0)} = {\\\\mathbf {e}}$ with ${\\\\mathbf {e}}^o = \\\\mathbf {0}$ , we arrive at: $\\\\small \\\\langle {\\\\mathbf {w}}, \\\\mathbf {\\\\delta }^{(L)}\\\\rangle \\\\le A \\\\Vert \\\\mathbf {W}_1^{mT} {\\\\mathbf {e}}^m\\\\Vert \\\\le A \\\\max _j |\\\\langle \\\\mathbf {W}^m_{1}(:,j) , {\\\\mathbf {e}}^m \\\\rangle | < \\\\epsilon ,$ which completes the proof.\",\n \"It is interesting to note that $A=1$ is attained when unit-norm filters (columns of $\\\\mathbf {W}_l$ ) are orthogonal, which can be imposed by using orthogonality regularization [35].\"\n ],\n [\n \"Experimental results\",\n \"We implemented all the algorithms in Pytorch 1.0.0 on a single GPU.\",\n \"Implementation details are reported in Supplemental material, sections REF and REF .\",\n \"The code is available at https://github.com/ccaiafa/SimultCodClass.\",\n \"Synthetic datasets: We synthetically generated $I=11,000$ ($10,000$ training $+$ $1,000$ test) $K$ -sparse data vectors ${\\\\mathbf {x}}_i \\\\in {R}^{100}$ using a dictionary $\\\\mathbf {D}\\\\in {R}^{100 \\\\times 200}$ obtained from a Gaussian distribution with normalized atoms, i.e.\",\n \"$\\\\Vert \\\\mathbf {D}(:,j)\\\\Vert =1, \\\\forall j$ .\",\n \"A random hyperplane $\\\\lbrace {\\\\mathbf {w}}, b\\\\rbrace $ with ${\\\\mathbf {w}}\\\\in {R}^N$ , $b\\\\in {R}$ was randomly chosen dividing data vectors into two classes according to the sign of the expression $\\\\langle {\\\\mathbf {w}}, {\\\\mathbf {x}}_i \\\\rangle +b$ , which defined the label $y_i$ .\",\n \"We also controlled the degree of separation between classes by discarding all data vectors with distances to the hyperplane lower than a pre-specified threshold, i.e.\",\n \"$|\\\\langle {\\\\mathbf {w}}, {\\\\mathbf {x}}_i \\\\rangle +b| < d$ .\",\n \"We used $n=10$ repetitions of each experiment with different masks and input data in order to compute statistics.\",\n \"We applied our simultaneous method (Simult.)\",\n \"with hyperparameters $\\\\lambda _1$ and $\\\\lambda _2$ in the cost function (REF ) tuned via cross-validation to train a logistic regression classifier on incomplete datasets with randomly distributed missing features.\",\n \"Then, we computed the classification accuracy on the complete test dataset and compared the results against the following standard sequential methods: Sequential Sparsity based (Seq.\",\n \"Sp.\",\n \"): reconstructions are obtained by finding the sparsest representation compatible with the observations solving a LASSO problem.\",\n \"We used Algorithm REF as shown in the Supp.\",\n \"material; Zero Fill (ZF): missing features are filled with zeros, which is equivalent to ignore unknown values; Mean Unsupervised (MU): missing features are filled with the mean computed on the available values; Mean Supervised (MS): as in the previous case but the mean is computed on the same class vectors only; K-Nearest Neighbor (KNN): as in the previous case but the mean is computed on the K-nearest neighbors of the same class vectors only.\",\n \"To compare the performance of classifiers, we computed the mean accuracy $\\\\pm $ standard error of the mean (s.e.m.\",\n \"), with $n=10$ , on complete test datasets using all the methods for two levels of separation between classes ($d=0.0, 0.2$ ), two levels of sparsity ($K=4,32$ ) and missing features in the training dataset ranging from $25\\\\%$ to $95\\\\%$ as shown in Fig.\",\n \"REF .\",\n \"Our results show that the simultaneous algorithm clearly outperforms all the sequential methods.\",\n \"A t-test was performed to evaluate the statistical significance with $p < 0.05$ of the difference between our algorithm and MS.\",\n \"It is interesting to note that, when classes has some degree of separation ($d=0.2$ ), using the simple MS method, can give good results but not better than our algorithm.\",\n \"Figure: Experimental results on synthetic dataset with random masks using our algorithm (red) and compared to various sequential methods.\",\n \"Test accuracy (mean ±\\\\pm s.e.m with n=10n=10) is shown as a function of the percentage of missing features for separation of classes d=0.0,0.2d=0.0, 0.2 and levels of sparsity K=4,32K=4,32.\",\n \"Statistical significance for the difference between Simult.\",\n \"and MS is shown (p<0.05p < 0.05).In the second experiment, we generated $I =10,000$ $K$ -sparse data vectors ${\\\\mathbf {x}}_i \\\\in {R}^{100}$ using $\\\\mathbf {D}\\\\in {R}^{100 \\\\times 100}$ and we evaluated the sufficient condition of equation (REF ) on $n=10$ repetitions of the experiment with $95\\\\%$ missing features and separation $d=0.0$ .\",\n \"Fig.\",\n \"REF clearly shows that the sufficient condition is mostly met in practice, especially for highly sparse representations of input data (small $K$ ).\",\n \"This means that in practice it is not necessary to accurately reconstruct the input vectors, it is enough to capture the intrinsic characteristics of the classes such that the distances of reconstructions to the separating hyperplane satisfy the sufficient condition (REF ).\",\n \"Figure: Verification of the sufficient condition () for various levels of sparsity KK: 2D-histogram of ϵ\\\\epsilon versus g=|〈𝐰 m ,𝐱 m 〉|+|〈𝐰 m ,𝐱 ^ m 〉|g = |\\\\langle {\\\\mathbf {w}}^{m}, {\\\\mathbf {x}}^m\\\\rangle | + |\\\\langle {\\\\mathbf {w}}^{m}, \\\\hat{{\\\\mathbf {x}}}^m \\\\rangle |.\",\n \"Mean + s.e.m (n=10n=10) percentage of correctly classified data samples are shown for ϵ>g\\\\epsilon > g and ϵ \\\\epsilon _i >0$ and $\\\\epsilon _i > 2 \\\\Vert {\\\\mathbf {w}}^m_i\\\\Vert _1\\\\sqrt{\\\\frac{K}{1-\\\\delta ^i_K}},$ then the full data vector ${\\\\mathbf {x}}_i$ is also perfectly separated with this classifier, in other words: $f({\\\\mathbf {x}}_i) = \\\\langle {\\\\mathbf {w}}_i, {\\\\mathbf {x}}_{i} \\\\rangle + b > 0$ ($\\\\le 0$ ) if $y_i =1$ ($y_i=0$ ).\",\n \"Let us prove that the sufficient condition (REF ) is met under the hypothesis of Theorem REF .\",\n \"Taking into account that ${\\\\mathbf {x}}_i^m = \\\\mathbf {D}_i^m {\\\\mathbf {s}}_i$ , we can write $\\\\small | \\\\langle {\\\\mathbf {w}}^{m}, \\\\mathbf {D}_i^m{\\\\mathbf {s}}_i \\\\rangle | & = \\\\Big |\\\\sum _{j=1}^{N-M_i} {\\\\mathbf {w}}^m(j) \\\\sum _{n=1}^N\\\\mathbf {D}_i^m(j,n) {\\\\mathbf {s}}_i(n) \\\\Big | \\\\nonumber \\\\\\\\& \\\\le \\\\sum _{j=1}^{N-M_i} |{\\\\mathbf {w}}^m(j)| \\\\sum _{n=1}^N |\\\\mathbf {D}_i^m(j,n)| |{\\\\mathbf {s}}_i(n)|.$ Since we assumed normalized vectors $\\\\Vert {\\\\mathbf {x}}_i\\\\Vert \\\\le 1$ , by applying the left-hand side of the RIP we obtain: $\\\\Vert {\\\\mathbf {s}}_i\\\\Vert \\\\le 1/\\\\sqrt{1-\\\\delta ^i_K}$ , and, taking into account that $\\\\Vert {\\\\mathbf {s}}_i\\\\Vert _1 \\\\le \\\\sqrt{K}\\\\Vert {\\\\mathbf {s}}_i\\\\Vert $ and $ |\\\\mathbf {D}_i^m(j,n)| \\\\le 1$ (columns of $\\\\mathbf {D}$ are unit-norm), we obtain: $\\\\small | \\\\langle {\\\\mathbf {w}}^{m}, \\\\mathbf {D}_i^m{\\\\mathbf {s}}_i \\\\rangle | \\\\le \\\\sqrt{\\\\frac{K}{1-\\\\delta ^i_K}} \\\\sum _{j=1}^{N-M_i} |{\\\\mathbf {w}}^m(j)| = \\\\sqrt{\\\\frac{K}{1-\\\\delta ^i_K}}\\\\Vert {\\\\mathbf {w}}^m_i\\\\Vert _1.$ Similarly, using $\\\\hat{{\\\\mathbf {x}}_i}^m = \\\\mathbf {D}_i^{\\\\prime m} {\\\\mathbf {s}}_i^{\\\\prime }$ , we can obtain that $| \\\\langle {\\\\mathbf {w}}^{m}, \\\\mathbf {D}_i^{\\\\prime m}{\\\\mathbf {s}}_i^{\\\\prime } \\\\rangle | \\\\le \\\\sqrt{\\\\frac{K}{1-\\\\delta ^i_K}}\\\\Vert {\\\\mathbf {w}}^m_i\\\\Vert _1.$ Putting equations (REF ) and (REF ) together with equation (REF ) complete the proof of the sufficient condition (REF ).\"\n ],\n [\n \"Implementation\",\n \"We implemented all the algorithms in Pytorch 1.0.0 on a single GPU.\",\n \"The code is available athttps://github.com/ccaiafa/SimultCodClass.. Initializations of dictionary $\\\\mathbf {D}$ and coefficients ${\\\\mathbf {s}}_i$ were made at random.\",\n \"However, we think some improvements in convergence could be achieved by using some dedicated dictionaries such as the case of Wavelet or Cosine Transform matrices.\",\n \"To update NN weights ($\\\\Theta $ ), we used standard Stochastic Gradient Descent (SGD) with learning rate $\\\\sigma _{\\\\Theta }$ and momentum $m$ , while for updating dictionary $\\\\mathbf {D}$ and vector coefficients ${\\\\mathbf {s}}_i$ , we used fixed update rate $\\\\sigma = \\\\sigma _{\\\\mathbf {D}} = \\\\sigma _{{\\\\mathbf {s}}}$ .\",\n \"It is noted that we used different update rates for training and testing stages.\",\n \"In Table REF , we report the settings used for experiments for MNIST and CIFAR10 datasets, which includes Number of iterations $N_{iter}$ , batch size $bs$ , learning rate $\\\\sigma _{\\\\Theta }$ , momentum $m$ , update rate $\\\\sigma $ (training and test).\"\n ],\n [\n \"Hyperparameter tuning\",\n \"In Table REF we present the results of the grid search for hyper-parameter tuning on MNIST and CIFAR10 datasets.\",\n \"We fit our model to the training dataset for a range of values of parameters $\\\\lambda _1$ and $\\\\lambda _2$ and apply it to a validation data set.\",\n \"Figure REF shows the validation accuracy obtained with different classifiers and levels of missing entries for MNIST dataset.\",\n \"Table: Experimental settings for MNIST and CIFAR10 datasets: Number of iterations N iter N_{iter}, batch size bsbs, learning rate σ Θ \\\\sigma _{\\\\Theta }, momentum mm, update rate σ\\\\sigma (training and test)\"\n ],\n [\n \"Additional visual results\",\n \"To visually evaluate our results, additional randomly selected examples of original (complete) images of the test dataset in MNIST and Fashion, together with their given incomplete observations and obtained reconstructions, are shown in Figure REF and Figure REF Table: Hyper-parameter tuning: crossvalidated hyperparameters λ 1 \\\\lambda _1 and λ 2 \\\\lambda _2 obtained for MNIST and CIFAR10 datasets with the classifiers used in our experiments.Figure: Test accuracy in the grid search for hyper-parameter tuning in MNIST dataset: λ 1 \\\\lambda _1 and λ 2 \\\\lambda _2 were tuned by cross-validation for various levels of missing entries: 25%, 50% and 75%.Figure: Reconstructions of incomplete test MNIST dataset images by applying our simultaneous classification and coding algorithm with the CNN4 architecture.Figure: Reconstructions of incomplete test Fashion dataset images by applying our simultaneous classification and coding algorithm with the CNN4 architecture with Batch Normalization (BN).\"\n ]\n]"},"id":{"kind":"string","value":"2011.14047"}}}],"truncated":false,"partial":false},"paginationData":{"pageIndex":27,"numItemsPerPage":100,"numTotalItems":1867602,"offset":2700,"length":100}},"jwt":"eyJhbGciOiJFZERTQSJ9.eyJyZWFkIjp0cnVlLCJwZXJtaXNzaW9ucyI6eyJyZXBvLmNvbnRlbnQucmVhZCI6dHJ1ZX0sImlhdCI6MTc1MDcxNTQ2Miwic3ViIjoiL2RhdGFzZXRzL2hvd2V5L3VuYXJYaXZlIiwiZXhwIjoxNzUwNzE5MDYyLCJpc3MiOiJodHRwczovL2h1Z2dpbmdmYWNlLmNvIn0.qTRuFBeDoZpp8DdDBs-N3Jq7nVZH_ITImfJh32ChnYeB5JksKnrlc1FAsit8FvVKP7y7me6Mg62M0qb1bdBhAg","displayUrls":true},"discussionsStats":{"closed":0,"open":1,"total":1},"fullWidth":true,"hasGatedAccess":true,"hasFullAccess":true,"isEmbedded":false,"savedQueries":{"community":[],"user":[]}}">
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"Symmetric core-cohesive blockmodel in preschool children's interaction\n networks"
],
[
"Abstract Researchers have extensively studied the social mechanisms that drive the formation of networks observed among preschool children.",
"However, less attention has been given to global network structures in terms of blockmodels.",
"A blockmodel is a network where the nodes are groups of equivalent units (according to links to others) from a studied network.",
"Cugmas et al.",
"(2019) showed that mutuality, popularity, assortativity, and different types of transitivity mechanisms can lead the global network structure to the proposed asymmetric core-cohesive blockmodel.",
"Yet, they did not provide any evidence that such a global network structure actually appears in any empirical data.",
"In this paper, the symmetric version of the core-cohesive blockmodel type is proposed.",
"This blockmodel type consists of three or more groups of units.",
"The units from each group are internally well linked to each other while those from different groups are not linked to each other.",
"This is true for all groups, except one in which the units have mutual links to all other units in the network.",
"In this study, it is shown that the proposed blockmodel type appears in empirical interactional networks collected among preschool children.",
"Monte Carlo simulations confirm that the most often studied social network mechanisms can lead the global network structure to the proposed symmetric blockmodel type.",
"The units' attributes are not considered in this study."
],
[
"Abstract",
"Researchers have extensively studied the social mechanisms that drive the formation of networks observed among preschool children.",
"However, less attention has been given to global network structures in terms of blockmodels.",
"A blockmodel is a network where the nodes are groups of equivalent units (according to links to others) from a studied network.",
"Cugmas et al.",
"[1] showed that mutuality, popularity, assortativity, and different types of transitivity mechanisms can lead the global network structure to the proposed asymmetric core-cohesive blockmodel.",
"Yet, they did not provide any evidence that such a global network structure actually appears in any empirical data.",
"In this paper, the symmetric version of the core-cohesive blockmodel type is proposed.",
"This blockmodel type consists of three or more groups of units.",
"The units from each group are internally well linked to each other while those from different groups are not linked to each other.",
"This is true for all groups, except one in which the units have mutual links to all other units in the network.",
"In this study, it is shown that the proposed blockmodel type appears in empirical interactional networks collected among preschool children.",
"Monte Carlo simulations confirm that the most often studied social network mechanisms can lead the global network structure to the proposed symmetric blockmodel type.",
"The units' attributes are not considered in this study.",
"One of the key attempts in sociology, and also in psychology, is to reveal the (social) mechanisms that are responsible for a given (social) output.",
"When the relationships among individuals are studied, the social output is a social network.",
"In social network analysis, there are different approaches to study the underlying social mechanisms of a given network.",
"The main focus of earlier studies was on social mechanisms in the context of empirical networks while less attention was paid to the social mechanisms in the context of specific global network structures.",
"Therefore, the general objective of the current study is to identify fundamental social mechanisms that guide the formation of a global network structure.",
"In this study, the global network structure is narrowed to a structure with three or more groups.",
"The units from the first group (called the core group) have symmetric links established with all units in the network, while the units from the other groups (called cohesive groups) are internally well linked.",
"The units from different cohesive groups are not linked to each other.",
"This global network structure (called symmetric core-cohesive blockmodel, described in more detail in subsection Global network structure) is proposed since it is a combination of cohesive and symmetric core-periphery global network structures and because these global network structures can arise from the well-known transitivity [2], [1] and popularity [1] mechanisms.",
"These two mechanisms were found to be present in the formation of many liking and friendship networks collected among preschoolers (see subsection Local mechanisms).",
"The assumption made in this study is that the proposed global network structure appears among preschool children.",
"Entrance to preschool brings a set of peers together who were previously unknown to one another.",
"This is rare in the natural world and, thus, the shift into preschool peer groups offers a unique opportunity to assess and understand the mechanisms behind peer group formation.",
"Preschool entry is also distinct from other social network settings in that it offers a closed network space in which peers interact.",
"Preschool also provides a unique developmental context in which children are motivated, perhaps for the first time, to form new and enduring social relationships with similar-age peers [3], [4].",
"This assumption (of the emergence of the proposed global network structure) is tested by using the blockmodeling approach [5] on the symmetrized networks previously analyzed by Schaefer et al.",
"[6].",
"Their study's main focus was the network dynamics rather than the global network structure.",
"They showed that the selected local network mechanisms are important in such networks and that the importance of different local network mechanisms change throughout the school year.",
"Building on the assumption (tested in this paper) that the proposed global network structure emerges in interactional preschool networks, the following research question is posed: can the proposed global network structure appear due to the selected local network mechanisms without considering the nodes' attributes?",
"Here, the same local network mechanisms are assumed as in the study of Schaefer et al.",
"[6] and other previous studies on preschool network dynamics.",
"The research question is addressed using Monte Carlo simulations, specifically, by applying the proposed model from the family of network evolution models.",
"The study is relevant since understanding of the local network mechanisms at play, in the context of global network structures, is important while studying real (empirically observed) networks.",
"Namely, the proposed global network structure's emergence at preschools raises very important developmental questions, e.g., how children in the core group differ from children in cohesive groups and what are the implications (if any) for their further individual development?",
"Should such a global network structure be encouraged or discouraged?",
"Is this a period where scholars may be able to document the emergence of social cliques and associated social norms?",
"Will some children be integrated into cohesive groups, while others are left with minimal peer affiliations in the global network [7]?",
"The paper is organized as follows: a new global network structure is formally defined and the local network mechanisms are proposed and described (section Global network structure).",
"Next, the global network structures of the empirical interactional preschool networks are analyzed (section The empirical case).",
"The main research question concerned with the proposed global network structure's emergence is addressed in section Simulation approach and some conclusions are outlined.",
"In the following section, the proposed global network structure is defined in the blockmodel context.",
"Different local network mechanisms that may drive the global network structure of preschool children's interactional networks towards the proposed one are discussed.",
"A blockmodel is a network in which the units are groups of equivalent units from a studied network [5].",
"The term reflects the fact that if a network is represented by a matrix, which is then split according to a partition (groups), blocks (submatrices) are formed in the matrix.",
"The term “block” refers to a submatrix showing the links among units from the same or different group(s).",
"Two selected units are structurally equivalent if they have the same pattern of links to the other units [8], [9].",
"The possible block types are identified through a selected definition of equivalence which is based on links among the units.",
"Structural equivalence [9] and its generalization, regular equivalence [10], are the most common.",
"When structural equivalence is used, only null and complete blocks are possible.",
"In ideal complete blocks, all possible links are present while no link exists in ideal null blocks.",
"A demonstration of blockmodeling according to structural equivalence is given in Fig REF .",
"The original network is visualized in matrix form in Fig REF A.",
"Here, each row and column represents a unit.",
"Gray colored cells in a matrix represent a link from the $i$ -th unit (row) to the $j$ -th unit (column).",
"Cells on the diagonal represent loops (a given unit is linked to itself).",
"The units are permuted (see Fig REF B) in such a way that those with the same pattern of links are placed together and form a cluster (group).",
"Two groups are shown in Fig REF B.",
"Figure: Example of an empirical network and its blockmodeling solution(A) empirical network, (B) empirical network drawn in line with the blockmodeling solution, (C) blockmodel, (D) blockmodeling solution with two inconsistenciesIn the blockmodeling context, the clusters of units are shrinked into nodes.",
"The blockmodel that is obtained is visualized in Fig REF C. The obtained blockmodel has two nodes (shrinked groups).",
"Here, two types of blocks appear (complete and null).",
"Complete blocks are on the diagonal of the matrix because the units from both groups are internally linked to each other.",
"Off-diagonal blocks refer to the relationships between different groups.",
"Since the units from different groups are not linked to each other, the off-diagonal blocks are null blocks.",
"The example represents an ideal case, meaning that there are all possible links in complete blocks and there is no link in the null blocks.",
"However, this is unrealistic for empirical networks.",
"In such networks, there are usually some non-links in complete blocks and some links in null blocks (see Fig REF D).",
"Such links are called errors or inconsistencies.",
"There are several well-known blockmodel types, with two being the cohesive and (symmetric or asymmetric) core-periphery blockmodel types.",
"The cohesive blockmodel type (Fig REF A) contains at least two groups of units where units from different groups are not linked to each other, while all units inside each cohesive group are linked to each other.",
"On the other hand, the symmetric core-periphery blockmodel (Fig REF B) is defined by two groups of units.",
"The units from the core group are internally well linked to each other and units from the periphery are not linked to each other.",
"The units from the core are also linked to the units from the periphery and vice versa (in the asymmetric case, the units from the periphery are linked to the core ones or vice versa).",
"Figure: Different representations of networks with a cohesive blockmodel, symmetric core-periphery blockmodel, and symmetric core-cohesive blockmodel(A) cohesive blockmodel, (B) symmetric core-periphery blockmodel, (C) symmetric core-cohesive blockmodelThe newly proposed symmetric core-cohesive blockmodel type (Fig REF C) is seen as a combination of a cohesive blockmodel and a symmetric core-periphery blockmodel.",
"A symmetric core-cohesive blockmodel consists of one core group of units to which all units in the network are linked, and where units from the core group are linked to all other units in the network.",
"The other units are classified into cohesive groups.",
"Units from each cohesive group are internally linked to each other, while units from different cohesive groups are not linked to each other.",
"The model can be extended in such a way that a group of units which are not linked to each other would also exist.",
"It has been hypothesized that the asymmetric core-cohesive blockmodel type might appear in networks observed among preschool children where the links are defined by “friend nominations” or “liking\" [1].",
"Although research is lacking on the global network structure's evolution in the blockmodel context, many studies address the mechanisms that affect the creation and dissolution of ties.",
"The social mechanisms of attraction most often discussed are mutuality (also known as reciprocity), popularity (also known as the Matthew effect or preferential attachment), transitivity, and assortativity (also known as assortative mixing or homophily) [11].",
"The last one may be considered through the assortativity of in-degree or other units' attributes, such as gender [12], [13].",
"Simulations confirm that an asymmetric core-cohesive blockmodel can appear as a result of the listed mechanisms.",
"Since conducting longitudinal sociometric interviews with a high level of reliability and validity among preschool children might be too demanding for both the children and the researcher, the data analyzed in such settings are often observational.",
"In such studies, a link is often operationalized as an interaction and therefore the observed links are undirected.",
"If such interactions are considered as an indicator of friendship, popularity or liking, the same mechanisms must be considered when testing for the emergence of the symmetric core-cohesive blockmodel type.",
"The following mechanisms are often discussed in the literature: Mutuality or reciprocity is defined through the reciprocation of ties and is one of the most fundamental social network mechanisms (besides creating links) and a basic feature of social life [14].",
"Analyzing 49- to 62-month-old preschool children, Snyder et al.",
"[15] not only found that children spend much time with selected friends and less with others, but also strong evidence of mutuality.",
"Observed mutual links in the empirical global network structures can also emerge since children prefer to interact with peers who are similar to themselves.",
"This tendency often fosters the emergence of mutual peer relationships during childhood [16], [17], [18], [6].",
"The researcher cannot indirectly study this mutuality when analyzing non-directed interactional empirical networks.",
"However, the mechanism can play a role in the process of creating the initiative for interactions (also see subsection The algorithm for generating networks).",
"Popularity is defined through an in-degree in social network analysis and is usually an operationalization of likeability or social status [14].",
"As a social network mechanism, popularity expresses the tendency to create links to others with a relatively high (in)degree.",
"This is especially the case for less popular ones who wish to increase their own popularity by creating links with those who are most popular [19].",
"The fact that some units become more popular than others can relate to their personal attributes (e.g., wealth, being good at something, etc.)",
"or positive or negative behavior [20].",
"Transitivity measures the tendency for triadic closure in networks - \"the friends of my friends are also my friends\".",
"Transitivity in peer groups may arise from the increased propinquity of individuals who share mutual friends, or from a psychological need for balance - a convergence of third parties' evaluations [6].",
"Many empirical studies highlight the importance of these mechanisms.",
"For example, Snyder et al.",
"[15] noticed that children spend considerable time with selected friends and less with others.",
"They also observed a strong mutual affiliation of friendships, which is subjected to the level of positive social consequences available from peers in the classroom.",
"Daniel et al.",
"[14] used ERGM [21] to study the mutuality, reciprocity, popularity, and transitivity mechanisms on the forming of affiliative ties in 19 Portuguese preschool peer groups.",
"They found that all of these mechanisms are important for forming affiliative ties.",
"Schaefer et al.",
"[6] studied the three most common network-formation mechanisms (reciprocity, popularity, and triadic closure) among preschool children throughout a school year in four waves using SIENA [22], [23], [24].",
"They found the reciprocity effect is constant over time while the popularity effect is most important midway through the school year.",
"The importance of the triadic closure effect increases over time, which is expected since very early on friendships are typically play-oriented dyads that primarily socialize children into group life [25].",
"When children gain more social contacts and greater confidence, they move into larger groups [26].",
"The hypothesis about the symmetric core-cohesive blockmodel being present in empirical interactional networks is tested in the subsections below.",
"To this end, the empirical data collected among preschool children are analyzed using generalized blockmodeling.",
"The data were collected as part of a bigger longitudinal study of young children's preparedness for school between 2004 and 2006 in Head Start preschools (the active consent to participate in the study was obtained from parents or guardians of children included in the study).",
"The data were also analyzed in the study by Schaefer et al.",
"[6].",
"The data are observational in nature, meaning that trained observers present in school classes recorded interactions among the children.",
"Specifically, observers were present for several hours in a classroom two to three days per week.",
"To ensure the greatest validity and reliability, two observers monitored the same children at the same time for 10 seconds.",
"The order in which the observers watched over the children was random.",
"When all children had been observed, the observers waited 5 minutes before repeating their observations (with a randomly reordered list of children).",
"Children were observed in different activities, e.g.",
"free play, talking, aggressive behavior, and others.",
"The observers coded the type of activity in which a given child was involved and up to five other children with whom the selected child was interacting.",
"Only the free-play data (data collected when children were able to play freely) are analyzed in this study.",
"Children had to be observed at least 13 times during the whole school year to be included in the analysis.",
"Based on the observational data, four complete networks are generated for each class.",
"Each network's construction is based on a two-month period, as presented in Table REF .",
"The networks are in matrix form in which each row and each column represents a child.",
"The number of a given child's (ego, in a given row) observed interactions with other children (alter, in a given column) is shown in the corresponding cells of the matrix.",
"The obtained networks were transformed from directed to undirected and binarized: there is a link between two children if the number of observed interactions is higher than the median (of the number of interactions between all possible pairs in the network) divided by two.",
"Table: Some basic descriptive statistics for the undirected networks.The number of children varies between 14 and 21 across all networks.",
"In the last period, the children were aged between 37 and 60 months and the share of males varied between 43% and 69%.",
"Binarized networks are blockmodeled to evaluate the global network structure.",
"Blockmodeling is a way of reducing a large, potentially incoherent network to a smaller, comprehensible, and interpretable structure [5].",
"In a blockmodeling procedure, a list of allowed and forbidden block types is given.",
"Since structural equivalence is used, these block types are null and complete.",
"In order to not constrain the blockmodeling procedure, the relationships between groups (image matrix) is not pre-specified.",
"The blockmodeling was done using the \"blockmodeling\" package [27] for the R programming language.",
"The number of iterations in the blockmodeling was 500 and 3 clusters were set for all networks.",
"Fig REF gives the matrix representation of the analyzed networks.",
"Each matrix corresponds to one network at a given time point.",
"Black dots denote links.",
"Children are ordered by rows and by columns in line with the solution from the blockmodeling.",
"It can be seen that the networks are very dense, which is expected since interactional networks were observed in a closed environment (classroom).",
"Some are almost complete.",
"Figure: Obtained blockmodel structures for each class (ID1 to ID11) and each time period.Undirected and binarized empirical networks are considered.",
"The obtained symmetric core-cohesive blockmodels are presented in the frame matrices.A symmetric core-periphery blockmodel structure (see the framed matrices) appears in almost all classes in at least one time period.",
"It does not appear in just two classes (ID2 and ID5) out of 11 classes.",
"In the other classes, the symmetric core-cohesive blockmodel appears in the 2nd time period (in 7 classes out of 11) or in the 3rd and 4th time periods (in 5 classes out of 11).",
"The group sizes vary - in some cases, the core group consists of only 2 children (ID3 in Feb-Mar 2005) while in some other cases the core group consists of almost half the children (e.g., ID4 in Nov-Dec 2004 and ID43 in Nov-Dec 2005).",
"Some of the blockmodels obtained are similar to the symmetric core-cohesive blockmodel type but are without links within the core (e.g., ID8 and ID11 in Sep-Oct 2005) or without one cohesive group (ID3 in Apr-May 2005).",
"It has been shown that the proposed symmetric core-cohesive blockmodel type appears in empirical networks – specifically, in interactional networks collected among preschool children.",
"The question of whether the most commonly studied local network mechanisms can lead the global network structure towards the symmetric core-cohesive is addressed in the next section.",
"Attributes of the units are not considered in this study.",
"It should be noted that simulations can never prove that certain mechanisms cause global structure in empirical networks, only that they could cause it.",
"Cugmas et al.",
"[1] have already shown that the mutuality, popularity, assortativity, and outgoing-two-path mechanisms can lead towards the asymmetric core-cohesive blockmodel type and that different combinations of the local network mechanisms lead to this global network structure.",
"The results were different when only one of the mechanisms was considered.",
"For example, when only the popularity mechanism was considered, the resulting blockmodel was an asymmetric core-periphery blockmodel, while, on the other hand, the transitivity mechanism plays a role while forming the cohesive groups.",
"Since the mechanisms are not independent, the role of the assortativity and mutuality mechanisms when considered together with popularity and transitivity in a blockmodel context is unclear and depends on the strengths of the other mechanisms.",
"In this paper, the symmetric interactional networks of preschool children are studied.",
"Therefore, the simulation approach proposed by Cugmas et al.",
"[1] for asymmetric networks is adapted to the symmetric case.",
"To evaluate whether the selected local network mechanisms can lead the global network structure towards the symmetric core-cohesive blockmodel, the adapted algorithm for generating networks is presented in the next subsection, and followed by definitions of the selected local network mechanisms.",
"Many networks are generated using the proposed algorithm.",
"In this generating process, different strengths of the mechanisms are considered.",
"The global network structures of the generated networks are then evaluated in the Results subsection by applying the concepts of inconsistent blocks and relative fit value, which are also described in subsection Simulation design.",
"A symmetric core-cohesive blockmodel may be generated in several ways by considering different local mechanisms.",
"Two distinct approaches are identified with regard to whether symmetric or asymmetric links are generated: Symmetric (non)links: here, it is assumed that all asymmetric links are reciprocated immediately.",
"This means that a symmetric tie will exist if at least one of the actors chooses that tie and will not exist if at least one of the actors does not want it.",
"The reciprocity mechanism is not considered in this case.",
"Asymmetric links: only asymmetric links can be formed at a time.",
"To achieve symmetric networks: the reciprocity mechanism must be considered.",
"Here, a symmetric tie will exist if both actors choose the tie and will not exist if neither actor wants it (an asymmetric link will exist if only one chooses the tie).",
"The generated networks can be asymmetric and therefore need to be analyzed as such or symmetrized before being further analyzed (e.g., by preserving all or only the symmetric links); and the reciprocity mechanism does not necessarily have to be considered, but the networks must be symmetrized before being further analyzed.",
"This means that a symmetric tie will exist if at least one actor chooses the tie and will not exist if neither actor wants it.",
"The observed interactional networks are symmetric by the definition of “interaction”, although the process which initiates interactions is asymmetric.",
"In such a process, an ego has to initiate an interaction, while an alter can either: (i) accept (and reciprocate), (ii) tolerate, or (iii) reject (i.e., actively avoid) interaction.",
"Even where an interaction is actively rejected by the alter, it can still be observed, although it is more likely to be recorded if it is either accepted or tolerated.",
"Therefore, the approach where asymmetric ties are formed (by considering the mutuality mechanism) and the network is symmetrized, before being further analyzed, is the closest representation of the emergence of empirical networks.",
"Networks are represented in the form of an adjacency matrix $X$ of size $n*n$ where $n$ is the number of units.",
"The possible values are 1s and 0s where 1s represent links, while 0s represent non-links.",
"Because loops are not present, the diagonal values are 0.",
"The proposed algorithm (see Algorithm REF ) comes from the family of network evolution models (NEM) [28] and can take initial networks with different blockmodels.",
"The algorithm for generating networks used in this study [1] initial network $X$ vector of strengths of the mechanisms $\\theta $ probability of establishing a link $q$ number of iterations $k$ $l$ in $1:k$ randomly select unit $i$ calculate network statistics according to the selected mechanisms for unit $i$ and all other units and save it in S calculate $\\phi = S\\theta ^T$ if $\\phi \\ge Q_3(\\phi )$ , classify unit $j$ into set $C$ , where $Q_3$ is 3rd quartile if $\\phi \\le Q_1(\\phi )$ , classify unit $j$ into set $F$ , where $Q_1$ is 1st quartile with probability $q$ set $i \\rightarrow j$ where $j$ is randomly selected from set $C$ with probability $1-q$ set $i \\lnot \\rightarrow j$ where $j$ is randomly selected from set $F$ generated network $X$ The algorithm is iterational where the number of iterations $k$ can be determined based on the desired number of changes in the global network structure.",
"Further, parameter $q$ must be set.",
"It reflects the tendency towards the creation of a link and can be estimated based on the density of the network with the expected blockmodel.",
"Yet, there is no guarantee the generated networks' density will equal $q$ since it depends on several factors, including the selected local mechanisms.",
"In the iterational process, a unit $i$ is randomly selected with probability $\\frac{1}{n}$ .",
"Then, the network statistics $S$ are calculated based on the operationalized selected mechanisms (see the next subsection).",
"These network statistics are weighted by the vector of strengths of local mechanisms $\\theta $ producing vector $\\phi = S\\theta ^T$ .",
"These units, for which it holds that their corresponding weighted network statistic is higher than or equal to the third quartile of all weighted network statistics, are classified in the set $C$ and are the candidates to accept the incoming tie from unit $i$ .",
"The other units, for which it holds that their corresponding weighted network statistic is lower than or equal to the first quartile of all weighted network statistics, are classified in the set $F$ and are candidates for being dissolved of an incoming tie by unit $i$ .",
"With probability $q$ , the link from $i$ to randomly selected $j$ from set $C$ is set and with probability $1-q$ a non-link from $i$ to randomly selected $j$ from set $F$ is set.",
"Since the unit can establish a link that already exists or dissolve a link that does not exist, there could be no visible change of a link upon a given iteration.",
"The mechanisms are operationalized by different network statistics defined on a binary network, and normalized so that the minimum corresponding values are 0 and the maximum values are 1.",
"These network statistics ($S$ ) are weighted (by considering $\\theta $ ) and summed to produce vector $\\phi $ as described in the previous subsection.",
"The local network mechanisms of which the network statistics are weighted with higher weights (in an absolute value) are more important in the network's evolution.",
"The interpretation of a given mechanism depends on the sign of a corresponding weight.",
"For example, positively weighted popularity statistics refers to the tendency to create a link to those with a relatively high in-degree.",
"On the contrary, a negative sign reflects the tendency to avoid establishing links to those with a relatively high in-degree.",
"The mechanisms are defined in the same way as in a study of the asymmetric core-cohesive blockmodel type [1].",
"Therefore, only a brief description of the proposed mechanisms is given here (the mechanisms are schematically shown in Fig REF , where dashed lines illustrate the links under evaluation appear, are confirmed, or disappear): Parameter $q$ (Fig REF A) reflects the tendency to have a link.",
"Since this is not a focal mechanism, it is implemented in the NEM algorithm as parameter q and is therefore technically not considered as a mechanism in this study.",
"The mutuality mechanism ($M$ ) (Fig REF B) reflects the tendency to reciprocate links.",
"The alter popularity mechanism ($P$ ) (Fig REF C) reflects the tendency to create links to the most popular ones.",
"The assortativity mechanism ($A$ ) (Fig REF D) reflects the tendency to create links to those units with the same level of popularity (in-degree).",
"The transitivity mechanism ($T$ ) (Fig REF E) is a tendency for a unit to directly connect to units, to which it is indirectly connected with (one or more) paths of length two (with more paths increasing the tendency).",
"The outgoing-shared-partner mechanism ($OSP$ ) (Fig REF F) represents a \"structural homophily\" effect which is traditionally based on similarity according to the units' attributes.",
"In the case of the OSP, it is defined by similar choices of partners [29].",
"Figure: Illustrations of different mechanisms considered.",
"(A) parameter qq, (B) mutuality mechanism, (C) popularity mechanism, (D) assortativity mechanism, (E) transitivity mechanism, (F) outgoing-shared-partners mechanismThe described NEM algorithm is used to generate the networks by considering the selected social mechanisms.",
"Since different mechanism strengths are to be considered, 300 randomly selected $\\theta $ are generated.",
"The random values are generated by first sampling five values from the standard normal distribution $\\Phi $ and then multiplying them by a scalar [30], [31] (after such normalization, the sum of the squared elements of $\\theta $ equals 1).",
"$\\theta = \\frac{\\Phi }{\\sqrt{\\sum \\Phi ^2_i}}$ Within the NEM algorithm, parameter $q$ is set to 5/9 and a total of 116,490 iterations are applied.",
"Parameter $q$ , which indicates the tendency to create a link, is set arbitrarily but with reference to generating asymmetric core-cohesive blockmodels.",
"Initial networks are empty with 24 units.",
"The 30 networks are generated for each $\\theta $ .",
"Generalized blockmodeling for binary networks (on symmetrized generated networks, structural equivalence is used) is done after the selected number of iterations of the algorithm.",
"More precisely, the intermediate number of iterations $m$ , at which the global network structure is analyzed, is determined as $m_i=m_{i-1}*1.9$ , where $m_1=100$ [1].",
"This approach is used since most changes in the structure of the links happen at a lower number of iterations.",
"Based on the generalized blockmodeling solution, the number of inconsistent blocks is calculated and used as the fit function.",
"It is defined as the number of different blocks between the symmetric core-cohesive blockmodel with three groups and the empirically obtained blockmodel with three groups.",
"Some $\\theta $ s that generate networks with the lowest number of inconsistent blocks are selected and further analyzed.",
"For each network generated by the selected $\\theta $ s, the relative fit (RF) [1] is calculated as $RF = 1 - \\frac{P^m}{\\frac{1}{k} \\sum _{i=1}^{k} P^r_i}$ where $P^m$ is the value of the criterion function [32], [33] obtained on the empirical network and $P_i^r$ is the value of the criterion function obtained on the $i$ -th randomized network.",
"There are $k$ randomized networks.",
"The mean value of the criterion function in the case of random networks is estimated by simulations.",
"RF is a more detailed measure of the fit of a given blockmodel to the empirical data and its use is most valid when the presence of a given blockmodel type is confirmed by non-specified blockmodeling.",
"Higher values indicate a better fit (the value of 1 indicates a perfect fit) and the expected value of the RF measure in the case of a random network is zero.",
"There are six different $\\theta $ s generating networks without any inconsistent block at the end of the iterations.",
"Further, 76 different $\\theta $ s generate networks with the mean number of inconsistent blocks less than or equal to 0.5, and 109 different $\\theta $ s generate networks with the mean number of inconsistent blocks less than or equal to 1.",
"The $\\theta $ s that generate each network with a symmetric core-cohesive blockmodel are shown in Table REF along with the number of inconsistent blocks at a different number of iterations and the mean RF value of the generated networks.",
"Although all the generated networks have the same blockmodel, they differ largely in the level of errors, expressed by RF.",
"Table: Mean number of inconsistent blocks and mean RF values with the corresponding parameter values.",
"For those θ\\theta s which generated networks with the mean RF at the end of the iterations equal to zero.",
"Initial is an empty network.A more detailed insight into RF for a selected $\\theta $ is given in Fig REF .",
"The mean RF values are calculated for the symmetric core-cohesive blockmodel type, cohesive blockmodel type, and symmetric core-periphery blockmodel type.",
"All RF values are close to zero at the first 190 iterations.",
"At such a low number of iterations, there are insufficient links to enable any of the considered blockmodel types to emerge.",
"However, at 361 iterations, a global network structure, close to cohesive, can be visually recognized on the generated networks Fig REF .",
"Since there is a relatively high level of errors in null and complete blocks, the corresponding mean RF is very low.",
"With a higher number of iterations (until 1,303 iterations), the mean RF, corresponding to all considered blockmodel types, is decreasing.",
"At this step, the links among different groups are established yet, in some cases, links within the core units are not present.",
"Moreover, there is a high level of errors in the null and complete blocks.",
"After 1,303 iterations, the mean RF value for the core-cohesive and cohesive blockmodel is only increasing until 61,311 iterations.",
"Figure: Mean RF for each blockmodel type visualized by lines and the distribution of the density visualized by boxplots.The networks are generated by considering θ={M=-0.18,P=0.74,A=0.37,T=-0.35,OSP=0.42}\\theta =\\lbrace M=-0.18,P=0.74,A=0.37,T=-0.35,OSP=0.42\\rbrace , q=5/9q=5/9, d 0 =0d_0=0.Figure: Some networks generated.The networks are generated by considering θ={M=-0.18,P=0.74,A=0.37,T=-0.35,OSP=0.42}\\theta =\\lbrace M=-0.18,P=0.74,A=0.37,T=-0.35,OSP=0.42\\rbrace , q=5/9q=5/9, d 0 =0d_0=0.",
"The networks are drawn in line with the blockmodels obtained by generalized blockmodeling (non-specified model).",
"Networks for different repetitions of the algorithm for generating networks are drawn in lines for different numbers of iterations.The mean RF, corresponding to the symmetric core-cohesive blockmodel type, is close to 1 at the end of the iterations, indicating the global network structure is the desirable one with almost no error in null and complete blocks (as confirmed in Fig REF ).",
"The mean RF for the cohesive blockmodel is lower while the mean RF for the symmetric core-periphery blockmodel type is highly negative, indicating that the randomized networks fit this blockmodel type much more than the networks generated by using the proposed algorithm.",
"Interactional networks collected in preschool classrooms are studied in this paper.",
"In such an environment, children start to form groups.",
"Children within the individual groups spend more time with each other than they do with children from other groups.",
"At the same time, a group of children is formed which spends a considerable amount of time with all the others from any group.",
"This leads to the newly proposed blockmodel type, i.e.",
"symmetric core-cohesive.",
"It consists of one group of units which are called core units and two or several other groups of units which are called cohesive groups.",
"The units from all groups are internally linked to each other.",
"The units from all cohesive groups are linked to the units from the core group and vice versa.",
"The units from different cohesive groups are not linked to each other.",
"The existence of this blockmodel type is evaluated on empirical data.",
"The data were collected within a larger longitudinal study among preschool children in the United States between 2004 and 2006.",
"The interactions among the children in classrooms were recorded and complete networks were formed.",
"The symmetric core-cohesive blockmodel was found to be present in almost all analyzed classes in at least one time period.",
"This proves that the proposed global structure (blockmodel type) is relevant for such data.",
"The most common local network mechanisms (popularity, assortativity, transitivity, and outgoing-shared-partners mechanism) are considered.",
"Attributes of the units are not taken into account and the initial networks are empty.",
"The adapted version of the algorithm proposed by Cugmas et al.",
"[1] is used to generate the networks by considering the local network mechanisms.",
"The results of the Monte Carlo simulations confirm that the selected mechanisms can generate networks with the symmetric core-cohesive blockmodel.",
"The results do not imply that the global network structures of the empirical preschool networks collected in the 11 classes in the United States emerged due to the studied local network mechanisms.",
"To address this question, a different methodology should be applied.",
"The study is important in several ways given that understanding the emergence of peer network structure holds important implications for directing adaptive (prosocial) and redirecting maladaptive (bullying) peer network dynamics via intervention and prevention strategies.",
"First, blockmodeling is shown to be an efficient way to describe and analyze empirical interactional network global structures.",
"Second, understanding the link between the global network structure and the local network mechanisms in a given context is necessary for studying (e.g., modelling) the empirically obtained networks.",
"It has been shown that the selected local network mechanisms are important in the formation of the symmetric core-cohesive blockmodel even without considering any further attributes of the units."
]
] | 1906.04566 |
[
[
"Sharing of vulnerability information among companies -- a survey of\n Swedish companies"
],
[
"Abstract Software products are rarely developed from scratch and vulnerabilities in such products might reside in parts that are either open source software or provided by another organization.",
"Hence, the total cybersecurity of a product often depends on cooperation, explicit or implicit, between several organizations.",
"We study the attitudes and practices of companies in software ecosystems towards sharing vulnerability information.",
"Furthermore, we compare these practices to contemporary cybersecurity recommendations.",
"This is performed through a questionnaire-based qualitative survey.",
"The questionnaire is divided into two parts: the providers' perspective and the acquirers' perspective.",
"The results show that companies are willing to share information with each other regarding vulnerabilities.",
"Sharing is not considered to be harmful neither to the cybersecurity nor their business, even though a majority of the respondents consider vulnerability information sensitive.",
"However, the companies, despite being open to sharing, are less inclined to proactively sharing vulnerability information.",
"Furthermore, the providers do not perceive that there is a large interest in vulnerability information from their customers.",
"Hence, the companies' overall attitude to sharing vulnerability information is passive but open.",
"In contrast, contemporary cybersecurity guidelines recommend active disclosure and sharing among actors in an ecosystem."
],
[
"Introduction",
"Software products are rarely developed from scratch, nor by a single company or organization.",
"Third party components can be both open source software (OSS) and purchased parts, and might depend on continuously available services from others to work as intended.",
"The resulting product is the combined efforts from a software ecosystem [1].",
"Any component of the software can have vulnerabilities, whether developed internally or acquired.",
"Vulnerabilities are “...`flaws' or `mistakes' in computer-based systems that may be exploited to compromise the network and information security of affected systems” [2].",
"The total cybersecurity cannot be handled by any one organization alone.",
"Rather, it depends on the cooperation of several organizations.",
"We define cybersecurity as measures taken to prevent, detect, and react to actions that may compromise confidentiality, integrity, or availability to data or devices, primarily in the context of Internet connectivity.",
"There are new vulnerabilities disclosed every day.",
"Between 2005 and 2016, the number of new vulnerabilities reported in the National Vulnerability Database (NVD) [3] ranged between 4 000-8 000 per year.",
"In 2017 and 2018, around 14 700 and 16 500 new vulnerabilities respectively were reported.",
"It requires both effort and knowledge to be able to evaluate these vulnerabilities, sometimes found in several different public databases, non-trivial to fuse when there is conflicting information [4].",
"The situation is further aggravated by the fact that information on a vulnerability alone is often not enough to assess whether there is an impact on the cybersecurity of a product.",
"The configuration of the product, how the vulnerability can be exploited, the environment in which the product is running, etc., impact the damage of a vulnerability.",
"An acquired component might use OSS with vulnerabilities.",
"Hence, there is a need to understand not just the components developed internally but also those acquired.",
"Even if a product might be susceptible to an exploit, it might still be more economical from a business perspective not to fix the product.",
"For example, if the fixing and updating is expensive and the risk of the product being exploited is low.",
"At the same time, the customers' best interest must also be considered, and it might be morally questionable to leave vulnerabilities in the software without informing them.",
"Previous research suggests appropriate timing is an important aspect both for vulnerability disclosure by vendors [5] and patching by their customers [6].",
"In this paper, we surveyed companies' attitudes towards sharing information on their vulnerabilities within the value chain.",
"Specifically, we studied the two perspectives of organizations providing and organizations acquiring software.",
"What is the attitude towards sharing vulnerability information within the software ecosystem?",
"How do contemporary cybersecurity recommendations align with companies' preferences for vulnerability disclosure for (IoT) ecosystems?",
"A company can either have an acquirer role or a provider role in a software ecosystem, or it has both roles.",
"An acquirer receives software or uses a service from another company or organization, e.g.",
"for a smart home, an acquirer acquires intelligent light switches with hardware, software, and radio.",
"The acquirer in turn provides a product or service to others.",
"A provider might also acquire parts of the software they provide.",
"For example, the provider of intelligent light switches might develop the hardware and software for the light switches, but not the radio component.",
"The rest of the paper is organized as follows: Background and related work is found in Section .",
"Section outlines the research method.",
"The results from the survey are found in Section and the discussion is found in Section .",
"The paper is concluded with Section .",
"The contemporary recommendations are to share information within the software ecosystem to ensure the cybersecurity of the system as a whole.",
"However, as pointed out above, the optimal timing of disclosures is non-trivial [5], [7], [8].",
"The U.S. Department of Homeland Security released a document in 2016 with non-binding principles and best practices for security in connected devices [9].",
"They recommends that vulnerability disclosure should involve developers, manufacturers, and service providers.",
"It should also include information regarding any vulnerabilities reported to a computer security incident response team (CSIRT).",
"The U.S. Food and Drug Administration (FDA) has released similar recommendations [10], [11].",
"The European Union Agency for Network and Information Security (ENISA) released a report in November 2017 [12], recognizing the need for coordinated vulnerability disclosure and the importance of participating in information sharing platforms.",
"These recommendations are also acknowledged by national bodies, e.g., the Swedish Civil Contingencies Agency (MSB) in Sweden [13].",
"The Broadband Internet Technical Advisory Group (BITAG) has released recommendations stating that manufacturers should report information on software vulnerabilities that pose security or privacy threats to consumers [14].",
"The IoT Security Foundation (IoTSF) state in their guidelines that an organization should have a mechanism for issuing security advisories for informing users when a problem is fixed [15], [16].",
"The Online Trust Alliance (OTA) – an initiative within the Internet Society (ISOC) – has released an IoT Security & Privacy Trust Framework [17].",
"They recommended responsible remediate of vulnerabilities and threats."
],
[
"Related work",
"Mufti et al.",
"performed a systematic mapping study and a set of case studies where they develop and evaluate a “readiness model” for security requirements engineering [18].",
"The mapping study included a literature study with 104 primary studies on security requirements engineering.",
"The resulting model describes readiness in maturity levels from `initial' with no security requirements to `monitored' with security requirements for prevention and long-term goals.",
"In our previous work, we present an interview-based qualitative survey [19].",
"The purpose of that survey was to understand how security vulnerabilities, especially in third party software in IoT systems, are managed by industrial organizations.",
"We saw that companies can be characterized according to their role in the system development value chain, from component developer to system integrator, i.e., a classification which can be used to describe sharing of information, and is similar to that in this paper.",
"Another interview study on security and IoT is presented by Asplund et al. [20].",
"They investigate the degree to which security is seen as important by practitioners.",
"They found that legacy systems enhanced by IoT solutions were often highly critical for society, which could slow down the process of transforming them into an IoT architecture, and they also found that system availability in general is more important than confidentiality of data.",
"That is, they saw that engineers in different domains value (aspects of) security differently."
],
[
"Research method",
"The purpose of this qualitative survey [21] was to explore how companies reason about sharing information on vulnerabilities either in their own software or other companies' with whom they have a business relationship – bilateral or other.",
"The survey was partly descriptive and partly exploratory [22].",
"It was descriptive in that the ecosystems and the companies are described.",
"It was exploratory in that we want to build up an understanding of scenarios of how companies reason about information on vulnerabilities in general and sharing of vulnerability related information within their software ecosystem in particular."
],
[
"Study design",
"In the previous work, we identified that there is a need to further understand vulnerability management within software ecosystems [19].",
"In the current study, we focused on the B2B relationships among companies with either a provider or an acquirer role in the software ecosystem.",
"We developed the first version of the questionnaire based on related work.",
"We iterated the development of the questionnaire twice, with internal reviews among the authors in each iteration.",
"We also asked a colleague external to the project to review the questionnaire after the second iteration.",
"The final version of the questionnaire consisted of four parts:The two versions of the questionnaire, including all Likert items [21] used can be found at https://sics.box.com/s/daok9em2txe1625o2ubzl99gj5nrwora Five context questions to characterize the companies.",
"Five groups of Likert itemsA Likert item is one statement that the respondent should judge whether they agree or disagree to it.",
"on vulnerability management and related practices for providers.",
"Five groups of Likert item on vulnerability management and related practices for acquirers.",
"Four general fix-answers questions on the companies' handling of cybersecurity.",
"The attitude towards being more open to share information regarding vulnerabilities was investigated.",
"Hence, a high sum total score on the Likert items should indicate a willingness to share information with others and a low score that the companies are less inclined to do so.",
"The two parts for providers and acquirers are mirrored as far as possible.",
"If there is a Likert item for the provider part, it is usually also found in the acquirer part.",
"This means we can compare the perspectives.",
"Companies can be both providers and acquirers at the same time, though this is not always the case.",
"The five groups of Likert items make up the Likert scales found in Table REF .",
"Table: The groups of Likert items and their corresponding identifiers in the provider and acquirer part – see also Figure and Figure .We decided to have five fixed-alternative expressions, ranging from “strongly disagree” to “strongly agree” [21].",
"We also decided to include a \"do not know/not applicable\" alternative as it is plausible that not all respondents can answer all items in the Likert questions [23].",
"Lastly, three items in each of the two parts were deliberately phrased negatively in the sense that a disagreement should be treated as an agreement and vice verse."
],
[
"Execution",
"In the first data collection round, we asked members of a cybersecurity project to fill in the survey.",
"In this step, we selected the companies purposely [21] as we knew their interest and maturity in cybersecurity.",
"The respondents were security experts from the large and medium sized companies and managers from the small companies.",
"Based on the first round, we made some adjustments to the questionnaire.",
"Specifically, we added one Likert item to two groups for the provider part (6f and 9c) and to the two corresponding groups for the acquirer part (12f and 15c).",
"In addition, we removed one Likert item from one question for the provider part REF (13d in the first version of the questionnaire).",
"As the changes are minor, we see no risk in using those answers along with the rest of the answers.",
"For the second round, we first used a systematic sample with large companies in SwedenWe selected the companies registered at the Swedish stock exchange on the large capital list.",
"At the time of the study, it was 94 companies.",
"However, we excluded pure investment companies.",
"Therefore our sample frame was 84 companies.",
"as our sample frame [23].",
"We approached the companies through their web page and contact information which could be found there.",
"We asked to get in touch with CTOs, business managers, etc., within the companies.",
"We employed a person on an hourly basis to elicit the answers from the companies.",
"They contacted companies both via e-mail and by phone.",
"Once a contact was established – beyond the generic switchboard – we reminded them every second week and we sent them up to 5 reminders.",
"We got answers from managers either specifically for security or for IT.",
"In the third round, we used the same questionnaire as in the second round.",
"To complement the answers gathered in the first and second round, we utilized our professional networks to elicit additional answers, i.e., convenience sampling [23].",
"The respondents were more mixed but overall individuals with good insight into the software development of their companies but not necessarily cybersecurity."
],
[
"Sample companies",
"In total, we got answers from 17 companies of different sizes and various domains, see Table REF .",
"5 companies answered only the provider part, 4 companies answered only the acquirer part and 8 companies answered both parts.",
"Table: List of companies and their size, in terms of number of employees.The large company in round one also provided an answer in round two.",
"When the answer is removed, we have answers from 9 large companies and in total 16 companies, cf.",
"Section REF , second to last paragraph.",
"In the first round, there were 4 companies .",
"We got answers for all of them, as expected since we knew they had an interest in the subject.",
"In the second round, 84 companies from our sample frame contacted.",
"18 companies could not find an appropriate person who was willing to answer the questionnaire.",
"52 companies never replied at all.",
"Our initial email bounced for 9 companies and we could not find other ways to contact them.",
"Three companies explicitly said they would not answer the survey as they neither provide nor acquire software-intense products relevant for our study.",
"We got answers from 3 companies.",
"In the third round, we used convenience sampling.",
"In total, we contacted 22 companies.",
"This resulted in 10 answers.",
"The large company from round one also provided an answer in round two.",
"However, the answer was provided by different individuals, though from the same business unit.",
"The answers are very similar.",
"Therefore, we decided to keep the answer from round two, as round two is from a methodology perspective more rigorous.",
"All of the companies are active in Sweden, operating on an international market.",
"All of the large companies have development units both within and outside Sweden whereas the non-large companies have their development work in Sweden."
],
[
"Final Likert scale",
"In order to select the items of the Likert scale to use in the analysis, we perform an item analysis to identify which of the items that have discriminative power [21].",
"(As three items in each part of the survey are phrased negatively, we first inverted those answers before performing the item analysis.)",
"Those items with the largest discriminative power are those that best answer the overall question of the attitude of sharing vulnerability information.",
"The following items are removed: The items related to the business ecosystem both for the provider (group 7) and the acquirer part (group 13) have a low discriminative power.",
"One item from the OSS group in the provider part has no counterpart in the acquirer part and low discriminative power (item 6e).",
"For release management and software updates, one item (10b) in the provider part has no corresponding item in the acquirer part and a low discriminative power.",
"Two items (10c and 10e and the corresponding 16c and 16d) have a low discriminative power in both parts.",
"One item in the acquirer part (12e) has no corresponding item in the provider part but a high discriminative power.",
"That item is retained in the final scale.",
"The final Likert scale is found in Figure REF .",
"Figure: Providers and acquirers, stacked bar charts representing the frequency of answers per Likert item (i.e.",
"each item sums up to 100%).",
"Note that the answers are inverted for 6 items as the questions are phrased negatively, see Section .",
"Also note that item 12e does not have a corresponding item in the provider part.Figure: Large and non-large companies, stacked bar charts representing the frequency of answers per Likert item (i.e.",
"each item sums up to 100%).",
"(Non-large are the medium, small and micro companies.)",
"Note that the answers are inverted for 6 items as the questions are phrased negatively, see Section ."
],
[
"Practices and prevalence of OSS",
"OSS is a significant part of the products for many companies in the study (cf.",
"6a and 12a in Figure REF ).",
"However, it is a larger part in the non-large (medium, small, and micro) companies than in large companies, see Figure REF .",
"Hence, smaller companies seem to utilize OSS more than larger companies in the survey.",
"However, the companies are less inclined to actively contribute to OSS projects (6b, 6c, 12b and 12c).",
"Non-large companies seem slightly more inclined to contribute than large companies.",
"The difference is smaller than the difference on prevalence on OSS.",
"Another similarity between providers and acquirers is that non-large companies seldom have dedicated individuals who monitor forums on cybersecurity for OSS, e.g., NVD and exploit-DB (6d and 12d in Figure REF ).",
"It seems like larger companies in the survey are more inclined to have dedicated individuals monitoring vulnerabilities databases, independent of role (6d and 12d in Figure REF ).",
"Acquirers use more third-party software or service to keep track of vulnerabilities in OSS used (6f and 12f) than providers.",
"Also, the acquirers seem to demand that providers keep their OSS component updated with the latest version – more than the providers are inclined to provide it (6g and 12g).",
"There is no difference between large companies and non-large companies in the survey."
],
[
"Information sharing in the value chain",
"The providers answer more positive to sharing vulnerability information with their customer (8a).",
"The acquirers indicate that they do not get security information from their providers (14a) – with a similar pattern for critical vulnerabilities (8c and 14c in Figure REF ).",
"Non-large acquirers answered, in the median, around “Neither agree nor disagree\", see Figure REF .",
"Both acquirers and providers are inclined to ask for information on vulnerabilities.",
"The acquirers seem less inclined to ask the providers for information on security whereas the providers comply with requests from acquirers on security (8b and 14b in Figure REF ) – no difference whether a large or non-large company.",
"This is in line with that the providers perceive an interest in cybersecurity from their customers (8e and 14e in Figure REF ) – which acquirers also answer.",
"Both providers and acquirers are aligned in the perception that cybersecurity information is sensitive and that vulnerabilities addressed should be communicated in, e.g., release notes (8f, 8g, 14f and 14g in Figure REF ) – no difference between large and non-large companies."
],
[
"Effects of information sharing in the value chain",
"The items related to the information sharing practices – groups 8 and 14 – are contrasted by the items related to the effect of information sharing in the value chain – groups 9 and 15.",
"For the latter, the attitude in general is more positive to sharing than compared to the actual practices from the former.",
"Most answers are either neutral or positive, cf.",
"Figure REF (note that 9a and 15a are negatively posed and should be interpreted inversely).",
"The exception is the item asking whether sharing cybersecurity information is harmful (9a), where the attitude from the providers is more balanced.",
"Furthermore, non-large providers are less inclined to consider vulnerability information to be harmful than larger companies, see 9a in Figure REF .",
"The profile is similar between providers and acquirers, albeit the latter is less extreme in their answers."
],
[
"Practices related to software update and release management",
"Both acquirers and providers indicate in their answers that updating software is costly (10d and 16c).",
"However, the acquirers still have an ambition to keep the software up to date (16a).",
"Providers, when asked if they update their software frequently (daily or weekly) indicated that they do not (10a).",
"However, as that item is expressed using the extreme value of daily or weekly, the results should be interpreted with care.",
"Lastly in the group on release management, the acquirers are more positive to pay for updates than the providers perceive (10f and 16e).",
"The answers from large and non-large companies are, in the median, similar."
],
[
"General questions on cybersecurity",
"41% (7) answered that they are using a third-party service to handle cybersecurity and 23% (4) answered that they have not considered it (question 17).",
"In terms of cybersecurity competence, 9 answered that they have competence within the company to handle cybersecurity and 8 that they do not (question 18).",
"Cyber insurance is only used by 2 of the companies in the study and 10 of the companies have not even considered it (question 19).",
"This is not surprising, as it is known that cyber insurance uptake in Sweden is still low [24].",
"Lastly, 9 of the companies find cybersecurity to be very important whereas 8 do not prioritize it (question 20).",
"There is no correlation among these questions (17-20).",
"Half of the companies in the survey answered that they only have a basic understanding of cybersecurity, which seems to be uncorrelated to how important cybersecurity is perceived.",
"The sample is likely skewed, however, as it is plausible that people are more inclined to participate in the survey if they have an interest in cybersecurity.",
"This is further reinforced by our difficulties to elicit answers, especially in round two.",
"Despite a substantial investment with an hourly employed person to elicit answers, we only got a 3.6% response rate in this round, even though several of the companies indicate that they find cybersecurity important.",
"We see three possible explanations for the difficulties to elicit answers, in addition to survey-fatigue and scarcity of time that affects all surveys: 1) even though the spontaneous answer is that cybersecurity is important, in reality respondents are not prepared to actually invest in it, 2) the topic of the survey is advanced and many organizations simply lack the experience and competence to answer the questionnaire, and 3) the topic is considered too sensitive to answer.",
"The first two explanations might indicate a need to raise the cybersecurity awareness and competence in industry.",
"The prevalence and approach to OSS is similar between providers and acquirers.",
"However, as the acquirers are not updating the software with the latest OSS version themselves, they expect the providers to take the cost.",
"Also, as the acquirers are likely less capable of handling vulnerabilities on the OSS components, they are more inclined to use third-party tools or services to monitor the security.",
"Hence, the awareness of the importance of having a proactive attitude to updated software seems to falter."
],
[
"RQ1 What is the attitude towards sharing vulnerability information within the software ecosystem?",
"In general, companies seem reactive rather than proactive when it comes to information sharing of vulnerabilities.",
"There is a willingness to share information, at least with business partners (8a, 8b, 14a, and 14b), however, it does not seem to be proactive and planned upfront (8c, 8d, 14d).",
"At the same time, the attitude to sharing is positive as seen in groups 9 and 15.",
"This indicates that the perception of wanted position and the actual reality in the companies is not aligned.",
"This is similar to the handling of OSS.",
"The companies commonly use OSS (items 6a and 12a), however, are less inclined to contribute back or actively participating in the community (items 6b, 6c, 12b and 12c).",
"This indicates that the companies do not have an elaborate OSS or software ecosystem strategy, whether large or non-large.",
"Rather, it seems as if they want to get access to assets but do not see the benefit from giving away assets without monetary compensation.",
"This indicates that the companies in the survey have not yet seen the benefit of sharing, indicating an immaturity.",
"Interestingly, this is true even for non-large companies, which we expected to be more active and involved in OSS communities.",
"Both providers and acquirers consider vulnerability information to be sensitive (items 8f and 14f).",
"When asked whether they do not share cybersecurity information because sharing it can be harmful (items 9a and 15a), the attitude seems contradictory to the previous question.",
"We interpret this as an indication that respondents are unsure how sensitive it is to share vulnerability information.",
"There might also be an influence from the wording – and therefore the interpretation – of the questions on the answers.",
"Interestingly, the attitude to being transparent and open with vulnerabilities is overall positive (9b and 15b).",
"This can be contrasted to the perception that the counterparts in the software ecosystem are negative to disclosing vulnerability information (9f and 15f respectively).",
"We speculate that this is due to self-serving bias, where oneself is considered overly mature and counterparts overly immature.",
"Overall, in relation to the research question, we hypothesize that there is a lack of maturity and therefore practices for how to handle vulnerability disclosure.",
"In relation to RQ1, we interpret the answers that overall the companies do not have explicit nor elaborate practices regarding sharing of vulnerability information.",
"Furthermore, the companies seem, in general, to consider vulnerability information to be sensitive, though this is not a well-informed opinion."
],
[
"RQ2 How do contemporary cybersecurity recommendations align with companies' preference for vulnerability disclosure for (IoT) ecosystems?",
"As outlined in the background section (Section REF ), government agencies and industry interest groups in general promote disclosure and sharing of vulnerability information in the software ecosystem.",
"Both American and European government agencies recommend (proactive) and even coordinated disclosure [9], [12].",
"The idea is that if the actors in a software ecosystem cooperate, cybersecurity will be improved.",
"The BITAG and ISOC organizations go even further and indicate that threats should be reported to the consumers as well [14], [17].",
"In our survey, providers tend to be less proactive in providing information but do provide it if requested.",
"On the other hand, acquirers are not that interested in cybersecurity.",
"Also, vulnerability information is considered sensitive – at least by providers.",
"Hence, there seems to be a tendency to keep information confidential as a way to achieve “security through obscurity\".",
"Therefore, even the respondents who rate themselves as competent in the area of cybersecurity seem to be misaligned with contemporary recommendations.",
"We speculate that this is a combination of lack of competence in cybersecurity as well as a lack of understanding how to communicate around vulnerabilities as it can be seen as negative from the market [25] and customer perspectives.",
"Indeed, vulnerability disclosure could be a tragedy of the commons, where the recommendations are correct that overall cybersecurity would increase if there was more information sharing, but individual companies still could be worse off implementing such practices, at least in the short run.",
"However, it could also be that information sharing is good for individual companies even in the short run, but that they fail to appreciate this, e.g., because the investments needed to become more mature are more tangible than the benefits entailed.",
"These lines of reasoning imply that it is not just about increasing the cybersecurity knowledge in the software development parts of a company but a more general problem that requires a broader approach outside the technical roles.",
"There are also recommendations, e.g., from ISOC [17], on the software update release process.",
"Acquirers seem quite willing to update often and even pay for updates.",
"The providers seem less inclined to providing updates with the same frequency.",
"Most of the cost might be on the provider, explaining this.",
"It can also be, however, that providers have a deeper and more nuanced insight into the actual need for updates from a technical perspective, whereas acquirers simply always want “latest and greatest”.",
"This discrepancy indicates that there is a mismatch in the software ecosystem in understanding each other in terms of incentives and willingness to pay for updated software.",
"For cybersecurity, this might imply that the lack of communication and mutual understanding of the actual need for addressing vulnerabilities lead to unnecessary cost and efforts.",
"At the same time, the acquirers are somewhat inclined to trust their providers (12e in Figure REF ).",
"This implies that there is a lack of practices and an underlying assumption that the counterparts in the software ecosystem can be trusted."
],
[
"Threats to validity and reliability",
"Here the validity of the conducted research is discussed based on commonly considered validity threats, e.g., as listed in [26].",
"Content validity concerns how appropriate the contents of the survey questions are to the respondents.",
"The questions asked concern cybersecurity, which requires some level of competence to answer.",
"Because of that, measures were taken not only to formulate the questions in a clear and understandable way with terms used in industry, but also to find the right persons in the organizations to answer the questions.",
"In the first round, questions were answered by respondents participating in a research project on cybersecurity, which means that uncertainties in the questions could be solved.",
"In the later rounds initial contacts were often taken with subjects in management positions, but they were asked to get support by experts in their organizations.",
"During the research it was found that this type of questions can be difficult to answer, at least when seeking an answer for the entire company.",
"This emphasizes the importance of formulating the questions as clearly as possible, and spending effort on finding the right respondents.",
"We believe the measures we have taken appropriately address the threats to content validity.",
"Construct validity concern the degree to which the constructs investigated are actually measured by the questions.",
"In this type of study this is a threat since there is a risk that people do not interpret the questions in the way intended by the researchers, especially when they cover aspects such as cybersecurity that requires some specific competence to understand.",
"Well formulated questions increase both the number of possible respondents and the likelihood that they understand the content of the questions.",
"The questions were developed in iterations, which we believe resulted in questions with lowered risk of misinterpretations.",
"Also, in the first round they were answered by respondents in a research project on cybersecurity, which we believe made these respondents more motivated to understand the questions and to give feedback than would be the case if we had started with respondents from the later rounds.",
"Reliability concerns the degree to which a respondent would hypothetically give the same answers if they answered the same questions again under the same conditions, or if two persons would give the same answers, e.g., measured by a measure of rater agreement.",
"We have not measured the agreement between respondents since the sample represents different views.",
"If the sample would be significantly larger it may have been possible to study groups of respondents, but with this sample size it is not realistic.",
"The validity of a survey is, of course, affected by the possibility to generalize from the sample to the entire sample frame.",
"We cannot argue to have a statistically rigorous sampling approach (cf.",
"Section ).",
"However, our main aim is not to quantitatively survey and make generalization statements of a population.",
"Rather, our aim is to understand the diversity and different approaches, i.e.",
"the same reason we do not investigate rater agreement when we analyze reliability.",
"Hence, we are not attempting to generalize the findings to an entire population.",
"However, at the same time, the threats to generalization should not be exaggerated [27].",
"As we have covered different company sizes, several domains, both technical and non-technical ones, and many locations in Sweden, we believe the findings to be, at least to some extent, relevant for providers and acquirers of software-intense products.",
"We should have some coverage of relevant phenomena in that domain even though we cannot make statements of the statistical significance.",
"For the larger companies there may be several more or less independent parts of the companies.",
"Hence, for those cases, the answers are more from that part of the company rather than for the entire company.",
"Again, this of course is a threat to generalization.",
"At the same time, even though results are not statistically significant, we do believe the findings in general are relevant for the entire population."
],
[
"Conclusion",
"The attitude towards open communities and sharing of information is, in general, positive.",
"However, when asked specific questions, respondents' attitude seems to be that it is OK to use information and source code from others but the companies are less positive to actually contributing themselves, whether to OSS development or to vulnerability information disclosure.",
"In combination with what seems to be a lack of established practices and procedures on vulnerability disclosure and communication, we identify a need to further improve the competences required for this.",
"Furthermore, we call for improved processes and decision support within product management and market communication, etc., as vulnerability handling is not isolated to purely technical issues.",
"Lastly, there might be a further need to adapt and establish recommendations from government agencies and industry interest groups.",
"The practical implication of this study is that there is a need for companies to increase their knowledge of vulnerability management in general and specifically understanding their own technical environment to be able to make informed decisions on how to analyze and share vulnerability information.",
"There also seems to be a role for a trusted third party to facilitate the sharing of vulnerability information.",
"However, we believe there is a need to better understand the vulnerability sharing in order to provide validated guidelines.",
"Future research should address business models related to vulnerability information sharing.",
"Furthermore, we believe there is a need to better understand how individual vulnerabilities in individual parts of a larger software system configuration impacts the overall cybersecurity.",
"Lastly, there is a need to better understand how to upfront design complex systems made up of components from several companies and OSS communities to allow for analyzability of the resulting cybersecurity."
],
[
"Acknowledgements",
"We would like to thank all the participating companies and Vinnova (grant 2016-00603 and grant 2018-03965) for funding our research.",
"U. Franke was partially supported by the Swedish Civil Contingencies Agency, MSB (agreement no.",
"2015-6986).",
"We would also like to thank Daniel Wisenhoff for his work in collecting answers."
]
] | 1906.04424 |
[
[
"Chinese Embedding via Stroke and Glyph Information: A Dual-channel View"
],
[
"Abstract Recent studies have consistently given positive hints that morphology is helpful in enriching word embeddings.",
"In this paper, we argue that Chinese word embeddings can be substantially enriched by the morphological information hidden in characters which is reflected not only in strokes order sequentially, but also in character glyphs spatially.",
"Then, we propose a novel Dual-channel Word Embedding (DWE) model to realize the joint learning of sequential and spatial information of characters.",
"Through the evaluation on both word similarity and word analogy tasks, our model shows its rationality and superiority in modelling the morphology of Chinese."
],
[
"Introduction",
"Word embeddings are fixed-length vector representations for words [14], [5].",
"In recent years, the morphology of words is drawing more and more attention [4], especially for Chinese whose writing system is based on logogramshttps://en.wikipedia.org/wiki/Logogram.",
"UTF8gbsn With the gradual exploration of the semantic features of Chinese, scholars have found that not only words and characters are important semantic carriers, but also strokehttps://en.wikipedia.org/wiki/Stroke_(CJK_character) feature of Chinese characters is crucial for inferring semantics [2].",
"Actually, a Chinese word usually consists of several characters, and each character can be further decomposed into a stroke sequence which is certain and changeless, and this kind of stroke sequence is very similar to the construction of English words.",
"In Chinese, a particular sequence of strokes can reflect the inherent semantics.",
"As shown in the upper half of Figure REF , the Chinese character “驾\" (drive) can be decomposed into a sequence of eight strokes, where the last three strokes together correspond to a root character “马\" (horse) similar to the root “clar\" of English word “declare\" and “clarify\".",
"Moreover, Chinese is a language originated from Oracle Bone Inscriptions (a kind of hieroglyphics).",
"Its character glyphs have a spatial structure similar to graphs which can convey abundant semantics [21].",
"Additionally, the critical reason why Chinese characters are so rich in morphological information is that they are composed of basic strokes in a 2-D spatial order.",
"However, different spatial configurations of strokes may lead to different semantics.",
"As shown in the lower half of Figure 1, three Chinese characters “入\" (enter), “八\" (eight) and “人\" (man) share exactly a common stroke sequence, but they have completely different semantics because of their different spatial configurations.",
"Figure: The upper part is an example for illustrating the inclusion relationship hidden in strokes order and character glyphs.",
"The lower part reflects that a common stroke sequence may form different Chinese characters if their spatial configurations are different.In addition, some biological investigations have confirmed that there are actually two processing channels for Chinese language.",
"Specifically, Chinese readers not only activate the left brain which is a dominant hemisphere in processing alphabetic languages [20], [10], [15], but also activate the areas of the right brain that are responsible for image processing and spatial information at the same time [23].",
"Therefore, we argue that the morphological information of characters in Chinese consists of two parts, i.e., the sequential information hidden in root-like strokes order, and the spatial information hidden in graph-like character glyphs.",
"Along this line, we propose a novel Dual-channel Word Embedding (DWE) model for Chinese to realize the joint learning of sequential and spatial information in characters.",
"Finally, we evaluate DWE on two representative tasks, where the experimental results exactly validate the superiority of DWE in capturing the morphological information of Chinese."
],
[
"Morphological Word Representations",
"Traditional methods on getting word embeddings are mainly based on the distributional hypothesis [8]: words with similar contexts tend to have similar semantics.",
"To explore more interpretable models, some scholars have gradually noticed the importance of the morphology of words in conveying semantics [13], [17], and some studies have proved that the morphology of words can indeed enrich the semantics of word embeddings [18], [19], [4].",
"More recently, Wieting et al.",
"wieting2016charagram proposed to represent words using character n-gram count vectors.",
"Further, Bojanowski et al.",
"bojanowski2017enriching improved the classic skip-gram model [14] by taking subwords into account in the acquisition of word embeddings, which is instructive for us to regard certain stroke sequences as roots in English."
],
[
"Embedding for Chinese Language",
"The complexity of Chinese itself has given birth to a lot of research on Chinese embedding, including the utilization of character features [3] and radicals [22], [26], [27].",
"Considering the 2-D graphic structure of Chinese characters, Su and Lee su2017learning creatively proposed to enhance word representations by character glyphs.",
"Lately, Cao et al.",
"cao2018cw2vec proposed that a Chinese word can be decomposed into a sequence of strokes which correspond to subwords in English, and Wu et al.",
"wu2019glyce designed a Tianzige-CNN to model the spatial structure of Chinese characters from the perspective of image processing.",
"However, their methods are either somewhat loose for the stroke criteria or unable to capture the interactions between strokes and character glyphs."
],
[
"DWE Model",
"As we mentioned earlier, it is reasonable and imperative to learn Chinese word embeddings from two channels, i.e., a sequential stroke n-gram channel and a spatial glyph channel.",
"Inspired by the previous works [3], [6], [21], [25], we propose to combine the representation of Chinese words with the representation of characters to obtain finer-grained semantics, so that unknown words can be identified and their relationship with other known Chinese characters can be found by distinguishing the common stroke sequences or character glyph they share.",
"UTF8gbsn Our DWE model is shown in Figure REF .",
"For an arbitrary Chinese word $w$ , e.g., “驾车\", it will be firstly decomposed into several characters, e.g., “驾\" and “车\", and each of the characters will be further processed in a dual-channel character embedding sub-module to refine its morphological information.",
"In sequential channel, each character can be decomposed into a stroke sequence according to the criteria of Chinese writing system as shown in Figure REF .",
"After retrieving the stroke sequence, we add special boundary symbols $<$ and $>$ at the beginning and end of it and adopt an efficient approach by utilizing the stroke n-gram method [2]We apply a richer standard of strokes (32 kinds of strokes) than they did (only 5 kinds of strokes).",
"to extract strokes order information for each character.",
"More precisely, we firstly scan each character throughout the training corpus and obtain a stroke n-gram dictionary $G$ .",
"Then, we use $G(c)$ to denote the collection of stroke n-grams of each character $c$ in $w$ .",
"While in spatial channel, to capture the semantics hidden in glyphs, we render the glyph $I_c$ for each character $c$ and apply a well-known CNN structure, LeNet [11], to process each character glyph, which is also helpful to distinguish between those characters that are identical in strokes.",
"After that, we combine the representation of words with the representation of characters and define the word embedding for $w$ as follows: $\\textbf {w} = \\textbf {w}_{ID} \\oplus \\frac{1}{N_c}(\\sum _{c \\in w} {\\sum _{g \\in G(c)} \\textbf {g} \\ast CNN(I_c)}),$ where $\\oplus $ and $\\ast $ are compositional operationThere are a variety of options for $\\oplus $ and $\\ast $ , e.g., addition, item-wise dot product and concatenation.",
"In this paper, we uses the addition operation for $\\oplus $ and item-wise dot product operation for $\\ast $ .. $\\textbf {w}_{ID}$ is the word ID embedding and $N_c$ is the number of characters in $w$ .",
"Figure: An illustration of our Dual-channel Word Embedding (DWE) model.According to the previous work [14], we compute the similarity between current word $w$ and one of its context words $e$ by defining a score function as $s(w, e) = \\textbf {w} \\cdot \\textbf {e}$ , where $\\textbf {w}$ and $\\textbf {e}$ are embedding vectors of $w$ and $e$ respectively.",
"Following the previous works [14], [1], the objective function is defined as follows: $\\small \\begin{split}\\mathcal {L} = \\sum _{w \\in D} \\sum _{e \\in T(w)} log \\sigma (s(w, e)) + \\lambda \\mathbb {E}_{e^{\\prime }\\sim P} [log \\sigma (-s(w, e^{\\prime }))],\\end{split}$ where $\\lambda $ is the number of negative samples and $\\mathbb {E}_{e^{\\prime }\\sim P}[\\cdot ]$ is the expectation term.",
"For each $w$ in training corpus $D$ , a set of negative samples $T(w)$ will be selected according to the distribution $P$ , which is usually set as the word unigram distribution.",
"And $\\sigma $ is the sigmoid function.",
"Table: Performance on word similarity and word analogy task.",
"The dimension of embeddings is set as 300.",
"The evaluation metric is ρ\\rho for word similarity and accuracy percentage for word analogy."
],
[
"Dataset Preparation",
"We download parts of Chinese Wikipedia articles from Large-Scale Chinese Datasets for NLPhttps://github.com/brightmart/nlp_chinese_corpus.",
"For word segmentation and filtering the stopwords, we apply the jiebahttps://github.com/fxsjy/jieba toolkit based on the stopwords tablehttps://github.com/YueYongDev/stopwords.",
"Finally, we get 11,529,432 segmented words.",
"In accordance with their work [3], all items whose Unicode falls into the range between 0x4E00 and 0x9FA5 are Chinese characters.",
"We crawl the stroke information of all 20,402 characters from an online dictionaryhttps://bihua.51240.com/ and render each character glyph to a 28 $\\times $ 28 1-bit grayscale bitmap by using Pillowhttps://github.com/python-pillow/Pillow."
],
[
"Experimental Setup",
"We choose adagrad [7] as our optimizing algorithm, and we set the batch size as 4,096 and learning rate as 0.05.",
"In practice, the slide window size $n$ of stroke $n$ -grams is set as $3 \\le n \\le 6$ .",
"The dimension of all word embeddings of different models is consistently set as 300.",
"We use two test tasks to evaluate the performance of different models: one is word similarity, and the other is word analogy.",
"A word similarity test consists of multiple word pairs and similarity scores annotated by humans.",
"Good word representations should make the calculated similarity have a high rank correlation with human annotated scores, which is usually measured by the Spearman's correlation $\\rho $ [28].",
"The form of an analogy problem is like “king\":“queen\" = “man\":“?",
"\", and “woman\" is the most proper answer to “?\".",
"That is, in this task, given three words $a$ , $b$ , and $h$ , the goal is to infer the fourth word $t$ which satisfies “$a$ is to $b$ that is similar to $h$ is to $t$ \".",
"We use $3CosAdd$ [14] and $3CosMul$ function [12] to calculate the most appropriate word $t$ .",
"By using the same data used in [3] and [2], we adopt two manually-annotated datasets for Chinese word similarity task, i.e., wordsim-240 and wordsim-296 [9] and a three-groupcapitals of countries, (China) states/provinces of cities, and family relations.",
"dataset for Chinese word analogy task."
],
[
"Baseline Methods",
"We use gensimhttps://radimrehurek.com/gensim/ to implement both CBOW and Skipgram and apply the source codes pulished by the authors to implement CWEhttps://github.com/Leonard-Xu/CWE, JWEhttps://github.com/hkust-knowcomp/jwe, GWEhttps://github.com/ray1007/GWE and GloVehttps://github.com/stanfordnlp/GloVe.",
"Since Cao et al.",
"cao2018cw2vec did not publish their code, we follow their paper and reproduce cw2vec in mxnethttps://mxnet.apache.org/ which we also use to implement sisg [1]http://gluon-nlp.mxnet.io/master/examples/word_embed-ding/word_embedding_training.html and our DWE.",
"To encourage further research, we will publish our model and datasets."
],
[
"Experimental Results",
"UTF8gbsn The experimental results are shown in Table REF .",
"We can observe that our DWE model achieves the best results both on dataset wordsim-240 and wordsim-296 in the similarity task as expected because of the particularity of Chinese morphology, but it only improves the accuracy for the family group in the analogy task.",
"Actually, it is not by chance that we get these results, because DWE has the advantage of distinguishing between morphologically related words, which can be verified by the results of the similarity task.",
"Meanwhile, in the word analogy task, those words expressing family relations in Chinese are mostly compositional in their character glyphs.",
"For example, in an analogy pair “兄弟\" (brother) : “姐妹\" (sister) = “儿子\" (son) : “女儿\" (daughter), we can easily find that “兄弟\" and “儿子\" share an exactly common part of glyph “儿\" (male relative of a junior generation) while “姐妹\" and “女儿\" share an exactly common part of glyph “女\" (female), and this kind of morphological pattern can be accurately captured by our model.",
"However, most of the names of countries, capitals and cities are transliterated words, and the relationship between the morphology and semantics of words is minimal, which is consistent with the findings reported in [21].",
"For instance, in an analogy pair “西班牙\" (Spain) : “马德里\" (Madrid) = “法国\" (France) : “巴黎\" (Paris), we cannot infer any relevance among these four words literally because they are all translated by pronunciation.",
"In summary, since different words that are morphologically similar tend to have similar semantics in Chinese, simultaneously modeling the sequential and spatial information of characters from both stroke n-grams and glyph features can indeed improve the modeling of Chinese word representations substantially."
],
[
"Conclusions",
"In this article, we first analyzed the similarities and differences in terms of morphology between alphabetical languages and Chinese.",
"Then, we delved deeper into the particularity of Chinese morphology and proposed our DWE model by taking into account the sequential information of strokes order and the spatial information of glyphs.",
"Through the evaluation on two representative tasks, our model shows its superiority in capturing the morphological information of Chinese."
]
] | 1906.04287 |
[
[
"Gait modeling and optimization for the perturbed Stokes regime"
],
[
"Abstract Many forms of locomotion, both natural and artificial, are dominated by viscous friction in the sense that without power expenditure they quickly come to a standstill.",
"From geometric mechanics, it is known that for swimming at the \"Stokesian\" (viscous; zero Reynolds number) limit, the motion is governed by a reduced order \"connection\" model that describes how body shape change produces motion for the body frame with respect to the world.",
"In the \"perturbed Stokes regime\" where inertial forces are still dominated by viscosity, but are not negligible (low Reynolds number), we show that motion is still governed by a functional relationship between shape velocity and body velocity, but this function is no longer linear in shape change rate.",
"We derive this model using results from singular perturbation theory, and the theory of noncompact normally hyperbolic invariant manifolds (NHIMs).",
"Using the theoretical properties of this reduced-order model, we develop an algorithm that estimates an approximation to the dynamics near a cyclic body shape change (a \"gait\") directly from observational data of shape and body motion.",
"This extends our previous work which assumed kinematic \"connection\" models.",
"To compare the old and new algorithms, we analyze simulated swimmers over a range of inertia to damping ratios.",
"Our new class of models performs well on the Stokesian regime, and over several orders of magnitude outside it into the perturbed Stokes regime, where it gives significantly improved prediction accuracy compared to previous work.",
"In addition to algorithmic improvements, we thereby present a new class of models that is of independent interest.",
"Their application to data-driven modeling improves our ability to study the optimality of animal gaits, and our ability to use hardware-in-the-loop optimization to produce gaits for robots."
],
[
"Introduction",
"In this paper, we study how animals and robots move through space by deforming the “shape” of their body — typically in a cyclic fashion — to propel that body.",
"We call such motion-producing cyclic shape deformations gaits.",
"We study a class of locomotion which includes swimming and crawling in viscous media, in which the viscous damping forces are large compared to the inertia of the body.",
"A classic exposition of such locomotors “living life at low Reynolds number” is given in [33].",
"An important aspect of our work is that we consider the perturbed Stokes regime [9] in which the inertia-damping ratio (or Reynolds number) is small but nonzero, as opposed to previous geometric mechanics literature addressing only the viscous or Stokesian limit which formally assumes the inertia-damping ratio is zero [25], [24], [16], [17], [1].",
"We note that our methods are related to the realization of nonholonomic constraints as a limit of friction forces [4], [23], [8].",
"For both scientific and engineering purposes, it is often of interest to ask whether a particular gait is optimal with respect to a goal function.",
"For animal locomotion, explicit equations of motion are nigh impossible to come by, and therefore directly testing animal gait optimality via analytical tools like the calculus of variations is not an option.",
"However, if a model can be obtained from experimental data for the local dynamics on a tubular neighborhood of the gait cycle — i.e.",
"a model valid for small variations in the gait cycle — then local optimality tests can be formulated and evaluated on these models.",
"Such an approach was taken in [1], which introduced an algorithm informed by both geometric mechanics and data-driven techniques for studying oscillators [35], [34], [36].",
"One limitation of [1] was the assumption that motion was entirely kinematic, effectively assuming that the inertia-damping ratio is zero by assuming a viscous connection-based model as introduced by [24] and to be discussed more below.",
"The real-world systems we are interested in have small — but always nonzero — inertia-damping ratio, and therefore we are interested in the extent to which the algorithm of [1] can be improved.",
"By applying normally hyperbolic invariant manifold (NHIM) theory [11], [12], [13], [19], [14], [7] in a singular perturbation context, we show that an exponentially stable invariant slow manifold exists for small inertia-damping ratio (this was also shown in [9]).",
"Furthermore, this slow manifold is close to the viscous connection (viewed geometrically as a subbundle — hence as a submanifold — of state space), and therefore the dynamics restricted to the slow manifold are close to those assumed in the purely viscous case [25], [24], [16], [1], and reduce to those in the zero inertia-damping ratio limit.",
"Aside from its theoretical appeal, this result also has practical implications: it is possible to explicitly compute “correction terms” which, when added to the purely-viscous connection model, yield the dynamics restricted to the slow manifold.",
"The slow-manifold dynamics are provably more accurate than those of the idealized viscous connection model.",
"Additionally, they still enjoy the same useful properties of reduced dimension and symmetry under the group.",
"The computation of such correction terms is a fundamental technique in geometric singular perturbation theory [14], [20], and has been used, e.g., to compute reduced-order models of robots with flexible joints [37].",
"Given an algorithm that produces a data-driven local model of dynamics near a gait, we could conduct variational tests for local optimality of that gait with respect to any cost functional that the model allows us to evaluate.",
"Thus we have in mind two classes of application for the approach we present below: a biological application — verification of whether a postulated goal function is optimized for an observed animal gait, and an engineering application — optimization of robot gaits with “hardware-in-the-loop” by iteratively modeling and improving the gait with respect to a goal functional without the need for precise models of the robot or its interactions with the environment.",
"It is clear why our approach would be a boon to biology.",
"In most cases we cannot cajole animals to vary their gaits and observe whether that improves them.",
"Additionally, we rarely have detailed enough models of animal-environment interaction to allow gait optimality to be assessed from a model.",
"The value to gait optimization of robots comes from the fact that a gait, being a periodic continuous function of shape, is an infinite-dimensional object.",
"Thus, gait parameterizations are unavoidably of high dimension.",
"Any gradient calculation for optimization of a gait thus requires many tests to identify the influence of these many parameters.",
"Combined with the high practical cost of hardware experiments in terms of time and robot wear-and-tear, this renders hardware-in-the-loop optimization nigh infeasible.",
"We propose that by producing a tractably computable local model, we can resolve this problem.",
"The high-dimensional gradients can be computed by simulating the (local) model instead of directly using the hardware, decoupling the dimension of the gait parameterization from the number of experiments conducted on hardware.",
"It is our hope that, through a combination of geometric mechanics and NHIM theory, we can develop an algorithm which can serve the purposes of both biologists and engineers."
],
[
"Acknowledgements",
"The authors were supported by NSF CMMI 1825918 and ARO grants W911NF-14-1-0573 and W911NF-17-1-0306 to Revzen.",
"Kvalheim would like to thank Jaap Eldering for introducing him to the relevance of NHIM theory to locomotion, for helpful comments and suggestions regarding the global asymptotic stability of the slow manifold of Thm.",
"REF , and for other useful suggestions."
],
[
"Background",
"In studying locomotion, we will consider dissipative Lagrangian mechanical systems on a product configuration space $Q = S \\times G$ with coordinates $(r,g)$ , and with a Lagrangian of the form kinetic minus potential energy.",
"Here $S$ is the shape space of the locomoting body, and $G$ is a Lie group (typically a subgroup of the Euclidean group $\\mathsf {SE}(3)$ of rigid motions) representing the body's position and orientation in the world.In a formal sense, one may start with generalized coordinates $Q$ and the action of $G$ , and define $S$ as a quotient manifold $Q/G$ .",
"The details of this construction are not germane to our argument.",
"Instead, for simplicity we postulate the separation of configuration into “shape” and “body-frame” here, with the more general case treated in the appendices.",
"We assume throughout this paper that $S$ is compact.",
"We will also assume that this system is subjected to external viscous drag forces which are linear in velocity.We make this assumption for simplicity.",
"In principle, it should be possible to relax this assumption to derive modified but similar results for a force depending nonlinearly on velocities, as long as the linear approximation (with respect to velocities) of this force satisfies the same assumptions that we impose on our assumed linear force.",
"If the physics of locomotion are independent of the body's position and orientation, then the Lagrangian $L(r,g,\\dot{r},\\dot{g})$ is independent of $g,\\dot{g}$ and the viscous drag force $F_R(r,g,\\dot{r},\\dot{g})$ is equivariant in $g$ (on the $g,\\dot{g}$ components).",
"Under this symmetry assumption, [25] derived general equations of motion satisfied by $g$ and by the body momentumHere $\\mathfrak {g}^*$ is the vector space dual of the Lie algebra $\\mathfrak {g}$ of $G$ .",
"$p\\in \\mathfrak {g}^*$ ; these equations are essentially special cases of those derived in [3].",
"For a detailed statement and derivations of these equations, see §.",
"Let us suppose that the kinetic energy metric of the body is scaled by a dimensionless inertial parameter $m > 0$ , that the viscous drag force $F_R$ is scaled by a dimensionless damping parameter $c > 0$ , and define $\\epsilon \\frac{m}{c}$ the dimensionless ratio of the two which is (up to scale) the Reynolds number in the case of fluid dynamics.",
"[25] showed that in the limit $\\epsilon \\rightarrow 0$ , the equation of motion for $g$ becomes independent of $p$ .",
"Defining the body velocityThe body velocity is often written $g^{-1}\\dot{g}$ by an abuse of notation which is only defined on matrix Lie groups where the product of a tangent vector and a group element is naturally defined.",
"For a general definition note that $\\dot{g} \\in g G$ , and the derivative of the left action ${L}_{g^{-1}}$ restricts to a map $g G \\rightarrow e G \\cong \\mathfrak {g}$ .",
"Hence the definition above.",
"$\\accentset{\\scriptstyle \\circ }{g} {L}_{g^{-1}}\\dot{g}$ , they obtained $\\accentset{\\scriptstyle \\circ }{g}=-A_{\\textnormal {visc}}(r) \\cdot \\dot{r},$ where $A_{\\textnormal {visc}}$ is called the local viscous connection.",
"Away from the Stokes limit, [9] studied the perturbed Stokes regime in which $\\epsilon $ is assumed to be small but nonzero.",
"For $\\epsilon $ sufficiently small they showed there is an exponentially stable invariant slow manifold $M_\\epsilon $ , to which the dynamics converge.",
"We derive similar results tailored for our applications in §.",
"Using an asymptotic series expansion for the slow manifold, in § we also prove that the equations of motion for trajectories within $M_\\epsilon $ take the form given by Thm.",
"REF below.",
"Hence trajectories of the full dynamics converge to solutions of Eqn.",
"(REF ) below, after a transient duration that goes to zero with $\\epsilon $ .",
"Theorem 1 Assume that the shape space $S$ is compact.",
"For sufficiently small $\\epsilon > 0$ , there exist smooth fields of linear maps $B(r)$ and bilinear maps $G(r)$ such that the dynamics restricted to the slow manifold $M_\\epsilon $ satisfy $\\accentset{\\scriptstyle \\circ }{g} = -A_{\\textnormal {visc}}(r) \\cdot \\dot{r} + \\epsilon B(r)\\cdot \\ddot{r} +\\epsilon G(r)\\cdot (\\dot{r},\\dot{r}) + \\mathcal {O}(\\epsilon ^2).$ Remark 1 The bilinear maps or $(1,2)$ tensors $G(r)$ are not, in general, symmetric: e.g., they are unlike Hessians.",
"[1] developed a data-driven algorithm for approximating the equations of motion of a locomotion system assuming the model of Eqn.",
"(REF ).",
"Here we define and study an extension of their approach to models of the form of Eqn.",
"(REF ).",
"We examine the efficacy of this extension in modeling motion in the perturbed Stokes regime, in which $\\epsilon $ is allowed to be small but nonzero."
],
[
"Estimating Data-Driven Models in the Perturbed Stokes Regime",
"In this section, we develop a data-driven algorithm for estimating the dynamics Eqn.",
"(REF ) in a neighborhood of an exponentially stable periodic orbit.",
"We assume that the image of this periodic orbit is contained in the slow manifold $M_\\epsilon $ of Thm.",
"REF , and for simplicity we assume that — on the slow manifold — $\\ddot{r}=f(r,\\dot{r})$ can be written autonomously as a function of $r$ and $\\dot{r}$ .",
"Letting $\\gamma (t)$ denote the shape (or $r$ ) component of this periodic orbit, we refer to $\\gamma $ as a gait."
],
[
"Determination of regressors for estimation of the dynamics",
"In this section we closely follow the approach of [1] to produce a data driven model of the dynamics from an ensemble of noisy trajectories near $\\Gamma \\text{Im } \\gamma $ .",
"We extensively use the Einstein summation convention in the regression equations below.",
"Let $T$ be the period of $\\gamma $ .",
"Since we assume that that the exponentially stable periodic orbit is contained in the slow manifold on which $\\ddot{r}$ is of the form $\\ddot{r}=f(r,\\dot{r})$ , it follows that there is an asymptotic phase map $\\phi \\colon S̰ \\rightarrow [0,T)$ whose derivative along trajectories is equal to one [15].",
"Given trajectory data $(r(t),\\dot{r}(t)),~t\\in [t_0,t_1]$ , we assign asymptotic phase values $\\phi _t \\phi (r(t),\\dot{r}(t))$ to each data point using an algorithm such as that of [35].In principle, any circle-valued “phase” function of state whose derivative along trajectories is positive could be used instead of asymptotic phase.",
"We chose to use asymptotic phase because it is dynamically meaningful and there exist algorithms to compute it.",
"After grouping data points according to their phase values, we construct Fourier series models of $\\gamma ,\\dot{\\gamma },\\ddot{\\gamma }$ as functions of phase.In practice the Fourier series models of $\\gamma ,\\dot{\\gamma },\\ddot{\\gamma }$ might be computed from their own noisy data sets, and in this case the resulting Fourier models need not be derivatives of one another.",
"We find that the use of matched filters is helpful in mitigating this issue; see [1], [34] for more details.",
"Next, we select $M$ evenly spaced values of phase, $\\phi _1, \\ldots , \\phi _M$ , to obtain values $\\gamma _{m} := \\gamma (\\phi _m), \\dot{\\gamma }_m := \\dot{\\gamma }(\\phi _m),\\ddot{\\gamma }_m := \\ddot{\\gamma }(\\phi _m)$ — the shapes, shape velocities, and shape accelerations of a system that is following the gait cycle precisely.",
"For each $m$ we collect from our trajectory data all triples $(r_n,\\dot{r}_n,\\ddot{r}_n)(r(t_{n}),\\dot{r}(t_{n}),\\ddot{r}(t_{n}))$ that are sufficiently close to $(\\gamma _m,\\dot{\\gamma }_m,\\ddot{\\gamma }_m)$ , i.e., such that $\\Vert r_{n} - \\gamma _m\\Vert ,\\Vert \\dot{r}_{n} - \\dot{\\gamma }_m\\Vert ,\\Vert \\ddot{r}_{n} - \\ddot{\\gamma }_m\\Vert <\\kappa $ for allThe astute experimentalist realizes that since the derivative terms contain $dt$ and $dt^2$ in their units, a certain degree of numerical conditioning can be obtained by judicious choice of units for time.",
"$n$ , and we also collect the corresponding $\\accentset{\\scriptstyle \\circ }{g}_n$ values.",
"We define the offsets $\\delta _n := r_n - \\gamma _m$ , $\\dot{\\delta }_n\\dot{r}_n-\\dot{\\gamma }_m$ , $\\ddot{\\delta }_n\\ddot{r}_n-\\ddot{\\gamma }_m$ .",
"Note that the range of $n$ depends on $m$ , but for notational simplicity we do not display this.",
"Introducing coordinates and Taylor expanding, [1] obtained from Eqn.",
"(REF ) the following expression (no sum over $m$ or $n$ ): $\\accentset{\\scriptstyle \\circ }{g}^k_n \\approx - \\underbrace{A^k_{m,i}\\dot{\\gamma }_m^i}_{C^k_{0,m}} - \\underbrace{A^k_{m,i}}_{C^k_{1,m}}\\dot{\\delta }^i_n - \\underbrace{\\frac{\\partial A^k_{m,i}}{\\partial r^j}\\dot{\\gamma }_m^i}_{C^k_{2,m}}\\delta ^j_n - \\underbrace{\\frac{\\partial A^k_{m,i}}{\\partial r^j}}_{C^k_{3,m}}\\delta ^j_n \\dot{\\delta }^i_n.$ Omitted here are higher-order terms, the subscript of $A_{\\textnormal {visc}}$ , and the nonlinear $\\gamma $ dependence of the local expression $A^k_i$ .",
"They then operationalized Eqn.",
"(REF ) as a least-squares problem, written in matrix form as follows (for each $k$ and $m$ ; indices $k$ and $m$ elided below for clarity): $\\begin{bmatrix} \\accentset{\\scriptstyle \\circ }{g}_1 \\\\ \\vdots \\\\ \\accentset{\\scriptstyle \\circ }{g}_N \\end{bmatrix} =\\begin{bmatrix}1, & \\delta _1, & \\dot{\\delta }_1, & \\delta _{1}\\otimes \\dot{\\delta }_{1} \\\\\\vdots & \\vdots & \\vdots & \\vdots \\\\1, & \\delta _{N}, & {\\dot{\\delta }}_{N}, & \\delta _{N}\\otimes \\dot{\\delta }_{N} \\end{bmatrix}\\cdot \\begin{bmatrix}\\widehat{C}_0 \\\\ \\widehat{C}_1 \\\\ \\widehat{C}_2 \\\\ \\widehat{C}_3\\end{bmatrix}$ where $\\widehat{~}$ indicates “estimated” and $\\otimes $ is the outer product.",
"For a $d$ -dimensional shape space, the row of unknowns on the right consists of $1+d+d+d^2$ elements.",
"Once they have computed a least squares model for every $m$ , they construct Fourier series so that the $\\widehat{C}_i$ may be smoothly interpolated at any phase value.",
"The result is a local model of Eqn.",
"(REF ).",
"In the perturbed Stokes regime which we seek to model, we follow a similar approach by expanding Eqn.",
"(REF ) instead of Eqn.",
"(REF ).",
"We obtain (no sum over $m$ or $n$ ): $\\begin{split}\\accentset{\\scriptstyle \\circ }{g}^k_n &\\approx - A^k_{m,i}\\dot{\\gamma }^i_m - A^k_{m,i}\\dot{\\delta }^i_n - \\frac{\\partial A^k_{m,i}}{\\partial r^j}\\delta ^j_n \\dot{\\gamma }^i_m - \\frac{\\partial A^k_{m,i}}{\\partial r^j}\\delta ^j_n \\dot{\\delta }^i_n + \\epsilon \\left( B^k_{m,i}\\ddot{\\gamma }^i_m + B^k_{m,i}\\ddot{\\delta }^i_n + \\frac{\\partial B^k_{m,i}}{\\partial r^j}\\delta ^j_n \\ddot{\\gamma }^i_m \\right.",
"\\\\ \\ldots & + \\frac{\\partial B^k_{m,i}}{\\partial r^j}\\delta ^j_n \\ddot{\\delta }^i_n + G^k_{m,i,j}\\dot{\\gamma }^i_m \\dot{\\gamma }^j_m + G^k_{m,i,j}\\dot{\\gamma }^i_m \\dot{\\delta }^j_n + G^k_{m,i,j}\\dot{\\delta }^i_n \\dot{\\gamma }^j_m + G^k_{m,i,j}\\dot{\\delta }^i_n \\dot{\\delta }^j_n \\\\\\ldots & + \\left.",
"\\frac{\\partial G^k_{m,i,j}}{\\partial r^{\\ell }}\\delta ^\\ell _n \\dot{\\gamma }^i_m \\dot{\\gamma }^j_m + \\frac{\\partial G^k_{m,i,j}}{\\partial r^{\\ell }}\\delta ^\\ell _n \\dot{\\gamma }^i_m \\dot{\\delta }^j_n + \\frac{\\partial G^k_{m,i,j}}{\\partial r^{\\ell }}\\delta ^\\ell _n \\dot{\\delta }^i_n \\dot{\\gamma }^j_m + \\frac{\\partial G^k_{m,i,j}}{\\partial r^{\\ell }}\\delta ^\\ell _n \\dot{\\delta }^i_n \\dot{\\delta }^j_n \\right).\\end{split}$ Partitioning these terms according to their dependence on the observations $\\delta $ , $\\dot{\\delta }$ , and $\\ddot{\\delta }$ , we obtained $\\begin{split}\\accentset{\\scriptstyle \\circ }{g}^k_n &\\approx \\left(- A^k_{m,i}\\dot{\\gamma }^i_m + \\epsilon B^k_{m,i}\\ddot{\\gamma }^i_m + \\epsilon G^k_{m,i,j}\\dot{\\gamma }^i_m \\dot{\\gamma }^j_m\\right) +\\left(-\\frac{\\partial A^k_{m,j}}{\\partial r^i} \\dot{\\gamma }^j_m + \\epsilon \\frac{\\partial B^k_{m,j}}{\\partial r^i}\\ddot{\\gamma }^j_m + \\epsilon \\frac{\\partial G^k_{m,j,\\ell }}{\\partial r^{i}}\\dot{\\gamma }^j_m \\dot{\\gamma }^\\ell _m \\right)\\delta ^i_n \\\\\\ldots & + \\left(- A^k_{m,i} + \\epsilon G^k_{m,j,i}\\dot{\\gamma }^j_m + \\epsilon G^k_{m,i,j} \\dot{\\gamma }^j_m\\right)\\dot{\\delta }^i_n + \\left(- \\frac{\\partial A^k_{m,j}}{\\partial r^i} + \\epsilon \\frac{\\partial G^k_{m,\\ell , j}}{\\partial r^{i}}\\dot{\\gamma }^\\ell _m + \\epsilon \\frac{\\partial G^k_{m,j,\\ell }}{\\partial r^{i}} \\dot{\\gamma }^\\ell _m\\right)\\delta ^i_n\\dot{\\delta }^j_n\\\\\\ldots & + \\epsilon \\left( B^k_{m,i}\\,\\ddot{\\delta }^i_n + \\frac{\\partial B^k_{m,j}}{\\partial r^i} \\,\\delta ^i_n\\ddot{\\delta }^j_n+ G^k_{m,i,j}\\,\\dot{\\delta }^i_n\\dot{\\delta }^j_n + \\frac{\\partial G^k_{m,j,\\ell }}{\\partial r^{i}} \\,\\delta ^i_n \\dot{\\delta }^j_n \\dot{\\delta }^\\ell _n\\right),\\end{split}$ giving a similar least squares problem written in matrix form as follows (for each $k$ and $m$ ; indices $k$ and $m$ elided below for clarity): $\\begin{bmatrix} \\accentset{\\scriptstyle \\circ }{g}_1 \\\\ \\vdots \\\\ \\accentset{\\scriptstyle \\circ }{g}_N \\end{bmatrix} =\\begin{bmatrix}1, & \\delta _1, & \\dot{\\delta }_1, & \\ddot{\\delta }_1 & \\delta _{1}\\otimes \\dot{\\delta }_{1} & \\delta _{1}\\otimes \\ddot{\\delta }_{1} & \\dot{\\delta }_{1}\\otimes \\dot{\\delta }_{1} & \\delta _{1}\\otimes \\dot{\\delta }_{1}\\otimes \\dot{\\delta }_{1} \\\\\\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\vdots \\\\1, & \\delta _{N}, & {\\dot{\\delta }}_{N}, & \\ddot{\\delta }_N & \\delta _{N}\\otimes \\dot{\\delta }_{N} & \\delta _{N}\\otimes \\ddot{\\delta }_{N} & \\dot{\\delta }_{N}\\otimes \\dot{\\delta }_{N} & \\delta _{N}\\otimes \\dot{\\delta }_{N} \\otimes \\dot{\\delta }_{N} \\end{bmatrix}\\cdot \\begin{bmatrix}\\widehat{C}_0 \\\\ \\widehat{C}_1 \\\\ \\widehat{C}_2 \\\\ \\widehat{C}_3 \\\\ \\widehat{C}_4 \\\\ \\widehat{C}_5 \\\\ \\widehat{C}_6 \\\\ \\widehat{C}_7\\end{bmatrix}$ For a $d$ -dimensional shape space, the row of unknowns on the right consists of $1+d+d+d+d^2+d^2+d^2+d^3$ elements.",
"Once we have computed a least squares model for every $m$ , we similarly construct Fourier series so that the $\\widehat{C}_i$ may be smoothly interpolated at any phase value.",
"The result is a local model of Eqn.",
"(REF ).",
"Because it is the only term of order $\\kappa ^3$ , we find that in practice the 3-index regressor $\\delta \\otimes \\dot{\\delta } \\otimes \\dot{\\delta }$ can often be omitted if $\\kappa >0$ is sufficiently small.",
"In the remainder of this paper, we refer to the regressors of Eqn.",
"(REF ) (with the 3-index term excluded) as the “perturbed Stokes regressors”, and refer to those used in the [1] algorithm as the “Stokes regressors.” Remark 2 All tensors appearing in Eqn.",
"(REF ) and Eqn.",
"(REF ) are not necessarily symmetric, and therefore the order of terms matters.",
"Remark 3 Examining Eqn.",
"(REF ), we see that there are some constraints that the regression does not enforce.",
"Namely, $C_0 = \\left[C_1\\right]_i \\dot{\\gamma }^i$ and $C_2 = \\left[C_3\\right]_i \\dot{\\gamma }^i$ .",
"When we performed regressions ignoring these implicit constraints, we found that the constraints are not respected in the results.",
"However, an important consequence of Eqn.",
"(REF ) is that, for systems operating in the perturbed Stokes regime, such a mismatch is actually to be expected — this is because some independent new terms appear in $C_1,\\ldots ,C_3$ which break the constraints."
],
[
"Local models enable optimality testing and optimization",
"The data-driven models computed by the process described above have predictive power locally, in a neighborhood of a gait cycle.",
"For any shape trajectory inside this neighborhood, we can used the local model to predict the trajectory of the body in the world.",
"We assume that we are interested in some $\\mathbb {R}$ -valued goal functional $\\tilde{\\phi }(\\gamma ,g_\\gamma )$ defined on an appropriate space of trajectories.",
"Here the group trajectory $g_\\gamma (t)$ is determined by the gait $\\gamma (t)$ via Eqn.",
"(REF ), and therefore we may consider the goal functional $\\phi (\\gamma ) := \\tilde{\\phi }(\\gamma ,g_\\gamma )$ to be a function of $\\gamma $ alone."
],
[
"Testing for Optimality",
"— We can test the gait of an organism for optimality by checking that $0 = \\frac{\\partial }{\\partial s}\\phi (\\gamma _s)|_{s=0}$ for all smooth variations $\\gamma _s$ of a gait $\\gamma $ (where $\\gamma _0 = \\gamma $ ).",
"This condition is necessary for local optimality, but depending on the choice of $\\phi $ it is often possible to argue on physical grounds that its satisfaction is also sufficient for optimality.",
"While this variational condition can be used to derive a PDE via the Euler-Lagrange approach, a more computationally straightforward approach is to consider a finite- (but often high-) dimensional family $\\gamma _p$ with $p \\in \\mathbb {R}^N$ , and numerically computing the gradient $\\nabla _p \\phi (\\gamma _p)$ .",
"When this gradient is sufficiently small at some parameter $p_*$ , then it might be possible to argue that the gait is nearly extremal (or possibly optimal) with respect to $\\phi $ .In some cases this procedure is provably correct.",
"Furthermore, suitable finite-dimensional families that provide these guarantees always exist [31].",
"We do not discuss these technicalities any further here.",
"Since we can compute $\\phi $ using a data-driven model around $\\gamma _p$ , we can compute $\\nabla _p \\phi (\\gamma _p)$ .",
"We can do so directly from observation and without need for any general model of body-environment interactions, so long as use of Thm.",
"REF can be justified.",
"— We can use the gradient $\\nabla _p \\phi (\\gamma _p)$ to iteratively improve the gait of a robot whose dynamics satisfy Thm.",
"REF without requiring any further details of the physics.",
"Taking parameter set $p$ we compute the next iterate $p^{\\prime } := p + \\alpha \\nabla _p \\phi (\\gamma _p)$ , with the step-size scaling $\\alpha >0$ chosen to ensure that $p^{\\prime }$ is within the domain for which our local model of $\\phi $ is valid, using the approach of [1].",
"For each gait $\\gamma _p$ , we only require enough experimental data for building a good local model of $\\phi $ near $\\gamma _p$ — a dataset whose size does not depend on the dimension of the representation $p$ .",
"We plan to use this decoupling to perform hardware-in-the-loop optimization to produce rapid adaptation of robot motions in the face of foreign environments, mechanical failures, and more."
],
[
"Performance Comparison of the Two Data-Driven Models",
"One of the primary contributions of this paper is the introduction of new regressors based on Thm.",
"REF , which we use to augment the regressors used in the algorithm of [1] for estimating the dynamics near a gait.",
"These allow us to extend the domain of validity of their algorithm from the Stokesian limit to include the perturbed Stokes regime.",
"To demonstrate this, we constructed a swimming model which we simulated at various Reynolds numbers, and tested the ability of the two types of local models to predict the results of the fully nonlinear simulation.All of these simulations did not account for fluid-fluid interactions; as such we make no claim that they are physically meaningful at the higher Reynolds number in the ranges shown."
],
[
"Modeling a swimmer",
"We tested the prediction quality of both models on a swimming model.",
"The system shown in Fig.",
"REF had uniformly distributed mass along a central body, with two paddles comprising chains of massless links extending from the center of the body.",
"Each paddle could be broken up into an arbitrary number $\\frac{n}{2}$ ($n$ even) of equally spaced links, which sum to a constant total length independent of $n$ .",
"This allowed us to vary the behavior of the system from one reminiscent of a boat with oars (for $n = 2$ ) to one more like a bacterial cell with flagella (for $n$ large).",
"The system moves in a homogeneous and isotropic plane.",
"Its configuration space is $S\\times G = \\mathbb {T}^{n} \\times \\mathsf {SE}(2)$ : the $n$ -torus and the special Euclidean group of planar rigid motions $\\mathsf {SE}(2)$ .",
"We assume the dynamics are equivariant under $\\mathsf {SE}(2)$ .",
"The group element $g \\in \\mathsf {SE}(2)$ provides the position and orientation of the central body in world coordinates with respect to a fixed inertial reference frame.",
"Hereon we represent $g$ as a column vector $g = [x,y,\\theta ]^T$ , and similarly represent $\\dot{g}$ as a column vector.",
"We define the body velocity $\\accentset{\\scriptstyle \\circ }{g} = \\begin{bmatrix} \\cos (\\theta ) & \\sin (\\theta ) & 0 \\\\-\\sin (\\theta ) & \\cos (\\theta ) &0 \\\\ 0 & 0 & 1 \\end{bmatrix} \\dot{g}.$ We treat the link at the main body (length $L$ ) and the links comprising the paddles (length $d$ ) as slender members, and model their drag forces according to Cox theory [5] using the drag matrices $C_{\\frac{d}{n}} =c \\begin{bmatrix}C_x \\frac{d}{n} & 0 & 0\\\\ 0 & C_y \\frac{d}{n}& 0\\\\0 & 0 & \\frac{1}{12} (\\frac{d}{n})^3 C_y\\end{bmatrix}, \\quad C_{_L} =c \\begin{bmatrix}C_x L & 0 & 0\\\\ 0 & C_y L& 0\\\\0 & 0 & \\frac{1}{12}L^3 C_y\\end{bmatrix},$ where the factor $c > 0$ is explicitly written for later scaling purposes.",
"The drag coefficient ratio $C_y/C_x$ has a maximum value of 2 corresponding to the limit of infinitesimally thin segments, and we will assume this limiting ratio here (c.f.",
"[17]).",
"Given these drag matrices, the wrench on the central link can be written as $F_{\\textnormal {body}} = c \\bar{F}_{\\textnormal {body}} = -C_{_L}\\accentset{\\scriptstyle \\circ }{g}.$ The wrench that the segments (denoted $i$ ) apply on the body can be written as $F_{i} = c \\bar{F}_i= -W_iC_{\\frac{d}{n}}V_i\\begin{bmatrix}\\accentset{\\scriptstyle \\circ }{g} \\\\ \\dot{\\alpha }\\end{bmatrix},$ where the linear map $W_i(g,\\alpha )\\colon \\mathfrak {se}(2)^*\\rightarrow \\mathfrak {se}(2)^*$ maps a wrench on link $i$ to a wrench on the body and the linear map $V_i(g,\\alpha )\\colon \\mathfrak {se}(2)\\rightarrow \\mathfrak {se}(2)$ maps a velocity in the body frame to a velocity in the link frame.",
"Let $R_\\beta $ denote the counterclockwise rotation of the plane by angle $\\beta $ , define $e_2[0,1]^T$ , and write $\\accentset{\\scriptstyle \\circ }{g} = [\\accentset{\\scriptstyle \\circ }{g}_{x,y}^T,\\dot{\\theta }]^T$ .",
"Then, for the $n$ -segment model (recall that $n$ must be even), for $i \\in \\lbrace 1,\\ldots , n\\rbrace $ the linear maps $V_i$ and $W_i$ are given by $\\begin{split}V_i \\cdot \\begin{bmatrix}\\accentset{\\scriptstyle \\circ }{g}\\\\\\dot{\\alpha }\\end{bmatrix} &= \\begin{bmatrix}R_{\\alpha _*+\\cdots +\\alpha _i}^{-1}\\accentset{\\scriptstyle \\circ }{g}_{x,y}+ \\left(\\frac{d}{2n}\\left(\\dot{\\theta }+\\sum _{k=*}^i\\dot{\\alpha }_k\\right)+\\frac{d}{n}\\sum _{k=*}^{i-1}\\left(\\dot{\\theta }+\\sum _{j=*}^k\\dot{\\alpha }_j\\right)R^{-1}_{\\alpha _{k+1}+\\cdots +\\alpha _i}\\right) e_2 \\\\ \\dot{\\theta } + \\sum _{k=*}^i\\dot{\\alpha }_k\\end{bmatrix}\\\\W_i \\cdot \\begin{bmatrix}f\\\\\\tau \\end{bmatrix} &= \\begin{bmatrix}R_{\\alpha _*+\\cdots +\\alpha _i}f\\\\ \\tau + e_2^T\\left(\\frac{d}{2n}I_{2\\times 2} + \\frac{d}{n}\\sum _{k=*+1}^{i}R_{\\alpha _k+\\alpha _{k+1}+\\cdots + \\alpha _i} \\right)\\cdot f\\end{bmatrix},\\end{split}$ where $* 1+ i / \\frac{n}{2}\\cdot \\frac{n}{2}\\in \\lbrace 1,\\frac{n}{2}+1\\rbrace $ , $f = [f_1,f_2]^T$ , and where a summation is understood to be zero if the lower bound of its index set exceeds its upper bound.",
"These wrenches act on the body (which has uniformly distributed mass $m$ and moment of inertia $I = m\\bar{I}$ about its midpoint) yielding the following equations of motion in world coordinates: $ \\ddot{g} = \\begin{bmatrix} \\ddot{x} \\\\ \\ddot{y} \\\\ \\ddot{\\theta } \\end{bmatrix} =\\frac{1}{\\epsilon } \\begin{bmatrix}1 & 0 & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & \\frac{1}{\\bar{I}}\\end{bmatrix}\\begin{bmatrix} \\cos (\\theta ) & -\\sin (\\theta ) & 0 \\\\\\sin (\\theta ) & \\cos (\\theta ) &0 \\\\ 0 & 0 & 1 \\end{bmatrix}\\Bigg (\\bar{F}_{\\textnormal {body}} + \\sum _{i=1}^n\\bar{F}_{i}\\Bigg ),$ where $\\epsilon \\frac{m}{c}$ is the dimensionless inertia-damping ratio.",
"In keeping with our earlier conventions that $m$ , $c$ , and $\\epsilon $ are all dimensionless we think of the “1” terms on the diagonal in Eqn.",
"(REF ) as having units of inverse time.",
"Upon inspection of Eqn.",
"(REF ), we see that by modifying $\\epsilon $ we can directly adjust the ratio of inertial to viscous forces in the swimming model.",
"The Stokesian limit corresponds to $\\epsilon \\rightarrow 0$ ; on the other hand, the $\\epsilon \\rightarrow \\infty $ limit corresponds to a fully “momentum-dominated” regime, wherein viscous effects are negligible and motion is governed by conservation of momentum via Noether's theorem (see Corollary REF §REF ).",
"In the following §REF we simulate the swimming model at a variety of $\\epsilon $ values, and compare the performance of the two algorithms for estimating the dynamics near a gait cycle."
],
[
"Comparison of the estimated models",
"In all simulations in this section, we used the parameter values $L = 1$ , $d = 0.5$ , $C_x = 1$ , $C_y = 2$ , and $\\bar{I} = 1$ .",
"The only remaining free variable is $\\epsilon $ , which governs both the ratio of inertial to viscous forces and the rate of attraction to the slow manifold.",
"The procedure we used for generating simulations for experiments in this section is identical to that described in [1].",
"Briefly, an experiment consists of 30 cycles of a numerically integrated stochastic differential equation (SDE) representing shape space dynamics consisting of a deterministic oscillator perturbed by system noise (see [1] for precise details on the SDE, parameter values used, etc.).",
"We used these noisy shape dynamics to drive the body momentum and group dynamics via the full equations of motion Eqn.",
"(REF ) derived in §REF .",
"For each simulation we recorded a “ground truth” body velocity trajectory $\\accentset{\\scriptstyle \\circ }{g}_G$ .",
"We used this record to evaluate the accuracy of the data-driven approximations.",
"We denoted the body velocity computed with the perturbed Stokes regressors by $\\accentset{\\scriptstyle \\circ }{g}_{p}$ , and those computed with the Stokes regressors by $\\accentset{\\scriptstyle \\circ }{g}_{s}$ .",
"As a “zeroth-order” phase model of the dynamics, we constructed a Fourier series model of $\\accentset{\\scriptstyle \\circ }{g}_G$ with respect to the estimated phase (see §REF ), which we denote by $\\accentset{\\scriptstyle \\circ }{g}_a$ .",
"For any data point, the zeroth-order model prediction is $\\accentset{\\scriptstyle \\circ }{g}_a(\\varphi )$ for the phase $\\varphi $ of that data point.",
"We computed the RMS errors $e^k_*$ for each component $k$ of the body velocity and each model $* = p,s,a$ by $e^k_* := \\langle |\\accentset{\\scriptstyle \\circ }{g}^k_*-\\accentset{\\scriptstyle \\circ }{g}^k_G|^2\\rangle ^{1/2}$ .",
"Since the numerical value of these errors means little, we defined the metric $\\Gamma ^k_* := 1 - e^k_* / e^k_a$ for $* = p,s$ to indicate how much better the regression models were performing compared to the zeroth-order phase model $\\accentset{\\scriptstyle \\circ }{g}_a$ .",
"A $\\Gamma ^k_*$ of 0 indicates doing no better than the zeroth order model whereas a 1 indicates a perfect model.",
"To further highlight the difference in prediction quality, we also plot $\\Delta ^k := \\Gamma ^k_p-\\Gamma ^k_s$ .",
"Figure: Comparison of model prediction quality when using the perturbed Stokes regressors versus the Stokes regressors on three gaits, in terms of the Γ\\Gamma and Δ\\Delta quality metrics.",
"We have plotted the components of Δ\\Delta , representing the relative advantage of perturbed Stokes regressors (top row; (A)), and Γ\\Gamma , representing model prediction quality (bottom row; (B)), against 6 orders of magnitude variation in the inertial to viscosity ratio ϵ\\epsilon (logarithmic scale; sampled at 25 values (vertical gray lines).",
"We present three gaits, whose shape space loci are in-phase paddle angle (which leads to anti-phase paddle motions; “Twist in Place”; left column; blue line in shape-space plot), anti-phase paddle angle (bilaterally symmetric paddle motions; “Symmetric Flap”; middle column; green line in shape-space plot), and quarter-cycle out of phase paddle angles (“Circle Amp.",
"1”; right column; red line in shape-space plot).",
"All three gaits have paddle angles ranging between -1-1 and 1 radians.",
"For each value of ϵ\\epsilon we performed 8 simulation trials each consisting of 30 (noisy) gait cycles, and plotted mean and standard deviation of Δ\\Delta and Γ\\Gamma for each component of the 𝔰𝔢(2)\\mathfrak {se}(2) body motion (XX blue; YY orange; θ\\theta red; saturated for Δ\\Delta and Γ p \\Gamma _p, pale for Γ s \\Gamma _s).",
"Consistently for all components and gaits, the perturbed Stokes regressors provide a better model for an order of magnitude or wider range of ϵ\\epsilon around ϵ=1\\epsilon =1.",
"For Twist in Place and Symmetric Flap gaits, both models are accurate for large and small ϵ\\epsilon (Γ\\Gamma close to 1); for the Circle Amplitude 1 gait, the prediction is only accurate for the Stokes regime (small ϵ\\epsilon ).Figure: Comparison of model prediction quality when using the perturbed Stokes regressors versus the Stokes regressors on two extremal gaits, in terms of the Γ\\Gamma and Δ\\Delta quality metrics.",
"Plots consist of the same types as those in Fig. .",
"We only plot the XX (blue) and YY (orange) components of Γ\\Gamma (middle column; saturated color Γ p \\Gamma _p; pale colors Γ s \\Gamma _s) and Δ\\Delta (right column).",
"We selected the gait to maximize either the XX component of total body frame motion (top row) or the YY component (bottom row).",
"The gaits are extremal in the Stokes regime (ϵ=0\\epsilon =0) and selected by taking the zero level set of the connection curvature (method from , ).",
"Following their approach, we plot the connection of the coordinate being optimized as a vector field over the shape-space (black arrows; left column), with the shape-space gait locus plotted over it (diamond shapes in left column, colored by coordinate optimized).",
"Results show that both models are most accurate for small ϵ\\epsilon (the Stokes regime; Γ\\Gamma closer to 1), with the perturbed Stokes regressors providing improvements across the entire range.",
"Over the two order of magnitude range of 10 -0.5 <ϵ<10 1.5 10^{-0.5}<\\epsilon <10^{1.5} this advantage is noticeably more pronounced (the perturbed Stokes regime; bump in Δ\\Delta plots).",
"Also note that the XX extremal gait shows much greater Δ x \\Delta ^x; the YY extremal gait shows much greater Δ y \\Delta ^y.Figure: Comparison of model prediction quality when using the perturbed Stokes regressors versus the Stokes regressors on paddles with different dimensions of the shape space, shown in terms of the Γ\\Gamma and Δ\\Delta quality metrics.",
"Plots consist of the same types as those in Fig. .",
"We plotted Γ\\Gamma and Δ\\Delta of three swimmers with different numbers of paddle segments: one segment per paddle (light blue), two segments (blue), and three segments (purple); see Fig.",
"for schematic.",
"We used a symmetric flapping gait (see Fig.",
"; small cartoons above).",
"The paddles moved symmetrically with total angles of all joints summing up to a sinusoid of amplitude π\\pi .We plot the XX components of Γ\\Gamma (left column; one plot per model; saturated colors Γ p \\Gamma _p; pale colors Γ s \\Gamma _s) and Δ\\Delta (right column).",
"Results show that over the two order of magnitude range of 10 -0.5 <ϵ<10 1.5 10^{-0.5}<\\epsilon <10^{1.5}, the perturbed Stokes regressors consistently provide improvements.The relative improvement Δ\\Delta increased markedly with shape space dimension, by as much as 0.50.5 in Δ\\Delta ."
],
[
"Algorithm comparison using manually selected gaits",
"We chose to first test the modeling approaches on a collection of simple manually selected behaviors.",
"These include behaviors we term “twist in place” and “symmetric flapping” gaits, both of which initialize with paddles aligned at a quarter turn away from the body (as depicted in the two-segment model in Figure REF ), and respectively involve anti-symmetric and symmetric sinusoidal movement of the paddles with amplitude 1.",
"The “symmetric flapping gait” primarily moves in the direction of the $x$ body axis, while the “twist in place gait” primarily changes the $\\theta $ body coordinate.",
"Finally, we considered a “circle” gait which also initializes the paddles at a quarter turn away from the body and moves them sinusoidally with amplitude 1, but has a quarter cycle phase offset between them.",
"This gait tends to move the system in a way that changes all three body coordinates throughout its execution.",
"We selected these three gaits because they are simple to describe and span a range of resultant body motions.",
"For single link paddles, the body shape space is 2D, and these gaits are represented by loci that are diagonal lines with slopes 1, $-1$ , and a circle (see Fig.",
"REF ).",
"We simulated the gaits and plotted mean and variance of $\\Gamma _s$ , $\\Gamma _p$ and $\\Delta $ for each value of $\\epsilon $ (Fig.",
"REF ).",
"The plot shows that for all three gaits tested and for all three body coordinates, over a range spanning an order of magnitude or more around $\\epsilon =1$ , the perturbed Stokes models are better by $\\Delta >0.05$ or more."
],
[
"Algorithm comparison using extremal gaits",
"Arbitrarily selected gaits such as those examined in the previous section are not expected to exhibit any special properties with respect to our modeling approach.",
"In particular, with respect to a goal function $\\phi (\\cdot )$ , they are expected to be regular points of $\\phi (\\cdot )$ .",
"However, $\\phi $ -optimal gaits have $\\nabla _p\\phi = 0$ and thus have additional structure that might interact with the modeling approach.",
"We chose goal functionals $\\int \\accentset{\\scriptstyle \\circ }{g}^x(t)\\,\\mathrm {d}t$ and $\\int \\accentset{\\scriptstyle \\circ }{g}^y(t)\\,\\mathrm {d}t$ (where superscripts denote components) corresponding to displacement in the $x$ and $y$ coordinates as measured in the body frame of the paddleboat.",
"This is not the same as actual $x$ or $y$ displacement in the world, since boat orientation changes over time.",
"Using the methods of [17], we determined the extremal gaits for these goal functionals in the Stokes regime with high accuracy.",
"Plotted in the shape-space (and superimposed on the “connection vector fields” [16], [17] of the appropriate goal functional) they are diamond shaped (Fig.",
"REF ).",
"We also plotted $\\Gamma $ and $\\Delta $ , revealing that again, perturbed Stokes regressors improve performance ($\\Delta >0.15$ ) over a range of two orders of magnitude in $\\epsilon $ .",
"Unlike the arbitrary gaits of the previous section, the extremal gaits have $\\Gamma >0.1$ for all $\\epsilon >1$ for both model types.",
"This suggests that even outside the perturbed Stokes regime the addition of regressors improves upon the zeroth order phase model.",
"It is also notable that in the extremal $x$ gait, $\\Delta ^x$ is significantly better than $\\Delta ^y$ , whereas in the extremal $y$ gait the converse is true."
],
[
"Performance gains grow with shape space dimension",
"Thus far we have only presented results for systems having 2D shape spaces.",
"Because data-driven methods are often handicapped by their inability to scale with model dimensionality, we chose also to test our approach on systems of higher dimension by extending each paddle into a multi-segmented model.",
"We selected a gait similar to that of the symmetric flapping gait, but with the additional feature that the bending angle of a paddle was uniformly distributed through the joints it contains.",
"In particular, the relative angles between adjacent segments were equal and of amplitude $\\pi /N$ , where $N$ is the number of joints.",
"We plotted $\\Gamma ^x_p$ , $\\Gamma ^x_s$ and $\\Delta ^x$ for paddles with 1, 2 and 3 segments (Fig.",
"REF ).",
"The $\\Delta ^x$ shows a marked improvement in the 4D and 6D models, suggesting that as shape-space complexity increased, the advantage of perturbed Stokes regressors became comparatively more significant."
],
[
"Discussion",
"The results of §REF show that for all versions of the swimming model and all gaits that we tested there exists a sizable window of $\\epsilon $ values wherein the perturbed Stokes regressors provide models of superior quality when compared to the Stokes regressors.",
"In particular, the improvement is consistently present in the region $\\log _{10} \\epsilon \\in [0,1]$ , suggesting that this range of $\\epsilon $ might be the range for which the predicted slow manifold is both present and sufficiently simple to be captured by the new regressors.",
"As noted in §REF , the perturbed Stokes regressors seem to improve prediction performance more in the direction in which the gait was extremal.",
"We hypothesize that this is because extremal gaits have already exhausted any first-order improvements available, i.e.",
"gradients are zero.",
"With the first-order terms close to zero, the presence of more high-order terms among the perturbed Stokes regressors may have a greater effect on the relative prediction error.",
"It is interesting to note the large magnitude of improvement in $\\Delta $ as the shape space dimension increased in Fig.",
"REF .",
"Whether this is an artifact of the particular model and/or gait, or a more general feature, remains to be determined.",
"At the lower end $\\epsilon $ magnitudes studied here, the systems are near the Stokesian limit, and therefore we expect relatively little improvement from adding regressors designed for the perturbed Stokes regime.",
"This is consistent with our experimental results in all figures which show for $\\epsilon $ small both small values of $\\Delta $ and large values of $\\Gamma $ for both sets of regressors.",
"For very large values of $\\epsilon $ , the predictive quality of both algorithms is hindered by at least three factors, although only the first two can be observed here.",
"The $\\mathcal {O}(\\epsilon ^2)$ term in Thm.",
"REF becomes more significant as $\\epsilon $ increases.",
"This issue is insurmountable if we restrict ourselves to Stokes regressors.",
"If we do not, it is possible to compute correction terms which are higher order in $\\epsilon $ and which can inform the selection of additional regressors for addition to our algorithm.",
"It is one possible direction for future work.",
"For $\\epsilon $ sufficiently large, we expect a bifurcation in which the slow manifold (whose existence is guaranteed by Thm.",
"REF in §) ceases to exist.",
"For such values of $\\epsilon $ , the hypotheses of Thm.",
"REF are not satisfied, and a reduced-order model may not exist.",
"This is a mathematical expression of the physical reality of inertial effects playing a dominant role as $\\epsilon $ increases, and eventually requiring momentum states to be added to the models.",
"For sufficiently large values of $\\epsilon $ the full complications of fluid-fluid interactions to come into play, and the linear viscous friction model we used becomes less and less accurate.",
"We conjecture that for many systems this effect will not have significant influence until after $\\epsilon $ is already sufficiently large for the slow manifold to have disappeared.",
"It would be interesting to explore this issue further."
],
[
"Conclusion",
"We have shown that the accuracy of data-driven models motivated from geometric mechanics can be improved by using a collection of regressors derived from an asymptotic series approximation of an attracting invariant manifold in the small parameter $\\epsilon $ representing the ratio of inertial to viscous forces (a Reynolds-number-like parameter).",
"The existence of such an invariant manifold was previously known in similar situations,But see the discussion preceding Thm.",
"REF in §, which details how our result differs from that of [9].",
"as were the approximation techniques we employed, but the combination of these together for producing data-driven models of locomotion is a novel contribution.",
"In simulations where we tested geometrically similar motions over 6 orders of magnitude of $\\epsilon $ , we obtained improvements of 5–$65\\%$ (depending on the specific system and gait) compared to previous work, suggesting that these better-informed models can indeed capture the perturbed Stokes regime more accurately.",
"Furthermore, the results of one of our experiments showed further improvements as the shape-space dimension of the locomoting system increased; this suggests that higher-dimensional systems might be modeled effectively using our approach.",
"Future work will include application of our algorithm to questions of locomotion optimality in animals, and to hardware-in-the-loop optimization of robot motions.",
"An additional direction for future work is the selection of regressors and regression techniques for hybrid dynamical systems, and for non-viscous dissipation models."
],
[
"Appendix A — Derivation of the Equations of Motion",
"In this and the following section we consider systems more general than those considered earlier, and in so doing assume that the reader is familiar with some basic concepts in geometric mechanics and differential geometry: Lie groups, group actions, and principal bundles.",
"We refer the reader to [26], [29], [27], [2] for the relevant standard definitions related to Lie groups and group actions, and we refer the reader to [26], [30], [28], [2] for material on bundles.",
"We consider a mechanical system on a configuration space $Q$ whose Lagrangian is of the form kinetic minus potential energy.",
"We will also consider this system to be subjected to external viscous forcing arising from a Rayleigh dissipation function, and also subjected to an external force exerted by the locomoting body.",
"We are interested in the situation that we have a smooth action $\\theta \\colon G \\times Q \\rightarrow Q$ of a Lie group $G$ on $Q$ , such that the Lagrangian, viscous forces, and external force are all symmetric under the action.",
"In this case, we say that $G$ is a symmetry group.",
"In §REF , we will define some geometric quantities on $Q$ which encode information about the symmetry and the dynamics.",
"Working in coordinates induced by a local trivialization, in §REF we derive the equations of motion in terms of these quantities.",
"In §REF , we recall how the equations become governed by the so-called viscous connection in the Stokesian limit [25], [9], which will set the stage for our derivation in § of a corrected reduced-order model for the perturbed Stokes regime."
],
[
"The mechanical and viscous connections",
"In this section, we define the mechanical and viscous (or Stokes) connections, roughly following [25].",
"We consider a Lagrangian $L\\colon Q̰ \\rightarrow \\mathbb {R}$ which is invariant under the lifted action $_g$ of $G$ on $Q̰$ (here $ denotes the derivative or pushforward).We assume the Lagrangian to be of the form kinetic minus potential energy, where kinetic energy is given by $ m2k$, where $ m > 0$ is a dimensionless mass parameter, $ k$ is a smooth symmetric bilinear form, and $ m k$ is the \\textit {kinetic energy metric}.In what follows, we assume that $ k$ is positive definite when restricted to tangent spaces to $ G$ orbits, but \\emph {not} necessarily that $ k$ is positive definite on all tangent vectors.\\footnote {This does not affect any of the following derivations and results.However, this generality is merely a convenience ensuring that our results apply to certain idealized examples, e.g., linkages with some links having zero mass (c.f.",
"§\\ref {sec:performance}).",
"Of course such examples are not physical and, e.g., must be supplemented with assumptions to ensure that the massless links have well-defined dynamics.",
"}Denoting by $ g$ the Lie algebra of $ G$ and $ g*$ its dual, we define the (Lagrangian) \\textit {momentum map} $ JQ̰ g*$ via\\begin{equation}\\langle J(v_q), \\xi \\rangle = \\langle \\mathbb {F}L(v_q), \\xi _Q(q) \\rangle = mk_q(v_q, \\xi _Q(q)),\\end{equation}where $ v q Q$ and $ g$.Here $ FLQ̰* Q$ is the \\textit {fiber derivative} of $ L$ given by $ FL(vq)(wq)s|s=0 L(vq + $s$ wq)$,and the smooth vector field $ Q$ on $ Q$ is the \\textit {infinitesimal generator} defined by $ Q(q)s|s=0(s)(q)$.We define the \\textit {mechanical connection} $ mechQ̰ g$ via $ mech(vq)I-1(q)J(vq)$, where $ I(q)gg*$ is the \\textit {locked inertia tensor} defined via\\begin{equation}\\langle I(q) \\xi , \\eta \\rangle \\langle \\mathbb {F}L(\\xi _Q(q)),\\eta _Q(q)\\rangle = mk_q(\\xi _Q(q),\\eta _Q(q)),\\end{equation}where $ ,g$.$ We now follow an analogous procedure to define the viscous connection $\\Gamma _{\\textnormal {visc}}\\colon Q̰ \\rightarrow \\mathbb {R}$ .",
"We consider a Rayleigh dissipation function $R \\colon Q̰ \\rightarrow \\mathbb {R}$ defined in terms of a $G$ -invariant smooth symmetric bilinear form $\\nu $ on $Q$ : $R(v_q)\\frac{c}{2}\\nu _q(v_q,v_q)$ , where $c>0$ is a dimensionless parameter representing the amount of damping or dissipation in the system due to viscous forces.",
"As with $k$ , we assume that $\\nu $ is positive definite when restricted to tangent spaces to $G$ orbits, but not necessarily that $\\nu $ is positive definite on all tangent vectors.This generality simply allows for, e.g., the situation of a linkage in which not all links are subject to viscous forces.",
"The corresponding force field $F_R\\colon Q̰ \\rightarrow * Q$ is given by minus the fiber derivative of $R$ , $F_R\\mathbb {F}(-R)$ .",
"We define a map $K\\colon Q̰ \\rightarrow \\mathfrak {g}^*$ , analogous to the momentum map $J$ , via $\\langle K(v_q), \\xi \\rangle = \\langle F_R(v_q), \\xi _Q(q) \\rangle = -c\\nu _q(v_q, \\xi _Q(q)),$ where $v \\in q Q$ and $\\xi \\in \\mathfrak {g}$ .",
"We define the viscous connection or Stokes connection $\\Gamma _{\\textnormal {visc}}\\colon Q̰ \\rightarrow \\mathfrak {g}$ via $\\Gamma _{\\textnormal {visc}}(v_q){-1}(q)K(v_q)$ , where $q)\\colon \\mathfrak {g}\\rightarrow \\mathfrak {g}^*$ is defined via $\\langle q) \\xi , \\eta \\rangle \\langle F_R(\\xi _Q(q)),\\eta _Q(q)\\rangle = -c\\nu _q(\\xi _Q(q),\\eta _Q(q)),$ where $\\xi ,\\eta \\in \\mathfrak {g}$ .",
"Using the $G$ -invariance of $L$ and $\\nu $ , a calculation shows that $\\Gamma _{\\textnormal {mech}}$ and $\\Gamma _{\\textnormal {visc}}$ are equivariant with respect to the adjoint action of $G$ on $\\mathfrak {g}$ : $\\forall g \\in G\\colon \\Gamma _{\\textnormal {mech}}\\circ _g = \\textnormal {Ad}_g \\circ \\Gamma _{\\textnormal {mech}}, \\quad \\Gamma _{\\textnormal {visc}}\\circ _g = \\textnormal {Ad}_g \\circ \\Gamma _{\\textnormal {visc}}$ Hence if the natural projection $\\pi _Q\\colon Q \\rightarrow Q/G$ from $Q$ to the space of orbits $Q/G$ of points in $Q$ is a principal $G$ -bundle, then the mechanical and viscous connections $\\Gamma _{\\textnormal {mech}}$ and $\\Gamma _{\\textnormal {visc}}$ are indeed principal connections; this justifies their titles.",
"Now in order for our system to move itself through space, we also allow there to be a $G$ -equivariant external force $F_E\\colon \\mathbb {R}\\times Q̰\\rightarrow *Q$ exerted by the locomoting body, subject to the requirement that $F_E$ takes values in the annihilator of $\\ker _Q$ , the distribution tangent to group orbits.",
"This requirement reflects the physically reasonable assumption that the locomoting body can exert only “internal forces” which directly affect only its shape $r\\in Q/G$ (c.f.",
"[9] and [3]).",
"For future use, we now prove the following Proposition 1 The derivative of $J$ along trajectories of the $G$ -symmetric mechanical system is given by $\\dot{J} = K,$ making the canonical identifications $J \\mathfrak {g}\\cong \\mathfrak {g}$ .",
"We compute in a local trivialization on $Q̰$ induced by a chart for $Q$ , so that we may write a trajectory as $(q,\\dot{q})$ .",
"Note that in such local coordinates, $\\mathbb {F}L(q,\\dot{q})(v_q) = \\frac{\\partial L(q,\\dot{q})}{\\partial \\dot{q}}v_q$ .",
"Hence $\\begin{split}\\langle \\dot{J}(q,\\dot{q}), \\xi \\rangle &= \\frac{d}{dt}\\left(\\frac{\\partial L(q(t),\\dot{q}(t))}{\\partial \\dot{q}}\\xi _Q(q(t))\\right)\\\\&= \\left(\\frac{d}{dt}\\frac{\\partial L}{\\partial \\dot{q}}\\right)\\xi _Q(q) + \\frac{\\partial L}{\\partial \\dot{q}}_Q(q)\\dot{q}\\\\&= \\left(\\frac{\\partial L}{\\partial q} + F_R + F_E \\right)\\xi _Q(q) + \\frac{\\partial L}{\\partial \\dot{q}}_Q(q)\\dot{q},\\end{split}$ where we obtained the last line using $\\frac{d}{dt}\\frac{\\partial L}{\\partial \\dot{q}} - \\frac{\\partial L}{\\partial q}=F_R + F_E,$ which follows from the Lagrange-d'Alembert principle [2].",
"Since $F_E$ annihilates tangent vectors to group orbits, $\\langle F_E, \\xi _Q(q)\\rangle = 0$ .",
"Hence rearranging and letting $\\Phi _\\xi ^s$ denote the flow of $\\xi _Q$ , we find $\\begin{split}\\langle \\dot{J}(q,\\dot{q}), \\xi \\rangle &= \\frac{\\partial }{\\partial s} L\\left(\\Phi _\\xi ^s(q(t)), _\\xi ^s(q(t))\\dot{q}(t)\\right) + \\langle F_R(q,\\dot{q}), \\xi _Q(q)\\rangle \\\\&= \\frac{\\partial }{\\partial s} L\\left(\\Phi _\\xi ^s(q(t)), _\\xi ^s(q(t))\\dot{q}(t)\\right) + \\langle K(q,\\dot{q}), \\xi \\rangle .\\end{split}$ The derivative term is zero due to the invariance of $L$ under the action of $G$ , so from the arbitrariness of $\\xi \\in \\mathfrak {g}$ we obtain the desired result.",
"As a corollary, we obtain a slight generalization of the classical Noether's theorem.",
"Corollary 1 (Noether's theorem) Consider a mechanical system given by a $G$ -invariant Lagrangian of the form kinetic minus potential energy.",
"Assume that the only external forces take values in the annihilator of the distribution tangent to the $G$ orbits.",
"Then the derivative of the momentum map $J$ along trajectories satisfies $\\dot{J}=0.$ Set $K = 0$ in Proposition REF ."
],
[
"Local form of the equations of motion",
"Assuming that the action of $G$ on $Q$ is free and proper [27] so that $\\pi _Q\\colon Q\\rightarrow Q/G$ is a principal $G$ -bundle, we now derive the equations in a local trivialization, following [25].",
"In a local trivialization $U \\times G$ , $\\pi _Q$ simply becomes projection onto the first factor and the $G$ action is given by left multiplication on the second factor.",
"We define $S Q/G$ to be the shape space representing all possible shapes of a locomoting body, and we write a point in the local trivialization as $(r,g)\\in U\\times G$ where $U\\subset S$ .",
"We assume that $U$ is the domain of a chart for $S$ , so that we have induced coordinates $(r,\\dot{r})$ for $Ṵ$ .",
"Defining the body velocityAs mentioned in the main text, the body velocity is often written $g^{-1}\\dot{g}$ by an abuse of notation which is only defined on matrix Lie groups where the product of a tangent vector and a group element is naturally defined.",
"We use the alternative notation $\\accentset{\\scriptstyle \\circ }{g}$ as a matter of personal preference.",
"$\\accentset{\\scriptstyle \\circ }{g} {L}_{g^{-1}}\\dot{g}$ , the equivariance property (REF ) of the connection forms $\\Gamma _{\\textnormal {mech}}, \\Gamma _{\\textnormal {visc}}$ imply that they may be written in the trivialization as $\\begin{split}\\Gamma _{\\textnormal {mech}}(r,g)\\cdot (\\dot{r},\\dot{g}) &= \\textnormal {Ad}_g\\left(\\accentset{\\scriptstyle \\circ }{g}+A_{\\textnormal {mech}}(r)\\cdot \\dot{r}\\right)\\\\\\Gamma _{\\textnormal {visc}}(r,g)\\cdot (\\dot{r},\\dot{g}) &= \\textnormal {Ad}_g\\left(\\accentset{\\scriptstyle \\circ }{g}+A_{\\textnormal {visc}}(r)\\cdot \\dot{r}\\right),\\end{split}$ where $A_{\\textnormal {mech}}\\colon Ṵ \\rightarrow \\mathfrak {g}$ and $A_{\\textnormal {visc}}\\colon Ṵ \\rightarrow \\mathfrak {g}$ are respectively the local mechanical connection and local viscous connection.",
"We define a diffeomorphism $(r,\\dot{r},g,\\dot{g})\\mapsto (r,\\dot{r},g,p)$ , with $p$ the body momentum defined by $p\\textnormal {Ad}_g^*{J} \\in \\mathfrak {g}^*.$ Here $\\textnormal {Ad}_g^*$ is the dual of the adjoint action $\\textnormal {Ad}_g$ of $G$ on $\\mathfrak {g}$ .",
"We additionally define $\\begin{split}\\mathbb {I}_{\\textnormal {loc}}&\\textnormal {Ad}_g^* I\\textnormal {Ad}_g \\colon \\mathfrak {g}\\rightarrow \\mathfrak {g}^*\\\\\\mathbb {V}_{\\textnormal {loc}}&\\textnormal {Ad}_g^* _g \\colon \\mathfrak {g}\\rightarrow \\mathfrak {g}^*\\end{split}$ to be the local forms of $I$ and $.We note that the invariance of the Lagrangian $ L$ and Rayleigh dissipation function $ R$ under $ G$, together with the general identity $ g Q(q) = (Adg)Q(g(q))$, imply that $ Iloc(r),Vloc(r)$ depend on the shape variable $ r$ only.$ Rearranging (REF ), using the expressions (REF ), (REF ), and using Proposition REF , we obtain the equations of motion $\\begin{split}\\accentset{\\scriptstyle \\circ }{g} &= -A_{\\textnormal {mech}}\\cdot \\dot{r} + \\mathbb {I}_{\\textnormal {loc}}^{-1}p\\\\\\dot{p} &= \\mathbb {V}_{\\textnormal {loc}}(A_{\\textnormal {visc}}- A_{\\textnormal {mech}})\\cdot \\dot{r} + \\mathbb {V}_{\\textnormal {loc}}\\mathbb {I}_{\\textnormal {loc}}^{-1} p + \\textnormal {ad}^*_{\\mathbb {I}_{\\textnormal {loc}}^{-1}p}p-\\textnormal {ad}^*_{A_{\\textnormal {mech}}\\cdot \\dot{r}}p,\\end{split}$ where we have suppressed the $r$ -dependence of $A_{\\textnormal {mech}},A_{\\textnormal {visc}},\\mathbb {I}_{\\textnormal {loc}},\\mathbb {V}_{\\textnormal {loc}}$ for readability.",
"Notice that the $\\dot{p}$ equation is completely decoupled from $g$ .",
"In this paper, we are interested in the effect of shape changes on body motion, and not on the generation of shape changes themselves.",
"Hence we have suppressed the equations for $\\dot{r},\\ddot{r}$ from (REF ), simply viewing $r, \\dot{r}$ as inputs in those equations, but see [3] for more details on the specific form of the equations.",
"We merely note that, if the kinetic energy metric is positive-definite, then the Lagrangian is hyperregular and our assumption of $G$ -equivariance of the exerted force $F_E$ implies that $\\ddot{r} = f(t,r,\\dot{r},\\mathbb {I}_{\\textnormal {loc}}^{-1} p)$ for some function $f$ which depends on the local trivialization.",
"If the kinetic energy metric is not positive-definite (for use in toy examples like those in §; see the precise assumptions in §REF , and the footnote there), then we assume that $\\ddot{r}$ is given by (REF )."
],
[
"Reduction in the Stokesian limit",
"From the definitions (), (REF ) of $\\mathbb {I}_{\\textnormal {loc}}, \\mathbb {V}_{\\textnormal {loc}}$ , we see that we may define $\\bar{\\mathbb {I}}_{\\textnormal {loc}}, \\bar{\\mathbb {V}}_{\\textnormal {loc}}$ by $\\mathbb {I}_{\\textnormal {loc}}(r) m \\bar{\\mathbb {I}}_{\\textnormal {loc}}(r) ~~~~\\mathbb {V}_{\\textnormal {loc}}(r) c \\bar{\\mathbb {V}}_{\\textnormal {loc}}(r).$ Defining the dimensionless parameter $\\epsilon \\frac{m}{c}$ and multiplying both sides of (REF ) by $\\mathbb {I}_{\\textnormal {loc}}\\mathbb {V}_{\\textnormal {loc}}^{-1}$ , we obtain the rewritten equations of motion $\\begin{split}\\accentset{\\scriptstyle \\circ }{g} &= -A_{\\textnormal {mech}}\\cdot \\dot{r} + \\frac{1}{m}\\bar{\\mathbb {I}}_{\\textnormal {loc}}^{-1}p\\\\\\epsilon \\bar{\\mathbb {I}}_{\\textnormal {loc}}\\bar{\\mathbb {V}}_{\\textnormal {loc}}^{-1}\\dot{p} &= m\\bar{\\mathbb {I}}_{\\textnormal {loc}}(A_{\\textnormal {visc}}- A_{\\textnormal {mech}})\\cdot \\dot{r} + p + \\epsilon \\bar{\\mathbb {I}}_{\\textnormal {loc}}\\bar{\\mathbb {V}}_{\\textnormal {loc}}^{-1} \\textnormal {ad}^*_{\\mathbb {I}_{\\textnormal {loc}}^{-1}p}p- \\epsilon \\bar{\\mathbb {I}}_{\\textnormal {loc}}\\bar{\\mathbb {V}}_{\\textnormal {loc}}^{-1}\\textnormal {ad}^*_{A_{\\textnormal {mech}}\\cdot \\dot{r}}p.\\end{split}$ In considering the limit in which viscous forces dominate the inertia of the locomoting body, [25] formally set $\\epsilon =0$ in (REF ) to obtain $p = m\\bar{\\mathbb {I}}_{\\textnormal {loc}}(A_{\\textnormal {mech}}-A_{\\textnormal {visc}})\\cdot \\dot{r}$ from the second equation.",
"Substituting this into the first equation of (REF ), they derive the following form of the equations of motion: $\\accentset{\\scriptstyle \\circ }{g}=-A_{\\textnormal {visc}}\\cdot \\dot{r}.$ In the language of differential geometry, (REF ) states that in the Stokesian limit trajectories are horizontal with respect to the viscous connection.",
"We will see in the next section that this reduction can be extended away from the $\\epsilon \\rightarrow 0$ limit."
],
[
"Appendix B — Reduction in the Perturbed Stokes Regime",
"In [9], the argument of [25] was explained in more detail using the theory of normally hyperbolic invariant manifolds (NHIMs) in the context of geometric singular perturbation theory [14], [20], [22].",
"The idea is to show that for $\\epsilon > 0$ sufficiently small, the dynamics (REF ) possess an exponentially attractive invariant slow manifold $M_\\epsilon $ , such that the dynamics restricted to $M_\\epsilon $ approach (REF ) as $\\epsilon \\rightarrow 0$ .",
"We give an alternative argument which yields a result differing from that of [9] in two ways.",
"[9] give an argument for general mechanical systems without symmetry under the assumption that the configuration space $Q$ is compact, although they do indicate that compactness can be replaced with uniformity conditions using noncompact NHIM theory [7].",
"Our argument assumes symmetry but allows $G$ to be noncompact, though we do require that $SQ/G$ be compact.",
"This enables application of our result to locomotion systems with noncompact symmetry groups, such as the Euclidean group of planar rigid motions $\\mathsf {SE}(2)$ as in the systems of §.",
"[9] consider the limit $m\\rightarrow 0$ while holding $c$ and the force exerted by the locomoting body fixed.",
"This makes sense, because if the exerted force were held fixed while taking $c \\rightarrow \\infty $ , then trivial dynamics would result in the singular limit: the system would not move at all.",
"Rather than holding the exerted force fixed, we will consider the differential equation prescribing the dynamics of the shape variable to be fixed.This implicitly assumes that the locomoting body is capable of exerting $\\mathcal {O}(c)$ forces.",
"Under this assumption, we show that the dynamics depend only on the ratio $\\epsilon = \\frac{m}{c}$ , and in particular the dynamics obtained in the two singular limits $m \\rightarrow 0$ and $c \\rightarrow \\infty $ are the same.",
"Before stating Theorem REF , we need the following definition.",
"Definition 1 ($C^k_b$ time-dependent vector fields) Let $M$ be a compact manifold with boundary, and let $f\\colon \\mathbb {R}\\times M \\rightarrow M̰$ a $C^{k \\ge 0}$ time-dependent vector field.",
"Let $(U_i)_{i=1}^n$ be a finite open cover of $M$ and $(V_i, \\psi _i)_{i=1}^n$ be a finite atlas for $M$ such that $\\bar{U}_i \\subset V_i$ for all $i$ , and for each $i$ define $f_i (_i \\circ f\\circ (\\textnormal {id}_\\mathbb {R}\\times \\psi _i^{-1}))$ .",
"We define an associated $C^k$ norm ${f}_k$ of $f$ via ${f}_{ k}\\max _{1\\le i\\le n}\\max _{\\begin{array}{c}0 \\le j \\le k\\\\x\\in \\psi _i(\\bar{U}_i)\\end{array}}{j f_i(x)},$ where ${j f_i(x)}$ denotes the norm of a $j$ -linear map; here $j f$ includes partial derivatives with respect to time as well as the spatial variables.",
"If ${f}_k< \\infty $ , we say that $f$ is $C^k$ -bounded and write $f\\in C^k_b$ .",
"The norm ${\\cdot }_k$ makes the $C^k_b$ time-dependent vector fields into a Banach space.",
"The norms induced by any two such finite covers of $M$ are equivalent, and thereby induce a canonical $C^k_b$ topology on the space of $C^k_b$ time-dependent vector fields.",
"Remark 4 Definition REF defines the $C^k_b$ topology on the space of $C^k_b$ time-dependent vector fields on a compact manifold.",
"As discussed in [7], this $C^k_b$ topology is finer than the $C^k$ weak Whitney topology and coarser than the $C^k$ strong Whitney topology [18], but all of these topologies induce the same topology on the subspace of time-independent vector fields due to compactness.",
"Definition REF is a special case of the definition in [7] for the $C^k_b$ topology on $C^k_b$ vector fields on Riemannian manifolds of bounded geometry, and on $C^k_b$ maps between such manifolds.",
"The following theorem concerns a $G$ -symmetric dynamical system on $Q̰$ whose equations of motion are consistent with our assumptions so far: i.e., they are given in local trivializations by (REF ) and an equation of the form (REF ).",
"Theorem 2 Assume that $S=Q/G$ is compact.",
"Let $2 \\le k < \\infty $ , and let $X^\\epsilon $ be a $C^k$ family of $G$ -symmetric time-dependent vector fields on $Q̰$ with the following properties: For every compact neighborhood with $C^k$ boundary $K_0 \\subset Q̰$ and $\\epsilon > 0$ , $X^\\epsilon |_{\\mathbb {R}\\times K_0}\\in C^k_b$ (Definition REF ).",
"There exists a compact connected neighborhood $K\\subset S̰$ of the zero section of $S̰$ with $C^k$ boundary, such that $N_Q^{-1}(K) \\subset Q̰$ is positively invariant for $X^\\epsilon $ , for all sufficiently small $\\epsilon > 0$ .",
"$X^\\epsilon $ is given in each local trivialization $U\\times G)$ , where $U$ is a chart for $S$ , by (REF ) and (REF ): $\\begin{split}\\ddot{r} &= f\\left(t,r,\\dot{r},\\frac{1}{m}\\bar{\\mathbb {I}}_{\\textnormal {loc}}^{-1}p\\right)\\\\\\epsilon \\bar{\\mathbb {I}}_{\\textnormal {loc}}\\bar{\\mathbb {V}}_{\\textnormal {loc}}^{-1}\\dot{p} &= m\\bar{\\mathbb {I}}_{\\textnormal {loc}}(A_{\\textnormal {visc}}- A_{\\textnormal {mech}})\\cdot \\dot{r} + p + \\epsilon \\bar{\\mathbb {I}}_{\\textnormal {loc}}\\bar{\\mathbb {V}}_{\\textnormal {loc}}^{-1} \\textnormal {ad}^*_{\\mathbb {I}_{\\textnormal {loc}}^{-1}p}p- \\epsilon \\bar{\\mathbb {I}}_{\\textnormal {loc}}\\bar{\\mathbb {V}}_{\\textnormal {loc}}^{-1}\\textnormal {ad}^*_{A_{\\textnormal {mech}}\\cdot \\dot{r}}p\\\\\\accentset{\\scriptstyle \\circ }{g} &= -A_{\\textnormal {mech}}\\cdot \\dot{r} + \\frac{1}{m}\\bar{\\mathbb {I}}_{\\textnormal {loc}}^{-1}p\\end{split}$ for some function $f$ which depends on the local trivialization but is independent of $\\epsilon $ .",
"Then for all sufficiently small $\\epsilon > 0$ , there exists a $C^k$ noncompact normally hyperbolic invariant manifold with boundary $M_\\epsilon \\subset \\mathbb {R}\\times N \\subset \\mathbb {R}\\times Q̰$ for the extended dynamics given by the extended vector field $(1,X_\\epsilon )$ on $\\mathbb {R}\\times Q̰$ .",
"Additionally, $M_\\epsilon $ is uniformly (in time and space) globally asymptotically stable and uniformly locally exponentially stable (with respect to the distance induced by any complete $G$ -invariant Riemannian metric on $Q̰$ ) for the extended dynamics restricted to $\\mathbb {R}\\times N$ .",
"Finally, there exists $\\epsilon _0 > 0$ such that, for each local trivialization $U\\times G$ , there exists a $C^k$ map $h_\\epsilon \\colon \\mathbb {R}\\times (Ṵ \\cap K) \\times (0,\\epsilon _0)\\rightarrow \\mathfrak {g}^*$ such that $M_\\epsilon \\cap _Q^{-1}(Ṵ \\cap K)$ corresponds to $\\lbrace (t,r,\\dot{r},p,g): p = h_\\epsilon (t,r,\\dot{r},\\epsilon )\\rbrace ,$ $h_\\epsilon (t,r,\\dot{r},\\epsilon ) = \\mathbb {I}_{\\textnormal {loc}}\\left[(A_{\\textnormal {mech}}(r)-A_{\\textnormal {visc}}(r))\\cdot \\dot{r} + \\mathcal {O}(\\epsilon )\\right]$ (with $p$ defined by (REF )), and $h_\\epsilon $ together with its partial derivatives of order $k$ or less are bounded uniformly in time.",
"If $f(t,r,\\dot{r},\\mathbb {I}_{\\textnormal {loc}}^{-1} p)$ is independent of $t$ , then $h_\\epsilon $ and $M_\\epsilon $ are independent of $t$ , and $M_\\epsilon $ can be interpreted as a compact NHIM for the (non-extended) dynamics restricted to $N$ .",
"Remark 5 Note that even if we assume $f\\in C^\\infty $ , we can generally only obtain $C^k$ NHIMs $M_\\epsilon $ for $k$ finite.",
"This is because we obtain $M_\\epsilon $ as a perturbation of a NHIM $M_0$ , and perturbations of $C^\\infty $ NHIMs are generally only finitely smooth because the maximum perturbation size $\\epsilon $ required to obtain degree of smoothness $k$ for $M_\\epsilon $ generally depends on $k$ in such a way that $\\epsilon \\rightarrow 0$ as $k\\rightarrow \\infty $ .",
"See [7] and [38] for more discussion.",
"Remark 6 By replacing compactness of $Q/G$ with uniformity conditions, it should be possible to generalize Theorem REF to the situation of $Q$ noncompact where either $Q/G$ is noncompact, or where there is no symmetry at all.",
"This was pointed out in [9].",
"This observation seems important for the consideration of dissipative mechanical systems which are only approximately symmetric under a group $G$ , which seems to be a more realistic assumption.",
"Remark 7 By taking $\\epsilon \\rightarrow 0$ in Theorem REF , we find that $p = \\mathbb {I}_{\\textnormal {loc}}(A_{\\textnormal {mech}}-A_{\\textnormal {visc}})\\cdot \\dot{r}$ in the limit.",
"Substituting this into the first equation of (REF ), we obtain Equation (REF ) as in [25].",
"Preparation of the equations of motion.",
"Throughout the proof, we consider the dynamics in local trivializations of the form $U\\times G$ for $Q$ , where $U$ is the domain of a chart for $S$ , so that we have induced coordinates $(r,\\dot{r})$ for $Ṵ$ .",
"In such a local trivialization we would like to use (REF ) to analyze the dynamics, but there are two (related) problems with this.",
"First, the definition of $p$ depends on $m$ , and this will cause difficulties in verifying Definition REF to check that certain vector fields are close in the $C^k_b$ topology.",
"Second, we would like to analyze (REF ) in a singular perturbation framework, but this is difficult to do directly because $m$ explicitly appears, and the size of $m$ may or may not be commensurate with the size of $\\epsilon $ .",
"To remedy this situation, we change variables via the diffeomorphism $(r,\\dot{r},p,g)\\mapsto (r,\\dot{r},\\Omega ,g)$ of $Ṵ \\times \\mathfrak {g}^* \\times G \\rightarrow Ṵ \\times \\mathfrak {g}\\times G$ where $\\Omega \\in \\mathfrak {g}$ is defined by $\\Omega \\mathbb {I}_{\\textnormal {loc}}^{-1}p = \\textnormal {Ad}_{g^{-1}}\\Gamma _{\\textnormal {mech}}(\\dot{g},\\dot{r})=\\accentset{\\scriptstyle \\circ }{g} + A_{\\textnormal {mech}}\\cdot \\dot{r}.$ Sometimes $\\Omega $ is referred to as the (body) locked angular velocity [3].",
"Differentiating $\\mathbb {I}_{\\textnormal {loc}}\\Omega = p$ , using (REF ), and rearranging yields $\\begin{split}\\dot{t} &= 1\\\\\\dot{r} &= v\\\\\\dot{v} &= f(t,r,v,\\Omega )\\\\\\epsilon \\dot{\\Omega } &= -\\epsilon \\bar{\\mathbb {I}}_{\\textnormal {loc}}^{-1} \\left(\\frac{d}{dt}\\bar{\\mathbb {I}}_{\\textnormal {loc}}\\right)\\Omega + \\bar{\\mathbb {I}}_{\\textnormal {loc}}^{-1}\\bar{\\mathbb {V}}_{\\textnormal {loc}}(A_{\\textnormal {visc}}- A_{\\textnormal {mech}})\\cdot v + \\bar{\\mathbb {I}}_{\\textnormal {loc}}^{-1}\\bar{\\mathbb {V}}_{\\textnormal {loc}}\\Omega + \\epsilon \\bar{\\mathbb {I}}_{\\textnormal {loc}}^{-1}\\textnormal {ad}^*_{\\accentset{\\scriptstyle \\circ }{g}}\\bar{\\mathbb {I}}_{\\textnormal {loc}}\\Omega ,\\end{split}$ where we have introduced the variable $v\\dot{r}$ .",
"We have written $\\textnormal {ad}^*_{\\accentset{\\scriptstyle \\circ }{g}}$ for space reasons, but note that the $\\dot{\\Omega }$ equation is independent of $g$ since $\\accentset{\\scriptstyle \\circ }{g} = -A_{\\textnormal {mech}}\\cdot \\dot{r} + \\Omega ,$ and this implies that $\\textnormal {ad}^*_{\\accentset{\\scriptstyle \\circ }{g}} = \\textnormal {ad}^*_{\\Omega }-\\textnormal {ad}^*_{A_{\\textnormal {mech}}\\cdot \\dot{r}}$ .",
"We see that (REF ) is split into slow $(t,r,v)$ and fast $(\\Omega )$ variables, which is the appropriate setup for a singular perturbation analysis.",
"The remainder of the proof consists of two parts: (i) proving that the NHIM $M_\\epsilon $ exists, and (ii) establishing the stability properties of $M_\\epsilon $ .",
"Proof that $M_\\epsilon $ exists.",
"Introducing the “fast time” $\\tau \\frac{1}{\\epsilon } t$ and denoting a derivative with respect to $\\tau $ by a prime, after the time-rescaling we obtain the regularized equations $\\begin{split}t^{\\prime } &= \\epsilon \\\\r^{\\prime } &= \\epsilon v\\\\v^{\\prime } &= \\epsilon f(t,r,v,\\Omega )\\\\\\Omega ^{\\prime } &= -\\epsilon \\bar{\\mathbb {I}}_{\\textnormal {loc}}^{-1} \\left(\\frac{d}{dt}\\bar{\\mathbb {I}}_{\\textnormal {loc}}\\right)\\Omega + \\bar{\\mathbb {I}}_{\\textnormal {loc}}^{-1}\\bar{\\mathbb {V}}_{\\textnormal {loc}}(A_{\\textnormal {visc}}- A_{\\textnormal {mech}})\\cdot v + \\bar{\\mathbb {I}}_{\\textnormal {loc}}^{-1}\\bar{\\mathbb {V}}_{\\textnormal {loc}}\\Omega + \\epsilon \\bar{\\mathbb {I}}_{\\textnormal {loc}}^{-1}\\textnormal {ad}^*_{\\accentset{\\scriptstyle \\circ }{g}}\\bar{\\mathbb {I}}_{\\textnormal {loc}}\\Omega .\\end{split}$ This rescaling of time is equivalent to replacing the vector field $(1,X_\\epsilon )$ on $\\mathbb {R}\\times Q̰$ by $(\\epsilon ,\\epsilon X_\\epsilon )$ .",
"We see from (REF ) and (REF ) that there is a well-defined $C^k$ time-dependent vector field $\\tilde{X}_0$ given by the pointwise limit $\\tilde{X}_0\\lim _{\\epsilon \\rightarrow 0} \\epsilon X_\\epsilon $ .",
"Given any $G$ -symmetric time-dependent vector field $Y$ on $Q̰$ , we let $Y/G$ denote the corresponding reduced vector field on $(Q̰)/G$ .",
"Hence (REF ) shows that the extended vector field $(1,\\tilde{X}_0/G)$ has a smooth embedded submanifold $(M_0/G)$ of critical points whose intersection with a locally trivializable neighborhood is given by $\\lbrace (r,v,\\Omega ) \\in Ṵ \\times \\mathfrak {g}: \\Omega = (A_{\\textnormal {mech}}-A_{\\textnormal {visc}})\\cdot v\\rbrace ,$ and it is readily seen that $M_0/G$ is described globally as the quotient of the Ehresmann connection $M_0 \\ker \\Gamma _{\\textnormal {visc}}$ by the lifted action of $G$ on $Q̰$ .",
"Furthermore, $M_0/G$ is a globally exponentially stable NHIM for the $\\epsilon = 0$ system.",
"To see this, first note that in any local trivialization $t, r, v$ are constants when $\\epsilon = 0$ , and hence $\\Omega ^{\\prime }$ is of the form $\\Omega ^{\\prime } = \\bar{\\mathbb {I}}_{\\textnormal {loc}}^{-1} \\bar{\\mathbb {V}}_{\\textnormal {loc}}\\Omega + b$ for a constant $b$ , and therefore has a globally exponentially stable equilibrium provided that all eigenvalues of $\\bar{\\mathbb {I}}_{\\textnormal {loc}}^{-1}\\bar{\\mathbb {V}}_{\\textnormal {loc}}$ have negative real part.",
"To see that this is the case, fix a basis of $\\mathfrak {g}$ and corresponding dual basis for $\\mathfrak {g}^*$ , and first consider the product $I^{-1} .",
"With respect to our chosen basis, $ I, and their inverses $I^{-1}, {-1}$ are respectively represented by $r$ -dependent matrices $I_{ij}, V_{ij}$ and their inverses $I^{ij}, V^{ij}$ .",
"It is immediate from the definitions () and (REF ) that $I_{ij}$ and $V_{ij}$ are respectively positive definite and negative definite symmetric matrices (this is why we required the bilinear forms $k, \\nu $ to be positive definite when restricted to vectors tangent to $G$ orbits).",
"Since $I_{ij}$ is symmetric positive definite, we may let $(\\sqrt{I})_{ij}$ be a matrix square root of $I_{ij}$ and let $(\\sqrt{I})^{ij}$ be its inverse.",
"But then the product $I^{ik}V_{kj}$ is similar to the symmetric negative definite matrix $(\\sqrt{I})^{ik}V_{k\\ell }(\\sqrt{I})^{\\ell j}$ (Einstein summation implied).",
"Hence $I^{-1} has only eigenvalues with negative real part, and the same is true of $ Iloc-1Vloc$ because of the similarity $ Iloc-1Vloc= Adg-1 I-1g$.$ Let $\\tilde{\\pi }\\colon (Q̰)/G \\rightarrow S̰$ denote the projection induced by $_Q$ .",
"Equation (REF ) implies that $M_0/G$ is the image of a section $\\sigma _0\\colon S̰ \\rightarrow (Q̰)/G$ of $\\tilde{\\pi }$ .",
"Hence $(M_0/G)\\cap \\tilde{\\pi }^{-1}(K) = \\sigma _0(K)$ is compact, and $M_0/G$ intersects $\\tilde{\\pi }^{-1}(\\partial K)$ transversely.",
"Furthermore, the assumption that $X^\\epsilon |_{\\mathbb {R}\\times K_0} \\in C^k_b$ for any compact neighborhood with $C^k$ boundary $K_0 \\subset Q̰$ implies that all partial derivatives of $f$ are bounded on compact sets uniformly in time.",
"This makes it clear that for any compact $K_1\\subset (Q̰)/G$ , $(\\epsilon X_\\epsilon /G)|_{\\mathbb {R}\\times K_1}$ can be made arbitrarily close to $(\\tilde{X}_0/G)|_{\\mathbb {R}\\times K_1}$ in the $C^k_b$ topology (Definition REF ) by taking $\\epsilon > 0$ sufficiently small.",
"Hence by the noncompact NHIM results of [7], it follows that $(M_0/G)\\cap \\tilde{\\pi }^{-1}(K)$ persists in extended state space $\\mathbb {R}\\times N$ to a nearby attracting NHIM $M_\\epsilon /G$ with boundary for $(\\epsilon , \\epsilon X_\\epsilon /G)$ .$M_\\epsilon /G$ is unique up to the choice of a cutoff function used to modify the dynamics near the boundary of a slightly enlarged neighborhood of $\\tilde{\\pi }^{-1}(K)$ , used in order to render a slightly enlarged version of $(M_0/G)\\cap \\tilde{\\pi }^{-1}(K)$ overflowing invariant [7].",
"See [10] and [21] for more details on such boundary modifications.",
"Furthermore, $M_\\epsilon /G$ is the image of a section $\\sigma _\\epsilon \\colon \\mathbb {R}\\times K \\rightarrow (Q̰)/G$ of $\\tilde{\\pi }$ , and is given in each local trivialization of $(Q̰)/G$ by the graph of a function $\\Omega = \\tilde{h}_\\epsilon (t,r,\\dot{r},\\epsilon )$ which is $C^k$ bounded uniformly in time.",
"By symmetry, the preimage $M_\\epsilon = \\pi _{Q̰}^{-1}(M_\\epsilon /G)$ of $M_\\epsilon /G$ via the quotient $\\pi _{Q̰}\\colon Q̰ \\rightarrow (Q̰)/G$ yields a NHIM $M_\\epsilon $ for $(\\epsilon , \\epsilon X_\\epsilon )$ (and hence also for $(1,X_\\epsilon )$ ) on the subset $\\mathbb {R}\\times N$ of $\\mathbb {R}\\times Q̰$ , and $M_\\epsilon $ is given in each local trivialization by the graph of the same function $\\Omega = \\tilde{h}_\\epsilon $ as $M_\\epsilon /G$ but augmented with trivial dependence on $g$ .",
"The function $h_\\epsilon $ from the theorem statement is given by $h_\\epsilon = \\mathbb {I}_{\\textnormal {loc}}\\tilde{h}_\\epsilon $ .",
"Proof of the stability properties of $M_\\epsilon $.",
"Fix any complete $G$ -invariant Riemannian metric onFor example, take the Sasaki metric on $Q̰$ induced by any complete $G$ -invariant metric on $Q$ .",
"$Q̰$ , so that it descends to a metric on $(Q̰)/G$ making $\\pi _{Q̰}\\colon Q̰ \\rightarrow (Q̰)/G$ into a Riemannian submersion [6].",
"We have distance functions $\\tilde{d}$ and $d$ on $Q̰$ and $(Q̰)/G$ induced by these metrics.",
"For $t \\in \\mathbb {R}$ , we let $M_\\epsilon (t)M_\\epsilon \\cap (\\lbrace t\\rbrace \\times N)$ and $M_\\epsilon (t)/G\\pi _{Q̰}(M_\\epsilon (t))$ .",
"Given $w\\in Q̰$ and its orbit $\\pi _{Q̰}(w) \\in (Q̰)/G$ , it follows that for all $t\\in \\mathbb {R}$ , $\\tilde{d}(w, M_\\epsilon (t)) = d(\\pi _{Q̰}(w), M_\\epsilon (t)/G)$ .To prove this, first note that $d(\\pi _{Q̰}(w), M_\\epsilon (t)/G)\\le \\tilde{d}(w, M_\\epsilon (t))$ because the length $\\ell (\\tilde{\\gamma })$ of any curve $\\tilde{\\gamma }\\colon [0,1]\\rightarrow Q̰$ satisfies $\\ell (\\pi _{Q̰}\\circ \\tilde{\\gamma })\\le \\ell (\\tilde{\\gamma })$ .",
"But if $\\gamma :[0,1]\\rightarrow (Q̰)/G$ is any curve joining $\\pi _{Q̰}(w)$ to $M_\\epsilon /G$ , then its horizontal lift $\\tilde{\\gamma }$ is a curve joining $w$ to $M_\\epsilon $ such that $\\ell (\\tilde{\\gamma })=\\ell (\\gamma )$ .",
"Taking the infimum over all such $\\gamma $ shows that $\\tilde{d}(w, M_\\epsilon (t)) = d(\\pi _{Q̰}(w), M_\\epsilon (t)/G)$ .",
"Hence it suffices to prove that $M_\\epsilon /G$ is uniformly globally asymptotically stable and locally exponentially stable for the vector field $(1, X_\\epsilon /G)$ on $\\mathbb {R}\\times \\tilde{\\pi }^{-1}(K) = \\mathbb {R}\\times \\pi _{Q̰}(N)$ , and to do this it suffices to prove the same for $(\\epsilon , \\epsilon X_\\epsilon /G)$ .",
"Fixing an inner product $\\langle \\,\\cdot \\,, \\,\\cdot \\,\\rangle $ and associated norm ${\\,\\cdot \\,}$ on $\\mathfrak {g}$ , we accomplish this in two steps.",
"First, we show that there exists a compact neighborhood $K_0 \\subset \\pi _{Q̰}(N)$ of $M_\\epsilon /G$ such that $K_0$ is positively invariant for the time-dependent flow of $X_\\epsilon $ , and such that any other compact neighborhood $K_1\\subset \\pi _{Q̰}(N)$ of $M_\\epsilon /G$ flows into $K_0$ after some finite time depending on $K_1$ but independent of the initial time.",
"Second, we show that all trajectories in $K_0$ converge to $M_\\epsilon /G$ at a uniform exponential rate.",
"To achieve this second step, we show that in the intersection of each local trivialization with $K_0$ , ${\\Omega -\\tilde{h}_\\epsilon (t,r,v)}$ decreases at an exponential rate.",
"Since $(Q̰)/G$ is covered by finitely many local trivialization (by compactness of $S$ ), and since all Riemannian metrics are uniformly equivalent on compact setsLet ${\\,\\cdot \\,}, {\\,\\cdot \\,}^{\\prime }$ denote the Finslers (norms) induced by two Riemannian metrics, and $K_0$ our compact set.",
"Since all norms are equivalent on finite-dimensional vector spaces, we have that the restrictions of these norms to the tangent space of a single point $x$ satisfy $\\frac{1}{c(x)}{\\,\\cdot \\,} \\le {\\,\\cdot \\,}^{\\prime } \\le c(x) {\\,\\cdot \\,}$ .",
"Defining $\\bar{c}\\sup _{x\\in K_0}c(x)$ , we obtain the uniform equivalence $\\frac{1}{\\bar{c}}{\\,\\cdot \\,} \\le {\\,\\cdot \\,}^{\\prime } \\le \\bar{c} {\\,\\cdot \\,}$ on all of $K_0$ .",
"If $K_0$ is a connected submanifold and we give it the restricted metrics, then by considering the lengths of curves in $K_0$ this implies the uniform bound $\\frac{1}{\\bar{c}}d \\le d^{\\prime } \\le \\bar{c}d$ on the Riemannian distances between points in $K_0$ with respect to the restricted metrics., this will establish uniform exponential convergence of points in $K_0$ with respect to the distance induced by any Riemannian metric, and in particular the distance $d$ .",
"Consider a local trivialization $U\\times G$ of $Q$ and the associated form (REF ) of the dynamics restricted to $\\tilde{\\pi }^{-1}(K \\cap Ṵ)$ .",
"Differentiating ${\\Omega }^2$ using the last equation of (REF ), it is easy to check that $\\frac{d}{d\\tau }{\\Omega }^2 \\rightarrow -\\infty $ as ${\\Omega }^2 \\rightarrow \\infty $ , uniformly in $(t,r,v,\\epsilon )$ for $\\epsilon $ sufficiently small.",
"(This follows from the negative definiteness of $\\mathbb {I}_{\\textnormal {loc}}^{-1}\\mathbb {V}_{\\textnormal {loc}}$ and the compactness of $K$ .)",
"Hence we see that there exists $k_0 > 0$ such that for all $\\epsilon $ sufficiently small, $\\frac{d}{d\\tau }{\\Omega }^2 \\le -1$ when ${\\Omega }^2 \\ge k_0^2$ .",
"Now $k_0$ might depend on the local trivialization, but we can replace $k_0$ with the largest such constant selected from finitely many fixed local trivializations covering $Q$ .",
"Hence there exists a compact subset $K_0 \\subset \\pi _{Q̰}(N)$ given by $\\lbrace {\\Omega } \\le k_0\\rbrace $ in each of these fixed local trivializations, such that $K_0$ is positively invariant for the time-dependent flow of $X_\\epsilon $ and such that any other compact neighborhood $K_1 \\subset \\pi _{Q̰}(N)$ of $M_\\epsilon /G$ flows into $K_0$ after some finite time independent of the initial time.",
"It remains only to establish the uniform exponential rate of convergence of trajectories in $K_0$ to $M_\\epsilon $ .",
"For each local trivialization $U \\times G$ of $Q$ , we define the translated variable $\\tilde{\\Omega }\\Omega - \\tilde{h}_\\epsilon (t,r,v,\\epsilon )$ .",
"Since $M_\\epsilon /G$ is invariant, we must have $\\tilde{\\Omega }^{\\prime } = 0$ whenever $\\tilde{\\Omega } = 0$ .",
"Differentiating $\\tilde{\\Omega }$ using (REF ), we therefore find that $\\begin{split}\\tilde{\\Omega }^{\\prime } &= \\left[-\\epsilon \\bar{\\mathbb {I}}_{\\textnormal {loc}}^{-1} \\left(\\frac{d}{dt}\\bar{\\mathbb {I}}_{\\textnormal {loc}}\\right) + \\epsilon \\bar{\\mathbb {I}}_{\\textnormal {loc}}^{-1}\\textnormal {ad}^*_{\\accentset{\\scriptstyle \\circ }{g}}\\bar{\\mathbb {I}}_{\\textnormal {loc}}+ \\epsilon \\zeta (t,r,v,\\tilde{\\Omega }) + \\bar{\\mathbb {I}}_{\\textnormal {loc}}^{-1}\\bar{\\mathbb {V}}_{\\textnormal {loc}}\\right]\\tilde{\\Omega }\\\\&\\left[\\epsilon A(t,r,v,\\tilde{\\Omega }) + \\bar{\\mathbb {I}}_{\\textnormal {loc}}^{-1}\\bar{\\mathbb {V}}_{\\textnormal {loc}}(r) \\right]\\tilde{\\Omega },\\end{split}$ since all of the terms which do not vanish when $\\tilde{\\Omega } = 0$ must cancel.",
"Here $\\zeta $ is defined via Hadamard's lemma [32]: $\\zeta (t,r,v,\\tilde{\\Omega }) \\frac{\\partial }{\\partial v}\\tilde{h}_\\epsilon (t,r,v) \\int _{0}^{1}\\frac{\\partial }{\\partial \\Omega }f(t,r,v,\\tilde{h}_\\epsilon (t,r,v) + s \\tilde{\\Omega })\\,ds,$ so that $\\zeta (t,r,v,\\tilde{\\Omega })\\tilde{\\Omega } = \\tilde{h}_\\epsilon (t,r,v) f(t,r,v,\\tilde{h}_\\epsilon + \\tilde{\\Omega })$ .",
"As previously mentioned, the $C^k$ boundedness of $X_\\epsilon $ on compact subsets of $Q̰$ implies that $\\tilde{h}_\\epsilon $ , $f$ , and their first $k$ partial derivatives are uniformly bounded on sets of the form $\\mathbb {R}\\times K_2$ with $K_2$ compact.",
"Hence whenever ${\\Omega }\\le k_0$ and $(r,v) \\in U \\cap K$ , ${A(t,r,v,\\tilde{\\Omega })} \\le L$ for some constant $L$ depending on the local trivialization; we replace $L$ with the largest such constant chosen from finitely many local trivializations covering $Q$ .",
"Integrating both sides of (REF ), taking norms using the triangle inequality, and applying Grönwall's Lemma therefore yields $\\begin{split}{\\tilde{\\Omega }(\\tau )} &\\le e^{-\\lambda (\\tau -\\tau _0)} e^{\\int _{\\tau _0}^{\\tau }\\epsilon {A(t(s),r(s),v(s),\\tilde{\\Omega }(s)}\\,ds }{\\tilde{\\Omega }(\\tau _0)}\\\\& \\le e^{\\left[-\\lambda + \\epsilon L \\right](\\tau -\\tau _0)} {\\tilde{\\Omega }(\\tau _0)}.\\end{split}$ where $-\\lambda < 0$ is defined via $-\\lambda \\sup _{r \\in S} \\max \\, \\textnormal {spec}(\\bar{\\mathbb {I}}_{\\textnormal {loc}}^{-1}\\bar{\\mathbb {V}}_{\\textnormal {loc}}(r))$ , and is strictly negative since $S$ is compact.",
"By the previous discussion, requiring $\\epsilon > 0$ to be sufficiently small so that $-\\lambda + \\epsilon L < 0$ completes the proof.",
"Theorem REF and Remark REF show that, to zeroth order in $\\epsilon $ , the dynamics restricted to the slow manifold $M_\\epsilon $ are given by the viscous connection model (REF ).",
"The following theorem shows that the dynamics restricted to $M_\\epsilon $ can be explicitly computed to higher order in $\\epsilon $ .",
"We compute the restricted dynamics to first order in $\\epsilon $ .",
"Higher order terms in $\\epsilon $ can also be computed recursively, but we choose not to pursue this here.",
"Theorem 3 Assume the same hypotheses as in Theorem REF .",
"Then the dynamics restricted to the slow manifold $M_\\epsilon $ are given in a local trivialization by $\\accentset{\\scriptstyle \\circ }{g}= -A_{\\textnormal {visc}}\\cdot \\dot{r} + \\epsilon \\bar{\\mathbb {V}}_{\\textnormal {loc}}^{-1} \\left(\\left(\\frac{\\partial }{\\partial _r} \\bar{h}_0\\right)\\dot{r} + \\left(\\frac{\\partial }{\\partial \\dot{r}}\\bar{h}_0\\right) \\ddot{r} - \\textnormal {ad}^*_{\\accentset{\\scriptstyle \\circ }{g}}(\\bar{h}_0)\\right) + \\mathcal {O}(\\epsilon ^2),$ where $\\bar{h}_0(r,\\dot{r}) \\frac{1}{m}h_0(r,\\dot{r}) = \\bar{\\mathbb {I}}_{\\textnormal {loc}}(A_{\\textnormal {mech}}(r)-A_{\\textnormal {visc}}(r))\\cdot \\dot{r},$ where we are using the definition $\\bar{\\mathbb {I}}_{\\textnormal {loc}}\\frac{1}{m}\\mathbb {I}_{\\textnormal {loc}}$ .",
"Alternatively, we may write $\\accentset{\\scriptstyle \\circ }{g}= -A_{\\textnormal {visc}}\\cdot \\dot{r} + \\epsilon \\bar{\\mathbb {V}}_{\\textnormal {loc}}^{-1} \\left(\\left(\\frac{\\partial }{\\partial r} \\bar{h}_0\\right)\\dot{r} + \\left(\\frac{\\partial }{\\partial \\dot{r}}\\bar{h}_0\\right) f(t,r,\\dot{r},\\bar{\\mathbb {I}}_{\\textnormal {loc}}^{-1}\\bar{h}_0) - \\textnormal {ad}^*_{\\accentset{\\scriptstyle \\circ }{g}}(\\bar{h}_0)\\right) + \\mathcal {O}(\\epsilon ^2),$ for a different $\\mathcal {O}(\\epsilon ^2)$ term.",
"Remark 8 Notice the presence, in the second term of (REF ), of $\\bar{h}_0$ rather than $h_0$ of (REF ).",
"This is important because the expression for $h_0$ contains an $\\mathbb {I}_{\\textnormal {loc}}= m \\bar{\\mathbb {I}}_{\\textnormal {loc}}$ factor.",
"Because of the possibility that the size of $m$ is commensurate with $\\epsilon $ , this means that $h_0$ could be $\\mathcal {O}(\\epsilon )$ .",
"However, $\\bar{h}_0$ is $\\mathcal {O}(1)$ , ensuring that the second term is $\\mathcal {O}(\\epsilon )$ but not $\\mathcal {O}(\\epsilon ^2)$ .",
"Remark 9 Equations (REF ) and (REF ) can be viewed as adding $\\mathcal {O}(\\epsilon )$ correction terms to the viscous connection model (REF ), valid in the limit $\\epsilon \\rightarrow 0$ , to account for the more realistic situation that the inertia-damping ratio $\\frac{m}{c} = \\epsilon $ is small but nonzero.",
"[Proof of Theorem REF ] Consider the function $\\tilde{h}_\\epsilon (t,r,\\dot{r},\\epsilon ) \\mathbb {I}_{\\textnormal {loc}}^{-1} h_\\epsilon = (A_{\\textnormal {mech}}(r)-A_{\\textnormal {visc}}(r))\\cdot \\dot{r} + \\mathcal {O}(\\epsilon )$ from the proof of Theorem REF , and define $\\bar{h}_\\epsilon \\bar{\\mathbb {I}}_{\\textnormal {loc}}\\tilde{h}_\\epsilon = \\frac{1}{m}h_\\epsilon $ .",
"Since $\\bar{h}_\\epsilon , \\tilde{h}_\\epsilon \\in C^k$ , we may expand them as asymptotic series $\\begin{split}\\bar{h}_\\epsilon &= \\bar{h}_0 + \\epsilon \\bar{h}_1 + \\ldots + \\epsilon ^{k} \\bar{h}_{k} + \\mathcal {O}(\\epsilon ^{k+1})\\\\\\tilde{h}_\\epsilon &= \\tilde{h}_0 + \\epsilon \\tilde{h}_1 + \\ldots + \\epsilon ^{k} \\tilde{h}_{k} + \\mathcal {O}(\\epsilon ^{k+1}),\\end{split}$ where for all $i$ , $\\bar{h}_i = \\bar{\\mathbb {I}}_{\\textnormal {loc}}\\tilde{h}_i$ .",
"We also already know from Theorem REF that $\\tilde{h}_0 =(A_{\\textnormal {mech}}-A_{\\textnormal {visc}})\\cdot \\dot{r}$ , and therefore $\\tilde{h}_0(t,r,\\dot{r}) \\equiv \\tilde{h}_0(r,\\dot{r})$ has no explicit $t$ -dependence.",
"We now compute $\\tilde{h}_1$ via a standard technique [20].",
"Differentiating both sides of the equation $\\Omega = \\tilde{h}_\\epsilon (t,r,\\dot{r},\\epsilon )$ with respect to time (using (REF ) to differentiate the left hand side), substituting the second equation of (REF ) for $\\Omega $ in the resulting expression, and retaining terms only up to $\\mathcal {O}(\\epsilon )$ we obtain $-\\epsilon \\bar{\\mathbb {I}}_{\\textnormal {loc}}^{-1} \\left(\\frac{d}{dt}\\bar{\\mathbb {I}}_{\\textnormal {loc}}\\right)\\tilde{h}_0 + \\bar{\\mathbb {I}}_{\\textnormal {loc}}^{-1}\\bar{\\mathbb {V}}_{\\textnormal {loc}}(A_{\\textnormal {visc}}- A_{\\textnormal {mech}})\\cdot \\dot{r} + \\bar{\\mathbb {I}}_{\\textnormal {loc}}^{-1}\\bar{\\mathbb {V}}_{\\textnormal {loc}}\\left(\\tilde{h}_0+\\epsilon \\tilde{h}_1 \\right) + \\epsilon \\bar{\\mathbb {I}}_{\\textnormal {loc}}^{-1}\\textnormal {ad}^*_{\\accentset{\\scriptstyle \\circ }{g}}\\bar{\\mathbb {I}}_{\\textnormal {loc}}\\tilde{h}_0 = \\epsilon \\dot{\\tilde{h}}_0 + \\mathcal {O}(\\epsilon ^2).$ Equating the coefficients of $\\epsilon $ yields $\\tilde{h}_1 &= \\bar{\\mathbb {V}}_{\\textnormal {loc}}^{-1} \\left(\\frac{d}{dt}\\bar{\\mathbb {I}}_{\\textnormal {loc}}\\right)\\tilde{h}_0+ \\bar{\\mathbb {V}}_{\\textnormal {loc}}^{-1}\\bar{\\mathbb {I}}_{\\textnormal {loc}}\\dot{\\tilde{h}}_0 - \\bar{\\mathbb {V}}_{\\textnormal {loc}}^{-1}\\textnormal {ad}^*_{\\accentset{\\scriptstyle \\circ }{g}}\\bar{\\mathbb {I}}_{\\textnormal {loc}}\\tilde{h}_0\\\\&= \\bar{\\mathbb {V}}_{\\textnormal {loc}}^{-1}\\frac{d}{dt}\\left(\\bar{\\mathbb {I}}_{\\textnormal {loc}}\\tilde{h}_0 \\right) - \\bar{\\mathbb {V}}_{\\textnormal {loc}}^{-1}\\textnormal {ad}^*_{\\accentset{\\scriptstyle \\circ }{g}}\\bar{\\mathbb {I}}_{\\textnormal {loc}}\\tilde{h}_0.$ Since $h_1 = \\mathbb {I}_{\\textnormal {loc}}\\tilde{h}_1$ and $\\bar{h}_0 = \\bar{\\mathbb {I}}_{\\textnormal {loc}}\\tilde{h}_0$ , we find $h_1 = \\mathbb {I}_{\\textnormal {loc}}\\bar{\\mathbb {V}}_{\\textnormal {loc}}^{-1}\\frac{d}{dt}\\left(\\bar{h}_0 \\right) - \\mathbb {I}_{\\textnormal {loc}}\\bar{\\mathbb {V}}_{\\textnormal {loc}}^{-1}\\textnormal {ad}^*_{\\accentset{\\scriptstyle \\circ }{g}}\\left(\\bar{h}_0\\right),$ and therefore (substituting $\\ddot{r} = f(t,r,\\dot{r},\\mathbb {I}_{\\textnormal {loc}}^{-1} p) = f(t,r,\\dot{r},\\tilde{h}_0) + \\mathcal {O}(\\epsilon )$ and differentiating $\\bar{h}_0(r,\\dot{r})$ via the chain rule), $\\begin{split}h_\\epsilon (t,r,\\dot{r},\\epsilon ) &= \\mathbb {I}_{\\textnormal {loc}}(A_{\\textnormal {mech}}-A_{\\textnormal {visc}})\\cdot \\dot{r} \\\\ & + \\epsilon \\mathbb {I}_{\\textnormal {loc}}\\bar{\\mathbb {V}}_{\\textnormal {loc}}^{-1} \\left(\\left(\\frac{\\partial }{\\partial r} \\bar{h}_0\\right)\\dot{r} + \\left(\\frac{\\partial }{\\partial \\dot{r}}\\bar{h}_0\\right) f(t,r,\\dot{r},\\tilde{h}_0) - \\textnormal {ad}^*_{\\accentset{\\scriptstyle \\circ }{g}}(\\bar{h}_0)\\right) + \\mathbb {I}_{\\textnormal {loc}}\\mathcal {O}(\\epsilon ^2).\\end{split}$ Notice that, since $\\tilde{h}_0$ is a function of $r,\\dot{r}$ only, the $\\mathcal {O}(\\epsilon )$ portion of the right hand side of (REF ) is a function of $t,r,\\dot{r}$ alone and not $p$ .",
"This is required since $h_\\epsilon $ is required to be a function of $t,r,\\dot{r},\\epsilon $ alone, and is the reason that we needed to replace $\\ddot{r}$ by $f(t,r,\\dot{r},\\tilde{h}_0)$ in the $\\mathcal {O}(\\epsilon )$ term.",
"Substituting (REF ) into the first equation of (REF ) yields Equation (REF ).",
"Finally, making the substitution $f(t,r,\\dot{r},\\tilde{h}_0) = \\ddot{r} + \\mathcal {O}(\\epsilon )$ in Equation (REF ) yields Equation (REF ).",
"The following theorem makes clearer the functional form of the dynamics (REF ), and it removes the $\\accentset{\\scriptstyle \\circ }{g}$ dependence of the right hand side of (REF ).",
"Theorem REF $^{\\prime }$ Assume the hypotheses of Theorem REF .",
"For sufficiently small $\\epsilon > 0$ , then for each local trivialization there exist smooth fields of linear maps $B(r)$ and $(1,2)$ tensors $G(r)$ such that the dynamics restricted to the slow manifold $M_\\epsilon $ in the local trivialization satisfy $\\accentset{\\scriptstyle \\circ }{g} = -A_{\\textnormal {visc}}(r) \\cdot \\dot{r} + \\epsilon B(r)\\cdot \\ddot{r} +\\epsilon G(r)\\cdot (\\dot{r},\\dot{r}) + \\mathcal {O}(\\epsilon ^2).$ Remark 10 The (1,2) tensors $G(r)$ are not generally symmetric, which is clear from Equation (REF ) below.",
"Using the properties of $\\textnormal {ad}^*$ , we may write $\\textnormal {ad}^*_{\\accentset{\\scriptstyle \\circ }{g}}(\\bar{h}_0) = (C\\cdot \\bar{h}_0)\\cdot (\\accentset{\\scriptstyle \\circ }{g})$ for an appropriate ($r$ -independent) linear map $C\\colon \\mathfrak {g}^* \\rightarrow \\text{End}(\\mathfrak {g})$ , and hence we may rewrite (REF ) as $(\\textnormal {id}_\\mathfrak {g}+ \\epsilon \\bar{\\mathbb {V}}_{\\textnormal {loc}}^{-1} (C\\cdot \\bar{h}_0) )\\cdot (\\accentset{\\scriptstyle \\circ }{g}) = -A_{\\textnormal {visc}}\\cdot \\dot{r} +\\epsilon \\bar{\\mathbb {V}}_{\\textnormal {loc}}^{-1} \\left(\\left(\\frac{\\partial }{\\partial r} \\bar{h}_0\\right)\\dot{r} + \\left(\\frac{\\partial }{\\partial \\dot{r}}\\bar{h}_0\\right)\\ddot{r}\\right) + \\mathcal {O}(\\epsilon ^2).$ For sufficiently small $\\epsilon $ , we may use the identity $(\\textnormal {id}_\\mathfrak {g}+ \\epsilon \\bar{\\mathbb {V}}_{\\textnormal {loc}}^{-1} (C\\cdot \\bar{h}_0) )^{-1} = \\textnormal {id}_\\mathfrak {g}- \\epsilon \\bar{\\mathbb {V}}_{\\textnormal {loc}}^{-1} (C\\cdot \\bar{h}_0) + \\mathcal {O}(\\epsilon ^2)$ to obtain $\\accentset{\\scriptstyle \\circ }{g} = -A_{\\textnormal {visc}}\\cdot \\dot{r} + \\epsilon \\bar{\\mathbb {V}}_{\\textnormal {loc}}^{-1}(C\\cdot \\bar{h}_0)\\cdot A_{\\textnormal {visc}}\\cdot \\dot{r} + \\epsilon \\bar{\\mathbb {V}}_{\\textnormal {loc}}^{-1} \\left(\\frac{\\partial }{\\partial r} \\bar{h}_0\\right)\\dot{r} + \\epsilon \\bar{\\mathbb {V}}_{\\textnormal {loc}}^{-1} \\left(\\frac{\\partial }{\\partial \\dot{r}}\\bar{h}_0\\right)\\ddot{r} + \\mathcal {O}(\\epsilon ^2).$ Since $\\bar{h}_0(r,\\dot{r})= \\bar{\\mathbb {I}}_{\\textnormal {loc}}(r)(A_{\\textnormal {mech}}(r)-A_{\\textnormal {visc}}(r))\\cdot \\dot{r}$ is linear in $\\dot{r}$ , it follows that the second and third terms are bilinear in $\\dot{r}$ , and the fourth term is linear in $\\ddot{r}$ .",
"Hence we may take $B(r)\\bar{\\mathbb {V}}_{\\textnormal {loc}}^{-1} \\left(\\frac{\\partial }{\\partial \\dot{r}}\\bar{h}_0\\right)$ and $G(r)\\cdot (\\dot{r},\\dot{r})\\bar{\\mathbb {V}}_{\\textnormal {loc}}^{-1}(C\\cdot \\mathbb {I}_{\\textnormal {loc}}(A_{\\textnormal {mech}}-A_{\\textnormal {visc}})\\cdot \\dot{r})\\cdot A_{\\textnormal {visc}}\\cdot \\dot{r} + \\epsilon \\bar{\\mathbb {V}}_{\\textnormal {loc}}^{-1} \\frac{\\partial }{\\partial r}\\left( \\mathbb {I}_{\\textnormal {loc}}(A_{\\textnormal {mech}}-A_{\\textnormal {visc}})\\cdot \\dot{r}\\right)\\cdot \\dot{r}.$"
]
] | 1906.04384 |
[
[
"Relationship-Embedded Representation Learning for Grounding Referring\n Expressions"
],
[
"Abstract Grounding referring expressions in images aims to locate the object instance in an image described by a referring expression.",
"It involves a joint understanding of natural language and image content, and is essential for a range of visual tasks related to human-computer interaction.",
"As a language-to-vision matching task, the core of this problem is to not only extract all the necessary information (i.e., objects and the relationships among them) in both the image and referring expression, but also make full use of context information to align cross-modal semantic concepts in the extracted information.",
"Unfortunately, existing work on grounding referring expressions fails to accurately extract multi-order relationships from the referring expression and associate them with the objects and their related contexts in the image.",
"In this paper, we propose a Cross-Modal Relationship Extractor (CMRE) to adaptively highlight objects and relationships (spatial and semantic relations) related to the given expression with a cross-modal attention mechanism, and represent the extracted information as a language-guided visual relation graph.",
"In addition, we propose a Gated Graph Convolutional Network (GGCN) to compute multimodal semantic contexts by fusing information from different modes and propagating multimodal information in the structured relation graph.",
"Experimental results on three common benchmark datasets show that our Cross-Modal Relationship Inference Network, which consists of CMRE and GGCN, significantly surpasses all existing state-of-the-art methods.",
"Code is available at https://github.com/sibeiyang/sgmn/tree/master/lib/cmrin_models"
],
[
"Introduction",
"A fundamental capability of AI for bridging humans and machines in the physical world is comprehending natural language utterances and their relationship with visual information.",
"This capability is required by many challenging tasks, among which, grounding referring expressions [2], [3] is an essential one.",
"The task of grounding referring expressions needs to locate a target visual object in an image by understanding multimodal semantic concepts as well as relationships between referring natural language expressions (e.g.",
"“the man with sun glasses”, “the dog near a white car”) and the image content.",
"Identifying the object proposal referred to by an expression from a set of proposals in an image is a typical formulation of grounding referring expressions [4].",
"Recent methods adopt the combination of Convolutional Neural Networks (CNN) [5] and Long Short-Term Memory Neural Networks (LSTM) [6] to process multimodal information in images and referring expressions.",
"CNNs extract visual features of single objects, global visual contexts [3], [7] and pairwise visual differences [8], [4], [9], [10] while LSTMs encode global language contexts [8], [11], [3], [9], [10] and language features of decomposed phrases [12], [4], [13].",
"In addition, the cooperation between CNNs and LSTMs captures the context of object pairs [12], [14], [13].",
"However, existing work cannot extract all the required information (i.e.",
"individual objects; first-order relationships or multi-order relationships) accurately from referring expressions and the captured contexts in such work also have discrepancies with the contexts described by referring expressions.",
"In this paper, we refer to [13] and define the “context” as the objects as well as their attributes and relationships mentioned in the expression that help distinguish a referred object from other objects.",
"Figure: Cross-Modal Relationship Inference Network.",
"Given an expression and image, Cross-Modal Relationship Extractor constructs the language-guided visual relation graphs (spatial relation graph as an example, the attention scores of proposals and edges' types are visualized inside green dashed box).",
"The Gated Graph Convolutional Network capture semantic context and computes the matching scores between context of proposals and context of expression (the matching scores of proposals are shown inside blue dashed box).",
"Warmer color indicates higher scores of pixels and darker blue indicates higher scores of edges' types.To solve the problem of grounding referring expressions, the accurate extraction of all required information (i.e.",
"objects and the relationships among them in the image and referring expressions) is crucial for any given pair of expression and image.",
"Because of the unpredictability and flexibility of an expression describing the scene in an image [3], the proposed model needs to extract the information adaptively.",
"For example, if “The man holding a red balloon” is located in an image with two or more men, the nouns/noun phrases (“man” and “red balloon”) and the relation word “holding” need to be extracted from the natural language expression; meanwhile, proposals for “man” and “red balloon” and the visual relationship (`holding”) linking them together should be identified in the image.",
"“The parking meter on the left of the man holding a red balloon” is a more complicated example, which involves an additional object “parking meter” and additional relational information “left”.",
"In this example, on one hand, there are three individual objects (i.e.",
"“man”, “red balloon” and “parking meter”), that need to be recognized in both the image and expression.",
"Object proposals in the image can be either obtained with an object detector [15] or provided as part of the dataset [9], [3].",
"Nouns and noun phrases in the expression need to be extracted, and words in the same phrase should refer to the same object.",
"Unfortunately, existing methods only consider individual words and softly parsed phrases [12], [4], [13], but words in the same softly parsed phrase cannot be constrained to the same object.",
"On the other hand, the second-order relationship between the target and the “red balloon” via the “man” need to be inferred from either the detected direct semantic relationship “holding” or spatial relationship between object pairs “on the left of”.",
"Unfortunately, existing work either does not support relationship modeling or only considers first-order relationships among objects [12], [14], [13].",
"Theoretically, visual relation detectors [16], [17], [18] and natural language parsers can help achieve that goal by detecting relational information in the image and parsing grammatical relations among the words in the expression.",
"However, existing visual relation detectors, which focus only on the extraction of semantic relationship, cannot deliver satisfactory and sufficient clues for highly unrestricted scene compositions [13], and existing language parsers have adverse effects on performance for grounding referring expressions due to their parsing errors [4], [13].",
"Moreover, the target object is distinguished from other objects on the basis of their contexts and the context of the expression [14], [9], [13]; therefore, accurate and consistent representation of contextual information in the referring expression and object proposals is essential.",
"Nevertheless, existing methods for context modeling either cannot represent the context accurately or cannot achieve high-level consistency between both types of contexts mentioned above, and the reasons are given below.",
"First, noisy information introduced by existing work on global language context modeling [8], [11], [3], [9], [10] and global visual context modeling [3], [7] makes it hard to align and match these two types of contexts.",
"Second, pairwise visual differences computed in existing work [8], [4], [9], [10] can only represent instance-level visual differences among objects of the same category.",
"Third, existing work on context modeling for object pairs [12], [14], [13] only considers first-order relationships instead of multi-order relationships (e.g., they directly extract the relationship between the pairs of (target, “man”) and (target, “balloon”) without considering the “man” is “holding the balloon” when extracting the relationship between the target “parking meter” and “the man”).",
"In addition, multi-order relationships are actually structured information, which cannot be modeled by the context encoders adopted by existing work on grounding referring expressions.",
"Given the limitations of existing methods, our proposed end-to-end Cross-Modal Relationship Inference Network (CMRIN) aims to overcome the aforementioned difficulties.",
"CMRIN consists of two modules, i.e., the Cross-Modal Relationship Extractor (CMRE) and the Gated Graph Convolutional Network (GGCN).",
"An example is illustrated in Fig.",
"REF .",
"The CMRE extracts all the required information adaptively (i.e., nouns/noun phrases and relationship words from the expressions, and object proposals and their visual relationships from the image) for constructing a language-guided visual relation graph with cross-modal attention.",
"First, CMRE constructs two scene graphs (a spatial relation graph as well as a semantic relation graph) for the image.",
"Second, it extracts noun phrases in the expression using a constituency tree, meanwhile, it learns to classify the words in the expression into four types and further assign the words/phrases to the vertices and edges in each scene graph.",
"Finally, it constructs the language-guided visual relation graph from the normalized attention distribution of words/phrases over vertices and edges of each scene graph.",
"The GGCN fuses information from different modes and propagates the fused information in the language-guided visual relation graph to obtain semantic contexts of the expression by performing the following two steps.",
"First, it fuses the contexts in the expression into the visual relation graph to form a multimodal relation graph, which includes the spatial/semantic relationships, visual information and language contexts; Second, gated graph convolutional operations are applied to the multimodal relation graph to obtain the semantic contexts.",
"We have tested our proposed CMRIN on three common benchmark datasets, including RefCOCO [9], RefCOCO+ [9] and RefCOCOg [3], for grounding referring expressions.",
"Experimental results show that our proposed network outperforms all other state-of-the-art methods.",
"In summary, this paper has the following contributions: Cross-Modal Relationship Extractor (CMRE) is proposed to convert the pair of input expression and image into a language-guided visual relation graph.",
"For any given pair of expression and image, CMRE highlights objects as well as spatial and semantic relationships among them with a cross-modal attention mechanism by considering the words and phrases in the expression as guidance.",
"Gated Graph Convolutional Network (GGCN) is proposed to capture multimodal semantic context with multi-order relationships.",
"GGCN fuses information from different modes and propagates fused information in the language-guided visual relation graph.",
"CMRE and GGCN are integrated into Cross-Modal Relationship Inference Network (CMRIN), which outperforms all existing state-of-the-art methods on grounding referring expressions using the ground-truth proposals.",
"In addition, CMRIN shows its robustness using the detected proposals.",
"This paper is an extended version of [1], it provides a more complete introduction and analysis to the proposed cross-modal relationship inference network for referring expression comprehension, providing additional insights and relevant research discussion, verification of the effectiveness of framework components, network parameter analysis and more elaborated experimental comparisons.",
"Furthermore, we propose to add phrase parsing for the expression and apply it to enhance the representation of language-guided visual relation graphs, which helps to better align linguistic words with visual objects.",
"Second, to complement the spatial relation graph, we have also extracted semantic relations and use them as another guidance in edge gate computation for multi-order relationship inference in our proposed GGCN.",
"Experimental results show that by introducing phrase decomposition for referring expressions and semantic relationship modeling for images, it can bring different levels of performance improvement and make the algorithm more complete and robust.",
"Grounding referring expression and referring expression generation [3] are dual tasks.",
"The latter is to generate an unambiguous text expression for a target object in an image, and the former selects the corresponding object according to the image content referred by a text expression.",
"To address the problem of grounding referring expression, some previous work [8], [11], [3], [10], [9], [19] extracts visual object features from CNN and treat an expression as a whole to encode language feature through an LSTM.",
"Among them, some methods [11], [3], [9] learn to maximize the posterior probability of the target object given the expression and the image, and the others [8], [10] model the joint probability of the target object and the expression directly.",
"Specifically, MMI [3] applies the same CNN-LSTM network architecture for grounding referring expressions and referring expression generation respectively, and jointly optimize those two parts together.",
"Speaker [9] improves MMI by taking more consideration between the comparisons on objects of the same type in the image, and it encodes visual appearance differences and relative spatial (i.e.",
"location and size) differences between object and surrounding objects of the same object category.",
"Speaker-Listener-Reinforcer [10] proposes an Reinforcer module to sample more discriminative expressions for helping the training of the Speaker.",
"Attr [8] suggests that the attributes of objects help to distinguish the target object from other ones, and it learns the attributes of objects and encodes the features from the learned attributes and visual features.",
"A-ATT [19] adopts joint attention mechanism on query, image and objects multiply round to obtain the communication among the three different types of information.",
"However, all of the above methods independently encode the images and expressions without considering the interactions between them, and the learned monolithic representations in the two modes are not practical to the semantic-rich visual scenes and complex expressions.",
"Different from the methods above, Neg Bag [14] proposes to feed the concatenation of visual object representation, visual context representation and the word embedding to an LSTM model.",
"Recent methods [12], [4], [13] learn to decompose an expression into different components and compute the language-vision matching scores of each module for objects.",
"Specially, CMN [12] learns to parse the expression into a fixed form of subject-object-relationship; MAttNet [4] decomposes the expression into subject, location and relationship modules, and the module weights are computed for combining those three modules.",
"VC [13] obtains the context-cue language features and referent-cue language features for both single objects and pairwise objects.",
"However, all of the existing works are based on simple expression decomposition and match directly with the detected object features and the additionally computed relationship features [16], [17], [18], without considering the cross-modal alignment of multi-order relationship among objects and attributes, they are therefore arduous to adapt to the referring of objects in highly unrestricted scenes.",
"Our Cross-Modal Relationship Extractor also learns to parse the expression, but we treat the parsed words as the guidance to highlight all the objects and their relationships described in the expression automatically to build the language-guided visual relation graphs which are further enhanced by a tailor-designed gated graph neural network for cross-modal multi-order context reasoning and alignment."
],
[
"Context Modeling",
"Context modeling has been applied in many visual recognition tasks, e.g., object detection [20], [21], [22], semantic segmentation [23], [24] and saliency detection [25], [26].",
"For example, ION [20] uses four directional Recurrent Neural Networks (RNNs) to compute the context features on feature maps from four spatial directions.",
"Ren et al.",
"[21] propose Recurrent Rolling Convolution architecture to gradually aggregate context among the feature maps with different resolutions.",
"Context Encoding Module [24] encodes the global semantic context by learning an inherent codebook which is a set of visual centers.",
"Recently, Structure Inference Network [27] formulates the context modeling task as a graph structure inference problem [28], [29], [30], and it obtains the scene context by applying RNN to proposals in image.",
"As contextual information helps to distinguish the target from other objects, previous work on grounding referring expressions has also attempted to captured the context.",
"For example, some early works [3], [7] propose to encode the entire image as a visual context, but that global contextual information usually cannot accurately match with the local context described by the expression.",
"Other works [8], [4], [9], [10] capture the visual difference between the objects belonging to the same category in an image, but the visual difference of the object's appearance is often insufficient to distinguish the target from other objects.",
"In fact, the visual difference between the context including appearance and relationship is essential, e.g., “Man holding a balloon”, the necessary information to locate the “man” is not only the appearance of the “man” but the “holding” relation with the “balloon”.",
"There are also some works [12], [14], [13] which model the context from the context of object pairs, but they only consider the context with the first-order relationship between the objects.",
"Inspired by Graph Convolutional Network [29] for classification, our Gated Graph Convolutional Network flexibly capture the context referring to the expression by message passing, and the context with multi-order relationships can be captured.",
"Figure: An overview of our Cross-Modal Relationship Inference Network for grounding referring expressions (better view in color).",
"We use color to represent semantics, i.e.",
"yellow denotes “person”, green denotes “green shirt”, blue denotes “umbrella”, purple means “white T-shirt”, brown means “wearing” and dark grey refers to “held by”.",
"It includes a Cross-Modal Relationship Extractor (CMRE) and a Gated Graph Convolutional Network (GGCN).",
"First, CMRE constructs (a) a spatial relation graph from the visual features of object proposals and spatial relationships between proposals.",
"Second, CMRE parses the expression into a constituency tree and extracts the valid noun phrases.",
"Third, CMRE highlights the vertices (red bounding boxes) and edges (solid lines) to generate (b) a language-guided visual relation graph using cross-modal attention between words/phrases in the referring expression and the spatial relation graph's vertices and edges.",
"Fourth, GGCN fuses the context of words into the language-guided visual relation graph to obtain (c) a multimodal (language, visual and spatial information) relation graph.",
"Fifth, GGCN captures (d) the multimodal semantic context with first-order relationships by performing gated graph convolutional operations in the relation graph.",
"By performing gated graph convolutional operations multiple iterations, (e) semantic context with multi-order relationships can be computed.",
"Finally, CMRIN calculates the matching scores between semantic context of proposals and the global context of the referring expression.",
"The triplet loss with online hard negative mining is adopted during training and the proposal with the highest matching score is chosen."
],
[
"Vision-Language",
"The combination of language and vision has been extensively studied in the last few years due to its significance for building AI systems.",
"Besides grounding referring expressions, image/video captioning [31], [32] and visual question answering [33] are two popular and fundamental tasks.",
"Image caption is to generate image-relevant textual descriptions for given images.",
"Early approaches [31], [34] extract visual concepts (i.e., objects and attributes) from images and format the sentences from those visual concepts and templates.",
"Recently, some work [35], [36] starts to encode the image as visual representations (e.g., single visual representation for the whole image [35] and a set of visual representations for different sub-regions of the image [36]) by applying CNN, and then decode the visual representations into language descriptions through LSTM.",
"The attention mechanism is adopted to attend the most relevant part of visual information [36] of it with the already generated text [37] in every time step of LSTM.",
"Some of the recent approaches [38], [39] use the additional information (e.g.",
"phrases and semantic words) extracted from image or text to help generate high quality sentences.",
"There are also some works which focus on the description of a specified object in an image, a.k.a referring expression generation [3], which is a dual problem of visual grounding.",
"It can be applied as an auxiliary component to referring expression comprehension to enhance the performance of cross-modal matching by computing the semantic distance between the generated statement and the given expression [3].",
"However, as the description of the object is varied and involves complex context information and relationships with other objects, the performance improvement for referring expression comprehension of unrestricted complex scenes is limited.",
"Visual question answering is to correctly infer the answer for a given pair of image and textual question.",
"Most of the existing work [40], [41], [42], [43], [44], [45], [46] extracts the visual features from the image through CNN and encodes the question to language representation by passing the question into LSTM.",
"And then, the answer is predicted by cooperation between those two types of representations.",
"The cooperation is implemented by approaches, like learning a common embedding space for visual and language representations [40], [41], [42], or attending the most discriminate regions of image by applying different attention mechanisms on both representations [43], [44], [45], [46], or using both of them together.",
"Besides the direct prediction of the answers, interpreting the reasoning procedure is important as well.",
"The reasoning procedures are modeled from three different perspectives (i.e.",
"relation-based modeling [47], [48], attention-based modeling [42], [49] and module-based modeling [50], [51]).",
"Although visual question answering and referring expression comprehension have different problem definitions and solving goals, visual grounding is the key to endowing VQA with interpretability, which helps to ground their answers to relevant regions in the image [52], [51].",
"On the other hand, cross-modal feature fusion and semantics reasoning are equally effective and important for both issues.",
"The study of the two problems can be integrated and learned from each other [42], [53]."
],
[
"Graph Neural Networks",
"blackGraph Neural Networks (GNNs) which are widely used to model the relational dependencies among elements of a graph through message passing [54], [29], [55], have been successfully applied to various context-aware visual tasks, e.g., semi-supervised classification [29], zero-shot recognition[56] and object detection [27].",
"blackGraph-structured representations and GNNs have also been introduced to the tasks of language and vision understanding.",
"The methods in [57], [58], [59], [60] for VQA and image captioning represent an image as a graph structure where the vertices represent visual regions of an image and the edges are relationships among them, and then capture the visual context of each region node by GNN propagation.",
"Specifically, [57] and [60] encode the contextual features at vertices by using the graph networks based on the recurrent unit [61] and the graph convolutional network (GCN) respectively.",
"Their graph networks operate in the modes of vision and language independently.",
"Different with them, our graph network performs on the top of multi-modal graph to learn the language-guided contexts at vertices.",
"The recent works, [58] and [59], also obtain the convolved graph representations over the language-conditioned graphs: the former identifies neighbors for a vertex as its K most similar vertices and update feature at the vertex as sum of the learned features of its neighbors weighted by the learned weighting factors in each convolution layer, and the latter considers relationships between any pairs of vertices and aggregates the relational features for vertices using max pooling operator.",
"Different with the above methods, we define gates of vertices and edges to implement the different influences of neighbors and relationships, and the gates are learned globally.",
"To the best of our knowledge, we are the first to incorporate the graph convolutional networks in referring expressions comprehension for multi-order relationships representation learning."
],
[
"Cross-Modal Relationship Inference Network",
"Our proposed Cross-Modal Relationship Inference Network (CMRIN) relies on relationships among objects and context captured in the multimodal relation graph to choose the target object proposal in the input image referred to by the input expression.",
"First, CMRIN constructs a language-guided visual relation graph using the Cross-Modal Relationship Extractor.",
"Second, it captures multimodal context from the relation graph based on the Gated Graph Convolutional Network.",
"Finally, a matching score is computed for each object proposal according to its multimodal context and the context of the input expression.",
"The overall architecture of our CMRIN for grounding referring expressions is illustrated in Fig.",
"REF .",
"In the rest of this section, we elaborate all the modules in this network."
],
[
"Cross-Modal Relationship Extractor",
"The Cross-Modal Relationship Extractor (CMRE) adaptively constructs the language-guided visual relation graph according to each given pair of image and expression using a cross-modal attention mechanism.",
"Our CMRE considers both the word level and the phrase level.",
"At the word level, it softly classify the words in the expression into four types (i.e., entity, relation, absolute location, and unnecessary words) according to the context of the words.",
"At the phrase level, it extracts noun phrases, which are directly taken as entity phrases.",
"Meanwhile, the context of the entire expression can be computed from the context of each individual word.",
"In addition, a spatial relation graph of the image is constructed by linking object proposals in the image according to their size and locations and a semantic relation graph is constructed by an off-the-shelf object relationship detector [62].",
"Next, CMRE generates the language-guided visual relation graph by highlighting the vertices and edges of the relation graphs.",
"Highlighting is implemented as computing cross-modal attention between the words/phrases in the expression and the vertices and edges in the relation graphs.",
"blackExploring spatial relations and semantic relations among object proposals within an image is necessary for grounding referring expressions, because they are frequently occurs in referring expressions.",
"Thus, we construct two different graphs by exploring two different types of relationships, i.e., spatial relation graph and semantic relation graph.",
"blackFor spatial relation graph, we obtain the spatial relationship between each pair of object proposals according to their size and locations, which bears resemblance to the approach in [63].",
"For a given image $I$ with $K$ object proposals (bounding boxes), $O = \\lbrace o_i\\rbrace _{i=1}^{K}$ , the location of each proposal $o_i$ is denoted as $loc_i = (x_i, y_i, w_i, h_i)$ , where $(x_i, y_i)$ are the normalized coordinates of the center of proposal $o_i$ , and $w_i$ and $h_i$ are the normalized width and height.",
"The spatial feature $\\mathbf {p}_i$ is defined as $\\mathbf {p}_i = [x_i, y_i, w_i, h_i, w_ih_i]$ .",
"For any pair of proposals $o_i$ and $o_j$ , the spatial relationship $r_{ij}$ between them is defined as follows.",
"We compute the relative distance $d_{ij}$ , relative angle $\\theta _{ij}$ (i.e.",
"the angle between the horizontal axis and vector $(x_i-x_j, y_i-y_j)$ ) and Intersection over Union $u_{ij}$ between them.",
"If $o_i$ includes $o_j$ , $r_{ij}$ is set to “inside”; if $o_i$ is covered by $o_j$ , $r_{ij}$ is set to “cover”; if none of the above two cases is true and $u_{ij}$ is larger than $0.5$ , $r_{ij}$ is set to “overlap”; otherwise, when the ratio between $d_{ij}$ and the diagonal length of the image is larger than $0.5$ , $r_{ij}$ is set to “no relationship”.",
"In the rest of the cases, $r_{ij}$ is assigned to one of the following spatial relationships, “right”, “top right”, “top”, “top left”, “left”, “bottom left”, “bottom” and “bottom right”, according to the relative angle $\\theta _{ij}$ .",
"The details are shown in Fig.",
"REF .",
"Figure: All types of spatial relationships between proposal o i o_i (green box) and proposal o j o_j (blue box).",
"The number following the relationship is the label.The directed spatial relation graph $G^s=(V, E, \\mathbf {X}^s)$ is constructed from the set of object proposals $O$ and the set of pairwise relationships $R = \\lbrace r_{ij}\\rbrace _{i,j=1}^{K}$ , where $V = \\lbrace v_i\\rbrace _{i=1}^{K}$ is the set of vertices and vertex $v_i$ corresponds to proposal $o_i$ ; $E = \\lbrace e_{ij}\\rbrace _{i,j=1}^{K}$ is the set of edges and $e_{ij}$ is the index label of relationship $r_{ij}$ ; $\\mathbf {X}^s = \\lbrace \\mathbf {x}_i^s\\rbrace _{i=1}^K$ is the set of features at vertices and $\\mathbf {x}^s_i \\in \\mathbb {R}^{D_x}$ is the visual feature of proposal $o_i$ , where $D_x$ is the dimension of visual feature.",
"$\\mathbf {x}^s_i$ is extracted using a pretrained CNN model.",
"A valid index label of $E$ ranges from 1 to $N_e = 11$ (the label of “no relationship” is 0).",
"blackSimilar to the spatial relation graph $G^s = (V, E, \\mathbf {X}^s)$ , the semantic relation graph $G^{sem} = (V, \\mathbf {E}^{sem}, \\mathbf {X}^s)$ shares the same sets of vertices and features at vertices as $G^s$ , but instead the set of edges $\\mathbf {E}^{sem}$ is extracted by a pretrained object relationship detector [62].",
"blackThe spatial relation graph $G^s$ and semantic relation graph $G^{sem}$ , which are constructed from the image, involves the visual features of the proposals as well as their spatial relationships or semantic relationships.",
"They are further transformed into the language-guided visual relation graph based on the guidance from the expression, which will be detailed in Section REF .",
"blackTo simplify the description and focus on the pipeline design of the proposed method, we adopt the $G^s$ as the example for remaining part in Section .",
"And the detail implementation for the semantic branch will be described in Section REF ."
],
[
"Phrase",
"Parsing the phrases in the expression is paramount as it helps to accurately highlight the vertices of graph $G^s$ referred to by the expression.",
"For example, if the expression is “the umbrella held by a lady wearing a green skirt”, it is necessary to recognize the noun phrase (i.e.",
"“green skirt”), and the words in this phrase refer to the same vertex.",
"We only extract noun phrases in this paper since they are most relevant to the objects in the image.",
"Specifically, Our CMRE follows the three steps below to extract the noun phrases.",
"First, it parses the given expression into a constituency tree, which breaks the expression into sub-phrases.",
"Second, it locates candidate noun phrases (i.e.",
"“the umbrella”, “a lady” and “a green skirt”) from the leaves to the root.",
"On each path from the leaves to the root, it extracts the first noun phrase and ignores the other noun phrases.",
"Third, it eliminates determiners and words indicating absolute location in the extracted noun phrase candidates.",
"A candidate phrase is valid if the number of remaining words is at least two.",
"Thus, “green shirt” is a valid noun phrase.",
"For a given expression $L = \\lbrace l_t\\rbrace _{t=1}^T$ ($T$ is the number of words), we denote the set of extracted noun phrases as $Q = \\lbrace q_m\\rbrace _{m=1}^{M}$ , where $M$ is the number of phrases."
],
[
"Language Representation",
"Inspired by the attention weighted sum of word vectors over different modules in [12], [13], [4], our CMRE defines attention distributions of words/phrases over the vertices and edges of the spatial relation graph $G^s$ .",
"In addition, different words in a referring expression may play different roles.",
"For referring expressions, words can usually be classified into four types (i.e entity, relation, absolute location and unnecessary words), and the type for noun phrases is entity.",
"By parsing the expression into different types of words and distributing words/phrases over the vertices and edges of graph $G^s$ , the language embedding of every vertex and edge can be captured, and the global language context can also be obtained.",
"Given an expression $L = \\lbrace l_t\\rbrace _{t=1}^T$ , CMRE first learns a $D_f$ -dimensional embedding for each word, $\\mathbf {F}^l = \\lbrace \\mathbf {f}^l_t \\in R^{D_f}\\rbrace _{t=1}^{T}$ , and then applies a bi-directional LSTM [64] to encode the context of words.",
"The context of word $l_t$ is the concatenation of its forward and backward hidden vectors, denoted as $\\mathbf {h}^l_t \\in \\mathbb {R}^{D_h}$ .",
"The weight $\\mathbf {m}_t$ of each type (i.e.",
"entity, relation, absolute location and unnecessary word) for word $l_t$ is defined as follows.",
"$\\mathbf {m}_t = \\text{softmax}( \\mathbf {W}_{l1}\\sigma (\\mathbf {W}_{l0}\\mathbf {h}^l_t + \\mathbf {b}_{l0}) + \\mathbf {b}_{l1}),$ where $\\mathbf {W}_{l0} \\in \\mathbb {R}^{D_{l0} \\times D_{h}}$ , $\\mathbf {b}_{l0} \\in \\mathbb {R}^{D_{l0} \\times 1}$ , $\\mathbf {W}_{l1} \\in \\mathbb {R}^{4 \\times D_{l0}}$ and $\\mathbf {b}_{l1} \\in \\mathbb {R}^{4 \\times 1}$ are learnable parameters, $D_{l0}$ and $D_h$ are hyper-parameters and $\\sigma $ is the activation function.",
"The feature vector of a phrase is computed as the mean embedding feature (context) of words appearing in the phrase.",
"The set of features for all phrases in the expression is denoted as $\\mathbf {F}^q = \\lbrace \\mathbf {f}^q_m\\rbrace _{m=1}^M$ (contextual embeddings $\\mathbf {H}^q = \\lbrace \\mathbf {h}^q_m \\in \\mathbb {R}^{D_h}\\rbrace _{m=1}^M$ ).",
"Next, CMRIN computes the language context of every vertex in graph $G^s$ from both words and phrases.",
"When words are considered, on the basis of the word embedding $\\mathbf {F}^l = \\lbrace \\mathbf {f}^l_t\\rbrace _{t=1}^T$ and the entity weights of words $\\lbrace \\mathbf {m}_t^{(0)}\\rbrace _{t=1}^T$ , a weighted normalized attention distribution over the vertices of graph $G^s$ is defined as follows.",
"$\\begin{aligned}\\alpha ^{l}_{t,i} &= \\mathbf {W}^{l}_n[\\text{tanh}(\\mathbf {W}^{l}_{v}\\mathbf {x}_i^s + \\mathbf {W}^{l}_{f}\\mathbf {f}^{l}_t)], \\\\\\lambda ^l_{t,i} &= \\mathbf {m}_t^{(0)}\\frac{\\text{exp}(\\alpha ^l_{t,i})}{\\sum _i^K {\\text{exp}(\\alpha ^l_{t,i})}},\\end{aligned}$ where $\\mathbf {W}^l_{n} \\in \\mathbb {R}^{1 \\times D_{n}}$ , $\\mathbf {W}^l_{v} \\in \\mathbb {R}^{D_{n} \\times D_{x}}$ and $\\mathbf {W}^l_{f} \\in \\mathbb {R}^{D_{n} \\times D_{h}}$ are transformation matrices and $D_n$ is hyper-parameter.",
"$\\lambda _{t,i}$ is the weighted normalized attention, indicating the probability that word $l_t$ refers to vertex $v_i$ .",
"Likewise, CMRIN computes $\\alpha ^{q}_{m,i}$ for the phrases $Q=\\lbrace q\\rbrace _i^M$ on the basis of their features $\\mathbf {F}^q = \\lbrace \\mathbf {f}^q_m\\rbrace _{m=1}^M$ , and the normalized distribution over the vertices are computed as follows.",
"$\\begin{aligned}\\lambda ^q_{m,i} &= \\frac{\\text{exp}(\\alpha ^q_{m,i})}{\\sum _i^K {\\text{exp}(\\alpha ^q_{m,i})}},\\end{aligned}$ The language context $\\mathbf {c}_i$ at vertex $v_i$ is computed by aggregating all attention weighted word contexts and phrase contexts.",
"$\\mathbf {h}_i = \\frac{\\sum _{t=1}^{T}\\lambda ^l_{t,i}\\mathbf {h}^l_t + \\sum _{m=1}^{M}\\lambda ^q_{m,i}\\mathbf {h}^q_m}{\\sum _{t=1}^{T}\\lambda ^l_{t,i}+\\sum _{m=1}^{M}\\lambda ^q_{m,i}}$ Then, the global language context $\\mathbf {h}_g$ of graph $G^s$ is calculated as follows.",
"$\\mathbf {h}_g = \\sum _{t=0}^{T}(\\mathbf {m}_t^{(0)} + \\mathbf {m}_t^{(1)} + \\mathbf {m}_t^{(2)})\\mathbf {h}_t^{l}$ where the entity weight, relation weight and absolute location weight are the first three elements of $\\mathbf {m}_t$ .",
"CMRIN computes the global context only from word contexts because phrases are only used for improving the accuracy of vertex highlighting in the relation graphs."
],
[
"Language-Guided Visual Relation Graph",
"Different object proposals and different relationships between proposals do not have equal contributions in solving grounding referring expressions.",
"The proposals and relationships mentioned in the referring expression should be given more attention.",
"Our CMRE highlights the vertices and edges of the spatial relation graph $G^s$ , that have connections with the referring expression, to generate the language-guided visual relation graph $G^v$ .",
"The highlighting operation is implemented by designing a gate for each vertex and edge in graph $G^s$ .",
"The gate $p^{v}_i$ for vertex $v_i$ is defined as the sum over the weighted probabilities that individual words and phrases in the expression refer to vertex $v_i$ , $p^{v}_i = \\sum _{t=1}^{T}\\lambda ^{l}_{t,i} + \\sum _{m=1}^{M}\\lambda ^{q}_{m,i}$ Each edge has its own type and the gates for edges are formulated as the gates for edges' types.",
"The weighted normalized distribution of words over the edges of graph $G^s$ is defined as follows.",
"$\\mathbf {w}^{e}_{t} = \\text{softmax}( \\mathbf {W}_{e1}\\sigma (\\mathbf {W}_{e0}\\mathbf {h}^{l}_t + \\mathbf {b}_{e0}) + \\mathbf {b}_{e1})\\mathbf {m}_t^{(1)},$ where $\\mathbf {W}_{e0} \\in \\mathbb {R}^{D_{e0} \\times D_{h}}$ , $\\mathbf {b}_{e0} \\in \\mathbb {R}^{D_{e0} \\times 1}$ , $\\mathbf {W}_{e1} \\in \\mathbb {R}^{N_e \\times D_{e0}}$ and $\\mathbf {b}_{e1} \\in \\mathbb {R}^{N_e \\times 1}$ are learnable parameters, and $D_{e0}$ is hyper-parameter.",
"$w_{t,j}^{e}$ is the $j$ -th element of ${\\mathbf {w}^{e}_t}$ , which is the weighted probability of word $l_t$ referring to edge type $j$ .",
"And the gate $p^{e}_j$ for edges with type $j \\in \\lbrace 1,2,..N^e\\rbrace $ is the sum over all the weighted probabilities that individual words in the expression refer to edge type $j$ , $p^{e}_j = \\sum _{t=1}^{T}w_{t,j}^{e}.$ The language-guided visual relation graph is defined as $G^v = (V, E, \\mathbf {X}, P^{v}, P^{e})$ , where $P^{v} = \\lbrace p^{v}_i\\rbrace _{i=1}^K$ , and $P^{e} = \\lbrace p^{e}_j\\rbrace _{j=1}^{N_e}$ ."
],
[
"Multimodal Context Modeling",
"Our proposed Gated Graph Convolutional Network (GGCN) further fuses the language context into the language-guided visual relation graph to generate multimodal relation graph $G^m$ , and computes a multimodal semantic context for every vertex by performing gated graph convolutional operations on the graph $G^m$ ."
],
[
"Language-Vision Feature",
"As suggested by visual relationships detection [16], [18], the spatial locations together with the appearance features of objects are the key indicators of visual relationship, and the categories of objects is highly predictive of relationship.",
"Our GGCN fuses the language context of vertices into the language-guided visual relation graph $G^v$ ($G^v$ encodes the spatial relationships and appearance features of proposals) to generate multimodal relation graph $G^m$ , which forms the basis for computing the semantic context of vertices.",
"We define feature $\\mathbf {x}_i^m$ at vertex $v_i$ in $G^m$ to be the concatenation of the visual feature $\\mathbf {x}_i^s$ at vertex $v_i$ in the language-guided visual relation graph and the language context $\\mathbf {h}_i$ at vertex $v_i$ , i.e.",
"$\\mathbf {x}_i^m = [\\mathbf {x}_i^s, \\mathbf {h}_i]$ .",
"The multimodal graph is defined as $G^m = (V, E, \\mathbf {X}^m, P^{v}, P^{e})$ , where $\\mathbf {X}^m = \\lbrace \\mathbf {x}^m_i\\rbrace _{i=1}^{K}$ ."
],
[
"Semantic Context Modeling",
"Multi-order relationships may exist in referring expressions.",
"We obtain semantic context representing multi-order relationships through message passing.",
"On one hand, semantic features are obtained by learning to fuse the spatial relations, visual features and language features.",
"On the other hand, context representing multi-order relationships is computed by propagating pairwise context in graph $G^m$ .",
"Inspired by Graph Convolutional Network (GCN) for classification [29], [56], our GGCN adopts graph convolutional operations in multimodal relation graph $G^m$ for computing semantic context.",
"Different from GCN operating in unweighted graphs, GGCN operates in weighted directed graphs with extra gate operations.",
"The $n$ -th gated graph convolution operation at vertex $v_i$ in graph $G^m = (V, E, \\mathbf {X}^m, P^{v}, P^{e})$ is defined as follows.",
"$\\begin{aligned}\\overrightarrow{\\mathbf {x}_i}^{(n)} &= \\sum _{e_{i,j} > 0} {p^{e}_{e_{i,j}}} (\\overrightarrow{\\mathbf {W}}^{(n)}\\hat{\\mathbf {x}}_j^{(n-1)} p^{v}_j + \\mathbf {b}^{(n)}_{e_{i,j}}), \\\\\\overleftarrow{\\mathbf {x}_i}^{(n)} &= \\sum _{e_{j,i} > 0} {p^{e}_{e_{j,i}}} (\\overleftarrow{\\mathbf {W}}^{(n)}\\hat{\\mathbf {x}}_j^{(n-1)} p^{v}_j + \\mathbf {b}^{(n)}_{e_{j,i}}), \\\\\\tilde{\\mathbf {x}}_i^{(n)} &= \\widetilde{\\mathbf {W}}^{(n)}\\hat{\\mathbf {x}}^{(n-1)}_i + \\widetilde{\\mathbf {b}}^{(n)}, \\\\\\hat{\\mathbf {x}}^{(n)}_i & = \\sigma (\\overrightarrow{\\mathbf {x}_i}^{(n)} + \\overleftarrow{\\mathbf {x}_i}^{(n)} + \\tilde{\\mathbf {x}}_i^{(n)}),\\end{aligned}$ where $\\hat{\\mathbf {x}}^{(0)}_i = \\mathbf {x}_i^m$ , $\\overrightarrow{\\mathbf {W}}^{(n)}, \\overleftarrow{\\mathbf {W}}^{(n)}, \\widetilde{\\mathbf {W}}^{(n)} \\in \\mathbb {R}^{D_e \\times (D_x + D_h)}$ $\\lbrace \\mathbf {b}_j^{(n)}\\rbrace _{j=1}^{N_e}, \\widetilde{\\mathbf {b}}^{(n)} \\in \\mathbb {R}^{D_e \\times 1}$ are learnable parameters, and $D_e$ is hyper-parameter.",
"$\\overrightarrow{\\mathbf {x}_i}^{(n)}$ and $\\overleftarrow{\\mathbf {x}_i}^{(n)}$ are encoded features for out- and in- relationships respectively.",
"$\\tilde{\\mathbf {x}}_i^{(n)}$ is the updated feature for itself.",
"The final encoded feature $\\hat{\\mathbf {x}}_i^{(n)}$ is the sum of the above three features and $\\sigma $ is the activation function.",
"By performing the gated graph convolution operation multiple iterations ($N$ ), semantic context representing multi-order relationships among vertices can be computed.",
"Such semantic context are denoted as $\\mathbf {X}^c = \\lbrace \\mathbf {x}^{c}_{i} = \\hat{\\mathbf {x}}^{(N)}_i\\rbrace _{i=1}^K$ .",
"Finally, for each vertex $v_i$ , we concatenate its encoded spatial feature $\\mathbf {p}_i$ mentioned before and its language-guided semantic context $\\mathbf {x}_i^c$ to obtain the multimodal context $\\mathbf {x}_i = [\\mathbf {W}_p\\mathbf {p}_i, \\mathbf {x}_i^c]$ , where $\\mathbf {W}_p \\in \\mathbb {R}^{D_p \\times 5}$ and $D_p$ is hyper-parameter."
],
[
"Loss Function",
"The matching score between proposal $o_i$ and expression $L$ is defined as follows, $s_i = \\text{L2Norm}(\\mathbf {W}_{s0}\\mathbf {x}_i) \\odot \\text{L2Norm}(\\mathbf {W}_{s1}\\mathbf {h}_g),$ where $\\mathbf {W}_{s0} \\in \\mathbb {R}^{D_s \\times (D_p + D_x)}$ and $\\mathbf {W}_{s0} \\in \\mathbb {R}^{D_s \\times D_h}$ are transformation matrices, and $D_s$ is hyper-parameter.",
"Inspired by the deep metric learning algorithm for face recognition in [65], we adopt the triplet loss with online hard negative mining to train our CMRIN model.",
"The triplet loss is defined as $loss = \\text{max}(s_{neg} + \\Delta - s_{gt}, 0),$ where $s_{gt}$ and $s_{neg}$ are the matching scores of the ground-truth proposal and the negative proposal respectively.",
"The negative proposal is randomly chosen from the set of online hard negative proposals, $\\lbrace o_j | s_j + \\Delta - s_{gt} > 0\\rbrace $ , where $\\Delta $ is the margin.",
"During testing, we predict the target object by choosing the object proposal with the highest matching score."
],
[
"Datasets",
"We have evaluated our CMRIN on three commonly used benchmark datasets for referring expression comprehension (i.e., RefCOCO [9], RefCOCO+ [9] and RefCOCOg [3]).",
"In RefCOCO, there are 50,000 target objects, collected from 19,994 images in MSCOCO [66], and 142,210 referring expressions, collected from an interactive game interface [2].",
"RefCOCO is split into train, validation, test A, and test B, which has 120,624, 10,834, 5,657 and 5,095 expression-target pairs, respectively.",
"Test A includes images of multiple people while test B contains images with multiple other objects.",
"RefCOCO+ has 49,856 target objects collected from 19,992 images in MSCOCO, and 141,564 expressions collected from an interactive game interface.",
"Different from RefCOCO, RefCOCO+ does not contain descriptions of absolute location in the expressions.",
"It is split into train, validation, test A, and test B, which has 120,191, 10,758, 5,726 and 4,889 expression-target pairs, respectively.",
"RefCOCOg includes 49,822 target objects from 25,799 images in MSCOCO, and 95,010 long referring expressions collected in a non-interactive setting.",
"RefCOCOg [14] has 80,512, 4,896 and 9,602 expression-target pairs for training, validation, and testing, respectively."
],
[
"Evaluation and Implementation",
"The Precision@1 metric (the fraction of correct predictions) is used for measuring the performance of a method for grounding referring expressions.",
"A prediction is considered to be a true positive when the Intersection over Union between the ground-truth proposal and the top predicted proposal for a referring expression is larger than 0.5.",
"For a given dataset, we count the number of occurrences of each word in the training set.",
"If a word appears more than five times, we add it to the vocabulary.",
"Each word in the expression is initially an one-hot vector, which is further converted into a word embedding.",
"We parse the expression into a constituency tree by Stanford CoreNLP toolkit [67].",
"Annotated regions of object instances are provided in RefCOCO, RefCOCO+ and RefCOCOg.",
"The target objects in the three datasets belong to the 80 object categories in MSCOCO, but the referring expressions may include objects beyond the 80 categories.",
"In order to make the scope of target objects consistent with referring expressions, it is necessary to recognize objects in expressions, even when they are not in the 80 categories.",
"Inspired by the Bottom-Up Attention Model in [68] for image captioning and visual question answering, we train ResNet-101 based Faster R-CNN [69], [15] over selected 1,460 object categories in the Visual Genome dataset [70], excluding the images in the training, validation and testing sets of RefCOCO, RefCOCO+ and RefCOCOg.",
"We combine the detected objects and the ground-truth objects provided by MSCOCO to form the final set of objects in the images.",
"We extract the visual features of objects as the 2,048-dimensional output from the pool5 layer of the ResNet-101 based Faster R-CNN model.",
"Since some previous methods use VGG-16 as the feature extractor, we also extract the 4,096-dimensional output from the fc7 layer of VGG-16 for fair comparison.",
"We set the mini-batch size to 64.",
"The Adam optimizer [71] is adopted to update network parameters with the learning rate set to 0.0005 initially and reduced to 0.0001 after 5 epochs.",
"Margin $\\Delta $ is set to $0.1$ ."
],
[
"Comparison with the State of the Art",
"We compare the performance of our proposed CMRIN against the state-of-the-art methods, including MMI [3], Neg Bag [14], CG [11], Attr [8], CMN [12], Speaker [9], Listener [10], VC [13], A-ATT [19] and MAttNet [4]."
],
[
"Quantitative Evaluation",
"Table REF shows quantitative evaluation results on RefCOCO, RefCOCO+ and RefCOCOg datasets.",
"Our proposed CMRIN consistently outperforms existing methods across all the datasets by a large margin, which indicates that our CMRIN performs well on datasets with different characteristics.",
"Specifically, CMRIN improves the average Precision@1 over validation and testing sets achieved by the existing best-performing algorithm by 1.80%, 5.17% and 3.14% respectively on the RefCOCO, RefCOCO+ and RefCOCOg datasets when VGG-16 is used as the backbone network.",
"Our CMRIN significantly improves the precision in the person category (test A of RefCOCO and RefCOCO+), which indicates that casting appearance attributes (e.g.",
"shirt, glasses and shoes) of a person as external relationships between person and appearance attributes can effectively distinguish the target person from other persons.",
"After we switch to the visual features extracted by ResNet-101 based Faster R-CNN, the Precision@1 of our CMRIN is further improved by another black$\\sim $ 4.40%.",
"It improves the average Precision@1 over validation and testing sets achieved by MAttNet [4] by black1.62%, 5.03% and 3.13% respectively on the three datasets.",
"Note that our CMRIN only uses the 2048-dimensional features from pool5 while MattNet uses the feature maps generated from the last convolutional layers of both the third and fourth stages.",
"Figure: Qualitative results showing initial attention score (gate) maps and final matching score maps.",
"Warmer color indicates higher score."
],
[
"Qualitative Evaluation",
"Visualizations of some samples along with their attention maps and matching scores are shown in Fig.",
"REF .",
"They are generated from our CMRIN using ResNet-101 based Faster R-CNN features.",
"Without relationship modeling, our CMRIN can identify the proposals appearing in the given expression (second columns), and it achieves this goal on the basis of mentioned objects in the given sentence (e.g.",
"the parking meter in Fig.",
"REF (a) and the elephant in full view in Fig.",
"REF (d) have higher attention scores).",
"After fusing information from different modes and propagating multimodal information in the structured relation graph, it is capable of learning semantic context and locating target proposals (third columns) even when the target objects do not attract the most attention at the beginning.",
"It is worth noting that our CMRIN learns semantic relationships (“behind”) for pairs of proposals with different spatial relationships (“bottom right” between “car” and “parking meter” in Fig.",
"REF (a); “top” between “green plant” and “lady's head” in Fig.",
"REF (b)), which indicates that CMRIN is able to infer semantic relationships from the initial spatial relationships.",
"In addition, CMRIN learns the context for target “elephant” (Fig.",
"REF (d)) from “two other elephants” by considering the relations from multiple elephants together.",
"Moreover, multi-order relationships are learned through propagation in CMRIN, e.g., the relationships (“right” in Fig.",
"REF (c)) between object pairs are propagated gradually to the target proposal (most “right man”).",
"Fig.",
"REF demonstrates more qualitative results from our proposed CMRIN.",
"In order to better visualize the pixels covered by multiple proposals generated from the same object (e.g., the pixels covered by proposal “man” and proposal “shirt weared by the man”), we compute the score of a pixel in attention score maps as the sum of the scores of all covering proposals.",
"And in order to distinguish different objects, we set the score of a pixel in matching score maps to be the maximum among scores of all covering objects.",
"We exclude negative scores and normalize the range of each score map to $[0, 1]$ ."
],
[
"Ablation Study",
"blackWe evaluate the proposed CMRIN in five different aspects: 1) the effectiveness of the two modules in our proposed network architecture, i.e., CMRE and GGCN modules; 2) the impact of different training schemes on the performance of CMRIN; 3) the necessity of phrases; 4) the impact of variants of spatial relation graphs used in CMRIN; 5) we explore the effectiveness of incorporating semantic relation graph and detail its implementation.",
"In the following experiments, features computed using ResNet-101 based Faster R-CNN are adopted."
],
[
"Variants of Network Architecture",
"Our proposed CMRIN includes CMRE and GGCN modules.",
"To demonstrate the effectiveness and necessity of each module and further compare each module against its variants, we have trained five additional models for comparison.",
"The results are shown in Table REF .",
"As a baseline (row 1), we use the concatenation of instance-level visual features of objects and the location features as the visual features, and use the last hidden state of the LSTM based expression encoder as the language feature, and then compute the matching scores between the visual features and the language feature.",
"In comparison, a simple variant (row 2) that relies on a global visual context, which is computed by applying graph convolutional operations to the spatial relation graph, already outperforms the baseline.",
"This demonstrates the significance of visual context.",
"Another variant (row 3) with a visual context computed in the language-guided visual relation graph outperforms the above two versions.",
"It captures the context by considering cross-modal information.",
"By fusing the context of words into the language-guided visual relation graph, the semantic context can be captured by applying gated graph convolutional operations (row 6, the final version of CMRIN).",
"Finally, we explore the number of gated graph convolutional layers used in CMRIN.",
"The 1-layer CMRIN (row 4) performs worse than the 2-layer CMRIN because it only captures context with first-order relationships.",
"The 3-layer CMRIN (row 5) does not further improve the performance.",
"One possible reason is that third-order relationships merely occur in the expressions."
],
[
"Different Training Schemes",
"In this section, we evaluate the impact of different loss function designs on the performance of the proposed CMRIN.",
"As shown in Table REF , CMRIN is robust with respect to different loss function settings (i.e., the softmax loss and triplet loss with different parameter settings) and consistently outperforms existing state-of-the-art models on all the three commonly used benchmark datasets (i.e.",
"RefCOCO, RefCOCO+ and RefCOCOg).",
"Specifically, we compare different loss functions, hyperparameters and sampling strategies in the triplet loss for the training of CMRIN.",
"We optimize the proposed CMRIN with the softmax loss (row 1), which is commonly adopted by existing works [12], [11], [9], [13].",
"The performance of CMRIN using the softmax loss performs worse than that of CMRIN using the triplet loss (row 7) because the matching score between the context of an expression and the context of a proposal is not always exactly zero or one.",
"For example, in the image associated with expression “the umbrella held by a lady wearing a green skirt”, there are three umbrellas held by three different ladies and only one of them wears a green skirt.",
"The context of two umbrellas held by the ladies without wearing a green skirt partially matches the context (“the umbrella held by a lady”) of the expression.",
"In addition, we explore the effects of different margins (i.e.",
"0.1, 0.2 and 0.5) in the triplet loss.",
"CMRIN trained using the triplet loss with a 0.5 margin achieves worse performance (row 3) than that with other margins (row 2 and 7) over all the three datasets.",
"Moreover, the performance of CMRIN using the triplet loss with a 0.5 margin fluctuates during training.",
"The models trained by the triplet loss with margin 0.1 and 0.2 have similar performance.",
"Meanwhile, noting that sampling strategies for the triplet loss are essential in face recognition [65], [72], we also sample triplets using different online sampling strategies, including random sampling with one hard negative proposal (row 7), random sampling with two hard negative proposals (row 4), hardest negative mining (row 5) and random semi-hard negative mining (row 6; semi-hard negative proposals can be hard and some of their matching scores are smaller than the matching score of the ground-truth proposal).",
"CMRINs using the triplet loss with one or two negative proposals have similar performance.",
"Their differences in average precision over the three testing sets (RefCOCO, RefCOCO+ and RefCOCOg) are -0.19%, -0.04% and 0.06%, respectively.",
"CMRINs trained using the triplet loss with three different definitions of negative proposals have similar performance except the triplet loss with hardest negative mining on the RefCOCO+ dataset.",
"Their differences in precision over the validation sets and testing sets are within $\\pm $ 0.65%, which demonstrates the robustness of our proposed CMRIN with respect to different sampling strategies.",
"We report the performance of CMRIN (row 7: triplet loss with a margin of 0.1, one negative proposal and random hard negative mining) as the final version of our algorithm, which is chosen according to its performance on the validation sets."
],
[
"Necessity of Phrases",
"blackWe discuss the necessity of phrases in this section and the results are shown in Table REF .",
"The performance of the variant using words to highlight the vertices of the spatial relation graph (row 1) is worse than that of the final version using both words and phrases (row 3), which demonstrates the effectiveness of phrases in improving the accuracy of vertex highlighting.",
"It is worth noting that we implicitly considered the word context (phrase-level information) in our conference version (row 2) by using the contextual features of words to attend the vertices instead of using the word embeddings.",
"However, the contextual features of words introduce the global noise of the expressions, which increases the difficulty of learning the correspondence between words and vertices.",
"The performance of CMRIN with implicit phrases is worse than that of it with explicit phrases in the object category (i.e., test B), because the visual contents of object category is sensitive to contextual noise.",
"In addition, explicit use of phrases can help align between the linguistic words and visual objects."
],
[
"Variants of Spatial Relation Graph",
"blackWe conduct experiments for CMRINs with different spatial relation graphs to evaluate the effects of different designs for spatial relation graphs, and those designs come from two perspectives.",
"blackOn one hand, we adopt three types of designs for edges, i.e, “type(11)”, “type(7)” and “soft”.",
"Specifically, “type(11)” is the 11 types of edges introduced in Section REF ; the “type(7)” is a coarse-grained version of “type(11)” and its 7 types of edges are “inside”, “cover”, “overlap”, “right”', “top”, “left” and “bottom”; the “soft” is a fine-grained version of “type(11)” and it directly encodes the edges as relative location representations [4] by calculating the offsets and area ratios between objects.",
"blackAs shown in Table REF , the performance of CMRIN with “type(7)” (row 1) is slightly worse than that of it with “type(11)” (row 7), because the design of “type(7)” is coarse than the design of “type(11)”.",
"The CMRIN with “soft” (row 2) and “type(11)” (row 3) have similar performance on RefCOCO and RefCOCOg datasets, but the performance of latter is better than that of the former on RefCOCO+ dataset, which indicates that “type(11)” is fine enough to capture spatial relationships.",
"In addition, “type(11)” is more memory- and computation-efficient than “soft”.",
"blackOn the other hand, we evaluate different conditions for connecting between objects, i.e, “edge num”, “axis dis” and “center dis”.",
"In particular, the “edge num(5)” constraints the maximum out-degrees of each vertices to 5 and a vertex is connected to its 5 nearest nodes based on the distances between their normalized center coordinates (i.e., center distances) [4]; the “axis dis(0.15)” connects each pair of objects as long as the relative distances between them in axes are smaller than 15% of the length and width of the image respectively [73]; the “center dis(threshold)” creates a edge for each pairs of objects whose center distance is smaller than the threshold.",
"As shown in Table REF , CMRIN with “center dis(0.7)” (row 6) has relative lower precision than that with other conditions (row 3, 4, 5 and 7), because the “center dis(0.7)” covers several redundant edges which introduces noisy information.",
"The CMRIN with remaining conditions have similar performance, and their differences in average precision over the validation and testing sets on RefCOCO, RefCOCO+ and RefCOCOg datasets are within $\\pm $ 0.07%, $\\pm $ 0.27% and $\\pm $ 0.55%, respectively."
],
[
"Semantic Relation Graph Branch",
"blackIt is intuitive to encode the semantic relationships among objects, in this section, we explore the effectiveness and detail the implementation of semantic relation graph branch.",
"Table: blackExperimental results of spatial/semantic relation graph branch on RefCOCO, RefCOCO+ and RefCOCOg.Effectiveness.",
"blackWe compare the CMRINs with single spatial relation graph branch, with single semantic relation graph branch and joint model including both branches.",
"And the results are shown in the Table REF .",
"The performance of single semantic branch is worse than that of single spatial branch, because the object relationship detector cannot recognize the semantic relations completely accurately in the highly unrestricted scenes.",
"Moreover, the spatial relation graph branch also implicitly captures the semantic relationships as described in Section REF .",
"In addition, the model including both branches achieves the best precision, which indicate the possibility of cooperation between the spatial and semantic relationship representations.",
"Implementation.",
"blackTo implement the branch using semantic relationship graph, we first use a visual relationship detector (the relationship detection part of [62]) trained on the Visual Genome datasets excluding the images in RefCOCO, RefCOCO+ and RefCOCOg datasets to extract the semantic relationships among objects.",
"And, for each predicted edge, we compute its features as the probability-weighted embedding of the relationship categories, and the probabilities to relationship categories are predicted by the visual relationship detector.",
"Since we represent each edge as a type in spatial branch and represent each edge as a feature in semantic branch, the implementation for semantic branch has some minor differences with that of spatial branch.",
"In particular, 1) we attend the language representations over the features of edges instead of over the types of edges; 2) We learn the bias vectors for edges by using fully connected layers to encode the edge features in each gated graph convolutional layers instead of learning the bias vectors for types of edges.",
"blackFor model including both branches, the semantic branch and spatial branch share the same language representations and the vertex attention computation, but have their individual attention learning between language representations and visual edges, gated graph convolutional layers and matching computations.",
"The final score of the two-branched model is the mean of matching scores from those two branches.",
"The whole model is end-to-end trained by the triplet loss with online hard negative mining.",
"We have also evaluated the performance of the proposed CMRIN for grounding referring expressions using automatically detected objects in the three datasets.",
"The detected objects are provided by [4], and they were detected with a pretrained Faster R-CNN in COCO's training images with the images in the validation and testing sets of RefCOCO, RefCOCO+ and RefCOCOg excluded.",
"The results are shown in Table REF .",
"The proposed CMRIN outperforms existing state-of-the-art models, which demonstrates the robustness of CMRIN with respect to object detection results.",
"Specifically, CMRIN improves the average precision in the person category achieved with the existing best-performing method by 3.80%, and it improves the Precision@1 on RefCOCO+'s test A and test B by 5.50% and 1.27%, respectively.",
"Table: Comparison with the state-of-the-art methods on RefCOCO, RefCOCO+ and RefCOCOg when using detected proposals.",
"The best performing method is marked in bold."
],
[
"Effectiveness on multi-order relationships subsets",
"To evaluate the effectiveness of our method on multi-order relationships alone, we identify the subset of expressions with indirect references in the RefCOCOg's test set.",
"There are 2,507 expressions with multiple verbs/location words and Precision@1 on this subset is 81.07% while the number of remaining expression is 7,095 and Precision@1 on them is 80.22%.",
"In contrast, the Precision@1 of MattNet[4] (existing best method) is 76.31% and 79.30% respectively on these two subsets.",
"This result demonstrates that our method can handle indirect references equally well as other simpler cases."
],
[
"Conclusions",
"In this paper, we focus on the task of referring expression comprehension in images, and demonstrate that a feasible solution for this task needs to not only extract all the necessary information in both the image and referring expressions, but also compute and represent multimodal contexts for the extracted information.",
"In order to overcome the challenges, we propose an end-to-end Cross-Modal Relationship Inference Network (CMRIN), which consists of a Cross-Modal Relationship Extractor (CMRE) and a Gated Graph Convolutional Network (GGCN).",
"CMRE extracts all the required information adaptively for constructing language-guided visual relation graphs with cross-modal attention.",
"GGCN fuses information from different modes and propagates the fused information in the language-guided relation graphs to obtain multi-order semantic contexts.",
"Experimental results on three commonly used benchmark datasets show that our proposed method outperforms all existing state-of-the-art methods.",
"[Figure: NO_CAPTION [Figure: NO_CAPTION [Figure: NO_CAPTION"
]
] | 1906.04464 |
[
[
"Actions of solvable Baumslag-Solitar groups on hyperbolic metric spaces"
],
[
"Abstract We give a complete list of the cobounded actions of solvable Baumslag-Solitar groups on hyperbolic metric spaces up to a natural equivalence relation.",
"The set of equivalence classes carries a natural partial order first introduced by Abbott-Balasubramanya-Osin, and we describe the resulting poset completely.",
"There are finitely many equivalence classes of actions, and each equivalence class contains the action on a point, a tree, or the hyperbolic plane."
],
[
"Introduction",
"The Baumslag-Solitar groups $BS(m,n)$ are a classically-studied family of groups defined by the particularly straightforward presentations $\\langle a, t \\mid ta^mt^{-1}=a^n\\rangle $ .",
"In the case $m=n=1$ , $BS(1,1)$ is isomorphic to the abelian group $\\mathbb {Z}^2$ .",
"In general, for $n\\ge 2$ , $BS(1,n)$ is a nonabelian solvable group via the isomorphism $BS(1,n)\\cong \\mathbb {Z}\\left[\\frac{1}{n}\\right]\\rtimes \\mathbb {Z}$ .",
"The group $BS(1,n)$ admits several natural actions on hyperbolic metric spaces.",
"The Cayley graph of $BS(1,n)$ with respect to the generating set $\\lbrace a^{\\pm 1},t^{\\pm 1}\\rbrace $ consists of a number of “sheets,” each of which is quasi-isometric to the hyperbolic plane $\\mathbb {H}^2$ .",
"The sheets are glued together along complements of horoballs in a pattern described by the $(n+1)$ -regular tree.",
"The result is pictured in Figure REF for the case $n=2$ .",
"Collapsing each sheet down to a vertical geodesic line gives a projection from the Cayley graph to the $(n+1)$ -regular tree.",
"Moreover, the action of $BS(1,n)$ on its Cayley graph permutes the fibers of this projection, so that we obtain an action of $BS(1,n)$ on the $(n+1)$ -regular tree.",
"Similarly, the idea of “collapsing the sheets down to a single sheet” gives an action on the hyperbolic plane, although this idea is a bit harder to formalize.",
"Figure: Two natural actions of the group BS(1,2)BS(1,2) via projections.Formally, the action of $BS(1,n)$ on $\\mathbb {H}^2$ is given by the representation $BS(1,n)\\rightarrow \\operatorname{PSL}(2,\\mathbb {R}),$ where $ a\\mapsto \\begin{pmatrix} 1 & 1 \\\\ 0 & 1 \\end{pmatrix}, \\quad t \\mapsto \\begin{pmatrix} \\sqrt{n} & 0 \\\\ 0 & 1/\\sqrt{n} \\end{pmatrix}.$ Another way to obtain the natural action on the $(n+1)$ -regular tree is by expressing $BS(1,n)$ as an HNN extension over the subgroup $\\langle a \\rangle \\cong \\mathbb {Z}$ and considering the action of $BS(1,n)$ on the Bass-Serre tree of the resulting one-edge graph of groups.",
"In addition to these actions, $BS(1,n)$ admits an obvious homomorphism $BS(1,n)\\rightarrow \\mathbb {Z}$ defined by $a\\mapsto 0$ , $t\\mapsto 1$ .",
"This defines an action of $BS(1,n)$ on (the hyperbolic metric space) $\\mathbb {R}$ via integral translations.",
"This action may also be obtained by collapsing either the hyperbolic plane or the $(n+1)$ -regular tree onto a vertical geodesic in a height-respecting manner and noting that $BS(1,n)$ permutes the fibers of the resulting projection.",
"Even more trivially, any group admits an action on a point (which is a hyperbolic metric space).",
"All of these actions are cobounded in the sense that the orbit of a point under the action admits the entire space as a bounded neighborhood.",
"One may naturally wonder whether these four actions (the actions on a tree, the hyperbolic plane, the line, and a point) give a complete list of the nontrivial cobounded actions of $BS(1,n)$ on hyperbolic metric spaces up to equivalence.",
"In this paper we show that this is indeed the case if $n$ is prime.",
"In the case that $n$ is not prime, we show that $BS(1,n)$ admits actions on certain other Bass-Serre trees.",
"These other Bass-Serre trees correspond to expressions of $BS(1,n)$ as HNN extensions over (fairly nonobvious) subgroups of $\\mathbb {Z}\\left[\\frac{1}{n}\\right]$ which are defined using certain ideals in the ring $\\mathbb {Z}_n$ of $n$ -adic integers.",
"Combined with the canonical actions described in the above paragraphs, we show that this gives a complete list of the cobounded hyperbolic actions of $BS(1,n)$ (up to an equivalence relation, described below).",
"Before stating these results precisely, we need to introduce some terminology."
],
[
"Hyperbolic structures on groups",
"In groups that admit interesting actions on hyperbolic metric spaces, it is natural to wonder whether it is possible to describe all such actions explicitly.",
"Unfortunately, this (slightly naive) goal is currently unattainable for almost all commonly studied groups.",
"For instance, using the machinery of combinatorial horoballs introduced by Groves–Manning in [6], one may produce uncountably many parabolic actions of any countable group on hyperbolic metric spaces, i.e.",
"actions with a fixed point on the boundary and all group elements acting elliptically or parabolically.",
"For this reason, we restrict to considering cobounded actions on hyperbolic metric spaces, which are never parabolic.",
"Moreover, we must use some notion of equivalence for the actions of a fixed group on different hyperbolic spaces, as it is quite easy to modify an action on a given hyperbolic space equivariantly to produce an action on a quasi-isometric space.",
"The equivalence relation we consider, introducted in [1], is roughly equivariant quasi-isometry.",
"See Section for more details.",
"Having restricted to cobounded actions up to equivalence, it is usually still quite difficult to describe explicitly all of the equivalence classes of actions of a given group on different hyperbolic metric spaces.",
"For instance, in [1] Abbott–Balasubramanya–Osin considered the hyperbolic actions of acylindrically hyperbolic groups, a very wide class of groups all displaying some features of negative curvature.",
"There they showed that any acylindrically hyperbolic group admits uncountably many distinct equivalence classes of actions on hyperbolic spaces.",
"But for groups which don't display strong features of negative curvature, it may be possible to give a complete list of their cobounded hyperbolic actions.",
"For instance, in [1] Abbott–Balasubramanya–Osin give a complete list of the equivalence classes of cobounded actions of $\\mathbb {Z}^n$ on hyperbolic metric spaces.",
"More recently, in [3], Balasubramanya gives a complete list of the actions of lamplighter groups on hyperbolic spaces.",
"Our work draws inspiration from her strategy.",
"In these cases, it is possible to say even more about the actions of a fixed group on different hyperbolic metric spaces.",
"We are interested in the question of when one action “retains more information” about the group than another.",
"This question leads to a partial order on the set of equivalence classes of actions of a group $G$ on hyperbolic spaces $X$ , defined in [1].",
"Roughly, we say that $G\\curvearrowright X$ dominates $G\\curvearrowright Y$ when the action $G\\curvearrowright Y$ may be obtained by equivariantly coning off certain subspaces of $X$ .",
"See Section REF for the precise definition.",
"This partial order descends to a partial order on equivalence classes of actions.",
"Hence, for a group $G$ , the set of equivalence classes of actions of $G$ on different hyperbolic metric spaces is a poset $\\mathcal {H}(G)$ .",
"In this paper we give a complete description of the poset $\\mathcal {H}(BS(1,n))$ .",
"Let $n=p_1^{n_1}p_2^{n_2}\\dots p_k^{n_k}$ be the prime factorization of $n$ and let $\\mathcal {K}_n$ be the poset $2^{\\lbrace 1,\\dots ,k\\rbrace }\\setminus \\lbrace \\emptyset \\rbrace $ , with the partial order given by inclusion.",
"Theorem 1.1 For any $n\\ge 2$ , $\\mathcal {H}(BS(1,n))$ has the following structure: $\\mathcal {H}_{qp}(BS(1,n))$ , the subposet of quasi-parabolic structures, contains a copy of $\\mathcal {K}_n$ and a single additional structure which is incomparable to every element of $\\mathcal {K}_n$ ; every quasi-parabolic structure dominates a single lineal structure, which dominates a single elliptic structure (see Figure REF ).",
"Figure: The poset ℋ(BS(1,n))\\mathcal {H}(BS(1,n)).",
"The subposet circled in red is isomorphic to the poset 2 {1,⋯,k} 2^{\\lbrace 1,\\dots , k\\rbrace }, which is a lattice.Following [1], we say a group $G$ is $\\mathcal {H}$ –accessible if $\\mathcal {H}(G)$ contains a largest element.",
"Otherwise, we say $G$ is $\\mathcal {H}$ –inaccessible.",
"Corollary 1.2 $BS(1,n)$ is $\\mathcal {H}$ –inaccessible."
],
[
"About the proof",
"Any group action on a hyperbolic space falls into one of finitely many types (elliptic, parabolic, lineal, quasi-parabolic, or general type) depending on the number of fixed points on the boundary and the types of isometries defined by various group elements.",
"See Section for precise definitions.",
"Since $BS(1,n)$ is solvable, it contains no free subgroup and therefore by the Ping-Pong Lemma, any action of $BS(1,n)$ on a hyperbolic metric space must have a fixed point on the boundary.",
"Since we consider only cobounded actions, we see that $BS(1,n)$ can have only lineal or quasi-parabolic cobounded actions on hyperbolic metric spaces.",
"Hence we are left to consider such actions.",
"We crucially use the fact that $BS(1,n)$ can be written as a semidirect product $\\mathbb {Z}\\left[\\frac{1}{n}\\right]\\rtimes _\\alpha \\mathbb {Z}$ .",
"In [4] Caprace–Cornulier–Monod–Tessera classified the lineal and quasi-parabolic actions of certain groups $H\\rtimes \\mathbb {Z}$ in the language of confining subsets of $H$ under the action of $\\mathbb {Z}$ (see Section ).",
"Using techniques developed in [4], we show that the lineal and quasi-parabolic actions of $BS(1,n)$ naturally correspond to confining subsets of $\\mathbb {Z}\\left[\\frac{1}{n}\\right]$ under the actions of $\\alpha $ and $\\alpha ^{-1}$ , and so we are led to try to classify such subsets.",
"The confining subsets under the action of $\\alpha ^{-1}$ are easy to classify, and in fact they all correspond to (the equivalence class of) the action of $BS(1,n)$ on $\\mathbb {H}^2$ .",
"On the other hand, the classification of confining subsets under the action of $\\alpha $ is more complicated.",
"We show that such subsets correspond in a natural way to ideals in the ring of $n$ -adic integers $\\mathbb {Z}_n$ .",
"To see how such ideals arise, we consider confining subsets $Q\\subset \\mathbb {Z}\\left[\\frac{1}{n}\\right]$ under the action of $\\alpha $ and write elements $a\\in Q$ in base $n$ : $a=\\pm a_r\\cdots a_0.a_{-1} \\cdots a_{-s},$ allowing any number of 0's at the end of this expression.",
"We then consider the set of all $\\ldots x_2 x_1 x_0\\in \\mathbb {Z}_n$ such that for any $s$ , the sequence $x_sx_{s-1}\\ldots x_0$ appears as the sequence of digits to the right of the decimal point in some element of $Q$ .",
"That is, we require that there is a number of the form $a_r \\cdots a_0 .x_s \\cdots x_0$ in $Q$ for arbitrarily large $s$ .",
"We show that the set of $n$ -adic integers $\\ldots x_s x_{s-1} \\ldots x_0$ obtained in this manner is an ideal of $\\mathbb {Z}_n$ .",
"We also show that this process can be reversed, to associate confining subsets of $H$ to ideals of $\\mathbb {Z}_n$ .",
"With this correspondence in hand, we describe how all of the resulting actions are equivalent to the actions of $BS(1,n)$ on certain Bass-Serre trees."
],
[
"Other results and results in the literature",
"The poset described in Theorem REF is interesting because of its asymmetry (when $n$ is not a power of a prime).",
"In [3] Balasubramanya described $\\mathcal {H}(L_n)$ where $L_n$ is the Lamplighter group $\\mathbb {Z}_n \\wr \\mathbb {Z}=\\mathbb {Z}_n \\wr \\langle t \\rangle $ .",
"In this case, $\\mathcal {H}(L_n)$ splits into two isomorphic sub-posets of quasi-parabolic structures (corresponding to actions in which the fixed point of $L_n$ is the attracting fixed point of $t$ , and respectively the repelling fixed point of $t$ ) which each dominate a single lineal structure which in turn dominates an elliptic structure.",
"Forthcoming work of the authors in [2] shows that this structure also holds for semidirect products $\\mathbb {Z}^2 \\rtimes _\\alpha \\mathbb {Z}$ where $\\alpha \\in SL(2,\\mathbb {Z})$ is an Anosov matrix.",
"It would be interesting to see what extra properties of a semidirect product $G\\rtimes \\mathbb {Z}$ are needed to ensure this kind of symmetry.",
"In [3] Balasubramanya considers hyperbolic actions of general wreath products $G \\wr \\mathbb {Z}=\\left(\\bigoplus _{n \\in \\mathbb {Z}} G\\right) \\rtimes \\mathbb {Z}$ and shows that $\\mathcal {H}(G\\wr \\mathbb {Z})$ always contains two copies of the poset of subgroups of $G$ .",
"In the case $G=\\mathbb {Z}/n\\mathbb {Z}$ we have $G\\wr \\mathbb {Z}=L_n$ and this suffices to describe all of $\\mathcal {H}(L_n)$ .",
"In Section we consider a general algebraic construction of quasi-parabolic structures on semidirect products $H\\rtimes \\mathbb {Z}$ .",
"We show that in the case of $\\mathbb {Z}\\wr \\mathbb {Z}$ , this construction suffices to produce a countable chain of quasi-parabolic structures."
],
[
"Actions on hyperbolic spaces",
"Given a metric space $X$ , we denote by $d_X$ the distance function on $X$ .",
"A map $f\\colon X\\rightarrow Y$ between metric spaces $X$ and $Y$ is a quasi-isometric embedding if there is a constant $C$ such that for all $x,y\\in X$ , $\\frac{1}{C} d_X(x,y)-C\\le d_Y(f(x),f(y))\\le Cd_X(x,y)+C.$ If, in addition, $Y$ is contained in the $C$ –neighborhood of $f(X)$ then $f$ is called a quasi-isometry.",
"If a group $G$ acts (by isometries) on $X$ and $Y$ , then a map $f\\colon X\\rightarrow Y$ is coarsely $G$ –equivariant if for every $x\\in X$ we have $\\sup _{g\\in G} d_Y(f(gx),gf(x))<\\infty .$ We will assume that all actions are by isometries.",
"The action of a group $G$ on a metric space $X$ is cobounded if there exists a bounded diameter subspace $B\\subseteq X$ such that $X=\\bigcup _{g\\in G}gB$ .",
"Given an action $G\\curvearrowright X$ of $G$ on a hyperbolic space, an element $g\\in G$ is elliptic if it has bounded orbits; loxodromic if the map $\\mathbb {Z}\\rightarrow X$ given by $n\\mapsto g^n\\cdot x_0$ for some (equivalently, any) $x_0\\in X$ is a quasi-isometric embedding; and parabolic otherwise.",
"Any group action on a hyperbolic space falls into one of finitely many types depending on the number of fixed points on the boundary and the types of isometries defined by various group elements.",
"This classification was described by Gromov in [5]: the action $G\\curvearrowright X$ (where $X$ is hyperbolic) is elliptic if $G$ has a bounded orbit in $X$ ; lineal if $G$ fixes two points in $\\partial X$ ; parabolic if $G$ fixes a unique point of $\\partial X$ and no element of $G$ acts as a loxodromic isometry of $X$ ; quasi-parabolic if $G$ fixes a unique point of $\\partial X$ and at least one element of $G$ acts as a loxodromic isometry; and general type if $G$ doesn't fix any point of $\\partial X$ and at least one element of $G$ acts as a loxodromic isometry."
],
[
"Hyperbolic structures",
"Fix a group $G$ .",
"For any (possibly infinite) generating set $S$ of $G$ , let $\\Gamma (G,S)$ be the Cayley graph of $G$ with respect to the generating set $S$ , and let $\\Vert \\cdot \\Vert _S$ denote the word norm on $G$ with respect to $S$ .",
"Given two generating sets $S,T$ of a group $G$ , we say $T$ is dominated by $S$ , written $T\\preceq S$ , if $\\sup _{g\\in S}\\Vert g\\Vert _T<\\infty .$ It is clear that $\\preceq $ is a preorder on the set of generating sets of $G$ and so induces an equivalence relation: $S\\sim T$ if and only if $T\\preceq S$ and $S\\preceq T$ .",
"Let $[S]$ be the equivalence class of a generating set.",
"Then the preorder $\\preceq $ induces a partial order $\\preccurlyeq $ on the set of all equivalence classes of generating sets of $G$ via $[S]\\preccurlyeq [T]$ if and only if $S\\preceq T$ .",
"Definition 2.1 Given a group $G$ , the poset of hyperbolic structures on $G$ is defined to be $\\mathcal {H}(G):= \\lbrace [S]\\mid G=\\langle S\\rangle \\textrm { and } \\Gamma (G,S) \\textrm { is hyperbolic}\\rbrace ,$ equipped with the partial order $\\preccurlyeq $ .",
"Notice that since hyperbolicity is a quasi-isometry invariant of geodesic metric spaces, the above definition is independent of the choice of representative of the equivalence class $[S]$ .",
"Every element $[S]\\in \\mathcal {H}(G)$ gives rise to a cobounded action on a hyperbolic space, namely $G\\curvearrowright \\Gamma (G,S)$ .",
"Moreover, given a cobounded action on a hyperbolic space $G\\curvearrowright X$ , a standard Schwarz–Milnor argument (see [1]) provides a generating set $S$ of $G$ such that $\\Gamma (G,S)$ is equivariantly quasi-isometric to $X$ .",
"We say that two actions $G\\curvearrowright X$ and $G\\curvearrowright Y$ are equivalent if there exists a coarsely $G$ –equivariant quasi-isometry $X\\rightarrow Y$ .",
"By [1], there is a one-to-one correspondence between equivalence classes $[S]\\in \\mathcal {H}(G)$ and equivalence classes of cobounded actions $G\\curvearrowright X$ with $X$ hyperbolic.",
"We denote the set of equivalence classes of cobounded elliptic, lineal, quasi-parabolic, and general-type actions by $\\mathcal {H}_e,\\mathcal {H}_\\ell ,\\mathcal {H}_{qp},$ and $\\mathcal {H}_{gt}$ , respectively.",
"Since parabolic actions cannot be cobounded, we have for any group $G$ , $\\mathcal {H}(G)=\\mathcal {H}_e(G)\\sqcup \\mathcal {H}_\\ell (G)\\sqcup \\mathcal {H}_{qp}(G)\\sqcup \\mathcal {H}_{gt}(G).$ A lineal action of a group $G$ on a hyperbolic space $X$ is orientable if no element of $G$ permutes the two limit points of $G$ on $\\partial X$ .",
"We denote the set of equivalence classes of orientable lineal actions of $G$ by $\\mathcal {H}_\\ell ^+(G)$ .",
"The action of a group $G$ on a hyperbolic metric space $X$ is focal if it fixes a boundary point $\\xi \\in \\partial X$ and if some element of $G$ acts as a loxodromic isometry.",
"If $[S]\\in \\mathcal {H}(G)$ is a focal action, then $[S]\\in \\mathcal {H}_\\ell ^+(G)\\sqcup \\mathcal {H}_{qp}(G)$ .",
"Quasi-characters.",
"A map $q\\colon G\\rightarrow \\mathbb {R}$ is a quasi-character (also called a quasimorphism) if there exists a constant $D$ such that for all $g,h\\in G$ , we have $|q(gh)-q(g)-q(h)|\\le D$ .",
"We say that $q$ has defect at most $D$.",
"If, in addition, the restriction of $q$ to every cyclic subgroup is a homomorphism, then $q$ is called a pseudocharacter (or homogeneous quasimorphism).",
"Every quasi-character $q$ gives rise to a pseudocharacter $\\rho $ defined by $\\rho (g)=\\lim _{n\\rightarrow \\infty }\\frac{q(g^n)}{n}$ ; we call $\\rho $ the homogenization of $q$.",
"Every pseudocharacter is constant on conjugacy classes.",
"If $q$ has defect at most $D$ , then it is straightforward to check that $|q(g)-\\rho (g)|\\le D$ for all $g\\in G$ .",
"Let $G\\curvearrowright X$ be an action on a hyperbolic space with a global fixed point $\\xi \\in \\partial X$ .",
"For any sequence $\\mathbf {x}=(x_n)$ in $X$ converging to $\\xi $ and any fixed basepoint $s\\in X$ , we define the associated quasi-character $q_{\\bf x}\\colon G\\rightarrow \\mathbb {R}$ as follows.",
"For all $g\\in G$ , $ q_\\mathbf {x}(g)=\\limsup _{n\\rightarrow \\infty }(d_X(gs,x_n)-d_X(s,x_n)).",
"$ Its homogenization $\\rho _\\mathbf {x}\\colon G\\rightarrow \\mathbb {R}$ is the Busemann pseudocharacter.",
"It is known that for any two sequences $\\mathbf {x},\\mathbf {y}$ converging to $\\xi $ , $\\sup _{g\\in G}|q_\\mathbf {x}(g)-q_\\mathbf {y}(g)|<\\infty $ , and thus we may drop the subscript $\\mathbf {x}$ in $\\rho _\\mathbf {x}$ .",
"If $\\rho $ is a homomorphism, then the action $G\\curvearrowright X$ is called regular.",
"Lemma 2.2 ([4]) Let $G\\curvearrowright X$ be an action on a hyperbolic space with a global fixed point $\\xi \\in \\partial X$ .",
"Then the (possibly empty) set of loxodromic isometries of the action is $\\lbrace g\\in G\\mid \\rho (g)\\ne 0\\rbrace $ , and the set of those with attracting fixed point $\\xi $ is $\\lbrace g\\in G\\mid \\rho (g)>0\\rbrace $ .",
"In particular, the action of $G$ is elliptic or parabolic if and only if $\\rho \\equiv 0$ , and lineal or quasi-parabolic otherwise."
],
[
"Confining subsets",
"Consider a group $G=H\\rtimes _{\\alpha } \\mathbb {Z}$ where $\\alpha \\in \\operatorname{Aut}(H)$ acts by $\\alpha (h)=tht^{-1}$ for any $h\\in H$ , where $t$ is a generator of $\\mathbb {Z}$ .",
"Let $Q$ be a symmetric subset of $H$ .",
"The following definition is from [4].",
"Definition 2.3 The action of $\\alpha $ is (strictly) confining $H$ into $Q$ if it satisfies the following three conditions.",
"$\\alpha (Q)$ is (strictly) contained in $Q$ ; $H=\\bigcup _{k\\ge 0} \\alpha ^{-k}(Q)$ ; and $\\alpha ^{k_0}(Q\\cdot Q)\\subseteq Q$ for some $k_0\\in \\mathbb {Z}_{\\ge 0}$ .",
"Remark 2.4 The definition of confining subset given in [4] does not require symmetry of the subset $Q\\subset H$ .",
"However, according to [4], to classify regular quasi-parabolic structures on a group it suffices to consider only confining subsets which are symmetric.",
"See also Proposition REF in this paper.",
"Remark 2.5 By the discussion after the statement of [4], if there is a subset $Q\\subseteq H$ such that the action of $\\alpha $ is confining $H$ into $Q$ but not strictly confining, then $[Q\\cup \\lbrace \\alpha ^{\\pm 1}\\rbrace ]\\in \\mathcal {H}_\\ell ^+(G)$ .",
"If the action is strictly confining, then $[Q\\cup \\lbrace \\alpha ^{\\pm 1}\\rbrace ]\\in \\mathcal {H}_{qp}(G)$ .",
"In this paper, we will focus primarily on describing subsets $Q$ of $H$ into which the action of $\\alpha $ is (strictly) confining $H$ .",
"For brevity, we will refer to such $Q$ as (strictly) confining under the action of $\\alpha $ .",
"To see an example of how confining subsets arise, it is useful to consider the action $BS(1,n)\\curvearrowright \\mathbb {H}^2$ described in the introduction.",
"In this action, the conjugates of $t$ act as loxodromic isometries whereas the elements of $H=\\mathbb {Z}\\left[\\frac{1}{n}\\right]$ act as parabolic isometries.",
"Consider the subset $Q\\subset H$ of isometries that translate a given point $p$ (say $i$ in the upper half plane model) by some bounded distance (say 1).",
"If $g\\in H$ then we consider the action of a conjugate $t^{-k}gt^k$ where $k\\gg 0$ .",
"Considering the actions of $t^k$ , $g$ , and $t^{-k}$ in turn we see that: $t^k$ first translates $p$ vertically by a very large distance, $g$ shifts $t^kp$ within the horocycle based at $\\infty $ containing it by a very small distance, $t^{-k}$ takes this horocycle isometrically back to the horocycle containing $p$ .",
"In other words, $t^{-k}gt^k$ is a parabolic isometry which moves $p$ by a much smaller distance than $g$ itself does.",
"In particular, if $k$ is large enough, then $t^{-k}gt^k\\in Q$ .",
"Furthermore, the infimal $k$ with $t^{-k}gt^k\\in Q$ depends only on how far $g$ moves the original point $p$ .",
"Using these facts it is easy to see that $Q$ is a confining subset of $H$ .",
"Thus there is a correspondence of quasi-parabolic structures on $BS(1,n)$ and confining subsets of $H$ .",
"Precisely, we have the following result, which is a minor modification of [3] and [4].",
"It is proved in Section REF .",
"propqpconfining A hyperbolic structure $[T]$ is an element of $\\mathcal {H}_{qp}(BS(1,n))$ if and only if there exists a symmetric subset $Q\\subset \\mathbb {Z}\\left[\\frac{1}{n}\\right]$ which is strictly confining under the action of $\\alpha $ or $\\alpha ^{-1}$ such that $[T]=[Q\\cup \\lbrace t^{\\pm 1}\\rbrace ]$ ."
],
[
"$n$ –adic integers",
"Definition 2.6 An $n$ –adic integer is an infinite series $\\sum _{i=0}^\\infty a_in^i,$ where $a_i\\in \\lbrace 0,1,\\dots ,n-1\\rbrace $ .",
"Such an element can also be written in its base $n$ expansion as $\\dots a_3a_2a_1a_0.$ We denote the set of $n$ –adic integers by $\\mathbb {Z}_n$ .",
"We define operations of addition and multiplication on $\\mathbb {Z}_n$ , which gives it the structure of a ring.",
"Definition 2.7 Let $a=\\sum _{i=0}^\\infty a_in^i$ and $b=\\sum _{i=0}^\\infty b_in^i$ be elements of $\\mathbb {Z}_n$ .",
"Then the sum $a+b$ is the $n$ –adic integer $c=\\sum _{i=0}^\\infty c_in^i$ defined inductively as follows.",
"Let $c_0=a_0+b_0\\mod {n}$ (where we identify $\\mathbb {Z}/n\\mathbb {Z}$ with $\\lbrace 0,\\dots ,n-1\\rbrace $ ) and $t_0=\\left\\lfloor \\frac{a+b}{n}\\right\\rfloor $ , so that $a_0+b_0=c_0+t_0n$ .",
"Assume that $c_0,\\dots , c_{i-1}$ and $t_0,\\dots ,t_{i-1}$ have been defined, and let $c_i=a_i+b_i+t_{i-1}\\mod {n}$ and $t_i= \\left\\lfloor \\frac{a_i+b_i+t_{i-1}}{n}\\right\\rfloor ,$ so that $a_i+b_i=c_i+t_in$ .",
"The product $ab$ is the $n$ –adic number $d=\\sum _{i=o}^\\infty d_in^i$ defined inductively as follows.",
"Let $d_0=a_0b_0\\mod {n}$ , and let $s_0=\\left\\lfloor \\frac{a_0b_0}{n}\\right\\rfloor $ , so that $a_0b_0=d_0+s_0n$ .",
"Assume that $d_1,\\dots , d_{i-1}$ and $s_1,\\dots , s_{i-1}$ have been defined, and let $d_i=\\sum _{j=0}^ia_jb_{i-j}+s_{i-1}\\mod {n}$ and $s_i=\\left\\lfloor \\frac{\\sum _{j=0}^ia_jb_{n-j}+s_{i-1}}{n}\\right\\rfloor ,$ so that $\\sum _{j=0}^ia_jb_{n-j}+s_{i-1}=d_i+s_in.$ In the above operations, we think of $t_i$ and $s_i$ as the amounts that are “carried\" at each step, analogous to how we carry digits when adding and multiplying in base 10.",
"An element $a=\\dots a_2a_1a_0$ of the ring $\\mathbb {Z}_n$ is a unit if and only if $a_0$ is relatively prime to $n$ .",
"Let $\\varphi _l\\colon \\mathbb {Z}_n\\rightarrow \\mathbb {Z}/n^l\\mathbb {Z}$ be the ring homomorphism which identifies an element of $\\mathbb {Z}_n$ with its $l$ –th partial sum: $\\varphi _l(a)=\\varphi _l\\left(\\sum _{i=0}^\\infty a_in^i\\right)=\\sum _{i=0}^{l-1} a_in^i.$ These homomorphisms are compatible in the following sense.",
"For any $k\\le l$ , let the map ${}_n^{}T^l_k\\colon \\mathbb {Z}/n^l\\mathbb {Z}\\rightarrow \\mathbb {Z}/n^k\\mathbb {Z}$ be reduction modulo $n^k$ .",
"Then for any $a\\in \\mathbb {Z}_n$ , we have ${}_n^{}T^l_k(\\varphi _l(a))=\\varphi _k(a)$ .",
"In fact, any infinite sequence $(a^i)$ of elements $a^i\\in \\mathbb {Z}/n^i\\mathbb {Z}$ which satisfies ${}_nT^l_k(a^l)=a^k$ defines an element $a\\in \\mathbb {Z}_n$ .",
"Namely, identify $a^l$ with the unique representative $b^l$ of its congruence class in the set $ \\lbrace 0,1,\\ldots ,n^i-1\\rbrace $ .",
"We may write $b^l=\\sum _{i=0}^{l-1} b^l_i n^i$ .",
"Then for any $i\\ge 0$ and $l,k>i$ we have $b^l_i=b^k_i$ and hence we may write $a_0=b^1_0=b^2_0=\\cdots , \\quad a_1=b^2_1=b^3_1=\\cdots ,\\quad \\ldots $ and then, writing $a=\\sum _{i=0}^\\infty a_i n^i$ we have $\\varphi _i(a)=a^i$ for all $i$ .",
"In particular, this shows that $\\mathbb {Z}_n$ is isomorphic to the inverse limit $\\varprojlim \\mathbb {Z}/n^i\\mathbb {Z}$ .",
"There is a metric $d$ on $\\mathbb {Z}_n$ , called the $n$ –adic metric, defined by $d(x,y)= n^{-q}$ if $q$ is maximal such that $n^q$ divides $x-y$ , i.e.",
"if the first $q$ digits of $x-y$ are zero and the $(q+1)$ –st digit is non-zero.",
"Lemma 2.8 With the topology coming from the $n$ –adic metric, $\\mathbb {Z}_n$ is a compact space.",
"Suppose we have an infinite sequence $\\lbrace x^i\\rbrace _{i\\in \\mathbb {Z}_{>0}}$ such that for each $i$ , we have $x^i=\\dots x^i_3x^i_2x^i_1x^i_0$ .",
"By the pigeon-hole principle, there is some $y_0\\in \\lbrace 0,\\dots , n-1\\rbrace $ such that $x^i_0=y_0$ for infinitely many $i$ .",
"The collection of these $x^i$ forms a subsequence $\\lbrace x^{i_{0,j}}\\rbrace _{j \\in \\mathbb {Z}_{>0}}$ .",
"Repeating this construction iteratively, we have a sequence of subsequences $\\lbrace \\lbrace x^{i_{k,j}}\\rbrace _j\\rbrace _k$ and a number $y=\\dots y_3y_2y_1y_0\\in \\mathbb {Z}_n$ such that for each $k$ every element of $\\lbrace x^{i_{k,j}}\\rbrace _j$ agrees with $y$ in its first $k+1$ digits.",
"Moreover, $\\lbrace x^{i_{k+1,j}}\\rbrace _j$ is a subsequence of $\\lbrace x^{i_{k,j}}\\rbrace _j$ .",
"Thus the diagonal subsequence $\\lbrace x^{i_{k,k}}\\rbrace _k$ of $\\lbrace x^i\\rbrace _i$ converges to $y$ .",
"Consequently, $\\mathbb {Z}_n$ is sequentially compact.",
"Since $\\mathbb {Z}_n$ is a metric space, it is also compact.",
"We now consider the ring structure of $\\mathbb {Z}_n$ .",
"Lemma 2.9 Let $n=p_1^{n_1}p_2^{n_2}\\dots p_k^{n_k}$ .",
"There is an isomorphism $ \\mathbb {Z}_n\\cong \\mathbb {Z}_{p_1^{n_1}}\\times \\cdots \\times \\mathbb {Z}_{p_k^{n_k}}.$ Let us give a description of this isomorphism.",
"First, suppose $n=n_1n_2$ , where $n_1$ and $n_2$ are relatively prime.",
"To define a map on $\\mathbb {Z}_n$ , we use the identification of an element of $a\\in \\mathbb {Z}_m$ with the sequence $(\\varphi _l(a))\\in \\varprojlim \\mathbb {Z}/m^l\\mathbb {Z}$ .",
"Let $f\\colon \\mathbb {Z}_n\\rightarrow \\mathbb {Z}_{n_1}\\times \\mathbb {Z}_{n_2}$ be defined by $f(a)=(x_l,y_l)=(\\varphi _l(a)\\mod {n}_1^l,\\varphi _l(a)\\mod {n}_2^l).$ It is clear that the sequence $(x_l)$ (respectively, $(y_l)$ ) satisfies ${}_{n_1}^{}T^l_k(x_l)=x_k$ (respectively, ${}_{n_2}^{}T^l_k(y_l)=y_k$ ) for any $k\\le l$ , and so $(x_l)$ (respectively, $(y_l)$ ) determines a unique point in $\\mathbb {Z}_{n_1}$ (respectively, $\\mathbb {Z}_{n_2}$ ).",
"The fact that $f$ is an isomorphism follows from the Chinese remainder theorem.",
"The isomorphism (REF ) now follows by repeatedly applying $f$ to distinct pairs of prime factors.",
"Moreover, there is an isomorphism $g\\colon \\mathbb {Z}_{p^j}\\rightarrow \\mathbb {Z}_p$ defined as follows.",
"The number $g(a)$ is $a$ with each coefficient expanded to $a_i=a_{i,j-1} p^{j-1}+\\dots +a_{i,1}p+a_{i,0}$ , where $a_{i,k}\\in \\lbrace 0,\\dots , p-1\\rbrace $ .",
"Composing $g$ with the isomorphism (REF ) shows that, in fact, there is an isomorphism $\\mathbb {Z}_n\\rightarrow \\mathbb {Z}_{p_1}\\times \\cdots \\times Z_{p_k}$ .",
"We use the isomorphism $g$ solely to describe the ideals of $\\mathbb {Z}_{p^j}$ .",
"The non-zero ideals of $\\mathbb {Z}_p$ are exactly $p^i\\mathbb {Z}_p=\\lbrace \\sum _{j=i}^\\infty a_jp^j\\mid a_j\\in \\lbrace 0,\\dots , p-1\\rbrace \\rbrace $ .",
"Using the above isomorphism, it is clear that the non-zero ideals of $\\mathbb {Z}_{p^j}$ are exactly $g^{-1}(p^i\\mathbb {Z}_p)=p^i\\mathbb {Z}_{p^j}$ .",
"We now give a technical description of when elements of $\\mathbb {Z}_{p^j}$ and, more generally, $\\mathbb {Z}_n$ are contained in a particular ideal.",
"On a first reading the reader may want to skip Lemmas REF and REF and simply read Example REF and Remark REF instead.",
"The following lemma describes when an element $a\\in \\mathbb {Z}_{p^j}$ is contained in an ideal $p^i\\mathbb {Z}_{p^j}$ .",
"By the above discussion, this occurs when the image of the element under the isomorphism $g$ is contained in $g(p^i\\mathbb {Z}_{p^j})=p^i\\mathbb {Z}_p$ .",
"An element $x\\in \\mathbb {Z}_p$ is in the ideal $p^i\\mathbb {Z}_p$ exactly when $\\varphi _i(x)\\equiv 0\\mod {p}^{i-1}$ .",
"Since $g$ expands the coefficients of $a$ , the definition of $L$ in the statement of the lemma is simply the smallest positive integer such that the expansion of $\\varphi _L(a)$ contains $\\varphi _i(g(a))$ .",
"Equivalently, it is the smallest positive integer such that the expansion of the $p^{(L-1)j}$ term in $a$ contains $p^{i-1}$ , which is the largest power of $p$ appearing in $\\varphi _i(g(a))$ .",
"The largest power of $p$ in the expansion of the $p^{(L-1)j}$ term of $a$ is $p^{(L-1)j+j-1}=p^{Lj-1}$ , and thus $L$ is the smallest positive integer such that $Lj-1\\ge i-1$ .",
"Lemma 2.10 For any $a\\in \\mathbb {Z}_{p^j}$ , $ a\\in p^i\\mathbb {Z}_{p^j} \\qquad \\iff \\qquad \\varphi _{L}(a)\\equiv 0\\mod {p}^{i-1},$ where $L=\\left\\lceil \\frac{i}{j}\\right\\rceil $ .",
"Moreover, $a\\in (0) \\qquad \\iff \\qquad a=0\\qquad \\iff \\qquad \\varphi _s(a)=0,$ for all $s$ .",
"For the first statement, notice that $a\\in p^i\\mathbb {Z}_{p^j}$ if and only if $g(a)\\in p^i\\mathbb {Z}_p$ if and only if $\\varphi _{i}(g(a))\\equiv 0\\mod {p}^{i-1}$ if and only if $\\varphi _{L}(a)\\equiv 0\\mod {p}^{i-1}$ .",
"The first two “if and only if\" statements are clear, while the last follows from the following calculation: $ \\varphi _L(a)&=a_{L-1}p^{(L-1)j}+\\cdots +a_1p^j+a_0 \\\\&=a_{L-1,j-1}p^{Lj-1}+\\cdots +a_{L-1,k}p^{i-1}+\\cdots + a_{L-1,0}p^{(L-1)j} +a_{L-2,j-1}p^{(L-1)j-1} +\\cdots +a_{0,1}p+a_{0,0} \\\\&=a_{L-1,j-1}p^{Lj-1}+\\cdots +a_{L-1,k+1}p^{i}+\\varphi _{i}(g(a)),$ where $k\\in \\lbrace 0,\\dots , j-1\\rbrace $ is such that $(L-1)j+k=i-1$ .",
"Now, if $\\varphi _i(g(a))\\equiv 0\\mod {p}^{i-1}$ , then it is clear that $\\varphi _L(a)\\equiv 0\\mod {p}^{i-1}$ .",
"Similarly, if $\\varphi _L(a)\\equiv 0\\mod {p}^{i-1}$ , then since $a_{L-1,j-1}p^{(L-1)j}+\\cdots +a_{L-1,k}p^{i}\\equiv 0\\mod {p}^{i-1}$ , we must have $\\varphi _i(g(a))\\equiv 0\\mod {p}^{i-1}$ .",
"The second statement is just the definition of the zero ideal.",
"Example 2.11 We give two examples of explicit descriptions of ideals in $\\mathbb {Z}_{2^3}$ .",
"First consider the ideal $2\\mathbb {Z}_{2^3}$ .",
"Then $a=\\dots a_2a_1a_0\\in 2\\mathbb {Z}_{2^3}\\quad \\iff \\quad g(a)\\in 2\\mathbb {Z}_2=\\left\\lbrace \\sum _{i=1}^\\infty b_i2^i\\,\\vert \\,b_i\\in \\lbrace 0,\\dots , p-1\\rbrace \\right\\rbrace .$ Therefore, the only restriction on the partial sums of $a$ is that $\\varphi _1(a)=a_0=a_{0,2}2^2+a_{0,1}2+0,$ i.e., $a_0\\equiv 0\\mod {2}$ .",
"Next, consider the ideal $2^{13}\\mathbb {Z}_{2^3}$ .",
"Then $a=\\dots a_2a_1a_0\\in 2^{13}\\mathbb {Z}_{2^3} \\iff g(a)\\in 2^{13}\\mathbb {Z}_2=\\left\\lbrace \\sum _{i=13}^\\infty b_i2^i\\,\\vert \\, b_i\\in \\lbrace 0,\\dots , p-1\\rbrace \\right\\rbrace .$ In this case, we must have $\\varphi _5(a)=a_42^{12}+a_32^9+a_22^6+a_12^3+a_0=a_{4,2}\\cdot 2^{14}+a_{4,1}\\cdot 2^{13}+0\\cdot 2^{12}+0\\cdot 2^{11}+0\\cdot 2^{10}+\\dots +0\\cdot 2+0,$ i.e., $\\varphi _5(a)\\equiv 0\\mod {2}^{13}$ .",
"Note that this also shows that $\\varphi _i(a)\\equiv 0\\mod {2}^{3i-1}$ for all $i\\le 4$ (for example, $\\varphi _2(a)=a_12^3+a_0=0\\cdot 2^5+0\\cdot 2^4+0\\cdot 2^3+0\\cdot 2^2+0\\cdot 2+0\\equiv 0\\mod {2}^5$ ).",
"Using the isomorphism in (REF ), any ideal $a$ of $\\mathbb {Z}_n$ can be written as $a=a_1\\times \\cdots \\times a_k,$ where $a_i=p_i^{a_i}\\mathbb {Z}_{p_i^{n_i}}$ for some $a_i$ or $a_i=(0)$ .",
"The next lemma gives a precise description of when an element $a\\in \\mathbb {Z}_n$ is contained in an ideal $a=a_1\\times \\cdots \\times a_k$ .",
"The conditions are similar to those in Lemma REF .",
"For each $i$ such that $a_i=p^{a_i}\\mathbb {Z}_{p_i^{n_i}}$ , we have a constant $L_i=\\lceil \\frac{a_i}{n_i}\\rceil $ as in Lemma REF , and one needs only check a condition on a single partial sum, namely that $\\varphi _{L_i}(a)\\equiv 0\\mod {p}_i^{a_i}$ .",
"On the other hand, for each $i$ such that $a_i=(0)$ , one needs to check that every partial sum of $a$ satisfies an appropriate condition.",
"Lemma 2.12 Let $a =a_1\\times \\cdots \\times a_k$ , where for each $i$ , $a_i=p_i^{a_i}\\mathbb {Z}_{p_i^{n_i}}$ or $a_i=(0)$ .",
"For each $i$ , let $L_i=\\left\\lceil \\frac{a_i}{n_i}\\right\\rceil $ .",
"Then for any $a=\\dots a_3a_2a_1a_0\\in \\mathbb {Z}_n$ , $ a\\in a \\quad \\iff \\quad & \\cdot \\varphi _s(a)\\equiv 0\\mod {p}_{i}^{k} \\quad \\textrm {for all s, all k\\le s-1, and all i such that }a_i=(0); and\\\\ & \\cdot \\varphi _s(a)\\equiv 0\\mod {p}_i^{a_i}, \\textrm {for any s such that } s=L_i \\textrm { for some i}.$ For any $i$ such that $a_i=p_i^{a_i}\\mathbb {Z}_{p_i^{n_i}}$ , this follows immediately from the definition of the isomorphism $\\mathbb {Z}_n\\rightarrow \\mathbb {Z}_{p_1^{n_1}}\\times \\cdots \\times \\mathbb {Z}_{p_k^{n_k}}$ and Lemma REF .",
"Fix $i$ such that $a_i=(0)$ .",
"Then by the isomorphism in (REF ), we have that any $a\\in \\mathbb {Z}_n$ can be written as $a=(a_1,\\dots , a_k)$ where, considering $\\mathbb {Z}_n$ and $\\mathbb {Z}_{p_i^{n_i}}$ as $\\varprojlim \\mathbb {Z}/n^l\\mathbb {Z}$ and $\\varprojlim \\mathbb {Z}/p_i^{ln_i}\\mathbb {Z}$ , respectively, $a_i$ is given by the sequence $(\\varphi _s(a)\\mod {p}_i^{sn_i})_{s=1}^\\infty $ .",
"Now, $a_i\\in (0)\\subset \\mathbb {Z}_{p_i^{n_i}}$ if and only if $g(a_i)\\in (0)\\subset \\mathbb {Z}_{p_i}$ , which is the case if and only if $\\varphi _s(a)\\mod {p}_i^{sn_i}\\equiv 0\\mod {p}_i^{(s+1)n}$ if and only if $\\varphi _s(a)\\equiv 0\\mod {p}_i^{s+1}$ .",
"The result follows.",
"Remark 2.13 We point out one particular case of this lemma which will be important in later sections: if some $a_i=(0)$ , then $a_0\\equiv 0\\mod {p}_i^{n_i}$ .",
"We now describe a partial order on the set of ideals of $\\mathbb {Z}_n$ .",
"Definition 2.14 Define a relation $\\le $ on ideals of $\\mathbb {Z}_n$ by $\\mathfrak {a}\\le \\mathfrak {b}$ if $n^k\\mathfrak {a}\\subseteq \\mathfrak {b}$ for some $k$ .",
"Define an equivalence $\\mathfrak {a}\\sim \\mathfrak {b}$ if $\\mathfrak {a}\\le \\mathfrak {b}$ and $\\mathfrak {b}\\le \\mathfrak {a}$ .",
"Definition 2.15 An ideal $\\mathfrak {a}=\\mathfrak {a}_1\\times \\cdots \\times \\mathfrak {a}_k$ is full if $\\mathfrak {a}_j$ is either $(0)$ or $\\mathbb {Z}_{p_j^{n_j}}$ for every $j=1,\\dots , k$ .",
"Lemma 2.16 For any $n$ , there is a unique full ideal in each equivalence class of ideals of $\\mathbb {Z}_n$ .",
"Let $a=a_1\\times \\cdots \\times a_k$ be an ideal of $\\mathbb {Z}_n$ , and consider the full ideal $b=b_1\\times \\cdots \\times b_k$ where $b_i=(0)$ if and only if $a_i=(0)$ .",
"Recall that if $b_i\\ne (0)$ , then $b_i=\\mathbb {Z}_{p_i^{n_i}}$ .",
"We claim that $a\\sim b$ .",
"It is clear that $a \\subseteq b$ , and thus $a\\le b$ .",
"For any $i$ such that $a_i\\ne (0)$ , let $a_i=p^{a_i}\\mathbb {Z}_{p_i^{n_i}}$ , and let $A=\\max _i\\lbrace a_i\\rbrace $ .",
"We will show that $n^Ab\\subset a$ .",
"We have $n^Ab =n^Ab_1\\times \\cdots \\times n^Ab_k,$ and, for each $i$ , $n^Ab_i=p_1^{An_1}\\cdots p_k^{An_k}\\mathbb {Z}_{p_i^{n_i}}.$ For all $j\\ne i$ , the element $p_j^{An_j}$ is a unit in $\\mathbb {Z}_{p_i^{n_i}}$ , and thus $p_j^{n_j}\\mathbb {Z}_{p_i^{n_i}}=\\mathbb {Z}_{p_i^{n_i}}$ .",
"Therefore, we have $n^Ab_i=p_i^{An_i}\\mathbb {Z}_{p_i^{n_i}},$ and since $A\\ge a_i$ by definition, it follows that $n^Ab_i=p_i^{An_i}\\mathbb {Z}_{p_i^{n_i}}\\subseteq p_i^{a_i}\\mathbb {Z}_{p_i^{n_i}}=a_i.$ Consequently, $n^Ab\\subseteq a,$ which implies that $b\\le a$ .",
"Therefore, $a\\sim b$ .",
"Suppose next that there are two distinct full ideals, $b=b_1\\times \\cdots \\times b_k$ and $c=c_1\\times \\cdots \\times c_k$ with $b\\sim a\\sim c$ .",
"Then $b\\sim c$ , which immediately implies that $b_i=(0)$ if and only if $c_i=(0)$ .",
"Indeed, if there is an $i$ such that (without loss of generality) $b_i=(0)$ but $c_i\\ne (0)$ , then there is no power $C$ of $n$ such that $p_i^Cc_i\\subset b_i$ , and so $b\\lnot \\sim c$ , which is a contradiction.",
"Thus by the definition of full ideals, we conclude that $b=c$ ."
],
[
"$BS(1,n)$",
"Fix $n=p_1^{n_1}p_2^{n_2}\\dots p_k^{n_k}$ , and recall that $BS(1,n)=\\langle a,t\\mid tat^{-1}=a^n\\rangle $ .",
"Let $\\tau : BS(1,n)\\rightarrow \\mathbb {Z}$ be the homomorphism $a\\mapsto 0, t\\mapsto 1$ .",
"Then there is a short exact sequence $0\\rightarrow H\\rightarrow BS(1,n)\\xrightarrow{}\\mathbb {Z}\\rightarrow 0,$ where $H:=\\ker (\\tau )\\cong \\mathbb {Z}\\left[\\frac{1}{n}\\right]$ .",
"This gives rise to an isomorphism $BS(1,n)\\cong \\mathbb {Z}\\left[\\frac{1}{n}\\right] \\rtimes _\\alpha \\mathbb {Z}$ , where $\\alpha (x)=n\\cdot x$ for $x\\in \\mathbb {Z}\\left[\\frac{1}{n}\\right]$ .",
"For the rest of this paper we will make the identifications $H=\\mathbb {Z}\\left[\\frac{1}{n}\\right],$ and $BS(1,n)=H\\rtimes _\\alpha \\mathbb {Z}=\\langle H, t \\mid txt^{-1}=\\alpha (x) \\text{ for } x\\in H\\rangle .$ In addition to the standard representation of elements of $H$ as Laurent polynomials in $n$ , we also represent elements by their $n$ –ary expansion; e.g.",
"$\\frac{1}{n}=0.1$ while $n+\\frac{1}{n}+\\frac{1}{n^4}=10.1001$ .",
"We switch between these representations interchangeably.",
"Given an element $x=\\pm x_kx_{k-1}\\cdots x_2x_1x_0.x_{-1}x_{-2}\\cdots x_{-m}\\in H$ , we have $0\\le x_i < n$ for all $-m\\le i\\le k$ .",
"We call $-m$ the leading negative place of place of $x$, which we denote by $p(x)=-m.$ We call $x_{-m}$ the leading negative term of $x$, which we denote by $c(x)=x_{-m}.$ The automorphism $\\alpha $ acts on $H$ by multiplication by $n$ , which has the effect of adding one to each index, so that the $i$ –th term of the image of $x$ is the $(i+1)$ –st term of $\\alpha (x)$ .",
"For example, $\\alpha (21.021311)=210.21311$ (here we are assuming that $n\\ge 4$ ).",
"Lemma 2.17 Let $Q\\subseteq H$ be such that $Q\\cup \\lbrace t^{\\pm 1}\\rbrace $ is a generating set of $BS(1,n)$ and $\\alpha (Q)\\subset Q$ .",
"Then any element $w\\in BS(1,n)$ can be written as $w=t^{-r} x_1\\ldots x_m t^s$ where $r,s\\ge 0$ , $x_i\\in Q$ for all $i$ , and $r+s+m= \\Vert w\\Vert _{Q\\cup \\lbrace t^\\pm 1\\rbrace }$ .",
"Moreover, if $w\\in H$ then $r=s$ .",
"Write $w$ as a reduced word in $Q\\cup \\lbrace t^{\\pm 1}\\rbrace $ .",
"By using the relations $tx=\\alpha (x)t$ and $xt^{-1}=t^{-1}\\alpha (x)$ for $x\\in Q$ , we may move all copies of $t$ in $w$ to the right and all copies of $t^{-1}$ to the left without increasing the word length of $w$ in $Q\\cup \\lbrace t^{\\pm 1}\\rbrace $ .",
"The result is an expression of $w$ as a reduced word, $w=t^{-r} x_1\\ldots x_m t^s$ where $r,s\\ge 0$ and $x_i\\in Q$ for all $i$ .",
"Since the word length of $w$ has not changed, we have $r+s+m= \\Vert w\\Vert _{Q\\cup \\lbrace t^\\pm 1\\rbrace }$ .",
"The second statement is clear."
],
[
"Confining subsets of $H$",
"We first describe two particular subsets of $H= \\mathbb {Z}\\left[\\frac{1}{n}\\right]$ which are strictly confining under the action of $\\alpha $ and $\\alpha ^{-1}$ , respectively.",
"Lemma 3.1 The subset $ Q^+=\\lbrace x\\in H\\mid x= \\pm x_kx_{k-1}\\cdots x_2x_1x_0 \\textrm { for some } k\\in \\mathbb {N}\\rbrace = \\mathbb {Z}\\subset H$ is strictly confining under the action of $\\alpha $ .",
"The subset $ Q^-=\\lbrace x\\in H\\mid x=\\pm 0.x_{-1}x_{-2}\\cdots x_{-m} \\textrm { for some } m\\in \\mathbb {N}\\rbrace \\subset H$ is strictly confining under the action of $\\alpha ^{-1}$ .",
"We will verify that Definition REF holds for $Q^-$ ; the proof for $Q^+$ is analogous.",
"We have $\\alpha ^{-1}(Q^-)=\\lbrace x\\in H\\mid x= \\pm 0.0x_{-1}x_{-2} \\cdots x_{-m} \\mid \\textrm { for some } m\\in \\mathbb {Z}_{>0}\\rbrace .$ Thus $\\alpha ^{-1}(Q^-)\\subset Q^-$ , so (a) holds.",
"Moreover, it is clear that $\\bigcup _{n\\ge 0}\\alpha ^n(Q^-)=H$ .",
"Indeed, let $x=\\pm x_kx_{k-1}\\cdots x_0.x_{-1}x_{-2}\\cdots x_{-m}$ be any element of $H$ .",
"Since $\\alpha ^{-(k+1)}(x)\\in Q^-$ , it follows that $x\\in \\alpha ^{k+1}(Q^-)$ , and thus (b) holds.",
"Finally, let $x,y\\in Q^-$ .",
"Then we have $x+y=\\pm z_0.z_{-1} \\cdots z_{-m}$ where each $z_i \\in \\lbrace 0,\\ldots ,n-1\\rbrace $ .",
"Hence $\\alpha ^{-1}(x+y)\\in Q^-$ and (c) holds with $k_0=1$ .",
"Thus $Q^-$ is confining under the action of $\\alpha ^{-1}$ .",
"To see that $Q^-$ is strictly confining, note that $0.1\\in Q \\setminus \\alpha ^{-1}(Q)$ .",
"The following lemma appears as [3]; we include a proof here for completeness.",
"Recall that given two (possibly infinite) generating sets $S,T$ of a group $G$ , we say $[S]=[T]$ if $\\sup _{g\\in S}\\Vert g\\Vert _T<\\infty $ and $\\sup _{h\\in T}\\Vert h\\Vert _S<\\infty $ .",
"Lemma 3.2 Suppose $Q$ is a symmetric subset of $H$ which is confining under the action of $\\alpha $ .",
"Let $S$ be a symmetric subset of $H$ such that there exists $K\\in \\mathbb {Z}_{\\ge 0}$ with $\\alpha ^K(g)\\in Q$ for all $g\\in S$ .",
"Then $\\overline{Q}=Q\\cup \\bigcup _{i\\ge 0} \\alpha ^i(S)$ is confining under the action of $\\alpha $ and $ [Q\\cup \\lbrace t^{\\pm 1}\\rbrace ]=[\\overline{Q}\\cup \\lbrace t^{\\pm 1}\\rbrace ].$ We note that this lemma applies, for example, to all finite symmetric subsets $S$ of $H$ .",
"[Proof of Lemma REF ] First we prove that $\\overline{Q}$ is confining under the action of $\\alpha $ .",
"Conditions (a) and (b) from Definition REF are clear (using that $Q\\subseteq \\overline{Q}$ for condition (b)).",
"To see that condition (c) holds, note that for any $i\\ge 0$ , and any $g\\in S$ , $\\alpha ^K(\\alpha ^i(g))=\\alpha ^i(\\alpha ^K(g))\\in \\alpha ^i(Q)\\subseteq Q.$ We also have $\\alpha ^K(g)\\in Q$ for any $g\\in \\overline{Q}$ .",
"Hence, if $g,h\\in \\overline{Q}$ we have $\\alpha ^K(g)\\in Q$ and $\\alpha ^K(h)\\in Q$ , and therefore $\\alpha ^{K+k_0}(g+h)=\\alpha ^{k_0}(\\alpha ^K(g)+\\alpha ^K(h))\\in \\alpha ^{k_0}(Q + Q)\\subseteq Q \\subseteq \\overline{Q},$ where $k_0$ is large enough so that $\\alpha ^{k_0}(Q+Q)\\subseteq Q$ .",
"Therefore (c) holds with constant $K+k_0$ .",
"To see that $[Q\\cup \\lbrace t^{\\pm 1}\\rbrace ]=[\\overline{Q}\\cup \\lbrace t^{\\pm 1}\\rbrace ]$ , note first of all that clearly $[\\overline{Q}\\cup \\lbrace t^{\\pm 1}\\rbrace ]\\preccurlyeq [Q \\cup \\lbrace t^{\\pm 1}\\rbrace ]$ .",
"On the other hand, by our above observation, $\\overline{Q}$ is really just a finite union: $\\overline{Q}=Q\\cup \\bigcup _{i=0}^{K-1} \\alpha ^i(S).$ For each $i$ between 0 and $K-1$ and each $g\\in S$ , we have $\\alpha ^i(g)=\\alpha ^{-(K-i)}(\\alpha ^K(g))=t^{-(K-i)}\\alpha ^K(g)t^{(K-i)}$ and $\\alpha ^K(g)\\in Q$ .",
"Hence $\\Vert \\alpha ^i(g) \\Vert _{Q\\cup \\lbrace t^{\\pm 1}\\rbrace } \\le 2(K-i)+1\\le 2K+1.$ In other words, any element of $\\overline{Q}\\cup \\lbrace t^{\\pm 1}\\rbrace $ has word length at most $2K+1$ with respect to $Q\\cup \\lbrace t^{\\pm 1}\\rbrace $ , so $[Q \\cup \\lbrace t^{\\pm 1}\\rbrace ] \\preccurlyeq [\\overline{Q} \\cup \\lbrace t^{\\pm 1}\\rbrace ]$ .",
"Lemma 3.3 For any $Q\\subseteq H$ which is confining under the action of $\\alpha $ , we have $[Q\\cup \\lbrace t^{\\pm 1}\\rbrace ] \\preccurlyeq [Q^+ \\cup \\lbrace t^{\\pm 1}\\rbrace ]$ .",
"We show that every element of $Q^+=\\mathbb {Z}$ has bounded word length with respect to $Q\\cup \\lbrace t^{\\pm 1}\\rbrace $ .",
"First, we apply Lemma REF with $S=\\lbrace \\pm 1\\rbrace $ to pass to $\\overline{Q}\\supset Q$ such that $\\lbrace \\pm 1\\rbrace \\subseteq \\overline{Q}$ and $[Q\\cup \\lbrace t^{\\pm 1}\\rbrace ]=[\\overline{Q} \\cup \\lbrace t^{\\pm 1}\\rbrace ]$ .",
"We begin by showing that every element of $\\mathbb {Z}_{>0}=\\lbrace 1,2,\\ldots \\rbrace $ has bounded word length with respect to $\\overline{Q}\\cup \\lbrace t^{\\pm 1}\\rbrace $ .",
"Choose $k_1$ such that $\\alpha ^{k_1}(\\overline{Q} + \\overline{Q})\\subseteq \\overline{Q}$ .",
"We actually initially prove that every element of $\\alpha ^{k_1}(\\mathbb {Z}_{>0})=\\lbrace n^{k_1},2n^{k_1},3n^{k_1},\\ldots \\rbrace $ has bounded word length with respect to $\\overline{Q}\\cup \\lbrace t^{\\pm 1}\\rbrace $ .",
"If every such element has word length at most $ L$ then every element of $\\mathbb {Z}_{>0}$ has word length less than $ L+n^{k_1}$ with respect to $\\overline{Q}\\cup \\lbrace t^{\\pm 1}\\rbrace $ because such an element can be written as $a n^{k_1}+\\underbrace{1+\\cdots +1}_{< n^{k_1} \\text{ times}}, \\quad \\textrm {where} \\quad a\\in \\mathbb {Z}_{>0}$ and $1\\in \\overline{Q}$ .",
"The proof of this weaker statement is by induction.",
"First, note that $\\alpha ^{k_1}(1)=n^{k_1}\\in \\overline{Q}$ .",
"Hence every element of the set $\\lbrace n^{k_1},2n^{k_1},\\ldots ,n^{2k_1}=n^{k_1}\\cdot n^{k_1}\\rbrace $ has word length at most $ n^{k_1}$ with respect to $\\overline{Q}\\cup \\lbrace t^{\\pm 1}\\rbrace $ .",
"Suppose for induction that every element of $\\lbrace n^{(l-1)k_1},n^{(l-1)k_1}+n^{k_1},n^{(l-1)k_1}+2n^{k_1},\\ldots ,n^{lk_1}\\rbrace =\\lbrace an^{k_1}\\mid a\\in \\mathbb {Z}_{>0}\\rbrace \\cap [n^{(l-1)k_1},n^{lk_1}]$ has word length at most $ n^{k_1}$ with respect to $\\overline{Q}\\cup \\lbrace t^{\\pm 1}\\rbrace $ .",
"Enumerate the elements of this set as $x_0=n^{(l-1)k_1},x_1=n^{(l-1)k_1}+n^{k_1},\\ldots , x_s=n^{lk_1}.$ Consider an element $y\\in \\lbrace n^{lk_1},n^{lk_1}+n^{k_1},n^{lk_1}+2n^{k_1},\\ldots ,n^{(l+1)k_1}\\rbrace =\\lbrace an^{k_1}\\mid a\\in \\mathbb {Z}_{>0}\\rbrace \\cap [n^{lk_1},n^{(l+1)k_1}].$ Such an element $y$ satisfies $n^{k_1}x_j \\le y\\le n^{k_1}x_{j+1}=n^{k_1}(x_j+n^{k_1})=n^{k_1}x_j + n^{2k_1}$ for some $j$ .",
"Hence we have $ y=n^{k_1}x_j+an^{k_1},$ where $0\\le a\\le n^{k_1}$ .",
"Since $x_j$ has word length at most $ n^{k_1}$ , we may write $x_j=g_1+\\cdots +g_{n^{k_1}}$ where all $g_i\\in \\overline{Q}$ and $g_i=0$ for all $i>\\Vert x_j\\Vert _{\\overline{Q}\\cup \\lbrace t^{\\pm 1}\\rbrace }$ .",
"Thus we can rewrite equation (REF ) as $\\begin{tabular}{l l l}\\end{tabular}$ $ & $ =$ & $ nk1(g1+...+gnk1)+nk1(1++1ank1 times)$ \\\\& $ =$ & $ nk1(g1+1)+nk1(g2+1)++nk1(ga+1)+nk1(ga+1)++nk1(gnk1)$.",
"$ $ In this last sum, every term is an element of $ Q$, and there are $ nk1$ terms.",
"Thus $ yQ{t1}nk1$.",
"This completes the induction.$ We have shown so far that every element of $\\mathbb {Z}_{>0}$ has bounded word length with respect to $\\overline{Q}$ .",
"A completely analogous argument using multiples of $-n^{k_1}$ proves that every element of $\\mathbb {Z}_{<0}=\\lbrace -1,-2,\\ldots \\rbrace $ has bounded word length with respect to $\\overline{Q}$ .",
"Hence we have shown $[Q\\cup \\lbrace t^{\\pm 1}\\rbrace ]=[\\overline{Q}\\cup \\lbrace t^{\\pm 1}\\rbrace ]\\preccurlyeq [\\mathbb {Z}\\cup \\lbrace t^{\\pm 1}\\rbrace ] = [Q^+\\cup \\lbrace t^{\\pm 1}\\rbrace ].$"
],
[
"Subsets confining under the action of $\\alpha $ and ideals of {{formula:cbbe6979-be3e-464c-bfa2-15c3b3f9f1b0}}",
"In this subsection, we describe the connections between subsets of $H$ which are confining under the action of $\\alpha $ and ideals of $\\mathbb {Z}_n$ ."
],
[
"From confining subsets to ideals",
"We begin by describing a way to associate an ideal of $\\mathbb {Z}_n$ to a symmetric subset $Q$ of $H$ which is confining under the action of $\\alpha $ .",
"We define $\\mathcal {L}(Q)=\\left\\lbrace \\:{\\ldots x_2 x_1 x_0 \\in \\mathbb {Z}_n \\,\\Bigg \\vert \\,\\begin{aligned} & \\text{for any } t\\ge 0, \\exists a \\in Q \\text{ with } a=a_r \\cdots a_0.",
"x_t \\cdots x_0 \\\\ &\\text{ for some }a_r,\\dots , a_0\\in \\lbrace 0,\\dots , n-1\\rbrace \\end{aligned}}\\:\\right\\rbrace .$ That is, an element $\\ldots x_2 x_1 x_0$ is in $\\mathcal {L}(Q)$ if for any $t$ , there exists a positive element of $Q$ whose fractional part is $0.x_t\\cdots x_0$ .",
"Note in particular that $\\mathcal {L}(Q)$ is nonempty for any $Q$ as above.",
"To see this, first notice that $Q$ always contains a positive integer $a=a_r\\cdots a_0$ .",
"We may equivalently write $a=a_r\\cdots a_0.",
"\\underbrace{0 \\cdots \\cdots 0}_{t \\text{ times}}$ and since $t$ is arbitrary, this shows that $0 \\ (= \\ldots 0 0 0 ) \\in \\mathcal {L}(Q)$ .",
"Lemma 3.4 The set $\\mathcal {L}(Q)\\subseteq \\mathbb {Z}_n$ is closed.",
"Let $x\\in \\overline{\\mathcal {L}(Q)}$ and write $x=\\ldots x_2 x_1 x_0$ .",
"Then for any $t\\ge 0$ , there exists $y\\in \\mathcal {L}(Q)$ with $y=\\ldots y_2 y_1 y_0$ and $y_i=x_i$ for $i\\le t$ .",
"By definition of $\\mathcal {L}(Q)$ there exists $a\\in Q$ with $a=a_r\\cdots a_0.y_t \\cdots y_0$ .",
"But then of course we also have $a=a_r\\cdots a_0.",
"x_t \\cdots x_0$ .",
"Since $t$ is arbitrary, this implies that $x\\in \\mathcal {L}(Q)$ .",
"Lemma 3.5 In the notation above, $\\mathcal {L}(Q)$ is an ideal of $\\mathbb {Z}_n$ .",
"First we show that $\\mathcal {L}(Q)$ is closed under addition.",
"Let $x=\\dots x_2 x_1 x_0 \\text{ and } y=\\dots y_1 y_1 y_0 \\in \\mathcal {L}(Q)$ and $ \\begin{array}{l l l l l}& \\cdots & x_2 & x_1 & x_0 \\\\+ & \\cdots & y_2 & y_1 & y_0 \\\\\\hline & \\cdots & z_2 & z_1 & z_0\\\\\\end{array}$ Let $k_0$ be large enough that $\\alpha ^{k_0}(Q+Q)\\subseteq Q$ .",
"By definition of $\\mathcal {L}(Q)$ , for any $t\\ge 0$ there exist (positive numbers) $ \\begin{array}{l}a=a_r\\cdots a_0.",
"x_{t+k_0} x_{t+k_0-1} \\cdots x_0 \\\\b=b_s\\cdots b_0.",
"y_{t+k_0} y_{t+k_0-1} \\cdots y_0 \\end{array}$ in $Q$ .",
"We see immediately that $a+b$ is given by $c_u\\cdots c_0.",
"z_{t+k_0} z_{t+k_0-1} \\cdots z_0$ for some $c_u, \\ldots , c_0 \\in \\lbrace 0,\\ldots , n-1\\rbrace $ .",
"This implies that $\\alpha ^{k_0}(a+b)=c_u\\cdots c_0 z_{t+k_0} z_{t+k_0-1}\\cdots z_{t+1}.",
"z_t \\cdots z_0\\in Q.$ Since $t$ is arbitrary, this implies that $z\\in \\mathcal {L}(Q)$ .",
"Now we show that $\\mathcal {L}(Q)$ is closed under multiplication by elements of $\\mathbb {Z}_n$ .",
"Let $x \\in \\mathcal {L}(Q) \\text{ and } p=\\ldots p_2 p_1 p_0 \\in \\mathbb {Z}_n.$ For every $t\\ge 0$ we have $p_t \\ldots p_1 p_0 \\cdot x = \\underbrace{x+ \\cdots + x}_{p_t \\ldots p_1 p_0 \\text{ times}}\\in \\mathcal {L}(Q)$ by the above paragraph.",
"Note that $p\\cdot x$ is the limit of the sequence $\\lbrace p_t\\ldots p_0 \\cdot x\\rbrace _{t=0}^\\infty \\subseteq \\mathcal {L}(Q)$ .",
"But by Lemma REF , $\\mathcal {L}(Q)$ is closed, so this implies that $p\\cdot x\\in \\mathcal {L}(Q)$ as well."
],
[
"From ideals to confining subsets",
"We next describe how to associate a subset of $H$ which is confining under the action of $\\alpha $ to an ideal of $\\mathbb {Z}_n$ .",
"For any ideal $\\mathfrak {b}$ of $\\mathbb {Z}_n$ , let $\\mathcal {S}(b)= \\left\\lbrace \\:{(-1)^\\delta x_r\\cdots x_0.",
"x_{-1} \\cdots x_{-s}\\in H \\,\\Bigg \\vert \\,\\begin{aligned} & \\delta \\in \\lbrace 0,1\\rbrace \\text{ and }\\exists b\\in \\mathfrak {b} \\text{ with } b= \\ldots c_2 c_1 x_{-1} \\ldots x_{-s} \\\\&\\text{ for some } c_1, c_2, \\ldots \\in \\lbrace 0,\\ldots , n-1\\rbrace \\end{aligned}}\\:\\right\\rbrace .$ Thus $\\mathcal {S}(\\mathfrak {b})$ is the set of elements of $H$ whose fractional parts appear as the tail end of digits of some element of the ideal $b$ .",
"Remark 3.6 Since $0\\in b$ for any ideal $b$ , it follows that $\\mathcal {S}(b)$ must contain $\\mathbb {Z}$ .",
"Lemma 3.7 In the notation above, $\\mathcal {S}(\\mathfrak {b})$ is confining under the action of $\\alpha $ .",
"We will check the conditions of Definition REF .",
"Let $x=x_s\\cdots x_1x_0.x_{-1}x_{-2}\\cdots x_{p(x)}\\in \\mathcal {S}(b).$ By definition of $\\mathcal {S}(b)$ , there is an element $b=\\dots c_2 c_1x_{-1}x_{-2}\\dots x_{p(x)}\\in b,$ and so $\\alpha (x)=x_s\\cdots x_0x_{-1}.x_{-2}\\cdots x_{p(x)}\\in \\mathcal {S}(b).$ Thus $\\alpha (\\mathcal {S}(b))\\subseteq \\mathcal {S}(b)$ , and Definition REF (a) holds.",
"Since $\\mathbb {Z}\\subseteq \\mathcal {S}(b)$ by Remark REF , we have $\\bigcup _{i=1}^\\infty \\alpha ^{-i}(\\mathcal {S}(b))=H$ , and so Definition REF (b) holds.",
"Let $x,y\\in \\mathcal {S}(b)$ .",
"We first deal with the case that $x$ and $y$ are both positive.",
"We want to show that $x+y\\in \\mathcal {S}(b)$ .",
"Let $x=x_r\\cdots x_0.x_{-1}\\cdots x_{p(x)},$ and $y=y_s\\cdots y_0.y_{-1}\\cdots y_{p(y)}$ , and assume without loss of generality that $p(x)\\le p(y)$ .",
"Then $x+y=z$ , where $z$ is given by $\\begin{array}{ccccccccccccccccccc}&&x_r&\\cdots & x_s&\\cdots &x_0 & .x_{-1}& x_{-2} & \\cdots & x_{p(y)} &\\cdots & \\cdots &\\cdots & x_{p(x)} \\\\+&& &&y_s&\\cdots &y_0& .y_{-1} & y_{-2} & \\cdots & y_{p(y)} & 0&\\cdots &0 &0\\\\ \\hline &z_t&\\cdots &\\cdots &\\cdots &\\cdots &z_0 &.z_{-1}&z_{-2}&\\cdots &z_{p(y)} &\\cdots &\\cdots &\\cdots &z_{p(z)},\\end{array}$ where here we've assumed without loss of generality that $r\\ge s$ (the same argument works if $r<s$ ).",
"By the definition of $\\mathcal {S}(b)$ , there exist $a,b\\in b$ with $a=\\dots x_{-1}x_{-2}\\dots x_{p(x)}$ and $b=\\dots y_{-1}y_{-2}\\dots y_{p(y)}$ .",
"Since $b$ is an ideal, $n^{p(y)-p(x)}b=\\dots y_{-1}y_{-2}\\dots y_{p(y)}\\underbrace{0\\dots \\dots \\dots \\dots 0}_{p(y)-p(x) \\textrm { times}}\\in b$ and $a+n^{p(y)-p(x)}b\\in b,$ where $a+n^{p(y)-p(x)}b$ is given by $\\begin{array}{cccccccccc}&\\dots & x_{-1}& x_{-2} & \\dots & x_{p(y)} & \\dots & \\dots & \\dots & x_{p(x)} \\\\+&\\dots & y_{-1} & y_{-2} & \\dots & y_{p(y)} & 0&\\dots &\\dots &0 \\\\ \\hline &\\dots &z_{-1}&z_{-2}&\\dots &z_{p(y)} &\\dots &\\dots &\\dots &z_{p(z)}.\\end{array}$ Therefore, by the definition of $\\mathcal {S}(b)$ , $z=x+y\\in \\mathcal {S}(b)$ .",
"If $x,y \\in \\mathcal {S}(b)$ are both negative, then we show in a completely analogous way that $x+y\\in \\mathcal {S}(b)$ .",
"We now consider the case that one of $x,y$ is positive and the other is negative.",
"By possibly multiplying $x+y$ by $-1$ , we assume without loss of generality that $x=x_r\\cdots x_0 .",
"x_{-1} \\cdots x_{p(x)}$ and $y=-y_s \\cdots y_0.",
"y_{-1}\\cdots y_{p(y)}$ with $x\\ge |y|$ so that also $r\\ge s$ .",
"Then $x+y=z$ , where $z$ is given by $\\begin{array}{cccccccccccccccc} & x_r & \\cdots &x_t & \\cdots & x_s &\\cdots & x_0 & .x_{-1} & x_{-2} & \\cdots & x_{p(y)} & \\cdots & \\cdots & x_{p(x)} \\\\- & & & & & y_s & \\cdots & y_0 & .y_{-1} & y_{-2} & \\cdots & y_{p(y)} & 0 & \\cdots & 0 \\\\\\hline & & & z_t & \\cdots & z_s & \\cdots & z_0 & .z_{-1} & z_{-2} & \\cdots & z_{p(y)} & \\cdots & \\cdots & z_{p(x)}\\end{array}$ (note, here we are assuming that $p(x)\\le p(y)$ but the argument is easily modified if $p(x)>p(y)$ ).",
"By definition of $\\mathcal {S}(b)$ , there exist elements $c,d\\in b$ with $c=\\ldots c_2 c_1 x_{-1} \\ldots x_{p(x)}$ and $d=\\ldots d_2 d_1 y_{-1} \\ldots y_{p(y)}$ .",
"Then $c-n^{p(y)-p(x)}b\\in b$ is given by $\\begin{array}{ccccccccc}\\end{array}& \\ldots & c_1 & x_{-1} & \\ldots & x_{p(y)} & \\ldots & \\ldots &x_{p(x)} \\\\-& \\ldots &d_1 & y_{-1} & \\ldots & y_{p(y)} & 0 & \\ldots & 0 \\\\\\hline & \\ldots & \\ldots & z_{-1} & \\ldots & z_{p(y)} & \\ldots & \\ldots & z_{p(x)}.",
"\\\\ $ $ Hence we see that $ S (b)$, as desired.$ By the above discussion, Definition REF (c) holds with $k_0=0$ .",
"We conclude that $\\mathcal {S}(b)$ is confining under the action of $\\alpha $ .",
"Lemma 3.8 Let $Q\\subseteq H$ be confining under the action of $\\alpha $ .",
"Then there exists $K>0$ such that $\\alpha ^K(\\mathbb {Z})\\subseteq Q$ .",
"By Lemma REF , every element of $\\mathbb {Z}=Q^+$ has uniformly bounded word length with respect to the the generating set $Q\\cup \\lbrace t^{\\pm 1}\\rbrace $ of $H$ .",
"Consider an element $w\\in \\mathbb {Z}$ .",
"By Lemma REF we may write $w$ as a reduced word $w=t^{-r} x_1\\ldots x_m t^r,$ where $r\\ge 0$ and $x_i\\in Q$ for all $i$ .",
"This gives us $w=\\alpha ^{-r}(x_1)+\\cdots +\\alpha ^{-r}(x_m).$ Since $\\Vert w\\Vert _{Q\\cup \\lbrace t^{\\pm 1}\\rbrace }$ is uniformly bounded, we have both $r$ and $m$ are uniformly bounded, say $r,m\\le R$ .",
"Hence we have $\\alpha ^R(w)=\\alpha ^{R-r}(x_1)+\\cdots + \\alpha ^{R-r}(x_m),$ and $\\alpha ^{R-r}(x_i)\\in Q$ for each $i$ .",
"Thus, $\\alpha ^R(w)\\in Q^R$ , where $Q^R$ represents the words of length at most $ R$ in $Q$ .",
"Consequently, $\\alpha ^{Rk_0}(\\alpha ^R(w))\\in \\alpha ^{Rk_0}(Q^R)\\subseteq Q$ where the last inclusion follows by Definition REF (c).",
"Thus we see that $\\alpha ^K(\\mathbb {Z})\\subseteq Q$ , where $K=Rk_0+R$ .",
"Lemma 3.9 Let $Q\\subseteq H$ be confining under the action of $\\alpha $ .",
"Then there exists $M>0$ such that $\\mathcal {S}(\\mathcal {L}(Q))\\subseteq \\alpha ^{-M}(Q)$ .",
"Let $a\\in \\mathcal {S}(\\mathcal {L}(Q))$ .",
"Since $Q$ and $\\mathcal {S}(\\mathcal {L}(Q))$ are symmetric, we may suppose that $a=a_r \\cdots a_0.a_{-1} \\cdots a_{-s}$ is positive.",
"By definition of $\\mathcal {S}(\\mathcal {L}(Q))$ there exists an element $x=\\ldots x_2 x_1 a_{-1} \\ldots a_{-s} \\in \\mathcal {L}(Q).$ Then by definition of $\\mathcal {L}(Q)$ , there exists an element $b=b_t \\cdots b_0.",
"a_{-1} \\cdots a_{-s}\\in Q.$ We may add an integer $c$ to $b$ to obtain $c+b=a_r\\cdots a_0.a_{-1} \\cdots a_{-s}=a,$ and by Lemma REF , we have $\\alpha ^K(c)\\in Q$ .",
"Thus, $\\alpha ^K(a)=\\alpha ^K(c+b)=\\alpha ^K(c)+\\alpha ^K(b)\\in Q+Q.$ Let $k_0$ be large enough that $\\alpha ^{k_0}(Q+Q)\\subseteq Q$ .",
"Then we have $\\alpha ^{K+k_0}(a)=\\alpha ^{k_0}(\\alpha ^K(a))\\in Q$ so the result holds with $M=K+k_0$ .",
"Recall that two ideals $a,b$ in $\\mathbb {Z}_n$ are equivalent (written $a\\sim b$ ) if there exists a constant $k$ such that $n^ka\\subseteq b$ and $n^kb\\subseteq a$ (see Definition REF ).",
"Lemma 3.10 Let $a,b$ be ideals of $\\mathbb {Z}_n$ such that $a\\sim b$ .",
"Then $\\mathcal {S}(a)=\\mathcal {S}(b)$ .",
"By definition, the ideal $b$ determines only the fractional parts of the elements in $\\mathcal {S}(b)$ , and if $b=\\dots b_2b_1 b_0\\in b$ , then there are elements $\\mathcal {S}(b)$ with fractional part $0.b_k\\cdots b_0$ for each $k\\ge 1$ and arbitrary integral part.",
"From this description, it is clear that for any $k$ , the elements $\\dots b_2b_1b_0 \\qquad \\textrm { and } \\qquad \\dots b_2b_1b_0\\underbrace{0\\dots 0}_{k \\textrm { times}}$ define the same set of fractional parts of elements in $\\mathcal {S}(b)$ .",
"Since there exists $k$ such that $n^k{b}\\subset {a}$ , we see that $\\mathcal {S}({b})\\subseteq \\mathcal {S}({a})$ .",
"By a symmetric argument, we also have $\\mathcal {S}({a})\\subseteq \\mathcal {S}({b})$ ."
],
[
"Actions on Bass-Serre trees",
"Lemma 3.11 For any ideal $a\\subseteq \\mathbb {Z}_n$ , the confining subset $\\mathcal {S}(\\mathfrak {a})$ is a subgroup of $H$ .",
"It follows from the proof of Lemma REF that $\\mathcal {S}(\\mathfrak {a})$ is closed under addition.",
"Moreover, by definition it is closed under additive inverses.",
"We consider a particular ascending HNN extension of $\\mathcal {S}(\\mathfrak {a})$ : $G(\\mathfrak {a})=\\langle \\mathcal {S}(\\mathfrak {a}), s \\mid sxs^{-1} = \\alpha (x) \\text{ for } x\\in \\mathcal {S}(\\alpha )\\rangle .$ Lemma 3.12 For any ideal $a\\subseteq \\mathbb {Z}_n$ , $BS(1,n)\\cong G(\\mathfrak {a}).$ By Remark REF , $\\mathbb {Z}\\subseteq \\mathcal {S}(a)$ .",
"There is an obvious homomorphism $f:G(\\mathfrak {a})\\rightarrow BS(1,n)$ defined by $x\\mapsto x \\text{ for } x\\in \\mathcal {S}(\\mathfrak {a}), \\qquad s\\mapsto t.$ This homomorphism is surjective because $BS(1,n)$ is generated by $\\mathbb {Z}\\subseteq \\mathcal {S}(\\mathfrak {a})$ and $t$ .",
"We now show that $f$ is injective.",
"Let $g \\in \\ker (f)$ .",
"By Lemma REF we can find an expression of $g$ as a minimal length word in the generating set $\\mathcal {S}(a)\\cup \\lbrace t^{\\pm 1}\\rbrace $ of the form $g=s^{-i}(x_1+\\cdots +x_w)t^j,$ where $i,j\\ge 0$ and $x_i\\in \\mathcal {S}(a)$ .",
"By Lemma REF we may write $x_1+\\dots +x_w=x\\in \\mathcal {S}(a)$ , and the result is that $g=s^{-i}xs^j$ .",
"We have $f(g)=t^{-i}xt^j=\\alpha ^{-i}(x)t^{j-i}=1.$ Since $\\alpha ^{-i}(x)\\in H$ , we obtain a contradiction unless $j=i$ .",
"In this case we have $f(g)=\\alpha ^{-i}(x)=0$ in $H$ , and since $\\alpha $ is an automorphism, $x=0$ .",
"But then $g=s^{-i} 0 s^i=\\alpha ^{-i}(0)=0$ in $G(\\mathfrak {a})$ .",
"This proves the statement.",
"Hence we have a description of $BS(1,n)$ as an HNN extension over the subgroup $\\mathcal {S}(\\mathfrak {a})\\le H$ and therefore an action of $BS(1,n)$ on the standard Bass-Serre tree associated to this HNN extension.",
"Denote this tree by $T(\\mathfrak {a})$ .",
"Proposition 3.13 The action of $BS(1,n)$ on $\\Gamma (BS(1,n),\\mathcal {S}(\\mathfrak {a})\\cup \\lbrace t^{\\pm 1}\\rbrace )$ is equivalent to the action of $BS(1,n)$ on $T(\\mathfrak {a})$ .",
"We apply the standard Schwarz-Milnor Lemma (see [1]).",
"The Bass-Serre tree $T(\\mathfrak {a})$ admits the following description.",
"The vertices of $T(\\mathfrak {a})$ are the left cosets of $\\mathcal {S}(\\mathfrak {a})$ , and two cosets $a\\mathcal {S}(\\mathfrak {a})$ and $b\\mathcal {S}(\\mathfrak {a})$ are joined by an edge if $a\\mathcal {S}(\\mathfrak {a})=bxt\\mathcal {S}(\\mathfrak {a}) \\quad \\text{ or }\\quad a\\mathcal {S}(\\mathfrak {a})=bxt^{-1}\\mathcal {S}(\\mathfrak {a}) \\quad \\text{ for some } x\\in \\mathcal {S}(\\mathfrak {a}).$ Consider the vertex $v=\\mathcal {S}(a)$ and the edge $E=[\\mathcal {S}(a),t\\mathcal {S}(a)]$ containing $v$ .",
"Clearly we have $\\bigcup _{g\\in BS(1,n)} gE=T(\\mathfrak {a})$ .",
"Hence by the Schwarz-Milnor lemma, the action of $BS(1,n)$ on $T(\\mathfrak {a})$ is equivalent to the action of $BS(1,n)$ on $\\Gamma (BS(1,n),S)$ where $S=\\lbrace g\\in BS(1,n)\\mid d(v,gv)\\le 3\\rbrace $ .",
"Note that $\\mathcal {S}(\\mathfrak {a})\\subseteq S$ since it fixes the vertex $v$ , and thus $[S\\cup \\lbrace t^{\\pm 1}\\rbrace ]\\preccurlyeq [\\mathcal {S}(\\mathfrak {a})\\cup \\lbrace t^{\\pm 1}\\rbrace ]$ .",
"We will show that also $[\\mathcal {S}(\\mathfrak {a})\\cup \\lbrace t^{\\pm 1}\\rbrace ]\\preccurlyeq [S\\cup t^{-1}] $ which will prove the proposition.",
"By the description of the vertices of $T(\\mathfrak {a})$ as cosets of $\\mathcal {S}(\\mathfrak {a})$ , any vertex in the radius 3 neighborhood of $v$ has the form $xt^\\delta v$ where $x\\in \\mathcal {S}(\\mathfrak {a})$ and $\\delta \\in \\lbrace \\pm 1\\rbrace $ or $x_1t^{\\delta _1} x_2 t^{\\delta _2} v$ where $x_i\\in \\mathcal {S}(\\mathfrak {a})$ for $i=1,2$ and $\\delta _i\\in \\lbrace \\pm 1\\rbrace $ for $i=1,2$ or $x_1t^{\\delta _1}x_2 t^{\\delta _2}x_3t^{\\delta _3}v$ where $x_i \\in \\mathcal {S}(\\mathfrak {a})$ for $i=1,2,3$ and $\\delta _i \\in \\lbrace \\pm 1\\rbrace $ for $i=1,2,3$ .",
"If $g\\in S$ , then it therefore sends $v$ to a vertex of one of the above three forms.",
"We deal with the last case explicitly, showing that $g$ has bounded word length in the generating set $\\mathcal {S}(\\mathfrak {a})\\cup \\lbrace t^{\\pm 1}\\rbrace $ .",
"The other two cases are entirely analogous.",
"If $gv=x_1t^{\\delta _1}x_2 t^{\\delta _2}x_3t^{\\delta _3}v$ then $(x_1t^{\\delta _1}x_2 t^{\\delta _2}x_3t^{\\delta _3})^{-1}g \\in \\operatorname{Stab}_{BS(1,n)}(v)=\\mathcal {S}(\\mathfrak {a}).$ Hence $g=x_1t^{\\delta _1}x_2t^{\\delta _2}x_3t^{\\delta _3}y$ for some $y\\in \\mathcal {S}(\\mathfrak {a})$ , and this shows that $g$ has word length at most 7 in the generating set $\\mathcal {S}(\\mathfrak {a})\\cup \\lbrace t^{\\pm 1}\\rbrace $ ."
],
[
"Subsets confining under the action of $\\alpha ^{-1}$",
"Let $Q^-$ be the subset defined in (REF ).",
"Lemma 3.14 Let $Q\\subseteq H$ be confining under the action of $\\alpha ^{-1}$ .",
"Then $\\alpha ^{-k}(Q^-)\\subseteq Q$ for some $k\\ge 0$ .",
"Let $k_0\\in \\mathbb {Z}_{\\ge 0}$ be large enough that $\\alpha ^{-k_0}(Q+Q)\\subseteq Q$ .",
"We may suppose that $k_0>0$ , for otherwise $Q$ is a subgroup of $H$ and it is easy to check that in fact $Q=H$ .",
"Since $H=\\bigcup _{k\\in \\mathbb {Z}_{\\ge 0}} \\alpha ^k(Q)$ , we may also choose $k_1$ large enough that $\\alpha ^{-k_1}(a)=an^{-k_1}\\in Q$ for any $a\\in \\lbrace 0,\\ldots , n-1\\rbrace $ .",
"We claim that any number of the form $\\sum _{i=0}^r a_i n^{-k_1-(i+1)k_0} \\text{ with } a_i \\in \\lbrace 0,\\ldots ,n-1\\rbrace \\text{ for all } 0\\le i\\le r$ (for any $r\\ge 0$ ) lies in $Q$ .",
"In other words, any number $0.\\underbrace{0\\cdots \\cdots \\cdots 0}_{k_1+k_0-1 \\text{ times}} a_0 \\underbrace{0\\cdots \\cdots 0}_{k_0 -1\\text{ times}} a_1 \\underbrace{0\\cdots \\cdots 0}_{k_0-1 \\text{ times}}a_2 \\underbrace{0\\cdots \\cdots 0}_{k_0-1 \\text{ times}} \\cdots a_r$ between 0 and 1 which may be written in base $n$ with $k_1+k_0-1$ 0's after the decimal point and then $k_0-1$ 0's between any consecutive potentially nonzero digits lies in $Q$ .",
"We will prove this by induction on $r$ .",
"The base case, when $r=0$ , follows since $a_0 n^{-k_1}\\in Q$ for any $a_0 \\in \\lbrace 0,\\ldots ,n-1\\rbrace $ and therefore $\\alpha ^{-k_0}(a_0 n^{-k_1})=a_0 n^{-k_1-k_0}\\in Q$ because $Q$ is closed under the action of $\\alpha ^{-1}$ .",
"Suppose that the claim is true for all $r<s$ .",
"Consider a number $x=\\sum _{i=0}^s a_i n^{-k_1-(i+1)k_0}$ with $a_i\\in \\lbrace 0,\\ldots ,n-1\\rbrace $ for all $i$ .",
"We may write $x=a_0n^{-k_1-k_0}+\\sum _{i=1}^s a_in^{-k_1-(i+1)k_0}.$ We have that $a_0n^{-k_1}\\in Q$ and by induction, $\\sum _{i=1}^s a_i n^{-k_1-ik_0}=\\sum _{i=0}^{s-1} a_{i+1}n^{-k_1-(i+1)k_0} \\in Q.$ Hence, $x= a_0n^{-k_1-k_0}+\\sum _{i=1}^s a_in^{-k_1-(i+1)k_0}=\\alpha ^{-k_0}\\left(a_0n^{-k_1}+\\sum _{i=1}^s a_in^{-k_1-ik_0}\\right)\\in Q.$ This proves the claim.",
"Now consider a number of the form $x=\\sum _{i=0}^r a_i n^{-k_1-k_0-i}$ .",
"In other words, $x=0.\\underbrace{0\\cdots \\cdots \\cdots 0}_{k_1 +k_0-1 \\text{ times}}a_0 a_1 \\cdots a_r.$ We may write $x=\\sum _{j=0}^{k_0-1} \\sum _{\\begin{array}{c} i\\in \\mathbb {Z}_{\\ge 0} \\\\ j+ik_0\\le r\\end{array}} a_{j+ik_0} n^{-(k_1+(i+1)k_0+j)}=\\sum _{j=0}^{k_0-1} x_j,$ where $x_j=\\sum _{\\begin{array}{c} i\\in \\mathbb {Z}_{\\ge 0} \\\\ j+ik_0\\le r\\end{array}} a_{j+ik_0} n^{-(k_1+(i+1)k_0+j)}.$ In other words, we are writing $\\begin{tabular}{l l l l l l l l l l l l l l l l l}x & = & 0& .",
"& 0& \\cdots & 0 & a_0 &0 & 0 & \\cdots & 0 & 0 & a_{k_0} & 0 &0 & \\cdots \\\\& + & 0& .",
"& 0& \\cdots & 0 & 0 & a_{1} &0 & \\cdots & 0 & 0& 0 & a_{1+k_0}&0 & \\cdots \\\\& + & 0 & .",
"& 0 & \\cdots & 0 & 0 & 0 & a_{2} & \\cdots & 0 & 0 & 0 & 0 & a_{2+k_0} & \\cdots \\\\& + & \\cdots & & & & & & & & & & & & & & \\\\& + & 0 & .",
"& 0 & \\cdots & 0 & 0 & 0 & 0 & \\cdots & 0 & a_{(k_0-1)} &0 & 0 & 0 & \\cdots \\\\&\\end{tabular}$ with the summands being $x_0,x_1,x_2,\\ldots ,$ and $x_{k_0-1}$ , respectively.",
"Notice that $x_j=\\sum _{\\begin{array}{c}i \\in \\mathbb {Z}_{\\ge 0} \\\\j+ik_0\\le r\\end{array}} a_{j+ik_0} n^{-(k_1+(i+1)k_0+j)}=\\alpha ^{-j}\\left(\\sum _{\\begin{array}{c}i \\in \\mathbb {Z}_{\\ge 0} \\\\j+ik_0\\le r\\end{array}} a_{j+ik_0} n^{-(k_1+(i+1)k_0)}\\right)=\\alpha ^{-j}(y_j),$ where $y_j=\\sum _{\\begin{array}{c}i \\in \\mathbb {Z}_{\\ge 0} \\\\j+ik_0\\le r\\end{array}} a_{j+ik_0} n^{-(k_1+(i+1)k_0)}$ and by our claim, $y_j\\in Q$ for each $j\\in \\lbrace 0,\\ldots ,k_0-1\\rbrace $ .",
"Since $Q$ is closed under the action of $\\alpha ^{-1}$ , we have $x_j\\in Q$ for each $j$ .",
"Therefore $x\\in Q^{k_0}$ .",
"Using that $\\alpha ^{-k_0}(Q+Q)\\subseteq Q$ , we have $\\alpha ^{-k_0^2}(x)=\\alpha ^{-k_0 \\cdot k_0}(x)\\in Q$ .",
"Finally, for any positive number $y=\\sum _{i=0}^r a_i n^{-i-1} \\in Q^-$ , we have $\\alpha ^{-k_1-k_0+1}(y)=\\sum _{i=0}^r a_i n^{-k_1-k_0-i}.$ By our work in the last paragraph, this proves that $\\alpha ^{-k_0^2}(\\alpha ^{-k_1-k_0+1}(y))\\in Q.$ In other words, $\\alpha ^{-k_0^2-k_0-k_1+1}(y)\\in Q$ .",
"We have shown that $\\alpha ^{-k}(y)\\in Q$ for any positive $y\\in Q^-$ where $k=k_0^2+k_0+k_1-1$ .",
"Since $Q^-$ and $Q$ are symmetric, this proves that for any negative $y\\in Q^-$ we have $\\alpha ^{-k}(y)=-\\alpha ^{-k}(-y)\\in Q$ .",
"In other words, $\\alpha ^{-k}(Q^-)\\subset Q$ where $k=k_0^2+k_0+k_1-1$ .",
"Proposition 3.15 Let $Q\\subseteq H$ be strictly confining under the action of $\\alpha ^{-1}$ .",
"Then $[Q\\cup \\lbrace t^{\\pm 1}\\rbrace ]=[Q^-\\cup \\lbrace t^{\\pm 1}\\rbrace ]$ .",
"By the above, we have $[Q\\cup \\lbrace t^{\\pm 1}\\rbrace ] \\preccurlyeq [Q^-\\cup \\lbrace t^{\\pm 1}\\rbrace ]$ .",
"Suppose that the inequality is strict.",
"We will show then that $[Q\\cup \\lbrace t^{\\pm 1}\\rbrace ]=\\left[H\\cup \\lbrace t^{\\pm 1}\\rbrace \\right]$ and this will contradict that $Q$ is strictly confining.",
"By Lemma REF and Lemma REF , we may suppose that $Q^-\\subseteq Q$ .",
"If there exists $k$ large enough that $\\alpha ^{-k}(x)\\in Q^-$ for all $x\\in Q$ , then we have $[Q\\cup \\lbrace t^{\\pm 1}\\rbrace ]=[Q^-\\cup \\lbrace t^{\\pm 1}\\rbrace ]$ , as desired.",
"Otherwise, there exist numbers $x=a_r\\cdots a_0.a_{-1} \\cdots a_{p(x)}\\in Q$ with $r$ arbitrarily large and $a_r\\ne 0$ .",
"We will prove in this case that $n^t\\in Q$ for $t$ arbitrarily large.",
"This will complete the proof, for in this case, for any $a\\in \\lbrace 0,\\ldots ,n-1\\rbrace $ we will also have $an^t\\in Q$ for any $t$ , by the standard arguments.",
"Hence given a positive number $y=\\sum _{i=-p}^q a_i n^i\\in H$ we have $\\sum _{i=-p}^q a_i n^{i+k_0(p+q+1)}\\in Q^{p+q+1}$ and therefore $\\alpha ^{-(p+q+1)k_0}\\left(\\sum _{i=-p}^q a_i n^{i+k_0(p+q+1)}\\right)=y\\in Q.$ Since $Q$ is symmetric, this will prove that $H\\subseteq Q$ .",
"Consider a number $x=a_r\\cdots a_0.a_{-1} \\cdots a_{p(x)}\\in Q$ with $r\\gg 0$ (how large will be determined later in the proof).",
"We clearly have $n^{r-1} \\le x<n^{r+1}$ .",
"For any $s>0$ we have $n^sx=a_r\\cdots a_0 \\cdots a_{-s}.",
"a_{-s-1}\\cdots a_{p(x)}\\ge n^{r+s}=1\\underbrace{0\\ldots \\dots 0}_{r+s \\text{ times}}$ (with the decimal point to the right of $a_{p(x)}$ if $s>p(x)$ ).",
"We claim that there exists $c\\in \\lbrace 0,1,\\ldots , n^s\\rbrace $ such that $cx=1\\underbrace{0\\cdots \\cdots 0}_{s-2 \\text{ times}} b_r b_{r-1} \\cdots b_0.b_{-1} \\cdots b_{p(cx)}.$ If this is not the case, then since $n^sx \\ge n^{r+s}$ there exists some $c\\in \\lbrace 0,\\ldots ,n^s\\rbrace $ with $cx<1\\underbrace{0\\ldots \\dots \\dots 0}_{r+s-1 \\text{ times}}\\text{ but } (c+1)x\\ge 1\\underbrace{0\\ldots \\dots 0}_{s-3 \\text{ times}} 1 \\underbrace{0 \\ldots \\dots 0}_{r+1 \\text{ times}}.$ From these inequalities we see that $x=(c+1)x-cx \\ge 1\\underbrace{0 \\ldots \\dots 0}_{r+1 \\text{ times}}=n^{r+1}$ which is a contradiction.",
"So choose $c\\in \\lbrace 0,1,\\ldots ,n^s\\rbrace $ with $cx=1\\underbrace{0\\cdots \\cdots 0}_{s-2 \\text{ times}} b_r b_{r-1} \\cdots b_0.b_{-1} \\cdots b_{p(cx)}.$ Assuming that $r> n^sk_0$ , we have $\\alpha ^{-ck_0}(cx)=1\\underbrace{0\\cdots \\cdots 0}_{s-2 \\text{ times}} b_r b_{r-1} \\cdots b_{ck_0}.",
"b_{ck_0-1}\\cdots b_0b_{-1} \\cdots b_{p(cx)}\\in Q.$ Since $Q$ is closed under the action of $\\alpha ^{-1}$ , we then also have $1\\underbrace{0\\cdots \\cdots 0}_{s-2 \\text{ times}}.b_r b_{r-1} \\cdots b_{p(cx)}\\in Q.$ Since $Q^-\\subseteq Q$ , we then have $10\\ldots 0=n^{s-2}\\in Q+Q$ and therefore $n^{s-2-k_0}\\in Q$ .",
"But since $s$ is arbitrarily large, we then have $n^t\\in Q$ for $t$ arbitrarily large.",
"This completes the proof."
],
[
"The action on $\\mathbb {H}^2$",
"There is a well known action of $BS(1,n)$ on $\\mathbb {H}^2$ defined via the representation $BS(1,n)\\rightarrow \\operatorname{PSL}(2,\\mathbb {R}),$ where $a \\mapsto \\begin{pmatrix} 1 & 1 \\\\ 0 & 1 \\end{pmatrix},\\quad t\\mapsto \\begin{pmatrix} n^{1/2} & 0 \\\\ 0 & n^{-1/2} \\end{pmatrix}.$ The restriction of this representation to $H=\\langle t^k a t^{-k} \\mid k\\in \\mathbb {Z}\\rangle $ is given by $r\\mapsto \\begin{pmatrix} 1 & r \\\\ 0 & 1 \\end{pmatrix}$ where we identify $H$ as a subset of $\\mathbb {R}$ in the usual way.",
"Proposition 3.16 The action of $BS(1,n)$ on $\\mathbb {H}^2$ is equivalent to the action of $BS(1,n)$ on $\\Gamma (BS(1,n),Q^- \\cup \\lbrace t^{\\pm 1}\\rbrace )$ .",
"We will apply the Schwarz-Milnor Lemma.",
"See [1], Lemma 3.11.",
"We consider the upper half plane model of $\\mathbb {H}^2$ .",
"We show first that the orbit of $i$ under $BS(1,n)$ is $(\\log (n)+1)$ –dense in $\\mathbb {H}^2$ .",
"For this, note first that the orbit of $i$ under $\\langle a \\rangle $ is 1–dense in the horocycle $\\lbrace z\\in \\mathbb {H}^2\\mid \\Im (z)=1\\rbrace $ .",
"This follows easily from the fact that $d(i,ai)=d(i,1+i)<1$ .",
"Now, for any $k\\in \\mathbb {Z}$ , $t^k$ takes the horocycle $\\lbrace z\\in \\mathbb {H}^2\\mid \\Im (z)=1\\rbrace $ isometrically to the horocycle $\\lbrace z\\in \\mathbb {H}^2 \\mid \\Im (z)=n^k\\rbrace $ .",
"Hence the orbit of $i$ is 1–dense in the horocycle $\\lbrace z\\in \\mathbb {H}^2 \\mid \\Im (z)=n^k\\rbrace $ for any $k\\in \\mathbb {Z}$ .",
"Moreover, for any $k\\in \\mathbb {Z}$ , the distance between the horocycles $\\lbrace z\\in \\mathbb {H}^2 \\mid \\Im (z)=n^k\\rbrace $ and $\\lbrace z\\in \\mathbb {H}^2 \\mid \\Im (z)=n^{k+1}\\rbrace $ is exactly $\\log (n)$ .",
"Hence, any $z\\in \\mathbb {H}^2$ has distance at most $ \\log (n)$ from a horocycle $\\lbrace z\\in \\mathbb {H}^2 \\mid \\Im (z)=n^k\\rbrace $ for some $k\\in \\mathbb {Z}$ .",
"By the triangle inequality, $z$ has distance at most $ \\log (n)+1$ from some point in the orbit of $i$ .",
"This clearly proves that $\\bigcup _{g\\in BS(1,n)}g B_{\\log (n)+1}(i)=\\mathbb {H}^2$ .",
"Therefore by [1], the action $BS(1,n)\\curvearrowright \\mathbb {H}^2$ is equivalent to the action $BS(1,n)\\curvearrowright \\Gamma (BS(1,n),S)$ where $S=\\lbrace g\\in BS(1,n) \\mid d_{\\mathbb {H}^2}(i,gi)\\le 2\\log (n)+3\\rbrace .$ We will prove that $[S\\cup \\lbrace t^{\\pm 1}\\rbrace ]=[Q^- \\cup \\lbrace t^{\\pm 1}\\rbrace ]$ , which will finish the proof.",
"Let $g\\in S$ .",
"We may write $g=rt^k$ where $r\\in H$ and $k\\in \\mathbb {Z}$ .",
"Observe that $d(i,gi)\\ge |k|\\log (n)$ .",
"Since $g\\in S$ , we have $|k|\\log (n)\\le 2\\log (n)+3,$ and hence $|k|\\le \\frac{2\\log (n)+3}{\\log (n)}\\le \\frac{2\\log (2)+3}{\\log (2)}<7.$ We have $gi=n^ki+r$ , and hence $d(i,gi)=2\\operatorname{arcsinh}\\left(\\frac{1}{2} \\sqrt{\\frac{r^2+(n^k-1)^2}{n^k}}\\right)\\ge 2\\operatorname{arcsinh}\\left(\\frac{1}{2} \\sqrt{\\frac{r^2}{n^7}}\\right).$ This clearly defines an upper bound on $|r|$ , and hence there exists a uniform $l>0$ (that is, independent of $r$ ) such that $|\\alpha ^{-l}(r)|<1$ .",
"For such $l$ , we clearly have $\\alpha ^{-l}(r)\\in Q^-$ .",
"To summarize, we have $g=rt^k$ , where $|k|<7$ and $\\alpha ^{-l}(r)\\in Q^-$ .",
"In other words, there exists $s\\in Q^-$ such that $g=\\alpha ^l(s)t^k=t^lst^{-l}t^k.$ This proves that $\\Vert g\\Vert _{Q^-\\cup \\lbrace t^{\\pm 1}\\rbrace }\\le 2l+|k|+1<2l+8$ .",
"Thus, $[Q^-\\cup \\lbrace t^{\\pm 1}\\rbrace ]\\preccurlyeq [S\\cup \\lbrace t^{\\pm 1}\\rbrace ].$ On the other hand, any element $s\\in Q^-$ has $d(i,si)< 1<2\\log (n)+3$ , so we automatically have $s\\in S$ .",
"So $Q^-\\cup \\lbrace t^{\\pm 1}\\rbrace \\subseteq S\\cup \\lbrace t^{\\pm 1}\\rbrace $ and this proves $[S\\cup \\lbrace t^{\\pm 1}\\rbrace ] \\preccurlyeq [Q^- \\cup \\lbrace t^{\\pm 1}\\rbrace ].$"
],
[
"Quasi-parabolic structures",
"Lemma 4.1 The commutator subgroup of $BS(1,n)$ has index $n-1$ in $H$ .",
"The abelianization of $BS(1,n)$ is given by the obvious homomorphism $BS(1,n)=\\langle a, t : tat^{-1}=a^n\\rangle \\rightarrow \\langle a, t:[a,t]=1, a=a^n\\rangle = \\mathbb {Z}\\oplus \\mathbb {Z}/((n-1)\\mathbb {Z}).$ The kernel of this homomorphism is $[BS(1,n),BS(1,n)]$ whereas $H$ is the kernel of the composition $BS(1,n)\\rightarrow \\mathbb {Z}\\oplus \\mathbb {Z}/((n-1)\\mathbb {Z}) \\rightarrow \\mathbb {Z}.$ The lemma follows easily from this.",
"The proof of Proposition REF is a modification to the proofs of [3] and [4].",
"We recall the statement for the reader's convenience: * Given a strictly confining subset $Q$ , a quasi-parabolic structure is constructed in [4].",
"It remains to prove the forward direction.",
"Let $[T]\\in \\mathcal {H}_{qp}(BS(1,n))$ .",
"Fix a sequence $\\mathbf {x}=(x_n)$ in $\\Gamma (G,T)$ , let $q_\\mathbf {x}\\colon G\\rightarrow \\mathbb {R}$ be the associated quasi-character, and let $\\rho \\colon G\\rightarrow \\mathbb {R}$ be the Busemann pseudocharacter (see Section REF for definitions).",
"Since $BS(1,n)$ is amenable, $\\rho $ is a homomorphism and by Lemma REF , $\\rho (h)=0$ for all $h\\in H$ .",
"Moreover, we must have $\\rho (t)\\ne 0$ , else $\\rho (g)=0$ for all $g\\in BS(1,n)$ .",
"Claim 4.2 There exist constants $r_0,n_0\\ge 0$ such that $Q=\\bigcup _{i=1}^{n_0-1}\\alpha ^i(B(1,r_0)\\cap H)$ is confining under the action of $\\alpha $ or $\\alpha ^{-1}$ .",
"[Proof of Claim] As $[T]\\in \\mathcal {H}_{qp}(BS(1,n))$ , the group $BS(1,n)$ fixes a single point $\\xi \\in \\partial \\Gamma (BS(1,n),T)$ .",
"As $\\rho (t)\\ne 0$ , $t$ acts as a hyperbolic isometry of $\\Gamma (G,T)$ , and thus either $t$ or $t^{-1}$ has $\\xi $ as its repelling point.",
"Assume without loss of generality that it is $t$ ; we will show that $Q$ is strictly confining under the action of $\\alpha $ .",
"On the other hand, if we assume that $t^{-1}$ has $\\xi $ as its repelling point, then an analogous proof will show that $Q$ is strictly confining under the action of $\\alpha ^{-1}$ .",
"First note that the sequence $(1,t^{-1},t^{-2},\\dots )$ defines a $K$ -quasi-geodesic ray in $\\Gamma (G,T)$ for some $K$ , and thus so does the sequence $(g,gt^{-1},gt^{-2},\\dots )$ for any $g\\in BS(1,n)$ .",
"Recall that there is a constant $r_0$ , depending only on the hyperbolicity constant of $\\Gamma (G,T)$ and $K$ such that any two $K$ -quasi-geodesic rays with the same endpoint on $\\partial \\Gamma (G,T)$ are eventually $r_0$ –close to each other.",
"In particular, if $\\rho (g)\\le C$ , then there is a constant $n_0=n_0(d_T(1,g))$ such that $ d_T(t^{-n},gt^{-n})\\le r_0+C$ for all $n\\ge n_0$ .",
"Fix $n_0=n_0(r_0)$ , and define $Q=\\bigcup _{i=0}^{n_0-1}\\alpha ^i(B(1,r_0)\\cap H).$ Choose $r_1$ such that $Q\\subseteq B(1,r_1)$ .",
"Note that such an $r_1$ exists since for any $i$ and any $h\\in B(1,r_0)\\cap H$ we have $\\Vert \\alpha ^i(h)\\Vert _T=\\Vert t^iht^{-i}\\Vert _T \\le r_0 +2i\\Vert t\\Vert _T.$ Let $n_1=n_0(2r_1)$ .",
"For any $h\\in B(1,r_0)\\cap H$ , we have $d_T(1,h)\\le r_0$ and $\\rho (h)=0$ , and so it follows from (REF ) that for all $n\\ge n_0$ , $d_T(\\alpha ^n(h),1)=d_T(t^{n}ht^{-n},1)=d_X(ht^{-n},t^{-n})\\le r_0.$ Therefore, $\\alpha ^n(h)\\in B(1,r_0)\\cap H$ , and thus for all $n\\ge n_0$ , $\\alpha ^{n}(B(1,r_0)\\cap H)\\subseteq B(1,r_0)\\cap H.$ We now check the conditions of Definition REF .",
"Let $h\\in Q$ , so that $h\\in \\alpha ^i(B(1,r_0)\\cap H)$ for some $0\\le i\\le n_0-1$ .",
"If $i<n_0-1$ , then $\\alpha (h)\\in \\alpha ^{i+1}(B(1,r_0)\\cap H)\\subseteq Q$ .",
"On the other hand, if $i=n_0-1$ , then $\\alpha (h)\\in \\alpha ^{n_0}(B(1,r_0)\\cap H)\\subseteq B(1,r_0)\\cap H\\subseteq Q$ .",
"Therefore condition (a) holds.",
"For any $h\\in H$ , there is a constant $n_h=n_0(d_T(1,h))$ such that $\\alpha ^{n_h}(h)\\in B(1,r_0)\\cap H\\subseteq Q$ .",
"Therefore $H=\\bigcup _{i=0}^\\infty \\alpha ^{-i}(Q)$ and (b) holds.",
"Finally, $Q+Q\\subseteq B(1,2r_1)$ , and so $\\alpha ^{n_1}(Q+Q)\\subseteq B(1,r_0)\\cap H\\subseteq Q$ , and (c) holds with constant $n_1$ .",
"Therefore, $Q$ is confining under the action of $\\alpha $ , concluding the proof of the claim.",
"Let $S=Q\\cup \\lbrace t^{\\pm 1}\\rbrace $ .",
"We will show that the map $\\iota \\colon (BS(1,n),d_S)\\rightarrow (BS(1,n),d_T)$ is a quasi-isometry.",
"Since $\\sup _{s\\in S}d_X(1,s)<\\infty $ , the map $\\iota $ is Lipschitz.",
"Thus it suffices to show that for any bounded subset $B\\subseteq \\Gamma (BS(1,n),T)$ , we have $\\sup _{b\\in B}d_S(1,b)<\\infty $ .",
"Fix any $M>0$ and let $B\\subseteq \\Gamma (BS(1,n),T)$ be such that $d_T(1,b)\\le M$ for all $b\\in B$ .",
"For each $b\\in B$ , we have $b=ht^k$ for some $h\\in H$ and some $k$ .",
"By the definition of $q_\\mathbf {x}$ (see (REF )), we have $q_\\mathbf {x}(g)\\le d_T(1,g)$ , and since there exists a constant $D$ (the defect of $q_\\mathbf {x}$ ) such that $|\\rho (g)-q_\\mathbf {x}(g)|\\le D$ , we have $-d_T(1,g)-D\\le \\rho (g)\\le d_T(1,g)+D.$ Consequently $\\rho $ maps bounded subsets of $\\Gamma (G,T)$ to bounded subsets of $\\mathbb {R}$ .",
"Since $\\rho (ht^n)=\\rho (h)+n\\rho (t)=n\\rho (t)$ , for all $n$ , it follows that there is a constant $K$ such that for any $b$ with $d_T(1,b)=d_T(1,ht^k)\\le M$ we have $k\\le K$ .",
"This implies that $d_T(1,h)\\le d_T(1,ht^k)+kd_T(1,t)\\le M+Kd_T(1,t).$ Hence by (REF ), there is a uniform $N$ such that $\\alpha ^N(h)\\in Q$ and therefore $d_S(1,h)\\le 2N+1$ .",
"Therefore, $d_S(1,b)=d_S(1,ht^k)\\le d_S(1,h)+k\\le 2N+1+K.$ Since the map $\\iota $ is the identity map on vertices, it is clearly surjective, and therefore it is a quasi-isometry.",
"Finally, $Q$ is strictly confining under the action of $\\alpha $ .",
"Indeed, if $Q$ is confining but not strictly confining under the action of $\\alpha $ , then $[Q\\cup \\lbrace t^{\\pm 1}\\rbrace ]=[T]\\in \\mathcal {H}_\\ell (BS(1,n))$ , which is a contradiction.",
"Recall that given an ideal $a$ of $\\mathbb {Z}_n$ , there is an associated subset $\\mathcal {S}(a)\\subseteq H$ defined in (REF ) which is confining under the action of $\\alpha $ .",
"Recall also that $\\mathfrak {a}=\\mathfrak {a}_1\\times \\cdots \\mathfrak {a}_k$ is full if $\\mathfrak {a}_j$ is either $(0)$ or $\\mathbb {Z}_{p_j}$ for every $j=1,\\dots , k$ (see Definition REF ).",
"Lemma 4.3 Let $a,b$ be full ideals of $\\mathbb {Z}_n$ such that $a\\lnot \\le b$ and $b\\lnot \\le a$ .",
"Then $[\\mathcal {S}(a)\\cup \\lbrace t^{\\pm 1}\\rbrace ]$ and $[\\mathcal {S}(b)\\cup \\lbrace t^{\\pm 1}\\rbrace ]$ are incomparable.",
"Since $a=a_1\\times \\cdots \\times a_k$ and $b=b_1\\times \\cdots \\times b_k$ are full ideals of $\\mathbb {Z}_n$ , there exist $1\\le i,j\\le k$ such that $a_i=(0)$ , $a_j=\\mathbb {Z}_{p_j^{n_j}}$ , $b_i=\\mathbb {Z}_{p_i^{n_i}}$ , and $b_j=(0)$ .",
"Consider $\\mathcal {S}(a)$ and $\\mathcal {S}(b)$ .",
"For any $x=\\pm x_r\\cdots x_0.x_{-1}\\cdots x_{-p(x)}\\in \\mathcal {S}(a)$ , there is an element $a=\\dots x_{-1}\\dots x_{-p(x)}\\in a$ .",
"Similarly, for any $y=\\pm y_m\\cdots y_0.y_{-1}\\cdots y_{-p(y)}\\in \\mathcal {S}(b)$ , there is an element $b=\\dots y_{-1}\\dots y_{-p(y)} \\in b$ .",
"Since $a_i=(0)$ , $x_{-p(x)}\\equiv 0\\mod {p}_i^{n_i},$ and since $b_j=(0)$ , $y_{-p(y)}\\equiv 0\\mod {p}_j^{n_j},$ by Lemma REF .",
"For any $K\\ge 0$ , choose $x\\in \\mathcal {S}(a)$ such that $p(x)=K$ and $c(x)\\lnot \\equiv 0\\mod {p}_j^{n_j}$ , which is possible since $a_j=\\mathbb {Z}_{p_j^{n_j}}$ .",
"By Lemma REF we can find an expression of $x$ as a minimal length word in the generating set $\\mathcal {S}(b)\\cup \\lbrace t^{\\pm 1}\\rbrace $ of the form $x=t^{-u}(g_1+\\cdots +g_w)t^u.$ By Lemma REF we may write $g_1+\\dots +g_w=g\\in \\mathcal {S}(b)$ , and the result is that $x=\\alpha ^{-u}(g)$ .",
"Now, since $g\\in \\mathcal {S}(b)$ and $b_j=(0)$ , it follows that if $p(g)<0$ , then $c(g)\\equiv 0\\mod {p}_j^{n_j}$ .",
"But this implies that $c(x)=c(\\alpha ^{-u}(g))\\equiv 0\\mod {p}_j^{n_j}$ , which is a contradiction.",
"Consequently, we must have $p(g)\\ge 0$ , which implies that $u\\ge K$ .",
"Therefore $\\Vert x\\Vert _{\\mathcal {S}(b)\\cup \\lbrace t^{\\pm 1}\\rbrace }\\ge 2K+1.$ Since $K$ was arbitrary, it follows that $\\sup _{x\\in \\mathcal {S}(a)\\cup \\lbrace t^{\\pm 1}\\rbrace }\\Vert x\\Vert _{\\mathcal {S}(b)\\cup \\lbrace t^{\\pm 1}\\rbrace }=\\infty ,$ and thus $[\\mathcal {S}(a)\\cup \\lbrace t^{\\pm 1}\\rbrace ]\\lnot \\preccurlyeq [\\mathcal {S}(b)\\cup \\lbrace t^{\\pm 1}\\rbrace ].$ A similar argument shows that $[\\mathcal {S}(a)\\cup \\lbrace t^{\\pm 1}\\rbrace ]\\lnot \\succcurlyeq [\\mathcal {S}(b)\\cup \\lbrace t^{\\pm 1}\\rbrace ],$ and the result follows.",
"Lemma 4.4 Let $a,b$ be full ideals of $\\mathbb {Z}_n$ , and suppose $a\\lneq b$ .",
"Then $[\\mathcal {S}(a)\\cup \\lbrace t^{\\pm 1}\\rbrace ]\\succnsim [\\mathcal {S}(b)\\cup \\lbrace t^{\\pm 1}\\rbrace ]$ .",
"Since $a\\lneq b$ and $a=a_1\\times \\cdots \\times a_k,b=b_1\\times \\cdots \\times b_k$ are both full ideals of $\\mathbb {Z}_n$ , we have $a\\subsetneq b$ .",
"Therefore, there exists some $1\\le i\\le k$ such that $a_i=(0)$ while $b_i=\\mathbb {Z}_{p_i^{n_i}}$ .",
"Consequently, for all $a=\\dots a_2a_1a_0\\in a$ , we have $a_0\\equiv 0\\mod {p}_i^{n_i}$ by Lemma REF .",
"We first show that $\\mathcal {S}(a)\\subseteq \\mathcal {S}(b)$ .",
"Let $x=\\pm x_r\\cdots x_1x_0.x_{-1}\\cdots x_{-p(x)}\\in \\mathcal {S}(a)$ .",
"Then there exists some $a\\in a$ such that $a=\\dots x_{-1}\\dots x_{-p(x)}$ .",
"Since $a\\subseteq b$ , we have $a\\in b$ , and so by the definition of $\\mathcal {S}(b)$ , it follows that $x\\in \\mathcal {S}(b)$ .",
"Therefore $S(a)\\cup \\lbrace t^{\\pm 1}\\rbrace \\subseteq \\mathcal {S}(b)\\cup \\lbrace t^{\\pm 1}\\rbrace $ , and $[S(a)\\cup \\lbrace t^{\\pm 1}\\rbrace ]\\succcurlyeq [\\mathcal {S}(b)\\cup \\lbrace t^{\\pm 1}\\rbrace ].$ Finally, an argument analogous to the proof of Lemma REF shows that $[\\mathcal {S}(a)\\cup \\lbrace t^{\\pm 1}\\rbrace ]\\ne [\\mathcal {S}(b)\\cup \\lbrace t^{\\pm 1}\\rbrace ]$ , completing the proof.",
"Recall that given a subset $Q\\subseteq H$ which is confining under the action of $\\alpha $ , there is an associated ideal $\\mathcal {L}(Q)$ of $\\mathbb {Z}_n$ defined by (REF ).",
"Lemma 4.5 For any confining subset $Q$ under the action of $\\alpha $ , we have $[Q\\cup \\lbrace t^{\\pm 1}\\rbrace ]= [\\mathcal {S}(\\mathcal {L}(Q))\\cup \\lbrace t^{\\pm 1}\\rbrace ]$ .",
"By Lemma REF , we have that $\\mathcal {S}(\\mathcal {L}(Q)) \\subseteq \\alpha ^{-M}(Q)$ for some $M$ .",
"Note that this implies that $[Q\\cup \\lbrace t^{\\pm 1}\\rbrace ]\\preccurlyeq [\\mathcal {S}(\\mathcal {L}(Q))\\cup \\lbrace t^{\\pm 1}\\rbrace ].$ We will show that $[\\mathcal {S}(\\mathcal {L}(Q))\\cup \\lbrace t^{\\pm 1}\\rbrace ]\\preccurlyeq [Q\\cup \\lbrace t^{\\pm 1}\\rbrace ]$ .",
"Suppose this is not the case.",
"Then $Q\\lnot \\subseteq \\alpha ^{-k}(\\mathcal {S}(\\mathcal {L}(Q)))$ for any $k$ and hence there exist elements $a=a_r \\cdots a_0.",
"a_{-1} \\cdots a_{-s}\\in Q$ with $\\inf \\lbrace k : \\alpha ^k(a)\\in \\mathcal {S}(\\mathcal {L}(Q))\\rbrace $ arbitrarily large.",
"Let $a$ be an element as above, let $\\ell =\\inf \\lbrace k : \\alpha ^k(a)\\in \\mathcal {S}(\\mathcal {L}(Q))\\rbrace ,$ and assume $\\ell >k_0$ , where $k_0$ is large enough that $\\alpha ^{k_0}(Q+Q)\\subseteq Q$ .",
"Let $t\\le s$ be largest with the property that there does not exist an element of the form $\\dots a_{-t} \\dots a_{-s} \\in \\mathcal {L}(Q).$ Note that $t> \\ell $ .",
"For otherwise, we would have $\\alpha ^{\\ell -1}(a)= a_r \\cdots a_0 a_{-1} \\cdots a_{-\\ell +1}.a_{-\\ell } \\cdots a_{-s}\\in \\mathcal {S}(\\mathcal {L}(Q)),$ contradicting the definition of $\\ell $ as an infimum.",
"We consider two cases: If $t=s$ , then by definition there does not exist an element of $\\mathcal {L}(Q)$ with one's digit $a_{-s}$ .",
"In other words, there does not exist an element of the form $\\ldots a_{-s}$ in $\\mathcal {L}(Q)$ .",
"On the other hand suppose that $t<s$ .",
"Then by definition of $t$ , there exists an element $x= \\dots x_2 x_1 x_0 a_{-t-1} \\dots a_{-s}\\in \\mathcal {L}(Q).$ Let $y=\\dots y_2 y_1 y_0\\in \\mathcal {L}(Q)$ be the additive inverse of $x$ .",
"That is, $\\begin{array}{l l l l l l}& \\cdots & x_0 & a_{-t-1} & \\cdots & a_{-s} \\\\+ & \\cdots & y_{s-t} & y_{s-t-1} & \\cdots & y_0 \\\\\\hline & \\cdots & 0 & 0 & \\cdots & 0 \\\\\\end{array}$ By definition of $\\mathcal {L}(Q)$ , there exists an element $b=b_u \\cdots b_0.y_{s-1} y_{s-2} \\cdots y_0 \\in Q.$ We have $a+b=c\\in Q+Q$ , where $c=c_v\\cdots c_0.c_{-1} \\cdots c_{-t}$ is given by $\\begin{array}{l l l l l l l l l l l l l}& & & & a_r & \\cdots & a_0 .",
"& a_{-1} & \\cdots & a_{-t} & a_{-t-1} & \\cdots & a_{-s} \\\\+ & & b_u & \\cdots & b_r & \\cdots & b_0 .",
"& y_{s-1} & \\cdots & y_{s-t} & y_{s-t-1} & \\cdots & y_0 \\\\\\hline c_v & \\cdots & c_u & \\cdots & c_r & \\cdots & c_0.",
"& c_{-1} & \\cdots & c_{-t} & 0 & \\cdots & 0 \\\\\\end{array}$ (note, we are assuming in the above expression that $u\\ge r$ , but the case $u< r$ is identical).",
"Therefore $\\alpha ^{k_0}(c)=c_v \\cdots c_0 c_{-1} \\cdots c_{-k_0}.c_{-k_0-1} \\cdots c_{-t} \\in Q.$ Note that there does not exist an element of $\\mathcal {L}(Q)$ whose one's digit is $c_{-t}$ .",
"For suppose that $z=\\dots z_2 z_1 z_0 c_{-t}$ is such an element.",
"Then we also have $n^{s-t}z= \\dots z_2 z_1 z_0 c_{-t} \\underbrace{0 \\dots \\dots \\dots 0}_{s-t\\text{ times}}\\in \\mathcal {L}(Q)$ and hence $n^{s-t}z-y\\in \\mathcal {L}(Q)$ is given by $\\begin{array}{l l l l l l}& \\cdots & c_{-t} & 0 & \\cdots & 0 \\\\- & \\cdots & y_{s-t} & y_{s-t-1}& \\cdots & y_0 \\\\\\hline & \\cdots & a_{-t} & a_{-t-1} & \\cdots & a_{-s} \\\\\\end{array}$ contradicting the definition of $t$ .",
"Taking $d=a$ in Case (1) above or $d=\\alpha ^{k_0}(c)=c_v \\cdots c_{-k_0}.",
"c_{-k_0-1} \\cdots c_{-t}$ in Case (2) above, we have shown so far that there are elements $d_w \\cdots d_0.d_{-1} \\cdots d_{-u}\\in Q$ with $u$ arbitrarily large (at least $ \\ell -k_0$ ) and with the property that there does not exist any element of the form $\\ldots d_{-u}$ in $\\mathcal {L}(Q)$ .",
"That is, there exists a sequence $u_i \\rightarrow \\infty $ and a sequence $\\lbrace d^i\\rbrace _{i=1}^\\infty \\subset Q$ with $p(d^i)=-u_i$ , and $c(d^i)=d^i_{-u_i}$ with the property that there does not exist an element of the form $\\ldots d^i_{-u_i}$ in $\\mathcal {L}(Q)$ .",
"Writing $d^i=d_{w_i}^i \\cdots d_0^i .",
"d_{-1}^i \\cdots d_{-u_i}^i$ we may pass to a subsequence to assume that the sequence of integers $d_{-1}^i \\cdots d_{-u_i}^i \\in \\mathbb {Z}\\subseteq \\mathbb {Z}_n$ converges to a number $\\ldots e_2 e_1 e_0\\in \\mathbb {Z}_n$ .",
"We claim that $\\ldots e_2 e_1 e_0\\in \\mathcal {L}(Q)$ .",
"To prove the claim, note that given any $t\\ge 0$ and all large enough $i$ , the number $d_{-1}^i \\cdots d_{-u_i}^i$ has $d^i_{-u_i}=e_0,\\,\\,\\, d^i_{-u_i+1}=e_1,\\,\\,\\, \\ldots \\,\\,\\, d^i_{-u_i+t}=e_t.$ We have $\\begin{array}{l l l} \\alpha ^{u_i-t-1}(d^i) & =& d_{w_i}^i \\cdots d_0^i d_{-1}^i \\cdots d_{-u_i+t+1}^i .",
"d_{-u_i+t}^i d_{-u_i+t-1}^i \\cdots d_{-u_i}^i \\\\ &= & d_{w_i}^i \\cdots d_{-u_i+t+1}^i.",
"e_t e_{t-1} \\cdots e_0\\end{array}$ for all such $i$ .",
"Since $\\alpha ^{u_i-t-1}(d^i)\\in Q$ and $t$ is arbitrary, this proves the claim.",
"However, this is a contradiction, as we have $d_{-u_i}^i=e_0$ for all large enough $i$ , while there does not exist an element of the form $\\ldots d_{-u_i}^i$ in $\\mathcal {L}(Q)$ for any $i$ .",
"This completes the proof.",
"Define $\\mathcal {J}_n$ to be the poset $2^{\\lbrace 1,\\dots , k\\rbrace }\\setminus \\lbrace \\lbrace 1,\\dots , k\\rbrace \\rbrace $ with the partial order given by inclusion.",
"Recall that $\\mathcal {K}_n$ is the poset $2^{\\lbrace 1,\\ldots k\\rbrace } \\setminus \\lbrace \\emptyset \\rbrace $ .",
"We have that $\\mathcal {K}_n$ is isomorphic to the opposite poset of $\\mathcal {J}_n$ .",
"We now define a map $\\Phi \\colon \\mathcal {J}_n\\rightarrow \\mathcal {H}_{qp}(BS(1,n))$ as follows.",
"Given $A\\in \\mathcal {J}_n$ , let $a=a_1\\times \\ldots \\times a_k$ be the full ideal of $\\mathbb {Z}_n$ defined by $a_i=(0)$ if and only if $i\\in A$ , and let $ \\Phi (A)=[\\mathcal {S}(a)\\cup \\lbrace t^{\\pm 1}\\rbrace ].$ Proposition 4.6 The map $\\Phi \\colon \\mathcal {J}_n\\rightarrow \\mathcal {H}_{qp}(BS(1,n))$ defined in (REF ) is an order-reversing injective map.",
"Hence $\\Phi $ induces an injective homomorphism of posets $\\mathcal {K}_n \\rightarrow \\mathcal {H}_{qp}(BS(1,n)).$ Moreover, $\\mathcal {H}_{qp}(BS(1,n))$ contains exactly one additional structure which is incomparable to every $[Y]\\in \\Phi (\\mathcal {J}_n)$ .",
"Lemmas REF and REF show that the map $\\Phi $ is an injective order-reversing map of posets.",
"By Proposition REF , if $[S]\\in \\mathcal {H}_{qp}(BS(1,n))$ , then there exists a $Q\\subseteq H$ which is strictly confining under the action of $\\alpha $ or $\\alpha ^{-1}$ and such that $[S]=[Q\\cup \\lbrace t^{\\pm 1}\\rbrace ]$ .",
"Fix $[S]\\in \\mathcal {H}_{qp}(BS(1,n))$ such that the corresponding subset $Q$ is strictly confining under the action of $\\alpha $ , and consider the ideal $\\mathcal {L}(Q)$ .",
"By Lemma REF , $[S]= [\\mathcal {S}(\\mathcal {L}(Q))\\cup \\lbrace t^{\\pm 1}\\rbrace ]$ .",
"Moreover, by Lemma REF , there is a proper full ideal $a\\sim \\mathcal {L}(Q)$ , and $\\mathcal {S}(\\mathcal {L}(Q))=\\mathcal {S}(a)$ by Lemma REF .",
"Thus $[S]= [\\mathcal {S}(a)\\cup \\lbrace t^{\\pm 1}\\rbrace ]$ .",
"Let $A=\\lbrace 1\\le i\\le k\\mid a_i=(0)\\rbrace \\subseteq \\mathcal {J}_n$ .",
"Then $[S]=\\Phi (A)$ , and so every quasi-parabolic structure whose associated subset is strictly confining under the action of $\\alpha $ is in the image of $\\Phi $ .",
"By Proposition REF , $\\mathcal {H}_{qp}(BS(1,n))$ has a single additional element, $[Q^-\\cup \\lbrace t^{\\pm 1}\\rbrace ]$ , where $Q^-$ , defined in (REF ), is strictly confining under the action of $\\alpha ^{-1}$ .",
"It remains to show that $[Q^-\\cup \\lbrace t^{\\pm 1}\\rbrace ]$ is incomparable to all $[S]\\in \\mathcal {H}_{qp}(BS(1,n))\\setminus \\lbrace [Q^-\\cup \\lbrace t^{\\pm 1}\\rbrace ]\\rbrace $ .",
"To see this last fact, note that the action $BS(1,n)\\curvearrowright \\Gamma (BS(1,n),Q^-\\cup \\lbrace t^{\\pm 1}\\rbrace )$ is equivalent to the action $BS(1,n)\\curvearrowright \\mathbb {H}^2$ by Proposition REF .",
"Hence in this action, the common fixed point of all elements of $BS(1,n)$ is the attracting fixed point of $t$ .",
"On the other hand, for $[S]\\in \\mathcal {H}_{qp}(BS(1,n))\\setminus \\lbrace [Q^-\\cup \\lbrace t^{\\pm 1}\\rbrace ]\\rbrace $ , $BS(1,n)\\curvearrowright \\Gamma (BS(1,n),S)$ is equivalent to the action of $BS(1,n)$ on one of the Bass-Serre trees described in Section REF .",
"Hence in the action $BS(1,n)\\curvearrowright \\Gamma (BS(1,n),S)$ , the common fixed point of all elements of $BS(1,n)$ is the repelling fixed point of $t$ .",
"If we had, for example, $[Q^-\\cup \\lbrace t^{\\pm 1}\\rbrace ]\\preccurlyeq [S]$ then this would imply that every element of $BS(1,n)$ would fix the repelling fixed point of $t$ in $\\partial \\Gamma (BS(1,n), Q^- \\cup \\lbrace t^{\\pm 1}\\rbrace )$ as well as the attracting fixed point of $t$ .",
"This would imply that the action $BS(1,n)\\curvearrowright \\Gamma (BS(1,n), Q^- \\cup \\lbrace t^{\\pm 1}\\rbrace )$ is lineal, which is a contradiction."
],
[
"Proof of Theorem ",
"Proposition REF gives a complete description of $\\mathcal {H}_{qp}(BS(1,n))$ .",
"We now turn our attention to other hyperbolic structures.",
"We first show that for any $n\\ge 2$ , $|\\mathcal {H}_l(BS(1,n))|=1$ .",
"Consider an action $BS(1,n)\\curvearrowright X$ with $X$ hyperbolic.",
"Then every element of $H$ must act elliptically or parabolically.",
"For if $g\\in H$ then $tgt^{-1}$ and $g$ have the same asymptotic translation length.",
"However, $tgt^{-1}=g^n$ has asymptotic translation length equal to $n$ times the asymptotic translation length of $g$ .",
"This is only possible if the asymptotic translation length of $g$ is 0.",
"Hence the induced action of $H$ on $X$ is either elliptic or parabolic.",
"According to [1], since $H\\trianglelefteq BS(1,n)$ , if $H\\curvearrowright X$ is parabolic, then $BS(1,n)\\curvearrowright X$ is parabolic or quasi-parabolic.",
"In particular, if $BS(1,n)\\curvearrowright X$ is a lineal action then $H\\curvearrowright X$ is elliptic.",
"This shows that if $[S] \\in \\mathcal {H}_l(BS(1,n))$ then $[S]\\preccurlyeq [H\\cup \\lbrace t^{\\pm 1}\\rbrace ]$ .",
"But by [1], if $[A]\\preccurlyeq [B]$ and both structures are lineal, then $[A]=[B]$ .",
"Therefore $[S]=[H\\cup \\lbrace t^{\\pm 1}\\rbrace ]$ .",
"Every quasi-parabolic structure dominates the lineal structure defined by its Busemann quasi-morphism.",
"Since $|\\mathcal {H}_l(BS(1,n))|=1$ , it follows that every element of $\\mathcal {H}_{qp}(BS(1,n))$ dominates this single lineal structure.",
"For any $n\\ge 2$ , $BS(1,n)$ is solvable, and so contains no free subgroups.",
"Thus by the Ping-Pong lemma, $H_{gt}(BS(1,n))=\\emptyset $ .",
"Finally, for any group $G$ , $\\mathcal {H}_e(G)$ has a single element which is the smallest element in $\\mathcal {H}(G)$ , completing the proof of Theorem REF ."
],
[
"Generating confining subsets",
"In this section, we give a general method for constructing confining subsets of groups of the form $H\\rtimes \\mathbb {Z}$ and show that when $H=\\mathbb {Z}[x,\\frac{1}{x}]$ (the additive group of the Laurent polynomial ring over $\\mathbb {Z}$ ) this construction yields a countable chain of quasi-parabolic structures on $\\mathbb {Z}\\wr \\mathbb {Z}\\cong \\mathbb {Z}[x,\\frac{1}{x}]\\rtimes \\mathbb {Z}$ .",
"This result should be compared to [3] which shows that $\\mathbb {Z}_n\\wr \\mathbb {Z}\\cong \\mathbb {Z}_n\\left[x,\\frac{1}{x}\\right]\\rtimes \\mathbb {Z}$ has only finitely many quasi-parabolic structures.",
"Consider a group $G=H\\rtimes \\mathbb {Z}$ where $\\mathbb {Z}=\\langle t\\rangle $ acts by $tht^{-1}=\\alpha (h)$ for any $h\\in H$ , where $\\alpha \\in \\operatorname{Aut}(H)$ .",
"Let $S$ be a symmetric subset of $H$ with the following properties: $\\alpha (S)\\subseteq S$ $\\bigcup _{n\\ge 0} \\alpha ^{-n}(S)$ generates $H$ .",
"Define a subset $Q^i\\subseteq H$ by $Q^i=Q^i_0\\cup Q^i_1\\cup Q^i_2\\cup \\ldots $ where $Q^i_0=S$ and $Q^i_{n+1}=Q^i_n\\cup \\alpha ^i(Q^i_n\\cdot Q^i_n)$ for all $n\\ge 0$ .",
"In other words, $Q^i$ is the smallest subset of $H$ containing $S$ and with the property that $\\alpha ^i(Q^i \\cdot Q^i)\\subset Q^i$ .",
"Lemma 5.1 $Q^i$ is a confining subset of $H$ under the action of $\\alpha $ .",
"First, we prove by induction that $\\alpha (Q^i_n)\\subseteq Q^i_n$ for all $n$ .",
"The base case $n=0$ holds by point (i).",
"Suppose for induction that $\\alpha (Q^i_n)\\subseteq Q^i_n$ .",
"Now if $x\\in Q^i_{n+1}$ then $x\\in Q^i_n$ or $x\\in \\alpha ^i(Q^i_n\\cdot Q^i_n)$ .",
"In the first case we have $\\alpha (x)\\in Q^i_n\\subseteq Q^i_{n+1}$ .",
"Otherwise, we may write $x=\\alpha ^i(yz)$ where $y,z\\in Q^i_n$ .",
"Then we have $\\alpha (x)=\\alpha ^i(\\alpha (y)\\alpha (z))$ and since $\\alpha (y),\\alpha (z)\\in Q^i_n$ we have $\\alpha (x)\\in \\alpha ^i(Q^i_n\\cdot Q^i_n)$ .",
"Since $\\alpha (Q_n^i)\\subset Q_n^i$ for all $n$ , we have $\\alpha (Q^i)\\subset Q^i$ .",
"Now to see that $\\alpha ^i(Q^i\\cdot Q^i)\\subseteq Q^i$ simply use the fact that $\\alpha ^i(Q^i_n\\cdot Q^i_n)\\subseteq Q^i_{n+1}$ and $Q^i$ is the union of the sets $Q^i_n$ for $n\\ge 0$ .",
"Finally, we prove that $H=\\bigcup _{n\\ge 0} \\alpha ^{-n}(Q^i)$ .",
"We use fact (ii) above and induction on the word length of an element $h\\in H$ in the semigroup generating set $\\bigcup _{n\\ge 0}\\alpha ^{-n}(S)$ .",
"If $h$ has word length one with respect to this generating set then $h=\\alpha ^{-n}(s)$ for some $s\\in S$ .",
"Hence, $\\alpha ^n(h)=s\\in Q^i_1$ .",
"If $h$ has word length at most $k$ , then $\\alpha ^n(h)\\in Q^i$ for some $n\\ge 0$ .",
"Let $h$ have word length $k+1$ .",
"Then we may write $h=\\alpha ^{-n}(s)h^{\\prime }$ where $s\\in Q^i$ and $h^{\\prime }$ has word length $k$ .",
"By induction there exists $m$ such that $\\alpha ^m(h^{\\prime })\\in Q^i$ .",
"We have $\\alpha ^{m+n}(h)=\\alpha ^m(s)\\alpha ^{m+n}(h^{\\prime }).$ Since $\\alpha (Q^i)\\subseteq Q^i$ and $\\alpha ^m(h^{\\prime })\\in Q^i$ , we have $\\alpha ^{m+n}(h^{\\prime })\\in Q^i$ .",
"Similarly $\\alpha ^m(s)\\in Q^i$ .",
"Choose $r$ large enough such that $\\alpha ^m(s),\\alpha ^{m+n}(h^{\\prime })\\in Q^i_r$ .",
"Then we have $\\alpha ^{m+n+i}(h)=\\alpha ^i(\\alpha ^m(s)\\alpha ^{m+n}(h^{\\prime }))\\in \\alpha ^i(Q^i_r \\cdot Q^i_r)\\subseteq Q^i_{r+1}.$ This completes the induction.",
"By [4], $[Q^i\\cup \\lbrace t^{\\pm 1}\\rbrace ]\\in \\mathcal {H}_{qp}(G)\\cup \\mathcal {H}_\\ell (G)$ for all $i$ .",
"Clearly we have $ [Q^1\\cup \\lbrace t^{\\pm 1}\\rbrace ] \\preccurlyeq [Q^2\\cup \\lbrace t^{\\pm 1}\\rbrace ] \\preccurlyeq [Q^3\\cup \\lbrace t^{\\pm 1}\\rbrace ] \\preccurlyeq \\ldots .$ However in general it is possible that these inequalities are not strict.",
"Proposition 5.2 Let $H=\\mathbb {Z}\\left[x,\\frac{1}{x}\\right]$ (the additive group of the Laurent polynomial ring over $\\mathbb {Z}$ ) where $t$ acts on $H$ by $t\\cdot p(x)=xp(x)$ .",
"Set $S=\\lbrace \\pm 1 ,\\pm x,\\pm x^2,\\ldots \\rbrace $ .",
"Define $Q^i$ using $S$ as above.",
"Then we have $[Q^i\\cup \\lbrace t^{\\pm 1}\\rbrace ] \\ne [Q^j\\cup \\lbrace t^{\\pm 1}\\rbrace ]$ for $i<j$ .",
"Moreover, $[Q^i \\cup \\lbrace t^{\\pm 1}\\rbrace ]\\in \\mathcal {H}_{qp}(H\\rtimes \\mathbb {Z})$ for each $i$ .",
"Note the following, which can be proven inductively: $Q^i_r$ consists of polynomials in $x$ (with no terms of negative degree), if $p(x)\\in Q^i_r \\setminus Q^i_{r-1}$ then every term of $p(x)$ has degree at least $ri$ , the largest coefficient of a term of $p(x)\\in Q^i_r$ is $2^r$ .",
"Hence we have the following table: Table: NO_CAPTIONHere the entry under $x^k$ denotes the largest absolute value of the coefficient of the degree $k$ term of any polynomial $p(x)\\in Q^i$ .",
"A similar table holds for $Q^j$ .",
"In particular, we see that if $p(x)\\in Q^j$ and $p(x)$ contains a term of degree $k$ , then the absolute value of the coefficient of $x^k$ is at most $2^{\\lfloor k/j\\rfloor }$ .",
"Note in particular that the sequence $1, 2x^i=\\alpha ^i(x+x), 4x^{2i}=\\alpha ^i(2x^i+2x^i), 8x^{3i}=\\alpha ^i(4x^{2i}+4x^{2i}),\\ldots $ is contained in $Q^i$ .",
"Hence all have word length 1 in the generating set $Q^i\\cup \\lbrace t\\rbrace $ .",
"We claim that the word length of $2^r x^{ri}$ in the generating set $Q^j\\cup \\lbrace t\\rbrace $ goes to infinity as $r\\rightarrow \\infty $ .",
"Let $\\Vert \\cdot \\Vert _j$ denote word length with respect to the generating set $Q^j\\cup \\lbrace t^{\\pm 1}\\rbrace $ .",
"To prove the claim, write $2^rx^{ri}=g_1\\ldots g_n$ where $g_1\\ldots g_n$ is a word in $Q^j\\cup \\lbrace t^{\\pm 1}\\rbrace $ of minimal length $n=\\Vert 2^r x^{ri}\\Vert _j$ representing $2^rx^{ri}$ .",
"Each $g_i$ is either $t$ , $t^{-1}$ , or a polynomial $p(x)\\in Q^j$ .",
"By an argument analogous to that of Lemma REF , we may rewrite $2^r x^{ri}$ as a word of length $n$ of the form $2^rx^{ri}=t^{-k} (p_1(x)+\\ldots + p_m(x))t^l.$ Since the word on the right represents a polynomial, we must in fact have $k=l$ and we have $2^rx^{ri}=\\alpha ^{-k}(p_1(x)+\\ldots +p_m(x)) ,$ and therefore $p_1(x)+\\ldots +p_m(x) = 2^rx^{ri+k}.$ It follows that $n=2k+m$ .",
"Since each $p_*(x)$ contains $x^{ri+k}$ as a term with coefficient at most $2^{\\lfloor (ri+k)/j\\rfloor }$ , we must have $m \\ge \\frac{2^r}{2^{\\lfloor (ri+k)/j\\rfloor }}=2^{r-\\lfloor (ri+k)/j\\rfloor }\\ge 2^{r-(ri+k)/j}=2^{(1-i/j)r-k/j}.$ So to bound $n$ from below, it suffices to minimize $2k+m$ subject to the condition $m=2^{(1-i/j)r-k/j}.$ Rewriting $2k+m$ in terms of $k$ yields $2k+2^{(1-i/j)r-k/j}.$ Defining a function $f(k)=2k+2^{(1-i/j)r-k/j}$ , we see that $f$ has a unique minimum at the unique zero of its derivative.",
"The derivative with respect to $k$ is $f^{\\prime }(k)=2-\\frac{1}{j}2^{(1-i/j)r-k/j}.$ Solving the equation $f^{\\prime }(k)=0$ for $k$ yields $k=(j-i)r-j\\log _2(j) -j.$ Since $j-i>0$ we must have $k\\rightarrow \\infty $ as $r\\rightarrow \\infty $ and, in particular, $n\\rightarrow \\infty $ as $r\\rightarrow \\infty $ .",
"In other words, $\\Vert 2^rx^{ri}\\Vert _j\\rightarrow \\infty $ as $r\\rightarrow \\infty $ .",
"For the final sentence, simply note that each $Q^i$ is strictly confining since $1\\notin \\alpha (Q^i)$ for any $i$ .",
"Remark 5.3 This argument doesn't work for the wreath product $\\mathbb {Z}_n\\wr \\mathbb {Z}=\\mathbb {Z}_n[x,\\frac{1}{x}]\\rtimes \\mathbb {Z}$ since the generator of $\\mathbb {Z}_n$ doesn't have infinite order.",
"Carolyn R. Abbott Department of Mathematics, Columbia University, New York, NY 10027.",
"E-mail: [email protected] Alexander J. Rasmussen Department of Mathematics, Yale University, New Haven, CT 06520.",
"E-mail: [email protected]"
]
] | 1906.04227 |
[
[
"Discovery of the first heavily obscured QSO candidate at $z>6$ in a\n close galaxy pair"
],
[
"Abstract While theoretical arguments predict that most of the early growth of supermassive black holes (SMBHs) happened during heavily obscured phases of accretion, current methods used for selecting $z>6$ quasars (QSOs) are strongly biased against obscured QSOs, thus considerably limiting our understanding of accreting SMBHs during the first Gyr of the Universe from an observational point of view.",
"We report the $Chandra$ discovery of the first heavily obscured QSO candidate in the early universe, hosted by a close ($\\approx5$ kpc) galaxy pair at $z=6.515$.",
"One of the members is an optically classified type 1 QSO, PSO167-13.",
"The companion galaxy was first detected as a [C II] emitter by ALMA.",
"An X-ray source is significantly ($P=0.9996$) detected by $Chandra$ in the 2-5 keV band, with $<1.14$ net counts in the 0.5-2 keV band, although the current positional uncertainty does not allow a conclusive association with either PSO167-13 or its companion galaxy.",
"From X-ray photometry and hardness-ratio arguments, we estimated an obscuring column density of $N_H>2\\times10^{24}\\,\\mathrm{cm^{-2}}$ and $N_H>6\\times10^{23}\\,\\mathrm{cm^{-2}}$ at $68\\%$ and $90\\%$ confidence levels, respectively.",
"Thus, regardless of which of the two galaxies is associated with the X-ray emission, this source is the first heavily obscured QSO candidate at $z>6$."
],
[
"Introduction",
"The discovery of accreting supermassive black holes (SMBHs) with masses of $10^9-10^{10}\\,M_\\odot $ shining as quasars (QSOs) at $z>6$ [26], [1], [2] when the Universe was less than 1 Gyr-old challenges our understanding of SMBH formation and growth in the early universe, and is one of the major open issues in modern astrophysics [37], [50].",
"Different classes of theories have been proposed to explain the formation of the BH seeds that eventually became SMBHs.",
"The two most popular classes of models involve the formation of “light seeds” ($M \\approx 10^2\\,M_\\odot $ ), as remnants of the first Pop III stars, and “heavy seeds\" ($M \\approx 10^4-10^6\\,M_\\odot $ ), perhaps formed during the direct collapse of giant pristine gas clouds (e.g., [46], [41], [40], [50], and references therein).",
"To match the masses of the SMBHs discovered at $z>6$ , all such models require continuous, nearly Eddington-limited or even super-Eddington accretion phases during which the growing SMBH is expected to be heavily obscured by the same accreting material with large column densities, even exceeding the Compton-thick level ($N_H=1.5\\times 10^{24}\\,\\mathrm {cm^{-2}}$ ; e.g., [32], [33]).",
"“Wet” (i.e., gas-rich) galaxy mergers are expected to provide both a large amount of gas and the mechanisms to drive it toward the galaxy nuclear regions, thus allowing efficient SMBH accretion [14].",
"Indeed, high-redshift QSOs are usually found in overdense environments in simulations [6], [4], [13], but no consensus has yet been reached among observational works [3], [21], [31].",
"Currently, approximately 180 quasars have been discovered at $z> 6$ (e.g., [1] and references therein; [22], [19], [20], [47], [11], [36]), up to $z = 7.54$ (ULAS J1342+0928; [2]).",
"However, these rare QSOs have been selected from wide-field optical/near-infrared(NIR) surveys such as, for example, SDSS, CFHQS, and PanSTARRS-1, and thus are, by selection, optically type 1 (i.e., broad emission-line QSOs with blue UV continua).",
"The selection of $z>6$ QSO candidates typically relies on the detection of the blue power-law UV continuum, absorbed at $\\lambda <1216$ Å by the $Ly\\alpha $ forest, and suppressed at wavelengths shorter than the $Ly\\alpha $ break at 912 Å, due to absorption by intervening neutral hydrogen.",
"Therefore, the census of accreting SMBHs in the early universe is currently missing, by selection, the key population of obscured systems, thereby strongly limiting our understanding of the early phases of SMBH growth.",
"Currently, the highest redshift, Compton-thick QSO candidate is XID403 at $z=4.76$ [12], [5], an X-ray-selected QSO in the Chandra Deep Field-South [51], [17].",
"In this Letter, we report the discovery in the X-ray band of the first heavily obscured QSO candidate at $z>6$ , in a close ($\\approx 5$ kpc) pair of galaxies at $z\\approx 6.515$ .",
"One of the two galaxies hosts an optically classified type-1 QSO, PSO J167.6415–13.4960 (hereafter PSO167–13).",
"Evidence for interaction between the two galaxies is reported in [23].",
"Errors and limits are reported at the $68\\%$ confidence level, unless otherwise noted.",
"We adopt a flat cosmology with $H_0=67.7\\,\\mathrm {km\\,s^{-1}}$ and $\\Omega _m=0.307$ [34].",
"Figure: Soft (left panel) and hard (right panel) band Chandra images of a 20 '' ×20 '' 20^{\\prime \\prime }\\times 20^{\\prime \\prime } region around the UV position of PSO167–13 (cyan cross), smoothed with a Gaussian function with a three-pixel kernel radius.",
"The black cross (not shown in the right panel for clarity) marks the ALMA position of the companion galaxy (Fig.",
", right panel).The X-ray source is significantly (P=0.9996P=0.9996) detected in the hard band, while no counts are detected in the soft band.",
"The black circle (R=1R=1 arcsec) is centered on the hard-band emission centroid and is used for photometry evaluation."
],
[
"Target description and data analysis",
"PSO167-13 was first selected as a high-redshift QSO candidate on the basis of its colors in the PanSTARRS-1 survey ([43], see Fig.",
"REF , left panel), and was then confirmed spectroscopically to lie at $z=6.515$ both in the rest-frame UV [43] and sub-millimeter with Atacama large millimeter array (ALMA), via detection of the [C II] ($158\\,\\mu m$ ) emission line [8].",
"An investigation of the ALMA data-cube at frequencies near the [C II] emission line [49] revealed the presence of a close companion, separated by $0.9^{\\prime \\prime }$ ($\\approx 5$ kpc in projection at the redshift of the QSO) from the rest-frame UV and [C II] position of the QSO, and by $\\Delta v\\approx -270\\,\\mathrm {km\\,s^{-1}}$ (corresponding to $\\Delta z \\approx 0.007$ ) in velocity space (based on the frequency of the [C II] emission peaks).",
"The companion galaxy thus forms a physical pair with PSO167–13.",
"Its existence was recently confirmed by a deep HST/WFC3 observation in the F140W ($\\approx 1.4\\,\\mu m$ ) band (with AB magnitude F140W$=25.5$ ) and new high-resolution ($\\approx 0.25\"$ ) ALMA imaging (Fig.",
"REF , center and right panels; [23], [29]).",
"No rest-frame UV spectrum is currently available for this galaxy.",
"Similar companions have been found in about a quarter of the $z>6$ QSOs observed with ALMA [7].",
"We observed PSO167–13 for 59 ks with Chandra as part of a larger program aimed at making exploratory observations of a statistically significant sample of ten $z>6$ QSOs (Vito et al., in prep.",
").Chandra observations of the remaining 9 targets have been completed and the analysis is ongoing.",
"PSO167–13 is the only source showing significant evidence of obscuration.",
"We reprocessed the Chandra observations with the chandra_repro script in CIAO 4.10,http://cxc.harvard.edu/ciao/ using CALDB v4.8.1,http://cxc.harvard.edu/caldb/ setting the option check_vf_pha=yes in the case of observations taken in Very Faint mode, and extracted the response matrix and ancillary file using the specextract tool.",
"The astrometry for all instruments has been consistently locked on the PanSTARRS-1 frame, using six common sources in the field for Chandra (we used the CIAO wcs_match and wcs_update tools), and the position of PSO167–13 itself for HST and ALMA.",
"We detected significant emission in the hard (2–5 keV) band using a standard circular extraction region of 1 arcsec radius (Fig.",
"REF ).",
"In particular, we detected three counts, with an expected background level of 0.14 counts, corresponding to a number of net counts of $2.86_{-1.44}^{+2.14}$ and a false-detection probability (i.e., that the detected emission is due to a background fluctuation) of only $P=4\\times 10^{-4}$ [48].",
"The corresponding flux is $F_{2-5\\,\\mathrm {keV}}=8.1_{-3.9}^{+5.9}\\times 10^{-16}\\,\\mathrm {erg\\,cm^{-2}\\,s^{-1}}$ .",
"As a check on the detection significance, after having masked bright sources including PS167–13, we performed aperture photometry using $R=1$ arcsec regions randomly centered over $10^5$ positions across the field in the $2-5$ keV band, and detected $\\ge 3$ counts for 52 of them ($P=5\\times 10^{-4}$ ).",
"Moreover, 10 of these 52 regions are also coincident with the positions of PanSTARRS galaxies, and therefore could be real X-ray sources, increasing the agreement with the false-source probability reported above.",
"All of the three detected counts have energies in the range $2.5\\lesssim \\frac{E}{\\mathrm {keV}}\\lesssim 3.5$ , which is not surprising since the effective area of Chandra drops at high energies.",
"If we restrict the detection to the $2-4$ keV band, thus excluding the background-dominated higher energies, we derive an even higher detection significance ($P=2\\times 10^{-4}$ ).",
"We detected zero counts in the soft (0.5–2 keV) band at the UV position of PSO167–13 (cyan cross in Fig.",
"REF ), thus setting an upper limit on the net counts of $<1.14$ [48].",
"In order to evaluate the significance of the soft-band nondetection, we assumed a standard $\\Gamma =1.9$ power-law model, suitable for high-redshift luminous QSOs (e.g., [39], [28]), normalized to the observed net-count rate in the hard band.",
"Accounting for Galactic absorption, the expected background in the extraction region ($\\approx 0.1$ counts), and the Chandra effective area at the position of the target, the expected number of soft-band counts is 6.59.",
"Given this expectation, the Poisson probability of detecting zero counts is $P(x=0,\\mu =6.59)=1.37\\times 10^{-3}$ .",
"Conservatively assuming a rather flat slope ($\\Gamma =1.6$ , based on the uncertainties on the average photon index in [39], [28]), the source nondetection in the soft band remains significant ($P=6.1\\times 10^{-3}$ ).",
"The centroid of the hard-band emission is shifted from the optical and sub-millimeter position of PSO167–13 (cyan cross in Fig.",
"REF ) by $0.97$ arcsec and by $0.15$ arcsec from the [C II] position of the companion galaxy (black cross).",
"We computed the positional uncertainty via 1000 MARX 5.3.3https://space.mit.edu/ASC/MARX/ simulations of a source with three counts in the hard band at the position of PSO167–13, accounting for the real instrumental configuration, and including a (negligible) residual astrometry uncertainty.",
"We found a positional uncertainty of $0.73$ arcsec and $1.17$ arcsec at 68% and 90% confidence levels, respectively.",
"The observed offset between the X-ray source and the optical position of PSO167–13 is significant at $\\approx 1.5\\sigma $ only, such that the hard-band X-ray emission is consistent with being produced by the type-1 QSO."
],
[
"Results and discussion",
"The measured X-ray photometry corresponds to a hardness ratio$HR=(H-S)/(H+S)$ , where $S$ and $H$ are source net counts in the soft and hard bands, respectively.",
"This quantity is widely used to characterize the spectra of X-ray sources with limited photon statistics.",
"of $HR>0.47$ and an effective power-law photon index at rest-frame $4-38$ keV of $\\Gamma <0.55$ , computed accounting for the effective area at the position of the X-ray source and Galactic absorption.",
"These extremely hard values for an object at $z=6.515$ strongly suggest the source is heavily obscured.",
"We estimated the column density required to retrieve such values through spectral simulations with XSPEC, assuming an intrinsic power-law spectrum with $\\Gamma =1.9$ and accounting for Galactic absorption, and obtained $N_H>2\\times 10^{24}\\,\\mathrm {cm^{-2}}$ and $N_H>6\\times 10^{23}\\,\\mathrm {cm^{-2}}$ at the $68\\%$ and $90\\%$ confidence levels, respectively.",
"The column density cannot be constrained at $\\gtrsim 95\\%$ confidence level, due to the combination of the number of detected counts and the photoelectric cut-off shifting outside the Chandra band for low column densities.",
"We estimated the rest-frame $2-10$ keV luminosity of this source from the detected counts in the observed-frame hard band assuming $\\Gamma =1.9$ to be in the range $L_{2-10\\mathrm {keV}}=[6.6_{-3.2}^{+4.5}, 7.4_{-3.6}^{+5.4}]\\times 10^{44}\\,\\mathrm {erg\\,s^{-1}}$ , where the lower and upper limits are computed assuming $N_H=[0, 2]\\times 10^{24}\\,\\mathrm {cm^{-2}}$, respectively.",
"The derived luminosity does not vary significantly for very different values of $N_H$ , as the high rest-frame energies (i.e., $15-38$ keV) probed at $z=6.52$ are not strongly affected by even moderately Compton-thick obscuration.",
"Figure: Rest-frame UV spectrum of PSO167-13 (red line, adapted from ) obtained with VLT/FORS2 and Magellan/FIRE, compared with the average QSO spectrum of .",
"A zoom in the C IV region is shown in the inset, where we also report the best fit to the C IV line, and associated redshift and equivalent width.",
"Tentative BAL features might be present at wavelengths bluer than the C IV emission line, which is also particularly weak.",
"PSO167–13 may thus be a BAL QSO and/or a WLQ.Considering the hard-band positional accuracy (see the green circle in the center and right panels of Fig.",
"REF ,), the source of the X-ray emission could be either PSO167–13 or its companion galaxy.",
"Assuming that the QSO is the source of the hard-band emission with a somewhat large X-ray offset (0.97 arcsec), the upper limit we derived above on the number of soft-band counts corresponds to an observed soft-band X-ray emission $\\gtrsim 4$ times weaker than that expected from its UV luminosity [15].",
"The intrinsic (i.e., corrected for absorption) luminosity estimated in the previous paragraph would be consistent within a factor of two with the expectations based on the QSO UV luminosity [15].",
"Several physical processes could explain why an optically classified type-1 QSO is heavily obscured in the X-rays.",
"For instance, approximately $50\\%$ of the Weak Emission-Line QSOs (WLQs; e.g., [10]) are associated with weak and hard X-ray emission [16], [30], possibly linked to the presence of thick accretion disks with large column density on small scales that prevent ionizing radiation from reaching the broad-line region.",
"Moreover, WLQs are usually found to be fast-accreting QSOs [16], [18], as is PSO167–13 ($\\lambda _{\\mathrm {Edd}}\\gtrsim 1$ ; [22]).",
"Similarly, in broad-absorption-line QSOs (BALQSOs), small-scale screening material may absorb the ionizing UV/X-ray radiation, thus allowing the acceleration of the outflowing wind producing the BALs [35], but still allowing the detection of the blue UV continuum.",
"[38] reported the emergence of BALs on timescales of $\\sim 100$ days, that is, shorter than the rest-frame time that passed from the UV spectral observation to the X-ray imaging of PSO167–13, which was 6 months.",
"In the currently available rest-frame UV spectrum of PSO167–13 (Fig.",
"REF ) the C IV line is relatively weak ($EW_{CIV}\\approx 12\\,\\mathrm {Å}$ ), and some BAL features might also be present at bluer wavelengths.",
"The C IV line is blueshifted by $\\sim -5800\\,\\mathrm {km\\,s^{-1}}$ with respect to the Mg II emission line, similarly to some hyper-luminous QSOs [44] and other $z>6$ QSOs [2], [25].",
"Such extreme blueshifts are also seen in WLQs [16], [30].",
"We also note that [27] detected significant spectral variability (from $N_H\\approx 0\\,\\mathrm {cm^{-2}}$ to $N_H\\approx 5\\times 10^{23}\\,\\mathrm {cm^{-2}}$ ) in two distinct X-ray observations of the $z>6$ QSO SDSS J1030+0524.",
"A similar increase of the obscuration might have taken place also for PSO167–13 between its observations in rest-frame UV and X-rays.",
"Additional rest-frame UV spectroscopic observations are also needed to investigate such a possibility, as well as to help characterize this object as a possible WLQ or BALQSO.",
"Alternatively, since the X-ray centroid is consistent with the position of the companion galaxy, this could host a heavily obscured QSO, in a close and interacting [23] pair with PSO167–13.",
"In this scenario, only its proximity to the optically type-1 QSO allowed us to discover it with Chandra, as high-redshift obscured QSOs are missed by UV surveys, and the lack of strong detection of X-ray emission from PSO167–13 can be explained by a moderate intrinsic X-ray weakness (a factor of $\\ge 4$ ).",
"Similar pairs of QSOs have been discovered at redshifts as high as $z\\approx 5$ [24], although with larger separation, but beyond $z\\approx 3.3$ none are known to include an obscured QSO [45].",
"To summarize, if PSO167–13, optically classified as a type-1 QSO, were found to be responsible for the high-energy emission, it would be an intrinsically X-ray normal but heavily obscured QSO, and the causes of the UV/X-ray misclassification would need to be investigated.",
"Alternatively, if the companion galaxy were found to be the X-ray source, it would be a heavily obscured QSO in an interacting pair with PSO167–13.",
"In this case, PSO167–13 would be intrinsically X-ray weak by a factor of $\\ge 4$ .",
"Thus, regardless of which of the two members of the system produced the hard-band detection, it represents the first heavily obscured QSO candidate in the early universe.",
"Deeper X-ray observations are required to better constrain the column density and to improve the positional accuracy of the hard-band X-ray source, thereby allowing a confident association with either PSO167–13 or its companion galaxy, and confirmation or rejection of the QSO-pair nature of this system.",
"We thank the anonymous referee for their useful comments and suggestions.",
"FV acknowledges financial support from CONICYT and CASSACA through the Fourth call for tenders of the CAS-CONICYT Fund.",
"WNB acknowledges Chandra X-ray Center grant G08-19076X.",
"BL acknowledges financial support from the National Key R&D Program of China grant 2016YFA0400702 and National Natural Science Foundation of China grant 11673010.",
"We acknowledge financial contribution from CONICYT grants Basal-CATA AFB-170002 (FV, FEB), the Ministry of Economy, Development, and Tourism's Millennium Science Initiative through grant IC120009, awarded to The Millennium Institute of Astrophysics, MAS (FEB), and the agreement ASI-INAF n.2017-14-H.O."
]
] | 1906.04241 |
[
[
"Thermodynamic Bounds on Coherent Transport in Periodically Driven\n Conductors"
],
[
"Abstract Periodically driven coherent conductors provide a universal platform for the development of quantum transport devices.",
"Here, we lay down a comprehensive theory to describe the thermodynamics of these systems.",
"We first focus on moderate thermo-electrical biases and low driving frequencies.",
"For this linear response regime, we establish generalized Onsager-Casimir relations and an extended fluctuation-dissipation theorem.",
"Furthermore, we derive a family of thermodynamic bounds proving that any local matter or heat current puts a non-trivial lower limit on the overall dissipation rate of a coherent transport process.",
"These bounds do not depend on system-specific parameters, are robust against dephasing and involve only experimentally accessible quantities.",
"They thus provide powerful tools to optimize the performance of mesoscopic devices and for thermodynamic inference, as we demonstrate by working out three specific applications.",
"We then show that physically transparent extensions of our bounds hold also for strong biases and high frequencies.",
"These generalized bounds imply a thermodynamic uncertainty relation that fully accounts for quantum effects and periodic driving.",
"Moreover, they lead to a universal and operationally accessible bound on entropy production that can be readily used for thermodynamic inference and device engineering far from equilibrium.",
"Connecting a broad variety of topics that range from thermodynamic geometry over thermodynamic uncertainty relations to quantum engineering, our work provides a unifying thermodynamic theory of coherent transport that can be tested and utilized with current technologies."
],
[
"Introduction",
"Transport is a thermodynamic process, where gradients in intensive parameters such as chemical potential and temperature drive currents of extensive quantities like matter and energy.",
"In macroscopic systems at high temperatures, this phenomenon can be understood as a result of frequent collisions between classical particles, which lead to random but biased changes of their direction of motion.",
"This mechanism is know as diffusive transport [1].",
"Reducing the temperature of the system increases the mean free path that particles can travel between consecutive collisions.",
"When this length scale becomes comparable to the dimensions of the conductor, as occurs in nano-scale structures at millikelvin temperatures, coherent transport sets in [2].",
"This regime is governed by the laws of quantum mechanics and can no longer be describe in terms of collisions between particles with well-defined positions and momenta.",
"Instead, coherent transport can be seen as arising from the unitary propagation of beams of carriers that are emitted and absorbed by distant thermal reservoirs and undergo elastic scattering within the conductor.",
"This approach goes back to the pioneering work of Landauer [3] and has since evolved into a standard theoretical tool of mesoscopic physics [2], [4].",
"In particular, it has been extended to systems that are subject to oscillating electromagnetic fields, where the scattering of beams is still coherent but no longer elastic, since carriers can exchange discrete amounts of energy with the driving fields [5], [6], [7], [8].",
"On the experimental side, technological progress has made it possible to realize and control periodically driven coherent conductors with a high degree of precision.",
"Today, these systems provide us with a versatile platform to test the basic principles of thermodynamics at small-length and energy scales and to develop new mesoscopic devices such as parametric quantum pumps, which can be used to realize dynamical single-electron sources [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19] as well as for metrological applications [20], [21], [22], and adiabatic quantum motors, which may provide motive power to future nano-machines [23], [24], [25], [26].",
"These endeavors will require a powerful theoretical framework to describe the thermodynamics of coherent transport in the presence of both thermo-chemical biases and periodic driving.",
"A suitable starting point for such a theory is provided by Onsager's irreversible thermodynamics [27], [28], [29].",
"The key idea of this approach is to describe irreversible processes in terms of two types of variables: thermodynamic forces, which drive the process, and currents, which correspond to the system's response.",
"This concept is universal in that it can be applied to macroscopic [27], [28], [29] and mesoscopic [30], [31] systems alike.",
"Moreover, it can be consistently expanded to include periodic driving.",
"Specifically, in the context of coherent transport, this generalization can be achieved by introducing an additional force, which is proportional to the frequency of the applied fields, and an additional current, which corresponds to the flux of photons that is absorbed by the carriers inside the conductor [32].",
"Here, we show that this framework can further be underpinned by rigorous generalizations of two cornerstone results of classical irreversible thermodynamics: the Onsager-Casimir relations [27], [28], [33], [29], which explain the interdependence between linear-response coefficients as a consequence of microscopic time-reversal symmetry, and the fluctuation-dissipation theorem, which connects these coefficients to equilibrium current fluctuations [34], [35].",
"Focusing on moderate electric and thermal biases and slowly varying driving fields, we then set out to derive our first key result, the relations $\\sigma \\ge \\frac{N}{4 K^{xx} (N-1)} (J^x_\\alpha )^2,$ which bound the overall dissipation $\\sigma $ caused by a coherent transport process in an $N$ -terminal conductor in terms of any period averaged matter current $J^\\rho _\\alpha $ or heat current $J^q_\\alpha $ .",
"The coefficients $K^{\\rho \\rho }$ and $K^{qq}$ thereby depend only on the equilibrium temperature and chemical potential of the conductor.",
"These bounds are stronger than the second law, which only requires $\\sigma \\ge 0$ , and universal in that they do not involve any system-specific parameters.",
"In the second part of this article, we extend our theory to systems that are driven far away from equilibrium.",
"This endeavor leads to our second key result, the relation $\\sigma \\ge \\sqrt{(P^{\\rho \\rho }_{\\alpha \\alpha }+2T_\\alpha /h)^2+2\\psi ^\\ast (J^\\rho _\\alpha )^2}-(P^{\\rho \\rho }_{\\alpha \\alpha }+2T_\\alpha /h),$ which makes it possible to bound the total dissipation rate $\\sigma $ by measuring the average $J^\\rho _\\alpha $ and the zero-frequency noise $P^{\\rho \\rho }_{\\alpha \\alpha }$ of a single matter current along with the temperature $T_\\alpha $ of the corresponding reservoir; $h$ denotes Planck's constant and $\\psi ^\\ast \\simeq 8/9$ is a numerical factor.",
"Quite remarkably, the bound (REF ) holds for any coherent multi-terminal conductor, arbitrary strong thermo-chemical biases and arbitrary fast periodic driving fields.",
"Covering both thermal and mechanical driving, the relations (REF ) and (REF ) lead to non-trivial bounds on the figures of merit of cyclic nano-machines based on coherent conductors.",
"In this respect, they advance an active line of research, which has so far mainly focused on steady-state devices and is driven by two major motivations [36], [37], [38], [39], [40], [41], [42], [43], [44], [45], [46], [47], [48], [49].",
"First, universal bounds on figures such as efficiency or power consumption make it possible to quantitatively compare and optimize different theoretical models of mesoscopic devices.",
"Second, such relations can be used in experiments to estimate quantities that cannot be measured directly, a strategy known as thermodynamic inference [50].",
"As we show by working out three specific applications, our theory provides powerful tools for both of these purposes.",
"In particular, we provide a detailed analysis of parametric quantum pumps, for which we uncover a close connection between our approach and the concepts of thermodynamic geometry, a framework that recently proved very useful for optimizing slowly driven quantum thermal machines [51], [52], [53], [54], [55].",
"We proceed as follows.",
"In the next section, we review the essentials of the scattering approach to coherent transport in periodically driven conductors and show how it can be furnished with a thermodynamic structure.",
"We then work out the linear-response theory for this framework and lay down the corresponding generalizations of the Onsager-Casimir relations and the fluctuation dissipation theorem in Sec. .",
"In Sec.",
", we derive the key relation (REF ) and discuss its range of validity.",
"The purpose of Sec.",
"is to demonstrate the versatile applicability of our linear-response results.",
"To this end, we consider three different mesoscopic devices.",
"As an introductory example, we analyze a simple model of a quantum generator, whereby we prove that our new bounds are tight.",
"We then move on to parametric quantum pumps, for which we derive an explicit optimization principle by connecting our theory to the framework of thermodynamic geometry.",
"Furthermore, we show how our bounds can be used for thermodynamic inference.",
"Finally, we further illustrate this technique by applying it to adiabatic quantum motors.",
"In the second major part of this article, Sec.",
", we show how the bounds (REF ) can be extended beyond the limits of the linear-response regime and derive our second key relation (REF ).",
"We then put these results in context with recent developments on thermodynamic uncertainty relations.",
"To this end, we work out a case study, which proves that our bounds on entropy production go significantly beyond earlier results.",
"Finally, we discuss the implications of our theory for autonomous coherent conductors.",
"We summarize our work in Sec.",
"."
],
[
"Scattering Approach",
"Scattering theory provides an elegant tool to describe coherent transport in periodically driven mesoscopic systems.",
"In this approach, the conductor is divided into a sample region, where carriers may be subject to a periodically modulated potential and an external magnetic field, and a set of ideal leads.",
"Each lead is connected to a reservoir, which injects a continuous beam of thermalized, non-interacting carriers.",
"These beams propagates coherently through the system before being reabsorbed by the reservoirs see Fig.",
"REF .",
"The emerging matter and energy currents are thus determined by the inelastic scattering amplitudes of the driven sample and the chemical potentials and temperatures of the reservoirs.",
"In the following, we provide a brief review of this framework and its thermodynamic interpretation.",
"Further details may be found in Refs.",
"[5], [6], [7], [8]."
],
[
"Mean Currents and Fluctuations",
"In the Heisenberg picture, the matter and energy currents that enter the system through the terminal $\\alpha $ correspond to time dependent operators $\\hat{J}^{\\rho }_{\\alpha ,t}$ and $\\hat{J}^{\\varepsilon }_{\\alpha ,t}$ .",
"The mean values and fluctuations of these currents are given by the general expressions $J^u_\\alpha & = \\lim _{t\\rightarrow \\infty } \\frac{1}{t}\\int _0^t\\!\\!\\!",
"dt^{\\prime }\\bigl \\langle \\hat{J}^u_{\\alpha ,t^{\\prime }}\\bigr \\rangle \\quad \\text{and}\\\\P^{uv}_{\\alpha \\beta } &= \\lim _{t\\rightarrow \\infty }\\frac{1}{t} \\int _0^t\\!\\!\\!",
"dt^{\\prime } \\!",
"\\int _0^t\\!\\!\\!",
"dt^{\\prime \\prime }\\bigl \\langle \\bigl (\\hat{J}^u_{\\alpha ,t^{\\prime }}-J^u_\\alpha \\bigr )\\bigl (\\hat{J}^v_{\\beta ,t^{\\prime \\prime }}-J^v_\\beta \\bigr )\\bigr \\rangle ,$ where $u,v=\\rho ,\\varepsilon $ and angular brackets indicate the average over all quantum states of the injected carriers.",
"Since the carriers are non-interacting, this average can be evaluated by treating the incoming beams as ideal Fermionic quantum gases, which leads to the generalized Landauer-Büttiker formula $J^u_\\alpha &= \\frac{1}{h}\\int _0^\\infty \\!\\!\\!",
"dE\\sum \\nolimits _\\beta \\sum \\nolimits _n\\Bigl (\\xi _E^u \\delta _{\\alpha \\beta }\\delta _{n0}-\\xi _{E\\raisebox {-0.7pt}{{{\\scriptsize n}}}}^u\\bigl | S^{\\alpha \\beta }_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}},E} \\bigr |^2\\Bigr )f^\\beta _E$ for the mean currents with $\\xi ^\\rho _E = 1$ and $\\xi ^\\varepsilon _E =E$ .",
"Here, we have introduced the shorthand notation $E_n \\equiv E + n\\hbar \\omega $ , where $\\hbar =h/2\\pi $ denotes the reduced Planck constant and $\\omega \\equiv 2\\pi / the frequency of periodic driving fields acting onthe sample.Thermodynamics enters Eq.~(\\ref {TwMeanCurrents}) via the Fermifunctions\\begin{equation}f^\\alpha _E \\equiv \\frac{1}{1+\\exp [(E-\\mu _\\alpha )/T_\\alpha ]},\\end{equation}where $$ and $ T$ are the chemical potential and temperatureof the reservoir $$ and Boltzmann^{\\prime }s constant is set to $ 1$throughout.The properties of the sample are encoded in the Floquet scatteringamplitudes $ SE$n$,E$.These objects describe the transmission of an incoming carrier withenergy $ E$ from the terminal $$ to the terminal $$ underthe absorption of $ n$ photons with energy $$.Note that, for simplicity, we assume that each lead supports only onetransport channel.Furthermore, we use the convention that the photon-counting index runsover all integers and that the Floquet scattering amplitudes are zeroif one of their energy arguments is negative.$ The current fluctuations () can be evaluated in the same way as the mean currents.",
"The resulting formula involves two contributions, $P^{uv}_{\\alpha \\beta }=D^{uv}_{\\alpha \\beta } + R^{uv}_{\\alpha \\beta }$ , which are given by $D^{uv}_{\\alpha \\beta } &= \\frac{1}{h}\\int _0^\\infty \\!\\!\\!",
"dE\\sum \\nolimits _n\\bigl (A^{uv,\\alpha \\beta }_{n,E} + A^{vu,\\beta \\alpha }_{n,E}+\\delta _{\\alpha \\beta }B^{uv,\\alpha }_{n,E}\\bigr ),\\\\R^{uv}_{\\alpha \\beta } &= \\frac{1}{2h}\\!\\int _0^\\infty \\!\\!\\!",
"dE\\sum \\nolimits _{\\gamma \\delta }\\sum \\nolimits _nC^{u\\alpha ,\\gamma \\delta }_{n,E}C^{v\\beta ,\\gamma \\delta \\ast }_{n,E}$ with $A^{uv,\\alpha \\beta }_{n,E} &\\equiv \\xi ^u_E\\xi ^v_E \\delta _{\\alpha \\beta }\\delta _{n0}f^{\\prime \\alpha }_E-\\xi ^u_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}}}\\xi ^v_E \\bigl | S^{\\alpha \\beta }_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}},E} \\bigr |^2f^{\\prime \\beta }_E,\\\\B^{uv,\\alpha }_{n,E} &\\equiv \\sum \\nolimits _\\gamma \\xi ^u_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}}}\\xi ^v_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}}}\\bigl | S^{\\alpha \\gamma }_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}},E} \\bigr |^2(f^{\\prime \\gamma }_E -f^{\\prime \\alpha }_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}}}),\\\\C^{u\\alpha ,\\beta \\gamma }_{n,E} &\\equiv \\sum \\nolimits _m \\xi ^u_{E\\raisebox {-0.7pt}{{{\\scriptsize m}}}}S^{\\alpha \\gamma }_{E\\raisebox {-0.7pt}{{{\\scriptsize m}}},E\\raisebox {-0.7pt}{{{\\scriptsize n}}}}S^{\\alpha \\beta \\ast }_{E\\raisebox {-0.7pt}{{{\\scriptsize m}}},E}(f^\\beta _E-f^\\gamma _{E\\raisebox {-0.7pt}{{{\\scriptsize n}}}})$ and $f^{\\prime \\alpha }_E \\equiv f^\\alpha _E(1-f^\\alpha _E)$ .",
"The thermal, or Nyquist-Johnson, noise $D^{uv}_{\\alpha \\beta }$ thereby arises from thermal fluctuations in the injected beams of carriers and vanishes in the zero-temperature limit.",
"By contrast, the shot noise $R^{uv}_{\\alpha \\beta }$ stems from the probabilistic nature of carrier transmissions through the sample, and therefore persists at zero temperature."
],
[
"Unitarity and Time-Reversal Symmetry",
"The Floquet scattering amplitudes generally depend on the structure of the sample and the applied driving protocols $\\mathbf {V}_t$ , where $\\mathbf {V}\\equiv \\lbrace V_j\\rbrace $ denotes the set of external control parameters.",
"Still, they obey two universal relations, which follow from fundamental principles.",
"First, the unitarity conditions $& \\sum \\nolimits _\\alpha \\sum \\nolimits _nS^{\\alpha \\beta }_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}},E}S^{\\alpha \\gamma \\ast }_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}},E\\raisebox {-0.7pt}{{{\\scriptsize m}}}}=\\delta _{\\beta \\gamma }\\delta _{m0}\\quad \\text{and}\\\\[3pt]& \\sum \\nolimits _\\alpha \\sum \\nolimits _nS^{\\beta \\alpha }_{E,E\\raisebox {-0.7pt}{{{\\scriptsize n}}}}S^{\\gamma \\alpha \\ast }_{E\\raisebox {-0.7pt}{{{\\scriptsize m}}},E\\raisebox {-0.7pt}{{{\\scriptsize n}}}}=\\delta _{\\beta \\gamma }\\delta _{m0}$ ensure the conservation of probabilities in individual scattering events.",
"Second, the invariance of Schrödinger's equation under time reversal implies the symmetry $S^{\\alpha \\beta }_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}},E} = \\mathsf {T}_\\mathbf {B}\\mathsf {T}_{\\mathbf {V}}S^{\\beta \\alpha }_{E,E\\raisebox {-0.7pt}{{{\\scriptsize n}}}},$ where the symbolic operators $\\mathsf {T}_\\mathbf {B}$ and $\\mathsf {T}_{\\mathbf {V}}$ indicate the reversal of external magnetic fields and driving protocols, respectively.",
"Note that, while the unitarity conditions (REF ) apply to the scattering amplitudes of any given sample, the symmetry relation (REF ) connects the scattering amplitudes of two different systems that are related to each other by time reversal."
],
[
"Thermodynamics",
"The thermodynamics of coherent transport can be developed from the conservation laws $\\sum \\nolimits _\\alpha J^\\rho _\\alpha = 0 \\quad \\text{and}\\quad \\Pi _{{{\\rm ac}}} + \\sum \\nolimits _\\alpha J^\\varepsilon _\\alpha = 0,$ which can be easily verified using generalized Landauer-Büttiker formula (REF ) and the unitarity conditions (REF ).",
"They reflect the fact that neither matter nor energy are accumulated in the sample over a full cycle.",
"The quantity $\\Pi _{{{\\rm ac}}}\\equiv -\\sum \\nolimits _\\alpha J^\\varepsilon _\\alpha = \\frac{1}{\\int _0^\\infty \\!\\!\\!",
"dE\\sum \\nolimits _{\\alpha \\beta }\\sum \\nolimits _nn\\bigl | S^{\\alpha \\beta }_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}},E} \\bigr |^2 f^\\beta _E}thereby corresponds to the average mechanical power that is absorbedby the carriers from the driving fields.$ Combining the conservation laws (REF ), leads to the first law of thermodynamics $\\Pi _{{{\\rm ac}}} + \\Pi _{{{\\rm el}}} = -\\sum \\nolimits _\\alpha J^q_\\alpha .$ Here, the electrical power $\\Pi _{{{\\rm el}}}$ , which is generated by matter currents flowing in the direction of chemical potential gradients, and the heat currents entering the conductor from the reservoirs, $J^q_\\alpha $ , are given by $\\Pi _{{{\\rm el}}}\\equiv \\sum \\nolimits _\\alpha \\mu _\\alpha J^\\rho _\\alpha \\quad \\text{and}\\quad J^q_\\alpha \\equiv J^\\varepsilon _\\alpha - \\mu _\\alpha J^\\rho _\\alpha .$ Since the transfer of carriers through the system is coherent, and thus reversible, dissipation occurs only in the reservoirs due to the in- and outflux of heat.",
"Hence, the total rate of entropy production is given by $\\sigma &\\equiv -\\sum \\nolimits _\\alpha J^q_\\alpha /T_\\alpha \\\\&= \\frac{1}{h}\\int _0^\\infty \\!\\!\\!",
"dE\\sum \\nolimits _{\\alpha \\beta }\\sum \\nolimits _n\\left(\\frac{E\\raisebox {-0.7pt}{{{\\scriptsize n}}}-\\mu _\\alpha }{T_\\alpha } -\\frac{E-\\mu _\\beta }{T_\\beta }\\right)\\bigl | S^{\\alpha \\beta }_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}},E} \\bigr |^2 f^\\beta _E.\\nonumber $ This expression is non-negative for any temperature and chemical potential profiles and any set of Floquet scattering amplitudes [8].",
"The scattering formalism is therefore consistent with the second law of thermodynamics, which requires $\\sigma \\ge 0$ .",
"In irreversible thermodynamics, transport is described in terms of thermodynamic forces, or affinities, which correspond to gradients of intensive variables, such as temperature, and currents of extensive quantities like energy [29].",
"Every affinity forms a conjugate pair with a specific current such that the products of these pairs add up to the total rate of entropy production in the system.",
"Close to equilibrium, the currents become linear functions of the affinities with the corresponding response coefficients obeying two universal relations: the Onsager-Casimir symmetry, which connects reciprocal coefficients [27], [28], [33], and the fluctuation-dissipation theorem, which relates them to equilibrium current fluctuations [34], [35].",
"These results are well-established for stationary coherent transport in mesoscopic systems [30], [31], [56].",
"In the following, we show how they can be extended to systems with periodic driving by further developing the approach that was proposed in Ref.",
"[32]."
],
[
"Affinities",
"The affinities for the matter and heat currents are given by the thermo-chemical gradients $F^\\rho _\\alpha \\equiv (\\mu _\\alpha -\\mu )/T \\quad \\text{and}\\quad F^q_\\alpha \\equiv 1/T - 1/T_\\alpha ,$ where $\\mu $ and $T$ are the reference chemical potential and temperature.",
"Using these definitions and the conservation laws (REF ), the total rate of entropy production (REF ) can be written as $\\sigma = \\Pi _{{{\\rm ac}}}/T + \\sum \\nolimits _\\alpha \\sum \\nolimits _xF^x_\\alpha J^x_\\alpha $ with $x=\\rho ,q$ .",
"Upon recalling the expression (REF ) for the average mechanical power, this result suggests that we introduce the photon current $J^\\omega \\equiv \\Pi _{{{\\rm ac}}}/\\hbar \\omega =\\frac{1}{h}\\int _0^\\infty \\!\\!\\!",
"dE\\sum \\nolimits _{\\alpha \\beta }\\sum \\nolimits _nn \\bigl | S^{\\alpha \\beta }_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}},E} \\bigr |^2 f^\\beta _E$ and the corresponding affinity $F^\\omega \\equiv \\hbar \\omega /T$ such that $\\sigma $ assumes the canonical bilinear form $\\sigma = J^\\omega F^\\omega +\\sum \\nolimits _\\alpha \\sum \\nolimits _xJ^x_\\alpha F^x_\\alpha = \\sum \\nolimits _AJ_A F_A.$ Here, we use capital roman letters to denote compound indices covering both thermo-chemical and mechanical quantities, i.e., $A=\\lbrace (x,\\alpha )\\rbrace ,\\omega $ .",
"Two remarks are in order.",
"First, the interpretation of $J^\\omega $ as flux of photons derives from the fact that the carriers and the driving fields exchange only discrete amounts of energy.",
"This phenomenon, which, on the technical level, is a consequence of the Floquet theorem [8], is a manifestation of the laws of quantum mechanics and has no counterpart in classical mechanics [57].",
"Second, to achieve a thermodynamic unification of steady-state and periodic driving, the driving frequency is treated as a thermodynamic force in Eq.",
"(REF ).",
"This approach, which was proposed in Ref.",
"[32], is more suitable for coherent transport than using the amplitude of the time-dependent fields as an effective affinity, a scheme that has proved very useful for systems obeying stochastic dynamics [58], [59], [60], [61], [62].",
"In particular, as we show next, the frequency-based approach enables a non-trivial linear-response theory, while a perturbation theory in the driving strength only leads to a trivial decoupling of thermo-chemical currents and mechanical driving, see App.",
"REF for details."
],
[
"Kinetic Coefficients",
"The kinetic coefficients that govern the relation between currents and affinities in the linear-response regime are defined as $L_{AB} = \\partial _{F_B}J_A|_{{{\\rm eq}}}$ where the notation $\\cdots |_{{{\\rm eq}}}$ indicates the limit $F_A\\rightarrow 0$ .",
"To calculate the coefficients (REF ), we first observe that the Fermi functions of the reservoirs are given by $f^\\alpha _E \\simeq f_E + \\bigl (F^\\rho _\\alpha +(E-\\mu )F^q_\\alpha \\bigr )f^{\\prime }_E\\\\$ with $f_E\\equiv f^\\alpha _E|_{{{\\rm eq}}}$ and $f^{\\prime }_E\\equiv f^{\\prime \\alpha }_E|_{{{\\rm eq}}}= f_E(1-f_E)$ , up to second-order corrections in the thermo-chemical affinities.",
"Second, we recall that the Floquet scattering amplitudes admit the low-frequency expansion [7], [63] $S^{\\alpha \\beta }_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}},E\\raisebox {-0.7pt}{{{\\scriptsize m}}}}\\simeq \\frac{1}{ \\!\\left(\\mathcal {S}^{\\alpha \\beta }_{E,\\mathbf {V}\\raisebox {-0.7pt}{{{\\scriptsize t}}}} +\\hbar \\omega \\frac{n+m}{2}\\partial _E^{\\phantom{\\beta }}\\mathcal {S}^{\\alpha \\beta }_{E,\\mathbf {V}\\raisebox {-0.7pt}{{{\\scriptsize t}}}}+\\hbar \\omega \\mathcal {A}^{\\alpha \\beta }_{E,t}\\right)e^{i(n-m)\\omega t}.",
"}$ Here, the frozen scattering amplitudes $\\mathcal {S}^{\\alpha \\beta }_{E,\\mathbf {V}}$ describe the transmission of carriers with energy $E$ at fixed parameters $\\mathbf {V}$ .",
"The correction term $\\mathcal {A}^{\\alpha \\beta }_{E,t}$ is required to ensure that the right-hand side of Eq.",
"(REF ) obeys the unitarity condition (REF ).",
"In general, the approximation (REF ) is applicable if the driving fields vary only slightly during the average dwell time $\\tau _{{{\\rm dw}}}$ of carriers inside the sample.",
"This time scale is connected to the typical energy range $\\delta _E$ over which the frozen scattering amplitudes change by the relation $\\tau _{{{\\rm dw}}}= \\hbar /\\delta _E$ , for details see [64], [7], [65].",
"Using Eqs.",
"(REF ) and (REF ), the kinetic coefficients (REF ) can be determined as $L_{\\alpha \\beta }^{xy} & = \\frac{1}{h}\\int _0^\\infty \\!\\!\\!",
"dE\\; \\zeta ^x_E\\zeta ^y_E\\Bigl (\\delta _{\\alpha \\beta }-\\bigl \\bigl |\\mathcal {S}^{\\alpha \\beta }_{E,\\mathbf {V}}\\bigr |^2\\bigr \\Bigr )f^{\\prime }_E ,\\\\L_\\alpha ^{x\\omega } & = \\frac{1}{h} \\int _0^\\infty \\!\\!\\!",
"dE\\; \\zeta ^x_E\\zeta ^\\omega \\sum \\nolimits _\\beta {{{\\rm Im}}}\\Bigl [\\bigl \\dot{\\mathcal {S}}_{E,\\mathbf {V}}^{\\alpha \\beta }\\mathcal {S}_{E,\\mathbf {V}}^{\\alpha \\beta \\ast }\\bigr \\Bigr ]f^{\\prime }_E ,\\\\L_\\alpha ^{\\omega x} & = \\frac{1}{h} \\int _0^\\infty \\!\\!\\!",
"dE\\; \\zeta ^x_E\\zeta ^\\omega \\sum \\nolimits _\\beta {{{\\rm Im}}}\\Bigl [\\bigl \\mathcal {S}_{E,\\mathbf {V}}^{\\beta \\alpha }\\dot{\\mathcal {S}}_{E,\\mathbf {V}}^{\\beta \\alpha \\ast }\\bigr \\Bigr ]f^{\\prime }_E ,\\\\L^{\\omega \\omega } & = \\frac{1}{2h} \\int _0^\\infty \\!\\!\\!",
"dE\\; (\\zeta ^\\omega )^2\\sum \\nolimits _{\\alpha \\beta }\\;\\bigl \\bigl |\\dot{\\mathcal {S}}^{\\alpha \\beta }_{E,\\mathbf {V}}\\bigr |^2\\bigr f^{\\prime }_E$ with $\\zeta ^\\rho _E\\equiv 1,\\;\\zeta ^q_E\\equiv E-\\mu ,\\;\\zeta ^\\omega \\equiv 1/\\omega $ , dots indicating time derivatives and double brackets denoting the time average over one period, $\\cdots \\equiv \\frac{1}{\\cdots , for details seeApp.~\\ref {Appx_KinCoeff_Ad}.Notably, the expressions (\\ref {TwKinCoeffExp}) do not depend on thecorrections \\mathcal {A}^{\\alpha \\beta }_{E,t} as a result of the unitaritycondition (\\ref {TwSumRulesSM}).Instead, they involve only the frozen scattering amplitudes\\mathcal {S}^{\\alpha \\beta }_{E,\\mathbf {V}}, which are generally much easier to obtain thanthe full Floquet scattering amplitudes.",
"}The linear response regime with respect to the affinities $ F$,$ Fq$ and $ F$ is defined by the three conditions\\begin{equation}F_\\alpha ^\\rho \\ll \\mu /T, \\quad F_\\alpha ^q \\ll 1/T\\quad \\text{and}\\quad F^\\omega \\ll \\delta _E/T,\\end{equation}under which the currents obey the kinetic equations\\begin{equation}J_A = \\sum \\nolimits _BL_{AB} F_B\\end{equation}This result extends the conventional framework of linear-irreversiblethermodynamics to periodically driven coherent conductors.We note that, at low temperatures, the function $ f'E$ in theEqs.~(\\ref {TwKinCoeffExp}) is sharply peaked around $$.The transmission of carriers then occurs only at energies close to theFermi edge.It is therefore typically sufficient to require that the slow-drivingcondition $ FE/$ is obeyed at $ E$ for the kineticequations (\\ref {TwARKinEq}) to be valid.$"
],
[
"Onsager-Casimir Relations",
"The Onsager-Casimir, or reciprocal, relations between linear-response coefficients follow from the symmetry of microscopic dynamics under time reversal.",
"This principle enters stationary scattering theory through the property $\\mathcal {S}^{\\alpha \\beta }_{E,\\mathbf {V}} = \\mathsf {T}_\\mathbf {B}\\mathcal {S}^{\\beta \\alpha }_{E,\\mathbf {V}}$ of the frozen scattering amplitudes [8], which, together with the formulas (REF ), implies the relations $\\mathsf {T}_{\\mathbf {B}}L^{xy}_{\\alpha \\beta } = L^{yx}_{\\beta \\alpha }\\quad \\text{and}\\quad \\mathsf {T}_{\\mathbf {B}}L^{\\omega x}_\\alpha = - L^{x\\omega }_\\alpha ;$ recall Sec.",
"(REF ) for the definition of $\\mathsf {T}_\\mathbf {B}$ and $\\mathsf {T}_{\\mathbf {V}}$ .",
"Hence, while the thermo-chemical coefficients obey the conventional Onsager-Casimir symmetry, the cross-coefficients that couple either to the mechanical affinity or the photon current are anti-symmetric.",
"The original symmetry can, however, be restored for all kinetic coefficients by reversing both magnetic fields and driving protocols.",
"That is, we have the generalized Onsager-Casimir relations Note that $& \\mathsf {T}_{\\mathbf {V}}\\bigl \\bigl | \\mathcal {S}^{\\alpha \\beta }_{E,\\mathbf {V}} \\bigr |^2\\bigr =\\bigl \\bigl | \\mathcal {S}^{\\alpha \\beta }_{E,\\mathbf {V}} \\bigr |^2\\bigr \\quad \\text{and}\\\\& \\mathsf {T}_{\\mathbf {V}}\\bigl \\dot{\\mathcal {S}}^{\\alpha \\beta }_{E,\\mathbf {V}}\\mathcal {S}^{\\alpha \\beta \\ast }_{E,\\mathbf {V}}\\bigr =-\\bigl \\dot{\\mathcal {S}}^{\\alpha \\beta }_{E,\\mathbf {V}}\\mathcal {S}^{\\alpha \\beta \\ast }_{E,\\mathbf {V}}\\bigr ,$ since $\\mathsf {T}_{\\mathbf {V}}\\mathcal {S}^{\\alpha \\beta }_{E,\\mathbf {V}\\raisebox {-0.7pt}{{{\\scriptsize t}}}} = \\mathcal {S}^{\\alpha \\beta }_{E,\\mathbf {V}_{-t}}$ .",
"$\\mathsf {T}_{\\mathbf {B}}\\mathsf {T}_{\\mathbf {V}}L_{AB} = L_{BA}$ Notably, this result implies that, if the driving protocols are symmetric, i.e., if $\\mathbf {V}_t = \\mathbf {V}_{-t}$ , the mechanical and thermo-chemical currents and affinities decouple, since $L^{x\\omega }_\\alpha = L^{\\omega x}_\\alpha =0$ ."
],
[
"Fluctuation-Dissipation Theorem",
"The fluctuation-dissipation theorem provides a link between kinetic coefficients and equilibrium current fluctuations.",
"For periodically driven coherent conductors, this connection can be established as follows.",
"First, we note that, with respect to the fluctuations of matter and energy currents, $P^{uv}_{\\alpha \\beta }$ , which are spelled out in Eqs.",
"(REF ) and (REF ), the joint fluctuations of matter and heat currents are given by $P^{xy}_{\\alpha \\beta } = \\sum \\nolimits _{uv} c^{xu}_\\alpha c^{yv}_\\beta P^{uv}_{\\alpha \\beta }$ with $c^{\\rho u}_\\alpha \\equiv \\delta _{u\\rho }$ and $c^{q u}_\\alpha \\equiv \\delta _{u\\varepsilon }-\\mu _\\alpha \\delta _{u\\rho }$ [8].",
"Next, we observe that the shot noise () vanishes in equilibrium, $R^{uv}_{\\alpha \\beta }|_{{{\\rm eq}}}=0$ .",
"Therefore, we have $P^{xy}_{\\alpha \\beta }|_{{{\\rm eq}}}= \\sum \\nolimits _{uv}(c^{xu}_\\alpha c^{yv}_\\beta D^{uv}_{\\alpha \\beta })|_{{{\\rm eq}}} = L^{xy}_{\\alpha \\beta } + L^{yx}_{\\beta \\alpha }$ as can be easily verified by inspection.",
"For systems without magnetic fields, we thus recover the standard result $P^{xy}_{\\alpha \\beta }=2L^{xy}_{\\alpha \\beta }$ by using the symmetry (REF ).",
"To derive a fluctuation-dissipation relation for the coefficients $L^{\\omega x}_\\alpha $ , $L^{x\\omega }_\\alpha $ and $L^{\\omega \\omega }$ , we have to consider the fluctuations involving the photon current, which, due to energy conservation, can be obtained from the sum rules $P^{u\\omega }_\\alpha &= -\\sum \\nolimits _\\beta P^{u\\varepsilon }_{\\alpha \\beta }/\\hbar \\omega \\;\\;\\text{and}\\;\\;P^{\\omega \\omega } = \\sum \\nolimits _{\\alpha \\beta }P^{\\varepsilon \\varepsilon }_{\\alpha \\beta }/(\\hbar \\omega )^2.$ Inserting the expressions (REF ) and (REF ) for the thermal and the shot noise and using the unitarity conditions (REF ) for the Floquet scattering amplitudes yields the explicit results $P^{u\\omega }_\\alpha = D^{u\\omega }_\\alpha +R^{u\\omega }_\\alpha $ and $P^{\\omega \\omega }= D^{\\omega \\omega } + R^{\\omega \\omega }$ with $D_\\alpha ^{u\\omega } & = \\frac{1}{h}\\int _0^\\infty \\!\\!\\!",
"dE\\sum \\nolimits _\\beta \\sum \\nolimits _n\\bigl ( \\xi ^u_E A^{\\omega 1,\\beta \\alpha }_{n,E}-\\xi ^u_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}}} A^{\\omega 1,\\alpha \\beta }_{n,E}\\bigr ),\\\\D^{\\omega \\omega } &=\\frac{1}{h}\\int _0^\\infty \\!\\!\\!",
"dE\\sum \\nolimits _{\\alpha \\beta }\\sum \\nolimits _nA^{\\omega 2,\\alpha \\beta }_{n,E},\\\\R^{u\\omega }_\\alpha &= \\frac{1}{2h}\\int _0^\\infty \\!\\!\\!",
"dE\\sum \\nolimits _{\\gamma \\delta }\\sum \\nolimits _nC^{u\\alpha ,\\gamma \\delta }_{n,E} C^{\\omega ,\\gamma \\delta \\ast }_{n,E},\\\\R^{\\omega \\omega } &= \\frac{1}{2h}\\int _0^\\infty \\!\\!\\!",
"dE\\sum \\nolimits _{\\gamma \\delta }\\sum \\nolimits _nC^{\\omega ,\\gamma \\delta }_{n,E}C^{\\omega ,\\gamma \\delta \\ast }_{n,E}$ and $A^{\\omega k,\\alpha \\beta }_{n,E} & \\equiv n^k \\bigl | S^{\\alpha \\beta }_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}},E} \\bigr |^2 f^{\\prime \\beta }_E\\\\C^{\\omega ,\\beta \\gamma }_{n,E} &\\equiv \\sum \\nolimits _\\alpha \\sum \\nolimits _m m S^{\\alpha \\gamma }_{E\\raisebox {-0.7pt}{{{\\scriptsize m}}},E\\raisebox {-0.7pt}{{{\\scriptsize n}}}}S^{\\alpha \\beta \\ast }_{E\\raisebox {-0.7pt}{{{\\scriptsize m}}},E}(f^\\beta _E- f^\\gamma _{E\\raisebox {-0.7pt}{{{\\scriptsize n}}}}).$ The expression (REF ) shows that $R^{u\\omega }_\\alpha |_{{{\\rm eq}}}=0$ .",
"After switching from energy to heat currents, we are therefore left with Recall that $&\\sum \\nolimits _nn \\bigl | S^{\\alpha \\beta }_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}},E} \\bigr |^2 \\rightarrow \\zeta ^\\omega {{{\\rm Im}}}\\Bigl [\\bigl \\mathcal {S}^{\\alpha \\beta }_{E,\\mathbf {V}} \\dot{\\mathcal {S}}^{\\alpha \\beta \\ast }_{E,\\mathbf {V}}\\bigr \\Bigr ] \\quad \\text{and}\\\\&\\sum \\nolimits _nn^2 \\bigl | S^{\\alpha \\beta }_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}},E} \\bigr |^2 \\rightarrow (\\zeta ^\\omega )^2\\bigl \\dot{\\mathcal {S}}^{\\alpha \\beta }_{E,\\mathbf {V}}\\dot{\\mathcal {S}}^{\\alpha \\beta \\ast }_{E,\\mathbf {V}}\\bigr $ in the limit $\\omega \\rightarrow 0$ .",
"$P^{x\\omega }_\\alpha |_{{{\\rm eq}}}& = \\sum \\nolimits _u (c^{xu}_\\alpha D^{u\\omega }_\\alpha )|_{{{\\rm eq}}}= L^{x\\omega }_\\alpha +L^{\\omega x}_\\alpha \\quad \\text{and}\\\\P^{\\omega \\omega }|_{{{\\rm eq}}}& = D^{\\omega \\omega }|_{{{\\rm eq}}}= 2 L^{\\omega \\omega }.$ Hence, quite remarkably, the mechanical kinetic coefficients and the equilibrium fluctuations of the photon current obey the same relations as their thermo-chemical counterparts.",
"This result is summarized by the extended fluctuation-dissipation theorem $P_{AB}|_{{{\\rm eq}}}= L_{AB} + L_{BA},$ which completes our linear-response framework.",
"Notably, it implies, together with the symmetries (REF ) and (REF ), that $P^{x\\omega }_\\alpha |_{{{\\rm eq}}}=0$ for systems without a magnetic field or with symmetric driving protocols.",
"That is, equilibrium correlations between the photon current and the thermo-chemical currents are ultimately a result of broken time-reversal symmetry.",
"According to the second law, the rate of entropy production $\\sigma $ , which provides a measure for the thermodynamic cost of irreversible transport, cannot be negative.",
"However, the laws of thermodynamics do not determine how much entropy must be generated to sustain a given current as the following argument shows.",
"In linear response, the currents $J_A$ can be divided into an irreversible and a reversible contributions given by [38] $J_A^{{{\\rm irr}}}\\equiv \\sum \\nolimits _B\\frac{L_{AB}+L_{BA}}{2}F_B,\\quad J_A^{{{\\rm rev}}}\\equiv \\sum \\nolimits _B\\frac{L_{AB}-L_{BA}}{2}F_B.$ Using these variables, the rate of entropy production (REF ), can be expressed as $\\sigma = \\sum \\nolimits _AJ_A^{{{\\rm irr}}}F_A.$ Hence, the reversible currents, which, due to the generalized Onsager-Casimir relation (REF ), exist only in systems with broken time reversal symmetry, do not contribute to $\\sigma $ .",
"As a result, transport without dissipation seems to be possible in situations where $J_A^{{{\\rm irr}}}=0$ and at least one reversible current is finite [37], [68].",
"This a priori surprising observation prompts the question whether their might be stronger bounds on thermal currents than the second law.",
"In the following, we first derive such bounds for coherent transport and then prove their robustness against dephasing."
],
[
"Coherent Transport",
"Our new bounds follow from the unitarity conditions for the frozen scattering amplitudes [7], $\\sum \\nolimits _\\alpha \\mathcal {S}^{\\alpha \\beta }_{E,\\mathbf {V}}\\mathcal {S}^{\\alpha \\gamma \\ast }_{E,\\mathbf {V}}=\\sum \\nolimits _\\alpha \\mathcal {S}^{\\beta \\alpha }_{E,\\mathbf {V}}\\mathcal {S}^{\\gamma \\alpha \\ast }_{E,\\mathbf {V}} = \\delta _{\\beta \\gamma },$ which ensure probability conservation, and the sum rule $\\sum \\nolimits _{\\alpha \\beta }\\bigl \\dot{\\mathcal {S}}^{\\alpha \\beta }_{E,\\mathbf {V}}\\mathcal {S}^{\\alpha \\beta \\ast }_{E,\\mathbf {V}}\\bigr =0,$ which plays the role of a gauge condition fixing the global phase of the frozen scattering amplitudes, see Lemma REF of App. .",
"Together, they lead to the sum rules $\\sum \\nolimits _\\alpha L^{xy}_{\\alpha \\beta }=0$ and $\\sum \\nolimits _\\alpha L^{x\\omega }_\\alpha =0$ for the kinetic coefficients (REF ), which imply the conservation laws $\\sum \\nolimits _\\alpha J^\\rho _\\alpha = 0 \\quad \\text{and}\\quad \\sum \\nolimits _\\alpha J^q_\\alpha =0.$ Note that the conservation law for the heat currents is consistent with the first law (REF ) in linear response, since the electrical and the mechanical power, $\\Pi _{{{\\rm el}}}$ and $\\Pi _{{{\\rm ac}}}$ , are of second order in the affinities.",
"As a technical tool, we now define the quadratic form $\\Xi \\equiv \\sigma +\\sum \\nolimits _\\alpha \\sum \\nolimits _xJ^x_\\alpha G^x_\\alpha +\\sum \\nolimits _\\alpha \\sum \\nolimits _{xy}K^{xy}G^x_\\alpha G^y_\\alpha ,$ where the $G^x_\\alpha $ are real but otherwise arbitrary variables and the coefficients $K^{xy}\\equiv \\frac{1}{2h}\\int _0^\\infty \\!\\!\\!",
"dE\\; \\zeta ^x_E\\zeta ^y_E f^{\\prime }_E$ have been chosen such that $\\Xi $ is positive semi-definite.",
"To verify this property, we expand the rate of entropy production $\\sigma $ and the thermo-chemical currents $J^x_\\alpha $ in the affinities using Eq.",
"(REF ) and the kinetic equations ().",
"Upon inserting the expressions (REF ) for the kinetic coefficients, and applying the unitarity conditions (REF ), Eq.",
"(REF ) can thus be rewritten in the form $\\Xi = \\frac{1}{2h}\\int _0^\\infty \\!\\!\\!",
"dE\\;\\sum \\nolimits _{\\alpha \\beta }\\bigl \\bigl | X^{\\alpha \\beta }_E\\mathcal {S}^{\\alpha \\beta }_{E,\\mathbf {V}}- iX^\\omega \\dot{\\mathcal {S}}^{\\alpha \\beta }_{E,\\mathbf {V}} \\bigr |^2\\bigr f^{\\prime }_E$ with $X^{\\alpha \\beta }_E\\equiv \\sum \\nolimits _x\\zeta ^x_E (F^x_\\alpha - F^x_\\beta +G^x_\\alpha )$ and $X^\\omega \\equiv \\zeta ^w F^\\omega $ , which proves that $\\Xi $ cannot be negative, since $f^{\\prime }_E\\ge 0$ .",
"This result leads to a whole family of bounds on $\\sigma $ , which can be extracted as follows.",
"We first set $G^q_\\alpha =0$ and minimize the right-hand side of Eq.",
"(REF ) with respect to the variables $G^\\rho _\\alpha $ .",
"After repeating this step with the roles of $G^q_\\alpha $ and $G^\\rho _\\alpha $ interchanged, we arrive at the cumulative bound $\\sigma \\ge \\frac{1}{4 K^{xx}}\\sum \\nolimits _\\alpha (J^x_\\alpha )^2.$ To obtain bounds that involve only a single current, we rewrite Eq.",
"(REF ) as $\\sigma \\ge \\frac{(J^x_\\alpha )^2}{4K^{xx}}+ M^x_\\alpha \\quad \\text{with}\\quad M^x_\\alpha \\equiv \\frac{1}{4 K^{xx}}\\sum \\nolimits _{\\beta \\ne \\alpha } (J^x_\\beta )^2.$ Minimizing $M^x_\\alpha $ with respect to the currents $J^x_\\beta $ while taking into account the conservation laws (REF ) as a constraint yields $M^x_\\alpha \\ge (J^x_\\alpha )^2/4K^{xx}(N-1)$ and thus $\\sigma \\ge \\frac{N}{4K^{xx}(N-1)} (J^x_\\alpha )^2,$ where $N$ is the number of terminals of the conductor and the inequality holds for any pair of indices $x$ and $\\alpha $ .",
"This relation constitutes a key result of this paper.",
"Going beyond the second law, it shows that, regardless of the behavior of the system under time reversal, any matter or heat current comes at the price of a minimal rate of entropy production that is proportional to the square of this current.",
"The positive coefficients $K^{xx}$ thereby depend only on the equilibrium properties of the reservoirs.",
"Specifically, they are given by $K^{\\rho \\rho }&= \\frac{T\\varphi }{2h(1+\\varphi )}\\le \\frac{T}{2h}\\quad \\text{and}\\\\K^{qq} &=\\frac{\\pi ^2 T^3}{6h}\\begin{aligned}[t]&-\\frac{T^3(\\ln [\\varphi ])^2}{2h(1+\\varphi )}-\\frac{T^3\\ln [\\varphi ]\\ln [1+1/\\varphi ]}{h}\\\\&+\\frac{T^3{{{\\rm Li}}}_2[-1/\\varphi ]}{h}\\le \\frac{\\pi ^2 T^3}{6h},\\end{aligned}$ where ${{{\\rm Li}}}_2$ denotes the dilogarithm, $\\varphi \\equiv \\exp [\\mu /T]$ the equilibrium fugacity of the reservoirs and the inequalities are saturated in the limit $\\varphi \\rightarrow \\infty $ , which is practically realized in mesoscopic conductors."
],
[
"Dephasing",
"Coherent transport is characterized by a fixed phase relation between incoming and outgoing carriers.",
"Under realistic conditions, however, phase-breaking mechanisms such as carrier-carrier or carrier-phonon interactions can hardly be completely suppressed.",
"Probe terminals provide an elegant way to account for such effects [69].",
"In this approach, virtual reservoirs are attached to the conductor, whose temperature and chemical potential are adjusted such that they do not exchange matter or heat with the remaining system on average, but rather act as a source of dephasing.",
"On the technical level, the virtual reservoirs differ from physical ones only in that their affinities are fixed by the conditions of zero mean currents.",
"The bounds (REF ), however, were derived without any assumptions on the affinities.",
"They therefore apply also to systems with arbitrary many probe terminals, where only the currents between physical reservoirs contribute to the sum on the right; those into the virtual reservoirs are zero by construction.",
"Since the conservation laws (REF ) are likewise not affected by the probe terminals, also the bounds (REF ) remain valid with $N$ referring to the number of real terminals.",
"Hence, our new bounds are robust against dephasing and hold even in the limit of fully incoherent transmission [69]."
],
[
"Applications",
"In this section we discuss three different quantum devices to explore the practical implications of our new bounds on coherent transport and their potential as tools of thermodynamic inference.",
"As a first example, we consider a basic model of a magnetic-flux driven quantum generator, which was proposed in Ref. [36].",
"This case study serves as a simple illustration of our general theory and shows that our bounds are tight.",
"We then move on to parametric quantum pumps, which make it possible to move a well-defined amount of carriers from one reservoir to another in a given cycle time.",
"Such devices can be realized for instance with tunable-barrier quantum dots [70], [71], and, owing to their high accuracy, are promising candidates for experimentally accessible quantum representations of the ampere [20], [21], [22].",
"Here, we show that, in the slow-driving regime, the energy that is required to move a given amount of carriers is subject to a fundamental lower bound, which depends only on the cycle time.",
"We further derive an explicit optimization principle for adiabatic quantum pumps by connecting our theory with the geometric approach to parametric pumping [72], [73], [74], [75], [76], [77], [78] and the notion of thermodynamic length [79], [80], [81], [82], [83].",
"As a third application of our theory, we derive a universal trade-off relation between the efficiency and the power consumption of adiabatic quantum motors, that is, devices that convert an electric current into motive power of mesoscopic mechanical objects like nano-paddle wheels or conveyor belts [23], [24], [25], [26]."
],
[
"Quantum Generator",
"The setup of Fig.",
"REF provides a simple realization of a quantum generator.",
"The frozen scattering amplitudes for this system are $\\mathcal {S}^{11}_{E,\\phi } &= e^{i(\\chi _E+\\phi )}, \\quad \\mathcal {S}^{12}_{E,\\phi } = \\mathcal {S}^{21}_{E,\\phi } =0 \\quad \\text{and}\\\\\\mathcal {S}^{22}_{E,\\phi } &= e^{i(\\chi _E-\\phi )},\\nonumber $ where the irrelevant dynamical phase $\\chi _E$ is determined by the circumference of the loop and the Aharonov-Bohm phase $\\phi \\equiv e\\Phi /\\hbar c$ plays the role of an external control parameter; here, $\\Phi $ is the tunable magnetic flux through the ring, $e$ is the carrier charge and $c$ the speed of light [84].",
"We use the right reservoir as a reference.",
"Hence, the chemical affinity is $F^\\rho _1\\equiv F^\\rho $ and the electric current $J^\\rho _1\\equiv J^\\rho $ flows from left to right.",
"For a linearly increasing flux, i.e., for $\\phi _t\\equiv \\omega t$ , the kinetic coefficients are $L^{\\rho \\rho } & = 0, \\quad L^{\\rho \\omega } =- L^{\\omega \\rho } = \\frac{T\\varphi }{h(1+\\varphi )} \\quad \\text{and}\\\\L^{\\omega \\omega } & = \\frac{T\\varphi }{h(1+\\varphi )},\\nonumber $ according to the formulas (REF ), where $\\varphi =\\exp [\\mu /T]$ is the fugacity of the right reservoir.",
"The electric current and the rate of entropy production are thus given by $J^\\rho = L^{\\rho \\omega }F^\\omega $ and $\\sigma = L^{\\omega \\omega }(F^\\omega )^2$ .",
"Upon recalling the expressions (REF ) for the coefficient $K^{\\rho \\rho }$ , it is now straightforward to verify that the bound (REF ) is saturated, that is, $\\sigma = (J^\\rho )^2/2 K^{\\rho \\rho }$ for any $F^\\rho $ and $F^\\omega $ .",
"This result shows that our bounds are tight.",
"On the microscopic level, the saturation of the bound (REF ) is a consequence of the working mechanism of the quantum generator, which is described in Fig.",
"REF .",
"Every transmitted carrier leads to the net dissipation of one quantum of energy $\\hbar \\omega $ .",
"As a result, the irreversible part of the photon current, $J^\\omega _{{{\\rm irr}}}= L^{\\omega \\omega } F^\\omega $ , is equal to the electric current $J^\\rho $ and proportional to the rate of entropy production $\\sigma =F^\\omega J^\\omega _{{{\\rm irr}}}$ .",
"By contrast, the reversible photon current, $J^\\omega _{{{\\rm rev}}}=L^{\\omega \\rho }F^\\rho $ , is decoupled from the dissipation rate $\\sigma $ , which is independent of the chemical affinity $F^\\rho $ .",
"In particular, for $F^\\omega =0$ and $F^\\rho \\ne 0$ , we have $\\sigma =0$ and $J^\\omega =J^\\omega _{{{\\rm rev}}}\\ne 0$ .",
"This observation does not imply the occurrence of dissipationless transport, since the electric current vanishes for $F^\\omega =0$ .",
"It shows, however, that no general bound of the form (REF ) exists for the photon current.",
"A parametric quantum pump can be described as a two-terminal conductor, whose potential landscape is changed periodically to generate a flow of carriers between two reservoirs with the same chemical potential and temperature.",
"One pumping cycle requires the energy input $U\\equiv _{{{\\rm ac}}}$ and moves the amount of carriers $\\mathcal {Q}\\equiv J̰^\\rho $ from the first reservoir to the second, see Fig. .",
"Our bound (REF ) implies that, in the slow-driving regime, these two figures are connected by the trade-off relation $U\\ge \\frac{T \\mathcal {Q}^2}{2K̰^{\\rho \\rho }} \\ge \\hbar \\omega \\mathcal {Q}^2,$ where we have used that $U=T̰\\sigma $ and the second inequality follows from Eq.",
"(REF ).",
"This result is quite remarkable as it puts a universal lower bound on the energy that must be provided to generate a given pump flux $\\mathcal {Q}$ in a given cycle time $.Hence, Eq.~(\\ref {ApplAQP_NB}) makes it possible to estimate theenergy consumption of a quantum pump, even in situations, where onlythe flux $ Q$ can be measured and the scattering amplitudesof the sample are unknown.$"
],
[
"Geometry and Optimal Driving Speed",
"Finding optimal driving protocols for a quantum pump is a difficult task, which typically requires the solution of involved variational problems.",
"In the slow-driving regime, it is, however, possible to derive a universal optimization principle for the parameterization $\\gamma $ of the closed path $\\Gamma $ that is mapped out by the vector of control parameters $\\mathbf {V}_t$ during the cycle.",
"To this end, we recall that, by using Eq.",
"(), the pump flux can be written as a line integral in the space of control parameters [72], $\\mathcal {Q} = L̰^{\\rho w}F^\\omega =\\oint _{\\Gamma }\\sum \\nolimits _{j} \\mathcal {A}^j_\\mathbf {V}dV_j,$ which proves that $\\mathcal {Q}$ is independent of the parametrization of $\\Gamma $ .",
"The objects $\\mathcal {A}^j_\\mathbf {V}\\equiv \\frac{1}{2\\pi T} \\int _0^\\infty \\!\\!\\!",
"dE\\sum \\nolimits _\\beta {{{\\rm Im}}}\\Bigl [S^{1\\beta \\ast }_{E,\\mathbf {V}} \\partial _{V_j} S^{1\\beta }_{E,\\mathbf {V}}\\Bigr ] f^{\\prime }_E$ are thereby considered as the components of a vector field $\\mathcal {A}_{\\mathbf {V}}$ , which, in analogy to the theory of geometric phases, is called the Berry potential [85].",
"Second, the input $U$ can be expressed in the form $U = T L̰^{\\omega \\omega }(F^\\omega )^2= h \\!\\sum \\nolimits _{ij} g^{ij}_{\\mathbf {V}_t}\\dot{V}_{i,t} \\dot{V}_{j,t},$ where we have used Eq.",
"() and introduced the thermodynamic metric Note that the coefficients $g^{ij}_{\\mathbf {V}}$ form a symmetric, positive semi-definite matrix and can therefore be consistently identified with a, possibly degenerate, metric in the space of control parameters.",
"$g^{ij}_\\mathbf {V}\\equiv \\frac{1}{8\\pi ^2 T}\\int _0^\\infty \\!\\!\\!",
"dE\\sum \\nolimits _{\\alpha \\beta }{{{\\rm Re}}}\\Bigl [\\bigl (\\partial _{V_i} S^{\\alpha \\beta }_{E,\\mathbf {V}}\\bigr )\\bigl (\\partial _{V_j} S^{\\alpha \\beta \\ast }_{E,\\mathbf {V}}\\bigr )\\Bigr ]f^{\\prime }_E.$ The optimal parameterization $\\gamma ^\\ast _t$ of the path $\\Gamma $ , which minimizes the input $U$ , can be determined from the objective functional $O[\\dot{\\hat{\\gamma }}_t]\\equiv \\!",
"\\left(\\sum \\nolimits _{ij}g^{ij}_{\\mathbf {V}_t}\\dot{V}_{i,t} \\dot{V}_{j,t}\\bigl /\\dot{\\hat{\\gamma }}_t-\\Lambda \\dot{\\hat{\\gamma }}_t\\right),$ where $\\Lambda $ is a Lagrange multiplier accounting for the boundary condition $\\gamma _{ - \\gamma _0 = and \\dot{\\hat{\\gamma }}_t is thederivative of the inverse function \\hat{\\gamma }_t of \\gamma _t\\footnote {Note that \\gamma _t is a monotonically increasing and thereforeinvertible function.",
"}.Solving the Euler-Lagrange equation for this functional yields thecondition\\begin{equation}t = \\frac{{\\mathcal {L}} \\int _0^{\\gamma ^\\ast _t} \\!\\!\\!",
"ds \\;\\sqrt{\\sum \\nolimits _{ij} g^{ij}_{\\mathbf {V}_s}\\dot{V}_{i,s} \\dot{V}_{j,s}},}{w}here\\begin{equation}\\mathcal {L}\\equiv \\oint _\\Gamma \\sqrt{\\sum \\nolimits _{ij}g^{ij}_{\\mathbf {V}} dV_i dV_j}\\end{equation}denotes the thermodynamic length of \\Gamma .Replacing the protocols {\\mathbf {V}}_t with {\\mathbf {V}}_{\\gamma ^\\ast _t} minimizesthe energy consumption of the pump without changing its flux\\mathcal {Q}, which depends only on the path \\Gamma .Inserting the minimal energy uptake U^\\ast =\\hbar \\omega \\mathcal {L}^2 intoEq.~(\\ref {ApplAQP_NB}) yields the remarkably simple relation\\begin{equation}\\mathcal {L}\\ge |\\mathcal {Q}|\\end{equation}between thermodynamic length and pump flux, which connects ourthermodynamic bounds with the geometric theory of slowly drivenquantum pumps.\\end{equation}\\subsubsection {Tunable-Barrier Pump}\\begin{figure}\\includegraphics [scale=1.05]{FigS_2.png}\\caption {Parametric quantum pump.\\textbf {Top:}Sketch of a generic setup.A narrow conductor connects two reservoirs with the same chemicalpotential and temperature.Periodically changing the gate voltages V_1 and V_2 creates twooscillating potential barriers driving the pump current J^\\rho .\\textbf {Middle:}The left plot shows the Berry curvature \\mathcal {B}_\\mathbf {V} for\\chi _\\mu =\\pi /4.Circles indicate the control path that is determined by theprotocols (\\ref {FrTBDrPt}) for \\rho =1/4, \\; 1/2, \\; 3/4, \\;1.As the path expands into the positive peak of \\mathcal {B}_{\\mathbf {V}},the pump flux \\mathcal {Q} first increases sharply and thenapproaches a limit value, which depend on the parameter \\chi _\\mu .\\textbf {Bottom:}The left panel shows the optimal driving speed \\dot{\\gamma }^\\ast _t for\\chi _\\mu =\\pi /4 and \\rho =1.The horizontal line corresponds to constant speed.On the right, the energy uptake is plotted against the pump flux forconstant and optimal driving speed, U and U^\\ast respectively,where \\chi _\\mu =\\pi /4 and \\rho varies between 0 and 5.The shaded area indicates the bound (\\ref {ApplAQP_NB}).",
"}\\end{figure}}We now consider a simple model of a quantum pump, where the potentialinside the conductor consists of two delta-barriers withdimensionless strengths $ V1$ and $ V2$.The single-particle Hamiltonian of this system reads\\begin{equation}H_{\\mathbf {V}}= \\frac{p^2}{2M} + \\frac{\\hbar ^2V_1}{Md}\\delta _{r}+ \\frac{\\hbar ^2 V_2}{Md}\\delta _{r-d},\\end{equation}where $ V(V1,V2)$, $ p$ and $ r$ are the momentum and positionof the carrier, $ d$ denotes the distance between the two barriers and$ M$ the carrier mass, see Fig.~\\ref {Fig_QPump}.The corresponding frozen scattering amplitudes are\\cite {Moskalets2004}\\begin{subequations}{\\begin{@align}{1}{-1}&\\mathcal {S}^{12}_{E,\\mathbf {V}} = \\mathcal {S}^{21}_{E,\\mathbf {V}}=\\mathcal {Z}_{E,\\mathbf {V}} \\chi ^2_E e^{i\\chi _E},\\\\&\\mathcal {S}^{11}_{E,\\mathbf {V}} =\\mathcal {Z}_{E,\\mathbf {V}}\\bigl (V_1(V_2-i\\chi _E)-V_2(V_1+i\\chi _E)e^{2i\\chi _E}\\bigr ),\\\\&\\mathcal {S}^{22}_{E,\\mathbf {V}} =\\mathcal {Z}_{E,\\mathbf {V}}\\bigl (V_2(V_1-i\\chi _E)-V_1(V_2+i\\chi _E)e^{2i\\chi _E}\\bigr )\\end{@align}}\\end{subequations}with $ ZE,V1/(V1V2e2iE -(V1-iE)(V2-iE))$and $ Ed2ME/$.In the following, we focus on the low-temperature limit, where thefunction $ f'E$ is sharply peaked around $ E=$ and can therefore bereplaced with $ TE-$ in the Eqs.~(\\ref {ApplAQP_GeoCharge})and (\\ref {ApplAQP_Metric}).$ To find a suitable control path, we recall that the expression (REF ) for the pump flux can be rewritten as an area integral with the help of Stokes' theorem [72], $\\mathcal {Q} = \\int _{S_\\Gamma } \\!\\!\\!",
"dS \\; \\mathcal {B}_{\\mathbf {V}},$ where $\\mathcal {B}_\\mathbf {V}\\equiv \\partial _{V_1}\\mathcal {A}^2_\\mathbf {V}-\\partial _{V_2}\\mathcal {A}^1_{\\mathbf {V}}$ is the Berry curvature corresponding to the potential $\\mathcal {A}_\\mathbf {V}$ and $S_\\Gamma $ denotes the area encircled by $\\Gamma $ .",
"As shown in Fig.",
", the function $\\mathcal {B}_{\\mathbf {V}}$ features two antisymmetric peaks.",
"Hence, to generate a large flux $\\mathcal {Q}$ , the path $\\Gamma $ has to cover the positive peak while avoiding the negative one.",
"This condition is met by the circles with the parameterization $V_{1,t} = V_0 - \\rho \\cos [\\omega t], \\;\\;\\;\\;V_{2,t} = V_0 - \\rho \\sin [\\omega t],$ where $V_0\\equiv \\rho /\\sqrt{2}-\\chi _\\mu \\cot [\\chi _\\mu ]/2$ and the radius $\\rho $ determines the amplitude of the driving.",
"Using the protocols (REF ), we numerically calculate the flux $\\mathcal {Q}$ , the input $U$ , the optimal driving speed $\\dot{\\gamma }^\\ast $ and the minimized input $U^\\ast $ .",
"The results of these calculations are plotted in Fig. .",
"Two observations stand out.",
"First, the energy consumption of the pump can indeed be significantly reduced by optimizing the driving speed.",
"Second, our bound (REF ) underestimates the energy uptake by at least a factor of 6 for constant, and at least a factor of 4 for optimal driving speed.",
"Microscopically, the deviations arise from idle scattering events, where carriers pick up energy from the external driving without contributing to the pump flux.",
"This effect becomes more and more dominant as the amplitude $\\rho $ of the potential modulations increases.",
"Still, at least for moderate amplitudes, our bound (REF ) predicts the correct order of magnitude for the energy uptake of the device."
],
[
"Bound on Efficiency",
"A quantum motor can be described in terms of two components: a mesoscopic conductor hosting an electric current $J^\\rho $ between two reservoirs with the same temperature but different chemical potentials and a mechanical rotor that couples to the dynamics of the carriers [24], see Fig.",
"REF .",
"Provided that the rotor is much heavier than the carriers, it can be treated as a classical degree of freedom, which creates a slowly and periodically changing potential inside the conductor [88].",
"In this Born-Oppenheimer picture, the motive power of the rotor, that is, the output of the motor, is given by $\\Pi _{{{\\rm m}}}\\equiv -\\Pi _{{{\\rm ac}}}= - T F^\\omega J^\\omega $ .",
"The electric power $\\Pi _{{{\\rm el}}}=TF^\\rho J^\\rho $ is the input of the motor and its efficiency is defined as $\\eta _{{{\\rm m}}} \\equiv \\Pi _{{{\\rm m}}}/\\Pi _{{{\\rm el}}}\\le 1$ for $\\Pi _{{{\\rm m}}}>0$ .",
"The upper bound 1 thereby follows from the second law, which requires $\\sigma = \\Pi _{{{\\rm el}}}/T-\\Pi _{{{\\rm m}}}/T \\ge 0$ .",
"Going beyond this trivial result, our bound (REF ) implies that the efficiency and the input of the device are connected by the universal relation $\\eta _{{{\\rm m}}}\\le 1-\\frac{\\Pi _{{{\\rm el}}}}{2T K^{\\rho \\rho }(F^\\rho )^2}\\le 1- \\frac{h\\Pi _{{{\\rm el}}}}{\\Delta \\mu ^2}= 1- \\frac{hJ^\\rho }{\\Delta \\mu },$ where the second inequality follows from Eq.",
"(REF ).",
"Depending only on the electric current $J^\\rho $ and the chemical potential bias $\\Delta \\mu =\\mu _1-\\mu $ , both of which are generally easy to access in experiments, the bound (REF ) has a key practical implication: it makes it possible to put a non-trivial upper limit on the efficiency of an adiabatic quantum motor without measuring the motive power of the rotor or invoking a specific model."
],
[
"Paddle Wheel Motor",
"To test the accuracy of the bound (REF ), we consider a simple quantum motor, which consists of a rotating nano-paddle wheel creating a sliding delta-barrier inside a narrow conductor, see Fig.",
"REF .",
"The single-particle Hamiltonian of this model is $H_{\\mathbf {V}} = \\frac{p^2}{2M} + \\frac{\\hbar ^2 V}{Md} \\delta _{r-da}$ and the frozen scattering amplitudes are given by $\\mathcal {S}^{12}_{E,\\mathbf {V}}&=\\mathcal {S}^{21}_{E,\\mathbf {V}}=i\\chi _E e^{i\\chi _E}/(i\\chi _E -V),\\\\\\mathcal {S}^{11}_{E,\\mathbf {V}}&= Ve^{2i a\\chi _E}/(i\\chi _E -V), \\\\\\mathcal {S}^{22}_{E,\\mathbf {V}}&= Ve^{2i(1-a)\\chi _E}/(i\\chi _E -V)$ with $\\mathbf {V}\\equiv (V,a)$ and $\\chi _E\\equiv d\\sqrt{2ME}/\\hbar $ .",
"Here, $d$ denotes the length of the contact region between the conductor and the paddle wheel and the dimensionless control parameters $V$ and $a\\in [0,1]$ correspond to the strength and the position of the barrier.",
"In the low-temperature limit, the electric and the motive power are given by $\\Pi _{{{\\rm el}}}/\\hbar \\omega ^2 &= \\bar{L}^{\\rho \\rho }v^2+\\bar{L}^{\\rho \\omega }v \\quad \\text{and}\\\\\\Pi _{{{\\rm m}}}/\\hbar \\omega ^2 &= -\\bar{L}^{w\\rho }v-\\bar{L}^{\\omega \\omega },$ where $v\\equiv \\Delta \\mu /\\hbar \\omega $ and we have introduced the rescaled kinetic coefficients $\\bar{L}^{\\rho \\rho } &=\\frac{1}{2\\pi }\\bigl \\bigl | \\mathcal {S}^{12}_{\\mu ,\\mathbf {V}} \\bigr |^2\\bigr \\\\\\bar{L}^{\\rho \\omega } &=-\\bar{L}^{\\omega \\rho }=\\frac{1}{2\\pi \\omega }\\sum \\nolimits _\\beta {{{\\rm Im}}}\\bigr [\\bigr \\mathcal {S}^{\\beta 1}_{\\mu ,\\mathbf {V}}\\dot{\\mathcal {S}}_{\\mu ,\\mathbf {V}}^{\\beta 1\\ast }\\bigr \\bigr ],\\\\\\bar{L}^{\\omega \\omega } &= \\frac{1}{4\\pi \\omega ^2}\\sum \\nolimits _{\\alpha \\beta }\\bigl \\bigl | \\dot{\\mathcal {S}}^{\\alpha \\beta }_{\\mu ,\\mathbf {V}} \\bigr |^2\\bigr .$ We now calculate $\\Pi _{{{\\rm el}}}$ and $\\Pi _{{{\\rm m}}}$ for the protocols $V_t= V_0\\sin ^2[\\omega t/2] \\quad \\text{and}\\quad a_t = t/{{{\\rm mod}}}\\; 1,$ which mimic the motion of the paddle wheel, assuming that its radius is much larger than $d$ .",
"The bias $v$ is fixed by maximizing the efficiency of the motor (REF ).",
"That is, we set $v=v^\\ast \\equiv \\bar{L}^{\\omega \\omega }\\bigl (1+\\sqrt{Z_{{{\\rm m}}}+1}\\bigr )\\bigl /\\bar{L}^{\\rho \\omega }$ , whereby the efficiency becomes $\\eta ^\\ast _{{{\\rm m}}}=\\frac{\\sqrt{Z_{{{\\rm m}}}+1}-1}{\\sqrt{Z_{{{\\rm m}}}+1}+1}$ with $Z_{{{\\rm m}}}\\equiv (\\bar{L}^{\\rho \\omega })^2/\\bar{L}^{\\rho \\rho }\\bar{L}^{\\omega \\omega }$ playing the role of a figure of merit.",
"Our results are plotted in Fig.",
"REF .",
"For $V_0\\lesssim 10$ , the bound (REF ) overestimates the efficiency of the paddle-wheel motor by about a factor of 2.",
"As $V_0$ increases, the deviation decays to approximately a factor of $3/2$ , since fewer carriers tunnel through the moving barrier without transmitting energy to the paddle-wheel.",
"The same qualitative behavior is observed if the bias $v$ is fixed independently of the kinetic coefficients."
],
[
"Far From Equilibrium",
"Going beyond the first and the second law, our bounds (REF ) and (REF ) provide strong universal constraints on currents in periodically driven coherent conductors.",
"They were derived within the framework of linear response theory, which describes the arguably most relevant regime of operation of mesoscopic devices.",
"Still, the question remains whether similar bounds apply also far from equilibrium.",
"In this section, we show how such a generalization of our theory can be achieved.",
"We first derive a family of new bounds on currents in coherent conductors that hold for arbitrary driving frequencies and thermal gradients.",
"We then discuss the interpretation of these bounds in the context of thermodynamic uncertainty relations [89], [90] and show how they can be used for thermodynamic inference.",
"To illustrate the general picture, we revisit the quantum generator of Sec.",
"REF , whose Floquet scattering amplitudes can be calculated exactly for arbitrary driving frequencies.",
"We then put our results in the context of recent developments and conclude this section by summarizing the main implications of our theory for systems without time dependent driving."
],
[
"Derivation",
"To extend the approach of Sec.",
"REF into the non-linear regime, we consider the quadratic form $\\Xi _\\psi \\equiv \\sigma +\\psi \\sum \\nolimits _\\alpha \\sum \\nolimits _xJ^x_\\alpha G^x_\\alpha +\\psi \\sum \\nolimits _\\alpha \\sum \\nolimits _{xy}\\hat{K}^{xy}_\\alpha G^x_\\alpha G^y_\\alpha .$ Here, $\\psi $ is a yet undetermined positive number and the coefficients $\\hat{K}^{xy}_\\alpha $ are defined as $\\hat{K}^{xy}_\\alpha \\equiv \\frac{1}{4h}\\int _0^\\infty \\!\\!\\!",
"dE\\begin{aligned}[t]&\\sum \\nolimits _\\beta \\sum \\nolimits _n\\zeta ^{x,\\alpha }_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}}}\\zeta ^{y,\\alpha }_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}}}\\bigl | \\hat{S}^{\\alpha \\beta }_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}},E} \\bigr |^2\\\\&\\times \\bigl (f^\\alpha _{E\\raisebox {-0.7pt}{{{\\scriptsize n}}}}(1-f^\\beta _E)+f^\\beta _E(1-f^\\alpha _{E\\raisebox {-0.7pt}{{{\\scriptsize n}}}})\\bigr )\\end{aligned}$ with $\\zeta ^{\\rho ,\\alpha }_E\\equiv 1$ , $\\zeta ^{q,\\alpha }_E\\equiv E-\\mu _\\alpha $ and the modified Floquet scattering amplitudes $\\hat{S}^{\\alpha \\beta }_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}},E}\\equiv (1-\\delta _{n0}\\delta _{\\alpha \\beta })S^{\\alpha \\beta }_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}},E}.$ This ansatz is essentially found by inspection but can be motivated a posteriori as we shall see in Sec.",
"REF .",
"We now recall the formulas (REF ), (REF ) and (REF ) and use the unitarity conditions (REF ) to express the thermo-chemical currents and the rate of entropy production as $J^x_\\alpha & = \\frac{1}{h}\\int _0^\\infty \\!\\!\\!",
"dE\\sum \\nolimits _\\beta \\sum \\nolimits _n\\zeta ^{x,\\alpha }_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}}}\\bigl | \\hat{S}^{\\alpha \\beta }_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}},E} \\bigr |^2 (f^\\alpha _{E\\raisebox {-0.7pt}{{{\\scriptsize n}}}}-f^\\beta _E),\\\\\\sigma & =\\begin{aligned}[t]\\frac{1}{h}\\int _0^\\infty \\!\\!\\!",
"dE&\\sum \\nolimits _{\\alpha \\beta }\\sum \\nolimits _n\\bigl | \\hat{S}^{\\alpha \\beta }_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}},E} \\bigr |^2\\\\&\\times \\bigl ((\\nu ^\\beta _E-\\nu ^\\alpha _{E\\raisebox {-0.7pt}{{{\\scriptsize n}}}})f^\\beta _E -g^\\beta _E +g^\\alpha _{E\\raisebox {-0.7pt}{{{\\scriptsize n}}}}\\bigr )\\end{aligned}$ with $\\nu ^\\alpha _E\\equiv (\\mu _\\alpha -E)/T_\\alpha $ and $g^\\alpha _E\\equiv \\nu ^\\alpha _E-\\ln [f^\\alpha _E]$ .",
"Inserting these expressions and the definition (REF ) into Eq.",
"(REF ) gives $\\Xi _\\psi = \\frac{1}{h}\\int _0^\\infty \\!\\!\\!",
"dE\\sum \\nolimits _{\\alpha \\beta }\\sum \\nolimits _n\\bigl | \\hat{S}^{\\alpha \\beta }_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}},E} \\bigr |^2 \\Xi ^{\\alpha \\beta }_{\\psi ,n,E},$ where $\\Xi ^{\\alpha \\beta }_{\\psi ,n,E}\\equiv \\begin{aligned}[t]&(\\nu ^\\beta _E-\\nu ^\\alpha _{E\\raisebox {-0.7pt}{{{\\scriptsize n}}}})f^\\beta _E - g^\\beta _E + g^\\alpha _{E\\raisebox {-0.7pt}{{{\\scriptsize n}}}}\\\\&+2\\psi X^{\\alpha }_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}}} (f^\\alpha _{E\\raisebox {-0.7pt}{{{\\scriptsize n}}}}-f^\\beta _E)\\\\&+\\psi (X^{\\alpha }_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}}})^2\\bigl (f^\\alpha _{E\\raisebox {-0.7pt}{{{\\scriptsize n}}}}(1-f^\\beta _E)+f^\\beta _E(1-f^\\alpha _{E\\raisebox {-0.7pt}{{{\\scriptsize n}}}})\\bigr )\\end{aligned}$ is a quadratic form in the variables $X^\\alpha _E\\equiv \\sum _x\\zeta ^{x,\\alpha }_E G^x_\\alpha /2$ .",
"As has been shown in Ref.",
"[47], this quadratic form is positive semidefinite if $\\psi \\le \\psi ^\\ast \\equiv \\min _{z\\in \\mathbb {R}}\\frac{(1-e^z+ze^z)(e^z+1)}{(e^z-1)^2}\\simeq 0.89612.$ Consequently, we have $\\Xi _{\\psi ^\\ast }\\ge 0$ for any choice of the auxiliary variables $G^x_\\alpha $ .",
"From here, we can proceed as in Sec.",
"REF .",
"Minimizing $\\Xi _{\\psi ^\\ast }$ , first with respect to the $\\lbrace G^\\rho _\\alpha \\rbrace $ and then with respect to the $\\lbrace G^q_\\alpha \\rbrace $ while setting the respectively remaining auxiliary variables to zero, yields the cumulative bound $\\sigma \\ge \\psi ^\\ast \\sum \\nolimits _\\alpha \\frac{(J^x_\\alpha )^2}{4\\hat{K}^{xx}_\\alpha },$ which can be regarded as the counterpart of Eq.",
"(REF ).",
"For the individual matter and heat currents, we obtain the bounds $\\sigma &\\ge \\frac{\\psi ^\\ast (1+\\theta ^\\rho _\\alpha )}{4\\hat{K}^{\\rho \\rho }_\\alpha }(J^\\rho _\\alpha )^2\\quad \\text{and}\\\\\\sigma &\\ge \\frac{\\psi ^\\ast }{4\\hat{K}^{qq}_\\alpha }\\bigl ((J^q_\\alpha )^2+ (J^q_\\alpha +\\Pi _{{{\\rm ac}}}+\\Pi _{{{\\rm el}}})^2\\theta ^q_\\alpha \\bigr )$ with $\\theta ^x_\\alpha \\equiv \\hat{K}^{xx}_\\alpha \\bigl /\\sum _{\\beta \\ne \\alpha }\\hat{K}^{xx}_\\beta \\ge 0$ , which can be derived from Eq.",
"(REF ) by repeating the steps that led from Eq.",
"(REF ) to Eq.",
"(REF ); for the heat currents, the conservation law $\\sum \\nolimits _\\alpha J^q_\\alpha =0$ , which holds only in linear response, must thereby be replaced with the first law (REF ).",
"We note that, in linear response, the bounds (REF ) are weaker than our previous bound (REF ) by a factor $\\psi ^\\ast $ .",
"Specifically, upon neglecting third-order corrections in the affinities, Eqs.",
"(REF ) imply $\\sigma \\ge \\frac{\\psi ^\\ast (1+\\theta ^x_\\alpha |_{{{\\rm eq}}})}{4\\hat{K}^{xx}_\\alpha |_{{{\\rm eq}}}} (J^x_\\alpha )^2\\ge \\frac{\\psi ^\\ast N}{4 K^{xx} (N-1)}(J^x_\\alpha )^2,$ where the second inequality follows by noting that $\\hat{K}^{xx}_\\alpha |_{{{\\rm eq}}}&= K^{xx}- \\frac{1}{2h}\\int _0^\\infty \\!\\!\\!",
"dE\\;(\\zeta ^x_E)^2\\bigl | \\bigl \\mathcal {S}^{\\alpha \\alpha }_{E,\\mathbf {V}}\\bigr \\bigr |^2 f^{\\prime }_E\\\\&\\le K^{xx}\\nonumber $ and therefore $\\frac{1+\\theta ^x_\\alpha |_{{{\\rm eq}}}}{\\hat{K}^{xx}_\\alpha |_{{{\\rm eq}}}}= \\frac{1}{\\hat{K}^{xx}_\\alpha |_{{{\\rm eq}}}}+ \\frac{1}{\\sum _{\\beta \\ne \\alpha }\\hat{K}^{xx}_\\beta |_{{{\\rm eq}}}}\\ge \\frac{N}{K^{xx}(N-1)}.$ This observation shows that the bounds (REF ) cannot be saturated close to equilibrium."
],
[
"Thermodynamic Uncertainty Relations",
"A thermodynamic uncertainty relation describes the trade-off between precision and entropy production in a given class of thermodynamic processes [89], [90].",
"For classical matter transport in multi-terminal systems without time-dependent driving, for example, the relation $\\sigma (\\epsilon ^\\rho _\\alpha )^2 \\ge \\psi ^\\ast $ has been derived in Ref. [47].",
"The reduced zero-frequency noise, or squared relative uncertainty, $(\\epsilon ^\\rho _\\alpha )^2 \\equiv P^{\\rho \\rho }_{\\alpha \\alpha }/(J^\\rho _\\alpha )^2$ , thereby provides a measure for the accuracy at which a given amount of particles is extracted from the reservoir $\\alpha $ in a given time.",
"As we show in the following, our bounds (REF ) make it possible to extend this relation into the quantum regime and to include heat currents as well as periodic driving.",
"We first recall that, according to the formulas (REF ) and (REF ), the diagonal thermal and the shot noise of the thermo-chemical currents are given by $D^{xx}_{\\alpha \\alpha }&= \\sum \\nolimits _{uv} c^{xu}_\\alpha c^{xv}_\\alpha D^{uv}_{\\alpha \\alpha }\\\\&\\begin{aligned}[t]=\\frac{1}{h}\\int _0^\\infty \\!\\!\\!",
"dE\\sum \\nolimits _\\beta \\sum \\nolimits _n\\big ((&\\zeta ^{x,\\alpha }_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}}})^2 \\bigl | S^{\\alpha \\beta }_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}},E} \\bigr |^2(f^{\\prime \\alpha }_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}}} + f^{\\prime \\beta }_E)\\\\&-2\\delta _{\\alpha \\beta }\\zeta ^{x,\\alpha }_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}}}\\zeta ^{x,\\alpha }_E\\bigl | S^{\\alpha \\alpha }_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}},E} \\bigr |^2f^{\\prime \\alpha }_E\\big ),\\end{aligned}\\nonumber \\\\[3pt]R^{xx}_{\\alpha \\alpha } &=\\sum \\nolimits _{uv} c^{xu}_\\alpha c^{xv}_\\alpha R^{uv}_{\\alpha \\alpha }\\\\&= \\frac{1}{2h}\\int _0^\\infty \\!\\!\\!",
"dE\\sum \\nolimits _{\\gamma \\delta }\\sum \\nolimits _n\\Bigl |\\sum \\nolimits _u c^{xu}_\\alpha C^{u\\alpha ,\\gamma \\delta }_{n,E}\\Bigr |^2\\nonumber $ with $c^{\\rho u}_\\alpha \\equiv \\delta _{u\\rho }$ , $c^{q u}_\\alpha \\equiv \\delta _{u\\varepsilon }-\\mu _\\alpha \\delta _{u\\rho }$ and $f^{\\prime \\alpha }_E\\equiv f_E^\\alpha (1-f^\\alpha _E)$ .",
"Next, comparing Eq.",
"(REF ) with Eq.",
"(REF ) shows that the coefficients $\\hat{K}^{xx}_\\alpha $ can be decomposed as $4\\hat{K}^{xx}_\\alpha = D^{xx}_{\\alpha \\alpha } + \\Psi ^{xx}_\\alpha + \\Omega ^{xx}_\\alpha .$ The Fermi correction $\\Psi ^{xx}_\\alpha \\equiv \\frac{1}{h}\\int _0^\\infty \\!\\!\\!",
"dE\\sum \\nolimits _\\beta \\sum \\nolimits _n(\\zeta ^{x,\\alpha }_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}}})^2\\bigl | \\hat{S}^{\\alpha \\beta }_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}},E} \\bigr |^2(f^\\alpha _{E\\raisebox {-0.7pt}{{{\\scriptsize n}}}}-f^\\beta _E)^2$ thereby accounts for Pauli blocking between incoming and outgoing carriers; it vanishes in equilibrium and in the quasi-classical limit, where second-order terms in the fugacities $\\varphi _\\alpha \\equiv \\exp [\\mu _\\alpha /T_\\alpha ]$ can be neglected and the exclusion principle becomes irrelevant [29].",
"The second correction in Eq.",
"(REF ), $\\Omega ^{xx}_\\alpha \\equiv \\frac{2}{h}\\int _0^\\infty \\!\\!\\!",
"dE\\sum \\nolimits _n\\zeta ^{x,\\alpha }_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}}}\\zeta ^{x,\\alpha }_E\\bigl | \\hat{S}^{\\alpha \\alpha }_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}},E} \\bigr |^2f^{\\prime \\alpha }_E,$ arises from inelastic reflections of carriers at the sample and vanishes if the external driving is turned off, i.e., if $S^{\\alpha \\alpha }_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}},E} = \\delta _{n0} S^{\\alpha \\alpha }_E$ and thus $\\hat{S}^{\\alpha \\alpha }_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}},E}=0$ .",
"Finally, since the shot noise () is non-negative, Eq.",
"(REF ) implies that $\\hat{K}^{xx}_\\alpha \\le P^{xx}_{\\alpha \\alpha }+\\Psi ^{xx}_\\alpha +\\Omega ^{xx}_\\alpha $ , where $P^{xx}_{\\alpha \\alpha }=D^{xx}_{\\alpha \\alpha }+R^{xx}_{\\alpha \\alpha }$ is the zero-frequency noise of the thermo-chemical currents.",
"Inserting this bound into the Eqs.",
"(REF ) and setting $\\theta ^x_\\alpha =0$ yields the thermodynamic uncertainty relation $\\sigma (\\epsilon ^x_\\alpha )^2\\ge \\frac{\\psi ^\\ast }{1+\\Psi ^{xx}_\\alpha /P^{xx}_{\\alpha \\alpha }+\\Omega ^{xx}_\\alpha /P^{xx}_{\\alpha \\alpha }},$ where $(\\epsilon ^x_\\alpha )^2\\equiv P^{xx}_{\\alpha \\alpha }/(J^x_\\alpha )^2$ .",
"This result can be interpreted as follows.",
"The corrections $\\Psi ^{\\rho \\rho }_\\alpha $ and $\\Omega ^{\\rho \\rho }_\\alpha $ are non-negative.",
"Therefore, the relation (REF ) shows that Pauli blocking and periodic driving are both sources of precision, which reduce the minimal amount of entropy that must be produced to generate an arbitrary matter current $J^\\rho _\\alpha $ with given uncertainty $\\epsilon ^\\rho _\\alpha $ .",
"These two mechanisms of reducing the thermodynamic cost of precision have been described separately before for different classes of systems, see for instance Refs.",
"[47], [91], [92] and [93], [94], [95], [48], [96], [97], [98], [99], [100], respectively.",
"Our uncertainty relation (REF ) now accounts for both of them in a unified manner through additive corrections.",
"For heat currents, the situation is slightly different.",
"The corresponding Pauli corrections $\\Psi ^{qq}_\\alpha $ are non-negative and therefore universally suppress the lower bound on the product $\\sigma (\\epsilon ^q_\\alpha )^2$ compared to the quasi-classical limit, where $\\Psi ^{qq}_\\alpha =0$ ; this observation is in line with earlier results, see Ref. [101].",
"The sign of the driving corrections $\\Omega ^{qq}_\\alpha $ , however, depends on the structure of the Floquet scattering amplitudes of the sample.",
"Periodic driving can thus either reduce or increase the minimal thermodynamic cost of heat transport at given precision.",
"We note that the bound (REF ), to which the relation (REF ) reduces in the quasi-classical limit and without periodic driving, can be asymptotically saturated in systems with infinitely many terminals and strong biases [47].",
"Hence, for matter currents, also the uncertainty relation (REF ) and the stronger bound (REF ) can be regarded as tight.",
"Whether these bounds are also tight for heat currents is an open question."
],
[
"Thermodynamic Inference",
"Although the corrections $\\Psi ^{xx}_\\alpha $ and $\\Omega ^{xx}_\\alpha $ admit a transparent physical interpretation, it is not clear how they can be measured.",
"Before the generalized uncertainty relation (REF ) can be used for thermodynamic inference, it is therefore necessary to link these quantities to experimentally accessible observables.",
"For matter currents, such a connection is provided by the bounds $\\Psi ^{\\rho \\rho }_\\alpha \\le \\sigma /2\\quad \\text{and}\\quad \\Omega ^{\\rho \\rho }_\\alpha \\le 2T_\\alpha /h$ which ultimately follow from the unitarity conditions (REF ), see Lemmas REF and REF of App.",
"for details.",
"Inserting these bounds into Eq.",
"(REF ) yields $\\frac{\\sigma (P^{\\rho \\rho }_{\\alpha \\alpha }+\\sigma /2+2T_\\alpha /h)}{(J^\\rho _\\alpha )^2}\\ge \\psi ^\\ast .$ This relation provides a powerful tool for thermodynamic inference.",
"Specifically, upon measuring an arbitrary matter current $J^\\rho _\\alpha $ , its zero-frequency noise $P^{\\rho \\rho }_{\\alpha \\alpha }$ and the temperature $T_\\alpha $ of the corresponding reservoir, one obtains the lower bound $\\sigma \\ge \\sigma ^\\ast _\\alpha \\equiv \\bigl | J^\\rho _\\alpha \\bigr |\\left(\\sqrt{\\phi _\\alpha ^2 +2\\psi ^\\ast }-|\\phi _\\alpha |\\right)$ on the dissipation rate $\\sigma $ , where $\\phi _\\alpha \\equiv (hP^{\\rho \\rho }_{\\alpha \\alpha }+2T_\\alpha )/h J^\\rho _\\alpha $ is a modified Fano factor.",
"This bound holds, within the limits of the scattering approach to quantum transport, for any sample and driving protocols, arbitrary driving frequencies, arbitrary voltage and temperature biases and in presence of magnetic fields.",
"It therefore makes it possible to derive universal constraints on otherwise unaccessible figures of merit of coherent transport devices.",
"For instance, upon recalling the setups of Secs.",
"REF and REF , the energy uptake $U$ of a parametric quantum pump and the efficiency of a quantum motor $\\eta _{{{\\rm m}}}$ can be bounded as $U & \\ge T{J^\\rho }\\left(\\sqrt{\\phi ^2+2\\psi ^\\ast }-|\\phi |\\right)\\quad \\text{and}\\\\\\eta _{{{\\rm m}}} &\\le 1- \\frac{T}{\\Delta \\mu }\\Bigl (\\sqrt{\\phi ^2 +2\\psi ^\\ast }-|\\phi |\\Bigr )$ where the parameter $\\phi \\equiv (hP^{\\rho \\rho }_{11}+2T)/J^\\rho $ is experimentally accessible.",
"These relations can be regarded as far-from-equilibrium counterparts of our linear-response bounds (REF ) and (REF )."
],
[
"Quantum Generator Revisited",
"To illustrate our far-from-equilibrium theory, we now return to the quantum generator of Sec.",
"REF .",
"The Floquet scattering amplitudes for this system can be found exactly by solving the corresponding Schrödinger equation, see App. .",
"It is thus straightforward to numerically calculate the quantities $& Q^{(1)} \\equiv \\frac{4\\sigma \\hat{K}^{\\rho \\rho }_1}{(1+\\theta ^\\rho _1)(J^\\rho _1)^2},\\\\[3pt]& Q^{(2)} \\equiv \\sigma (\\epsilon ^\\rho _1)^2(1+\\Psi ^{\\rho \\rho }_1/P^{\\rho \\rho }_{11}+\\Omega ^{\\rho \\rho }_1/P^{\\rho \\rho }_{11}),\\\\[3pt]& Q^{(3)} \\equiv \\frac{\\sigma (P^{\\rho \\rho }_{11} + \\sigma /2 +2T/h)}{(J^\\rho _1)^2}\\quad \\text{and}\\\\[3pt]& \\sigma ^\\ast _1 = \\bigl | J^\\rho _1 \\bigr |\\left(\\sqrt{\\phi _1^2+2\\psi ^\\ast }-\\phi _1\\right)$ which are plotted Fig.",
"REF .",
"According to the relations (REF ), (REF ) and (REF ), the dimensionless coefficients $Q^{(1)}$ , $Q^{(2)}$ and $Q^{(3)}$ are bounded from below by $\\psi ^\\ast $ .",
"The coefficient $Q^{(1)}$ indeed comes close to this bound for small biases and driving frequencies.",
"In fact, we have $Q^{(1)}\\rightarrow 1$ for $\\Delta \\mu ,\\omega \\rightarrow 0$ , which confirms our previous observation that the linear-response counterpart (REF ) of the bound (REF ) is saturated for the quantum generator.",
"After passing through its minimum, $Q^{(1)}$ grows monotonically with the driving frequency.",
"Hence, the bound (REF ) becomes less and less tight when the system is driven faster.",
"The coefficients $Q^{(2)}$ and $Q^{(3)}$ show a qualitatively similar dependence on the driving frequency.",
"However, their minimums do not become smaller than 2 and 4, respectively.",
"Finally, we find that the figure $\\sigma ^\\ast _1$ , at best, underestimates the actual dissipation rate by about a factor 4.",
"This result proves that the bound (REF ) is generally strong enough to predict the correct order of magnitude for $\\sigma $ ."
],
[
"Comparison with Earlier Results",
"Thermodynamic uncertainty relations are currently a subject of active research in stochastic thermodynamics, see Refs.",
"[89], [90] for recent reviews.",
"To put our bounds in context with these developments, we compare them with the four relations $Q_{{{\\rm cl}}}^{\\rho ,\\alpha } &\\equiv \\frac{\\sigma P^{\\rho \\rho }_{\\alpha \\alpha }}{\\psi ^\\ast (J^\\rho _\\alpha )^2}\\ge 1,\\\\[3pt]Q_\\omega ^{x,\\alpha } &\\equiv \\frac{\\sigma P^{xx}_{\\alpha \\alpha }}{2(J^x_\\alpha -\\omega \\partial _\\omega J^x_\\alpha )^2}\\ge 1,\\\\[3pt]Q_{{{\\rm FT}}}^{x,\\alpha } &\\equiv \\frac{(\\exp []-1)P^{xx}_{\\alpha \\alpha }}{2J^x_\\alpha )^2}\\ge 1\\quad \\text{and}\\\\[3pt]Q_{{{\\rm hys}}}^{x,\\alpha } &\\equiv \\frac{(\\exp [\\sigma +\\tilde{\\sigma })/2]-1)(P^{xx}_{\\alpha \\alpha }+\\tilde{P}^{xx}_{\\alpha \\alpha })}{(J^x_\\alpha +\\tilde{J}^x_\\alpha )^2}\\ge 1,$ which were derived earlier for different classes of systems [47], [98], [102], [103], [104], [105].",
"For concreteness, we focus again on the quantum generator of Sec.",
"REF , for which the coefficients $Q_{{{\\rm cl}}}^{\\rho ,1}$ , $Q_\\omega ^{\\rho ,1}$ , $Q_{{{\\rm FT}}}^{\\rho ,1}$ and $Q_{{{\\rm hys}}}^{\\rho ,1}$ can be easily calculated.",
"The results of this analysis, which are summarized in Fig.",
"REF , lead to the following conclusions.",
"First, the bound (REF ), which was derived for classical ballistic transport without periodic driving [47], is violated at low frequencies.",
"In fact, the coefficient $Q_{{{\\rm cl}}}^{\\rho 1}$ goes to zero for $\\Delta \\mu ,\\omega \\rightarrow 0$ .",
"This behavior can be understood by observing that the driving corrections (REF ) do, in contrast to the Pauli corrections (REF ), not vanish in adiabatic equilibrium.",
"Instead, we have $\\Omega ^{xx}_\\alpha |_{{{\\rm eq}}} = \\frac{2}{h}\\int _0^\\infty \\!\\!\\!",
"dE\\;(\\zeta ^x_E)^2\\Bigl (\\bigl \\bigl | \\mathcal {S}^{\\alpha \\alpha }_{E,\\mathbf {V}} \\bigr |^2\\bigr -\\bigl | \\bigl \\mathcal {S}^{\\alpha \\alpha }_{E,\\mathbf {V}}\\bigr \\bigr |^2\\Bigr )f^{\\prime }_E$ and thus $\\Omega ^{\\rho \\rho }_1|_{{{\\rm eq}}}=2T\\varphi /h(1+\\varphi )>0$ with $\\varphi =\\exp [\\mu /T]$ the quantum generator.",
"We recall that the notation $\\cdots |_{{{\\rm eq}}}$ indicates the limit $F^x_\\alpha ,\\omega \\rightarrow 0$ .",
"In general, the corrections $\\Omega ^{xx}_\\alpha $ vanish only if the driving fields are switched off, that is, if their amplitudes rather than their frequency go to zero.",
"In this limit, the quantum generator does not produce any current and the bound (REF ) becomes trivial.",
"The second relation () was originally derived for periodically modulated Markov jump processes [98].",
"In linear response, it holds also for coherent transport, provided that the frozen scattering amplitudes obey the symmetry $\\mathcal {S}^{\\alpha \\beta }_{E,\\mathbf {V}}=\\mathcal {S}^{\\beta \\alpha }_{E,\\mathbf {V}}$ , as is the case for the quantum generator, cf.",
"Eq.",
"(REF ); for details, see App. .",
"Far from equilibrium, however, the bound () can be violated as the second plot in Fig.",
"REF proves.",
"The fluctuation theorem uncertainty relation () follows from a general symmetry argument.",
"Specifically, it holds if the joint probability distribution of the entropy production per cycle $$ and the accumulated current $J̰^x_\\alpha $ obeys a strong detailed fluctuation theorem [102].",
"In general, such a relation holds only if no magnetic fields are applied to the system and the driving protocols are symmetric under time reversal.",
"These conditions do not apply to the quantum generator.",
"As a result, the third plot in Fig.",
"REF shows clear violations of the bound () at low frequencies.",
"The hysteretic uncertainty relation () provides a generalization of Eq. ().",
"The restriction to systems without magnetic fields and symmetric driving protocols is thereby removed by considering the actual thermodynamic process together with its time-reversed counterpart, which is denoted by a tilde [103], [104]; in coherent transport, quantities with a tilde are obtained from original ones by replacing $S^{\\alpha \\beta }_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}},E}$ with $\\mathsf {T}_\\mathbf {B}\\mathsf {T}_{\\mathbf {V}}S^{\\alpha \\beta }_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}},E}=S^{\\beta \\alpha }_{E,E\\raisebox {-0.7pt}{{{\\scriptsize n}}}}$ .",
"The last plot in Fig.",
"REF shows that the hysteretic uncertainty relation () is not violated for the quantum generator.",
"Whether or not it holds for coherent transport in general remains an open question at this point.",
"In any case, the relation () has only limited predictive power in practice.",
"First, it is inapplicable if the time-reversed process cannot be realized.",
"Second, since it involves only the symmeterized variable $\\sigma +\\tilde{\\sigma }$ , it cannot be used to bound the entropy production $\\sigma $ of the actual transport process.",
"Third, if $\\tilde{J}^x_\\alpha = -J^x_\\alpha $ , as is the case for $J^\\rho $ in the setup of the quantum generator, Eq.",
"() reduces to the trivial bound $\\sigma +\\tilde{\\sigma }\\ge 0$ .",
"By contrast, our relation (REF ) still yields a non-trivial lower bound on $\\sigma $ in this situation.",
"In summary, our case study shows that the new bounds (REF ), (REF ) and (REF ), which fully incorporate quantum effects and do not rely on special symmetries, go beyond the earlier results (REF ).",
"In particular, Eq.",
"(REF ) provides a universal bound on entropy production that depends only on parameters that can be measured in a given setup without having to realize the time-reversed transport process."
],
[
"Autonomous Systems",
"In the previous section, we have considered our far-from-equilibrium bounds on coherent transport in the context of earlier results for periodically driven thermodynamic processes.",
"However, our theory also has profound implications for autonomous systems, which can be established as follows.",
"If no time dependent fields are applied to the sample, the driving correction (REF ) is zero.",
"The bounds (REF ) and (REF ) then imply the new relation $\\frac{\\sigma (P^{\\rho \\rho }_{\\alpha \\alpha }+\\sigma /2)}{(J^\\rho _\\alpha )^2}= \\sigma (\\epsilon ^\\rho _\\alpha )^2 + \\frac{\\sigma ^2}{2(J^\\rho _\\alpha )^2}\\ge \\psi ^\\ast .$ This result shows that Pauli-blocking as a source of precision can be incorporated into the classical uncertainty relation (REF ), which otherwise can be violated in the quantum regime [47], [91], [92], through the universal correction $\\sigma ^2/2(J^\\rho _\\alpha )^2$ .",
"Quite remarkably, this quantum shift does not depend on any additional parameters.",
"As a result, the relation (REF ) leads to a bound on entropy production, $\\sigma \\ge \\bigl | J^\\rho _\\alpha \\bigr |\\left(\\sqrt{\\mathrm {F}_\\alpha ^2 +2\\psi ^\\ast }-|\\mathrm {F}_\\alpha |\\right),$ that involves only the current $J^\\rho _\\alpha $ and the standard zero-frequency Fano factor $\\mathrm {F}_\\alpha \\equiv P^{\\rho \\rho }_{\\alpha \\alpha }/J^\\rho _\\alpha $ .",
"Like the relation (REF ), this bound holds for any coherent multi-terminal conductor, arbitrary electric and thermal biases and in presence of external magnetic fields.",
"Finally, we note that, because the bound (REF ) on the driving corrections $\\Omega ^{\\rho \\rho }_\\alpha $ is independent of the driving strength, the relations (REF ) and (REF ) cannot be recovered from their more general counterparts (REF ) and (REF ).",
"This observation may indicate that the bounds (REF ) and (REF ) can be further optimized.",
"We leave it to future research to probe whether such refined bounds exist.",
"Furthermore, it remains yet an open problem whether operationally accessible bounds on entropy production, similar to the ones given in Eqs.",
"(REF ) and (REF ), can be formulated in terms of individual heat currents."
],
[
"Summary",
"Starting from the scattering approach to quantum transport, we have developed a universal thermodynamic framework for coherent conductors that are driven by thermo-chemical gradients and periodically changing electromagnetic fields, whose frequency plays the role of an additional thermodynamic force.",
"Focusing on the linear-response regime, we have shown that this framework can be equipped with consistent generalizations of the Onsager-Casimir relations and the fluctuation-dissipation theorem.",
"As our first key result, we have derived a family of thermodynamic bounds on matter and heat currents, which go beyond the second law, hold for arbitrary samples and driving protocols and involve only experimentally accessible quantities.",
"From a conceptual point of view, these bounds prove that transport without dissipation is impossible in conventional coherent conductors, even in the presence of reversible currents, which generically occur in systems with broken time reversal symmetry and do not contribute to the average entropy production.",
"From a practical perspective, they provide powerful tools of thermodynamic inference.",
"In particular, they make it possible to determine model-independent lower bounds on the total dissipation rate, which is generally difficult to access experimentally, by measuring the electric currents in the individual terminals of the conductor.",
"When applied to mesoscopic devices, our bounds lead to non-trivial relations between key figures of merit, which provide both universal benchmarks for theoretical models and a new avenue to estimate the performance of experimental realizations of nano-machines, whose output or input cannot be measured directly.",
"We have illustrated this method for parametric quantum pumps and adiabatic quantum motors, where in both cases a mechanical quantity was bounded in terms of an easy-to-measure electric current.",
"Beyond these examples, our results are applicable to any system that can be described as a coherent multi-terminal conductor, including thermoelectric heat engines and refrigerators, which have gained much attention in recent years [36], [37], [38], [39], [40], [41], [42], [43], [44], [45], [46], [47], [48], [49].",
"Our work thus provides a versatile toolbox for both theoretical and experimental studies seeking to develop powerful and efficient quantum transport devices.",
"Going beyond the linear-response regime, in the second part of this article, we have derived a family of thermodynamic bounds on matter and heat currents that hold for arbitrary thermo-chemical gradients and driving frequencies.",
"These relations imply a thermodynamic uncertainty relation for coherent transport, which accounts for quantum effects and periodic driving in a unified manner, and a fully universal bound on entropy production that depends only on experimentally accessible parameters.",
"This bound, which is our second key result, can be directly used for thermodynamic inference far from equilibrium.",
"In fact, it can be determined by measuring a single electric current, its zero-frequency noise and, for systems with periodic driving, the temperature of the corresponding reservoir.",
"Thus, we are now able to pass the baton to the experimentalists to test our theoretical results and to utilize their wide-ranging applicability to further explore the quantum thermodynamics of coherent transport devices.",
"E.P.",
"acknowledges helpful comments from V. Kashcheyevs.",
"K.B.",
"thanks K. Saito for insightful discussions.",
"E.P.",
"acknowledges support from the Vilho, Yrjö and Kalle Foundation of the Finnish Academy of Science and Letters through the grant for doctoral studies.",
"C.F.",
"acknowledges support from the Academy of Finland (Projects No.",
"308515 and No.",
"312299).",
"M.M.",
"acknowledges the warm hospitality of Aalto University, support from the Aalto Science Institute through its Visiting Fellow Programme, and support from the Ministry of Education and Science of Ukraine (project No.",
"0119U002565).",
"K.B.",
"acknowledges support from Academy of Finland (Contract No.",
"296073), the Japan Society for the Promotion of Science through a Postdoctoral Fellowship for Research in Japan (Fellowship ID: P19026), the University of Nottingham through a Nottingham Research Fellowship and from UK Research and Innovation through a Future Leaders Fellowship (Grant Reference: MR/S034714/1).",
"Authors from Aalto University are associated with the local Centre for Quantum Engineering."
],
[
"Kinetic Coefficients",
"In this appendix, we derive the formulas (REF ) for the linear-response coefficients in the frequency picture, that is, with the driving frequency playing the role of an additional affinity.",
"We then compare this approach with the amplitude picture, where the mechanical affinity corresponds to the strength of the applied periodic fields."
],
[
"Frequency Picture",
"We first recall that the thermo-chemical currents and the photon current are given by $&J^x_\\alpha = \\frac{1}{h}\\int _0^\\infty \\!\\!\\!",
"dE\\sum \\nolimits _\\beta \\sum \\nolimits _n\\Bigl (\\zeta ^{x,\\alpha }_E\\delta _{\\alpha \\beta }\\delta _{n0}-\\zeta ^{x,\\alpha }_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}}}\\bigl | S^{\\alpha \\beta }_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}},E} \\bigr |^2\\Bigr )f^\\beta _E\\nonumber \\\\[3pt]&\\text{with}\\quad \\begin{aligned}[t]\\zeta ^{x,\\alpha }_E &=\\zeta ^x_E-\\delta _{xq}TF^\\rho _\\alpha ,\\\\[3pt]\\zeta ^{x,\\alpha }_E &=\\zeta ^x_E-\\delta _{xq}T(F^\\rho _\\alpha -nF^\\omega )\\quad \\text{and}\\end{aligned}\\\\[3pt]& J^\\omega =\\frac{1}{h}\\int _0^\\infty \\!\\!\\!",
"dE\\sum \\nolimits _{\\alpha \\beta }\\sum \\nolimits _nn \\bigl | S^{\\alpha \\beta }_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}},E} \\bigr |^2 f^\\beta _E,$ where $\\zeta ^\\rho _E\\equiv 1$ , $\\zeta ^q_E\\equiv E-\\mu $ and $\\zeta ^\\omega =1/\\omega $ .",
"Inserting the expansion of the Fermi function (REF ) and the expansions $&\\sum \\nolimits _n\\bigl | S^{\\alpha \\beta }_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}},E} \\bigr |^2 = \\mathcal {X}_0+ \\frac{TF^\\omega }{2}\\mathcal {X}_1,\\\\[3pt]&\\sum \\nolimits _nn\\bigl | S^{\\alpha \\beta }_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}},E} \\bigr |^2 = \\mathcal {Y}_0+ \\frac{TF^\\omega }{2}\\mathcal {Y}_1$ and collecting first-order contributions in the affinities then yields $F^x_\\alpha $ and $F^\\omega $ yields $L^{xy}_{\\alpha \\beta } &= \\frac{1}{h}\\int _0^\\infty \\!\\!\\!",
"dE\\;\\zeta ^x_E\\zeta ^y_E\\big (\\delta _{\\alpha \\beta }-\\mathcal {X}_0\\big )f^{\\prime }_E,\\\\[3pt]L^{x\\omega }_\\alpha &= -\\frac{T}{2h}\\int _0^\\infty \\!\\!\\!",
"dE\\sum \\nolimits _\\beta \\bigl (2\\mathcal {Y}_0\\delta _{xq}+\\zeta ^x_E\\mathcal {X}_1\\big )f_E,\\\\[3pt]L^{\\omega x}_\\beta &=\\frac{1}{h}\\int _0^\\infty \\!\\!\\!",
"dE\\; \\zeta ^x_E \\sum \\nolimits _\\alpha \\mathcal {Y}_0 f^{\\prime }_E,\\\\[3pt]L^{\\omega \\omega } &= \\frac{T}{2h}\\int _0^\\infty \\!\\!\\!",
"dE\\; \\sum \\nolimits _{\\alpha \\beta }\\mathcal {Y}_1 f_E.$ Note that the zeroth-order terms vanish as can be shown with the help of the unitarity conditions (REF ) for the frozen scattering amplitudes.",
"Upon inserting the expressions ()-() for $\\mathcal {X}_0,\\mathcal {X}_1,\\mathcal {Y}_0$ and $\\mathcal {Y}_1$ in the Eqs.",
"(REF ), we arrive at $L^{xy}_{\\alpha \\beta } &=\\frac{1}{h}\\int _0^\\infty \\!\\!\\!",
"dE\\; \\zeta ^x_E\\zeta ^y_E \\Big (\\delta _{\\alpha \\beta }-\\bigl \\bigl | \\mathcal {S}^{\\alpha \\beta }_{E,\\mathbf {V}} \\bigr |^2\\bigr \\Big )f^{\\prime }_E,\\\\[3pt]L^{\\omega x}_\\alpha &= \\frac{T}{2h}\\int _0^\\infty \\!\\!\\!",
"dE\\sum \\nolimits _\\beta \\Big ( 2\\zeta ^\\omega {{{\\rm Im}}}\\Bigl [\\bigl \\dot{\\mathcal {S}}^{\\alpha \\beta }_{E,\\mathbf {V}}\\mathcal {S}^{\\alpha \\beta \\ast }_{E,\\mathbf {V}}\\bigr \\Bigr ]\\delta _{xq}-4\\zeta ^x_E{{{\\rm Re}}}\\Bigl [\\bigl \\mathcal {S}^{\\alpha \\beta \\ast }_{E,\\mathbf {V}}\\mathcal {A}^{\\alpha \\beta }_E\\bigr \\Bigr ]+\\zeta ^x_E \\zeta ^\\omega \\partial _E^{\\phantom{\\beta }}{{{\\rm Im}}}\\Bigl [\\bigl \\dot{\\mathcal {S}}^{\\alpha \\beta }_{E,\\mathbf {V}}\\mathcal {S}^{\\alpha \\beta \\ast }_{E,\\mathbf {V}}\\bigr \\Bigr ]\\Bigr )f_E\\\\&= \\frac{T}{h}\\int _0^\\infty \\!\\!\\!",
"dE\\sum \\nolimits _\\beta \\Bigl (\\zeta ^\\omega {{{\\rm Im}}}\\Bigl [\\bigl \\dot{\\mathcal {S}}^{\\alpha \\beta }_{E,\\mathbf {V}}\\mathcal {S}^{\\alpha \\beta \\ast }_{E,\\mathbf {V}}\\bigr \\Bigr ]\\delta _{xq}+\\zeta ^x_E\\zeta ^\\omega \\partial _E^{\\phantom{\\beta }}{{{\\rm Im}}}\\Bigl [\\bigl \\dot{\\mathcal {S}}^{\\alpha \\beta }_{E,\\mathbf {V}}\\mathcal {S}^{\\alpha \\beta \\ast }_{E,\\mathbf {V}}\\bigr \\Bigr ]\\Bigr )f_E\\nonumber \\\\&= \\frac{1}{h}\\int _0^\\infty \\!\\!\\!",
"dE\\;\\zeta ^x_E\\zeta ^\\omega \\sum \\nolimits _\\beta {{{\\rm Im}}}\\Bigr [\\bigl \\dot{\\mathcal {S}}^{\\alpha \\beta }_{E,\\mathbf {V}}\\mathcal {S}^{\\alpha \\beta \\ast }_{E,\\mathbf {V}}\\bigr \\Bigr ]f^{\\prime }_E,\\nonumber \\\\[3pt]L^{x\\omega }_\\alpha &= \\frac{1}{h}\\int _0^\\infty \\!\\!\\!",
"dE\\; \\zeta ^x_E \\zeta ^\\omega \\sum \\nolimits _\\beta {{{\\rm Im}}}\\Bigl [\\bigl \\dot{\\mathcal {S}}^{\\beta \\alpha \\ast }_{E,\\mathbf {V}}\\mathcal {S}^{\\beta \\alpha }_{E,\\mathbf {V}}\\bigr \\Bigr ],\\\\[3pt]L^{\\omega \\omega }&= \\frac{T}{2h}\\int _0^\\infty \\!\\!\\!",
"dE\\;\\zeta ^\\omega \\sum \\nolimits _{\\alpha \\beta }\\Bigl (4{{{\\rm Im}}}\\Bigl [\\bigl \\dot{\\mathcal {S}}^{\\alpha \\beta \\ast }_{E,\\mathbf {V}}\\mathcal {A}^{\\alpha \\beta }_E\\bigr \\Bigr ]+\\zeta ^\\omega \\partial _E^{\\phantom{\\beta }}\\bigl \\bigl | \\dot{\\mathcal {S}}^{\\alpha \\beta }_{E,\\mathbf {V}} \\bigr |^2\\bigr \\Bigr )f_E\\\\&= \\frac{1}{2h}\\int _0^\\infty \\!\\!\\!",
"dE\\; (\\zeta ^\\omega )^2\\sum \\nolimits _{\\alpha \\beta }\\bigl \\bigl | \\dot{\\mathcal {S}}^{\\alpha \\beta }_{E,\\mathbf {V}} \\bigr |^2\\bigr f^{\\prime }_E.",
"\\nonumber $ Here, we have used the sum rule $4\\sum \\nolimits _\\beta {{{\\rm Re}}}\\Bigl [\\bigl \\mathcal {S}^{\\alpha \\beta \\ast }_{E,\\mathbf {V}}\\mathcal {A}^{\\alpha \\beta }_E\\bigr \\Bigr ]&= -i\\zeta ^\\omega \\sum \\nolimits _\\beta \\partial _E^{\\phantom{\\beta }}\\bigl \\dot{\\mathcal {S}}^{\\alpha \\beta \\ast }_{E,\\mathbf {V}}\\mathcal {S}^{\\alpha \\beta }_{E,\\mathbf {V}}\\bigr \\\\&= -\\zeta ^\\omega \\sum \\nolimits _\\beta \\partial _E^{\\phantom{\\beta }}{{{\\rm Im}}}\\Bigl [\\bigl \\dot{\\mathcal {S}}^{\\alpha \\beta }_{E,\\mathbf {V}}\\mathcal {S}^{\\alpha \\beta \\ast }_{E,\\mathbf {V}}\\bigr \\Bigr ], \\nonumber $ which follows from Eq.",
"(), in the first step of Eq.",
"(); in the second step, we have applied an integration by parts with respect to $E$ noting that $\\partial _E \\zeta ^x_E = \\delta _{xq}$ and $\\partial _E f_E = - f^{\\prime }_E/T$ .",
"In Eq.",
"(), we have used that $\\sum \\nolimits _{\\alpha \\beta }{{{\\rm Im}}}\\Big [\\bigl \\dot{\\mathcal {S}}^{\\alpha \\beta \\ast }_{E,\\mathbf {V}}\\mathcal {A}^{\\alpha \\beta }_E\\bigr \\Big ]=0$ as a consequence of the sum rule () and the fact that the period average is a linear operation; finally we have integrated the remaining term by parts."
],
[
"Amplitude Picture",
"The amplitude picture provides an alternative way of extending the framework of irreversible thermodynamics to periodically driven coherent conductors.",
"This theory, which is complementary to the one discussed in Sec.",
", can be developed along the same lines as for stochastic systems, see Refs.",
"[58], [59], [60], [61].",
"We first divide the single-carrier Hamiltonian that describes the dynamics inside the conductor into two contributions, $H_t \\equiv H_0 + \\Delta U_t,$ where free the Hamiltonian $H_0$ is time-independent, the dynamical scattering potential $U_t=U_{t+ accounts for the periodic drivingand the parameter \\Delta denotes to the amplitude of thisperturbation.The Floquet scattering amplitudes can thus be decomposed as\\begin{equation}S^{\\alpha \\beta }_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}},E} =\\delta _{n0} S^{\\alpha \\beta }_E+ \\Delta Z^{\\alpha \\beta }_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}},E}\\end{equation}with the first contribution describing elastic scattering events andthe second one accounting for inelastic events, which are induced bythe time-dependent driving.We now introduce the affinity F^w\\equiv \\Delta /T and the work flux\\begin{equation}J^w\\equiv \\Pi _{{{\\rm ac}}}/\\Delta = \\frac{\\Delta }{\\int _0^\\infty \\!\\!\\!",
"dE\\sum \\nolimits _{\\alpha \\beta }\\sum \\nolimits _nn\\bigl | Z^{\\alpha \\beta }_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}},E} \\bigr |^2f^\\beta _E,}where the second expression is obtained by insertingEq.~(\\ref {App2ScattExp}) into Eq.~(\\ref {TwAcPower}).With these definitions, the total rate of entropy production(\\ref {TwEntProdAux}) assumes the canonical bilinear form\\begin{equation}\\sigma =J^wF^w +\\sum \\nolimits _\\alpha \\sum \\nolimits _xJ^x_\\alpha F^x_\\alpha .\\end{equation}\\end{equation}The frequency and the amplitude picture are equivalent on the generallevel.Their corresponding linear-response theories, however, describe twophysically different regimes, where either the speed or the strengthof the periodic driving is small.This difference becomes particularly clear from the kineticcoefficients in the amplitude picture, which are given by\\footnote {To derive these expressions, insert the first-orderexpansions (\\ref {TwFermiFunctExp}) andS^{\\alpha \\beta }_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}},E}=\\delta _{n0}S^{\\alpha \\beta }_E +\\Delta \\mathcal {Z}^{\\alpha \\beta }_{En,E}into the Eqs.",
"(\\ref {AppAKinCurr}) and (\\ref {AppAWorkFl}) and use thesum rule\\begin{equation*}\\sum \\nolimits _\\beta {{{\\rm Re}}}\\Bigl [ S^{\\alpha \\beta }_E \\mathcal {Z}^{\\alpha \\beta \\ast }_{E,E}\\Bigr ]=0,\\end{equation*}which follows from the unitarity condition (\\ref {TwSumRulesSMr}).",
"}{\\begin{@align}{1}{-1}G^{xy}_{\\alpha \\beta }&\\equiv \\partial _{F^y_\\beta }J^x_\\alpha |_{{{\\rm eq}}}= \\frac{1}{h}\\int _0^\\infty \\!\\!\\!",
"dE\\; \\zeta ^x_E \\zeta ^y_E \\Bigl (\\delta _{\\alpha \\beta } - \\bigl | S^{\\alpha \\beta }_E \\bigr |^2\\Bigr ) f^{\\prime }_E,\\\\[3pt]G^{wx}_\\alpha &\\equiv \\partial _{F^w}J^x_\\alpha |_{{{\\rm eq}}}= 0,\\\\[3pt]G^{xw}_\\alpha &\\equiv \\partial _{F^x_\\alpha } J^w|_{{{\\rm eq}}}=0, \\\\[3pt]G^{ww} &\\equiv \\partial _{F^w}J^w|_{{{\\rm eq}}}= \\frac{T}{\\int _0^\\infty \\!\\!\\!",
"dE\\sum \\nolimits _{\\alpha \\beta }n \\bigl | \\mathcal {Z}^{\\alpha \\beta }_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}},E} \\bigr |^2f_E}\\end{@align}with \\mathcal {Z}^{\\alpha \\beta }_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}},E}\\equiv Z^{\\alpha \\beta }_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}},E}|_{\\Delta =0}.Hence, the thermo-chemical variables, F^x_\\alpha and J^x_\\alpha decouplefrom the mechanical ones, F^w and J^w, in the weak-driving regime.By contrast, this coupling persists in slow-driving regime, providedthat the driving protocols are not symmetric under time reversal, cf.Sec.~\\ref {Sec_OCRel}.",
"}\\section {Some useful Lemmas}In this appendix, we collect a series of sum rules for the Floquetscattering amplitudes in the slow-driving regime along withsketches of their proofs.Further details on the derivations may be found inRefs.~\\cite {Moskalets2012,Ludovico2015b}.",
"}\\begin{lemma}Let S^{\\alpha \\beta }_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}},E} be the Floquet scattering amplitudes of amulti-terminal conductor that is subject to the periodic drivingfields \\mathbf {V} with frequency \\omega \\equiv 2\\pi /, then\\begin{equation}\\sum \\nolimits _n\\bigl | S^{\\alpha \\beta }_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}},E} \\bigr |^2 = \\mathcal {X}_0+ \\frac{\\hbar \\omega }{2}\\mathcal {X}_1, \\quad \\sum \\nolimits _nn\\bigl | S^{\\alpha \\beta }_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}},E} \\bigr |^2 = \\mathcal {Y}_0+ \\frac{\\hbar \\omega }{2}\\mathcal {Y}_1\\end{equation}up to second-order corrections in \\hbar \\omega with\\begin{subequations}{\\begin{@align}{1}{-1}&\\mathcal {X}_0=\\bigl \\bigl | \\mathcal {S}^{\\alpha \\beta }_{E,\\mathbf {V}} \\bigr |^2\\bigr ,\\\\[3pt]&\\mathcal {X}_1=4{{{\\rm Re}}}\\Bigr [\\bigl \\mathcal {S}^{\\alpha \\beta \\ast }_{E,\\mathbf {V}}\\mathcal {A}^{\\alpha \\beta }_E\\bigr \\Bigr ]-\\frac{1}{\\omega }\\partial _E^{\\phantom{\\beta }}{{{\\rm Im}}}\\Bigl [\\bigl \\dot{\\mathcal {S}}^{\\alpha \\beta }_{E,\\mathbf {V}}\\mathcal {S}^{\\alpha \\beta \\ast }_{E,\\mathbf {V}}\\bigr \\Bigr ],\\\\[3pt]&\\mathcal {Y}_0=- \\frac{1}{\\omega }{{{\\rm Im}}}\\Bigr [\\bigl \\dot{\\mathcal {S}}^{\\alpha \\beta }_{E,\\mathbf {V}}\\mathcal {S}^{\\alpha \\beta \\ast }_{E,\\mathbf {V}}\\bigr \\Bigr ],\\\\[3pt]&\\mathcal {Y}_1=\\frac{4}{\\omega }{{{\\rm Im}}}\\Bigl [\\bigl \\dot{\\mathcal {S}}^{\\alpha \\beta \\ast }_{E,\\mathbf {V}}\\mathcal {A}^{\\alpha \\beta }_E\\bigr \\Bigr ]+\\frac{1}{\\omega ^2}\\partial _E^{\\phantom{\\beta }}\\bigl \\bigl | \\dot{\\mathcal {S}}^{\\alpha \\beta }_{E,\\mathbf {V}} \\bigr |^2\\bigr .\\end{@align}}Here, \\mathcal {S}^{\\alpha \\beta }_{E,\\mathbf {V}} denotes the frozen scattering amplitudes,\\mathcal {A}^{\\alpha \\beta }_E the non-adiabatic corrections defined inEq.~(\\ref {TwAdApproxFSA}) and \\cdots indicates thetime average over one period .\\end{subequations}\\end{lemma}\\begin{proof}Insert the low-frequency expansion (\\ref {TwAdApproxFSA}) forS^{\\alpha \\beta }_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}},E} and collect all zeroth and first order terms in\\hbar \\omega .Perform the sum over all integers n by using the symbolic identity\\begin{equation}\\sum \\nolimits _nn^k e^{i n\\omega (t-t^{\\prime })}=\\frac{{(i\\omega )^k}\\partial ^k_t\\delta _{t-t^{\\prime }},}{w}here k=0,1,\\dots and t,t^{\\prime }\\in [0,.Integrate by parts with respect to t as needed.\\end{equation}\\end{proof}\\begin{lemma}The frozen scattering amplitudes and the non-adiabatic correctionsobey the joint sum rules\\begin{subequations}{\\begin{@align}{1}{-1}&\\begin{aligned}[t]& 2i\\omega \\sum \\nolimits _\\alpha \\bigl (\\mathcal {S}^{\\alpha \\beta }_{E,\\mathbf {V}\\raisebox {-0.7pt}{{{\\scriptsize t}}}}\\mathcal {A}^{\\alpha \\gamma \\ast }_{E,t}+\\mathcal {S}^{\\alpha \\gamma \\ast }_{E,t}\\mathcal {A}^{\\alpha \\beta }_{E,\\mathbf {V}\\raisebox {-0.7pt}{{{\\scriptsize t}}}}\\bigr )\\\\&\\hspace*{28.45274pt}=\\sum \\nolimits _\\alpha \\bigl (\\dot{\\mathcal {S}}^{\\alpha \\beta }_{E,\\mathbf {V}\\raisebox {-0.7pt}{{{\\scriptsize t}}}}\\partial _E^{\\phantom{\\beta }}\\mathcal {S}^{\\alpha \\gamma \\ast }_{E,\\mathbf {V}\\raisebox {-0.7pt}{{{\\scriptsize t}}}}-\\dot{\\mathcal {S}}^{\\alpha \\gamma \\ast }_{E,\\mathbf {V}\\raisebox {-0.7pt}{{{\\scriptsize t}}}}\\partial _E^{\\phantom{\\beta }}\\mathcal {S}^{\\alpha \\beta }_{E,\\mathbf {V}\\raisebox {-0.7pt}{{{\\scriptsize t}}}}\\bigr ),\\end{aligned}\\\\[3pt]&\\begin{aligned}[t]& 2i\\omega \\sum \\nolimits _\\alpha \\bigl (\\mathcal {S}^{\\beta \\alpha }_{E,\\mathbf {V}\\raisebox {-0.7pt}{{{\\scriptsize t}}}}\\mathcal {A}^{\\gamma \\alpha \\ast }_{E,t}+\\mathcal {S}^{\\gamma \\alpha \\ast }_{E,\\mathbf {V}\\raisebox {-0.7pt}{{{\\scriptsize t}}}}\\mathcal {A}^{\\beta \\alpha }_{E,t}\\bigr )\\\\&\\hspace*{28.45274pt} =\\sum \\nolimits _\\alpha \\bigl (\\dot{\\mathcal {S}}^{\\gamma \\alpha \\ast }_{E,\\mathbf {V}\\raisebox {-0.7pt}{{{\\scriptsize t}}}}\\partial _E^{\\phantom{\\beta }}\\mathcal {S}^{\\beta \\alpha }_{E,\\mathbf {V}\\raisebox {-0.7pt}{{{\\scriptsize t}}}}-\\dot{\\mathcal {S}}^{\\beta \\alpha }_{E,\\mathbf {V}\\raisebox {-0.7pt}{{{\\scriptsize t}}}}\\partial _E^{\\phantom{\\beta }}\\mathcal {S}^{\\gamma \\alpha \\ast }_{E,\\mathbf {V}\\raisebox {-0.7pt}{{{\\scriptsize t}}}}\\bigr ),\\end{aligned}\\\\[3pt]&\\sum \\nolimits _{\\alpha \\beta }{{{\\rm Im}}}\\Bigl [\\dot{\\mathcal {S}}^{\\alpha \\beta \\ast }_{E,\\mathbf {V}\\raisebox {-0.7pt}{{{\\scriptsize t}}}}\\mathcal {A}^{\\alpha \\beta }_{E,t}\\Bigr ]=0\\end{@align}}\\end{subequations}for any t\\in [0,.\\end{lemma}\\begin{proof}We begin with the sum rule (\\ref {Lem_2a}).Inserting the expansion (\\ref {TwAdApproxFSA}) into the unitaritycondition (\\ref {TwSumRulesSMl}), collecting first-order terms in\\hbar \\omega and carrying out the sum over n usingEq.~(\\ref {Lem_1Aux}) shows that{\\begin{@align}{1}{-1}\\Bigl \\lbrace \\sum \\nolimits _\\alpha \\Bigl (& 2\\dot{\\mathcal {S}}^{\\alpha \\beta }_{E,\\mathbf {V}\\raisebox {-0.7pt}{{{\\scriptsize t}}}}\\partial _E^{\\phantom{\\beta }}\\mathcal {S}^{\\alpha \\gamma \\ast }_{E,\\mathbf {V}\\raisebox {-0.7pt}{{{\\scriptsize t}}}}\\\\& + \\mathcal {S}^{\\alpha \\beta }_{E,\\mathbf {V}\\raisebox {-0.7pt}{{{\\scriptsize t}}}}\\partial _E^{\\phantom{\\beta }}\\dot{\\mathcal {S}}^{\\alpha \\gamma \\ast }_{E,\\mathbf {V}\\raisebox {-0.7pt}{{{\\scriptsize t}}}}+ \\mathcal {S}^{\\alpha \\gamma \\ast }_{E,\\mathbf {V}\\raisebox {-0.7pt}{{{\\scriptsize t}}}}\\partial _E^{\\phantom{\\beta }}\\dot{\\mathcal {S}}^{\\alpha \\beta }_{E,\\mathbf {V}\\raisebox {-0.7pt}{{{\\scriptsize t}}}}\\nonumber \\\\& -2i\\omega \\bigl (\\mathcal {S}^{\\alpha \\beta }_{E,\\mathbf {V}\\raisebox {-0.7pt}{{{\\scriptsize t}}}}\\mathcal {A}^{\\alpha \\gamma \\ast }_{E,t}+\\mathcal {S}^{\\alpha \\gamma \\ast }_{E,\\mathbf {V}\\raisebox {-0.7pt}{{{\\scriptsize t}}}}\\mathcal {A}^{\\alpha \\beta }_{E,t}\\bigr )\\Bigr )\\Bigr \\rbrace e^{im\\omega t}=0,\\nonumber \\end{@align}}for every integer m and any indices \\beta and \\gamma .This condition can only be met if the expression inside the curlybrackets vanishes for every t\\in [0,, that is, if\\begin{equation}2i\\omega (\\mathbb {A}^\\dagger \\mathbb {S}+ \\mathbb {A}\\mathbb {S}^\\dagger )=2\\mathbb {S}^{\\prime \\dagger }\\dot{\\mathbb {S}} +\\dot{\\mathbb {S}}^{\\prime \\dagger }\\mathbb {S}+\\mathbb {S}^\\dagger \\dot{\\mathbb {S}}^{\\prime },\\end{equation}where we introduced the matrices \\mathbb {S} and \\mathbb {A} with elements(\\mathbb {S})_{\\alpha \\beta }\\equiv \\mathcal {S}^{\\alpha \\beta }_{E,\\mathbf {V}\\raisebox {-0.7pt}{{{\\scriptsize t}}}} and (\\mathbb {A})_{\\alpha \\beta }\\equiv \\mathcal {A}^{\\alpha \\beta }_{E,t} to simplify the notation and primes indicatederivatives with respect to E.Next, we note that the matrix \\mathbb {S} is unitary, i.e.,\\mathbb {S}^\\dagger \\mathbb {S}=\\mathbb {S}\\mathbb {S}^\\dagger =\\mathbb {1}, owing to theconditions (\\ref {TwARSumRulesSM}).Therefore, we have \\partial _t\\partial _E\\mathbb {S}^\\dagger \\mathbb {S}=0 and writingout the derivatives gives\\begin{equation}\\dot{\\mathbb {S}}^{\\prime \\dagger }\\mathbb {S}+\\mathbb {S}^\\dagger \\dot{\\mathbb {S}}^{\\prime }\\\\= -\\mathbb {S}^{\\prime \\dagger }\\dot{\\mathbb {S}}-\\dot{\\mathbb {S}}^\\dagger \\mathbb {S}^{\\prime }.\\end{equation}Inserting this relation into Eq.~(\\ref {Lem_2Aux2}) yields the result\\begin{equation}2i\\omega (\\mathbb {A}^\\dagger \\mathbb {S}+\\mathbb {A}\\mathbb {S}^\\dagger )=\\mathbb {S}^{\\prime \\dagger }\\dot{\\mathbb {S}}-\\dot{\\mathbb {S}}^\\dagger \\mathbb {S}^{\\prime },\\end{equation}which is equivalent to the first sum rule (\\ref {Lem_2a}).The second sum rule (\\ref {Lem_2b}) can be derived along the same linesstarting with the unitarity condition (\\ref {TwSumRulesSMr}).\\end{proof}To derive the third sum rule (\\ref {Lem_2c}), we first note that{\\begin{@align}{1}{-1}2i\\sum \\nolimits _{\\alpha \\beta }{{{\\rm Im}}}\\Bigl [\\dot{\\mathcal {S}}^{\\alpha \\beta \\ast }_{E,\\mathbf {V}\\raisebox {-0.7pt}{{{\\scriptsize t}}}}\\mathcal {A}^{\\alpha \\beta }_{E,t}\\Bigr ]& = {{{\\rm tr}}}\\bigl \\lbrace \\dot{\\mathbb {S}}^\\dagger \\mathbb {A}-\\mathbb {A}^\\dagger \\dot{\\mathbb {S}}\\bigr \\rbrace \\\\& = {{{\\rm tr}}}\\bigl \\lbrace \\mathbb {A}\\dot{\\mathbb {S}}^\\dagger \\mathbb {S}\\mathbb {S}^\\dagger -\\mathbb {A}^\\dagger \\dot{\\mathbb {S}}\\mathbb {S}^\\dagger \\mathbb {S}\\bigr \\rbrace \\nonumber \\\\& = {{{\\rm tr}}}\\bigl \\lbrace \\bigl (\\mathbb {S}^\\dagger \\mathbb {A}+\\mathbb {A}^\\dagger \\mathbb {S}\\bigr )\\dot{\\mathbb {S}}^\\dagger \\mathbb {S}\\bigr \\rbrace ,\\nonumber \\end{@align}}where we have used that $ SS=-SS$,since $ S$ is unitary.Second, eliminating the matrix $ A$ with the help of the relation(\\ref {Lem_2Aux3}) gives{\\begin{@align}{1}{-1}-4\\omega \\sum \\nolimits _{\\alpha \\beta }{{{\\rm Im}}}\\Bigl [\\dot{\\mathcal {S}}^{\\alpha \\beta \\ast }_{E,\\mathbf {V}\\raisebox {-0.7pt}{{{\\scriptsize t}}}}\\mathcal {A}^{\\alpha \\beta }_{E,t}\\Bigr ]&= {{{\\rm tr}}}\\bigl \\lbrace \\bigl (\\mathbb {S}^{\\prime \\dagger }\\dot{\\mathbb {S}}-\\dot{\\mathbb {S}}^\\dagger \\mathbb {S}^{\\prime }\\bigr )\\dot{\\mathbb {S}}^\\dagger \\mathbb {S}\\bigr \\rbrace \\\\&= {{{\\rm tr}}}\\bigl \\lbrace \\bigl (\\mathbb {S}^\\dagger \\mathbb {S}^{\\prime }\\dot{\\mathbb {S}}^\\dagger \\mathbb {S}-\\dot{\\mathbb {S}}^\\dagger \\mathbb {S}^{\\prime }\\bigr )\\dot{\\mathbb {S}}^\\dagger \\mathbb {S}\\bigr \\rbrace \\nonumber \\\\&= 0,\\nonumber \\end{@align}}where the second line follows by noting that $ SS'= -S'S$ and $ SS=-SS$.$ Lemma 1 The frozen scattering amplitudes can be chosen such that they obey the sum rule $\\sum \\nolimits _{\\alpha \\beta }\\bigl \\dot{\\mathcal {S}}^{\\alpha \\beta }_{E,\\mathbf {V}}\\mathcal {S}^{\\alpha \\beta \\ast }_{E,\\mathbf {V}}\\bigr =0.$ The frozen scattering matrix admits the spectral decomposition $\\mathbb {S}=\\sum _k |\\phi _k\\rangle \\langle \\phi _k| e^{i\\phi _k}$ , where the $\\phi _k$ are real and the $|\\phi _k\\rangle $ are normalized orthogonal eigenvectors.",
"Since $\\mathbb {S}$ is a $-periodic function of time, the phases$ k$ further have to obey the condition$ k=Mk$ with $ Mk$ being an integer.It follows that{\\begin{@align}{1}{-1}\\sum \\nolimits _{\\alpha \\beta }\\bigl \\dot{\\mathcal {S}}^{\\alpha \\beta }_{E,\\mathbf {V}}\\mathcal {S}^{\\alpha \\beta \\ast }_{E,\\mathbf {V}}\\bigr & = \\bigl {{{\\rm tr}}}\\bigl \\lbrace \\dot{\\mathbb {S}}\\mathbb {S}^\\dagger \\bigr \\rbrace \\bigr \\\\& = \\sum \\nolimits _k \\bigl i\\dot{\\phi }_k+ \\partial _t \\langle \\phi _k|\\phi _k\\rangle \\bigr \\nonumber \\\\& = i\\omega \\sum \\nolimits _k M_k\\equiv i \\omega M,\\nonumber \\end{@align}}Thus, the sum rules (\\ref {Lem_3a}) can be enforced by applying thegauge transformation $ SE,VSE,Ve-iMt$ to the frozen scattering amplitudes.$ Lemma 2 Let $S^{\\alpha \\beta }_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}},E}$ be a set of Floquet scattering amplitudes that obey the unitarity conditions (REF ) and $f^\\alpha _E$ the Fermi function (), then $\\sigma &\\equiv \\frac{1}{h}\\int _0^\\infty \\!\\!\\!",
"dE\\sum \\nolimits _{\\alpha \\beta }\\sum \\nolimits _n\\left(\\frac{E_n-\\mu _\\alpha }{T_\\alpha }-\\frac{E-\\mu _\\beta }{T_\\beta }\\right)\\bigl | S^{\\alpha \\beta }_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}},E} \\bigr |^2 f^\\beta _E\\nonumber \\\\&\\ge \\frac{2}{h}\\int _0^\\infty \\!\\!\\!",
"dE\\sum \\nolimits _{\\alpha \\beta }\\sum \\nolimits _n\\bigl | S^{\\alpha \\beta }_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}},E} \\bigr |^2(f^\\alpha _{E\\raisebox {-0.7pt}{{{\\scriptsize n}}}}-f^\\beta _E)^2.$ We first observe that, using the conditions (REF ), $\\sigma $ can be expressed as [8] $\\sigma = \\frac{1}{h}\\int _0^\\infty \\!\\!\\!",
"dE\\sum \\nolimits _{\\alpha \\beta }\\sum \\nolimits _n\\bigl | S^{\\alpha \\beta }_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}},E} \\bigr |^2H^{\\alpha \\beta }_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}},E}$ with $H^{\\alpha \\beta }_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}},E} \\equiv \\eta [f^\\alpha _{E\\raisebox {-0.7pt}{{{\\scriptsize n}}}}]-\\eta [f^\\beta _E]-\\eta ^{\\prime }[f^\\alpha _{E\\raisebox {-0.7pt}{{{\\scriptsize n}}}}]\\bigl (f^\\alpha _{E\\raisebox {-0.7pt}{{{\\scriptsize n}}}}-f^\\beta _E\\bigr ),$ $\\eta [x]\\equiv -x\\ln [x] -(1-x)\\ln [1-x]$ and primes indicating derivatives.",
"Next, we note that, by Taylor's theorem, there exists a $g$ between $f^\\alpha _{E\\raisebox {-0.7pt}{{{\\scriptsize n}}}}$ and $f^\\beta _E$ such that $H^{\\alpha \\beta }_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}},E} = - \\frac{\\eta ^{\\prime \\prime }[g]}{2}(f^\\alpha _{E\\raisebox {-0.7pt}{{{\\scriptsize n}}}}-f^\\beta _E)^2.$ Since the Fermi function () takes only values between 0 and 1, it follows that $g\\in [0,1]$ .",
"Therefore, we have $-\\eta ^{\\prime \\prime }[g]= 1/g+1/(1-g)\\ge 4$ and $H^{\\alpha \\beta }_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}},E}\\ge 2 (f^\\alpha _{E\\raisebox {-0.7pt}{{{\\scriptsize n}}}}-f^\\beta _E)^2$ .",
"Inserting this bound into Eq.",
"(REF ) completes the proof.",
"Lemma 3 Let $S^{\\alpha \\beta }_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}},E}$ be a set of Floquet scattering amplitudes that obey the unitarity conditions (REF ) and $f^\\alpha _E$ the Fermi function ().",
"Then, for $\\hat{S}^{\\alpha \\beta }_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}},E}\\equiv (1-\\delta _{n0}\\delta _{\\alpha \\beta })S^{\\alpha \\beta }_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}},E}$ and $f^{\\prime \\alpha }_E\\equiv f^\\alpha _E(1-f^\\alpha _E)$ , we have $\\Psi ^{\\rho \\rho }_\\alpha \\equiv \\frac{1}{h}\\int _0^\\infty \\!\\!\\!",
"dE\\sum \\nolimits _\\beta \\sum \\nolimits _n\\bigl | \\hat{S}^{\\alpha \\beta }_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}},E} \\bigr |^2(f^\\alpha _{E\\raisebox {-0.7pt}{{{\\scriptsize n}}}}-f^\\beta _E)^2\\le \\frac{\\sigma }{2}$ and $\\Omega ^{\\rho \\rho }_\\alpha \\equiv \\frac{2}{h}\\int _0^\\infty \\!\\!\\!",
"dE\\sum \\nolimits _n\\bigl | \\hat{S}^{\\alpha \\alpha }_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}},E} \\bigr |^2f^{\\prime \\alpha }_E\\le \\frac{2T_\\alpha }{h},$ where $\\sigma $ is defined in Eq.",
"(REF ).",
"For Eq.",
"(REF ), observe that $2\\Psi ^{\\rho \\rho }_\\alpha &\\le \\frac{2}{h}\\int _0^\\infty \\!\\!\\!",
"dE\\sum \\nolimits _{\\alpha \\beta }\\sum \\nolimits _n\\bigl | \\hat{S}^{\\alpha \\beta }_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}},E} \\bigr |^2(f^\\alpha _{E\\raisebox {-0.7pt}{{{\\scriptsize n}}}}-f^\\beta _E)^2\\\\&= \\frac{2}{h}\\int _0^\\infty \\!\\!\\!",
"dE\\sum \\nolimits _{\\alpha \\beta }\\sum \\nolimits _n\\bigl | S^{\\alpha \\beta }_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}},E} \\bigr |^2(f^\\alpha _{E\\raisebox {-0.7pt}{{{\\scriptsize n}}}}-f^\\beta _E)^2\\nonumber $ and use Lemma REF .",
"For Eq.",
"(REF ), note that $f^{\\prime \\alpha }_E\\ge 0$ and, by the unitarity conditions (REF ), $\\sum _n\\bigl | \\hat{S}^{\\alpha \\alpha }_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}},E} \\bigr |^2\\le \\sum \\nolimits _\\beta \\sum \\nolimits _n\\bigl | S^{\\beta \\alpha }_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}},E} \\bigr |^2=1$ such that $\\Omega ^{\\rho \\rho }_\\alpha \\le \\frac{2}{h}\\int _0^\\infty \\!\\!\\!",
"dE\\; f^{\\prime }_E = \\frac{2T_\\alpha }{h}\\frac{\\varphi _\\alpha }{1+\\varphi _\\alpha }\\le \\frac{2T_\\alpha }{h},$ where $\\varphi _\\alpha \\equiv \\exp [\\mu _\\alpha /T_\\alpha ]$ ."
],
[
"Thermodynamic Uncertainty Relation in Linear Response",
"In this appendix, we show that the thermodynamic uncertainty relation $\\frac{\\sigma P^{xx}_{\\alpha \\alpha }}{(J^x_\\alpha -\\omega \\partial _\\omega J^x_\\alpha )^2}\\ge 2,$ which was derived in Ref.",
"[98] for periodically driven Markov jump processes, holds for coherent transport in linear response if the frozen scattering amplitudes obey the symmetry $\\mathcal {S}^{\\alpha \\beta }_{E,\\mathbf {V}}=\\mathcal {S}^{\\beta \\alpha }_{E,\\mathbf {V}}.$ We first recall Sec.",
"REF and note that the symmetry (REF ) implies $L^{x\\omega }_\\alpha =-L^{\\omega x}_\\alpha $ .",
"As a result, the total rate of entropy production (REF ) can be divided into two contributions, $\\sigma = \\sigma _{{{\\rm th}}}+\\sigma _{{{\\rm ac}}}$ with $\\sigma _{{{\\rm th}}} &\\equiv \\sum \\nolimits _{\\alpha \\beta }\\sum \\nolimits _{xy}L^{xy}_{\\alpha \\beta } F^x_\\alpha F^y_\\beta \\\\&= \\frac{1}{2h}\\int _0^\\infty \\!\\!\\!",
"dE\\sum \\nolimits _{\\alpha \\beta }\\bigl \\bigl | \\hat{\\mathcal {S}}_{E,\\mathbf {V}}^{\\alpha \\beta } \\bigr |^2\\bigr \\bigl (Y^{\\alpha \\beta }_E\\bigr )^2 f^{\\prime }_E\\ge 0\\quad \\text{and}\\nonumber \\\\[3pt]\\sigma _{{{\\rm ac}}} &\\equiv L^{\\omega \\omega } (F^\\omega )^2 \\ge 0,$ where $\\hat{\\mathcal {S}}^{\\alpha \\beta }_{E,\\mathbf {V}}\\equiv (1-\\delta _{\\alpha \\beta })\\mathcal {S}^{\\alpha \\beta }_{E,\\mathbf {V}}$ , $Y^{\\alpha \\beta }_E\\equiv \\sum _x \\zeta ^x_E (F^x_\\alpha -F^x_\\beta )$ and the second line in Eq.",
"(REF ) follows by using the unitarity conditions (REF ).",
"We now consider the quadratic form $\\Xi ^x_\\alpha &\\equiv \\sigma _{{{\\rm th}}}+2G\\sum \\nolimits _\\beta \\sum \\nolimits _yL^{xy}_{\\alpha \\beta }F^y_\\beta + G^2L^{xx}_{\\alpha \\alpha }\\\\&= \\frac{1}{2h}\\int _0^\\infty \\!\\!\\!",
"dE\\sum \\nolimits _{\\beta \\ne \\alpha }\\Bigl \\lbrace \\begin{aligned}[t] &\\sum \\nolimits _{\\gamma \\ne \\alpha }\\bigl \\bigl | \\hat{\\mathcal {S}}^{\\beta \\gamma }_{E,\\mathbf {V}} \\bigr |^2\\bigr \\bigl (Y^{\\alpha \\beta }_E\\bigr )^2\\\\&+2\\bigl \\bigl | \\hat{\\mathcal {S}}^{\\alpha \\beta }_{E,\\mathbf {V}} \\bigr |^2\\bigr \\bigl (\\zeta ^x_E G +Y^{\\alpha \\beta }_E\\bigr )^2\\Bigr \\rbrace f^{\\prime }_E,\\end{aligned}\\nonumber $ where $G$ is real and otherwise arbitrary.",
"The second line in Eq.",
"(REF ), which follows from the unitarity conditions (REF ) and the symmetry (REF ), proves that $\\Xi ^x_\\alpha $ is positive semi-definite.",
"Next, recalling the kinetic equations () and the fluctuation-dissipation theorem (REF ), yields $\\sum \\nolimits _\\beta \\sum \\nolimits _yL^{xy}_{\\alpha \\beta } F^y_\\beta = J^x_\\alpha - \\omega \\partial _\\omega J^x_\\alpha ,\\quad L^{xx}_{\\alpha \\alpha } = P^{xx}_{\\alpha \\alpha }|_{{{\\rm eq}}}/2.$ Inserting these identities into the definition of $\\Xi ^x_\\alpha $ and noting that $\\sigma _{{{\\rm th}}}\\le \\sigma $ since $\\sigma _{{{\\rm ac}}}\\ge 0$ shows that $\\sigma +2G(J^x_\\alpha - \\omega \\partial _\\omega J^x_\\alpha )+G^2 P^{xx}_{\\alpha \\alpha }|_{{{\\rm eq}}}/2\\ge \\Xi ^x_\\alpha \\ge 0$ for any $G$ .",
"Minimizing the left-hand side of this inequality with respect to $G$ finally gives the thermodynamic uncertainty relation (REF )."
],
[
"Quantum Generator",
"In this appendix, we derive the exact Floquet scattering amplitudes for the quantum generator discussed in Secs.",
"REF and REF by solving the corresponding Floquet-Schrödinger equation in position representation [8].",
"We further provide explicit formulas for the quantities entering the relations (REF ) and (REF )."
],
[
"Scattering Amplitudes",
"The system is parameterized according to Fig.",
"REF .",
"An incoming carrier with energy $E>0$ in the terminal $\\alpha $ is described by the scattering state $|\\phi ^\\alpha _{E,t}\\rangle $ .",
"On the leads, the wave function of this state is given by $\\phi ^\\alpha _{E,t}[r_1]&= \\delta _{\\alpha 1}w^-_E[r_1]+\\sum \\nolimits _nC^{1\\alpha }_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}},E}w^+_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}}}[r_1] e^{-in\\omega t},\\\\[3pt]\\phi ^\\alpha _{E,t}[r_2]&= \\delta _{\\alpha 2}w^-_E[r_2]+\\sum \\nolimits _nC^{2\\alpha }_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}},E}w^+_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}}}[r_2] e^{-i n\\omega t},$ where the summations run over all integers, the $C^{\\alpha \\beta }_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}},E}$ are yet undetermined complex coefficients and $w^\\pm _E[r]\\equiv w_E \\exp [\\pm i k_E r]$ denotes a normalized incoming $(-)$ or outgoing $(+)$ plane wave with $w_E\\equiv \\sqrt{M/2\\pi k_E \\hbar ^2}$ and $k_E\\equiv \\sqrt{2ME/\\hbar ^2}$ [8]; we recall that $M$ denotes the carrier mass.",
"On the loop, the scattering wave function obeys the Floquet-Schrödinger equation $E\\phi ^\\alpha _{E,t}[\\rho ]=-\\frac{\\hbar ^2}{2M}\\bigl (\\partial _\\rho -i\\omega t/l\\bigr )^2\\phi ^\\alpha _{E,t}[\\rho ]-i\\hbar \\partial _t\\phi ^\\alpha _{E,t}[\\rho ],$ with respect to the boundary conditions $&\\phi ^1_{E,t}[\\rho ]\\big |_{\\rho =0} = w_E+\\sum \\nolimits _nC^{21}_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}},E}w_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}}} e^{-i n\\omega t}\\\\[3pt]&\\phi ^1_{E,t}[\\rho ]\\big |_{\\rho =l} =\\sum \\nolimits _nC^{11}_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}},E} w_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}}} e^{-i n\\omega t},\\\\[3pt]&\\hat{P}_t\\phi ^1_{E,t}[\\rho ]\\big |_{\\rho =0}= w^{\\prime }_E-\\sum \\nolimits _nC^{21}_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}},E} w^{\\prime }_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}}} e^{-in\\omega t},\\\\[3pt]&\\hat{P}_t\\phi ^1_{E,t}[\\rho ]\\big |_{\\rho =l}= \\sum \\nolimits _nC^{11}_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}},E} w^{\\prime }_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}}} e^{-in\\omega t}$ and $&\\phi ^2_{E,t}[\\rho ]\\big |_{\\rho =0} =\\sum \\nolimits _nC^{22}_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}},E} w_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}}} e^{-in\\omega t},\\\\[3pt]&\\phi ^2_{E,t}[\\rho ]\\big |_{\\rho =l} = w_E+\\sum \\nolimits _nC^{12}_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}},E} w_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}}} e^{-in\\omega t}\\\\[3pt]&\\hat{P}_t\\phi ^2_{E,t}[\\rho ]\\big |_{\\rho =0} = -\\sum \\nolimits _nC^{22}_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}},E} w^{\\prime }_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}}} e^{-in\\omega t},\\\\[3pt]&\\hat{P}_t\\phi ^2_{E,t}[\\rho ]\\big |_{\\rho =l}= -w^{\\prime }_E + \\sum \\nolimits _nC^{12}_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}},E} w^{\\prime }_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}}} e^{-in\\omega t},$ where $w^{\\prime }_E\\equiv w_E k_E$ and $\\hat{P}_t\\equiv -i\\partial _\\rho -\\omega t/l$ .",
"Here, we assume that the magnetic flux $\\Phi $ increases linearly in time, that is, $\\Phi _t = \\hbar \\omega c t/e$ , where $c$ denotes the speed of light and $e$ the carrier charge.",
"The parameter $E$ corresponds to the Floquet energy on the loop.",
"The boundary conditions (REF ) and (REF ) are determined by the beam splitter that connects the loop with the leads To obtain boundary conditions (REF ) and (REF ), we require that both $\\phi ^\\alpha _{E,t}$ and $p\\phi ^\\alpha _{E,t}$ are continuous at the beam splitter, where the second condition ensures the continuity of probability currents [109].",
"We recall that the momentum operator has the position representation $p-i\\hbar \\partial _{r_\\alpha }$ on the leads and $p-i\\hbar \\partial _\\rho -\\hbar \\omega t/l=\\hat{P}_t/\\hbar $ on the loop.",
"The additional factor $(-1)$ on the right-hand sides of the Eqs.",
"(REF ) and (REF ) accounts for the fact that the parameterizations of the leads and the loop run in opposite directions at the connections $\\rho =0\\rightarrow r_1=0$ and $\\rho =0\\rightarrow r_2=0$ .. A general solution of Eq.",
"(REF ) that is compatible with the boundary conditions (REF ) and (REF ) is given by $\\phi ^\\alpha _{E,t}[\\rho ]= \\sum \\nolimits _n\\bigl (a^\\alpha _n {{{\\rm Ai}}}_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}}}[\\rho ]+b^\\alpha _n {{{\\rm Bi}}}_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}}}[\\rho ]\\bigr )e^{i(\\rho /l-n)\\omega t} ,$ where $a^\\alpha _n$ and $b^\\alpha _n$ are arbitrary complex coefficients.",
"The modified Airy functions are thereby defined as ${{{\\rm Xi}}}_{E}[\\rho ] \\equiv {{{\\rm Xi}}}\\bigl [(\\hbar \\omega /\\mathcal {E})^\\frac{1}{3}(\\rho /l-E/\\hbar \\omega )\\bigr ]$ in terms of the standard Airy functions ${{{\\rm Xi}}}\\equiv {{{\\rm Ai}}},{{{\\rm Bi}}}$ [108].",
"The parameter $\\mathcal {E}\\equiv \\hbar ^2/2M l^2$ sets the natural energy scale of the system.",
"Inserting the solution (REF ) into the boundary conditions (REF ) and (REF ) and collecting Fourier components yields two sets of linear equations, which can be written compactly as $&\\mathbb {A}_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}}}\\mathbf {a}_n^1 = \\delta _{n0}\\mathbf {1} + C^{21}_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}},E}\\hat{\\mathbf {1}},\\\\[3pt]&\\mathbb {A}_{E\\raisebox {-0.7pt}{{{\\scriptsize n-1}}}}\\mathbf {a}_n^1=C^{11}_{E\\raisebox {-0.7pt}{{{\\scriptsize n-1}}},E}\\mathbf {1}$ and $& \\mathbb {A}_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}}}\\mathbf {a}^2_n=C^{22}_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}},E}\\hat{\\mathbf {1}},\\\\[3pt]& \\mathbb {A}_{E\\raisebox {-0.7pt}{{{\\scriptsize n-1}}}}\\mathbf {a}^2_n= \\delta _{n1}\\hat{\\mathbf {1}} + C^{12}_{E\\raisebox {-0.7pt}{{{\\scriptsize n-1}}},E}\\mathbf {1}.$ Here, we have used the relation ${{{\\rm Xi}}}_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}}}[l]={{{\\rm Xi}}}_{E\\raisebox {-0.7pt}{{{\\scriptsize n-1}}}}[0]$ .",
"Furthermore, we have introduced the vectors $\\mathbf {a}^\\alpha _n\\equiv (a^\\alpha _n,b^\\alpha _n)^{{{\\rm t}}}$ , $\\mathbf {1}\\equiv (1,1)^{{{\\rm t}}}$ , $\\hat{\\mathbf {1}}\\equiv (1,-1)^{{{\\rm t}}}$ and the matrix $\\mathbb {A}_E\\equiv \\frac{1}{w_E}\\left(\\!\\begin{array}{cc}{{{\\rm Ai}}}_E[0] & {{{\\rm Bi}}}_E[0]\\\\[3pt]-i{{{\\rm Ai}}}^{\\prime }_E[0] & -i{{{\\rm Bi}}}^{\\prime }_E[0]\\end{array}\\!\\right),$ where ${{{\\rm Xi}}}^{\\prime }_E[\\rho ]& \\equiv \\partial _\\rho {{{\\rm Xi}}}_E[\\rho ]/k_E\\\\&=(\\hbar \\omega /\\mathcal {E})^{\\frac{1}{3}}(\\mathcal {E}/E)^{\\frac{1}{2}}{{{\\rm Xi}}}^{\\prime }\\bigl [(\\hbar \\omega /\\mathcal {E})^{\\frac{1}{3}}(\\rho /l-E/\\hbar \\omega )\\bigr ]\\nonumber $ with ${{{\\rm Xi}}}^{\\prime }\\equiv {{{\\rm Ai}}}^{\\prime },{{{\\rm Bi}}}^{\\prime }$ denoting the derivatives of the standard Airy functions [108].",
"Solving the linear systems (REF ) and (REF ) yields $& C^{11}_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}},E} = \\delta _{n(-1)}\\frac{2}{\\mathbf {1}^{{{{\\rm t}}}}\\mathbb {A}_E\\mathbb {A}^{-1}_{E\\raisebox {-0.7pt}{{{\\scriptsize -1}}}}\\mathbf {1}},\\\\[3pt]& C^{21}_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}},E} = \\delta _{n0} \\frac{\\hat{\\mathbf {1}}^{{{{\\rm t}}}}\\mathbb {A}_E\\mathbb {A}^{-1}_{E\\raisebox {-0.7pt}{{{\\scriptsize -1}}}}\\mathbf {1}}{\\mathbf {1}^{{{{\\rm t}}}}\\mathbb {A}_E\\mathbb {A}^{-1}_{E\\raisebox {-0.7pt}{{{\\scriptsize -1}}}}\\mathbf {1}},\\\\[3pt]& C^{22}_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}},E} = \\delta _{n1}\\frac{2}{\\hat{\\mathbf {1}}^{{{{\\rm t}}}}\\mathbb {A}_E\\mathbb {A}^{-1}_{E\\raisebox {-0.7pt}{{{\\scriptsize 1}}}}\\hat{\\mathbf {1}}},\\\\[3pt]& C^{12}_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}},E} = \\delta _{n0}\\frac{\\mathbf {1}^{{{{\\rm t}}}}\\mathbb {A}_E\\mathbb {A}^{-1}_{E\\raisebox {-0.7pt}{{{\\scriptsize 1}}}}\\hat{\\mathbf {1}}}{\\hat{\\mathbf {1}}^{{{{\\rm t}}}}\\mathbb {A}_E\\mathbb {A}^{-1}_{E\\raisebox {-0.7pt}{{{\\scriptsize 1}}}}\\hat{\\mathbf {1}}}.$ The Floquet scattering amplitudes can now be determined by inserting these expressions into the ansatz (REF ) and comparing the result with the asymptotic boundary conditions for the Floquet scattering wave functions [8], which are given by $&\\phi ^\\alpha _{E,t}[r_1]\\big |_{r_1\\rightarrow \\infty }= \\delta _{\\alpha 1}w^-_E[r_1]+\\sum \\nolimits _nS^{1\\alpha }_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}},E}w^+_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}}}[r_1] e^{-in\\omega t},\\\\[3pt]&\\phi ^\\alpha _{E,t}[r_2]\\big |_{r_2\\rightarrow \\infty }= \\delta _{\\alpha 2}w^-_E[r_2]+\\sum \\nolimits _nS^{2\\alpha }_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}},E}w^+_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}}}[r_2] e^{-i n\\omega t}.$ Upon observing that the outgoing plane waves become decaying exponentials for negative energies, that is, $w^+_E[r]\\propto \\exp [-k_{|E|}r]$ for $E\\le 0$ , we thus arrive at $& S^{11}_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}},E} =0, && S^{21}_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}},E} = C^{21}_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}},E}&& (E\\le \\hbar \\omega );\\\\& S^{11}_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}},E} = C^{11}_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}},E},&& S^{21}_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}},E} = C^{21}_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}},E}&& (E>\\hbar \\omega );\\\\& S^{22}_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}},E} = C^{22}_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}},E},&& S^{12}_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}},E} = C^{12}_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}},E}&& (E>0).$ This result leads to the compact expressions $\\bigl | S^{11}_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}},E} \\bigr |^2 &= \\delta _{n(-1)}\\Theta _{E\\raisebox {-0.7pt}{{{\\scriptsize -1}}}}R_{E\\raisebox {-0.7pt}{{{\\scriptsize -1}}}},\\\\[3pt]\\bigl | S^{21}_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}},E} \\bigr |^2 &= \\delta _{n0}+\\delta _{n0}\\Theta _{E\\raisebox {-0.7pt}{{{\\scriptsize -1}}}}(T_{E\\raisebox {-0.7pt}{{{\\scriptsize -1}}}}-1),\\\\[3pt]\\bigl | S^{22}_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}},E} \\bigr |^2 &= \\delta _{n1} R_{E},\\\\[3pt]\\bigl | S^{12}_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}},E} \\bigr |^2 &= \\delta _{n0} T_{E},$ for the reflection and transmission probabilities, which were used in Sec.",
"REF .",
"Here, $\\Theta $ denotes the Heaviside step function and the reflection and transmission functions are given by $& R_E\\equiv \\frac{4}{\\bigl | \\mathbf {1}^{{{\\rm t}}}\\mathbb {A}_{E\\raisebox {-0.7pt}{{{\\scriptsize 1}}}}\\mathbb {A}^{-1}_E\\mathbf {1} \\bigr |^2}= \\frac{4}{\\bigl | \\hat{\\mathbf {1}}^{{{\\rm t}}}\\mathbb {A}_E\\mathbb {A}^{-1}_{E\\raisebox {-0.7pt}{{{\\scriptsize 1}}}}\\hat{\\mathbf {1}} \\bigr |^2},\\\\[3pt]& T_E\\equiv \\frac{\\bigl | \\hat{\\mathbf {1}}^{{{\\rm t}}}\\mathbb {A}_{E\\raisebox {-0.7pt}{{{\\scriptsize 1}}}}\\mathbb {A}^{-1}_E\\mathbf {1} \\bigr |^2}{\\bigl | \\mathbf {1}^{{{\\rm t}}}\\mathbb {A}_{E\\raisebox {-0.7pt}{{{\\scriptsize 1}}}}\\mathbb {A}^{-1}_E\\mathbf {1} \\bigr |^2}=\\frac{\\bigl | \\mathbf {1}^{{{\\rm t}}}\\mathbb {A}_E\\mathbb {A}^{-1}_{E\\raisebox {-0.7pt}{{{\\scriptsize 1}}}}\\hat{\\mathbf {1}} \\bigr |^2}{\\bigl | \\hat{\\mathbf {1}}^{{{\\rm t}}}\\mathbb {A}_E\\mathbb {A}^{-1}_{E\\raisebox {-0.7pt}{{{\\scriptsize 1}}}}\\hat{\\mathbf {1}} \\bigr |^2}$ and obey the sum rule $T_E+R_E=1$ .",
"Using this relation, it is straightforward to verify that the Floquet scattering amplitudes (REF ) obey the unitarity conditions (REF ).",
"For illustration, the transmission function $T_E$ is plotted in Fig.",
"REF for different energies."
],
[
"Thermodynamic Quantities",
"Upon inserting the scattering amplitudes (REF ) into the general expressions (REF ), it is now straightforward to determine the mean values of the matter currents $J^\\rho _\\alpha $ and the heat currents $J^q_\\alpha = J^\\varepsilon _\\alpha -\\mu _\\alpha J^\\rho _E$ , which are given by $J^\\rho _1 &= - J^\\rho _2 = \\frac{1}{h}\\int _0^\\infty \\!\\!\\!",
"dE\\;\\Bigl \\lbrace f^1_E-R_E f^1_{E\\raisebox {-0.7pt}{{{\\scriptsize 1}}}}-T_E f^2_E\\Bigr \\rbrace ,\\\\[3pt]J^q_1 &=\\frac{1}{h}\\int _0^\\infty \\!\\!\\!",
"dE\\;\\zeta ^{q,1}_E\\Bigl \\lbrace f^1_E-R_E f^1_{E\\raisebox {-0.7pt}{{{\\scriptsize 1}}}}-T_E f^2_E\\Bigr \\rbrace ,\\\\[3pt]J^q_2 &= \\frac{1}{h}\\int _0^\\infty \\!\\!\\!",
"dE\\;\\Bigl \\lbrace \\zeta ^{q,2}_E(f^2_E-f^1_E)+\\zeta ^{q,2}_{E\\raisebox {-0.7pt}{{{\\scriptsize 1}}}}R_E (f^1_{E\\raisebox {-0.7pt}{{{\\scriptsize 1}}}}-f^2_E)\\Bigr \\rbrace ,$ Analogously, the expressions (REF ) yields the thermal fluctuations of the matter and heat currents, $D^{\\rho \\rho }_{\\alpha \\alpha }$ and $D^{qq}_{\\alpha \\alpha }= \\sum \\nolimits _{uv}(\\delta _{u\\varepsilon }-\\mu _\\alpha \\delta _{u\\rho })(\\delta _{v\\varepsilon }-\\mu _\\alpha \\delta _{v\\rho })D^{uv}_{\\alpha \\alpha }$ , $D^{\\rho \\rho }_{11} &= D^{\\rho \\rho }_{22}= \\frac{1}{h}\\int _0^\\infty \\!\\!\\!",
"dE\\; \\Bigl \\lbrace f^{\\prime 1}_E + T_E f^{\\prime 2}_E-R_E f^{\\prime 1}_{E\\raisebox {-0.7pt}{{{\\scriptsize 1}}}}\\Bigr \\rbrace ,\\\\[3pt]D^{qq}_{11} &= \\frac{1}{h}\\int _0^\\infty \\!\\!\\!",
"dE\\; \\zeta ^{q,1}_E\\Bigl \\lbrace \\begin{aligned}[t]& \\zeta ^{q,1}_E( f^{\\prime 1}_E + T_E f^{\\prime 2}_E - R_E f^{\\prime 1}_{E\\raisebox {-0.7pt}{{{\\scriptsize 1}}}})\\\\& -2\\hbar \\omega R_E f^{\\prime 1}_{E\\raisebox {-0.7pt}{{{\\scriptsize 1}}}}\\Bigr \\rbrace ,\\end{aligned}\\\\[3pt]D^{qq}_{22} &= \\frac{1}{h}\\int _0^\\infty \\!\\!\\!",
"dE\\; \\Bigl \\lbrace \\begin{aligned}[t]& (\\hbar \\omega )^2 R_E f^{\\prime 2}_E-(\\zeta ^{q,2}_{E\\raisebox {-0.7pt}{{{\\scriptsize 1}}}})^2 R_E f^{\\prime 1}_{E\\raisebox {-0.7pt}{{{\\scriptsize 1}}}}\\\\& +(\\zeta ^{q,2}_E)^2(f^{\\prime 1}_E + T_E f^{\\prime 2}_E)\\Bigr \\rbrace .\\end{aligned}$ Finally, the shot-noise contributions to the current fluctuations, $R^{\\rho \\rho }_{\\alpha \\alpha }$ and $R^{qq}_{\\alpha \\alpha }= \\sum \\nolimits _{uv}(\\delta _{u\\varepsilon }-\\mu _\\alpha \\delta _{u\\rho })(\\delta _{v\\varepsilon }-\\mu _\\alpha \\delta _{v\\rho })R^{uv}_{\\alpha \\alpha }$ are obtained from Eq.",
"() as $R^{\\rho \\rho }_{11}&= R^{\\rho \\rho }_{22} = \\frac{1}{h}\\int _0^\\infty \\!\\!\\!",
"dE\\;T_E R_E (f^1_{E\\raisebox {-0.7pt}{{{\\scriptsize 1}}}}-f^2_E)^2,\\\\[3pt]R^{qq}_{11} &= \\frac{1}{h}\\int _0^\\infty \\!\\!\\!",
"dE\\;(\\zeta ^{q,1}_E)^2 T_E R_E (f^1_{E\\raisebox {-0.7pt}{{{\\scriptsize 1}}}}-f^2_E)^2,\\\\[3pt]R^{qq}_{22} &= \\frac{1}{h}\\int _0^\\infty \\!\\!\\!",
"dE\\;(\\zeta ^{q,2}_{E\\raisebox {-0.7pt}{{{\\scriptsize 1}}}})^2 T_E R_E (f^1_{E\\raisebox {-0.7pt}{{{\\scriptsize 1}}}}-f^2_E)^2.$ In order to evaluate the coefficient () for $x=\\rho $ and $\\alpha =1$ , we further need the mean currents $\\tilde{J}^x_\\alpha $ as well as the thermal and shot-noise fluctuations $\\tilde{D}^{\\rho \\rho }_{11}$ and $\\tilde{R}^{\\rho \\rho }_{11}$ for the time-reversed system.",
"These quantities, which are found by replacing $S^{\\alpha \\beta }_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}},E}$ with $\\mathsf {T}_{\\mathbf {B}}\\mathsf {T}_{\\mathbf {V}}S^{\\alpha \\beta }_{E\\raisebox {-0.7pt}{{{\\scriptsize n}}},E} =S^{\\beta \\alpha }_{E,E\\raisebox {-0.7pt}{{{\\scriptsize n}}}}$ in Eqs.",
"(REF ) and (REF ), are given by $\\tilde{J}^{\\rho }_1 &= -\\tilde{J}^{\\rho }_2 = \\frac{1}{h}\\int _0^\\infty \\!\\!\\!",
"dE\\;\\Bigl \\lbrace T_E f^1_E +R_E f^2_{E\\raisebox {-0.7pt}{{{\\scriptsize 1}}}}- f^2_E\\Bigr \\rbrace ,\\\\[3pt]\\tilde{J}^{q}_1 &= \\frac{1}{h}\\int _0^\\infty \\!\\!\\!",
"dE\\;\\Bigl \\lbrace \\zeta ^{q,1}_E (f^1_E-f^2_E)+ \\zeta ^{q,1}_{E\\raisebox {-0.7pt}{{{\\scriptsize 1}}}} R_E (f^2_{E\\raisebox {-0.7pt}{{{\\scriptsize 1}}}}-f^1_E)\\Bigr \\rbrace ,\\\\\\tilde{J}^{q}_2 &= \\frac{1}{h}\\int _0^\\infty \\!\\!\\!",
"dE\\; \\zeta ^{q,2}_E\\Bigl \\lbrace f^2_E - R_Ef^2_{E\\raisebox {-0.7pt}{{{\\scriptsize 1}}}} -T_E f^1_E\\Bigr \\rbrace $ and $\\tilde{D}^{\\rho \\rho }_{11} &= \\frac{1}{h}\\int _0^\\infty \\!\\!\\!",
"dE\\;\\Bigl \\lbrace f^{\\prime 2}_E+ T_Ef^{\\prime 1}_E -R_E f^{\\prime 2}_{E\\raisebox {-0.7pt}{{{\\scriptsize 1}}}}\\Bigr \\rbrace ,\\\\[3pt]\\tilde{R}^{\\rho \\rho }_{11} &= \\frac{1}{h}\\int _0^\\infty \\!\\!\\!",
"dE\\;T_E R_E (f^1_E -f^2_{E\\raisebox {-0.7pt}{{{\\scriptsize 1}}}})^2.$"
]
] | 1906.04297 |
[
[
"Recent progress on the Dirichlet problem for the minimal surface system\n and minimal cones"
],
[
"Abstract This is a very brief report on recent developments on the Dirichlet problem for the minimal surface system and minimal cones in Euclidean spaces.",
"We shall mainly focus on two directions: (1) Further systematic developments after Lawson-Osserman's paper \\cite{l-o} on the Dirichlet problem for minimal graphs of high codimensions.",
"Aspects including non-existence, non-uniqueness and irregularity properties of solutions have been explored from different points of view.",
"(2) Complexities and varieties of area-minimizing cones in high codimensions.",
"We shall mention interesting history and exhibit some recent results which successfully furnished new families of minimizing cones of different types."
],
[
"Introduction",
"Introduction"
],
[
"Plateau problem",
"The problem is to consider minimal surfaces spanning a given contour.",
"Roughly speaking, there are two kinds depending on desired minimality.",
"One is the “minimizing\" setting for finding global minimizers for area functionals under various boundary or topological constraints; while the other is the “minimal\" setting for critical points.",
"Stories of the problem can trace back to J.-L. Lagrange who, in 1768, considered graphs with minimal area over some domain $D$ of $\\mathbb {R}^2$ .",
"A necessary condition is the Euler-Lagrange equation, for $z=z(x,y)$ , $(1+z_y^2)z_{xx}-2z_xz_yz_{xy}+(1+z_x^2)z_{yy}=0.$ From then on, the theory of minimal surfaces (with vanishing mean curvature) soon launched an adventure journey.",
"Lots of great mathematicians, including Monge, J. Meusnie, A.-M. Legendre, S. Poisson, H. Scherk, E. Catalan, O. Bonnet, H. Schwarz, S. Lie and many others, entered this filed and made it flourishing for more than one century.",
"During the mathematical developments, Belgian physicist J. Plateau did a good number of intriguing experiments with soap films (not merely using wires) and in [30] gained some explanation about the phenomena of stability and instability, i.e., whether or not small deformations of the film can decrease area.",
"By laws of surface tension, an observable soap film bounded by a given simple closed curve is stable minimal.",
"Thus Plateau provided physical solutions to the question in $\\mathbb {R}^3$ in the minimal setting, and the problem was named after Plateau since then.",
"However, it took more time for rigorous mathematical arguments.",
"In 1930, J. Douglas [10] and T. Radó [31] affirmatively answered the problem in $\\mathbb {R}^3$ , respectively, in the minimizing setting.",
"General cases were subsequently studied and a big portion were solved due to Federer and Fleming's celebrated compactness theorems of normal currents and integral currents [12] in expanded territories."
],
[
"Dirichlet problem for minimal graph of condimension one",
"It can be seen that Plateau problem (for minimal surfaces with given simple closed boundary curves) is actually beyond the scope of Lagrange's original question.",
"If $D$ is a bounded domain of $\\mathbb {R}^{n+1}$ with $C^2$ boundary and $\\phi :\\partial D\\rightarrow {\\mathbb {R}}^{1}$ , then the Dirichlet problem for minimal surface equation is asking for solution $f: D\\rightarrow {\\mathbb {R}}^{1}$ to satisfy following generalization of (REF ) $(1+|\\nabla f|^2)\\triangle f- \\sum _{i,j=1}^{n+1}f_i f_j f_{ij}=0$ and $f|_{\\partial D}=\\phi $ .",
"Hence the Dirichlet problem can be regarded as a special kind of Plateau problem, which searches for graph solutions for graph boundary data.",
"For $n+1=2$ and convex $D$ , the Dirichlet problem is solvable for any continuous boundary data, see [32].",
"In general situation, by efforts of Jenkins-Serrin [18] and later Bombieri-De Giorgi-Miranda [4], the Dirichlet problem turns out to be well posed (i.e., having a unique solution) for any continuous boundary function if and only if $\\partial D$ is everywhere mean convex.",
"Moreover, if solution exists, it must be $C^\\omega $ due to de Giorgi [7] (also see [40] and [26]); and its graph is absolutely area-minimizing (see [11]), which means any competitor sharing the same boundary possesses larger volume.",
"Dirichlet problem for minimal graphs of high codimensions will be discussed in §."
],
[
"Bernstein problem",
"In his paper [2], Bernstein showed that every solution to (REF ) for $n=1$ or (REF ) in the entire $\\mathbb {R}^2$ (with no boundary requirement at infinity) has to be affine.",
"Fleming [14] suggested a new idea for this problem which also works for $n\\ge 2$ via De Giorgi's improvement [8].",
"The principle states that the existence of a non-affine solution over $\\mathbb {R}^{n+1}$ implies the existence of a non-planar area-minimizing hypercone in $\\mathbb {R}^n$ .",
"Almgren [1] followed this line and gained the same conclusion for $n=2$ .",
"In [37] J. Simons greatly extended the results by showing no non-planar stable hypercones in $\\mathbb {R}^{n+1}$ for $n\\le 6$ .",
"In $\\mathbb {R}^8$ , he discovered stable minimal hypercones $C_{k,k}=C\\left(S^{k}\\left(\\sqrt{\\frac{1}{2}}\\right)\\times S^{k}\\left(\\sqrt{\\frac{1}{2}}\\right)\\right)\\subset \\mathbb {R}^{2(k+1)}\\ \\text{ when } k\\ge 3.$ Here, for a set $E$ in unit sphere, the cone over $E$ is defined to be $C(E):=\\lbrace tx:x\\in E,\\ t\\in (0,\\infty )\\rbrace $ .",
"Then he naturally raised the question whether $C_{k,k}$ for $k\\ge 3$ in $\\mathbb {R}^{2(k+1)}$ , nowadays called Simons cones, are area-minimizing.",
"Immediately, the celebrated article [3] by Bombieri-De Giorgi-Giusti confirmed that all Simons cones are area-minimizing and constructed a non-planar minimal graph over $\\mathbb {R}^8$ in $\\mathbb {R}^9$ which has $C_{3,3}\\times \\mathbb {R}$ as its tangent cone at infinity.",
"As a result, the yes-no answer to the Bernstein problem got complete: there exist no non-planar minimal graphs over $\\mathbb {R}^{n+1}$ in $\\mathbb {R}^{n+2}$ when $n\\le 6$ ; but there are such creatures when $n\\ge 7$ .",
"Still, lots of interesting subtle behaviors are mysterious to us, such as what types of entire minimal graphs can occur?",
"Right after [3], H. B. Lawson, Jr. considered equivariant plateau problems in [20] and obtained almost all homogeneous area-minimizing hypercones (see [49]).",
"P. Simoes [34], [35] added that $C_{2,4}$ is also minimizing.",
"R. Hardt and L. Simon [15] discovered characterization foliations for area-minimizing hypercones.",
"D. Ferus and H. Karcher [13] showed, by constructing characterization foliation, that every cone over the minimal isoparametric hypersurface of an inhomogeneous isoparametric foliation on a sphere is area-minimizing.",
"G. Lawlor [19] completed the classification of all homogeneous area-minimizing hypercones.",
"Hence one can get a classification of all isoparametric homogeneous area-minimizing hypercones accordingly.",
"Actually, for each $C$ of these minimizing hypercones, L. Simon [36] gave a beautiful construction of minimal graph with tangent cone $C\\times \\mathbb {R}$ at infinity, thus creating a huge variety of solutions to the Bernstein problem."
],
[
"Dirichlet problem for minimal surfaces of high codimensions",
"Dirichlet problem for minimal surfaces of high codimensions Given an open bounded, strictly convex $\\Omega \\subset \\mathbb {R}^{n+1}$ and $\\phi :\\partial \\Omega \\rightarrow {\\mathbb {R}}^{m+1}$ , the Dirichlet problem (cf.",
"[18], [4], [7], [26], [21]) searches for weak solutions $f\\in C^0(\\bar{\\Omega })\\cap Lip(\\Omega )$ such that $\\left\\lbrace \\begin{array}{cc}\\sum \\limits _{i=1}^{n+1}\\frac{\\partial }{\\partial x^i}(\\sqrt{g}g^{ij})=0, & j=1,\\cdots ,n+1,\\\\\\sum \\limits _{i,j=1}^{n+1}\\frac{\\partial }{\\partial x^i}(\\sqrt{g}g^{ij}\\frac{\\partial f^\\alpha }{\\partial x^j})=0, & \\alpha =1,\\cdots ,m+1,\\end{array}\\right.$ where $g_{ij}=\\delta _{ij}+\\sum \\limits _{\\alpha =1}^{m+1}\\frac{\\partial f^\\alpha }{\\partial x^i}\\frac{\\partial f^\\alpha }{\\partial x^j}$ , $(g^{ij})=(g_{ij})^{-1}$ and $g=\\det (g_{ij})$ , and further $f|_{\\partial \\Omega }=\\phi .$ Note that $F(x)\\mapsto (x, f(x))$ being harmonic (i.e., (REF )) is equivalent to its (or its graph) being minimal with respect to the induced metric from Euclidean space.",
"When $m=0$ , (REF ) can be reduced to the classical (REF ) to which many literatures were devoted as mentioned in §.",
"In this section we shall talk about the case of $m\\ge 1$ .",
"An astonishing pioneering work was done by Lawson and Osserman in [21], in which $\\Omega $ is always assumed to be a unit disk $\\mathbb {D}^{n+1}$ .",
"In particular, they exhibited the following remarkable differences.",
"(1) For $n=1$ , $m\\ge 1$ , real analytic boundary data can be found so that there exist at least three different analytic solutions to the Dirichlet problem.",
"Moreover, one of them has unstable minimal graph.",
"(2) For $n\\ge 3$ and $n-1\\ge m\\ge 2$ , the problem is in general not solvable.",
"A non-existence theorem is that, for each $C^2$ map $\\eta :S^{n}\\rightarrow S^{m}$ that is not homotopic to zero under the dimension assumption, there exists a positive constant $R_\\eta $ depending only on $\\eta $ , such that the problem is unsolvable for the boundary data $\\phi =R\\cdot \\eta $ , where $R$ is a (vertical rescaling) constant no less than $R_\\eta $ .",
"(3) For certain boundary data, there exists a Lipschitz solution to the Dirichlet problem which is not $C^1$ .",
"The ideas are briefly summarized as follows.",
"(1) is based on a classical result by Radó for $n=1$ case, which says that every solution to the Plateau problem for boundary data given by a graph over boundary of a convex domain in some 2-dimensional plane has to be a graph over that domain.",
"In fact, Lawson and Osserman were able to construct an action invariant boundary data of graph type for a $Z_4$ -action in the total ambient Euclidean space $\\mathbb {R}^{3+m}$ (for $m\\ge 1$ ), such that under action of a generator of this $Z_4$ -action each geometric solution to the Plateau problem (in the minimizing setting) cannot be fixed.",
"Namely, one gains two distinct geometric solutions to that boundary, and therefore, according to Radó, two essentially different solutions to the corresponding Dirichlet problem.",
"Then by [25] and [33] there exists an unstable minimal solution of min-max type to the same boundary.",
"Such boundary condition violates the uniqueness of solution and the minimizing property of solution graph.",
"In particular, they constructed boundary supports at least three analytic solutions to the Dirichlet problem.",
"It seems that more than 3 solutions may be created for certain boundary data, if one considered symmetry by a discrete action of higher order group and actions of the entire group in some more subtle way.",
"(2) is due to a nice special volume expression and the well-known density monotonicity for minimal varieties in Euclidean space.",
"The proof of this meaningful result was achieved through a contradiction argument.",
"Roughly speaking, the former can provide an upper bound for volume of graphs of solutions (as long as existed); while the latter guarantee a lower bound.",
"Combined with the dimension assumption, these two bounds together lead to a contradiction when the rescaling factor becomes big.",
"However, it is still completely mysterious and quite challenging to us how to figure out the exact maximal value of stretching factor with existence of solution(s).",
"(3) is stimulated by (2).",
"After establishing the non-existence result (2), Lawson-Osserman realized that, for a map satisfying both the dimension and homotopy conditions, if one rescaled the vertical stretching factor by a tiny number, then Dirichlet problem is solvable due to the Implicit Functional Theorem, e.g.",
"see [28]; however if by a quite large number, then no Lipschitz solution can ever exist to the rescaled boundary functions.",
"So a natural philosophy by Lawson-Osserman states that there should exist $R_0$ such that the boundary condition $R_0\\cdot \\eta $ supports a singular solution.",
"For first concrete examples of such kind, they considered the three noted Hopf maps between unit spheres.",
"Expressed in complex coordinates $\\eta (z_1,z_2)=(|z_1|^2-|z_2|^2,2z_1\\bar{z}_2)$ is the first.",
"They looked for a minimal cone $C=C(\\text{graph of }\\phi )$ over graph $\\phi =R_0\\cdot \\eta $ .",
"Therefore, if existed, $C$ is also a graph with a link of “spherical graph\" type $L:=C\\bigcap S^6=\\left\\lbrace (\\alpha x,\\sqrt{1-\\alpha ^2}\\eta (x):x\\in S^3\\right\\rbrace .$ Since a cone is minimal if and only if its link is a minimal variety in the unit sphere, it only needs to determine when $L$ is minimal.",
"If one uses quaternions, then, isometrically up to a sign, $\\eta (q)=qi\\bar{q}$ for $q$ of unit length of $\\mathbb {H}$ into pure imaginary part of $\\mathbb {H}$ , and $L$ can be viewed as an orbit through $((1,0,0,0), i)$ under action $Sp(1)\\cong S^3$ with $q\\cdot (\\alpha x,\\sqrt{1-\\alpha ^2}\\eta (x))=(\\alpha qx,\\sqrt{1-\\alpha ^2}q\\eta (x)\\bar{q})$ .",
"As a result, the orbit of maximal volume, corresponding to $\\alpha ={\\frac{2}{3}}$ , is minimal in $S^6$ .",
"Hence slope $R_0$ can take value $\\frac{\\sqrt{1-\\alpha ^2}}{\\alpha }=\\frac{\\sqrt{5}}{2}$ .",
"Similar procedures can be done for the other two Hopf maps.",
"Inspired by the above, in recent joint work [46], we attacked the question directly by generalizing (REF ).",
"We introduced Definition 2.1 A $C^2$ map $\\eta : S^{n}\\rightarrow S^{m}$ is called an Lawson-Osserman map (LOM) if there exists $\\theta \\in (0,\\frac{\\pi }{2})$ , s.t.",
"$F(x) :=(\\cos \\theta \\cdot x,\\sin \\theta \\cdot \\eta (x))$ gives a mininmal submanifold in $S^{m+n+1}$ .",
"The cone C(Image(F)) is called associated Lawson-Osserman cone (LOC).",
"Remark 2.2 For $\\phi =\\tan \\theta \\cdot \\eta $ , $C(Graph(\\phi ))=C(Image(F))$ is a mininmal graph.",
"So there is a singular solution given by $f(x)={\\left\\lbrace \\begin{array}{ll}|x|\\cdot \\tan \\theta \\cdot \\eta (\\frac{x}{|x|}), & x\\ne 0; \\\\0, & x=0.\\end{array}\\right.",
"}$ By Remark REF it is clear that each Lawson-Osserman map induces a boundary function $\\phi $ which supports a cone-type singular solution.",
"Then how many LOMs?",
"In [46] we give a characterization.",
"Theorem 2.3 A $C^2$ map $\\eta :\\ S^{n}{\\rightarrow } S^{m}$ is LOM if and only if the followings hold for standard metrics $g_{m+n+1}, g_m, g_n$ of unit spheres ${\\left\\lbrace \\begin{array}{ll}\\eta :(S^n,F^*g_{m+n+1})\\rightarrow (S^m,g_m)\\text{ is harmonic};\\\\\\sum _{i=1}^n\\dfrac{1}{\\cos ^2\\theta +\\lambda _i^2\\sin ^2\\theta }=n,\\text{ where }\\lambda _i^2 \\text{ are diagonals of }\\eta ^*g_{m+n+1} \\text{ to } g_n.\\end{array}\\right.",
"}$ In order to better understand the second condition in (REF ), we put a strong restriction.",
"Definition 2.4 $\\eta $ is called an LOMSE, if it is an LOM and in addition, for each $x\\in S^n$ , all nonzero singular values of $(\\eta _*)_x$ are equal, i.e., $\\lbrace \\lambda _1,\\cdots ,\\lambda _n\\rbrace =\\lbrace 0,\\lambda \\rbrace .$ Remark 2.5 Let $p,\\, n-p$ be the multiplicities for $\\lambda $ and 0.",
"Then the second in (REF ) becomes $\\frac{n-p}{\\cos ^2\\theta }+\\frac{p}{\\cos ^2\\theta +\\lambda ^2\\sin ^2\\theta }=n.$ From the equality one can easily deduce that $p$ and $\\lambda $ have to be independent of point $x$ .",
"So how many these LOMSEs?",
"There turns out to be a constellation of uncountably many, even under the severe restriction!",
"In [46] we derived a structure theorem.",
"Theorem 2.6 $\\eta $ is an LOMSE if and only if $\\eta =i\\circ \\pi $ where $\\pi $ is a Hopf fibration to $(\\mathbb {P}^p,h)$ and $i:(\\mathbb {P}^p,\\lambda ^2h) {\\looparrowright } (S^m,g_m)$ is an isometric minimal immersion.",
"Remark 2.7 $\\pi $ gives a countably many levels and in most levels the moduli space of isometric minimal immersions from projective spaces into standard spheres form (a sequence of) compact convex bodies of vector spaces of high dimensions.",
"[46] perfectly embeds the relevant theory (see [9], [45], [29], [44], [42], [43]) into the construction of LOMSEs.",
"Remark 2.8 In particular, using coordinates of ambient Euclidean spaces, $\\eta $ can be expressed as $(\\eta _1,\\cdots ,\\eta _{m+1})$ .",
"All $\\eta _i$ are spherical harmonic polynomials sharing a common even degree $k$ .",
"Moreover $\\lambda =\\sqrt{\\frac{k(k+n-1)}{p}}$ .",
"We call such an LOMSE of ${\\bf (n,p,k)}$ type.",
"Besides singular solutions, we cared about smooth solutions as well.",
"By Morrey's famous regularity result [24], a $C^1$ solution to (REF ) is automatically $C^\\omega $ .",
"In particular, a preferred variation of LOC associated to an LOM $\\eta $ can be $M=M_{\\rho ,\\eta }:=\\lbrace (rx,\\rho (r)\\eta (x)):x\\in S^n, r\\in (0,\\infty )\\rbrace \\subset \\mathbb {R}^{m+n+2}.$ Its being minimal is equivalent to two conditions (similar to that of (REF ), see [46] for details).",
"When $\\eta $ is an LOMSE, one of the conditions holds for free and the other gives the following.",
"Theorem 2.9 For an LOMSE $\\eta $ , $M$ above is minimal if and only if $\\frac{\\rho _{rr}}{1+\\rho _r^2}+\\frac{(n-p)\\rho _r}{r}+\\frac{p(\\frac{\\rho _r}{r}-\\frac{\\lambda ^2\\rho }{r^2})}{1+\\frac{\\lambda ^2\\rho ^2}{r^2}}=0.$ By introducing $\\varphi :=\\frac{\\rho }{r}$ and $t:=\\log r$ , (REF ) transforms to $\\left\\lbrace \\begin{array}{ll}\\varphi _t=\\psi ,\\\\\\psi _t=-\\psi -\\Big [\\big (n-p+\\frac{p}{1+\\lambda ^2\\varphi ^2}\\big )\\psi +\\big (n-p+\\frac{(1-\\lambda ^2)p}{1+\\lambda ^2\\varphi ^2}\\big )\\varphi \\Big ]\\big [1+(\\varphi +\\psi )^2\\big ].\\end{array}\\right.$ This system is symmetric about the origin and owns exact 3 fixed points $(0,0),\\, P(\\varphi _0, 0)$ and $-P$ , where $\\varphi _0=\\tan \\theta $ .",
"Through linearization, it can be seen that the origin is always a saddle point and $P$ has two types: $P$ is a stable center when $(n,p,k)=(3,2,2), (5,4,2), (5,4,4)$ or $n\\ge 7$ ; $P$ is a stable spiral point when $(n,p)=(3,2)$ , $k\\ge 4$ or $(n,p)=(5,4)$ , $k\\ge 6$ .",
"By very careful analysis including excluding limit circles, there exists a special orbit emitting from the origin and approaching to $P$ for the system (REF ) and for $t\\in (-\\infty , +\\infty )$ .",
"Figure: NO_CAPTIONTranslated back to the $r\\rho $ -plane, the illustration graphs would be $\\begin{minipage}[c]{0.5}\\includegraphics [scale=0.45]{I3N2019.eps}\\end{minipage}\\begin{minipage}[c]{0.5}\\includegraphics [scale=0.45]{I4N2019.eps}\\end{minipage}$ Therefore, we got minimal graphs (other than cone) in $\\mathbb {R}^{m+n+2}$ defined everywhere away from the origin of $\\mathbb {R}^{n+1}$ .",
"Since $\\frac{d\\rho }{dr}=\\varphi +\\psi $ , intense attentions were given to orbits emitting from the origin in $\\varphi \\psi $ -plane.",
"They produce minimal surfaces which are $C^1$ at $r=0$ .",
"So the natural $C^0$ extension is in fact $C^\\omega $ according to Morrey's regularity result.",
"This is the way how we constructed entire $C^\\omega $ minimal graphs with LOCs as tangent cones at infinity.",
"Type (II) contains interesting information.",
"Note that the fixed point orbit $P$ stands for the LOC and the ray in the $r\\rho $ -plane with constant slope $\\varphi _0=\\tan \\theta $ .",
"Since vertical line $\\varphi =\\varphi _0$ intersects the orbit infinitely many times, there are corresponding intersections of the solution curve and the LOC ray in $r\\rho $ -plane.",
"Each intersection point gives us a minimal graph $G_i$ over a disk of radius $r_i$ .",
"Rescale $G_i$ by $\\frac{1}{r_i}$ and denote new graphs by $\\tilde{G}_i$ .",
"Then $\\tilde{G}_i$ are mutually different minimal graphs over the unit disk $\\mathbb {D}^{n+1}$ with the same boundary $-$ graph of $\\tan \\theta \\cdot \\eta $ over unit sphere $S^n$ .",
"Hence we see that there exist boundary data which support infinitely many $C^\\omega $ solutions and at least one singular solution to the Dirichlet problem!",
"This extended Lawson-Osserman's non-uniqueness result (1) from finiteness to infiniteness.",
"More can be read off.",
"Clearly, by density monotonicity, volumes of $\\tilde{G}_i$ strictly increase to that of the truncated LOC.",
"So none of the LOCs of Type (II) are area-minimizing.",
"Actually, recently we showed in a joint work [27] that LOC of Type (II) are even not stable.",
"They bring unstable singular solutions to the Dirichlet problem (cf.",
"(1) for Lawson-Osserman construction).",
"Since the solution curve oscillates between rays of slopes $\\varphi _1$ and $\\varphi _2$ , it is not always the case that once we have singular solution for slope $\\varphi _0$ , then solutions suddenly vanish immediately for $\\varphi >\\varphi _0$ .",
"It is a question for what kind of $\\varphi $ outside $[0,\\varphi _1]$ the problem can be solved?",
"Union of the set of such value and $[0,\\varphi _1]$ is called slope-existence range of $\\eta $ for the Dirichlet problem.",
"To extend $[0,\\varphi _1]$ , maybe the first difficulty is to figure out whether the orbit between the origin and point $(\\varphi _1,0)$ gives a stable compact minimal graph.",
"In the opposite direction on non-existence, a recent preprint [48] confirmed that the slope-existence range should usually be contained in a compact set of $\\mathbb {R}_{\\ge 0}$ .",
"More precisely, we prove Theorem 2.10 For every LOMSE $\\eta $ of either Type (I) or Type (II), there exists positive constant $R_\\eta $ such that when constant $R\\ge R_\\eta $ , the Dirichlet problem has no solutions for $\\phi =R\\cdot \\eta $ ."
],
[
"Minimal cones",
"As briefly mentioned before, it is useful to see if a local structure is stable or not for observation, and it is also quite important to know structures of minimizing currents.",
"Minimal cones are infinitesimal structures of minimal varieties, while minimizing cones are infinitesimal structures of minimizing currents.",
"Both determine, in some sense, local diversities of certain geometric objects.",
"In fact we naturally encountered many examples of minimal cones.",
"For example, Lawson-Osserman [21] constructed three minimal cones for singular solutions to the Dirichlet problem.",
"The first cone was shown to be coassociative in $\\mathbb {R}^7$ and hence area-minimizing by the fundamental theorem of calibrated geometries in the milestone paper [16].",
"However, it was unknown for 40 years if the other two are minimizing or not.",
"In our recent joint work [47] we proved that all LOCs of $(n, p, 2)$ type (for which case moduli space of $i$ to each Laplacian eigenvalue is a single point) are area-minimizing.",
"Since the other two original Lawson-Osserman cones are of $(7,4,2)$ type and $(15,8,2)$ type respectively, so the long-standing question got settled.",
"Area-minimizing cones of $(n,p,2)$ type are all homeomorphic to Euclidean spaces.",
"For other kind of area-minimizing cones, we considered those associated to isoparametric foliations of unit spheres.",
"There are two natural classes of minimal surfaces $-$ minimal isoparametric hypersurfaces and focal submanifolds.",
"By virtue of a successful combination of Lawlor's curvature criterion and beautiful structure of isoparametric foliations, we were able in [41] to show that, except in low dimensions, cones overs the “minimal products\" (defined therein) among these two classes are area-minimizing.",
"These provide a large number of new area-minimizing cones with various links of rich complexities.",
"Note that none of them can be split as product of (area-minimizing) cones of lower dimensions.",
"It is currently unknown to the author whether minimal products of links of general area-minimizing cones can always span an area-minimizing cone.",
"In [50] we considered a realization problem, first attacked by N. Smale [38], [39] in later 1990s.",
"Can any area-minimizing cone be realized as a tangent cone at a point of some homologically area-minimizing compact singular submanifold?",
"N. Smale constructed first such examples by applying many tools in geometric analysis and geometric measure theories in [38], while ours seems a bit simpler through the theory of calibrations with necessary understandings on Lawlor's work [19].",
"We showed Theorem 3.1 Every oriented area-minimizing cone in [19] can be realized to the above question.",
"Remark 3.2 Prototypes can be all the newly-discovered oriented area-minimizing cones in TZ, x-y-z and all Cheng's examples of homogeneous area-minimizing cones of codimension 2 in [6] (e.g.",
"minimal cones over $\\text{U}(7)/\\text{U}(1)\\times \\text{SU}(2)^3$ in $\\mathbb {R}^{42}$ , $\\text{Sp}(n)\\times \\text{Sp}(3)/\\text{Sp}(1)^3\\times \\text{Sp}(n-3)$ in $\\mathbb {R}^{12n}$ for $n\\ge 4$ , and $\\text{Sp}(4)/\\text{Sp}(1)^4$ in $\\mathbb {R}^{27}$ ) via a variation of our arguments in [50].",
"All the above cones have smooth links.",
"It could be highly useful if one can derive an effective way to study cones with non-smooth links.",
"As for stability and instability, in [27] we borrowed ideas [5], [16], [20] for orbit space.",
"We focused on a preferred subspace associated to given LOMSE and its quotient space.",
"With a canonical metric $\\sigma _0^2\\cdot \\left[\\left(r^2+\\lambda ^2\\rho ^2\\right)^p\\cdot r^{2(n-p)}\\right]\\cdot [dr^2+d\\rho ^2]$ where $\\sigma _0$ is the volume of $n$ -dimensional unit sphere, the length of any curve in the quotient space equals the volume of corresponding submanifold in $\\mathbb {R}^{m+n+2}$ .",
"Hence, the infinitely many $C^\\omega $ solution curves in the $r\\rho $ -plane for Type (II) in § determine geodesics connecting $Q:=(1,\\tan \\theta )$ and the origin in the quotient space.",
"Figure: NO_CAPTIONWe showed Theorem 3.3 The line segment $\\overline{0Q}$ is stable for Type (I) and unstable for Type (II).",
"Remark 3.4 In fact, $\\overline{0Q}$ is minimizing for Type (I).",
"However, the difficulty is whether it is possible to lift the stability or even the area-minimality property back to LOCs in $\\mathbb {R}^{m+n+2}$ ."
],
[
"Open questions",
"Besides several open questions in previous sections, we want to emphasize a few more in this section.",
"1.",
"Systematic study about LOMs beyond LOMSEs.",
"It still remains unclear how to construct LOMs for other type distribution of critical values, for instance $\\lbrace 0,\\, \\lambda _1,\\, \\lambda _2\\rbrace $ or $\\lbrace \\lambda _1,\\, \\lambda _2\\rbrace $ where $\\lambda _1$ and $\\lambda _2$ are different positive numbers.",
"These may involve more complicated dynamic systems and perhaps chaos phenomena.",
"2.",
"It seems unknown in general if the cones over image of $i$ itself in Remark REF is area-minimizing or not.",
"This would need certain systematic understandings about second fundamental form of $i$ .",
"It can also help us a lot for a complete classification of which Lawson-Osserman cones associated to LOMSEs are area-minimizing.",
"3.",
"How about situation for the finite left cases in [41].",
"The most famous one may be the cone over the image of Veronese map in $S^4$ , a focal submanifold for the isoparametric foliation with $g=3$ and $m=1$ .",
"It is still open whether the cone over the image, a minimal embedded $\\mathbb {R}P^2$ of constant curvature, is a minimizing current mod 2 (see [51])."
],
[
"Aknowlegement",
"The author would like to thank MPIM at Bonn for warm hospitality.",
"This work was sponsored in part by the NSFC (Grant No.",
"11601071) and a Start-up Research Fund from Tongji University."
]
] | 1906.04558 |
[
[
"Performance Analysis and Characterization of Training Deep Learning\n Models on Mobile Devices"
],
[
"Abstract Training deep learning models on mobile devices recently becomes possible, because of increasing computation power on mobile hardware and the advantages of enabling high user experiences.",
"Most of the existing work on machine learning at mobile devices is focused on the inference of deep learning models (particularly convolutional neural network and recurrent neural network), but not training.",
"The performance characterization of training deep learning models on mobile devices is largely unexplored, although understanding the performance characterization is critical for designing and implementing deep learning models on mobile devices.",
"In this paper, we perform a variety of experiments on a representative mobile device (the NVIDIA TX2) to study the performance of training deep learning models.",
"We introduce a benchmark suite and tools to study performance of training deep learning models on mobile devices, from the perspectives of memory consumption, hardware utilization, and power consumption.",
"The tools can correlate performance results with fine-grained operations in deep learning models, providing capabilities to capture performance variance and problems at a fine granularity.",
"We reveal interesting performance problems and opportunities, including under-utilization of heterogeneous hardware, large energy consumption of the memory, and high predictability of workload characterization.",
"Based on the performance analysis, we suggest interesting research directions."
],
[
"Introduction",
"Deep learning models have been widely deployed on mobile devices (e.g., mobile phones and smart home hub) to process on-board sensing data and enable a variety of mobile applications (e.g., machine translation, speech recognition, cognitive assistance and street navigation) [1], [2], [3].",
"Those models are deployed for model inference (not for model training).",
"The existing work has been conducted to analyze the performance and resource utilization of deep learning workloads on mobile devices when those models are deployed for model inference [4], [5], [6], [7], [8], [9], [10].",
"Those studies are important for optimizing the performance of deep learning models on mobile devices.",
"Besides model inference, training deep learning models on mobile devices recently becomes possible, because of increasing computation power on mobile hardware and the advantages of enabling high user experiences.",
"In particular, training deep learning models opens up a new approach to utilize the computational resources.",
"As the hardware of mobile devices is increasingly powerful and domain-specific, especially with the emergence of artificial intelligence (AI) chipsets and powerful mobile GPU [11], [12], [13], [14], training deep learning models is moving from the cloud to mobile devices to leverage these decentralized computational resources.",
"Furthermore, training deep learning models on mobile devices can avoid transmitting user data to the cloud as in the traditional method.",
"The traditional method can cause the breach of user privacy (even using an anonymous dataset and mixing it with other data).",
"For applications where the training objective is specified on the basis of data available on each mobile device, training on mobile devices can significantly reduce privacy and security risks by limiting the attack surface to only the device.",
"Most of the existing work on machine learning at mobile devices is focused on the inference of deep learning models (particularly convolutional neural network (CNN) and recurrent neural network (RNN)), but not training.",
"The performance characterization of training deep learning models on mobile devices is largely unexplored, although understanding the performance characterization is critical for designing and implementing deep learning models on mobile devices.",
"The recent work studies the performance of training deep learning models on servers [15].",
"However, training on mobile devices and on servers have different requirements, and have to be studied separately.",
"First, training deep learning networks should not interfere with the user's regular operations on mobile devices; The interference can manifest as unexpected shorter battery life because of large energy consumption caused by training deep learning networks, or the extended latency of the user's operations.",
"Second, the training data is likely to be collected and used for training on a daily basis.",
"For example, to train a deep learning network for image classification, the system can use images collected per day as training samples and train the model every day.",
"Third, a mobile device usually has a small memory capacity (compared with servers), hence some large deep learning models with tens of GB memory footprint (e.g., ResNet201 and VGG19) are not suitable to be trained on mobile devices.",
"In this paper, we perform a variety of experiments on a representative mobile device (the NVIDIA TX2) to study the performance of training deep learning models.",
"Our study provides insightful observations and reveals potential research opportunities.",
"In particular, this paper aims to answer the following research questions.",
"First, is training a deep learning network on a mobile device even possible?",
"The existing work on federated learning has shown preliminary success of training some machine learning models on mobile devices [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26].",
"Different from a typical deep learning model, those machine learning models are small in terms of memory consumption and model size.",
"Training deep learning models is known to be compute-intensive and memory-intensive.",
"Traditionally deep learning models are trained on servers with GPU with thousands of cores and high memory bandwidth.",
"However, mobile devices are under recourse constraint, e.g., limited computation power and relatively small memory capacity.",
"It is unknown whether and which deep learning models are trainable.",
"Second, how does training various deep learning models in mobile devices differ?",
"Deep learning models have shown success in a broad range of application domains.",
"Many deep learning models, such as DenseNet, Inception, ResNet, SqueezeNet and XceptionNet are related to image classification, which is one of the most common application domains for deep learning models.",
"Other kinds of deep learning models, such as reinforcement learning, Generative Adversarial Network (GAN) that are used in other application domains such as robot controls, image generation and natural language processing, should also be taken into consideration.",
"We aim to explore a diverse set of deep learning models in our study.",
"Third, what are the major performance problems when we train deep learning models on mobile devices?",
"Are those problems on mobile devices different from those on servers?",
"Answering the two questions is useful to identify research problems and train deep learning models more efficiently on mobile devices.",
"By conducting extensive experiments with various deep learning models on a specific mobile device, the NVIDIA TX2, we find many insightful observations.",
"In summary, we make the following contributions.",
"We introduce a benchmark suite for studying the workload of training deep learning models on mobile devices.",
"The benchmark suite includes four application domains and includes ten common deep learning models.",
"Those benchmarks are chosen with the consideration of possible resource constrained on mobile devices.",
"We make our benchmark suite open-source and intend to continually expand it to support various mobile devices.",
"We introduce a tool to study performance of training deep learning models on mobile devices, from the perspectives of the memory consumption, hardware utilization, and power consumption.",
"More importantly, the tool can correlate performance results with fine-grained operations in deep learning models, providing capabilities to capture performance variance and problems at a fine granularity.",
"We reveal interesting performance problems and opportunities, including under-utilization of heterogeneous hardware, large energy consumption of memory, and high predictability of workload characterization.",
"Based on the performance analysis, we suggest interesting research directions."
],
[
"Training Deep Learning Models on Mobile Devices",
"Deep learning is a general-purpose method that can be used to learn and model complicated linear and non-linear relationships between input datasets and output.",
"Many deep learning models can be represented as a directed acyclic graph where nodes of the graph are connected neurons.",
"Embedded in the graph, there are a number of parameters (e.g., “weights” and “bias”).",
"Those neurons and parameters are organized as layers.",
"The process of obtaining the optimal values of those parameters to reach high prediction accuracy is called “training”.",
"Training involves many iterations of computation (sometimes millions of iterations), in order to acquire the optimal parameters.",
"Each iteration is a training step, and consists of forward and backward passes.",
"During the backward pass, a backpropagation algorithm is used to optimize the parameters.",
"The backpropagation algorithm calculates the gradient of each parameter.",
"Also, the number of parameters and the corresponding gradients are equal.",
"The intermediate results, which are generated by each layer, are dominated by feature maps.",
"The forward pass generates feature maps, and the backward pass uses them to update the parameters.",
"Hence, feature maps need to be temporarily stored in the memory during the forward pass and before the backward pass can consume them.",
"The modern machine learning frameworks, e.g., TensorFlow [27] and PyTorch [28], employ a dataflow graph where the deep learning models training is modeled as a directed graph composed of a set of nodes (operations).",
"By decomposing a deep learning model graph (discussed above) into a set of nodes or fine-grained operations (e.g., SoftMax, Sigmoid and MatMul), these frameworks greatly improve hardware utilization and system throughput.",
"Training a deep learning model easily involves a large number of operations.",
"In a single training step of training deep learning models, there can be tens of different operations.",
"Each operation can be invoked hundreds of times, each of which is an operation instance."
],
[
"Methods",
"In this section, we introduce the metrics and tools we use to evaluate the performance of training deep learning models.",
"Figure: Profiling tools and profiling workflow."
],
[
"Evaluation Metrics",
"CPU utilization.",
"This metric quantifies how frequently CPU is utilized during deep learning models training.",
"This metric is defined in Equations REF and REF .",
"$CPU\\_Core\\_Utilization = \\frac{ T_{active}^{C} \\times 100}{T_{total}}\\%$ $CPU\\_Avg\\_Utilization = \\frac{\\sum _{i}^{n} ( CPU\\_Core\\_Utilization^{i})}{n}$ Equation REF is used to calculate the utilization of an individual CPU core.",
"In Equation REF , $T_{total}$ denotes the total training time; $T_{active}^{C}$ indicates the active time of the CPU core.",
"Equation REF is used to calculate the average CPU utilization of all CPU cores.",
"In Equation REF , $n$ is the total number of CPU cores for training deep learning models, and ${i}$ is the index of the CPU core.",
"A larger value of $CPU\\_Avg\\_Utilization$ indicates higher CPU utilization.",
"We want high CPU utilization to achieve high throughout processing of training operations.",
"GPU utilization.",
"This metric quantifies how frequently GPU is utilized during deep learning model training.",
"This metric is defined in Equation REF .",
"$GPU\\_Utilization = \\frac{T_{active}^{G} \\times 100}{T_{total}}\\%$ As shown in Equation REF , the GPU utilization is defined similar to the CPU utilization in Equation REF .",
"We also want high GPU utilization to achieve high throughout processing of training operations.",
"Peak memory consumption.",
"Training deep learning models can be memory-consuming, as a large amount of parameters, derivatives and temporary variables use the memory space.",
"Some popular deep learning models involve a large number of parameters.",
"For example, VGG-16 and Resnet-50 have 138 million and 25 million parameters respectively, consuming 6.3 GB and 5.8 GB memory (the batch size is 64); SqueezeNet, a small deep learning model designed for mobile devices has 5 million parameters, consuming 5.7 GB memory (the batch size is 64).",
"The memory consumption of a deep learning model sets up a constraint on whether training the model on a mobile device is feasible.",
"The peak memory consumption is defined as the maximum memory usage during the training process.",
"Energy consumption.",
"Since a mobile device has limited battery life, reporting energy consumption of training deep learning models is critical to determine if the training is feasible within the battery life.",
"Energy consumption is calculated based on Equation REF .",
"During the model training, we collect power consumption of the mobile device periodically.",
"In Equation REF , $time\\_interval$ defines how frequently we collect power data, and $Power\\_Consumption_{i}$ is the whole system power of the mobile device collected in a power sample data $i$ .",
"$Energy = \\sum _{i} time\\_interval \\times Power\\_Consumption_{i}$ Throughput.",
"This metric is used to evaluate the efficiency of the training process.",
"Throughput in this paper is defined as how many training samples can be processed and used for training in one second.",
"For example, when we train DenseNet40 using the batch size of 4, we can finish five training steps in one second, and each training step processes four images (samples).",
"Hence, the throughput for training DenseNet40 is 20 samples per second.",
"Idle state ratio for a core.",
"During the training, some CPU cores can be idle (i.e., the utilization is 0).",
"Idle state ratio for a core is the percentage of the total training time that the core is idle."
],
[
"Profiling Tools",
"We use the existing tools, Nvprof [29], Tegrastats [30] and TensorFlow Profiler [31], for performance analysis and characterization of training deep learning models on the NVIDIA TX2.",
"Nvprof is a profiling tool that collects execution time of computation on CPU and GPU.",
"Nvprof can be used to identify idle or busy states of CPU and GPU.",
"When used for GPU, Nvprof can also be used to identify GPU kernel names (hence the names of operations running on GPU).",
"Tegrastats is a tool that collects hardware utilization of CPU and GPU, power consumption of hardware components (CPU, GPU, and memory) and memory consumption.",
"TensorFlow Profiler [31] is a tool integrated into TensorFlow runtime to perform operations statistics, including operations execution time and dependency between operations.",
"Nvprof and Tegrastats do not provide APIs that allow the programmer to trigger or stop profiling within the application.",
"Nvprof and Tegrastats can run continuously as a system daemon and collect system-wide information at any moment.",
"Simply using Nvprof and Tegrastats cannot meet the user's needs, because sometimes the user wants to correlate the profiling results (energy and memory consumption) with operations during the training process.",
"The training process for deep learning models easily involves a large number of operations (thousands or even millions of operations in a single time step).",
"Collecting the profiling results for operations is challenging.",
"We develop a tool to address the above challenge.",
"In particular, during the training process, we record the start and end times of all operations at each layer; We also periodically examine the execution information (including CPU and GPU utilization, power consumption of hardware components) every 10ms (we choose 10ms to control the profiling overhead).",
"The execution information is dumped into a file with the implicit timestamp information, due to our periodical profiling method.",
"After the training process, we associate the execution information with operations based on timing information (i.e., the start and end times of all operations).",
"Figure REF shows our tools and profiling workflow.",
"In Section (the section of evaluation results), the results are presented for all operations as a whole, because that allows us to easily present the results."
],
[
"Training Deep Learning Models on NVIDIA TX2",
"We use the NVIDIA TX2 as our evaluation platform.",
"This platform is an embedded system-on-module (SoM) with a dual-core NVIDIA Denver2 plus a quad-core ARM Cortex-A57 (six CPU cores in total), eight GB LPDDR4 and integrated 256-core Pascal GPU (mobile GPU).",
"The GPU has two streaming multiprocessors (SM), and each SM has 128 cores.",
"The eight GB memory is shared between CPU and GPU.",
"The peak power consumption of TX2 is just 15 Watt.",
"TX2 is a representative mobile platform.",
"It has been commonly used in self-driving cars, robotics and drones.",
"Many other common mobile devices, such as Snapdragon 855, Movidius Myriad2 and Nvidia Xavier, have comparable computation capability and memory capacity.",
"Table REF summarizes the major hardware features of the NVIDIA TX2."
],
[
"Evaluation Results",
"In this section, we present the evaluation results and highlight major observations."
],
[
"Experiment Setup",
"Table REF summarizes the deep learning models we use for evaluation.",
"The table also lists those deep learning models that cannot be successfully trained on TX2 because of the memory constraint.",
"Among those models, DenseNet100 and NMT can train for a few time steps, but have segmentation faults later on; VGG19, ResNet101, ResNet152 and BERT cannot get started on training at all.",
"We use TensorFlow v1.13 to train the deep learning models.",
"Unless indicated otherwise, we use the default configruations for TensorFlow.",
"Note that we use TensorFlow instead of TensorFlow Lite, although TensorFlow Lite targets on mobile devices, because of the following reasons.",
"(1) TensorFlow Lite only supports model inference, not training.",
"Currently, there is no training framework especially targeting on training deep learning models on mobile devices.",
"(2) TensorFlow and TensorFlow Lite have common implementations for many operations (e.g., convolution, matrix multiplication and max-pooling), especially those operations in the forward pass of some deep learning models.",
"When reporting the performance, we skip the first three training steps, because they are often used by the TensorFlow runtime system to explore hardware architectures (e.g., cache capacities and memory access latency) for performance optimization.",
"The performance of the first three training steps is not representative of other training steps.",
"Table: The Specifications of NVIDIA TX2Table: Descriptions for deep learning models in our evaluation"
],
[
"Performance Analysis",
"We study the training performance from the following perspectives: hardware (CPU and GPU) utilization, power consumption, and peak memory consumption.",
"Hardware Utilization Figure REF shows the CPU and GPU utilization when we train the deep learning model Inception V1.",
"The figure shows the hardware utilization for three training steps.",
"Since the NVIDIA TX2 includes 6 CPU cores, we use six subgraphs to show the utilization of each of the six cores: the first two subgraphs show the utilization of two Denver2 cores, and the rest of them shows the utilization of four A57 cores.",
"We have the following observations.",
"Observation 1: The GPU utilization is generally much higher than the CPU utilization.",
"In most of the times, the GPU utilization is close to 100%, while each CPU core utilization ranges from 0% to 60%.",
"Also, when the GPU utilization is high, the CPU utilization tends to be low, and vice versa.",
"This indicates that the workload is not balanced well between CPU and GPU.",
"There seems a lack of effective coordination between CPU and GPU.",
"This observation is general and exists in many deep learning models (e.g., Inception V1, DCGAN, Resnet50, Xception).",
"Our further investigation reveals that when the GPU utilization is low, CPU is either busy with data fetching from storage (SSD) to main memory, or working on small operations that are not worth to be offloaded to GPU due to the large data copy overhead; When the GPU utilization is high, CPU is working on a few small operations, and most of CPU cycles are wasted.",
"Such an observation also exists in servers, but the difference is that the utilization difference between CPU and GPU on servers tends to be larger [49], because GPU on servers are much more powerful than CPU on servers and hence more operations (after kernel fusing) tend to be scheduled on GPU.",
"Observation 2: The utilization of GPU and each core in CPU is predictable.",
"The utilization shows a periodical pattern where busy cycles alternate with less busy cycles.",
"A period of the pattern corresponds to one time step.",
"Across time steps, such a pattern repeatedly appears.",
"This indicates that the computation across time steps remains stable and hence is highly predictable.",
"This observation is consistent with the existing work that leverages predictability of deep learning workloads for performance optimization [50], [51].",
"Such an observation also exists in servers.",
"Since this observation is determined by the process of training deep learning models that repeatedly goes through a computation graph (and not hardware architecture-related), this observation is general and independent of hardware architectures.",
"Figure: CPU and GPU utilization of different models.Figure: Idle state ratio of six CPU cores for different models.Figure: Energy consumption of different models.Figure: Power usage of different models in three iterations.Observation 3: The GPU utilization is sensitive to the batch size, while the CPU utilization is not.",
"Figure REF shows the CPU and GPU utilization when the batch size changes.",
"For DenseNet, the GPU utilization increases from $81.6\\%$ to $96.4\\%$ as the batch size changes from 4 to 64.",
"For SqueezeNet and ResNet50, the GPU utilization increase from 71.4% to 84.5% and from 85.6% to 95.6% respectively, as we increase the batch size.",
"However, for the CPU utilization, there is only 2.1% difference on average across models.",
"As we increase the batch size, the memory footprint increases and computation for operations also increases.",
"Since GPU works on most computation-intensive operations during the training, its utilization also increases as more computation requires more thread-level parallelism.",
"The CPU utilization does not increase very much, because CPU works on small operations and most of data objects in those operations can be in caches.",
"Slight increase of memory footprint due to the increase of the batch size does not cause extra cache misses and dos not significantly impact execution time.",
"Such an observation also exists in servers.",
"But the variance of GPU utilization on servers does not change as much as that on mobile devices, because GPU on servers have more cores and hence offers more thread-level parallelism to work on increased computation as we increase the batch size.",
"[51] Observation 4: Different cores have different utilization during the training.",
"TX2 has six heterogeneous cores: Two of them are Denver2 and four of them are A57.",
"As Figure REF shows, the utilization of each core changes differently, as the batch size increases.",
"There is no obvious correlation between the changes of the utilization across cores.",
"We also notice that the two Denver2 cores have the highest idle state ratio (as high as 65%) among all CPU cores, which indicates a large room for performance improvement.",
"We do not have the above observation on servers, because servers (especially x86 servers) usually do not have heterogeneous CPU cores [52].",
"FigureREF shows the GPU utilization of different deep learning models from different application domains.",
"The models from the domain of computer vision have the similar GPU utilization, hence we show Inception V1 as a representative of this domain.",
"We choose other models including LSTM and DCGAN to represent different application domains.",
"Observation 5: The GPU utilization varies on different application domains.",
"In Figure REF , we find the LSTM model (the domain of natural language processing) has a low GPU utilization (only about $25\\%$ ), while the models from the domain of computer vision (e.g., ResNet) have higher GPU utilization (95%).",
"Those models from computer vision has high GPU utilization, because they often employ convolution which is easy to leverage SIMT (Single Instruction Multiple Thread) on GPU and reach high GPU utilization.",
"In LSTM, operations often have dependency and there is lack of available thread-level parallelism.",
"The observation is consistent with the existing work[53], [54], [55] that LSTM has lower utilization than computer vision models.",
"Such an observation also exists in servers.",
"Since the GPU utilization is heavily impacted by the application domain, Observation 5 is general and independent of hardware architectures [15].",
"Power and Energy Consumption Figures REF and REF show power and energy consumption of GPU, CPU, and memory.",
"Figure REF shows how power consumption changes for three training steps.",
"Figure REF shows energy consumption for one training step, when the batch size changes.",
"Energy consumption is calculated based on Equation REF with the time interval of 5ms.",
"Observation 6: GPU is a power-consuming hardware component, but for some deep learning model, the memory consumes more power than GPU.",
"Figure REF shows that for the domain of computer vision, GPU is the most time-consuming hardware component when we train deep learning models (especially CNN models) such as ResNet50 and VGG16.",
"In those models, GPU consumes $4\\times $ and $2\\times $ of power consumption of CPU and memory respectively.",
"GPU consumption can take up to $57.4\\%$ of the whole system power.",
"Different from the above examples, the memory is the most power-consuming hardware component (not GPU), when we train LSTM.",
"Compared with the CNN models, LSTM has relatively bad data locality and causes more intensive memory accesses.",
"As a result, the memory, shared between GPU and CPU, draws large power consumption.",
"Such an observation does not exist in servers.",
"In servers, GPU (including its global memory) is the most power-consuming (e.g., NVIDIA V100 takes up to 250 Watt (more than half of the system power) when training LSTM, while the memory (main memory) takes only 20%-30% of the total system power.)",
"Observation 7: The power consumption across training steps is predictable.",
"Similar to the hardware utilization, the power consumption of hardware components shows a periodical pattern.",
"This pattern is highly predictable across training steps.",
"Figure REF shows such results.",
"On servers, we have the similar observations.",
"Observation 8: As we increase the batch size, the energy consumption increases as well, but not in a proportional way.",
"Figure REF shows the results to support this observation.",
"As we increase the batch size from 4 to 64 (16x increase), the energy consumption of the whole system increases as well.",
"However, the increase of the energy consumption is at least 2.2x and at most 10.5x, less than 16x when we change the batch size.",
"Also, different models show quite different energy consumption.",
"Among the five deep learning models for computer vision, DenseNet40 is the most energy-consuming one, while the Squeezenet is the most energy efficient one.",
"The above conclusion is true as we run the training to completion (including all time steps).",
"The above observation also exists in servers.",
"Consistent with the results of power consumption, we notice that for some models (e.g., DenseNet40), GPU is the most energy-consuming one, while for LSMT, the memory is the most energy-consuming one.",
"Peak Memory Consumption The memory is one of the key limiters for deep learning models training on mobile devices.",
"Some large models, e.g., ResNet101 and VGG19, consume more than 10 GB memory for training, while TX2 only has 8 GB memory.",
"Those models cannot be trained on TX2.",
"In our study, we aim to study the impact of the batch size on the memory consumption of deep learning models.",
"Different from on servers, on mobile devices we must carefully choose the batch size, not only for good training accuracy as on servers, but also for acceptable memory consumption.",
"For training (especially CNN and RNN), the memory is consumed by the following critical variables: parameters (including weights and bias), gradients, input data, and intermediate data.",
"Among them, the intermediate data is the most memory consuming.",
"The intermediate data includes the work space and feature map.",
"The work space is the memory consumed by the machine learning framework (e.g., TensorFlow or PyTorch).",
"The memory consumption of the work space varies for different frameworks.",
"The feature map, sitting in the middle of two neighbor layers of a CNN or RNN model, is generated by one layer, and used as input of the next layer.",
"Observation 9: Choosing a good batch size is critical to be able to train deep learning models on mobile devices.",
"Figure REF shows memory usage as we change the batch size.",
"As expected, parameters, input data and gradients remain constant, as we increase the batch size.",
"But the memory consumption of intermediate data increases significantly, as we increase the batch size.",
"For example, for DenseNet40, when the batch size increases from 4 to 64, the memory consumption of intermediate data increases from 2.2 GB to 5.9 GB.",
"When we use larger batch sizes (12nd 256), we run out of memory for all models.",
"Figure: Throughput of deep learning models.Figure: Memory usage of deep learning models.Figure: The accuracy variance as we increase training samples at a daily base to train DenseNet40.Throughput To quantify throughput, we employ a common metric, training samples per second, instead of using images per iteration (training step) as in some deep learning models, because of the following two reasons.",
"First, our collection of deep learning models includes CNN, RNN and Deep reinforcement learning models, which means that the training samples for some models are not images.",
"For example, the training samples are sentences for some RNNs (e.g., seq2seq).",
"Second, as we change the batch size for evaluation, the number of training samples for a training step changes as well, which indicates that the execution time per training step (iteration) changes.",
"Hence, using “second” (the time metric) instead of “iteration” makes more sense.",
"Observation 10: Throughput increases as the batch size increases.",
"Figure REF shows the throughput as we change the batch size.",
"For all models, the throughput increases as the batch size increases.",
"For example, for ResNet50, the throughput increases from 9 to 55 samples per second as the batch size increases from 4 to 64.",
"The above observation can also be seen in servers [15].",
"Observation 11: Across models, throughput changes differently, as we increase the batch size.",
"In Figure REF , the throughput of the deep reinforcement learning model increases from 889 to 13,618 samples per second as the batch size increases from 4 to 64 (15.3x speedup).",
"However, for DenseNet and ResNet50, such throughput speedup is 1.7x and 6.1x, respectively.",
"The deep reinforcement learning model has big throughput speedup as we increase the batch size.",
"This is because the training time of the deep reinforcement learning does not change too much, as we increase the batch size.",
"As a result, the throughput increases significantly, as we increase the batch size.",
"The above observation also exists on servers [15].",
"Study on modeling accuracy Training a deep learning model on a mobile device is different from that on a server, because training samples can be dynamically generated when the mobile device is used.",
"For example, training DenseNet40 can be based on training samples (images) collected at the user's day time.",
"In this evaluation, we evaluate a scenario where the user uses a mobile device to generate 64 images per day, and those images are used to train a deep learning model (DenseNet40).",
"We also assume that the model, before started to be trained, is already trained on a server, but needs to be trained further, using the user's new training samples.",
"Figure REF shows the variance of the accuracy, as we use the above training method for 17 days.",
"As the day 0, the training accuracy is 60.65%, because the model is already trained on a server.",
"Observation 12: Training a deep learning model on a mobile device can slowly increase training accuracy.",
"Figure REF reveals that the accuracy of DenseNet40 increases from 60.65% to 61.32% (using three different batch sizes).",
"The above observation reveals that using the above method does slowly increases the accuracy.",
"In this special scenario, depending on whether the user has high requirement on the model accuracy, the training on the mobile device can continue as more training samples are collected or stop."
],
[
"Discussion and Future Research Directions",
"Feasibility of training deep learning models on mobile devices.",
"Our work demonstrates the feasibility of training some deep learning models on a mobile device.",
"Most of the models we study are traditionally trained on a server, and seldom trained on any mobile device.",
"Those deep learning models come from various application domains, and have potential to provide new services for mobile users.",
"Our observation 9 reveals that choosing an appropriate batch size has a big impact on whether training a deep learning model is feasible.",
"Furthermore, it is well known that the batch size has an impact on the model accuracy.",
"Hence, there is a non-trivial tradeoff between the training feasibility and accuracy on mobile devices.",
"Such a tradeoff deserves further study.",
"Hardware utilization.",
"Mobile devices often offer rich hardware heterogeneity (e.g., there are two types of CPU cores in the NIVIDA TX2), richer than x86 servers.",
"However, such hardware heterogeneity is not leveraged well in the current machine learning frameworks.",
"This fact is pronounced by two observations: (1) The utilization of all CPU cores is relatively low (comparing with GPU); (2) The heterogeneity of CPU cores is completely ignored.",
"As a result of such fact, the CPU cycles are wasted and the computation power of specialized CPU cores (e.g., ARM Cortex-A57) is not fully utilized.",
"The recent work from Facebook [49] reveals that the performance difference between CPU and GPU on mobile devices is smaller than that on servers.",
"Based on this work and our observations, we see great opportunities to improve the current scheduling mechanism in the machine learning frameworks.",
"Only using GPU for computation-intensive operations may not be a good scheduling strategy.",
"Instead, balancing workloads on CPU and GPU to maximize system throughput (for finishing operations) is a better one.",
"Energy consumption.",
"Mobile devices are more sensitive to energy consumption than servers.",
"Training deep learning models on mobile devices raises concerns on whether the battery life is good enough to support training.",
"Although the recent work suggests to train deep learning models when the mobile device is charged [16], [56], [57], the charging time can be longer than the training time.",
"The batter may still be needed to finish training.",
"Hence, minimizing energy consumption during training is critical.",
"Our observation reveals that the memory can be more energy-consuming than CPU and GPU, when we train some deep learning networks.",
"Reducing energy consumption of the memory is necessary for mobile devices.",
"How to reduce energy consumption of the memory without impacting performance (execution time) is an open topic.",
"Predictability of workload characterization.",
"The workload of training deep learning networks is predictable, which means execution time, hardware utilization and power consumption show a repetitive pattern across training steps.",
"Such predictability allows us to apply dynamic profiling on a few training steps to collect workload characterization, based on which we guide operation scheduling and power management in the future training steps.",
"Predictability of execution time during the training has been leveraged in the existing work [51], [50].",
"We expect to leverage the predictability of other characterization in the future work."
],
[
"Related Work",
"Performance optimization of deep learning model training.",
"Some recent works [19], [20], [21], [22], [23], [24], [25], [26], [16], [17], [18], [58] have demonstrated the promise of training neural networks (NN) on mobile devices.",
"They are focused on exploring performance optimization in the perspectives of algorithm and system.",
"For example, Mao et al.",
"[58] implement a distributed mobile learning system that trains a neural network by multiple devices of the same local network in parallel.",
"They design a scheduler to adapt the training configuration for heterogeneous mobile resources and network circumstances.",
"Bonawitz et al.",
"[16] develop a federated learning system to achieve NNs training on mobile platforms using TensorFlow.",
"Some practical issues have been addressed, e.g., local data distribution, unreliable device connectivity and limited on-board resources.",
"Konečnỳ et al.",
"[19] use parameter compression techniques to reduce the uplink communication costs in federated learning.",
"This paper is orthogonal to the above works.",
"Our comprehensive model profiling and analysis can be used to develop more efficient NN training schemes on mobile devices.",
"Zhu et al.",
"[15] study the training performance and resource utilization of eight deep learning model models implemented on three machine learning frameworks running on servers (not mobile devices) across different hardware configurations.",
"However, they do not consider power and energy efficiency.",
"In contrast, our work is focused on deep learning models training on mobile devices.",
"Profiling of deep neural network inference.",
"Many works have been conducted to analyze the performance and resource utilization of machine learning workloads (inference, not training) on mobile devices [4], [5], [6], [7], [8], [9], [10].",
"Lu et al.",
"[4] measure the performance and resource usage of each layer in CNNs running on mobile CPUs and GPUs.",
"Based on the results of profiling and modeling, they implement a modeling tool to estimate the compute time and resource usage of CNNs.",
"However, they only consider CNNs, but not RNNs or reinforcement learning models which are also important for mobile applications.",
"Hanhirova et al.",
"[5] profile the performance of multiple CNN-based models for object recognition and detection on both embedded mobile processors and high-performance server processors.",
"They find that there exists significant latency–throughput trade-offs.",
"Unfortunately, the above works only study the inference of CNNs.",
"On the contrary, we profile and analyze the performance and resource requirements of CNNs, RNNs and deep reinforcement learning models training on mobile devices."
],
[
"Conclusions",
"Training deep learning networks on mobile devices is emerging because of increasing computation power on mobile hardware and the advantages of enabling high user experiences.",
"The performance characterization of training deep learning models on mobile devices is largely unexplored, although understanding the performance characterization is critical for designing and implementing deep learning models on mobile devices.",
"This paper is the first work that comprehensively studies the performance of training deep learning network on a mobile device.",
"Our study is based on a set of profiling tools on mobile devices, and uses a set of representative deep learning models from multiple application domains.",
"We reveal many research opportunities as a result of our study.",
"We hope that our study can motivate future study on optimizing performance of training deep learning networks on mobile devices."
]
] | 1906.04278 |
[
[
"A nonlinear Lazarev-Lieb theorem: $L^2$-orthogonality via motion\n planning"
],
[
"Abstract Lazarev and Lieb showed that finitely many integrable functions from the unit interval to $\\mathbb{C}$ can be simultaneously annihilated in the $L^2$ inner product by a smooth function to the unit circle.",
"Here we answer a question of Lazarev and Lieb proving a generalization of their result by lower bounding the equivariant topology of the space of smooth circle-valued functions with a certain $W^{1,1}$-norm bound.",
"Our proof uses a relaxed notion of motion planning algorithm that instead of contractibility yields a lower bound for the $\\mathbb{Z}/2$-coindex of a space."
],
[
"Introduction",
"In 1965 Hobby and Rice established the following result: Theorem 1.1 (Hobby and Rice [4]) Let $f_1, \\ldots , f_{n} \\in L^{1}([0, 1]; \\mathbb {R})$ .",
"Then there exists ${h\\colon [0, 1] \\rightarrow \\lbrace \\pm 1\\rbrace }$ with at most $n$ sign changes, such that for all $j$ , $\\int _{0}^{1}f_{j}(x)h(x)dx = 0.$ If we restrict the $f_{j}$ to lie in $L^{2}([0, 1]; \\mathbb {R})$ , we can view this as an orthogonality result in the $L^{2}$ inner product.",
"The Hobby–Rice theorem and its generalizations have found a multitude of applications, ranging from mathematical physics [6] and combinatorics [1] to the geometry of spatial curves [2].",
"The theorem also holds for $f_{1}, \\ldots , f_{n} \\in L^{1}([0, 1];\\mathbb {C})$ , provided $h$ is allowed $2n$ sign changes, by splitting the $f_{j}$ into real and imaginary parts.",
"Lazarev and Lieb showed that for complex-valued $f_{j}$ , the function $h$ can be chosen in $C^{\\infty }([0, 1];S^{1})$ , where $S^{1}$ denotes the unit circle in $:$ Theorem 1.2 (Lazarev and Lieb [5]) Let $f_{1}, \\ldots , f_{n} \\in L^{1}([0, 1]; $ .",
"Then there exists $h \\in C^{\\infty }([0, 1]; S^{1})$ such that for all $j$ , $\\int _{0}^{1}f_{j}(x)h(x)dx = 0.$ If $h$ is obtained by smoothing the function $h_{0}$ guaranteed by Theorem REF , then we would expect its $W^{1, 1}$ -norm, given by $\\Vert h\\Vert _{W^{1, 1}} = \\int _{0}^{1}|h(x)|dx + \\int _{0}^{1}|h^{\\prime }(x)|dx$ to be approximately $1 + 2 \\pi n$ , since $|h(x)| = 1$ , and each sign change of $h_{0}$ contributes approximately $\\pi $ to $\\int _{0}^{1}|h^{\\prime }(x)|dx$ .",
"However, Lazarev and Lieb did not establish any bound on the $W^{1, 1}$ -norm of $h$ and left this as an open problem; this was accomplished by Rutherfoord [9], who established a bound of $1 + 5 \\pi n$ .",
"Here we improve this bound to $1 + 2 \\pi n$ ; see Corollary REF .",
"The Hobby–Rice theorem has a simple proof due to Pinkus [8] via the Borsuk–Ulam theorem, which states that any map $f\\colon S^{n} \\rightarrow \\mathbb {R}^{n}$ with $f(-x) = -f(x)$ for all $x \\in S^{n}$ has a zero.",
"Lazarev and Lieb asked whether there is a similar proof of their result and write: “There seems to be no way to adapt the proof of the Hobby–Rice Theorem (which involves a fixed-point argument).” Rutherfoord [9] offered a simplified proof of Theorem REF based on Brouwer's fixed point theorem.",
"Here we give a proof using the Borsuk–Ulam theorem directly, which adapts Pinkus' proof of the Hobby–Rice theorem.",
"The advantage of this approach is that our main result gives a nonlinear extension of the result of Lazarev and Lieb; see Section for the proof: Theorem 1.3 Let $\\psi \\colon C^{\\infty }([0, 1]; S^{1}) \\rightarrow \\mathbb {R}^{n}$ be continuous with respect to the $L^{1}$ -norm such that $\\psi (-h)= - \\psi (h)$ for all $h \\in C^{\\infty }([0, 1]; S^{1})$ .",
"Then there exists $h \\in C^{\\infty }([0, 1]; S^{1})$ with $\\psi (h) = 0$ and $\\Vert h\\Vert _{W^{1, 1}} \\le 1 + \\pi n$ .",
"This is a non-linear extension of Theorem REF since for given $f_{1}, \\ldots , f_{n} \\in L^{1}([0, 1]; $ the map $\\psi (h) = (\\int _{0}^{1}f_{j}(x)h(x)dx)_{j}$ is continuous (see Section ) and linear, so in particular, $\\psi $ satisfies $\\psi (-h) = -\\psi (h)$ .",
"Using the $L^{1}$ -norm is no restriction; as we show in the next section, the $L^{p}$ norms on $C^{\\infty }([0, 1]; S^{1})$ for $1 \\le p < \\infty $ are all equivalent, so we could replace $L^{1}$ with any such $L^{p}$ .",
"In fact, the only relevant feature of the $L^{1}$ -norm is that functions $h_{1}, h_{2}$ are close in the $L^{1}$ -norm if $h_{1}, h_{2}$ are uniformly close outside of a set of small measure.",
"As a consequence, we recover the result of Lazarev and Lieb, with a $W^{1,1}$ -norm bound of $1 + 2 \\pi n$ since $\\psi $ takes values in ${n} \\cong \\mathbb {R}^{2n}$ ; see Section for the proof: Corollary 1.4 Let $f_{1}, \\ldots , f_{n} \\in L^{1}([0, 1]; $ .",
"Then there exists $h \\in C^{\\infty }([0, 1]; S^{1})$ with $\\Vert h\\Vert _{W^{1, 1}} \\le 1 + 2\\pi n$ such that for all $j$ , $\\int _{0}^{1}f_{j}(x)h(x)dx = 0.$ Given a space $Z$ with a $\\mathbb {Z}/2$ -action $\\sigma \\colon Z \\rightarrow Z$ , the largest integer $n$ such that the $n$ -sphere $S^n$ with the antipodal $\\mathbb {Z}/2$ -action (i.e.",
"$x \\mapsto -x$ ) admits a continuous map $f\\colon S^n \\rightarrow Z$ with $f(-x) = \\sigma (f(x))$ for all $x \\in S^n$ is called the $\\mathbb {Z}/2$ -coindex of $Z$ , denoted $\\operatorname{\\mathrm {coind}}Z$ .",
"We show that the coindex of the space of smooth $S^1$ -valued functions in the $L^1$ -norm with $W^{1,1}$ -norm at most $1 + \\pi n$ is between $n$ and $2n-1$ ; see Theorem REF .",
"Determining the coindex exactly remains an interesting open problem.",
"Our proof proceeds by constructing $\\mathbb {Z}/2$ -maps from $S^n$ , i.e., commuting with the antipodal $\\mathbb {Z}/2$ -actions, via elementary obstruction theory, that is, inductively dimension by dimension.",
"We find it illuminating to phrase our proof using the language of motion planning algorithms.",
"A motion planning algorithm (mpa) for a space $Z$ is a continuous choice of connecting path for any two endpoints in $Z$ ; see Section for details and Farber [3] for an introduction.",
"An mpa for $Z$ exists if and only if $Z$ is contractible.",
"Here we introduce the notion of (full) lifted mpa, which does not imply contractibility but is sufficiently strong to establish lower bounds for the coindex of $Z$ ; see Theorem REF .",
"We refer to Section for details."
],
[
"Relationship between topologies on $C^{\\infty }([0, 1]; S^{1})$",
"We now make precise our introductory comments about the topologies on $C^{\\infty }([0, 1]; S^{1})$ induced by the various $L^{p}$ -norms and the $d_{0,\\infty }$ metric.",
"Proposition 2.1 The $L^{p}$ -norms for $1 \\le p < \\infty $ induce equivalent topologies on $C^{\\infty }([0, 1]; S^{1})$ .",
"For $1 \\le p < \\infty $ , let $Z_{p}$ be $C^{\\infty }([0, 1]; S^{1})$ , equipped with the topology induced by the $L^{p}$ -norm.",
"Note that $\\Vert h\\Vert _{p} < \\infty $ for all $h \\in C^{\\infty }([0, 1]; S^{1})$ , so the identity maps $1_{p,q}\\colon Z_{p} \\rightarrow Z_{q}$ are well-defined as functions.",
"It suffices to show that $1_{p, q}$ is continuous for all $p, q \\in [1, \\infty )$ .",
"It is a standard fact that $1_{p, q}$ is continuous for $p \\ge q$ when the domain has finite measure, as is the case here for $[0, 1]$ .",
"For $p < q$ , we have $\\Vert h_{2} - h_{1}\\Vert _{q} &= \\left(\\int _{0}^{1}|h_{2}(x) -h_{1}(x)|^{q}dx\\right)^{1/q}\\\\&\\le \\left(\\int _{0}^{1}|h_{2}(x) - h_{1}(x)|^{p} \\cdot (\\text{diam}(S^{1}))^{q-p}dx\\right)^{1/q}\\\\&\\le (\\text{diam}(S^{1}))^{(q-p)/q} \\cdot \\Vert h_{2} - h_{1}\\Vert _{p}^{p/q}$ Since $S^{1}$ is bounded, $1_{p, q}$ is continuous.",
"Hence the $Z_{p}$ are all homeomorphic.",
"In the introduction, we claimed that “the only relevant feature of the $L^{1}$ -norm is that functions $h_{1}, h_{2}$ are close in the $L^{1}$ -norm if $h_{1}, h_{2}$ are uniformly close outside of a set of small measure.” To give content to this statement, we define a metric $d_{0, \\infty }$ on $C^{\\infty }([0, 1]; S^{1})$ by $d_{0, \\infty }(h_{1}, h_{2}) &= \\inf \\lbrace \\delta > 0 : |h_{2}(x) - h_{1}(x)| <\\delta \\text{ for all } x \\in [0, 1] \\setminus S,\\\\&\\qquad \\qquad \\text{ for some } S \\subseteq [0, 1] \\text{ with } \\mu (S) <\\delta \\rbrace .$ Proposition 2.2 The function $d_{0, \\infty }$ is a metric.",
"By the continuity of maps in $C^{\\infty }([0, 1]; S^{1})$ , we have $d_{0,\\infty }(h_{1}, h_{2}) = 0$ iff $h_{1} = h_{2}$ .",
"For the triangle inequality, suppose: $|h_{2}(x) - h_{1}(x)| < \\delta _{1}$ for all $x \\in [0, 1]\\setminus S_{1}$ , where $\\mu (S_{1}) < \\delta _{1}$ .",
"$|h_{3}(x) - h_{2}(x)| < \\delta _{2}$ for all $x \\in [0, 1]\\setminus S_{2}$ , where $\\mu (S_{2}) < \\delta _{2}$ Then $|h_{3}(x) - h_{1}(x)| < \\delta _{1} + \\delta _{2}$ for all $x \\in [0,1]\\setminus (S_{1} \\cup S_{2})$ , and $\\mu (S_{1}\\cup S_{2}) < \\delta _{1} +\\delta _{2}$ .",
"Hence $d_{0, \\infty }(h_{1}, h_{3}) \\le \\delta _{1} + \\delta _{2}$ .",
"Taking the infimum over $\\delta _{1}, \\delta _{2}$ , we obtain $d_{0,\\infty }(h_{1}, h_{3}) \\le d_{0, \\infty }(h_{1}, h_{2}) + d_{0, \\infty }(h_{2},h_{3})$ .",
"Proposition 2.3 The metric $d_{0, \\infty }$ and the norm $\\Vert \\cdot \\Vert _{1}$ induce equivalent topologies on $C^{\\infty }([0, 1]; S^{1})$ .",
"Let $Z_{0, \\infty }$ be $C^{\\infty }([0, 1]; S^{1})$ , equipped with the topology induced by $d_{0, \\infty }$ ; it suffices to show that the identity maps between $Z_{0, \\infty }, Z_{1}$ are continuous.",
"For the identity map $1\\colon Z_{0, \\infty } \\rightarrow Z_{1}$ , suppose $d_{0, \\infty }(h_{1}, h_{2}) < \\delta $ , so that there exists $S \\subseteq [0,1]$ with $\\mu (S) < \\delta $ such that $|h_{2}(x) - h_{1}(x)| < \\delta $ on $[0,1] \\setminus S$ .",
"Then $\\int _{0}^{1}|h_{2}(x) - h_{1}(x)|dx \\le \\int _{S}\\text{diam}(S^{1})dx +\\int _{[0, 1] \\setminus S}\\delta dx \\le \\delta (\\text{diam}(S^{1}) + 1).$ This shows that $1\\colon Z_{0, \\infty } \\rightarrow Z_{1}$ is continuous.",
"For the identity map $1\\colon Z_{1} \\rightarrow Z_{0, \\infty }$ , let $\\varepsilon > 0$ and suppose $\\Vert h_{2} - h_{1}\\Vert _{1} < \\delta $ for $\\delta =\\varepsilon ^{2}$ .",
"If $d_{0, \\infty }(h_{1}, h_{2}) \\ge \\varepsilon $ , then $|h_{2}(x) - h_{1}(x)| \\ge \\varepsilon $ on a set $S$ with $\\mu (S) \\ge \\varepsilon $ , implying $\\Vert h_{2} - h_{1}\\Vert _{1} \\ge \\varepsilon ^{2}$ , a contradiction.",
"Hence $d_{0, \\infty }(h_{1}, h_{2}) < \\varepsilon $ , and $1\\colon Z_{1} \\rightarrow Z_{0, \\infty }$ is continuous.",
"Now we expand our view to consider $L^{p}$ spaces under other measures $\\mu $ .",
"We show that finite, absolutely continuous measures can only produce coarser topologies than Lebesgue measure: Proposition 2.4 Let $\\mu $ be a finite measure on $[0, 1]$ that is absolutely continuous with respect to Lebesgue measure.",
"Let $Z_{1}$ be $C^{\\infty }([0, 1]; S^{1})$ , equipped with the topology induced by the $L_{1}$ -norm with respect to Lebesgue measure, and let $Z_{1, \\mu }$ be $C^{\\infty }([0, 1]; S^{1})$ , equipped with the topology induced by the $L_{1}$ -norm with respect to $\\mu $ .",
"Then the identity function $1\\colon Z_{1} \\rightarrow Z_{1, \\mu }$ is continuous.",
"By Proposition REF , it suffices to show that $1\\colon Z_{0, \\infty }\\rightarrow Z_{1, \\mu }$ is continuous.",
"The argument is similar to the argument that $1\\colon Z_{0, \\infty } \\rightarrow Z_{1}$ is continuous.",
"Using $\\lambda $ to denote Lebesgue measure, suppose $d_{0, \\infty }(h_{1}, h_{2}) <\\delta $ , so that there exists $S \\subseteq [0, 1]$ with $\\lambda (S) < \\delta $ such that $|h_{2}(x) - h_{1}(x)| < \\delta $ on $[0, 1] \\setminus S$ .",
"Then $\\int _{[0, 1]}|h_{2}(x) - h_{1}(x)|d\\mu &\\le \\int _{S}\\text{diam}(S^{1})d\\mu + \\int _{[0, 1] \\setminus S}\\delta d\\mu \\\\&\\le \\text{diam}(S^{1})\\mu (S) + \\delta \\mu ([0, 1])$ Note that since $\\mu $ is finite, we have $\\mu ([0, 1]) < \\infty $ .",
"As $\\delta \\rightarrow 0$ , we have $\\lambda (S) \\rightarrow 0$ , so $\\mu (S) \\rightarrow 0$ by absolute continuity, hence the right side approaches 0.",
"This shows the desired continuity.",
"The relationships between the topologies on $C^{\\infty }([0, 1]; S^{1})$ can be summarized as follows, where $1 < p_{1} < p_{2} < \\infty $ and $\\mu $ is a finite measure on $[0, 1]$ which is absolutely continuous with respect to Lebesgue measure: Z [hook]r Zp2 [leftrightarrow]r [hook]d Zp1 [leftrightarrow]r [hook]d Z1 [leftrightarrow]r [hook]d Z0, [hook]ld Zp2, [leftrightarrow]r Zp1, [leftrightarrow]r Z1, Therefore, when establishing the continuity of $\\psi $ for the sake of applying Theorem REF , we may use any $L^{p}$ norm on $C^{\\infty }([0, 1]; S^{1})$ , with respect to any finite measure $\\mu $ on $[0, 1]$ which is absolutely continuous with respect to Lebesgue measure.",
"(If we use a measure $\\mu $ other than Lebesgue measure, we can precompose $\\psi $ with $1\\colon Z_{1} \\rightarrow Z_{1, \\mu }$ before applying Theorem REF .)",
"With these results in hand, we can now deduce Corollary REF from Theorem REF : Let $\\psi \\colon C^{\\infty }([0, 1]; S^{1}) \\rightarrow {n}$ be given by component maps $\\psi _{j}\\colon h \\mapsto \\int _{0}^{1}f_{j}(x)h(x)dx.$ We claim $\\psi _{j}$ is continuous.",
"Since $f_{j} \\in L^{1}([0, 1]; $ , $f_{j}$ induces a finite measure $\\mu _{f}$ which is absolutely continuous with respect to Lebesgue measure, given by $\\mu _{f}(S) = \\int _{0}^{1}|f_{j}(x)|dx.$ By the above, we may view $C^{\\infty }([0, 1]; S^{1})$ as having the topology induced by the $L^{1}$ -norm $\\Vert \\cdot \\Vert _{1}$ with respect to $\\mu _{f}$ .",
"Then $|\\psi _{j}(h_{2}) - \\psi _{j}(h_{1})| &\\le \\int _{0}^{1}|f_{j}(x)|\\cdot |h_{2}(x) - h_{1}(x)|dx\\\\&\\le \\int _{[0, 1]}|h_{2} - h_{1}|d\\mu _{f}\\\\&\\le \\Vert h_{2} - h_{1}\\Vert _{1}.$ Therefore, $\\psi _{j}$ is continuous, so $\\psi $ is continuous.",
"Viewing the codomain ${n}$ of $\\psi $ as $\\mathbb {R}^{2n}$ , we may apply Theorem REF and get $\\Vert h\\Vert _{W^{1, 1}} \\le 1 + 2 \\pi n$ ."
],
[
"Lifts of motion planning algorithms and the coindex",
"Our proof of Theorem REF makes use of motion planning algorithms; see Farber [3].",
"We use $Y, Z$ in the following definitions to match our notation later: Definition 3.1 Let $Z$ be a topological space, and let $PZ$ be the space of continuous paths $\\gamma \\colon [0, 1] \\rightarrow Z$ , equipped with the compact-open topology.",
"Then a motion planning algorithm (or mpa) is a continuous map $s\\colon Z \\times Z \\rightarrow PZ$ , such that $s(z_{0}, z_{1})(0) =z_{0}$ and $s(z_{0}, z_{1})(1) = z_{1}$ .",
"For $Z$ a locally compact Hausdorff space, using the compact-open topology for $PZ$ ensures that a function $s\\colon Z\\times Z \\rightarrow PZ$ is continuous if and only if its uncurried form $\\widetilde{s}\\colon Z \\times Z \\times [0, 1] \\rightarrow Z$ given by $(z_{0}, z_{1}, t) \\mapsto s(z_{0}, z_{1})(t)$ is continuous; see Munkres [7].",
"One basic fact is that an mpa for $Z$ exists if and only if $Z$ is contractible [3].",
"We weaken the definition above for our purposes: Definition 3.2 Let $Y, Z$ be topological spaces, and let $\\phi \\colon Y \\rightarrow Z$ be continuous.",
"Let $(\\preceq )$ be a preorder on $Y$ , and let $Y^{2}_{\\preceq } =\\lbrace (y_{0}, y_{1}) \\in Y^{2} : y_{0} \\preceq y_{1}\\rbrace $ , giving $Y^{2}$ the product topology and $Y^{2}_{\\preceq }$ the resulting subspace topology.",
"A lifted motion planning algorithm (or lifted mpa) for $(Y,Z, \\phi , \\preceq )$ is a family of maps $s_{w}\\colon Y^{2}_{\\preceq }\\rightarrow PY$ for $w \\in (0, 1]$ with $s_{w}(y_{0}, y_{1})(0) = y_{0}$ and $s_{w}(y_{0}, y_{1})(1) = y_{1}$ , assembling into a continuous map $s\\colon (0, 1] \\times Y^{2}_{\\preceq } \\rightarrow PY$ , with the following continuity property: $&\\text{For all $y \\in Y$ and all neighborhoods $V$ of $\\phi (y) \\in Z$,}\\\\&\\qquad \\text{there exists a neighborhood $U$ of $\\phi (y) \\in Z$ and $\\delta > 0$ such that:}\\\\&\\qquad \\qquad \\text{if}\\quad \\phi (y_{0}), \\phi (y_{1}) \\in U, \\quad w <\\delta ,\\\\&\\qquad \\qquad \\text{then}\\quad \\phi (s_{w}(y_{0}, y_{1})(t)) \\in V \\text{ forall } t \\in [0, 1].$ Definition 3.3 A lifted mpa $s\\colon (0, 1] \\times Y^{2}_{\\preceq } \\rightarrow PY$ for $(Y, Z, \\phi , \\preceq )$ is full if $y_{0}\\preceq y_{1}$ for all $y_{0}, y_{1} \\in Y$ .",
"In this case we say $s$ is a full lifted mpa for $(Y, Z, \\phi )$ , omitting $(\\preceq )$ .",
"The continuity property essentially says that if two points $y_{1}, y_{2} \\in Y$ have images in $Z$ close to $\\phi (y) \\in Z$ , then $s_{w}$ carries $(y_{0},y_{1})$ to a path whose image under $\\phi $ is a path that stays close to $\\phi (y)$ , provided $w$ is small.",
"Note that an mpa $s\\colon Z \\times Z \\rightarrow PZ$ satisfying $s(z, z) =c_{z}$ for all $z \\in Z$ extends to a full lifted mpa for $(Z, Z, 1_{Z})$ by taking $s_{w} = s$ for all $w$ ; the continuity property just restates the continuity of $s$ at diagonal points $(z, z) \\in Z \\times Z$ .",
"This relaxed notion of mpa still provides lower bounds for the (equivariant) topology of $Z$ that are weaker than contractibility.",
"Recall that for a topological space $Z$ with $\\mathbb {Z}/2$ -action generated by $\\sigma \\colon Z \\rightarrow Z$ the $\\mathbb {Z}/2$ -coindex of $Z$ denoted by $\\operatorname{\\mathrm {coind}}Z$ is the largest integer $n$ such that there is a $\\mathbb {Z}/2$ -map $f\\colon S^n \\rightarrow Z$ , that is, a map satisfying $f(-x) = \\sigma (f(x))$ .",
"Definition 3.4 Let $x \\in S^{k}$ , and let $x = (x_{1}, \\ldots , x_{k+1})$ .",
"We say that $x$ is positive if its last nonzero coordinate is positive, and negative otherwise.",
"Our main tool in proving Theorem REF will be the following theorem: Theorem 3.5 Let $Y, Z$ be topological spaces, equip $Y$ with a $\\mathbb {Z}$ -action generated by $\\rho \\colon Y \\rightarrow Y$ , and equip $Z$ with a $\\mathbb {Z}/2$ -action generated by $\\sigma \\colon Z \\rightarrow Z$ .",
"Let $\\phi \\colon Y \\rightarrow Z$ be continuous and equivariant, i.e., $\\sigma \\circ \\phi = \\phi \\circ \\rho $ .",
"Let $(\\preceq )$ be a preorder on $Y$ and $s\\colon (0, 1] \\times Y^{2}_{\\preceq } \\rightarrow PY$ a lifted mpa for $(Y, Z, \\phi ,\\preceq )$ such that: $y \\preceq \\rho (y)$ .",
"$\\rho (y_{0}) \\preceq \\rho (y_{1})$ if and only if $y_{0} \\preceq y_{1}$ .",
"$y_{0} \\preceq y_{1}$ implies $y_{0} \\preceq s_{w}(y_{0},y_{1})(t) \\preceq y_{1}$ , for all $w \\in (0, 1]$ , $t \\in [0, 1]$ .",
"Then for each integer $n\\ge 0$ , there exists a $\\mathbb {Z}/{2}$ -map $\\beta _{n}\\colon S^{n} \\rightarrow Z$ .",
"Moreover, for any choice of initial point $y^{*} \\in Y$ , the maps $\\beta _{n}$ can be chosen such that $\\beta _{n}$ maps each positive point of $S^{n}$ to a point in $Z$ of the form $\\phi (y)$ , with $y^{*} \\preceq y \\preceq \\rho ^{n}(y^{*})$ , that is, the subspace of these points $\\phi (y)$ and their antipodes $\\sigma (\\phi (y))$ in $Z$ has coindex at least $n$ .",
"We will apply Theorem REF by taking $Z$ to be $C^{\\infty }([0, 1]; S^{1})$ with the topology induced by the $L^{1}$ -norm, and $Y$ to be $C^{\\infty }([0,1]; \\mathbb {R})$ with the $L^{1}$ -norm, restricted to increasing functions.",
"Using lifted mpa's allows us to reason about paths in $Y$ , which are simpler than paths in $Z$ .",
"The theorem encapsulates the inductive construction of a function $\\alpha _{n}\\colon S^{n} \\rightarrow Y$ , from which we produce ${\\beta _{n}\\colon S^{n} \\rightarrow Z}$ ; the continuity property of a lifted mpa is needed for this construction to work.",
"The last part of the theorem will give us the $W^{1,1}$ -norm bound.",
"We will inductively construct a function $\\alpha _{n}\\colon S^{n} \\rightarrow Y$ and then take $\\beta _{n} = \\phi \\circ \\alpha _{n}$ .",
"We will allow $\\alpha _{n}$ to be discontinuous on the equator of $S^{n}$ , but in such a way that $\\phi \\circ \\alpha _{n}$ is continuous everywhere.",
"Specifically, let $\\alpha _{k}\\colon S^{k} \\rightarrow Y$ be a function, not necessarily continuous.",
"Let $m \\colon S^{k} \\rightarrow S^{k}$ be given by $(x_{1}, \\ldots , x_{k}, x_{k+1}) \\mapsto (x_{1}, \\ldots , x_{k}, -x_{k+1})$ , so that $m$ mirrors points across the plane perpendicular to the last coordinate axis.",
"Then we say that $\\alpha _{k}$ is good if For $x$ positive, $y^{*} \\preceq \\alpha _{k}(x) \\preceq \\rho ^{k}(y^{*})$ , and $\\alpha _{k}(-x) = \\rho (\\alpha _{k}(x))$ .",
"For $x$ in the open upper hemisphere, $\\alpha _{k}(x) \\preceq \\alpha _{k}(m(x))$ .",
"$\\alpha _{k}$ is continuous on the open upper hemisphere.",
"$\\phi \\circ \\alpha _{k}$ is continuous.",
"Let $u, l \\colon B^{k+1} \\rightarrow S^{k}$ be the projections to the closed upper and lower hemispheres, that is, $u(x)$ is the unique point in the closed upper hemisphere sharing its first $k$ coordinates with $x$ , and similarly for $l(x)$ for the lower hemisphere.",
"Then we have the following claim: Claim If $\\alpha _{k}\\colon S^{k} \\rightarrow Y$ is good, then $\\alpha _{k}$ extends to $\\widetilde{\\alpha }_{k} \\colon B^{k+1} \\rightarrow Y$ , such that: For all $x \\in B^{k+1}$ , we have $y^{*} \\preceq \\widetilde{\\alpha }_{k}(x)\\preceq \\rho ^{k+1}(y^{*})$ .",
"For all $x \\in B^{k+1}$ , we have $\\alpha _{k}(u(x)) \\preceq \\widetilde{\\alpha }_{k}(x) \\preceq \\alpha _{k}(l(x))$ .",
"$\\widetilde{\\alpha }_{k}$ is continuous in the interior of $B^{k+1}$ .",
"$\\phi \\circ \\widetilde{\\alpha }_{k}$ is continuous.",
"Let $E \\subset S^{k}$ be the equator, the set of points neither in the open upper or lower hemisphere.",
"The set $E$ is compact, so the distance $d(x, E)$ for $x\\in B^{k+1}$ is well-defined and nonzero for $x \\notin E$ .",
"Define $\\widetilde{\\alpha }_{k} \\colon B^{k+1} \\rightarrow X_{k+1}$ by $&\\widetilde{\\alpha }_{k}(x) = {\\left\\lbrace \\begin{array}{ll}\\alpha _{k}(x) & x \\in E\\\\s_{w(x)}(\\alpha _{k}(u(x)), \\alpha _{k}(l(x)))(t(x)) & x \\notin E\\end{array}\\right.",
"}\\\\&\\qquad \\text{where }w(x) = \\min (d(x, E), t(x), 1 - t(x))\\\\&\\qquad \\hphantom{\\text{where }}\\,\\, t(x) = \\frac{d(u(x), x)}{d(u(x),l(x))}$ Note that $l(x) = m(u(x))$ , so ($\\alpha $ -2) implies $\\alpha _{k}(u(x))\\preceq \\alpha _{k}(l(x))$ , so $s_{w(x)}(\\alpha _{k}(u(x)),\\alpha _{k}(l(x)))$ is well-defined, and (3) gives $\\alpha _{k}(u(x)) \\preceq \\widetilde{\\alpha }(x) \\preceq \\alpha _{k}(l(x))$ , establishing ($\\widetilde{\\alpha }$ -2).",
"By ($\\alpha $ -1), we have $\\rho (y^{*}) \\preceq \\rho (\\alpha _{k}(x)) \\preceq \\rho ^{k+1}(y^{*})$ for $x$ negative, so $y^{*} \\preceq \\alpha _{k}(x)\\preceq \\rho ^{k+1}(y^{*})$ for all $x \\in S^{k}$ .",
"Along with the inequality above, this implies $y^{*} \\preceq \\widetilde{\\alpha }_{k}(x) \\preceq \\rho ^{k+1}(y^{*})$ , establishing ($\\widetilde{\\alpha }$ -1).",
"The function $\\widetilde{\\alpha }_{k}$ is continuous for $x \\notin E$ , since $u(-)$ , $l(-)$ , $d(-, -)$ , $d(-, E)$ are all continuous, $u(x), l(x) \\notin E$ , and $\\alpha _{k}$ is continuous on the open upper (and hence lower) hemisphere.",
"In particular, $\\widetilde{\\alpha }_{k}$ is continuous in the interior of $B^{k+1}$ , establishing ($\\widetilde{\\alpha }$ -3).",
"It remains to show $\\phi \\circ \\widetilde{\\alpha }_{k}$ is continuous at $x\\in E$ .",
"Let $V$ be a neighborhood of $\\phi (\\widetilde{\\alpha }_{k}(x)) =\\phi (\\alpha _{k}(x)) \\in Z$ , and obtain $\\delta > 0$ and a neighborhood $U$ of $\\phi (\\alpha _{k}(x)) \\in Z$ as in the lifted mpa definition.",
"Since $u(-), l(-), d(-, E)$ are continuous, there exists a neighborhood $W\\subseteq B^{k+1}$ of $x$ such that for all $x^{\\prime } \\in W$ we have $d(x^{\\prime }, E) <\\delta $ and $u(x^{\\prime }), l(x^{\\prime }) \\in (\\phi \\circ \\alpha _{k})^{-1}(U)$ , using the continuity of $\\phi \\circ \\alpha _{k}$ given by ($\\alpha $ -4).",
"Then $\\phi (\\alpha _{k}(u(x^{\\prime }))), \\phi (\\alpha _{k}(l(x^{\\prime }))) \\in U$ , so the lifted mpa property implies $\\phi (\\widetilde{\\alpha }_{k}(x)) \\in V$ , which shows $\\phi \\circ \\widetilde{\\alpha }_{k}$ is continuous at $x$ , establishing ($\\widetilde{\\alpha }$ -4).",
"We use the claim above to inductively construct $\\alpha _{k}\\colon S^{k}\\rightarrow Y$ , by extending each $\\alpha _{k}$ to a map $\\widetilde{\\alpha }_{k} \\colon B^{k+1} \\rightarrow Y$ , using $\\widetilde{\\alpha }_{k}$ for the upper hemisphere of $\\alpha _{k+1}$ , and extending to the negative hemisphere via $\\alpha _{k+1}(-x) =\\rho (\\alpha _{k+1}(x))$ .",
"Specifically, we have the following claim: Claim For all $k \\ge 0$ there exists $\\alpha _{k}\\colon S^{k} \\rightarrow Y$ , not necessarily continuous, such that $\\alpha _{k}$ is good.",
"We use induction.",
"For the base case, use $\\pm 1$ to denote the points of $S^{0}$ ; then let $\\alpha _{0}$ map $\\pm 1$ to $y^{*}, \\rho (y^{*})$ , respectively.",
"Then $\\alpha _{0}$ is good.",
"Given $\\alpha _{k}$ good and $\\widetilde{\\alpha }_{k}$ obtained through the previous claim, we now construct $\\alpha _{k+1} \\colon S^{k+1} \\rightarrow Y$ .",
"Let $\\pi \\colon S^{k+1}_{\\ge 0} \\rightarrow B^{k+1}$ be the projection of the closed upper hemisphere onto the first $k + 1$ coordinates.",
"We define maps on the two closed hemispheres as follows: $(\\alpha _{k+1})_{\\ge 0} \\colon S^{k+1}_{\\ge 0} \\rightarrow Y &\\qquad x\\mapsto \\widetilde{\\alpha }_{k}(\\pi (x))\\\\(\\alpha _{k+1})_{\\le 0} \\colon S^{k+1}_{\\le 0} \\rightarrow Y &\\qquad x\\mapsto \\rho (\\widetilde{\\alpha }_{k}(\\pi (-x)))$ Finally, we define $\\alpha _{k+1}$ by $x \\mapsto (\\alpha _{k+1})_{\\ge 0}(x)$ for $x$ positive and $x \\mapsto (\\alpha _{k+1})_{\\le 0}(x)$ for $x$ negative.",
"For $\\alpha _{k+1}$ , ($\\alpha $ -1) holds by construction, due to ($\\widetilde{\\alpha }$ -1).",
"Next, since $\\widetilde{\\alpha }_{k}$ is continuous in the interior of $B^{k+1}$ , we have that $(\\alpha _{k+1})_{\\ge 0}$ is continuous on the open upper hemisphere, hence $\\alpha _{k+1}$ is also, so ($\\alpha $ -3) holds also.",
"Since $\\widetilde{\\alpha }_{k}$ satisfies $\\widetilde{\\alpha }_{k}(-x) =\\rho (\\widetilde{\\alpha }_{k}(x))$ for positive $x$ on the boundary sphere $S^{k}\\subset B^{k+1}$ , we have $(\\alpha _{k+1})_{\\le 0}(x) =\\rho ^{2}((\\alpha _{k+1})_{\\ge 0}(x))$ for positive $x$ on the equator $S^{k}\\subset S^{k+1}$ , and $(\\alpha _{k+1})_{\\le 0}(x) =(\\alpha _{k+1})_{\\ge 0}(x)$ for negative $x$ on the equator.",
"Hence $\\phi \\circ (\\alpha _{k+1})_{\\ge 0}, \\phi \\circ (\\alpha _{k+1})_{\\le 0}$ agree on the equator, since $\\phi \\circ \\rho ^{2} = \\sigma ^{2} \\circ \\phi = \\phi $ .",
"Moreover, both composites are continuous; for the second, we have $\\phi \\circ (\\alpha _{k+1})_{\\le 0} = \\phi \\circ \\rho \\circ \\widetilde{\\alpha }_{k} \\circ \\pi \\circ (-) = \\sigma \\circ (\\phi \\circ \\widetilde{\\alpha }_{k}) \\circ \\pi \\circ (-)$ and $\\sigma , \\phi \\circ \\widetilde{\\alpha }_{k}, \\pi , (-)$ are continuous.",
"Hence ($\\alpha $ -4) holds.",
"Before showing ($\\alpha $ -2), we show that ($\\widetilde{\\alpha }$ -2) implies $\\widetilde{\\alpha }_{k}(x) \\preceq \\rho (\\widetilde{\\alpha }_{k}(-x))$ for all $x \\in B^{k+1}$ not on the equator.",
"For such $x$ , $u(-x)$ is on the open upper hemisphere and hence is positive.",
"By ($\\widetilde{\\alpha }$ -2), we have $\\widetilde{\\alpha }_{k}(x) \\preceq \\alpha _{k}(l(x)) = \\alpha _{k}(-u(-x)) =\\rho (\\alpha _{k}(u(-x))) \\preceq \\rho (\\widetilde{\\alpha }_{k}(-x)).$ This proves the inequality above.",
"Now we show ($\\alpha $ -2).",
"For $x \\in S^{k+1}$ in the open upper hemisphere, we have $\\alpha _{k + 1}(x) = \\widetilde{\\alpha }_{k}(\\pi (x)) \\preceq \\rho (\\widetilde{\\alpha }_{k}(-\\pi (x))) =\\rho (\\widetilde{\\alpha }_{k}(\\pi (-x))) = \\alpha _{k + 1}(m(x))$ by the inequality above.",
"Hence ($\\alpha $ -2) holds.",
"Taking $\\beta _{n} = \\phi \\circ \\alpha _{n}$ , Theorem REF follows from the claims above.",
"To see that $\\beta _{n}$ is a $\\mathbb {Z}/2$ -map, note that for $x\\in S^{n}$ positive, we have $\\beta _{n}(-x) = \\phi (\\alpha _{n}(-x)) = \\phi (\\rho (\\alpha _{n}(x))) =\\sigma (\\phi (\\alpha _{n}(x))) = \\sigma (\\beta _{n}(x))$ The other conclusions of the theorem are clear.",
"For a full lifted mpa, the preorder conditions of Theorem REF are trivially satisfied, so we get: Corollary 3.6 Let $Y, Z$ be topological spaces, equip $Y$ with a $\\mathbb {Z}$ -action generated by $\\rho \\colon Y \\rightarrow Y$ , and equip $Z$ with a $\\mathbb {Z}/2$ -action generated by $\\sigma \\colon Z \\rightarrow Z$ .",
"Let $\\phi \\colon Y \\rightarrow Z$ be continuous and equivariant, i.e., $\\sigma \\circ \\phi = \\phi \\circ \\rho $ .",
"If there is a full lifted mpa for $(Y, Z, \\phi )$ , then there exists a $\\mathbb {Z}/2$ -map $\\beta _{n}\\colon S^{n} \\rightarrow Z$ for all integers $n \\ge 0$ ."
],
[
"Constructing a lifted mpa",
"The goal of this section is to prove our main result, Theorem REF , by constructing a lifted mpa satisfying the conditions of Theorem REF .",
"As a warm-up, we use Theorem REF to prove the Hobby-Rice theorem, Theorem REF : The idea is to lift the space of functions with range in $\\lbrace \\pm 1\\rbrace $ to nondecreasing functions with range in $\\mathbb {Z}$ .",
"By describing a continuous map from pairs of such functions to paths between them, we will produce a lifted mpa, which will imply the result by Theorem REF .",
"Let $Y$ be the space of nondecreasing functions $g\\colon [0, 1] \\rightarrow \\mathbb {Z}$ with finite range, and let $Z$ be the space of functions $h\\colon [0, 1] \\rightarrow \\lbrace \\pm 1\\rbrace $ .",
"Equip $Y, Z$ with the $L^{1}$ -norm, and define $\\rho (g) = g + 1$ , $\\sigma (h) = -h$ , and $\\phi (g)(x) = {\\left\\lbrace \\begin{array}{ll}1 & g(x) \\text{ even}\\\\-1 & g(x) \\text{ odd}\\end{array}\\right.",
"}$ Let $g_{0} \\preceq g_{1}$ if $g_{0}(x) \\le g_{1}(x)$ for all $x \\in [0, 1]$ .",
"Finally, for $g_{0} \\preceq g_{1}$ define $s_{w}(g_{0}, g_{1})$ to be the path (in $t$ ) of functions following $g_{0}$ on $[0, 1 - t)$ and $g_{1}$ on $[1 - t, 1]$ : $s_{w}(g_{0}, g_{1})(t)(x) = {\\left\\lbrace \\begin{array}{ll}g_{0}(x) & x < 1 - t\\\\g_{1}(x) & x \\ge 1 - t\\end{array}\\right.",
"}$ Note that $s_{w}$ is independent of $w$ .",
"The conditions of Theorem REF are straightforward to check, except perhaps the continuity property in the lifted mpa definition, which we check now.",
"We are given $g \\in Y$ , and we may assume $V$ is a basis set, so that $V$ consists of all $h \\in Z$ with $\\Vert h - \\phi (g)\\Vert < \\varepsilon $ for some $\\varepsilon > 0$ .",
"By our choice of $U$ we may ensure that $g_{0}, g_{1} \\in Y$ have the same parity as $g$ except on a sets $S_{0}, S_{1}$ with $\\mu (S_{i}) < \\varepsilon / 4$ .",
"Then functions $g^{\\prime }$ along the path $s_{w}(g_{0}, g_{1})$ have the same parity as $g$ except on $S_{0} \\cup S_{1}$ , where $\\mu (S_{0} \\cup S_{1}) < \\varepsilon / 2$ , which implies $\\Vert \\phi (g^{\\prime }) - \\phi (g)\\Vert < \\varepsilon $ .",
"Hence the conditions of Theorem REF are satisfied, so we obtain a $\\mathbb {Z}/2$ -map $\\beta _{n}\\colon S^{n} \\rightarrow Z$ .",
"Applying the Borsuk–Ulam theorem to $\\psi \\circ \\beta _{n}\\colon S^{n} \\rightarrow \\mathbb {R}^{n}$ , where $\\psi \\colon h\\mapsto (\\int _{0}^{1}f_{j}(x)h(x)dx)_{j}$ , we obtain $x \\in S^{n}$ with $\\psi (\\beta _{n}(x)) = 0$ .",
"Hence also $\\psi (\\beta _{n}(-x)) = 0$ , so we may assume $x$ is positive.",
"Taking $y^{*} = 0$ in the last part of Theorem REF , we may ensure that $\\beta _{n}$ maps each positive point of $S^{n}$ to a point in $Z$ of the form $\\phi (g)$ with $0 \\le g \\le n$ , so that $\\phi (g)$ has at most $n$ sign changes.",
"This completes the proof.",
"Now we prove our main result, Theorem REF : Consider the space $C^{\\infty }([0, 1]; \\mathbb {R})$ with the $L^{1}$ -norm, and let $Y$ be the subspace of nondecreasing functions in $C^{\\infty }([0, 1]; \\mathbb {R})$ , equipped with the action $\\rho \\colon g \\mapsto g + \\pi $ .",
"Let $Z$ be $C^{\\infty }([0, 1]; S^{1})$ with the $L^{1}$ -norm, equipped with the action $\\sigma \\colon h \\mapsto -h$ .",
"Define $\\phi \\colon Y \\rightarrow Z$ by $\\phi (g)(x) = e^{ig(x)}$ ; then $\\phi $ is continuous since $x \\mapsto e^{ix}$ is 1-Lipschitz: $\\Vert \\phi (g_{2}) - \\phi (g_{1})\\Vert _{1} &= \\int _{0}^{1}|e^{ig_{2}(x)} -e^{ig_{1}(x)}|dx\\\\&\\le \\int _{0}^{1}|g_{2}(x) - g_{1}(x)|dx\\\\&\\le \\Vert g_{2} - g_{1}\\Vert _{1}.$ Define $(\\preceq )$ on $Y$ as $(\\le )$ pointwise.",
"Then properties (1) and (2) of Theorem REF and the commutativity property $\\phi \\circ \\rho = \\sigma \\circ \\phi $ evidently hold.",
"It remains to construct the lifted mpa $s$ .",
"Let $\\tau \\colon \\mathbb {R}\\rightarrow [0, 1]$ be a smooth, nondecreasing function with $\\tau (x) = 0$ for $x \\le -1$ , and $\\tau (x) = 1$ for $x \\ge 1$ .",
"(For example, take an integral of a mollifier.)",
"Then define $s_{w}\\colon Y_{\\preceq }^{2} \\rightarrow PY$ by $s_{w}(g_{0}, g_{1})(t)(x) = \\left(1 - \\tau \\left(\\frac{x - (1 -t)}{w}\\right)\\right)g_{0}(x) + \\tau \\left(\\frac{x - (1 -t)}{w}\\right)g_{1}(x).$ Since $\\tau $ is smooth, and since $x \\mapsto (x - (1 - t))/w$ is smooth for $w \\ne 0$ , the function $s_{w}(g_{0}, g_{1})(t)\\colon [0, 1] \\rightarrow \\mathbb {R}$ is smooth.",
"Also, $s_{w}(g_{0}, g_{1})(t)$ is nondecreasing: $&\\frac{d}{dx}[s_{w}(g_{0}, g_{1})(t)(x)]\\\\&\\qquad = -\\frac{1}{w} \\cdot \\tau ^{\\prime }\\left(\\frac{x - (1 - t)}{w}\\right) \\cdot g_{0}(x) + \\left(1 - \\tau \\left(\\frac{x - (1 - t)}{w}\\right)\\right) \\cdot g_{0}^{\\prime }(x)\\\\&\\qquad \\qquad + \\frac{1}{w}\\cdot \\tau ^{\\prime }\\left(\\frac{x - (1 -t)}{w}\\right) \\cdot g_{1}(x) + \\tau \\left(\\frac{x - (1 - t)}{w}\\right) \\cdot g_{1}^{\\prime }(x)\\\\&\\qquad \\ge \\frac{1}{w}\\cdot \\tau ^{\\prime }\\left(\\frac{x - (1 - t)}{w}\\right) \\cdot (g_{1}(x) - g_{0}(x))\\\\&\\qquad \\ge 0.$ Therefore, $s_{w}(g_{0}, g_{1})$ takes values in $PY$ .",
"Since $g_{0} \\le g_{1}$ , we have $g_{0} \\le s_{w}(g_{0}, g_{1})(t) \\le g_{1}$ , so property (3) of Theorem REF holds.",
"Next we show $s_{w}(g_{0}, g_{1})(t)$ is continuous in $w, g_{0}, g_{1}, t$ .",
"First we establish a helpful result.",
"Let $B$ be the subspace of $L^{\\infty }([0, 1]; \\mathbb {R})$ consisting of smooth functions, and let $\\widetilde{Y}$ be the space $L^{1}([0, 1]; \\mathbb {R})$ , of which $Y$ is a subspace; then pointwise multiplication $(b, g) \\mapsto b \\cdot g$ defines a continuous map $B \\times \\widetilde{Y} \\rightarrow \\widetilde{Y}$ , via the following inequality, using Hölder's inequality: $\\Vert b_{2}g_{2} - b_{1}g_{1}\\Vert _{1} &\\le \\Vert b_{2}(g_{2} - g_{1})\\Vert _{1} +\\Vert g_{1}(b_{2} - b_{1})\\Vert _{1}\\\\&\\le \\Vert b_{2}\\Vert _{\\infty } \\cdot \\Vert g_{2} - g_{1}\\Vert _{1} + \\Vert g_{1}\\Vert _{1} \\cdot \\Vert b_{2} - b_{1}\\Vert _{\\infty }.$ Since $(w, g_{0}, g_{1}, t) \\mapsto g_{0}$ , $(w, g_{0}, g_{1}, t) \\mapsto g_{1}$ are continuous maps $(0, 1] \\times Y \\times Y \\times [0, 1]\\rightarrow Y$ , by the result above it suffices to show that $(w, g_{0}, g_{1}, t) \\mapsto \\left(x \\mapsto \\tau \\left(\\frac{x - (1 -t)}{w}\\right)\\right)$ is a continuous map to $B$ ; the subtraction from 1 in the first term is handled by virtue of the fact that $B$ is a normed linear space, so that pointwise addition and scalar multiplication by $-1$ each define a continuous map.",
"Since $\\tau $ is constant outside of the compact set $[-1, 1]$ , $\\tau $ is uniformly continuous, hence it suffices to prove that $(w, g_{0}, g_{1}, t) \\mapsto \\left(x \\mapsto \\frac{x - (1 - t)}{w}\\right)$ is a continuous map to $B$ .",
"Note that $\\sup _{x \\in [0, 1]}\\left|\\frac{x}{w_{2}} - \\frac{x}{w_{1}}\\right| =\\left|\\frac{1}{w_{2}} - \\frac{1}{w_{1}}\\right|$ Since $w \\mapsto 1/w$ is a continuous map $\\mathbb {R}\\setminus \\lbrace 0\\rbrace \\rightarrow \\mathbb {R}$ , the map $(w, g_{0}, g_{1}, t) \\mapsto (x \\mapsto x/w)$ is a continuous map to $B$ , as is $(w, g_{0}, g_{1}, t) \\mapsto (x\\mapsto -(1 - t)/w)$ , so the map above is indeed a continuous map to $B$ .",
"Hence $s_{w}(g_{0}, g_{1})(t)$ is continuous in $w, g_{0}, g_{1}, t$ .",
"It remains to show the continuity property for a lifted mpa.",
"Let $g \\in Y$ , then for $g_{0}, g_{1} \\in Y$ we have $&\\Vert \\phi (s_{w}(g_{0}, g_{1})(t)) - \\phi (g)\\Vert _{1}\\\\&\\qquad = \\int _{0}^{1 - t - w}|\\phi (g_{0})(x) - \\phi (g)(x)|dx + \\int _{1 - t+ w}^{1}|\\phi (g_{1})(x) - \\phi (g)(x)|dx\\\\&\\qquad \\qquad + \\int _{1 - t - w}^{1 - t + w}|\\phi (s_{w}(g_{0},g_{1})(t))(x) - \\phi (g)(x)|dx\\\\&\\qquad \\le \\Vert \\phi (g_{0}) - \\phi (g)\\Vert _{1} + \\Vert \\phi (g_{1}) - \\phi (g)\\Vert _{1} +4w,$ where we use the fact that $S^{1}$ has diameter 2 in the last step.",
"This inequality implies the continuity property for a lifted mpa.",
"Therefore, we may apply Theorem REF to obtain a $\\mathbb {Z}/2$ -map $\\beta _{n}\\colon S^{n} \\rightarrow Z$ .",
"Then $\\psi \\circ \\beta _{n}\\colon S^{n}\\rightarrow \\mathbb {R}^{n}$ is a $\\mathbb {Z}/2$ -map, so by the Borsuk–Ulam theorem, we have $\\psi (\\beta _{n}(x)) = 0$ for some $x \\in S^{n}$ , and we may assume $x$ is positive.",
"Taking $y^{*} = c_{0}$ in the last part of Theorem REF , we have $\\rho ^{n}(y^{*}) = c_{n}$ , so we may ensure that $h = \\beta _{n}(x)$ is of the form $\\phi (g)$ for $g \\in Y$ , where $g$ is an increasing function with range in $[0, \\pi n]$ .",
"This gives the desired $W^{1, 1}$ -norm bound: $\\int _{0}^{1}\\left|\\frac{d}{dx}[e^{ig(x)}]\\right|dx = \\int _{0}^{1}|g^{\\prime }(x)|dx =g(1) - g(0) \\le \\pi n,$ which implies $\\Vert h\\Vert _{W^{1, 1}} \\le 1 + \\pi n$ ."
],
[
"Improving the bound further",
"In the introduction we argued that a $W^{1,1}$ -norm bound of $1+2\\pi n$ in Theorem REF might be expected from smoothing the Hobby–Rice theorem.",
"In this section, we show an improved bound for Theorem REF in the case where the $f_{j}$ are real-valued.",
"The idea is to modify the $S^{1}$ step of our construction so that some functions in the image of $\\alpha _{k}$ have smaller range within $[0, \\pi k]$ , and to modify the later steps so that functions $h$ in the image of $\\alpha _{k}$ with large range have $\\psi (\\phi (h))\\ne 0$ .",
"Theorem 5.1 Let $f_{1}, \\ldots , f_{n} \\in L^{1}([0, 1]; \\mathbb {R})$ .",
"Then there exists $h \\in C^{\\infty }([0, 1]; S^{1})$ such that for all $j$ , $\\int _{0}^{1}f_{j}(x)h(x)dx = 0.$ Moreover, for any $\\varepsilon > 0$ , $h$ can be chosen such that $\\Vert h\\Vert _{W^{1,1}} < 1 + \\pi (2n - 1) + \\varepsilon .$ Define $Y, Z, \\rho , \\sigma , \\phi , s$ as in the proof of Theorem REF , let $y^{*} = c_{0}$ , and let $(\\preceq )$ be $(\\le )$ .",
"We will produce $\\alpha _{n}\\colon S^{n} \\rightarrow Y$ and $\\beta _{n}\\colon S^{n} \\rightarrow Z$ by the inductive construction in the proof of Theorem REF , but we modify the first step by defining $\\alpha _{1}\\colon S^{1} \\rightarrow Y$ by $e^{ix} \\mapsto c_{x}$ for $x \\in [0, 2\\pi )$ .",
"This $\\alpha _{1}$ differs from the $\\alpha _{1}$ obtained in the proof of Theorem REF , which only gives constant functions at $\\pm 1 \\in S^{1}$ , but is still good in the sense introduced in the proof of Theorem REF .",
"Using this $\\alpha _{1}$ as our base case, we inductively construct $\\alpha _k$ as before with the following additional condition: $&\\text{For $\\delta > 0$ (depending on $k$ and the $f_{j}$), $\\alpha _{k}$may be chosen such that for all $x$:}\\\\&\\qquad \\text{Re}[e^{i\\alpha _{k}(x)(t)}] = \\pi _{1}(x)\\qquad \\text{for $t \\in [0, 1] \\setminus S$, where $\\mu _{f}(S) < \\delta $}\\qquad (P_{\\alpha _{k},\\delta })$ Here $\\mu _{f}$ is as in the proof of Corollary REF , that is, $\\mu _{f}(S) = \\int _{0}^{1}|f_{j}(x)|dx,$ and $\\pi _{1}\\colon S^{k}\\rightarrow [-1, 1]$ is the projection to the first coordinate.",
"The condition $(P_{\\alpha _{k}, \\delta })$ holds for $k = 1$ and all $\\delta > 0$ by our definition of $\\alpha _{1}$ .",
"To show that the condition carries through the inductive step, it suffices to show that given $\\delta > 0$ , there exists $\\delta ^{\\prime } > 0$ such that given $\\alpha _{k}$ such that $(P_{\\alpha _{k}, \\delta ^{\\prime }})$ holds, we can extend $\\alpha _{k}$ to $\\widetilde{\\alpha }_{k}$ as in the first claim in the proof of Theorem REF such that $(P_{\\widetilde{\\alpha }_{k}, \\delta })$ holds.",
"We accomplish this by modifying the definition of $\\widetilde{\\alpha }_{k}$ in the first claim in the proof of Theorem REF to impose a universal upper bound on $w(x)$ .",
"Since $\\mu _{f}$ is absolutely continuous with respect to Lebesgue measure $\\lambda $ , for $\\delta ^{\\prime \\prime } > 0$ there exists $\\delta ^{\\prime \\prime \\prime } > 0$ such that $\\lambda (S) \\le 2\\delta ^{\\prime \\prime \\prime }$ implies $\\mu _{f}(S) < \\delta ^{\\prime \\prime }$ .",
"Then we use $\\delta ^{\\prime \\prime \\prime }$ as our upper bound on $w(x)$ : $&\\widetilde{\\alpha }_{k}(x) = {\\left\\lbrace \\begin{array}{ll}\\alpha _{k}(x) & x \\in E\\\\s_{w(x)}(\\alpha _{k}(u(x)), \\alpha _{k}(l(x)))(t(x)) & x \\notin E\\end{array}\\right.",
"}\\\\&\\qquad \\text{where }w(x) = \\min (d(x, E), t(x), 1 - t(x), \\delta ^{\\prime \\prime \\prime })\\\\&\\qquad \\hphantom{\\text{where }}\\,\\, t(x) = \\frac{d(u(x), x)}{d(u(x), l(x))}$ This ensures that functions in the image of $\\widetilde{\\alpha }_{k}$ are equal to one of the functions $\\alpha _{k}(u(x)), \\alpha _{k}(l(x))$ except on a set $S$ with $\\mu _{f}(S) < \\delta ^{\\prime \\prime }$ .",
"Hence we may take $\\delta ^{\\prime } = \\delta ^{\\prime \\prime } = \\delta / 2$ ; then $(P_{\\widetilde{\\alpha _{k}}}, \\delta )$ holds as desired.",
"This shows that for any $\\delta > 0$ , $\\alpha _{k}$ may be chosen such that $(P_{\\alpha _{k}, \\delta })$ holds.",
"Now we apply the Borsuk–Ulam theorem as before.",
"We have the following diagram: $S^{2n} \\xrightarrow[(\\mathbb {Z}/2)]{\\phi \\circ \\alpha _{2n}} Z\\xrightarrow[(\\mathbb {Z}/2)]{\\psi } \\mathbb {C}^{n}$ The composition $\\psi \\circ \\phi \\circ \\alpha _{2n}$ is a $\\mathbb {Z}/{2}$ -map, so the Borsuk–Ulam theorem implies that it has a zero; that is, there exists $x \\in S^{2n}$ such that for all $j$ , we have $\\int _{0}^{1}f_{j}(t)e^{i\\alpha _{2n}(x)(t)}dt = 0.$ Moreover, we may assume $x \\in S^{2n}$ is positive.",
"But by the above, we have for the real parts, for all $j$ , $&\\mathrm {Re}\\left[\\int _{0}^{1}f_{j}(t)e^{i\\alpha _{2n}(x)(t)}dt\\right]\\\\&\\qquad = \\int _{0}^{1}f_{j}(t) \\cdot \\mathrm {Re}[e^{i\\alpha _{2n}(x)(t)}]dt\\\\&\\qquad = \\pi _{1}(x) \\cdot \\int _{0}^{1}f_{j}(t)dt +\\int _{S}f_{j}(t)(\\mathrm {Re}[e^{i\\alpha _{2n}(x)(t)}] - \\pi _{1}(x))dx.$ We can bound the last term as follows: $\\left|\\int _{S}f_{j}(t)(\\mathrm {Re}[e^{i\\alpha _{2n}(x)(t)}] -\\pi _{1}(x))dx\\right| &\\le \\int _{S}|\\mathrm {Re}[e^{i\\alpha _{2n}(x)(t)}] -\\pi _{1}(x)|d\\mu _{f}\\\\&\\le 2\\mu _{f}(S).$ Now if all $\\int _{0}^{1}f_{j}(t)dt$ are 0, then we may take $h$ to be an arbitrary constant, which gives ${\\Vert h\\Vert _{W^{1, 1}} = 1}$ .",
"Hence we may assume that some $\\int _{0}^{1}f_{j}(t)dt$ is nonzero.",
"In this case, we may ensure that for the $x$ with $(\\psi \\circ \\phi \\circ \\alpha _{2n})(x) = 0$ guaranteed by the Borsuk–Ulam theorem, $\\pi _{1}(x)$ is smaller than any constant we like, by taking $\\delta $ small in $(P_{\\alpha _{2n}, \\delta })$ .",
"In particular, choose $\\delta $ sufficiently small such that $|\\mathrm {Re}[e^{i\\theta }]| < \\delta $ implies $|\\theta -\\pi / 2| < \\varepsilon ^{\\prime }$ for $\\theta \\in [0, \\pi ]$ .",
"Now we analyze the ranges of functions $\\alpha _{k}(x)\\colon [0, 1]\\rightarrow \\mathbb {R}$ with $x$ positive and $|\\pi _{1}(x)| < \\delta $ , using the fact that functions $\\alpha _{k + 1}(x)$ are produced as transition functions between two functions $\\alpha _{k}(x^{\\prime }), \\alpha _{k}(x^{\\prime \\prime })$ with $\\pi _{1}(x^{\\prime }) = \\pi _{1}(x^{\\prime \\prime }) = \\pi _{1}(x)$ .",
"For $k = 1$ , $\\alpha _{k}(x)$ has range in $[\\pi / 2 - \\varepsilon ^{\\prime }, \\pi / 2 + \\varepsilon ^{\\prime }]$ , and each increment of $k$ extends the right end of this interval by $\\pi $ .",
"Hence $\\alpha _{2n}(x)$ has range in $[\\pi / 2 - \\varepsilon ^{\\prime }, \\pi / 2 + \\pi (2n - 1) + \\varepsilon ^{\\prime }].$ Hence taking $h = \\phi (\\alpha _{2n}(x))$ gives $\\Vert h\\Vert _{W^{1, 1}} \\le 1 + \\pi (2n - 1) + 2\\varepsilon ^{\\prime }$ .",
"Choosing $\\varepsilon ^{\\prime } < \\varepsilon / 2$ gives the desired result."
],
[
"A lower bound",
"We ask whether $\\Vert h\\Vert _{W^{1,1}} \\le 1 + 2 n \\pi $ is the best possible bound in Theorem REF .",
"We prove a lower bound of $1 + n \\pi $ in the case that the $f_{j}$ are real-valued, which implies the same lower bound in the case that the $f_{j}$ are complex-valued.",
"Theorem 6.1 There exist $f_{1}, \\ldots , f_{n} \\in L^{1}([0, 1]; \\mathbb {R})$ , such that for any $h \\in C^{1}([0, 1]; S^{1})$ with $\\int _{0}^{1}f_{j}(x)h(x)dx = 0\\qquad j = 1, \\ldots , n$ we have $\\Vert h\\Vert _{W^{1,1}} > \\pi n + 1$ .",
"Consider the case $n = 1$ , and take $f_{1}$ constant and nonzero.",
"Suppose for contradiction that $\\Vert h\\Vert _{W^{1,1}} \\le \\pi + 1$ , and write $h(x)$ as $e^{ig(x)}$ for $g \\in C^{1}([0, 1]; \\mathbb {R})$ , so that $\\int _{0}^{1}|g^{\\prime }(x)|dx \\le \\pi $ .",
"Since $g$ is continuous, $g$ attains its minimum $m$ and maximum $M$ on $[0, 1]$ .",
"By adding a constant to $g$ , we may assume $m = 0$ ; then we have $M \\le \\pi $ .",
"Since $f_{1}$ is constant, we have $\\int _{0}^{1}h(x)dx = 0$ , so $\\int _{0}^{1}\\text{Im}(h(x))dx = 0$ .",
"But $\\text{Im}(h(x))$ is continuous in $x$ and nonnegative, so $\\text{Im}(h(x)) = 0$ for all $x$ .",
"Hence $h$ is constant at either 1 or $-1$ , but this contradicts $\\int _{0}^{1}h(x)dx =0$ .",
"Therefore, $\\Vert h\\Vert _{W^{1,1}} > \\pi + 1$ for $n = 1$ .",
"Now allow $n$ arbitrary, and take each $f_{j}$ to be the indicator function on a disjoint interval $I_{j}$ .",
"If $\\Vert h\\Vert _{W^{1,1}} \\le \\pi n + 1$ , then $\\int _{I_{j}}|g^{\\prime }(x)|dx \\le \\pi $ for some $j$ , and we obtain a contradiction as above.",
"Therefore, $\\Vert h\\Vert _{W^{1,1}} > \\pi n + 1$ .",
"This $W^{1,1}$ -norm bound establishes an upper bound for the coindex of the space of smooth circle-valued functions with norm at most ${1+\\pi n}$ : Theorem 6.2 For integer $n \\ge 1$ let $Y_n$ denote the space of $C^\\infty $ -functions $f\\colon [0,1] \\rightarrow S^1$ with $\\Vert f\\Vert _{W^{1,1}} \\le 1+ \\pi n$ .",
"Then $n \\le \\operatorname{\\mathrm {coind}}Y_n \\le 2n-1.$ In the proof of Theorem REF we constructed a $\\mathbb {Z}/2$ -map $\\beta _n \\colon S^n \\rightarrow Y_n$ , which shows that ${\\operatorname{\\mathrm {coind}}Y_n \\ge n}$ .",
"Let $f_1, \\dots , f_n$ be chosen as in Theorem REF .",
"Then the map $\\psi \\colon Y_n \\rightarrow \\mathbb {R}^{2n}$ given by $\\psi (h) = (\\int _{0}^{1}f_{j}(x)h(x)dx)_j$ has no zero and is a $\\mathbb {Z}/2$ -map.",
"Thus $\\psi $ radially projects to a $\\mathbb {Z}/2$ -map $Y_n \\rightarrow S^{2n-1}$ .",
"A $\\mathbb {Z}/2$ -map $S^{2n} \\rightarrow Y_n$ would compose with $\\psi $ to a $\\mathbb {Z}/2$ -map $S^{2n} \\rightarrow S^{2n-1}$ , contradicting the Borsuk–Ulam theorem.",
"This implies $\\operatorname{\\mathrm {coind}}Y_n \\le 2n-1$ .",
"Problem 6.3 Determine the homotopy type of $Y_n$ ."
],
[
"Acknowledgements",
"The first author would like to thank Marius Lemm for bringing [5] to his attention."
]
] | 1906.04417 |
[
[
"Longitudinal Wobbling Motion in $^{187}$Au"
],
[
"Abstract The rare phenomenon of nuclear wobbling motion has been investigated for the nucleus $^{187}$Au.",
"A longitudinal wobbling-bands pair has been identified and clearly distinguished from the associated signature-partner band on the basis of angular distribution measurements.",
"Theoretical calculations in the framework of the Particle Rotor Model (PRM) are found to agree well with the experimental observations.",
"This is the first experimental evidence for longitudinal wobbling bands where the expected signature partner band has also been identified, and establishes this exotic collective mode as a general phenomenon over the nuclear chart."
],
[
"Longitudinal Wobbling Motion in $^{187}$ Au N. Sensharma U. Garg Physics Department, University of Notre Dame, Notre Dame, IN 46556, USA Q.",
"B. Chen Physik-Department, Technische Universität München, D-85747 Garching, Germany S. Frauendorf D. P. Burdette J. L. Cozzi K. B. Howard Physics Department, University of Notre Dame, Notre Dame, IN 46556, USA S. Zhu National Nuclear Data Center, Brookhaven National Laboratory, Upton, NY 11973, USA M. P. Carpenter P. Copp F. G. Kondev T. Lauritsen J. Li D. Seweryniak J. Wu Physics Division, Argonne National Laboratory, Argonne, IL 60439, USA A. D. Ayangeakaa D. J. Hartley Department of Physics, United States Naval Academy, Annapolis, MD 21402, USA R. V. F. Janssens Department of Physics and Astronomy, University of North Carolina Chapel Hill, NC 27599, USA Triangle Universities Nuclear Laboratory, Duke University, Durham, NC 27708, USA A. M. Forney W. B. Walters Department of Chemistry and Biochemistry, University of Maryland, College Park, MD 20742, USA S. S. Ghugre UGC-DAE Consortium for Scientific Research, Kolkata 700 064, India R. Palit Department of Nuclear and Atomic Physics, Tata Institute of Fundamental Research, Mumbai 400 005, India The rare phenomenon of nuclear wobbling motion has been investigated in the nucleus $^{187}$ Au.",
"A longitudinal wobbling-bands pair has been identified and clearly distinguished from the associated signature-partner band on the basis of angular distribution measurements.",
"Theoretical calculations in the framework of the Particle Rotor Model (PRM) are found to agree well with the experimental observations.",
"This is the first experimental evidence for longitudinal wobbling bands where the expected signature partner band has also been identified, and establishes this exotic collective mode as a general phenomenon over the nuclear chart.",
"27.70.+q, 23.20.-g, 23.20.En, 23.20.Gq, 21.60.Ev The shape of a nucleus, determined via specific characteristic spectroscopic features observed in experiments, is a manifestation of the self-organization of a finite fermionic system.",
"Studying the appearance of various shapes with changing of neutron-to-proton ratio or increasing angular momentum reveals new insights into fundamental principles governing finite fermionic systems in general.",
"The range of shapes that nuclei can assume encompasses spherical symmetry and axial deformation near the ground state, discussed in the textbooks [1], or coexistence of both shapes in one nucleus [2], in analogy to stereo isomers of molecules.",
"Triaxial nuclei (shaped like an ellipsoid with all three axes unequal), being rare in the ground state [3], have drawn considerable attention over the years.",
"The experimental observation of this unusual geometry is aided by two fingerprints: chirality and wobbling.",
"Chirality, although not prevalent in nuclear physics because nuclei have rather simple shapes, has been observed, nonetheless, in a number of nuclei across the nuclear periodic table [4].",
"Nuclear wobbling motion, the other principal signature of triaxiality, is a collective mode that appears when the moments of inertia ${\\cal J}_i$ of all three principal axes of the nuclear density distribution are unequal, which is a clear signal for a triaxial nuclear shape.",
"The mode is well known in classical mechanics.",
"For a given angular momentum, uniform rotation about the axis with the largest moment of inertia (the m-axis in Fig.",
"REF ) corresponds to minimal energy.",
"At a somewhat larger energy, this axis precesses (wobbles) about the space-fixed angular-momentum axis $\\vec{J}$ .",
"In a quantal system, such as the nucleus (or a molecule), $\\vec{J}$ wobbles about the medium (m-) axis in the body fixed frame (as illustrated in Fig.",
"REF (a)).",
"This mode manifests itself in the appearance of rotational bands that correspond to successive excitations of wobbling phonons, n$_{\\omega }$ , and alternating signature $\\alpha $ = $\\alpha _0$ + n$_{\\omega }$ , which determines the spin sequence I = $\\alpha $ + even number.",
"Adjacent wobbling bands n$_{\\omega +1}$ and n$_{\\omega }$ are connected by $\\Delta $ I = 1 transitions with a collectively-enhanced E2 component, which is generated by the wobbling motion of the entire charged body.",
"Figure: (Color online) Angular momentum geometry of (a) simple wobbler, (b) signature partner, (c) longitudinal and (d) transverse wobblerin the body fixed frame, where l, m, and s correspond to the long-, medium-, and short axis, respectively.",
"RR, jj and JJ are the rotor, odd particle, and total angular momentum, respectively.Microscopic calculations give ratios between the three moments of inertia that are close to the ratios of irrotational flow [5].",
"The reason is that collective rotation about a symmetry axis is not possible for a system of identical fermions.",
"Accordingly, the medium axis (m-) has the largest moment of inertia, because deviation from axial symmetry is maximal.",
"Although predicted quite sometime ago [1], there is only fragmentary evidence for simple wobbling (Fig.",
"REF (a)) in even-even nuclei [6].",
"Instead, wobbling has been demonstrated for a few odd-A nuclei: $^{105}$ Pd [7], $^{135}$ Pr [8], [9], $^{133}$ La [10], $^{161}$ Lu [11], $^{163}$ Lu [12], [13], $^{165}$ Lu [14], $^{167}$ Lu [15] and $^{167}$ Ta [16].",
"All these nuclei have an odd nucleon (neutron in the case of $^{105}$ Pd [7], and proton for all the other cases) occupying a high-$j$ orbital.",
"Depending on the particle (hole) nature of the odd quasiparticle arising from the bottom (top) of a deformed $j$ shell, its angular momentum gets aligned with the short, s- (long, l-) axes of the triaxial rotor, because this maximizes (minimizes) the overlap of its density distribution with the triaxial core, which minimizes the energy.",
"If the quasiparticle arises from the middle of the $j$ shell, it tends to align its angular momentum along the m-axis.",
"As the presence of the odd quasiparticle modifies the wobbling motion considerably, Frauendorf and Dönau [17] classified it as “longitudinal wobbling” (LW) and “transverse wobbling” (TW) when, the odd nucleon aligns its angular momentum along the m-axis, or along one of the perpendicular (s- or l-) axes, respectively (Figs.",
"REF (c) and (d)).",
"The wobbling energy (E$_\\textrm {wobb}$ ), which is the energy of the n$_{\\omega }$ = 1 wobbling band relative to the n$_{\\omega }$ = 0 band, is defined in Ref.",
"[17] as: $E_\\textrm {wobb} = E(I, n_{\\omega } = 1) - \\\\ \\left[\\frac{E(I+1, n_{\\omega } = 0) + E(I-1, n_{\\omega } = 0)}{2}\\right]$ For a qualitative understanding of its $I$ -dependence, let us assume that ${\\cal J}_\\parallel $ refers to the axis of uniform rotation, and the moments of inertia of the two perpendicular axes are equal to ${\\cal J}_\\bot $ (this is the case for irrotational flow and the triaxiality parameter $\\gamma =30^\\circ $ ).",
"In accordance with Frozen Alignment (FA) approximation of Ref.",
"[17], the quasiparticle angular momentum $\\vec{j}$ is assumed to be rigidly aligned with the axis.",
"With $A_i=1/2{\\cal J}_i$ , the rotor energy is given by $A_\\parallel R^2_\\parallel + A_\\bot R_\\bot ^2$ .",
"Exciting the first wobbling quantum corresponds to changing $J_\\parallel $ from $I$ to $I-1$ , for fixed $I$ .",
"The geometry of the precession cones in Fig.",
"REF implies a change of $R^2_\\bot $ from 0 to $\\approx 2I$ and of $R^2_\\parallel $ from $(I-j)^2$ to $(I-j-1)^2$ , which gives an increase in the rotor energy by: $E_\\textrm {wobb} = (A_\\bot -A_\\parallel )2I + 2\\bar{j}A_\\parallel ,~~~\\bar{j}=j+1/2$ In case of the simple and the longitudinal wobbler (Fig.",
"REF (a) and (c)) the precession cone revolves about the m-axis with the largest moment of inertia.",
"As $A_\\parallel <A_\\bot $ , the wobbling energy $E_\\textrm {wobb} $ increases with $I$ .",
"For the case of transverse wobbling (Fig.",
"REF (d)), the precession cone revolves about the s- (or l-) axis, which has a smaller moment of inertia than that for rotation about the m-axis.",
"In this case, $A_\\parallel >A_\\bot $ and the wobbling energy $E_\\textrm {wobb} $ decreases with $I$ until zero, where the mode becomes unstable.",
"The expression, $E_\\textrm {wobb}=\\sqrt{\\left((A_{\\bot 1}-A_\\parallel )2I+2\\bar{j}A_\\parallel \\right)\\left((A_{\\bot 2}-A_\\parallel )2I+2\\bar{j}A_\\parallel \\right)}$ obtained in Ref.",
"[17] for three different moments of inertia ${\\cal J}_{\\bot 1}, ~{\\cal J}_{\\bot 2}, ~{\\cal J}_{\\parallel }$ , is the geometric mean value of the wobbling energies given in Eq.",
"REF .",
"It should be understood that the frozen alignment scenario discussed here is an idealization to illustrate the longitudinal and transverse coupling schemes in a transparent way.",
"The odd particle responds to the inertial forces, changing its orientation to a certain degree.",
"Nevertheless, the qualitative classification remains valid.",
"Wobbling is characterized by collectively enhanced $I\\rightarrow I-1$ , E2 transitions from the wobbling to the yrast band, where the wobbling energy increases (decreases) for LW (TW).",
"The signature-partner bands represent another type of excitation involving a partial de-alignment of the odd particle with respect to its preferred axis (Fig.",
"REF (b)); for those, the connecting $\\Delta I$ = 1 transitions are of predominant M1 character, with very little, if any, E2 admixture.",
"In all of the cases mentioned above (except $^{133}$ La[10]), the wobbling bands have been identified as corresponding to TW, because E$_{\\textrm {wobb}}$ decreases with increasing angular momentum.",
"In this Letter, we report on the observation of band structures corresponding to longitudinal wobbling motion in the nucleus $^{187}$ Au.",
"This is the first case of observation of bands corresponding to longitudinal wobbling, clearly distinguished from the associated signature-partner band.",
"Further, these results open up a new mass region, and a different set of orbitals, where this exotic collective motion is established.",
"Occurrence of triaxiality at low spins has been established in this mass region by observation of chiral band pairs in several nuclei [18], [19], [20] and suggested by large-scale, mean-field calculations (see, for example, Refs.",
"[21], [3], [22]).",
"Also, earlier studies of the coupling of an odd number of particles to a rotor had revealed substantial deviations from an axial shape [23], [24].",
"To populate the levels of interest in $^{187}$ Au, a $^{19}$ F beam was used with an enriched $^{174}$ Yb target (13 mg/cm$^{2}$ -thick foil with a 33 mg/cm$^{2}$ $^{208}$ Pb backing) at the ATLAS facility of the Argonne National Laboratory.",
"Data were collected with the Gammasphere array in two separate runs using the same beam and target combination.",
"For the first run, a total of 57 Compton-suppressed Germanium detectors of the Gammasphere array were employed and the beam energy was 105 MeV.",
"For the second, the number of detectors was 73, and the beam energy 115 MeV.",
"Data were acquired in the triple-coincidence mode, with the combined total of three- and higher-fold $\\gamma $ -ray coincidence events being 1.08 $\\times $ 10$^{9}$ .",
"To take advantage of higher statistics, the data from both measurements were combined, and the analyses performed using the RADWARE suite of codes [25].",
"Energy and efficiency calibrations were performed for the added data set and the calibrated data was sorted into $\\gamma $ -$\\gamma $ coincidence matrices and $\\gamma $ -$\\gamma $ -$\\gamma $ coincidence cubes.",
"A partial level scheme for $^{187}$ Au relevant to the focus of this work is presented in Fig.",
"REF ; additional information on the level structure, along with details of the coincidence relationships, as well as the relevant coincidence spectra, will be presented in a forthcoming publication [26].",
"The arrangement of Gammasphere detectors into 17 different angular rings around the beam line has enabled high statistics angular distribution measurements for the relevant transitions of Fig.",
"REF .",
"The analysis procedure followed for these measurements is the same as that described in Refs.",
"[8], [9].",
"The validity of the method has been established by examining the angular distributions for two known stretched E2 transitions (333.8- and 413.7-keV) in the Yrast band (shown in Figs.",
"REF (g) and (h)).",
"As expected, the mixing ratios extracted for these stretched E2 transitions are extremely small: $\\delta $ = -0.04(1) and -0.03(1), respectively.",
"Further details, including a discussion of the various factors that might affect the extraction of final results—efficiency corrections, spin alignment, attenuation coefficients etc.—will be provided in Ref.",
"[26].",
"Figure: (Color online) Partial level scheme of 187 ^{187}Au relevant to the focus of present work.",
"Shown are the Yrast band, the n ω _{\\omega } = 1 wobbling band (LW) and the signature partner band (SP).",
"Newly identified transitions are marked with an asterisk (*).",
"All the stretched E2 transitions are shown in green, pure M1 transitions in blue, and the M1 + E2 mixed transitions in gold color.",
"The lowest level shown is a 9/2 - ^{-} isomeric level with E x _\\text{x} = 121.0 keV.Spins and parities of the bandheads as well as some low-lying levels of Bands (1) and (2) had been established previously [27], [28].",
"In addition to these, the present work has identified a new band [Band (3)] built on an 11/2$^{-}$ state at 386.3 keV.",
"The spins and parities for the levels in Band (3) have been assigned on the basis of angular distribution measurements as well as coincidence relationships, details of which will be provided in Ref.",
"[26].",
"Band (2) in Fig.",
"REF is found to decay to Band (1) (the Yrast band) via six $\\Delta I$ = 1 transitions.",
"Previous work [29], [30] has identified this band as the unfavored signature partner of Band (1).",
"However, based on the high-statistics angular distribution measurements for the connecting transitions between the two bands, this sequence has been identified in the present work as the first wobbling (n$_{\\omega }$ = 1) band.",
"Figs.",
"REF (a) – (d) provide the angular distributions for the four lowest n$_{\\omega }$ = 1 $\\rightarrow $ Yrast (n$_{\\omega }$ = 0) connecting transitions.",
"The mixing ratio, $\\delta $ , and the percentage of E2 mixing are noted on each plot.",
"These transitions have an 87%–93% E2 component, clearly identifying them as $\\Delta I$ = 1, E2 in nature, which is the hallmark of wobbling bands [12].",
"The present work has not been able to identify any other wobbling bands corresponding to higher phonon numbers (n$_{\\omega }$ = 2 and above).",
"The absence of these bands can be attributed to reasons discussed previously in Ref.",
"[17].",
"Figure: (Color online) Angular distribution plots for the four lowest LW →\\rightarrow Yrast linking transitions [(a)–(d)], two SP →\\rightarrow Yrast linking transitions [(e),(f)], and two Yrast in-band transitions [(g),(h)].",
"The experimental points are shown by black squares and the solid red lines are fits to the angular distributions.",
"The dashed blue lines [in (a)–(f)] and the dotted green lines [in (g),(h)] represent the expected angular distribution for pure Δ\\Delta I = 1, M1 and Δ\\Delta I = 2, E2 transitions respectively.Band (3) is found to decay to the yrast band via two $\\Delta I$ = 1 transitions (265.3- and 436.5-keV).",
"The angular distributions for these transitions (Figs.",
"REF (e) and (f) ) reveal a very small E2 component ($\\approx $ 0.4% and 1.0% E2 admixture, respectively), identifying these transitions as being essentially of a pure M1 character.",
"Angular distribution measurements for the in-band transitions have revealed a stretched E2 character.",
"Moreover, a $\\Delta $ I = 2 crossover transition (429.2-keV) was also identified, connecting the 15/2$^{-}$ level in Band (2) to the 11/2$^{-}$ level in Band (3).",
"The spin and parity of the 15/2$^{-}$ level in Band (2) have been established previously [27], [28].",
"The observed $\\Delta I$ = 2 nature of the in-band transitions, as also of the 429.2-keV transition, along with the pure $\\Delta I$ = 1 nature of the two connecting transitions (265.3- and 436.5-keV) have led to the spin and parity assignments in Fig.",
"REF .",
"Band (3), with its two almost pure M1 connecting transitions to the Yrast band, has, thus, been identified as the unfavored signature partner (SP) of the Yrast band.",
"Fig.",
"REF (d) displays the variation of E$_\\textrm {wobb}$ with spin.",
"The increasing trend clearly identifies $^{187}$ Au as a longitudinal wobbler.",
"This is different from all the other known wobblers, which have been identified as being of the transverse type.",
"Thus, $^{187}$ Au represents the first clear observation of longitudinal wobbler motion in nuclei.",
"To further understand the nature of wobbling, we have carried out calculations in the framework of the Particle Rotor Model (PRM) [17], [31], [32] for the $h_{9/2}$ band structures, with the deformation parameters $\\beta =0.23, \\gamma =23^\\circ $ , the pairing gap $\\Delta = 0.88~\\textrm {MeV}$ , and the chemical potential located at $\\lambda =-1.32~\\textrm {MeV}$ , 0.38 MeV below the second level of the $h_{9/2}$ shell.",
"The $h_{9/2}$ proton is described by a single-$j$ shell Hamiltonian.",
"Including the $f_{7/2}$ orbital into the PRM calculations provided results that agree with Figs.",
"REF and REF within the shown accuracy, because the admixture of the $f_{7/2}$ proton to the eigenstates is lower than 5%.",
"We have also carried out cranking calculations based on the configuration-fixed covariant density functional PC-PK1 [33], [34], [35], [36], [37].",
"The equilibrium deformations changed only slightly from $\\beta =0.28,~\\gamma =22^\\circ $ at $I=9/2$ to $\\beta =0.28,~\\gamma =25^\\circ $ at $I=29/2$ , which justifies the assumption of a constant deformation for the PRM calculations.",
"The analogue calculations for the $^{186}$ Pt core provided quite similar equilibrium deformations.",
"As input for the PRM, we used the smaller deformation $\\beta =0.23, \\gamma =23^\\circ $ found by the HFB-D1S calculations for $^{186}$ Pt [38] because it accounts for the experimental values of B(E2,$13/2^-\\rightarrow 9/2^-$ ) = 1.49 $(eb)^2$ for $^{187}$ Au [39] and B(E2,$2^+ \\rightarrow 0^+$ ) = 0.84 $(eb)^2$ for $^{186}$ Pt [40].",
"The moments of inertia are of the irrotational-flow type: $\\mathcal {J}_k=\\mathcal {J}_0\\sin ^2(\\gamma -2k\\pi /3)$ , with $\\mathcal {J}_0=38.0~\\hbar ^2/\\textrm {MeV}$ .",
"The PRM calculations locate the chemical potential in the middle of the $h_{9/2}$ shell.",
"Frauendorf and Dönau [17] had predicted the appearance of LW for quasiparticles from half-filled shells.",
"Figure: (Color online) Experimental level energies minus a rotor contribution for the (a) Yrast (b) LW and (c) SP bands.",
"(d) Wobbling energy plot for the LW band as a function of spin.",
"Also shown for comparison are results from PRM calculations (dashed, dotted, and dash-dotted lines).",
"The line through the experimental points in (d) is drawn to guide the eye.Figures REF (a) – (c) display the experimental level energies minus a rotor contribution for the Yrast, LW and the SP bands.",
"The general agreement between the experimental level energies and those calculated in the PRM is satisfactory.",
"The wobbling energy, as predicted by PRM (shown in Fig.",
"REF (d)), reproduces the increasing trend, but the value is slightly underestimated as compared to the experiment.",
"Also indicative of wobbling bands is a high reduced E2 transition probability, B(E2), for the n$_{\\omega }$ = 1 $\\rightarrow $ n$_{\\omega }$ = 0 connecting transitions.",
"In Figs.",
"REF (a) and (b), we present the ratios of the transition probabilities B(E2)$_\\text{out}$ /B(E2)$_\\text{in}$ and B(M1)$_\\text{out}$ /B(E2)$_\\text{in}$ , respectively, for these connecting transitions.",
"The measured B(E2)$_\\text{out}$ /B(E2)$_\\text{in}$ ratios are large (up to $\\sim $ 0.7), indicating that the band exhibits the character of a collective quadrupole excitation, which further strengthens our argument that Band (2) is, indeed, the wobbling band.",
"PRM calculations reproduce the B(E2)$_\\text{out}$ /B(E2)$_\\text{in}$ ratios well but the B(M1)$_\\text{out}$ /B(E2)$_\\text{in}$ ratios are overestimated somewhat (note the very different vertical scales in (a) and (b)).",
"Figure: (Color online) (a) B(E2) out _\\text{out}/B(E2) in _\\text{in} ratios vs. spin and (b) B(M1) out _\\text{out}/B(E2) in _\\text{in} vs. spin for the LW →\\rightarrow Yrast connecting transitions.",
"(c) and (d): same as (a) and (b), but for SP →\\rightarrow Yrast connecting transitions; the vertical scale for these is on the right side.",
"The experimental points are shown by black squares connected by black solid line and the blue dashed lines correspond to results from PRM calculations.The results for the transitions from the SP to the Yrast band are presented in Figs.",
"REF (c) and (d).",
"The B(E2)$_\\text{out}$ /B(E2)$_\\text{in}$ ratios for these transitions are much smaller than those for LW $\\rightarrow $ Yrast linking transitions, which further supports the wobbling and signature-partner interpretations for the LW and SP bands.",
"The PRM calculations overestimate the B(M1)/B(E2)$_\\text{in}$ ratios for the LW band and underestimate it for the SP bands.",
"This may be attributed to an incorrect mixing between the wobbling and SP states, which is sensitive to the excitation energies and the ratios between the moments of inertia.",
"In summary, we have observed wobbling motion in the nucleus $^{187}$ Au.",
"Two rotational bands have been identified as corresponding to n$_{\\omega }$ = 0 and n$_{\\omega }$ = 1 wobbling-bands pair.",
"The signature partner of the n$_{\\omega }$ = 0 (yrast) band has also been identified.",
"An increasing wobbling energy, E$_\\textrm {wobb}$ , with spin establishes $^{187}$ Au as the first nucleus in which longitudinal wobbling motion has been observed and clearly distinguished from the signature-partner band.",
"Results from PRM calculations are in good agreement with experimental observations.",
"These results open the A$\\sim $ 190 region as a new arena where this exotic collective mode has now been observed and establish this mode as a general phenomenon over the nuclear chart encompassing many different nuclear orbitals.",
"Continuing experimental efforts are warranted to explore this behavior further.",
"UG acknowledges travel support from CUSTIPEN (China-U.S.",
"Theory Institute for Physics with Exotic Nuclei) which was instrumental in the initiation of theoretical collaboration with QBC.",
"This work has been supported in part by the U.S. National Science Foundation [Grants No.",
"PHY-1713857 (UND) and No.",
"PHY-1203100 (USNA)], and by the U. S. Department of Energy, Office of Science, Office of Nuclear Physics [Contract No.",
"DE-AC02-06CH11357 (ANL), No.",
"DE-FG02-95ER40934 (UND), No.",
"DE-FG02-97ER41033 (UNC), DE-FG02-97ER41041 (TUNL), No.",
"DE-FG02-94ER40834 (Maryland), and No.",
"DE-SC0009971(CUSTIPEN)].",
"The work of QBC was supported by Deutsche Forschungsgemeinschaft (DFG) and National Natural Science Foundation of China (NSFC) through funds provided to the Sino-German CRC 110 “Symmetries and the Emergence of Structure in QCD\" (DFG Grant No.",
"TRR110 and NSFC Grant No.",
"11621131001).",
"This research used resources of ANL's ATLAS facility, which is a DOE Office of Science User Facility."
]
] | 1906.04408 |
[
[
"Online Object Representations with Contrastive Learning"
],
[
"Abstract We propose a self-supervised approach for learning representations of objects from monocular videos and demonstrate it is particularly useful in situated settings such as robotics.",
"The main contributions of this paper are: 1) a self-supervising objective trained with contrastive learning that can discover and disentangle object attributes from video without using any labels; 2) we leverage object self-supervision for online adaptation: the longer our online model looks at objects in a video, the lower the object identification error, while the offline baseline remains with a large fixed error; 3) to explore the possibilities of a system entirely free of human supervision, we let a robot collect its own data, train on this data with our self-supervise scheme, and then show the robot can point to objects similar to the one presented in front of it, demonstrating generalization of object attributes.",
"An interesting and perhaps surprising finding of this approach is that given a limited set of objects, object correspondences will naturally emerge when using contrastive learning without requiring explicit positive pairs.",
"Videos illustrating online object adaptation and robotic pointing are available at: https://online-objects.github.io/."
],
[
"Introduction",
"One of the biggest challenges in real world robotics is robustness and adaptability to new situations.",
"A robot deployed in the real world is likely to encounter a number of objects it has never seen before.",
"Even if it can identify the class of an object, it may be useful to recognize a particular instance of it.",
"Relying on human supervision in this context is unrealistic.",
"Instead if a robot can self-supervise its understanding of objects, it can adapt to new situations when using online learning.",
"Online self-supervision is key to robustness and adaptability and arguably a prerequisite to real-world deployment.",
"Moreover, removing human supervision has the potential to enable learning richer and less biased continuous representations than those obtained by supervised training and a limited set of discrete labels.",
"Unbiased representations can prove useful in unknown future environments different from the ones seen during supervision, a typical challenge for robotics.",
"Furthermore, the ability to autonomously train to recognize and differentiate previously unseen objects as well as to infer general properties and attributes is an important skill for robotic agents.",
"In this work we focus on situated settings (i.e.",
"an agent is embedded in an environment), which allows us to use temporal continuity as the basis for self-supervising correspondences between different views of objects.",
"We present a self-supervised method that learns representations to disentangle perceptual and semantic object attributes such as class, function, and color.",
"We automatically acquire training data by capturing videos with a real robot; a robot base moves around a table to capture objects in various arrangements.",
"Assuming a pre-existing objectness detector, we extract objects from random frames of a scene containing the same objects, and let a metric learning system decide how to assign positive and negative pairs of embeddings.",
"Representations that generalize across objects naturally emerge despite not being given groundtruth matches.",
"Unlike previous methods, we abstain from employing additional self-supervisory training signals such as depth or those used for tracking.",
"The only input to the system are monocular videos.",
"This simplifies data collection and allows our embedding to integrate into existing end-to-end learning pipelines.",
"We demonstrate that a trained Object-Contrastive Network (OCN) embedding allows us to reliably identify object instances based on their visual features such as color and shape.",
"Moreover, we show that objects are also organized along their semantic or functional properties.",
"For example, a cup might not only be associated with other cups, but also with other containers like bowls or vases.",
"Fig.",
"REF shows the effectiveness of online self-supervision: by training on randomly selected frames of a continuous video sequence (top) OCN can adapt to the present objects and thereby lower the object identification error.",
"While the supervised baseline remains at a constant high error rate (52.4%), OCN converges to a 2.2% error.",
"The graph (bottom) shows the object identification error obtained by training on progressively longer sub-sequences of a 200 seconds video.",
"The key contributions of this work are: (1) a self-supervising objective trained with contrastive learning that can discover and disentangle object attributes from video without using any labels; (2) we leverage object self-supervision for online adaptation: the longer our online model looks at objects in a video, the lower the object identification error, while the offline baseline remains with a large fixed error; (3) to explore the possibilities of a system entirely free of human supervision: we let a robot collect its own data, then train on this data with our self-supervised training scheme, and show the robot can point to objects similar to the one presented in front of it, demonstrating generalization of object attributes."
],
[
"Related Work",
"Object discovery from visual media.",
"Identifying objects and their attributes has a long history in computer vision and robotics [44].",
"Traditionally, approaches focus on identifying regions in unlabeled images to locate and identify objects [42], [38], [2], [10], [20].",
"Discovering objects based on the notion of 'objectness' instead of specific categories enables more principled strategies for object recognition [45], [37].",
"Several methods address the challenge to discover, track, and segment objects in videos based on supervised [48] or unsupervised [22], [40], [12] techniques.",
"The spatio-temporal signal present in videos can also help to reveal additional cues that allow to identify objects [49], [17].",
"In the context of robotics, methods also focus on exploiting depth to discover objects and their properties [27], [19].",
"Many recent approaches exploit the effectiveness of convolutional deep neural networks to detect objects [36], [26], [24] and to even provide pixel-precise segmentations [13].",
"While the detection efficiency of these methods is unparalleled, they rely on supervised training procedures and therefore require large amounts of labeled data.",
"Self-supervised methods for the discovery of object attributes mostly focus on learning representations by identifying features in multi-view imagery [6], [23] and videos [49], or by stabilizing the training signal through domain randomization [7], [52].",
"Some methods not only operate on RGB images but also employ additional signals, such as depth [9], [34] or egomotion [1] to self-supervise the learning process.",
"It has been recognized, that contrasting observations from multiple views can provide a view-invariant training signal allowing to even differentiate subtle cues as relevant features that can be leveraged for instance categorization and imitation learning tasks [41].",
"Unsupervised representation learning.",
"Unlike supervised learning techniques, unsupervised methods focus on learning representations directly from data to enable image retrieval [32], transfer learning [53], image denoising [47], and other tasks [8], [21].",
"Using data from multiple modalities, such as imagery of multiple views [41], sound [29], [3], or other sensory inputs [5], along with the often inherent spatio-temporal coherence [7], [35], can facilitate the unsupervised learning of representations and embeddings.",
"For example, [51] explore multiple architectures to compare image patches and [31] exploit temporal coherence to learn object-centric features.",
"[11] rely of spatial proximity of detected objects to determine attraction in metric learning, OCN operates similarly but does not require spatial proximity for positive matches, it does however take advantage of the likely presence of a same object in any pair of frames within a video.",
"[54] also take a similar unsupervised metric learning approach for tracking specific faces, using tracking trajectories and heuristics for matching trajectories and obtain richer positive matches.",
"While our approach is simpler in that it does not require tracking or 3D matching, it could be augmented with extra matching signals.",
"In robotics and other real-world scenarios where agents are often only able obtain sparse signals from their environment, self-learned embeddings can serve as an efficient representation to optimize learning objectives.",
"[30] introduce a curiosity-driven approach to obtain a reward signal from visual inputs; other methods use similar strategies to enable grasping [33] and manipulation tasks [41], or to be pose and background agnostic [15].",
"[28] jointly uses 3D synthetic and real data to learn a representation to detect objects and estimate their pose, even for cluttered configurations.",
"[16] learn semantic classes of objects in videos by integrating clustering into a convolutional neural network.",
"Figure: Object-Contrastive Networks (OCN): by attracting nearest neighbors in embedding space and repulsing others using metric learning, continuous object representations naturally emerge.",
"In a video collected by a robot looking at a table from different viewpoints, objects are extracted from random pairs of frames.",
"Given two lists of objects, each object is attracted to its closest neighbor while being pushed against all other objects.",
"Noisy repulsion may occur when the same object across viewpoint is not matched against itself.",
"However the learning still converges towards disentangled and semantically meaningful object representations."
],
[
"Learning of Object Representations",
"We propose a model called Object-Contrastive Network (OCN) trained with a metric learning loss (Fig.",
"REF ).",
"The approach is very simple: 1) extract object bounding boxes using a general off-the-shelf objectness detector [36], 2) train a deep object model on each cropped image extracted from any random pair of frames from the video, using the following training objective: nearest neighbors in the embedding space are pulled together from different frames while being pushed away from the other objects from any frame (using n-pairs loss [43]).",
"This does not rely on knowing the true correspondence between objects.",
"The fact that this works at all despite not using any labels might be surprising.",
"One of the main findings of this paper is that given a limited set of objects, object correspondences will naturally emerge when using metric learning.",
"One advantage of self-supervising object representation is that these continuous representations are not biased by or limited to a discrete set of labels determined by human annotators.",
"We show these embeddings discover and disentangle object attributes and generalize to previously unseen environments.",
"We propose a self-supervised approach to learn object representations for the following reasons: (1) make data collection simple and scalable, (2) increase autonomy in robotics by continuously learning about new objects without assistance, (3) discover continuous representations that are richer and more nuanced than the discrete set of attributes that humans might provide as supervision which may not match future and new environments.",
"All these objectives require a method that can learn about objects and differentiate them without supervision.",
"To bootstrap our learning signal we leverage two assumptions: (1) we are provided with a general objectness model so that we can attend to individual objects in a scene, (2) during an observation sequence the same objects will be present in most frames (this can later be relaxed by using an approximate estimation of ego-motion).",
"Given a video sequence around a scene containing multiple objects, we randomly select two frames $I$ and $\\hat{I}$ in the sequence and detect the objects present in each image.",
"Let us assume $N$ and $M$ objects are detected in image $I$ and $\\hat{I}$ , respectively.",
"Each of the $n$ -th and $m$ -th cropped object images are embedded in a low dimensional space, organized by a metric learning objective.",
"Unlike traditional methods which rely on human-provided similarity labels to drive metric learning, we use a self-supervised approach to mine synthetic similarity labels."
],
[
"Objectness Detection",
"To detect objects, we use Faster-RCNN [36] trained on the COCO object detection dataset [25].",
"Faster-RCNN detects objects in two stages: first generate class-agnostic bounding box proposals of all objects present in an image (Fig.",
"REF ), second associate detected objects with class labels.",
"We use OCN to discover object attributes, and only rely on the first objectness stage of Faster-R-CNN to detect object candidates."
],
[
"Metric Loss for Object Disentanglement",
"We denote a cropped object image by $x \\in \\mathcal {X}$ and compute its embedding based on a convolutional neural network $f(x): \\mathcal {X} \\rightarrow K$ .",
"Note that for simplicity we may omit $x$ from $f(x)$ while $f$ inherits all superscripts and subscripts.",
"Let us consider two pairs of images $I$ and $\\hat{I}$ that are taken at random from the same contiguous observation sequence.",
"Let us also assume there are $n$ and $m$ objects detected in $I$ and $\\hat{I}$ respectively.",
"We denote the $n$ -th and $m$ -th objects in the images $I$ and $\\hat{I}$ as $x_n^{I}$ and $x_m^{\\hat{I}}$ , respectively.",
"We compute the distance matrix $D_{n,m} = \\sqrt{(f_{n}^{I} - f_{m}^{\\hat{I}}})^2,~n\\in 1..N,~m\\in 1..M$ .",
"For every embedded anchor $f_{n}^{I},~n\\in 1..N$ , we select a positive embedding $f_{m}^{\\hat{I}}$ with minimum distance as positive: $f_{n+}^{\\hat{I}} = argmin(D_{n,m})$ .",
"Given a batch of (anchor, positive) pairs $\\lbrace (x_i, x_i^+)\\rbrace _{i=1}^N$ , the n-pair loss is defined as follows [43]: $\\mathcal {L}_{N-pair}\\big (\\lbrace (x_i, x_i^+)\\rbrace _{i=1}^N;f\\big ) = \\\\\\frac{1}{N} \\sum _{i=1}^N log \\Big (1 + \\sum _{j \\ne i} exp(f_i^\\intercal f_j^+ - f_i^\\intercal f_i^+) \\Big )$ The loss learns embeddings that identify ground truth (anchor, positive)-pairs from all other (anchor, negative)-pairs in the same batch.",
"It is formulated as a sum of softmax multi-class cross-entropy losses over a batch, encouraging the inner product of each (anchor, positive)-pair ($f_i$ , $f_i^+$ ) to be larger than all (anchor, negative)-pairs ($f_i$ , $f_{j\\ne i}^+$ ).",
"The final OCN training objective over a sequence is the sum of npairs losses over all pairs of individual frames: $\\mathcal {L}_{OCN} = \\mathcal {L}_{N-pair}\\big (\\lbrace (x_n^{I}, x_{n+}^{\\hat{I}})\\rbrace _{n=1}^N;f\\big ) \\\\+ \\mathcal {L}_{N-pair}\\big (\\lbrace (x_m^{\\hat{I}}, x_{m+}^{I})\\rbrace _{m=1}^M;f\\big )$"
],
[
"Architecture and Embedding Space",
"OCN takes a standard ResNet50 architecture until layer global_pool and initializes it with ImageNet pre-trained weights.",
"We then add three additional ResNet convolutional layers and a fully connected layer to produce the final embedding.",
"The network is trained with the n-pairs metric learning loss as discussed in Sec.",
"REF .",
"Our architecture is depicted in Fig.",
"REF and Fig.",
"REF .",
"Figure: Models and baselines: for comparison purposes all models evaluated in Sec.",
"share the same architecture of a standard ResNet50 model followed by additional layers.",
"While the architectures are shared, the weights are not across models.",
"While the unsupervised model (left) does not require supervision labels, the 'softmax' baseline as well as the supervised evaluations (right) use attributes labels provided with each object.",
"We evaluate the quality of the embeddings with two types of classifiers: linear and nearest neighbor.Object-centric Embeding Space: By using multiple views of the same scene and by attending to individual objects, our architecture allows us to differentiate subtle variations of object attributes.",
"Observing the same object across different views facilitates learning invariance to scene-specific properties, such as scale, occlusion, lighting, and background, as each frame exhibits variations of these factors.",
"The network solves the metric loss by representing object-centric attributes, such as shape, function, or color, as these are consistent for (anchor, positive)-pairs, and dissimilar for (anchor, negative)-pairs."
],
[
"Discussion",
"One might expect that this approach may only work if it is given an initialization so that matching the same object across multiple frames is more likely than random chance.",
"While ImageNet pretraining certainly helps convergence as shown in Tab.",
"REF , it is not a requirement to learn meaningful representations as shown in Sec. .",
"When all weights are random and no labels are provided, what can drive the network to consistently converge to meaningful embeddings?",
"We estimate that the co-occurrence of the following hypotheses drives this convergence: (1) objects often remains visually similar to themselves across multiple viewpoints, (2) limiting the possible object matches within a scene increases the likelihood of a positive match, (3) the low-dimensionality of the embedding space forces the model to generalize by sharing abstract features across objects, (4) the smoothness of embeddings learned with metric learning facilitates convergence when supervision signals are weak, and (5) occasional true-positive matches (even by chance) yield more coherent gradients than false-positive matches which produce inconsistent gradients and dissipate as noise, leading over time to an acceleration of consistent gradients and stronger initial supervision signal."
],
[
"Experiments",
"Online Results: we quantitatively evaluate the online adaptation capabilities of our model through the object identification error of entirely novel objects.",
"In Fig.",
"REF we show that a model observing objects for a few minutes from different angles can self-teach to identify them almost perfectly while the offline supervised approach cannot.",
"OCN is trained on the first 5, 10, 20, 40, 80, and 160 seconds of the 200 seconds video, then evaluated on the identification error of the last 40 seconds of the video for each phase.",
"The supervised offline baseline stays at a 52.4% error, while OCN improves down to 2% error after 80s, a 25x error reduction.",
"Robotic Experiments: here we let a robot collect its own data by looking at a table from multiple angles (Fig.",
"REF and Fig.",
"REF ).",
"It then trains itself with OCN on that data, and is asked to point to objects similar to the one presented in front of it.",
"Objects can be similar in terms of shape, color or class.",
"If able to perform that task, the robot has learned to distinguish and recognize these attributes entirely on its own, from scratch and by collecting its own data.",
"We find in Tab.",
"REF that the robot is able to perform the pointing task with 72% recognition accuracy of 5 classes, and 89% recognition accuracy of the binary is-container attribute.",
"Offline Analysis: to analyze what our model is able to disentangle, we quantitatively evaluate performance on a large-scale synthetic dataset with 12k object models (e.g.",
"Fig.",
"REF ), as well as on a real dataset collected by a robot and show that our unsupervised object understanding generalizes to previously unseen objects.",
"In Tab.",
"REF we find that our self-supervised model closely follows its supervised equivalent baseline when trained with metric learning.",
"As expected the cross-entropy/softmax supervised baseline approach performs best and establishes the error lower bound while the ResNet50 baseline are upper-bound results."
],
[
"Data Collection and Training",
"We generated three datasets of real and synthetic objects for our experiments.",
"For the real data we arrange objects in table-top configurations and use frames from continuous camera trajectories.",
"The labeled synthetic data is generated from renderings of 3D objects in a similar configuration.",
"Details about the datasets are reported in Tab.",
"REF ."
],
[
"Real Data for Online Training",
"For the online adaptation experiment, we captured videos of table-top object configurations in the 5 environments (categories): kids room, kitchen, living room, office, and work bench (Figs.",
"REF , REF , and REF ).",
"We show objects common to each environment (e.g.",
"toys for kids room, tools for work bench) and arrange them randomly; we captured 3 videos for each environment and used 75 unique objects.",
"To allow capturing the objects from multiple view points we use a head-mounted camera and interact with the objects (e.g.",
"turning or flipping them).",
"Additionally, we captured 5 videos of more challenging object configurations (referred to as `challenging') with cluttered objects or where objects are not permanently in view.",
"Finally, we selected 5 videos from the Epic-Kitchens [4] dataset to show that OCN can also operate on even more realistic video sequences.",
"From all these videos we take the first 200 seconds and sample the sequence with 15 FPS to extract 3,000 frames.",
"We then use the first 2,400 frames (160s) for training OCN and the remaining 600 frames (40s) for evaluation.",
"We manually select up to 30 reference objects (those we interacted with) as cropped images for each video in order of their appearance from the beginning of the video (Fig.",
"REF ).",
"Then we use object detection to find the bounding boxes of these objects in the video sequence and manually correct these boxes (add, delete) in case object detection did not identify an object.",
"This allows us to prevent artifacts of the object detection to interfere with the evaluation of OCN..",
"Figure: We use 187 unique object instance in the real world experiments: 110 object for training (left), 43 objects for test (center), and 34 objects for validation (right).",
"The degree of similarity makes it harder to differentiate these objects."
],
[
"Automatic Real Data Collection",
"To explore the possibilities of a system entirely free of human supervision we automated the real world data collection by using a mobile robot equipped with an HD camera (Fig.",
"REF ).",
"For this dataset we use 187 unique object instances spread across six categories including `balls', `bottles & cans', `bowls', `cups & mugs', `glasses', and `plates'.",
"Tab.",
"REF provides details about the number of objects in each category and how they are split between training, test, and validation.",
"Note that we distinguish between cups & mugs and glasses categories based on whether it has a handle.",
"Fig.",
"REF shows our entire object dataset.",
"At each run, we place about 10 objects on the table and then trigger the capturing process by having the robot rotate around the table by 90 degrees (Fig.",
"REF ).",
"On average 130 images are captured at each run.",
"We select random pairs of frames from each trajectory for training OCN.",
"We performed 345, 109, and 122 runs of data collection for training, test, and validation dataset.",
"In total 43,084 images were captured for OCN training and 15,061 and 16,385 were used for test and validation, respectively."
],
[
"Synthetic Data Generation",
"To generate diverse object configurations we use 12 categories (airplane, car, chair, cup, bottle, bowl, guitars, keyboard, lamp, monitor, radio, vase) from ModelNet [50].",
"The selected categories cover around 8k models of the 12k models available in the entire dataset.",
"ModelNet provides the object models in a 80-20 split for training and testing.",
"We further split the testing data into models for test and validation, resulting in a 80-10-10 split for training, validation, and test.",
"For validation purposes, we manually assign each model labels describing the semantic and functional properties of the object, including the labels `class', `has lid', `has wheels', `has buttons', `has flat surface', `has legs', `is container', `is sittable', `is device'.",
"We randomly define the number of objects (up to 20) in a scene (Fig.",
"REF ).",
"Further, we randomly define the positions of the objects and vary their sizes, both so that they do not intersect.",
"Additionally, each object is assigned one of eight predefined colors.",
"We use this setup to generate 100K scenes for training, and 50K scenes for each, validation and testing.",
"For each scene we generate 10 views and select random combination of two views for detecting objects.",
"In total we produced 400K views (200K pairs) for training and 50K views (25K pairs) for each, validation and testing."
],
[
"Training",
"OCN is trained based on two views of the same synthetic or real scene.",
"We randomly pick two frames of a video sequence and detect objects to produce two sets of cropped images.",
"The distance matrix $D_{n,m}$ (Sec.",
"REF ) is constructed based on the individually detected objects for each of the two frames.",
"The object detector was not specifically trained on any of our datasets.",
"As the number of detected objects per view varies, we reciprocally use both frames to find anchors and their corresponding positives as discussed in Sec.",
"REF .",
"Across our experiments, we observed an embeddings size of 32-64 provides optimal results; the OCN training converged after 600k-1.2M iterations."
],
[
"Experimental Results",
"We evaluated the effectiveness of OCN embeddings on identifying objects through self-supervised online training, a real robotics pointing tasks, and large-scale synthetic data."
],
[
"Online Object Identification",
"Our self-supervised online training scheme enables to train and to evaluate on unseen objects and scenes.",
"This is of utmost importance for robotic agents to ensure adaptability and robustness in real world scenes.",
"To show the potential of our method for these situations we use OCN embeddings to identify instances of objects across multiple views and over time.",
"Figure: Evaluation of online adaptation: we train an OCN on the first 5, 10, 20, 40, 80, and 160 seconds of each 200 second test video and then evaluate on the remaining 40 seconds.",
"Here we report the lowest average error of all videos (over 1000K iterations) of online adaptation.",
"Results are shown for 5 and 7 categories and compared to the ResNet50 baseline.We use sequences of videos showing objects in random configurations in different environments (Sec.",
"REF , Fig.",
"REF ) and train an OCN on the first 5, 10, 20, 40, 80, and 160 seconds of a 200 seconds video.",
"Our dataset provides object bounding boxes and unique identifiers for each object as well as reference objects and their identifiers.",
"The goal of this experiment is to assign the identifier of a reference object to the matching object detected in a video frame.",
"We evaluate the identification error (ground truth index vs. assigned index) of objects present in the last 40 seconds of each video and for each training phase to then compare our results to a ResNet50 (2048-dimensional vectors) baseline.",
"We train an OCN for each video individually.",
"Therefore, we only split our dataset into validation and testing data.",
"For the categories kids room, kitchen, living room, office, and work bench we use 2 videos for validation and 1 video for testing; for the categories `challenging' and epic kitchen we use 3 videos for validation and 2 for testing.",
"We jointly train on the validation videos to find meaningful hyperparameters across the categories and use the same hyperparameters for the test videos.",
"Fig.",
"REF shows the same video frames of two scenes from our dataset.",
"Objects with wrongly matched indices are shown with a red bounding box, correctly matched objects are shown with random colors.",
"In Fig.",
"REF and Tab.",
"REF we report the average error of OCN object identification across our videos compared to the ResNet50 baseline.",
"As the supervised model cannot adapt to unknown objects OCN outperforms this baseline by a large margin.",
"Furthermore, the optimal result among the first 50K training iterations closely follows the overall optimum obtained after 1000K iterations.",
"We report results for 5 categories (kids room, kitchen, living room, office, work bench), that we specifically captured for evaluating OCN and the whole dataset (7 categories).",
"The latter data also shows cluttered objects which are more challenging to detect.",
"To evaluate the degree of how object detection is limiting application of OCN we counted the number of manually added bounding boxes of the evaluation sequences.",
"On average the evaluation sequences of the 5 categories have 5,122 boxes (468 added, 9.13%), while the whole dataset (7 categories) has 5,002 boxes on average (1183 added, 25.94%).",
"Table: Evaluation of online adaptation: we report the lowest error among 50K and 1000K iterations of online adaptation in %.",
"[S], [A] = average error for 5 and 7 categories.Fig.",
"REF illustrates how objects of one view (anchors) are matched to the objects of another view.",
"We can find the nearest neighbors (positives) in the scene through the OCN embedding space as well as the closest matching objects with descending similarity (negatives).",
"For our synthetic data we report the quality of finding corresponding objects in Tab.",
"REF and differentiate between `attribute errors', that indicate a mismatch of specific attributes (e.g.",
"a blue cup is associated to a red cup), and `object matching errors', which measure when objects are not of the same instance.",
"An OCN embedding significantly improves detecting object instances across multiple views.",
"Table: Object correspondences errors: attribute error indicates a mismatch of an object attribute, while an object matching error is measured when the matched objects are not the same instance."
],
[
"Robot Experiment",
"To evaluate OCN for real world robotics scenarios we defined a robotics pointing task.",
"The goal of the task is to enable a robot to point to an object that it deems most similar to the object directly in front of it (Fig.",
"REF ).",
"The objects on the rear table are randomly selected from the object categories (Tab.",
"REF ).",
"We consider two sets of these target objects.",
"The quantitative experiment in Tab.",
"REF uses three query objects per category and is ran three times for each combination of query and target objects ($3 \\times 2 \\times 18 = 108$ experiments performed).",
"The full set for one of the three runs is shown in Fig.",
"REF .",
"A quantitative evaluation of OCN performance for this experiment is shown in Tab.",
"REF .",
"We report on errors related to `class' and `container' attributes.",
"While the trained OCN model is performing well on the most categories, it has difficulty on the object classes `cups & mugs' and `glasses'.",
"These categories are generally mistaken with the category `bowls'.",
"As a result the network performs much better in the attribute `container' since all the three categories `bowls', `bottles & cans', and 'glasses' refer to the same attribute.",
"At the beginning of each experiment the robot captures a snapshot of the scene.",
"We then split the captured image into two images: the upper portion of the image that contains the target object set and the lower portion of the image that only contains the query object.",
"We detect the objects and find the nearest neighbor of the query object in the embedding space to find the closest match.",
"Figure: The robot experiment of pointing to the best match of a query object (placed in front of the robot on the small table).",
"The closest match is selected from two sets of target objects, placed on the table behind the query object.",
"The first and the second row correspond to the experiment for the first and second target set.",
"Images with green frame indicate cases where both the `class' and `container' attributes are matched correctly.",
"Blue frames show where only the `container' attribute is matched correctly and red frames indicate neither attribute is matched."
],
[
"Object Attribute Classification",
"One way to evaluate the quality of unsupervised embeddings is to train attribute classifiers on top of the embedding using labeled data.",
"Note however, that this may not entirely reflect the quality of an embedding because it is only measuring a discrete and small number of attributes while an embedding may capture more continuous and larger number of abstract concepts.",
"Classifiers: we consider two types of classifiers to be applied on top of existing embeddings in this experiment: linear and nearest-neighbor classifiers.",
"The linear classifier consists of a single linear layer going from embedding space to the 1-hot encoding of the target label for each attribute.",
"It is trained with a range of learning rates and the best model is retained for each attribute.",
"The nearest-neighbor classifier consists of embedding an entire `training' set, and for each embedding of the evaluation set, assigning to it the labels of the nearest sample from the training set.",
"Nearest-neighbor classification is not a perfect approach because it does not necessarily measure generalization as linear classification does and results may vary significantly depending on how many nearest neighbors are available.",
"It is also less subject to data imbalances.",
"We still report this metric to get a sense of its performance because in an unsupervised inference context, the models might be used in a nearest-neighbor fashion (e.g.",
"as in Sec.",
"REF ).",
"Figure: An OCN embedding organizes objects along their visual and semantic features.",
"For example, a red bowl as query object is associated with other similarly colored objects and other containers.",
"The leftmost object (black border) is the query object and its nearest neighbors are listed in descending order.",
"The top row shows renderings of our synthetic dataset, while the bottom row shows real objects.",
"Please note that these are the nearest neighbors among all objects in the respective dataset.Baselines: we compare multiple baselines (BL) in Tab.",
"REF and Tab.",
"REF .",
"The `Softmax' baseline refers to the model described in Fig.",
"REF , i.e.",
"the exact same architecture as for OCN except that the model is trained with a supervised cross-entropy/softmax loss.",
"The `ResNet50' baseline refers to using the unmodified outputs of the ResNet50 model [14] (2048-dimensional vectors) as embeddings and training a nearest-neighbor classifier as defined above.",
"We consider `Softmax' and `ResNet50' baselines as the lower and upper error-bounds for standard approaches to a classification task.",
"The `OCN supervised' baseline refers to the exact same OCN training described in Fig.",
"REF , except that the positive matches are provided rather than discovered automatically.",
"`OCN supervised' represents the metric learning upper bound for classification.",
"Finally we indicate as a reference the error rates for random classification.",
"Table: Attributes classification errors: using attribute labels, we train either a linear or nearest-neighbor classifier on top of existing fixed embeddings.",
"The supervised OCN is trained using labeled positive matches, while the unsupervised one decides on positive matches on its own.",
"All models here are initialized and frozen with ImageNet-pretrained weights for the ResNet50 part of the architecture, while the additional layers above are random and trainable.Results: we quantitatively evaluate our unsupervised models against supervised baselines on the labeled synthetic datasets (train and test) introduced in Sec.",
"REF .",
"Note that there is no overlap in object instances between the training and the evaluation set.",
"The first take-away is that unsupervised performance closely follows its supervised baseline when trained with metric learning.",
"As expected the cross-entropy/softmax approach performs best and establishes the error lower bound while the ResNet50 baseline are upper-bound results.",
"In Fig.",
"REF and Sec.",
", we show results of nearest neighbor objects discovered by OCN."
],
[
"Conclusion and Future Work",
"We introduced a self-supervised objective for object representations able to disentangle object attributes, such as color, shape, and function.",
"We showed this objective can be used in online settings which is particularly useful for robotics to increase robustness and adaptability to unseen objects.",
"We demonstrated a robot is able to discover similarities between objects and pick an object that most resembles one presented to it.",
"In summary, we find that within a single scene with novel objects, the more our model looks at objects, the more it can recognize them and understand their visual attributes, despite never receiving any labels for them.",
"Current limitations include relying on all objects to be present in all frames of a video.",
"Relaxing this limitation will allow to use the model in unconstrained settings.",
"Additionally, the online training is currently not real-time as we first set out to demonstrate the usefulness of online-learning in non-real-time.",
"Real-time training requires additional engineering that is beyond the scope of this research.",
"Finally, the model currently relies on an off-the-self object detector which might be noisy, an avenue for future research is to back-propagate gradients through the objectness model to improve detection and reduce noise."
],
[
"Supplementary Material",
"In the following we provide details on our datasets and report additional experiments and results."
],
[
"Random Weights",
"We find in Tab.",
"REF that models that are not pretrained with ImageNet supervision perform worse but still yield reasonable results.",
"This indicates that the approach does not rely on a good initialization to bootstrap itself without labels.",
"Even more surprisingly, when freezing the weights of the ResNet50 base of the model to its random initialization, results degrade but still remain far below chance as well as below the 'ResNet50 embeddings' baseline.",
"Obtaining reasonable results with random weights has already been observed in prior work such as [18], [39] and [46].",
"Table: Results with random weights (no ImageNet pre-training)"
]
] | 1906.04312 |
[
[
"The Demand Query Model for Bipartite Matching"
],
[
"Abstract We introduce a `concrete complexity' model for studying algorithms for matching in bipartite graphs.",
"The model is based on the \"demand query\" model used for combinatorial auctions.",
"Most (but not all) known algorithms for bipartite matching seem to be translatable into this model including exact, approximate, sequential, parallel, and online ones.",
"A perfect matching in a bipartite graph can be found in this model with O(n^{3/2}) demand queries (in a bipartite graph with n vertices on each side) and our main open problem is to either improve the upper bound or prove a lower bound.",
"An improved upper bound could yield \"normal\" algorithms whose running time is better than the fastest ones known, while a lower bound would rule out a faster algorithm for bipartite matching from within a large class of algorithms.",
"Our main result is a lower bound for finding an approximately maximum size matching in parallel: A deterministic algorithm that runs in n^{o(1)} rounds, where each round can make at most n^{1.99} demand queries cannot find a matching whose size is within n^{o(1)} factor of the maximum.",
"This is in contrast to randomized algorithms that can find a matching whose size is $99\\%$ of the maximum in O(\\log n) rounds, each making n demand queries."
],
[
"Introduction",
"In the (unweighted, bipartite) matching problem, a bipartite graph with $n$ left vertices and $n$ right vertices is given, and the problem is to find a maximum-size matching: a set of edges of the graph of largest cardinality such that no two of which share a vertex.",
"This problem has numerous applications and has been studied extensively in a variety of computational models including sequential, parallel, online, and approximate.",
"Many extensions of the problem (e.g.",
"to weighted or to non-bipartite graphs) have been studied as well.",
"The fastest known deterministic algorithm for the problem [8] is close to half a century old and runs in time $O(n^{5/2})$ .",
"It is a long standing open problem whether this running time may be improved, but a faster randomized algorithm whose running time is $n^\\omega $ (where $\\omega =2.3...$ is the matrix multiplication exponent) was given in [16].",
"Another long standing open problem is whether there exists a deterministic NC algorithm (a parallel algorithm running in poly-logarithmic time using a polynomial number of processors) for finding a maximum matching.",
"Randomized such algorithms were given in [9], [17].",
"While an enormous amount of algorithmic work was done on many variants of the matching problem, there is currently no hope for proving hardness results: we simply lack any tools that can prove lower bounds for general algorithms.",
"One approach for progress is to define a concrete model of computation – one that is strong enough to capture many of the known algorithms – and try to understand the complexity in that model.",
"This is our approach here."
],
[
"The Demand Query Model",
"Our computational model is inspired by an economic point of view.",
"We consider the right vertices of our bipartite graph to be items and the left vertices to be agents, each who is interested in acquiring exactly a single item from some subset of items – those that are adjacent to it in the graph.",
"In a general economic setting, a “demand query” (see e.g., [3]) asks an economic agent the following type of question: suppose that each item $j$ , from some set of items, could be purchased for price $p_j$ – which set of items would you demand to purchase?",
"In our setting, we assume that our agents are “unit demand”, i.e.",
"desire a single item, and as our graphs are unweighted our agent will simply choose the least expensive item.",
"Thus in our setting we consider the following type of query on a biprtite graph.",
"Definition 1 A Demand Query accepts a left vertex $v$ and an order on the right vertices $(u_1...u_{n})$ and returns the index of the first vertex $i$ in the order such that there is an edge $(v,u_i)$ in the graph, or 0 to denote that none exists.For compatibility with the notion of demand queries, we only allow $v$ to be a left vertex.",
"In the graph context it may be natural to consider also variants of our model where $v$ can be any vertex in the graph.",
"It turns out that our main lower bound applies to this stronger model as well.",
"Desiring a concrete model, the only cost of an algorithm that we will count is the number of demand queries required.",
"(It will turn out that none of our upper bounds have any other significant algorithmic costs.)",
"In particular every problem in this model has an upper bound of $O(n^2)$ as a single demand query can certainly tell us whether an edge $(v,u)$ exists in the graph.",
"We first observe that many of the classic algorithms for matching can be “implemented” in this model, including augmenting paths methods and primal-dual auction-like algorithms [2], [6].",
"These can give us the following basic upper bound: Theorem 1 A maximum size matching in a bipartite graph with $n$ left and $n$ right vertices can be found using $O(n^{3/2})$ demand queries."
],
[
"From the Demand Query Model to General Algorithms",
"When converting an algorithm in the demand query model into a “real algorithm” that does not have a demand query as a primitive, but rather must implement it over a normal computation model, each demand query can be trivially implemented in $O(n)$ time by going over the edges of the queried vertex and finding the one with minimum rank in the list.",
"Using this $O(n^{3/2})$ bound with this simulation matches the currently best $O(n^{5/2})$ algorithm for bipartite matching [8].One may certainly attempt to obtain faster simulations of the demand query model by normal algorithms using, say, clever data structures, but we have not been able to do so in general.",
"While we generally focus on the case of dense graphs, we note that for sparse graphs with $m << n^2$ edges, one may also obtain the improved known upper bound of $O(mn^{1/2})$ , as a demand query to vertex $v$ can in fact be simulated in $O(d_v)$ time, where $d_v$ is the degree of $v$ in the graph.Getting the $O(mn^{1/2})$ total running time is slightly non-trivial since this requires that high-degree vertices are not queried too often by the simulated algorithm.",
"In case that they are, some data structure work will be needed to even things out in an amortized sense [18].",
"We also find that the complexity of finding an approximately maximal matching, one whose size is at least $(1-\\epsilon )$ of the maximal in the demand query model is only $O(n/\\epsilon )$ .",
"Again, with the direct simulation of demand queries by “normal” algorithms, this implies the (known) $O(n^2)$ algorithm (linear in the input size) for finding a matching of size at least $99\\%$ of maximal.",
"While most algorithms for bipartite matching seem to be translatable into this model, a known algorithmic technique that does not seem to fall under this model is an algebraic one relying on matrix multiplication.",
"A simple example is solving the decision problem of whether a perfect matching exists using randomization to test whether the symbolic determinant of the graph is identically zero [14] which can be done in time $O(n^\\omega )$ (where $\\omega =2.3...$ is the matrix multiplication exponent).",
"A randomized algorithm with the same running time that actually finds a maximum matching is given in [16].The distinction between the decision problem and the search problem is not significant for anything in this paper though: all upper bounds actually find a matching, and the lower bound applies even for the decision variant.",
"These algorithms are all randomized and it is not clear whether deterministic algorithms can match this performance.",
"Another known technique that dose not seem to fall in this model is using interior point methods, which in [15] was used for giving an $O(m^{10/7})$ algorithm, where $m$ is the total number of edges in the graph, which beats the $O(n^{5/2})$ bound for very sparse graphs."
],
[
"Lower Bounds?",
"A lower bound of $\\Omega (n)$ for bipartite matching in the demand query model is trivial, even for the approximate version of the problemE.g.",
"for distinguishing between a maximum matching of any positive size and zero size, i.e.",
"whether the graph is empty.. Our main open problem is to close the gap between this trivial lower bound and the $O(n^{3/2})$ upper bound: either improve the upper bound, which would likely imply faster normal algorithmsIn principle, as the demand query model formally allows “non-uniform” algorithms, this simulation by normal algorithms is not a formal theorem, but we do not know of any example where this non-uniformity is used., or prove a lower bound, which would rule out faster algorithms from a rather wide class of algorithms.",
"Open Problem 1: Does there exists a faster algorithm for finding a maximal matching in the demand query model?",
"(I.e.",
"one that uses $o(n^{3/2})$ queries, or perhaps even $O(n)$ queries?)",
"A hint that an answer may be nontrivial is that the nondeterministic (certificate) complexity of maximum matching in this model is only $\\Theta (n)$ : a maximum matching can be given by listing the $n$ edges in it, where each edge can be certified by a single demand query, and, by Hall's theorem, the maximality of said matching of size $k$ can be certified by exhibiting a set vertices (left and right) of total size $2n-k$ with no edges between them, which can be certified using one demand query for each left vertex in the set."
],
[
"Related Models",
"The demand query model lies between two well studied models: Boolean decision trees [5] and communication complexity [12].",
"In the Boolean decision tree model the allowed queries are only to individual edges, i.e.",
"may ask whether an edge $(v,u)$ exists in the graph.",
"In the communication complexity model the left vertices are treated as agents that each “knows” the set of edges connected to it and considers the amount of communication needed between these agents.",
"This would be equivalent to a decision tree model that allows arbitrary queries about the set of egdes adjacent to any single left vertex.",
"In the decision tree model, it is not difficult to see that the complexity of even determining whether a perfect matching exists is $\\Theta (n^2)$ , even non-deterministically (for certifying none existence).",
"In the communication complexity model what is known is identical to what was described above in the demand query model, and one may consider the demand query model as an intermediate step towards understanding the complexity in the communication complexity model.",
"A model that turns out to be equivalent, up to log factors, to the demand query model is a model that allows “OR” queries on the edges of a single left vertex.",
"I.e.",
"an “OR” query specifies a left vertex $v$ and a set $S$ of right vertices and asks whether there exists an edge between $v$ and some vertex in $S$ .",
"This binary query is clearly weaker than the demand query, but it is not difficult to use binary search to simulate demand queries using $O(\\log n)$ “OR” queries.",
"It will be convenient to prove our lower bound in the binary “OR” query model which them implies the lower bound in the demand query model.",
"A slightly stronger model (which is incomparable to the communication complexity model, but generally seems rather weaker) would allow asking about the “OR” of an arbitrary set of edges, not just those connected to a single vertex.",
"I.e.",
"such a query can present an arbitrary subset of $n^2$ possible edges and ask whether at least one of them exists in the graph.",
"It is known [13] that such a model is equivalent, up to log factors, to the following complexity measure in Boolean decision trees: the maximal number of 1-answers on the path to any leaf of the tree.",
"An even stronger model would allow asking for an $OR$ of a subset of edges and their negations (e.g.",
"“is either $(u_1,v_1)$ an edge in the graph OR $(u_2,v_2)$ not an edge?”).",
"It can be shown that this model captures, up to a log factor, the logarithm of minimum Boolean decision tree size.",
"We do not know any improved bounds in these models and proving an improved upper bound in one of them is an open problem as well."
],
[
"Parallel Algorithms in the Demand Query Model",
"Up to this point we have introduced our general model, its motivation, and the main open problem.",
"We now move onto the issue of the parallel complexity in this model, which is our main technical result.",
"A parallel algorithm in the demand query model proceeds in rounds, where at each round multiple demand queries can be made.",
"The point is that the identity of all the queries within a single round must all be determined only based on the outcomes of queries from the previous rounds.",
"The two basic parameters in a parallel algorithm are thus the number of rounds and the (maximum allowed) number of queries per round.",
"Any sequential algorithm that makes a total of $q$ queries can be viewed as a parallel algorithm with $q$ rounds, each of 1 query.",
"Every parallel algorithm with $r$ rounds of $q$ queries each, can be trivially converted to a sequential algorithm with $r \\cdot q$ queries, but the converse is not necessarily true.",
"Any problem (on graphs with with $n^2$ possible edges) can be trivially solved with a single round of $n^2$ queries.",
"The question that we ask is whether a maximum matching in a bipartite graph (or at least an approximation thereof) can be found with “few” rounds or “few” queries each, say is $n^{o(1)}$ rounds of $n^{1+o(1)}$ demand queries per round.",
"(Recall that the trivial lower bound on the total number of queries is $\\Omega (n)$ .)",
"In [7] a randomized algorithm was presented that operates within fully within this model and within $O(\\log n / \\epsilon ^2)$ rounds, each of $n$ demand queries (a single query per left vertex) outputs, with high probability, a matching whose size is at least $(1-\\epsilon )$ fraction of the maximum matching.",
"Open Problem 2: Does there exists a randomized parallel algorithm that uses “few” (even $O(\\log n)$ ) rounds each with “few” (even $O(n)$ ) demand queries and outputs, with high probability, a perfect matching if one exists."
],
[
"Main Result: A Lower Bound for Deterministic Parallel Algorithms",
"Our main result is a lower bound showing that randomization is crucial here: no deterministic algorithm that uses “few” rounds of “few” queries can get a decent approximation to the maximum matching size.",
"In fact our lower bounds show that the algorithms cannot even distinguish between graphs with a perfect matching and those with a small maximum matching size.",
"We provide a general tradeoff between the number of rounds, the number of queries per round, and the level of approximation possible which in particular implies: Instance of main theorem with a specific choice of parameters: Deterministic algorithms with $n^{1/7}$ -rounds each using $n^{8/7}$ demand queries cannot approximate the maximum matching size within a factor of $n^{1/7}$ .",
"Beyond tightening our tradeoffs, a natural challenge is to extend our lower bound to the stronger communication complexity model.",
"Our lower bound uses a direct adversary techniqe which does not seem to extend to the communication model.",
"Open Problem 3: Does there exists a deterministic communication protocol between the $n$ agents (left vertices) that uses “few” (even $O(\\log n)$ ) rounds of communication where in each round each player sends “few” (even $O(1)$ ) bits of communication and outputs, with high probability, a perfect matching if one exists.",
"Improving upon [7], [1], in [4] a nearly logarithmic lower bound on the number of rounds was proven if each round allows, say, $polylog(n)$ bits of communication per player.",
"This bound is exponentially lower than the query bound that we obtain, and unlike our bound, applies also to randomized protocols.",
"A gap between deterministic and randomized simultaneous communication protocols was exhibited in [7]."
],
[
"Warmup: The Greedy Online Algorithm",
"We start by looking at the simplest greedy algorithm that also has the advantage of being “online” [10]: we can have the left vertices come one after another in an online fashion, and as each vertex arrives we immediately match him to an arbitrary yet unmatched right vertex.",
"This can be done using a single demand query per vertex: the query will place all the unmatched vertices (in an arbitrary order) before all the matched ones.",
"This is known to produce a matching whose size is at least 1/2 of the maximum size.",
"A randomized variant fixes a random order on the right vertices and allocates the first unmatched vertex rather in this order than an arbitrary one (still a single demand query per vertex, but now with the fixed random order on the yet unmatched vertices), and it is known that this produces, in expectation, a $(1-1/e)$ -approximation to the maximum matching [10].",
"Proposition 1 There is a deterministic online algorithm for maximum matching that uses 1-demand query per vertex and provides a 1/2-approximately maximum matching.",
"There is a randomized online algorithm for maximum matching that uses 1-demand query per vertex and provides a (1-1/e)-approximately maximum matching."
],
[
"The Ascending Auction Algorithm",
"We now present the “ascending auction” algorithm of [6], [2] which is naturally described in the demand-query model, and may be viewed as motivating this model.This algorithm is quite naturally interpreted as an ascending auction that can be used also in more general scenarios [11] and in our scenario turns out to also has nice incentive properties [6].",
"It will be beneficial to present the algorithm as an approximation algorithm that is parametrized by the desired approximation ratio.",
"Theorem 2 For every parameter $0 < \\epsilon < 1$ , there exists an algorithm that uses $O(n/\\epsilon )$ demand queries and produces a matching whose size is at least $(1-\\epsilon )$ fraction of the maximum size matching.",
"Here is the algorithm essentially due to [6]: For each left vertex $u$ initialize “prices” $p_u=0$ .",
"Initialize the matching $M=\\emptyset $ .",
"Initialize the set of “TBD” vertices $A$ to be the set of all right vertices.",
"While $A$ is not empty: Pick an arbitrary vertex $v \\in A$ and remove it from $A$ .",
"Ask a demand query on $v$ with the order on $u$ 's induced by increasing values of $p_u$ (lowest price first, and ties broken arbitrarily), and let $u$ be the answer to that query.",
"If $p_u < 1$ : If for some vertex $v^{\\prime }$ we have that $(v^{\\prime },u)\\in M$ , insert $v^{\\prime }$ into $A$ and remove $(v^{\\prime },u)$ from $M$ .",
"Insert $(v,u)$ into $M$ .",
"Increase $p_u$ by $\\epsilon $ Output $M$ For completeness, the correctness of this algorithm is sketched below.",
"(For ease of exposition let us assume that $\\epsilon = 1/t$ for some $t$ .)",
"The invariant that we keep is that for every left vertex $v \\notin A$ we have one of the following two cases: either (1) for all existing edges $(v,u)$ we have that $p_u = 1$ or (2) for some edge $(v,u)$ we have that there exists a unique right vertex $u$ such that $(v,u)\\in M$ and for all other edges $(v,w)$ it holds that $p_w \\ge p_u-\\epsilon $ .",
"For an arbitrary matching $N$ let us define $util(v,N)=1-p_u$ for $(v,u)\\in N$ and $util(v,N)=0$ is $v$ is unmatched in $N$ .",
"It is easy to see that our invariant ensures that for any matching $N$ we have that $util(v,N) \\le util(v,M) + \\epsilon $ (where $M$ was the matching produced by the algorithm), and in fact for $v$ that is unmatched in $N$ we don't even loose $\\epsilon $ : $util(v,N) = 0 \\le util(v,M)$ .",
"Let us now sum $util(v,N)$ up over all left vertices $v$ : we get $|N|$ times 1 minus $\\sum p_u$ over all $u$ that are matched in $N$ which is bounded from above by $\\sum p_u$ over all right vertices $u$ .",
"For the case of the matching $M$ found by the algorithm any unmatched right vertex $u$ still has $p_u=0$ so we get exactly $\\sum p_u$ over all left vertices $u$ .",
"We thus have $|N|-\\sum _u p_u \\le \\sum _v util(v,N) \\le \\sum _v util(v,M) + |N|\\epsilon = |M| - \\sum _u p_u + |N|\\epsilon $ .",
"It follows that $|M| \\ge (1-\\epsilon )|N|$ .",
"In terms of the running time, every iteration of the main loop makes a single demand query and either increases the price $p_u$ of some right vertex by $\\epsilon $ or removes a left vertex (forever) from $A$ .",
"There can clearly be at most $n$ iterations that remove a vertex and, since the price of any right vertex never increases above 1, there can be at most $n/\\epsilon $ price increase iterations, for a total running time of $O(n/\\epsilon )$ .",
"In terms of obtaining the perfectly maximum matching, one could take $\\epsilon = 1/(n+1)$ which due to the integrality of the matching size would imply that the algorithm produces the maximum matching.",
"This makes sense as a way of obtaining a “usual” $O(n^3)$ -time algorithm for maximum matching (when each demand query is executed in $O(n)$ time), but in our model this would require $O(n^2)$ demand queries which in our model is trivial."
],
[
"Augmenting Paths",
"We now present a variant of the directed connectivity problem, which is used as a step (“augmenting path”) in many algorithms for bipartite matching: We are given some mapping from the right vertices to the left vertices $\\pi :R \\rightarrow (L \\cup \\lbrace \\Lambda \\rbrace )$ and are given a subset $S$ of left vertices.",
"Our goal is to find a directed path $(v_1, u_1, v_2, u_2, ... , v_k, u_k)$ such that $v_1 \\in S$ , $\\pi (u_k)=\\Lambda $ , and for every $1 \\le i < k$ we have that the edge $(v_i, u_i)$ is in the graph and $\\pi (u_i) = v_{i+1}$ , or say that none exists.",
"Lemma 1 For every $S$ and $\\pi $ , this problem can be solved with $O(n)$ demand queries.",
"We will implement breadth first search in the demand query model: Initialize a FIFO Que $Q$ with all left vertices in $S$ , and initialize a set of “discarded” left vertices $D=\\emptyset $ .",
"Repeat until $Q = \\emptyset $ : Let $v$ be the first vertex in $Q$ .",
"Pick an arbitrary order of the right vertices such that all vertices $u$ with $\\pi (u)=\\Lambda $ appear before all others and then appear all vertices with $\\pi (u) \\notin (D \\cup Q)$ and last appear those with $\\pi (u) \\in (D \\cup Q)$ , and ask a demand query from $v$ in this order.",
"If $\\pi (u)=\\Lambda $ Then output the path leading to $v$ and then $u$ and halt.",
"If $\\pi (u) \\notin (D \\cup Q)$ Then enque $\\pi (u)$ into $Q$ , attaching to it the path first leading to $v$ , then $u$ and then $\\pi (u)$ .",
"If $\\pi (u) \\in (D \\cup Q)$ Then remove $v$ from $Q$ and insert $v$ into $D$ .",
"If a path was not found and the loop terminated due to $Q$ being empty, there is no such path.",
"In each round we make a single demand query and either insert a new vertex into $Q$ (which we can do at most $n$ times) or remove an element from $Q$ (which again we can do at most $n$ times).",
"Note: We could have alternatively simulated depth first search.",
"It is not difficult to prove a matching lower bound for this problem.",
"This version of the connectivity problem is exactly what is needed for an augmenting path step used in many matching algorithms: A partial matching defines our mapping $\\pi $ by having $\\pi (u)$ be defined as the vertex that is matched to it in the partial matching (and $\\Lambda $ is $u$ is not matched) and defines the set $S$ of unmatched left vertices.",
"As it is known that a matching in a bipartite graph has maximum size if and only it admits no augmenting path, this gives a $O(n)$ algorithm for testing whether a given matching has maximum size."
],
[
"An $O(n^{3/2})$ -query algorithm",
"We now have all the ingredients for describing our best algorithm for maximum matching in this model.",
"Theorem 3 There exists an algorithm for maximum matching in a bipartite graph that uses $O(n^{3/2})$ demand queries.",
"This is obtained by first running the ascending auction algorithm with $\\epsilon =1/\\sqrt{n}$ .",
"This requires $O(n^{3/2})$ demand queries and produces a matching whose size is at least $(1-1/\\sqrt{n})$ times the maximum size.",
"We then run a sequence of augmenting path steps, each of which requires $O(n)$ additional demand queries.",
"Notice that every augmenting path step increases the matching size by 1, and we started with a matching that can be smaller than the maximum matching by at most an additive $O(\\sqrt{n})$ , at most $\\sqrt{n}$ augmenting path steps are needed before the maximum matching is obtained, for a total of at most $O(n^{3/2})$ demand queries."
],
[
"A Randomized Parallel Algorithm",
"In the parallel verion of our model we proceed by rounds, where at each round a set of queries is asked, a set that may be determined by the answers to the queries from previous rounds.",
"An $r$ -round $q$ -query-per-round protocol is one with at most $r$ rounds, where at each round at most $q$ demand queries are made.",
"Theorem 4 (Dobzinski-Nisan-Oren) for any $\\delta >0$ there exists a randomized $O(\\log n/\\delta ^2)$ -round protocol where at each round there is a single demand query for every left vertex (for a total of $n$ demand queries per round) which returns a matching of size at least $(1-\\delta )$ -fraction of the optimal matching.",
"The algorithm is a randomized parallel variant of the ascending auction algorithm: For each left vertex $u$ initialize “prices” $p_u=0$ .",
"Initialize the matching $M=\\emptyset $ , and the discarded vertices $D=\\emptyset $ .",
"Repeat $O(\\log n/\\delta ^2)$ times: For each vertex $v \\notin D$ that is currently unmatched in $M$ , in parallel: Ask a demand query on $v$ with the order on $u$ 's induced by increasing values of $p_u$ , with ties broken randomly, and let $u_v$ be the answer to that query.",
"If $p_{u_v} > 1$ : insert $v$ into $D$ .",
"For each $u$ such for some $v$ we have that $u=u_v$ , increase the price $p_u$ by $\\delta $ and pick an arbitrary $v$ such that $u=u_v$ and insert $(v,u)$ into the matching $M$ , removing any previous edge matched to $u$ , if any."
],
[
"The Lower Bound",
"It will be more convinient to prove our lower bound with a slightly weaker query, the OR query, which turns out to be essentially equivalent to a demand query, (up to log factors).",
"Definition 2 An OR Query accepts a left vertex $v$ and a subset $S$ of the right vertices and returns whether there exists a vertex $u \\in S$ such that $(v,u)$ is an edge in the graph.",
"While it is clear that a demand query is stronger than an OR query, it turns out that the gap between them is not large: Lemma 2 A demad query can be simulated by $\\log _2 (n+1)$ OR queries.",
"We simulate each demand query $(v,(u_1...u_n))$ by a binary search that uses OR queries: we start by asking whether there is an edge between $v$ and $\\lbrace u_1...u_{n/2}\\rbrace $ , according to the answer we then either focus on the first half of the variables (asking whether there is an edge to $\\lbrace u_1...u_{n/4}\\rbrace $ ) or the second half (asking whether there is an edge to $\\lbrace u_1...u_{3n/4}\\rbrace $ ), etc.",
"The overhead in the simulation is clearly optimal since a demand query has $n$ possible answers.",
"We will proceed by providing our lower bounds in the OR query model, which as we have just shown implies the same lower bounds – with a log factor loss – in the demand query model.",
"Theorem 5 Every deterministic $r$ -round $q$ -query (per round) algorithm that uses OR queries cannot distinguish between grpahs with a perfect matching (of size $n$ ) and those with a matching of size at most $\\alpha n$ for $\\alpha = r \\sqrt{q r \\log n} / n$ .",
"Corollary 1 Deterministic algorithms with polylog rounds of polylog demand queries per-player can only find matching of size $\\tilde{O}(\\sqrt{n})$ in a graph that has a perfect matching.",
"Corollary 2 Deterministic $r$ -round algorithms require at least $q = n^2/(r^3 \\log ^4 n)$ demand queries per round in order to find a a maximum matching in a bipartite graph or even find a matching whose size is a constant fraction of the maximum size matching.",
"Corollary 3 Deterministic $n^{1/7}$ -round $n^{8/7}$ queries-per-round algorithms cannot approximate the maximum matching within a factor of $n^{1/7}$ .",
"We will describe an adversary algorithm for answering the rounds of queries.",
"At every round, our adversary will decide (based on the queries in this round) on the existence or lack of existence of some subset of the edges of the graph, in a way that the answers to all queries in this round are determined by these choices.",
"The edges that were decided to exist in this round will be called \"YES\" egdes of the round, and those that were decided not to exist will be called \"NO\" edges of the round.",
"For any query to a set $S$ of edges that contains some edge that was already decided to exist (a \"YES\" edge from some previous round), the answer to this query is already determined so the adversary can simply ignore this query in this round (and thus the discussion below can assume that such sets are never queried).",
"Similarly any query to a set $S$ that includes some subset of \"NO\" edges from previous rounds may be considered by the adversary as though it is a query only to the subset of edges that were previously not answered.",
"The rest of the discussion can thus assume without loss of generality that all edges queried in all queries in this round are “new” ones for which the adversary has not committed to an answer yet.",
"Specifically, for every $OR$ query on a set $S$ of edges made in this round the adversary will either fix all the edges in $S$ to be \"NO\" edges (which fixes a negative answer to this query), or will fix (at least) one edge in $S$ to be \"YES\" (which fixes a positive anser to this query).",
"The adversary's goal is to make sure that after fixing all these \"YES\" and \"NO\" edges in all $r$ rounds, the underlying graph can still either have a perfect matching or not have any matching of size greater than $\\alpha n$ .",
"The adversary will maintain this property by maintaining the following constraints on the \"YES\" and \"NO\" edges of each round: For every vertex in the graph, strictly less than $n/(2r)$ edges adjacent to it are \"NO\" edges of the round.",
"The set of vertices that are adjacent to any \"YES\" edge of the round is of size at most $O(\\alpha n /r)$ .",
"Given these conditions, the algorithm cannot distinguish between the following two extreme cases: (a) all edges that were not decided in any round in fact do not exist, in which case the maximum possible matching is of size $O(\\alpha n)$ since every edge in it must be adjacent to a \"YES\" edge of some round, and there are at most $O(\\alpha n / r)$ such vertices in every round (b) all edges that were not decided yet do exist in which case a perfect matching exists (which follows from the fact that the degree of every vertex is strictly more that $n/2$ since in each round strictly less than $n/(2r)$ edges adjacent to any vertex were set to be “NO” edges.)",
"Here is how the adversary makes decisions for a given round that makes $q$ $OR$ queries on sets $S_1 ... S_q$ (each of which contains only edges that were not answered to be “YES” or “NO” in any previous round).",
"The adversary will handle separately the $OR$ queries on sets that contain at least $\\theta = n \\sqrt{\\log n}/\\sqrt{2qr}$ edges (“big queries”) and those of size at most $\\theta $ edges (“small queries”).",
"Big queries: the adversary will pick a set of size $O(n\\log n / \\theta )$ of vertices with the property that it intersects each of the “Big” queries $S_i$ and answer that all edges connected to them (those which were not already decided) exist, i.e are “YES” edges for this round.",
"This suffices for answering all the big queries with a positive answer.",
"The existence of such a set is given by a random construction: choose every vertex at random to be in this set with probability $O(\\log n / \\theta )$ : for a fixed big query the probability that at non of its edges connects to this chosen set is at most $(1-\\log n /\\theta )^\\theta < 1/n^2$ and since there are certainly less than $n^2$ queries, with high probability the chosen set intersects all big queries simultaneously.",
"Small queries: At this point the adversary has already ensured a positive answer to all big queries, and perhaps also to some small queries, so let us focus on the small queries for which the answer is not yet fixed.",
"There are at most $q \\theta $ edges in all of the these small queries combined.",
"Since each edge contains two vertices, there can be at most $(2 \\cdot q \\theta ) / (n/(2r)) = 2 \\sqrt{2qr\\log n}$ vertices in the graph that are each adjacent to more than $n/(2r)$ of these edges, which will be called “heavy” vertices.",
"The adversary will answer “YES” to all edges that are adjacent to one of these heavy vertices and “NO” to all other edges.",
"So we only need to show that there are at most $O(\\alpha n /r)$ such heavy vertices, which is true since $\\alpha n / r = \\sqrt{q r \\log n}$ as needed."
]
] | 1906.04213 |
[
[
"Decay spectroscopy of $^{50}$Sc and $^{50m}$Sc to $^{50}$Ti"
],
[
"Abstract The $\\beta$ decay of the isomeric and ground state of $^{50}$Sc to the semi-magic nucleus $^{50}_{22}$Ti$_{28}$ has been studied using a $^{50}$Ca beam delivered to the GRIFFIN $\\gamma$-ray spectrometer at the TRIUMF-ISAC facility.",
"$\\beta$-decay branching ratios are reported to 16 excited states with a total of 38 $\\gamma$-ray transitions linking them.",
"These new data significantly expands the information available over previous studies.",
"Relative intensities are measured to less than 0.001$\\%$ that of the strongest transition with the majority of $\\gamma$-ray transitions observed here in $\\beta$ decay for the first time.",
"The data are compared to shell-model calculations utilizing both phenomenologically-derived interactions employed in the ${\\it pf}$ shell as well as a state-of-the-art, ${\\it ab~initio}$ based interaction built in the valence-space in-medium similarity renormalization group framework."
],
[
"Introduction",
"Nuclei in the vicinity of magic neutron (N) and proton (Z) numbers, often display simple patterns of low-energy excitations which can be well described in a spherical shell model approach.",
"The structure of these lowest-lying excited states may be deduced by considering the behavior of a single nucleon or pair of nucleons occupying just a few single-particle orbits near the Fermi surface in a spherically-symmetric potential.",
"However, this simple picture does not describe the nature of all excitations observed at low energies.",
"Most notably, the limitations in the number of basis states included in this approach means that it usually does not capture deformed configurations or collective behaviors which can coexist along side the spherical single-particle structures and typically involve breaking the core.",
"Particle-hole excitations across major shell gaps are energetically costly, requiring several MeV of energy.",
"However, the additional correlation energy, coming primarily from the neutron-proton quadrupole-quadrupole interaction [1], which becomes possible with this release of particles from the core makes such cross-shell excitations energetically competitive with the lowest-lying states [1], [2].",
"Recently, the rapid development of new theoretical methods and the availability of increased computational power have extended the reach of ab initio methods to medium-mass nuclei [3], [4], [5], [6].",
"The need to include three-nucleon forces in the interactions for an accurate description of excitations has become evident [7].",
"In order to support the further development of these methods, detailed spectroscopic information of excited nuclear states and transitions is necessary.",
"The even-even N=28 isotones above $^{48}$ Ca provide a good example, where protons fill the $0f_{7/2}$ orbital.",
"The seniority=2 (V=2), J=$2^+$ ,$4^+$ ,$6^+$ states in $^{50}_{22}$ Ti, $^{52}_{24}$ Cr and $^{54}_{26}$ Fe with E$_x$ $\\approx $ 1.5-3 MeV show very little change in excitation energy as additional pairs of protons are added to the $0f_{7/2}$ orbital.",
"In contrast the J=$0_2^+$ ,$2_2^+$ excited states originating from neutron two-particle, two-hole ($\\nu $ 2p-2h) excitations across the N=28 shell gap to the $1p_{3/2}$ orbital show an abrupt change beyond $^{50}$ Ti.",
"There is a rapid lowering in excitation energy as more proton pairs are added to increase the attractive correlation strength, and these states begin to intrude upon the $\\pi 0f_{7/2}$ ground state structure at $\\le $ 3 MeV in $^{52}$ Cr and $^{54}$ Fe [8].",
"In addition, below $^{48}$ Ca, the N=28 shell gap has been shown to vanish upon the significant removal of protons [9].",
"The electric quadrupole transition strength B(E2; $0_1^+$ $\\rightarrow $ $2_1^+$ ) is frequently used to probe the evolution of collectivity near closed shells, appearing enhanced at mid-shell and at a minimum at the shell closure, for example the B(E2) for the $2^+_1 \\rightarrow 0^+_1$ transition measured in $^{52}$ Cr is around twice that of $^{50}$ Ti [10].",
"In general, constraints on the decay intensities observed between the mainly non-yrast states and the V=2 seniority states are valuable targets for experiments aiming to understand the microscopic behaviors in A$\\approx $ 50 semi-magic nuclei, particularly the B(E2; $0_2^+$ $\\rightarrow $ $2_2^+$ ) transition strength for the $\\nu $ 2p-2h configuration.",
"A deeper insight into the interplay between configurations will be obtained from a detailed comparison between calculations and experimental data.",
"In this article we report on the most sensitive study of the $\\beta $ decay of $^{50}$ Sc to $^{50}$ Ti performed to date, using the GRIFFIN spectrometer at TRIUMF-ISAC [11], [12], [13], [14].",
"The analysis of the excited states in $^{50}$ Sc populated from the $\\beta $ decay of the $^{50}$ Ca beam in this work have been previously reported in Ref.",
"[15].",
"The new data for $^{50}$ Ti presented here are compared to shell-model calculations utilizing both phenomenologically-derived interactions employed in the pf shell as well as an ab initio based interaction built in the valence-space in-medium similarity renormalization group framework."
],
[
"Experimental details",
"The isotope $^{50}$ Ca ($T_{1/2}$ = 13.9(6) s [16]) was produced from reactions induced in a 22.49 g/cm$^2$ Ta target by a 500 MeV proton beam delivered by the TRIUMF Cyclotron [17].",
"The position of the 60 $\\mu $ A proton beam on the ISOL target was continuously rastered such that a tighter proton beam spot could be used to induce a higher localized power density in the Ta material.",
"The calcium atoms created in the target that diffused out of the material were ionized using resonant-laser ionization and accelerated to 20 keV, mass separated and delivered to the experimental station.",
"The typical beam intensity of $^{50}$ Ca was $\\sim 10^6$ ions/s.",
"Small amounts of surface-ionized $^{50}$ K ($T_{1/2}$ =472(4) ms [18]) and $^{150}$ Tb ($T_{1/2}$ =3.5 hrs [19], [20]) were also present in the beam.",
"The ions were stopped in a mylar tape at the central focus of the Gamma-Ray Infrastructure For Fundamental Investigations of Nuclei (GRIFFIN) spectrometer [11], [12], [13], [14].",
"GRIFFIN consists of an array of 16 high-purity Germanium (HPGe) clover detectors coupled to a series of ancillary detectors.",
"Fifteen HPGe clovers were used in the present work.",
"An array of plastic scintillator paddles (SCEPTAR) was used for the detection of $\\beta $ particles.",
"Four cerium-doped lanthanum bromide (LaBr$_3$ (Ce)) scintillators were installed in the array but the data from them was not used in this work.",
"The GRIFFIN clovers were positioned at a source-to-detector distance of 11 cm from the implantation point.",
"A 20 mm thick delrin plastic absorber shell was placed around the vacuum chamber to prevent $\\beta $ particles from reaching the HPGe detectors while minimizing the flux of Bremsstrahlung photons created as the $\\beta $ particles were brought to rest.",
"In order to study the longer-lived $^{50}$ Sc daughter (T$_{1/2}$ = 102.5(5) s) activity, the beam was continuously delivered to the experimental station with the tape stationary.",
"Data was collected in this way for a period of 5 hours.",
"As previously reported [15], a series of short cycles were collected to clearly distinguish the activity of the $^{50}$ Ca beam.",
"In addition, a longer tape cycle of 1 min $^{50}$ Ca implantation and 10 min decay was collected for a short time and used in the current analysis.",
"Energy and timing signals were collected from each detector using the GRIFFIN digital data acquisition system [13], operated in a triggerless mode.",
"HPGe energy and efficiency were calibrated using standard radioactive sources of $^{133}$ Ba, $^{152}$ Eu, $^{60}$ Co and $^{56}$ Co with the necessary corrections for coincidence summing applied."
],
[
"Gamma-ray energy spectra",
"Events observed by individual GRIFFIN clover detectors were time-correlated to produce $\\gamma $ -ray addback spectra and provided the principal tool for offline analysis of $^{50}$ Sc decay.",
"Time-correlated $\\gamma $ -ray addback hits were used to construct $\\gamma $ -$\\gamma $ matrices in order to establish branching ratios and coincidence relationships in $^{50}$ Ti, with the option of requiring coincidences with $\\beta $ electrons detected in SCEPTAR.",
"An addback energy spectrum is shown in Figure REF for $\\gamma $ -ray energies below 1.6 MeV, encompassing the most intense transitions observed following the decay of $^{50}$ Ca and $^{50}$ Sc.",
"The intensity of the 1554 keV 2$^{+}$ $\\rightarrow $ 0$^{+}$ transition in $^{50}$ Ti yields a total of $\\approx $ 2.9 $\\times $ 10$^{9}$ $^{50}$ Sc decays.",
"The fraction of contaminant nuclei in the beam was assessed using $\\gamma $ rays emitted following $\\beta $ -decay and $\\beta $ -delayed neutron emission of $^{50}$ K and the EC decay of $^{150}$ Tb, transmitted to GRIFFIN as $Q=1^{+}$ and 3$^{+}$ charge states, respectively.",
"The total contaminant activity detected in the chamber was $\\approx $ 1 $\\%$ relative to the 1554 keV transition.",
"Figure: γ\\gamma -ray addback energy spectrum collected with GRIFFIN following the decay of 50 ^{50}Ca.",
"Internal transitions belonging to 50 ^{50}Sc ( ∘ ^{\\circ }) and 50 ^{50}Ti ( * ^{*}) are indicated.",
"Room background lines are also labelled, including those from the 138 ^{138}La natural radioactivity present in the LaBr 3 _3(Ce) detectors.Figure: γ\\gamma -ray gated addback energy spectra showing coincidence gates with the (a) 1130 keV transition feeding the 4310 keV 2 + 2^+ state, (b) 2618 keV γ\\gamma ray and the (c) 1554 keV transition.",
"A γ\\gamma -ray singles energy spectrum comprised of the sum of all addback hits is shown in the inset of panel (c) for a subset of energies, where additional transitions at 4035 and 4070 keV are observed following the β\\beta and β\\beta -n decay of 50 ^{50}K, respectively.",
"Peaks labelled with an asterisk are transitions identified in 50 ^{50}Ti.",
"Summing (s.) and escape (esc.)",
"peaks are also indicated."
],
[
"Level scheme",
"The level scheme observed in the decay of $^{50}$ Sc to $^{50}$ Ti was constructed on the basis of a $\\gamma - \\gamma $ coincidence analysis.",
"Examples of $\\gamma $ -ray gated addback energy spectra are shown in Figure REF .",
"Figure REF shows the placement of $\\gamma $ rays observed in the current work into the $^{50}$ Ti level scheme.",
"The width of the arrows indicates the intensity relative to that of the 1554 keV transition.",
"In addition to the 1554 keV decay from the first excited state, only one other transition is observed to decay directly to the ground state, de-exciting the $J^\\pi $ =$2_2^+$ state at 4310 keV (Figure REF (a) gated on the 1130 keV feeding transition).",
"Otherwise all excited states eventually feed the yrast, $J^+$ =$2^{+}$ ,$4^{+}$ and $6^{+}$ states.",
"For this reason, gates placed upon the $\\gamma $ ray of interest (i.e.",
"gating from above) were often the most useful concerning placement in the level scheme.",
"Transitions were placed according to their observed coincidence (or non-coincidence) with the strongest transitions in $^{50}$ Ti, notably the 524, 1121 and 1554 keV $\\gamma $ rays, and via comparison of $\\gamma $ -ray energies with the energy difference $\\Delta E$ between previously known excited states.",
"Feeding transitions were observed in a few cases above the 3199 keV $6^+$ state (at 4147, 4172 and 4310 keV) helping to constrain the measurement of direct $\\beta $ decay branching to these states.",
"The 3132 keV $\\gamma $ ray de-exciting the 5807 keV state to the 2675 keV $4^+$ state is observed to interfere with a different transition (the 2618 keV transition de-exciting the yrast $3^+$ state at 4172 keV) through an energy-coincidence of the associated single-escape peak at 2621 keV.",
"This was confirmed by gating on the 2618 keV transition (Figure REF b) where strong coincidences are observed with both a 511 keV escape photon and the 1121 keV transition.",
"The intensity of the 2618 keV $\\gamma $ ray has been corrected for this contribution (Section REF ).",
"There is good agreement between the current work and that from a previous $\\beta $ -decay study reported by Alburger et al.",
"[21] as well as with Ruyl et al.",
"[22] who utilized thermal neutron capture on metallic titanium targets.",
"In the present work, the level scheme has been expanded with a number of levels and transitions observed for the first time in $\\beta $ decay.",
"This is discussed in detail in the following Sections.",
"Figure: Level scheme of the levels observed in 50 ^{50}Ti following the β\\beta decay of 50 ^{50}Sc.",
"Line width is proportional to the observed γ\\gamma -ray intensity.",
"Shown in red are the unobserved intra-band transitions of a band built on the 2-particle-2-hole neutron excitation across the N=28N=28 shell gap."
],
[
"Gamma-ray relative intensities",
"Intensities of $\\gamma $ rays observed following the decay of $^{50}$ Sc are given in Table REF .",
"Branching ratios of $\\gamma $ rays from each excited state are also provided.",
"$\\gamma - \\gamma $ coincidence requirements have been applied where possible in order to isolate the transitions of interest and obtain the optimum peak-to-background ratio for extracting relative intensities.",
"This is especially important for weaker branches obscured by the Compton-scattered background arising from more intense transitions (mainly the 1121 and 1554 keV $\\gamma $ rays).",
"A procedure similar to the `gating from below' method employed by Kulp et al.",
"[23] has been used here to obtain the normalized intensity $I_\\gamma $ for the transitions of interest.",
"Modifications to the overall detection efficiency due to coincidence timing restrictions and the angular coverage of GRIFFIN compared to singles data are assumed to be $\\approx $ 1.0 (see for example, Ref.",
"[24]).",
"The gating transition was chosen to de-excite states below that of the transition of interest with the 524, 1121 and 1554 keV $\\gamma $ rays being the most common choice due to their well-characterized branching ratios.",
"The transition of interest directly feeds the level depopulated by the gating transition in all cases.",
"All intensities provided in Table REF have been corrected for summing effects using an empirical method described in Ref.",
"[14].",
"In cases where transition intensities are extracted from $\\gamma $ -ray addback hits without any coincidence conditions applied (i.e.",
"at 2756, 3426, 3950, 3993, 4308 and 4601 keV), summing corrections are obtained using a $\\gamma - \\gamma $ -addback coincidence matrix with the requirement that HPGe clovers be located at 180 $^\\circ $ with respect to each other.",
"A normalization factor is included to reconcile the difference in combinatorial efficiency between the intensity obtained in singles and the summing correction.",
"This may arise due to asymmetries in the detector array where the availability of single crystals (clovers) differs from the number of crystal (clover) pairs at 180 $^\\circ $ .",
"A similar method is employed to extract summing corrections for the remaining transitions in Table REF extracted from coincidence measurements.",
"In the coincidence case the summing correction factors are specific to the transition of interest as well as the choice of the gating transition.",
"Much care must be taken when constructing the necessary coincidence matrices used to determine these factors in order that the same experimental conditions are applied to them as to the experimental data.",
"Data in Table REF are compared to previously reported measurements where available.",
"A total of 38 transitions have been identified in $^{50}$ Ti.",
"The vast majority of the $\\gamma $ -ray intensity ($\\approx $ 99 $\\%$ ) is contained in the 524, 1121 and 1554 keV transitions.",
"The remaining $\\approx $ 1 $\\%$ includes many weak transitions, 25 of which are observed here in $\\beta ^{-}$ decay for the first time, with around half of these transitions having not been reported in any previous experiment.",
"Relative intensities are determined to below 1$\\times $ $10^{-3}$ $\\%$ that of the 1554 keV transition to the ground state, which is a factor of $\\approx $ 15 lower than that of the 3826 keV transition identified by Alburger et al [21].",
"Table: Energies, intensities and branching ratios of γ\\gamma rays measured in the beta decay of 50 ^{50}Sc.",
"I γ I_{\\gamma } is expressed relative to the 1554 keV transition to the ground state of 50 ^{50}Ti.",
"Data are compared to previous measurements where available.",
"See text for details."
],
[
"Beta-decay branching ratios",
"$\\beta $ -decay branching ratios to excited states in $^{50}$ Ti were determined based on the coincidence relationships established in the current work and the observed intensities of $\\gamma $ rays with the appropriate corrections for internal conversion.",
"The $\\beta $ -decay branching ratios from this work are reported in Table REF .",
"Conversion coefficients were calculated using the BrIcc-FO (frozen orbitals) formalism [26] with transition multipolarities inferred from the spin-parities of the initial and final states summarized in Table REF .",
"While mixing ratios compiled in Ref.",
"[25] are used where available, the leading order multipolarity is assumed to dominate for most mixed transitions.",
"It should be noted however that the corrections are small: of the order $10^{-4} - 10^{-5}$ in $^{50}$ Ti.",
"Table: β\\beta -branching ratios and calculated log(ft) values observed in the β\\beta decay of 50 ^{50}Sc as compared to previously reported values.",
"Spin and parities are assigned in the current work by comparison with both existing assignments and with the typical values for log(ft) compiled in Ref.",
".",
"In some cases further restrictions have been imposed upon spin and parity assignments by considering the initial and final states in 50 ^{50}Ti connected by internal γ\\gamma ray transitions (see text for details).",
"The excited states at 6237 and 6625 keV are reported here for the first time.The ground state spin of $^{50}$ Sc is established as $J^\\pi =5^+$ and therefore a direct $\\beta $ decay to the $0^+$ ground state of $^{50}$ Ti would be a fourth forbidden transition.",
"We therefore make the assumption that there is negligible direct feeding to the $0^+$ ground state, as was also done by Alburger et al.",
"[21].",
"An upper limit for the $\\beta $ decay branch from the 257 keV isomeric state of $^{50}$ Sc feeding the 1554 keV $2^+$ state in $^{50}$ Ti was reported by Alburger et al.",
"[21].",
"In our previous reporting from the present work, the upper limit for this decay branch was reduced from $<$ 2.5% to $<$ 1% [15] by observation of the number of 1554 keV $\\gamma $ rays collected in singles relative to those in coincidence with the 1121 keV transition, extracted from short-decay cycles.",
"Direct $\\beta $ decay from the $2^+$ isomeric state in $^{50}$ Sc has been observed unequivocally in the current work through a detailed examination of the $\\beta $ feeding and log($ft$ ) values.",
"Evidence of $\\beta $ feeding from the isomer is found in the case of the 1554 keV $2^+$ state, the 4172 keV $3^+$ state and the 4310 keV $2^+$ state and are detailed in Table REF .",
"The branching ratios to these states are reported as upper limits because of possible unobserved $\\gamma $ -ray transitions feeding these levels.",
"The $\\beta $ -decay branching ratio of the 257 keV isomeric state in $^{50}$ Sc is found to have an upper limit of $\\le $ 0.092% at the 95% confidence level."
],
[
"log(",
"Table REF lists the log(ft) values obtained using the log(ft) calculator of Ref.",
"[28] for the decay of the $5^+$ ground state and $2^+$ isomeric state in $^{50}$ Sc.",
"The $\\beta $ -branching ratios determined in the current work are used as inputs to the calculation in addition to excitation energies of states populated in $^{50}$ Ti, the half-lives of the $5^+$ ground state and $2^+$ isomeric state in $^{50}$ Sc and the $\\beta $ -decay Q value of 6.890(16) MeV [25].",
"The log(ft) values from this work are plotted in Figure REF in comparison to the range of typical values for $\\beta ^{-,+}$ and $EC$ decaying nuclei compiled by Singh et al.",
"[27].",
"A good agreement is found for log(ft) values obtained in the current work compared to those included in the 2011 nuclear data evaluation by Elekes et al.",
"[25] and support the assigned spin and parities given in column 4 of Table REF .",
"As is evident in Figure REF , $^{50}$ Sc $5^+$ ground state $\\beta $ decay to known $4^+, 5^+$ and $6^+$ levels proceeds with an experimental weighted-mean log(ft) of $\\approx $ 7, consistent with a dominance of L=0 allowed Fermi and Gamow-Teller $\\beta $ transitions.",
"The $^{50m}$ Sc $2^+$ isomeric state $\\beta $ decay to known $2^+$ and $3^+$ levels also have log($ft$ ) values consistent with allowed $\\Delta J=0,1$ transitions."
],
[
"Previously known excited states",
"A number of $\\gamma $ rays are observed in the current work to de-excite levels previously identified by Alburger et al.",
"[21] (and references therein) but were not originally placed in the $^{50}$ Ti level scheme determined from $^{50}$ Sc $\\beta ^-$ decay.",
"This includes excited states in Table REF with energies 4881, 5441 and 5806 keV where at least one transition was previously assigned depopulating each level.",
"For example, a transition observed in the current work with energy 733 keV is assigned de-exciting the 4881 keV state while the 1130, 1269 and 3885 keV transitions are assigned to the 5441 keV level.",
"In addition the 1635 and 4252 keV transitions are assigned de-exciting the 5806 keV state.",
"In each of these cases the additional decay intensity introduced in the current work does not dramatically alter the previously established log(ft) values.",
"In cases where no additional transitions are placed de-exciting a level (for example the $4^+$ states at 4147 and 5380 keV) our log(ft) values are similar to those of Ref.",
"[25] which incorporates the work of Alburger et al.",
"[21].",
"The relative intensities of $\\gamma $ rays de-exciting the 5380 keV state show good agreement with that of Ref.",
"[25] although a significantly larger intensity for the 3825 keV transition (around 40 $\\%$ of the 2705 keV transition) was reported in Ref.",
"[21].",
"Consequently a weaker $\\beta $ branch of 0.17 $\\%$ is determined in the current work for the 5380 keV state.",
"The $\\beta $ branch to the 2675 keV $4_1^+$ state is reduced compared to the value reported in Ref.",
"[21] from around 8 $\\%$ to 3$\\%$ .",
"This is due to the higher relative intensity of the 524 keV transition observed in the current work.",
"Additional information regarding the spin and parity assignments of some previously observed excited states may be obtained through the placement of internal $\\gamma $ ray transitions in $^{50}$ Ti.",
"For example the 5441 and 5806 keV states have been assigned J$^{\\pi }$ =4$^+$ in Table REF as each state depopulates via transitions that feed the 1554 keV $2_1^+$ state.",
"Figure: Histograms summarizing log(ft) values for 50 ^{50}Sc β\\beta decay determined from experimental branching ratios.",
"The hatched histogram indicates β\\beta transitions from the 5 + ^+ ground state whereas the open-area histogram corresponds to transitions from both the ground state and the 257 keV 2 + ^+ isomeric state.",
"The open-area histogram corresponds to Δ\\Delta J=0,1 transitions except where shaded blue (Δ\\Delta J=2) or red (Δ\\Delta J=3) and has been scaled by 0.85 for clarity.",
"Lower-limits associated with individual bins are indicated by arrows.",
"Data are compared to typical log(ft) values compiled in Ref.",
"for different Δ\\Delta J (Δπ\\Delta \\pi =no).",
"See text for details."
],
[
"Discussion",
"In this work many excited states are observed, several of which have been observed in $\\beta $ decay for the first time.",
"These states require additional comment."
],
[
"The 4172 keV doublet",
"It should be noted that the 4171.9 keV $3^+$ state is an energy doublet appearing in close proximity to a 4172.5 keV state with a tentative spin-parity assignment of $(2)^+$ [25], the latter decaying to the 1554 keV $2_{1}^{+}$ via a 2618 kev $\\gamma $ ray.",
"The $3^{+}$ state also decays to the 1554 keV $2_{1}^{+}$ state via a 2618 kev $\\gamma $ ray and is assigned an additional 1497 keV decay pathway to the 2675 keV $4_{1}^{+}$ state.",
"Both the 1497 and 2618 keV transitions are observed here with intensities that are in good agreement with neutron-capture measurements [22].",
"Both the measurement of $\\gamma $ -ray energies 2618.0(2), 1497.1(2) keV and intensities are consistent with dominant $\\beta $ -feeding to the 4171.968(17) keV $3^+$ state.",
"The 1208 keV transition was originally reported feeding a state in $^{50}$ Ti with excitation energy 4171.8 keV and a spin-parity of $3^+$ or $4^+$ (Ref.",
"[21] and references therein).",
"The same transition is reported here in addition to 5 weak feeding transitions observed at 1015, 1269, 1635, 1983 and 2065 keV, decaying from states with E$_x$ $\\ge $ 5186 keV.",
"These are placed in the $^{50}$ Ti level scheme for the first time (Figure REF ), feeding the 4171.9 keV $3^+$ state and are observed in coincidence with the 1497 and 2618 keV transitions.",
"The newly-placed transitions together with the known 1208 keV transition accounts for $\\sim $ 46$\\%$ of the observed total intensity of the $\\gamma $ rays depopulating the $3^+$ state.",
"The resulting direct $\\beta $ -feeding of the 4171.9 keV $3^+$ level is found to be $\\sim $ 0.07%.",
"This result precludes direct feeding from the $^{50}$ Sc $5^+$ ground state, where the corresponding log($ft$ ) ($\\approx $ 8) does not agree with the value expected for a $J$ =2 transition ($\\approx $ 12) [27].",
"It is unlikely that there is an overestimation of the $\\beta $ branch intensity due to contributions from unobserved feeding.",
"In such a scenario more than 200 additional feeding transitions (assuming an upper limit for the detection of very weak transitions in the current work of $I_\\gamma \\le $ 0.0003% such as for the 1983 keV $\\gamma $ ray in Table REF ) would be required to bring the experimental log($ft$ ) value into line with the typical range Ref.",
"[27].",
"It is therefore much more likely that the 4171.9 keV $3^+$ level is fed by the 257 keV $2^+$ isomeric state in $^{50}$ Sc ($T_{1/2}$ = 0.35(4) s).",
"A series of time-gated $\\gamma $ -ray addback energy spectra were produced using the decay portion of the long tape cycle.",
"Each spectrum contained data integrated over 5 s with cuts placed from $\\approx $ 0-300 s after the cessation of $^{50}$ Ca implantation upon the tape.",
"The intensity of the 2618 keV peak was obtained as a function of time and compared to that of the neighboring 2675 keV $\\gamma $ -ray energy sum peak (incorporating both the 1554 and 1121 keV transitions).",
"The decay curve gated on the 2675 keV transition is consistent with the decay of the $^{50}$ Sc ground state.",
"The decay curve gated on the 2618 keV transition, however, shows a significant decrease in intensity after around 100 s: around a factor of 2 lower compared to that of the 2675 keV transition.",
"Note that the contaminant 3132 keV single-escape peak at 2618 keV was subtracted from the data.",
"In summary, evidence was obtained in the current work for the direct feeding of the 4171.9 keV $3^+$ level by the 257 keV isomeric state in $^{50}$ Sc."
],
[
"The 4310 keV state",
"The excited state at 4310 keV was known previously, for example it is reported as a 4322(20) keV state in the $^{49}$ Ti(d,p) stripping reaction [29].",
"It was assigned a spin of J=2 based on the L=1 transfer strength observed following analysis of the differential cross section.",
"The observation of $\\gamma $ rays from this state feeding the 0$^{+}$ and $2_{1}^{+}$ states in a $^{49}$ Ti(n,$\\gamma $) study [25] also dissuades assignment of a large spin for this state.",
"The 2756 and 4308 keV $\\gamma $ rays are observed here in $\\beta $ decay below a relative intensity of 0.01$\\%$ with respect to the 2$_{1}^{+}$ $\\rightarrow $ 0$_{1}^{+}$ transition.",
"The observed branching ratio of the weaker 4308 keV transition is 30.6(41), slightly above the value reported in Ref.",
"[25] of 19.6(21).",
"In the present study we have constrained the $\\beta $ branching ratio to this state from observation of a feeding transition with energy 1130 keV.",
"A $\\gamma $ -ray coincidence addback energy spectrum gated on this feeding transition is shown in Figure REF .",
"An estimate of the branching ratio for the unobserved $2_3^+$ $\\rightarrow $ $0_2^+$ 440 keV $E2$ transition is reported here via consideration of the 2314 keV transition (not observed in the present work) which is placed de-exciting the $0_{2}^{+}$ state [22].",
"We assume that the intensity of the 2314 keV transition is directly analogous to the 440 since (i) no additional transitions are placed feeding the $0_{2}^{+}$ state, (ii) the $0_{2}^{+} \\rightarrow 2_{1}^{+}$ decay has a branching ratio equal to unity and (iii) direct $\\beta $ feeding to the $0_{2}^{+}$ state is unlikely.",
"Analysis of the unobserved 2314 keV transition focused on the spectrum shown in Figure REF (a) gated on the 1130 keV transition and centered on energies near 2314 and 2756 keV.",
"The background level observed near 2314 keV yields an upper limit of $\\approx $ 25 counts while around 450 counts are recorded in the same spectrum for the 2756 keV peak.",
"This corresponds to an intensity of $\\le $ 5$\\%$ for the 2314 keV transition relative to the 2756 keV peak or equivalently for the 440 keV transition, a $\\le $ 4$\\%$ $\\gamma $ -ray branch from the 4310 keV state.",
"The upper limit on the 440 keV $\\gamma $ -ray branch from the 4310 keV state could potentially be improved in future experiments via the use of bremsstrahlung- and Compton-suppressed clovers in GRIFFIN in addition to a higher time-integrated $^{50}$ Ca beam intensity.",
"The difference in the observed intensity between the feeding and depopulating transitions yields a $\\beta $ branch for this state of 0.002$\\%$ .",
"This corresponds to a log(ft) value around 12 for the 5$^+$ $\\rightarrow $ 2$^+$ $\\beta $ transition from the $^{50}$ Sc ground state: well below the expected hindrance of this unique second-forbidden $\\Delta $ J=3 transition [27].",
"It is likely therefore that the obtained $\\beta $ branch incorporates the 2$^+$ $\\rightarrow $ 2$^+$ $\\beta $ transition from the 257 keV $^{50}$ Sc isomer (yielding a log(ft) of around 7).",
"For this reason an upper limit is adopted for the $\\beta $ branch given in Table REF .",
"Note that this value includes the intensity limit established for the unobserved 440 keV $\\gamma $ ray."
],
[
"The 5186 keV state",
"The 5186 keV state was identified via the observation of $\\gamma $ rays of 1015 keV, 2511 keV and 3631 keV, which represent transitions to the $3^+$ member of the 4172 keV doublet, $4^+_1$ 2675 keV state and $2^+_1$ 1554 keV state respectively.",
"The relative branching ratios were determined using coincidence gates applied to the 1121 and 1554 keV transitions and are in good agreement with previous $^{49}$ Ti(n,$\\gamma $) measurements [25].",
"This state was previously assigned a tentative spin of $J$ =$(3)$ or $(4)$ based on the results of $\\gamma $ -ray angular distributions following the capture of polarized thermal neutrons on a polarized $^{49}$ Ti target [22].",
"A positive parity was inferred from $L$ =1 transfer observed with $^{49}$ Ti(d,p) [29].",
"In the absence of any observed feeding transitions, a log(ft) of 6.94(2) was obtained for this state from a $\\beta $ -branching ratio of $\\approx $ 0.1 $\\%$ , confirming the $4^+$ spin-parity assignment adopted in Table REF ."
],
[
"The 5548 keV state",
"The 5548 keV state was observed to decay by a single $\\gamma $ ray of 3993 keV with an intensity of $<$ 0.001$\\%$ .",
"It appears in coincidence with the 1554 keV transition to the $^{50}$ Ti ground state.",
"A $\\gamma $ -ray coincidence spectrum gated on the 1554 keV transition is shown in Figure REF , panel (c).",
"The same spectrum without any coincidence requirements is shown inset.",
"While no $\\gamma $ rays are observed to feed this state, two additional de-excitation branches were reported in Ref.",
"[22] with energies of 2348.4 and 2872.8 keV.",
"The 2348.4 keV $\\gamma $ ray was a tentative assignment but the 2872.8 keV $\\gamma $ ray was reported with an appreciable intensity relative to the 3993 keV branch.",
"A possible explanation for their non-observation in the present work is that they are obscured by the Compton edges of more intense transitions, namely the 2675 keV $4_1^+$ $\\rightarrow $ $2_1^+$ $\\rightarrow $ $0_1^+$ sum peak and the 3132 keV transition from the 5806 keV state.",
"The background level near the 3993 keV transition is an order of magnitude lower despite a similar contribution from the Compton edge of the 4252 keV transition.",
"In addition, a coincidence gate placed on the 1554 keV $\\gamma $ ray does not exclude any of the most intense transitions or their associated backgrounds from incomplete energy collection.",
"The unobserved $\\gamma $ rays affect the determination of the $\\beta $ branching ratio to the 5548 keV state.",
"Thus we have corrected the total decay intensity using the literature branching ratios in combination with the observed intensity in this study.",
"This correction yields a $\\beta $ branching ratio which is around a factor of two higher than the value obtained when no correction is applied.",
"The calculated log($ft$ ) value decreases from $\\approx $ 8.57(6) to 8.17(4) but is still at the upper limit of the tail of the $\\Delta J$ =$\\pm 1$ range of typical values found in Ref.",
"[27] Since the tentative 2348 keV transition to the $6^+$ state at 3199 keV is not confirmed in the present work it is possible that the 5548 keV level has a spin of ($2,3$ )$^+$ and is populated by the $^{50m}$ Sc decay.",
"Considering only the 3993 and the (unobserved) 2872.8 keV transition intensities, a $\\beta $ -feeding intensity of 0.00175 is obtained corresponding to a log($ft$ )=6.08 (See Tab.",
"REF )."
],
[
"The 6123 keV state",
"Branching ratios of $\\gamma $ rays de-exciting the 6123 keV state with 1242, 1976, 2924 and 3448 keV feeding the $5^+$ (4881 keV), $4^+$ (4147 keV), $6^+$ (3199 keV) and $4^+$ (2675 keV) states were established from $\\gamma - \\gamma $ coincidence gates placed at 1121 keV to isolate the 2924 and 3448 keV transitions, 1472 keV for the 1976 keV transition, and at 2206 keV in order to obtain a value for the 1242 keV transition.",
"The data presented in Table REF are in reasonable agreement with results from neutron-capture measurements, where the 6123 keV state was assigned J$^{\\pi }$ =4$^+$ [22], although the branching ratio of the 1976 keV $\\gamma $ ray is higher than the reported lower limit of $<15\\%$ of the 1242 keV transition.",
"We note that the uncertainties provided in the literature are in general quite large for the weaker branches from the 6123 keV state.",
"No evidence of the reported 1636 keV transition is observed here, despite a significant intensity of 85 $\\%$ relative to the 1242 keV $\\gamma $ ray reported in Ref.",
"[22].",
"It is possible that the 1636 keV $\\gamma $ ray is obscured by the Compton scattered $6_1^+$ $\\rightarrow $ $4_1^+$ $\\rightarrow $ $2_1^+$ sum peak observed here at 1644 keV.",
"However, the 1242 keV $\\gamma $ ray is observed despite a 10-fold increase in background (dominated by backscattered 1554 keV $\\gamma $ rays) and only a modest increase in efficiency ($\\approx $ 13$\\%$ ).",
"A search for additional gating transitions that might be used to isolate the unobserved 1636 keV $\\gamma $ ray (for example, the 2933 and 4487 keV $\\gamma $ rays which de-excite the 4487 keV $2^+$ state) was not successful.",
"We also note that the 1636 keV $\\gamma $ ray is not listed as unambiguously assigned in Table 3 of Ref.",
"[22] and therefore no further corrections to the total decay intensity from the 6123 keV state have been applied.",
"No $\\gamma $ -ray transitions feeding this state were identified.",
"The assigned spin and parity of $J^+$ =$(4^+)$ is fully consistent with the extracted log(ft) value in Table REF of 5.63(4).",
"However since it was not possible to confirm the 1636 keV transition to the 4987 keV, $2^+$ level in the present work then this spin and parity assignment remains tentative."
],
[
"The 6156 keV state",
"Four $\\gamma $ rays of 1983, 2009 keV, 3481 keV and 4601 keV are found to decay from the 6156 keV state and populate the $3^+$ (4172 keV), $4_2^+$ (4147 keV), $4_1^+$ (2675 keV) and $2_1^+$ (1554 keV) states, respectively.",
"No feeding $\\gamma $ -ray transitions were observed for this state.",
"The 4601 keV $\\gamma $ ray was previously the only transition identified as de-exciting this state [22], although here it is in fact the weakest branch at around 10$\\%$ of the intensity of the 2009 keV transition.",
"All three transitions are observed here following $\\beta $ decay for the first time.",
"A tentative range of spins J$^{\\pi }$ =$(2,3,4)$ and positive parity were originally assigned to the 6156.0 keV state in Ref.[22].",
"With the total decay intensity of the 3 de-exciting transitions taken into account, a $\\beta $ -branching ratio of $\\approx $ 0.01$\\%$ is obtained yielding a log(ft) of 6.29(4), in agreement for $\\Delta $ J=0,$\\pm $ 1 transitions.",
"A J$^{\\pi }$ =4$^+$ assignment is therefore confirmed.",
"It is noted that if the 4601 keV $\\gamma $ ray was indeed the only transition observed in the present experiment, then the transition rate would decrease by more than an order of magnitude (log(ft) $\\approx $ 7.5) but would remain in the tail of the typical $\\Delta $ J=0,$\\pm $ 1 values of Ref.",
"[27]."
],
[
"The 6237 keV state",
"The 6237.0 keV state was identified from the observation of $\\gamma $ rays of 2065 keV feeding the $3^+$ member of the 4172 keV doublet, 3038.0 keV feeding the $6_1^+$ (3199 keV) state and 3561.0 keV populating the $4_1^+$ (2675.0 keV) state.",
"It is one of only two states observed in the present study that has not been reported elsewhere.",
"The 3561 keV $\\gamma $ ray is observed in coincidence with the 1121 keV and 1554 keV transitions to the $2_1^+$ (1554 keV) state and the ground state, respectively, whereas the 3038 keV $\\gamma $ ray is observed in coincidence with both of these transitions plus the 524 keV $\\gamma $ decay from the $6_1^+$ (3199 keV) state.",
"No $\\gamma $ -ray transitions were observed to feed this state.",
"A log(ft) value of 5.73(4) is calculated for this state as shown in Table REF .",
"A constraint is placed on the spin and parity of this state of $J^\\pi $ =$(4-5)^+$ based on the log(ft) value and the observation of the de-populating 2065 keV transition feeding the $3^+$ member of the 4172 keV doublet."
],
[
"The 6625 keV state",
"The 6625.0 keV state has not been observed in any previous work and was identified here via the observation of $\\gamma $ rays at 3427 keV decaying to the $6_1^+$ (3199 keV) state and 3950 keV to the $4_1^+$ (2675 keV) state.",
"The 3950 keV $\\gamma $ ray is observed in coincidence with the 1121 keV and 1554 keV transitions to the $2_1^+$ and $0_1^+$ states, respectively, while for the 3427 keV $\\gamma $ ray an additional strong coincidence is found with the 524 keV transition to the $4_1^+$ state.",
"No feeding transitions were observed, although the excitation energy lies close to the $\\beta $ -decay $Q$ value (similarly for the 6237 keV state).",
"A log(ft) value of 5.39(10) is calculated and from this an assignment of a positive parity with a spin value in the range $J$ =$(4-6)$ is provided in Table REF .",
"Several excited states in $^{50}$ Ti with $E_x$ $\\approx $ 6.3-6.8 MeV are proposed in Ref.",
"[22] with (tentatively) assigned spin and parities which would mean they are accessible via allowed Gamow-Teller $\\beta $ transitions.",
"A search was performed of the data for the associated decay $\\gamma $ rays including examining $\\gamma $ -$\\gamma $ -$\\gamma $ triple coincidences between the 524, 1121 or 1554 keV transitions to reduce the background contribution from Compton scattered sum peaks which may obscure very weak transitions.",
"No additional $\\gamma $ rays were identified.",
"The 6625.0 keV state represents the highest observed excitation energy populated in the $\\beta ^-$ decay of the $^{50}$ Sc."
],
[
"Shell model calculations",
"The stable $^{50}$ Ti nucleus sits just above doubly-magic $^{48}$ Ca with a closed-shell of 28 neutrons and 2 valence protons outside the magic proton shell closure of 20.",
"It is expected that the spherical shell model will reproduce the excitations in $^{50}$ Ti with good accuracy.",
"Shell model calculations using both phenomenological and ab initio based interactions have been performed for comparison with the experimental data.",
"Shell model calculations were performed with the NuShellX@MSU shell-model code [30] using the phenomenological KB3G [10] and GXPF1A [31] interactions in the $pf$ valence space ($0f_{7/2}$ , $1p_{3/2}$ , $0f_{5/2}$ , $1p_{1/2}$ ), known to well reproduce experimental data in this region.",
"In addition, we derive ab initio shell-model Hamiltonians within the valence-space in-medium similarity renormalization group (VS-IMSRG) framework [32], [33], [34], [35], [36], based on two-nucleon (NN) and three-nucleon (3N) forces derived from chiral effective field theory [3], [4].",
"The details of the particular input NN+3N interaction used here (EM1.8_2.0), and developed in Refs.",
"[37], [38], [39], as well as the specifics of the Hamiltonian are described in Ref.",
"[15].",
"Starting in a single-particle spherical harmonic oscillator (HO) basis with energy $\\hbar \\omega =16$ MeV, we first transform the input Hamiltonian to the Hartree-Fock (HF) basis, then use the Magnus formulation of the VS-IMSRG [40], [41], with the ensemble normal ordering procedure [35], which captures the bulk effects of residual 3N forces among valence nucleons, to produce an approximate unitary transformation which decouples the $^{40}$ Ca core.",
"A second transformation is performed to decouple a specific $pf$ -shell valence-space Hamiltonian appropriate for $^{50}$ Ti.",
"These results are well converged within the basis size $e=2n+l \\le e_{\\mathrm {max}}=12$ and $e_1 + e_2 + e_3 \\le E_{\\mathrm {3max}} = 16 $ .",
"We finally use the approximate unitary transformation to decouple an effective valence-space $E2$ or $M1$ operator consistent with the valence-space Hamiltonian [42].",
"Table: Transition strength, B(E2)B(E2), values are given in e 2 fm 4 e^2fm^4 and quadrupole moments, QQ, are given in e 2 fm 2 e^2fm^2.Three distinct sets of states emerge from the calculations.",
"Firstly the ground-state band where a pair of protons occupy primarily the $\\pi 0f_{7/2}$ orbital ($J^\\pi = 0^+, 2^+, 4^+, 6^+$ ) and the neutron orbitals are filled up to the $N=28$ shell gap, secondly a neutron 1-particle-1-hole ($\\nu 0f_{7/2}^{-1}-1p_{3/2}^{1}$ ) configuration ($J^\\pi = 2^+, 3^+, 4^+, 5^+$ ) and finally a neutron 2-particle-2-hole ($\\nu 0f_{7/2}^{-2}-1p_{3/2}^{2}$ ) configuration ($J^\\pi = 0^+, 2^+$ ).",
"In both of these cross-shell excitations the particles occupy primarily the $1p_{3/2}$ orbital above $N=28$ .",
"A detailed comparison of every experimentally observed excited state is difficult due to the level density and in some cases the lack of firm spin assignments.",
"Instead we concentrate here on an examination of these three low-lying structures.",
"Select properties from the three calculations are shown in Figure REF and are given in Table REF in comparison to the experimental data.",
"While the first excited 2$^+$ energy predicted in the VS-IM-SRG is several hundred keV higher than experiment, possibly arising from a too-large shell gap, the spacing between higher-lying states agrees well with experiment.",
"The degree of collectivity as evidenced in the magnitude of the $B(E2)$ values is also underpredicted by the VS-IM-SRG.",
"This is expected because of the limitations placed on the model space used for the transformation.",
"Despite this, the spectroscopic information such as the pattern of $B(E2)$ values and quadrupole moments ($Q$ ) are well reproduced, as was first noted in Ref.",
"[43].",
"All three calculations predict a positive quadrupole moment in the ground state and $2p-2h$ bands while the $1p-1h$ configuration is predicted to be negative, all with similar magnitudes.",
"It is interesting that the $B(E2)$ values predicted for the ground-state band are consistent between the interactions but that the predictions for the $2p-2h$ band are quite different, ranging from 10 to 70% that of the $2^+_1\\rightarrow 0^+_1$ transition.",
"This distinction might be used in the future to distinguish between the calculations once an experimental value for the strength of this transition becomes available.",
"Figure: Selected experimentally observed levels in 50 ^{50}Ti are compared to the results of three different theory calculations.",
"Excited states belonging to the ground state band (full-width levels), 2p2h2p2h (quarter-width, left) and 1p1h1p1h particle-hole excitations (quarter-width, right) are included.",
"B(E2) transition strengths are provided in units of e 2 fm 4 e^2fm^4 and are proportional to the widths of the arrows."
],
[
"Conclusion",
"The GRIFFIN $\\gamma $ -ray spectrometer at the TRIUMF-ISAC facility has been used to study the $\\beta $ decay of $^{50}$ Sc to $^{50}$ Ti using a radioactive beam of $^{50}$ Ca.",
"$\\beta $ -decay branching ratios from both the ground state and isomeric state of $^{50}$ Sc to 16 excited states in $^{50}$ Ti are determined from a total of 38 $\\gamma $ -ray transitions which significantly expands the information available over previous studies.",
"Relative intensities are measured to less than 0.001$\\%$ that of the strongest transition with the majority of $\\gamma $ -ray transitions observed here in $\\beta $ decay for the first time.",
"The data are compared to shell-model calculations utilizing both phenomenologically-derived interactions employed in the pf shell as well as a state-of-the-art, ab initio based interaction built in the valence-space in-medium similarity renormalization group framework.",
"The differences in predictions for cross-shell excitations across the $N=28$ neutron shell closure are discussed.",
"We would like to thank the operations and beam delivery staff at TRIUMF for providing the radioactive beam.",
"C.E.S.",
"acknowledges support from the Canada Research Chairs program.",
"The GRIFFIN spectrometer was jointly funded by the Canadian Foundation for Innovation (CFI), TRIUMF, and the University of Guelph.",
"TRIUMF receives federal funding via a contribution agreement through the National Research Council Canada (NRC).",
"This work was supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC)."
]
] | 1906.04245 |
[
[
"Coupled Variational Recurrent Collaborative Filtering"
],
[
"Abstract We focus on the problem of streaming recommender system and explore novel collaborative filtering algorithms to handle the data dynamicity and complexity in a streaming manner.",
"Although deep neural networks have demonstrated the effectiveness of recommendation tasks, it is lack of explorations on integrating probabilistic models and deep architectures under streaming recommendation settings.",
"Conjoining the complementary advantages of probabilistic models and deep neural networks could enhance both model effectiveness and the understanding of inference uncertainties.",
"To bridge the gap, in this paper, we propose a Coupled Variational Recurrent Collaborative Filtering (CVRCF) framework based on the idea of Deep Bayesian Learning to handle the streaming recommendation problem.",
"The framework jointly combines stochastic processes and deep factorization models under a Bayesian paradigm to model the generation and evolution of users' preferences and items' popularities.",
"To ensure efficient optimization and streaming update, we further propose a sequential variational inference algorithm based on a cross variational recurrent neural network structure.",
"Experimental results on three benchmark datasets demonstrate that the proposed framework performs favorably against the state-of-the-art methods in terms of both temporal dependency modeling and predictive accuracy.",
"The learned latent variables also provide visualized interpretations for the evolution of temporal dynamics."
],
[
"Introduction",
"With the explosive growth of online information, recommender systems have been pervasively used in real-world business services and widely studied in literature [31], [32].",
"Upon classical static settings, in real-world applications, data are often grown in a streaming fashion and evolving with time.",
"For example, Snapchat users share over 400 million snaps [40] and Facebook users upload 300 million photos per day [39].",
"The ever-growing data volume along with rapidly evolved data properties puts the demand of time aware and online recommender systems, which could incorporate the temporal information to handle the data temporality and update in a streaming manner to alleviate the burden of data complexity.",
"Deep learning techniques have been widely conducted in exploiting temporal dynamics to improve the recommendation performance [52], [18], [50], [2].",
"Despite the prominence shown recently in deep recommender systems [13], [43], [18], deep frameworks also have their own limitations.",
"One of the well-known facts is that deep recommender systems are usually deterministic approaches, which only output point estimations without taking the uncertainty into account.",
"It significantly limits their power in modeling the randomness of the measurement noises [35] and providing predictions of the missing or unobserved interactions in recommender systems.",
"As probabilistic approaches, especially Bayesian methods, provide solid mathematical tools for coping with the randomness and uncertainty, it motivates us to conduct streaming recommendations from the view of Deep Bayesian Learning (DBL) to conjoin the advantages of probabilistic models and deep learning models.",
"Though some recent attempts have been made on integrating probabilistic approaches with deep autoencoder architecture for recommendation tasks [16], [24], [33], they are still underpinned the static recommendation setting, which allows them to be retrospective to all the historical data during the updates.",
"Simply applying DBL to streaming recommendations is a non-trivial task due to the following challenges.",
"First, coordinating the temporal dynamics is difficult given the continuous-time discrete-event recommendation process along with the protean patterns on both user and item modes.",
"A user's preference on certain items may evolve rapidly, while on others maintaining a long-term fix.",
"Second, the high velocity of streaming data requires an updatable model, which could expeditiously extract the prior knowledge from former time steps and effectively digest it for current predictions.",
"Also, since the data occurrence is, in fact, continuous-valued, taking the continues time information into consideration could be potentially helpful for the knowledge distillation [2].",
"Third, the DBL frameworks are usually expensive in terms of both time and space complexities.",
"Existing optimization algorithms often require a huge amount of computation to infer and update especially under streaming setting such as Sequential Monte Carlo, which is usually infeasible for large-scale recommendations.",
"To tackle the aforementioned challenges, in this paper, we propose to investigate the ways to conduct streaming recommendation by leveraging the advantages of both deep models and probabilistic processes.",
"We stick to the factorization-based approaches due to their popularity and superiority among all collaborative filtering techniques [17].",
"Specifically, we study: (1) How to model the streaming recommender system with an updatable probabilistic process?",
"(2) How to incorporate deep architectures into the probabilistic framework?",
"(3) How to efficiently learn and update the joint framework with streaming Bayesian inference?",
"Through answering these three questions, we propose a Coupled Variational Recurrent Collaborative Filtering (CVRCF) framework.",
"CVRCF incorporates deep architectures into the traditional factorization-based model and encodes temporal relationships with a coupled variational gated recurrent network, which is optimized through sequential variational inference.",
"The main contributions are summarized as follows: Propose a novel streaming recommender system CVRCF, which incorporates deep models and the general probabilistic framework for streaming recommendations; Build up a linkage between probabilistic process and deep factorization based model under a streaming setting with sequential variational inference leveraging a continues-time discrete-event cross RNN model; Empirically validate the effectiveness of CVRCF on different real-world datasets comparing with the state-of-the-art, explore the temporal drifting patterns learned from CVRCF, and analyze the model sensitivities."
],
[
"Preliminaries",
"Notations: Before discussing the proposed framework CVRCF for streaming recommendations, we first introduce the mathematical notations.",
"We consider the streaming interactions as a continues-time discrete-event process.",
"Equipped with this viewpoint, we denote $T \\in \\mathbb {N}$ as the discrete time step and the inputs of a streaming recommender system can be denoted as a list of user-item interactions $\\lbrace x_{ij}^T\\rbrace $ with their occurrence time $\\lbrace \\tau _{ij}^T\\rbrace $ , where $x_{ij}^T$ denotes the interaction event of the $i^\\text{th}$ user and the $j^\\text{th}$ item occurred between time step $T-1$ and $T$ , $\\tau _{ij}^T$ denotes the concrete time that $x_{ij}^T$ occurs.",
"The time interval between two consecutive time steps is called granularity, which does not need to be fixed in practice.",
"All interactions arrived before the $T^\\text{th}$ time step are denoted as $\\lbrace x_{ij}^{\\le T} \\rbrace $ (or $\\lbrace x^{\\le T} \\rbrace $ ).",
"Without loss of generality, interactions are regarded as ratings throughout this paper.",
"Problem Statement: Based on these notations, the streaming recommendation problem we studied in this paper is defined as: for any $T=1,2,\\ldots $ , given the sequence of historical user-item interactions $\\lbrace x_{ij}^{\\le T-1}\\rbrace $ , with the actual time information $\\lbrace \\tau _{ij}^{\\le T-1}\\rbrace $ , we aim at predicting the upcoming interactions $\\lbrace x_{ij}^{T}\\rbrace $ in a streaming manner.",
"The streaming manner here means that the model should be streamingly updatable.",
"In another word, if we assume a model is achieved at $T=k-1$ , then at time $T=k$ , the model should be able to update based only on the data acquired between time $T=k-1$ and $T=k$ , i.e., $\\lbrace x_{ij}^{k}\\rbrace $ ."
],
[
"Coupled Variational Recurrent Collaborative Filtering",
"The core of CVRCF is a dynamic probabilistic factor-based model that consists of four components.",
"The first two formulate the user-item interactions and temporal dynamics, respectively.",
"Each of them incorporates a probabilistic skeleton induced by deep architectures.",
"The third component is a sequential variational inference algorithm, which provides an efficient optimization scheme for streaming updates.",
"The last component allows us to generate rating predictions based on the up-to-date model."
],
[
"Interaction Network",
"Factor-based models are widely adopted in recommendation modelings.",
"They have shown a great success in multiple recommendation tasks [25].",
"Most of them follow the traditional matrix factorization setting, in which users and items are modeled as latent factors; and their interactions are defined as the linear combinations of these factors.",
"However, such simple linear combinations are often insufficient to model complex user-item interactions [17].",
"Thus, we consider a deep probabilistic matrix factorization setting as follow $\\small x_{ij}^T|\\mathbf {u}_i^T,\\mathbf {v}_j^T,\\sigma _{i,j,T}^2\\sim \\mathcal {N}(f_1(\\mathbf {u}_i^T,\\mathbf {v}_j^T), f_2(\\mathbf {u}_i^T,\\mathbf {v}_j^T,\\sigma ^2_{i,j,T})),$ where both $f_1(\\cdot )$ and $f_2(\\cdot )$ are represented by deep neural networks.",
"We represent the latent vectors of user $i$ and item $j$ at time step $T$ as $\\mathbf {u}_i^T$ and $\\mathbf {v}_j^T$ , respectively.",
"The rating $x_{ij}^T$ is modeled as a Gaussian random variable whose location and scale values are the output of the deep networks.",
"The environmental noise $\\sigma _{i,j,T}^2$ could either be predefined as a hyperparameter [25] or jointly learned.",
"It is worth pointing out that we assume the variance of $x_{ij}^T$ depends on both the latent vectors and the environmental noises, which is slightly different from the conventional probabilistic setting [25]."
],
[
"Temporal Drifting Process",
"The temporal dynamics of a recommender system depend on the drifting of users' preferences and item popularities [30], [9].",
"A user's tastes for a certain type of items may change over time while the popularity of an item may also vary with time goes by.",
"To capture the inherent dynamics, we intend to encode the drifting processes into user and item latent factors based on three hypotheses: We assume the latent factors of both user and item can be decomposed as the combination of a stationary term ($\\mathbf {u}^s_i$ ) and a dynamic term ($\\Delta \\mathbf {u}_i^T$ ) [50].",
"The stationary factor captures the long-term preference, which varies slowly over time.",
"The dynamic factor encodes the short-term changes, which evolves rapidly.",
"An illustrative example is shown in Figure REF , where a user's dynamic factor evolves between two consecutive time steps, causing his preference drifted from $\\mathbf {u}_i^{T-1}$ to $\\mathbf {u}_i^{T}$ .",
"We assume the two factors are independent of each other for simplicity.",
"The dynamic factors of a user or an item follows a Markov process [5].",
"The intuition of using a Markov process comes from the observation that the changing of a user's current preference could be highly affected by his former preference.",
"The changing of latent factors of a particular user $i$ (or item $j$ ) between two consecutive time steps $T-1$ and $T$ depends on the time interval between the last events before these two time steps, which involves this user (or item), i.e., $\\Delta \\tau _{u,i}^{T} = \\tau _{u,i}^{T}-\\tau _{u,i}^{T-1}$ , where $\\tau _{u,i}^{T-1}$ and $\\tau _{u,i}^{T}$ denote the actual time of the two last interactions of user $i$ before time step $T-1$ and $T$ , respectively.",
"Intuitively, the longer the interval is, the larger the drifting may happen.",
"$ \\tau _{u,i}^{T}$ is defined to be equal to $\\tau _{u,i}^{T-1}$ if no interactions happens between time step $T-1$ and $T$ .",
"Upon these hypotheses, we model the evolution of hidden topics of a user (or an item), via spatiotemporal Gaussian priors, which is mathematically formulated as follows: $ \\begin{aligned}{\\left\\lbrace \\begin{array}{ll}\\mathbf {u}_i^T=\\mathbf {u}_i^s+ \\Delta \\mathbf {u}_i^T,\\\\ \\\\\\mathbf {u}_i^s \\sim \\mathcal {N}(\\mathbf {0}, \\sigma ^2_U\\mathbf {I}),\\\\ \\\\\\Delta \\mathbf {u}_i^T|\\Delta \\mathbf {u}_i^{T-1} \\sim \\mathcal {N}({\\mu }_{u,i,T}, {\\Sigma }_{u,i,T}).\\end{array}\\right.",
"}\\end{aligned}$ It is worth pointing out that only the users, which have interactions between time $T$ and $T-1$ , need to be considered here while factors of users who do not have interactions are assumed to be unchanged till their next interaction happens.",
"We place the zero-mean spherical Gaussian prior on the stationary factors [25], where $\\sigma _U$ denotes the scale hyperparameter.",
"For dynamic factors, the kernel matrix ${\\Sigma }_{u,i,T}$ is defined as a diagonal matrix here for simplicity, i.e., ${\\Sigma }_{u,i,T} \\triangleq diag({\\sigma }_{u,i,T}^2)$ .",
"Motivated by the recent advances in deep kernel learning, which combines the non-parametric flexibility of kernel approaches with the structural properties of deep architectures [49], we further define the kernel as an output of a deep neural network $f_3(\\cdot )$ to enhance its generality, i.e., ${\\sigma }_{u,i,T}^2=f_3(\\Delta \\mathbf {u}_i^{T-1},\\Delta \\tau ^T_{u,i})$ .",
"Coping with the last two hypotheses, this spatiotemporal kernel takes $\\Delta \\mathbf {u}_i^{T-1}$ , which represents the user's dynamic preference at last time step, as a spatial effect to decide the drifting uncertainty and it is stationary for temporal effect, which means ${\\Sigma }_{u,i,T}$ depends on the time internal $\\Delta \\tau _{u, i}^T$ rather than the concrete time $\\tau _{u,i}^T$ and $\\tau _{u,i}^{T-1}$ .",
"For a more unified representation, we can further define ${\\mu }_{u,i,T}=f_4(\\Delta \\mathbf {u}_i^{T-1},\\Delta \\tau ^T_{u,i})$ , where $f_4(\\cdot )$ denotes a predefined deep neural network.",
"The definition of the whole drifting prior obeys the Markov property for the discrete events on the continues timeline, which implies that the current state depends only on the former state.",
"It is also applicable to employ other state dependency correlations and network structures.",
"Similar prior with corresponding notations is defined for items.",
"Figure: CVRCF prediction network."
],
[
"Deep Sequential Variational Inference",
"The third component of the CVRCF framework is the inference model.",
"It composites the two former components with a sequential Bayesian skeleton and associates them with the last prediction component for streaming recommendations."
],
[
"Joint Distribution",
"The joint distribution of all observations up to time $T$ and the latent factors is defined as follows: $\\begin{aligned}& p(x^{\\le T}, \\mathbf {U}^{\\le T}, \\mathbf {V}^{\\le T}) = p(x^{\\le T}, \\mathbf {U}^{s}, \\mathbf {V}^{s},\\Delta \\mathbf {U}^{\\le T}, \\Delta \\mathbf {V}^{\\le T})\\\\= ~ & p(x^{\\le T}, \\Delta \\mathbf {U}^{\\le T}, \\Delta \\mathbf {V}^{\\le T}|\\mathbf {U}^{s}, \\mathbf {V}^{s})p(\\mathbf {U}^{s})p( \\mathbf {V}^{s})\\\\= ~ & p(\\mathbf {U}^{s})p( \\mathbf {V}^{s}) \\Big [ \\prod _{t\\le T} p(x^t|x^{<t},\\Delta \\mathbf {U}^{\\le t}, \\Delta \\mathbf {V}^{\\le t}, \\mathbf {U}^{s}, \\mathbf {V}^{s}) \\\\& \\qquad \\quad \\times p(\\Delta \\mathbf {U}^{t}, \\Delta \\mathbf {V}^{t}|x^{<t},\\Delta \\mathbf {U}^{<t}, \\Delta \\mathbf {V}^{<t},\\mathbf {U}^{s}, \\mathbf {V}^{s})\\Big ],\\end{aligned}$ where $\\mathbf {U}$ and $\\mathbf {V}$ are the matrices of the latent factors for existing users and items.",
"Our goal is to infer the posterior distribution of latent factors for every $t$ , i.e., $p(\\mathbf {U}^{t},\\mathbf {V}^{t}|x^{\\le t}), \\forall t\\le T$ .",
"However, it is intractable for direct inferences based on the current model assumptions.",
"To overcome this challenge, existing works usually focus on two types of approaches - Sequential Monte Carlo methods (SMC) [11] and Variational Inference methods (VI) [3].",
"The traditional sequential Bayesian updating usually uses SMC methods (a.k.a., particle filtering) to deal with intractable target posterior distributions.",
"Although this approach is very accurate when suitable proposal distributions and enough particle samples are presented, the sampling process is often too slow to apply to high dimensional and large-scale data [34].",
"On the other hands, the variational inference is much faster compared to SMC.",
"However, the accuracy highly depends on the approximation distribution, especially in streaming settings [42].",
"Although there are hybrid models combine both algorithms together [15], [27], the computational complexity makes it prohibited for large-scale recommender systems.",
"To trade-off the model scalability and accuracy, we consider the streaming variational inference framework [3] by leveraging deep neural networks as the variational approximator to obtain more flexible posteriors."
],
[
"Sequential Variational Inference Network",
"Before introducing the deep architectures, we first assume the latent factors can be partitioned into independent units followed by the traditional mean-field approximation: $\\begin{aligned}&q(\\Delta \\mathbf {U}^{\\le T}, \\Delta \\mathbf {V}^{\\le T}| x^{\\le T},\\mathbf {U}^{s},\\mathbf {V}^{s}) = q(\\Delta \\mathbf {U}^{\\le T}|x^{\\le T})q(\\Delta \\mathbf {V}^{\\le T}|x^{\\le T}),\\end{aligned}$ where $q$ denotes the approximated variational posterior.",
"Further, each user (or item) is placed by a Gaussian variational posterior as follows: $q(\\Delta \\mathbf {u}_i^t| \\Delta \\mathbf {u}_i^{\\le {t-1}},x_i^{\\le t}) = \\mathcal {N}({\\mu }_{u,i,t}^\\ast , {\\Sigma }^\\ast _{u,i,t}), \\forall 1\\le t \\le T,$ where ${\\Sigma }_{u,i,t}$ is diagonal with the similar definition as the priors defined in Equ.",
"(REF ).",
"$x_i^{\\le t}$ denotes all the interactions related to user $i$ before time step $t$ .",
"To infer the variational posterior, we propose a Coupled Variational Gated Recurrent Network structure (CVGRN) leveraging two variational Gated Recurrent Units (GRUs) for users and items, respectively.",
"fig:infer demonstrates the key idea of the proposed inference network.",
"Blocks represent the inputs of two GRUs at different time steps.",
"$q_{u,i}^{t}$ and $q_{v,j}^{t}$ represent the approximated posterior distribution $q(\\Delta \\mathbf {u}_i^{t}| \\Delta \\mathbf {u}_i^{\\le t-1},x_i^{\\le {t}}) $ and $q(\\Delta \\mathbf {v}_j^{t}| \\Delta \\mathbf {v}_j^{\\le t-1},x_j^{\\le t }) $ , which are inferred based on the GRUs output states $\\textbf {h}_{u,i}^{t-1}$ and $\\textbf {h}_{v,j}^{t-1}$ and the interactions elated to user $i$ and item $j$ between time step $t-1$ and $t$ , i.e., $\\lbrace x_i^{t}\\rbrace $ and $\\lbrace x_j^{t}\\rbrace $ .",
"Specifically, assume a user and a movie interact with each other at time $t$ .",
"The red and blue blocks denote the inputs of the user chain and item chain at time step $t$ , respectively, which are denoted as $\\mathbf {y}^{t}_{u,i}$ and $\\mathbf {y}^{t}_{v,j}$ .",
"These two inputs are constructed based on user $i$ 's or item $j$ 's interactions between time steps $t-1$ and $t$ , respectively.",
"For example, $\\mathbf {y}_{u,i}^{t}$ is defined as $\\mathbf {y}_{u,i}^{t}= [\\mathbf {W}_u\\cdot \\mathbf {x}_{u,i}^{t}, \\log (\\Delta \\tau _{u,i}^{t}), 1_{u, \\text{new}} ]$ , where $\\mathbf {x}_{u,i}^{t}$ denotes a sparse vector consisting of the ratings $\\lbrace x_i^{t}\\rbrace $ given by user $i$ in time interval $\\Delta \\tau _{u,i}^{t}$ .",
"$\\mathbf {W}_u$ is an embedding matrix, which is employed to reduce the length of GRUs inputs for alleviating intermediate data explosion.",
"$1_{u, \\text{new}}$ indicates whether a user is a new user or not [50].",
"The log interval $\\log (\\Delta \\tau _{u,i}^{t})$ is concatenated into the inputs to encode continues-time information [2].",
"Inferring $q_{u,i}^{t}$ is equivalent to inferring ${\\mu }_{u,i,t}^\\ast $ and ${\\Sigma }^\\ast _{u,i,t}$ in Equ.",
"(REF ), which are calculated as: $[{\\mu }_{u,i,t}^\\ast , {\\Sigma }^\\ast _{u,i,t}] = f_5 (\\mathbf {h}_{u,i}^{t-1}, \\mathbf {y}_{u,i}^{t})$ .",
"$f_5$ is a deep neural network.",
"Since all of the users (or items) share the same RNN chain, the model size could be largely reduced.",
"Moreover, to further reduce the number of latent variables, the conditioned prior distributions of the dynamic factors $\\Delta \\mathbf {u}_i^T|\\Delta \\mathbf {u}_i^{T-1}$ , which is defined in Equ.",
"(REF ), are assumed to be parameterized by the latent states, i.e., $[{\\mu }_{u,i,t}, {\\Sigma }_{u,i,t}] = [f_4(\\mathbf {h}_{u,i}^{t-1},\\Delta \\tau ^T_{u,i}), f_3(\\mathbf {h}_{u,i}^{t-1},\\Delta \\tau ^T_{u,i})]$ .",
"To further encode the temporal information, we exponentially decay the latent state variables at each time step [26] as $\\mathbf {h}_{u,i}^{t} \\leftarrow \\mathbf {h}_{u,,i}^{t} \\cdot e^{ \\frac{\\Delta \\tau _{u,i}^{t}}{\\lambda } }$ , where $\\lambda $ is a predefined decay rate."
],
[
"Objective Function",
"Considering RNN as a graphical model, we leverage the conditionally independency between current latent state and future inputs, and have $\\mathbf {h}^{t}\\rotatebox [origin=c]{90}{\\models }x^{>t} | \\mathbf {h}^{t-1},x^{t}$ .",
"Then Equ.",
"(REF ) could be written as: $\\begin{aligned}q(& \\Delta \\mathbf {U}^{\\le T},\\Delta \\mathbf {V}^{\\le T}|x^{\\le T},\\mathbf {U}^{s},\\mathbf {V}^{s})\\\\&=\\prod _{t\\le T}q(\\Delta \\mathbf {U}^t|x^{\\le t},\\Delta \\mathbf {U}^{<t})q(\\Delta \\mathbf {V}^t|x^{\\le t},\\Delta \\mathbf {V}^{<t}).\\end{aligned}$ To obtain the objective function, we try to follow the traditional variational autoencoder to derive a variant variational lower bound.",
"We start from the joint log likelihood and drive the objective function as follows: $\\small \\begin{aligned}& \\log p(x^{\\le T}, \\mathbf {U}^{s},\\mathbf {V}^{s}) = \\log p(x^{\\le T}| \\mathbf {U}^{s},\\mathbf {V}^{s}) + \\log p(\\mathbf {U}^{s} ) + \\log p(\\mathbf {V}^{s} ) \\\\& = \\int \\log p(x^{\\le T},\\Delta \\mathbf {U}^{\\le T},\\Delta \\mathbf {V}^{\\le T} | \\mathbf {U}^{s},\\mathbf {V}^{s}) d \\Delta \\mathbf {U}^{\\le T} d \\Delta \\mathbf {V}^{\\le T} + \\log p(\\mathbf {U}^{s} ) + \\log p(\\mathbf {V}^{s} )\\\\& \\ge \\int q(\\Delta \\mathbf {U}^{\\le T},\\Delta \\mathbf {V}^{\\le T}|x^{\\le T}) \\log \\frac{p(x^{\\le T},\\Delta \\mathbf {U}^{\\le T},\\Delta \\mathbf {V}^{\\le T} | \\mathbf {U}^{s},\\mathbf {V}^{s} )}{q(\\Delta \\mathbf {U}^{\\le T},\\Delta \\mathbf {V}^{\\le T}|x^{\\le T})} d \\Delta \\mathbf {U}^{\\le T} d \\Delta \\mathbf {V}^{\\le T} \\\\& + \\log p(\\mathbf {U}^{s} ) + \\log p(\\mathbf {V}^{s} ) \\\\& = \\sum _{t\\le T}\\Big \\lbrace E_{q(\\Delta \\mathbf {U}^t|x^{\\le t},\\Delta \\mathbf {U}^{<t}),q(\\Delta \\mathbf {V}^t|x^{\\le t},\\Delta \\mathbf {V}^{<t})} [\\log p(x^{t}|x^{<t}, \\mathbf {U}^{\\le t}, \\mathbf {V}^{\\le t} )]\\\\& -KL(q(\\Delta \\mathbf {U}^t|x^{\\le t},\\Delta \\mathbf {U}^{<t}) || p(\\Delta \\mathbf {U}^{t}|\\Delta \\mathbf {U}^{<t})) \\\\& -KL(q(\\Delta \\mathbf {V}^t|x^{\\le t},\\Delta \\mathbf {V}^{<t}) || p(\\Delta \\mathbf {V}^{t}|\\Delta \\mathbf {V}^{<t})) \\Big \\rbrace \\\\& + \\log p(\\mathbf {U}^{s} ) + \\log p(\\mathbf {V}^{s} ).\\end{aligned}$ To further simply the expression, we denote the probabilities $q(\\Delta \\mathbf {U}^t|x^{\\le t},\\Delta \\mathbf {U}^{<t})$ , $q(\\Delta \\mathbf {V}^t|x^{\\le t},\\Delta \\mathbf {V}^{<t})$ , $p(\\Delta \\mathbf {U}^{t}|\\Delta \\mathbf {U}^{<t})$ , $p(\\Delta \\mathbf {V}^{t}|\\Delta \\mathbf {V}^{<t})$ , and $p(x^{t}|x^{<t},\\mathbf {U}^{\\le t},\\mathbf {V}^{\\le t} )$ , as $q_u^t$ , $q_v^t$ , $p_u^t$ , $p_v^t$ , and $p_{x}^t$ , respectively.",
"Based on the former definitions, the objective function is defined as a timestep-wise variational lower bound as follows: $\\begin{aligned}\\mathcal {L} = & \\sum _{t\\le T}\\Big \\lbrace \\mathbb {E}_{q_u^t,q_v^t} [ \\log p^t_{x}] - \\text{KL}(q_u^t || p_u^t) - \\text{KL}(q_v^t|| p_v^t) \\Big \\rbrace \\\\& ~~ + \\log p(\\mathbf {U}^s) + \\log p(\\mathbf {V}^s).\\end{aligned}$ It is worth pointing out that the expectation term is calculated based on sampling, i.e., $ \\mathbb {E}_{q_u,q_v} [\\log p^t_{x}] \\simeq \\frac{1}{L} \\sum _{l=1}^L \\log p(x^{t}|x^{<t},\\mathbf {U}^{\\le t,l},\\mathbf {V}^{\\le t,l} )$ , where $L$ is the number of samples we wish to use to estimate the quantity.",
"We specifically set $L=1$ for every iteration in the implementation following the setting in conventional Variational Auto-Encoder [21] and adopt the reparameterization trick for feasible optimization.",
"As the rating sequence of each user or item could be infinite long under the streaming setting, which makes it infeasible to feed the whole sequences into the RNNs, this step-wise objective function allows us to truncate the sequences into multiple segmentations for a streaming inference.",
"In another words, assume $q_u^T$ , $q_v^T$ , $p_u^T$ , $p_v^T$ , $\\mathbf {U}^s$ and $\\mathbf {V}^s$ are achieved at time step $T$ , they could be treated as the prior distribution of the latent variables at time step $T+1$ and updated based on the new interactions $\\lbrace x_{ij}^{k}\\rbrace $ , the CVRCF framework, and the following step-wise objective function: $\\begin{aligned}\\mathcal {L} = & \\Big \\lbrace \\mathbb {E}_{q_u^{T+1},q_v^{T+1}} [ \\log p^{T+1}_{x}] - \\text{KL}(q_u^{T+1} || p_u^{T+1}) - \\text{KL}(q_v^{T+1}|| p_v^{T+1}) \\Big \\rbrace \\\\& ~~ + \\log p(\\mathbf {U}^s) + \\log p(\\mathbf {V}^s).\\end{aligned}$ It is worth pointing that as stated in Section REF , we assume the stationary factors $\\mathbf {U}^s$ and $\\mathbf {V}^s$ represent long-term users' preferences and item popularities.",
"Thus, they should also be updated at each time-step.",
"However, they remain the same between two consecutive time steps while the dynamic factors keep evolving."
],
[
"Prediction Network",
"The prediction model is based on the generation model described in fig:pred.",
"At any testing time between time steps $T-1$ and $T$ , to predict a specific ratings of a user $i$ to an item $j$ , we first calculate the expectations of the current latent representations $\\mathbf {u}_i^{T}$ and $\\mathbf {v}_j^{T}$ based on the prior distributions $p_{u,i}^T$ and $p_{v,j}^T$ , and the stationary factors $\\mathbf {u}_i^s$ and $\\mathbf {v}_j^s$ .",
"The ratings is then predicted based on the distribution parameterized by the interaction network in Equ.",
"(REF ), i.e., $\\mathbb {E}(x_{ij}^{T}|\\cdot ) =f_1(\\mathbb {E}(\\mathbf {u}_i^{T} ), \\mathbb {E}(\\mathbf {v}_j^{T}))$ .",
"Similarly, the variance could also be predicted as: $V(x_{ij}^{T}|\\cdot )= f_2(\\mathbb {E}(\\mathbf {u}_i^{T}),\\mathbb {E}(\\mathbf {v}_j^{T}), \\sigma ^2_{i,j,T} )$ .",
"$\\sigma ^2_{i,j,T}$ is assumed to be learnable as a function of the hidden states $\\mathbf {h}_{u,i}^{T-1}$ and $\\mathbf {h}_{v,j}^{T-1}$ in our implementation.",
"In this section, we empirically evaluate the performance of CVRCF framework by analyzing three major aspects.",
"Q1: What are the general performance of CVRCF compared with the other baselines?",
"Q2: What are the temporal drifting dynamics of users and items we could learned?",
"Q3: What are the sensitivities of the model to the key hyperparameters?",
"The code of CVRCF is available at GitHub: https://github.com/song3134/CVRCF."
],
[
"Datasets",
"Three widely-adopted benchmark datasets shown in Figure REF are employed in our experiments.",
"Detailed statistics of them are elaborated as follows: MovieTweetings (MT) [10]: It is a benchmark dataset consisting of movies ratings that were contained in well-structured tweets on Twitter.",
"It contains $696,531$ ratings (0-10) provided by $53,275$ users to $30,686$ movies.",
"All ratings are time-associated spanning from $02/28/2013$ to $04/07/2018$ .",
"The granularity is defined as four weeks.",
"MovieLens-10M (ML-10M) [14]: It contains ten million ratings to $10,681$ movies by $71,567$ users spanning from 1995 to 2009.",
"The granularity is defined as four weeks.",
"Netflix [28]: The Netflix challenge dataset consists of 100 million ratings by $480,189$ users to $17,700$ movies from 1999 to 2006.",
"The granularity is defined as two weeks.",
"Table: An overview of all experimental methods."
],
[
"Baselines",
"As our main focus is factorization-based approaches, five representative factorization-based baseline algorithms, including two batch algorithms and three streaming algorithms are selected for comparison from different perspectives shown in Table REF .",
"Brief descriptions of these methods are listed as follows.",
"PMF [25]: Probabilistic Matrix Factorization is a conventional recommendation algorithm, which does not consider temporal information.",
"TimeSVD++ [22]: The temporal-envoled variation of the classical static factor-based algorithm SVD++.",
"We implement it with Graphchi [23] C++ pacakge.",
"sD-PMF: A streaming version of the PMF model combined with the deep interaction network, which is employed in the CVRCF Framework.",
"This model is used to test the effectiveness of the dynamic factors optimized with the RNN structure in CVRCF.",
"sRec [5]: Streaming Recommender System is the state-of-the-art shallow dynamic recommendation model.",
"It is a probabilistic factor-based model optimized with a recursive mean-field approximation.",
"sRRN [50]: A streaming variation of Recurrent Recommender Network (RRN), which is a state-of-the-art deep heuristic streaming recommendation model."
],
[
"Experimental Setup",
"For each dataset, we segment the data along timeline into three parts with ratios $4:1:5$ serving as training, validation, and testing sets, respectively."
],
[
"Training Settings",
"During the training phase, the training and validation sets serve as the historical datasets to decide the best hyperparameters for all methods.",
"As each user or movie may have too many ratings, to reduce and memory and protect the feasible use of GRU structures, we truncate the training sequences along the timeline into batches for the user and movie chain, respectively.",
"This will affect the RNN effectiveness to some extent, but by varying the number of training epoch, it does not have an obvious influence on the experimental results during our experiments.",
"Moreover, to protect the stationary factor get faster trained, in each epoch, every truncated batch is processed with multiple iterations.",
"The number of this iteration hyperparameter used in the training phase is set based on validation and will be further analyzed in hyperparameter analysis section."
],
[
"Testing Settings",
"During the testing phase, at each time step $t$ , the testing is first done to get the prediction of the upcoming ratings $\\lbrace x^{t+1} \\rbrace $ , and then these ratings are assumed to arrive and be used to update the models.",
"Different from dynamic methods, at each update, the static methods are reconstructed from scratch using all the previously arrived testing ratings including the training ratings, while the streaming models only employ the current-step arrived ratings for the current update.",
"Based on this setting, no later data is used to predict any former data and no temporal overlapping is existed between each pair of testing intervals.",
"Besides, for fair comparisons, at each testing step, only ratings for existing users and items are used for testing since some baselines (e.g., PMF and time-SVD++) cannot explicitly cope with new users and items.",
"All the experimental results are the arithmetic average of ten different times runs to ensure the reliability.",
"The performance is evaluated via the root mean square error (RMSE).",
"Table: The RMSE results on the three datasets."
],
[
"Parameter Setting",
"Settings of the hyperparameters for all the baselines follow the original papers, which result in their best performance.",
"Hyperparameters in all the methods are selected based on cross-validation using the training and validation sets.",
"For the static baselines PMF and timeSVD++, all of their regularization parameters are chosen over $\\lbrace 10^{-4},10^{-3},\\ldots , 10^2\\rbrace $ and the sizes of their latent factors are chosen over $\\lbrace 20,40, 60, 80, 100\\rbrace $ .",
"For streaming methods, the size of the stationary factors for sRRN and CVRCF are chosen to be 20 for all the datasets.",
"The stationary factors for sD-PMF is chosen over $\\lbrace 20, 40, 60, 80, 100\\rbrace $ .",
"The size of the dynamic factors of CVRCF is chosen to be 40 including the sizes of both mean and variance parameters.",
"The size of the dynamic factors and the length of the RNN inputs for sRRN is chosen to be the same as CVRCF for fair comparisons.",
"The size of the latent states ($\\mathbf {h}_u$ & $\\mathbf {h}_v$ ) of CVRCF is set to be 20 which is half of the length we used in sRRN.",
"The exponential decay factors are set to be 1 week and 4 weeks for the user RNN and movie RNN, respectively.",
"In the training phase, the truncation hyperparameters of all the RNN-based models are set to be 20, 20, and 10 weeks for the three datasets, respectively, to alleviate the intermediate data explosion."
],
[
"General Evaluation Results",
"We first analyze the general performance of CVRCF model by comparing it with different categories of baselines based on the RMSE results shown in Table REF and Figure REF .",
"From Table REF , three conclusions could be drawn as follows.",
"First, CVRCF outperforms all baselines on all datasets.",
"Although time-SVD++ could achieve comparable performance on MT and ML-10M dataset, it has to be reconstructed from scratch using all of the historical data at each update.",
"Second, CVRCF highly outperforms sD-PMF, which confirms the effectiveness of the dynamic factors employed in CVRCF for capturing the temporal relationships during the streaming process.",
"Third, comparing with shallow probabilistic model sRec, CVRCF displays prominent improvement, which demonstrate the effectiveness of deep architectures in modeling complex drifting interactions.",
"To further analyze the time-varying pattern of each method and their performance consistency on different datasets, we display the RMSE changing curves of the four representative methods on two larger datasets ML-10M and Netflix in Figure REF .",
"From the figure, we could observe that on each dataset the performance of all methods shows similar varying patterns and starting form the first testing step, CVRCF consistently achieves the best performance across two datasets with the evolving of the system.",
"Since MovieLens-10M has the longest testing timeline among all three testing datasets, Figure REF illustrates that CVRCF has stable effectiveness on the dataset with strong temporal relationships in long-term evaluation.",
"By comparison, Netflix is a much larger dataset in terms of users and interactions.",
"Results in Figure REF confirms the superiority of the proposed method on large-scale datasets.",
"Finally, as sRRN could be treated as an ablation method of CVRCF without the probabilistic component, the relative improvement of the proposed method on the general performance validates the effectiveness of combining probabilistic approach in capturing the prospective process of streaming data generation."
],
[
"Evaluation of Temporal Dynamics",
"To analyze the temporal drifting dynamics learned from CVRCF, we visualize the learned latent factors including the location factors ($\\mathbf {u}_i^T$ and $\\mathbf {v}_j^T$ ) and uncertainty factors (${\\sigma }_{u,i,T}$ and ${\\sigma }_{v,j,T}$ ).",
"We conduct exploration on the ML-10M dataset and update the models every half a year during testing."
],
[
"Drifting of the Location Factors",
"We first visualize the drifting of the average location factors $\\mathbf {u}_i^T$ and $\\mathbf {v}_j^T$ with heatmap shown in subfig:loc.",
"The X-axis denotes the index of the latent factors and the Y-axis denotes the timeline.",
"Each factor is adjusted with centralization for joint visualization.",
"From the figure, we could discover that the users' preference factors change more smoothly than movies' popularity factors, which display a block-wise changing patterns.",
"As we update the model every half a year, the stationary factors of movies especially for the new movies are only updated or learned every half a year, which is consistent with the length of the blocks.",
"Thus, the block-wise structure, which appears only on the movie factors, could be explained as: the movie drifting is more likely to be captured by the stationary factors, while the drifting pattern of the users is more likely to be captured by dynamic factors.",
"Since the dynamic factors and stationary factors are defined to capture the short-term and long-term preference, respectively, the finding is also consistent with the fact that users preference usually change more frequently compared to movie popularities."
],
[
"Drifting of the Uncertainty Factors",
"subfig:var displays the drifting of the average uncertainty factors learned from CVRCF.",
"Each column is first normalized with $L_{\\infty }$ -norm.",
"There are two major observations we could find from subfig:var.",
"From an overall perspective, with the evolving of the system, the variances of the learned dynamic factors decrease.",
"This is because the incremental ratings provide more information for each user and item, and reduce the uncertainties of the whole system during the testing phase.",
"From the local perspective, at some time steps, the variance of the latent factors are sharply increased and then slowly decreased.",
"This is because, at some time steps, users and movies increase are dramatically.",
"The cold-start problem introduced by the incremental users and items may raise the uncertainties of the system within a short time but would be alleviated with time goes by.",
"In other words, although new users and items are continually enrolled, the number of ratings related to them could be deficient at first and then increasing over time.",
"Figure: Drifting of variance factors of users & movies.Figure: Effects of the granularities."
],
[
"Hyperparameter Sensitivity Analysis",
"Finally, we study the sensitivity of CVRCF to different hyperparameters using the ML-10M dataset.",
"We pick five hyperparameters, which are the most influential ones in our experiments, and analyze their effects by coupling some of them.",
"These pairwise effects are displayed in Figure REF ."
],
[
"Training Epochs & Training Batch Iterations",
"We first analyze the pairwise effects of the training epoch and the training batch iterations.",
"Figure REF shows that these two parameters highly affects the learning process and may cause overfitting or underfitting when the product of them are too large or too small.",
"With the number of training batch iteration increasing, less epoch should be adopted to protect the testing effectiveness.",
"This may be because: since the stationary factors are outside the RNNs and have high degrees of freedom, they may get overtrained when the batch iteration is setting too large given fixed training epoch.",
"Thus, early stopping should be employed via limiting the number of epochs to prevent the RNN structures not further learning effectively.",
"On the contrary, insufficient batch iterations would limit the power of stationary factors in capturing long-term preferences."
],
[
"Testing Batch Iterations & Testing Update Interval",
"Secondly, we focus on the testing phase and analyze the influence of the testing batch iterations and the length of the model updating interval.",
"As shown in Figure REF , for a fixed testing update interval, with the increasing of the testing batch iterations, the testing performance first decreases and then increases.",
"This might because: in the testing phase, new ratings, users, and items never stop to arrive.",
"Insufficient testing batch iterations would highly affect the learning of latent factors especially for the stationary factors of new users or items.",
"On the contrary, superfluous iterations would also lead to overfitting as in the training phase described above.",
"Besides, with the enlarging of the testing update interval, ratings in each batch increase which requires more updating iterations under the same remaining settings."
],
[
"Granularities",
"Finally, we explore the effect of the granularities.",
"We assume the two granularities defined for users and movies could be different for a more general treatment.",
"From fig:gran, we can see that although different granularities do affect the results, their influences are shown to be very trivial based on the scale of the $Z$ -axis.",
"Moreover, user granularity seems to have larger effects than movie granularity and its optimal value is shown to be lower than movie granularity.",
"This may illustrate that the users' preferences are varying more frequently than the items' popularities."
],
[
"Related Work",
"Streaming Recommender Systems.",
"Beyond traditional static settings, streaming recommender systems have attracted widespread concerns in coping with the high data velocity and their naturally incremental properties [1], [5].",
"Different from static time-aware models [22], [12], [48], [20], [19], which only take account of temporal dynamics without updating in an streaming fashion, streaming recommender systems dynamically encode temporal information and generate response instantaneously [7], [38], [6], [36].",
"Some existing works focus on extending classical memory-based recommendation algorithms into online fashions to address the streaming challenges such as [4] and [41].",
"Besides memory-based methods, model-based methods [30], [9], [8] are becoming more and more popular in recent years, which conducts recommendation based on well-trained models rather than explicitly aggregating and prediction based on the similarity relationships.",
"Diaz-Aviles et al.",
"leverage the active learning strategy to sample and maintain a delicately designed reservoir, thus providing a pairwise matrix factorization approach for streaming recommendation.",
"Chang et al.",
"[5] exploit continuous Markov process (Brownian motion) to model the temporal drifting of users and items, which introduces a principled way to model data streams.",
"Wang et al.",
"[46] propose a streaming ranking-based framework based on Bayesian Personalized Ranking [29] to address the user interest drifting as well as system overload problem.",
"Although many recent advances based on deep neural networks especially RNNs have been made to model streaming inputs and capture the complex temporal dynamics [18], [50], [2], most of them overlook the causality inherited in the data generation process, which is one of the main aspects considered in our framework via the deep Bayesian learning.",
"Deep Recommender Systems.",
"Deep learning techniques have brought vast vitality and achieve dramatic improvement in recommender systems [52].",
"They have been adopted in various recommendation tasks as well as accommodating different data sources [43], [13], [50].",
"From the perspective of the general framework, deep recommender systems could be categorized into solely deep models, which conduct recommendations based only on deep frameworks [13], [37], [17]; and integration models, which integrate deep techniques with traditional recommender systems [43], [44], [45].",
"From the perspective of deep frameworks, these models could also be divided into: (1) single deep models, which are built upon single neural building blocks such as multi-layer perceptron [17], convolutional neural network [47], and recurrent neural network [50]; and (2) composite models, which are constructed with different deep learning techniques [51].",
"From the first viewpoint of devision, the proposed framework could be categorized as an integration model, which combines and probabilistic recommender systems with deep learning models.",
"It is also a hybrid deep models, which jointly incorporates RNN and MLP structures.",
"The coupled variational inference structure also provides its' uniqueness comparing to other streaming deep recommender systems."
],
[
"Conclusion and Future Work",
"In this paper, we focus on the recommendation problem under streaming setting and propose a deep streaming recommender system - CVRCF.",
"CVRCF incorporates deep architectures into traditional factorization-based model and encodes the temporal relationship with Gaussian-Markov components.",
"Standing upon the sequential variational inference, CVRCF is optimized leveraging a cross variational GRU network and could continually update under the streaming setting.",
"By conducting experiments on various real-world benchmark datasets, we empirically validate the effectiveness our proposed framework, explore the learned drifting patterns, and validate the stability of our framework.",
"Future work will center on exploring different assumptions of stochastic processes of the dynamic factors and incorporate other deep learning structures, such as graph neural networks, into the proposed framework."
],
[
"Acknowledgments",
"The authors thank the anonymous reviewers for their helpful comments.",
"This work is, in part, supported by DARPA under grant #W911NF-16-1-0565 and #FA8750-17-2-0116, and NSF under grant #IIS-1657196 and #IIS-1718840.",
"The views and conclusions contained in this paper are those of the authors and should not be interpreted as representing any funding agencies."
]
] | 1906.04386 |
[
[
"Analysis of linear systems over idempotent semifields"
],
[
"Abstract In this paper, we present and analyze methods for solving a system of linear equations over idempotent semifields.",
"The first method is based on the pseudo-inverse of the system matrix.",
"We then present a specific version of Cramer's rule which is also related to the pseudo-inverse of the system matrix.",
"In these two methods, the constant vector plays an implicit role in solvability of the system.",
"Another method is called the normalization method in which both the system matrix and the constant vector play explicit roles in the solution process.",
"Each of these methods yields the maximal solution if it exists.",
"Finally, we show the maximal solutions obtained from these methods and some previous methods are all identical."
],
[
"Introduction",
"Solving linear systems of equations is an important aspect of many mathematical problems and their applications in various areas of science and engineering.",
"Recently, idempotent semifields have been finding applications in computer science, optimization theory and control theory (see [1], [6], [5]).",
"This makes linear system solution over idempotent semifields ever so important.",
"In this work, we intend to extend the solution methods presented in [3] and [7], namely, the pseudo-inverse method, the extended Cramer's rule, and the normalization method, to idempotent semifields.",
"The notion of semirings was first introduced by Vandiver [12] in 1934 as a generalization of rings, where division and subtraction are not necessarily defined.",
"A semiring $(S,\\oplus ,\\otimes ,0,1)$ is an algebraic structure in which $(S, \\oplus )$ is a commutative monoid with an identity element 0 and $(S,\\otimes )$ is a monoid with an identity element 1, connected by ring-like distributivity.",
"The additive identity 0 is multiplicatively absorbing, and $0 \\ne 1$ .",
"A commutative semiring in which every nonzero element is multiplicatively invetrible is called a semifield.",
"We obtain maximal solutions through the proposed methods for solving linear systems over idempotent semifields if they exist.",
"A maximal solution is determined with respect to the defined total order on the idempotent semifield.",
"Importantly, we prove that the maximal solutions obtained from the presented methods are all identical.",
"In Section 3, we first present a necessary and sufficient condition based on the pseudo-inverse of the system matrix, whenever the determinant of the system matrix in its idempotent semifield is nonzero.",
"We also propose the extended Cramer's rule to find the maximal solution.",
"Moreover, by introducing the normalization method over idempotent semifields, we present an equivalent condition on column minimum elements of the associated normalized matrix of a linear system to determine the maximal solution.",
"Section 4 concerns the comparison of the maximal solutions obtained from the presented methods in Section 3.",
"We show that the pseudo-inverse method and the normalization method are equivalent.",
"Finally, we prove that the column minimum elements of the associated normalized matrix of a linear system are the same as the maximal solution obtained from the $ LU $ -method, which is introduced in [4]."
],
[
"Definitions and preliminaries",
"In this section, we give some definitions and preliminary notions.",
"For convenience, we use $ \\mathbb {N} $ , and $\\underline{n}$ to denote the set of all positive integers, and the set $\\lbrace 1,2,\\cdots ,n\\rbrace $ for $n \\in \\mathbb {N}$ , respectively.",
"Definition 1 (See [2]) A semiring $ (S,\\oplus ,\\otimes , 0,1)$ is an algebraic system consisting of a nonempty set $ S $ with two binary operations, addition and multiplication, such that the following conditions hold: $ (S, \\oplus ) $ is a commutative monoid with identity element 0; $ (S, \\otimes ) $ is a monoid with identity element 1; Multiplication distributes over addition from either side, that is $ a\\otimes (b\\oplus c)= (a\\otimes b)\\oplus (a\\otimes c) $ and $ (b \\oplus c)\\otimes a=(b\\otimes a) \\oplus (c\\otimes a) $ for all $ a, b, c \\in S $ ; The neutral element of $ S $ is an absorbing element, that is $ a\\otimes 0 =0= 0 \\otimes a $ for all $ a \\in S $ ; $ 1 \\ne 0 $ .",
"A semiring $ S $ is called commutative if $ a\\otimes b = b \\otimes a $ .",
"It is zerosumfree if $a\\oplus b=0$ implies that $a=0=b$ , for all $ a, b \\in S $ .",
"Definition 2 A semiring $S$ is called additively idempotent if $a\\oplus a=a $ , for every $ a\\in S $ .",
"Definition 3 A commutative semiring $(S,\\oplus ,\\otimes ,0,1)$ is called a semifield if every nonzero element of $S$ is multiplicatively invetrible.",
"Definition 4 The semifield $(S,\\oplus , \\otimes , 0,1)$ is idempotent if it is an additively idempotent, totally ordered, and radicable semifield.",
"Note that the radicability implies the power $a^{q}$ be defined for any $a \\in S \\setminus \\lbrace 0\\rbrace $ and $q \\in \\mathbb {Q}$ (rational numbers).",
"In particular, for any non-negative integer $p$ we have $a^{0}=1 ~~~~~~~ a^{p}=a^{p-1}a~~~~~ a^{-p}=(a^{-1})^{p} $ The totally ordered operator “$\\le _{S} $ \" on an idempotent semifield defines the following partial order: $a \\le _{S} b\\Longleftrightarrow a\\oplus b = b $ , that is induced by additive idempotency.",
"Notice that the last partial order equips addition to an extremal property in the form of the following inequalities $ a \\le _{S} a \\oplus b ,~~~~~~ b\\le _{S} a \\oplus b ,$ which makes idempotent semifields zerosumfree, since we have $a \\ge _{S} 0$ for any $ a \\in S$ .",
"The total order defined on $ S $ is compatible with the algebraic operations, that is: $ (a \\le _{S} b \\wedge c \\le _{S} d) \\Longrightarrow a\\oplus c \\le _{S} b \\oplus d, $ $ (a \\le _{S} b \\wedge c \\le _{S} d) \\Longrightarrow a\\otimes c \\le _{S} b \\otimes d,$ for any $ a,b,c,d \\in S $ .",
"Throughout this article, we consider the notation $``\\ge _{S}\" $ as the converse of the order $ ``\\le _{S}\" $ , which is a totally ordered operator satisfying $ a \\ge _{S} b $ if and only if $ b\\le _{S} a $ , for any $ a, b \\in S $ .",
"Furthermore, we use $ ``a <_{S} b\" $ whenever, $ a\\le _{S} b $ and $ a \\ne b$ .",
"The notation $ ``>_{S} \" $ is defined similarly.",
"Definition 5 Let $ (S, \\oplus , \\otimes , 0_{S}, 1_{S} ) $ be a semifield.",
"A semivector space over a semifield $ S $ is a commutative monoid $ (\\mathcal {V},+) $ with identity element $ 0_{\\mathcal {V}}$ for which we have a scalar multiplication function $S \\times \\mathcal {V} \\longrightarrow \\mathcal {V} $ , denoted by $ (s, v)\\mapsto sv $ , which satisfies the following conditions for all $ s, s^{\\prime } \\in S $ and $ v, v^{\\prime } \\in \\mathcal {V} $ : $ (s\\otimes s^{\\prime })m=s(s^{\\prime }v) $ ; $ s(v+v^{\\prime })= sv +sv^{\\prime }$ ; $ (s\\oplus s^{\\prime })v=sv +s^{\\prime }v $ ; $ 1_{S}v=v$ ; $ s0_{\\mathcal {V}}=0_{\\mathcal {V}}=0_{S}v$ .",
"The elements of a semivector space $ \\mathcal {V} $ are called vectors.",
"Let $ \\mathcal {W} $ be a nonempty subset of $ \\mathcal {V} $ .",
"Then the expression $ \\displaystyle {\\sum _{i=1}^{n}}s_{i}\\omega _{i} $ with $ \\omega _{i} \\in \\mathcal {W} $ , for any $ i \\in \\underline{n}$ is a linear combination of the elements of $ \\mathcal {W} $ .",
"Indeed, the nonempty subset $ \\mathcal {W} $ of vectors is called linearly independent if no vector of $ \\mathcal {W} $ is a linear combination of other vectors of $ \\mathcal {W} $ .",
"Otherwise, it is a linearly dependent subset (see [10]).",
"Let $S$ be an idempotent semifield.",
"We denote the set of all $m \\times n$ matrices over $S$ by $M_{m \\times n}(S)$ .",
"For $A \\in M_{m \\times n} (S)$ , we denote by $a_{ij}$ and $A^{T}$ the $(i,j)$ -entry of $A$ and the transpose of $A$ , respectively.",
"For any $A, B \\in M_{m \\times n}(S)$ , $C \\in M_{n \\times l}(S)$ and $\\lambda \\in S$ , we define: $A+B = (a_{ij} \\oplus b_{ij})_{m \\times n},$ $AC=(\\bigoplus _{k=1}^{n} (a_{ik} \\otimes b_{kj}))_{m \\times l},$ and $\\lambda A=(\\lambda \\otimes a_{ij})_{m\\times n}.$ It is easy to verify that $M_{n}(S):=M_{n \\times n}(S)$ forms a semiring with respect to the matrix addition and the matrix multiplication.",
"Definition 6 (See [13]) The set of all linear combinations of columns (rows) of an arbitrary matrix $ A $ , denoted by $ Col(A) ~(Row(A)) $ , is the column (row) space of $ A $ .",
"The smallest $ n \\in \\mathbb {N} $ for which there exists a set of generator of $ Col(A) ~(Row(A)) $ with cardinality $ n $ is the column (row) rank of $ A$ .",
"The column rank and the row rank of a matrix over an idempotent semifield are not necessarily equal.",
"Let $A \\in M_{n}(S)$ , $\\mathcal {S}_{n}$ be the symmetric group of degree $n \\ge 2$ , and $\\mathcal {A}_{n}$ be the alternating group on $n$ such that $\\mathcal {A}_{n}= \\lbrace \\sigma \\vert \\sigma \\in \\mathcal {S}_{n} ~\\text{and}~\\sigma \\text{~is~an~even~permutation}\\rbrace .$ The positive determinant, $\\vert A \\vert ^{+}$ , and negative determinant, $\\vert A \\vert ^{-}$ , of $A$ are $\\vert A \\vert ^{+}=\\bigoplus _{\\sigma \\in \\mathcal {A}_{n}} \\bigotimes _{i=1}^{n} a_{i\\sigma (i)},$ and $\\vert A \\vert ^{-}= \\bigoplus _{\\sigma \\in \\mathcal {S}_{n} \\backslash \\mathcal {A}_{n}} \\bigotimes _{i=1}^{n} a_{i\\sigma (i)}.",
"$ Clearly, if $S$ is a commutative ring, then $\\vert A \\vert =\\vert A \\vert ^{+} - \\vert A \\vert ^{-}$ .",
"Definition 7 Let $S$ be a semiring.",
"A bijection $\\varepsilon $ on $S$ is called an $\\varepsilon $ -function of $S$ if $\\varepsilon (\\varepsilon (a))=a$ , $\\varepsilon (a \\oplus b)= \\varepsilon (a) \\oplus \\varepsilon (b)$ , and $\\varepsilon (a \\otimes b)=\\varepsilon (a) \\otimes b=a \\otimes \\varepsilon (b)$ for all $a, b \\in S$ .",
"Consequently, $\\varepsilon (a) \\otimes \\varepsilon (b)=a \\otimes b$ and $\\varepsilon (0)=0$ .",
"The identity mapping: $a \\mapsto a$ is an $\\varepsilon $ -function of $S$ that is called the identity $\\varepsilon $ -function.",
"Remark 1 Any semiring $S$ has at least one $\\varepsilon $ -function since the identical mapping of $S$ is an $\\varepsilon $ -function of $S$ .",
"If $S$ is a ring, then the mapping : $a \\mapsto -a$ , $( a \\in S)$ is an $\\varepsilon $ -function of $S$ .",
"Definition 8 Let $S$ be a commutative semiring with an $\\varepsilon $ -function $\\varepsilon $ , and $A \\in M_{n}(S)$ .",
"The $\\varepsilon $ -determinant of $A$ , denoted by $\\det _{\\varepsilon }(A)$ , is defined by $ det_{\\varepsilon }(A)= \\bigoplus _{\\sigma \\in \\mathcal {S}_{n}}\\varepsilon ^{\\tau (\\sigma )}(a_{1\\sigma (1)} \\otimes a_{2\\sigma (2)} \\otimes \\cdots \\otimes a_{n\\sigma (n)})$ where $\\tau (\\sigma )$ is the number of the inversions of the permutation $\\sigma $ , and $\\varepsilon ^{(k)}$ is defined by $\\varepsilon ^{(0)}(a)=a$ and $\\varepsilon ^{(k)}(a)=\\varepsilon ^{(k-1)}(\\varepsilon (a))$ for all positive integers $k$ .",
"Since $\\varepsilon ^{(2)}(a)=a$ , $\\det _{\\varepsilon }(A)$ can be rewritten in the form of $det_{\\varepsilon }(A)=\\vert A \\vert ^{+} \\oplus \\varepsilon (\\vert A \\vert ^{-})$ .",
"Let $(S,\\oplus , \\otimes , 0,1)$ be an idempotent semifield, $A \\in M_{m\\times n}(S)$ , and $b \\in S^{m}$ be a column vector.",
"Then the $i-$ th equation of the linear system $AX=b$ is $ \\bigoplus _{j=1}^{n}( a_{ij} \\otimes x_{j})= (a_{i1}\\otimes x_{1})\\oplus (a_{i2}\\otimes x_{2}) \\oplus \\cdots \\oplus (a_{in} \\otimes x_{n}) =b_{i}.$ Definition 9 A solution $X^{*}$ of the system $AX=b$ is called maximal if $X \\le X^{*}$ for any solution $X$ .",
"Definition 10 A vector $b\\in S^{m}$ is called regular if it has no zero element.",
"Without loss of generality, we can assume that $b$ is regular in the system $AX=b$ .",
"Otherwise, let $b_{i}=0$ for some $i \\in \\underline{n}$ .",
"Then in the $i-$ th equation of the system, we have $a_{ij}\\otimes x_{j}=0$ for any $j \\in \\underline{n}$ , since $S$ is zerosumfree.",
"As such, $x_{j}=0$ if $a_{ij}\\ne 0$ .",
"Consequently, the $i-$ th equation can be removed from the system together with every column $A_{j}$ where $a_{ij} \\ne 0$ , and the corresponding $x_{j}$ can be set to 0.",
"Definition 11 Let the linear system of equations $ AX=b $ have solutions.",
"Suppose that $ A_{j_{1}}, A_{j_{2}}, \\cdots , A_{j_{k}} $ are linearly independent columns of $ A $ , and $ b $ is a linear combination of them.",
"Then the corresponding variables, $ x_{j_{1}}, x_{j_{2}}, \\cdots , x_{j_{k}} $ , are called leading variables and other variables are called free variables of the system $ AX= b$ .",
"The degrees of freedom of the linear system $ AX=b $ , denoted by $ \\mathcal {D}_{f}$ , is the number of free variables."
],
[
"Methods for solving linear systems over idempotent semifields ",
"In this section, we present methods to solve a linear system of equations over an idempotemt semifield.",
"Those methods are pseudo-inverse method, extended Cramer's rule and normalization method, which determine the maximal solution of the system.",
"Throughout this section, we consider $ S $ as an idempotent semifield.",
"Definition 12 Let $A \\in M_{n}(S)$ and $\\varepsilon $ be an $\\varepsilon $ -function on $ S $ .",
"The $\\varepsilon $ -adjoint matrix $A$ , denoted by $adj_{\\varepsilon }(A)$ , is defined as follows.",
"$ adj_{\\varepsilon }(A)=((\\varepsilon ^{(i+j)} det_{\\varepsilon }(A(i | j)))_{n \\times n})^{T}, $ where $ A(i|j) $ is the $ (n-1) \\times (n-1) $ submatrix of $ A $ obtained from $ A $ by removing the $ i $ -th row and the $ j $ -th column.",
"Theorem 1 (See [9]) Let $A \\in M_{n}(S)$ .",
"We have $Aadj_{\\varepsilon }(A)=(\\det _{\\varepsilon }(A_{r}(i \\Rightarrow j)))_{n \\times n}$ , $adj_{\\varepsilon }(A)A=(\\det _{\\varepsilon }(A_{c}(i \\Rightarrow j)))_{n \\times n}$ , where $A_{r}(i \\Rightarrow j)$ ($A_{c}(i \\Rightarrow j)$ ) denotes the matrix obtained from $A$ by replacing the $j$ -th row (column) of $A$ by the $i$ -th row (column) of $A$ .",
"Definition 13 Let $A \\in M_{n}(S)$ and $\\det _{\\varepsilon }(A)$ be a nonzero element of $ S $ .",
"Then the square matrix $ \\det _{\\varepsilon }(A)^{-1} adj_{\\varepsilon }(A)$ , denoted by $ A^{-}$ , is called the pseudo-inverse of $A$ .",
"Corollary 1 Let $ A\\in M_{n}(S)$ .",
"Then the elements of the multiplication matrix $ AA^{-} $ are $ (AA^{-})_{ij}= det_{\\varepsilon }(A)^{-1} det_{\\varepsilon }(A_{r}(i \\Rightarrow j)).",
"$ In particular, the diagonal entries of the matrix $AA^{-}$ are 1.",
"Furthermore, the entries of the matrix $ A^{-}A $ are defined analogusly.",
"$(AA^{-})_{ii} &\\ = det_{\\varepsilon }(A)^{-1}(Aadj_{\\varepsilon }(A))_{ii}\\\\&\\ = det_{\\varepsilon }(A)^{-1} det_{\\varepsilon }(A_{r}(i \\Rightarrow i))\\\\&\\ = det_{\\varepsilon }(A)^{-1} det_{\\varepsilon }(A)\\\\&\\ = 1$"
],
[
"First we introduce the pseudo-inverse method to obtain a maximal solution of the linear system.",
"In the following theorem, we present a necessary and sufficient condition on the system matrix over idempotent semifields which is an extension of Theorem 3 in [3] for “$ \\max -plus$ algebra \".",
"Theorem 2 Let $ AX=b $ , where $ A \\in M_{n}(S) $ , and $ b $ be a regular vector of size $ n $ .",
"Then the system $ AX=b $ has the maximal solution $X^{*}=A^{-}b $ with $X^{*}=(x_{i}^{*})_{i=1}^{n}$ if and only if $(AA^{-})_{ij}\\otimes b_{j} \\le _{S} b_{i} $ for any $i,j \\in \\underline{n}$ .",
"Assume that $X^{*}=A^{-}b $ is a maximal solution of the system $ AX=b $ .",
"Then $ AA^{-}b=b $ , that is in the $i$ -th equation of $ AA^{-}b=b $ for any $i \\in \\underline{n}$ we have $ ((AA^{-})_{i1} \\otimes b_{1}) \\oplus ((AA^{-})_{i2} \\otimes b_{2})\\oplus ((AA^{-})_{in} \\otimes b_{n})= b_{i}.",
"$ The induced partial order on the idempotent semifield $ S $ implies that $(AA^{-})_{ij}\\otimes b_{j} \\le _{S} b_{i} $ for any $ j \\in \\underline{n}$ .",
"Conversely, suppose that $(AA^{-})_{ij}\\otimes b_{j} \\le _{S} b_{i} $ for any $i,j \\in \\underline{n}$ .",
"We prove that $X^{*}=A^{-}b $ is a solution of the system and it is maximal corresponding to the total order $ \\le _{S} $ .",
"Clearly, the $ i $ -th component of the vector $ AX^{*} $ is $(AX^{*})_{i} \\nonumber &= (AA^{-}b)_{i}\\\\&=\\bigoplus _{j=1}^{n}((AA^{-})_{ij} \\otimes b_{j})\\\\&=\\bigoplus _{{{\\scriptstyle \\begin{matrix}j=1 \\\\ j \\ne i\\end{matrix}}}}^{n} ((AA^{-})_{ij} \\otimes b_{j}) \\oplus ((AA^{-})_{ii} \\otimes b_{i})\\\\&=b_{i}.",
"\\nonumber $ The equalities $(3.1)$ and $(3.2)$ are obtained from Corollary REF and the assumption $(AA^{-})_{ij}\\otimes b_{j} \\le _{S} b_{i} $ for any $ i,j \\in \\underline{n}$ , respectively.",
"As such, $ X^{*} $ is a solution of the system $ AX=b $ .",
"Now, we show that $ X^{*}=A^{-}b $ is the maximal solution of $ AX=b $ .",
"Since $ AX^{*}=b $ , the $ i $ -th equation of the system $A^{-}AX^{*}=X^{*}$ in the form $ ((A^{-}A)_{i1} \\otimes x^{*}_{1}) \\oplus \\cdots \\oplus x^{*}_{i} \\oplus \\cdots \\oplus ((A^{-}A)_{in} \\otimes x^{*}_{n})= x^{*}_{i} $ yields the inequalities $(A^{-}A)_{ij} \\otimes x^{*}_{j} \\le _{S} x^{*}_{i}$ for any $j \\ne i$ .",
"We now consider another solution of the system such as $ Y=(y_{i})_{i}^{n}$ , that is $AY=b$ and therefore $ A^{-}AY=X^{*} $ .",
"There is no loss of generality to assume that $ y_{i} \\ne x^{*}_{i} $ for some $ i \\in \\underline{n} $ and $ y_{j} = x^{*}_{j} $ for any $ j \\ne i $ .",
"Note that the $ i $ -th equation of the system $ A^{-}AY=X^{*} $ is $ ((A^{-}A)_{i1} \\otimes x^{*}_{1}) \\oplus \\cdots \\oplus (A^{-}A)_{ii} \\otimes y_{i} \\oplus \\cdots \\oplus ((A^{-}A)_{in} \\otimes x^{*}_{n})= x^{*}_{i}.",
"$ Due to the induced partially ordered operator $`` \\le _{S}\" $ over idempotent semifields, we have $ (A^{-}A)_{ii} \\otimes y_{i} \\le _{S} x^{*}_{i} $ which implies $ y_{i} <_{S} x^{*}_{i} $ , since $ (A^{-}A)_{ii}=1 $ and $ y_{i} \\ne x^{*}_{i} $ .",
"Moreover, $ Y $ is a solution of the system $ AX=b $ , so the inequalities REF cannot be all proper.",
"If all of them are proper, then $ ((A^{-}A)_{i1} \\otimes x^{*}_{1}) \\oplus \\cdots \\oplus (A^{-}A)_{ii} \\otimes y_{i} \\oplus \\cdots \\oplus ((A^{-}A)_{in} \\otimes x^{*}_{n}) <_{S} x^{*}_{i}, $ which is a contradiction.",
"In fact, if all of the inequalities (REF ) are proper, then $ X^{*} $ is the unique solution.",
"Otherwise, suppose that $ (A^{-}A)_{ij} \\otimes x^{*}_{j}= x^{*}_{i} $ for some $ j \\ne i $ .",
"Then $ Y $ is a solution of the system that satisfies $ Y \\le _{S} X^{*} $ .",
"Hence $ X^{*} $ is a maximal solution.",
"In the following example, we use the necessary and sufficient condition $(AA^{-})_{ij}\\otimes b_{j} \\le _{S} b_{i} $ for the system $ AX=b $ to have the maximal solution $ X^{*}=A^{-}b $ .",
"Example 1 Let $ A\\in M_{4}(S) $ , where $ S=\\mathbb {R}_{\\max ,\\times }=(\\mathbb {R_{+}} \\cup \\lbrace 0\\rbrace , \\max ,\\times ,0,1) $ , and $\\mathbb {R_{+}} $ be the set of all positive real numbers.",
"Then the defined total order on “$\\max -\\rm times$ algebra\" is the standard less than or equal relation $ `` \\le \" $ over $\\mathbb {R}$ and the multiplication $ a \\times b^{-1} $ , denoted by $ \\dfrac{a}{b}$ , is the usual real numbers division.",
"Consider the following system $ AX=b $ : $\\left[\\begin{array}{cccc}5&7&9&10\\\\4&2&0&7\\\\3&0&3&5\\\\1&8&1&6\\end{array}\\right]\\left[\\begin{array}{c}x_{1}\\\\x_{2}\\\\x_{3}\\\\x_{4}\\end{array}\\right]=\\left[\\begin{array}{c}27\\\\16\\\\12\\\\24\\end{array}\\right],$ where $ \\det _{\\varepsilon }(A)=a_{13} \\times a_{24} \\times a_{31} \\times a_{42} = 1512 $ .",
"Due to Corollary REF , we have $ (AA^{-})_{ij}=\\dfrac{\\det _{\\varepsilon }(A_{r}(i \\Rightarrow j)) }{\\det _{\\varepsilon }(A)} $ , for any $i,j \\in \\lbrace 1,\\cdots ,4 \\rbrace $ .",
"As such, $ AA^{-} $ is $AA^{-}=\\left[\\begin{array}{cccc}1&\\frac{10}{7}&\\frac{40}{21}&\\frac{7}{8}\\\\\\frac{4}{9}&1&\\frac{4}{3}&\\frac{7}{18}\\\\\\frac{1}{3}&\\frac{5}{7}&1&\\frac{7}{24}\\\\\\frac{8}{21}&\\frac{6}{7}&\\frac{8}{7}&1\\\\\\end{array}\\right].$ By Theorem REF , we must check the condition $(AA^{-})_{ij} \\times b_{j} \\le b_{i} $ , for any $i,j \\in \\lbrace 1,\\cdots ,4 \\rbrace $ .",
"In order to simplify the computation procedure, we check $ (AA^{-})_{ij} \\le \\dfrac{b_{i}}{b_{j}} \\le \\dfrac{1}{(AA^{-})_{ji}} $ , for any $ 1 \\le i\\le j \\le 4 $ .",
"Since these inequalities hold, for instance $ (AA^{-})_{12} \\le \\dfrac{27}{16} \\le \\dfrac{1}{(AA^{-})_{21}} $ , the system $ AX=b $ has the maximal solution $ X^{*}=A^{-}b$ : $X^{*}=\\left[\\begin{array}{cccc}\\frac{1}{9}&\\frac{5}{21}&\\frac{1}{3}&\\frac{7}{72}\\\\\\frac{1}{21}&\\frac{3}{28}&\\frac{1}{7}&\\frac{1}{8}\\\\\\frac{1}{9}&\\frac{10}{63}&\\frac{40}{189}&\\frac{7}{72}\\\\\\frac{4}{63}&\\frac{1}{7}&\\frac{4}{21}&\\frac{1}{18}\\end{array}\\right]\\left[\\begin{array}{c}27\\\\16\\\\12\\\\24\\end{array}\\right]=\\left[\\begin{array}{c}4\\\\3\\\\3\\\\\\frac{16}{7}\\end{array}\\right].$"
],
[
"We know that in classic linear algebra over fields, commutative rings and in a fortiori over commutative semirings (see [11]), Cramer's rule is a useful method for determining the unique solution of a linear system whenever the system matrix is invertible.",
"Furthermore, Sararnrakskul proves that a square matrix $A$ over a semifield is invertible if and only if every row and every column of $A$ contains exactly one nonzero element ( see [8]).",
"As such, Cramer's rule can be used only for this type of matrices over an idempotent semifield.",
"Now, we introduce a new version of Cramer's rule to obtain the maximal solution of a linear system, based on the pseudo-inverse of the system matrix.",
"In the next theorem, we present the extended Cramer's rule over idempotent semifields whenever the $ \\varepsilon $ -determinant of the system matrix is nonzero, which is an extension of Theorem 4 in [3].",
"Theorem 3 Let $ A \\in M_{n}(S) $ , $ b $ be a regular vector of size $ n $ , and $ \\det _{\\varepsilon }(A) \\ne 0$ .",
"Then the following statements are equivalent: the system $ AX= b $ has the maximal solution $ X^{*}=(\\det _{\\varepsilon }(A)^{-1} \\otimes \\det _{\\varepsilon }(A_{[i]}))_{i=1}^{n}$ , and $(AA^{-})_{ij}\\otimes b_{j} \\le _{S} b_{i} $ for any $i,j \\in \\underline{n}$ , where $A_{[i]} $ is the matrix obtained from replacing the $ i $ -th column of $ A $ by the column vector $b $ .",
"Due to Theorem REF , the inequalities $ (AA^{-})_{ij}\\otimes b_{j} \\le _{S} b_{i} $ for every $i,j \\in \\underline{n}$ are necessary and sufficient conditions for the system $ AX=b $ to have the maximal solution $ X^{*}=A^{-}b$ .",
"Consequently, $x^{*}_{i} \\nonumber &= (A^{-}b)_{i} \\\\ \\nonumber &=det_{\\varepsilon }(A)^{-1} \\otimes (adj_{\\varepsilon }(A)b)_{i} \\\\ \\nonumber &=det_{\\varepsilon }(A)^{-1} \\otimes ( \\bigoplus _{j=1}^{n}((adj_{\\varepsilon }(A))_{ij} \\otimes b_{j} ) ) \\\\ \\nonumber &=det_{\\varepsilon }(A)^{-1} \\otimes ( \\bigoplus _{j=1}^{n}(det_{\\varepsilon }(A(j|i)) \\otimes b_{j} ) )\\\\&=det_{\\varepsilon }(A)^{-1} det_{\\varepsilon }(\\left[\\begin{array}{ccccccc}a_{11}&\\cdots &a_{1(j-1)}&b_{1}&a_{1(j+1)}\\cdots &a_{1n}\\\\\\vdots &~&\\vdots &\\vdots &\\vdots &\\vdots &~\\\\a_{n1}&\\cdots &a_{n(j-1)}&b_{n}&a_{n(j+1)}\\cdots &a_{nn}\\end{array}\\right] ) \\\\&=det_{\\varepsilon }(A)^{-1} \\otimes det_{\\varepsilon }(A_{[i]}), \\nonumber $ where the equality $ (3.4) $ is obtained from Laplace's theorem for semirings (see Theorem 3.3 in [9]).",
"The normalization method for solving a system of linear equations over “$\\max -\\rm plus$ algebra\" is already introduced in [7].",
"Here, we extend this method to idempotent semifields.",
"To this end, we start by defining the following normalization method."
],
[
"Consider the linear system $ AX=b $ , where $ A \\in M_{m \\times n}(S) $ , $ b $ is a regular vector of size $ m $ and $ X $ is an unknown vector of size $ n $ .",
"Let the $ j$ -th column of $ A $ , denoted by $ A_{j}$ , be a regular vector of size $ m $ , for any $ j \\in \\underline{n}$ .",
"In fact, the system matrix, $ A $ , contains no zero element.",
"Then the normalized matrix of $ A $ is $\\tilde{A}=\\left[\\begin{array}{c|c|c|c}\\hat{A}_{1}^{-1} A_{1} & \\hat{A}_{2}^{-1} A_{2} &\\cdots & \\hat{A}_{n}^{-1} A_{n}\\end{array}\\right],$ where $ \\hat{A}_{j}= \\@root m \\of {a_{1j}\\otimes \\cdots \\otimes a_{mj}} $ , for any $ j \\in \\underline{n}$ .",
"The normalized vector of the regular vector $ b $ is defined similarly as follows.",
"$\\tilde{b} = \\hat{b}^{-1} b ,$ where $ \\hat{b}= \\@root m \\of {b_{1}\\otimes \\cdots \\otimes b_{m}}.$ As such, the normalized system corresponding to the system $ AX=b $ , denoted by $ \\tilde{A}Y=\\tilde{b} $ , is obtained as follows.",
"$AX=b&\\ \\Rightarrow \\bigoplus _{j=1}^{n}( A_{j}x_{j} ) =b \\\\&\\ \\Rightarrow \\bigoplus _{j=1}^{n}( A_{j}(\\hat{A}_{j}^{-1} \\otimes \\hat{A}_{j} \\otimes x_{j}) ) =(\\hat{b} \\otimes \\hat{b}^{-1})b \\\\&\\ \\Rightarrow \\bigoplus _{j=1}^{n}( \\tilde{A}_{j}( \\hat{A}_{j}\\otimes x_{j}) ) =\\hat{b}\\tilde{b} \\\\&\\ \\Rightarrow \\bigoplus _{j=1}^{n}( \\tilde{A}_{j}( \\hat{A}_{j} \\otimes \\hat{b}^{-1}\\otimes x_{j}) ) =\\tilde{b} \\\\&\\ \\Rightarrow \\bigoplus _{j=1}^{n}(\\tilde{A}_{j} y_{j} ) =\\tilde{b} \\\\&\\ \\Rightarrow \\tilde{A}Y=\\tilde{b},$ where $ Y= (\\hat{A}_{j}\\otimes \\hat{b}^{-1} )X $ .",
"The $ i$ -th equation of the normalized system $ \\tilde{A}Y=\\tilde{b} $ is $ (\\tilde{a}_{i1} \\otimes y_{1}) \\oplus \\cdots \\oplus (\\tilde{a}_{in} \\otimes y_{n})=\\tilde{b}_{i}, $ which implies $ y_{j} \\le _{S} \\tilde{b}_{i}$ , for any $ i \\in \\underline{m}$ and $ j \\in \\underline{n} $ .",
"The associated normalized matrix of the system $ AX=b $ can be defined as $ Q= (q_{ij}) \\in M_{m\\otimes n}(S) $ with $ q_{ij}= \\tilde{b}_{i} \\otimes \\tilde{a}_{ij}^{-1} $ .",
"We choose $ y_{j} $ as the minimum element of the $ j$ -th column of $ Q $ with respect to the order $ `` \\le _{S} \" $ .",
"Note further that in the normalization process of column $ A_{j} $ , we disregard zero elements, that is if $ a_{ij}=0 $ for some $ i \\in \\underline{m} $ and $ j \\in \\underline{n} $ , then $ \\hat{A}_{j}= \\@root m \\of {a_{1j}\\otimes \\cdots \\otimes a_{(i-1)j} \\otimes a_{(i+1)j}\\otimes \\cdots \\otimes a_{mj}}.",
"$ and $ \\tilde{a}_{ij}= 0 \\otimes \\hat{A}_{j}^{-1} = 0 $ .",
"As such, we set $ q_{ij} := 0^{-} $ , where $ a <_{S} 0^{-}$ for any $ a \\in S $ .",
"The $ j$ -th column minimum element of $ Q $ is therefore determined regardless of $q_{ij} $ .",
"Thus, without loss of generality, we consider every column of the system matrix to be regular.",
"In the next theorem, we present a necessary and sufficient condition on the column minimum elements of the associated normalized matrix to solve the systems $ \\tilde{A}Y=\\tilde{b} $ and consequently $ AX=b$ , since we choose $ y_{j} $ as the $ j$ -th column minimum element, for any $ j \\in \\underline{n} $ .",
"Theorem 4 Let $ AX=b $ be a linear system of equations with $ A \\in M_{m \\times n }(S) $ and the regular $ m$ -vector $ b $ .",
"Then the system $ AX=b $ has solutions if and only if every row of the associated normalized matrix $ Q $ contains at least one column minimum element.",
"Let the system $ AX=b $ have solutions.",
"Assume that the $ i$ -th row of $ Q $ contains no column minimum element, that is $ y_{j} \\ne q_{ij} $ and so $ y_{j} <_{S} \\tilde{b}_{i} \\otimes \\tilde{a}_{ij}^{-1} $ , for any $ j \\in \\underline{n} $ .",
"Thus, in the $ i$ -th equation of the system $ \\tilde{A}Y=\\tilde{b} $ we have $ \\bigoplus _{j=1}^{n}(\\tilde{a}_{ij} \\otimes y_{j}) <_{S} \\tilde{b}_{i}, $ which implies the system $\\tilde{A}Y=\\tilde{b} $ and consequently the system $ AX=b$ have no solution and is a contradiction.",
"Conversely, suppose that every row of $ Q $ contains at least one column minimum element, that is for every $ i \\in \\underline{m} $ , there exist some $ k \\in \\underline{n} $ such that $ y_{k} = \\tilde{b}_{i} \\otimes \\tilde{a}_{ik}^{-1} $ .",
"Then the $ i$ -th equation of the system $ \\tilde{A}Y=\\tilde{b} $ , for every $ i \\in \\underline{m} $ is as follows $\\bigoplus _{j=1}^{n}(\\tilde{a}_{ij} \\otimes y_{j}) \\nonumber &= \\bigoplus _{{{\\scriptstyle \\begin{matrix}j=1 \\\\ j \\ne k\\end{matrix}}}}^{n}(\\tilde{a}_{ij} \\otimes y_{j}) \\oplus (\\tilde{a}_{ik} \\otimes y_{k}) \\\\&=\\bigoplus _{{{\\scriptstyle \\begin{matrix}j=1 \\\\ j \\ne k\\end{matrix}}}}^{n}(\\tilde{a}_{ij} \\otimes y_{j}) \\oplus \\tilde{b}_{i} \\\\ \\nonumber &=\\tilde{b}_{i}, \\nonumber $ where the equality $ (3.5) $ is obtained from choosing $ y_{j} $ as the $ j$ -th column minimum element of the matrix $ Q $ .",
"That means the system $ \\tilde{A}Y=\\tilde{b} $ and consequently, the system $ AX=b $ has solutions.",
"Indeed, the obtained solution $ X=((\\hat{A}_{i} \\otimes \\hat{b}_{i}^{-1})\\otimes y_{i})_{i=1}^{n} $ of the system $ AX=b $ is maximal.",
"Remark 2 The normalization method and the associated normalized matrix of a linear system provide comprehensive information which enables us to compute the degrees of freedom and determine the column rank and the row rank of a matrix.",
"These applications over idempotent semifields are similar to the descriptive methods stated in [7] over $``\\max -plus~algebra\"$ ."
],
[
"Let $ A, A^{\\prime } \\in M_{m\\times n}(S) $ .",
"We say $ A$ is equivalent to $ A^{\\prime }$ if the $ j$ -th column of $ A^{\\prime } $ is a scalar multiple of the $ j$ -th column of $ A $ , for any $ j \\in \\underline{n} $ , that is $A \\sim A^{\\prime } \\Longleftrightarrow A^{\\prime }= \\left[\\begin{array}{c|c|c}\\alpha _{1}A_{1}&\\cdots &\\alpha _{n}A_{n}\\end{array}\\right],$ for some $ \\alpha _{1}, \\cdots , \\alpha _{n} \\in S \\setminus \\lbrace 0 \\rbrace $ .",
"As such, the equivalence class of $ A $ is $\\left[ A \\right]= \\lbrace A^{\\prime }\\in M_{m \\times n}(S) \\vert A \\sim A^{\\prime }\\rbrace .",
"$ The equivalence relation on vectors is defined analogously."
],
[
"Equivalance of the solution methods",
"In this section, we show that the presented solution methods in the previous section are equivalent and their maximal solutions are identical.",
"In the next theorem, we intend to show the equivalence of the pseudo-inverse method and the normalization method for solving a linear system.",
"Theorem 5 Let $ AX= b $ , where $ A \\in M_{n}(S), $ and $ b $ be a regular vector of size $ n $ .",
"If $ (AA^{-})_{ij} \\otimes b_{j} \\le _{S} b_{i} $ , for any $i,j \\in \\underline{n} $ , then $ (A^{-}b)_{k} = b_{i} \\otimes a_{ik}^{-1} $ , for some $ i \\in \\underline{n} $ and for any $ k \\in \\underline{n} $ .",
"By Theorem REF , the inequalities $ (AA^{-})_{ij} \\otimes b_{j} \\le _{S} b_{i} $ implies that the system $ AX=b $ have solutions.",
"Due to Theorem REF , every row of the associated normalized matrix $ Q $ contains at least one column minimum element.",
"As such, the maximal solution of the system $ AX=b $ through the normalization method is $ X^{\\bullet }= (x^{\\bullet }_{k})_{k=1}^{n}$ , where $ x^{\\bullet }_{k}= \\tilde{b}_{i} \\otimes \\tilde{a}_{ik}^{-1} \\otimes \\hat{b} \\otimes \\hat{A}_{k}^{-1}= b_{i} \\otimes a_{ik}^{-1}, $ for some $ i \\in \\underline{n}$ .",
"Let $ A^{-}= (a^{-}_{ij}) $ , then the inequalities $ (AA^{-})_{ij} \\otimes b_{j} \\le _{S} b_{i} $ implies that $&\\ \\nonumber \\bigoplus _{k=1}^{n} (a_{ik}\\otimes a^{-}_{kj}) \\otimes b_{j} \\le _{S} b_{i},\\qquad ~~~~ \\forall i,j \\in \\underline{n} \\\\ \\nonumber &\\ \\Rightarrow \\bigoplus _{k=1}^{n} (a_{ik}\\otimes a^{-}_{kj} \\otimes b_{j}) \\le _{S} b_{i},\\qquad \\forall i,j \\in \\underline{n} \\\\ \\nonumber &\\ \\Rightarrow a^{-}_{kj} \\otimes b_{j} \\le _{S} b_{i} \\otimes a_{ik}^{-1},\\qquad ~~~~~~~ \\forall i,j,k \\in \\underline{n} \\\\ \\nonumber &\\ \\Rightarrow \\bigoplus _{j=1}^{n} (a^{-}_{kj} \\otimes b_{j}) \\le _{S} b_{i} \\otimes a_{ik}^{-1},\\qquad \\forall i,k \\in \\underline{n} \\\\&\\ \\Rightarrow (A^{-}b)_{k} \\le _{S} b_{i} \\otimes a_{ik}^{-1},\\qquad ~~~~~~~~ \\forall i,k \\in \\underline{n}.$ On the other hand, $X^{*}=A^{-}b $ is the maximal solution of the system $ AX=b $ .",
"As such, $ x^{\\bullet }_{k}= b_{i} \\otimes a_{ik}^{-1} \\le _{S} (A^{-}b)_{k} $ , for any $ k \\in \\underline{n}$ .",
"This inequality and (4.1) lead to $ (A^{-}b)_{k} = b_{i} \\otimes a_{ik}^{-1} $ , for some $ i \\in \\underline{n} $ and any $ k \\in \\underline{n} $ .",
"Consequently, these two methods have the same maximal solution, $X^{\\bullet } =X^{*} $ Example 2 Consider the linear system $ AX=b $ given in Example REF .",
"We want to solve the system through the normalization method.",
"To this end, we must construct the associated normalized matrix $ Q=(\\tilde{b}_{i} \\otimes \\tilde{a}_{ij}^{-1}) $ of the system $ AX=b $ .",
"Since the column minimum elements of $ Q $ form the maximal solution of the normalized system $ \\tilde{A}Y=\\tilde{b} $ and the maximal solution of the system $ AX=b $ is $ X=(x_{i})_{i=1}^{4} $ , where $ x_{i}=(\\hat{b}\\otimes \\hat{A}_{i}^{-1}) \\otimes y_{i} $ , for any $ i \\in \\lbrace 1, \\cdots , 4 \\rbrace $ .",
"Without loss of generality, we can consider the matrix $ Q^{\\prime }= ((\\tilde{b}_{i} \\otimes \\tilde{a}_{ij}^{-1}) \\otimes (\\hat{b}\\otimes \\hat{A}_{i}^{-1}))=(b_{i} \\otimes a_{ij}^{-1}) $ which is equivalent to $ Q $ with coefficients $ \\hat{b}\\otimes \\hat{A}_{i}^{-1} \\in S\\setminus \\lbrace 0 \\rbrace $ , for any $ i \\in \\lbrace 1, \\cdots , 4 \\rbrace $ .",
"It suffices to check the condition of Theorem REF on the column minimum elements of $ Q^{\\prime }$ : $Q^{\\prime }=\\left[\\begin{array}{cccc}\\frac{27}{5}&\\frac{27}{7}&\\boxed{3}&\\frac{27}{10}\\\\\\boxed{4}&8&0^{-}&\\boxed{\\frac{16}{7}}\\\\\\boxed{4}&0^{-}&4&\\frac{12}{5}\\\\24&\\boxed{3}&24&4\\end{array}\\right].$ Clearly, every row of $ Q^{\\prime } $ contains at least one column minimum element.",
"Now, the boxed entries of $ Q^{\\prime } $ form the maximal solution of the system $AX=b $ as $X^{*}=\\left[\\begin{array}{c}4 \\\\3 \\\\3 \\\\\\frac{16}{7}\\end{array}\\right],$ which is the same as the maximal solution of pseudo-inverse method.",
"Remark 3 Theorems REF and REF imply that the maximal solutions obtained from the normalization method and the extended Cramer's rule should be the same.",
"The notion of generalized $LU$ -factorization and the method for solving linear systems through $LU$ -factorization over idempotent semifields are presented in [4].",
"The following theorem proves that the maximal solutions of $LU$ -method and the normalization method are the same.",
"Theorem 6 Let $ A \\in M_{n}(S)$ , $ A $ have generalized $LU$ -factorization and $ b $ be a regular $ n$ -vector.",
"If $ a_{ik} \\otimes a_{kk}^{-1} \\le _{S} b_{i} \\otimes b_{k}^{-1} $ and $ a_{(n-j)l} \\otimes a_{ll}^{-1} \\le _{S} b_{(n-j)} \\otimes b_{l}^{-1} $ for every $2\\le i\\le n$ , $1\\le k\\le i-1$ , $1\\le j\\le n-1$ , and $n-j+1\\le l\\le n$ , then the $ k $ -th column minimum element of the associated normalized matrix $ Q $ is $ \\tilde{b}_{k} \\otimes \\tilde{a}_{kk}^{-1} $ .",
"Due to the extended version of Theorem 8 in [4], the assumed inequalities imply that the linear system $ AX=b $ have the maximal solution $ X^{*}=(b_{k}\\otimes a_{kk}^{-1})_{k=1}^{n} $ .",
"By Theorem REF , every row of $ Q $ contains at least one column minimum element.",
"The assumption $ a_{ik} \\otimes a_{kk}^{-1} \\le _{S} b_{i} \\otimes b_{k}^{-1} $ , for any $ 2\\le i\\le n $ and $ 1\\le k\\le i-1 $ implies that $ b_{k} \\otimes a_{kk}^{-1} \\le _{S} b_{i} \\otimes a_{ik}^{-1}$ and therefore $ (\\hat{b}^{-1}\\otimes b_{k}) \\otimes (\\hat{A}_{k} \\otimes a_{kk}^{-1}) \\le _{S} (\\hat{b}^{-1}\\otimes b_{i}) \\otimes (\\hat{A}_{k} \\otimes a_{ik}^{-1}), $ which means $\\tilde{b}_{k} \\otimes \\tilde{a}_{kk}^{-1} \\le _{S} \\tilde{b}_{i} \\otimes \\tilde{a}_{ik}^{-1},$ for any $ 2\\le i\\le n $ and $ k < i $ , since the total order $ `` \\le _{S} \" $ is compatible with multiplication.",
"It suffices to show that the inequality (REF ) holds for any $ k > i $ .",
"Letting $ i:= n-j $ and $ k:= l $ , we can rewrite the inequalities $ a_{(n-j)l} \\otimes a_{ll}^{-1} \\le _{S} b_{(n-j)} \\otimes b_{l}^{-1} $ as the inequalities $ a_{ik} \\otimes a_{kk}^{-1} \\le _{S} b_{i} \\otimes b_{k}^{-1} $ , for any $ 1 \\le i \\le n-1 $ and $ k > i $ .",
"Now, the compatibility with multiplication leads to $ b_{k} \\otimes a_{kk}^{-1} \\le _{S} b_{i} \\otimes a_{ik}^{-1} $ and $ \\tilde{b}_{k} \\otimes \\tilde{a}_{kk}^{-1} \\le _{S} \\tilde{b}_{i} \\otimes \\tilde{a}_{ik}^{-1}.",
"$ Hence, $ \\tilde{b}_{k} \\otimes \\tilde{a}_{kk}^{-1} $ is the $ k$ -th column minimum element of $ Q $ .",
"Remark 4 Note that the maximal solution of the linear system $ AX=b $ obtained from the normalization method in Theorem REF is $ X= ( \\tilde{b}_{k} \\otimes \\tilde{a}_{kk}^{-1} \\otimes \\hat{b} \\otimes \\hat{A}_{k}^{-1} )_{k=1}^{n}=(b_{k}\\otimes a_{kk}^{-1})_{k=1}^{n}, $ which is the same as the maximal solution obtained from the $LU$ -method.",
"Example 3 Let $ A\\in M_{4}(S) $ where $S=\\mathbb {R}_{\\min , \\times }=(\\mathbb {R_{+}} \\cup \\lbrace +\\infty \\rbrace , \\min ,\\times ,+\\infty ,1)$ and $\\mathbb {R_{+}} $ be the set of all positive real numbers.",
"Consider the following system $ AX=b $ : $\\left[\\begin{array}{cccc}1&6&9&8\\\\6&2&7&5\\\\9&7&1&7\\\\8&5&6&3\\end{array}\\right]\\left[\\begin{array}{c}x_{1}\\\\x_{2}\\\\x_{3}\\\\x_{4}\\end{array}\\right]=\\left[\\begin{array}{c}4\\\\6\\\\1\\\\6\\end{array}\\right],$ which is given in Example 6 of [4].",
"The system has already been solved through the $LU$ -method and its maximal solution is $X^{*}= \\left[\\begin{array}{c}4\\\\3\\\\1\\\\2\\end{array}\\right].$ Now, we solve the system by the normalization method.",
"Without loss of generality, we consider the matrix $ Q^{\\prime }= (b_{i} \\otimes a_{ij}^{-1} ) $ , which is equivalent to the associated normalized matrix of the system $ AX=b $ .",
"Its column minimum elements determine the maximal solution of the system: $\\left[\\begin{array}{cccc}\\boxed{4}&\\frac{2}{3}&\\frac{4}{9}&\\frac{1}{2}\\\\1&\\boxed{3}&\\frac{6}{7}&\\frac{6}{5}\\\\\\frac{1}{9}&\\frac{1}{7}&\\boxed{1}&\\frac{1}{7}\\\\\\frac{3}{4}&\\frac{6}{5}&\\boxed{1}&\\boxed{2}\\end{array}\\right],$ where the column minimum elements of $ Q^{\\prime } $ are boxed with respect to the total order $`` \\le _{S}\" $ on “$\\min -\\rm times$ algebra\" as the standard greater than or equal relation $ `` \\ge \" $ over $\\mathbb {R}$ .",
"Thus, these elements form the maximal solution of the system, which is the same as the solution obtained from the $ LU $ -method."
],
[
"Concluding Remarks",
"In this paper, we extended the solution methods such as the pseudo-inverse method, the extended Cramer's rule and the normalization method to idempotent semifields.",
"Applying each of these methods, we determined the maximal solution of a linear system.",
"Importantly, we proved that the maximal solution obtained from the $ LU $ -method and the above-mentioned methods are identical."
]
] | 1906.04543 |
[
[
"Contextual Documentation Referencing on Stack Overflow"
],
[
"Abstract Software engineering is knowledge-intensive and requires software developers to continually search for knowledge, often on community question answering platforms such as Stack Overflow.",
"Such information sharing platforms do not exist in isolation, and part of the evidence that they exist in a broader software documentation ecosystem is the common presence of hyperlinks to other documentation resources found in forum posts.",
"With the goal of helping to improve the information diffusion between Stack Overflow and other documentation resources, we conducted a study to answer the question of how and why documentation is referenced in Stack Overflow threads.",
"We sampled and classified 759 links from two different domains, regular expressions and Android development, to qualitatively and quantitatively analyze the links' context and purpose, including attribution, awareness, and recommendations.",
"We found that links on Stack Overflow serve a wide range of distinct purposes, ranging from citation links attributing content copied into Stack Overflow, over links clarifying concepts using Wikipedia pages, to recommendations of software components and resources for background reading.",
"This purpose spectrum has major corollaries, including our observation that links to documentation resources are a reflection of the information needs typical to a technology domain.",
"We contribute a framework and method to analyze the context and purpose of Stack Overflow links, a public dataset of annotated links, and a description of five major observations about linking practices on Stack Overflow.",
"We further point to potential tool support to enhance the information diffusion between Stack Overflow and other documentation resources."
],
[
"Introduction",
"The knowledge-intensive nature of current-day software engineering means that software developers are continually in search of knowledge.",
"A popular model for knowledge sharing on the Internet is the community question answering site, with Stack Overflow [1] serving as the de facto forum for most programmers [2].",
"On Stack Overflow, registered users can post questions, answer posted questions, and comment on questions and answers by other users, which can then be viewed by anyone.",
"As of December 2019, Stack Overflow archives 19M questions, 28M answers, and 72M comments.",
"At this scale, Stack Overflow constitutes a major information broker between posters, contributors, and non-contributing readers (so-called “lurkers”).",
"Stack Overflow, however, does not exist in isolation—the site is only one of many sources of programmer knowledge in a software documentation ecosystem.",
"Past research has extensively characterized the strengths and weaknesses of Stack Overflow (e.g., good at “how-to” documentation [3], bad at completeness [4]) compared to other sources, such as API documentation (e.g., good at structure [5], bad at scenarios [6]).",
"Meng et al.",
"'s observational study corroborates that developers seek a diversity of documentation content when solving programming tasks [7].",
"With these complementary strengths and weaknesses, it is only natural that links exist from one source to another.",
"In fact, previous studies found that link sharing is a significant phenomenon on Stack Overflow that make the site part of a larger interconnected network of online resources used and referenced by developers [8], [9].",
"Given the crucial role that on-line resources play in developers' quest for technical knowledge, it is important to know how information is diffused between resources types so we can facilitate this quest (see Section ).",
"We conducted a multi-case study to answer the question of how and why documentation is referenced in Stack Overflow threads.",
"We sampled 759 links from two different domains (Java regular expressions and Android development), classified and qualitatively analyzed them, and then used the resulting data to derive association rules and build logistic regression models to identify properties of Stack Overflow questions that attract links to documentation resources.",
"Our main findings include that links on Stack Overflow serve widely diverse purposes that range from simple pointers to API documentation over links to concept descriptions on Wikipedia to suggestions of software components and background readings.",
"This purpose spectrum (see Section ) allows us to modulate Stack Overflow's requirement to add context for links [10].",
"We also find that links to documentation resources are a reflection of the information needs typical to a technology domain, with significant differences between the two domains in our multi-case study.",
"Our main contributions are: (1) a framework and method to analyze the context and purpose of documentation links on Stack Overflow, (2) a public dataset with 759 annotated links that other researchers can use, and (3) a description of five major observations about linking practices on Stack Overflow, with detailed links to evidence, implications, and a conceptual framework to capture the relations between the five observations.",
"The remainder of this paper is structured as follows: We provide additional background and motivation in Section and outline our study design in Section .",
"Section describes our method for link sampling and classification, Sections and describe our qualitative and quantitative analyses, respectively.",
"Section presents the major findings derived from these analyses, Section describes threats to validity.",
"We conclude the paper in Section ."
],
[
"Background and Motivation",
"This work is a systematic investigation of current information diffusion (link sharing) practices on Stack Overflow, with the goal of informing the development of advanced technology to facilitate this diffusion.",
"This research takes place in the context of previous studies on information diffusion in on-line developer communities."
],
[
"Information Diffusion on Stack Overflow",
"Stack Overflow explicitly encourages the inclusion of links to external resources in answers, but requests that users add context so that “fellow users will have some idea what it is and why it's there.” [10].",
"This advice is overly general.",
"Not all link targets need to be quoted, and in some cases, the context for a link is obvious.",
"However, deciding when and how to include links to other documentation sources in Stack Overflow posts requires differentiating common linking practices and understanding their unique characteristics.",
"The following examples illustrate the richness and diversity of linking practices on Stack Overflow.",
"When considering the potential value of links on Stack Overflow, the best case scenario is the recommendation of specific information relevant to the thread (links are in bold): ...have a look at Greedy, Reluctant, and Possessive Quantifiers section of the Java RegEx tutorial... [11] In this case, a contributor provided a comment to point the original poster to a section of a tutorial introducing the concept of regular expression quantifiers and explaining how to use them.",
"These “ideal” links provide clear value added to the thread, and form a type of information that can even be automatically mined to improve information discovery [12].",
"However, the reality of linking practices goes broadly beyond this expected scenario.",
"For example, links to obvious documentation resources can be introduced defensively by the original poster themselves, to avoid having a question downvoted [13]: I've already tried this solution (http://developer.android .com/training/articles/security-ssl.html) but I still have the same error:...[14] Other links bind a reference to library classes to its documentation.",
"This can be useful to help make code fragments more self-explanatory [15], but we observed that such links are also provided for well-known, pervasive classes: When you want to return more than one result, you need to return an array (String[]) or a Collection like an ArrayList, for example.",
"[16] From the point of view of links as mechanisms to increase the flow of valuable software development knowledge, degenerate practices include providing links to comic strips (such as xkcd) and similar sites: ...reminds me of this xkcd[17] As these examples show, linking practices on Stack Overflow are diverse and the intrinsic value of a link as a carrier of relevant technical information is not uniform.",
"The first example link, to a specific section of a tutorial, has an obvious purpose and value.",
"The link to a comic strip is clearly noise.",
"Between these extremes lies a gray zone where links play different roles in different contexts.",
"As illustrated above, links in on-line developer forums can fulfill the important mission of complementing documentation with explanations of concepts or descriptions of code elements.",
"However, a manual linking process is prone to omissions.",
"A number of techniques have been proposed to automatically enhance on-line resources through linking and recommendation.",
"The idea of automatically enhancing information diffusion is clearly captured by Gao et al.",
"'s proposal to automatically add links to recognized entities in Stack Overflow posts from a database of popular URLs, and taking into account the context in which the entity appears [18].",
"A different take on the problem is offered by Li et al.",
"[12], who built a collaborative filtering recommender system to recommend other learning resources, based on co-occurrences of links in Stack Overflow posts.",
"In approaches that are based on existing link data, the automatic linking system relies on the assumption that the underlying linking practices are sound.",
"Our study sheds light on the linking practices that are used as foundations for collaborative filtering.",
"Xu et al.",
"'s deep-learning-based approach for predicting semantically linkable knowledge in developer forums [19] avoids the issue of relying on existing links.",
"This is an important advancement for improving information diffusion in knowledge networks.",
"However linking that is based on the semantics of the text may not necessarily take into account the purpose for linking a knowledge unit (e.g., the comic strip mentioned above).",
"Our study focuses specifically on eliciting the purpose of links so that it is possible to account for it when enabling information diffusion though automated approaches.",
"Content is one concern for documentation ecosystems, but quality is another important one.",
"Previous work has attempted to automatically identify high-quality posts using features based on the number of edits on a question [20], author popularity [21], and code readability [22].",
"In their conceptual framework of success factors for Stack Overflow questions, Calefato et al.",
"[23] considered the presence of links as one aspect of a question's presentation quality.",
"However, they did not find a significant effect of the fact that a question contained a link on the success of that question, that is whether it attracted an accepted answer.",
"A direction of future work is to consider not only the presence of a links, but also their purpose and targets, as enabled by our study.",
"There have been different studies investigating individual aspects of link usage on Stack Overflow.",
"Gomez et al.",
"[8] conducted a preliminary study of the links found on Stack Overflow.",
"Their study focused on the different types of links in posts (not comments) and it did not factor in a distinction based on the domain.",
"In this article, we investigate two specific domains, which allows us to understand the data in a specific context.",
"Moreover, we integrate an analysis of the purpose of the information sharing that goes beyond a basic description of its nature.",
"Vincent et al.",
"[9] analyzed the usage of Wikipedia by Stack Overflow authors.",
"They found that 1.28% of all Stack Overflow posts contain links to Wikipedia.",
"Using version 2018-07-31 of the SOTorrent dataset [24], we identified 1.94% of all threads, but only 0.85% of all posts, to contain links to Wikipedia.",
"Also considering links in comments, which Vincent et al.",
"did not, the ratio of threads with links to Wikipedia increases to 2.55%.",
"Ye et al.",
"contributed a study of link sharing on Stack Overflow that focuses on the sharing of links to other Stack Overflow posts[25].",
"In contrast, our study covers links to all external resources, and for this reason an important part of our study design addresses the problem of categorizing the types of documentation referenced.",
"The part of the Ye et al.",
"study that is the most complementary to ours is their analysis of the purpose of links.",
"However, because the study focuses on internal links, their classification does not include purposes that would be exclusive to resources outside of Stack Overflow itself.",
"Their classification is also more abstract, with four categories of purpose (excluding the “other” category), whereas we analyze the link purposes at a finer granularity.",
"In addition to Stack Overflow, studies have also investigated linking practices in other context.",
"Hata et al.",
"[26] studied the role of links contained in source code comments in terms of prevalence, link targets, purposes, decay, and evolutionary aspects.",
"They report that links can be fragile since link targets change frequently or disappear.",
"Links are also shared as part of code review.",
"Jiang et al.",
"contribute a study of link sharing in review comments [27], reporting that roughly half the links they identified refer to resources outside the project.",
"This observation further motivates our study in that the observations we make about information diffusion may also be applicable to contexts other than question and answer forums."
],
[
"Study Design",
"To investigate how and why documentation resources are referenced in Stack Overflow threads, we conducted a mixed-methods study involving a qualitative analysis of 759 links from 742 different threads and a quantitative analysis using association rule mining and logistic regression models."
],
[
"Research Questions",
"The overall goal of the study is to discover the roles that links to documentation play in Stack Overflow threads and thus pave the way for a more systematic treatment of documentation references on Q&A sites for software developers.",
"We split our research questions into two sub-questions: RQ1 What is the context around documentation links in Stack Overflow threads?",
"With this question we study how links are provided.",
"RQ2 What is the purpose that documentation links in Stack Overflow threads serve?",
"With this question we study why links are provided.",
"With these questions, our aim was to collect specific insights about linking practices on Stack Overflow, that can support actionable implications for authors and readers of Q&A forums and for the development of technology based on the analysis of such forums.",
"Our first research question was motivated by the fact that Stack Overflow encourages users to provide context for links [10], in particular by quoting external sources [28].",
"We qualitatively analyzed whether users follow this advice (see Section ), but we also built logistic regression models capturing different features of Stack Overflow posts to quantitatively analyze which of those features are related to the presence of documentation links (see Section ).",
"As the examples in Section illustrate, links on Stack Overflow serve diverse purposes.",
"To conduct a structured analysis of those purposes, we first built a classifier that was able to identify links to the most frequently referenced documentation resources (see Section ).",
"Based on a stratified sample of documentation links identified using the classifier, all three authors independently coded the purpose of 759 links using a jointly developed coding guide (see Section ).",
"We mined the resulting data for association rules between documentation resources and assigned purposes and then used our qualitative and quantitative results to corroborate five major findings about linking practices on Stack Overflow (see Section ).",
"Because even a cursory inspection of Stack Overflow threads shows clear differences in the use of references to external documentation, we structured our research as a multi-case study of linking practices for two different domains: use of regular expressions in Java (Regex), and Android development (Android).",
"We bounded our investigation to clearly-defined domains to support a richer analysis of linking practices in the context of the wider documentation ecosystem they integrate.",
"We selected Regex and Android because they constituted two very different domains (library vs. framework, small vs. large, integrated in the programming language vs. third-party, theoretically vs. practically grounded), and because we were familiar with both technologies.",
"The importance of this latter aspect is not to be underestimated as a contributor to the meaningfulness of qualitative data analysis.",
"Despite the ready availability of structured data from Stack Overflow, generating reliable insights about linking practices requires an extensive combination of analytical processing and manual inspection.",
"Figure REF outlines the general process we followed.",
"The research proceeded sequentially: we first completed an entire iteration for Regex (referenced as number 1 on the figure), and then repeated the process for the second case (Android), referenced as number 2.",
"In the following description, numbers refer to the step in the process overview (indicated after the period in Figure REF ).",
"The first step was to retrieve all Stack Overflow threads related to each case ($N$ .1).",
"For this purpose we utilized the SOTorrent dataset [24].",
"For the Regex case, we retrieved all threads with tags and , and for the Android case, the threads with tags and .",
"For each case, we used the most recent release at the time (2018-05-04 for the Regex case [29] and 2018-07-31 for the Android case [30]).",
"The second step was to process the links to determine what they were linking to, and to abstract the target of the links to one of a small set of documentation resource categories (e.g., links to other Stack Overflow threads vs. links to API documentation).",
"We built a URL mapper to classify links to such documentation resources using the 25 most frequently referenced root domains for each case (Section and $N$ .2 in Figure REF ).",
"The classification of links was necessary to create a stratified sample for detailed analysis, i.e., a sample guaranteed to contain links to all different types of resources.",
"The third step was then to draw samples containing links to all identified documentation resources and qualitatively analyze their context and purpose (see Section and $N$ .3 in Figure REF ).",
"This step involved extensive manual inspection and labeling of links in their context.",
"In step four, to investigate the motivation behind linking to documentation resources of a certain type, we used association rule mining [31] to investigate the relationship between resource type and purpose (see Section and $N$ .4 in Figure REF ).",
"Finally, we built logistic regression models to analyze which properties capturing the question context attract links to documentation resources in comments and answers (see Section and $N$ .5 in Figure REF ).",
"In these models, we treat question features as independent variables and the presence of a link to a particular resource as dependent variable.",
"To support the complete replicability of this process and the verification of the results presented in this paper, we provide our coding guide, samples, and the analysis and data retrieval scripts as supplementary material [32]."
],
[
"Link Classification and Sampling",
"Links on Stack Overflow may point to resources other than documentation, e.g., tools or images.",
"To be able to study links to documentation resources on Stack Overflow, we built a URL-based classifier that takes as input a link and outputs either one of 12 documentation resource categories that best describes the target of the link, or marks the link as NotDocumentation (see Table REF ).",
"Those 12 categories emerged during an iterative analysis of the most frequently linked domains.",
"We used the classifier to categorize all links in the two cases and then sampled links from each category of documentation links for our qualitative analysis."
],
[
"Building the Classifier",
"As mentioned above, we built the link categorization and corresponding classifier following a grounded, iterative approach.",
"First, we ranked all referenced root domains according to the number of posts in which they were referenced (the root domain of , for example, is ).",
"Starting with the most frequently referenced root domain, we inspected the extracted links and either decided that they form a new resource category or assigned them to an existing one.",
"Tables REF and REF show the five most frequently referenced root domains, meaning that those were the first five domains we derived resource categories from.",
"For each of the analyzed domains, we started by investigating the different paths that were linked from the Stack Overflow posts retrieved for the particular case.",
"For both cases, the most frequent link target was the platform itself.",
"Because such links are internal to platform, we created a dedicated documentation resource category StackOverflow.",
"However, we soon realized that not all links to can be considered software documentation links, because the linked paths included user profiles (e.g., ) or internal help pages (e.g., ).",
"Instead of excluding the paths that we did not consider documentation targets, we followed a whitelisting approach.",
"We first built regular expressions matching the paths of the domains that we identified as pointing to documentation resources (e.g., another Stack Overflow post).",
"After integrating those regular expressions in our link classifier, we executed the classification and analyzed the links to the current domain that had not been classified yet.",
"We then refined the regular expressions and repeated the process until all links to documentation resources were classified either as Documentation, NotDocumentation, or InvalidOrDead (see also Table REF ).",
"This process was performed by two authors who continuously discussed the emerging resource categories and associated regular expressions.",
"All decisions in the process were made unanimously.",
"The source code of the classifier, including the regular expressions for all documentation resources, is available on GitHubhttps://github.com/sbaltes/condor and archived on Zenodo [33].",
"To conclude the above example, for the root domain we decided to only match links to questions, answers, post revisions, and comments—but not links to user profiles or pages with tips on how to write questions and answers (see above).",
"To illustrate this classification approach, we briefly describe the path matching for this domain.",
"As mentioned above, we modeled internal links as a separate documentation resource.",
"The regular expressions for the corresponding StackOverflow documentation resource all start with: ^https?://((www|pt|ru|es)\\\\.",
")?stackoverflow\\\\.com This prefix is followed by expressions matching the different paths we determined to point to documentation resources: /(a|q|questions)/[\\\\d]+.",
"* /revisions.",
"* /posts/\\\\d+/revisions.",
"* /posts/comments.",
"* All other paths for the root domain are automatically classified as .",
"Root domains that we have not analyzed yet are automatically labeled as , root domains that our tool determined to be invalid or dead are automatically classified as .",
"This allowed us to track our progress.",
"We continued with the next root domain once we could not find paths anymore that were incorrectly labeled as .",
"We repeated the classification process for the 25 most frequently referenced root domains in both samples, which enabled us to classify 78.5% of all active links in the Regex sample and 68.9% of all active links in the Android sample.",
"The ratio of classified active links can be derived from the data in Table REF as follows: ${(\\text{Documentation} + \\text{NotDocumentation})}{(\\text{All} - \\text{Dead})}$ .",
"Because we conducted our analysis of the Android case after the Regex case had been completed, the classifier for Android links was built by extending the preliminary Regex link classifier.",
"Note that, as a last step, we re-ran the final classifier for the Regex case.",
"Table REF shows the documentation resources we extracted for both cases.",
"In the following, we briefly describe which kinds of documentation resources we assigned to the different categories together with exemplary links.",
"StackOverflow: This documentation resource consists of Stack Overflow questions, answers, post revisions, and comments (see details above).",
"OtherForum: We used this category to capture links to non-Stack-Overflow forum posts or threads including certain subpages of and .",
"{Java$\\vert $ Android$\\vert $ Other}{Reference$\\vert $ API}: The resource category JavaReference represents official Java documentation except for the Java API documentation, which is represented by JavaAPI.",
"OtherReference, AndroidReference, OtherAPI, and AndroidAPI are analogously defined.",
"Examples for OtherAPI include API documentation hosted on , , and .",
"Examples for OtherReference includes the subpages of , certain reports on , and different GitHub Pages.https://pages.github.com/ AndroidIssue: Since Android issue descriptions were quite frequently referenced in the Android case, we created a dedicated category for them.",
"Those links typically point to subpages of or .",
"IndependentTutorial: Links in this category point to independent tutorials.",
"By `independent', we mean tutorials not provided by authoritative entities such as Oracle for Java or Google for Android.",
"Examples include , , and .",
"Wikipedia: We assigned links to Wikipedia pages in various languages to this category.",
"YouTube: Especially in the Android case, Stack Overflow users frequently referenced YouTube videos.",
"We assigned such links to this category.",
"Section provides further examples for specific documentation resources, together with associated purposes we identified.",
"Table REF lists the five most frequently referenced root domains for Regex, together with the number of links to those domains and the assigned resource categories.",
"Table REF lists this information for Android.",
"Because of the high effort involved in reviewing each link manually, we produced a sample of links to documentation resources for the qualitative analysis.",
"We randomly sampled (up to) 40 links per documentation resource: We selected 20 links from questions (10 from question posts and 10 from question comments) and 20 links from answers (10 from answer posts and 10 from answer comments).",
"Because some documentation resources had insufficient links to fulfill all of those selection constraints, the Regex sample contained only 279 links (and not $8 \\cdot 40 = 320$ ).",
"The Android sample contained $12 \\cdot 40 = 480$ links, because we added four additional documentation resources that were only exhibited in that domain (see Table REF ).",
"Section discusses implications of this sampling approach."
],
[
"Qualitative Analysis",
"We qualitatively analyzed all links in our samples to build a first layer of interpretation for linking practices.",
"Following our research questions, we organized the coding [34] along two dimensions, context and purpose.",
"For analyzing the context, much information is already available directly in the posts (e.g., the text surrounding the links).",
"For context, we designed the coding task to complement this information with insights that are impossible to extract automatically, namely, whether the text in the context includes a quote or a summary of the link target—or whether the link is provided without any context.",
"For purpose, we were interested in producing an abstraction of the purpose of the link as it would appear to a third party who read the corresponding thread."
],
[
"Development of the Coding Guide",
"We developed a coding guide by considering the context and purpose dimensions separately.",
"For the context, creating the coding guide amounted to agreeing on what constituted a quote, a summary, and a link without context.",
"The task was thus to indicate, for each link in the sample, or as values for the attributes Quote and Summary.",
"The attribute Quote indicates the presence of non-trivial content that has been copied without modification from the linked documentation resource into the Stack Overflow post or comment, the attribute Summary indicates that the Stack Overflow author provided at least one key insight from the linked documentation resource in their own words.",
"The third context code LinkOnly was assigned in case only the URL was provided (including anchor text) without any additional information surrounding it.",
"Note that while the codes Quote and Summary can be assigned independent of the purpose codes, LinkOnly makes deriving a purpose impossible because no context is provided.",
"Therefore we modeled the former two as independent binary codes, as outlined above (see also Tables REF and REF ).",
"Figure REF illustrates the difference between the three context codes we assigned.",
"The development of a reliable coding guide for a link's purpose was much more challenging, and required multiple iterations.",
"In an initial coding phase, we built a coding guide using a subset of the links for Regex.",
"During the initial coding, all three authors coded 80 links in four tasks of 20 each, discussing emerging categories after completing each task, until a stable coding guide emerged.",
"Prior to starting with the Android sample, all three authors coded 50 links and then discussed if changes to the coding guide were required, which only led to one minor addition.",
"Note that, while the codes are not mutually exclusive, the coders always assigned one code that they considered to most accurately describe the link purpose.",
"Table REF lists the codes with a brief description.",
"The full description can be found in the supplementary material.",
"The modification that was required for the Android case was simply to add “watching a video” to the code BackgroundReading, because of the new documentation resource YouTube.",
"Table: Code catalog for link context and purpose (summary).We used the coding guide in a focused coding phase to go over all links in the sample and code them according to the guide, which we provide as supplementary material.",
"All three authors used the coding guide to independently code the links by opening the Stack Overflow thread in a web browser, locating the link, and analyzing the surrounding context.",
"We coded the links in sets of up to 100 links, computing inter-rater agreement and discussing results after each set to ensure there were no major divergences or misunderstandings of the coding guide.",
"To measure our inter-rater agreement, we calculated a three-way Cohen's kappa ($\\kappa $ ) [35] for each set.",
"Table REF presents the agreement data.",
"Table: Inter-rater agreement for link purpose coding, with number of items in the set (#) and corresponding κ\\kappa value.The task of identifying the purpose of a link turns out to be very challenging.",
"In some cases, the purpose can be ambiguous or opaque.",
"The difficulty of the task is reflected in the kappa values.",
"Although they increase towards the end as we became more proficient, values in the 0.65-0.80 range, although usable, are indicative of a non-negligible amount of residual flexibility of interpretation.",
"The difficulty of the coding task is the reason we opted for the unusual and very labor intensive practice of coding every single item in our data set in triplicate.",
"This decision significantly mitigates the threats of bias in the coding task, since we were able to systematically detect links with ambiguous purpose and resolve disagreements by applying the following formal process: After each coding iteration, we merged the purpose and LinkOnly codes by selecting the code which at least two investigators used (majority vote), and assigned the code Other if there was no agreement, which happened for 14 Regex links (5%) and for 13 Android links (2.7%).",
"The binary codes capturing the link context were assigned a value of if at least two investigators considered the link to be accompanied by a Quote or Summary respectively.",
"Tables REF and REF show the frequency of each code per documentation resource for both cases.",
"Examples for the three context codes can be found in Figure REF .",
"Section presents examples for the purpose codes together with related developer resources.",
"While our URL mapper was able to detect most invalid or dead links, we still noticed some broken links in the samples (coded as N/A).",
"We also coded links as N/A if they were not rendered on Stack Overflow's website, but present in the Markdown source of the posts or comments, which we used to extract the links from.",
"Table: Documentation resources and corresponding codes (purpose and context) for Regex case.Table: Documentation resources and corresponding codes (purpose and context) for Android case."
],
[
"Quantitative Analysis",
"The qualitative analysis provides the foundation that enabled three quantitative analyses to better understand linking practices: A systematic comparison of code distributions between our two cases, to relate differences to their context.",
"The mining of association rules to detect correspondences between a resource type and a link purpose.",
"The building of logistic regression models, using question features as independent variables and presence of a link to a particular resource as dependent variable, to determine the characteristics of a Stack Overflow question that are related to the features of documentation links in an answer or a comment."
],
[
"Code Frequency Comparison",
"Figure REF shows the relative frequency of the purpose codes we assigned.Our use of stratified random sampling precludes the calculation of confidence intervals, which rely on an assumption of simple random sampling.",
"As stated in the main text, the figure thus documents the code we assigned, without the implication that they would generalize to a population.",
"This is consistent with our research goal and use of a case study method, whereby we sought to understand the phenomenon of link sharing as broadly as possible for two specific topics, as opposed to drawing implications for an entire dataset.",
"The bar charts reveals two major differences: in our sample, the code Awareness was about twice as common in the Android case than in the Regex case (31.0% vs. 16.8%).",
"The reverse was true for the code Concept, which was about twice as common in the Regex case (13.3% vs. 6.3%).",
"Both difference were significant according to a two-tailed Fisher's exact test [36] with a significance level of $\\alpha =0.01$ .The p-values were $0.0001$ for the Awareness frequency difference and $0.0014$ for the Concept frequency difference.",
"Both of these differences can be directly linked to salient aspects of the technological environment of the cases analyzed.",
"The sample for the Regex case exhibits twice as many Concept-related links, which can be explained by the theoretical nature of the domain.",
"The links we coded are to concepts such as context-free grammar and regular language.",
"As for Android, the extensive use of links for Awareness purposes can be explained by the huge size of this technology ecosystem, where many users end up posting answers and comments simply to point out relevant resources to each other.",
"To distill the main motivation behind linking to documentation resources of a certain type, we mined association rules between resources and assigned purpose codes.",
"We first transformed the documentation resource categories as well as the purpose and LinkOnly codes into binary properties of the links, added the Quote and Summary codes, and then applied the apriori algorithm [37] as implemented in the R package https://cran.r-project.org/web/packages/arules/index.html to retrieve binary rules.",
"Table: Binary association rules between documentation resource type and purpose/context codes in the Regex sample.Table: Binary association rules between documentation resource type and purpose/context in the Android sample; rules only present in this sample are highlighted with a gray background.We note that the maximum support of a mined association rule is limited by the fact that we only sampled up to 40 links per documentation resource.",
"The Regex sample, for example, contained 279 links in total (see Table REF ).",
"If a rule is true for all 40 links to one particular resource, the support would still only be ${40}{279} = 0.14$ .",
"In our analysis, we considered rules with at least 10% of the maximum possible support, which was ${0.14}{10} = 0.014$ for the regex sample and ${0.08}{10} = 0.008$ for the Android sample.",
"Moreover, we excluded rules with less than 25% confidence, meaning that a rule must be true in at least 1 out of 4 cases, and we further excluded rules involving the code Other.",
"Tables REF and REF show the binary association rules between the documentation resource types and the purpose/context codes.",
"In the following, we discuss those rules and provide illustrating examples.",
"The purpose Concept was clearly associated with the resource Wikipedia, having the highest and second highest confidence in the two samples, respectively.",
"A typical usage scenario was to mention a concept related to the question and then use the first mention of the concept as link anchor pointing to the corresponding page on Wikipedia: I think you're using * as if it's the Kleene star, not * as Java, JavaScript, & co. interpret * in regexps.",
"[38] This observation provides a clear characterization of the extent to which Wikipedia is leveraged to avoid defining concepts.",
"The observation directly corroborates that of Vincent et al.",
"[9], who found that “on SO, Wikipedia supports answers in the form of links and quoted text.",
"Answers often use technical terms or acronyms and include a Wikipedia link in lieu of defining these terms.” A second dominant group of association rules are related to Recommendations, which often pointed directly to the API documentation of a recommended software component.",
"This is represented by the rule OtherAPI $\\rightarrow $ Recommendation in the regex sample and JavaAPI/OtherAPI $\\rightarrow $ Recommendation in the Android sample.",
"You could use Apache Commons Lang for that... [39] A main use case of reference documentation was providing readers with pointers to resources for BackgroundReading.",
"This relationship is also reproduced in the association rules we identified, since JavaReference were associated with BackgroundReading in both samples.",
"Moreover, AndroidReference was associated with this purpose in the second sample.",
"An example for BackgroundReading is provided below: Instead of asking people to code your regular expressions for you, try reading the Java Regular Expressions Tutorial.",
"...docs.oracle.com/javase/tutorial/... [40] The above example illustrates the difference between the codes Recommendation and BackgroundReading.",
"We used Recommendation to highlight that the authors' primary intention was to recommend a specific tool or library (like in the example).",
"BackgroundReading, on the other hand, indicates that the author recommends a certain resource describing background knowledge relevant for the topic of the particular thread (see also descriptions of the codes in Table REF ).",
"Other rules for link purposes were not as insightful because they rather confirmed the definition of our codes than indicated a particular linking practice.",
"For example, although StackOverflow $\\rightarrow $ Awareness was a strong rule for both cases, it is hardly surprising that people will link to a Stack Overflow post to make others aware of it.",
"Regarding the context of links, we only identified two rules that were present in both samples: Attribution $\\rightarrow $ Quote and Reference $\\rightarrow $ Summary.",
"The former indicates an obvious relationship between content copied from external sources and the purpose of attributing that content.",
"The latter indicates that especially for reference documentation, Stack Overflow authors felt the need to summarize key insights instead of copying content as-is.",
"Overall, quoting content was not very common in the posts and comments we analyzed.",
"In the Regex sample, 7.5% of the links referred to content being quoted, in the Android sample only 3.1% (see Table REF ).",
"The quoted content ranged from complete code snippets to small parts of the reference documentation.",
"Summarizing linked resources was more common than quoting (17.9% in Regex and 7.1% in Android).",
"However, there was neither a summary nor a quote for 203 Regex (72.8%) and 400 Android links (83.3%), which can become a problem once the links are dead.",
"To investigate which properties of a Stack Overflow question might explain whether it will attract documentation links, we built separate logistic regression models for the Regex and Android cases.",
"For each of the two cases (Regex and Android), the input data for the model building were three samples, each containing 100 Stack Overflow threads: Documentation links: One sample with threads that attracted links to documentation resources.",
"To identify such threads, we relied on our previous classification and randomly selected 100 threads with at least one answer or comment containing a link classified as pointing to one of the documentation resources (see Table REF ).",
"Non-documentation links: One sample with threads that attracted links, but not to documentation resources.",
"We randomly selected 100 threads with at least one answer or comment containing a non-classified or non-documentation link (see Table REF ).",
"No links: One sample with threads that did not attract links at all.",
"To draw this sample, we utilized the SOTorrent dataset and selected only threads without any links in answers and comments (no records in tables and ).",
"Our data retrieval and sampling scripts are available as part of the supplementary material.",
"Two of the authors independently analyzed all 600 threads to verify that they are indeed a representative of the corresponding class.",
"In case we found contradicting evidence (e.g., a link to a documentation resource in one of the non-documentation samples), we excluded those threads and then sampled and analyzed replacements.",
"In the course of analyzing the two non-documentation samples, we also coded the purposes of those links.",
"In the Regex sample, the most common purposes of non-documentation links were referring to a (regex) tool (46), source code (19), or websites with posting recommendationsExamples: http://whathaveyoutried.com/ or http://sscce.org/ (16).",
"In the Android sample, the most common purposes were linking source code (28), an online tool (22, e.g., JSON or XML validators), or an image file (19, e.g., icons or screenshots).",
"Table: Features of Stack Overflow posts used as independent variables in the logistic regression models.Table REF shows the features used as independent variables in the logistic regression models.",
"The set of features consists of numeric features that can be extracted from the question, such as LengthText or CodeBlockCount.",
"Note that we excluded features that would be unknown at the time when the question was posted, such as how many views the question attracted or its score.",
"We retrieved the data for the features from the SOTorrent dataset, which contains the content of Stack Overflow posts separated into text and code blocks, collects links from posts and questions, and provides the metadata from the official Stack Overflow data dump.",
"For the textual features, shown in the bottom part of Table REF , we treated each token as a separate feature and used token frequency as feature values.",
"We separated text into tokens using whitespace, and we removed stopwordsWe used the “Long Stopword List” from https://www.ranks.nl/stopwords and punctuation as well as special characters.",
"All tokens were stemmed using the Porter stemming algorithm [41].",
"We discarded features consisting of a single character such as a single digit, and we limited the set of features to tokens whose frequency in our dataset exceeded a minimum threshold.",
"We used the goodness of fit (measured using McFadden's pseudo-$R^2$ [42]) to determine the best threshold for each dataset, resulting in a threshold of 15 for the Regex dataset (McFadden's pseudo-$R^2$ = 0.549) and 22 for the Android dataset (McFadden's pseudo-$R^2$ = 0.592).",
"This led to a total of 138 features for the Regex dataset and 203 features for the Android dataset.",
"Table REF shows the number of features resulting from each textual property.",
"The interpretation of logistic regression models may be misleading if the metrics that are used to construct them are correlated [43].",
"As Table REF shows, some of our features are likely to be correlated, e.g., LineCountText and LengthText.",
"To mitigate correlated metrics, we used AutoSpearman [44], an automated metric selection approach based on correlation analyses, with a threshold of 0.7.",
"Following the advice of Tantithamthavorn and Hassan [43], we used ANOVA Type-II importance scores to interpret our logistic regression models after constructing them using the glm function in R. Table: Most important features for explaining whether a Stack Overflow question related to regular expressions will attract a particular type of documentation link.",
"All p-values << 0.001.Table: Most important features for explaining whether a Stack Overflow question related to Android will attract a particular type of documentation link.",
"All p-values << 0.001.We built logistic regression models for specific types of documentation resources.",
"Note that we do not treat type of documentation resource as a categorical variable since posts can contain links to multiple different documentation resources.",
"While we did not have enough data to allow the construction of models for all types of resources, Tables REF and REF show the five most important features (as determined by the ANOVA Type-II test) for a subset of resource types for the Regex and Android datasets.",
"Table REF indicates that Regex questions about parsing and patterns are associated with a higher chance of attracting a link to Wikipedia.",
"In contrast, questions about specific problems are associated with a lower likelihood.",
"For Android, questions about devices are associated with a higher chance of attracting Wikipedia links while questions about converting are associated with attracting links to the JavaAPI.",
"As shown in Table REF , links to the AndroidReference documentation are associated with questions asked by users with a higher reputation.",
"Interestingly, a manual inspection of the corresponding questions suggests that many of these high-reputation users are outsiders whose expertise is in areas other than Android.",
"We provide an interpretation of these results as part of the discussion in the next section.",
"Ultimately, such models could be used to recommend the inclusion of different types of links in Stack Overflow posts."
],
[
"Findings",
"Our systematic analysis of the context (RQ1) and purpose (RQ2) of documentation links led to five major findings about linking practices on Stack Overflow.",
"In this section, we detail the evidence for each finding and discuss its main implication.",
"Figure: Relationships between findings about linking practices on Stack Overflow."
],
[
"Purpose Spectrum",
"Our qualitative analysis has shown that documentation links on Stack Overflow serve a variety of purposes.",
"Figure REF shows a rich diversity of purposes with eight of ten categories showing relative frequency above 5%.",
"Manually reviewing all the links (through the coding process) also showed that the different categories of link purposes can be positioned on a spectrum bounded by the concepts of Citation and Recommendation, where citations are not meant to be consulted whereas recommendations are explicit entreaties to follow the link.",
"Figure REF positions every link purpose category except for Other along this axis.",
"Figure: Purpose codes arranged on the purpose spectrum from citation to recommendation.Citation links include the ones labeled as Attribution and LinkedMention.",
"The purpose of Attribution links is to credit the source of content copied into Stack Overflow, which can help users meet Stack Overflow's requirement for attribution [45].",
"The purpose of the LinkedMention links is to uniquely identify a software artifact without the need to provide further context.",
"Often, users add such LinkedMention references as inline links, which underlines their peripheral role: Is there a regex that would work with String.split() to break a String into contiguous characters...?",
"[46] We place Consulted and Concept in the middle of the spectrum because they are open to interpretation.",
"Consulted links are typically added for context, but in some cases this context is simply to show due diligence (closer to citation) and in some cases it is to point to an unclear document to be explained, e.g.,: I am trying to understand the regular expression in Solr and came across this Java doc where explains... having a hard time understanding what it really means.",
"[47] As for Concept links, they are useful for readers who want to learn more about a mentioned concept, but they are usually also peripheral to the actual content of the post or comment (reproduced from Section ).",
"I think you're using * as if it's the Kleene star, not * as Java, JavaScript, & co. interpret * in regexps.",
"[38] Closer towards Recommendation we place Awareness links that steer users' attention towards related resources, without particularly endorsing them, as well as Reference links that users include to make statements verifiable and more trustworthy by pointing to documentation resources supporting their claims.",
"One purpose of links towards the Recommendation end of the spectrum is to explicitly guide readers to BackgroundReading.",
"Such links are especially helpful for users who are new to a topic or domain since they support them in identifying relevant background knowledge: There is a good detailed description of lookarounds (look-behind and look-ahead) as well as a lot of other regex “magic” here[48] Finally, we find explicit Recommendation links.",
"They allow readers to retrieve a specific software component recommended by a Stack Overflow author using the provided link (reproduced from Section ).",
"You could use Apache Commons Lang for that... [39] Implication: Forum users add links to documentation for a variety of purposes.",
"This purpose may not be clear to the reader.",
"Links whose purpose is not clear may confuse or waste the time of inexperienced users, who are surmised to visit more links as they navigate web sites [49].",
"Automated analysis of link data (e.g., [8]) may miss opportunities for additional interpretation if link purpose is not taken into account.",
"In the two cases we studied, mined association rules show consistent relations between a resource type (e.g., Wikipedia, Stack Overflow) and a link's purpose.",
"Links to Wikipedia, for example, often serve to define Concepts, an observation consistent with previous work [9].",
"Links to the documentation of software components and tools are often included to recommend the tool rather than to refer to the linked document specifically (Recommendation).",
"Implication: For technology domains where certain resource types can be strongly associated with a link purpose, it may be possible to automatically recommend links to enhance a post, or infer the purpose of a linked resource.",
"The distribution of link purposes shown in Figure REF and detailed in Tables REF –REF shows remarkable consistency between cases except for two major differences: Purpose code Concept is about twice as common in case Regex and purpose code Awareness is about twice as common in case Android.",
"For the other codes, the relative frequency differs not more than 3 percentage points.",
"Both differences mentioned above are significant at the level $\\alpha =0.01$ (see Section ).",
"From this we conjectured that the higher proportion of Concept links is explained by the theoretical nature of the domain, which involves concepts such as “parsing”, “context-free grammar”, “pattern”, etc.",
"This observation is corroborated by the regression model, which shows that one of the dominant features for explaining whether a Stack Overflow question related to regular expressions will attract a particular type of documentation link include such theoretical concepts, namely “parsing” and “pattern”.",
"As for Android, the extensive use of links for Awareness purposes can be explained by the size of this technology ecosystem.",
"As mentioned above, we added the documentation resource Youtube while adapting the classifier for the Android case.",
"This is another manifestation of domain-specific link usage, because in the Regex case, only 26 posts pointed to Youtube (0.09% of all posts containing links), while in the Android case, linking Youtube videos was much more common (1,822 posts or 0.8% of all posts containing links).",
"The difference was significant according to a two-tailed Fisher's exact test [36] with a significance level of $\\alpha =0.01$ ($\\text{p-value}<2.2\\times 10^{-16}$ ).",
"Typical use cases of linking Youtube videos include pointing to tutorialsExample tutorial: https://youtu.be/fn5OlqQuOCk or conference talks.Example conference talk: https://youtu.be/N6YdwzAvwOA Implication: Links to documentation resources are a reflection of the information needs typical to a technology domain.",
"Details on the distribution of purpose links for a domain can thus assist in the design of documentation.",
"Even though Stack Overflow encourages users to provide context for links [10], they are rarely accompanied by a Quote [28] or a Summary.",
"Our analysis shows that, for 72.8% of the analyzed links, authors did not provide a quote and for 83.3% of the links they did not provide a summary.",
"Although in some situations this lack of context may render links worthless once their target is unavailable, our analysis also revealed valid use cases for links without context, as links at the Citation end of the purpose spectrum do not necessarily need context.",
"However, links towards the Recommendation end of the spectrum should always be accompanied by additional information to preserve that information in case the linked resources becomes unavailable.",
"Implication: Our link Purpose Spectrum observation allows us to modulate the requirement to add context for links, given that our data shows the context to be self-explanatory for links whose purpose is akin to a citation.",
"We hypothesize that the importance of context for orienting users is proportional to a link's position on the purpose spectrum.",
"Missing context is thus not necessarily a problem for links whose purpose is citation.",
"The logistic regression analysis shows that users with a high reputation score are not necessarily more familiar with reference documentation than lower reputation users.",
"Links to the AndroidReference documentation are associated with questions asked by users with a higher reputation.",
"The median user reputation of users asking questions which attract links to the AndroidReference documentation in the dataset used for the logistic regression analysis is $1063.5$ , while the corresponding median for the remaining questions is 86.",
"A manual inspection of the corresponding questions suggests that many of these high-reputation users are outsiders whose expertise is, based on the questions they typically answer, in areas other than Android (often iOS).",
"Similarly, links to Wikipedia are also associated with questions asked by users with a higher reputation.",
"Implication: In previous research efforts, researchers have often treated an individual's reputation on Stack Overflow as a proxy for this individual's general programming knowledge (e.g., [50]).",
"Our results indicate that this operationalization may not be valid in all scenarios, because Stack Overflow authors' knowledge is domain-specific.",
"The findings described in the previous paragraphs build on each other to form a small conceptual framework defined in terms of logical implications.",
"Figure REF summarizes the findings and their relationships.",
"Our primary finding concerns the variety of linking purposes we elicited and the observation that linking purpose types span a spectrum that characterizes to what extent a link is intended to be followed (Purpose Spectrum).",
"We also collected evidence of a notable correspondence between a resource type (e.g., Wikipedia) and a link's purpose (Purpose–Resource Correspondence), and that link usage may be specific to a technology domain (Domain-Specific Link Usage).",
"Both of these observations are consequences of Purpose Spectrum in the sense that it is the observed richness of linking purposes that enables the elicitation of specific linking practices.",
"A fourth observation is the extent to which links in Stack Overflow threads lack context, despite the presence of guidelines explicitly requesting such context (Missing Link Context).",
"To a certain extent, this observed problem can be mitigated by Purpose–Resource Correspondence because this correspondence supports partial inference of a link's purpose.",
"Finally, our analysis reveals a pattern that would be counter-intuitive at first glance: users with high reputation attract answers with links to the reference documentation, which can also be construed a symptom of lack of expertise (Reputation-Expertise Mismatch).",
"This finding is enabled by the Purpose–Resource Correspondence which relates links to documentation resources with a type of information need."
],
[
"Threats to Validity",
"The external validity of our results may be limited due to our choice of the two specific domains Regex and Android, both of which were taken from the Java domain.",
"While Java is one of the most popular programming languages today,TIOBE Index for December 2019, https://www.tiobe.com/tiobe-index/, verified 16 December 2019. its documentation ecosystem may differ from other languages.",
"The documentation resources we identified, such as API documentation [51] and Wikipedia [9], are, however, likely to also play an important role for other languages and domains.",
"Another threat is that our URL mapper was only able to classify 78.5% of all active links in the Regex sample and 68.9% of all active links in the Android sample (see Section ).",
"Note that a classification of the remaining links would only add more documentation resources, but not invalidate the ones we have already identified.",
"Also, the number of posts containing a link to the corresponding root domain considerably drops after the top five (Regex) respectively the top six (Android) root domains (see Figure REF ).",
"This also means that the marginal profit of analyzing additional root domains drops considerably after analyzing the most frequently referenced root domains.",
"While iteratively building the classifier, two authors continuously discussed the emerging documentation resource categories and corresponding sub-pages of the root domains.",
"Since we followed a whitelisting approach based on regular expressions matching certain sub-paths of the domains, making all decisions unanimously, the false positive rate of our approach is very low.",
"We may, however, have missed certain sub-paths, marking them as NotDocumentation when they were in fact documentation resources (false negatives).",
"We mitigated this bias by manually inspecting the links marked as NotDocumentation after each iteration, filtering out links that clearly did not point to documentation resources until no links were left to analyze.",
"The stratified sampling strategy we used to select documentation links for our analyses represents a threat to the external validity of our results.",
"Note that in a random sample, the top three documentation resources, especially the internal Stack Overflow links, would overshadow the less frequent documentation resources (see Table REF ).",
"Our sampling strategy allowed us to analyze a broader and more diverse sample of documentation resources not dominated by those very frequent link targets.",
"In the association rule analysis we conducted, support and confidence only hold for our samples—they would differ in non-stratified samples.",
"In Section , we described how to interpret those values considering the stratification.",
"Moreover, the fact that all rules derived from the Regex sample were also present in the Android sample further supports their credibility.",
"The purpose distribution would likely differ in a random sample.",
"However, in a random sample, frequently referenced documentation resources such as Stack Overflow, JavaAPI, and AndroidAPI would dominate the analysis.",
"The stratification allowed us to consider a more diverse range of resources and purposes.",
"Qualitative data analysis always depends on the imagination and perception of the researcher.",
"To mitigate this threat, all three authors conducted the qualitative analysis independently.",
"We coded links in sets of up to 100 links and thoroughly discussed our results after finishing each set.",
"After assessing our inter-rater agreement ($\\kappa $ values between 0.65 and 0.80, see Table REF ), we only assigned a code if at least two researchers agreed on it."
],
[
"Conclusion",
"Over the past decade, the community question answering platform Stack Overflow has become extremely popular among programmers for finding and sharing knowledge.",
"However, the site does not exist in isolation, and users frequently link to other documentation sources, such as API documentation and encyclopedia articles, from within questions, answers, or comments on Stack Overflow.",
"To understand how and why documentation is referenced from Stack Overflow threads, we conducted a multi-case study of links in two different technology domains, regular expressions and Android development.",
"We used qualitative and quantitative research methods to systematically investigate the context and purpose of a sample of 759 documentation links.",
"We identified a spectrum of purposes for which links are included in Stack Overflow threads, ranging from Attribution and LinkedMention on the citation end of the spectrum to BackgroundReading and Recommendation of software artifacts on the recommendation side.",
"Citations are not necessarily meant to be consulted whereas recommendations are explicit requests to follow a link.",
"This observation relates to Stack Overflow's recommendation to add context to every link: While adding context in the form of summaries or quotes is important for links on the recommendation end of the purpose spectrum, it is less important for links primarily included for citation purposes.",
"We also found that links to documentation resources are a reflection of the information needs typical to a technology domain.",
"For example, Concept links were twice as common in threads about regular expressions compared to Android, while we found the opposite for Awareness links.",
"These insights can inform the design and customization of documentation for different technology domains.",
"Our work forms a first step towards understanding how and why documentation resources are referenced on Stack Overflow, with the ultimate goal of improving the efficiency of information diffusion between Stack Overflow and the broader software documentation ecosystem, as motivated in Section .",
"In the short term, Stack Overflow authors can use our results to reflect on the intended purpose before posting a link, and to learn how they can make their post more valuable by providing context.",
"Another direction for future work is developing tool support for guiding Stack Overflow users to enhance (potential) information diffusion.",
"One tool could assist readers of Stack Overflow threads by automatically classifying links in posts or comments along the purpose spectrum we presented in this paper.",
"Such a tool could be implemented as a browser plugin visualizing the determined purpose of the link, helping users to judge whether the link it is worth following based on their particular needs.",
"Another idea is to extend the models we presented in Section to be able to recommend Stack Overflow authors to include a certain type of link while creating or revising Stack Overflow posts.",
"[Figure: NO_CAPTION [Figure: NO_CAPTION [Figure: NO_CAPTION"
]
] | 1906.04357 |
[
[
"Continual Reinforcement Learning deployed in Real-life using Policy\n Distillation and Sim2Real Transfer"
],
[
"Abstract We focus on the problem of teaching a robot to solve tasks presented sequentially, i.e., in a continual learning scenario.",
"The robot should be able to solve all tasks it has encountered, without forgetting past tasks.",
"We provide preliminary work on applying Reinforcement Learning to such setting, on 2D navigation tasks for a 3 wheel omni-directional robot.",
"Our approach takes advantage of state representation learning and policy distillation.",
"Policies are trained using learned features as input, rather than raw observations, allowing better sample efficiency.",
"Policy distillation is used to combine multiple policies into a single one that solves all encountered tasks."
],
[
"Introduction",
"In realistic real-life reinforcement learning scenarios, involving for instance service robots, tasks evolve over time either because the context of one task changes or because new tasks appear [7].",
"Our end goal is therefore to have an embodied agent in real-life that learns incrementally as time passes.",
"One example would be a robot tasked with wrapping gifts.",
"Most gifts are rectangular packages (cuboids), so the robot would first learn to wrap cuboids.",
"Then if a soccer ball appears, the robot would have to learn how to wrap a sphere while still being able to wrap cuboids.",
"Indeed, the robot should add additional knowledge to his repertoire.",
"Moreover, even if it would be easier to learn how to wrap spheres and cuboids before test time, there are potentially many other shapes that have to be considered, and thus, learning continually seems more natural and convenient than trying to learn all skills at once.",
"Continual Learning (CL) and State Representation Learning (SRL) are essential to build agents that face such challenge.",
"SRL allows to build strong representations of the world since agents should be able to understand their surroundings, and extract general concepts from sensory inputs of complex scenes.",
"An agent better sees a chair as an object, not as a bunch of pixels together in an image.",
"CL allows to learn such representation without forgetting in settings where the distribution of data change through time and is needed for agents that learn in the real-world and are required to adapt to changes.",
"Combining CL and SRL would then allow to create strong representation robust to catastrophic forgetting.",
"Reinforcement learning (RL) is a popular approach to learn robot controllers that also has to face the CL challenges, and can take advantage of SRL to learn faster and to produce more robust policies.",
"Therefore we perform our experiments (Fig.",
"REF ) in a setup where tasks are encountered sequentially and not all at once.",
"Note that it differs from a setting where we can pick and shuffle experiences, often encountered in the multi-task RL literature (cf Section REF ).",
"Figure: Having access to task 1 only first, and then task 2 only, we learn a single policy that solves two real-life navigation tasks using policy distillation and sim2real transfer.In our approach we aim to take advantage of simulations to create this scenario.",
"We demonstrate that deploying a policy in real-life which has continually learned two tasks in simulation is successful with our approach.",
"Our contribution consists on applying two major paradigms for robotics in real life: a state representation learning approach for compact and efficient representation that facilitates learning a policy, and a policy that learns continually in a sequential manner.",
"The approach is deployed in a real robot thanks to policy distillation and sim2real transfer.",
"Furthermore, in opposition to most methods in reinforcement learning, the solution we propose does not need a task indicator at test time.",
"Indeed, the information about the task to be solved can be found from a different color tag in the image.",
"The rest of the article is structured as follows.",
"Section introduces state representation learning, multi-task RL and continual learning paradigms in an RL setting, Sec.",
"details the robotics settings and tasks performed; Sec.",
"details the methods utilized and Sec.",
"concludes with future insights from our experiments."
],
[
"State representation learning (SRL)",
"Scaling end-to-end reinforcement learning to control real robots from vision presents a series of challenges, in particular in terms of sample efficiency.",
"Against end-to-end learning, SRL [15] can help learn a compact, efficient and relevant representation of states.",
"Previous works such as [9], [40], [38], [13], [31], [35], [25] have shown that SRL can speed up policy learning, reducing the number of samples needed while additionally being easier to interpret."
],
[
"Multi-task RL",
"Multi-task RL aims at constructing one single policy module that can solve a number of different tasks.",
"The CURIOUS algorithm [6] selects through exploration the tasks to be learned that improve an absolute learning progress metric the most.",
"Policy distillation [28] can also be used to merge different policies into one module/network.",
"The Distral algorithm [34] is one successful example of such approach: a shared policy distills common behaviours from task-specific policies.",
"Then, the distilled policy is used to guide task-specific policies via regularization using a Kullback-Leibler (KL) divergence.",
"Other approaches like SAC-X [27] or HER [2] take advantage of Multi-task RL by learning auxiliary tasks in order to help the learning of an objective task."
],
[
"Continual Learning",
"Continual learning (CL) is the ability of a model to learn new skills without forgetting previous knowledge.",
"In our context, it means learning several tasks sequentially and being able to solve any task at the end of the sequence.",
"This differs from the easier multi-task scenario, where tasks can be experienced all at once.",
"Most CL approaches can be classified into four main methods that differ in the way they handle the memory from past tasks.",
"The first method, referred to as rehearsal, keeps samples from previous tasks [26], [21].",
"The second approach consists in applying regularization, either by constraining weight updates in order to maintain knowledge from previous tasks [12], [42], [19], or by keeping an old model in memory and distilling knowledge [11] into it later to remember [18], [30], [28].",
"The third category of strategies, dynamic network architectures, maintains past knowledge thanks to architectural modifications while learning [29], [8], [18], [8].",
"The fourth and more recent method is generative replay [32], [14], [41], [16], where a generative model is used as a memory to produce samples from previous tasks.",
"This approach has also been referred to as pseudo-rehearsal.",
"Note that all four type of methods can used for classification as well as for generation."
],
[
"RL in real-life",
"Applying RL to real-life scenarios is a major challenge that has been studied widely.",
"Most attempts fall into two categories: games and robotics.",
"For games, AlphaGo Zero [33] has mastered the game of Go from scratch without any human supervision by combining RL, self-play and Monte Carlo Tree Search [4].",
"AlphaStar [39] and OpenAI Five [22] were both able to get competitive results against professional human players on the game Starcraft and DOTA2, respectively.",
"Both solutions are based on RL, and current research is still investigating how to master the game with the same constraints as humans (e.g.",
"same FPS).",
"In robotics, there is a plethora of successful attempts at deploying RL on real robots.",
"One common approach is training policies in simulation and then deploying them in real-life hoping that they will successfully transfer, considering the gap in complexity between simulation and the real world.",
"Such approaches are termed Sim2Real [10], and have been successfully applied [5], [20] in many scenarios.",
"In order to cope with the unpredictable nature of the real world, one can use Domain Randomization [36], which we use in our approach.",
"This technique trains policies in numerous simulations that are randomly different from each other (different background, colors, etc.).",
"Using this technique, the transfer to real life is easier.",
"Others have tried to train a policy directly on real robots, facing the hurdle of the lack of sample efficiency that RL suffers from.",
"SAC-X [27] is one example where a successful policy is learned directly on the real robot.",
"One can also find applications of reinforcement learning in other domains: TacTex'13 [37], relying on online RL, is an autonomous broker agent that maximizes profit through energy trading; and [17] propose using policy gradients for dialogue generation using a set of reward functions designed to increase the diversity and length of generated responses.",
"In education, a faster teaching policy by POMDP (partially-observable Markov decision process) planning [23] leverages a probabilistic learner model in order to achieve a long-term teaching objective.",
"In the literature, most approaches focus on the single-task or simultaneous multi-task scenario.",
"In this paper, we attempt to train a policy on several tasks sequentially and deploy it in real life.",
"Hence, we attempt to apply RL in real life in a continual learning setting."
],
[
"Methods",
"In this section we present the method proposed to combine state representation learning (SRL) and continual learning (CL) in a real life reinforcement learning setting.",
"First we present how a single task is learned and how the SRL part works, secondly we explain how to learn continually and thirdly, we explain how we evaluate learning in the different phases of the learning sequence."
],
[
"Learning on one task",
"arrows decorations.markings Figure: SRL Combination model: combines the prediction of an image II's reconstruction loss and an inverse dynamic model loss in a state representation ss.",
"Arrows represent inference, dashed frames represent losses computations, rectangles are state representations, circles are real observed data, and squares are model predictions, tt represents the timestepEach task $i$ is learned following to procedure we describe here.",
"First, as we use an SRL approach, we need to learn a state representation encoder.",
"We sample data from the environment $Env_i$ (where $i$ refers to the task) with an agent guided by a random policy.",
"We call this dataset $D_{Random,i}$ .",
"$D_{Random,i}$ is then used to train an SRL model composed of an inverse model and an auto-encoder.",
"This architecture is inspired from [25], and illustrated in Fig.",
"REF .",
"Figure: Summary of the experimental setup.",
"Step 1 and 2 correspond to learning policies for task 1 and 2, and using those to create distillation datasets.",
"Step 3 is the distillation of the two policies into a single policy which can be deployed in simulation and on the real robot.Once the SRL model is trained, we use it's encoder $E_i$ to provide with features for learning a reinforcement learning policy $\\pi _i$ with the model $M(\\theta )$ ($\\theta $ represents the model parameters).",
"Once $\\pi _i$ is learned, we use it to generate sequences of on-policy observations with associated actions, which will eventually be used for distillation (Fig.",
"REF , right).",
"We call this the distillation dataset $D_{\\pi i}$ .",
"We generate $D_{\\pi i}$ the following way: we randomly sample a starting position and then let the agent generate a trajectory.",
"At each step we save both the observation and associated action.",
"We stop the sequence when enough reward is gathered (see Section ).",
"From each task is only kept the dataset $D_{\\pi _i}$ .",
"As soon as we change task, $D_{Random,i}$ and $Env_i$ are not available anymore.",
"In our setting, in order to decrease training time, we generate $D_{Random,i}$ in simulation and learn $\\pi _i$ also in simulation.",
"However, at the end of $T$ tasks, $\\pi _{D_0,...,D_{T-1}}$ are tested in a real robot.",
"In order to pass the reality gap, the datasets generated are augmented with different luminosity variations."
],
[
"Learning continually",
"To learn continually we use a distillation algorithm [28].",
"Once we learned several tasks, we can aggregate the distillation datasets $D_{\\pi _i}$ and distill the knowledge into a new model $M_Di(\\theta ^{\\prime })$ to produce a single flexible policy (Fig.",
"REF , right).",
"$\\theta ^{\\prime }$ are the parameters of the distillation model.",
"The distillation consists in learning in a supervised fashion the action probability associated to an observation at a timestep $t$ .",
"Each dataset $D_{\\pi _i}$ allows to distill the policy $\\pi _i$ into a new network.",
"We name the distilled policy $\\pi _{D_i}$ .",
"With the aggregation of several distillation datasets, we can distill several policies into the same network.",
"By extension of the previous nomenclature we call a model where policy 1 and policy 2 have been distilled in, $\\pi _{D_{1,2}}$ .",
"At test time, we do not need a task indicator; however, we assume that the observations and state space visually allow to recognize the current task.",
"In the context of continual RL, the task signal is mandatory if the observation does not give any clue about the policy to be run.",
"In our setting, as the policy can be inferred from a different color target tag, we do not need it.",
"The method presented allows to learn continually several policies without forgetting.",
"On the other hand, $M(\\theta )$ also learn on the sequence of task but without any memorization mechanism, its leads to catastrophic forgetting.",
"The dataset $D_{\\pi _i}$ contains 10k samples per task, which allows to learn the distillation very quickly (a few minutes are needed to learn $\\pi _{D_i}$ while several hours are needed to learn $\\pi _i$ )."
],
[
"Evaluation",
"The main evaluation is the performance of the single and final policy, which can supposedly achieve all previous tasks, as well as being deployed in real life.",
"For that, we report the mean and standard error on 5 runs of the policy on each task in simulation REF , and provide videos to show the behaviour of the final policy.",
"On the other hand we also would like to analyze the learning process.",
"In order to have an insight on the evolution of the distilled model, we save distillation datasets at different checkpoints in the sequence of tasks.",
"Those checkpoints are saved regularly during the RL training.",
"By distilling and evaluating at several time steps, we are able to evaluate the evolution of learning and forgetting on all environments, both separately and jointly.",
"At each checkpoint, we evaluate the actual policy $\\pi _i$ on past tasks to assess forgetting and compare it to $\\pi _{D_{0,..,t}}$ .",
"It is important to note that, even if we consider $Env_i$ as not available anymore at task $i+1$ , we did use it for evaluation purposes at any time.",
"Figure: Comparison between performance (normalized mean reward and standard error) of policy trained on one task only to distilled student policy on the two tasks.",
"The student policy has similar performance on both tasks.",
"Left: Target Reaching (TR).",
"Right: Target Circling (TC) task."
],
[
"Experimental setup",
"We apply our approach to learn continually two 2D navigation tasks on a real mobile robot."
],
[
"Robotic setup",
"The experiments consists of 2D navigation tasks using a 3 wheel omni-directional robot.",
"It is similar to the 2D random target mobile navigation ([24], identical reward setting and possibility of movement).",
"The robot is identified by a black QR code and the scene is recorded from above.",
"We are able to simulate the experiment, since the robot's input is a fixed RGB image of the scene recorded from above.",
"The robot uses 4 high level discrete actions (move left/right, move up/down in a cartesian plane relative to the robot) rather than motor commands.",
"The room where the real-life robotic experiments are to be performed is subject to illumination changes.",
"The input image is a top-down view of the floor, which is lighted by surroundings windows and artificial illumination of the room.",
"Hence, the illumination changes depending on the weather and time of the day.",
"We use domain randomization [36] to improve the chances of the policies learned in simulation to better transfer to the real world, by being robust to weather and time conditions.",
"During RL training, at each timestep, the color of the background is randomly changed."
],
[
"Continual learning setup",
"Our continual learning scenario is composed of two similar environments, where the robot is rewarded according to the associated task.",
"In both environments, the robot is free to navigate for up to 250 steps, performing only discrete actions within the boundaries identified by a red square.",
"In environment 1, the robot gets at each timestep $t$ a positive reward +1 for reaching the target identified by a red square marker (task 1), a negative reward $R_{t, bump}=-1$ for bumping into the boundaries, and no reward otherwise.",
"In environment 2, the robot gets at each timestep $t$ a reward $R_t$ (where $z_t$ is the 2D coordinate position with respect to the center of the circle, see eq.",
"REF ), which is highest when the agent is both keeping a distance to the target equal to a radius $r_{circle}$ (see eq.",
"REF ), and has been moving for the previous $k$ steps (see eq.",
"REF ).",
"An additional penalty term $R_{t, bump}=-1$ is added to the reward function in case of bump with the boundaries, and a coefficient $\\lambda =10$ is introduced to balance the behaviour.",
"$R_t$ is designed for agents to learn the task of circling around a central blue tag (task 2).",
"$R_{t, circle} = 1 - \\lambda (\\Vert z_t\\Vert - r_{circle}) ^2$ $R_{t, movement} = \\Vert z_t -z_{t-k} \\Vert _{2}^2$ $R_t = \\lambda R_{t, circle} * R_{t, movement} + \\lambda ^ 2 R_{t, bump}$ It is important to note that as the tags associated to each scenario's target are of different color, the algorithm can automatically infer which policy it needs to run and thus, does not need task labels at test time.",
"Moreover, while generating on-policy datasets $D_\\pi 1$ (see Section REF ) for task 1, we allow the robot a limited number of contacts with the target to reach ($N_{contacts}=10$ ) in order to mainly preserve the frames associated with the correct reaching behaviour.",
"There are no such additional constraints when recording for task 2, the limit is the standard episode size, i.e.",
"250 time-steps.",
"The main software related to our experimental setting can be found at the url: https://github.com/kalifou/robotics-rl-srl/tree/circular_movement_omnibot"
],
[
"Main result",
"Our main result is the continual learning of a single policy that solves both tasks in simulation, as presented in Fig.",
"REFThe deployment and evaluation in real life is part of future work.",
"The two teacher policies are learnt separately (i.e.",
"independently) on each environment.",
"Then, distillation is used to combine the two teacher policies into a single policy that can solve the two tasks.",
"Fig.",
"REF demonstrates the efficiency of our approach.",
"We can see that the single student distilled policy achieves close to maximum reward in both tasks."
],
[
"Evaluation of distillation",
"We performed a more explicit evaluation of distillation in the task 2 (target circling (TC) around).",
"While we train a policy using RL, we save the policy every 200 episodes (50K timesteps), and distill it into a new student policy which we test.",
"This is illustrated in Fig.",
"REF .",
"Both curves are very close, which indicates distillation works as intended.",
"It is able to transfer a policy using only a limited distillation dataset, with limited loss in the policy performance.",
"Figure: Demonstration of the effectiveness of distillation.",
"Blue: RL training curve of PPO2 on target circling task.",
"Green: Mean and std performance on 8 seeds of distilled student policy.",
"The blue policy is distilled into a student policy at regular time-step (1 episode = 250 timesteps).",
"Both curves are very close, which indicates distillation works as intended."
],
[
"Discussion and future work",
"Our work is preliminary and offers many possibilities for improvement.",
"Our roadmap include having not only a policy learned in a continual way, but also the SRL model associated.",
"We would need to update the SRL model as new tasks are presented sequentially.",
"One possible approach would be to use Continual SRL methods like S-TRIGGER [3] or VASE [1].",
"We also expect to encounter issues when scaling continual learning approaches to more tasks or environments.",
"Indeed, the agent should not accumulate knowledge blindly, but rather make connections between different types of information (i.e.",
"generalize) and/or selectively forget non-useful knowledge.",
"Moreover, we intend to soon provide with supplementary quantitative results and videos of these tasks deployed in the real-life setup.",
"We would like to train policies directly on the real robot, as it is the end goal scenario for this research.",
"One promising approach would be to use model-based RL while learning the SRL modelto improve sample efficiency.",
"The final goal would be to learn the policy on a real robot in a reasonable amount of time."
],
[
"Conclusion",
"In this paper we provide preliminary results towards a proper real life continual learning setup, where a real robot would encounter tasks presented in a sequence and be asked to accumulate knowledge in a scalable manner.",
"The building blocks for achieving a single policy that solves all presented tasks consists of RL that uses state representation learning models, and distillation into a single policy.",
"This model shows to be a good candidate for transfer to real life and future work should evaluate it in more and more complex tasks."
],
[
"Acknowledgement",
"This work is supported by the EU H2020 DREAM project (Grant agreement No 640891)."
]
] | 1906.04452 |
[
[
"Identifying Visible Actions in Lifestyle Vlogs"
],
[
"Abstract We consider the task of identifying human actions visible in online videos.",
"We focus on the widely spread genre of lifestyle vlogs, which consist of videos of people performing actions while verbally describing them.",
"Our goal is to identify if actions mentioned in the speech description of a video are visually present.",
"We construct a dataset with crowdsourced manual annotations of visible actions, and introduce a multimodal algorithm that leverages information derived from visual and linguistic clues to automatically infer which actions are visible in a video.",
"We demonstrate that our multimodal algorithm outperforms algorithms based only on one modality at a time."
],
[
"Introduction",
"There has been a surge of recent interest in detecting human actions in videos.",
"Work in this space has mainly focused on learning actions from articulated human pose [10], [47], [54] or mining spatial and temporal information from videos [39], [48].",
"A number of resources have been produced, including Action Bank [36], NTU RGB+D [37], SBU Kinect Interaction [53], and PKU-MMD [26].",
"Most research on video action detection has gathered video information for a set of pre-defined actions [11], [31], [23], an approach known as explicit data gathering [14].",
"For instance, given an action such as “open door,” a system would identify videos that include a visual depiction of this action.",
"While this approach is able to detect a specific set of actions, whose choice may be guided by downstream applications, it achieves high precision at the cost of low recall.",
"In many cases, the set of predefined actions is small (e.g., 203 activity classes in [11]), and for some actions, the number of visual depictions is very small.",
"An alternative approach is to start with a set of videos, and identify all the actions present in these videos [6], [1].",
"This approach has been referred to as implicit data gathering, and it typically leads to the identification of a larger number of actions, possibly with a small number of examples per action.",
"In this paper, we use an implicit data gathering approach to label human activities in videos.",
"To the best of our knowledge, we are the first to explore video action recognition using both transcribed audio and video information.",
"We focus on the popular genre of lifestyle vlogs, which consist of videos of people demonstrating routine actions while verbally describing them.",
"We use these videos to develop methods to identify if actions are visually present.",
"The paper makes three main contributions.",
"First, we introduce a novel dataset consisting of 1,268 short video clips paired with sets of actions mentioned in the video transcripts, as well as manual annotations of whether the actions are visible or not.",
"The dataset includes a total of 14,769 actions, 4,340 of which are visible.",
"Second, we propose a set of strong baselines to determine whether an action is visible or not.",
"Third, we introduce a multimodal neural architecture that combines information drawn from visual and linguistic clues, and show that it improves over models that rely on one modality at a time.",
"By making progress towards automatic action recognition, in addition to contributing to video understanding, this work has a number of important and exciting applications, including sports analytics [12], human-computer interaction [30], and automatic analysis of surveillance video footage [20].",
"The paper is organized as follows.",
"We begin by discussing related work, then describe our data collection and annotation process.",
"We next overview our experimental set-up and introduce a multimodal method for identifying visible actions in videos.",
"Finally, we discuss our results and conclude with general directions for future work."
],
[
"Related Work",
"There has been substantial work on action recognition in the computer vision community, focusing on creating datasets [41], [22], [38], [11] or introducing new methods [17], [3], [9], [46].",
"Table REF compares our dataset with previous action recognition datasets.Note that the number of actions shown for our dataset reflects the number of unique visible actions in the dataset and not the number of action classes, as in other datasets.",
"This is due to our annotation process (see §).",
"The largest datasets that have been compiled to date are based on YouTube videos [11], [31], [23].",
"These actions cover a broad range of classes including human-object interactions such as cooking [34], [7], [35] and playing tennis [22], as well as human-human interactions such as shaking hands and hugging [15].",
"Similar to our work, some of these previous datasets have considered everyday routine actions [11], [31], [23].",
"However, because these datasets rely on videos uploaded on YouTube, it has been observed they can be potentially biased towards unusual situations [23].",
"For example, searching for videos with the query “drinking tea\" results mainly in unusual videos such as dogs or birds drinking tea.",
"This bias can be addressed by paying people to act out everyday scenarios [38], but this can end up being very expensive.",
"In our work, we address this bias by changing the approach used to search for videos.",
"Instead of searching for actions in an explicit way, using queries such as “opening a fridge” or “making the bed,” we search for more general videos using queries such as “my morning routine.” This approach has been referred to as implicit (as opposed to explicit) data gathering, and was shown to result in a greater number of videos with more realistic action depictions [14].",
"Although we use implicit data gathering as proposed in the past, unlike [14] and other human action recognition datasets, we search for routine videos that contain rich audio descriptions of the actions being performed, and we use this transcribed audio to extract actions.",
"In these lifestyle vlogs, a vlogger typically performs an action while also describing it in detail.",
"To the best of our knowledge, we are the first to build a video action recognition dataset using both transcribed audio and video information.",
"Another important difference between our methodology and previously proposed methods is that we extract action labels from the transcripts.",
"By gathering data before annotating the actions, our action labels are post-defined (as in [14]).",
"This is unlike the majority of the existing human action datasets that use pre-defined labels [38], [11], [31], [23], [15], [7], [35], [27].",
"Post-defined labels allow us to use a larger set of labels, expanding on the simplified label set used in earlier datasets.",
"These action labels are more inline with everyday scenarios, where people often use different names for the same action.",
"For example, when interacting with a robot, a user could refer to an action in a variety of ways; our dataset includes the actions “stick it into the freezer,” “freeze it,” “pop into the freezer,” and “put into the freezer,” variations, which would not be included in current human action recognition datasets.",
"In addition to human action recognition, our work relates to other multimodal tasks such as visual question answering [19], [49], video summarization [16], [40], and mapping text descriptions to video content [21], [33].",
"Specifically, we use an architecture similar to [19], where an LSTM [18] is used together with frame-level visual features such as Inception [43], and sequence-level features such as C3D [46].",
"However, unlike [19] who encode the textual information (question-answers pairs) using an LSTM, we chose instead to encode our textual information (action descriptions and their contexts) using a large-scale language model ELMo [29].",
"Similar to previous research on multimodal methods [24], [52], [50], [19], we also perform feature ablation to determine the role played by each modality in solving the task.",
"Consistent with earlier work, we observe that the textual modality leads to the highest performance across individual modalities, and that the multimodal model combining textual and visual clues has the best overall performance."
],
[
"Data Collection and Annotation",
"We collect a dataset of routine and do-it-yourself (DIY) videos from YouTube, consisting of people performing daily activities, such as making breakfast or cleaning the house.",
"These videos also typically include a detailed verbal description of the actions being depicted.",
"We choose to focus on these lifestyle vlogs because they are very popular, with tens of millions having been uploaded on YouTube; tab:nbresultssearchqueries shows the approximate number of videos available for several routine queries.",
"Vlogs also capture a wide range of everyday activities; on average, we find thirty different visible human actions in five minutes of video.",
"By collecting routine videos, instead of searching explicitly for actions, we do implicit data gathering, a form of data collection introduced by [14].",
"Because everyday actions are common and not unusual, searching for them directly does not return many results.",
"In contrast, by collecting routine videos, we find many everyday activities present in these videos.",
"Table: Approximate number of videos found when searching for routine and do-it-yourself queries on YouTube."
],
[
"Data Gathering",
"We build a data gathering pipeline (see Figure REF ) to automatically extract and filter videos and their transcripts from YouTube.",
"The input to the pipeline is manually selected YouTube channels.",
"Ten channels are chosen for their rich routine videos, where the actor(s) describe their actions in great detail.",
"From each channel, we manually select two different playlists, and from each playlist, we randomly download ten videos.",
"The following data processing steps are applied: Transcript Filtering.",
"Transcripts are automatically generated by YouTube.",
"We filter out videos that do not contain any transcripts or that contain transcripts with an average (over the entire video) of less than 0.5 words per second.",
"These videos do not contain detailed action descriptions so we cannot effectively leverage textual information.",
"Extract Candidate Actions from Transcript.",
"Starting with the transcript, we generate a noisy list of potential actions.",
"This is done using the Stanford parser [4] to split the transcript into sentences and identify verb phrases, augmented by a set of hand-crafted rules to eliminate some parsing errors.",
"The resulting actions are noisy, containing phrases such as “found it helpful if you” and “created before up the top you.” Segment Videos into Miniclips.",
"The length of our collected videos varies from two minutes to twenty minutes.",
"To ease the annotation process, we split each video into miniclips (short video sequences of maximum one minute).",
"Miniclips are split to minimize the chance that the same action is shown across multiple miniclips.",
"This is done automatically, based on the transcript timestamp of each action.",
"Because YouTube transcripts have timing information, we are able to line up each action with its corresponding frames in the video.",
"We sometimes notice a gap of several seconds between the time an action occurs in the transcript and the time it is shown in the video.",
"To address this misalignment, we first map the actions to the miniclips using the time information from the transcript.",
"We then expand the miniclip by 15 seconds before the first action and 15 seconds after the last action.",
"This increases the chance that all actions will be captured in the miniclip.",
"Motion Filtering.",
"We remove miniclips that do not contain much movement.",
"We sample one out of every one hundred frames of the miniclip, and compute the 2D correlation coefficient between these sampled frames.",
"If the median of the obtained values is greater than a certain threshold (we choose 0.8), we filter out the miniclip.",
"Videos with low movement tend to show people sitting in front of the camera, describing their routine, but not acting out what they are saying.",
"There can be many actions in the transcript, but if they are not depicted in the video, we cannot leverage the video information.",
"Figure: Overview of the data gathering pipeline.Figure: Sample video frames, transcript, and annotations."
],
[
"Visual Action Annotation",
"Our goal is to identify which of the actions extracted from the transcripts are visually depicted in the videos.",
"We create an annotation task on Amazon Mechanical Turk (AMT) to identify actions that are visible.",
"We give each AMT turker a HIT consisting of five miniclips with up to seven actions generated from each miniclip.",
"The turker is asked to assign a label (visible in the video; not visible in the video; not an action) to each action.",
"Because it is difficult to reliably separate not visible and not an action, we group these labels together.",
"Each miniclip is annotated by three different turkers.",
"For the final annotation, we use the label assigned by the majority of turkers, i.e., visible or not visible / not an action.",
"To help detect spam, we identify and reject the turkers that assign the same label for every action in all five miniclips that they annotate.",
"Additionally, each HIT contains a ground truth miniclip that has been pre-labeled by two reliable annotators.",
"Each ground truth miniclip has more than four actions with labels that were agreed upon by both reliable annotators.",
"We compute accuracy between a turker's answers and the ground truth annotations; if this accuracy is less than 20%, we reject the HIT as spam.",
"After spam removal, we compute the agreement score between the turkers using Fleiss kappa [13].",
"Over the entire data set, the Fleiss agreement score is 0.35, indicating fair agreement.",
"On the ground truth data, the Fleiss kappa score is 0.46, indicating moderate agreement.",
"This fair to moderate agreement indicates that the task is difficult, and there are cases where the visibility of the actions is hard to label.",
"To illustrate, Figure REF shows examples where the annotators had low agreement.",
"Table REF shows statistics for our final dataset of videos labeled with actions, and Figure 2 shows a sample video and transcript, with annotations.",
"Table: Data statistics.Figure: An example of low agreement.",
"The table shows actions and annotations from workers #1, #2, and #3, as well as the ground truth (GT).",
"Labels are: visible - , not visible - x.",
"The bottom row shows screenshots from the video.",
"The Fleiss kappa agreement score is -0.2.For our experiments, we use the first eight YouTube channels from our dataset as train data, the ninth channel as validation data and the last channel as test data.",
"Statistics for this split are shown in Table REF .",
"Table: Statistics for the experimental data split."
],
[
"Discussion",
"The goal of our dataset is to capture naturally-occurring, routine actions.",
"Because the same action can be identified in different ways (e.g., “pop into the freezer”, “stick into the freezer\"), our dataset has a complex and diverse set of action labels.",
"These labels demonstrate the language used by humans in everyday scenarios; because of that, we choose not to group our labels into a pre-defined set of actions.",
"Table REF shows the number of unique verbs, which can be considered a lower bound for the number of unique actions in our dataset.",
"On average, a single verb is used in seven action labels, demonstrating the richness of our dataset.",
"The action labels extracted from the transcript are highly dependent on the performance of the constituency parser.",
"This can introduce noise or ill-defined action labels.",
"Some acions contain extra words (e.g., “brush my teeth of course”), or lack words (e.g., “let me just”).",
"Some of this noise is handled during the annotation process; for example, most actions that lack words are labeled as “not visible” or “not an action” because they are hard to interpret."
],
[
"Identifying Visible Actions in Videos",
"Our goal is to determine if actions mentioned in the transcript of a video are visually represented in the video.",
"We develop a multimodal model that leverages both visual and textual information, and we compare its performance with several single-modality baselines."
],
[
"Data Processing and Representations",
"Starting with our annotated dataset, which includes miniclips paired with transcripts and candidate actions drawn from the transcript, we extract several layers of information, which we then use to develop our multimodal model, as well as several baselines.",
"Action Embeddings.",
"To encode each action, we use both GloVe [28] and ELMo [29] embeddings.",
"When using GloVe embeddings, we represent the action as the average of all its individual word embeddings.",
"We use embeddings with dimension 50.",
"When using ELMo, we represent the action as a list of words which we feed into the default ELMo embedding layer.Implemented as the ELMo module in Tensorflow This performs a fixed mean pooling of all the contextualized word representations in each action.",
"Part-of-speech (POS).",
"We use POS information for each action.",
"Similar to word embeddings [28], we train POS embeddings.",
"We run the Stanford POS Tagger [45] on the transcripts and assign a POS to each word in an action.",
"To obtain the POS embeddings, we train GloVe on the Google N-gram corpushttp://storage.googleapis.com/books/ngrams/books/ datasetsv2.html using POS information from the five-grams.",
"Finally, for each action, we average together the POS embeddings for all the words in the action to form a POS embedding vector.",
"Context Embeddings.",
"Context can be helpful to determine if an action is visible or not.",
"We use two types of context information, action-level and sentence-level.",
"Action-level context takes into account the previous action and the next action; we denote it as Context$_A$ .",
"These are each calculated by taking the average of the action's GloVe embeddings.",
"Sentence-level context considers up to five words directly before the action and up to five words after the action (we do not consider words that are not in the same sentence as the action); we denote it as Context$_S$ .",
"Again, we average the GLoVe embeddings of the preceding and following words to get two context vectors.",
"Concreteness.",
"Our hypothesis is that the concreteness of the words in an action is related to its visibility in a video.",
"We use a dataset of words with associated concreteness scores from [2].",
"Each word is labeled by a human annotator with a value between 1 (very abstract) and 5 (very concrete).",
"The percentage of actions from our dataset that have at least one word in the concreteness dataset is 99.8%.",
"For each action, we use the concreteness scores of the verbs and nouns in the action.",
"We consider the concreteness score of an action to be the highest concreteness score of its corresponding verbs and nouns.",
"tab:concr1 shows several sample actions along with their concreteness scores and their visiblity.",
"Table: Visible actions with high concreteness scores (Con.",
"), and non-visible actions with low concreteness scores.",
"The noun or verb with the highest concreteness score is in bold.Table: NO_CAPTIONVideo Representations.",
"We use Yolo9000 [32] to identify objects present in each miniclip.",
"We choose YOLO9000 for its high and diverse number of labels (9,000 unique labels).",
"We sample the miniclips at a rate of 1 frame-per-second, and we use the Yolo9000 model pre-trained on COCO [25] and ImageNet [8].",
"We represent a video both at the frame level and the sequence level.",
"For frame-level video features, we use the Inception V3 model [43] pre-trained on ImageNet.",
"We extract the output of the very last layer before the Flatten operation (the “bottleneck layer\"); we choose this layer because the following fully connected layers are too specialized for the original task they were trained for.",
"We extract Inception V3 features from miniclips sampled at 1 frame-per-second.",
"For sequence-level video features, we use the C3D model [46] pre-trained on the Sports-1M dataset [22].",
"Similarly, we take the feature map of the sixth fully connected layer.",
"Because C3D captures motion information, it is important that it is applied on consecutive frames.",
"We take each frame used to extract the Inception features and extract C3D features from the 16 consecutive frames around it.",
"We use this approach because combining Inception V3 and C3D features has been shown to work well in other video-based models [19], [3], [23]."
],
[
"Baselines",
"Using the different data representations described in Section REF , we implement several baselines.",
"Concreteness.",
"We label as visible all the actions that have a concreteness score above a certain threshold, and label as non-visible the remaining ones.",
"We fine tune the threshold on our validation set; for fine tuning, we consider threshold values between 3 and 5.",
"Table REF shows the results obtained for this baseline.",
"Feature-based Classifier.",
"For our second set of baselines, we run a classifier on subsets of all of our features.",
"We use an SVM [5], and perform five-fold cross-validation across the train and validation sets, fine tuning the hyper-parameters (kernel type, C, gamma) using a grid search.",
"We run experiments with various combinations of features: action GloVe embeddings; POS embeddings; embeddings of sentence-level context (Context$_S$ ) and action-level context (Context$_A$ ); concreteness score.",
"The combinations that perform best during cross-validation on the combined train and validation sets are shown in Table REF .",
"Figure: Example of frames, corresponding actions, object detected with Yolo, and the object - word pair with the highest WUP similarity score in each frame.Figure: Overview of the multimodal neural architecture.",
"+ represents concatenation.LSTM and ELMo.",
"We also consider an LSTM model [18] that takes as input the tokenized action sequences padded to the length of the longest action.",
"These are passed through a trainable embedding layer, initialized with GloVe embeddings, before the LSTM.",
"The LSTM output is then passed through a feed forward network of fully connected layers, each followed by a dropout layer [42] at a rate of 50%.",
"We use a sigmoid activation function after the last hidden layer to get an output probability distribution.",
"We fine tune the model on the validation set for the number of training epochs, batch size, size of LSTM, and number of fully-connected layers.",
"We build a similar model that embeds actions using ELMo (composed of 2 bi-LSTMs).",
"We pass these embeddings through the same feed forward network and sigmoid activation function.",
"The results for both the LSTM and ELMo models are shown in Table REF .",
"Yolo Object Detection.",
"Our final baseline leverages video information from the YOLO9000 object detector.",
"This baseline builds on the intuition that many visible actions involve visible objects.",
"We thus label an action as visible if it contains at least one noun similar to objects detected in its corresponding miniclip.",
"To measure similarity, we compute both the Wu-Palmer (WUP) path-length-based semantic similarity [51] and the cosine similarity on the GloVe word embeddings.",
"For every action in a miniclip, each noun is compared to all detected objects and assigned a similarity score.",
"As in our concreteness baseline, the action is assigned the highest score of its corresponding nouns.",
"We use the validation data to fine tune the similarity threshold that decides if an action is visible or not.",
"The results are reported in Table REF .",
"Examples of actions that contain one or more words similar to detected objects by Yolo can be seen in Figure REF ."
],
[
"Multimodal Model",
"Each of our baselines considers only a single modality, either text or video.",
"While each of these modalities contributes important information, neither of them provides a full picture.",
"The visual modality is inherently necessary, because it shows the visibility of an action.",
"For example, the same spoken action can be labeled as either visible or non-visible, depending on its visual context; we find 162 unique actions that are labeled as both visible and not visible, depending on the miniclip.",
"This ambiguity has to be captured using video information.",
"However, the textual modality provides important clues that are often missing in the video.",
"The words of the person talking fill in details that many times cannot be inferred from the video.",
"For our full model, we combine both textual and visual information to leverage both modalities.",
"We propose a multimodal neural architecture that combines encoders for the video and text modalities, as well as additional information (e.g., concreteness).",
"Figure REF shows our model architecture.",
"The model takes as input a (miniclip $m$ , action $a$ ) pair and outputs the probability that action $a$ is visible in miniclip $m$ .",
"We use C3D and Inception V3 video features extracted for each frame, as described in Section REF .",
"These features are concatenated and run through an LSTM.",
"To represent the actions, we use ELMo embeddings (see Section REF ).",
"These features are concatenated with the output from the video encoding LSTM, and run through a three-layer feed forward network with dropout.",
"Finally, the result of the last layer is passed through a sigmoid function, which produces a probability distribution indicating whether the action is visible in the miniclip.",
"We use an RMSprop optimizer [44] and fine tune the number of epochs, batch size and size of the LSTM and fully-connected layers.",
"Table: Results from baselines and our best multimodal method on validation and test data.",
"Action G _G indicates action representation using GloVe embedding, and Action E _E indicates action representation using ELMo embedding.",
"Context S _S indicates sentence-level context, and Context A _A indicates action-level context."
],
[
"Evaluation and Results",
"Table REF shows the results obtained using the multimodal model for different sets of input features.",
"The model that uses all the input features available leads to the best results, improving significantly over the text-only and video-only methods.Significance is measured using a paired t-test: $p < 0.005$ when compared to the best text-only model; $p < 0.0005$ when compared to the best video-only model.",
"We find that using only Yolo to find visible objects does not provide sufficient information to solve this task.",
"This is due to both the low number of objects that Yolo is able to detect, and the fact that not all actions involve objects.",
"For example, visible actions from our datasets such as “get up\", “cut them in half\", “getting ready\", and “chopped up\" cannot be correctly labeled using only object detection.",
"Consequently, we need to use additional video information such as Inception and C3D information.",
"In general, we find that the text information plays an important role.",
"ELMo embeddings lead to better results than LSTM embeddings, with a relative error rate reduction of 6.8%.",
"This is not surprising given that ELMo uses two bidirectional LSTMs and has improved the state-of-the-art in many NLP tasks [29].",
"Consequently, we use ELMo in our multimodal model.",
"Moreover, the addition of extra information improves the results for both modalities.",
"Specifically, the addition of context is found to bring improvements.",
"The use of POS is also found to be generally helpful."
],
[
"Conclusion",
"In this paper, we address the task of identifying human actions visible in online videos.",
"We focus on the genre of lifestyle vlogs, and construct a new dataset consisting of 1,268 miniclips and 14,769 actions out of which 4,340 have been labeled as visible.",
"We describe and evaluate several text-based and video-based baselines, and introduce a multimodal neural model that leverages visual and linguistic information as well as additional information available in the input data.",
"We show that the multimodal model outperforms the use of one modality at a time.",
"A distinctive aspect of this work is that we label actions in videos based on the language that accompanies the video.",
"This has the potential to create a large repository of visual depictions of actions, with minimal human intervention, covering a wide spectrum of actions that typically occur in everyday life.",
"In future work, we plan to explore additional representations and architectures to improve the accuracy of our model, and to identify finer-grained alignments between visual actions and their verbal descriptions.",
"The dataset and the code introduced in this paper are publicly available at http://lit.eecs.umich.edu/downloads.html."
],
[
"Acknowledgments",
"This material is based in part upon work supported by the Michigan Institute for Data Science, by the National Science Foundation (grant #1815291), by the John Templeton Foundation (grant #61156), and by DARPA (grant #HR001117S0026-AIDA-FP-045).",
"Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the Michigan Institute for Data Science, the National Science Foundation, the John Templeton Foundation, or DARPA."
]
] | 1906.04236 |
[
[
"Psycholinguistics meets Continual Learning: Measuring Catastrophic\n Forgetting in Visual Question Answering"
],
[
"Abstract We study the issue of catastrophic forgetting in the context of neural multimodal approaches to Visual Question Answering (VQA).",
"Motivated by evidence from psycholinguistics, we devise a set of linguistically-informed VQA tasks, which differ by the types of questions involved (Wh-questions and polar questions).",
"We test what impact task difficulty has on continual learning, and whether the order in which a child acquires question types facilitates computational models.",
"Our results show that dramatic forgetting is at play and that task difficulty and order matter.",
"Two well-known current continual learning methods mitigate the problem only to a limiting degree."
],
[
"Introduction",
"Supervised machine learning models are incapable of continuously learning new tasks, as they forget how to perform the previously learned ones.",
"This problem, called catastrophic forgetting, is prominent in artificial neural networks [10].",
"Continual Learning (CL) addresses this problem by trying to equip models with the capability to continuously learn new tasks over time [15].",
"Catastrophic forgetting and CL have received considerable attention in computer vision [19], , but far less attention within Natural Language Processing (NLP).",
"Figure: Overview of our linguistically-informed CL setup for VQA.We investigate catastrophic forgetting in the context of multimodal models for Visual Question Answering [1] motivated by evidence from psycholinguistics.",
"VQA is the task of answering natural language questions about an image.",
"Evidence from child language acquisition indicates that children learn Wh-questions before polar (Yes/No) questions [12], [13].",
"Motivated by this finding, we design a set of linguistically-informed experiments: i) to investigate whether the order in which children acquire question types facilitates continual learning for computational models and, accordingly, the impact of task order on catastrophic forgetting; ii) to measure how far two well-known CL approaches help to overcome the problem [16], [7]Code and data are available at the link http://continual-vista.github.io/.."
],
[
"Contributions:",
"Our study contributes to the literature on CL in NLP.",
"In particular: i) we introduce a CL setup based on linguistically-informed task pairs which differ with respect to question types and level of difficulty; ii) we show the importance of task order, an often overlooked aspect, and observe asymmetric synergetic effects; iii) our results show that our VQA model suffers from extreme forgetting; rehearsal gives better results than a regularization-based method.",
"Our error analysis shows that the latter approach encounters problems even in discerning Task A after having been trained on Task B.",
"Our study opens the door to deeper investigations of CL on linguistic skills with different levels of difficulty based of psycholinguistics findings."
],
[
"Task Setup",
"As a first step towards understanding the connection between linguistic skills and the impact on CL, we design a set of experiments within VQA where tasks differ with respect to the type of question and the level of difficulty according to the psycholinguistics literature.",
"The overall setup is illustrated in Figure REF and described next."
],
[
"Dataset",
"CLEVR [5] allows to study the ability of VQA agents.",
"It requires compositional language and basic spatial reasoning skills.",
"Every question in CLEVR is derived by a Functional Program (FP) from a scene graph of the associated image.",
"The scene graph defines the objects and attributes in the image.",
"The FP contains functions corresponding to skills, e.g., querying object attributes or comparing values (see Fig.",
"REF , upper).",
"Questions are categorized by their type.",
"CLEVR consists of five question types whose answer labels range over 15 attributes, 10 numbers, and “yes”/“no” (in total 27 labels)."
],
[
"Multimodal Tasks",
"We select the CLEVR sub-tasks `query_attribute' and `equal_attribute' with attributes color, shape, material, and size.",
"The two types of questions differ by answer type $y \\in \\mathcal {Y}$ : [leftmargin=11pt,itemsep=0pt,topsep=0pt] Wh-questions (Wh-q): Questions about the attribute of an object, e.g., “What is the material of the large object...?”, where $y \\in \\lbrace blue, cube, small, \\ldots ,metal\\rbrace $ spans over $|color|=8$ , $|shape|=3$ , $|size|=2$ and $|material|=2$ (in total $|\\mathcal {Y}|=15$ ).",
"Yes/No questions (Y/N-q): Questions that compare objects with respect to an attribute, e.g., “Does the cyan ball have the same material as ...?”, with $y \\in \\lbrace yes, no \\rbrace $ (in total $|\\mathcal {Y}|=2$ )."
],
[
"Task Order",
"We learn Task A followed by Task B (TaskA$\\rightarrow $ TaskB), but experiment with both directions, i.e., by first assigning Wh-q to Task A and Y/N-q to Task B, and vice versa.",
"We expect that the inherent difficulty of a task and the order in which tasks are learned have an impact on CL."
],
[
"Single-head Evaluation",
"CL methods can be tested in two ways.",
"We opt for a single-head evaluation setup (see Fig.",
"REF , lower) with an output space over labels for all tasks (here: all CLEVR labels).",
"In contrast, in a multi-head setup predictions are restricted to task labels, as the task identifier is provided.",
"Single-head is more difficult yet more realistic [2]."
],
[
"VQA Model",
"We take the model proposed by yang2016:stacked as a starting point, using the code released by john:infe17 (LSTM+CNN+SA).",
"Questions are encoded with a recurrent neural network with Long Short-Term Memory (LSTM) units.",
"Images are encoded with a ResNet-101 Convolutional Neural Network (CNN) pre-trained on ImageNet [4].",
"The two representations are combined using Spatial Attention (SA) [17] to focus on the most salient objects and properties in the image and text.",
"The final answer distribution is predicted with a Multilayer Perceptron (MLP)."
],
[
"Baselines",
"In order to measure catastrophic forgetting, we first consider per-task baselines: A random baseline (i.e., random stratified sample of the label distribution per task) and the results of a model trained independently on each task (i.e., over task-specific $\\mathcal {Y}$ ).",
"For CL, we report again a random baseline (this time a random stratified sample drawing predictions according to the answer distribution of both tasks), and we consider the Naive and Cumulative baselines proposed by maltoni2018:continuous.",
"The Naive model is fine-tuned across tasks: It is first trained on Task A and then on Task B starting from the previously learned parameters.",
"The Cumulative model is trained from scratch on the training sets of both Task A and Task B.",
"This is a kind of upper bound, or performance that a CL model should achieve."
],
[
"Continual Learning Models",
"In CL there are two broad families of methods: Those that assume memory and access to explicit previous knowledge (instances), and those that have only access to compressed knowledge, such as previously learned parameters.",
"These two families correspond to rehearsal and regularization, respectively.",
"A widely-used regularization-based approach is Elastic Weight Consolidation ,kirkpatrick2017:overcoming.",
"A regularization term, parametrized by $\\lambda $ , is added to the loss function aiming the model to converge to parameters where it has a low error for both tasks.",
"In the Rehearsal approach [16], the model is first trained on Task A, then the parameters are fine-tuned through batches taken from a dataset containing a small number of examples of Task A and the training set of Task B.",
"The selection of training examples of Task A is done through uniform sampling."
],
[
"Data and Training Details",
"Since CLEVR has no published ground-truth answers for the test set, we split the original validation set into a validation and a test set.",
"To avoid performance impact due to different training data sizes, we downsample the training sets to the same size (Y/N-q data size), resulting in 125,654 training instances per task.",
"The validation and test sets contain, respectively, 26,960 and 26,774 data points for Wh-q and 13,417 and 13,681 data points for Y/N-q.",
"For the baselines, we select the model which reaches maximum accuracy on the validation set of each task.",
"For CL, we choose the model with the highest CL score computed according to the validation set of each task pair.",
"Details on hyper-parameters and evaluation metrics are provided in the supplementary material (SM)."
],
[
"Results and Analysis",
"The main results are provided in Table REF .",
"There are several take-aways."
],
[
"Task Difficulty",
"The results of the per-task models (cf.",
"first two rows in Table REF ) show that there is a large performance gap between the two tasks.",
"Wh-q is easier (.81) than Y/N-q (.52), regardless of the fact that a priori the latter should be easier (as shown by the respective task-specific random baselines).",
"The Y/N-q task-specific model performs only slightly above chance (.52, in line with what johnson2017:clevr report for `equal_attribute' questions).",
"This shows that despite the limited output space of the Y/N-q task, such type of questions in CLEVR are complex and require reasoning skills [5]."
],
[
"Catastrophic Forgetting",
"We observe that extreme forgetting is at play.",
"Naive forgets the previously learned skill completely: When tested on Task A after having been fine-tuned on Task B, it achieves 0.0 accuracy on the first task for both directions (I and II, cf.",
"Table REF lower).",
"The Cumulative model by nature cannot forget, since it is trained on both tasks simultaneously, achieving .81 and .74 on Wh-q and Y/N-q, respectively.",
"Interestingly, we observe an asymmetric synergetic effect.",
"Being exposed to the Wh-q task helps the Cumulative model improve on Y/N-q, reaching results beyond the task-specific model (from .52 to .74).",
"The effect is not symmetric as the accuracy on Wh-q does not further increase.",
"Table: Mean accuracy over 3 runs: Trained on each task independently(first two rows; per-task label space 𝒴\\mathcal {Y}) vs. CL setups (single-head label space over all 𝒴\\mathcal {Y})."
],
[
"Does CL Help?",
"Current CL methods show only limiting (or no) effect.",
"EWC performs bad overall: In the II) setup (y/n$\\rightarrow $ wh, harder task first), EWC does not yield any improvement over the Naive model; in the wh$\\rightarrow $ y/n setup, the model's result on Task A is above chance level (.25 vs. .04) but far off per-task performance (.81).",
"The Rehearsal model forgets less than Naive and EWC in both setups: In the y/n$\\rightarrow $ wh setup, it is above chance level (.51 vs. .25) reaching per-task random baseline results on Y/N questions (i.e., the model is able to identify Task A, despite the harder single-head setting, in contrast to the Naive and EWC models).",
"There is no boost derived from being exposed to the Wh-q task in any of the two setups."
],
[
"Task Order",
"The results in Table REF show that the order of tasks plays an important role: wh$\\rightarrow $ y/n facilitates CL more than the opposite order: less forgetting is at place when wh is learned first.",
"This confirms psycholinguistic evidence.",
"Overall, Rehearsal works better than EWC, but mitigates forgetting only to a limiting degree.",
"Figure: Analysis of the neuron activations on the penultimatehidden layer for the I) Wh →\\rightarrow Y/N setup.",
"“equal_{shape,color,material,size}” refers to Y/N-q, “query_{..}” refers to Wh-questions."
],
[
"Analysis",
"To get a deeper understanding of the models, we analyze the penultimate hidden layer on a sample of 512 questions from the test sets of both tasks (cf.",
"Fig.",
"REF ) and relate the representations to confusion matrices of the whole test sets (provided in the SM) and test results (Table REF ).",
"First of all, the model trained on Wh-q discriminates Wh-questions about different attributes very well, reflected in overall high accuracy (.81).",
"It otherwise clusters all instances from the other task (Y/N-q, which it has not been trained on) around Wh-questions related to size.",
"The Cumulative model, in contrast, is able to further tease the different kinds of Y/N questions apart.",
"Questions about different attributes become distinguishable in the plot, although overall Y/N questions remain closer together than the clusters for Wh-q.",
"This is in line with the lower performance of Cumulative on Y/N-q.",
"Our examination of the confusion matrices confirms that the two question types are never confused by the Cumulative model.",
"In contrast, the Naive model is very prone to this type of mistake (see plot in SM).",
"As for the CL models, Fig.",
"REF (two rightmost plots) shows that EWC learns representations which are rather similar to those learned by the model trained on Wh-q independently: Y/N questions result in a big hard-to-distinguish “blob”, and are confused with Wh-q about size, as visible in Fig.",
"REF and the confusion matrix analysis (in the SM).",
"In contrast, Rehearsal remembers how to distinguish among all kinds of Wh-q and between Wh-q and Y/N-q.",
"The error analysis confirms that the model hardly makes any mistakes related to task confusion.",
"However, despite the higher performance than EWC, Rehearsal is still not able to discern well between different kinds of Y/N-q."
],
[
"Related Work",
"Early work on life-long learning [3], [11] is related to ours, but typically concerns a single task (e.g., relation extraction).",
"lee:towa17 aims to transfer conversational skills from a synthetic domain to a customer-specific application in dialogue agents, while yogatama2019:learning show that current models for different NLP tasks are not able to properly reuse previously learned knowledge.",
"In general, continual learning has been mostly studied in computer vision.",
"To the best of our knowledge, little has been done on catastrophic forgetting in VQA.",
"A study on forgetting in the context of VQA and closest to ours is perez2018:film.",
"They show that their model forgets after being fine-tuned on data including images with objects of colors other than those previously seen.",
"We took this work as starting point and extended it to consider different types of questions and to test different CL methods beyond fine-tuning."
],
[
"Conclusion",
"We assessed to what extent a multimodal model suffers from catastrophic forgetting in a VQA task.",
"We built two tasks involving different linguistic characteristics which are known to be learned sequentially by children and on which multimodal models reach different performance.",
"Our results show that dramatic forgetting is at play in VQA, and for the tested task pairs we empirically found Rehearsal to work better than a regularization-based method (EWC).",
"More importantly, we show that the order in which models learn tasks is important, wh$\\rightarrow $ y/n facilitates continual learning more than the opposite order, thereby confirming psycholinguistic evidence.",
"Our error analysis highlights the importance of taking the kind of mistakes made by the models into account: A model that does not detect Task A after having been exposed to Task B should be penalized more than a model that answers Task A with wrong task-related labels, but is still capable of identifying the task.",
"Most importantly, our study revealed that differences in the inherent difficulty of the tasks at hand can have a strong impact on continual learning.",
"Regularization-based methods like EWC appear to work less well when applied to tasks with different levels of difficulty, as in our experiments.",
"We reserve a deeper investigation of this aspect to future research."
],
[
"Acknowledgements",
"We kindly acknowledge the support of NVIDIA Corporation with the donation of the GPUs used in our research to the University of Trento and IT University of Copenhagen.",
"R. Fernández was funded by the Netherlands Organisation for Scientific Research (NWO) under VIDI grant nr.",
"276-89-008, Asymmetry in Conversation."
]
] | 1906.04229 |
[
[
"A Document-grounded Matching Network for Response Selection in\n Retrieval-based Chatbots"
],
[
"Abstract We present a document-grounded matching network (DGMN) for response selection that can power a knowledge-aware retrieval-based chatbot system.",
"The challenges of building such a model lie in how to ground conversation contexts with background documents and how to recognize important information in the documents for matching.",
"To overcome the challenges, DGMN fuses information in a document and a context into representations of each other, and dynamically determines if grounding is necessary and importance of different parts of the document and the context through hierarchical interaction with a response at the matching step.",
"Empirical studies on two public data sets indicate that DGMN can significantly improve upon state-of-the-art methods and at the same time enjoys good interpretability."
],
[
"Introduction",
"Human-machine conversation is a long-standing goal of artificial intelligence.",
"Recently, building a chatbot for open domain conversation has gained increasing interest due to both availabilities of a large amount of human conversation data and powerful models learned with neural networks.",
"Existing methods are either retrieval-based or generation-based.",
"Retrieval-based methods respond to human input by selecting a response from a pre-built index [7], [23], while generation-based methods synthesize a response with a natural language model [14], [9].",
"In this work, we study the problem of response selection for retrieval-based chatbots, since retrieval-based systems are often superior to their generation-based counterparts on response fluency and diversity, are easy to evaluate.",
"Table: An example of document-grounded dialogueA key step in response selection is measuring the matching degree between a context (a message with a few turns of conversation history) and a response candidate.",
"Existing methods [20], [28] have achieved impressive performance on benchmarks [10], [20], but responses are selected solely based on conversation history.",
"Human conversations, on the other hand, are often grounded in external knowledge.",
"For example, in Reddit, discussion among users is usually along the document posted at the beginning of a thread which provides topics and basic facts for the following conversation.",
"Lack of knowledge grounding has become one of the major gaps between the current open domain dialog systems and real human conversations.",
"As a step toward bridging the gap, we investigate knowledge-grounded response selection in this work and specify the knowledge as unstructured documents that are common sources in practice.",
"The task is that given a document and a conversation context based on the document, one selects a response from a candidate pool that is consistent and relevant with both the conversation context and the background document.",
"Table REF shows an example from PERSONA-CHAT, a data set released recently by Facebook [25], to illustrate the task: given two speakers' profiles as documents and a conversation context, one is required to distinguish the true response from the false onesFor space limitation, we only show one false response here..",
"Intuitively, both documents and conversation contexts should participate in matching.",
"Since documents and contexts are highly asymmetric in terms of information they convey, and there exists complicated dependency among sentences in the documents and utterances in the contexts, challenges of the task include (1) how to ground conversation contexts with documents given that utterances in the contexts are not always related to the documents due to the casual nature of open domain conversation (e.g., the greetings in Table REF ); (2) how to comprehend documents with conversation contexts when information in the documents are rather redundant for proper response recognition (e.g., the description regarding to B's hobby in her profile in Table REF ); and (3) how to effectively leverage both information sources to perform matching.",
"To overcome the challenges, we propose a document-grounded matching network (DGMN).",
"DGMN encodes sentences in a document, utterances in a conversation context, and a response candidate through self-attention, and models context grounding and document comprehension by constructing a document-aware context representation and a context-aware document representation via an attention mechanism.",
"With the rich representations, DGMN distills matching information from each utterance-response pair and each sentence-response pair, where whether an utterance needs grounding, which parts of the document are crucial for grounding and matching, and which parts of the context are useful for representing the document are dynamically determined by a hierarchical interaction mechanism.",
"The final matching score is defined as an aggregation of matching signals from all pairs.",
"We conduct experiments on two public data sets: the PERSONA-CHAT data [25] and the CMU Document Grounded Conversation (CMUDoG) data [27].",
"Evaluation results indicate that on both data sets, DGMN can significantly outperform state-of-the-art methods.",
"Compared with Transformer, the best performing baseline on both data, absolute improvements from DGMN on $r@1$ (hits@1) are more than 13% on the PERSONA-CHAT data and more than 5% on the CMUDoG data.",
"Through both quantitative and qualitative analysis, we also demonstrate the effect of different representations to matching and how DGMN grounds conversation contexts with documents.",
"Our contributions in this work are three-fold: (1) proposal of a document-grounded matching network that performs response selection according to both conversation contexts and background knowledge; (2) empirical verification of the effectiveness of the proposed model on two public data sets; and (3) new state-of-the-art on the PERSONA-CHAT data without any pre-training on external resources."
],
[
"Document-Grounded Matching Network",
"In this section, we first formalize the document-grounded matching problem, and then introduce our model from an overview to details of components."
],
[
"Problem Formalization",
"Suppose that we have a data set $\\mathcal {D} = \\lbrace (D_i, c_i, y_i, r_i)\\rbrace _{i=1}^N$ where $D_i=\\lbrace d_{i, 1}, \\cdots , d_{i, m_i}\\rbrace $ is a document that serves as background knowledge for conversation with $d_{i,k}$ the $k$ -th sentence, $c_i=\\lbrace u_{i, 1}, \\cdots , u_{i, n_i}\\rbrace $ is a conversation context following $D_i$ with $u_{i,k}$ the $k$ -th utterance, $r_i$ is a response candidate, and $y_i \\in \\lbrace 0, 1\\rbrace $ is a label with $y_i=1$ indicating that $r_i$ is a proper response given $c_i$ and $D_i$ , otherwise $y_i=0$ .",
"The task is to learn a matching model $g(\\cdot ,\\cdot ,\\cdot )$ from $\\mathcal {D}$ , and thus for a new triple $(D,c,r)$ , $g(D,c,r)$ returns the matching degree between $c$ and $r$ under $D$ ."
],
[
"Model Overview",
"We define $g(D,c,r)$ as a document-grounded matching network.",
"Figure REF illustrates the architecture of the model.",
"In brief, DGMN consists of an encoding layer, a fusion layer, a matching layer, and an aggregation layer.",
"The encoding layer represents $D$ , $c$ , and $r$ via self-attention, and feeds the representations to the fusion layer where $D$ and $c$ are fused into the representations of each other as a document-aware context representation and a context-aware document representation.",
"Based on the representations given by the first two layers, the matching layer then lets each utterance in $c$ and each sentence in $D$ interact with $r$ , and distills matching signals from the interaction.",
"Matching signals in all pairs are finally aggregated as a matching score in the aggregation layer."
],
[
"Model Details",
"We elaborate each layer of the document-grounded matching network in this section."
],
[
"Encoding Layer",
"Given an utterance $u_i$ in a context $c$ , a sentence $d_j$ in a document $D$ , and a response candidate $r$ , the model first embeds $u_i$ , $d_i$ , and $r$ as $\\mathbf {E}_{u_i} = [\\mathbf {e}_{{u_i},1}, \\cdots , \\mathbf {e}_{{u_i}, l_u}]$ , $\\mathbf {E}_{d_j} = [\\mathbf {e}_{{d_j},1}, \\cdots , \\mathbf {e}_{{d_j}, l_d}]$ and $\\mathbf {E}_r = [\\mathbf {e}_{r,1}, \\cdots ,\\mathbf {e}_{r,l_r}]$ respectively by looking up a shared embedding table pre-trained with Glove [13] on the training data $\\mathcal {D}$ , where $\\mathbf {e}_{{u_i},k}$ , $\\mathbf {e}_{{d_j},k}$ and $\\mathbf {e}_{r,k}$ are representations of the $k$ -th words in $u_i$ , $d_j$ and $r$ respectively, and $l_u$ , $l_r$ , and $l_d$ are lengths of the three sequences.",
"$\\mathbf {E}_{u_i}$ , $\\mathbf {E}_{d_j}$ and $\\mathbf {E}_r$ are then processed by an attentive module to encode long-term dependency among words into the representations.",
"The attentive module simplifies the multi-head attention module in Transformer [17], and consists of a scaled dot-product attention component and a feed-forward component.",
"Without loss of generality, let $\\mathbf {Q} \\in \\mathbb {R}^{n_Q \\times d}$ , $\\mathbf {K} \\in \\mathbb {R}^{n_K \\times d}$ , and $\\mathbf {V} \\in \\mathbb {R}^{n_V \\times d}$ denote embedding matrices of a query, a key, and a value respectively, where $n_Q$ , $n_K$ , and $n_V$ are numbers of words in the input sequences, and $d$ stands for embedding size.",
"The scaled dot-product attention component is then defined as: $\\begin{aligned} \\text{Attention}(\\mathbf {Q},\\mathbf {K},\\mathbf {V}) = \\text{softmax}(\\mathbf {Q} \\mathbf {K}^{T}/\\sqrt{d})\\mathbf {V}.\\end{aligned}$ Intuitively, each entry of $\\mathbf {V}$ is weighted by a relevance score defined by the similarity of an entry of $\\mathbf {Q}$ and an entry of $\\mathbf {K}$ , and then an updated representation of $\\mathbf {Q}$ is formed by linearly combining the entries of $\\mathbf {V}$ with the weights.",
"In practice, we often let $\\mathbf {K}=\\mathbf {V}$ , and thus $\\mathbf {Q}$ is represented by similar entries of $\\mathbf {V}$ .",
"The feed-forward component takes $\\text{Attention}(\\mathbf {Q},\\mathbf {K},\\mathbf {V})$ as input, and transforms it to a new representation by two non-linear projections.",
"A residual connection [4] and a row-wise normalization [1] are applied to the result of each projection.",
"For ease of presentation, we denote the whole attentive module as $f_{\\text{ATT}}(\\mathbf {Q}, \\mathbf {K}, \\mathbf {V})$ .",
"$u_i$ , $d_j$ and $r$ are then represented by attending to themselves through $f_{\\text{ATT}}(\\cdot , \\cdot , \\cdot )$ : $\\mathbf {U}_i &= f_{\\text{ATT}}(\\mathbf {E}_{u_i}, \\mathbf {E}_{u_i}, \\mathbf {E}_{u_i}) \\\\\\mathbf {D}_j &= f_{\\text{ATT}}(\\mathbf {E}_{d_j}, \\mathbf {E}_{d_j}, \\mathbf {E}_{d_j}) \\\\\\mathbf {R} &= f_{\\text{ATT}}(\\mathbf {E}_r, \\mathbf {E}_r, \\mathbf {E}_r).$"
],
[
"Fusion Layer",
"The fusion layer grounds the conversation context by the document and fuses the information of the context into the document, which results in a document-aware context representation and a context-aware document representation.",
"Formally, the document-aware representation of $u_i$ is given by $\\mathbf {\\hat{U}}_{i}=[\\mathbf {\\hat{U}}_{i,1}, \\cdots , \\mathbf {\\hat{U}}_{i,m}]$ , where $m$ is the number of sentences in the document, and $\\forall j \\in \\lbrace 1,\\ldots , m\\rbrace $ , $\\mathbf {\\hat{U}}_{i,j}$ can be formulated as $\\mathbf {\\hat{U}}_{i,j} &= f_{\\text{ATT}}(\\mathbf {U}_i, \\mathbf {D}_j, \\mathbf {D}_j).$ Similarly, the context-aware representation of $d_j$ is defined as $\\mathbf {\\hat{D}}_{j}=[\\mathbf {\\hat{D}}_{j,1}, \\cdots , \\mathbf {\\hat{D}}_{j,n}]$ , where $n$ is the number of utterances in the context, and $\\forall i \\in \\lbrace 1,\\ldots , n\\rbrace $ , $\\mathbf {\\hat{D}}_{j,i}$ is calculated by $\\mathbf {\\hat{D}}_{j,i} &= f_{\\text{ATT}}(\\mathbf {D}_j, \\mathbf {U}_i, \\mathbf {U}_i).$ In $\\mathbf {\\hat{U}}_{i,j}$ , information in $d_j$ provides grounding to $u_i$ , and correlations between $d_j$ and $u_i$ will be distilled to enhance the original representation of $u_i$ .",
"The grounding is performed on a sentence-level rather than on a document-level (i.e., attention with a document vector).",
"This is motivated by the intuition that sentences in a document are differentially important to represent the semantics of an utterance in a context, and the importance should be dynamically recognized through interaction with a response in the matching step.",
"In a similar sense, by letting $d_j$ attend to $u_i$ in $\\mathbf {\\hat{D}}_{j,i}$ we attempt to highlight important parts of $d_j$ through their correlation with $u_i$ , and thus achieve better document understanding in matching.",
"As we have analyzed before, utterances in a context are not always related to the background document in chat.",
"To model this intuition, we append $\\mathbf {U}_i$ to $\\mathbf {\\hat{U}}_{i}$ as $\\mathbf {\\tilde{U}}_{i}=[\\mathbf {U}_i, \\mathbf {\\hat{U}}_{i,1}, \\cdots , \\mathbf {\\hat{U}}_{i,m}]$ and determine if an utterance needs grounding with the guide of response $r$ in the following matching layer.",
"Ideally, if an utterance does not need grounding, then only $\\mathbf {U}_i$ should participate in matching since other entries of $\\mathbf {\\tilde{U}}_{i}$ are noisy.",
"The weights of the entries of $\\mathbf {\\tilde{U}}_{i}$ will be learned from training data."
],
[
"Matching Layer",
"The matching layer pairs $\\mathbf {U}_i$ , $\\mathbf {\\tilde{U}}_{i}$ , $\\mathbf {\\hat{D}}_{j}$ with $\\mathbf {R}$ as $\\lbrace \\mathbf {U}_i,\\mathbf {R}\\rbrace $ , $\\lbrace \\mathbf {\\tilde{U}}_{i},\\mathbf {R}\\rbrace $ and $\\lbrace \\mathbf {\\hat{D}}_{j},\\mathbf {R}\\rbrace $ respectively, and extracts matching information from the pairs.",
"Different from existing matching models that are solely based on conversation contexts, $\\mathbf {\\tilde{U}}_{i}$ and $\\mathbf {\\hat{D}}_{j}$ now contain grounding information from multiple sentences (utterances).",
"Thus, the model needs to dynamically select important sentences (utterances) for grounding and even determine if grounding is necessary.",
"To tackle the new challenges, we propose a hierarchical interaction mechanism.",
"Take $\\lbrace \\mathbf {\\tilde{U}}_{i}, \\mathbf {R}\\rbrace $ as an example.",
"For ease of presentation, we define $\\mathbf {U}_i=\\mathbf {\\hat{U}}_{i,0}$ .",
"Let $\\mathbf {r}_j$ denote the $j$ -th entry of $\\mathbf {R}$ , then the first level interaction of $\\mathbf {\\tilde{U}}_{i}$ and $\\mathbf {R}$ happens between $\\mathbf {r}_j$ and each $\\mathbf {\\hat{U}}_{i,k}$ , $\\forall k\\in \\lbrace 0,\\ldots , m\\rbrace $ , and transforms $\\mathbf {\\hat{U}}_{i,k}$ into $h_{i,j,k}$ through $\\omega _{i, j, k, t} &= \\mathbf {v}_a^{\\top } \\text{tanh}(\\mathbf {w}_{a} [\\mathbf {\\hat{u}}_{i,k,t}; \\mathbf {r}_{j}] + \\mathbf {b}_{a}),\\\\\\alpha _{i, j, k, t} &= \\frac{\\exp (\\omega _{i,j,k,t})}{\\sum _{t=1}^{l_u} \\exp (\\omega _{i,j,k,t})},\\\\h_{i, j, k} &= \\sum \\nolimits _{t=1}^{l_u} \\alpha _{i,j,k,t} {\\mathbf {\\hat{u}}_{i,k,t}},$ where $\\mathbf {\\hat{u}}_{i,k,t}$ is the $t$ -th entry of $\\mathbf {\\hat{U}}_{i,k}$ , and $\\mathbf {w}_{a}$ , $\\mathbf {v}_{a}$ , and $\\mathbf {b}_{a}$ are parameters.",
"Through Eq.",
"(), the first level interaction tries to play emphasis on important words in each $\\mathbf {\\hat{U}}_{i,k}$ with respect to $\\mathbf {r}_j$ .",
"The second level interaction of $\\mathbf {\\tilde{U}}_{i}$ and $\\mathbf {R}$ then summarizes $[h_{i,j,0}, \\ldots , h_{i,j,m}]$ as $h_{i,j}$ by $\\omega ^{\\prime }_{i,j,k} &= \\mathbf {v^{\\prime }}_a^{\\top } \\text{tanh}(\\mathbf {w^{\\prime }}_{a} [h_{i,j,k}; \\mathbf {r}_{j}] + \\mathbf {b^{\\prime }}_{a}),\\\\\\alpha ^{\\prime }_{i,j,k} &= \\frac{\\exp (\\omega ^{\\prime }_{i,j,k})}{\\sum _{k=0}^{m} \\exp (\\omega ^{\\prime }_{i,j,k})},\\\\h_{i, j} &= \\sum \\nolimits _{k=0}^{m} \\alpha ^{\\prime }_{i,j,k} {h_{i,j,k}},$ where $\\mathbf {w^{\\prime }}_{a}$ , $\\mathbf {v^{\\prime }}_{a}$ , and $\\mathbf {b^{\\prime }}_{a}$ are parameters.",
"In the second level interaction, sentences in the document that can bring valuable grounding information for matching will play an important role in the formation of $h_{i,j}$ .",
"As a special case, when $\\alpha ^{\\prime }_{i,j,0}$ is much bigger than other weights, the model judges that $u_i$ does not need grounding from the document.",
"Finally, matching information between $\\mathbf {\\tilde{U}}_{i}$ and $\\mathbf {R}$ is stored in a matrix $\\mathbf {\\tilde{M}}_i = [\\mathbf {m}_{i,1} , \\cdots , \\mathbf {m}_{i,l_r}]$ .",
"$\\forall j \\in \\lbrace 1,\\ldots , l_r\\rbrace $ , $\\mathbf {m}_{i,j}$ is calculated by $\\mathbf {m}_{i,j}=\\text{ReLU}(\\mathbf {w}_p \\begin{bmatrix} (h_{i,j} - \\mathbf {r}_j) \\odot (h_{i,j} - \\mathbf {r}_j) \\\\ h_{i,j} \\odot \\mathbf {r}_j \\end{bmatrix} + \\mathbf {b}_p),$ where $\\mathbf {w}_{p}$ and $\\mathbf {b}_{p}$ are parameters, and $\\odot $ refers to element-wise multiplication.",
"Following the same procedure, we obtain $\\mathbf {\\hat{M}}_j$ as a matching matrix for $\\lbrace \\mathbf {\\hat{D}}_{j},\\mathbf {R}\\rbrace $ where utterances in the context that are helpful for representing $d_j$ are highlighted by $r$ .",
"Since $\\mathbf {U}_i$ is only made up of word representations (i.e., one-layer structure), the matching matrix $\\mathbf {M}_i$ for $\\lbrace \\mathbf {U}_i,\\mathbf {R}\\rbrace $ is calculated by one level interaction parameterized in a similar way as Eq.",
"(REF )-() and the same function as Eq.",
"(REF )."
],
[
"Aggregation Layer and Learning Method",
"The aggregation layer accumulates matching signals in $\\lbrace \\mathbf {M}_i\\rbrace _{i=1}^n$ , $\\lbrace \\mathbf {\\tilde{M}}_i\\rbrace _{i=1}^n$ , and $\\lbrace \\mathbf {\\hat{M}}_j\\rbrace _{j=1}^m$ as a matching score for $(D,c,r)$ .",
"Specifically, we construct a tensor from $\\lbrace \\mathbf {M}_i\\rbrace _{i=1}^n$ , and then apply a convolutional neural network [6] to the tensor to calculate a matching vector $\\mathbf {t}$ .",
"Similarly, we have matching vectors $\\mathbf {\\hat{t}}$ and $\\mathbf {\\tilde{t}}$ for $\\lbrace \\mathbf {\\hat{M}}_j\\rbrace _{j=1}^m$ and $\\lbrace \\mathbf {\\tilde{M}}_i\\rbrace _{i=1}^n$ , respectively.",
"The matching function $g(D, c, r)$ is defined as $ g(D, c, r) = \\sigma ([\\mathbf {t}; \\mathbf {\\hat{t}}; \\mathbf {\\tilde{t}}] \\mathbf {w}_o + \\mathbf {b}_o),$ where $\\mathbf {w}_o$ and $\\mathbf {b}_o$ are wights, and $\\sigma (\\cdot )$ is a sigmoid function.",
"Parameters of $g(D, c, r)$ are estimated from the training data $\\mathcal {D}$ by minimizing the following objective: $ \\small - \\sum _{i=1}^N \\Big ( y_i \\log g(D_i, c_i, r_i) + (1-y_i) \\log (1- g(D_i, c_i, r_i)) \\Big ).$"
],
[
"Experiments",
"We test our model on two public data sets."
],
[
"Experimental Setup",
"The first data we use is the PERSONA-CHAT data set published in [25].",
"The data is collected by requiring two workers on Amazon Mechanical Turk to chat with each other according to their assigned profiles.",
"Each profile is presented in a form of a document with an average of $4.49$ sentences.",
"The profiles define speakers' personas and provide characteristic knowledge for dialogues.",
"For each dialogue, there are both original profiles and revised profiles that are rephrased from the original ones by other crowd workers to force models to learn more than simple word overlap.",
"A revised profile shares the same number of sentences with its original one, and on average, there are $7.33$ words per sentence in the original profiles and $7.32$ words per sentence in the revised ones.",
"The data is split as a training set, a validation set, and a test set by the publishers.",
"In all the three sets, 7 turns before an utterance are used as conversation history, and the next turn of the utterance is treated as a positive response candidate.",
"Besides, each utterance is associated with 19 negative response candidates that are randomly sampled by the publishers.",
"More statistics of the three sets are shown in Table REF .",
"Following the insights in [25], we train models using revised profiles and test the models with both original and revised profiles.",
"In addition to PERSONA-CHAT, we also conduct experiments with CMUDoG data set published recently in [27].",
"Conversations in the data are collected from workers on Amazon Mechanical Turk and are based on movie-related wiki articles in two scenarios.",
"In the first scenario, only one worker has access to the provided document, and he/she is responsible for introducing the movie to the other worker; while in the second scenario, both workers know the document and they are asked to discuss the content of the document.",
"Since the data size for an individual scenario is small, we merge the data of the two scenarios in the experiments and filter out conversations less than 4 turns to avoid noise.",
"Each document consists of 4 sections and these sections are shown to the workers one by one every 3 turn (the first section lasts 6 turns due to initial greetings).",
"On average, each section contains $8.22$ sentences and $27.86$ words per sentence.",
"The data has been divided into a training set, a validation set, and a test set by the publishers.",
"In each set, we take 2 turns before an utterance as conversation history and the next turn of the utterance as a positive response candidate.",
"Since the data does not contain negative examples, we randomly sample 19 negative response candidates for each utterance from the same set.",
"Detailed statistics of the data is given in Table REF .",
"We employ $r@k$ as evaluation metrics where $k\\in \\lbrace 1,2,5\\rbrace $ .",
"For a single context, if the only positive candidate is ranked within top $k$ positions, then $r@k=1$ , otherwise, $r@k=0$ .",
"The final value of the metric is an average over all contexts in test data.",
"Note that in PERSONA-CHAT, $r@1$ is equivalent to hits@1 which is the metric used by [25] for model comparison.",
"Table: Statistics of the two data sets.Table: Evaluation results on the test sets of the PERSONA-CHAT data and the CMUDoG data.",
"Numbers in bold mean that improvement over the best baseline is statistically significant (t-test, pp-value <0.01<0.01)."
],
[
"Baseline Models",
"The following models are selected as baselines.",
"These models are the ranking models in [25] and [11] which perform much better than the generative models in [25] on the PERSONA-CHAT data.",
"Starspace: a supervised model in [21] that learns the similarity between a conversation context and a response candidate by optimizing task-specific embedding via the margin ranking loss.",
"The similarity is measured by the cosine of the sum of word embeddings.",
"Documents are concatenated to conversation contexts.",
"Profile Memory: the model in [25] that lets a conversation context attend over the associated document to produce a vector which is then combined with the context.",
"Cosine is used to measure the similarity between the output context representation and a response candidate.",
"KV Profile Memory: the best performing model in [25] which considers keys as dialogue history and values as the next dialogue utterances and uses a conversation context as input to perform attention over the keys in addition to the documents.",
"The past dialogues are stored in memory to help influence the prediction for the current conversation.",
"Transformer: a variant of the model proposed by [17] ([17]) for machine translation.",
"The model exhibits state-of-the-art performance on the PERSONA-CHAT data as reported in [11].",
"All baseline models are implemented with the code shared at https://github.com/facebookresearch/ParlAI/tree/master/projects/personachat and tuned on the validation sets.",
"We make sure that the baselines achieve the performance on the PERSONA-CHAT data as reported in [25] and [11].",
"Note that we do not include models pre-trained from large-scale external resources, such as the FT-PC model in [11], as baselines, since the comparison is unfair.",
"On the other hand, it is interesting to study if pre-train the proposed model on those large-scale external data (e.g., the Reddit data in [11] with over 5 million personas spanning more than 700 million conversations) can further improve its performance.",
"We leave the study as future work."
],
[
"Implementation Details",
"We set the size of word embedding as 300.",
"In PERSONA-CHAT, the number of sentences per document is limited to 5 (i.e., $m \\le 5$ ).",
"For each sentence in a document, each utterance in a context, and each response candidate, if the number of words is less than 20, we pad zeros, otherwise, we keep the latest 20 words (i.e., $l_u=l_r=l_d=20$ ).",
"In CMUDoG, we set $m \\le 20$ and $l_u=l_r=l_d=40$ following the same procedure.",
"In the matching layer of DGMN, the number of filters of CNN is set as 16, and the window sizes of convolution and pooling are both 3.",
"All models are learned using Adam [8] optimizer with a learning rate of $0.0001$ .",
"In training, we choose 32 as the size of mini-batches.",
"Early stopping on validation data is adopted as a regularization strategy."
],
[
"Evaluation Results",
"Table REF reports evaluation results on the two data sets.",
"We can see that on both data sets, DGMN outperforms all baselines over all metrics, and the improvement is statistically significant (t-test, $p$ -value $<0.01$ ).",
"Improvement from DGMN over Transformer on the CMUDoG data is smaller than that on the PERSONA-CHAT data.",
"The reason might be that Transformer can benefit from the wiki documents in CMUDoG that are longer and contain richer semantics than those handcrafted ones in PERSONA-CHAT.",
"Figure: Visualization of context grounding.",
"The first four graphs illustrate attention between the last utterance of the context and each sentence in the document.",
"The last one shows α i,j,k ' \\alpha ^{\\prime }_{i,j,k} in interaction.Table: Performance of DGMN across different length of grounded documents on all data sets."
],
[
"Discussions",
"In this section, we investigate how different representations affect the performance of DGMN by an ablation study, visualize the example in Table REF to illustrate how contexts are grounded by documents in DGMN, and check how the performance of DGMN changes with respect to document length."
],
[
"Ablation Study.",
"First, we calculate a matching score only with the self-attention based context representation and response representation and denote the model as DGMN($\\mathbf {t}$ ) which means only $\\mathbf {t}$ is kept in Eq.",
"(REF ).",
"Then, we take the document-aware context representation into account, and denote the model as DGMN($\\mathbf {t}$ +$\\mathbf {\\tilde{t}}$ ) in which both $\\mathbf {t}$ and $\\mathbf {\\tilde{t}}$ are used in Eq.",
"(REF ).",
"Based on $\\mathbf {t}$ +$\\mathbf {\\tilde{t}}$ , we further examine if the special configuration for utterances that do not need grounding matters to the performance of DGMN by removing $\\mathbf {U}_i$ from $\\mathbf {\\tilde{U}}_{i}$ .",
"The model is denoted as DGMN($\\mathbf {t}$ +$\\mathbf {\\tilde{t}}$ -NoGround).",
"Finally, the context-aware document representation is considered, and we have the full model of DGMN.",
"Table REF reports evaluation results on the two data sets.",
"We can conclude that (1) all representations are useful for matching; (2) some effect of the context-aware document representation might be covered by the document-aware context representation, as adding the former after the latter does not bring much gain; and (3) although simple, the special configuration for utterances that do not need grounding cannot be removed from DGMN."
],
[
"Visualization.",
"Second, to further understand how DGMN performs context grounding, we visualize the attention weights in formation of the document-aware context representation (i.e., $\\mathbf {\\hat{U}}_{i,j}$ ) and the weights in the second level of interaction (i.e., $\\alpha ^{\\prime }_{i,j,k}$ in Eq.",
"()) with the example in Table REF in Introduction.",
"Due to space limitation, we only visualize the last utterance of the context.",
"Figure REF shows the results.",
"It is interesting to see that words like “work” and “education” are highly correlated in the graph, and at the same time, weights between the utterance and irrelevant sentences in the profile, such as “I am very social and love those close to me”, are generally small.",
"Moreover, in the second level interaction, while most function words and punctuation point to the utterance itself (i.e., $u$ ), the word “student” indicates that information from “i use all of my time for my education.” is useful to recognize the relationship between the response candidate and the context.",
"The example explains why DGMN works well from one perspective."
],
[
"Performance Analysis in Terms of Document Length.",
"Finally, we study the relationship between the performance of DGMN and document length by binning text examples in both data into different buckets according to the document length.",
"Table REF reports the evaluation results.",
"On the PERSONA-CHAT data, both short profiles and long profiles lead to performance drop, while on the CMUDoG data, the longer the documents are, the better the performance of DGMN is.",
"The reason behind the difference might be that profiles in the PERSONA-CHAT data are handcrafted by crowd workers, and thus semantics among different sentences are relatively independent, while documents in the CMUDoG data come from Wikipedia, and there is rich semantic overlap among sentences.",
"Therefore, short profiles contain less useful information and long profiles contain more irrelevant information, and both will make the matching task more challenging.",
"On the other hand, the longer a wiki document is, the more relevant information it can provide to the matching task."
],
[
"Related Work",
"There are two groups of methods for building a chatbot.",
"The first group learns response generation models under an encoder-decoder framework [14], [18] with extensions to suppress generic responses [9], [12], [22], [15].",
"The second group learns a matching model of a human input and a response candidate for response selection.",
"Along this line, early work assumes that the input is a single message [19], [5].",
"Recently, conversation history is taken into account in matching.",
"Representative methods include the dual LSTM model [10], the deep learning to respond architecture [24], the multi-view matching model [26], the sequential matching network [20], the deep attention matching network [28], and the multi-representation fusion network [16].",
"Our work belongs to the second group.",
"The major difference we make is that in addition to conversation contexts, we also incorporate external documents as a kind of background knowledge into matching.",
"Before us, a few recent studies have considered grounding open domain dialogues with external knowledge.",
"For example, [3] ([3]) generalize the vanilla Seq2seq model by conditioning responses on both conversation history and external “facts”.",
"[25] ([25]) release a persona-based conversation data set where profiles created by crowd workers constrain speakers' personas in conversation.",
"[11] ([11]) further increase the scale of the persona-chat data with conversations extracted from Reddit.",
"[27] ([27]) publish a data set in which conversations are grounded in movie-related articles from Wikipedia.",
"[2] ([2]) release another document-grounded data set with wiki articles covering broader topics.",
"In this work, we study grounding retrieval-based open domain dialog systems with background documents and focus on building a powerful matching model with advanced neural architectures.",
"On the persona-chat data published in [25] ([25]) and the document-grounded conversation data set published in [27] ([27]), the model improves upon state-of-the-art methods with large margins."
],
[
"Conclusions",
"We propose a document-grounded matching network to incorporate external knowledge into response selection for retrieval-based chatbots.",
"Experimental results on two public data sets consistently show that the proposed model can significantly outperform state-of-the-art methods."
],
[
"Acknowledgments",
"This work was supported by the National Key Research and Development Program of China (No.",
"2017YFC0804001), the National Science Foundation of China (NSFC Nos.",
"61672058 and 61876196)."
]
] | 1906.04362 |
[
[
"Observation of Large Unidirectional Rashba Magnetoresistance in Ge(111)"
],
[
"Abstract Relating magnetotransport properties to specific spin textures at surfaces or interfaces is an intense field of research nowadays.",
"Here, we investigate the variation of the electrical resistance of Ge(111) grown epitaxially on semi-insulating Si(111) under the application of an external magnetic field.",
"We find a magnetoresistance term which is linear in current density j and magnetic field B, hence odd in j and B, corresponding to a unidirectional magnetoresistance.",
"At 15 K, for I = 10 $\\mu$A (or j = 0.33 A/m) and B = 1 T, it represents 0.5 % of the zero field resistance, a much higher value compared to previous reports on unidirectional magnetoresistance.",
"We ascribe the origin of this magnetoresistance to the interplay between the externally applied magnetic field and the current-induced pseudo-magnetic field in the spin-splitted subsurface states of Ge(111).",
"This unidirectional magnetoresistance is independent of the current direction with respect to the Ge crystal axes.",
"It progressively vanishes, either using a negative gate voltage due to carrier activation into the bulk (without spin-splitted bands), or by increasing the temperature due to the Rashba energy splitting of the subsurface states lower than $\\sim$58 k$_B$.",
"The highly developed technologies on semiconductor platforms would allow the rapid optimization of devices based on this phenomenon."
],
[
"pdfstartview=FitH, linkcolor=blue, citecolor=blue, filecolor=blue, urlcolor=blue Observation of Large Unidirectional Rashba Magnetoresistance in Ge(111) T. Guillet Univ.",
"Grenoble Alpes, CEA, CNRS, Grenoble INP, IRIG-SPINTEC, 38000 Grenoble, France C. Zucchetti LNESS-Dipartimento di Fisica, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy Q. Barbedienne Unité Mixte de Physique, CNRS, Thales, Univ.",
"Paris-Sud, Université Paris-Saclay, 91767, Palaiseau, France A. Marty Univ.",
"Grenoble Alpes, CEA, CNRS, Grenoble INP, IRIG-SPINTEC, 38000 Grenoble, France G. Isella LNESS-Dipartimento di Fisica, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy L. Cagnon Univ.",
"Grenoble Alpes, CNRS, Grenoble INP, Institut NEEL, 38000 Grenoble, France C. Vergnaud Univ.",
"Grenoble Alpes, CEA, CNRS, Grenoble INP, IRIG-SPINTEC, 38000 Grenoble, France N. Reyren Unité Mixte de Physique, CNRS, Thales, Univ.",
"Paris-Sud, Université Paris-Saclay, 91767, Palaiseau, France J.-M. George Unité Mixte de Physique, CNRS, Thales, Univ.",
"Paris-Sud, Université Paris-Saclay, 91767, Palaiseau, France A. Fert Unité Mixte de Physique, CNRS, Thales, Univ.",
"Paris-Sud, Université Paris-Saclay, 91767, Palaiseau, France M. Jamet Univ.",
"Grenoble Alpes, CEA, CNRS, Grenoble INP, IRIG-SPINTEC, 38000 Grenoble, France Relating magnetotransport properties to specific spin textures at surfaces or interfaces is an intense field of research nowadays.",
"Here, we investigate the variation of the electrical resistance of Ge(111) grown epitaxially on semi-insulating Si(111) under the application of an external magnetic field.",
"We find a magnetoresistance term which is linear in current density $j$ and magnetic field $B$ , hence odd in $j$ and $B$ , corresponding to a unidirectional magnetoresistance.",
"At ${15~\\textup {K}}$ , for ${I=10~\\text{µA}}$ (or $j=0.33~\\textup {A}\\,\\textup {m}^{-1})$ and ${B=1~\\textup {T}}$ , it represents $0.5\\%$ of the zero field resistance, a much higher value compared to previous reports on unidirectional magnetoresistance.",
"We ascribe the origin of this magnetoresistance to the interplay between the externally applied magnetic field and the current-induced pseudo-magnetic field in the spin-splitted subsurface states of Ge(111).",
"This unidirectional magnetoresistance is independent of the current direction with respect to the Ge crystal axes.",
"It progressively vanishes, either using a negative gate voltage due to carrier activation into the bulk (without spin-splitted bands), or by increasing the temperature due to the Rashba energy splitting of the subsurface states lower than $\\sim 58\\,k_B$ .",
"The highly developed technologies on semiconductor platforms would allow the rapid optimization of devices based on this phenomenon.",
"After decades of studies, spintronics has driven its most successful industrial revolutions in the read-out head of magnetic hard disk and in magnetic random access memory [1].",
"In both cases, a long-range magnetic order is the ultimate ingredient, since these applications rely on the giant magnetoresistance (GMR) effect [2], [3].",
"Due to the seek of magnetic ordering, the investigation of GMR has been wide and successful in ferromagnetic-based layers but still rare in semiconductors.",
"Since a connection with magnetism in semiconductors would be desirable, lots of interest has been devoted to doping semiconductors with magnetic impurities [4], [5].",
"The low solubility of magnetic ions [1] and Curie temperatures below ${200~\\textup {K}}$ [6] still limit the applicability of this technology.",
"An alternative to long-range magnetic order in semiconductors comes from spin-orbit coupling (SOC), the main core of the so-called spin-orbitronics field in semiconducting films and in topological insulators [7], [8].",
"Within this field, the investigation of magnetoresistance has recently moved over the standard ferromagnet-related effects [9], [10], [11], [12], [13], [14], [15], and a promising new type of magnetoresistance has been observed in the topological insulator Bi$_{2}$ Se$_{3}$ [13], and in the two-dimensional electron gas at the SrTiO$_3$ (111) surface [15].",
"Since, in both cases, no magnetic order is present, the effect has been related to the characteristic spin-momentum locking [13], [14], [15].",
"The detected magnetoresistance exhibits two characteristic features: it is unidirectional (i.e.",
"odd) and linear with the applied magnetic field and electrical current, therefore it has been classified among the unidirectional magnetoresistances (UMRs)[13], [14], [15].",
"Despite the same angular dependence, this SOC-related UMR has a different origin compared to another type of UMR recently investigated and involving a ferromagnetic layer [16], [17], [18], [19].",
"Here, we report the observation of UMR in Ge(111).",
"We ascribe its origin to the Rashba SOC, which generates spin-momentum locking inside the subsurface states of Ge(111).",
"Their presence and spin-texture have already been demonstrated exploiting angle and spin-resolved photoemission spectroscopy [20], [21], [22], [23].",
"Experimentally, we find that the UMR in the Ge(111) subsurface states is drastically larger compared to previous reports [13], [15].",
"We detect a maximum UMR value equivalent to $0.5\\%$ of the zero field resistance, when a magnetic field of ${1~\\text{T}}$ and a current of ${10~\\text{µA}}$ are applied at ${15~\\textup {K}}$ .",
"The effect progressively vanishes when increasing the temperature or applying a negative gate voltage due to carrier activation in the bulk valence bands of Ge and to the low value of the Rashba spin-orbit coupling ($\\sim 58\\,k_B$ )[20].",
"Figure: (color online) Sketch of the double Hall cross used for magnetoresistance 4-probe measurements.",
"The external magnetic field is applied along (θ,ϕ{\\theta ,\\varphi }) directions, θ\\theta and ϕ\\varphi being the polar and azimuth angles.",
"The current is applied along the [11 ¯0][1\\bar{1}0] crystal axis of Ge.",
"(b) 4-probe resistance versus temperature measured with an applied current of 10µA{10~\\text{µA}}.",
"The red curve corresponds to experimental data exhibiting a resistance saturation, the dashed black line shows the expected semiconducting behavior considering a thermal activation of 2.6 meV.",
"(c) Angular dependence in the (xy)(xy) plane of R l even R_{l}^{\\,\\textup {even}} at 15K{15~\\textup {K}}.",
"The applied magnetic field is 1 Tesla and the current is 10µA{10~\\text{µA}}.",
"The solid red line is a fit to the experimental data using a sine function.Figure: (color online) Angular dependence of R l odd R_{l}^{\\,\\textup {odd}} (black dots) and R t odd R_{t}^{\\,\\textup {odd}} (red dots) in: (a) the (xyxy) plane (θ=90 ∘ {\\theta =90^{\\circ }}), (b) the (yzyz) plane (ϕ=90 ∘ {\\varphi =90^{\\circ }}) and (c) the (xzxz) plane (ϕ=0 ∘ {\\varphi =0^{\\circ }}) respectively.",
"The temperature is 15K{15~\\textup {K}}, the applied current 10µA{10~\\text{µA}} and the magnetic field 1T{1~\\textup {T}}.",
"The solid black and red lines are fitting curves of R l odd R_{l}^{\\,\\textup {odd}} and R t odd R_{t}^{\\,\\textup {odd}} respectively using sine and cosine functions.",
"They superimpose to the experimental data.We perform magnetotransport measurements on a $2~\\text{µm}$ -thick Ge(111) using lithographically defined Hall bars (length ${\\ell =120~\\text{µm}}$ , width ${w=30~\\text{µm}}$ and aspect ratio $Z=\\ell /w=4$ ) as shown in Fig.",
"REF(a) (further details in the Supplementary Material).",
"We apply a DC charge current and measure the longitudinal ($R_{l}=U_{l}/I$ ) and transverse ($R_{t}=U_{t}/I$ ) resistances under the application of an external magnetic field $\\textbf {B}$ .",
"The direction of $\\textbf {B}$ is determined by its polar ($\\theta $ ) and azimuth ($\\varphi $ ) angles as shown in Fig.",
"REF(a).",
"In DC measurements, the UMR term is odd with respect to the applied current and thus defined as: ${R_{l}^{\\,\\textup {odd}}=\\,[R_{l}(I)-R_{l}(-I)]/2}$ .",
"We also measure ${R_{t}^{\\,\\textup {odd}}=\\,[R_{t}(I)-R_{t}(-I)]/2}$ and the longitudinal resistance which is even with respect to the applied current ${R_{l}^{\\,\\textup {even}}=\\,[R_{l}(I)+R_{l}(-I)]/2}$ .",
"All the measurements are carried out as a function of the temperature from ${15~\\text{K}}$ to ${295~\\text{K}}$ .",
"The conductivity is $p$ -type in the whole temperature range, at ${15~\\textup {K}}$ the carrier density reaches ${p\\approx 6\\times 10^{15}~\\text{cm}^{-3}}$ .",
"Figure: (color online) R UMR R_{\\textup {UMR}} normalized to the zero magnetic field resistance R l,0 R_{l,0} taken at ϕ=270 ∘ \\varphi =~270^{\\circ } (in %) as a function of (a) the applied current for B=1T{B=1~\\text{T}} and T=15K{T=15~\\textup {K}} (b) the magnetic field for I=10µA{I=10~\\text{µA}} and T=15K{T=15~\\textup {K}}, and (c) the temperature for B=1T{B=1~\\text{T}} and I=10µA{I=10~\\text{µA}}.",
"Red dotted lines are linear fits.",
"(d) Gate voltage dependence of R l,0 R_{l,0}, R l,max odd R_{l,\\textup {max}}^{\\textup {odd}} and ZR t,max odd {Z\\,R_{t,\\textup {max}}^{\\textup {odd}}}, R UMR =R l,max odd -ZR t,max odd {R_{\\textup {UMR}}=R_{l,\\textup {max}}^{\\,\\textup {odd}}-Z\\, R_{t,\\textup {max}}^{\\,\\textup {odd}}}.We report in Fig.",
"REF(b) the 4-probe temperature dependence of the zero magnetic field resistance $R_{l,0}$ .",
"The resistance saturation at low temperature is a fingerprint of a conduction channel in parallel with the bulk (black dashed line) which we attribute to the presence of subsurface states.",
"The angular dependence of $R_{l}^{\\,\\textup {even}}$ at ${15~\\textup {K}}$ in the $(xy)$ plane is shown in Fig.",
"REF(c) for ${I=10~\\text{µA}}$ .",
"This MR signal exhibits maxima (resp.",
"minima) for ${\\textbf {B}\\parallel \\hat{\\textbf {y}}}$ , $\\varphi =90^{\\circ }$ (resp.",
"${\\textbf {B}\\parallel \\hat{\\textbf {x}}}$ , $\\varphi =0^{\\circ }$ ).",
"Since the sign is not reversed when reversing the magnetic field direction, we call this term anisotropic magnetoresistance (AMR) by analogy with ferromagnets.",
"At ${15~\\textup {K}}$ , we find an AMR of ${0.4\\%}$ under a magnetic field of 1 T. The same behaviors are obtained for angular dependencies within $(zy)$ and $(zx)$ planes.",
"In Fig.",
"REF , we report the angular dependence of $R_{l}^{\\,\\textup {odd}}$ and $R_{t}^{\\,\\textup {odd}}$ in the ($xy$ ), ($xz$ ) and ($yz$ ) planes for ${B=1~\\text{T}}$ , ${I=10~\\text{µA}}$ at ${15~\\textup {K}}$ .",
"We observe a unidirectional behavior for both longitudinal and transverse resistances: the maximum (minimum) of $R_{l}^{\\,\\textup {odd}}$ is observed for ${\\textbf {B}\\parallel \\mathbin {\\hbox{{\\hfil $\\scriptscriptstyle -$\\hfil \\cr {\\hspace{-1.27501pt}}$\\scriptstyle ({+})$\\cr }}}}\\hat{\\textbf {y}}$$, and the maximum (minimum) of $ Rt odd$ is observed for $Bx$.",
"Thus, experimentally, $ Rl odd=-Rl,max odd ()()$ and $ Rt odd= -Rt,max odd ()()$.",
"These functions are shown as solid lines in Fig.~\\ref {Fig2}.The angular dependence of the transverse resistance reveals the presence of the Nernst effect due to a current-induced vertical temperature gradient (along $z$) in the Ge(111) film.",
"This effect generates spurious thermal UMR signal in the longitudinal resistance.",
"The Nernst effect contribution to $ Rl odd$ can be written as: $ Rl,max odd,Nernst=Z Rt,max odd$, with $ Z$ being the aspect ratio of the channel ($ Z=4$ in our case) \\cite {Avci2015}.",
"Hence, to remove the Nernst effect contribution from the longitudinal signal, we study $ RUMR=Rl,max odd-Z Rt,max odd$.",
"We find that the Nernst effect contribution is negligible at $ 15 K$ for low currents while it dominates when approaching room temperature and/or applying large currents (more details are given in the Supplementary Material).$ In Fig.",
"REF we investigate the dependencies of $R_{\\textup {UMR}}$ on the applied current [Fig.",
"REF(a)], magnetic field [Fig.",
"REF(b)], temperature [Fig.",
"REF(c)] and gate voltage [Fig.",
"REF(d)].",
"The signal is normalized with respect to the zero field resistance $R_{{l},0}$ at the corresponding current.",
"In agreement with previous reports on UMR generated by spin-momentum locking [16], [17] we observe a signal proportional to the current and the magnetic field.",
"${R_{\\textup {UMR}}/R_{l,0}}$ is maximum and almost constant at low temperature ($T<{20~\\text{K}}$ ) and sharply decreases when the temperature becomes comparable to the Rashba spin-splitting energy ($\\approx 60~\\text{K}$ ).",
"As shown in Fig.",
"REF(d), the application of a top gate voltage modulates the channel resistance $R_{l,0}$ .",
"In Fig.",
"REF(d), we also plot both the longitudinal and transverse odd resistance components as a function of the gate voltage.",
"The transverse component we attribute to the Nernst effect stays constant with the gate voltage.",
"This observation is consistent with the fact that this effect is due to vertical temperature gradient in the Ge(111) film and is almost unaffected by the top gate voltage.",
"By contrast, the longitudinal component attributed to the UMR effect is much affected by the gate voltage: it increases from $V_g=-10~\\text{V}$ to $V_g=+10~\\text{V}$ by a factor $\\approx 3$ .",
"$R_{\\textup {UMR}}$ cancels out at $V_g=-10~\\text{V}$ and increases from $-10~\\text{V}$ to $+10~\\text{V}$ .",
"Figure: (color online) (a) Schematics of the Ge(111) electronic band structure (in agreement with Ref.",
"Ohtsubo2013) showing the bulk conduction and valence bands.",
"The Fermi level is at a position corresponding to a pp-doped film.",
"Subsurface states are located just above the maximum of the bulk valence band and are crossed by the Fermi level.",
"They are spin-splitted by the Rashba and atomic spin-orbit interactions.",
"(b) Fermi contours of the subsurface states.",
"The outer (inner) contour is named C (D) with clockwise (counter-clockwise) spin helicity.",
"(c) Illustration of the combined effects of the applied magnetic field 𝐁\\textbf {B} and the current dependent pseudo-magnetic field 𝐁 E \\textbf {B}_{\\textup {E}} on the resistivity of subsurface states for a single contour (D here).",
"The contour is shifted by +Δ𝐤+~\\Delta \\textbf {k} due to the application of a current density 𝐣\\textbf {j} along +𝐱 ^+~\\hat{\\textbf {x}}.",
"The current direction and spin helicity set the pseudo-magnetic field 𝐁 E \\textbf {B}_{\\textup {E}}.To make a comparison with previous results on different systems, we can define a figure of merit $\\eta $ .",
"Since the UMR signal is proportional to the current and magnetic field, a natural definition is: ${\\eta =R_{\\textup {UMR}}/(R_{l,0}\\,j\\,B)}$ .",
"At ${15~\\text{K}}$ , in Ge(111), we obtain ${\\eta =2.5\\times 10^{-4}~\\textup {cm}^{2}/(\\textup {A\\,T})}$ when considering the charge current flowing in the whole Ge(111) film (${2~\\text{µm}}$ ) and ${\\eta =4.2\\times 10^{-7}~\\textup {cm}^{2}/(\\textup {A\\,T})}$ if we consider that the current completely flows within the spatial extension of the subsurface states (10 atomic layers from Ref. Aruga2015).",
"In the worst case scenario, the value of $\\eta $ obtained in Ge(111) is orders of magnitude larger than the one of SrTiO$_3$ at 7 K [${\\eta =2\\times 10^{-9}~\\textup {cm}^{2}/(\\textup {A\\,T})}$ from Ref.",
"He2018] and the one of Bi$_2$ Se$_3$ at ${60~\\text{K}}$ [${\\eta =2\\times 10^{-11}~\\textup {cm}^{2}/(\\textup {A\\,T})}$ from Ref. He2017].",
"In this second case, if we compare the results extrapolated at ${58~\\text{K}}$ for Ge(111) we still obtain a larger $\\eta $ value [${3\\times 10^{-8}~\\textup {cm}^{2}/(\\textup {A\\,T})}$ ].",
"At variance with previously reported systems [13], [15] the UMR is isotropic with respect to the direction of the current flow in the surface Brillouin zone (SBZ).",
"In fact, in the data shown in Figs.",
"REF$-$REF , the current flows along the $\\Gamma \\text{M}$ direction of the Ge(111) SBZ, but no difference, within the experimental error, is detected with the current flowing along other reciprocal lattice directions (see Supplementary Material).",
"In Refs.",
"He2017,He2018, the magnetoresistance is affected by the direction of the current flow in the SBZ, indicating that, in such a case, the UMR originates from the out-of-plane spin texture.",
"In the case of Ge, this contribution appears to be negligible.",
"We thus propose an alternative mechanism, in which the UMR in Ge(111), results from a combination of the applied magnetic field and the current-induced pseudo-magnetic field in the spin-splitted subsurface states of Ge(111) shown in Fig.",
"REF(a).",
"Ge(111) subsurface states are located close to the top of the valence bands and can only contribute to transport in $p$ -type Ge(111) [21].",
"This interpretation is supported by the fact that we do not observe this effect for $n$ -type Ge(111) (see Supplementary Material).",
"It also explains the gate voltage dependence of $R_{\\textup {UMR}}$ in Fig.",
"REF(d).",
"Applying negative gate voltage shifts the Fermi level down into the valence band which leads to the activation of bulk conduction and $R_{\\textup {UMR}}\\approx 0~\\Omega $ for $V_g=-10~\\text{V}$ .",
"At variance, by ramping the gate voltage from ${-10~\\text{V}}$ to ${+10~\\text{V}}$ , the Fermi level shifts into the subsurface states thus increasing $R_{\\textup {UMR}}$ .",
"Finally, this interpretation also explains the temperature dependence of the UMR.",
"By increasing the temperature, bulk conduction in the valence band is activated and shorts the subsurface states.",
"Moreover, the Rashba spin-orbit coupling of $\\sim 58\\,k_B$ in Ge subsurface states [21] becomes negligible with respect to $k_{\\textup {B}}T$ suppressing spin-momentum locking.",
"For the Fermi level crossing the subsurface states as shown in Fig.",
"REF(a), the Fermi contour is made of two concentric rings [C and D in Fig.",
"REF(b)] with opposite spin helicities.",
"To describe the magnetotransport inside the subsurface states, we consider the following model Hamiltonian : $ {H}=-\\frac{\\hbar ^{2}k^{2}}{2m^{*}}+\\alpha \\,(\\textbf {k}\\times \\mathbf {})\\cdot \\hat{\\textbf {z}}+g\\mu _{\\textup {B}}\\mathbf {}\\cdot \\textbf {B},$ with $\\hbar $ being the reduced Planck constant, $m^{*}$ the effective mass of holes in the subsurface states, $\\alpha $ the Rashba spin-orbit interaction, $\\mathbf {}$ the vector of Pauli matrices, $g$ the Landé factor and $\\mu _{\\textup {B}}$ the Bohr magneton.",
"When a 2D charge current density $\\textbf {j}$ flows in the subsurface states, in the Boltzmann approach, the momentum acquires an extra component ${\\Delta \\textbf {k}=\\beta \\textbf {j}}$ with ${\\beta =4\\pi /(e v_{\\textup {F}}k_{\\textup {F}})}$ , $v_{\\textup {F}}$ and $k_{\\textup {F}}$ the Fermi velocity and wavevector we consider ($e=\\vert e \\vert $ ).",
"A well-known consequence of such shifts of Rashba Fermi contours is the Rashba-Edelstein spin polarization [7] due to the unbalance between the opposite spin polarizations induced by the shifts in the same direction of the Rashba-splitted Fermi contours of opposite helicity.",
"In parallel with the Rashba-Edelstein effect, the shift $\\Delta \\textbf {k}$ introduces a current-induced out-of-equilibrium energy term which, from Eq.",
"REF , is equal to $\\alpha (\\Delta \\textbf {k}\\times \\mathbf {})\\cdot \\hat{\\textbf {z}}=\\alpha \\beta (\\textbf {z}\\times \\textbf {j})\\cdot \\mathbf {}$ and acts on the spins as a pseudo-magnetic field ${\\textbf {B}_{\\textup {E}}=(\\alpha \\beta /g\\mu _{\\textup {B}})\\,\\hat{\\textbf {z}}\\times \\textbf {j}}$ .",
"As illustrated in Fig.",
"REF(c), for a current along $\\pm ~\\hat{\\textbf {x}}$ with $\\alpha > 0$ , this field is directed along $\\pm ~\\hat{\\textbf {y}}$ and proportional to the current density.",
"In the presence of an applied magnetic field $\\textbf {B}$ , the spin of the subsurface states is submitted to $\\textbf {B}+\\textbf {B}_{\\textup {E}}$ , $\\textbf {B}_{\\textup {E}}$ increasing or decreasing the effect of the $y$ component of $\\textbf {B}$ for currents either along $\\textup {+}$ or $\\textup {-}$ $\\hat{\\textbf {x}}$ .",
"In the same way, still for $\\alpha > 0$ for $\\textbf {j}$ along $+\\hat{\\textbf {x}}$ and $\\textbf {B}_{\\textup {E}}$ along $\\hat{\\textbf {y}}$ , there is addition or subtraction of the effects of $\\textbf {B}$ and $\\textbf {B}_{\\textup {E}}$ for opposite orientations of $\\textbf {B}$ along $\\hat{\\textbf {y}}$ .",
"The physics of the UMR thus comes from the pseudo-field $\\textbf {B}_{\\textup {E}}$ induced by the out-of-equilibrium situation of a current flow and acting on the spins.",
"We can go a little further by assuming that the AMR term shown in Fig.",
"REF(c) (the only MR in the limit $j \\rightarrow 0$ ) is also due to the effect of $\\textbf {B}$ on the spins.",
"We thus follow Taskin $\\textit {et al.",
"}$ [29] who explain the AMR of Rashba systems by the re-introduction of some backscatterings by a partial re-alignement of the spins by $\\textbf {B}$ and we neglect contributions such as the effect of the Lorentz force on the trajectories.",
"Then, in the situation of finite $j$ , we add $\\textbf {B}_{\\textup {E}}$ to $\\textbf {B}$ in the $B^2$ term of the AMR to derive the expression of UMR.",
"The AMR term can be written as: $ {\\left(\\Delta R/R\\right)_{\\textup {AMR}}=-A\\,B^2\\,\\cos ^2{(\\varphi )}=AB_{y}^2-A\\,B^2}\\,$ Where ${A\\approx 0.004}$ .",
"Adding $B_{\\textup {E}y}=\\alpha \\beta j/g\\mu _{\\textup {B}}$ to $B_\\textup {y}$ , and keeping only the terms of first order in $j$ gives : $ \\Delta R/R=-AB^2\\textup {cos}^2(\\varphi )+2A(\\alpha \\beta /g\\mu _{\\textup {B}})\\,j\\,B\\,\\textup {sin}(\\varphi )\\,$ Where the second term, proportional to $j\\,B$ , is the UMR.",
"Our experimental results with an UMR proportional to $j\\,B\\,\\textup {sin}(\\varphi )$ , see [Fig.",
"REF ], correspond to a negative value of the Rashba coefficient $\\alpha $ , that is to the clockwise chirality of the spin orientation in the outer Fermi contour.",
"This chirality is in agreement with the chirality derived from spin-resolved ARPES measurements for the subsurface states inside Ge at Ge/Bi interfaces, as shown in Fig.",
"3a of [Ohtsubo2010].",
"Quantatively, taking reasonable values for the parameters in the expression of the UMR amplitude.",
"By setting ${B=1~\\text{T}}$ , ${j=0.33~\\textup {A}\\,\\textup {m}^{-1}}$ in the subsurface states, ${\\alpha =-0.2~\\textup {eV}\\cdot \\textup {Å}}$ (in [20], this value corresponds to Bi covered subsurface states, in our case it is probably an upper bound), ${k_{\\textup {F}}=0.025~\\textup {Å}^{-1}}$ (Rashba splitting $\\vert \\alpha k_{\\textup {F}}\\vert =5$ meV$\\sim 58\\,k_B$ ), $m^{*}=0.4\\,m_e$ [28], $m_e$ being the electron mass, ${v_{\\textup {F}}=\\hbar k_{\\textup {F}}/m^{*}}$ and ${g=2}$ , we find a UMR amplitude of ${\\approx 0.2\\%}$ .",
"This value is in good agreement with our low temperature experimental data.",
"We indeed find a maximum value of $0.5\\%$ at ${15~\\textup {K}}$ .",
"Therefore, by using simple arguments, we capture the physics of UMR in the Ge Rashba-splitted subsurface states.",
"In conclusion, we performed magnetoresistance measurements on Ge(111) and detected a unidirectional magnetoresistance (UMR) which scales linearly with both the current and the applied magnetic field.",
"We ascribe the UMR to the spin-momentum locking generated by the Rashba effect in the subsurface states of Ge(111) and interpret our results in a simple model relating the UMR to the Rashba coefficient and the characteristic parameters of the subsurface states.",
"Such unidirectional effects can be expected in any Rashba 2DEG and can be used to obtain information about the electronic structure details.",
"The amplitude of the detected UMR signal is much larger than the ones previously reported.",
"We also showed that this UMR is tunable by turning on and off the Rashba coupling in the conduction channel by applying a gate voltage.",
"Ultimately, these findings lead towards the development of a semiconductor-based spin transistor where the spin information can be manipulated by a gate-tunable Rashba field.",
"The authors acknowledge the financial support from the ANR project ANR-16-CE24-0017 TOPRISE.",
"One of us (AF) acknowledges fruitful discussions with A .Dyrdal and J. Barnas (Poznan University), as well as with S. Zhang (University of Arizona)."
]
] | 1906.04457 |
[
[
"Exact WKB and abelianization for the $T_3$ equation"
],
[
"Abstract We describe the exact WKB method from the point of view of abelianization, both for Schr\\\"odinger operators and for their higher-order analogues (opers).",
"The main new example which we consider is the \"$T_3$ equation,\" an order $3$ equation on the thrice-punctured sphere, with regular singularities at the punctures.",
"In this case the exact WKB analysis leads to consideration of a new sort of Darboux coordinate system on a moduli space of flat $\\mathrm{SL}(3)$-connections.",
"We give the simplest example of such a coordinate system, and verify numerically that in these coordinates the monodromy of the $T_3$ equation has the expected asymptotic properties.",
"We also briefly revisit the Schr\\\"odinger equation with cubic potential and the Mathieu equation from the point of view of abelianization."
],
[
"Exact WKB",
"The exact WKB method is a scheme for studying the monodromy (or bound states, or more generally Stokes data) of linear scalar differential equations.",
"This method was initiated in [1], [2], [3], [4] and subsequently developed in a large body of literature.",
"Its origin is in the study of Schrödinger equations, of the form $ \\left[ \\hbar ^{2} \\partial _z^2 + P(z) \\right] \\psi (z) = 0,$ where $P(z)$ is holomorphic or meromorphic; most of the literature is concerned with this case.",
"For some useful reviews see [5], [6], [7].",
"More recently the exact WKB investigation of higher-order analogues of Schrödinger equations has been taken up, e.g.",
"in [8], [9], [10], [11], [12], [13], [14]."
],
[
"Abelianization",
"The method of [15], [16] leads to a new geometric reformulation of exact WKB, both for Schrödinger operators and their higher-order analogues.",
"In this reformulation, the key step in exact WKB is a process of “abelianization” which replaces a flat ${\\mathrm {SL}}(K)$ -connection $\\nabla $ over a surface $C$ by a flat ${\\mathrm {GL}}(1)$ -connection $\\nabla ^\\mathrm {ab}$ over a $K$ -fold covering $\\Sigma \\rightarrow C$ .Throughout this paper ${\\mathrm {SL}}(K)$ means ${\\mathrm {SL}}(K,$ , and ${\\mathrm {GL}}(1)$ means ${\\mathrm {GL}}(1, = \\times $ .",
"Some aspects of this abelianization process and its relation to exact WKB have been further developed in [17], [18], [19], [20].",
"In sec:wkb-review we review the exact WKB method for Schrödinger operators, i.e.",
"the $K=2$ case, from the perspective of abelianization.",
"The aim is not to break any really new ground, but just to explain the theory from the abelianization point of view, which is a bit different from the conventional language of exact WKB."
],
[
"Voros symbols for Schrödinger equations",
"The exact WKB analysis of Schrödinger equations revolves around certain complex-valued functions $\\mathcal {X}_\\gamma (\\hbar )$ known as the Voros symbols.In the main text we will distinguish several different variants of the functions $\\mathcal {X}_\\gamma $ .",
"The functions $\\mathcal {X}_\\gamma ^{\\mathrm {intro}}$ we use in the introduction are related to those appearing in the main text by $\\mathcal {X}_\\gamma ^{\\mathrm {intro}}(\\hbar ) = \\mathcal {X}_\\gamma ^{\\vartheta = \\arg \\hbar }(\\hbar )$ .",
"In the language of abelianization, the $\\mathcal {X}_\\gamma (\\hbar )$ are the holonomies of the ${\\mathrm {GL}}(1)$ -connection $\\nabla ^\\mathrm {ab}$ around 1-cycles $\\gamma $ on the double cover $\\Sigma $ .",
"The $\\mathcal {X}_\\gamma (\\hbar )$ can be expressed as products of Wronskians of distinguished local solutions $\\psi _i(z,\\hbar )$ of (REF ).",
"The solutions $\\psi _i(z,\\hbar )$ have a dual role: On the one hand, the $\\psi _i(z,\\hbar )$ are produced by Borel resummation of the perturbative WKB series.",
"As a result, one has good control over their behavior as $\\hbar \\rightarrow 0$ , which gives good control over the behavior of $\\mathcal {X}_\\gamma (\\hbar )$ as $\\hbar \\rightarrow 0$ .",
"On the other hand, the $\\psi _i(z,\\hbar )$ can be characterized intrinsically: either as asymptotically decaying solutions as $z$ approaches a singularity, or as eigenvectors of the monodromy as $z$ goes around a loop.",
"This allows one to identify the $\\mathcal {X}_\\gamma (\\hbar )$ as familiar coordinate functions on a moduli space of flat ${\\mathrm {SL}}(2)$ -connections.",
"In a generic enough situation these are the “Fock-Goncharov coordinates” introduced in [21], as explained in [15], [7], [17], [22].",
"In less generic situations, as discussed in [15], [17], one can get the “exponentiated complexified Fenchel-Nielsen coordinates” studied in [23], [24], [25], or other slight variants.",
"The combination of these two points of view on the $\\mathcal {X}_\\gamma (\\hbar )$ is responsible for much of the power of the exact WKB method.",
"In this paper we revisit this story in two examples, again with the aim of showing how exact WKB works in the language of abelianization, and paving the way for the higher-order case: First, in sec:cubic, we discuss the Schrödinger equation with cubic potential.",
"This is an instance of (REF ) with $P(z) = z^3 - u$ .",
"We treat this example relatively briefly.",
"We consider only the choice $u = 1$ and real $\\hbar $ , for which the $\\mathcal {X}_\\gamma (\\hbar )$ are Fock-Goncharov coordinates.",
"Second, in sec:mathieu, we discuss the Mathieu equation.",
"This is an instance of (REF ) with $P(z) = \\frac{1}{z^3} - \\frac{2E-\\frac{1}{4} \\hbar ^2}{z^2} + \\frac{1}{z}$ .",
"We focus on the cases of real $\\hbar > 0$ and $E > 1$ or $E < -1$ .",
"For $E < -1$ the $\\mathcal {X}_\\gamma (\\hbar )$ turn out to be Fenchel-Nielsen coordinates, and we explain their application to the bound state problem for the modified Mathieu equation; for $E > 1$ the $\\mathcal {X}_\\gamma (\\hbar )$ turn out to be a slight variant of the Fock-Goncharov coordinates, and we explain their application to the quasiperiodic solutions of the ordinary Mathieu equation.",
"In either case we do not do anything really new, except perhaps that we give a new version of the exact quantization condition for the Mathieu equation, (REF ), and use it to derive the width of the gaps at small $\\hbar $ ."
],
[
"Exact WKB for order 3 equations",
"The next natural test bed is the case of ${\\mathrm {SL}}(3)$ -opers: this means order 3 equations of the general form $ \\left[ \\partial _z^3 + \\hbar ^{-2} P_2 \\partial _z + (\\hbar ^{-3} P_3 + \\frac{1}{2} \\hbar ^{-2} P^{\\prime }_2) \\right] \\psi (z) = 0.$ In sec:wkb-higher we describe an extension of the exact WKB method to this case, again in the language of abelianization.",
"This extension comes from combining the methods of [16] with the scaling limit of [26], now applied to families of ${\\mathrm {SL}}(3)$ -connections.",
"As in the order 2 case, the theory is founded on the existence of distinguished local solutions $\\psi _i(z,\\hbar )$ of (REF ), with $\\hbar \\rightarrow 0$ asymptotics given by the WKB series.",
"In contrast to the order 2 case, however, as far as we know, there are not yet theorems guaranteeing the existence of these local solutions.",
"Thus the higher-order exact WKB method is not yet on solid footing.",
"Nevertheless, we press on, making the assumption that the $\\psi _i(z,\\hbar )$ do exist.",
"Then, as before, one can use them to construct functions $\\mathcal {X}_\\gamma (\\hbar )$ , which we call spectral coordinates because of their relation to abelianization; one might also have called them higher-order Voros symbols.",
"Now the question arises: can we identify the $\\mathcal {X}_\\gamma (\\hbar )$ as some concrete coordinate functions on moduli of flat ${\\mathrm {SL}}(3)$ -connections — or, essentially equivalently, can we give an intrinsic characterization of the $\\psi _i(z,\\hbar )$ in terms of their monodromy properties?",
"For some examples of equations (REF ), the expected picture is well understood, and similar to the order 2 case.",
"One such situation arises if $C$ is a punctured surface, $P_2$ is meromorphic and generic with generic residues, and $P_3$ is small compared to $P_2$ .",
"In this case the $\\psi _i(z,\\hbar )$ in most of $C$ can be described by beginning with the filtrations induced by the asymptotic growth rate at the punctures, and then using the linear algebra of “snakes” as introduced by Fock-Goncharov [21], [27].",
"The $\\mathcal {X}_\\gamma (\\hbar )$ then turn out to be higher-rank Fock-Goncharov coordinates.",
"Another such situation arises if $(P_2, P_3)$ is a generalized Strebel pair of length-twist type as defined in [18]; then the $\\psi _i(z,\\hbar )$ can be characterized as monodromy eigenvectors, and the $\\mathcal {X}_\\gamma (\\hbar )$ turn out to be “higher length-twist” coordinates generalizing Fenchel-Nielsen.",
"In this paper we do not revisit these cases."
],
[
"The $T_3$ equation",
"Instead, in sec:t3, we turn our attention to the $T_3$ equation.",
"This is a specific instance of (REF ), defined on the Riemann surface $C = \\mathbb {CP}^1$ with three generic regular singularities (“full punctures” in the physics literature), and depending on a parameter $u \\in : namely, we takein (\\ref {eq:deg-3-oper-intro})\\begin{equation} P_2 = \\frac{9 \\hbar ^2 z}{(z^3 - 1)^2}, \\qquad P_3 = \\frac{u}{(z^3-1)^2}.\\end{equation}This equation is a particularly interesting test case.One way to understand this is to remark that this family of operscorresponds (in the sense of {sec:physics-intro} below)to a specific $ N=2$ superconformal quantum field theory, the Minahan-Nemeschanskytheory with flavor symmetry $ E6$ \\cite {Minahan:1996fg,Gaiotto:2009we},which is known to be difficult to study by conventional Lagrangianfield theory methods.$ As expected, in this case we meet new difficulties.",
"One source of these difficulties is that the Stokes graphs can be rather wild for general $(u,\\hbar )$ .",
"We thus restrict ourselves to only the simplest situation, which arises when $u^{\\prime } = u / \\hbar ^3$ is real and positive; in this case the Stokes graph is actually very simple.",
"It is shown in fig:circle-network below.",
"Then we find that the $\\psi _i(z,\\hbar )$ are solutions of an interesting linear algebra problem: relative to the local basis $\\lbrace \\psi _1, \\psi _2, \\psi _3 \\rbrace $ near $z=0$ , the monodromies $\\mathbf {A}$ , $\\mathbf {B}$ , $\\mathbf {C}$ around the three punctures must have zeroes in specific positions, $\\mathbf {A}&= \\begin{pmatrix} * & 0 & * \\\\ * & * & * \\\\ * & * & * \\end{pmatrix}, \\quad &\\mathbf {B}&= \\begin{pmatrix} * & * & * \\\\ * & * & 0 \\\\ * & * & * \\end{pmatrix}, &\\mathbf {C}&= \\begin{pmatrix} * & * & * \\\\ * & * & * \\\\ 0 & * & * \\end{pmatrix}, \\\\\\mathbf {A}^{-1} &= \\begin{pmatrix} * & * & * \\\\ 0 & * & * \\\\ * & * & * \\end{pmatrix}, &\\mathbf {B}^{-1} &= \\begin{pmatrix} * & * & * \\\\ * & * & * \\\\ * & 0 & * \\end{pmatrix}, \\quad &\\mathbf {C}^{-1} &= \\begin{pmatrix} * & * & 0 \\\\ * & * & * \\\\ * & * & * \\end{pmatrix}.$ The best approach we have found to this linear algebra problem involves a bit of algebraic geometry, as we describe in sec:enumerating-special-bases: we reduce the problem to finding fixed points of a certain degree 64 birational automorphism of $\\mathbb {CP}^2$ , and then identify these fixed points with singularities in the fibers of a certain rational elliptic surface.",
"At any rate, once this problem has been solved, we can then compute the spectral coordinates $\\mathcal {X}_\\gamma (\\hbar )$ for the $T_3$ equation.",
"The concrete formulas are given in (REF ) below, reproduced here: for a basis $\\lbrace \\gamma _A, \\gamma _B\\rbrace $ of $H_1(\\Sigma ,\\mathbb {Z})$ ,Here and elsewhere in this paper, unless explicitly noted, $\\sqrt{\\cdot }$ denotes the principal branch of the square root.",
"$\\mathcal {X}_{\\gamma _A} &= \\frac{[\\psi _2 , \\psi _3 , \\psi _1]}{[\\mathbf {C}^{-1} \\psi _3 , \\mathbf {A}\\psi _2 , \\psi _1]}, \\\\\\mathcal {X}_{\\gamma _B} &= \\sqrt{- \\frac{[\\mathbf {C}\\psi _1 , \\mathbf {B}^{-1} \\psi _2 , \\psi _3][\\mathbf {C}\\psi _1 , \\psi _1 , \\psi _3] [\\psi _2 , \\mathbf {A}\\psi _2 , \\psi _1] [\\mathbf {B}\\psi _3 , \\mathbf {A}^{-1} \\psi _1 , \\psi _2] [\\mathbf {B}\\psi _3 , \\psi _3 , \\psi _2]}{[\\psi _2 , \\mathbf {B}^{-1} \\psi _2 , \\psi _3][\\mathbf {C}^{-1} \\psi _3 , \\mathbf {A}\\psi _2 , \\psi _1] [\\mathbf {C}^{-1} \\psi _3 , \\psi _3 , \\psi _1] [\\psi _1 , \\psi _3 , \\psi _2] [\\psi _1 , \\mathbf {A}^{-1} \\psi _1 , \\psi _2]} } $ where $[\\psi , \\psi ^{\\prime }, \\psi ^{\\prime \\prime }]$ means the Wronskian of the three solutions $\\psi , \\psi ^{\\prime }, \\psi ^{\\prime \\prime }$ .",
"This gives a local Darboux coordinate system on the moduli space of flat ${\\mathrm {SL}}(3)$ -connections with unipotent holonomy on the thrice-punctured sphere.",
"As far as we know, this coordinate system has not been considered before.",
"What our computations say is that these particular coordinates arise naturally from the WKB analysis of the equation (REF ), ().",
"Combining our conjectures and computations, one can extract a concrete prediction: the quantities (REF ), computed from the monodromy of the $T_3$ equation, should have a specific asymptotic series expansion, with leading behavior $\\mathcal {X}_\\gamma \\sim \\exp (Z_\\gamma / \\hbar )$ , as $\\hbar \\rightarrow 0$ in an appropriate sector.",
"Here the constants $Z_\\gamma \\in are periods, $ Z= P31/3$,given explicitly in (\\ref {eq:T3-periods}) below.We have implemented this computation numerically and find very good agreement(see e.g.",
"{fig:X-numerics} below).We regard this as evidence that the higher-order exact WKB method indeedworks.$"
],
[
"Integral equations and analytic structures",
"A relatively recent development in the exact WKB method is the discovery that the functions $\\mathcal {X}_\\gamma (\\hbar )$ are, quite generally, solutions of integral equations in the $\\hbar $ -plane.",
"A general form of these integral equations was formulated in [26] (see (REF ) below), generalizing some cases which had been known before.",
"In particular, the equations closely resemble the thermodynamic Bethe ansatz, and some cases literally match with the high-temperature (chiral) limit of the thermodynamic Bethe ansatz for specific integrable models; these cases had been studied as part of the ODE-IM correspondence, explained in e.g.",
"[30], [31].",
"One way to motivate these equations is to argue that their solutions solve a certain Riemann-Hilbert problem: they have the same analytic structure and $\\hbar \\rightarrow 0$ asymptotic properties as the desired functions $\\mathcal {X}_\\gamma (\\hbar )$ have.",
"One hopes that these properties are sufficient to characterize $\\mathcal {X}_\\gamma (\\hbar )$ .For a more elementary example, if a function $x(\\hbar )$ is known to be holomorphic for $\\hbar \\in \\times $ , $x(\\hbar ) \\rightarrow c$ as $\\hbar \\rightarrow 0$ , and $x(\\hbar )$ is bounded as $\\hbar \\rightarrow \\infty $ , then we can conclude $x(\\hbar ) = c$ .",
"The general idea of determining the $\\mathcal {X}_\\gamma (\\hbar )$ from their analytic properties has appeared before in the exact WKB literature, e.g.",
"in [3] under the name “analytic bootstrap.” In another direction, the same Riemann-Hilbert problem has been studied recently in relation to the topological string [32].",
"In various sections of this paper we consider integral equations for our $\\mathcal {X}_\\gamma (\\hbar )$ : In sec:integral-equations-cubic we review the integral equations obeyed by $\\mathcal {X}_\\gamma (\\hbar )$ for the Schrödinger equation with cubic potential.",
"In this case the $\\mathcal {X}_\\gamma (\\hbar )$ are Fock-Goncharov coordinates.",
"This case is by now reasonably well understood in the literature; it was discussed already in [30], in [26], and more recently in [33].",
"In sec:integral-equations-mathieu we propose integral equations for $\\mathcal {X}_\\gamma (\\hbar )$ for the Mathieu equation, in the case where $\\mathcal {X}_\\gamma (\\hbar )$ are complexified exponentiated Fenchel-Nielsen coordinates.",
"This case is somewhat more difficult; we find definite equations, which do seem to be satisfied by the $\\mathcal {X}_\\gamma (\\hbar )$ in numerical experiments, but we are not able to use the equations to compute the $\\mathcal {X}_\\gamma (\\hbar )$ directly.",
"Finally in sec:integral-equations-t3 we write one version of the integral equations for the $\\mathcal {X}_\\gamma (\\hbar )$ of the $T_3$ equation.",
"Here, in order to determine the equations completely, one needs to find a closed formula for a certain transformation $\\mathbf {S}_{0,\\frac{\\pi }{3}}$ relating two different branches of $\\mathcal {X}_\\gamma (\\hbar )$ ; we formulate this problem carefully but do not solve it.",
"We also explain how one can approximate $\\mathbf {S}_{0,\\frac{\\pi }{3}}$ using some integer invariants previously computed in [34] (BPS indices in the Minahan-Nemeschansky $E_6$ theory), and give some numerical evidence that this approximation works.",
"All of these analyses just barely scratch the surface; there is much more to do here.",
"A closely related issue is that of the analytic structure of the maximal analytic continuation of $\\mathcal {X}_\\gamma (\\hbar )$ from a given initial $\\hbar $ .",
"Zeroes, poles, and branch cuts can all occur: In sec:cubic-analytic-cont we briefly recall the analytic properties of the Fock-Goncharov coordinates $\\mathcal {X}_\\gamma (\\hbar )$ for the Schrödinger equation with cubic potential.",
"These are relatively simple: the maximal analytic continuation is defined on a fivefold cover branched only at $\\hbar = 0$ , with a concrete monodromy action (REF ).",
"The $\\mathcal {X}_\\gamma (\\hbar )$ can also have poles or zeroes, which come from bound states of the Schrödinger equation; they occur in infinite discrete families.",
"In sec:mathieu-analytic-cont we describe the analytic properties of the complexified exponentiated Fenchel-Nielsen coordinates $\\mathcal {X}_\\gamma (\\hbar )$ for the Mathieu equation in the regime $E < -1$ .",
"These are a bit more complicated: there is infinite-order monodromy (REF ) around $\\hbar = 0$ , and also order-2 monodromy (REF ) around an infinite discrete family of other points.",
"The latter points are analytically continued versions of the edges of the bands/gaps in the Mathieu spectrum.",
"In this case we did not explore the positions of poles or zeroes.",
"In sec:t3-analytic-cont we consider the analytic properties of our new coordinates $\\mathcal {X}_\\gamma (\\hbar )$ for the $T_3$ equation.",
"The picture we find, through numerical experimentation, is that the maximal analytic continuation lives on a threefold cover, with order-3 monodromy around $\\hbar = 0$ , and order-2 monodromy around 6 other points.",
"In terms of the coordinate $u^{\\prime } = u / \\hbar ^3$ , the picture is simpler: there is only order-2 branching, around the points $u^{\\prime } = \\pm u^{\\prime }_*$ where $u^{\\prime }_* \\approx 0.041992794$ .",
"The $\\mathcal {X}_\\gamma (\\hbar )$ can also have poles or zeroes, which numerically do appear to occur, in infinite discrete families."
],
[
"Supersymmetric QFT",
"Over the last decade it has turned out that exact WKB is closely connected to ${\\mathcal {N}}=2$ supersymmetric quantum field theory in four dimensions.",
"This work was motivated by an attempt to understand these connections better.",
"They arose in two different ways: On the one hand, [35] discovered a new connection between Nekrasov's $\\Omega $ -background partition function $\\mathbf {Z}$ in ${\\mathcal {N}}=2$ theories and quantum integrable systems.",
"For ${\\mathcal {N}}=2$ theories of class $S$ , the AGT correspondence says $\\mathbf {Z}$ is related to Liouville conformal blocks on a Riemann surface $C$ [36], while the quantum integrable systems turned out to be spectral problems for Schrödinger equations (REF ) on $C$ .",
"The investigation of this connection between Schrödinger equations, Liouville conformal blocks and topological strings was carried out using WKB methods beginning in [37], [38].",
"This connection has led to a flow of ideas in both directions.",
"For example, it has been proposed that using exact WKB one can obtain “nonperturbative” information about $\\mathbf {Z}$ , e.g.",
"[39], [40], [41]; also, techniques from the study of $\\mathbf {Z}$ , such as the holomorphic anomaly equations, have been imported back to WKB, e.g.",
"[42], [43].",
"On the other hand, studying BPS states and supersymmetric defects in ${\\mathcal {N}}=2$ theories of class $S$ , [15], [16] were led to develop a version of exact WKB which applies to a slightly different sort of equation: instead of the 1-parameter families (REF ) or (REF ) parameterized by $\\hbar \\in \\times $ , [15], [16] treat a 2-parameter family of covariant constancy equations for flat ${\\mathrm {SL}}(K)$ -connections $\\nabla _{R,\\zeta }$ , parameterized by $R \\in \\mathbb {R}_+$ and $\\zeta \\in \\times $ : The family of flat connections $\\nabla _{R,\\zeta }$ arises from a solution $(D,\\varphi )$ of Hitchin's equations, through the formula $\\nabla _{R,\\zeta } = R \\zeta ^{-1} \\varphi + D + R \\zeta \\varphi ^\\dagger $ .",
"$ \\nabla _{R,\\zeta } \\psi (z) = 0.$ Despite the difference between (REF ) and (REF ), the geometric structures which appear in their exact WKB analysis are the same; in particular the Stokes graphs in exact WKB are the same as the spectral networks in [16].",
"A reason for this was conjectured in [26], as follows: in the case $K=2$ , taking the scaling limit $R \\rightarrow 0$ , $\\zeta \\rightarrow 0$ while holding $\\hbar = \\zeta / R$ fixed reduces the 2-parameter family of equations (REF ) to the 1-parameter family (REF ).",
"For general $K \\ge 2$ , this scaling limit similarly reduces (REF ) to a 1-parameter family of ${\\mathrm {SL}}(K)$ -opers, i.e.",
"order $K$ linear scalar ODEs.",
"This conjecture was proven in some cases in [44].",
"In this paper we mostly focus on questions internal to exact WKB, using these developments in physics only as motivation.",
"However, in the final section, sec:physics, we return briefly to the question of what our computations mean for ${\\mathcal {N}}=2$ field theory.",
"We propose that the construction of the functions $\\mathcal {X}_\\gamma $ provided by exact WKB is related to a construction of supersymmetric local operators in the field theory in $\\Omega $ -background, and comment on the expected relation of the $\\mathcal {X}_\\gamma $ to the Nekrasov $\\Omega $ -background partition function $\\mathbf {Z}$ , motivated by the ideas of [25]."
],
[
"Some questions",
"This project has raised, at least in our minds, many unanswered questions.",
"Here are some: In our study of the $T_3$ equation we consider only a specific Stokes graph, the simplest of infinitely many which occur at different points in the $u^{\\prime }$ -plane.",
"Even for this Stokes graph the monodromy properties of the local WKB solutions turn out to involve a complicated linear algebra problem, which seems to require real work to solve (in sec:enumerating-special-bases).",
"What kind of problem will appear at other points of the $u^{\\prime }$ -plane?",
"Is there some systematic way of solving all of them at once?",
"Similarly, what happens in other Minahan-Nemeschansky theories, like the $E_7$ or $E_8$ theories?",
"Is there a uniform way of describing the $\\mathcal {X}_\\gamma $ and their behavior, or do we have to treat each example separately?",
"In this paper we reformulate various aspects of exact WKB in the language of abelianization.",
"One notable exception is the “P/NP relation” discussed recently in the WKB literature, e.g.",
"[45], [46], [47], [48], [49], [42], [50], [33].",
"Does this part of the story have a useful geometric reformulation in the language of abelianization?",
"In our discussions of TBA-type integral equations in sec:integral-equations-mathieu and sec:integral-equations-t3 we make some progress, but do not attain the ultimate goal, which would be to completely determine the monodromy of the oper in terms of these integral equations.",
"It would be very interesting to push this project further.",
"For the $T_3$ equation the main obstruction to doing so is that we have not understood the coordinate transformation $\\mathbf {S}_{0,\\frac{\\pi }{3}}$ appearing in sec:integral-equations-t3.",
"Finding a closed form for this transformation would be very interesting in its own right since it would be equivalent to completely determining the BPS spectrum of the $E_6$ Minahan-Nemeschansky theory.",
"In sec:t3-analytic-cont we uncover an unexpectedly interesting analytic structure for the functions $\\mathcal {X}_\\gamma $ in the case of the $T_3$ equation.",
"It is natural to ask what is the physical meaning in the $E_6$ Minahan-Nemeschansky theory of the nonperturbative monodromy we find around the points $u^{\\prime } = \\pm u^{\\prime }_*$ .",
"(A similar question for pure ${\\mathcal {N}}=2$ supersymmetric $U(N)$ gauge theory was discussed in [51], where the relevant physics was proposed to be the appearance of new massless fields in the theory in $\\Omega $ -background; perhaps the monodromy we have found has a similar meaning.)",
"It would also be interesting to prove rigorously that there is no monodromy around any other points in the $u^{\\prime }$ -plane.",
"In this paper we treat only the case of the $T_3$ equation with unipotent monodromy, corresponding to the massless Minahan-Nemeschansky theory.",
"There is a natural perturbation to consider, taking semisimple monodromy instead of unipotent, corresponding to the mass-perturbed Minahan-Nemeschansky theory.",
"It would be interesting to study this case systematically — in particular, to see how the analytic structure of the $\\mathcal {X}_\\gamma $ is modified in this case.",
"(On general grounds we should expect that the structure could be more complicated; in the massless case the monodromy came ultimately from the fact that there were 4 discrete abelianizations of the $T_3$ equation; in the massive case there are 12 discrete abelianizations rather than 4.)",
"The exact WKB analysis we describe in this paper for equations of order $K > 2$ is still conjectural, mainly because it has not yet been proven that the local WKB series are Borel summable.",
"It would be very interesting to close this gap, perhaps by extending the approach of Koike-Schäfke from the $K=2$ case, or by using the integral equations of [52].",
"The Darboux coordinates we encounter on moduli of ${\\mathrm {SL}}(3)$ -connections over the thrice-punctured sphere are new as far as we know.",
"It would be interesting to understand explicitly their relation to other known Darboux coordinate systems on the same space, e.g.",
"the Fock-Goncharov coordinates [21], the coordinates introduced by Goldman [53], or the coordinates obtained from conformal field theory in [54].",
"Finally, as we discuss in sec:physics, the exact WKB computations we make here should have a precise meaning in terms of ${\\mathcal {N}}=2$ supersymmetric quantum field theories in the $\\Omega $ -background.",
"We make some proposals in this direction, but to put these proposals on a firm footing would seem to require new constructions of supersymmetric local operators and boundary conditions compatible with the $\\Omega $ -background.",
"It would be very interesting to develop this story further."
],
[
"Acknowledgements",
"We thank Dylan Allegretti, Tom Bridgeland, Gerald Dunne, Dan Freed, Marco Gualtieri, Kohei Iwaki, Saebyeok Jeong, Anton Leykin, Marcos Mariño, Nikita Nikolaev, Shinji Sasaki, Bernd Sturmfels and Joerg Teschner for useful and enlightening discussions.",
"LH's work is supported by a Royal Society Dorothy Hodgkin Fellowship.",
"AN's work on this paper was supported by NSF grant DMS-1711692 and by a Simons Fellowship in Mathematics."
],
[
"Exact WKB for Schrödinger equations",
"We consider a holomorphic Schrödinger equation, of the local formIn comparing to the ordinary Schrödinger equation on the real line we would have $P = 2(E-V)$ .",
"$ \\left[ \\hbar ^{2} \\partial _z^2 + P(z,\\hbar ) \\right] \\psi (z) = 0.$ The equation (REF ) can be given a global meaning on a Riemann surface $C$ equipped with a spin structure and complex projective structure.",
"In that case $\\psi (z)$ must be interpreted as a section of $K_C^{-\\frac{1}{2}}$ , and $P(z,\\hbar )$ as a meromorphic quadratic differential.",
"All of our considerations extend to this situation.",
"Nevertheless, most of the important constructions can be understood concretely in a single coordinate patch, and we will write them that way throughout."
],
[
"WKB solutions",
"The WKB method is often described in terms of distinguished local WKB solutions.",
"In this section we briefly recall the construction of these solutions.",
"(To forestall confusion we emphasize that the WKB solutions are exact solutions, not approximate solutions.)",
"Suppose we fix a contractible open set $U \\subset C$ , a local coordinate $z$ on $U$ , and a point $z_0 \\in U$ .",
"A WKB solution of (REF ) on $U$ means a solution of the form $ \\psi (z) = \\exp \\left( \\hbar ^{-1} \\int _{z_0}^z \\lambda (z) \\, \\mathrm {d}z \\right).$ For $\\psi (z)$ to be a solution of (REF ), $\\lambda $ must obey the Riccati equation, $ \\lambda ^2 + P + \\hbar \\partial _z \\lambda = 0.$ The first step in constructing such a $\\lambda $ is to build a formal series solution $\\lambda ^{\\mathrm {formal}}$ of (REF ) in powers of $\\hbar $ .",
"The order-$\\hbar ^0$ part of (REF ) is $ y^2 + p = 0,$ where $y$ (resp.",
"$p$ ) is the $\\hbar ^0$ term in $\\lambda $ (resp.",
"$P$ ).In the important special case of $\\hbar $ -independent $P$ , we just have $p = P$ .",
"Thus we have a two-fold ambiguity, resolved by choosing one of the two square roots of $-p$ .",
"It will be important to keep careful track of this choice of square root.",
"Thus we introduce the Riemann surface of $\\sqrt{-p}$ , $ \\Sigma = \\lbrace y^2 + p = 0\\rbrace .$ $\\Sigma $ is a branched double cover of $C$ .",
"A sheet of the covering $\\Sigma $ corresponds to a choice of $y$ obeying (REF ).",
"We use the generic labels $i,j$ to represent the sheets, and $y_i$ , $y_j$ for the corresponding square roots of $-p$ .In the WKB literature it is common to write the two square roots simply as $\\pm I\\sqrt{p}$ , and label the two sheets as $+$ , $-$ instead of $i$ , $j$ .",
"We now choose a sheet $i$ , and consider a formal series solution $\\lambda ^{\\mathrm {formal}}_i$ of (REF ), where we choose the $\\hbar ^0$ term to be $y_i$ .",
"The higher-order expansion of $\\lambda ^{\\mathrm {formal}}_i$ is then uniquely fixed by (REF ), taking the form $ \\lambda _i^{\\mathrm {formal}}= y_i + \\sum _{n = 1}^\\infty \\hbar ^n \\lambda _i^{{\\mathrm {formal}},n}.$ For example, if $P$ is $\\hbar $ -independent, this expansion is $ \\lambda _i^{\\mathrm {formal}}= y_i - \\hbar \\frac{P^{\\prime }}{4P} + \\hbar ^2 y_i \\frac{5 P^{\\prime 2} - 4 P P^{\\prime \\prime }}{32 P^3} + \\cdots .$ Note that although $\\lambda _i^{\\mathrm {formal}}$ is a formal solution of the differential equation (REF ), in writing this solution we do not have to do any integrals!",
"The series $\\lambda _i^{\\mathrm {formal}}$ in (REF ) is generally not convergent.",
"Nevertheless, one might hope that we could interpret $\\lambda _i^{\\mathrm {formal}}$ as an asymptotic series, and that there would be a unique actual solution $\\lambda _i$ with $\\lambda _i \\sim \\lambda _i^{\\mathrm {formal}}$ as $\\hbar \\rightarrow 0$ .",
"It turns out that the situation is more complicated.",
"There is no $\\lambda _i$ which has this asymptotic expansion, if $\\hbar $ is allowed to approach 0 from an arbitrary direction in the complex plane.",
"The best one can do in general is to ask for a solution $\\lambda _i^\\vartheta $ which has the expansion $\\lambda _i^\\vartheta \\sim \\lambda _i^{\\mathrm {formal}}$ as $\\hbar \\rightarrow 0$ while staying within a closed half-plane $\\mathbb {H}_\\vartheta = \\lbrace \\operatorname{Re}({\\mathrm {e}}^{-I\\vartheta } \\hbar ) \\ge 0\\rbrace .$ Such a $\\lambda _i^\\vartheta $ actually does existThis has been a folk-theorem for some time, at least for the case of $p$ with sufficiently generic residues, and a proof has been announced by Koike-Schäfke.",
"See [6] for an account., but only away from the $\\vartheta $ -Stokes curves of type $ij$, which we define next.",
"For simplicity we assume henceforward that $p(z)$ has only simple zeroes.",
"Then, from each zero of $p(z)$ there emanate three trajectories along which $\\int {\\mathrm {e}}^{-I\\vartheta } \\sqrt{-p(z)} \\mathrm {d}z$ is purely real; we call these $\\vartheta $ -Stokes curves.",
"The $\\vartheta $ -Stokes curves make up the Stokes graph $\\mathcal {W}(p, \\vartheta )$ .",
"Each Stokes curve is oriented away from the zero, and carries a label $ij$ , determined such that ${\\mathrm {e}}^{-I\\vartheta } (y_i - y_j) \\mathrm {d}z$ is positive along the oriented curve.Since $y_j = -y_i$ we could also have just written that ${\\mathrm {e}}^{-I\\vartheta } y_i$ is positive, and we could have labeled the curve just by the single index $i$ instead of the ordered pair $ij$ .",
"Our redundant-looking notation is chosen with an eye toward the generalization to higher-order equations, in sec:wkb-higher below.",
"See fig:sample-networks-combined for some examples of $\\vartheta $ -Stokes graphs in the case where $P(z)$ is a polynomial potential in the plane; many other such examples can be found e.g.",
"in [5], [15].",
"type=figure Figure: NO_CAPTION figureExamples of $\\vartheta $ -Stokes graphs at $\\vartheta =0$ , with $p(z) = z^n-1$ , for $n = 3, 4, 5$ .",
"The dashed lines denote branch cuts of the covering $\\Sigma \\rightarrow C$ ; the labels $i = 1,2$ are swapped when we cross a cut.",
"As long as the domain $U$ does not contain any $\\vartheta $ -Stokes curve of type $ij$ , $\\lambda _i^\\vartheta $ is defined on $U$ and can be integrated to give a WKB solution: $ \\psi _i^\\vartheta (z) = \\exp \\left( \\hbar ^{-1} \\int _{z_0}^z \\lambda _i^\\vartheta (z) \\, \\mathrm {d}z \\right).$ If $U$ does not contain any $\\vartheta $ -Stokes curve of either type $ij$ or $ji$ , then both $\\psi _i^\\vartheta $ and $\\psi _j^\\vartheta $ exist on $U$ , and give a basis of solutions of the Schrödinger equation (REF ).",
"If $U$ does contain a $\\vartheta $ -Stokes curve of type $ij$ , then we still get a basis of solutions on the complement of the Stokes curve, but $\\psi _i^\\vartheta $ jumps by a constant multiple of $\\psi _j^\\vartheta $ on crossing the Stokes curve."
],
[
"Abelianization",
"The WKB formula (REF ) has the awkward feature that it depends on the arbitrary choice of basepoint $z_0$ .",
"To see the content of (REF ) more clearly, we can observe that it represents a solution of the first-order equation $ \\left( \\partial _z - \\hbar ^{-1} \\lambda _i^\\vartheta (z) \\right) \\psi ^\\vartheta _i(z) = 0.$ (REF ) is much simpler than the original equation (REF ); a lot of the complexity of (REF ) has been swallowed into solving the Riccati equation to produce $\\lambda _i^\\vartheta (z)$ .",
"We interpret (REF ) as the condition that $\\psi ^\\vartheta _i (z)$ represents a flat section of a connection $\\nabla ^{\\mathrm {ab},\\vartheta }$ in a line bundle $\\mathcal {L}$ .",
"The line bundle $\\mathcal {L}$ lives not over the base $C$ but over the double cover $\\Sigma $ , since the function $\\lambda _i^\\vartheta $ depends on the sheet index $i$ .",
"The 1-form $- \\hbar ^{-1} \\lambda _i^\\vartheta \\mathrm {d}z$ represents $\\nabla ^{\\mathrm {ab},\\vartheta }$ relative to a local trivialization of $\\mathcal {L}$ ."
],
[
"Gluing across the Stokes graph",
"Consider a $\\vartheta $ -Stokes curve of type $ij$ .",
"$\\mathcal {L}$ and $\\nabla ^{\\mathrm {ab},\\vartheta }$ naively do not extend across the lift of this $\\vartheta $ -Stokes curve to sheet $i$ , because the solutions $\\psi _i^\\vartheta $ are different on the two sides.",
"We can nevertheless extend them “by hand” by giving a gluing map which takes $\\nabla ^{\\mathrm {ab},\\vartheta }$ -flat sections on one side to $\\nabla ^{\\mathrm {ab},\\vartheta }$ -flat sections on the other, i.e.",
"it maps $\\psi _i^{\\vartheta ,L}$ to some constant multiple of $\\psi _i^{\\vartheta ,R}$ .",
"There is a canonical and convenient choice: we glue $\\psi _i^{\\vartheta ,L}$ to the unique multiple of $\\psi _i^{\\vartheta ,R}$ which is of the form $\\psi _i^{\\vartheta ,L} + \\beta \\psi _j^{\\vartheta ,L}$ , and glue $\\psi _j^{\\vartheta ,L}$ to $\\psi _j^{\\vartheta ,L}$ .",
"This gluing prescription can be summarized asThe gluing rule (REF ) should be regarded as a version of the “WKB connection formula.” $ \\begin{pmatrix}\\psi _i^{L} \\\\\\psi _j^{L}\\end{pmatrix}\\mapsto \\begin{pmatrix} 1 & \\beta \\\\ 0 & 1 \\end{pmatrix} \\begin{pmatrix}\\psi _i^{L} \\\\\\psi _j^{L}\\end{pmatrix}=\\begin{pmatrix}\\frac{[\\psi _i^L, \\psi _j^L]}{[\\psi _i^R, \\psi _j^L]}\\psi _i^{R} \\\\\\frac{[\\psi _j^L, \\psi _i^L]}{[\\psi _j^R, \\psi _i^L]}\\psi _j^{R}\\end{pmatrix}$ where $[\\psi _1, \\psi _2]$ means the Wronskian of the two solutions.",
"An additional subtlety arises if a $\\vartheta $ -Stokes curve of type $ij$ coincides with a $\\vartheta $ -Stokes curve of type $ji$ , as e.g.",
"in the middle of fig:sample-networks-combined.",
"(This does not occur for generic values of $\\vartheta $ , but it can occur for special $\\vartheta $ , and in many of the examples we consider in this paper we take such a special $\\vartheta $ .)",
"In this case we have four distinct solutions $\\psi _i^{\\vartheta ,L}$ , $\\psi _j^{\\vartheta ,L}$ , $\\psi _i^{\\vartheta ,R}$ , $\\psi _j^{\\vartheta ,R}$ on $U$ , and we choose a gluing of the form $ \\begin{pmatrix}\\psi _i^{L} \\\\\\psi _j^{L}\\end{pmatrix}\\mapsto \\begin{pmatrix} \\rho & \\beta \\\\ \\alpha & \\rho \\end{pmatrix} \\begin{pmatrix}\\psi _i^{L} \\\\\\psi _j^{L}\\end{pmatrix}=\\begin{pmatrix}\\sqrt{\\frac{[\\psi _i^L , \\psi _j^L]}{[\\psi _i^R , \\psi _j^R]} \\frac{[\\psi _i^L , \\psi _j^R]}{[\\psi _i^R , \\psi _j^L]}}\\psi _i^{R} \\\\\\sqrt{\\frac{[\\psi _j^L , \\psi _i^L]}{[\\psi _j^R , \\psi _i^R]} \\frac{[\\psi _j^L , \\psi _i^R]}{[\\psi _j^R , \\psi _i^L]}}\\psi _j^{R}\\end{pmatrix}$ where $\\rho ^2 - \\alpha \\beta = 1$ .",
"We must make two technical comments about the gluing rule (REF ): In writing (REF ) we adopted the choice that the two diagonal entries of the gluing matrix should be equal.",
"We could alternatively have chosen e.g.",
"that the upper left entry of the gluing matrix should be 1, or the lower right entry should be 1.",
"These alternate choices also have their advantages: they arise naturally if one imagines that the two Stokes curves with labels $ij$ and $ji$ are infinitesimally displaced from one another, so that the gluing matrix arises as the product of an upper-triangular and a lower-triangular matrix.",
"This infinitesimal displacement was used in [16], [17] and was called “resolution” of the spectral network.",
"It appears naturally if we consider an infinitesimal perturbation of the phase $\\vartheta $ , either to $\\vartheta + \\epsilon $ or $\\vartheta - \\epsilon $ .",
"The choice we made in (REF ) is in some sense an average of these two resolutions, which avoids breaking symmetries.",
"The gluing matrix in (REF ) is determined only up to an overall sign.",
"To fix this ambiguity, we need to specify the branches of the square roots.",
"For this purpose (but only for this purpose!)",
"it is convenient to make a definite choice of the normalization of our solutions $\\psi _i^{L/R}$ , by choosing the basepoint $z_0$ in (REF ) to be on the Stokes curve.",
"Then we choose the principal branch for both square roots.",
"The motivation for this choice is that all four Wronskians appearing under the top square root asymptotically approach $2 \\hbar ^{-1} y_i$ as $\\hbar \\rightarrow 0$ , and similarly the four Wronskians under the bottom square root approach $2 \\hbar ^{-1} y_j$ , so both ratios approach 1.",
"After all this gluing, we get a line bundle $\\mathcal {L}$ with flat connection $\\nabla ^{\\mathrm {ab},\\vartheta }$ , defined over all of $\\Sigma $ except for the branch points.",
"It remains to consider the monodromy around the branch points.",
"By a short calculation (see e.g.",
"[17]), using the fact that the gluing matrices have determinant 1, one can show that $\\nabla ^{\\mathrm {ab},\\vartheta }$ has monodromy $-1$ on small loops encircling branch points.",
"We summarize this situation by saying that $\\nabla ^{\\mathrm {ab},\\vartheta }$ is an almost-flat connection over $\\Sigma $ ."
],
[
"$\\mathcal {W}$ -framings",
"The structure we have obtained from WKB can be encapsulated formally as follows.",
"The Schrödinger equation can be interpreted as a flat connection $\\nabla $ in the 1-jet bundle $J_1(K_C^{-\\frac{1}{2}})$ over $C$ : this is just the standard maneuver of replacing a second-order equation by a first-order equation with $2 \\times 2$ matrix coefficients, locally written as $\\underbrace{\\left[\\partial _z + \\hbar ^{-1} \\begin{pmatrix} 0 & -P(z) \\\\ 1 & 0 \\end{pmatrix}\\right]}_{\\nabla } \\underbrace{\\begin{pmatrix} - \\hbar \\psi ^{\\prime }(z) \\\\ \\psi (z) \\end{pmatrix}}_{J(\\psi )} = 0.$ Given a flat connection $\\nabla $ and a Stokes graph $\\mathcal {W}$ , one can formulate the notion of a $\\mathcal {W}$ -abelianization of $\\nabla $ , as in [17] (see also [20] for a more recent and mathematical treatment).",
"A $\\mathcal {W}$ -abelianization consists of: A flat ${\\mathrm {SL}}(2)$ -connection $\\nabla $ over $C$ , An almost-flat ${\\mathrm {GL}}(1)$ -connection $\\nabla ^\\mathrm {ab}$ over $\\Sigma $ , A flat isomorphism $\\iota : \\pi _* \\nabla ^\\mathrm {ab}\\simeq \\nabla $ away from the walls of $\\mathcal {W}$ (where $\\pi : \\Sigma \\rightarrow C$ is the projection), obeying the constraint that, at the walls of $\\mathcal {W}$ , $\\iota $ jumps by a unipotent transformation of the form (REF ) (for a wall of type $ij$ ) or (REF ) (for a wall of type $ij$ and $ji$ ).",
"Given the connection $\\nabla $ , to construct a $\\mathcal {W}$ -abelianization of $\\nabla $ amounts to producing projective bases of $\\nabla $ -flat sections in the various domains of $C \\setminus \\mathcal {W}$ , such that the relations between the bases in neighboring domains are given by matrices of the form (REF ) or (REF ).",
"This is ultimately a linear algebra problem determined by the combinatorics of $\\mathcal {W}$ and the monodromy and Stokes data of $\\nabla $ .",
"For any particular $\\mathcal {W}$ and $\\nabla $ , one can ask, how many $\\mathcal {W}$ -abelianizations of $\\nabla $ are there?",
"In the examples studied in [17], it turns out that there are just finitely many of them, and moreover they are in $1-1$ correspondence with some concrete extra data one can attach to $\\nabla $ , called $\\mathcal {W}$ -framings in [17].",
"For example, Suppose we consider Schrödinger equations on a Riemann surface $C$ , taking $P(z)$ meromorphic with $n$ second-order poles.",
"In this case, for generic $\\vartheta $ , the $\\vartheta $ -Stokes graph $\\mathcal {W}$ is a “Fock-Goncharov” network as described in [15], [17].",
"A $\\mathcal {W}$ -framing in this case is a choice of an eigenline of the monodromy around each of the $n$ punctures.",
"For generic $\\nabla $ , the monodromy at each puncture has 2 distinct eigenlines.",
"Thus $\\nabla $ admits $2^n$ distinct $\\mathcal {W}$ -framings.",
"Again, suppose we consider Schrödinger equations on a Riemann surface $C$ , taking $P(z)$ meromorphic with $n$ second-order poles.",
"For special $\\vartheta $ , the complement of the $\\vartheta $ -Stokes graph $\\mathcal {W}$ can include regions with the topology of an annulus.",
"For such a $\\vartheta $ , a $\\mathcal {W}$ -framing involves additional data: a choice of an eigenline of the monodromy around each annulus.",
"Thus $\\nabla $ admits $2^{n+m}$ distinct $\\mathcal {W}$ -framings, where $m$ is the number of annuli.",
"Now we come back to WKB.",
"The discussion of sec:wkb-solutions-sec:stokes-gluing above can be rephrased as follows: when $\\nabla $ is the flat ${\\mathrm {SL}}(2)$ -connection induced by a Schrödinger equation (REF ), and $\\mathcal {W}$ is the Stokes graph with phase $\\vartheta = \\arg \\hbar $ , exact WKB analysis constructs a distinguished $\\mathcal {W}$ -abelianization of $\\nabla $ .",
"This construction will be developed in more detail in [55].",
"It is somewhat remarkable that the WKB method automatically equips $\\nabla $ with a distinguished $\\mathcal {W}$ -framing.",
"In the cases above, this boils down to the statement that the local WKB solutions are automatically eigenvectors of the relevant monodromies of $\\nabla $ ."
],
[
"Spectral coordinates and their properties",
"Starting from the Schrödinger equation (REF ) and a choice of phase $\\vartheta $ , we have seen that exact WKB analysis gives rise to an almost-flat ${\\mathrm {GL}}(1)$ -connection $\\nabla ^{\\mathrm {ab},\\vartheta }$ over the surface $\\Sigma $ .",
"In particular, given any 1-cycle $\\gamma $ on $\\Sigma $ there is a corresponding holonomy, $\\mathcal {X}^\\vartheta _\\gamma = \\operatorname{Hol}_{\\gamma } \\nabla ^{\\mathrm {ab}, \\vartheta } \\in \\times $ As we have discussed in sec:intro-voros, the quantities $\\mathcal {X}^\\vartheta _\\gamma $ have various names, among them Voros symbols, spectral coordinates, and quantum periods.",
"They turn out to be extremely convenient for the analysis of the Schrödinger equation (REF ).",
"Here are a few of their expected properties: $\\mathcal {X}_\\gamma ^\\vartheta $ admits a complete asymptotic expansion as $\\hbar \\rightarrow 0$ in $\\mathbb {H}_\\vartheta $ , obtained by term-by-term integration of the formal WKB series (REF ): $ \\mathcal {X}_\\gamma ^\\vartheta \\sim \\exp \\left( \\hbar ^{-1} \\oint _\\gamma \\lambda ^{\\mathrm {formal}}\\, \\mathrm {d}z \\right).$ In particular, assuming $P(z)$ has no term of order $\\hbar $ , the leading asymptotic of $\\mathcal {X}_\\gamma ^\\vartheta $ is controlled by the classical period: if we define $ Z_\\gamma = \\oint _\\gamma y \\, \\mathrm {d}z$ then to leading order $ \\mathcal {X}^\\vartheta _\\gamma \\sim \\pm \\exp \\left( \\hbar ^{-1} Z_\\gamma \\right).$ The sign $\\pm $ in (REF ) is explicitly $\\exp \\oint _\\gamma \\frac{1}{4} \\frac{\\mathrm {d}p}{p} = (-1)^{\\frac{1}{2}w}$ , where $w$ is the number of zeroes of $p(z)$ enclosed by the projection of $\\gamma $ , counted with multiplicity.When $C$ is a compact Riemann surface of genus $g$ , to see that $(-1)^{\\frac{1}{2}w}$ does not depend on which side we call the “inside” of $\\gamma $ , we use the fact that a holomorphic quadratic differential has $4g-4$ zeroes, which is divisible by 4.",
"$\\mathcal {X}_\\gamma ^\\vartheta $ depends on $\\hbar $ , on the potential $P$ , and on the phase $\\vartheta $ .",
"As long as the topology of the $\\vartheta $ -Stokes graph does not change, the dependence of $\\mathcal {X}_\\gamma ^\\vartheta $ on $\\vartheta $ is trivial, while the dependence of $\\mathcal {X}_\\gamma ^\\vartheta $ on $\\hbar $ and $P$ is holomorphic.",
"There is a codimension-1 locus in the $(P,\\vartheta )$ parameter space where the topology of the $\\vartheta $ -Stokes graph does change; we call this the BPS locus.",
"When $(P,\\vartheta )$ crosses the BPS locus, the functions $\\mathcal {X}_\\gamma ^\\vartheta $ jump by a holomorphic transformation, called Stokes automorphism or Kontsevich-Soibelman transformation depending on the context.",
"This transformation can be computed from the Stokes graph at the BPS locus.In a generic situation the Stokes automorphisms which can occur are of the form $\\mathcal {X}_\\mu \\rightarrow \\mathcal {X}_\\mu (1 \\pm \\mathcal {X}_\\gamma )^{\\Omega (\\gamma ) \\langle \\gamma ,\\mu \\rangle }$ , where $\\Omega (\\gamma ) = +1$ for a “flip” of the Stokes graph and $\\Omega (\\gamma ) = -2$ for a “juggle” of the Stokes graph, in the terminology of [15].",
"The active rays corresponding to flips are typically isolated in the $\\hbar $ -plane, while juggles occur at the limit of infinite sequences of flips.",
"A general algorithm for computing the Stokes automorphism from a Stokes graph at the BPS locus is given in [16].",
"The asymptotic expansion (REF ) should hold as $\\hbar \\rightarrow 0$ in the half-plane $\\mathbb {H}_\\vartheta $ .",
"If $\\hbar $ is exactly in the middle of the half-plane $\\mathbb {H}_\\vartheta $ , i.e.",
"if $\\vartheta = \\arg \\hbar $ , then we can make a stronger conjecture, as follows.",
"If $(P,\\vartheta )$ is not on the BPS locus, $\\mathcal {X}_\\gamma ^\\vartheta $ is the Borel sum of the asymptotic expansion (REF ) along the ray ${\\mathrm {e}}^{I\\vartheta } \\mathbb {R}_+$ .",
"If $(P,\\vartheta )$ is on the BPS locus, then (REF ) may not be Borel summable along the ray ${\\mathrm {e}}^{I\\vartheta } \\mathbb {R}_+$ , because of singularities of the Borel transform.",
"In that case, our conjecture is that $\\mathcal {X}_\\gamma ^\\vartheta $ is obtained from (REF ) by Écalle's “median summation” (in the sense of [56], [57], also reviewed in [58] page 21.",
")This statement is sensitive to the particular gluing rule (REF ) which we chose.",
"Had we chosen a different rule, as described below (REF ), we would expect to get instead the “lateral summation” corresponding to perturbing $\\vartheta $ infinitesimally."
],
[
"Integral equations",
"Finally we come to one of the most interesting properties of the spectral coordinates of families of Schrödinger operators: this is the conjecture of [26] which says that they obey integral equations as functions of $\\hbar $ .",
"There is some choice involved in writing down the equations; one has to first choose some function $\\vartheta (\\arg \\hbar )$ , subject only to the constraint that $\\vert \\vartheta (\\arg \\hbar ) - \\arg \\hbar \\vert \\le \\frac{\\pi }{2}$ .",
"Then one considers the specialization $\\mathcal {X}_\\gamma ^{{\\mathrm {RH}}}(\\hbar ) = \\mathcal {X}_\\gamma ^{\\vartheta (\\arg \\hbar )}(\\hbar ).",
"$ $\\mathcal {X}_\\gamma ^{{\\mathrm {RH}}}$ is piecewise analytic in $\\hbar $ ; it jumps along some rays in the $\\hbar $ -plane, namely those rays at which the topology of the Stokes graph $\\mathcal {W}(p, \\vartheta (\\arg \\hbar ))$ jumps.",
"We call these active rays and denote them by $r$ .",
"When $\\hbar $ lies on an active ray $r$ , we let $\\mathcal {X}^{RH,r,\\pm }(\\hbar )$ denote the limit of $\\mathcal {X}^{{\\mathrm {RH}}}(\\hbar )$ as $\\arg \\hbar $ approaches the phase of $r$ from the $\\pm $ side.",
"The conjecture of [26] says that these functions are the unique solution of a system of coupled integral equations, of the form $ \\mathcal {X}_\\gamma ^{{\\mathrm {RH}}}(\\hbar ) = \\exp \\left[ \\frac{Z_\\gamma }{\\hbar } + \\frac{1}{4 \\pi I} \\sum _{r \\text{ active}} \\int _{r} \\frac{\\mathrm {d}\\hbar ^{\\prime }}{\\hbar ^{\\prime }} \\frac{\\hbar ^{\\prime } + \\hbar }{\\hbar ^{\\prime } - \\hbar } F_{r,\\gamma }(\\mathcal {X}^{RH,r,+}(\\hbar ^{\\prime })) \\right].$ This integral equation is similar to those appearing in the thermodynamic Bethe ansatz (TBA), and indeed (REF ) can be viewed as a generalization of the “ODE-IM correspondence” as we discussed in the introduction.",
"We are not aware of a completely rigorous proof of (REF ); morally the idea is that the $\\mathcal {X}_\\gamma ^{\\mathrm {RH}}(\\hbar )$ can be uniquely characterized in terms of their analytic properties in the $\\hbar $ -plane, and a solution of (REF ) would necessarily have the same analytic properties, so it must be $\\mathcal {X}_\\gamma ^{\\mathrm {RH}}(\\hbar )$ .",
"One direct argument which derives (REF ) from reasonable analytic assumptions is given in [33].",
"In another direction, [26] offers some reasons for optimism based on identifying (REF ) as the scaling limit of a better-behaved equation previously considered in [15].",
"For us, the strongest reason so far to believe (REF ) is a practical one: it has been checked to high precision in examples.",
"So far this has been done for various simple potentials, as reported e.g.",
"in [31], [26], [33], [59].",
"To formulate (REF ) completely, as we have explained, one needs to fix the choice of the function $\\vartheta (\\arg \\hbar )$ .",
"One canonical possibility is to take $ \\vartheta (\\arg \\hbar ) = \\arg \\hbar .",
"$ The resulting functions $\\mathcal {X}_\\gamma ^{\\mathrm {RH}}(\\hbar )$ are obtained by making WKB analysis for each $\\hbar $ using the Stokes graph adapted to the phase $\\vartheta = \\arg \\hbar $ .",
"This choice makes the functions $F_{r,\\gamma }$ relatively simple, at the cost that there may be many active rays (even infinitely many), and one has to consider all possible $\\vartheta $ -Stokes graphs.",
"See fig:uncollapsed-rays.",
"Figure: A sample picture of what the active raysin the ℏ\\hbar -plane can look like.",
"There are in general infinitelymany such rays, which can accumulate at discrete phases (as shown here)or even be dense in part or all of the ℏ\\hbar -plane.Another natural choice is to take $\\vartheta (\\arg \\hbar )$ to be piecewise constant; this has the effect of dividing the plane into sectors (of opening angle $\\le \\pi $ ) and collapsing all the active rays in each sector $S_i$ onto a single “aggregated” ray $r_i$ .",
"In this case the aggregated functions $F_{r_i,\\gamma }$ contain equivalent information to all of the functions $F_{r,\\gamma }$ for $r \\subset S_i$ .",
"In any case, to determine concretely the functions attached to the active rays, one can use the relation $ F_{r,\\gamma }(\\mathcal {X}_\\gamma ^{RH,r,+}) = \\log \\left( \\mathcal {X}_\\gamma ^{RH,r,+} / \\mathcal {X}_\\gamma ^{RH,r,-} \\right),$ if one knows the spectral coordinate systems $\\mathcal {X}_\\gamma ^\\vartheta $ for $\\vartheta $ on both sides of the active ray $r$ .",
"The most extreme possibility is to divide the plane into just two sectors, by fixing a phase $\\alpha $ and defining $ \\vartheta (\\arg \\hbar ) = {\\left\\lbrace \\begin{array}{ll} \\alpha & \\text{ for } \\hbar \\in \\mathbb {H}_\\alpha , \\\\\\alpha +\\pi & \\text{ for } \\hbar \\in \\mathbb {H}_{\\alpha +\\pi }.",
"\\end{array}\\right.",
"}$ Figure: Collapsing infinitely many active rays down to 2 by making the choice ().",
"Each active ray on the right carries functions F r,γ F_{r,\\gamma } which should be thought of as containing the same information as all the F r,γ F_{r,\\gamma } in thecorresponding half-plane on the left.In this case there are just 2 active rays $r$ , and we only have to consider two Stokes graphs, the $\\alpha $ -Stokes graph and the $(\\alpha +\\pi )$ -Stokes graph.",
"These two Stokes graphs are moreover identical except for an overall relabeling of all the Stokes lines, $ij \\rightarrow ji$ .",
"The function $F_{r,\\gamma }$ on each of the 2 active rays contains equivalent information to the “spectrum generator” discussed in [15].In the cluster algebra literature this object is called the “Donaldson-Thomas transformation” or “DT transformation” following [60].",
"When the $\\alpha $ -Stokes graph is of “Fock-Goncharov type,” the spectrum generator has been determined in [15]; these results were used in [26] to give several explicit examples of integral equations (REF )."
],
[
"Exact WKB for Schrödinger operators with cubic potential",
"The WKB method and exact WKB method have been explored rather thoroughly in the case of a Schrödinger equation in the plane with polynomial potential.",
"For the WKB method two important references are [61], [62]; for exact WKB see e.g.",
"the pioneering works [1], [3], [63], [64], and [5] for a clear recent treatment.",
"In this section we quickly touch on the very simplest example of this sort, the Schrödinger equation $ \\left[ \\hbar ^{2} \\partial _z^2 + (z^3-u) \\right] \\psi (z) = 0,$ for a constant $u \\in .This is an instance of (\\ref {eq:schrodinger}) with cubic potential\\begin{equation}P(z) = z^3 - u.\\end{equation}$"
],
[
"A Stokes graph",
"type=figure Figure: NO_CAPTION figure$\\vartheta $ -Stokes graph for the Schrödinger equation with cubic potential (REF ), at the phase $\\vartheta = 0$ , and $u = 1$ .",
"Two 1-cycles $\\gamma _A$ , $\\gamma _B$ on $\\Sigma $ are also shown.",
"Dashed orange segments denote branch cuts; on crossing a cut, the sheet labels are exchanged $1 \\leftrightarrow 2$ .",
"Orange crosses denote the turning points, zeroes of $p(z) = z^3 - 1$ .",
"The singularity at $z = \\infty $ is not shown.",
"Suppose we fix $u = 1$ and $\\vartheta = 0$ .",
"Then the $\\vartheta $ -Stokes graph is shown in fig:cubic-network.",
"This graph divides the plane up into 7 domains.",
"As we have reviewed in sec:wkb-review, there are canonical local solutions $\\lambda ^\\vartheta _i$ of the Riccati equation in each of these domains, and from these local solutions we can build local WKB solutions $\\psi _i^\\vartheta $ of (REF ) — or more invariantly, we can build an almost-flat ${\\mathrm {GL}}(1)$ -connection $\\nabla ^{\\mathrm {ab},\\vartheta }$ over the spectral curve $\\Sigma = \\lbrace y^2 + z^3 - 1 = 0\\rbrace $ .",
"The connection $\\nabla ^{\\mathrm {ab},\\vartheta }$ abelianizes the ${\\mathrm {SL}}(2)$ -connection $\\nabla $ in the $z$ -plane associated to (REF )."
],
[
"The spectral coordinates",
"Let $\\mathcal {X}_A$ (resp.",
"$\\mathcal {X}_B$ ) denote the monodromy of $\\nabla ^{\\mathrm {ab},\\vartheta }$ along the cycle $\\gamma _A$ (resp.",
"$\\gamma _B$ ) in fig:cubic-network.",
"The Stokes graph of fig:cubic-network is an example of a Fock-Goncharov network in the sense of [17], and $\\mathcal {X}_A$ , $\\mathcal {X}_B$ are Fock-Goncharov coordinates of the flat connection $\\nabla $ .",
"Let us explain this more concretely.",
"We first consider the local WKB solutions in each domain.",
"These turn out to have a simple and concrete characterization, as follows.",
"Let $\\ell _n$ ($n = 1, \\dots , 5$ ) denote the ray with phase $\\frac{2 \\pi }{5}(n + \\frac{1}{2})$ .",
"When $\\hbar \\in \\mathbb {R}_+$ , for each $n$ there exists a solution $\\psi ^{\\mathrm {sm}}_n$ such that $\\psi ^{\\mathrm {sm}}_n$ decays exponentially as $z \\rightarrow \\infty $ along $\\ell _n$ .",
"This $\\psi ^{\\mathrm {sm}}_n$ is unique up to scalar multiple.",
"Now let $U$ be one of the domains in the complement of the Stokes graph.",
"$U$ has two infinite “ends” which approach two of the five rays $\\ell _n$ .",
"As $z$ approaches $\\ell _n$ , the WKB solution $\\psi _j^\\vartheta $ is exponentially decaying, where $ij$ is the label on the $\\vartheta $ -Stokes curves asymptotic to $\\ell _n$ .This follows from the realization of $\\lambda _j^\\vartheta $ as the Borel summation of the WKB series, which implies that $\\lambda _j^\\vartheta \\mathrm {d}z$ is negative along $\\ell _n$ , since every term of the series has this property.",
"Thus up to scalar multiple $\\psi _j^\\vartheta $ is equal to $\\psi ^{\\mathrm {sm}}_n$ .",
"Now that we understand the local WKB solutions, we can use them to compute the spectral coordinates.",
"They turn out to be cross-ratios of the $\\psi ^{\\mathrm {sm}}_n$ , as follows (see app:computations for the computation): $ \\mathcal {X}_A = \\frac{[\\psi _3^{\\mathrm {sm}}, \\psi _2^{\\mathrm {sm}}]}{[\\psi _1^{\\mathrm {sm}}, \\psi _2^{\\mathrm {sm}}]} \\frac{[\\psi _1^{\\mathrm {sm}}, \\psi _5^{\\mathrm {sm}}]}{[\\psi _3^{\\mathrm {sm}}, \\psi _5^{\\mathrm {sm}}]}, \\qquad \\mathcal {X}_B = \\frac{[\\psi _3^{\\mathrm {sm}}, \\psi _4^{\\mathrm {sm}}]}{[\\psi _5^{\\mathrm {sm}}, \\psi _4^{\\mathrm {sm}}]} \\frac{[\\psi _5^{\\mathrm {sm}}, \\psi _1^{\\mathrm {sm}}]}{[\\psi _3^{\\mathrm {sm}}, \\psi _1^{\\mathrm {sm}}]}.$ As we promised above, these are Fock-Goncharov coordinates (or “complexified shear coordinates”) of the connection $\\nabla $ , in the sense of [21]."
],
[
"Analytic continuation",
"In our description of $\\mathcal {X}_A$ and $\\mathcal {X}_B$ above we used the conditions $u=1$ and $\\hbar \\in \\mathbb {R}_+$ .",
"It is interesting to consider the question of analytic continuation of these functions in $u$ and $\\hbar $ .",
"For this purpose a simple approach is to just start from the final formulas (REF ) and try to continue them directly.",
"The resulting analytic structure is very simple: First, as we vary $u$ , the $z \\rightarrow \\infty $ asymptotic behavior of the equation (REF ) does not change; for each $u$ we still have 5 decaying solutions $\\psi _n^{\\mathrm {sm}}$ , now depending on $u$ .",
"Since the equation (REF ) depends holomorphically on $u$ , so do its decaying solutions.",
"Thus the formula (REF ) defines single-valued analytic functions $(\\mathcal {X}_A, \\mathcal {X}_B)$ of $u \\in .$ These functions may have poles, because for general $u$ there is nothing preventing $\\psi _n^{\\mathrm {sm}}$ and $\\psi _{n^{\\prime }}^{\\mathrm {sm}}$ from coinciding, as long as $n$ and $n^{\\prime }$ are not consecutive.",
"Indeed, numerically one finds a discrete sequence of points $u = u_1, u_2, \\dots $ where $\\mathcal {X}_A$ has a simple pole ($\\psi _3^{\\mathrm {sm}}$ and $\\psi _5^{\\mathrm {sm}}$ become proportional) and conjugate points $u = u_1^*, u_2^*, \\dots $ where $\\mathcal {X}_B$ has a simple pole ($\\psi _3^{\\mathrm {sm}}$ and $\\psi _1^{\\mathrm {sm}}$ become proportional).",
"These poles can be thought of as “bound states” for the equation (REF ) along a complex contour asymptotic to $\\ell _n$ and $\\ell _{n^{\\prime }}$ .",
"The $u_i$ lie on the ray $\\arg u = -\\frac{4}{5} \\pi $ (but this is not trivial to see: it was proven in [65].)",
"Second, we can consider varying $\\hbar $ .",
"This leads to a slightly subtler analytic structure.",
"If we vary $\\arg \\hbar $ by an amount $\\beta $ , the distinguished rays $\\ell _n$ where we impose the exponential decay condition rotate counterclockwise in the plane by an angle $\\frac{2}{5} \\beta $ .",
"It follows that, when we go clockwise once around the singularity at $\\hbar = 0$ , the $\\psi _n^{\\mathrm {sm}}$ are permuted by $n \\mapsto n+2$ (mod 5); this transforms $(\\mathcal {X}_A, \\mathcal {X}_B)$ by $ (\\mathcal {X}_A, \\mathcal {X}_B) \\mapsto \\left( \\mathcal {X}_B^{-1} (1 - \\mathcal {X}_A^{-1})^{-1}, \\mathcal {X}_A \\right).$ Thus the maximal analytic continuation of the functions $(\\mathcal {X}_A, \\mathcal {X}_B)$ is defined on a 5-fold cover of the punctured plane $\\hbar \\in \\times $ .",
"In fact, the continuations in $u$ and $\\hbar $ are not unrelated: the continued functions actually depend only on the combination $u^{\\prime } = u / \\hbar ^{\\frac{6}{5}}$ , as one sees by dividing (REF ) by $\\hbar ^{\\frac{6}{5}}$ and then taking $z \\mapsto \\hbar ^{\\frac{2}{5}} z$ .",
"Note that the monodromy (REF ) acts by a symplectomorphism preserving the form $\\varpi = \\mathrm {d}\\log \\mathcal {X}_A \\wedge \\mathrm {d}\\log \\mathcal {X}_B$ .",
"This is a consistency check of the general story: $\\mathcal {X}_A$ and $\\mathcal {X}_B$ are local Darboux coordinates on a moduli space of ${\\mathrm {SL}}(2)$ -connections with irregular singularity at $z = \\infty $ .",
"We emphasize that the analytic continuation of $(\\mathcal {X}_A, \\mathcal {X}_B)$ which we have been discussing in this section is not given directly by WKB analysis; to make the WKB analysis directly at a given $(u,\\hbar )$ would require us to consider a different Stokes graph and spectral curve for each $(u, \\hbar )$ .",
"This would necessarily lead to single-valued functions of $(u, \\hbar )$ , but ones which are only piecewise analytic, jumping when the Stokes graph jumps.",
"These are the functions which we called $\\mathcal {X}_\\gamma ^{\\mathrm {RH}}$ above; we will discuss them more in sec:integral-equations-cubic below."
],
[
"The regular pentagon",
"One case worthy of special notice is the case $u = 0$ , where the potential $P(z)$ degenerates to the pure cubic, $P(z) = z^3$ .",
"At this point the equation (REF ) acquires an extra $\\mathbb {Z}/ 5\\mathbb {Z}$ symmetry which acts by $z \\rightarrow {\\mathrm {e}}^{2 \\pi I/ 5} z$ , and thus cyclically permutes the five rays $\\ell _n$ .",
"From this symmetry it follows that $(\\mathcal {X}_A, \\mathcal {X}_B)$ is a fixed point of the monodromy (REF ), which implies $\\mathcal {X}_A = \\mathcal {X}_B = x, \\qquad x^2 - x - 1 = 0.$ Numerically we find that the relevant solution of this quadratic isThe reader might wonder: what about the other solution, where $\\mathcal {X}_A = \\mathcal {X}_B = \\frac{1 + \\sqrt{5}}{2}$ ?",
"That one turns out to be associated to a Schrödinger equation with singular potential, $P(z) = z^3 - \\frac{3}{4} \\frac{\\hbar ^2}{z^2}$ .",
"The specific coefficient $-\\frac{3}{4} \\hbar ^2$ here ensures that the singularity at $z=0$ is only an “apparent singularity,” with trivial monodromy; thus this equation is still associated to a flat connection $\\nabla $ in the plane, and all our discussion of abelianization applies equally well to this case.",
"Moreover this equation still has the $\\mathbb {Z}/ 5\\mathbb {Z}$ symmetry (because the two terms $z^3$ and $1/z^2$ differ by a factor $z^5$ ), and numerically one checks that it has $\\mathcal {X}_A = \\mathcal {X}_B = \\frac{1 + \\sqrt{5}}{2}$ .",
"We thank Dylan Allegretti and Tom Bridgeland for several enlightening conversations about Schrödinger equations with apparent singularities.",
"$ \\mathcal {X}_A = \\mathcal {X}_B = \\frac{1 - \\sqrt{5}}{2}.$ Since $\\mathcal {X}_A$ and $\\mathcal {X}_B$ depend only on $u^{\\prime } = u / \\hbar ^{\\frac{6}{5}}$ , it follows that this fixed point also governs the $\\hbar \\rightarrow \\infty $ behavior for any constant $u$ ."
],
[
"Integral equations for spectral coordinates",
"Identifying the cross-ratios (REF ) as the spectral coordinates coming from WKB implies that they should have all the properties discussed in sec:spectral-coordinates-sec:integral-equations.",
"In particular, when they are extended to functions $\\mathcal {X}_\\gamma ^{\\mathrm {RH}}(\\hbar )$ as in sec:integral-equations, they should obey an integral equation of the form (REF ).",
"We make the canonical choice (REF ).",
"Then one direct way to identify the active rays is to use a computer to draw the $\\vartheta $ -Stokes graphs for various phases $\\vartheta $ ; the active rays are at the phases where the $\\vartheta $ -Stokes graph jumps.",
"It turns out that there are 6 such rays, as shown in fig:uncollapsed-rays-cubic.This corresponds to the well known BPS spectrum of the $(A_1,A_2)$ Argyres-Douglas field theory in its “maximal chamber,” discussed e.g.",
"in [66], [15].",
"Each of these rays $r$ has an associated class $\\mu \\in H_1(\\Sigma ,\\mathbb {Z})$ , and the function $F_{r,\\gamma }$ is $F_{r,\\gamma }(\\mathcal {X}) = \\langle \\gamma , \\mu \\rangle \\log (1 - \\mathcal {X}_\\mu )$ where $\\langle \\cdot ,\\cdot \\rangle $ is the intersection pairing on $H_1(\\Sigma ,\\mathbb {Z})$ .",
"type=figure Figure: NO_CAPTION figureThe 6 active rays in the $\\hbar $ -plane, each labeled by its charge $\\mu \\in H_1(\\Sigma ,\\mathbb {Z})$ .",
"These rays divide the $\\hbar $ -plane into 6 regions.",
"Each region is characterized by a different topology for the $\\vartheta $ -Stokes graph, where $\\vartheta = \\arg \\hbar $ .",
"In this case one can make a direct numerical test of the integral equation (REF ).",
"Namely, on the one hand we can solve (REF ) by numerical iteration, on the other hand we can determine $\\mathcal {X}_\\gamma ^{\\mathrm {RH}}(\\hbar )$ directly by numerical integration of the Schrödinger equation in the complex plane.",
"The two computations agree very well.",
"To give one concrete example, by direct numerical integration we obtain the estimates $\\mathcal {X}_A(\\hbar = 2 + I) &\\approx -0.230042356-0.324912345I, \\\\ \\mathcal {X}_B(\\hbar = 2 + I) &\\approx -0.288795812+0.476012574I,$ and each of these agrees with the result obtained from the integral equation (REF ), to the precision given.",
"Many similar computations for polynomial potentials have been made before, with similarly good numerical agreement, e.g.",
"already in [30] and more recently [26], [33], [59].",
"The appearance of the fixed point (REF ) at the $\\hbar \\rightarrow \\infty $ limit was already noticed in the very early TBA work [67]."
],
[
"Exact WKB for the Mathieu equation",
"Now let us recall how exact WKB analysis is applied to the Mathieu equation: $ \\left[ - \\hbar ^2 \\partial _x^2 + 2 \\cos (x) - 2 E \\right] \\psi (x) = 0.$ WKB analysis of this equation has been studied extensively; a review we found particularly helpful is [68], which covers many topics we will not touch here.",
"For other treatments of exact WKB for this equation see e.g.",
"[37], [69], [40], [42], and more broadly [70], [71], [72]."
],
[
"Exponential coordinate",
"Making the replacements $ z = {\\mathrm {e}}^{Ix}, \\qquad \\tilde{\\psi }(z) = \\left(Iz\\right)^{\\frac{1}{2}} \\psi (x)$ transforms (REF ) into an equation defined over $\\mathbb {CP}^1$ , with irregular singularities at $z = 0$ and $z = \\infty $ : $ \\left[ \\hbar ^2 \\partial _z^2 + P(z) \\right] \\tilde{\\psi }(z) = 0, \\qquad P(z) = \\frac{1}{z^3} - \\frac{2E-\\frac{1}{4} \\hbar ^2}{z^2} + \\frac{1}{z}.$ In what follows we will usually use the formulation (REF )."
],
[
"A simple Stokes graph",
"type=figure Figure: NO_CAPTION figure$\\vartheta $ -Stokes graph for the Mathieu equation, at the phase $\\vartheta = 0$ , and $E = -\\frac{9}{8}$ .",
"Two 1-cycles $\\gamma _A$ , $\\gamma _B$ on $\\Sigma $ are also shown.",
"Dashed orange segments denote branch cuts; on crossing a cut, the sheet labels are exchanged $1 \\leftrightarrow 2$ .",
"Orange crosses denote the turning points, zeroes of $P(z)$ .",
"The blue dot represents the singularity at $z=0$ ; the singularity at $z = \\infty $ is not shown.",
"We begin with real $E < -1$ and $\\vartheta = 0$ .",
"The $\\vartheta $ -Stokes graph is as shown in fig:mathieu-sn-1.",
"The Stokes curves divide the plane into 3 open domains: a simply connected domain near $z=0$ , another near $z=\\infty $ , and an annulus containing $z=1$ ."
],
[
"The spectral coordinates",
"Let $\\mathcal {X}_A$ (resp.",
"$\\mathcal {X}_B$ ) denote the monodromy of $\\nabla ^{\\mathrm {ab},\\vartheta }$ along the cycle $\\gamma _A$ (resp.",
"$\\gamma _B$ ) in fig:mathieu-sn-1.",
"This Stokes graph is an example of a Fenchel-Nielsen network in the sense of [17], and $\\mathcal {X}_A$ , $\\mathcal {X}_B$ are exponentiated Fenchel-Nielsen coordinates of the flat connection $\\nabla $ .",
"Let us explain this more concretely.",
"We first consider local WKB solutions in each of the three domains.",
"Let $\\psi _i$ be the local WKB solutions near $z=0$ .",
"$\\psi _1$ can be characterized as a solution which exponentially decays as $z \\rightarrow 0$ along the negative-$z$ ray, similarly to what we saw in sec:coords-cubic.",
"Let $\\psi ^{\\prime }_i$ be the local WKB solutions near $z = \\infty $ .",
"$\\psi ^{\\prime }_1$ can be characterized as a solution which exponentially decays as $z \\rightarrow \\infty $ along the negative-$z$ ray.",
"Let $\\psi ^{\\prime \\prime }_i$ be the local WKB solutions in some simply connected domain of the intermediate annulus.",
"These can be characterized as eigenvectors of the counterclockwise monodromy $M$ of $\\nabla $ .",
"At $E < -1$ and $\\hbar \\in \\mathbb {R}_+$ the eigenvalues of $M$ are real and negative, and we let $\\mu $ denote the eigenvalue which has $\\vert \\mu \\vert < 1$ ; then $\\psi ^{\\prime \\prime }_1$ is the eigenvector with eigenvalue $\\mu ^{-1}$ , while $\\psi ^{\\prime \\prime }_2$ is the one with eigenvalue $\\mu $ .",
"With the local WKB solutions understood, we can compute the spectral coordinates: $\\mathcal {X}_A$ is the smaller eigenvalue of monodromy of $\\nabla $ , $ \\mathcal {X}_A = \\mu .$ Indeed, the representative $\\gamma _A$ in fig:mathieu-sn-1 does not cross any Stokes curves, so the eigenvalue of monodromy of $\\nabla ^\\mathrm {ab}$ on sheet 2 agrees with the eigenvalue of monodromy of $\\nabla $ acting on $\\psi ^{\\prime \\prime }_2$ , which is $\\mu $ .",
"This is an exponentiated complexified Fenchel-Nielsen length coordinate, in the sense of [24], [17].",
"$\\mathcal {X}_B$ can be given in terms of Wronskians of the local WKB solutions in the three domains (see app:computations for the computation): $ \\mathcal {X}_B = \\frac{[\\psi _1 , \\psi ^{\\prime \\prime }_2]}{[\\psi _1 , \\psi ^{\\prime \\prime }_1]} \\frac{[\\psi ^{\\prime }_1 , \\psi ^{\\prime \\prime }_1]}{[\\psi ^{\\prime }_1 , \\psi ^{\\prime \\prime }_2]}.$ (In computing these Wronskians we have to evolve all the solutions to a common domain, which we do along the negative-$z$ ray.)",
"This is an exponentiated complexified Fenchel-Nielsen twist coordinate, in the sense of [24], [17]."
],
[
"Application: bound states",
"Now let us see one application of the spectral coordinates.",
"We return to the original Mathieu equation (REF ) and make the substitution $x = Ix^{\\prime } + \\pi $ with $x^{\\prime }$ real.",
"Then $(\\ref {eq:mathieu})$ becomes the modified Mathieu equation, $ \\left[ - \\hbar ^2 \\partial _{x^{\\prime }}^2 + 2 \\cosh (x^{\\prime }) + 2 E \\right] \\psi (x^{\\prime }) = 0.$ This is a Schrödinger equation with potential $V(x^{\\prime }) = \\cosh x^{\\prime }$ , for which we can formulate the usual bound state problem, i.e.",
"we look for $E$ such that there exists an $L^2$ solution of (REF ).",
"Such a solution exists only for countably many $E = E_1, E_2, \\dots $ .",
"With our sign conventions $E$ is minus the usual energy, so all $E_n < -1$ .",
"The condition for existence of a bound state is that $\\psi _1$ is proportional to $\\psi ^{\\prime }_1$ .",
"Substituting this condition in (REF ) gives simply $ \\mathcal {X}_B = 1.$ This is known as the “exact quantization condition” for the modified Mathieu bound states, discussed frequently in the literature, e.g.",
"[43], [69], [37].",
"To give some indication of how (REF ) can be used in practice, let us consider the leading term of the asymptotic expansion, $ \\mathcal {X}_B \\approx - \\exp (Z_B / \\hbar ).$ When $E < -1$ we have $Z_B \\in I\\mathbb {R}_-$ , and recall that $\\hbar > 0$ ; thus this leading approximation says that solutions of (REF ) will be found when $ Z_B \\approx 2 \\pi I\\left(n+\\frac{1}{2}\\right) \\hbar .$ To understand (REF ) more explicitly, we can expand $Z_B$ at large negative $E$ : one finds $Z_B(E) \\approx -4 I\\sqrt{-2 E} \\log (-E)$ .",
"Thus, for large negative $E$ and small $\\hbar $ , the desired bound states are approximately at $\\sqrt{-E} \\log (-E) \\approx \\frac{\\pi }{2 \\sqrt{2}} \\left(n+\\frac{1}{2}\\right) \\hbar .$ One can improve this estimate by including higher terms — either in the WKB expansion of $\\mathcal {X}_B$ in powers of $\\hbar $ , or in the expansion of $Z_B(E)$ in inverse powers of $E$ .",
"We will not explore these improvements here."
],
[
"Analytic continuation",
"So far we have considered the spectral coordinates $\\mathcal {X}_A$ and $\\mathcal {X}_B$ for $E<-1$ , $\\hbar >0$ , built using the exact WKB method.",
"It is also interesting to consider the analytic continuation of these coordinates to complex parameters.",
"To build this analytic continuation, we will build a $\\mathcal {W}$ -abelianization of $\\nabla $ which varies holomorphically with $(E,\\hbar )$ .",
"Said otherwise, we will build local solutions $\\psi _i, \\psi ^{\\prime }_i, \\psi ^{\\prime \\prime }_i$ which fit into a $\\mathcal {W}$ -abelianization and vary holomorphically with $(E,\\hbar )$ .",
"For general $(E,\\hbar )$ they will not necessarily be given by any kind of WKB analysis.",
"The local solutions $\\psi ^{\\prime \\prime }_i$ must be eigenvectors of the monodromy $M$ : to decide which one will be $\\psi ^{\\prime \\prime }_1$ and which will be $\\psi ^{\\prime \\prime }_2$ , we just require that our choice is continuously connected to the choice we got from WKB at $E < -1$ , $\\hbar > 0$ .",
"This gives a nice analytic continuation along any path in $(E,\\hbar )$ space, except at the codimension-1 locus where the eigenvalues of $M$ coincide.",
"Around this locus we have an order-2 monodromy exchanging $\\psi ^{\\prime \\prime }_1 \\leftrightarrow \\psi ^{\\prime \\prime }_2$ .",
"At our initial locus $(E < -1,\\hbar > 0)$ , $\\psi _1$ can be characterized by the property of exponential decay as $z \\rightarrow 0$ along the negative real axis.",
"As we vary $(E,\\hbar )$ we can define $\\psi _1$ by a similar condition, except that the negative real axis has to be replaced by a different path, which asymptotically has $z \\rightarrow 0$ with $\\arg z = 2 \\arg \\hbar + \\pi $ .",
"Similar comments apply to $\\psi ^{\\prime }_1$ except that we use a path with $z \\rightarrow \\infty $ and $\\arg z = - 2 \\arg \\hbar + \\pi $ .",
"This gives a nice analytic continuation of $\\psi _1$ and $\\psi ^{\\prime }_1$ along any path in $(E,\\hbar )$ space which avoids $\\hbar = 0$ .",
"Now we have to consider the possibility of monodromy around $\\hbar = 0$ .",
"As $\\arg \\hbar $ is continuously increased by $2\\pi $ , our paths into $z = 0$ and $z = \\infty $ wind around twice, in opposite directions.",
"The result is that as we go counterclockwise around $\\hbar = 0$ we have an infinite-order monodromy acting by $\\psi _1 \\mapsto M^{-2} \\psi _1$ , $\\psi ^{\\prime }_1 \\mapsto M^{2} \\psi ^{\\prime }_1$ .",
"(We might also wonder whether the eigenvectors $\\psi ^{\\prime \\prime }_i$ of $M$ are exchanged as $\\hbar $ goes around 0; this cannot occur, since the Mathieu equation depends only on $\\hbar ^2$ , so the monodromy around $\\hbar = 0$ is the square of an order-2 element, hence the identity.)",
"Using (REF ) and (REF ), the analytic structure of $\\mathcal {X}_A$ and $\\mathcal {X}_B$ follows from that of $\\psi _i$ , $\\psi ^{\\prime }_i$ , $\\psi ^{\\prime \\prime }_i$ ; we have unrestricted analytic continuation in $(E,\\hbar )$ , except that: Going around $\\hbar = 0$ counterclockwise we have the infinite-order monodromy $ (\\mathcal {X}_A, \\mathcal {X}_B) \\mapsto (\\mathcal {X}_A, \\mathcal {X}_A^8 \\mathcal {X}_B).$ Around the locus in $(E,\\hbar )$ space where the eigenvalues of $M$ coincide, we have the order-2 monodromy $ (\\mathcal {X}_A, \\mathcal {X}_B) \\mapsto (\\mathcal {X}_A^{-1}, \\mathcal {X}_B^{-1}).$ Note that both of these monodromies act by symplectomorphisms preserving the form $\\varpi = \\mathrm {d}\\log \\mathcal {X}_A \\wedge \\mathrm {d}\\log \\mathcal {X}_B$ .",
"This is a consistency check of the general story: $\\mathcal {X}_A$ and $\\mathcal {X}_B$ are local Darboux coordinates on the moduli space of ${\\mathrm {SL}}(2)$ -connections."
],
[
"Integral equations for spectral coordinates",
"As we have discussed in sec:integral-equations, one of the most interesting properties of spectral coordinates for families of Schrödinger equations is that they conjecturally obey integral equations of the form (REF ).",
"In the case of the Mathieu equation, integral equations for spectral coordinates were considered in [26].",
"There the function $\\vartheta (\\arg \\hbar )$ was chosen in the form (REF ), with $\\alpha $ a generic phase, collapsing all the active rays onto two aggregated rays.",
"In this case the Stokes graphs which appear are of Fock-Goncharov type in the terminology of [17], and the $\\mathcal {X}_\\gamma $ are Fock-Goncharov coordinates.",
"This example is thus qualitatively similar to the one we considered in sec:integral-equations-cubic above, though the details are more intricate.In particular, it seems to be harder to find a solution of the integral equations (REF ) directly by iteration in this case.",
"Instead one can start with a slightly different system of integral equations, those used in [73]; these one can solve by iteration; then one can take the limit $R \\rightarrow 0$ , $\\zeta \\rightarrow 0$ , $\\hbar = R / \\zeta $ , to get solutions of (REF ).",
"In this section we try something different: we try to find integral equations obeyed by the complexified Fenchel-Nielsen coordinates.",
"For this purpose we choose the very non-generic phase $\\alpha = 0$ , so that the aggregated rays are the positive and negative imaginary axes.",
"See fig:collapsing-mathieu.",
"type=figure Figure: NO_CAPTION figureCollapsing the infinitely many active rays down to 2 by making the choice (REF ) with $\\alpha = 0$ .",
"Each active ray on the right carries functions $F_{r,\\gamma }$ which should be thought of as containing the same information as all the $F_{r,\\gamma }$ in the corresponding half-plane on the left.",
"This is a particularly thorny case because the active rays on the left accumulate at the boundary of the half-planes.",
"Then, according to the recipe of sec:integral-equations, the functions $\\mathcal {X}_\\gamma ^{{\\mathrm {RH}}}$ are: $\\mathcal {X}_{\\gamma }^{{\\mathrm {RH}}}(\\hbar ) = {\\left\\lbrace \\begin{array}{ll} \\mathcal {X}_\\gamma ^{\\vartheta =0}(\\hbar ) & \\text{for } \\operatorname{Re}\\hbar > 0, \\\\\\mathcal {X}_\\gamma ^{\\vartheta =\\pi }(\\hbar ) & \\text{for } \\operatorname{Re}\\hbar < 0.",
"\\end{array}\\right.",
"}$ Now, to construct the functions $F_{r,\\gamma }$ appearing in the integral equation (REF ), we need to understand the discontinuity of $\\mathcal {X}_\\gamma ^{{\\mathrm {RH}}}$ across the imaginary axis.",
"It turns out that this discontinuity has a more complicated form than those we have previously considered: $\\mathcal {X}_\\gamma ^{{\\mathrm {RH}}}$ is continuous along some segments of the axis, and discontinuous along other segments.",
"Correspondingly the functions $F_{r,\\gamma }$ must be zero on some segments and nonzero on others, so in particular they cannot be holomorphic functions of $\\mathcal {X}_\\gamma $ .",
"This feature is related to the fact that each $r$ aggregates contributions from infinitely many rays which accumulate at the boundary of the half-plane, as shown in fig:collapsing-mathieu.",
"We can work out the discontinuities of the functions $\\mathcal {X}_\\gamma $ by keeping track of their symmetry properties.",
"First, we have $ \\mathcal {X}^{\\vartheta =\\pi }_\\gamma (-\\hbar ) = \\mathcal {X}_\\gamma ^{\\vartheta =0}(\\hbar )^{-1}.$ Second, $\\mathcal {X}_A(\\hbar )$ is real for $\\hbar > 0$ , which implies the reality property $\\mathcal {X}_A(\\bar{\\hbar }) = \\overline{\\mathcal {X}_A(\\hbar )}$ .",
"Combining this with (REF ) we get $\\mathcal {X}_A^{{\\mathrm {RH}}}(- \\bar{\\hbar }) = \\overline{\\mathcal {X}_A^{{\\mathrm {RH}}}(\\hbar )}^{-1}.$ It follows that the discontinuity of $\\mathcal {X}_A^{{\\mathrm {RH}}}$ at the imaginary axis is $\\mathcal {X}_A^{{\\mathrm {RH}}} \\mapsto \\overline{\\mathcal {X}_A^{{\\mathrm {RH}}}}^{-1} = \\mathcal {X}_A^{{\\mathrm {RH}}} \\times \\vert \\mathcal {X}_A^{{\\mathrm {RH}}}\\vert ^{-2}.$ For $\\mathcal {X}_B^{{\\mathrm {RH}}}$ it is similar except that the reality property has an extra sign, $\\mathcal {X}_B(\\bar{\\hbar }) = \\overline{\\mathcal {X}_B(\\hbar )}^{-1}$ , giving $\\mathcal {X}_B^{{\\mathrm {RH}}}(- \\bar{\\hbar }) = \\overline{\\mathcal {X}_B^{{\\mathrm {RH}}}(\\hbar )}.$ Thus the discontinuity of $\\mathcal {X}_B^{{\\mathrm {RH}}}$ is $\\mathcal {X}_B^{{\\mathrm {RH}}} \\mapsto \\overline{\\mathcal {X}_B^{{\\mathrm {RH}}}} = \\mathcal {X}_B^{{\\mathrm {RH}}} \\times \\frac{\\overline{\\mathcal {X}_B^{{\\mathrm {RH}}}}}{\\mathcal {X}_B^{{\\mathrm {RH}}}}.$ Substituting these discontinuities into the general form (REF ) using (REF ), we get integral equations which are most naturally written directly in terms of $x_\\gamma = \\log \\mathcal {X}_\\gamma $ : $ x_A^{{\\mathrm {RH}}}(\\hbar ) = \\frac{Z_A}{\\hbar } + \\frac{1}{2 \\pi I} \\int _0^{I\\infty } \\mathrm {d}\\hbar ^{\\prime } \\left( \\frac{2\\hbar }{\\hbar ^{\\prime 2} - \\hbar ^2} \\right) (-2 \\operatorname{Re}x_A^{{\\mathrm {RH}}}(\\hbar ^{\\prime }+0)),$ $ x_B^{{\\mathrm {RH}}}(\\hbar ) = \\frac{Z_B}{\\hbar } + \\frac{1}{2 \\pi I} \\int _0^{I\\infty } \\mathrm {d}\\hbar ^{\\prime } \\left( \\frac{2\\hbar }{\\hbar ^{\\prime 2} - \\hbar ^2} \\right) (-2 I\\operatorname{Im}x_B^{{\\mathrm {RH}}}(\\hbar ^{\\prime }+0)).$ Numerical experimentation gives us some confidence that (REF ), (REF ) do indeed hold.",
"These equations by themselves do not fully characterize $x_A^{{\\mathrm {RH}}}$ and $x_B^{{\\mathrm {RH}}}$ ; to see this it is enough to observe that they admit the “trivial” solutions $x_{\\gamma }^{{\\mathrm {RH}}}(\\hbar ) = \\frac{Z_{\\gamma }}{\\hbar }$ .",
"This is a bit disappointing when we compare to simpler examples like that of sec:integral-equations-cubic, where it is believed that the integral equations do characterize the spectral coordinates, and even give a useful way of computing them.",
"One hope remains; the actual functions $x_{\\gamma }^{{\\mathrm {RH}}}$ obey one more important condition: for $\\hbar \\in \\pm I\\mathbb {R}$ , the quantity $x_A^{{\\mathrm {RH}}} \\pm 2 x_B^{{\\mathrm {RH}}}$ is always either real or pure imaginary.",
"It would be interesting to know whether this property together with (REF ), (REF ) is enough to determine the functions $x^{{\\mathrm {RH}}}_{\\gamma }$ ."
],
[
"Another Stokes graph",
"To get good information about the region $E > -1$ from WKB, we switch to considering the $\\vartheta $ -Stokes graphs relevant for that region.",
"There are two possibilities, depending on whether $E \\in (-1,1)$ or $E>1$ .",
"Here we will just discuss $E > 1$ .",
"Then the $\\vartheta $ -Stokes graph for $\\vartheta = 0$ is shown in fig:mathieu-sn-3.",
"type=figure Figure: NO_CAPTION figure$\\vartheta $ -Stokes graph for the Mathieu equation, at the phase $\\vartheta = 0$ , and $E = \\frac{41}{40}$ .",
"All notation is as in fig:mathieu-sn-1."
],
[
"The spectral coordinates",
"Let $\\psi $ denote the unique solution of (REF ) which decays exponentially as $z \\rightarrow 0$ along the negative real axis, $\\psi ^{\\prime }$ the unique solution which decays exponentially as $z \\rightarrow \\infty $ along the negative real axis, and $M$ the operator of counterclockwise monodromy around $z = 0$ .",
"Then we have (see app:computations) $ \\mathcal {X}_A = \\pm \\sqrt{\\frac{[\\psi , M \\psi ^{\\prime }]}{[M \\psi , \\psi ^{\\prime }]}}, \\qquad \\mathcal {X}_B = \\frac{[\\psi , M \\psi ][\\psi ^{\\prime } , M \\psi ^{\\prime }]}{[\\psi , \\psi ^{\\prime }]^2}.",
"$ In particular, unlike sec:spectral-coords-mathieu, here there is no spectral coordinate which is equal to an eigenvalue of $M$ .",
"Nevertheless, we can express the trace of the monodromy in terms of spectral coordinates: $ \\operatorname{Tr}M = (\\mathcal {X}_A + \\mathcal {X}_A^{-1}) \\sqrt{1 - \\mathcal {X}_B}.$ One quick way to see (REF ) is to write $M$ relative to the basis $(\\psi ,\\psi ^{\\prime })$ as a matrix $\\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix}$ ; then (REF ) becomes $a + d = \\pm \\left(\\sqrt{\\frac{d}{a}} + \\sqrt{\\frac{a}{d}}\\right) (\\sqrt{1 + bc})$ which indeed holds, using the fact that $ad - bc = 1$ .",
"To fix the sign we use the facts that, at small $\\hbar $ , $\\sqrt{1 - \\mathcal {X}_B}$ is exponentially close to 1, and $\\mathcal {X}_A$ is exponentially close to an eigenvalue of $M$ ."
],
[
"Application: quasiperiodic solutions",
"Now we consider the application of these spectral coordinates to another classical spectral problem.",
"If we consider $x$ to be a real variable, then (REF ) is a Schrödinger equation with periodic potential, $V(x) = \\cos x$ .",
"The standard analysis of such equations involves fixing $\\nu \\in \\mathbb {R}/ 2 \\pi \\mathbb {Z}$ (quasimomentum) and looking at solutions obeying the quasiperiodic boundary condition $ \\psi (x + 2 \\pi ) = {\\mathrm {e}}^{I\\nu } \\psi (x).$ For fixed $\\nu $ , solutions of (REF ), (REF ) exist only at a countable set of energies $E = E_1, E_2, \\dots $ , with all $E_n > -1$ .",
"These can be thought of as analogues of the bound state energies for a confining potential on the real line.",
"Using (REF ) we can rewrite the quasiperiodicity condition (REF ) asThe minus sign on the right side in (REF ) arises because of the square-root cut in the transformation (REF ).",
"$ (\\mathcal {X}_A + \\mathcal {X}_A^{-1}) \\sqrt{1 - \\mathcal {X}_B} = -2 \\cos \\nu .$ This is another example of an “exact quantization condition” in the terminology of exact WKB (however, we have not found precisely (REF ) in the literature.)",
"When $E>1$ we have $Z_B \\in \\mathbb {R}_-$ , and the leading WKB asymptotic $\\mathcal {X}_B \\approx -\\exp (Z_B / \\hbar )$ , so the factor $\\sqrt{1 - \\mathcal {X}_B}$ in (REF ) gives an exponentially small correction.",
"As a first approach we could try neglecting this correction.",
"Then (REF ) reduces to $ \\mathcal {X}_A \\approx -{\\mathrm {e}}^{\\pm I\\nu }.$ To derive concrete predictions from (REF ) we can use the WKB series for $\\mathcal {X}_A$ .",
"For example, suppose we take the leading asymptotic $\\mathcal {X}_A \\approx -\\exp (Z_A / \\hbar )$ , and further take large $E$ , so that $Z_A \\approx 2 \\pi I\\sqrt{2 E}$ : then we get ${\\mathrm {e}}^{2 \\pi I\\sqrt{2 E} / \\hbar } \\approx {\\mathrm {e}}^{\\pm I\\nu },$ i.e.",
"$ E \\approx \\frac{\\hbar ^2}{2} \\left(n \\pm \\frac{\\nu }{2\\pi }\\right)^2.$ This is indeed the leading behavior of the energies at large $E$ and small $\\hbar $ ; in fact, in this limit we can approximate the quasiperiodic solutions with given $\\nu $ simply by the free-particle wavefunctions, $\\psi (x) \\approx {\\mathrm {e}}^{I(\\pm n + \\nu / 2\\pi ) x}$ .",
"To improve the accuracy one could include subleading terms in the WKB series of $\\mathcal {X}_A$ ; this gives perturbative corrections in a power series in $\\hbar $ .",
"Likewise one could take more terms in the expansion of $Z_A$ around large $E$ .",
"This would modify the relation between $E$ and $(n,\\nu )$ but preserve the basic feature that for every $E$ there is some corresponding $(n,\\nu )$ with $\\nu $ real.",
"Indeed, even if we used the exact $\\mathcal {X}_A$ in (REF ), we would still find that for every $E$ there is a corresponding $(n,\\nu )$ with $\\nu $ real; this follows from the fact that $\\vert \\mathcal {X}_A\\vert = 1$ for all large enough real $E$ , a consequence of (REF ).",
"Now, let us consider the nonperturbative correction $\\sqrt{1 - \\mathcal {X}_B}$ in (REF ).",
"This has a qualitatively new effect: when $\\mathcal {X}_A(E)$ is close to $\\pm 1$ , the LHS of (REF ) can have absolute value larger than 2.",
"For such an $E$ there is no solution to (REF ) for any real $\\nu $ ; the eigenvalues of the monodromy become complex.",
"This is the well-known phenomenon of “gaps” in the Mathieu spectrum.",
"It is known that the width of the gaps is exponentially suppressed by $\\frac{1}{2} Z_B / \\hbar $ ; see e.g.",
"[68] for discussion and references on this point.In this context the quantity $\\frac{1}{2} Z_B$ might be called a “1-instanton action” since it corresponds to the change in the exponent of a WKB solution upon integrating along a one-way path from one branch point to another, as opposed to $Z_B$ which is the integral over the round-trip path $\\gamma _B$ .",
"Let us see how to recover this fact from the exact quantization condition (REF ).",
"Taking $\\cos \\nu = -1$ , expanding $\\mathcal {X}_A = 1 + \\delta \\mathcal {X}_A$ and taking $\\mathcal {X}_B$ small, (REF ) gives $\\left(2 + (\\delta \\mathcal {X}_A)^2 \\right) \\left(1 - \\frac{1}{2} \\mathcal {X}_B\\right) \\approx 2,$ i.e.",
"the leading-order displacement of $\\mathcal {X}_A$ from the gap center is $\\delta \\mathcal {X}_A \\approx \\pm \\sqrt{\\mathcal {X}_B},$ and thus the leading-order displacement of $E$ from the gap center is $\\delta E \\approx \\pm \\frac{\\sqrt{\\mathcal {X}_B}}{\\partial \\mathcal {X}_A / \\partial E}.$ If we further take the leading $\\hbar \\rightarrow 0$ asymptotics of $\\mathcal {X}_A$ and $\\mathcal {X}_B$ , this becomesAs a check against blunders, we numerically computed the width of a few of the gaps and obtained reasonable agreement: for example, when $\\hbar = 0.2$ , there is a gap extending from $E_- \\approx 1.3836418$ to $E_+ \\approx 1.3838946$ , which thus has $\\delta E = \\frac{1}{2}(E_+ - E_-) \\approx 0.0001264$ , while the estimate (REF ) gives $\\delta E \\approx 0.0001278$ .",
"$ \\delta E \\approx \\pm \\hbar \\left( I\\frac{\\partial Z_A}{\\partial E} \\right)^{-1} \\exp \\left(\\frac{1}{2\\hbar } Z_B\\right).$ One could try to go beyond this leading-order estimate using the full $\\hbar $ expansions of $\\mathcal {X}_A$ and $\\mathcal {X}_B$ .",
"It would be interesting to know whether in this way one can recover the more detailed results on the gap widths explained in [68]."
],
[
"Exact WKB for higher order equations",
"So far we have been discussing order 2 differential equations (REF ).",
"We now move to the case of order 3 equations, involving two meromorphic “potentials” $P_2$ and $P_3$ : $ \\left[ \\partial _z^3 + \\hbar ^{-2} P_2 \\partial _z + (\\hbar ^{-3} P_3 + \\frac{1}{2} \\hbar ^{-2} P^{\\prime }_2) \\right] \\psi (z) = 0.$ The equation (REF ) can be given a global meaning on a Riemann surface $C$ with a complex projective structure, as with (REF ) above; in this case $\\psi (z)$ is a section of $K_C^{-1}$ , $P_2(z)$ is a meromorphic quadratic differential, and $P_3(z)$ is a meromorphic cubic differential.",
"In this section we explain how the exact WKB method is expected to extend to equations of the form (REF ).",
"In this situation there are no rigorous results yet, but there is a reasonable conjectural picture.",
"(The same picture is expected to work for equations of any order $K \\ge 2$ ; we stick to $K=3$ to be concrete, and because our main example has $K=3$ .)",
"Some numerical evidence supporting this conjectural picture in special cases has been obtained in [18], [59].",
"We will give more numerical evidence in the case of the $T_3$ equation in sec:t3-numerics and sec:integral-equations-t3 below.",
"All the formal structures in the story are parallel to the order 2 case, so this section is organized in parallel to sec:wkb-review, and we will be very brief."
],
[
"WKB solutions",
"WKB solutions of (REF ) are solutions of the form $ \\psi (z) = \\exp \\left(\\hbar ^{-1} \\int _{z_0}^z \\lambda (z) \\, \\mathrm {d}z \\right),$ where now $\\lambda $ must obey a higher analogue of the Riccati equation (REF ), $ \\lambda ^3 + 3 \\hbar \\lambda \\partial _z \\lambda + \\hbar ^2 \\partial _z^2 \\lambda + P_2 \\lambda + P_3 + \\frac{1}{2}\\hbar P_2^{\\prime } = 0.$ One again constructs WKB solutions $\\lambda _i^{\\mathrm {formal}}$ as power series in $\\hbar $ .",
"As before, one meets an ambiguity at order $\\hbar ^0$ ; this ambiguity is resolved by choosing a sheet $i$ of the 3-fold covering $\\Sigma = \\lbrace y^3 + p_2(z) y + p_3(z) = 0 \\rbrace .$ Now the conjectural picture is that, as in the order 2 case, there exist actual solutions of (REF ) which have the asymptotic behavior $\\lambda ^\\vartheta _i \\sim \\lambda _i^{\\mathrm {formal}}$ in the half-plane $\\mathbb {H}_\\vartheta $ , away from $\\vartheta $ -Stokes curves.",
"The $\\vartheta $ -Stokes curves carry labels $ij$ .",
"Along a $\\vartheta $ -Stokes curve of type $ij$ , ${\\mathrm {e}}^{-I\\vartheta } (y_i - y_j) \\mathrm {d}z$ is real and positive.",
"We make the simplifying assumption that all branch points of $\\Sigma $ are simple branch points, i.e.",
"only two $y_i$ collide at a time.",
"For the construction of the $\\vartheta $ -Stokes graph in this case see [16].",
"One key new feature of the higher-order case, first discovered in [74] and further investigated in e.g.",
"[75], [14], [16], is that Stokes curves of type $ik$ can be born from intersections of Stokes curves of types $ij$ and $jk$ .",
"See fig:sample-sn-higher for an example.",
"The local solution $\\lambda ^\\vartheta _i$ of (REF ) is supposed to exist away from $\\vartheta $ -Stokes curves of type $ij$ , as in the order 2 case.",
"On crossing a $\\vartheta $ -Stokes curve of type $ij$ , we conjecture that the local WKB solution $\\psi _i^\\vartheta $ jumps by a constant multiple of $\\psi _j^\\vartheta $ .Some evidence for this conjecture has been given in [76].",
"We thank Kohei Iwaki for pointing out this reference.",
"type=figure Figure: NO_CAPTION figureAn example of a $\\vartheta $ -Stokes graph at $\\vartheta =0$ , with $p_2(z) = 1$ and $p_3(z) = z^3-1$ ."
],
[
"Abelianization",
"As in the order 2 case, the WKB solutions of (REF ) can be thought of as solutions of a first-order equation over $\\Sigma $ , built using the $\\lambda _i^\\vartheta $ .",
"Thus, as before, exact WKB analysis of (REF ) leads to a line bundle $\\mathcal {L}$ with almost-flat connection $\\nabla ^{\\mathrm {ab},\\vartheta }$ over $\\Sigma $ , away from the $\\vartheta $ -Stokes curves."
],
[
"Gluing across the Stokes graph",
"Also as before, we can glue $\\mathcal {L}$ and $\\nabla ^{\\mathrm {ab},\\vartheta }$ across the $\\vartheta $ -Stokes curves.",
"At a $\\vartheta $ -Stokes curve of type $ij$ the gluing takes the form (cf.",
"(REF )) $ \\begin{pmatrix}\\psi _i^{L} \\\\\\psi _j^{L} \\\\\\psi _k^{L}\\end{pmatrix}\\mapsto \\begin{pmatrix} 1 & \\beta & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & 1 \\end{pmatrix} \\begin{pmatrix}\\psi _i^{L} \\\\\\psi _j^{L} \\\\\\psi _k^{L}\\end{pmatrix}=\\begin{pmatrix}\\frac{[\\psi _i^L , \\psi _j^L , \\psi _k^L]}{[\\psi _i^R , \\psi _j^L , \\psi _k^L]}\\psi _i^{R} \\\\\\frac{[\\psi _j^L, \\psi _k^L, \\psi _i^L]}{[\\psi _j^R, \\psi _k^L, \\psi _i^L]} \\psi _j^{R} \\\\\\frac{[\\psi _k^L, \\psi _i^L, \\psi _j^L]}{[\\psi _k^R, \\psi _i^L, \\psi _j^L]} \\psi _k^{R}\\end{pmatrix}$ .",
"If $\\vartheta $ -Stokes curves of type $ij$ and $ji$ coincide, we choose a gluing of the form (cf.",
"(REF ))As in the order 2 case (see sec:stokes-gluing) this is not the only possible choice, but it is the most invariant choice.",
"$ \\begin{pmatrix}\\psi _i^{L} \\\\\\psi _j^{L} \\\\\\psi _k^{L}\\end{pmatrix}\\mapsto \\begin{pmatrix} \\rho & \\beta & 0 \\\\ \\alpha & \\rho & 0 \\\\ 0 & 0 & 1 \\end{pmatrix} \\begin{pmatrix}\\psi _i^{L} \\\\\\psi _j^{L} \\\\\\psi _k^{L}\\end{pmatrix}=\\begin{pmatrix}\\sqrt{\\frac{[\\psi _i^L , \\psi _j^L , \\psi _k^L]}{[\\psi _i^R , \\psi _j^R , \\psi _k^L]} \\frac{[\\psi _i^L , \\psi _j^R , \\psi _k^L]}{[\\psi _i^R , \\psi _j^L , \\psi _k^L]}}\\psi _i^{R} \\\\\\sqrt{\\frac{[\\psi _j^L , \\psi _i^L , \\psi _k^L]}{[\\psi _j^R , \\psi _i^R , \\psi _k^L]} \\frac{[\\psi _j^L , \\psi _i^R , \\psi _k^L]}{[\\psi _j^R , \\psi _i^L , \\psi _k^L]}}\\psi _j^{R}\\\\ \\frac{[\\psi _k^L, \\psi _i^L, \\psi _j^L]}{[\\psi _k^R, \\psi _i^L, \\psi _j^L]} \\psi _k^{R}\\end{pmatrix},$ with $\\rho ^2 - \\alpha \\beta = 1$ .",
"(The branches of the square roots are fixed as was done above in the $K=2$ case.)",
"By this process we obtain a line bundle $\\mathcal {L}$ with almost-flat connection $\\nabla ^{\\mathrm {ab},\\vartheta }$ over $\\Sigma $ ."
],
[
"Spectral coordinates",
"Finally we can introduce higher-order versions of the spectral coordinates: as before, these are defined by $\\mathcal {X}_\\gamma ^\\vartheta = \\operatorname{Hol}_\\gamma \\nabla ^{\\mathrm {ab},\\vartheta } \\in \\times .$ The $\\mathcal {X}_\\gamma ^\\vartheta $ are expected to have all the same formal properties as in the order 2 case, discussed in sec:spectral-coordinates-sec:integral-equations; we will not repeat those here."
],
[
"Exact WKB for the $T_3$ equation",
"Now we consider a specific instance of (REF ), a third-order ODE over $\\mathbb {CP}^1$ with three regular singularities.",
"By convention we place the singularities at $\\lbrace 1,\\omega ,\\omega ^2\\rbrace $ where $\\omega = {\\mathrm {e}}^{2 \\pi I/ 3}$ :Our conventions here differ from those of [34] by the replacement $u \\rightarrow -u$ .",
"Sorry.",
"$ \\left[ \\partial _z^3 + \\hbar ^{-2} P_2 \\partial _z + (\\hbar ^{-3} P_3+\\frac{1}{2} \\hbar ^{-2} P^{\\prime }_2) \\right] \\psi (z) = 0, \\qquad P_2 = \\frac{9 \\hbar ^2 z}{(z^3 - 1)^2}, \\quad P_3 = \\frac{u}{(z^3-1)^2}.$ We call (REF ) the $T_3$ equation.",
"This equation actually does not depend on $u$ and $\\hbar $ separately, but only on the combination $u^{\\prime } = u / \\hbar ^3 \\in .$"
],
[
"A simple Stokes graph",
"The $\\vartheta $ -Stokes graphs for the $T_3$ equation were investigated in [34].",
"It was found there that the topology of the $\\vartheta $ -Stokes graph depends on the phase of the quantity $w = {\\mathrm {e}}^{-3 I\\vartheta } u$ .",
"For a generic phase of $w$ , it seems likely that the Stokes graph is “wild” — in particular, that it is dense at least in some parts of $\\mathbb {CP}^1$ .",
"WKB analysis involving such a wild Stokes graph may ultimately be very interesting, but we are not brave enough to try it today.In the order 2 case, some of the necessary analytic technology for dealing with wild Stokes graphs is developed in [77].",
"It would be exciting to develop the higher-rank analogue of this.",
"Instead, we focus on the non-generic situation where the Stokes graph is compact; this happens for a countable set of phases of $w$ .",
"We will not rederive the form of the Stokes graphs here, but simply lift them from [34].",
"The simplest Stokes graph arises when $w$ is real; to be completely concrete, we take $u > 0$ and $\\vartheta = 0$ .",
"See fig:circle-network.",
"type=figure Figure: NO_CAPTION figureThe $\\vartheta $ -Stokes graph for the $T_3$ equation, in case $u > 0$ and $\\vartheta = 0$ .",
"The three branch cuts emanating from the singularities meet at $z = \\infty $ .",
"Applying the higher-order exact WKB method is expected to produce a $\\mathcal {W}$ -abelianization of the $T_3$ equation.",
"Thus, we should begin by understanding concretely what this means.",
"We explained in sec:w-framings that $\\mathcal {W}$ -abelianizations of a meromorphic Schrödinger equation with second-order poles are in 1-1 correspondence with discrete data called $\\mathcal {W}$ -framings, and the choice of a $\\mathcal {W}$ -framing amounts to choosing one of the two eigenvectors of the monodromy around each singularity and each cylinder.",
"In the case of the $T_3$ equation, we will have a formally similar story, except that the linear-algebra problem one has to solve to find $\\mathcal {W}$ -abelianizations is more intricate: it does not just correspond to choosing eigenvectors of monodromy matrices."
],
[
"The abelianization problem for the $T_3$ equation",
"The local solutions of (REF ) in a neighborhood of $z=0$ form a 3-dimensional vector space $V$ .",
"In fig:base-cycles we show three cycles $A, B, C$ on $\\mathbb {CP}^1 \\setminus \\lbrace 1,\\omega ,\\omega ^2\\rbrace $ , beginning and ending at $z = 0$ .",
"type=figure Figure: NO_CAPTION figureThree cycles on $\\mathbb {CP}^1 \\setminus \\lbrace 1,\\omega ,\\omega ^2\\rbrace $ .",
"Let $\\mathbf {A}, \\mathbf {B}, \\mathbf {C}$ denote the maps $V \\rightarrow V$ induced by monodromy of (REF ) around these three cycles.",
"Note they satisfy $ \\mathbf {A}\\mathbf {B}\\mathbf {C}= 1.$ We say a basis $(\\psi _1, \\psi _2, \\psi _3)$ of $V$ is in special position if the following conditions are satisfied: $\\mathbf {C}\\psi _1, \\mathbf {B}^{-1} \\psi _2 \\in \\langle \\psi _1, \\psi _2\\rangle , \\\\\\mathbf {A}\\psi _2, \\mathbf {C}^{-1} \\psi _3 \\in \\langle \\psi _2, \\psi _3\\rangle , \\\\\\mathbf {B}\\psi _3, \\mathbf {A}^{-1} \\psi _1 \\in \\langle \\psi _3, \\psi _1\\rangle .",
"$ A concrete way to think about the special-position constraint is that relative to the basis $(\\psi _1,\\psi _2,\\psi _3)$ the monodromy endomorphisms must have zeroes in specific places: $\\mathbf {A}&= \\begin{pmatrix} * & 0 & * \\\\ * & * & * \\\\ * & * & * \\end{pmatrix}, \\quad &\\mathbf {B}&= \\begin{pmatrix} * & * & * \\\\ * & * & 0 \\\\ * & * & * \\end{pmatrix}, &\\mathbf {C}&= \\begin{pmatrix} * & * & * \\\\ * & * & * \\\\ 0 & * & * \\end{pmatrix}, \\\\\\mathbf {A}^{-1} &= \\begin{pmatrix} * & * & * \\\\ 0 & * & * \\\\ * & * & * \\end{pmatrix}, &\\mathbf {B}^{-1} &= \\begin{pmatrix} * & * & * \\\\ * & * & * \\\\ * & 0 & * \\end{pmatrix}, \\quad &\\mathbf {C}^{-1} &= \\begin{pmatrix} * & * & 0 \\\\ * & * & * \\\\ * & * & * \\end{pmatrix}.$ The special-position constraint is invariant under rescalings of the vectors $(\\psi _1, \\psi _2, \\psi _3)$ : it depends only on the projective basis of $V$ , which we can view as a 3-tuple of points in the projective space ${\\mathbb {P}}(V) \\simeq \\mathbb {CP}^2$ .",
"The point of this definition is the following, proven in app:computations: $\\mathcal {W}$ -abelianizations for the $T_3$ equation are in 1-1 correspondence with projective bases $(\\psi _1, \\psi _2, \\psi _3)$ of $V$ in special position.",
"Now the question arises: how can we enumerate the possible projective bases of $V$ in special position?",
"Note that (REF ) imposes 6 conditions on the basis, so a naive dimension count would suggest that bases obeying these constraints should occur discretely.",
"To enumerate them precisely is a problem of algebraic geometry, which we address in sec:enumerating-special-bases below.",
"The outcome is that when $\\mathbf {A}$ , $\\mathbf {B}$ , $\\mathbf {C}$ are unipotent and generic enough there are “$4+\\infty $ ” projective bases in special position: 4 occurring discretely plus a 1-parameter family."
],
[
"Projective bases in special position",
"In this section we consider the following question.",
"Suppose given unipotent endomorphisms $\\mathbf {A}$ , $\\mathbf {B}$ , $\\mathbf {C}$ of a 3-dimensional complex vector space $V$ , obeying $\\mathbf {A}\\mathbf {B}\\mathbf {C}= {\\mathbf {1}}$ .",
"Assume that $\\mathbf {A}$ , $\\mathbf {B}$ , $\\mathbf {C}$ are in general position; concretely this means that each of $\\mathbf {A}$ , $\\mathbf {B}$ , $\\mathbf {C}$ preserves a unique complete flag, and these flags are in general position.",
"How do we enumerate the projective bases of $V$ in special position?",
"We begin with an observation.",
"Let $\\langle e_{\\mathbf {A}}\\rangle $ denote the eigenline of $\\mathbf {A}$ and similarly for $\\mathbf {B}, \\mathbf {C}$ .",
"Suppose that $(\\psi _1,\\psi _2,\\psi _3)$ is a projective basis in special position.",
"Assume that $\\langle \\psi _1\\rangle \\ne \\langle e_\\mathbf {C}\\rangle $ .",
"Then $\\langle \\psi _1, \\mathbf {C}\\psi _1\\rangle $ is a plane, and (REF ) says this plane contains both $\\psi _2$ and $\\mathbf {B}^{-1} \\psi _2$ .",
"Equivalently, we have $\\psi _2 \\in \\langle \\psi _1, \\mathbf {C}\\psi _1\\rangle , \\qquad \\psi _2 \\in \\mathbf {B}\\langle \\psi _1, \\mathbf {C}\\psi _1\\rangle .$ Now, these two planes are not equal (if they were, then () would show that this plane also contains $\\psi _3$ , contradicting the linear independence of the $\\psi _i$ .)",
"Since both contain $\\psi _2$ , their intersection must be precisely $\\langle \\psi _2\\rangle $ : $ \\langle \\psi _2\\rangle = \\langle \\psi _1, \\mathbf {C}\\psi _1\\rangle \\cap \\mathbf {B}\\langle \\psi _1, \\mathbf {C}\\psi _1\\rangle .$ Let $X = \\mathbb {P}(V) \\simeq \\mathbb {CP}^2$ .",
"The relation (REF ) can be expressed as $\\psi _2 = \\Phi _{\\mathbf {B},\\mathbf {C}}(\\psi _1)$ where $\\Phi _{\\mathbf {B},\\mathbf {C}}: X \\dashrightarrow X$ is the product of two “quadratic transformations”A useful reference on quadratic transformations is [78].",
"$\\Phi _{\\mathbf {B},\\mathbf {C}} = \\Phi _{\\mathbf {B}^*} \\circ \\Phi _\\mathbf {C},$ with $\\Phi _\\mathbf {C}: X \\dashrightarrow X^*$ the quadratic transformation taking the line $\\langle \\psi \\rangle $ to the plane $\\langle \\psi ,\\mathbf {C}\\psi \\rangle $ , and $\\Phi _{\\mathbf {B}^*}: X^* \\dashrightarrow X$ the dual quadratic transformation taking a plane $p$ to the line $p \\cap \\mathbf {B}p$ .",
"Thus $\\Phi _{\\mathbf {B},\\mathbf {C}}$ is a birational map (Cremona transformation) of degree 4, i.e.",
"defined by three homogeneous degree 4 polynomials.",
"Thus $\\psi _2$ is determined by $\\psi _1$ .",
"Repeating this process using (), () shows $\\psi _3$ is determined by $\\psi _2$ , and $\\psi _1$ is determined by $\\psi _3$ : $ \\psi _3 = \\Phi _{\\mathbf {C},\\mathbf {A}}(\\psi _2), \\qquad \\psi _1 = \\Phi _{\\mathbf {A},\\mathbf {B}}(\\psi _3)$ Altogether, this means $\\psi _1$ is constrained to obey $\\psi _1 = {\\widehat{\\Phi }}(\\psi _1)$ where ${\\widehat{\\Phi }}: X \\dashrightarrow X$ is a degree 64 birational map ${\\widehat{\\Phi }}= \\Phi _{\\mathbf {A},\\mathbf {B}} \\circ \\Phi _{\\mathbf {C},\\mathbf {A}} \\circ \\Phi _{\\mathbf {B},\\mathbf {C}}.$ Thus, whenever $(\\psi _1,\\psi _2,\\psi _3)$ is a projective basis in special position, $\\langle \\psi _1\\rangle \\in X$ is a fixed point of ${\\widehat{\\Phi }}$ , and (REF ) then determines the rest of the basis.",
"This translates the job of finding projective bases in special position to the job of finding the fixed locus of ${\\widehat{\\Phi }}$ .",
"This problem is simplified by the observation that ${\\widehat{\\Phi }}$ preserves the ratio of two cubic forms.",
"Indeed, suppose we define a cubic form on $V$ by $F_{M,M^{\\prime }}(\\psi ) = [\\psi , M \\psi , M^{\\prime } \\psi ],$ and dually on $V^*$ $F^*_{M,M^{\\prime }}(\\eta ) = [\\eta , M^T \\eta , M^{\\prime T} \\eta ].$ Then we have an identity of sextic forms on $V$ ,We have no great insight into why this identity is true, although we have checked it in Mathematica; it is a specialization of a “remarkable identity” originally due to Zagier, given as equation 14 in [79].",
"$ F^*_{M,M^{\\prime }}(\\Phi _{M^{\\prime }}(\\psi )) = F_{M,M^{\\prime }}(\\psi ) F_{M^{\\prime -1},M}(\\psi ).$ Now we consider the ratio of cubic forms $r(\\psi ) = \\frac{F_{\\mathbf {C},\\mathbf {A}^{-1}}(\\psi )}{F_{\\mathbf {C}^{-1},\\mathbf {A}}(\\psi )}.$ Using (REF ) six times we obtain the desired invariance: $r({\\widehat{\\Phi }}(\\psi )) = r(\\psi ).$ (REF ) is equivalent to saying that ${\\widehat{\\Phi }}$ preserves a one-parameter family (pencil) of cubic curves $E_t \\subset X$ , $ E_t = \\lbrace F_{\\mathbf {C},\\mathbf {A}^{-1}} (\\psi ) + t F_{\\mathbf {C}^{-1},\\mathbf {A}} (\\psi ) = 0\\rbrace \\subset X.$ There are three points of $X$ which are common to all of the $E_t$ , or said otherwise, this pencil of cubic curves has a base locus supported at three points of $X$ .",
"Two of the base points are easy to spot: if $\\psi = e_\\mathbf {A}$ or $\\psi = e_\\mathbf {C}$ then $F_{\\mathbf {C},\\mathbf {A}^{-1}}(\\psi ) = F_{\\mathbf {C}^{-1},\\mathbf {A}}(\\psi ) = 0$ and so $\\psi $ lies on every $E_t$ .",
"The last base point is trickier: it is $p_\\mathbf {B}\\cap \\mathbf {C}^{-1} p_\\mathbf {B}$ where $p_\\mathbf {B}$ is the unique plane fixed by $\\mathbf {B}$ .",
"(Indeed if $\\psi \\in p_\\mathbf {B}\\cap \\mathbf {C}^{-1} p_\\mathbf {B}$ then $\\psi $ , $\\mathbf {C}\\psi $ and $\\mathbf {A}^{-1} \\psi $ all lie in $p_\\mathbf {B}$ , showing that $F_{\\mathbf {C},\\mathbf {A}^{-1}}(\\psi ) = 0$ ; similarly $F_{\\mathbf {C}^{-1},\\mathbf {A}}(\\psi ) = 0$ .)",
"These three base points lie on a line $\\ell \\subset X$ .",
"In fact the line $\\ell $ (with multiplicity 3) is equal to $E_t$ for some $t = t_*$ .",
"Any point of $\\ell $ is a fixed point of ${\\widehat{\\Phi }}$ (with the exception of the three base points, where ${\\widehat{\\Phi }}$ is not defined).",
"This gives a 1-parameter family of projective bases in special position.",
"Now we want to see if there are any other fixed points.",
"For this purpose the fact that ${\\widehat{\\Phi }}$ is not defined everywhere is technically inconvenient.",
"To resolve its indeterminacies we blow up the base locus.",
"This results in a singular surface, but by further blowing up the singular points, we obtain a smooth rational elliptic surface $\\widetilde{X}$ .",
"See fig:rational-surface.",
"type=figure Figure: NO_CAPTION figureA neighborhood of the $IV^*$ fiber (green) in the rational elliptic surface $\\widetilde{X}$ .",
"The preimage of each point of the base locus is a chain of three rational curves; two of the three are in the $IV^*$ fiber (green), while the last is a section of the elliptic fibration (black).",
"There are generically 4 other singular fibers elsewhere in $\\widetilde{X}$ (not shown).",
"$\\widetilde{X}$ has one fiber of Kodaira type $IV^*$ (affine $E_6$ configuration), which maps to the line $E_{t_*}$ through the base points in $X$ .",
"This fiber has Euler characteristic 8.",
"A smooth rational elliptic surface has Euler characteristic 12, and the smooth fibers do not contribute to the Euler characteristic, so there must be some other singular fibers in $\\widetilde{X}$ ; the most generic possibility is to have 4 more singular fibers, each of type $I_1$ (nodal torus), so that altogether $\\widetilde{X}$ has singular fibers $IV^* + 4 I_1$ .",
"For some special $\\mathbf {A}$ , $\\mathbf {B}$ , $\\mathbf {C}$ it may happen that some of the $I_1$ fibers collide.",
"In particular, in sec:t3-analytic-cont below we will meet a phenomenon where two $I_1$ fibers collide to make an $II$ fiber (cusp), so that $\\widetilde{X}$ has singular fibers $IV^* + II + 2 I_1$ .",
"The birational automorphism ${\\widehat{\\Phi }}$ of $X$ lifts to a regular map $\\widetilde{X}\\rightarrow X$ , so in particular ${\\widehat{\\Phi }}$ acts by an honest automorphism of each fiber except for the $IV^*$ fiber.",
"Since these fibers are (smooth or nodal) elliptic curves, their automorphism groups are easy to understand, and indeed by direct computations near a base point one can show that ${\\widehat{\\Phi }}$ is not trivial and not an inversion; thus it must act by a nontrivial translation on each fiber.",
"It follows that the only place a fixed point can occur is at a singularity of a fiber; in particular, in the generic $IV^* + 4 I_1$ case, the fixed points are exactly the 4 nodes.",
"Combining these with the continuous family we found before, we conclude finally that ${\\widehat{\\Phi }}$ has “$4 + \\infty $ ” fixed points, and thus there are “$4 + \\infty $ ” projective bases in special position, as we claimed above.",
"This description of the projective bases in special position gives some small insight into their nature and their number, but more importantly for us, it is efficient enough to be used for numerical computations: starting from $\\mathbf {A}$ , $\\mathbf {B}$ , $\\mathbf {C}$ , we use Mathematica to solve the polynomial system determining the singularities of the cubic curves (REF );In particular this seems to be much more efficient than trying to solve the coplanarity constraints (REF ) directly.",
"these give the desired basis elements $\\psi _1$ ; then we determine $\\psi _2$ and $\\psi _3$ using (REF ), (REF ).",
"Finally let us comment on the case of $\\mathbf {A}$ , $\\mathbf {B}$ , $\\mathbf {C}$ semisimple instead of unipotent.",
"(This case would arise if, instead of the conformally invariant Minahan-Nemeschansky theory, we considered its mass deformation.)",
"In this case the analysis is very similar to the above, except that the rational elliptic surface $\\widetilde{X}$ which appears is a bit different: it arises by blowing up 9 distinct points of $X$ (lying on a cubic curve), instead of 3 with multiplicity.",
"The result is that instead of singular fibers of type $IV^* + 4 I_1$ one generically gets $12 I_1$ , and so instead of “$4+\\infty $ ” $\\mathcal {W}$ -abelianizations there are generically just 12 $\\mathcal {W}$ -abelianizations."
],
[
"The spectral coordinates",
"Now we are in a position to decribe the spectral coordinates concretely.",
"type=figure Figure: NO_CAPTION figureCycles $\\gamma _A$ and $\\gamma _B$ on the 3-fold branched cover $\\Sigma $ .",
"Let $\\gamma _A$ , $\\gamma _B$ be the cycles on $\\Sigma $ shown in fig:circle-network-cycles.",
"Fix an ${\\mathrm {SL}}(3)$ -connection $\\nabla $ over $C$ , with unipotent holonomy, and fix a $\\mathcal {W}$ -abelianization of $\\nabla $ .",
"Let $(\\psi _1, \\psi _2, \\psi _3)$ be the corresponding basis of solutions near $z = 0$ .",
"As we have explained above, $(\\psi _1, \\psi _2, \\psi _3)$ are in special position.",
"Then, the spectral coordinates are (see app:computations) $\\mathcal {X}_A &= \\frac{[\\psi _2 , \\psi _3 , \\psi _1]}{[\\mathbf {C}^{-1} \\psi _3 , \\mathbf {A}\\psi _2 , \\psi _1]}, \\\\\\mathcal {X}_B &= \\sqrt{- \\frac{[\\mathbf {C}\\psi _1 , \\mathbf {B}^{-1} \\psi _2 , \\psi _3][\\mathbf {C}\\psi _1 , \\psi _1 , \\psi _3] [\\psi _2 , \\mathbf {A}\\psi _2 , \\psi _1] [\\mathbf {B}\\psi _3 , \\mathbf {A}^{-1} \\psi _1 , \\psi _2] [\\mathbf {B}\\psi _3 , \\psi _3 , \\psi _2]}{[\\psi _2 , \\mathbf {B}^{-1} \\psi _2 , \\psi _3][\\mathbf {C}^{-1} \\psi _3 , \\mathbf {A}\\psi _2 , \\psi _1] [\\mathbf {C}^{-1} \\psi _3 , \\psi _3 , \\psi _1] [\\psi _1 , \\psi _3 , \\psi _2] [\\psi _1 , \\mathbf {A}^{-1} \\psi _1 , \\psi _2]} }.",
"$"
],
[
"Spectral coordinates for the continuous family of abelianizations",
"In this section we record an interesting curiosity, not required for the rest of the paper.",
"Recall that there is a continuous family of $\\mathcal {W}$ -abelianizations, with the property that $\\psi _1$ is a linear combination of the eigenvectors of $\\mathbf {A}$ and $\\mathbf {C}$ , and similarly for $\\psi _2$ , $\\psi _3$ .",
"It turns out that the spectral coordinates $\\mathcal {X}_A$ and $\\mathcal {X}_B$ are independent of which member of the continuous family we take, so all of these $\\mathcal {W}$ -abelianizations are actually isomorphic, and in some sense they should be considered as just one abelianization.",
"Moreover, these spectral coordinates are Fock-Goncharov coordinates associated to an ideal triangulation of $\\mathbb {CP}^1$ .The triangulation is made up of 2 triangles, whose interiors are $\\lbrace \\vert z\\vert < 1\\rbrace $ and $\\lbrace \\vert z\\vert > 1\\rbrace $ .",
"Indeed, let $a_1$ be an eigenvector for $\\mathbf {A}$ , and $a_2$ another vector such that $\\langle a_1,a_2\\rangle $ is the unique plane preserved by $\\mathbf {A}$ ; likewise define $b_1, b_2$ and $c_1, c_2$ , and $d_1, d_2$ associated to the operator $\\mathbf {D}= \\mathbf {C}^{-1} \\mathbf {B}\\mathbf {C}$ .",
"Then the triple ratio and edge coordinate from [21] are $t = \\frac{[a_1 , a_2 , b_1] [b_1 , b_2 , c_1] [c_1 , c_2 , a_1]}{[a_1 , a_2 , c_1] [b_1 , b_2 , a_1] [c_1 , c_2 , b_1]}, \\qquad e = \\frac{[b_1 , c_1 , a_1][d_1 , a_2 , a_1]}{[a_2 , c_1 , a_1][b_1 , d_1 , a_1]}.$ These coordinates turn out to be related to the spectral coordinates for the continuous family of $\\mathcal {W}$ -abelianizations, by $\\mathcal {X}_A = e, \\qquad \\mathcal {X}_B = \\frac{t}{\\sqrt{e}}.$ It is not clear to us why the Fock-Goncharov coordinates appear as spectral coordinates for the Stokes graph $\\mathcal {W}$ .",
"In [27] it was shown that Fock-Goncharov coordinates do appear as spectral coordinates for a specific sort of spectral network associated to a triangulation, but that is a different spectral network from $\\mathcal {W}$ .",
"It would be interesting to understand this better.",
"At any rate, the Fock-Goncharov coordinates will not play much role in the rest of the paper; most of our attention will be focused instead on the 4 discrete abelianizations, since these are the ones which turn out to be directly related to WKB for the $T_3$ equation."
],
[
"The monodromy matrices",
"Relative to the projective basis $(\\psi _1, \\psi _2, \\psi _3)$ we can write the monodromy explicitly.",
"Its form depends on which $\\mathcal {W}$ -abelianization we take.",
"For the 4 discrete $\\mathcal {W}$ -abelianizations, it is (up to a diagonal gauge transformation) $\\mathbf {A}&= \\begin{pmatrix} -f(\\mathcal {X}_A) \\mathcal {X}_A & 0 & \\mathcal {X}_A \\mathcal {X}_B^{-1} \\sqrt{1 + f(\\mathcal {X}_A)^2 \\mathcal {X}_A} \\\\ (1 + f(\\mathcal {X}_A)^2 \\mathcal {X}_A) \\mathcal {X}_B & f(\\mathcal {X}_A) & -f(\\mathcal {X}_A) \\mathcal {X}_A \\sqrt{1 + f(\\mathcal {X}_A)^2 \\mathcal {X}_A} \\\\ f(\\mathcal {X}_A) \\mathcal {X}_B \\sqrt{1 + f(\\mathcal {X}_A)^2 \\mathcal {X}_A} & \\mathcal {X}_A^{-1} \\sqrt{1 + f(\\mathcal {X}_A)^2 \\mathcal {X}_A} & - f(\\mathcal {X}_A)^2 \\mathcal {X}_A \\end{pmatrix}, \\\\\\mathbf {B}&= \\begin{pmatrix} -f(\\mathcal {X}_A)^2 \\mathcal {X}_A & f(\\mathcal {X}_A) \\sqrt{1 + f(\\mathcal {X}_A)^2 \\mathcal {X}_A} & \\mathcal {X}_B^{-1} \\sqrt{1 + f(\\mathcal {X}_A)^2 \\mathcal {X}_A} \\\\ \\mathcal {X}_A \\sqrt{1 + f(\\mathcal {X}_A)^2 \\mathcal {X}_A} & -f(\\mathcal {X}_A) \\mathcal {X}_A & 0 \\\\ -f(\\mathcal {X}_A) \\mathcal {X}_B \\sqrt{1 + f(\\mathcal {X}_A)^2 \\mathcal {X}_A} & (1 + f(\\mathcal {X}_A)^2 \\mathcal {X}_A) \\mathcal {X}_B \\mathcal {X}_A^{-1} & f(\\mathcal {X}_A) \\end{pmatrix}, \\\\\\mathbf {C}&= \\begin{pmatrix} f(\\mathcal {X}_A) & -f(\\mathcal {X}_A)\\sqrt{1 + f(\\mathcal {X}_A)^2 \\mathcal {X}_A} & 1 + f(\\mathcal {X}_A)^2 \\mathcal {X}_A \\\\ \\sqrt{1 + f(\\mathcal {X}_A)^2 \\mathcal {X}_A} & - f(\\mathcal {X}_A)^2 \\mathcal {X}_A & f(\\mathcal {X}_A) \\mathcal {X}_A \\sqrt{1 + f(\\mathcal {X}_A)^2 \\mathcal {X}_A} \\\\ 0 & \\sqrt{1 + f(\\mathcal {X}_A)^2 \\mathcal {X}_A} & -f(\\mathcal {X}_A) \\mathcal {X}_A \\end{pmatrix},$ where $ f(\\mathcal {X}_A) = \\frac{1 - \\mathcal {X}_A \\pm \\sqrt{1 - 14 \\mathcal {X}_A + \\mathcal {X}_A^2}}{2 \\mathcal {X}_A}.$ The formulas (REF ) can be obtained directly by “nonabelianization:” we begin with $\\nabla ^\\mathrm {ab}$ and reconstruct $\\nabla $ from it, using only the constraint that the gluing matrices across Stokes curves are of the block form (REF ).",
"It we choose the $-$ sign in (REF ), then $f(\\mathcal {X}_A)$ is regular at $\\mathcal {X}_A = 0$ , with an expansion of the form $f(\\mathcal {X}_A) = 3 \\mathcal {X}_A + 12 \\mathcal {X}_A^2 + \\cdots $ This expansion played an important role in the analysis of BPS particles of the Minahan-Nemeschansky $E_6$ theory in [34]; its coefficients count BPS solitons in the Minahan-Nemeschansky theory coupled to a certain $\\frac{1}{2}$ -BPS surface defect.",
"The $-$ branch of the square root is also the one which appears for the $\\mathcal {W}$ -abelianization coming from exact WKB: when we take $u > 0$ and $\\hbar \\rightarrow 0$ with $\\arg \\hbar = 0$ , the WKB abelianization has $\\mathcal {X}_A$ exponentially small, and likewise $f(\\mathcal {X}_A)$ exponentially small.",
"On the other hand, when $\\mathcal {X}_A$ is not small, there is in general no canonical choice of branch in (REF ); both possibilities are possible.",
"This suggests that we should pay attention to the locus where the branches collide: this occurs when $1 - 14 \\mathcal {X}_A + \\mathcal {X}_A^2 = 0$ ie $\\mathcal {X}_A = 7 \\pm 4 \\sqrt{3}$ .",
"Indeed this locus will turn out to be important below."
],
[
"Testing the predictions of WKB",
"As we have described, when $u > 0$ and $\\hbar > 0$ , we conjecture that the higher-rank exact WKB method with $\\vartheta = 0$ furnishes a $\\mathcal {W}$ -abelianization of the ${\\mathrm {SL}}(3)$ -connection associated to the $T_3$ equation.",
"In fact we can go a bit further: since the $T_3$ equation depends only on $u^{\\prime } = u / \\hbar ^3$ , we could equally well study it by using exact WKB with $u > 0$ but $\\vartheta = \\arg \\hbar = \\frac{2 \\pi }{3}$ , or $\\vartheta = \\arg \\hbar = \\frac{4 \\pi }{3}$ .",
"The corresponding Stokes graphs $\\mathcal {W}^\\vartheta $ are not equal to $\\mathcal {W}= \\mathcal {W}^{\\vartheta =0}$ , but differ from $\\mathcal {W}$ only by cyclic permutations of the sheet labels $(123)$ .",
"Thus the $\\mathcal {W}^\\vartheta $ -abelianization provided by exact WKB can be converted to a $\\mathcal {W}$ -abelianization, by cyclically permuting the projective basis $(\\psi _1, \\psi _2, \\psi _3)$ .",
"In this way exact WKB should produce two additional $\\mathcal {W}$ -abelianizations $A_\\vartheta $ .",
"Altogether then, we expect that for $u^{\\prime } \\gg 0$ , among the $\\mathcal {W}$ -abelianizations of the $T_3$ equation we should find three coming from exact WKB, $A_\\vartheta $ ($\\vartheta = 0, \\frac{2\\pi }{3}, \\frac{4\\pi }{3}$ ).",
"The spectral coordinates associated to these three $\\mathcal {W}$ -abelianizations should have the small-$\\hbar $ asymptotic behavior $\\mathcal {X}_{\\gamma } \\approx \\exp \\left(Z_{\\gamma }(u) / \\hbar \\right), \\qquad \\arg \\hbar = \\vartheta .$ In fact, these asymptotics should hold not only for $\\arg \\hbar = \\vartheta $ but more generally for $\\arg \\hbar \\in (\\vartheta -\\frac{\\pi }{2}, \\vartheta +\\frac{\\pi }{2})$ .",
"It is convenient to rewrite these asymptotics in terms of the invariant parameter $u^{\\prime }$ , using the explicit formulas for the periods: $ Z_{A} = -M u^{\\frac{1}{3}}, \\qquad Z_{B} = -{\\mathrm {e}}^{-\\frac{2 \\pi I}{3}} M u^{\\frac{1}{3}},$ where $M = 2^{-\\frac{2}{3}} \\pi ^{-\\frac{1}{2}} \\Gamma \\left(\\frac{1}{3}\\right) \\Gamma \\left(\\frac{1}{6}\\right) \\approx 5.2999\\,.$ Then the prediction is $ \\mathcal {X}_{A} \\approx \\exp (-M {\\mathrm {e}}^{- I\\vartheta } u^{\\prime \\frac{1}{3}}), \\qquad \\mathcal {X}_{B} \\approx \\exp (-M {\\mathrm {e}}^{- I(\\vartheta + \\frac{2 \\pi }{3})} u^{\\prime \\frac{1}{3}}),$ This should hold for $u^{\\prime } \\gg 0$ but also more generally when $u^{\\prime }$ is analytically continued; in fact, since changing $\\arg \\hbar $ by $\\frac{\\pi }{2}$ changes $\\arg u^{\\prime }$ by $\\frac{3\\pi }{2}$ , the prediction (REF ) can be analytically continued to give the asymptotics as $u^{\\prime } \\rightarrow \\infty $ along an arbitrary ray.",
"We can test this prediction experimentally as follows: Numerically compute the monodromy matrices $\\mathbf {A}$ , $\\mathbf {B}$ , $\\mathbf {C}$ for the $T_3$ equation, for various values of $u^{\\prime }$ .",
"Use the method of sec:enumerating-special-bases to determine the $\\mathcal {W}$ -abelianizations for each $u^{\\prime }$ .",
"Use the formulas (REF ), () to compute the spectral coordinates $\\mathcal {X}_A$ and $\\mathcal {X}_B$ for each abelianization.",
"Check that 3 of the $\\mathcal {W}$ -abelianizations have the behavior (REF ) when $\\vert u^{\\prime }\\vert \\rightarrow \\infty $ .",
"Experimentally this indeed works; for a sample of the numerical evidence, see fig:X-numerics.",
"type=figure Figure: NO_CAPTION figureA numerical study of $\\mathcal {X}_A(u^{\\prime })$ and $\\mathcal {X}_B(u^{\\prime })$ for $\\arg u^{\\prime } = 0.2$ and $1<\\vert u^{\\prime }\\vert <80$ .",
"For each value of $u^{\\prime }$ , the values of $\\mathcal {X}_A$ and $\\mathcal {X}_B$ for all of the $\\mathcal {W}$ -abelianizations are plotted.",
"The 3 WKB asymptotic formulas are also plotted, with $\\vartheta = 0$ (orange), $\\vartheta = -\\frac{2\\pi }{3}$ (blue), $\\vartheta = \\frac{2\\pi }{3}$ (green).",
"In each case the curve plotted is the sum of the first three terms of the WKB asymptotic series.",
"Finally we consider what happens for $-u^{\\prime } \\gg 0$ .",
"We can reach this situation by taking $u > 0$ and $\\hbar < 0$ .",
"The resulting Stokes graph $\\mathcal {W}^{\\vartheta = \\pi }$ is identical to $\\mathcal {W}$ , except that the sheet labels are reversed.",
"Because all walls of $\\mathcal {W}$ are double, the notion of $\\mathcal {W}$ -abelianization is actually unaffected by this reversal of the sheet labels; a $\\mathcal {W}^{\\vartheta = \\pi }$ -abelianization is the same thing as a $\\mathcal {W}$ -abelianization.",
"Then, in parallel to $u^{\\prime } \\gg 0$ , exact WKB at the three phases $\\vartheta = \\arg \\hbar = \\pi , \\frac{5\\pi }{3}, \\frac{\\pi }{3}$ gives three $\\mathcal {W}$ -abelianizations $A_\\vartheta $ of the $T_3$ equation with $-u^{\\prime } \\gg 0$ ."
],
[
"Analytic continuation",
"Now let us consider the analytic continuation of the spectral coordinates $\\mathcal {X}_\\gamma $ in $u$ and $\\hbar $ .",
"The $\\mathcal {X}_\\gamma $ are really defined on the 4-fold cover given by the discrete $\\mathcal {W}$ -abelianizations; thus studying their monodromy is equivalent to studying the monodromy of the $\\mathcal {W}$ -abelianizations.",
"Since the $T_3$ equation depends only on $u^{\\prime } = u / \\hbar ^3$ this reduces to working out the monodromy in the $u^{\\prime }$ -plane.",
"We have not found an analytic way of computing this monodromy, but we have studied it numerically, by tracking the spectral coordinates $\\mathcal {X}_A$ and $\\mathcal {X}_B$ directly as functions of $u^{\\prime }$ .",
"Let us begin with large $\\vert u^{\\prime }\\vert $ .",
"As we have discussed above, at either $u^{\\prime } \\gg 0$ or $u^{\\prime } \\ll 0$ we have three $\\mathcal {W}$ -abelianizations $A_\\vartheta $ coming from WKB.",
"As we continue counterclockwise from one side to the other, these three $\\mathcal {W}$ -abelianizations continue as $A_\\vartheta \\rightarrow A_{\\vartheta + \\frac{\\pi }{3}}$ ; thus, going counterclockwise around a large circle in the $u^{\\prime }$ -plane induces the order-3 monodromy $A_\\vartheta \\rightarrow A_{\\vartheta + \\frac{2 \\pi }{3}}$ .",
"The behavior of $\\mathcal {X}_A$ as we go around the circle $\\vert u^{\\prime }\\vert =25$ is shown in fig:t3-x-u=25.",
"type=figure Figure: NO_CAPTION figureThe coordinate $\\log \\mathcal {X}_A(u^{\\prime })$ , plotted in $ 2 \\pi I\\mathbb {Z}$ , for $\\vert u^{\\prime }\\vert = 25$ .",
"The hue indicates the phase $\\arg u^{\\prime }$ .",
"For each value of $u^{\\prime }$ , there are 4 solid points on the plot, representing the values of $\\mathcal {X}_A(u^{\\prime })$ for the 4 discrete $\\mathcal {W}$ -abelianizations.",
"3 of these points lie on a large loop, while the fourth point lies on a smaller loop; the two loops come very close to one another.",
"As $\\arg u^{\\prime }$ advances by $2 \\pi $ , $\\mathcal {X}_A(u^{\\prime })$ moves one-third of the way around the large loop, or all the way around the small loop.",
"This reflects the fact that the monodromy permutes 3 of the discrete $\\mathcal {W}$ -abelianizations while leaving the fourth one invariant.",
"The hollow circles on the plot show the WKB asymptotic formula for $\\mathcal {X}_A(u^{\\prime })$ , analytically continued from $\\arg u^{\\prime } = 0$ to the region $-\\frac{3\\pi }{2} < \\arg u^{\\prime } < \\frac{3\\pi }{2}$ ; the fact that these points track closely with one of the 4 $\\mathcal {W}$ -abelianizations in this range confirms the prediction of WKB.",
"Now we can ask what happens in the interior of the $u^{\\prime }$ -plane.",
"By numerical exploration we found monodromy around just two points, located at $u^{\\prime } = \\pm u^{\\prime }_*$ , where $u^{\\prime }_* \\approx 0.041992794$ .",
"Coming in from $u^{\\prime } \\gg 0$ , we find that the two $\\mathcal {W}$ -abelianizations which we called $A_{\\frac{2\\pi }{3}}$ and $A_{\\frac{4\\pi }{3}}$ above collide at $u^{\\prime } = u^{\\prime }_*$ .",
"When they collide they have $\\mathcal {X}_A = 7 + 4 \\sqrt{3}$ and $\\vert \\mathcal {X}_B\\vert ^{-2} = \\mathcal {X}_A$ .",
"Traveling around a small loop around $u^{\\prime }_*$ , these two $\\mathcal {W}$ -abelianizations are exchanged.",
"Similarly, coming in from $u^{\\prime } \\ll 0$ , we find that the two $\\mathcal {W}$ -abelianizations we called $A_{\\frac{\\pi }{3}}$ and $A_{\\frac{5\\pi }{3}}$ are exchanged around $u^{\\prime } = -u^{\\prime }_*$ , with $\\mathcal {X}_A = 7 - 4 \\sqrt{3}$ there.",
"By numerical experimentation we have not found monodromy anywhere else in the $u^{\\prime }$ -plane.",
"Thus we conjecture that the only monodromy is around $\\pm u^{\\prime }_*$ .",
"It is straightforward to check that this gives a consistent global picture: the order-3 monodromy we found at large $\\vert u^{\\prime }\\vert $ can be factorized into the two order-2 monodromies around $\\pm u^{\\prime }_*$ .",
"It is interesting to compare the monodromy of the $\\mathcal {X}_\\gamma $ with that of the periods $Z_\\gamma $ .",
"At large $\\vert u^{\\prime }\\vert $ the two monodromies agree.",
"At small $\\vert u^{\\prime }\\vert $ the $Z_\\gamma $ have a single singularity at $u^{\\prime } = 0$ , while the $\\mathcal {X}_\\gamma $ have two singularities at $\\pm u^{\\prime }_*$ .",
"Since $\\mathcal {X}_\\gamma \\sim \\exp (Z_\\gamma / \\hbar )$ one might wonder whether one can globally take logs, to obtain a deformation $\\widetilde{Z}_\\gamma = \\hbar \\log \\mathcal {X}_\\gamma (\\hbar )$ .",
"Were this possible, we would just have two holomorphic functions $\\widetilde{Z}_\\gamma $ in the $u^{\\prime }$ -plane, transforming linearly under monodromy around the two points $\\pm u^{\\prime }_*$ .",
"Then it would be tempting to try to realize the $\\widetilde{Z}_\\gamma $ directly as periods of a globally defined 1-form on a family of deformed spectral curves.",
"The real situation is more delicate, because the analytically continued functions $\\mathcal {X}_\\gamma $ may have zeroes or poles at some values of $u^{\\prime }$ ; upon analytic continuation around such a $u^{\\prime }$ , $\\widetilde{Z}_\\gamma $ has an additive shift by $\\pm 2 \\pi I\\hbar $ .",
"To see examples of this kind of singularity concretely, we plot the spectral coordinates for all abelianizations on the line $u^{\\prime }>0$ : see fig:t3-x-ureal.",
"type=figure Figure: NO_CAPTION figureA numerical study of the spectral coordinate $\\mathcal {X}_A(u^{\\prime })$ for $0.01 < u^{\\prime } < 30$ .",
"Notation is as in fig:X-numerics.",
"Along the ray $u^{\\prime } > 0$ there appear to be infinitely many such singularities, with the first few at $u^{\\prime } \\approx 0.03013837, 0.23370955, 1.75819973, \\dots $ .",
"Similarly along the ray $u^{\\prime } \\in I\\mathbb {R}_+$ there are singularities which occur at $u^{\\prime } \\approx 0.4595I, \\dots $ So far we have been discussing the $\\mathcal {W}$ -abelianizations which occur discretely.",
"For the $\\mathcal {W}$ -abelianization which occurs in a continuous family, the situation is simpler: there is no monodromy mixing it with the other $\\mathcal {W}$ -abelianizations.",
"This matches with the fact from sec:t3-fg-coords that the corresponding spectral coordinates are the Fock-Goncharov coordinates, which are uniquely determined by the connection $\\nabla $ as long as each of $\\mathbf {A}$ , $\\mathbf {B}$ , $\\mathbf {C}$ preserves a unique flag."
],
[
"The uniformization point",
"It is also interesting to ask what happens at $u^{\\prime } = 0$ .",
"This point is a singularity for the periods $Z_\\gamma $ , but the $T_3$ equation at $u^{\\prime } = 0$ is perfectly regular.",
"Indeed, its monodromy representation can be described explicitly, because it has a simple interpretation: it is the image of the uniformization representation of the 3-punctured sphere, $\\pi _1(C) \\rightarrow \\Gamma _0(2) \\subset {\\mathrm {SL}}(2,\\mathbb {Z})$ , under the symmetric square $\\operatorname{Sym}^2: {\\mathrm {SL}}(2,\\mathbb {Z}) \\rightarrow {\\mathrm {SL}}(3,\\mathbb {Z})$ .",
"Thus it can be represented explicitly by the matrices $\\mathbf {A}&= \\operatorname{Sym}^2 \\begin{pmatrix} 1 & 2 \\\\ 0 & 1 \\end{pmatrix} = \\begin{pmatrix} 1 & 2 & 4 \\\\ 0 & 1 & 4 \\\\ 0 & 0 & 1 \\end{pmatrix}, \\\\ \\qquad \\mathbf {B}&= \\operatorname{Sym}^2 \\begin{pmatrix} 1 & 0 \\\\ -2 & 1 \\end{pmatrix} = \\begin{pmatrix} 1 & 0 & 0 \\\\ -4 & 1 & 0 \\\\ 4 & -2 & 1 \\end{pmatrix} , \\\\ \\qquad \\mathbf {C}&= (\\mathbf {A}\\mathbf {B})^{-1} = \\begin{pmatrix} 1 & -2 & 4 \\\\ 4 & -7 & 12 \\\\ 4 & -6 & 9 \\end{pmatrix}.$ The integrality of these matrices implies that the spectral coordinates $\\mathcal {X}_A$ , $\\mathcal {X}_B$ are algebraic.",
"Indeed, we have computed them explicitly: at $u^{\\prime } = 0$ two of the discrete $\\mathcal {W}$ -abelianizations have $(\\mathcal {X}_A, \\mathcal {X}_B) = \\left( \\frac{1}{5} (59 \\pm 24\\sqrt{6}), \\sqrt{ \\frac{1}{5} (59 \\mp 24\\sqrt{6}) } \\right)$ and the other two have the coincident value $(\\mathcal {X}_A, \\mathcal {X}_B) = \\left( -1, 1 \\right),$ while the continuous family of $\\mathcal {W}$ -abelianizations have $(\\mathcal {X}_A, \\mathcal {X}_B) = \\left( 1, 1 \\right).$ If we approach $u^{\\prime } = 0$ starting from $u^{\\prime } \\gg 0$ , the WKB abelianization $A_0$ smoothly approaches the one with $\\mathcal {X}_A = \\frac{1}{5}(59 - 24 \\sqrt{6}) \\approx 0.0424492$ (see the bottom curve in fig:t3-x-ureal)."
],
[
"Integral equations",
"Now let us consider the construction of an integral equation (REF ) obeyed by the spectral coordinates, following the scheme of sec:integral-equations.",
"For concreteness, we fix $u > 0$ (it is easy to restore more general $u$ dependence if needed.)",
"In the scheme of sec:integral-equations we have to choose a function $\\vartheta (\\arg \\hbar )$ .",
"It would be inconvenient in this example to choose $\\vartheta (\\arg \\hbar ) = \\arg \\hbar $ ; the results of [34] imply that there are infinitely many active rays, and indeed the active rays are everywhere dense.",
"We pick instead $\\vartheta (\\arg \\hbar ) = n \\frac{\\pi }{3} \\qquad \\text{ for } \\qquad \\arg \\hbar \\in \\left(n \\frac{\\pi }{3}-\\frac{\\pi }{6}, n \\frac{\\pi }{3}+\\frac{\\pi }{6} \\right).$ This choice has the effect of collapsing the infinitely many active rays down to 6 rays $r_n$ with phases $\\frac{\\pi }{6} + n \\frac{\\pi }{3}$ .",
"To write the integral equation (REF ) we need to determine the functions $F_{r_n,\\gamma }$ attached to those 6 rays.",
"According to (REF ), this amounts to determining the coordinate transformation which relates the spectral coordinates $\\mathcal {X}^{\\vartheta = n \\frac{\\pi }{3}}$ to the $\\mathcal {X}^{\\vartheta = (n+1) \\frac{\\pi }{3}}$ .",
"To be concrete let us focus on the ray $r_0$ , with phase $\\frac{\\pi }{6}$ ; the others are essentially the same.",
"The functions $x_\\gamma = \\mathcal {X}_\\gamma ^{\\vartheta = 0}(\\hbar )$ for $\\arg \\hbar = 0$ , and the functions $y_\\gamma = \\mathcal {X}_\\gamma ^{\\vartheta = \\frac{\\pi }{3}}(\\hbar )$ for $\\arg \\hbar = \\frac{\\pi }{3}$ , are associated to a $\\mathcal {W}$ -abelianization and a $\\mathcal {W}^{\\vartheta = \\frac{\\pi }{3}}$ -abelianization respectively.",
"We analytically continue $x$ and $y$ to a common sector $\\arg \\hbar \\in (-\\epsilon , \\frac{\\pi }{3} + \\epsilon )$ , which in particular contains $r_0$ .",
"In this sector $x_\\gamma $ and $y_\\gamma $ have the same asymptotics as $\\hbar \\rightarrow 0$ , but they are not the same; the “nonperturbative” difference between them, $F_{r_0,\\gamma } = y_\\gamma / x_\\gamma $ , is what we are after.",
"We can describe this difference a bit more concretely.",
"Just as in sec:t3-numerics, note that a $\\mathcal {W}^{\\vartheta = \\frac{\\pi }{3}}$ -abelianization also induces a $\\mathcal {W}$ -abelianization.",
"In fact, the $x_\\gamma $ are the spectral coordinates for the $\\mathcal {W}$ -abelianization $A_0$ , while the $y_\\gamma $ are obtained by applying a cyclic permutation of the basis cycles, $(A,B,-A-B) \\rightarrow (B,-A-B,A)$ , to the spectral coordinates for the $\\mathcal {W}$ -abelianization $A_{-\\frac{2\\pi }{3}}$ .",
"(For example, $x_B$ is given by the points along the orange curve in the lower part of fig:X-numerics, while $y_B$ is given by the points along the green curve in the upper part of that figure.)",
"We do not have a closed formula for the coordinate transformation $y = \\mathbf {S}_{0,\\frac{\\pi }{3}}(x)$ giving the $y_\\gamma $ as a function of the $x_\\gamma $ .",
"However, we do have some partial information.",
"As we vary $\\vartheta $ from 0 to $\\frac{\\pi }{3}$ , the $\\vartheta $ -Stokes graph jumps at a countable dense set of phases, and correspondingly $\\mathbf {S}_{0,\\frac{\\pi }{3}}$ admits a factorization into a countable product of Stokes automorphisms, of the form [80], [73], [16] $ \\mathbf {S}_{0,\\frac{\\pi }{3}} = \\mathbf {T}_{\\frac{\\pi }{3}}^{\\frac{1}{2}} \\circ \\left( \\prod ^\\text{\\Large $\\curvearrowright $}_{\\vartheta \\in (0,\\frac{\\pi }{3})} \\mathbf {T}_\\vartheta \\right) \\circ \\mathbf {T}_0^{\\frac{1}{2}}.$ In (REF ) the product over $\\vartheta $ is taken in decreasing order, $\\mathbf {T}_\\vartheta $ is a coordinate transformation of the form $ \\mathbf {T}_\\vartheta = \\prod _{\\gamma : \\arg (-Z_\\gamma ) = \\vartheta } \\mathcal {K}_\\gamma ^{\\Omega (\\gamma )}, \\quad \\mathcal {K}_\\gamma ^*\\mathcal {X}_\\mu = \\mathcal {X}_\\mu (1 - \\sigma (\\gamma ) \\mathcal {X}_\\gamma )^{\\langle \\mu ,\\gamma \\rangle },$ $\\sigma : H_1(\\Sigma ,\\mathbb {Z}) \\rightarrow \\lbrace \\pm 1\\rbrace $ is $\\sigma (a \\gamma _A + b \\gamma _B) = (-1)^{a + b + ab},$ and most crucially, there appear some integers $\\Omega (\\gamma ) \\in \\mathbb {Z}$ , determined by the jumping of the Stokes graphs.",
"In the relation to ${\\mathcal {N}}=2$ supersymmetric field theory, $\\Omega (\\gamma )$ is a helicity supertrace counting BPS particles with charge $\\gamma $ .",
"Note that $\\mathbf {T}_\\vartheta = 1$ except for countably many phases $\\vartheta $ ,The results of [34] show that $\\Omega (\\gamma ) \\ne 0$ for every primitive charge $\\gamma $ , so all of the countably many phases $\\vartheta $ which could give nontrivial $\\mathbf {T}_\\vartheta $ indeed do.",
"and for each such phase $\\mathbf {T}_\\vartheta $ is a countable product, so altogether the product in (REF ) involves a countably infinite number of $\\mathcal {K}_\\gamma $ .",
"The effect of the transformation $\\mathcal {K}_\\gamma $ is to multiply each $\\mathcal {X}_\\mu $ by some power of $(1 \\pm \\mathcal {X}_\\gamma )$ .",
"For the $\\mathcal {K}_\\gamma $ which contribute to $\\mathbf {S}_{0,\\frac{\\pi }{3}}$ we have $\\arg (-Z_\\gamma ) \\in [0,\\frac{\\pi }{3}]$ .",
"When $\\arg \\hbar \\in (-\\epsilon , \\frac{\\pi }{3} + \\epsilon )$ , these $\\mathcal {X}_\\gamma $ are exponentially suppressed like $\\exp (Z_\\gamma / \\hbar )$ as $\\hbar \\rightarrow 0$ , and thus $\\mathcal {K}_\\gamma $ acts by an exponentially small transformation on the coordinates.",
"In particular, if $\\arg \\hbar = \\frac{\\pi }{6}$ , then for $\\gamma = (a,b) = a \\gamma _A + b \\gamma _B$ , $\\operatorname{Re}(-Z_\\gamma / \\hbar )$ is proportional to $a-b$ ; all $(a,b)$ which contribute have $a-b > 0$ , and of those the least suppressed $\\mathcal {K}_{a,b}$ are the ones with $a-b = 1$ , next are the ones with $a-b = 2$ , and so on.",
"We do not know all of the $\\Omega (\\gamma )$ , but we do know some of them, by the results of [34]; in particular we know all of the $\\Omega (a,b)$ with $a-b \\le 3$ ; see fig:t3-bps-counts.",
"type=figure Figure: NO_CAPTION figureSome degeneracies of BPS particles in the $T_3$ theory with $u > 0$ .",
"Each green dot represents a charge $\\gamma = (a,b) = a \\gamma _A + b \\gamma _B$ , and is plotted at the point $-Z_\\gamma \\in ,and decorated by the BPS count $ () $\\mathbb {Z}$$.The charges shown are the ones with $ (-Z) [0, 3]$, and with the smallest values of $ Re(-Z/ )$ when $ 6$.$ Thus we can try approximating $\\mathbf {S}_{0,\\frac{\\pi }{3}}$ by just the contributions from these least-suppressed $\\Omega (a,b)$ ; this gives a sequence of approximations, $\\mathbf {S}_{0,\\frac{\\pi }{3}}^{(1)} &= \\mathcal {K}_{0,-1}^{\\frac{1}{2} 27} \\mathcal {K}_{1,0}^{\\frac{1}{2} 27} ,\\\\\\mathbf {S}_{0,\\frac{\\pi }{3}}^{(2)} &= \\mathcal {K}_{0,-1}^{\\frac{1}{2} 27} \\mathcal {K}_{0,-2}^{-\\frac{1}{2} 54} \\mathcal {K}_{1,-1}^{81} \\mathcal {K}_{1,0}^{\\frac{1}{2} 27} \\mathcal {K}_{2,0}^{-\\frac{1}{2} 54}, \\\\\\mathbf {S}_{0,\\frac{\\pi }{3}}^{(3)} &= \\mathcal {K}_{0,-1}^{\\frac{1}{2} 27} \\mathcal {K}_{0,-2}^{-\\frac{1}{2} 54} \\mathcal {K}_{0,-3}^{\\frac{1}{2} 240} \\mathcal {K}_{1,-2}^{432} \\mathcal {K}_{1,-1}^{81} \\mathcal {K}_{2,-1}^{432} \\mathcal {K}_{1,0}^{\\frac{1}{2} 27} \\mathcal {K}_{2,0}^{-\\frac{1}{2} 54} \\mathcal {K}_{3,0}^{\\frac{1}{2} 240},$ and so on.",
"To write the next approximation $\\mathbf {S}_{0,\\frac{\\pi }{3}}^{(4)}$ would require us to know the BPS count $\\Omega (3,-1)$ , which was not computed in [34], so for now we stop here.",
"We have tested these approximations numerically; for example, at $\\hbar = {\\mathrm {e}}^{\\frac{\\pi I}{6}}$ and $u=1$ , we find: retain-zero-exponent = true group-digits = false Table: NO_CAPTIONAs expected, the $\\mathbf {S}_{0,\\frac{\\pi }{3}}^{(k)}x$ are converging to $y$ as $k$ increases.",
"Also as expected, the speed of convergence increases as we increase $\\vert u\\vert $ ; for example, at $\\hbar = {\\mathrm {e}}^{\\frac{\\pi I}{6}}$ and $u=10$ , we find: Table: NO_CAPTIONWe regard these results as strong evidence for the consistency of the whole story.",
"We could also run this program in reverse: since we can compute $x$ and $y$ numerically for any given $u$ and in particular for large $\\vert u\\vert $ , we could try to determine the BPS counts $\\Omega (\\gamma )$ from the condition that $\\mathbf {S}_{0,\\frac{\\pi }{3}} x = y$ .",
"It is easy in this way to “discover” the fact that $\\Omega (1,0) = \\Omega (0,-1) = 27$ , and in principle one could iteratively determine the higher $\\Omega (a,b)$ by the same strategy.",
"As $a-b$ increases, so does the needed precision in the numerical computations of $x$ and $y$ .",
"With our confidence thus bolstered, we tried writing down approximate versions of the desired integral equation (REF ), taking $\\vartheta (\\arg \\hbar ) = \\arg \\hbar $ , but truncating as follows: we fix some $k$ , and then include only the $\\Omega (a,b)$ shown in fig:t3-bps-counts with $a-b \\le k$ , together with their images under the obvious $\\mathbb {Z}_6$ symmetry.",
"It is not clear a priori whether the resulting approximate equations have any right to work; nevertheless, we tried solving them numerically anyway, with the following results: $u=5$ : Table: NO_CAPTION$u=1$ : Table: NO_CAPTION$u=0.01$ : Table: NO_CAPTIONIn each of these tables, $\\mathcal {X}^{(k)}_\\gamma $ is the value computed numerically from the $k$ -th truncated integral equation, and $\\mathcal {X}_\\gamma $ is the value computed numerically from the monodromy of the $T_3$ equation.",
"These results offer some support for the conjecture that $\\lim _{k \\rightarrow \\infty } \\mathcal {X}^{(k)}_\\gamma = \\mathcal {X}_\\gamma $ .",
"Rather than studying these successive approximations, what would be really desirable would be to give a closed formula for $\\mathbf {S}_{0,\\frac{\\pi }{3}}$ ; then we could write down a version of the integral equation (REF ) which would compute the exact $\\mathcal {X}_\\gamma $ .",
"This remains as a problem for the future."
],
[
"Spectral problem",
"Finally, we briefly consider a spectral problem for the $T_3$ equation, analogous to those we considered for the Mathieu equation in sec:mathieu-bound-states and sec:mathieu-quasiperiodic: we search for those $u^{\\prime }$ such that the $T_3$ equation admits a discrete $\\mathcal {W}$ -abelianization with $ \\mathcal {X}_A = 1.$ We recall that for large $u^{\\prime }$ the asymptotics of 3 of the 4 discrete $\\mathcal {W}$ -abelianizations are given by (REF ).",
"Thus a natural first place to look for solutions of (REF ) at large $u^{\\prime }$ is at the $u^{\\prime }$ satisfying $1 = \\mathcal {X}_A \\approx \\exp (-M u^{\\prime \\frac{1}{3}}),$ where $u^{\\prime \\frac{1}{3}}$ is allowed to be any of the three cube roots.",
"This leads to potential solutions at $ u^{\\prime } \\approx 8 \\pi ^3 In^3 / M^3 = \\pm 1.666221I, \\pm 13.32977I, \\pm 44.9880I, \\pm 106.6381I, \\dots $ By numerical experimentation we find actual solutions at $u^{\\prime } \\approx \\pm 0.0610186I, \\pm 2.148003I, \\pm 14.24769I, \\pm 46.3655I, \\pm 108.4752I, \\dots $ which asymptotically indeed appear to approach the values (REF ).",
"The reader might find our choice of spectral problem a little unmotivated, since its very formulation involves the spectral coordinates $\\mathcal {X}_A$ .",
"It might be some comfort to know that the solutions of (REF ) can be alternatively described as points $u^{\\prime }$ for which $ \\operatorname{Tr}\\mathbf {A}\\mathbf {B}^{-1} - \\operatorname{Tr}\\mathbf {B}\\mathbf {A}^{-1} = \\pm 12 \\sqrt{3} I,$ as one sees by substituting (REF ) into the monodromy matrices (REF ).",
"In the parlance of exact WKB, one would say (REF ) is the “exact quantization condition” for the solutions of (REF ).",
"One could also go the other way, starting with one's favorite condition on the matrices $\\mathbf {A}$ , $\\mathbf {B}$ , $\\mathbf {C}$ and finding the corresponding exact quantization condition in terms of the spectral coordinates $\\mathcal {X}_A$ , $\\mathcal {X}_B$ ; we have not explored in this direction."
],
[
"Supersymmetric field theory",
"In the main part of this paper we have been exploring the exact WKB method for certain differential equations (opers) of order 2 and 3.",
"In this final section we consider the relation of our constructions to ${\\mathcal {N}}=2$ supersymmetric quantum field theories of class $S$ in four spacetime dimensions.",
"Our discussion here is somewhat open-ended; we hope to return to these questions in the future."
],
[
"Opers and QFT of class $S$",
"Fixing a Lie algebra $\\fg $ and a punctured Riemann surface $C$ with singularity data at the punctures determines an ${\\mathcal {N}}=2$ theory ${\\mathfrak {X}}(\\fg ,C)$ of class $S$ .",
"It has been known for some time that there is a connection between the theory ${\\mathfrak {X}}(\\fg ,C)$ and the space of $\\fg $ -opers on $C$ ; see e.g.",
"[35], [81], [18], [82] for various aspects of this connection.",
"In this section we describe a slightly different version of the connection.",
"The Coulomb branch of the theory ${\\mathfrak {X}}(\\fg ,C)$ is the base $\\mathcal {B}_0(\\fg ,C)$ of the Hitchin integrable system.",
"The algebra $\\mathcal {A}_0$ of chiral local operators in theory ${\\mathfrak {X}}(\\fg ,C)$ is canonically identified with the space of holomorphic functions on $\\mathcal {B}_0(\\fg ,C)$ .",
"Following [35], suppose we deform the theory by turning on the “$\\frac{1}{2}\\Omega $ -background” associated to a rotation in the $x_2$ -$x_3$ plane, with parameter $\\varepsilon = \\hbar $ .",
"This modification deforms $\\mathcal {A}_0$ into a new algebra $\\mathcal {A}_\\hbar $ , consisting of supersymmetric local operators inserted at the origin of the $x_2$ -$x_3$ plane, still free to move in the $x_0$ and $x_1$ directions.",
"$\\mathcal {A}_\\hbar $ can be thought of as the algebra of functions on a deformation $\\mathcal {B}_\\hbar (\\fg ,C)$ of $\\mathcal {B}_0(\\fg ,C)$ .",
"By studying the Hilbert space of the theory on $S^3$ and using the state-operator map, together with known facts about how $S$ -duality acts in the theory reduced on $S^1$ , one can show that the deformed space $\\mathcal {B}_\\hbar (\\fg ,C)$ is canonically isomorphic to the space of $\\fg $ -opers on $C$ .We thank David Ben-Zvi for explaining this point to us.",
"So, in short, turning on the $\\frac{1}{2}\\Omega $ -background deforms the Coulomb branch into the space of opers.",
"This deformation might sound a bit trivial since, when considered simply as complex manifolds, the Coulomb branch and the space of opers are isomorphic; however, the two spaces come equipped with natural presentations in terms of holomorphic functions, which are different in the two cases, as we will discuss below.",
"The three spaces of opers we considered in this paper correspond in this way to familiar quantum field theories: Table: NO_CAPTION"
],
[
"Spectral coordinates as vevs",
"The stars of this paper are the spectral coordinate functions $\\mathcal {X}_\\gamma (\\hbar )$ on $\\mathcal {B}_\\hbar (\\fg ,C)$ .",
"What is their meaning in the theory ${\\mathfrak {X}}(\\fg ,C)$ ?",
"The function $\\widetilde{Z}_{\\gamma }(\\hbar ) = \\hbar \\log \\mathcal {X}_\\gamma (\\hbar )$ is a deformation of the function $Z_\\gamma $ on $\\mathcal {B}_0(\\fg ,C)$ (if we momentarily ignore the multivaluedness of the $\\log $ ).",
"Since $Z_\\gamma $ is the vev of the vector multiplet scalar $a_\\gamma $ , we suspect that $\\widetilde{Z}_\\gamma (\\hbar )$ is likewise the vacuum expectation value of an operator $\\widetilde{a}_\\gamma (\\hbar )$ .",
"The operator $\\widetilde{a}_\\gamma (\\hbar )$ should be a deformation of $a_\\gamma $ which preserves supersymmetry in the $\\frac{1}{2}\\Omega $ -background.",
"Such a deformation might not be simple to construct; nevertheless, a posteriori, the WKB expansion (REF ) of $\\mathcal {X}_\\gamma (\\hbar )$ suggests that there is a universal $\\widetilde{a}_\\gamma (\\hbar )$ to all orders in $\\hbar $ .",
"What about going beyond series in $\\hbar $ ?",
"We have seen that the $\\mathcal {X}_\\gamma (\\hbar )$ can be defined beyond perturbation theory in various ways, corresponding to the different choices of spectral network.",
"One particularly interesting nonperturbative definition is the function we called $\\mathcal {X}_\\gamma ^{\\mathrm {RH}}(\\hbar )$ in §REF , with the canonical choice (REF ).",
"Thus we conjecture that this canonical choice corresponds to a canonical nonperturbative definition of $\\widetilde{a}_\\gamma (\\hbar )$ .",
"This canonical $\\widetilde{a}_\\gamma (\\hbar )$ must have some new features compared to $a_\\gamma $ : $\\widetilde{a}_\\gamma (\\hbar )$ should suffer from a nonperturbative discontinuity as a function of $\\hbar $ whenever there exists a BPS state whose central charge is aligned with $\\hbar $ , corresponding to the fact that the functions $\\mathcal {X}^{\\mathrm {RH}}_\\gamma (\\hbar )$ jump at the active rays.",
"We might interpret this as saying that the operators $\\widetilde{a}_\\gamma (\\hbar )$ are defined only in the IR (like the $a_\\gamma $ ), and the scale below which this IR description is appropriate goes to zero as $\\hbar $ approaches an active ray.",
"$\\widetilde{a}_\\gamma (\\hbar )$ should also suffer from an additive ambiguity, because $\\widetilde{Z}_\\gamma (\\hbar )$ has an ambiguity by shifts by $2 \\pi I\\hbar $ .",
"This ambiguity presumably comes from the possibility of shifting by a local operator built from background supergravity fields.",
"(After dimensional reduction to ${\\mathcal {N}}=(2,2)$ theory in the $x^0$ -$x^1$ plane, the rotation in the $x^2$ -$x^3$ plane becomes a global symmetry; then $\\hbar $ can be interpreted as a complex twisted mass for this global symmetry, and the ambiguity we are after would come from shifting by the scalar in the background vector multiplet.)",
"It would be very interesting to give a direct construction of the operator $\\widetilde{a}_\\gamma (\\hbar )$ and to understand more precisely why it has the above features."
],
[
"Scaling line defects",
"Although we do not have a direct construction of the operators $\\widetilde{a}_\\gamma (\\hbar )$ in hand, we can at least propose a construction which should yield the operators $\\exp (\\widetilde{a}_\\gamma (\\hbar ) / \\hbar )$ , as follows.",
"We recall that in an ${\\mathcal {N}}=2$ theory one has families of $\\frac{1}{2}$ -BPS line defects $L(\\zeta )$ labeled by a parameter $\\zeta \\in \\times $ .",
"It was argued in [83] that in the low-energy limit of the theory there exist distinguished $\\frac{1}{2}$ -BPS “IR line defects” $L_\\gamma $ .",
"The vacuum expectation values of these line defects on $\\mathbb {R}^3 \\times S^1$ are functions $\\hat{\\mathcal {X}}_\\gamma (R,\\zeta )$ which are close analogues of the functions $\\mathcal {X}_\\gamma ^{\\mathrm {RH}}(\\hbar )$ ; the precise relation was proposed in [26], $\\lim _{R \\rightarrow 0} \\hat{\\mathcal {X}}_\\gamma (R,\\zeta = \\hbar R) = \\mathcal {X}_\\gamma ^{\\mathrm {RH}}(\\hbar ).$ So far this is only a relation on the level of functions; can we promote it to the level of operators?",
"Here is a possible approach.",
"After the $\\Omega $ -background deformation in the $x_2$ -$x_3$ plane we expect that, for any $R>0$ , $L_\\gamma $ can be wrapped supersymmetrically around the circle $(x_2)^2 + (x_3)^2 = R^2$ .Here is a heuristic way to understand why $L_\\gamma $ can be wrapped supersymmetrically around the circle.",
"Suppose $\\hbar $ is real.",
"We imagine lifting the 4-dimensional theory to a 5-dimensional theory on an $\\mathbb {R}^4$ bundle over $S^1$ , where the $S^1$ base has length $\\rho $ , and the $x_2$ -$x_3$ plane in the fiber is rotated by an angle $\\rho \\hbar $ as we go around the $S^1$ base.",
"In the limit $\\rho \\rightarrow 0$ this gives rise to an effectively 4-dimensional theory, which can be identified with the $\\Omega $ -background deformation of the original theory.",
"On the other hand, this 5-dimensional background is locally Euclidean space, and in the 5-dimensional theory, we can put the line defect $L_\\gamma $ supersymmetrically on any straight line.",
"We choose a straight line in the $x_4$ direction, beginning at some point $(x_0,x_1,x_2,x_3,x_4 = 0)$ .",
"After going around the $S^1$ fiber this line will return to $(x_0,x_1,x^{\\prime }_2,x^{\\prime }_3,x_4 = 0)$ where $(x^{\\prime }_2,x^{\\prime }_3)$ is the image of $(x_2,x_3)$ under rotation by an angle $\\rho \\hbar $ .",
"If $\\rho \\hbar = \\frac{2\\pi }{N}$ , then after going around $N$ times, the line closes up to a loop, which pierces the $\\mathbb {R}^4$ fiber in $N$ points arranged around a circle in the $x_2$ -$x_3$ plane.",
"In the limit as $\\rho \\rightarrow 0$ ie $N \\rightarrow \\infty $ , these $N$ points just look like a line wrapped around the circle.",
"Taking the limit $R \\rightarrow 0$ then gives a supersymmetric local operator placed at the origin of the $x_2$ -$x_3$ plane, which we propose to identify with $\\exp (\\widetilde{a}_\\gamma (\\hbar ) / \\hbar )$ .",
"To get a different viewpoint on this construction, following [81], we can deform the $x_2$ -$x_3$ plane to a “cigar” metric and then compactify on the radial circle.",
"The result is a 3-dimensional theory on a half-space, with a boundary condition corresponding to the origin of the $x_2$ -$x_3$ plane.",
"At low energies the 3-dimensional theory is described by a sigma model into a moduli space $\\mathcal {M}(\\fg ,C,\\hbar )$ of flat $\\fg $ -connections on $C$ , and it was proposed in [81] that the boundary condition we get corresponds to a Lagrangian subspace $\\mathcal {L}_{\\mathrm {oper}}\\subset \\mathcal {M}(\\fg ,C,\\hbar )$ , whose points are the opers.By a change of variable introduced in [81], $\\mathcal {M}(\\fg ,C,\\hbar )$ can be identified with the moduli space of the theory without $\\Omega $ -background, compactified on a circle of radius $R = \\vert \\hbar \\vert ^{-1}$ .",
"This moduli space is hyperkähler, with complex structures labeled by $\\zeta \\in \\mathbb {CP}^1$ ; the boundary condition we get preserves the subalgebra labeled by $\\zeta = \\frac{\\hbar }{\\vert \\hbar \\vert }$ .",
"This is consistent with our proposal, as follows.",
"Wrapping $L_\\gamma $ around the compactification circle gives a local operator $O_\\gamma $ in the sigma model.",
"As we approach the boundary the radius of the compactification circle shrinks to zero, so at the boundary our proposal says $O_\\gamma $ should become identified with $\\exp (\\widetilde{a}_\\gamma (\\hbar ) / \\hbar )$ .",
"This is what the $\\mathcal {L}_{\\mathrm {oper}}$ boundary condition enforces: it requires that the $O_\\gamma $ obey the same relations as the $\\exp (\\widetilde{a}_\\gamma (\\hbar ) / \\hbar )$ ."
],
[
"Opers and instanton counting",
"Concretely, what are the relations obeyed by the local operators $\\widetilde{a}_\\gamma (\\hbar )$ , or by their vevs $\\widetilde{Z}_\\gamma (\\hbar )$ ?",
"The functions $Z_\\gamma $ on $\\mathcal {B}_0(\\fg ,C)$ obey well-known relations: choosing a symplectic basis $\\lbrace A_1, \\dots , A_r, B^1, \\dots , B^r\\rbrace $ for the charge lattice $\\Gamma $ , the $Z_B$ are determined by the $Z_A$ , via the formula $Z_{B^I} = \\partial \\mathcal {F}(Z_{A_1}, \\dots , Z_{A_r}) / \\partial Z_{A_I},$ for a locally defined holomorphic function $\\mathcal {F}$ called “prepotential.” The existence of such an $\\mathcal {F}$ reflects the fact that $Z$ gives a local Lagrangian embedding of the Coulomb branch $\\mathcal {B}_0$ into the symplectic vector space $\\Gamma ^* \\otimes .Physically, $$\\mathcal {F}$$ gives a Lagrangian description of the$ N=2$ theory on its Coulomb branch.$ At $\\hbar \\ne 0$ there is a very similar picture: any log spectral coordinate system $\\widetilde{Z}_\\gamma $ gives local Darboux coordinates on the moduli space $\\mathcal {M}(\\fg ,C,\\hbar )$ , and the fact that $\\mathcal {L}_{\\mathrm {oper}}$ is a Lagrangian subspace means that there is a locally defined $\\widetilde{\\mathcal {F}}$ for which $\\mathcal {L}_{\\mathrm {oper}}$ is given by the equationsIn conformal theories $\\widetilde{\\mathcal {F}}$ depends only on the $\\widetilde{Z}_{A_i}$ and not on $\\hbar $ .",
"In non-conformal theories there are complex parameters $m_i$ with the dimension of mass, and then $\\widetilde{\\mathcal {F}}$ depends on $\\hbar $ through the combinations $m_i / \\hbar $ .",
"$\\widetilde{Z}_{B^I} = \\partial {\\widetilde{\\mathcal {F}}}(\\widetilde{Z}_{A_1}, \\dots , \\widetilde{Z}_{A_r}, \\hbar ) / \\partial \\widetilde{Z}_{A_I}.$ Now it is natural to ask: what is the meaning of $\\widetilde{\\mathcal {F}}$ in the language of supersymmetric field theory?",
"In [25] this question was considered in the special case where $\\fg = A_1$ and the $\\widetilde{Z}_\\gamma $ are complexified Fenchel-Nielsen coordinates, like those we considered in sec:spectral-coords-mathieu above.",
"In this case (as long as $C$ has only regular punctures), the theory ${\\mathfrak {X}}(\\fg ,C)$ is a supersymmetric gauge theory [29], and so one can formulate the Nekrasov instanton partition function $\\mathbf {Z}(\\varepsilon _1, \\varepsilon _2; a)$ [84], [85].",
"The proposal of [25] is that $\\widetilde{\\mathcal {F}}$ is the $\\varepsilon _2 \\rightarrow 0$ limit of $\\mathbf {Z}$ , or more precisely, $ \\widetilde{\\mathcal {F}}\\left({\\widetilde{Z}}_A, \\hbar = \\varepsilon _1 \\right) = \\frac{1}{\\varepsilon _1} \\lim _{\\varepsilon _2 \\rightarrow 0} \\varepsilon _2 \\log \\mathbf {Z}(\\varepsilon _1,\\varepsilon _2 ; a = \\varepsilon _1 {\\widetilde{Z}}_A).$ The formula (REF ) is a direct link between two very different-looking objects: on the LHS the monodromy of ${\\mathrm {SL}}(2)$ -opers on the Riemann surface $C$ , on the RHS equivariant integrals over moduli of instantons in $\\mathbb {R}^4$ .",
"It has been extended in [18], [82] to a broader class of Lagrangian field theories of class $S$ ; in those cases the LHS involves monodromy of ${\\mathrm {SL}}(N)$ -opers on $C$ , expressed in terms of $\\widetilde{Z}_A$ which are higher-rank analogues of complexified Fenchel-Nielsen coordinates.",
"It is difficult to check (REF ) directly.",
"Nevertheless, in [25], [18] evidence for (REF ) has been given, and in [82] a proof in many cases.",
"The strategy is as follows.",
"In Lagrangian field theories of class $S$ one always has parameters $q_i$ which can be varied: from the field theory point of view these are gauge couplings, while from the point of view of $C$ they are moduli of the complex structure.",
"One considers a degeneration limit “$q_i \\rightarrow 0$ ”: in field theory this is a weak-coupling limit, and in the complex moduli space of $C$ it is a limit where $C$ maximally degenerates to a chain of three-punctured spheres.",
"Expanding both sides of (REF ) in powers of the $q_i$ , each term is a well-defined nonperturbative function of $\\varepsilon _1$ .",
"Thus the statement (REF ) is sensitive to the precise nonperturbative definition of $\\widetilde{Z}_\\gamma $ , and as is shown in [25], [18], [82], it holds only when one takes the $\\widetilde{Z}_\\gamma $ to be complexified Fenchel-Nielsen coordinates (or their higher-rank analogues).",
"In sec:t3 of this paper, we have been exploring a specific coordinate system $\\widetilde{Z}_\\gamma $ which arose naturally from the exact WKB analysis of the locus of opers associated to the $\\frac{1}{2}\\Omega $ -deformed $E_6$ Minahan-Nemeschansky theory.",
"One might ask whether some analogue of (REF ) holds in this setting.",
"To formulate this question sharply would require us to understand precisely how to define $\\mathbf {Z}$ in the non-Lagrangian Minahan-Nemeschansky theory.",
"We suspect that the proper formulation of $\\mathbf {Z}$ in a general non-Lagrangian field theory requires a choice of boundary condition, and that there is a natural class of boundary conditions corresponding to the different spectral coordinate systems $\\widetilde{Z}_\\gamma $ ; thus in a general theory the equality (REF ) could indeed hold, with both sides depending on this choice of boundary condition.",
"We hope to develop this story more fully in the future."
],
[
"Computations of spectral coordinates",
"In this appendix we give some computations omitted from the main text."
],
[
"Computations for the cubic potential",
"Computation of (REF ).",
"We will only describe the computation for $\\mathcal {X}_A$ ; that for $\\mathcal {X}_B$ is similar.",
"We need to compute the parallel transport of $\\nabla ^\\mathrm {ab}$ along a path in the homology class $\\gamma _A$ .",
"To compute concretely it is convenient to work relative to bases of $\\nabla ^\\mathrm {ab}$ -flat sections in each domain.",
"Each local $\\nabla ^\\mathrm {ab}$ -flat section corresponds to a local $\\nabla $ -flat section, and by continuation we can think of all these local flat sections as lying in a single 2-dimensional vector space $V$ , the space of global $\\nabla $ -flat sections over the plane.",
"See fig:cubic-computations.",
"type=figure Figure: NO_CAPTION figureThe Stokes graph from fig:cubic-network, with the local WKB bases shown in each domain.",
"To write the basis concretely as an ordered pair of solutions we have used the trivialization of the double cover $\\Sigma $ away from branch cuts; thus, in a domain containing a branch cut, we write two versions of the basis, one on each side of the cut.",
"Relative to these local bases, the parallel transport within each domain is just represented by 1, and the only nontrivial part is the gluing factor from (REF ): When we cross a single wall of type $ij$ on sheet $i$ , from side $L$ to side $R$ , we get a factor $\\frac{[\\psi _i^L, \\psi _j^L]}{[\\psi _i^R, \\psi _j^L]}.$ When we cross a single wall of type $ij$ on sheet $j$ , we also get a gluing factor, but this factor is just 1 if $\\psi _j^L = \\psi _j^R$ , which it always is in this example.",
"The representative of $\\gamma _A$ shown in fig:cubic-computations crosses six walls; multiplying the factors for these six crossings, starting from the eastmost region, gives $\\mathcal {X}_A = \\frac{[\\psi _5^{\\mathrm {sm}}, \\psi _1^{\\mathrm {sm}}]}{[\\psi _5^{\\mathrm {sm}}, \\psi _3^{\\mathrm {sm}}]} \\times 1 \\times 1 \\times \\frac{[\\psi _3^{\\mathrm {sm}}, \\psi _2^{\\mathrm {sm}}]}{[\\psi _1^{\\mathrm {sm}}, \\psi _2^{\\mathrm {sm}}]} \\times 1 \\times 1$ matching (REF ) as desired."
],
[
"Computations for the Mathieu equation",
"Computation of (REF ).",
"We need to compute the parallel transport of $\\nabla ^\\mathrm {ab}$ along a path in the homology class $\\gamma _B$ .",
"We use the path given in fig:mathieu-sn-1.",
"As above, it is convenient to work relative to bases of $\\nabla ^\\mathrm {ab}$ -flat sections in each domain.",
"See fig:computation-1.",
"Again by continuation we think of all these local flat sections as lying in a single 2-dimensional vector space $V$ .",
"In this case there is an added technical difficulty: the monodromy around $z = 0$ means there are no global $\\nabla $ -flat sections.",
"Instead we identify $V$ as the space of $\\nabla $ -flat sections on the complement of the blue dashed line (“monodromy cut”).",
"type=figure Figure: NO_CAPTION figureThe Stokes graph from fig:mathieu-sn-1, with the local WKB bases shown in each domain.",
"As before, to write the basis concretely as an ordered pair of solutions we have used the trivialization of the double cover $\\Sigma $ away from branch cuts; thus, in a domain containing a branch cut, we write two versions of the basis, one on each side of the cut.",
"When we cross the monodromy cut, the local WKB basis of $\\nabla ^\\mathrm {ab}$ -flat sections does not change, but the way we identify them with elements of $V$ does jump, by the action of the monodromy $M$ .",
"Again the only nontrivial part of the parallel transport is the gluing factors appearing in (REF ), (REF ), When we cross a double wall on sheet $i$ , from side $L$ to side $R$ , we get a factor $\\sqrt{\\frac{[\\psi _i^L , \\psi _j^L]}{[\\psi _i^R , \\psi _j^R]} \\frac{[\\psi _i^L , \\psi _j^R]}{[\\psi _i^R , \\psi _j^L]}},$ and when we cross a single wall of type $ij$ on sheet $i$ , from side $L$ to side $R$ , we get a factor $\\frac{[\\psi _i^L , \\psi _j^L]}{[\\psi _i^R , \\psi _j^L]}.$ We can further simplify these factors by choosing bases with $[\\psi _1, \\psi _2] = 1$ , $[\\psi ^{\\prime }_1 , \\psi ^{\\prime }_2] = 1$ , $[\\psi ^{\\prime \\prime }_1, \\psi ^{\\prime \\prime }_2] = 1$ .",
"Then starting from the southwest corner, the gluing factors we encounter are $\\mathcal {X}_B =\\sqrt{\\frac{[\\psi ^{\\prime }_2 , \\psi ^{\\prime \\prime }_1]}{[\\psi ^{\\prime \\prime }_2 , \\psi ^{\\prime }_1]}} \\times \\sqrt{\\frac{[\\psi ^{\\prime \\prime }_2 , \\psi _1]}{[\\psi _2 , \\psi ^{\\prime \\prime }_1]}} \\times 1 \\times \\sqrt{\\frac{[M\\psi _1 , \\psi ^{\\prime \\prime }_2]}{[\\psi ^{\\prime \\prime }_1 , M \\psi _2]}} \\times \\sqrt{\\frac{[\\psi ^{\\prime \\prime }_1 , M \\psi ^{\\prime }_2]}{[M \\psi ^{\\prime }_1 , \\psi ^{\\prime \\prime }_2]}} \\times 1.$ Using $M \\psi ^{\\prime \\prime }_1 = \\mu \\psi ^{\\prime \\prime }_1$ , $M \\psi ^{\\prime \\prime }_2 = \\mu ^{-1} \\psi _2$ , and $M \\psi _1 = \\psi _2$ , $M \\psi ^{\\prime }_1 = \\psi ^{\\prime }_2$ , this reduces to $\\mathcal {X}_B = \\frac{[\\psi _1 , \\psi ^{\\prime \\prime }_2]}{[\\psi _1 , \\psi ^{\\prime \\prime }_1]} \\frac{[\\psi ^{\\prime }_1 , \\psi ^{\\prime \\prime }_1]}{[\\psi ^{\\prime }_1 , \\psi ^{\\prime \\prime }_2]}$ which matches the desired (REF ).",
"Computation of (REF ).",
"Just as above, all we need to compute are the gluing factors along the paths $\\gamma _A$ and $\\gamma _B$ , with respect to the bases shown in fig:computation-2.",
"Figure: The Stokes graph from fig:mathieu-sn-3, with localWKB bases shown in each domain.",
"All notation is as in fig:computation-1 above.We can choose $[\\psi , \\psi ^{\\prime }] = 1$ to simplify.",
"In going around $\\gamma _A$ we only meet one wall, with the gluing factor $ \\mathcal {X}_A = \\pm \\sqrt{\\frac{}{}}{[M \\psi ^{\\prime } , \\psi ]}{[\\psi ^{\\prime } , M \\psi ]}.$ To fix the branch we would need to carefully implement the WKB prescription from sec:stokes-gluing, which we do not do here.",
"For $\\gamma _B$ the product of gluing factors, starting from the southeast, is $ \\mathcal {X}_B = \\frac{[M \\psi ^{\\prime } , \\psi ^{\\prime }]}{[\\psi , \\psi ^{\\prime }]} \\times 1 \\times \\frac{[M \\psi , \\psi ]}{[M \\psi , M \\psi ^{\\prime }]} \\times 1 = \\frac{[M \\psi ^{\\prime } , \\psi ^{\\prime }][M \\psi , \\psi ]}{[\\psi , \\psi ^{\\prime }]^2}.$ The results (REF ), (REF ) match the desired (REF )."
],
[
"Computations for the $T_3$ equation",
"Abelianizations and adapted bases.",
"Suppose we have a $\\mathcal {W}$ -abelianization of the $T_3$ equation.",
"Then we can choose bases compatible with the $\\mathcal {W}$ -abelianization in the various domains of fig:computations-3, as shown.",
"Figure: The Stokes graph from fig:circle-network, with local WKBbases shown in each domain.",
"The notation is as in the figures above.In writing the form of these bases we began by labeling the basis in the middle as $(\\psi _1,\\psi _2,\\psi _3)$ and then used the facts that: According to (REF ) the $k$ -th projective basis element does not change when we cross a wall of type $ij$ and $ji$ (this implies e.g.",
"that the first basis element in the northeast region must be $\\psi _1$ ), Crossing a branch cut of the covering $\\Sigma \\rightarrow C$ (orange in fig:computations-3) permutes the projective basis elements, The projective bases on the two sides of a monodromy cut (blue in fig:computations-3) differ by the monodromy ($\\mathbf {A}$ , $\\mathbf {B}$ or $\\mathbf {C}$ ) attached to the cut.",
"One key fact remains to be used: again by (REF ), for a wall of type $ij$ and $ji$ , the plane spanned by the $i$ -th and $j$ -th basis elements is the same on both sides of the wall.",
"Applying this to the northeast wall, which is of type 23 and 32, leads to the condition that $\\langle \\psi _2,\\psi _3\\rangle = \\langle \\mathbf {C}^{-1} \\psi _3, \\mathbf {A}\\psi _2\\rangle ,$ which is (); doing similarly for the other two walls gives the other two parts of (REF ).",
"Thus, the basis $(\\psi _1, \\psi _2, \\psi _3)$ is indeed a basis in special position.",
"Conversely, given a basis $(\\psi _1, \\psi _2, \\psi _3)$ in special position, the local bases shown in fig:computations-3 give a $\\mathcal {W}$ -abelianization.",
"This shows the claimed identification between $\\mathcal {W}$ -abelianizations and bases in special position.",
"Computation of (REF ).",
"As above, all we need to compute are the gluing factors along the paths representing $\\gamma _A$ and $\\gamma _B$ shown in fig:circle-network-cycles.",
"These factors are given by (REF ): for a wall of type $ij$ and $ji$ , and a path on sheet $i$ , the factor is $\\sqrt{\\frac{[\\psi _i^L , \\psi _j^L , \\psi _k^L]}{[\\psi _i^R , \\psi _j^R , \\psi _k^L]} \\frac{[\\psi _i^L , \\psi _j^R , \\psi _k^L]}{[\\psi _i^R , \\psi _j^L , \\psi _k^L]}}.$ Since all the walls are double, we will not need to use (REF ) anywhere.",
"For $\\gamma _A$ the computation is particularly simple: only two of the four crossings give a nontrivial factor, namely the places where the path crosses the 23-32 wall.",
"This gives directly $\\mathcal {X}_A &= \\sqrt{\\frac{[\\psi _2 , \\mathbf {A}\\psi _2 , \\psi _1]}{[\\mathbf {A}\\psi _2 , \\mathbf {C}^{-1} \\psi _3 , \\psi _1]} \\frac{[\\psi _2 , \\psi _3 , \\psi _1]}{[\\mathbf {A}\\psi _2 , \\psi _2 , \\psi _1]}} \\times \\sqrt{\\frac{[\\psi _3 , \\mathbf {C}^{-1} \\psi _3 , \\psi _1]}{[\\mathbf {C}^{-1} \\psi _3 , \\mathbf {A}\\psi _2 , \\psi _1]} \\frac{[\\psi _3 , \\psi _2 , \\psi _1]}{[\\mathbf {C}^{-1} \\psi _3 , \\psi _3 , \\psi _1]}} \\\\&= \\frac{[\\psi _2 , \\psi _3 , \\psi _1]}{[\\mathbf {C}^{-1} \\psi _3 , \\mathbf {A}\\psi _2 , \\psi _1]}$ matching (REF ) as desired.",
"The computation giving $\\mathcal {X}_B$ is similar but a little longer since three of the four crossings give nontrivial factors: thus we have altogether 6 factors in numerator and denominator; one common factor cancels, leaving the desired ()."
]
] | 1906.04271 |
[
[
"Determining the average prompt-fission-neutron multiplicity for\n $^{239}$Pu($n$,$f$) via a $^{240}$Pu($\\alpha$,$\\alpha^{\\prime}f$) surrogate\n reaction"
],
[
"Abstract The average prompt-fission-neutron multiplicity $\\bar{\\nu}$ is of significance in the areas of nuclear theory, nuclear nonproliferation, and nuclear energy.",
"In this work, the surrogate-reaction method has been used for the first time to indirectly determine $\\bar{\\nu}$ for $^{239}$Pu($n$,$f$) via $^{240}$Pu($\\alpha$,$\\alpha^{\\prime}f$) reactions.",
"A $^{240}$Pu target was bombarded with a beam of 53.9-MeV $\\alpha$ particles.",
"Scattered $\\alpha$ particles, fission products, and neutrons were measured with the NeutronSTARS detector array.",
"Values of $\\bar{\\nu}$ were obtained for a continuous range of equivalent incident neutron energies between 0.25--26.25~MeV, and the results agree well with direct neutron measurements."
],
[
"Introduction",
"The average prompt-fission-neutron multiplicity $\\bar{\\nu }$ following ($n$ ,$f$ ) reactions is important to both basic and applied physics.",
"In nuclear theory, measurements of $\\bar{\\nu }$ can be used to validate fission models and provide constraints on the fission process itself [1].",
"In the area of international safeguards and verification, nuclear materials are assayed with passive neutron-multiplicity counting, and here, $\\bar{\\nu }$ is needed to determine the amount of neutron-induced fission (or self-multiplication) in the sample [2], [3].",
"For proposed nuclear reactor concepts, such as accelerator-driven systems (ADS) and those based on the thorium-uranium cycle, there is interest in the $\\bar{\\nu }$ values for short-lived actinides, as the dependence of $\\bar{\\nu }$ on the incident neutron energy is important for determining the criticality, safety, and lifetime of these reactors [4], [5], [6].",
"In addition, $\\bar{\\nu }$ for short-lived actinides is also relevant to transmutation of radioactive waste with ADS [4], [5], [6].",
"Directly measuring $\\bar{\\nu }$ presents a number of experimental challenges, including producing high-flux neutron beams and addressing beam-related backgrounds.",
"For short-lived actinides, $\\bar{\\nu }$ data are particularly sparse due to the fact that target fabrication and high target activity are also issues.",
"These challenges can be bypassed with the surrogate-reaction method [7], an indirect measurement technique that has typically been used to determine the cross sections of reactions that proceed through a highly excited, statistically equilibrated compound nuclear state.",
"In a surrogate experiment, the desired compound nucleus (CN) is produced using an alternative (“surrogate”) reaction with a more experimentally accessible or preferable combination of projectile and target nucleus.",
"The surrogate method has been demonstrated to work well for determining ($n$ ,$f$ ) reaction cross sections of various actinides [8], [9], [10], [11], [12], [13]; the values obtained are within $\\sim $ 5–20% of direct neutron measurements.",
"The present work extends the applicability of this technique to determining $\\bar{\\nu }$ .",
"Benchmarking has been performed by using the surrogate reactions $^{240}$ Pu($\\alpha $ ,$\\alpha ^{\\prime }f$ ) and $^{242}$ Pu($\\alpha $ ,$\\alpha ^{\\prime }f$ ) to obtain $\\bar{\\nu }$ as a function of incident neutron energy for the reactions $^{239}$ Pu($n$ ,$f$ ) and $^{241}$ Pu($n$ ,$f$ ), respectively, for which direct-measurement data are available.",
"The results for $^{239}$ Pu($n$ ,$f$ ) are discussed in this paper, while those for $^{241}$ Pu($n$ ,$f$ ) can be found in Ref.",
"[14]."
],
[
"Surrogate-reaction technique",
"In the present work, the compound nucleus $^{240}$ Pu in the desired reaction $n + ^{239}\\text{Pu} \\rightarrow ^{240}\\text{Pu}^* \\rightarrow \\text{LF} + \\text{HF} + \\nu n$ is produced via the surrogate reaction $\\alpha + ^{240}\\text{Pu} \\rightarrow \\alpha ^\\prime + ^{240}\\text{Pu}^* \\rightarrow \\alpha ^\\prime + \\text{LF} + \\text{HF} + \\nu n,$ where LF and HF are the light and heavy fission fragments, respectively, and $\\nu $ is the prompt-fission-neutron multiplicity.",
"Assuming a statistically equilibrated CN, where the decay is independent of the method of formation [15], the ($n$ ,$f$ ) cross section for an incident neutron energy $E_n$ is given by the following Hauser-Feshbach [16], [17], [18] formula: $\\sigma _{n,f}(E_n) = \\sum _{J,\\pi }\\sigma _{n}^{CN}(E_\\text{ex},J,\\pi )G_{f}^{CN}(E_\\text{ex},J,\\pi ),$ where $\\sigma _{n}^{CN}(E_\\text{ex},J,\\pi )$ is the cross section for forming a CN with excitation energy $E_\\text{ex}$ , angular momentum $J$ , and parity $\\pi $ , and $G_{f}^{CN}(E_\\text{ex},J,\\pi )$ is the probability that the CN will fission.",
"In the Weisskopf-Ewing limit of Hauser-Feshbach theory, where the decay of the CN is independent of $J$ and $\\pi $ , Eq.",
"REF reduces to $\\sigma _{n,f}(E_n) = \\sigma _{n}^{CN}(E_\\text{ex})G_{f}^{CN}(E_\\text{ex}).$ Analogously, the ($\\alpha $ ,$\\alpha ^\\prime f$ ) cross section for an incident $\\alpha $ -particle energy $E_{\\alpha }$ is given by $\\sigma _{\\alpha ,\\alpha ^\\prime f}(E_\\alpha ) = \\sigma _{\\alpha ,\\alpha ^\\prime }^{CN}(E_\\text{ex})G_{f}^{CN}(E_\\text{ex}).$ In Eq.",
"REF and REF , $\\sigma _{n}^{CN}(E_\\text{ex})$ and $\\sigma _{\\alpha ,\\alpha ^\\prime }^{CN}(E_\\text{ex})$ are the $J\\pi $ -independent CN-formation cross sections and $G_{f}^{CN}(E_\\text{ex})$ is the $J\\pi $ -independent fission probability of the CN.",
"If the Weisskopf-Ewing approximation applies, then ($n$ ,$f$ ) and ($\\alpha $ ,$\\alpha ^\\prime f$ ) reactions that generate the same CN with excitation energy $E_\\text{ex}$ will have identical values of $G_{f}^{CN}(E_\\text{ex})$ and yield the same $\\bar{\\nu }$ .",
"The validity of this assumption is tested by comparing the $\\bar{\\nu }$ values obtained with the surrogate reaction $^{240}$ Pu($\\alpha $ ,$\\alpha ^{\\prime }f$ ) to those determined from direct $^{239}$ Pu($n$ ,$f$ ) measurements."
],
[
"Experiment",
"The experiment was performed in Cave 4 of the Texas A&M University Cyclotron Institute [19].",
"A $^{240}$ Pu target was loaded onto a target wheel [20] located at the center of the NeutronSTARS array [21] and bombarded with a 100-pA beam of 53.9-MeV alpha particles from the K150 Cyclotron; 4.75 days' worth of data was collected."
],
[
"Targets",
"The $^{240}$ Pu target was 99.995%-pure; it was fabricated by first epoxying a 100-$\\mu $ g/cm$^{2}$ -thick natural-carbon foil to an aluminum frame, and then electroplating plutonium onto the foil surface, covering a circular area 1.90 cm in diameter.",
"Properties of the target are given in Table REF .",
"The following calibration targets were included in the experiment: a $^{208}$ Pb foil to determine the beam energy; a natural-carbon foil, Mylar ((C$_{10}$ H$_{8}$ O$_{4}$ )$_\\text{n}$ ) foil, and empty aluminum frame to assess backgrounds due to $\\alpha $ interactions with carbon, oxygen and aluminum in the $^{240}$ Pu target.",
"Two phosphor targets were also used for beam alignment and observing the beam-spot size.",
"Table: Properties of the 240 ^{240}Pu targetused in the experiment.",
"In addition to 240 ^{240}Pu, a small amount of 238 ^{238}Pu was also present."
],
[
"Apparatus",
"The NeutronSTARS array is shown in Fig.",
"REF .",
"Charged particles, including inelastically scattered $\\alpha $ particles from $^{240}$ Pu($\\alpha $ ,$\\alpha ^{\\prime }f$ ) reactions, were detected with a silicon telescope located 19 mm downstream from the target and consisting of two Micron S2-type annular silicon detectors (a 152-$\\mu $ m-thick $\\Delta E$ detector and a 994-$\\mu $ m-thick $E$ detector) that were separated by 4 mm.",
"The energy loss in the two detectors was used for particle identification.",
"A 4.44-mg/cm$^2$ -thick aluminum-foil shield was placed between the target and the telescope to prevent fission fragments and $\\delta $ electrons produced in the target from damaging the $\\Delta E$ detector and degrading detector performance.",
"Fission fragments were detected with a third 146-$\\mu $ m-thick Micron S2 silicon detector located 19 mm upstream from the target.",
"The silicon detectors are segmented into 48 0.5-mm-wide rings on one side and 16 22.5$^\\circ $ -wide sectors on the other.",
"For this experiment, pairs of adjacent rings and sectors were bussed together to form 24 1-mm-wide rings and 8 45$^\\circ $ -wide sectors.",
"The silicon detectors are also coated with 27-$\\mu $ g/cm$^2$ aluminum contacts on the ring side and 500-$\\mu $ g/cm$^2$ gold contacts on the sector side.",
"The gold can significantly straggle the fission fragments, making energy separation between scattered $\\alpha $ particles and fission fragments difficult.",
"To minimize straggling, the fission detector was installed with the ring side facing downstream and the $^{240}$ Pu target was mounted with the electroplated surface facing upstream.",
"The target wheel and silicon detectors were mounted inside a vacuum chamber, which was surrounded by a neutron detector (referred to as “NeutronBall”) consisting of a tank filled with 3.5 tons of liquid scintillator.",
"The tank is segmented into six regions: four identical quadrants that make up the central cylinder and two endcaps.",
"Twenty photomultiplier tubes (PMTs), three on each quadrant and four on each endcap, are used to measure scintillation light.",
"At the time of the measurement, the central cylinder was filled with fresh EJ-335 liquid scintillator doped with 0.25-wt% of natural gadolinium [22]; however the two endcaps contained degraded liquid scintillator with poor optical transmission.",
"Therefore, in the present work, only events detected by the twelve PMTs on the central cylinder were included in the data analysis."
],
[
"Detector calibrations",
"For the $\\Delta E$ and $E$ detectors, the response of each ring and sector was calibrated with a $^{226}$ Ra $\\alpha $ point source that provided the following $\\alpha $ lines: 4784, 5304, 5489, 6002, and 7687 keV [23].",
"At 7687 keV, the resulting 1$\\sigma $ energy resolutions for the $\\Delta E$ detector and $E$ detector were approximately 40 keV and 24 keV, respectively.",
"The fission detector was calibrated with a $^{252}$ Cf spontaneous fission source.",
"The light and heavy fission-product mass peaks were used to gain match the response of the rings.",
"For NeutronBall, $^{60}$ Co and $^{228}$ Th $\\gamma $ -ray point sources provided calibration points at 1253 keV (the average energy of the 1173-keV and 1332-keV $\\gamma $ rays from $^{60}$ Co) and 2615 keV (from $^{208}$ Tl in the $^{228}$ Th decay chain) [23].",
"Another calibration point was provided by the 4440-keV $\\gamma $ rays [23] that were emitted following inelastic $\\alpha $ scattering with the natural-carbon target that promoted $^{12}$ C to its first excited state.",
"The energy resolution of the liquid scintillator at energy $E$ (in MeV) was $\\sigma (E)/E = 25\\%/\\sqrt{E}$ [21].",
"The efficiency for detecting a single neutron with the central cylinder of NeutronBall was determined to be 0.504(5) and was measured by placing a $^{252}$ Cf fission source at the target position.",
"More details will be given in Sec.",
"REF ."
],
[
"$\\alpha $ -particle beam",
"The $\\alpha $ -particle beam-spot size was approximately 3 mm in diameter and was observed with an in-vacuum camera that imaged the phosphor targets.",
"The exact beam energy provided by the K150 Cyclotron was determined from data collected for the $^{208}$ Pb target.",
"Scattering of $\\alpha $ particles to discrete states in $^{208}$ Pb was used as an in situ calibration.",
"The beam energy was determined to be 53.9(1) MeV.",
"This value allowed the excitation energy of the $^{208}$ Pb nucleus to be properly reconstructed after taking into account the energy deposition in the $\\Delta E$ -$E$ telescope, the energy loss in dead layers (i.e., the target, the aluminum-foil shield, and the gold and aluminum contacts on the surfaces of the silicon detectors), and the recoil energy of the $^{208}$ Pb nucleus.",
"The uncertainty in the beam energy was taken to be the 1$\\sigma $ width of the $\\alpha $ peak corresponding to elastic scattering.",
"Figure: (Color online) Cross-sectional views of (a) the NeutronSTARS detector array and(b) the inside of the target chamber (not to-scale); the α\\alpha -particle beamtravels from right to left.NeutronSTARS consists of a target chamber that sits at the center of a neutron detector.The latter is a large tank of gadolinium-doped liquid scintillator segmented intosix regions: four identical quadrants that make up the central cylinder and two endcaps.Three PMTs are attached to each quadrant and four are attached to each endcap.The target chamber contains a target wheel, a ΔE\\Delta E-EE telescope to measure scatteredα\\alpha particles, a fission detector to measure fission fragments (FF), and a δ\\delta shieldto prevent fission fragments and δ\\delta electrons from hitting the ΔE\\Delta E detector.A $^{240}$ Pu($\\alpha $ ,$\\alpha ^{\\prime }f$ ) interaction was indicated by a coincidence between an $\\alpha $ particle hitting the silicon telescope and a fission fragment hitting the fission detector.",
"For a $^{240}$ Pu CN with excitation energy $E_\\text{ex}$ , corresponding to an equivalent incident neutron energy $E_n$ , the average prompt-fission-neutron multiplicity was determined from $\\bar{\\nu }(E_n) = \\frac{N_{n}(E_n)}{N_{\\alpha -f}(E_n)\\epsilon _n},$ where $N_{\\alpha -f}(E_n)$ is the number of measured $^{240}$ Pu($\\alpha $ ,$\\alpha ^{\\prime }f$ ) $\\alpha $ -fission coincidences at $E_n$ , $N_n(E_n)$ is the number of detected prompt fission neutrons associated with these coincidences, and $\\epsilon _n$ is the single-neutron detection efficiency for the central cylinder of NeutronBall.",
"The analysis performed to obtain the quantities in Eq.",
"REF is discussed in this section, and the resulting $\\bar{\\nu }(E_n)$ distribution is given."
],
[
"Charged particles",
"For events in the silicon telescope, the energies deposited in the $\\Delta E$ and $E$ detectors ($E_{\\Delta E}$ and $E_{E}$ , respectively) were used for particle identification (PID).",
"Protons, deuterons, tritons, $^{3}$ He, and $\\alpha $ particles were distinguished by plotting the “linearized energy” $E_\\text{lin}$ [24] versus the total energy deposition in both the $\\Delta E$ and $E$ detectors, where $E_\\text{lin} = [(E_{\\Delta E} + E_{E})^{1.75} - E_{E}^{1.75}]^{1/1.75}.$ Alpha-particle events were isolated by generating a PID plot for each $\\Delta E$ -detector ring (e.g., Fig.",
"REF ) and gating on the region above $^3$ He ($E_\\text{lin}$ approximately between 16.5–24).",
"Figure: (Color online) Particle-identification plot for 53.9-MeV α\\alpha particles incident on 240 ^{240}Pu.The linearized energy versus the total energy deposited in both the ΔE\\Delta E and EE detectors is shownfor events hitting a chosen ring in the ΔE\\Delta E detector.Bands corresponding to protons (p), deuterons (d), tritons (t), 3 ^3He, and α\\alpha particles (α\\alpha )are indicated.",
"The diagonal streaks are due to high-energy charged particles that “punch through” theEE detector and therefore do not deposit all of their energy in the telescope."
],
[
"Fission",
"Fig.",
"REF shows the gain-matched spectrum measured by a single ring on the fission detector for $\\alpha $ particles incident on the $^{240}$ Pu target.",
"A double hump is present at higher energies due to heavy and light fission fragments hitting the detector.",
"The large peak at lower energies is primarily due to light ions from $^{240}$ Pu $\\alpha $ decay and $\\alpha $ -particle interactions with carbon and oxygen in the $^{240}$ Pu target, which was confirmed by analysis of the data collected for the natural-carbon and Mylar targets.",
"For each ring, fission events were selected and light-ion events removed by cutting above an energy deposition of 47 (arb.",
"units).",
"Figure: Gain-matched spectrum measured by a single ring on the fission detector for 53.9-MeV α\\alpha particles incident on 240 ^{240}Pu.",
"Peaks corresponding to heavy fission fragments (HF),light fission fragments (LF), and light ions are labeled.",
"A vertical line is drawn at theenergy cut used to separate fission fragments and light ions."
],
[
"$^{240}$ Pu({{formula:75bf9565-1799-4796-837b-002e360ae470}} ,{{formula:5749918d-759d-4b3f-9331-0f056452ebcf}} ) events",
"In Fig.",
"REF , the time difference between coincident $\\Delta E$ -$E$ $\\alpha $ -particle and fission-detector events is plotted; the energy deposited in the fission detector is given along the y axis.",
"A horizontal line is drawn at the energy cut-off used to isolate fission fragments from light ions.",
"Coincidences above the cut-off with a time difference between $-35$ ns and 86 ns (“prompt” region) were tagged as $^{240}$ Pu($\\alpha $ ,$\\alpha ^{\\prime }f$ ) events.",
"The small bursts of events present every 121 ns in Fig.",
"REF coincide with the K150 cyclotron frequency and are due to random coincidences such as an $\\alpha $ particle hitting the $\\Delta E$ -$E$ telescope and a fission fragment from a $^{240}$ Pu($\\alpha $ ,$f$ ) reaction in the target hitting the fission detector.",
"Figure: (Color online) Coincidences between an α\\alpha particle hitting the ΔE\\Delta E-EE telescope and an eventin the fission detector.",
"The energy deposited in the fission detector is plotted versus the fission-detectorevent time minus the α\\alpha -particle event time.",
"A horizontal line is drawn at the energy cut-off usedto isolate fission fragments from light ions.",
"Vertical lines indicate the gates used in the data analysisto identify and characterize prompt α\\alpha -fission coincidences (-35-35 ns to 86 ns) and random-coincidences(207 to 1901 ns)."
],
[
"Neutrons",
"PMT signals that arrived within a coincidence window of 200 ns were assumed to come from a single event in NeutronBall, e.g., a neutron capture on gadolinium, or an interaction of a room-background $\\gamma $ ray.",
"These signals were first gain matched, as described in Ref.",
"[21], then summed together to acquire the total energy deposited by the event.",
"Only events with energy greater than 2 MeV were included in the data analysis to exclude most of the contribution from backgrounds and electronic noise.",
"For the tagged $^{240}$ Pu($\\alpha $ ,$\\alpha ^{\\prime }f$ ) events, a timing gate was opened 50 $\\mu $ s before and closed 500 $\\mu $ s after the $\\alpha $ -fission coincidence.",
"The time difference between a NeutronBall event occurring within this gate and the $\\alpha $ -fission coincidence was plotted (Fig.",
"REF ).",
"The sharp peak around 0 $\\mu $ s in Fig.",
"REF is from the flash of prompt $\\gamma $ rays following fission and from proton recoils generated during thermalization of the neutron in the liquid scintillator.",
"The broad peak above 0 $\\mu $ s is attributed to prompt fission neutrons; its width is determined by the moderation time of the neutrons in the scintillator.",
"Both features lie on top of a flat background due to random coincidences.",
"Figure: (Color online) Time difference between an event in NeutronBall and an α\\alpha -fission coincidence tagged as a 240 ^{240}Pu(α\\alpha ,α ' f\\alpha ^{\\prime }f) event.",
"The peak due to prompt fission γ\\gamma rays andproton recoils is indicated.",
"The time windows used in the analysis to gate on prompt fissionneutrons (2 to 44 μ\\mu s) and random coincidences (-45-45 to -3-3 μ\\mu s) are also shown."
],
[
"Equivalent neutron energy",
"The excitation energy $E_\\text{ex}$ of $^{240}$ Pu following inelastic $\\alpha $ -particle scattering was determined from the beam energy $E_{\\alpha }$ , the scattered-$\\alpha $ -particle energy $E_{\\alpha ^{\\prime }}$ , and the $^{240}$ Pu recoil energy $E_r$ : $E_\\text{ex} = E_{\\alpha } - E_{\\alpha ^{\\prime }} - E_r.$ The value of $E_{\\alpha ^{\\prime }}$ was the total energy deposited in the $\\Delta E$ -$E$ telescope corrected for energy losses in the target, the $\\delta $ shield, and the inert gold and aluminum contacts on the surfaces of the silicon detectors.",
"The equivalent incident neutron energy $E_n$ was then determined from $E_n = \\frac{m_t+m_n}{m_t}(E_\\text{ex} - S_n),$ where $m_t$ is the mass of $^{239}$ Pu, $m_n$ is the neutron mass, and $S_n$ is the neutron separation energy for $^{240}$ Pu.",
"Fig.",
"REF shows the $E_n$ distribution for $^{240}$ Pu($\\alpha $ ,$\\alpha ^{\\prime }f$ ) events.",
"The corresponding $^{240}$ Pu excitation energy is also given.",
"Fission of $^{240}$ Pu starts to occur at ${E_n=-1.61}$ MeV (4.9-MeV $^{240}$ Pu excitation energy) [25].",
"The feature at ${E_n\\sim 5.5}$ MeV is due to $^{240}$ Pu second-chance fission [26], [27], and above ${E_n\\sim 18.5}$ MeV, the number of events tapers off quickly due to the $\\alpha $ -Pu Coulomb barrier.",
"Figure: The distribution of equivalent incident neutron energies (and corresponding 240 ^{240}Pu excitation energies)for 240 ^{240}Pu(α\\alpha ,α ' f\\alpha ^{\\prime }f) events; 0.5-MeV-wide energy bins are used."
],
[
"Average prompt-fission-neutron multiplicity",
"The average prompt-fission-neutron multiplicity was obtained with Eq.",
"REF for equivalent incident neutron energies ranging between 0.25 and 26.25 MeV.",
"The quantity $N_{\\alpha -f}(E_n)$ in Eq.",
"REF is the number of $\\alpha $ -fission coincidences in the (121-ns-wide) prompt region of Fig.",
"REF , corrected for the contribution from random coincidences.",
"This contribution was determined by taking the sum of $\\alpha $ -fission coincidences in the region 207–1901 ns and scaling down to a 121-ns-wide time window.",
"The number of neutrons $N_n(E_n)$ was obtained by taking the difference between the total counts in the time regions 2 to 44 $\\mu $ s and $-45$ to $-3$ $\\mu $ s in Fig.",
"REF .",
"The contribution from random $\\alpha $ -fission coincidences was determined from the time-difference spectrum for NeutronBall events associated with the 207–1901-ns region in Fig.",
"REF (scaled down to correspond to a 121-ns-wide $\\alpha $ -fission time window).",
"A single-neutron detection efficiency of ${\\epsilon _{n} = 0.504(5)}$ was obtained by first recording the time-difference between $^{252}$ Cf fission events in the fission detector and events in NeutronBall.",
"The total number of prompt neutrons measured was then determined and divided by the number of fission events and $\\bar{\\nu }$ for $^{252}$ Cf (i.e., 3.757) [28], [29].",
"The $\\bar{\\nu }(E_n)$ distribution obtained is given in Fig.",
"REF .",
"Each $\\bar{\\nu }$ value and its uncertainty is also provided in Table REF ; the uncertainty is dominated by the statistical uncertainties in the number of $\\alpha $ -fission coincidences and the number of detected neutrons.",
"Fig.",
"REF also shows that the results of the present work are consistent with direct neutron measurements for $^{239}$ Pu($n$ ,$f$ ) [30], [31], [32], [33], [34], [35], providing validation that the surrogate-reaction method can be used to determine $\\bar{\\nu }$ for actinides.",
"Figure: (Color online) The average prompt-fission-neutron multiplicity ν ¯\\bar{\\nu } as a function of incident neutron energyfor 239 ^{239}Pu(nn,ff).In the present work, ν ¯\\bar{\\nu } has been determined continuously from 0.25–26.25 MeV in 0.5-MeV-wide intervals.The results are compared with direct neutron measurements found in literature , , , , , .In the present work the uncertainties are primarily due to counting statistics; for the literature values,most of the uncertainties are smaller than the data markers.Table: Equivalent incident neutron energies E n E_n and correspondingν ¯\\bar{\\nu } values from the present work."
],
[
"Summary and Conclusions",
"$^{240}$ Pu($\\alpha $ ,$\\alpha ^{\\prime }f$ ) was used as a surrogate reaction to determine the $^{239}$ Pu($n$ ,$f$ ) prompt-fission-neutron multiplicity as a function of incident neutron energy from 0.25–26.25 MeV.",
"This is the first time $\\bar{\\nu }$ for $^{239}$ Pu($n$ ,$f$ ) has been obtained continuously over this neutron energy range in a single measurement.",
"The results of the present work are in good agreement with those from direct neutron measurements [30], [31], [32], [33], [34], [35].",
"Similar conclusions were drawn in Ref.",
"[14], where the surrogate reaction $^{242}$ Pu($\\alpha $ ,$\\alpha ^{\\prime }f$ ) was used to determine $\\bar{\\nu }$ for $^{241}$ Pu($n$ ,$f$ ).",
"The success of these two experiments opens the door to using surrogate reactions to obtain $\\bar{\\nu }(E_n)$ for a whole host of short-lived actinides that are currently inaccessible via direct methods."
],
[
"Acknowledgments",
"We thank the staff of the Texas A&M Cyclotron Institute for facilitating operations and facilities needed to perform this measurement.",
"This work was performed under the auspices of the U.S. Department of Energy National Nuclear Security Administration by Lawrence Livermore National Laboratory under Contract No.",
"DE-AC52-07NA27344, under Award No.",
"DE-NA0000979, and through the Nuclear Science and Security Consortium under Award No.",
"DE-NA-0003180."
]
] | 1906.04305 |
[
[
"Quantum corrections to the BTZ black hole extremality bound from the\n conformal bootstrap"
],
[
"Abstract Any unitary compact two-dimensional CFT with $c>1$ and no symmetries beyond Virasoro has a parametrically large density of primary states at large spin for $\\bar{h}>\\bar{h}_\\text{extr}\\sim \\frac{c-1}{24}$, of a universal form determined by modular invariance.",
"By including the contribution of light primary operators and multi-twist composites constructed from them in the modular bootstrap, we find that $\\bar{h}_\\text{extr}$ receives corrections in a large spin expansion, which we compute at finite $c$.",
"The analysis uses a formulation of the modular S-transform as a Fourier transform acting on the density of primary states.",
"For theories with gravitational duals, $\\bar{h}_\\text{extr}$ is interpreted as the extremality bound of rotating BTZ black holes, receiving quantum corrections which we compute at one loop by prohibiting naked singularities in the quantum-corrected geometry.",
"This gravity result is reproduced by modular bootstrap in a semiclassical $c\\to\\infty$ limit."
],
[
"=1 Department of Physics, University of California, Santa Barbara, CA 93106, USA"
]
] | 1906.04416 |
[
[
"NAS-FCOS: Fast Neural Architecture Search for Object Detection"
],
[
"Abstract The success of deep neural networks relies on significant architecture engineering.",
"Recently neural architecture search (NAS) has emerged as a promise to greatly reduce manual effort in network design by automatically searching for optimal architectures, although typically such algorithms need an excessive amount of computational resources, e.g., a few thousand GPU-days.",
"To date, on challenging vision tasks such as object detection, NAS, especially fast versions of NAS, is less studied.",
"Here we propose to search for the decoder structure of object detectors with search efficiency being taken into consideration.",
"To be more specific, we aim to efficiently search for the feature pyramid network (FPN) as well as the prediction head of a simple anchor-free object detector, namely FCOS, using a tailored reinforcement learning paradigm.",
"With carefully designed search space, search algorithms and strategies for evaluating network quality, we are able to efficiently search a top-performing detection architecture within 4 days using 8 V100 GPUs.",
"The discovered architecture surpasses state-of-the-art object detection models (such as Faster R-CNN, RetinaNet and FCOS) by 1.5 to 3.5 points in AP on the COCO dataset, with comparable computation complexity and memory footprint, demonstrating the efficacy of the proposed NAS for object detection."
],
[
"Introduction",
"Object detection is one of the fundamental tasks in computer vision, and has been researched extensively.",
"In the past few years, state-of-the-art methods for this task are based on deep convolutional neural networks (such as Faster R-CNN [20], RetinaNet [11]), due to their impressive performance.",
"Typically, the designs of object detection networks are much more complex than those for image classification, because the former need to localize and classify multiple objects in an image simultaneously while the latter only need to output image-level labels.",
"Due to its complex structure and numerous hyper-parameters, designing effective object detection networks is more challenging and usually needs much manual effort.",
"On the other hand, Neural Architecture Search (NAS) approaches [4], [17], [32] have been showing impressive results on automatically discovering top-performing neural network architectures in large-scale search spaces.",
"Compared to manual designs, NAS methods are data-driven instead of experience-driven, and hence need much less human intervention.",
"As defined in [3], the workflow of NAS can be divided into the following three processes: 1) sampling architecture from a search space following some search strategies; 2) evaluating the performance of the sampled architecture; and 3) updating the parameters based on the performance.",
"One of the main problems prohibiting NAS from being used in more realistic applications is its search efficiency.",
"The evaluation process is the most time consuming part because it involves a full training procedure of a neural network.",
"To reduce the evaluation time, in practice a proxy task is often used as a lower cost substitution.",
"In the proxy task, the input, network parameters and training iterations are often scaled down to speedup the evaluation.",
"However, there is often a performance gap for samples between the proxy tasks and target tasks, which makes the evaluation process biased.",
"How to design proxy tasks that are both accurate and efficient for specific problems is a challenging problem.",
"Another solution to improve search efficiency is constructing a supernet that covers the complete search space and training candidate architectures with shared parameters [15], [18].",
"However, this solution leads to significantly increased memory consumption and restricts itself to small-to-moderate sized search spaces.",
"To our knowledge, studies on efficient and accurate NAS approaches to object detection networks are rarely touched, despite its significant importance.",
"To this end, we present a fast and memory saving NAS method for object detection networks, which is capable of discovering top-performing architectures within significantly reduced search time.",
"Our overall detection architecture is based on FCOS [24], a simple anchor-free one-stage object detection framework, in which the feature pyramid network and prediction head are searched using our proposed NAS method.",
"Our main contributions are summarized as follows.",
"In this work, we propose a fast and memory-efficient NAS method for searching both FPN and head architectures, with carefully designed proxy tasks, search space and evaluation strategies, which is able to find top-performing architectures over $3,000$ architectures using 28 GPU-days only.",
"Specifically, this high efficiency is enabled with the following designs.",
"$-$ Developing a fast proxy task training scheme by skipping the backbone finetuning stage; $-$ Adapting progressive search strategy to reduce time cost taken by the extended search space; $-$ Using a more discriminative criterion for evaluation of searched architectures.",
"$-$ Employing an efficient anchor-free one-stage detection framework with simple post processing; Using NAS, we explore the workload relationship between FPN and head, proving the importance of weight sharing in head.",
"We show that the overall structure of NAS-FCOS is general and flexible in that it can be equipped with various backbones including MobileNetV2, ResNet-50, ResNet-101 and ResNeXt-101, and surpasses state-of-the-art object detection algorithms using comparable computation complexity and memory footprint.",
"More specifically, our model can improve the AP by $1.5\\sim 3.5$ points on all above models comparing to their FCOS counterparts."
],
[
"Object Detection",
"The frameworks of deep neural networks for object detection can be roughly categorized into two types: one-stage detectors [12] and two-stage detectors [6], [20].",
"Two-stage detection frameworks first generate class-independent region proposals using a region proposal network (RPN), and then classify and refine them using extra detection heads.",
"In spite of achieving top performance, the two-stage methods have noticeable drawbacks: they are computationally expensive and have many hyper-parameters that need to be tuned to fit a specific dataset.",
"In comparison, the structures of one-stage detectors are much simpler.",
"They directly predict object categories and bounding boxes at each location of feature maps generated by a single CNN backbone.",
"Note that most state-of-the-art object detectors (including both one-stage detectors [12], [16], [19] and two-stage detectors [20]) make predictions based on anchor boxes of different scales and aspect ratios at each convolutional feature map location.",
"However, the usage of anchor boxes may lead to high imbalance between object and non-object examples and introduce extra hyper-parameters.",
"More recently, anchor-free one-stage detectors [9], [10], [24], [29], [30] have attracted increasing research interests, due to their simple fully convolutional architectures and reduced consumption of computational resources."
],
[
"Neural Architecture Search",
"NAS is usually time consuming.",
"We have seen great improvements from $24,000$ GPU-days [32] to $0.2$ GPU-day [28].",
"The trick is to first construct a supernet containing the complete search space and train the candidates all at once with bi-level optimization and efficient weight sharing [13], [15].",
"But the large memory allocation and difficulties in approximated optimization prohibit the search for more complex structures.",
"Recently researchers [1], [5], [23] propose to apply single-path training to reduce the bias introduced by approximation and model simplification of the supernet.",
"DetNAS [2] follows this idea to search for an efficient object detection architecture.",
"One limitation of the single-path approach is that the search space is restricted to a sequential structure.",
"Single-path sampling and straight through estimate of the weight gradients introduce large variance to the optimization process and prohibit the search for more complex structures under this framework.",
"Within this very simple search space, NAS algorithms can only make trivial decisions like kernel sizes for manually designed modules.",
"Object detection models are different from single-path image classification networks in their way of merging multi-level features and distributing the task to parallel prediction heads.",
"Feature pyramid networks (FPNs) [4], [8], [11], [14], [27], designed to handle this job, plays an important role in modern object detection models.",
"NAS-FPN [4] targets on searching for an FPN alternative based on one-stage framework RetinaNet [12].",
"Feature pyramid architectures are sampled with a recurrent neural network (RNN) controller.",
"The RNN controller is trained with reinforcement learning (RL).",
"However, the search is very time-consuming even though a proxy task with ResNet-10 backbone is trained to evaluate each architecture.",
"Since all these three kinds of research ( [2], [4] and ours) focus on object detection framework, we demonstrate the differences among them that DetNAS [2] aims to search for the designs of better backbones, while NAS-FPN [4] searches the FPN structure, and our search space contains both FPN and head structure.",
"To speed up reward evaluation of RL-based NAS, the work of [17] proposes to use progressive tasks and other training acceleration methods.",
"By caching the encoder features, they are able to train semantic segmentation decoders with very large batch sizes very efficiently.",
"In the sequel of this paper, we refer to this technique as fast decoder adaptation.",
"However, directly applying this technique to object detection tasks does not enjoy similar speed boost, because they are either not in using a fully-convolutional model [11] or require complicated post processing that are not scalable with the batch size [12].",
"To reduce the post processing overhead, we resort to a recently introduced anchor-free one-stage framework, namely, FCOS [24], which significantly improve the search efficiency by cancelling the processing time of anchor-box matching in RetinaNet.",
"Compared to its anchor-based counterpart, FCOS significantly reduces the training memory footprint while being able to improve the performance."
],
[
"Our Approach",
"In our work, we search for anchor-free fully convolutional detection models with fast decoder adaptation.",
"Thus, NAS methods can be easily applied."
],
[
"Problem Formulation",
"We base our search algorithm upon a one-stage framework FCOS due to its simplicity.",
"Our training tuples $\\lbrace (\\mathbf {x}, Y)\\rbrace $ consist of input image tensors $\\mathbf {x}$ of size $(3\\times H\\times W)$ and FCOS output targets $Y$ in a pyramid representation, which is a list of tensors $\\mathbf {y}_l$ each of size $((K+4+1)\\times H_l\\times W_l)$ where $H_l\\times W_l$ is feature map size on level $p$ of the pyramid.",
"$(K+4+1)$ is the output channels of FCOS, the three terms are length-$K$ one-hot classification labels, 4 bounding box regression targets and 1 centerness factor respectively.",
"The network $g: \\mathbf {x}\\rightarrow \\hat{Y}$ in original FCOS consists of three parts, a backbone $b$ , FPN $f$ and multi-level subnets we call prediction heads $h$ in this paper.",
"First backbone $b: \\mathbf {x}\\rightarrow C$ maps the input tensor to a set of intermediate-leveled features $C = \\lbrace \\mathbf {c}_3, \\mathbf {c}_4, \\mathbf {c}_5\\rbrace $ , with resolution $(H_i\\times W_i) = (H/2^i \\times W/2^i)$ .",
"Then FPN $f: C\\rightarrow P$ maps the features to a feature pyramid $P=\\lbrace \\mathbf {p}_3, \\mathbf {p}_4, \\mathbf {p}_5, \\mathbf {p}_6, \\mathbf {p}_7\\rbrace $ .",
"Then the prediction head $h: \\mathbf {p}\\rightarrow \\mathbf {y}$ is applied to each level of $P$ and the result is collected to create the final prediction.",
"To avoid overfitting, same $h$ is often applied to all instances in $P$ .",
"Since objects of different scales require different effective receptive fields, the mechanism to select and merge intermediate-leveled features $C$ is particularly important in object detection network design.",
"Thus, most researches [16], [20] are carried out on designing $f$ and $h$ while using widely-adopted backbone structures such as ResNet [7].",
"Following this principle, our search goal is to decide when to choose which features from $C$ and how to merge them.",
"To improve the efficiency, we reuse the parameters in $b$ pretrained on target dataset and search for the optimal structures after that.",
"For the convenience of the following statement, we call the network components to search for, namely $f$ and $h$ , together the decoder structure for the objection detection network.",
"$f$ and $h$ take care of different parts of the detection job.",
"$f$ extracts features targeting different object scales in the pyramid representations $P$ , while $h$ is a unified mapping applied to each feature in $P$ to avoid overfitting.",
"In practice, people seldom discuss the possibility of using a more diversified $f$ to extract features at different levels or how many layers in $h$ need to be shared across the levels.",
"In this work, we use NAS as an automatic method to test these possibilities."
],
[
"Search Space",
"Considering the different functions of $f$ and $h$ , we apply two search space respectively.",
"Given the particularity of FPN structure, a basic block with new overall connection and $f$ 's output design is built for it.",
"For simplicity, sequential space is applied for $h$ part.",
"We replace the cell structure with atomic operations to provide even more flexibility.",
"To construct one basic block, we first choose two layers $\\mathbf {x}_1$ , $\\mathbf {x}_2$ from the sampling pool $X$ at id1, id2, then two operations op1, op2 are applied to each of them and an aggregation operation agg merges the two output into one feature.",
"To build a deep decoder structure, we apply multiple basic blocks with their outputs added to the sampling pool.",
"Our basic block $bb_t: X_{t-1}\\rightarrow X_t$ at time step $t$ transforms the sampling pool $X_{t-1}$ to $X_t = X_{t-1}\\cup \\lbrace \\mathbf {x}_t\\rbrace $ , where $\\mathbf {x}_t$ is the output of $bb_t$ .",
"Table: Unary operations used in the search process.The candidate operations are listed in Table REF .",
"We include only separable/depth-wise convolutions so that the decoder can be efficient.",
"In order to enable the decoder to apply convolutional filters on irregular grids, here we have also included deformable $3\\times 3$ convolutions [31].",
"For the aggregation operations, we include element-wise sum and concatenation followed by a $1\\times 1$ convolution.",
"The decoder configuration can be represented by a sequence with three components, FPN configuration, head configuration and weight sharing stages.",
"We provide detailed descriptions to each of them in the following sections.",
"The complete diagram of our decoder structure is shown in Fig.",
"REF .",
"Figure: A conceptual example of our NAS-FCOS decoder.",
"It consists of two sub networks, an FPN ff and a set of prediction heads hh which have shared structures.",
"One notable difference with other FPN-based one-stage detectors is that our heads have partially shared weights.",
"Only the last several layers of the predictions heads (marked as yellow) are tied by their weights.",
"The number of layers to share is decided automatically by the search algorithm.",
"Note that both FPN and head are in our actual search space; and have more layers than shown in this figure.",
"Here the figure is for illustration only."
],
[
"FPN Search Space",
"As mentioned above, the FPN $f$ maps the convolutional features $C$ to $P$ .",
"First, we initialize the sampling pool as $X_0 = C$ .",
"Our FPN is defined by applying the basic block 7 times to the sampling pool, $f:= bb_1^f\\circ bb_2^f \\circ \\cdots \\circ bb_7^f$ .",
"To yield pyramid features $P$ , we collect the last three basic block outputs $\\lbrace \\mathbf {x}_5, \\mathbf {x}_6, \\mathbf {x}_7\\rbrace $ as $\\lbrace \\mathbf {p}_3, \\mathbf {p}_4, \\mathbf {p}_5\\rbrace $ .",
"To allow shared information across all layers, we use a simple rule to create global features.",
"If there is some dangling layer $\\mathbf {x}_t$ which is not sampled by later blocks $\\lbrace bb_i^f|i > t\\rbrace $ nor belongs to the last three layers $t < 5$ , we use element-wise add to merge it to all output features $\\mathbf {p}^*_i = \\mathbf {p}_i + \\mathbf {x}_t, \\,\\, i\\in \\lbrace 3, 4, 5\\rbrace .$ Same as the aggregation operations, if the features have different resolution, the smaller one is upsampled with bilinear interpolation.",
"To be consistent with FCOS, $\\mathbf {p}_6$ and $\\mathbf {p}_7$ are obtained via a $3\\times 3$ stride-2 convolution on $\\mathbf {p}_5$ and $\\mathbf {p}_6$ respectively."
],
[
"Prediction Head Search Space",
"Prediction head $h$ maps each feature in the pyramid $P$ to the output of corresponding $\\mathbf {y}$ , which in FCOS and RetinaNet, consists of four $3\\times 3$ convolutions.",
"To explore the potential of the head, we therefore extend a sequential search space for its generation.",
"Specifically, our head is defined as a sequence of six basic operations.",
"Compared with candidate operations in the FPN structures, the head search space has two slight differences.",
"First, we add standard convolution modules (including conv1x1 and conv3x3) to the head sampling pool for better comparison.",
"Second, we follow the design of FCOS by replacing all the Batch Normalization (BN) layers to Group Normalization (GN) [25] in the operations sampling pool of head, considering that head needs to share weights between different levels, which causes BN invalid.",
"The final output of head is the output of the last (sixth) layer."
],
[
"Searching for Head Weight Sharing",
"To add even more flexibility and understand the effect of weight sharing in prediction heads, we further add an index $i$ as the location where the prediction head starts to share weights.",
"For every layer before stage $i$ , the head $h$ will create independent set of weights for each FPN output level, otherwise, it will use the global weights for sharing purpose.",
"Considering the independent part of the heads being extended FPN branch and the shared part as head with adaptive-length, we can further balance the workload for each individual FPN branch to extract level-specific features and the prediction head shared across all levels."
],
[
"Search Strategy",
"RL based strategy is applied to the search process.",
"We rely on an LSTM-based controller to predict the full configuration.",
"We consider using a progressive search strategy rather than the joint search for both FPN structure and prediction head part, since the former requires less computing resources and time cost than the latter.",
"The training dataset is randomly split into a meta-train $D_t$ and meta-val $D_v$ subset.",
"To speed up the training, we fix the backbone network and cache the pre-computed backbone output $C$ .",
"This makes our single architecture training cost independent from the depth of backbone network.",
"Taking this advantage, we can apply much more complex backbone structures and utilize high quality multilevel features as our decoder's input.",
"We find that the process of backbone finetuning can be skipped if the cached features are powerful enough.",
"Speedup techniques such as Polyak weight averaging are also applied during the training.",
"The most widely used detection metric is average precision (AP).",
"However, due to the difficulty of object detection task, at the early stages, AP is too low to tell the good architectures from the bad ones, which makes the controller take much more time to converge.",
"To make the architecture evaluation process easier even at the early stages of the training, we therefore use negative loss sum as the reward instead of average precision: $\\begin{split}R(a) = &- \\sum _{(x, Y)\\in D_v}(L_{cls}(x, Y|a) \\\\&+ L_{reg}(x, Y|a) + L_{ctr}(x, Y|a))\\end{split}$ where $L_{cls}$ , $L_{reg}$ , $L_{ctr}$ are the three loss terms in FCOS.",
"Gradient of the controller is estimated via proximal policy optimization (PPO) [22]."
],
[
"Searching Phase",
"We design a fast proxy task for evaluating the decoder architectures sampled in the searching phase.",
"PASCAL VOC is selected as the proxy dataset, which contains 5715 training images with object bounding box annotations of 20 classes.",
"Transfer capacity of the structures can be illustrated since the search and full training phase use different datasets.",
"The VOC training set is randomly split into a meta-train set with $4,000$ images and a meta-val set with 1715 images.",
"For each sampled architecture, we train it on meta-train and compute the reward (REF ) on meta-val.",
"Input images are resized to short size 384 and then randomly cropped to $384\\times 384$ .",
"Target object sizes of interest are scaled correspondingly.",
"We use Adam optimizer with learning rate 8e$-4$ and batch size 200.",
"Polyak averaging is applied with the decay rates of $0.9$ .",
"The decoder is evaluated after 300 iterations.",
"As we use fast decoder adaptation, the backbone features are fixed and cached during the search phase.",
"To enhance the cached backbone features, we first initialize them with pre-trained weights provided by open-source implementation of FCOShttps://tinyurl.com/FCOSv1 and then finetune on VOC using the training strategies of FCOS.",
"Note that the above finetuning process is only performed once at the begining of the search phase.",
"A progressive strategy is used for the search of $f$ and $h$ .",
"We first search for the FPN part and retain the original head.",
"All operations in the FPN structure have 64 output channels.",
"The decoder inputs $C$ are resized to fit output channel width of FPN via $1\\times 1$ convolutions.",
"After this step, a searched FPN structure is fixed and the second stage searching for the head will be started based on it.",
"Most parameters for searching head are identical to those for searching FPN structure, with the exception that the output channel width is adjusted from 64 to 128 to deliver more information.",
"For the FPN search part, the controller model nearly converged after searching over $2.8$ K architectures on the proxy task as shown in Fig.",
"REF .",
"Then, the top-20 best performing architectures on the proxy task are selected for the next full training phase.",
"For the head search part, we choose the best searched FPN among the top-20 architectures and pre-fetch its features.",
"It takes about 600 rounds for the controller to nearly converge, which is much faster than that for searching FPN architectures.",
"After that, we select for full training the top-10 heads that achieve best performance on the proxy task.",
"In total, the whole search phase can be finished within 4 days using 8 V100 GPUs.",
"Table: Results on test-dev set of MS COCO after full training.",
"R-50 and R-101 represents ResNet backbones and X-64x4d-101 represents ResNeXt-101 (64×464\\times 4d).",
"All networks share the same input image resolution.",
"FLOPs and parameters are being measured on 1088×8001088\\times 800, which is the median of the input size on COCO.",
"For RetinaNet and FCOS, we use official models provided by the authors.",
"For our NAS-FCOS, @128 and @256 means that the decoder channel width is 128 and 256 respectively.",
"@128-256 is the decoder with 128 FPN width and 256 head width.",
"The same improving tricks used on the newest FCOS version are used in our model for fair comparison.Figure: Performance of reward during the proxy task, which has been growing throughout the process, indicating that the model of reinforcement learning works."
],
[
"Full Training Phase",
"In this phase, we fully train the searched models on the MS COCO training dataset, and select the best one by evaluating them on MS COCO validation images.",
"Note that our training configurations are exactly the same as those in FCOS for fair comparison.",
"Input images are resized to short size 800 and the maximum long side is set to be 1333.",
"The models are trained using 4 V100 GPUs with batch size 16 for 90K iterations.",
"The initial learning rate is $0.01$ and reduces to one tenth at the 60K-th and 80K-th iterations.",
"The improving tricks are applied only on the final model (w/improv).",
"Figure: Our discovered FPN structure.",
"C 2 C_2 is omitted from this figure since it is not chosen by this particular structure during the search process."
],
[
"Search Results",
"The best FPN structure is illustrated in Fig.",
"REF .",
"The controller identifies that deformable convolution and concatenation are the best performing operations for unary and aggregation respectively.",
"From Fig.",
"REF , we can see that the controller chooses to use 4 operations (with two skip connections), rather than the maximum allowed 6 operations.",
"Note that the discovered “dconv + 1x1 conv” structure achieves a good trade-off between accuracy and FLOPs.",
"Compared with the original head, our searched head has fewer FLOPs/Params (FLOPs $79.24$ G vs. $89.16$ G, Params $3.41$ M vs. $4.92$ M) and significantly better performance (AP $38.7$ vs. $37.4$ ).",
"Figure: Our discovered Head structure.We use the searched decoder together with either light-weight backbones such as MobileNet-V2 [21] or more powerful backbones such as ResNet-101 [7] and ResNeXt-101 [26].",
"To balance the performance and efficiency, we implement three decoders with different computation budgets: one with feature dimension of 128 (@128), one with 256 (@256) and another with FPN channel width 128 and prediction head 256 (@128-256).",
"The results on the COCO test-dev with short side being 800 is shown in Table REF .",
"The searched decoder with feature dimension of 256 (@256) surpasses its FCOS counterpart by $1.5$ to $3.5$ points in AP under different backbones.",
"The one with 128 channels (@128) has significantly reduced parameters and calculation, making it more suitable for resource-constrained environments.",
"In particular, our searched model with 128 channels and MobileNetV2 backbone suparsses the original FCOS with the same backbone by $0.8$ AP points with only $1/3$ FLOPS.",
"The third type of decoder (@128-256) achieves a good balance between accuracy and parameters.",
"Note that our searched model outperforms the strongest FCOS variant by $1.4$ AP points ($46.1$ vs. $44.7$ ) with slightly smaller FLOPs and Params.",
"The comparison of FLOPs and number of parameters with other models are illustrated in Fig.",
"REF and Fig.",
"REF respectively.",
"Figure: Trend graph of head weight sharing during search.",
"The coordinates in the horizontal axis represent the number of the statistical period.",
"A period consists of 50 head structures.",
"The vertical axis represents the proportion of heads that fully share weights in 50 structures.In order to understand the importance of weight sharing in head, we add the number of layers shared by weights as an object of the search.",
"Fig.",
"REF shows a trend graph of head weight sharing during search.",
"We set 50 structures as a statistical cycle.",
"As the search deepens, the proportion of fully shared structures increases, indicating that on the multi-scale detection model, head weight sharing is a necessity.",
"Table: Comparison with other NAS methods.For NAS-FPN,the input size is 1280×12801280\\times 1280 andthe search cost should be timed by their number of TPUs used to train each architecture.Note thatthe FLOPs and AP of NAS-FPN @256 here are from Figure 11 in NAS-FPN ,and NAS-FPN 7@256 stacks the searched FPN structure 7 times.The input images are resized such that their shorter size is 800 pixels in DetNASNet and our models.Figure: Correlation between the search reward obtained on the VOC meta-val dataset and the AP evaluated on COCO-val.We also demonstrate the comparison with other NAS methods for object detection in Table REF .",
"Our method is able to search for twice more architectures than DetNAS [2] per GPU-day.",
"Note that the AP of NAS-FPN [4] is achieved by stacking the searched FPN 7 times, while we do not stack our searched FPN.",
"Our model with ResNeXt-101 (64x4d) as backbone outperforms NAS-FPN by $1.3$ AP points while using only $1/3$ FLOPs and less calculation cost.",
"Figure: Diagram of the relationship between FLOPs and AP with different backbones.",
"Points of different shapes represent different backbones.",
"NAS-FCOS@128 has a slight increase in precision which also gains the advantage of computation quantity.",
"One with 256 channels obtains the highest precision with more computation complexity.",
"Using FPN channel width 128 and prediction head 256 (@128-256) offers a trade-off.Figure: Diagram of the relationship between parameters and AP with different backbones.",
"Adjusting the number of channels in the FPN structure and head helps to achieve a balance between accuracy and parameters.We further measure the correlation between rewards obtained during the search process with the proxy dataset and APs attained by same architectures trained on COCO.",
"Specifically, we randomly sample 15 architectures from all the searched structures trained on COCO with batch size 16.",
"Since full training on COCO is time-consuming, we reduce the iterations to 60K.",
"The model is then evaluated on the COCO 2017 validation set.",
"As visible in Fig.",
"REF , there is a strong correlation between search rewards and APs obtained from COCO.",
"Poor- and well-performing architectures can be distinguished by the rewards on the proxy task very well.",
"Figure: Comparison of two different RL reward designs.",
"The vertical axis represents AP obtained from the proxy task on the validation dataset."
],
[
"Design of Reinforcement Learning Reward",
"As we discussed above, it is common to use widely accepted indicators as rewards for specific tasks in the search, such as mIOU for segmentation and AP for object detection.",
"However, we found that using AP as reward did not show a clear upward trend in short-term search rounds (blue curve in Fig.",
"REF ).",
"We further analyze the possible reason to be that the controller tries to learn a mapping from the decoder to the reward while the calculation of AP itself is complicated, which makes it difficult to learn this mapping within a limited number of iterations.",
"In comparison, we clearly see the increase of AP with the validation loss as RL rewards (red curve in Fig.",
"REF ).",
"Table: Comparisons between APs obtained under different search space with ResNet-50 backbone."
],
[
"Effectiveness of Search Space",
"To further discuss the impact of the search spaces $f$ and $h$ , we design three experiments for verification.",
"One is to search $f$ with the original head being fixed, one is to search $h$ with the original FPN being fixed and another is to search the entire decoder ($f$ +$h$ ).",
"As shown in Table REF , it turns out that searching $f$ brings slightly more benefits than searching $h$ only.",
"And our progressive search which combines both $f$ and $h$ achieves a better result."
],
[
"Impact of Deformable Convolution",
"As aforementioned, deformable convolutions are included in the set of candidate operations for both $f$ and $h$ , which are able to adapt to the geometric variations of objects.",
"For fair comparison, we also replace the whole standard $3\\times 3$ convolutions with deformable $3\\times 3$ convolutions in FPN structure of the original FCOS and repeat them twice, making the FLOPs and parameters nearly equal to our searched model.",
"The new model is therefore called DeformFPN-FCOS.",
"It turns out that our NAS-FCOS model still achieves better performance (AP $= 38.9$ with FPN search only, and AP $= 39.8$ with both FPN and Head searched) than the DeformFPN-FCOS model (AP $= 38.4$ ) under this circumstance."
],
[
"Conclusion",
"In this paper, we have proposed to use Neural Architecture Search to further optimize the process of designing object detection networks.",
"It is shown in this work that top-performing detectors can be efficiently searched using carefully designed proxy tasks, search strategies and model evaluation metrics.",
"The experiments on COCO demonstrates the efficiency of our discovered model NAS-FCOS and its flexibility to be used with various backbone architectures."
]
] | 1906.04423 |
[
[
"Sequential Source Coding for Stochastic Systems Subject to Finite Rate\n Constraints"
],
[
"Abstract In this paper, we revisit the sequential source coding framework to analyze fundamental performance limitations of discrete-time stochastic control systems subject to feedback data-rate constraints in finite-time horizon.",
"The basis of our results is a new characterization of the lower bound on the minimum total-rate achieved by sequential codes subject to a total (across time) distortion constraint and a computational algorithm that allocates optimally the rate-distortion for any fixed finite-time horizon.",
"This characterization facilitates the derivation of analytical, non-asymptotic, and finite-dimensional lower and upper bounds in two control-related scenarios.",
"(a) A parallel time-varying Gauss-Markov process with identically distributed spatial components that is quantized and transmitted through a noiseless channel to a minimum mean-squared error (MMSE) decoder.",
"(b) A time-varying quantized LQG closed-loop control system, with identically distributed spatial components and with a random data-rate allocation.",
"Our non-asymptotic lower bound on the quantized LQG control problem, reveals the absolute minimum data-rates for (mean square) stability of our time-varying plant for any fixed finite time horizon.",
"We supplement our framework with illustrative simulation experiments."
],
[
"Introduction",
"One of the fundamental characteristics of networked control systems (NCSs) [1] is the existence of an imperfect communication network between computational and physical entities.",
"In such setups, an analytical framework to assess impacts of communication and data-rate limitations on the control performance is strongly required.",
"In this paper, we adopt information-theoretic tools to analyze these requirements.",
"Specifically, we consider sequential coding of correlated sources initially introduced by [2] (see also [3]) (see Fig.",
"REF ), which is a generalization of the successive refinement source coding problem [4], [5], [6].",
"In successive refinement, source coding is performed in (time) stages where one first describes the given source within a few bits of information and, then, tries to “refine” the description of the same source (at the subsequent stages) when more information is available.",
"Sequential coding differs from successive refinement in that at the second stage, encoding involves describing a correlated (in time) source as opposed to improving the description of the same source.",
"To accomplish this task, sequential coding encompasses a spatio-temporal coding method.",
"Figure: Sequential coding of correlated sources.In addition, sequential coding is a temporally zero-delay coding paradigm since both encoding and decoding must occur in real-time.",
"The resulting zero-delay coding approach should not be confused with other existing works on zero-delay coding, see, e.g., [7], [8], [9], [10], [11], [12], because it relies on the use of a spatio-temporal coding approach (see Fig.",
"REF ) whereas the aforementioned papers rely solely on temporal coding approaches.",
"In what follows, we provide a detailed literature review on sequential source coding.",
"However, in order to shed more light on the historical route of this coding paradigm, we distinguish the work of [2] (see also [13], [14]) with the work of [3] because although their results complement each other, their underlining motivation has been different.",
"Indeed, [2] initiated this coding approach targeting video coding applications, whereas [3] aimed to develop a framework for delay-constrained systems and to study the communication theory in classical closed-loop control setups."
],
[
"Sequential coding via {{cite:a402ad206c70d633233677fc6a425860a755f06a}}",
"The authors of [2] characterized the minimum achievable rate-distortion region for two temporally correlated random variables with each being a vector of spatially independent and identically distributed ($\\mathop {\\mathrm {IID}}$ ) processes (also called “frames” or spatial vectors), subject to a coupled average distortion criterion.",
"The last decade, sequential coding approach of [2] was further studied in [13], [14], [15].",
"In [13], the authors used an extension of the framework of [2] to three time instants subject to a per-time distortion constraint to investigate the effect of sequential coding when possible coding delays occur within a multi-input multi-output system.",
"Around the same time, [14] generalized the framework of [2] to a finite number of time instants.",
"Compared to [2] and [13], their spatio-temporal source process is correlated over time whereas each frame is spatially jointly stationary and totally ergodic subject to a per-time average distortion criterion.",
"More recently, the same authors in [15] drew connections between sequential causal coding and predictive sequential causal coding, that is, for (first-order) Markov sources subject to a single-letter fidelity constraint, sequential causal coding and sequential predictive coding coincide.",
"For three time instants of an $\\mathop {\\mathrm {IID}}$ vector source containing jointly Gaussian correlated processes (not necessarily Markov) an explicit expression of the minimum achievable sum-rate for a per-time mean-squared error ($\\mathop {\\mathrm {MSE}}$ ) distortion is obtained in [16].",
"Inspired by the framework of [2], [13], Khina et al.",
"in [17] derived fundamental performance limitations in control-related applications.",
"In their work, they considered a multi-track system that tracks several parallel time-varying Gauss-Markov processes with $\\mathop {\\mathrm {IID}}$ spatial components conveyed over a single shared wireless communication link (possibly prone to packet drops) to a minimum mean-squared error ($\\mathop {\\mathrm {MMSE}}$ ) decoder.",
"In their Gauss-Markov multi-tracking scenario, they provided lower and upper bounds in finite-time and in the per unit time asymptotic limit for the distortion-rate region of time-varying Gauss-Markov sources subject to a mean-squared error ($\\mathop {\\mathrm {MSE}}$ ) distortion constraint.",
"Their lower bound is characterized by a forward in time distortion allocation algorithm operating with given data-rates at each time instant for a finite time horizon whereas their upper bound is obtained by means of a differential pulse-code modulation (DPCM) scheme using entropy coded dithered quantization (ECDQ) using one dimensional lattice constrained by data rates averaged across time (for details on this coding scheme, see, e.g., [18], [19]).",
"Subsequently, they used these bounds in a scalar-valued quantized linear quadratic Gaussian (LQG) closed-loop control problem to find similar bounds on the minimum cost of control.",
"A similar framework to [2] was independently introduced and developed by Tatikonda in [3] (see also [20]) in the context of delay-constrained and control-related applications.",
"Tatikonda in [3], introduced an information theoretic quantity called sequential rate distortion function ($\\mathop {\\mathrm {RDF}}$ ) that is attributed to the works of Gorbunov and Pinsker in [21], [22].",
"Using the sequential $\\mathop {\\mathrm {RDF}}$ , Tatikonda et al.",
"in [23] studied the performance analysis and synthesis of a multidimensional fully observable time-invariant Gaussian closed-loop control system when a communication link exists between a stochastic linear plant and a controller whereas the performance criterion is the classical linear quadratic cost.",
"The use of sequential $\\mathop {\\mathrm {RDF}}$ (also termed nonanticipative or causal $\\mathop {\\mathrm {RDF}}$ in the literature) in filtering applications is stressed in [24], [25], [26].",
"Analytical expressions of lower and upper bounds for the setup of [23] including the cases where a linear fully observable time-invariant plant is driven by $\\mathop {\\mathrm {IID}}$ non-Gaussian noise processes or when the system is modeled by time-invariant partially observable Gaussian processes are derived in [27].",
"Tanaka et al.",
"in [28], [29] studied the performance analysis and synthesis of a linear fully observable and partially observable Gaussian closed loop control problem when the performance criterion is the linear quadratic cost.",
"Moreover, they showed that one can derive lower bounds in finite time and in the per unit time asymptotic limit by casting the problems as semidefinite representable and thus numerically computable by known solvers.",
"An achievability bound on the asymptotic limit using a DPCM-based $\\mathop {\\mathrm {ECDQ}}$ scheme that uses one dimensional quantizer at each dimension was also proposed.",
"Lower and upper bounds for a general closed-loop control system subject to asymptotically average total data-rate constraints across the time are also investigated in [30], [31].",
"The lower bounds are obtained using sequential coding and directed information [32] whereas the upper bounds are obtained via a sequential $\\mathop {\\mathrm {ECDQ}}$ scheme using scalar quantizers."
],
[
"Contributions",
"In this paper, we first revisit the sequential coding framework developed by [2], [3], [13], [14] to obtain the following main results.",
"(1) Analytical non-asymptotic and finite-dimensional lower and upper bounds on the minimum achievable total-rates (per-dimension) for a multi-track communication scenario similar to the one considered in [17].",
"However, compared to [17], who derived distortion-rate bounds via forward recursions with given data rates across a finite time horizon, here we derive a lower bound subject to a dynamic reverse-waterfilling algorithm in which for a given distortion threshold $D>0$ we optimally assign the data-rates and the $\\mathop {\\mathrm {MSE}}$ distortions at each time instant for a finite time horizon (Theorem REF ).",
"We also implement our algorithm in Algorithm .",
"Our lower bound is the basis to derive our upper bound on the minimum achievable total-rates (per dimension) using a sequential $\\mathop {\\mathrm {DPCM}}$ -based $\\mathop {\\mathrm {ECDQ}}$ scheme that is constrained by total-rates for a finite time horizon.",
"For the specific rate constraint we use a dynamic reverse-waterfilling algorithm obtained from our lower bound to allocate the rate and the $\\mathop {\\mathrm {MSE}}$ distortion at each time instant for the whole finite time horizon.",
"This rate constraint is the fundamental difference compared to similar upper bounds derived in [17] and [30] (see also [31], [12]) that restrict their transmit rates to have fixed rates that are averaged across the time horizon or that are asymptotically averaged across the time.",
"(2) We obtain analogous bounds to (1) on the minimum achievable total (across time) cost-rate function of control (per-dimension) for a NCS with time-varying quantized $\\mathop {\\mathrm {LQG}}$ closed-loops operating with data-rate obtained subject to a solution of a reverse-waterfilling algorithm (Theorems REF , REF ).",
"Discussion of the contributions and additional results.",
"The non-asymptotic lower bound in (1) is obtained because for parallel processes all involved matrices in the characterization of the corresponding optimization problem commute by pairs [33] thus they are simultaneously diagonalizable by an orthogonal matrix [33] and the resulting optimization problem simplifies to one that resembles scalar-valued processes.",
"The upper bound in (1) is obtained because we are able to employ a lattice quantizer [19] using a quantization scheme with existing performance guarantees such as the $\\mathop {\\mathrm {DPCM}}$ -based $\\mathop {\\mathrm {ECDQ}}$ scheme and using existing approximations from quantization theory for high-dimensional but possibly finite-dimensional quantizers with a $\\mathop {\\mathrm {MSE}}$ performance criterion (see, e.g., [34]).",
"The non-asymptotic bounds derived in (2) are obtained using the so-called “weak separation principle” of quantized $\\mathop {\\mathrm {LQG}}$ control (for details, see §) and well-known inequalities that are used in information theory.",
"Interestingly, our lower bound in (2) also reveals the minimum allowable data rates on the cost-rate (or rate-cost) function in control at each time instant to ensure (mean square) stability of the plant (see e.g., [35] for the definition) for the specific NCS (Remark REF ).",
"Finally, for every bound in this paper, we derive the corresponding bounds in the infinite time horizon recovering several known results in the literature (see Corollaries REF -REF ).",
"This paper is organized as follows.",
"In § we give an overview of known results on sequential coding.",
"In § we derive non-asymptotic bounds and their corresponding per unit time asymptotic limits for a quantized state estimation problem.",
"In §, we use the results of § and the weak separation principle to derive non-asymptotic bounds and their corresponding per unit time asymptotic limits for a quantized $\\mathop {\\mathrm {LQG}}$ closed-loop control problem.",
"In § we discuss several open questions that can be answered based on this work and draw conclusions in §."
],
[
"$\\mathbb {R}$ is the set of real numbers, $\\mathbb {N}_{1}$ is the set of positive integers, and $\\mathbb {N}_1^n\\triangleq \\lbrace 1,\\ldots ,n\\rbrace $ , $n\\in \\mathbb {N}_1$ , respectively.",
"Let $\\mathbb {X}$ be a finite-dimensional Euclidean space, and ${\\cal B}(\\mathbb {X})$ be the Borel $\\sigma $ -algebra on $\\mathbb {X}$ .",
"A random variable ($\\mathop {\\mathrm {RV}}$ ) $X$ defined on some probability space ($\\Omega , {\\cal F}, {\\bf P}$ ) is a map $X : \\Omega \\mapsto \\mathbb {X}$ .",
"The probability distribution of a $\\mathop {\\mathrm {RV}}$ $X$ with realization $X=x$ on $\\mathbb {X}$ is denoted by ${\\bf P}_X\\equiv {p}(x)$ .",
"The conditional distribution of a $\\mathop {\\mathrm {RV}}$ $Y$ with realization $Y=y$ , given $X=x$ is denoted by ${\\bf Q}_{Y|X}\\equiv {q}(y|x)$ .",
"We denote the sequence of one-sided $\\mathop {\\mathrm {RVs}}$ by $X_{t,j} \\triangleq (X_{t}, X_{t+1}, \\ldots ,X_j),~{t}\\le {j},~(t,j)\\in {\\mathbb {N}}_1\\times \\mathbb {N}_1$ , and their values by $x_{t,j} \\in {\\mathbb {X}}_{t,j} \\triangleq \\times _{k={{t}}}^j {\\mathbb {X}}_k$ .",
"We denote the sequence of ordered $\\mathop {\\mathrm {RVs}}$ with “$i^{\\text{th}}$ ” spatial components by $X_{t,j}^i$ , so that $X_{t,j}^i$ is a vector of dimension “$i$ ”, and their values by $x_{t,j}^i \\in {\\mathbb {X}}_{t,j}^i \\triangleq \\times _{k={{t}}}^j {\\mathbb {X}}^i_k$ , where ${\\mathbb {X}}^i_k\\triangleq \\left(\\mathbb {X}_k(1),\\ldots \\mathbb {X}_k(i)\\right)$ .",
"The notation ${X}\\leftrightarrow {Y}\\leftrightarrow {Z}$ denotes a Markov Chain ($\\mathop {\\mathrm {MC}}$ ) which means that $p(x|y,z)=p(x|y)$ .",
"We denote the diagonal of a square matrix by $\\mathop {\\mathrm {diag}}(\\cdot )$ and the $p\\times {p}$ identity matrix by $I_p$ .",
"If $A\\in \\mathbb {R}^{p{\\times }{p}}$ , we denote by $A\\succeq {0}$ (resp., $A\\succ {0}$ ) a positive semidefinite matrix (resp., positive definite matrix).",
"We denote the determinant and trace of some matrix $A\\in \\mathbb {R}^{p\\times {p}}$ by $|A|$ and $\\mathop {\\mathrm {trace}}(A)$ , respectively.",
"We denote by $h(x)$ (resp.",
"$h(x|y)$ ) the differential entropy of a distribution $p(x)$ (resp.",
"$p(x|y)$ ).",
"We denote ${\\cal D}(P||Q)$ the relative entropy of probability distributions $P$ and $Q$ .",
"We denote by ${\\bf E}\\lbrace \\cdot \\rbrace $ the expectation operator and $||\\cdot ||_2$ the Euclidean norm.",
"Unless otherwise stated, when we say “total” distortion, “total-rate” or “total-cost” we mean with respect to time.",
"Similarly, by referring to “average total” we mean normalized over the total finite time horizon."
],
[
"Known Results on Sequential Coding",
"In this section, we give an overview of the sequential causal coding introduced and analyzed independently by [3] and [2], [13], [14].",
"We merge both frameworks because some results obtained in [13], [14] complement the results of [3] and vice versa.",
"In the following analysis, we will consider processes for a fixed time-span $t\\in \\mathbb {N}^n_1$ , i.e., ($X_1,\\ldots ,X_{n}$ ).",
"Following [13], [14], we assume that the sequences of $\\mathop {\\mathrm {RVs}}$ are defined on alphabet spaces with finite cardinality.",
"Nevertheless, these can be extended following for instance the techniques employed in [36] to continuous alphabet spaces as well (i.e., Gaussian processes) with $\\mathop {\\mathrm {MSE}}$ distortion constraints.",
"First, we use some definitions (with slight modifications to ease the readability of the paper) from [13] and [14].",
"Definition 1 (Sequential causal coding) A spatial order $p$ sequential causal code ${\\cal C}_p$ for the (joint) vector source ($X_1^p,X_2^p,\\ldots ,X_n^p$ ) is formally defined by a sequence of encoder and decoder pairs ($f^{(p)}_1,g^{(p)}_1$ ),$\\ldots $ ,($f^{(p)}_n,g^{(p)}_n$ ) such that $\\begin{split}f_t^{(p)}&:~\\mathbb {X}^p_{1,t}\\times \\underbrace{\\lbrace 0,1\\rbrace ^*\\times \\ldots \\times \\lbrace 0,1\\rbrace ^*}_{t-1~times}\\longrightarrow \\mathbb {\\lbrace }0,1\\rbrace ^*\\\\g_t^{(p)}&:~\\underbrace{\\lbrace 0,1\\rbrace ^*\\times \\ldots \\times \\lbrace 0,1\\rbrace ^*}_{t~times}\\longrightarrow \\mathbb {Y}^p_t,~t\\in \\mathbb {N}_1^n\\end{split},$ where $\\lbrace 0,1\\rbrace ^*$ denotes the set of all binary sequences of finite length satisfying the property that at each time instant $t$ the range of $\\lbrace f_t:~t\\in \\mathbb {N}_1^n\\rbrace $ given any $t-1$ binary sequences is an instantaneous code.",
"Moreover, the encoded and reconstructed sequences of $\\lbrace X_t^p:~t\\in \\mathbb {N}_1^n\\rbrace $ are given by $S_t=f_t(X^p_{1,t},S_{1,t-1})$ , with $S_t\\in \\mathbb {S}_t\\subset \\lbrace 0,1\\rbrace ^*$ , and $Y_t^p=g_t(S_{1,t})$ , respectively, with $|\\mathbb {Y}_t|<\\infty $ .",
"Moreover, the expected rate in bits per symbol at each time instant (normalized over the spatial components) is defined as $r_t\\triangleq \\frac{{\\bf E}|S_t|}{p},~t\\in \\mathbb {N}_1^n,$ where $|S_t|$ denotes the length of the binary sequence $S_t$ ."
],
[
"Distortion criterion",
"For each $t\\in \\mathbb {N}_1^n$ , we consider a total (in dimension) single-letter distortion criterion.",
"This means that the distortion between $X^p_t$ and $Y^p_t$ is measured by a function $d_t:~\\mathbb {X}^p_t\\times \\mathbb {Y}^p_t\\longrightarrow [0,\\infty )$ with maximum distortion $d_t^{\\max }=\\max _{x^p_t,y^p_t}d_t(x^p_t,y_t^p)<\\infty $ such that $d_t(x^p_t,y^p_t)\\triangleq \\frac{1}{p}\\sum _{i=1}^p{d}_t(x_t(i),y_t(i)).$ The per-time average distortion is defined as ${\\bf E}\\left\\lbrace d_t(X_t^p,Y_t^p)\\right\\rbrace \\triangleq \\frac{1}{p}\\sum _{i=1}^p{\\bf E}\\left\\lbrace {d}_t(X_t(i),Y_t(i))\\right\\rbrace .$ We remark that the following results are still valid even if the distortion function (REF ) has dependency on previous reproductions $\\lbrace Y^p_{1,t-1}:~t\\in \\mathbb {N}_1^n\\rbrace $ (see, e.g., [13]).",
"Definition 2 (Achievability) A rate-distortion tuple $(R_{1,n},D_{1,n})\\triangleq (R_1,\\ldots ,R_n,D_1,\\ldots ,D_n)$ for any “$n$ ” is said to be achievable for a given sequential causal coding system if for all $\\epsilon >0$ , there exists a sequential code $\\lbrace (f^{(p)}_t,g^{(p)}_t):~t\\in \\mathbb {N}_1^n\\rbrace $ such that there exists ${\\cal P}$ for which $\\begin{split}r_t&\\le {R_t}+\\epsilon ,\\\\{\\bf E}\\left\\lbrace d_t(X_t^p,Y_t^p)\\right\\rbrace &\\le {D_t}+\\epsilon ,~D_t\\ge {0},~\\forall {t}\\in \\mathbb {N}_1^n,\\end{split}$ holds $\\forall {p}\\ge {\\cal P}$ .",
"Moreover, let the set of all achievable rate-distortion tuples $(R_{1,n},D_{1,n})$ be denoted by ${\\cal R}^{*}$ .",
"Then, the minimum total-rate required to achieve the distortion tuple $(D_{1},~D_2,\\ldots ,D_n)$ is defined by the following optimization problem: ${\\cal R}^{\\mathop {\\mathrm {op}}}_{\\mathop {\\mathrm {sum}}}(D_{1,n})\\triangleq \\inf _{(R_{1,n},D_{1,n})\\in {\\cal R}^{*}}\\sum _{t=1}^nR_t.$ The finite alphabet source randomly generates symbols $X^p_{1,n}=x^p_{1,n}\\in \\mathbb {X}^p_{1,n}$ according to the following temporally correlated joint probability mass function ($\\mathop {\\mathrm {PMF}}$ ) ${p}(x_{1,n}^p)\\triangleq \\otimes _{i=1}^p{p}(x_1(i),\\ldots ,x_{n}(i)),$ where the joint process $\\lbrace (X_1(i),\\ldots ,X_n(i))\\rbrace _{i=1}^p$ is identically distributed.",
"This means that for each $i=1,\\ldots ,p$ , the temporally correlated joint process $(X_1(i),\\ldots ,X_{n}(i))$ is independent of every other temporally correlated joint process $(X_1(j),\\ldots ,X_{n}(j))$ , such that $i\\ne {j}$ .",
"Furthermore, each temporally correlated joint process $(X_1(i),\\ldots ,X_{n}(i))$ is spatially identically distributed.",
"Next, we characterize the achievable rate-distortion regions and the minimum achievable total-rate for the source model (REF ) with the distortion constraint (REF ).",
"The following lemma is given in [14].",
"Lemma 1 (Achievable rate-distortion region) Consider the source model (REF ) with the average distortion of (REF ).",
"Then, the “spatially” single-letter characterization of the rate-distortion region $(R_{1,n},~D_{1,n})$ is given by: $\\begin{split}{\\cal R}^{\\mathop {\\mathrm {IID}}}&=\\Bigg \\lbrace (R_{1,n},D_{1,n})\\Bigg {|}\\exists S_{1,n-1}, Y_{1,n},~\\lbrace g_t(\\cdot )\\rbrace _{t=1}^n,\\\\\\text{s.t.",
"}&\\quad R_1\\ge {I}(X_1;S_1),~\\mbox{(initial time)}\\\\&\\quad R_t\\ge {I}(X_{1,t};S_t|S_{1,t-1}),~t=2,\\ldots ,n-1,\\\\&\\quad R_n\\ge {I}(X_{1,n};Y_n|S_{1,n-1}),~\\mbox{(terminal time)},\\\\&\\quad D_t\\ge {\\bf E}\\left\\lbrace d_t(X_t,Y_t)\\right\\rbrace ,~t\\in \\mathbb {N}_1^n,\\\\&\\quad Y_1=g_1(S_1),~Y_{t}=g_t(S_{1,t}),~t=2,\\ldots ,n-1,\\\\&\\quad S_1\\leftrightarrow (X_{1})\\leftrightarrow {X}_{2,n},\\\\&\\quad S_t\\leftrightarrow (X_{1,t},S_{1,t-1})\\leftrightarrow {X}_{t+1,n},~t=2,\\ldots ,n-1\\Bigg \\rbrace ,\\end{split}$ where $\\lbrace S_{1,n-1},Y_{1,n}\\rbrace $ are the auxiliary (encoded) and reproduction $\\mathop {\\mathrm {RVs}}$ , respectively, taking values in some finite alphabet spaces $\\lbrace \\mathbb {S}_{1,n-1},\\mathbb {Y}_{1,n}\\rbrace $ , and $\\lbrace g_t(\\cdot ):~t\\in \\mathbb {N}_1^n\\rbrace $ are deterministic functions.",
"Remark 1 (Comments on Lemma REF ) In the characterization of Lemma REF , the spatial index is excluded because the rate and distortion regions are normalized with the total number of spatial components.",
"This point is also shown in [14].",
"Following [13] or [14], Lemma REF gives a set ${\\cal R}^{\\mathop {\\mathrm {IID}}}$ that is convex and closed (this can be shown by trivially generalizing the time-sharing and continuity arguments of [13] to $n$ time-steps).",
"This in turn means that ${\\cal R}^*={\\cal R^{\\mathop {\\mathrm {IID}}}}$ (see, e.g., [14]).",
"Thus, (REF ) can be reformulated to the following optimization problem: ${\\cal R}_{\\mathop {\\mathrm {sum}}}^{\\mathop {\\mathrm {IID}},\\mathop {\\mathrm {op}}}(D_{1,n})\\triangleq \\min _{(R_{1,n},D_{1,n})\\in {\\cal R}^{\\mathop {\\mathrm {IID}}}}\\sum _{t=1}^nR_t.$ In what follows, we state a lemma that gives a lower bound on ${\\cal R}_{\\mathop {\\mathrm {sum}}}^{\\mathop {\\mathrm {IID}},\\mathop {\\mathrm {op}}}(D_{1,n})$ .",
"The lemma is stated without a proof as it is already derived in various papers, e.g., [3], [30], [13] (for $n=3$ -time steps but can be trivially generalized to an arbitrary number of time-steps).",
"Lemma 2 (Lower bound on (REF )) For $p$ sufficiently large, the following lower bound holds: ${\\cal R}_{{\\mathop {\\mathrm {sum}}}}^{\\mathop {\\mathrm {IID}},\\mathop {\\mathrm {op}}}(D_{1,n})&\\ge {\\cal R}_{{\\mathop {\\mathrm {sum}}}}^{\\mathop {\\mathrm {IID}}}(D_{1,n})\\nonumber \\\\&\\triangleq \\min _{\\begin{array}{c}{\\bf E}\\left\\lbrace d_t(X_t,Y_t)\\right\\rbrace \\le {D}_t,~t\\in \\mathbb {N}_1^n\\\\Y_1\\leftrightarrow {X_{1}}\\leftrightarrow {X}_{2,n},\\\\~Y_t\\leftrightarrow (X_{1,t},Y_{1,t-1})\\leftrightarrow {X}_{t+1,n},~t=2,\\ldots ,n-1\\end{array}}I(X_{1,n};Y_{1,n}),$ where $I(X_{1,n};Y_{1,n})=\\sum _{t=1}^nI(X_{1,t};Y_t|Y_{1,t-1})$ is a variant of directed information [37], [32] obtained by the conditional independence constraints imposed in the constraint set of (REF ).",
"We note that the lower bound in Lemma REF is often encountered in the literature by the name nonanticipatory $\\epsilon -$ entropy and sequential or nonanticipative $\\mathop {\\mathrm {RDF}}$ .",
"Remark 2 (When do we achieve the lower bound in (REF )?)",
"It should be noted that in [14] it was shown via an algorithmic approach (see also [14] for an equivalent proof via a direct and converse coding theorem) that Lemma REF is achieved with equality if the number of $\\mathop {\\mathrm {IID}}$ spatial components tends to infinity, i.e., $p\\longrightarrow \\infty $ , which also means that the optimal minimizer or “test-channel” at each time instant in (REF ), corresponds precisely to the distribution generated by a sequential encoder, i.e., $S_t=Y_t$ , for any $t\\in \\mathbb {N}_1^n$ (see also the derivation of [13]).",
"In other words, the equality holds if the encoder (or quantizer for continuous alphabet sources) simulates exactly the corresponding “test-channel” distribution of (REF ).",
"This claim was also demonstrated via an application example for jointly Gaussian $\\mathop {\\mathrm {RVs}}$ and per-time $\\mathop {\\mathrm {MSE}}$ distortion in [13] and also stated as a corollary referring to an “ideal” $\\mathop {\\mathrm {DPCM}}$ -based $\\mathop {\\mathrm {MSE}}$ quantizer in [13].",
"In general, however, for any $p<\\infty $ , the equality in (REF ) is not achievable.",
"Next, we state the generalization of Lemma REF when the constrained set is subject to an average total distortion constraint defined as $\\frac{1}{n}\\sum _{t=1}^n{\\bf E}\\left\\lbrace d_t(X_t,Y_t)\\right\\rbrace \\le {D}$ with ${\\bf E}\\left\\lbrace d_t(X_t,Y_t)\\right\\rbrace $ given in (REF ).",
"This lemma was derived in [3].",
"Lemma 3 (Generalization of Lemma REF ) For $p$ sufficiently large, the following lower bound holds: $\\begin{split}{\\cal R}_{{\\mathop {\\mathrm {sum}}}}^{\\mathop {\\mathrm {IID}},\\mathop {\\mathrm {op}}}(D)&\\ge {\\cal R}_{{\\mathop {\\mathrm {sum}}}}^{\\mathop {\\mathrm {IID}}}(D)\\\\&=\\min _{\\begin{array}{c}\\frac{1}{n}\\sum _{t=1}^n{\\bf E}\\left\\lbrace d_t(X_t,Y_t)\\right\\rbrace \\le {D},~t\\in \\mathbb {N}_1^n\\\\Y_1\\leftrightarrow (X_{1})\\leftrightarrow {X}_{2,n},\\\\~Y_t\\leftrightarrow (X_{1,t},Y_{1,t-1})\\leftrightarrow {X}_{t+1,n},~t\\in \\mathbb {N}_2^{n-1}\\end{array}}I(X_{1,n};Y_{1,n}),\\end{split}$ Clearly, one can use the same methodology applied in [14] to demonstrate that the lower bound in (REF ) is achieved once $p\\longrightarrow \\infty $ (see the discussion in Remark REF ).",
"However, we once again point out that in general, (REF ) is a lower bound on the minimum achievable rates achieved by causal sequential codes.",
"Next, we state a few well-known structural results related to the bounds in Lemmas REF , REF .",
"In particular, if the temporally correlated joint $\\mathop {\\mathrm {PMF}}$ in (REF ) follows a finite-order Markov process, then, the description of the rate-distortion region in Lemma REF , and the corresponding bounds on the minimum achievable total-rate in Lemmas REF , REF can be simplified considerably following for instance the framework of [7], [20], [26].",
"For the important special case of first-order Markov process, (REF ) simplifies to ${\\cal R}^{\\mathop {\\mathrm {IID}},1}&=\\Bigg \\lbrace (R_{1,n},D_{1,n})\\Bigg {|}\\exists S_{1,n-1}, Y_{1,n},~\\lbrace g_t(\\cdot )\\rbrace _{t=1}^n,\\nonumber \\\\s.t.&~{R}_1\\ge {I}(X_1;S_1),~\\mbox{(initial time)}\\nonumber \\\\&~R_t\\ge {I}(X_{t};S_t|S_{1,t-1}),~t=2,\\ldots ,n-1,\\nonumber \\\\& ~R_n\\ge {I}(X_{n};Y_n|S_{1,n-1}),~\\mbox{(terminal time)},\\nonumber \\\\& ~D_t\\ge {\\bf E}\\left\\lbrace d_t(X_t,Y_t)\\right\\rbrace ,~t\\in \\mathbb {N}_1^n,\\nonumber \\\\& ~Y_1=g_1(S_1),~Y_{t}=g_t(S_{1,t}),~t=2,\\ldots ,n-1,\\nonumber \\\\& ~S_1\\leftrightarrow (X_{1})\\leftrightarrow {X}_{2,n},\\nonumber \\\\& ~S_t\\leftrightarrow (X_{t},S_{1,t-1})\\leftrightarrow (X_{1,t-1},{X}_{t+1,n})\\Bigg \\rbrace .$ Using (REF ), the minimum achievable total-rate can now be simplified to the following optimization problem: ${\\cal R}_{\\mathop {\\mathrm {sum}}}^{\\mathop {\\mathrm {IID}},\\mathop {\\mathrm {op}},1}(D_{1,n})\\triangleq \\min _{(R_{1,n},D_{1,n})\\in {\\cal R}^{\\mathop {\\mathrm {IID}},1}}\\sum _{t=1}^nR_t.$ Using the description of (REF ), we can simplify (REF ) and (REF ), respectively, as follows: $&{\\cal R}_{\\mathop {\\mathrm {sum}}}^{\\mathop {\\mathrm {IID}},\\mathop {\\mathrm {op}},1}(D_{1,n})\\ge {\\cal R}_{{\\mathop {\\mathrm {sum}}}}^{\\mathop {\\mathrm {IID}},1}(D_{1,n})=\\min _{\\begin{array}{c}{\\bf E}\\left\\lbrace d_t(X_t,Y_t)\\right\\rbrace \\le {D}_t,~t\\in \\mathbb {N}_1^n\\\\Y_1\\leftrightarrow {X_{1}}\\leftrightarrow {X}_{2,n},\\\\~Y_t\\leftrightarrow (X_{t},Y_{1,t-1})\\leftrightarrow (X_{1,t-1},{X}_{t+1,n}),~t\\in \\mathbb {N}_2^{n-1}\\end{array}}I(X_{1,n};Y_{1,n}),\\\\&{\\cal R}_{{\\mathop {\\mathrm {sum}}}}^{\\mathop {\\mathrm {IID}},\\mathop {\\mathrm {op}},1}(D)\\ge {\\cal R}_{{\\mathop {\\mathrm {sum}}}}^{\\mathop {\\mathrm {IID}},1}(D)\\triangleq \\min _{\\begin{array}{c}\\frac{1}{n}\\sum _{t=1}^n{\\bf E}\\left\\lbrace d_t(X_t,Y_t)\\right\\rbrace \\le {D},~t\\in \\mathbb {N}_1^n\\\\Y_1\\leftrightarrow {X_{1}}\\leftrightarrow {X}_{2,n},\\\\~Y_t\\leftrightarrow (X_{t},Y_{1,t-1})\\leftrightarrow (X_{1,t-1},~{X}_{t+1,n}),~t=2,\\ldots ,n-1\\end{array}}I(X_{1,n};Y_{1,n}),$ where $I(X_{1,n};Y_{1,n})=\\sum _{t=1}^n{I}(X_t;Y_t|Y_{1,t-1})$ .",
"In the sequel, we use the description of () to derive our main results."
],
[
"Application in Quantized State Estimation",
"In this section, we apply the sequential coding framework of the previous section to a state estimation problem and obtain new results in such applications.",
"We consider a similar scenario to [17] where a multi-track system estimates several “parallel” Gaussian processes over a single shared communication link as illustrated in Fig.",
"REF .",
"Following the sequential coding framework, we require the Gaussian source processes to have temporally correlated and spatially $\\mathop {\\mathrm {IID}}$ components, which are observed by an observer who collects the measured states into a single vector state.",
"Then, the observer/encoder maps the states as random finite-rate packets to a $\\mathop {\\mathrm {MMSE}}$ estimator through a noiseless link.",
"Compared to the result of [17] which derives a dynamic forward in time recursion of a distortion-rate allocation algorithm when the rate is given at each time instant, here we derive a dynamic rate-distortion reverse-waterfilling algorithm operating forward in time for which we only consider a given distortion threshold $D>0$ .",
"Figure: Multi-track state estimation system model.First, we describe the problem of interest.",
"State process.",
"Consider $p$ -parallel time-varying Gauss-Markov processes with $\\mathop {\\mathrm {IID}}$ spatial components as follows: $x_{t}(i)=\\alpha _{t-1}x_{t-1}(i)+w_{t-1}(i),~i\\in \\mathbb {N}_1^p,~t\\in \\mathbb {N}_1^n,$ where $x_1(i)\\equiv {x}_1$ is given, with $x_1\\sim {\\cal N}(0;\\sigma ^2_{x_1})$ ; the non-random coefficient $\\alpha _{t}\\in \\mathbb {R}$ is known at each time step $t$ , and $\\lbrace w_t(i)\\equiv {w}_t:~i\\in \\mathbb {N}_1^p\\rbrace $ , $w_t\\sim {\\cal N}(0;\\sigma ^2_{w_{t}})$ , is an independent Gaussian noise process at each $t$ , independent of $x_1, \\forall {i}\\in \\mathbb {N}_1^p$ .",
"Since (REF ) has $\\mathop {\\mathrm {IID}}$ spatial components it can be compactly written as a vector or frame as follows: $X_{t}=A_{t-1}X_{t-1}+W_{t-1},~X_1=\\text{given},~t\\in \\mathbb {N}_2^n,$ where $A_{t-1}=\\mathop {\\mathrm {diag}}(\\alpha _{t-1},\\ldots ,\\alpha _{t-1})\\in \\mathbb {R}^{p\\times {p}}$ , $X_t\\in \\mathbb {R}^p$ , and the independent Gaussian noise process $W_t\\in \\mathbb {R}^p\\sim {\\cal N}(0;\\Sigma _{W_t})$ , where $\\Sigma _{W_t}=\\mathop {\\mathrm {diag}}(\\sigma ^2_{w_t},\\ldots ,\\sigma ^2_{w_t})\\succ {0}\\in \\mathbb {R}^{p\\times {p}}$ independent of the initial state $X_1$ .",
"Observer/Encoder.",
"At the observer the spatially $\\mathop {\\mathrm {IID}}$ time-varying $\\mathbb {R}^p$ -valued Gauss-Markov processes are collected into a frame $X_t\\in \\mathbb {R}^p$ and mapped using sequential coding with encoded sequence: $S_t=f_t(X_{1,t},S_{1,t-1}),$ where at $t=1$ we assume $S_1=f_1(X_1)$ , and $R_t=\\frac{{\\bf E}|S_t|}{p}$ is the expected (random) rate (per dimension) at each time instant $t$ transmitted through the noiseless link.",
"MMSE Decoder.",
"The data packet $S_t$ is received using the following reconstructed sequence: $Y_t=g_t(S_{1,t}),$ where at $t=1$ we have $Y_1=g_1(S_1)$ .",
"Distortion.",
"We consider the average total $\\mathop {\\mathrm {MSE}}$ distortion normalized over all spatial components as follows: $\\frac{1}{n}\\sum _{t=1}^nD_t~\\mbox{with}~D_t\\triangleq \\frac{1}{p}{\\bf E}\\left\\lbrace ||X_t-Y_t||_2^2\\right\\rbrace .$ Performance.",
"The performance of the above system (per dimension) for a given $D>0$ can be cast to the following optimization problem: ${\\cal R}_{\\mathop {\\mathrm {sum}}}^{\\mathop {\\mathrm {IID}},\\mathop {\\mathrm {op}},1}(D)=\\min _{\\begin{array}{c}(f_t,~g_t):~t=1,\\ldots ,n\\\\ \\frac{1}{n}\\sum _{t=1}^nD_t\\le {D}\\end{array}}\\sum _{t=1}^n{R}_t.$ The next theorem is our first main result in this paper.",
"It derives a lower bound on the performance of Fig.",
"REF by means of a dynamic reverse-waterfilling algorithm.",
"Theorem 1 (Lower bound on (REF )) For the multi-track system in Fig.",
"REF , the minimum achievable total-rate for any “$n$ ” and any $p$ , however large, is ${\\cal R}_{\\mathop {\\mathrm {sum}}}^{\\mathop {\\mathrm {IID}},\\mathop {\\mathrm {op}},1}(D)=\\sum _{t=1}^n{R^{\\mathop {\\mathrm {op}}}_t}$ with the minimum achievable rate distortion at each time instant (per dimension) given by some $R^{\\mathop {\\mathrm {op}}}_t\\ge {R}^*_t$ such that $R_t^*=\\frac{1}{2}\\log _2\\left(\\frac{\\lambda _t}{D_{t}}\\right), $ where $\\lambda _t\\triangleq \\alpha ^2_{t-1}D_{t-1}+\\sigma ^2_{w_{t-1}}$ and $D_t$ is the distortion at each time instant evaluated based on a dynamic reverse-waterfilling algorithm operating forward in time.",
"The algorithm is as follows: $D_{t} &\\triangleq \\left\\lbrace \\begin{array}{ll} \\xi _t & \\mbox{if} \\quad \\xi _t\\le \\lambda _{t} \\\\\\lambda _{t} & \\mbox{if}\\quad \\xi _t>\\lambda _{t} \\end{array} \\right.,~\\forall {t},$ with $\\sum _{t=1}^n D_t= nD$ , and $\\xi _t={\\left\\lbrace \\begin{array}{ll}\\frac{1}{2b^2_t}\\left(\\sqrt{1+\\frac{2b^2_t}{\\theta }}-1\\right),~\\forall t\\in \\mathbb {N}_1^{n-1} \\\\\\frac{1}{2\\theta },~t=n\\end{array}\\right.",
"},$ where $\\theta >0$ is the Lagrangian multiplier tuned to obtain equality $\\sum _{t=1}^n D_t= nD$ , $b^2_t\\triangleq \\frac{\\alpha ^2_t}{\\sigma ^2_{w_t}}$ , and $D\\in (0, \\infty )$ .",
"See Appendix .",
"In the next remark, we discuss some technical observations regarding Theorem REF and draw connections with [13].",
"Remark 3 (1) The optimization problem in the derivation of Theorem REF suggests that $(A_t, \\Sigma _{W_t}, \\Delta _t, \\Lambda _t)$ commute by pairs[33] since they are all scalar matrices which in turn means that they are simultaneously diagonalizable by an orthogonal matrix [33] (in this case the orthogonal matrix is the identity matrix hence it is omitted from the characterization of the optimization problem).",
"(2) Theorem REF extends the result of [13] who found an explicit expression of the minimum total-rate $\\sum _{t=1}^nR_t^*$ for $n=3$ subject to a per-time $\\mathop {\\mathrm {MSE}}$ distortion, to a similar problem constrained by an average total-distortion that we solve using a dynamic reverse-waterfilling algorithm that allocates the rate and the distortion at each instant of time for a fixed finite time horizon.",
"Implementation of the dynamic reverse-waterfilling: It should be remarked that a way to implement the reverse-waterfilling algorithm in Theorem REF is proposed in [38].",
"A different algorithm using the bisection method (for details see, e.g., [39]) is proposed in Algorithm .",
"The method in Algorithm guarantees linear convergence with rate $\\frac{1}{2}$ .",
"On the other hand, [38] requires a specific proportionality gain factor $\\gamma \\in (0,1]$ chosen appropriately at each time instant.",
"The choice of $\\gamma $ affects the rate of convergence whereas it does not guarantee global convergence of the algorithm.",
"In Fig.",
"REF , we illustrate a numerical simulation using Algorithm by taking $a_t\\in (0,2)$ , $\\sigma ^2_{w_t}=1,$ for $t=\\lbrace 1,2,\\ldots ,200\\rbrace $ and $D=1$ .",
"Figure: Dynamic rate-distortion allocation for a time-horizon t={1,2...,200}t=\\lbrace 1,2\\ldots ,200\\rbrace for the system in Fig.",
".Dynamic reverse-waterfilling algorithm Initialize: number of time-steps $n$ ; distortion level $D$ ; error tolerance $\\epsilon $ ; nominal minimum and maximum value $\\theta ^{\\min }={0}$ and $\\theta ^{\\max }={\\frac{1}{2D}}$ ; initial variance ${\\lambda }_1=\\sigma ^2_{x_{1}}$ of the initial state $x_1$ , values $a_t$ and $\\sigma ^2_{w_t}$ of (REF ).",
"Set $\\theta =1/2D$ ; $\\text{flag}=0$ .",
"$\\text{flag}=0$ Compute $D_{t}~\\forall ~t$ as follows: $t=1:n$ Compute $\\xi _t$ according to (REF ).",
"Compute $D_t$ according to (REF ).",
"$t<n$ Compute $\\lambda _{t+1}$ according to $\\lambda _{t+1}\\triangleq \\alpha ^2_{t}D_{t}+\\sigma ^2_{w_{t}}$ .",
"$\\frac{1}{n}\\sum {D}_t-D\\ge \\epsilon $ Set $\\theta ^{\\min }=\\frac{\\theta }{n}$ .",
"Set $\\theta ^{\\max }=\\frac{\\theta }{n}$ .",
"$\\theta ^{\\max }-\\theta ^{\\min }\\ge \\frac{\\epsilon }{n}$ Compute $\\theta =\\frac{n(\\theta ^{\\min }+\\theta ^{\\max })}{2}$ .",
"$\\text{flag}\\leftarrow 1$ Output: $\\lbrace D_t:~t\\in \\mathbb {N}_1^n\\rbrace $ , $\\lbrace \\lambda _t:~t\\in \\mathbb {N}_1^n\\rbrace $ , for a given distortion level $D$ ."
],
[
"Steady-state solution of Theorem ",
"In this subsection, we study the steady-state case of the lower bound obtained in Theorem REF .",
"To do this, first, we restrict the state process of our setup to be time invariant, which means that in (REF ) the coefficients $\\alpha _{t-1}\\equiv {\\alpha },~\\forall {t}$ and $w_t\\sim {\\cal N}(0;\\sigma ^2_{w}),~\\forall {t}$ , or similarly, in (REF ) the matrix $A_{t-1}\\equiv {A}=\\mathop {\\mathrm {diag}}(\\alpha ,\\ldots ,\\alpha ),~\\forall {t}$ and $W_t\\sim {\\cal N}(0;\\Sigma _W),~\\forall {t}$ , where $\\Sigma _W=\\mathop {\\mathrm {diag}}(\\sigma ^2_{w},\\ldots ,\\sigma ^2_{w})\\succ {0}$ .",
"We also denote the steady-state average total rate and distortion as follows: $R_{\\infty }=\\limsup _{n\\longrightarrow \\infty }\\frac{1}{n}\\sum _{t=1}^nR_t,~~~~D_{\\infty }=\\limsup _{n\\longrightarrow \\infty }\\frac{1}{n}\\sum _{t=1}^nD_t.$ Steady-state Performance.",
"The minimum achievable steady-state performance of the multi-track system of Fig.",
"REF when the system is modeled by $p$ -parallel time-invariant Gauss-Markov processes (per dimension) can be cast to the following optimization problem: ${\\cal R}_{\\mathop {\\mathrm {sum}},ss}^{\\mathop {\\mathrm {IID}},\\mathop {\\mathrm {op}},1}(D)=\\min _{\\begin{array}{c}(f_t,~g_t):~t=1,\\ldots ,\\infty \\\\ D_{\\infty }\\le {D}\\end{array}}R_{\\infty }.$ The next corollary is a consequence of the lower bound derived in Theorem REF .",
"It states that the minimum achievable steady state total rate subject to steady-state total distortion constraint is equivalent to having the minimum achievable steady state total rate subject to a fixed distortion budget, i.e., $D_t=D,~\\forall {t}$ .",
"This result complements equivalent results derived in [40], [17].",
"Corollary 1 (Lower bound on (REF )) The minimum achievable steady state performance of (REF ), under a steady-state total distortion constraint $D_{\\infty }\\le {D}$ for any $p$ however larger, is bounded from below by ${\\cal R}_{\\mathop {\\mathrm {sum}},ss}^{\\mathop {\\mathrm {IID}},\\mathop {\\mathrm {op}},1}(D)\\ge {R}^*_{\\infty }$ , such that $R^*_{\\infty }=\\frac{1}{2}\\log _2\\left(\\alpha ^2+\\frac{\\sigma ^2_w}{D}\\right),$ where $R^*_{\\infty }\\triangleq \\lim _{n\\longrightarrow \\infty }\\frac{1}{n}\\sum _{t=1}^nR_t^*$ .",
"Consequently, assuming $D_t=D,~\\forall {t}$ , achieves (REF ) as $n\\longrightarrow \\infty $ .",
"To obtain our result, we first take the average total-rate, i.e., $\\frac{1}{n}\\sum _{t=1}^nR_t$ .",
"Then, we show the following inequalities: $\\frac{1}{n}\\sum _{t=1}^nR_t&\\stackrel{(a)}{\\ge }\\frac{1}{n}\\sum _{t=1}^nR_t^*\\nonumber \\\\&=\\frac{1}{n}\\sum _{t=1}^n\\frac{1}{2}\\log _2\\left(\\frac{\\lambda _t}{D_t}\\right)\\nonumber \\\\&\\stackrel{(b)}{=}\\frac{1}{n}\\sum _{t=1}^n\\left[\\frac{1}{2}\\log _2\\left(\\alpha ^2{D}_{t-1}+\\sigma ^2_w\\right)-\\frac{1}{2}\\log _2{D_t}\\right]\\nonumber \\\\&=\\frac{1}{2n}\\log _2\\left(\\frac{\\lambda _1}{D_n}\\right)+\\frac{1}{2n}\\sum _{t=1}^{n-1}\\log _2\\left(\\alpha ^2+\\frac{\\sigma ^2_w}{D_t}\\right)\\nonumber \\\\&\\stackrel{(c)}{\\ge }\\frac{1}{2n}\\log _2\\frac{\\lambda _1}{\\sum _{t=1}^nD_t}+\\frac{1}{2}\\frac{(n-1)}{n}\\log _2\\left(\\alpha ^2+\\frac{(n-1)\\sigma ^2_w}{\\sum _{t=1}^{n-1}D_t}\\right)\\nonumber \\\\&\\stackrel{(d)}{\\ge }\\frac{1}{2n}\\log _2\\left(\\frac{\\lambda _1}{nD}\\right)+\\frac{1}{2}\\frac{(n-1)}{n}\\log _2\\left(\\alpha ^2+\\frac{\\sigma ^2_w}{\\frac{nD}{n-1}}\\right),$ where $(a)$ follows from Theorem REF ; $(b)$ follows because for time-invariant processes $\\lambda _t=\\alpha ^2D_{t-1}+\\sigma ^2_{w}$ ; $(c)$ follows because in the first term $D_n\\le \\sum _{t=1}^nD_t$ and in the second term we apply Jensen's inequality [41]; $(d)$ follows because in the first term $\\sum _{t=1}^{n}D_t\\le {nD}$ and in the second term $\\sum _{t=1}^{n-1}D_t\\le \\sum _{t=1}^{n}D_t\\le {nD}$ since $D_n\\ge {0}$ .",
"We prove that ${\\cal R}_{\\mathop {\\mathrm {sum}},ss}^{\\mathop {\\mathrm {IID}},\\mathop {\\mathrm {op}},1}(D)\\ge {R}^*_{\\infty }$ where $R^*_{\\infty }$ is given by (REF ) by evaluating (REF ) in the limit $n\\longrightarrow \\infty $ and then minimizing both sides.",
"This is obtained because the first term equals to zero ($\\lambda _1, D$ are constants), in the second term $\\lim _{n\\longrightarrow \\infty }\\left(\\frac{n-1}{n}\\right)=1$ , $\\lim _{n\\longrightarrow \\infty }\\left(\\frac{n}{n-1}\\right)=1$ and then by taking the minimization in both sides.",
"This completes the proof.",
"Remark 4 (Connections to existing works) We note that the steady-state lower bound (per dimension) obtained in Corollary REF corresponds precisely to the solution of the time-invariant scalar-valued Gauss-Markov processes with per-time MSE distortion constraint derived in [23] and to the solution of stationary Gauss-Markov processes with MSE distortion constraint derived in [40], [21]."
],
[
"Upper bounds to the minimum achievable total-rate",
"In this section, we employ a sequential causal $\\mathop {\\mathrm {DPCM}}$ -based scheme using pre/post filtered $\\mathop {\\mathrm {ECDQ}}$ (for details, see, e.g., [19]) that ensures standard performance guarantees (achievable upper bounds) on the minimum achievable sum-rate ${\\cal R}_{\\mathop {\\mathrm {sum}}}^{\\mathop {\\mathrm {IID}},\\mathop {\\mathrm {op}},1}(D)=\\sum _{t=1}^nR^{\\mathop {\\mathrm {op}}}_t$ of the multi-track setup of Fig.",
"REF .",
"The reason for the choice of this quantization scheme is twofold.",
"First, it can be implemented in practice and, second, it allows to find analytical achievable bounds and approximations on finite-dimensional quantizers which generate near-Gaussian quantization noise and Gaussian quantization noise for infinite dimensional quantizers [42].",
"We first describe the sequential causal $\\mathop {\\mathrm {DPCM}}$ scheme using $\\mathop {\\mathrm {MMSE}}$ quantization for parallel time-varying Gauss-Markov processes.",
"Then, we bound the rate performance of such scheme using $\\mathop {\\mathrm {ECDQ}}$ and vector quantization followed by memoryless entropy coding.",
"This can be seen as a generalization of [13] to any finite time when the rate is allocated at each time instant.",
"Observe that because the state is modeled as a first-order Gauss-Markov process, the sequential causal coding is precisely equivalent to predictive coding (see, e.g., [12], [15]).",
"Therefore, we can immediately apply the standard sequential causal $\\mathop {\\mathrm {DPCM}}$ [18], [43] approach (with $\\mathbb {R}^p$ -valued $\\mathop {\\mathrm {MMSE}}$ quantizers) to obtain an achievable rate in our system.",
"DPCM scheme.",
"At each time instant $t$ the encoder or innovations' encoder performs the linear operation $\\widehat{X}_t=X_t-A_{t-1}Y_{t-1},$ where at $t=1$ we have $\\widehat{X}_{1}=X_1$ and also $Y_{t-1}\\triangleq {\\bf E}\\left\\lbrace X_{t-1}|S_{1,t-1}\\right\\rbrace $ , i.e., an estimate of $X_{t-1}$ given the previous quantized symbols $S_{1,t-1}$ .Note that the process $\\widehat{X}_t$ has a temporal correlation since it is the error of $X_t$ from all quantized symbols $S_{1,t-1}$ and not the infinite past of the source $X_{-\\infty ,t}=(X_{-\\infty },\\ldots ,X_t)$ .",
"Hence, $\\widehat{X}_t$ is only an estimate of the true process.",
"Then, by means of a $\\mathbb {R}^p$ -valued $\\mathop {\\mathrm {MMSE}}$ quantizer that operates at a rate (per dimension) $R_t$ , we generate the quantized reconstruction $\\widehat{Y}_t$ of the residual source $\\widehat{X}_t$ denoted by $\\widehat{Y}_t=Y_t-A_{t-1}Y_{t-1}$ .",
"Then, we send $S_t$ over the channel (the corresponding data packet to $\\widehat{Y}_t$ ).",
"At the decoder we receive $S_t$ and recover the quantized symbol $\\widehat{Y}_t$ of $\\widehat{X}_t$ .",
"Figure: DPCM \\mathop {\\mathrm {DPCM}} of parallel processes.Then, we generate the estimate $Y_t$ using the linear operation $Y_t=\\widehat{Y}_t+A_{t-1}Y_{t-1}.$ Combining both (REF ), (REF ), we obtain $X_t-Y_t=\\widehat{X}_t-\\widehat{Y}_t.$ MSE Performance.",
"From (REF ), we see that the error between $X_t$ and $Y_t$ is equal to the quantization error introduced by $\\widehat{X}_t$ and $\\widehat{Y}_t$ .",
"This also means that the $\\mathop {\\mathrm {MSE}}$ distortion (per dimension) at each instant of time satisfy $D_t=\\frac{1}{p}{\\bf E}\\lbrace ||X_t-Y_t||_2^2\\rbrace =\\frac{1}{p}{\\bf E}\\lbrace ||\\widehat{X}_t-\\widehat{Y}_t||_2^2\\rbrace .$ A pictorial view of the $\\mathop {\\mathrm {DPCM}}$ scheme is given in Fig.",
"REF .",
"The following theorem is another main result of this section.",
"Theorem 2 (Upper bound to $R_{\\mathop {\\mathrm {sum}}}^{\\mathop {\\mathrm {IID}},\\mathop {\\mathrm {op}},1}(D)$ ) Suppose that in (REF ) we apply a sequential causal $\\mathop {\\mathrm {DPCM}}$ -based $\\mathop {\\mathrm {ECDQ}}$ with a lattice quantizer.",
"Then, the minimum achievable total-rate ${\\cal R}_{\\mathop {\\mathrm {sum}}}^{\\mathop {\\mathrm {IID}},\\mathop {\\mathrm {op}},1}(D)=\\sum _{t=1}^n{R}^{\\mathop {\\mathrm {op}}}_t$ , where at each time instant ${R}^{\\mathop {\\mathrm {op}}}_t$ is upper bounded as follows: $R^{\\mathop {\\mathrm {op}}}_t\\le {R}^*_t+\\frac{1}{2}\\log _2\\left(2\\pi {e}G_p\\right)+\\frac{1}{p},~\\forall {t}, ~\\mbox{(bits/dimension)},$ where $R_t^*$ is obtained from Theorem REF , $\\frac{1}{2}\\log _2\\left(2\\pi {e}G_p\\right)$ is the divergence of the quantization noise from Gaussianity; $G_p$ is the dimensionless normalized second moment of the lattice [19] and $\\frac{1}{p}$ is the additional cost due to having prefix-free (instantaneous) coding.",
"See Appendix .",
"Next, we remark some technical comments related to Theorem REF , to better explain its novelty compared to the existing similar schemes in the literature.",
"Remark 5 (Comments on Theorem REF ) (1) The bound of Theorem REF allows the transmit rate to vary at each time instant for a finite time horizon while it achieves the $\\mathop {\\mathrm {MMSE}}$ distortion at each time step $t$ .",
"This is because our $\\mathop {\\mathrm {DPCM}}$ -based $\\mathop {\\mathrm {ECDQ}}$ scheme is constrained by total-rates that we find at each instant of time using the dynamic reverse-waterfilling algorithm of Theorem REF .",
"This loose rate-constraint is the new input of our bound compared to similar existing bounds in the literature (see, e.g., [17], [30], [12]) that assume fixed rates averaged across the time or asymptotically average total rate constraints hence restricting their transmit rate at each instant of time to be the same for any time horizon.",
"(2) Recently in [27] (see also [17]), it is pointed out that for discrete-time processes one can assume in the $\\mathop {\\mathrm {ECDQ}}$ coding scheme the clocks of the entropy encoder and the entropy decoder to be synchronized, thus, eliminating the additional rate-loss due to prefix-free coding.",
"This assumption, will give a better upper bound in Theorem REF because the term $\\frac{1}{p}$ will be removed.",
"Steady-state performance.",
"Next, we describe how to obtain an upper bound on (REF ).",
"Suppose that the system is modeled by $p$ -parallel time-invariant Gauss-Markov processes (per dimension) similar to §REF .",
"Corollary 2 (Upper bound on (REF )) Suppose that in (REF ) we apply a sequential causal $\\mathop {\\mathrm {DPCM}}$ -based $\\mathop {\\mathrm {ECDQ}}$ with a lattice quantizer assuming the system is time-invariant and that $D_t=D$ , $\\forall {t}$ .",
"Then, the minimum achievable steady-state performance ${\\cal R}_{\\mathop {\\mathrm {sum}},ss}^{\\mathop {\\mathrm {IID}},\\mathop {\\mathrm {op}},1}(D)={R}^{\\mathop {\\mathrm {op}}}_{\\infty }$ is upper bounded as follows: ${R}^{\\mathop {\\mathrm {op}}}_{\\infty }\\le {R}^*_\\infty +\\frac{1}{2}\\log _2\\left(2\\pi {e}G_p\\right)+\\frac{1}{p},~\\mbox{(bits/dimension)},$ where $R_\\infty ^*$ is given by (REF ).",
"This follows from Theorem REF and Corollary REF .",
"We note that Corollary REF is a known infinite time horizon bound derived in several paper in the literature, such as those discussed in Remark REF , (1)."
],
[
"Unfortunately, finding $G_p$ in (REF ) for good high-dimensional quantizers of possibly finite dimension is currently an open problem (although it can be approximated for any dimension using for example product lattices [34]).",
"Therefore, in what follows we propose existing computable bounds to the achievable upper bound of Theorem REF for any high-dimensional lattice quantizer.",
"Note that these bounds were derived as a consequence of the main result by Zador [34], namely, it is possible to reduce the $\\mathop {\\mathrm {MSE}}$ distortion normalized per dimension using higher-dimensional quantizers.",
"Toward this end, Zador introduced a lower bound on $G_p$ using the dimensionless normalized second moment of a $p$ -dimensional sphere, hereinafter denoted by $G(S_p)$ , for which it holds that: $G(S_p)=\\frac{1}{(p+2)\\pi }\\Gamma \\left(\\frac{p}{2}+1\\right)^{\\frac{2}{p}},$ where $\\Gamma (\\cdot )$ is the gamma function.",
"Moreover, $G_p$ and $G(S_p)$ are connected via the following inequalities: $\\frac{1}{2\\pi {e}}\\stackrel{(a)}{\\le }{G}(S_p)\\stackrel{(b)}{\\le }{G}_p\\stackrel{(c)}{\\le }\\frac{1}{12},$ where $(a), (b)$ holds with equality for $p\\longrightarrow \\infty $ ; $(c)$ holds with equality if $p=1$ .",
"Note that in [34], there is also an upper bound on $G_p$ due to Zador.",
"The bound is the following: $G_p\\le \\frac{1}{p\\pi }\\Gamma \\left(\\frac{p}{2}+1\\right)^{\\frac{2}{p}}\\Gamma \\left(1+\\frac{2}{p}\\right).$ Figure: Bounds on the minimum achievable total-rate.In Fig.",
"REF we illustrate two plots where we compute the bounds derived in Theorems REF , REF for two different scenarios.",
"In Fig.",
"REF , (a), we choose $t=\\lbrace 1,\\ldots ,20\\rbrace $ , $a_t\\in (0,1.5)$ , $\\sigma ^2_{w_t}=1$ , and $D=1$ , to illustrate the gap between the time-varying rate-distortion allocation obtained using the lower bound (REF ) and the upper bound (REF ) when the latter is approximated with the best known quantizer up to twenty four dimensions that is a lattice known as Leech lattice quantizer (for details see, e.g., [34]).",
"For this experiment the gap between the two bounds is approximately $0.126$ bits/dimension.",
"In Fig.",
"REF , (b), we perform another experiment assuming the same values for ($a_t,~\\sigma ^2_{w_t},~D$ ), whereas the quantization is performed for 500 dimensions.",
"We observe that the achievable bounds obtained via (REF ) and (REF ) are quite tight (they have a gap of approximately $0.0014$ bits/dimension) whereas the gap between the lower bound (REF ) with the achievable upper bound (REF ) approximated by (REF ) is $0.0097$ bits/dimension, and the one approximated by (REF ) is approximately $0.011$ bits/dimension.",
"Thus, compared to the first experiment where $p=24$ , the gap between the bounds on the minimum achievable rate $R^{\\mathop {\\mathrm {op}}}_t$ is considerably decreased because we increased the number of dimensions in the system.",
"Clearly, when the number of dimensions in the system increase, the gap between (REF ) and the high dimensional approximations of (REF ) will become arbitrary small.",
"The two bounds will coincide as $p\\longrightarrow \\infty $ , because then, the gap of coding noise from Gaussianity goes to zero (see, e.g., [44], [42]) and also because for $p\\longrightarrow \\infty $ , (REF ) is equal to (REF ) (see, e.g.,[34])."
],
[
"Application in NCSs",
"In this section, we demonstrate the sequential coding framework in the NCS setup of Fig.",
"REF by applying the results obtained in §.",
"Figure: Controlled system model.We first, describe each component of Fig.",
"REF .",
"Plant.",
"Consider $p$ parallel time-varying controlled Gauss-Markov processes as follows: $x_{t+1}(i)=\\alpha _{t}x_{t}(i)+\\beta _{t}u_t(i)+w_{t}(i),~i\\in \\mathbb {N}_1^p,~t\\in \\mathbb {N}_1^n,$ where $x_1(i)\\equiv {x}_1$ is given with $x_1\\sim {\\cal N}(0;\\sigma ^2_{x_1})$ , $\\forall {i}$ ; the non-random coefficients $(\\alpha _t,\\beta _t)\\in \\mathbb {R}$ are known to the system with $(\\alpha _t,\\beta _t)\\ne {0},~\\forall {t}$ ; $\\lbrace u_t(i):~i\\in \\mathbb {N}_1^p\\rbrace $ is the controlled process with $u_t(i)\\ne {u}_t(\\ell ),~$ for any $(i,{\\ell })\\in \\mathbb {N}_1^p$ ; $\\lbrace w_t(i)\\equiv {w}_t:~i\\in \\mathbb {N}_1^p\\rbrace $ is an independent Gaussian noise process such that $w_t\\sim {\\cal N}(0;\\sigma ^2_{w_{t}})$ , $\\sigma ^2_{w_t}>0$ , independent of $x_1$ , $\\forall {i}$ .",
"Again, similar to §, (REF ) can be compactly written as follows $X_{t+1}=A_{t}X_{t}+B_tU_t+W_{t},~X_1=\\mbox{given},~t\\in \\mathbb {N}_1^n,$ where $A_t=\\mathop {\\mathrm {diag}}\\left(\\alpha _{t},\\ldots ,\\alpha _t\\right)\\in \\mathbb {R}^{p\\times {p}}$ , $B_t=\\mathop {\\mathrm {diag}}\\left(\\beta _{t},\\ldots ,\\beta _t\\right)\\in \\mathbb {R}^{p\\times {p}}$ , $U_t\\in \\mathbb {R}^p$ , $W_t\\in \\mathbb {R}^p\\sim {\\cal N}(0;\\Sigma _{W_t})$ , $\\Sigma _{W_t}=\\mathop {\\mathrm {diag}}\\left(\\sigma ^2_{w_t},\\ldots ,\\sigma ^2_{w_t}\\right)\\succ {0}$ is an independent Gaussian noise process independent of $X_1$ .",
"Note that in this setup, the plant is fully observable for the observer that acts as an encoder but not for the controller due to the quantization noise (coding noise).",
"Observer/Encoder.",
"At the encoder the controlled process is collected into a frame $X_t\\in \\mathbb {R}^p$ from the plant and encoded as follows: $S_t=f_t(X_{1,t},S_{1,t-1}),$ where at $t=1$ we have $S_1=f_1(X_1)$ , and $R_t=\\frac{{\\bf E}|S_t|}{p}$ is the rate at each time instant $t$ available for transmission via the noiseless channel.",
"Note that in the design of Fig.",
"REF , the channel is noiseless, and the controller/decoder are deterministic mappings, thus, the observer/encoder implicitly has access to earlier control signals $U_{1,t-1}\\in \\mathbb {U}_{1,t-1}$ .",
"Decoder/Controller.",
"The data packet $S_t$ is received by the controller using the following reconstructed sequence: $U_t=g_t(S_{1,t}).~$ According to (REF ), when the sequence $S_{1,t}$ is available at the decoder/controller, all past control signals $U_{1,t-1}$ are completely specified.",
"Quadratic cost.",
"The cost of control (per dimension) is defined as ${\\mathop {\\mathrm {LQG}}}_{1,n}&\\triangleq \\frac{1}{p}{\\bf E}\\left\\lbrace {\\sum _{t = 1}^{n-1}\\left(X_t{Q}_tX_t + U_t{N}_tU_t\\right) + X_n{Q}_nX_n}\\right\\rbrace ,$ where $\\widetilde{Q}_t=\\mathop {\\mathrm {diag}}\\left(Q_t\\ldots ,Q_t\\right)\\succeq {0},~\\widetilde{Q}_t\\in \\mathbb {R}^{p\\times {p}}$ and $\\widetilde{N}_t=\\mathop {\\mathrm {diag}}\\left(N_t,\\ldots ,N_t\\right)\\succ {0},~\\widetilde{N}_t\\in \\mathbb {R}^{p\\times {p}}$ , are designing parameters that penalize the state variables or the control signals.",
"Performance.",
"The performance of Fig.",
"REF (per dimension) can be cast to a finite-time horizon quantized LQG control problem subject to all communication constraints as follows: ${\\Gamma ^{\\mathop {\\mathrm {IID}},\\mathop {\\mathrm {op}}}_{\\mathop {\\mathrm {sum}}}}(R)=\\min _{\\begin{array}{c}(f_t,~g_t):~t=1,\\ldots ,n\\\\ \\frac{1}{n}\\sum _{t=1}^nR_t\\le {R}\\end{array}}{\\mathop {\\mathrm {LQG}}}_{1,n}.$"
],
[
"Iterative Encoder/Controller Design",
"In general, as (REF ) suggests, the optimal performance of the system in Fig.",
"REF is achieved only when the encoder/controller pair is designed jointly.",
"This is a quite challenging task especially when the channel is noisy because information structure is non-nested in such cases (for details see, e.g., [45]).",
"There are examples, however, where the separation principle applies and the task comes much easier.",
"More precisely, the so-called certainty equivalent controller remains optimal if the estimation errors are independent of previous control commands (i.e., dual effect is absent) [46].",
"In our case, the optimal control strategy will be a certainty equivalence controller if we assume a fixed and given sequence of encoders $\\lbrace f^*_t:~t\\in \\mathbb {N}_1^n\\rbrace $ and the corresponding quantizer follows a predictive quantizer policy (similar to the $\\mathop {\\mathrm {DPCM}}$ -based $\\mathop {\\mathrm {ECDQ}}$ scheme proposed in §REF ), i.e., at each time instant it subtracts the effect of the previous control signals at the encoder and adds them at the decoder (see, e.g., [47], [48], [49]).",
"Moreover, the separation principle will also be optimal if we consider an $\\mathop {\\mathrm {MMSE}}$ estimate of the state (similar to what we have established in §), and an encoder that minimizes a distortion for state estimation at the controller.",
"The resulting separation principle is termed “weak separation principle” [48] as it relies on the fixed and given quantization policies.",
"This is different from the well-known full separation principle in the classical LQG stochastic control problem [50] where the problem separates naturally into a state estimator and a state feedback controller without any loss of optimality.",
"The previous analysis is described by a modified version of (REF ) as follows ${\\Gamma ^{\\mathop {\\mathrm {IID}},\\mathop {\\mathrm {op}}}_{\\mathop {\\mathrm {sum}}}(R)\\le \\Gamma ^{\\mathop {\\mathrm {IID}},\\mathop {\\mathrm {op}},ws}_{\\mathop {\\mathrm {sum}}}}=\\min _{\\begin{array}{c}(f^*_t,~g_t):~t=1,\\ldots ,n\\\\ \\frac{1}{n}\\sum _{t=1}^nR_t\\le {R}\\end{array}}{\\mathop {\\mathrm {LQG}}}_{1,n}.$ Next, we give the known solution of (REF ) in the form of a lemma that was first derived in [23], [48] for the more general setup of correlated vector-valued controlled Gauss-Markov processes with linear quadratic cost.",
"Lemma 4 (Weak separation principle for Fig.",
"REF ) The optimal controller that minimizes (REF ) is given by $U_t = -L_t{\\bf E}\\left\\lbrace X_t|S_{1,t}\\right\\rbrace ,$ where ${\\bf E}\\left\\lbrace X_t|S_{1,t}\\right\\rbrace $ are the fixed quantized state estimates obtained from the estimation problem in §; $\\widetilde{L}_t=\\mathop {\\mathrm {diag}}(L_t,\\ldots ,L_t)\\in \\mathbb {R}^p$ is the optimal $\\mathop {\\mathrm {LQG}}$ control (feedback) gain obtained as follows: $\\tilde{L}_t &=\\left(B^2_t\\widetilde{K}_{t+1}+\\widetilde{N}_t\\right)^{-1}B_t\\widetilde{K}_{t+1}A_t,$ and $\\widetilde{K}_t=\\mathop {\\mathrm {diag}}(K_t,\\ldots ,K_t)\\succeq {0}$ is obtained using the backward recursions: $\\widetilde{K}_t &=A_t^2\\left(\\widetilde{K}_{t+1}-\\widetilde{K}_{t+1}B^2_t(B_t^2\\widetilde{K}_{t+1}+\\widetilde{N}_t)^{-1}\\widetilde{K}_{t+1}\\right)+\\widetilde{Q}_t,$ with $\\widetilde{K}_{n+1} = 0$ .",
"Moreover, this controller achieves a minimum linear quadratic cost of $\\begin{split}{\\Gamma ^{\\mathop {\\mathrm {IID}},\\mathop {\\mathrm {op}},ws}_{\\mathop {\\mathrm {sum}}}}=&\\frac{1}{p}\\sum _{t = 1}^n\\Big \\lbrace \\mathop {\\mathrm {trace}}(\\Sigma _{W_t}\\widetilde{K}_t)+\\mathop {\\mathrm {trace}}(A_tB_t\\widetilde{L}_t \\widetilde{K}_{t+1}{\\bf E}\\lbrace ||X_t - Y_t||_2^2\\rbrace )\\Big \\rbrace ,\\end{split}$ where ${\\bf E}\\lbrace ||X_t - Y_t||_2^2\\rbrace $ is the $\\mathop {\\mathrm {MMSE}}$ distortion obtained using any quantization (coding) in the control/estimation system.",
"Before we prove our main theorem, we define the instantaneous cost of control as follows: $\\begin{split}&{\\mathop {\\mathrm {LQG}}}^{\\mathop {\\mathrm {op}}}_t\\triangleq \\frac{1}{p}\\Big \\lbrace \\mathop {\\mathrm {trace}}(\\Sigma _{W_t}\\widetilde{K}_t)+\\mathop {\\mathrm {trace}}(A_tB_t\\widetilde{L}_t \\widetilde{K}_{t+1}{\\bf E}\\lbrace ||X_t - Y_t||_2^2\\rbrace )\\Big \\rbrace ,~t\\in \\mathbb {N}_1^n.\\end{split}$ Next, we use Lemma REF to derive a lower bound on (REF ).",
"Theorem 3 (Lower bound on (REF )) For fixed coding policies, the minimum total-cost of control (per dimension) of (REF ), for any “$n$ ” and any $p$ , however large, is ${\\Gamma ^{\\mathop {\\mathrm {IID}},\\mathop {\\mathrm {op}},ws}_{\\mathop {\\mathrm {sum}}}}=\\sum _{t=1}^n{\\mathop {\\mathrm {LQG}}}^{\\mathop {\\mathrm {op}}}_{t}$ , with ${\\mathop {\\mathrm {LQG}}}^{\\mathop {\\mathrm {op}}}_{t}\\ge {\\mathop {\\mathrm {LQG}}}^*_t$ such that ${\\mathop {\\mathrm {LQG}}}_{t}^*=\\sigma ^2_{w_t}K_t + \\alpha _t\\beta _t L_t K_{t+1}D(R^*_t), $ where $D(R^*_t)$ is given by: $D(R^*_t)&\\triangleq \\left\\lbrace \\begin{array}{ll} \\frac{\\sigma ^2_{w_t}}{2^{2R^*_t}-\\alpha ^2_t}, & \\mbox{$\\forall {t}\\in \\mathbb {N}_1^{n-1}$} \\\\2^{-2R^*_n}, & \\mbox{for $t=n$} \\end{array} \\right.,$ with the pair $(D(R^*_t),R^*_t)$ given by (REF )-(REF ).",
"See Appendix .",
"In what follows, we include a technical remark related to the lower bound on the total cost-rate function of Theorem REF .",
"Remark 6 (Technical remarks on Theorem REF ) The expression of the lower bound in Theorem REF , can be reformulated for any $n$ , and any $p$ , to the equivalent expression of the total rate-cost function, denoted hereinafter by $\\sum _{t=1}^nR({\\mathop {\\mathrm {LQG}}}_t^*)$ , as follows $R({\\mathop {\\mathrm {LQG}}}_t^*)=\\frac{1}{2}\\log _2\\left(\\alpha _t^2+\\frac{\\alpha _t\\beta _tL_tK_{t+1}\\sigma ^2_{w_t}}{{\\mathop {\\mathrm {LQG}}}^*_t-\\sigma ^2_{w_t}K_t}\\right), ~t\\in \\mathbb {N}_1^{n-1},$ with $R({\\mathop {\\mathrm {LQG}}}_n^*)\\equiv {R}_n^*$ as it is independent of ${\\mathop {\\mathrm {LQG}}}_n^*$ .",
"Interestingly, one can observe that by substituting in (REF ) the per-dimension version of (REF ) we obtain ${R}({\\mathop {\\mathrm {LQG}}}_t^*)&=\\frac{1}{2}\\log _2\\left(\\alpha _t^2\\left(1+\\frac{\\frac{\\beta _t^2K^2_{t+1}\\sigma ^2_{w_t}}{\\beta _t^2K^2_{t+1}+N_t}}{\\mathop {\\mathrm {LQG}}^*_t-\\sigma ^2_{w_t}K_t}\\right)\\right),\\\\&=\\frac{1}{2}\\left[\\log _2(\\alpha _t^2)+\\log _2\\left(1+\\frac{\\frac{\\beta _t^2K^2_{t+1}\\sigma ^2_{w_t}}{\\beta _t^2K^2_{t+1}+N_t}}{\\mathop {\\mathrm {LQG}}^*_t-\\sigma ^2_{w_t}K_t}\\right)\\right].$ The bound in (REF ) extends the result of [27] from an asymptotically average total-rate cost function to the case of a total-rate cost function where at each instant of time the rate-cost function is obtained using an allocation of $\\mathop {\\mathrm {LQG}}_t^*$ obtained due to the rate-allocation of the quantized state estimation problem of Theorem REF .",
"Additionally, the expression in (REF ) reveals an interesting observation regarding the absolute minimum data rates for mean square stability of the plant (per dimension), i.e., $\\sup _t{\\bf E}\\lbrace (x_t)^2\\rbrace <\\infty $ (see, e.g., [35] for the definition) for a fixed finite time horizon.",
"In particular, (REF ) suggests that for unstable time-varying plants with arbitrary disturbances modeled as in (REF ), and provided that at each time instant the cost of control (per dimension) is with communication constraints, i.e., ${\\mathop {\\mathrm {LQG}}}_t^*>\\sigma ^2_{{w}_t}K_t$ (the derivation without communication constraints is well known as the separation principle holds without a loss and ${\\mathop {\\mathrm {LQG}}}_t^*=\\sigma ^2_{{w}_t}K_t,~\\forall {t}$ [50]), then, the minimum possible rates at each time instant $t$ , namely, ${R}({\\mathop {\\mathrm {LQG}}}_t^*)$ , cannot be lower than $\\log _2|\\alpha _t|$ , when $|\\alpha _t|>1$ .",
"This result extends known observations for time-invariant plants (see e.g., [27]) to parallel and (possibly unbounded) time-varying plants for any fixed finite time horizon.",
"Next, we use Theorem REF to find an upper bound on ${\\Gamma ^{\\mathop {\\mathrm {IID}},\\mathop {\\mathrm {op}},ws}_{\\mathop {\\mathrm {sum}}}}$ .",
"Theorem 4 (Upper bound on (REF )) Suppose that in the system of Fig.",
"REF , the fixed coding policies are obtained using the predictive coding scheme via sequential causal $\\mathop {\\mathrm {DPCM}}$ -based $\\mathop {\\mathrm {ECDQ}}$ coding scheme with an $\\mathbb {R}^p$ -valued lattice quantizer described in Theorem REF .",
"Then, ${\\Gamma ^{\\mathop {\\mathrm {IID}},\\mathop {\\mathrm {op}},ws}_{\\mathop {\\mathrm {sum}}}}=\\sum _{t=1}^n{\\mathop {\\mathrm {LQG}}}^{\\mathop {\\mathrm {op}}}_t$ for any $n$ , and any $p$ , with the instantaneous cost of control $\\lbrace \\mathop {\\mathrm {LQG}}_t:~t\\in \\mathbb {N}_1^{n-1}\\rbrace $ (per dimension) to be upper bounded as follows: ${\\mathop {\\mathrm {LQG}}}^{\\mathop {\\mathrm {op}}}_t\\le \\sigma ^2_{w_t}K_t+ \\alpha _t\\beta _t L_t K_{t+1}\\frac{4^{\\frac{1}{p}}(2\\pi {e}G_p)\\sigma ^2_{w_t}}{2^{2R^{\\mathop {\\mathrm {op}}}_t}-4^{\\frac{1}{p}}(2\\pi {e}G_p)\\alpha _t^2},$ whereas, at $t=n$ , ${\\mathop {\\mathrm {LQG}}}^{\\mathop {\\mathrm {op}}}_n=\\sigma ^2_{w_n}K_n$ and $R^{\\mathop {\\mathrm {op}}}_t$ is bounded above as in (REF ).",
"See Appendix .",
"Remark 7 (Comments on Theorem REF ) For infinitely large spatial components, i.e., $p\\longrightarrow \\infty $ , the upper bound in (REF ) approaches the lower bound in Theorem REF because $G_\\infty \\longrightarrow \\frac{1}{2\\pi {e}}$ (see e.g, [42]).",
"Moreover, one can easily obtain the equivalent inverse problem of the total rate-cost function for the upper bound in (REF ) similar to Remark REF .",
"Next, we note the main technical difference of both Theorems REF , REF compared to existing results in the literature.",
"Remark 8 (Connections to existing works) (1) Our bounds on $\\mathop {\\mathrm {LQG}}$ cost extend similar bounds derived in [17] to average total-rate constraints for any fixed finite time horizon.",
"This constraint requires the use of a dynamic reverse-waterfilling optimization algorithm (derived in Theorem REF ) to optimally assign the rates at each instant of time for the whole fixed finite time horizon.",
"In contrast, the fixed rate constraint (averaged across the time) assumed in [17] does not require a similar optimization technique because at each instant of time the transmit rate is the same.",
"Another structural difference compared to [17] is that in our bound we decouple the dependency of $D_{t-1}$ at each time instant.",
"(2) Our results also extend the steady-state bounds on $\\mathop {\\mathrm {LQG}}$ cost obtained in [30], [31], [28] to cost-rate functions constrained by total-rates obtained for any fixed finite time horizon.",
"By assumption, the rate constraint in those papers implies fixed (uniform) rates at each instant of time whereas our bounds require a rate allocation algorithm to assign optimally the rate at each time slot."
],
[
"Steady-state solution of Theorems ",
"In this subsection, we study the steady-state case of the bounds derived in Theorems REF , REF .",
"We start by making the following assumptions, i.e., (A1) we restrict the controlled process (REF ) to be time invariant, which means that $A_t\\equiv {A}=\\mathop {\\mathrm {diag}}\\left(\\alpha ,\\ldots ,\\alpha \\right)\\in \\mathbb {R}^{p\\times {p}}$ , $B_t\\equiv {B}=\\mathop {\\mathrm {diag}}\\left(\\beta ,\\ldots ,\\beta \\right)\\in \\mathbb {R}^{p\\times {p}}$ , $W_t\\in \\mathbb {R}^p\\sim {\\cal N}(0;\\Sigma _{W})$ , $\\Sigma _{W}=\\mathop {\\mathrm {diag}}\\left(\\sigma ^2_{w},\\ldots ,\\sigma ^2_{w}\\right)\\succ {0},~\\forall {t}$ ; (A2) we restrict the design parameters that penalize the control cost (REF ) to also be time invariant, i.e., $\\widetilde{Q}_t\\equiv \\mathop {\\mathrm {diag}}(Q,\\ldots ,Q)$ , $\\widetilde{N}_t\\equiv \\mathop {\\mathrm {diag}}(N, \\ldots , N)$ ; (A3) we fix $D_t\\equiv {D}$ , $\\forall {t}$ .",
"We denote the steady-state value of the total cost of control, (per dimension) as follows: ${\\mathop {\\mathrm {LQG}}}_{\\infty }=\\limsup _{n\\longrightarrow \\infty }\\frac{1}{n}\\sum _{t=1}^n{\\mathop {\\mathrm {LQG}}}_t.$ Steady-state Performance.",
"The minimum achievable steady-state performance (per dimension) of the quantized $\\mathop {\\mathrm {LQG}}$ control problem of Fig.",
"REF under the weak separation principle can be cast to the following optimization problem: $\\Gamma ^{\\mathop {\\mathrm {IID}},\\mathop {\\mathrm {op}},ws}_{\\mathop {\\mathrm {sum}},ss}=\\min _{\\begin{array}{c}(f^*_t,~g_t):~t=1,\\ldots ,\\infty \\\\R_\\infty \\le {R}\\end{array}}{\\mathop {\\mathrm {LQG}}}_{\\infty }.$ In the next two corollaries, we prove the lower and upper bounds on (REF ).",
"These bounds follow from the assumptions (A1)-(A3) and Corollaries REF , REF .",
"Corollary 3 (Lower bound on (REF )) The minimum achievable steady state performance of (REF ), under the assumptions (A1)-(A3), for any $p$ , is such that $\\Gamma ^{\\mathop {\\mathrm {IID}},\\mathop {\\mathrm {op}},ws}_{\\mathop {\\mathrm {sum}},ss}\\ge {\\mathop {\\mathrm {LQG}}}^{*}_{\\infty }$ , where ${\\mathop {\\mathrm {LQG}}}^{*}_{\\infty }=\\sigma ^2_{w}K_\\infty + \\alpha \\beta L_\\infty K_\\infty \\frac{\\sigma ^2_{w}}{2^{2R_{\\infty }^*}-\\alpha ^2},$ where ${\\mathop {\\mathrm {LQG}}}^*_{\\infty }\\triangleq \\lim _{n\\longrightarrow \\infty }\\frac{1}{n}\\sum _{t=1}^n{\\mathop {\\mathrm {LQG}}}_t^*$ , with $L_\\infty , K_\\infty $ given by (REF ) and (REF ), respectively.",
"The derivation follows from the assumptions (A1)-(A3).",
"In particular, $\\frac{1}{n}\\sum _{t=1}^n{\\mathop {\\mathrm {LQG}}}_t\\stackrel{(i)}{\\ge }&\\frac{1}{n}\\sum _{t=1}^n{\\mathop {\\mathrm {LQG}}}^{*}_t\\nonumber \\\\\\stackrel{(ii)}{=}&\\frac{1}{n}\\sum _{t=1}^n\\left(\\sigma ^2_{w}K_\\infty + \\alpha \\beta L_\\infty K_\\infty {D}\\right),$ where $(i)$ follows from Theorem REF ; $(ii)$ follows from the assumptions (A1)-(A3).",
"In particular, by imposing the assumptions (A1), (A2), in Lemma REF we obtain that the steady-steady optimal $\\mathop {\\mathrm {LQG}}$ control (feedback) gain (per dimension) becomes: ${L}_\\infty =\\frac{\\alpha \\beta {K_\\infty }}{\\beta ^2{K}_{\\infty }+{N}},$ where $K_\\infty $ is the positive solution of the quadratic equation: $\\beta ^2{K}_\\infty +\\left((1-\\alpha ^2)N-\\beta ^2{Q}\\right)K_\\infty -QN =0,$ given by the formula ${K}_\\infty =\\frac{1}{2\\beta ^2}\\left(\\sqrt{\\bar{f}^2+4\\beta ^2QN}-\\bar{f}\\right),$ with $\\bar{f}=(1-\\alpha ^2)N-\\beta ^2{Q}$ .",
"Finally by assumption (A3), we obtain from Corollary REF that $D\\equiv {D}(R_t^*)=\\frac{\\sigma ^2_{w}}{2^{2R_{\\infty }^*}-\\alpha ^2}$ , $\\forall {t}$ .",
"The result follows once we let in (REF ) $n\\longrightarrow \\infty $ .",
"This completes the derivation.",
"Corollary 4 (Upper bound on (REF )) The minimum achievable steady state performance of (REF ), under the assumptions (A1)-(A3), for any $p$ , is upper bounded as follows $\\Gamma ^{\\mathop {\\mathrm {IID}},\\mathop {\\mathrm {op}},ws}_{\\mathop {\\mathrm {sum}},ss}\\le \\sigma ^2_{w}K_\\infty + \\alpha \\beta L_\\infty K_\\infty \\frac{4^{\\frac{1}{p}}(2\\pi {e}G_p)\\sigma ^2_{w}}{2^{2R^{\\mathop {\\mathrm {op}}}_{\\infty }}-4^{\\frac{1}{p}}(2\\pi {e}G_p)\\alpha ^2},$ where $R^{\\mathop {\\mathrm {op}}}_\\infty $ is upper bounded by (REF ) and $K_\\infty $ , $L_\\infty $ are given by (REF ) and (REF ), respectively.",
"We omit the derivation because it is similar to the one obtained for the lower bound.",
"In contrast to the lower bound, here we make use of Theorem REF and Corollary REF .",
"Note that for Theorems REF , REF we can remark the following.",
"Remark 9 (Comments on Corollaries REF , REF ) The lower bound of Corollary REF (per dimension) is precisely the bound obtained by Tatikonda et al.",
"in [23] (see also [51]) for scalar time-invariant Gauss-Markov processes.",
"The upper bound of Corollary REF (per dimension) is similar to the upper bounds derived in [30], [31], [17].",
"It is also similar to the upper bound obtained in [28] albeit their space-filling term is obtained differently."
],
[
"Discussion and Open Questions",
"In this section, we discuss certain open problems that can be solved based on this work and discuss certain observations that stem from our main results."
],
[
"Dynamic reverse-waterfilling algorithm for multivariate Gaussian processes",
"Further to the technical observation raised in Remark REF , (1), it seems that the simultaneous diagonalization of $(A_t, \\Sigma _{W_t}, \\Delta _t, \\Lambda _t)$ by an orthogonal matrix is sufficient in order to extend the derivation of a dynamic reverse-waterfilling algorithm to the more general case of multivariate time-varying Gauss-Markov processes.",
"Our claim is further supported by the fact that for time-invariant multidimensional Gauss-Markov processes simultaneous diagonalization is shown to be sufficient for the derivation of a reverse-waterfilling algorithm in [52]."
],
[
"Non-Gaussian processes",
"Although not addressed in this paper, the non-asymptotic lower bounds derived in Theorems REF , REF can be extended to linear models driven by independent non-Gaussian noise processes ${w_t}\\sim (0;\\sigma ^2_{w_t})$ using entropy power inequalities [27]."
],
[
"Packet drops with instantaneous ACK",
"It would be interesting to extend our setup to the more practical scenario of communication links prone to packet drops.",
"In such case one needs to take into account the various packet erasure models (e.g., $\\mathop {\\mathrm {IID}}$ or Markov models) to study their impact on the non-asymptotic bounds derived for the two application examples of this paper.",
"Existing results for uniform (fixed) rate allocation are already studied in [17]."
],
[
"Conclusion",
"We revisited the sequential coding of correlated sources with independent spatial components to use it in the derivation of non-asymptotic, finite dimensional lower and upper bounds for two application examples in stochastic systems.",
"Our application examples included a parallel time-varying quantized state-estimation problem subject to a total $\\mathop {\\mathrm {MSE}}$ distortion constraint and a parallel time-varying quantized $\\mathop {\\mathrm {LQG}}$ closed-loop control system with linear quadratic cost.",
"For the latter example, its lower bound revealed the minimum possible rates for mean square stability of the plant at each instant of time when the system operates for a fixed finite time horizon."
],
[
"Proof of Theorem ",
"Since the source is modeled as a time-varying first-order Gauss-Markov process, then from () we obtain: $\\begin{split}&{\\cal R}_{{\\mathop {\\mathrm {sum}}}}^{\\mathop {\\mathrm {IID}},\\mathop {\\mathrm {op}},1}(D)\\ge {\\cal R}_{{\\mathop {\\mathrm {sum}}}}^{\\mathop {\\mathrm {IID}},1}(D),\\\\&=\\min _{\\begin{array}{c}\\frac{1}{n}\\frac{1}{p}\\sum _{t=1}^n{\\bf E}\\left\\lbrace ||X_t-Y_t||_2^2\\right\\rbrace \\le {D},\\\\Y_1\\leftrightarrow {X_{1}}\\leftrightarrow {X}_{2,n},\\\\~Y_t\\leftrightarrow (X_{t},Y_{1,t-1})\\leftrightarrow (X_{1,t-1},{X}_{t+1,n})\\end{array}}\\frac{1}{p}\\sum _{t=1}^nI(X_{t};Y_t|Y_{1,t-1})\\end{split}.$ It is trivial to see that the $\\mathop {\\mathrm {RHS}}$ term in (REF ) corresponds precisely to the sequential or $\\mathop {\\mathrm {NRDF}}$ obtained for parallel Gauss-Markov processes with a total $\\mathop {\\mathrm {MSE}}$ distortion constraint which is a simple generalization of the scalar-valued problem that has already been studied in [38].",
"Therefore, using the analysis of [38] we can obtain: $&{\\cal R}_{\\mathop {\\mathrm {sum}}}^{\\mathop {\\mathrm {IID}},1}(D)\\nonumber \\\\&\\stackrel{(a)}{=}\\min _{\\mbox{contraint in (\\ref {exam:eq:sumrate1})}}\\frac{1}{p}\\sum _{t=1}^n\\left\\lbrace h(X_t|Y_{1,t-1})-h(X_t|Y_{1,t})\\right\\rbrace \\nonumber \\\\&\\stackrel{(b)}{=}\\frac{1}{p}\\min _{\\begin{array}{c}\\Delta _t\\succeq {0},~t\\in \\mathbb {N}_1^n\\\\ \\frac{1}{n}\\frac{1}{p}\\sum _{t=1}^n\\mathop {\\mathrm {trace}}{(\\Delta _t)}\\le {D}\\end{array}} \\sum _{t=1}^n \\max \\left[0,\\frac{1}{2}\\log _2\\left(\\frac{|\\Lambda _t|}{|\\Delta _t|}\\right)\\right],\\nonumber \\\\&=\\min _{\\begin{array}{c}D_t\\ge {0},~t\\in \\mathbb {N}_1^n\\\\\\frac{1}{n}\\sum _{t=1}^nD_t\\le {D}\\end{array}} \\sum _{t=1}^n \\max \\left[0,\\frac{1}{2}\\log _2\\left(\\frac{\\lambda _t}{D_t}\\right)\\right],$ where $(a)$ follows by definition; $(b)$ follows from the fact that $h(X_t|Y_{1,t-1})=\\frac{1}{2}\\log _2(2\\pi {e})^p|\\Lambda _t|$ where $\\Lambda _t=\\mathop {\\mathrm {diag}}\\left(\\lambda _t,\\ldots \\lambda _t\\right)\\in \\mathbb {R}^{p\\times {p}}$ with $\\lambda _t=\\alpha ^2_{t-1}D_{t-1}+\\sigma ^2_{w_{t-1}}$ , and that $h(X_t|Y_{1,t})=\\frac{1}{2}\\log _2(2\\pi {e})^p|\\Delta _t|$ where $\\Delta _t=\\mathop {\\mathrm {diag}}\\left(D_t,\\ldots ,D_t\\right)\\in \\mathbb {R}^{p\\times {p}}$ for $D \\in [0, \\infty )$ .",
"The optimization problem of (REF ) is already solved in [38] and is given by (REF )-(REF )."
],
[
"Proof of Theorem ",
"In this proof we bound the rate performance of the $\\mathop {\\mathrm {DPCM}}$ scheme described in §REF at each time instant for any fixed finite time $n$ using an $\\mathop {\\mathrm {ECDQ}}$ scheme that utilizes the forward Gaussian test-channel realization that achieves the lower bound of Theorem REF .",
"In this scheme in fact we replace the quantization noise with an additive Gaussian noise with the same second moments.See e.g., [53] or [19] and the references therein.",
"First note that the Gaussian test-channel linear realization of the lower bound in Theorem REF is known to be[38] $Y_t=H_tX_t+(I_p-H_t)A_{t-1}Y_{t-1}+H^{\\frac{1}{2}}V_t,~V_t\\sim {\\cal N}(0;\\Delta _t),$ where $H_t\\triangleq {I_p-\\Delta _t\\Lambda ^{-1}_t}\\succeq {0}$ , $\\Delta _t\\triangleq \\mathop {\\mathrm {diag}}(D_t,\\ldots ,D_t)\\succ {0}$ , $\\Lambda _t=\\mathop {\\mathrm {diag}}\\left(\\lambda _t,\\ldots \\lambda _t\\right)\\succ {0}$ .",
"Pre/Post Filtered ECDQ with multiplicative factors for parallel sources.",
"[53] First, we consider a $p-$ dimensional lattice quantizer $Q_p$ [34] such that ${\\bf E}\\lbrace Z_tZ_t=\\Sigma _{V^c_t}, ~\\Sigma _{V_t^c}\\succ {0},\\nonumber $ where $Z_t\\in \\mathbb {R}^p$ is a random dither vector generated both at the encoder and the decoder independent of the input signals $\\widehat{X}_t$ and the previous realizations of the dither, uniformly distributed over the basic Voronoi cell of the $p-$ dimensional lattice quantizer $Q_p$ such that $V_t^c\\sim {Unif}(0;\\Sigma _{V_t^c})$ .",
"At the encoder the lattice quantizer quantize $H_{t}^{\\frac{1}{2}}\\widehat{X}_t+Z_t$ , that is, $Q_p(H_{t}^{\\frac{1}{2}}\\widehat{X}_t+Z_t)$ ,where $\\widehat{X}_t$ is given by (REF ).",
"Then, the encoder applies entropy coding to the output of the quantizer and transmits the output of the entropy coder.",
"At the decoder the coded bits are received and the output of the quantizer is reconstructed, i.e., $Q_p(H_{t}^{\\frac{1}{2}}\\widehat{X}_t+Z_t)$ .",
"Then, it generates an estimate by subtracting ${Z_t}$ from the quantizer's output and multiplies the result by $\\Phi _t$ as follows: $Y_t=\\Phi _t(Q_p(H_{t}^{\\frac{1}{2}}\\widehat{X}_t+Z_t)-Z_t),$ where $\\Phi _t=H_{t}^{\\frac{1}{2}}$ .",
"The coding rate at each instant of time of the conditional entropy of the $\\mathop {\\mathrm {MSE}}$ quantizer is given by[53] $H(Q_p|Z_t)&={I}(H^{\\frac{1}{2}}\\widehat{X}_t;H\\widehat{X}_t+H^{\\frac{1}{2}}V_t^c)\\nonumber \\\\&\\stackrel{(a)}{=}{I}(H^{\\frac{1}{2}}\\widehat{X}_t;H\\widehat{X}_t+H^{\\frac{1}{2}}V_t)+{\\cal D}(V^c_t||V_t)\\nonumber \\\\&\\qquad -{\\cal D}(H\\widehat{X}_t+H^{\\frac{1}{2}}V^c_t||H\\widehat{X}_t+H^{\\frac{1}{2}}V_t)\\nonumber \\\\&\\stackrel{(b)}{\\le }{I}(H^{\\frac{1}{2}}\\widehat{X}_t;H\\widehat{X}_t+H^{\\frac{1}{2}}V_t)+{\\cal D}(V^c_t||V_t)\\nonumber \\\\&\\stackrel{(c)}{\\le }{I}(H^{\\frac{1}{2}}\\widehat{X}_t;H\\widehat{X}_t+H^{\\frac{1}{2}}V_t)+\\frac{p}{2}\\log (2\\pi {e}G_p)\\nonumber \\\\&\\stackrel{(d)}{=}{I}(X_t;Y_t|Y_{1,t-1})+\\frac{p}{2}\\log (2\\pi {e}G_p)$ where $V_t^c\\in \\mathbb {R}^p$ is the (uniform) coding noise in the $\\mathop {\\mathrm {ECDQ}}$ scheme and $V_t$ is the corresponding Gaussian counterpart; $(a)$ follows because the two random vectors $V_t^c, V_t$ have the same second moments hence we can use the identity ${\\cal D}(x||x^{\\prime })=h(x^{\\prime })-h(x)$ ; $(b)$ follows because ${\\cal D}(H\\widehat{X}_t+H^{\\frac{1}{2}}V_t^c||H\\widehat{X}_t+H^{\\frac{1}{2}}V_t)\\ge {0}$ ; $(c)$ follows because the divergence of the coding noise from Gaussianity is less than or equal to $\\frac{p}{2}\\log (2\\pi {e}G_p)$ [42] where $G_p$ is the dimensionless normalized second moment of the lattice [19]; $(d)$ follows from data processing properties, i.e., $I(X_t;Y_t|Y_{1,t-1})\\stackrel{(\\ast )}{=}I(X_t;Y_t|Y_{t-1})\\stackrel{(\\ast \\ast )}{=}I(\\widehat{X}_t;\\widehat{Y}_t)\\stackrel{(\\ast \\ast \\ast )}{=}{I}(H^{\\frac{1}{2}}\\widehat{X}_t;H\\widehat{X}_t+H^{\\frac{1}{2}}V_t)$ where $(\\ast )$ follows from the realization of (REF ), $(\\ast \\ast )$ follows from the fact that $\\widehat{X}_t$ and $\\widehat{Y}_t$ (obtained by (REF )) are independent of $Y_{t-1}$ , and $(\\ast \\ast \\ast )$ follows from (REF ), (REF ) and the fact that $H$ is an invertible operation.",
"Since we assume joint (memoryless) entropy coding with lattice quantizers, then, the total coding rate per dimension is obtained as follows[41] $\\sum _{t=1}^n\\frac{{\\bf E}|S_t|}{p}&\\le \\frac{1}{p}\\sum _{t=1}^n\\left({H}(Q_p|Z_t)+1\\right)\\nonumber \\\\&\\stackrel{(e)}{\\le }\\frac{1}{p}\\sum _{t=1}^n{I}({X}_t;{Y}_t|Y_{1,t-1})+\\frac{n}{2}\\log (2\\pi {e}G_p)+\\frac{n}{p}\\nonumber \\\\&\\stackrel{(f)}{=}\\frac{1}{2p}\\sum _{t=1}^n\\log _2\\frac{|\\Lambda _t|}{|\\Delta _t|}+\\frac{n}{2}\\log (2\\pi {e}G_p)+\\frac{n}{p},$ where $(e)$ follows from (REF ); $(f)$ follows from the derivation of Theorem REF .",
"The derivation is complete once we minimize both sides of inequality in (REF ) with the appropriate constraint sets."
],
[
"Proof of Theorem ",
"Note that from (REF ) we obtain $\\begin{split}&{\\Gamma ^{\\mathop {\\mathrm {IID}},\\mathop {\\mathrm {op}},ws}}=\\sum _{t=1}^n{\\mathop {\\mathrm {LQG}}}^{\\mathop {\\mathrm {op}}}_t\\\\&=\\frac{1}{p}\\sum _{t=1}^n\\Big \\lbrace \\mathop {\\mathrm {trace}}(\\Sigma _{W_t}\\widetilde{K}_t)\\\\&+\\mathop {\\mathrm {trace}}(A_tB_t\\widetilde{L}_t \\widetilde{K}_{t+1}{\\bf E}\\lbrace ||X_t - Y_t||_2^2\\rbrace )\\Big \\rbrace \\\\&\\stackrel{(a)}{\\ge }\\frac{1}{p}\\sum _{t=1}^n\\Big \\lbrace \\mathop {\\mathrm {trace}}(\\Sigma _{W_t}\\widetilde{K}_t)\\\\&+\\mathop {\\mathrm {trace}}(A_tB_t\\widetilde{L}_t \\widetilde{K}_{t+1}{\\bf E}\\lbrace ||X_t - {\\bf E}\\lbrace X_t|S_{1,t}\\rbrace ||_2^2\\rbrace )\\Big \\rbrace \\\\&\\stackrel{(b)}{\\ge }\\frac{1}{p}\\sum _{t=1}^n\\Bigg \\lbrace \\mathop {\\mathrm {trace}}(\\Sigma _{W_t}\\widetilde{K}_t)+\\mathop {\\mathrm {trace}}\\Bigg (A_tB_t\\widetilde{L}_t \\widetilde{K}_{t+1}\\\\&{\\bf E}_{\\bar{S}_{1,t-1}}\\left\\lbrace \\frac{1}{2\\pi {e}}2^{\\frac{2}{p}h(X_t|S_{1,t-1}=\\bar{S}_{1,t-1})}\\right\\rbrace 2^{-2R^*_t}\\Bigg )\\Bigg \\rbrace \\\\&\\stackrel{(c)}{\\ge }\\frac{1}{p}\\sum _{t=1}^n\\Bigg \\lbrace \\mathop {\\mathrm {trace}}(\\Sigma _{W_t}\\widetilde{K}_t)+\\mathop {\\mathrm {trace}}\\Bigg (A_tB_t\\widetilde{L}_t \\widetilde{K}_{t+1}\\\\&\\left\\lbrace \\frac{1}{2\\pi {e}}2^{\\frac{2}{p}h(X_t|S_{1,t-1})}2^{-2R^*_t}\\right\\rbrace \\Bigg )\\Bigg \\rbrace ,\\\\&\\stackrel{(d)}{\\ge }\\sum _{t=1}^n\\Big \\lbrace \\sigma ^2_{w_t}K_t+\\alpha _t\\beta _tL_tK_{t+1}D(R_t^*)\\Big \\rbrace \\triangleq \\sum _{t=1}^n{\\mathop {\\mathrm {LQG}}}^*_t,\\end{split}$ where $(a)$ follows from the fact that $Y_t$ is $\\mathbb {S}_{1,t}-$ measurable and the $\\mathop {\\mathrm {MMSE}}$ is obtained for $Y_t={\\bf E}\\lbrace X_t|S_{1,t}\\rbrace $ ; $(b)$ follows from the fact that ${\\bf E}\\lbrace ||X_t - {\\bf E}\\lbrace X_t|S_{1,t}\\rbrace ||_2^2\\rbrace )={\\bf E}_{\\bar{S}_{1,t-1}}\\left\\lbrace {\\bf E}\\lbrace ||X_t - {\\bf E}\\lbrace X_t|S_{1,t}\\rbrace ||_2^2|S_{1,t-1}=\\bar{S}_{1,t-1}\\rbrace \\right\\rbrace $ , where ${\\bf E}_{\\bar{S}_{1,t}}\\lbrace \\cdot \\rbrace $ is the expectation with respect to some vector $\\bar{S}_{1,t-1}$ that is distributed similarly to $S_{1,t-1}$ , also from the $\\mathop {\\mathrm {MSE}}$ inequality in [41] and finally from the fact that $R^*_t\\ge {0}$ , where $R^*_t=\\frac{1}{p}\\left\\lbrace h^*(X_t|Y_{1,t-1})-h^*(X_t|Y_{1,t})\\right\\rbrace $ (see the derivation of Theorem REF , (1)) with $h^*(X_t|Y_{1,t-1})$ , $h^*(X_t|Y_{1,t})$ being the minimized values in (REF ); $(c)$ follows from Jensen's inequality [41], i.e., ${\\bf E}_{\\bar{S}_{1,t-1}}\\left\\lbrace 2^{\\frac{2}{p}h(X_t|S_{1,t-1}=\\bar{S}_{1,t-1})}\\right\\rbrace \\ge {2}^{\\frac{2}{p}h(X_t|S_{1,t-1})}$ ; $(d)$ follows from the fact that $\\lbrace h(X_t|S_{1,t-1})=h(A_{t-1}X_{t-1}+B_{t-1}U_{t-1}+W_{t-1}|S_{1,t-1}):~t\\in \\mathbb {N}_2^n\\rbrace $ is completely specified from the independent Gaussian noise process $\\lbrace W_{t-1}:~t\\in \\mathbb {N}_2^n\\rbrace $ because $\\lbrace U_{t-1}=g_{t}(S_{1,t-1}):~t\\in \\mathbb {N}_2^n\\rbrace $ (see (REF )) are constants conditioned on $S_{1,t-1}$ .",
"Therefore, $h(X_t|S_{1,t-1})$ is conditionally Gaussian thus equivalent to $h(X_t|Y_{1,t-1})$ .",
"This further means that $\\frac{1}{2\\pi {e}}{2}^{\\frac{2}{p}h(X_t|Y_{1,t-1})}2^{-2R^*_t}\\ge \\frac{1}{2\\pi {e}}{2}^{\\frac{2}{p}h^*(X_t|Y_{1,t-1})}2^{-2R^*_t}\\stackrel{(\\star )}{=}\\frac{1}{2\\pi {e}}{2}^{\\frac{1}{p}\\log _2(2\\pi {e})^p|\\Delta _t^*|}\\stackrel{(\\star \\star )}{=}\\min \\lbrace D_t\\rbrace \\equiv {D}(R_t^*)$ , where $(\\star )$ follows because $h^*(X_t|Y_{1,t})=\\frac{1}{2}\\log _2(2\\pi {e})^p|\\Delta _t^*|$ and $(\\star \\star )$ follows because $\\Delta _t^*=\\mathop {\\mathrm {diag}}(\\min \\lbrace D_t\\rbrace ,\\ldots ,\\min \\lbrace D_t\\rbrace )$ .",
"It remains to find $D(R_t^*)$ at each time instant in (REF ).",
"To do so, we reformulate the solution of the dynamic reverse-waterfilling solution in (REF ) as follows: $&nR\\equiv {\\cal R}_{\\mathop {\\mathrm {sum}}}^{\\mathop {\\mathrm {IID}},1}=\\sum _{t=1}^n{R^*_t}\\equiv \\frac{1}{2}\\sum _{t=1}^n \\log _2\\left(\\frac{\\lambda _t}{D_t}\\right)\\nonumber \\\\&=\\frac{1}{2}\\left\\lbrace \\underbrace{{0}{\\log _2(\\lambda _1)}}_{\\text{initial~step}}+\\sum _{t=1}^{n-1}\\log _2\\left(\\alpha ^2_t+\\frac{\\sigma ^2_{w_t}}{D_t}\\right)-\\underbrace{\\log _2 D_n}_{\\text{final~step}}\\right\\rbrace .$ From (REF ) we observe that at each time instant, the rate $R^*_t$ is a function of only one distortion $D_t$ since we have now decoupled the correlation with $D_{t-1}$ .",
"Moreover, we can assume without loss of generality, that the initial step is zero because it is independent of $D_0$ .",
"Thus, from (REF ), we can find at each time instant a $D_t\\in (0,\\infty )$ such that the rate is $R_t^*\\in [{0},\\infty )$ .",
"Since the rate distortion problem is equivalent to the distortion rate problem (see, e.g., [41]) we can immediately compute the total-distortion rate function, denoted by $D^{\\mathop {\\mathrm {IID}},1}_{\\mathop {\\mathrm {sum}}}(R)$ , as follows: $D^{\\mathop {\\mathrm {IID}},1}_{\\mathop {\\mathrm {sum}}}(R)\\triangleq \\sum _{t=1}^nD(R_t^*)=\\sum _{t=1}^{n-1} \\frac{\\sigma ^2_{w_t}}{2^{2R^*_t}-\\alpha _t^2}+2^{-2R^*_n}.$ Substituting $D(R_t^*)$ at each time instant in (REF ) the result follows.",
"This completes the proof."
],
[
"Proof of Theorem ",
"Note that from Lemma REF , (REF ), we obtain: $\\begin{split}&{\\Gamma ^{\\mathop {\\mathrm {IID}},\\mathop {\\mathrm {op}},ws}_{\\mathop {\\mathrm {sum}}}}=\\frac{1}{p}\\sum _{t = 1}^n\\Big \\lbrace \\mathop {\\mathrm {trace}}(\\Sigma _{W_t}\\widetilde{K}_t)\\\\&+\\mathop {\\mathrm {trace}}(A_tB_t\\widetilde{L}_t \\widetilde{K}_{t+1}{\\bf E}\\lbrace ||X_t - Y_t||_2^2\\rbrace )\\Big \\rbrace \\\\&=\\frac{1}{p}\\sum _{t = 1}^n\\Big \\lbrace \\mathop {\\mathrm {trace}}(\\Sigma _{W_t}\\widetilde{K}_t)\\\\&+\\mathop {\\mathrm {trace}}(A_tB_t\\widetilde{L}_t \\widetilde{K}_{t+1}D(R^{\\mathop {\\mathrm {op}}}_t))\\Big \\rbrace \\\\&\\stackrel{(a)}{\\le }\\sum _{t=1}^{n-1}\\left\\lbrace \\sigma ^2_{w_t}K_t+ \\alpha _t\\beta _t L_t K_{t+1}\\frac{4^{\\frac{1}{p}}(2\\pi {e}G_p)\\sigma ^2_{w_t}}{2^{2R^{\\mathop {\\mathrm {op}}}_t}-4^{\\frac{1}{p}}(2\\pi {e}G_p)\\alpha _t^2}\\right\\rbrace \\\\&+\\sigma ^2_{w_n}K_n,\\end{split}$ where $(a)$ is obtained in two steps.",
"As a first step, expand the inequality obtained in Theorem REF , (REF ) for the time horizon $n$ as follows $R^{\\mathop {\\mathrm {op}}}_1+\\ldots +R^{\\mathop {\\mathrm {op}}}_n\\le {R}_1^*+\\ldots +R_n^*+c$ where $c=\\frac{n}{p}\\log _2(2\\pi {e}G_p)+\\frac{n}{p}$ .",
"As a second step, we reformulate $\\lbrace R_t^*:~t\\in \\mathbb {N}_1^n\\rbrace $ similar to (REF ) (in the derivation of Theorem REF ) so that we decouple the dependence on $D_{t-1}$ at each time step.",
"Finally, for each $R^{\\mathop {\\mathrm {op}}}_t,~t=1,2\\ldots ,n,$ we solve the resulting inequality with respect to $D(R_t^{\\mathop {\\mathrm {op}}})$ which gives $\\begin{split}&D(R^{\\mathop {\\mathrm {op}}}_t)\\le \\frac{4^{\\frac{1}{p}}(2\\pi {e}G_p)\\sigma ^2_{w_{t}}}{2^{2R^{\\mathop {\\mathrm {op}}}_{t}}-4^{\\frac{1}{p}}(2\\pi {e}G_p)\\alpha _{t}^2},~t\\in \\mathbb {N}_1^{n-1}.\\end{split}$ Observe that the last step $t=n$ is not needed because in (REF ) we have ${K}_{n+1}=0$ .",
"This completes the proof."
],
[
"Acknowledgement",
"The authors wish to thank the Associate Editor and the anonymous reviewers for their valuable comments and suggestions.",
"They are also indebted to Prof. T. Charalambous for reading the paper and proposing the idea of bisection method for Algorithm .",
"They are also grateful to Prof. J. Østergaard for fruitful discussions on technical issues of the paper."
]
] | 1906.04217 |
[
[
"Schwinger-Dyson and loop equations for a product of square Ginibre\n random matrices"
],
[
"Abstract In this paper, we study the product of two complex Ginibre matrices and the loop equations satisfied by their resolvents (i.e.",
"the Stieltjes transform of the correlation functions).",
"We obtain using Schwinger-Dyson equation (SDE) techniques the general loop equations satisfied by the resolvents.",
"In order to deal with the product structure of the random matrix of interest, we consider SDEs involving the integral of higher derivatives.",
"One of the advantage of this technique is that it bypasses the reformulation of the problem in terms of singular values.",
"As a byproduct of this study we obtain the large $N$ limit of the Stieltjes transform of the $2$-point correlation function, as well as the first correction to the Stieltjes transform of the density, giving us access to corrections to the smoothed density.",
"In order to pave the way for the establishment of a topological recursion formula we also study the geometry of the corresponding spectral curve.",
"This paper also contains explicit results for different resolvents and their corrections."
],
[
"Introduction",
"The study of random matrices in mathematics can be traced back to the work of Hurwitz on the invariant measure for the matrix groups $U(N)$ and $SO(N)$ [32], [19].",
"In multivariate statistics another stream of random matrix theory was initiated with the work of Wishart [54] on estimating the covariance matrices of multivariate statistics when the number of variables is large.",
"In theoretical physics Wigner [53] used random matrices to model energy spectrum of Hamiltonians of highly excited states of heavy nuclei.",
"The works of physicists [46] on the large $N$ limit of $U(N)$ gauge theory provided yet another application to random matrices (and their generalized version often referred to as matrix models).",
"Since then random matrix theory and matrix models have been found useful in an overwhelming number of contemporary fields, for example communication engineering [50], the analysis of algorithms [47], and deep learning [43].",
"Many tools have been developed to understand the properties of different models and ensembles.",
"One of these tools is called loop equations, and has led to the now well-known Chekhov-Eynard-Orantin topological recursion formula [23], [13], [12].",
"In the realm of random matrix theory this formula allows for the systematic computation of correlation functions of random matrices, as series in $1/N$ .",
"However some random matrix ensembles are, in the existing literature, still out of the scope of these loop equations.",
"These are product ensembles, that is they are random matrices constructed out of a product of several random matrices.",
"In this paper we describe the loop equations for such a product ensemble, specifically considering the case of a random matrix constructed out of the product of two complex Ginibre matrices.",
"Such an ensemble was for instance considered in [10], with applications to the study of financial data, while a closely related product ensemble with applications to low energy QCD, was studied in [42] (see also the text book treatment [28]), allowing for insight into the poorly understood regime of non-zero baryon chemical potential.",
"More generally the product ensembles are found to have many applications.",
"Some of these applications are described in the thesis [34].",
"Among those, one finds applications to telecommunication problems where product ensembles provide a model of communication channels where the signal has to pass through different media [40].",
"One also finds applications to the study of spin chains with disorder [18], quantum transport [7], quantum information and random graph states [16], [17].",
"The product ensembles also relate to the study of neural networks.",
"Indeed information about the asymptotic behavior of such ensembles allows one to draw results about stability of gradient in a deep neural network with randomly initialized layers [31].",
"These product ensembles are also of interest for the study of the stability of large dynamical systems [8], [33].",
"As a consequence, finding mathematical and technical tools for investigating the properties of these ensembles can enable progress in these fields of study.",
"Yet another problem of importance is the one of Muttalib-Borodin ensembles.",
"These ensembles were first defined as invariant ensembles, via their eigenvalue probability density function (PDF) [41], and latter realized in terms of ensembles of random matrices with independent entries [14], [30].",
"Their joint PDF is proportional to, $\\prod _{l=1}^N e^{- V(\\lambda _l)} \\prod _{1 \\le i < j \\le N} (\\lambda _i - \\lambda _j)(\\lambda _i^\\theta - \\lambda _j^\\theta ),$ where $\\theta > 0$ is a parameter and $V(\\lambda _l)$ can be interpreted as a confining potential.",
"For general potential $V$ and $\\theta =2$ , this model relates to the $\\mathcal {O}(\\mathfrak {n})$ matrix model with $\\mathfrak {n}=-2$ , see [5], and it also relates to a particular model of disordered bosons [37].",
"A key structural interest in the Muttalib-Borodin ensembles is that they are biorthogonal ensembles.",
"That is they admit a family of biorthogonal polynomials and their correlation functions can be expressed in determinantal form, with a kernel that can be expressed in terms of the biorthogonal polynomials; see [11].",
"Although it is not immediately obvious, the singular values for the product of $M$ complex Ginibre matrices also give rise to biorthogonal ensembles [3], [36].",
"Moreover, in the asymptotic regime of large separation, the PDF for the squared singular values reduces to (REF ) with $\\theta = 1/M$ , and $V$ having the leading form $V(x) = - M x^{1/M}$ [27].",
"One attractive feature of both the Muttalib-Borodin ensemble, and the squared singular values of products of complex Ginibre matrices, is that in the global density limit the moments of spectral density are given by the Fuss-Catalan family of combinatorial numbers; see [44], [30].",
"Another is the special role played by particular special functions of the Meijer-G and Wright Bessel function class.",
"Underlying these special functions is a linear differential equation of degree $M +1$ .",
"Less well understood is the nonlinear differential system implied by the correlation kernel based on these special functions.",
"These are relevant to the study of gap probabilities; see [51], [38].",
"Other questions about products of random matrices have been investigated for instance in [21].",
"In this work, the authors are concerned about the behavior of traces of general words of Ginibre matrices.",
"In particular they show that the limiting square singular values distribution is a Fuss-Catalan distribution for any words.",
"In the work [20], the authors study the traces of the general words in an alphabet of random matrices constructed out of the marginals of a random tensor.",
"Using combinatorial techniques it is possible to show freeness of some marginals or to describe entirely the free cumulants when there is no freeness of the different marginals in the limit.",
"One interesting aspect is that using these products of marginals it is possible to find distribution interpolating between the square of a Marc̆enko-Pastur law and the free multiplicative square of a Marc̆enko-Pastur law.",
"However there is in general little technical tools to describe the lower order in $N$ observables of product ensembles.",
"Indeed free probability provides us with some useful techniques (free additive and multiplicative convolution), but those are restricted to the large $N$ limit, and comes in handy only for the study of the large $N$ density or the behavior of the large $N$ limit of the moments (with some extension to the fluctuations of the linear statistics via [15]).",
"In this paper we focus on describing the loop equations for the random matrix $S_2=X_1X_1^{\\dagger }X_2^{\\dagger }X_2$ , where $X_1, X_2$ are square complex Ginibre matrices.",
"In order to obtain these loop equations we start with Schwinger-Dyson identities and use them to obtain relations between moments, later translated in terms of equations on the resolvents of $S_2$ .",
"These equations on the resolvents are the loop equations.",
"One of the new features of the method presented here is that the starting point Schwinger-Dyson identities involve higher order derivatives.",
"This allows us to obtain relations between moments of the matrix $S_2$ only without having to deal with mixed quantities.",
"Thanks to the combinatorial interpretation of the moments of the matrix $S_2$ (that we also shortly describe), we show that the (connected) resolvents possess a $1/N$ expansion, which is the unique additional ingredient we need to be able to solve the loop equations recursively.",
"Using this data we illustrate the use of the obtained loop equations by computing the large $N$ limit of the resolvent $W_{0,1}(x)$ , thus recovering known results relating to the generating function of the moments.",
"We also compute $W_{0,2}(x_1,x_2)$ (that is the Stieltjes transform of the 2-point correlation function) and show that it takes the expected universal form once expressed in the correct variables, thus relating to the Bergmann kernel on the sphere.",
"We give explicit results for $W_{1,1}(x), W_{2,1}(x)$ (first and second correction to the large $N$ limit of the resolvent), $W_{1,2}(x_1,x_2)$ (first correction to $W_{0,2}(x_1,x_2)$ ), as well as $W_{0,3}(x_1,x_2,x_3)$ .",
"One interesting aspect of the obtained loop equations are their structural properties, that seem to generalize in a very natural way the usual bilinear loop equations for random matrices or matrix models.",
"In particular, the family of loop equations we obtain for this product of matrices are trilinear in the resolvents $W_{g,n}$ .",
"This is at the root of the appearance of the double ramification point of $W_{0,1}(x)$ and we expect that a topological recursion formula similar to the one obtained in [6] applies.",
"Moreover they contain generalizations of the derivative difference term usually appearing in the bilinear setting, as well as derivatives of first and second order.",
"Motivated by these interesting structural properties, we use the explicit computations to explore the analytical properties of the $W_{g,n}$ (or rather their analytic continuation on the associated spectral curve).",
"These explorations give further hint that there is a topological recursion formula to compute them systematically.",
"We expect that a similar technique allows to describe the loop equations for the product of $p\\ge 2$ rectangular Ginibre matrices $S_p=X_1X_2\\ldots X_p(X_1X_2\\ldots X_p)^{\\dagger }$ ; we leave this study, as well as the one of a topological recursion formula, to further works.",
"Note that, as a byproduct, we also expect that this technique applies to the interesting matrix models introduced in [1], [2] to generate hypergeometric Hurwitz numbers."
],
[
"Organisation of the paper.",
"The paper is organized as follows.",
"In section , we use the Wishart case (that is the case of one Ginibre matrix) as a pedagogical example.",
"It is used to sketch the combinatorial arguments allowing to show the existence of the $1/N$ expansion and to illustrate the Schwinger-Dyson equation technique in a simpler context.",
"The reader already accustomed to Schwinger-Dyson equations obtained using the matrix elements variables and knowledgeable on the associated combinatorics may consider skipping this section.",
"In section , we describe the heart of this paper, that is the derivation of the Schwinger-Dyson equations and loop equations for a product matrix of the form $S_2=X_1X_1^{\\dagger }X_2^{\\dagger }X_2$ .",
"The loop equations take the form of a family of equations on the resolvents, that is the Stieltjes transforms (denoted $W_n(x_1,\\ldots ,x_n)$ ) of the $n$ -point correlation functions.",
"We present the results step by step to make the method transparent to the reader and the first few special cases that are the loop equations for $W_1(x)$ , $W_2(x_1,x_2)$ and $W_3(x_1,x_2,x_3)$ are presented in details.",
"This section ends with the main result, that is the loop equations satisfied by any $W_{g,n}(x_1,\\ldots ,x_n)$ as shown on equation (REF ), where $W_{g,n}(x_1,\\ldots ,x_n)$ is the coefficient of order $g$ of the $1/N$ expansion of $W_n(x_1,\\ldots ,x_n)$ .",
"In section , we take on a geometrical point of view in order to compute the $W_{g,n}$ more effectively from the loop equations.",
"We describe in details the spectral curve geometry associated to the problem.",
"We compute after a change of variables, $W_{0,2}(x_1,x_2)$ , $W_{1,1}(x)$ , $W_{2,1}(x)$ , $W_{1,2}(x_1,x_2)$ and $W_{0,3}(x_1,x_2,x_3)$ (see equations (REF ), (REF ), (), (), (REF )).",
"We use these explicit computations to explore the analytic properties of the loop equations.",
"These properties are expected to be of importance to establish a topological recursion formula allowing to systematically compute every $W_{g,n}$ ."
],
[
"Acknowledgments",
"Stephane Dartois would like to thank Valentin Bonzom, Alexandr Garbali, Jesper Ipsen and Paul Zinn-Justin for useful discussions and technical help related to this work as well as for references.",
"This work was supported by the Australian Research Council grant DP170102028."
],
[
"One matrix case, Wishart ensemble",
"In this section, we illustrate the problem that is our interest in this paper on a simpler case, that is the (trivial) product of one matrix.",
"This is the case of a Wishart matrix.",
"We first recall the combinatorial representation of moments of a Wishart ensemble matrix.",
"We then show how we can compute the average resolvent of a Wishart matrix using the Schwinger-Dyson equation method.",
"It is only in the next section that we consider the case of the product of two Ginibre matrices.",
"Thus the technically knowledgeable reader can skip this section and start reading section ."
],
[
"Random Wishart matrices",
"In this paper we always consider square matrices.",
"In the Wishart matrices case it corresponds to setting the asymptotic size ratio parameter $c$ to 1.",
"Let $X\\in \\mathcal {M}_{N\\times N}(\\mathbb {C})$ be a Ginibre random matrix.",
"More concretely, $X$ is a random matrix whose entries are i.i.d.",
"complex Gaussian with zero mean, or more formally, the entries $X_{i,j}$ are distributed according to the density $\\frac{N}{2i\\pi }e^{-N\\vert X_{i,j}\\vert ^2}\\mathrm {d}\\bar{X}_{i,j}\\mathrm {d}X_{i,j}.$ In particular we denote, $\\mathrm {d}X^{\\dagger }\\mathrm {d}X=\\prod _{i,j}\\mathrm {d}\\bar{X}_{i,j}\\mathrm {d}X_{i,j},$ so that $X$ has the distribution $\\mathrm {d}\\mu (X)=\\frac{N^{N^2}}{(2i\\pi )^{N^2}}e^{-N\\mathrm {Tr}(XX^{\\dagger })}\\mathrm {d}X^{\\dagger }\\mathrm {d}X.$ A (complex) Wishart random matrix is the random variable defined as the product $S_1=XX^{\\dagger }$ .",
"Combinatorics of moments.",
"The moments $m_k$ of order $k$ of a Wishart random matrix are defined as $m_k=\\mathbb {E}\\left(\\mathrm {Tr}(S_1^k)\\right).$ Further, for any sequence of positive integers $k_1,\\ldots ,k_n$ we can define moments $m_{k_1,\\ldots ,k_n}$ of order $k_1,\\ldots ,k_n$ .",
"Similarly to the moments of order $k$ they are defined as the expectation of products of traces of powers of $S_1$ $m_{k_1,\\ldots ,k_n}=\\mathbb {E}\\left(\\prod _{i=1}^n\\mathrm {Tr}(S_1^{k_i})\\right).$ As is for instance explained in [20], the moments of order $k$ can be computed as a sum over labeled bicolored combinatorial maps $\\mathcal {M}$ with one black vertex.",
"This combinatorial representation of moments implies that the moments have a $1/N$ expansion.",
"That is $m_k=\\sum _{g\\ge 0}N^{1-2g}m_k^{[g]},$ where $m_k^{[g]}$ are the coefficients of this expansion.",
"This is a crucial point that allows one to solve the loop equations recursively.",
"Note also that this expansion is finite, that is here $g<k/2$ .",
"Let us be a bit more explicit on this point.",
"We recall the definition of labeled bicolored combinatorial maps with possibly more than one black vertex.",
"Definition 1 A labeled bicolored combinatorial map is a triplet $\\mathcal {M}=(E,\\sigma _{\\bullet },\\sigma _{\\circ })$ where, $E$ is the set of edges of $\\mathcal {M}$ $\\sigma _{\\bullet },\\sigma _{\\circ }$ are permutations on $E$ $\\mathcal {M}$ is said to be connected if and only if the group $\\langle \\sigma _{\\bullet }, \\sigma _{\\circ }\\rangle $ acts transitively on $E$ .",
"The cycles of $\\sigma _{\\circ }$ are called white vertices, the cycles of $\\sigma _{\\bullet }$ are called black vertices, and the cycles of $\\sigma _{\\bullet }\\sigma _{\\circ }$ are called faces.",
"Combinatorial maps can be represented graphically [20], [24] as they encode embeddings of graphs on surfaces.",
"We give a few examples in Fig.",
"REF .",
"Figure: Left: Map of genus 1 contributing to the computation of m 7 m_7.",
"Center: Connected map of genus 0 contributing to the computation of c 4,5 c_{4,5} and also to m 4,5 m_{4,5}.",
"Left: Disconnected map with two genus 0 components.",
"Contribute to the computation of m 2,2 m_{2,2}.We define the set of combinatorial maps $\\mathbb {M}_p=\\lbrace \\mathcal {M}=(E,\\sigma _{\\bullet },\\sigma _{\\circ })\\mid E=\\lbrace 1,\\ldots ,p\\rbrace , \\sigma _{\\bullet }=\\gamma =(123\\ldots p)\\rbrace $ .",
"One shows, using Wick-Isserlis theorem [52], [35], that the moments of order $k$ can be written as a sum over combinatorial maps $\\mathcal {M}\\in \\mathbb {M}_p$ (see [20] for details) $m_k=\\sum _{\\mathcal {M}\\in \\mathbb {M}_k}N^{V_{\\circ }(\\mathcal {M})-k+F(\\mathcal {M})},$ where $V_{\\circ }(\\mathcal {M})$ is the number of white vertices of $\\mathcal {M}$ and $F(\\mathcal {M})$ is the number of faces of $\\mathcal {M}$ .",
"Using the fact that $V_{\\bullet }+V_{\\circ }(\\mathcal {M})-k+F(\\mathcal {M})=2-2g(\\mathcal {M})$ , where $g(\\mathcal {M})$ is the genus of the combinatorial map (that is the genus of the surface in which the corresponding graph embedds), one can show equation (REF ).",
"Remark 1 Note that elements of $\\mathbb {M}_p$ are necessarily connected as $\\gamma $ acts transitively of $\\lbrace 1,\\ldots ,p\\rbrace $ .",
"We now define the relevant set of maps for studying the moments of order $k_1,\\ldots , k_n$ .",
"In this case we denote $p=\\sum _{i=1}^n k_i$ , $E=\\lbrace 1,\\ldots ,p\\rbrace $ and $\\gamma _{k_1,\\ldots , k_n}=(12\\ldots k_1)(k_1+1\\ldots k_2)\\ldots (k_{n-1}+1\\ldots k_n)$ $\\mathbb {M}_{k_1,\\ldots ,k_n}=\\lbrace \\mathcal {M}=(E,\\sigma _{\\bullet },\\sigma _{\\circ })\\mid \\sigma _{\\bullet }=\\gamma _{k_1,\\ldots , k_n}\\rbrace .$ The maps in $\\mathbb {M}_{k_1,\\ldots ,k_n}$ are possibly non-connected as $\\gamma _{k_1,\\ldots , k_n}$ does not act transitively on the set of edges.",
"Consequently we define the corresponding set of connected maps $\\mathbb {M}_{k_1,\\ldots ,k_n}^c=\\lbrace \\mathcal {M}=(E,\\sigma _{\\bullet },\\sigma _{\\circ })\\mid \\sigma _{\\bullet }=\\gamma _{k_1,\\ldots , k_n}, \\langle \\sigma _{\\bullet },\\sigma _{\\circ }\\rangle \\textrm { acts transitively on } E\\rbrace .$ We state without proofThe proof is very similar to the one black vertex case, already appearing in [20].",
"that $m_{k_1,\\ldots ,k_n}=\\sum _{\\mathcal {M}\\in \\mathbb {M}_{k_1,\\ldots ,k_n}} N^{V_{\\circ }(\\mathcal {M})-p+F(\\mathcal {M})},$ where $p=\\sum _i k_i$ .",
"We can define the associated cumulants $c_{k_1,\\ldots ,k_n}$ of the moments, through their relation to moments $m_{k_1,\\ldots ,k_n}=\\sum _{K\\vdash \\lbrace k_1,\\ldots ,k_n\\rbrace }\\prod _{\\kappa _i\\in K}c_{\\kappa _i}.$ This relation is just the moment-cumulant relation for the family of random variables $\\bigl \\lbrace R_{k_i}:=\\mathrm {Tr}(S_1^{k_i})\\bigr \\rbrace $ .",
"These cumulants can be expressed as sums over connected combinatorial maps $c_{k_1,\\ldots ,k_n}=\\sum _{\\mathcal {M}\\in \\mathbb {M}_{k_1,\\ldots ,k_n}^c} N^{V_{\\circ }(\\mathcal {M})-p+F(\\mathcal {M})}.$ Thanks to the connected condition, this sum is a polynomial in $1/N$ as long as $n>1$ .",
"That is to say we have $c_{k_1,\\ldots ,k_n}=\\sum _{g\\ge 0}N^{2-n-2g}c_{k_1,\\ldots ,k_n}^{[g]}.$ This last equation is shown starting from (REF ) and again using $V_{\\bullet }+V_{\\circ }(\\mathcal {M})-k+F(\\mathcal {M})=2-2g(\\mathcal {M})$ with $V_{\\bullet }=n$ .",
"Large $N$ limit of moments of a Wishart matrix.",
"Using (REF ), one can study the large $N$ limit of the moments of order $k$ of a Wishart matrix, that is one can compute the limit $\\lim _{N\\rightarrow \\infty } \\frac{1}{N} m_k=m^{[0]}_k.$ This limit is given by the number of planar, labeled, bicolored combinatorial maps with one black vertex and $k$ edges.",
"The number of such maps is given by the Catalan numberNote that one obtains Catalan numbers when the ratio parameter is set to $c=1$ , however for general values of $c$ one obtains the Narayana statistics on trees, that is polynomials in $c$ whose coefficients are Narayana numbers [22].",
"$C_k$ so that $m^{[0]}_k=C_k=\\frac{1}{k+1}\\binom{2k}{k}$ .",
"This allows to compute the large $N$ limit $W_{0,1}(x)$ of the moment generating function of the Wishart matrix $W_{0,1}(x):=\\lim _{N\\rightarrow \\infty }\\frac{1}{N} \\mathbb {E}\\left( \\mathrm {Tr}\\left((x-W)^{-1} \\right)\\right)=\\sum _{p\\ge 0}\\frac{m^{[0]}_p}{x^{p+1}} =\\frac{x-\\sqrt{x^2-4x}}{2x}.$ This last quantity is the Stieltjes transform of the limiting eigenvalues density of the Wishart matrix.",
"The knowledge of $W_{0,1}(x)$ allows in principleIn this specific case one can recover explicitly the limiting eigenvalue density via the inverse transformation.",
"However in general it can be more tedious to compute the inverse transform.",
"In the cases where the equation determining $W_{0,1}$ is an algebraic equation, one can deduce a system of polynomial equations on two quantities $u(x), v(x)$ , one of them being (proportional to) the large $N$ limit of the eigenvalue density $\\rho _{0,1}(x)$ .",
"We illustrate this fact in the later Remarks REF , REF .",
"to recover the limiting eigenvalues density via the inverse transformation.",
"Schwinger-Dyson equation method.",
"In this part we use an alternative method to compute $W_{0,1}(x)$ .",
"We use the Wishart case as a pedagogical example.",
"The Schwinger-Dyson equation method relies on the use of the simple identity $\\sum _{a,b=1}^N\\int \\frac{N^{N^2}}{(2i\\pi )^{N^2}}\\mathrm {d}X^{\\dagger }\\mathrm {d}X \\partial _{X^{\\dagger }_{ab}}\\left((X^{\\dagger }S_1^{k})_{ab}e^{-N\\mathrm {Tr}(XX^{\\dagger })}\\right)=0,$ after computing the derivatives explicitly we obtain the following set of relations between moments $\\sum _{\\begin{array}{c}p_1,p_2\\ge 0 \\\\ p_1+p_2=k\\end{array}}m_{p_1,p_2}-Nm_{k+1}=0.$ In order to continue this computation we define the $n$ -points resolvents $\\overline{W}_n(x_1,\\ldots ,x_n)$ and their connected counterpart $W_n(x_1,\\ldots ,x_n)$ $\\overline{W}_n(x_1,\\ldots ,x_n)&:=\\mathbb {E}\\left(\\prod _{i=1}^n\\mathrm {Tr}\\left((x_i-S_1)^{-1}\\right)\\right)=\\sum _{p_1,\\ldots ,p_n\\ge 0}\\frac{m_{p_1,\\ldots ,p_n}}{x_1^{p_1+1}\\ldots x_n^{p_n+1}} \\\\W_n(x_1,\\ldots ,x_n)&=\\sum _{p_1,\\ldots ,p_n\\ge 0}\\frac{c_{p_1,\\ldots ,p_n}}{x_1^{p_1+1}\\ldots x_n^{p_n+1}}.$ Note that we will often name the $n$ -points resolvents and their connected counterpart simply resolvents, unless the context makes it unclear which object we are discussing.",
"$W_{0,1}(x)$ is (up to normalization) the large $N$ limit of $W_1(x)$ .",
"We have the relation $\\overline{W}_n(x_1,\\ldots ,x_n)=\\sum _{K\\vdash \\lbrace 1,\\ldots ,n\\rbrace }\\prod _{K_i\\in K}W_{\\mid K_i\\mid }(x_{K_i}),$ where we used the notation $x_{K_i}=\\lbrace x_j\\rbrace _{j\\in K_i}$ .",
"The above relation is inherited from the moment-cumulant relation of equation (REF ).",
"Remark 2 Note that $\\overline{W}_1(x)=W_1(x)$ .",
"With these definitions in mind, one considers the equality $\\sum _{k\\ge 0}\\frac{1}{x^{k+1}}\\left(\\sum _{\\begin{array}{c}p_1,p_2\\ge 0 \\\\ p_1+p_2=k\\end{array}}m_{p_1,p_2}-Nm_{k+1}\\right)=0,$ leading after some rewriting to $\\overline{W}_2(x,x)-NW_1(x)+N^2/x=0,$ or only in terms of the connected resolvents $W_1(x)^2+W_2(x,x)-NW_1(x)+N^2/x=0.$ The (connected) resolvents inherit a $1/N$ expansion from the expansion of the cumulants, $W_n(x_1,x_2,\\ldots ,x_n)=\\sum _{g\\ge 0}N^{2-2g-n}W_{g,n}(x_1,x_2,\\ldots ,x_n)$ and thus we have $W_1(x)=\\sum _{g\\ge 0} N^{1-2g}W_{g,1}(x),\\quad W_2(x,x)=\\sum _{g\\ge 0}N^{-2g}W_{g,2}(x,x).$ In the large $N$ limit equation (REF ) reduces to an equation on $W_{0,1}(x)$ , $xW_{0,1}(x)^2-xW_{0,1}(x)+1=0.$ From which we select the solution which is analytic at infinity thus recovering expression (REF ).",
"Remark 3 From this last equation we can obtain a polynomial equation on $\\rho _{0,1}(x)$ , that is the corresponding limiting eigenvalue density.",
"To this aim, one introduces the two following operators acting on functions, $&\\delta f(x)=\\lim _{\\epsilon \\rightarrow 0^+}f(x+i\\epsilon )-f(x-i\\epsilon )\\\\&sf(x)=\\lim _{\\epsilon \\rightarrow 0^+}f(x+i\\epsilon )+f(x-i\\epsilon ).$ We have the following polarization property, that is for two functions $f_1, f_2$ , we have $&\\delta (f_1f_2)(x)=\\frac{1}{2}(\\delta f_1(x)sf_2(x)+sf_1(x)\\delta f_2(x))\\\\&s(f_1f_2)(x)=\\frac{1}{2}(\\delta f_1(x)\\delta f_2(x)+sf_1(x)sf_2(x))$ Starting from equation (REF ) one deduces the two equalities $&\\delta (xW_{0,1}(x)^2-xW_{0,1}(x)+1)=0\\\\&s(xW_{0,1}(x)^2-xW_{0,1}(x)+1)=0.$ After using the polarization formula, these equations boil down to the system on $u(x):=sW_{0,1}(x)$ and $v(x):=\\delta W_{0,1}(x)$ $&xu(x)-x=0\\\\&\\frac{x}{2}(u(x)^2+v(x)^2)-xu(x)+2=0.$ This in turn leads to $\\rho _{0,1}(x)=\\frac{1}{2i\\pi }v(x)=\\frac{1}{2\\pi }\\sqrt{\\frac{x-4}{x}}$ , where we choose the solution $v(x)$ that leads to a positive and normalized density."
],
[
"Loop equations for the product of two Ginibre matrices",
"In this section we consider the problem of computing $W_{0,1}(x)$ , $W_{0,2}(x_1,x_2)$ and $W_{1,1}(x)$ for a matrix $S_2=X_1X_1^{\\dagger }X_2^{\\dagger }X_2$ with $X_1, X_2$ two random $N\\times N$ complex matrices with normal entries of mean zero.",
"We compute these quantities by exclusive use of Schwinger-Dyson equation techniques.",
"More generally, we obtain the general equations satisfied by any $W_{g,n}$ for $(g,n)\\ge (0,1)$ .",
"In the first subsection, we briefly explain the combinatorics underlying the computation of the moments of the matrix $S_2$ that justifies the existence of a $1/N$ expansion for the $W_{g,n}$ .",
"In the second subsection we study in details the corresponding Schwinger-Dyson equations and obtain the loop equations satisfied by $W_{0,1}(x)$ , $W_{0,2}(x_1,x_2)$ and $W_{1,1}(x)$ in this context.",
"We show in particular that the loop equation satisfied by $W_{0,1}(x)$ is an algebraic equation of degree 3 in $W_{0,1}$ .",
"Finally we describe the loop equations satisfied by any $W_{g,n}$ ."
],
[
"Combinatorics of the moments of $S_2$ and existence of {{formula:f0c9f15c-fd28-4881-a5d5-d41234d7ad4b}} expansion",
"We describe here the combinatorics of the moments of the matrix $S_2$ .",
"This is a crucial point as this underlying combinatorics allows us to show that the cumulants of the random variables $\\left\\lbrace \\mathrm {Tr}(S_2^{i})\\right\\rbrace _{i=0}^{\\infty }$ have a $1/N$ expansion.",
"In the subsequent developments, we keep the same notation for the moments $m_k$ , $m_{k_1,\\ldots , k_n}$ but it should be clear that in this section and the following, the moments we consider are the moments of the matrix $S_2$ , and that is so, in both the one trace case, and the multiple traces case.",
"We have $m_k=\\mathbb {E}\\left(\\mathrm {Tr}(S_2^k)\\right), \\quad m_{k_1,\\ldots ,k_n}=\\mathbb {E}\\left(\\prod _{i=1}^n\\mathrm {Tr}(S_2^{k_i})\\right),$ where the expectation is taken with respect to the density $\\mathrm {d}\\mu (X_1,X_2)=\\left(\\frac{N^{N^2}}{(2i\\pi )^{N^2}}\\right)^2e^{-N\\mathrm {Tr}(X_1X_1^{\\dagger })}e^{-N\\mathrm {Tr}(X_2X_2^{\\dagger })}\\mathrm {d}X_1^{\\dagger }\\mathrm {d}X_1\\mathrm {d}X_2^{\\dagger }\\mathrm {d}X_2.$ By using the Wick-Isserlis theorem, it is possible to give a combinatorial interpretation to the moments of $S_2$ (see for instance [20]).",
"The moments $m_k$ of $S_2$ write as a sum over combinatorial maps with one black vertex, $2k$ edges of two different types, type I and type II, such that there are $k$ edges of type I and $k$ edges of type II.",
"Moreover the type of the edge alternates when going around the black vertex.",
"Finally the white vertices can only be incident to edges of one given type.",
"See Fig.",
"REF for examples.",
"Figure: Left: Example of a map with two types of edge contributing to the computation of m 4 m_4.",
"Right: Example of a map with two types of edge contributing to the computation of m 2,1 m_{2,1} and c 2,1 c_{2,1}.We denote the set made of these maps by $\\mathbb {M}_{2k}(2)$ .",
"In terms of permutations, these maps are such that $\\sigma _{\\bullet }=(12\\ldots 2k)$ and the action of $\\sigma _{\\circ }$ on the set of edges $E=\\lbrace 1,2,3,4,\\ldots ,2k\\rbrace $ factorizes over the odd and even subsets $E_o=\\lbrace 1,3,5,\\ldots , 2k-1\\rbrace , E_e=\\lbrace 2,4,6,\\ldots ,2k\\rbrace $ .",
"More formally we have the decomposition $\\sum _{\\mathcal {M}\\in \\mathbb {M}_{2k}(2)}N^{V_{\\circ }(\\mathcal {M})-2k+ F(\\mathcal {M})}.$ Similarly, for moments of order $k_1,\\ldots ,k_n$ , we have the set of maps $\\mathbb {M}_{2k_1, 2k_2, \\ldots , 2k_n}(2)$ , such that there are $n$ black vertices with degree distribution $2k_1, 2k_2, \\ldots , 2k_n$ and a total of $p=2\\sum _i k_i$ edges.",
"Types of edge alternate around each black vertex, and white vertices can only be incident to edges of the same type see Fig.",
"REF for examples.",
"We then have the decomposition $m_{k_1,\\ldots , k_n}=\\sum _{\\mathcal {M}\\in \\mathbb {M}_{2k_1, 2k_2, \\ldots , 2k_n}(2)} N^{V_{\\circ }(\\mathcal {M})-p+F(\\mathcal {M})}.$ Similarly we can express the cumulants $c_{k_1,\\ldots , k_n}$ for the family of random variables $\\left\\lbrace \\mathrm {Tr}(S_2^{i})\\right\\rbrace _{i=0}^{\\infty }$ as a sum over the set of connected maps $\\mathbb {M}^c_{2k_1, 2k_2, \\ldots , 2k_n}(2)$ $c_{k_1,\\ldots , k_n}=\\sum _{\\mathcal {M}\\in \\mathbb {M}^c_{2k_1, 2k_2, \\ldots , 2k_n}(2)} N^{V_{\\circ }(\\mathcal {M})-p+F(\\mathcal {M})}.$ The connected condition ensures that the $c_{k_1,\\ldots , k_n}$ have a $1/N$ expansion for $n\\ge 1$ .",
"This $1/N$ expansion as well as the definition of $c_{k_1,\\ldots , k_n}$ as the cumulants of the family $\\left\\lbrace \\mathrm {Tr}(S_2^{i})\\right\\rbrace _{i=0}^{\\infty }$ ensure that the resolvents for the matrix $S_2$ have the same structural properties than the resolvents of the Wishart matrix in equations (REF ), (REF ), that is we also have for the matrix $S_2$ $&\\overline{W}_n(x_1,\\ldots ,x_n)=\\sum _{K\\vdash \\lbrace 1,\\ldots ,n\\rbrace }\\prod _{K_i\\in K}W_{\\mid K_i\\mid }(x_{K_i}),\\\\&W_n(x_1,x_2,\\ldots ,x_n)=\\sum _{g\\ge 0}N^{2-2g-n}W_{g,n}(x_1,x_2,\\ldots ,x_n).$"
],
[
"Equation on $W_{1}$ and {{formula:90866b43-9b49-4a58-8b2f-cb146b01af8e}}",
"We now want to write Schwinger-Dyson equations for the moments of the matrix $S_2$ in order to obtain the loop equations for the resolvents.",
"We start with the set of identities $&\\sum _{a,b=1}^N\\int \\mathrm {d}X_1\\mathrm {d}X_1^{\\dagger } \\mathrm {d}X_2\\mathrm {d}X_2^{\\dagger }\\frac{\\partial }{\\partial X_{1,ab}^{\\dagger }}\\left(\\bigl [X_1^{\\dagger }X_2^{\\dagger }X_2 S_2^{k}\\bigr ]_{ab} e^{-N\\mathrm {Tr}(X_1X_1^{\\dagger })}e^{-N\\mathrm {Tr}(X_2X_2^{\\dagger })}\\right) =0 [1] \\\\&\\sum _{a,b=1}^N\\int \\mathrm {d}X_1\\mathrm {d}X_1^{\\dagger } \\mathrm {d}X_2\\mathrm {d}X_2^{\\dagger }\\frac{\\partial }{\\partial X_{2,ab}^{\\dagger }}\\left(\\bigl [S_2^{k}X_1X_1^{\\dagger }X_2^{\\dagger }\\bigr ]_{ab} e^{-N\\mathrm {Tr}(X_1X_1^{\\dagger })}e^{-N\\mathrm {Tr}(X_2X_2^{\\dagger })}\\right) =0.$ After evaluating explicitly the action of the derivatives, we obtain relations, $&\\sum _{\\begin{array}{c}p_1+p_2=k \\\\ p_1,p_2\\ge 0\\end{array}} \\mathbb {E}\\left( \\mathrm {Tr}(S_2^{p_1})\\mathrm {Tr}(S_2^{p_2}X_2^{\\dagger }X_2)\\right)-N\\mathbb {E}\\left( \\mathrm {Tr}(S_2^{k+1})\\right)=0\\\\&\\sum _{\\begin{array}{c}p_1+p_2=k \\\\ p_1,p_2\\ge 0\\end{array}} \\mathbb {E}\\left( \\mathrm {Tr}(S_2^{p_1}X_1X_1^{\\dagger })\\mathrm {Tr}(S_2^{p_2})\\right)-N\\mathbb {E}\\left( \\mathrm {Tr}(S_2^{k+1})\\right)=0,$ where for both equation, the first term comes from the evaluation of the derivative on the monomial, while the second term comes from the evaluation of the derivative on the exponential factor.",
"Note however that these equations contain mixed terms of the form $\\mathbb {E}\\left( \\mathrm {Tr}(S_2^{p_1})\\mathrm {Tr}(S_2^{p_2}X_2^*X_2)\\right)$ and $\\mathbb {E}\\left( \\mathrm {Tr}(S_2^{p_1}X_1X_1^*)\\mathrm {Tr}(S_2^{p_2})\\right)$ that cannot be expressed in terms of the moments of $S_2$ .",
"Thus these two equations do not close on the set of moments of $S_2$ .",
"In order to obtain a set of relations that closes over the set of moments of $S_2$ , we consider another identity involving higher derivatives.",
"This is, $\\int \\mathrm {d}X_1\\mathrm {d}X_1^{\\dagger } \\mathrm {d}X_2\\mathrm {d}X_2^{\\dagger }\\frac{\\partial }{\\partial X_{1,ab}^{\\dagger }}\\frac{\\partial }{\\partial X_{2,bc}^{\\dagger }}\\left( \\bigl [X_1^{\\dagger }X_2^{\\dagger }X_2 S_2^{k}X_1X_1^{\\dagger }X_2^{\\dagger }\\bigr ]_{ac}e^{-N\\mathrm {Tr}(X_1X_1^{\\dagger })}e^{-N\\mathrm {Tr}(X_2X_2^{\\dagger })}\\right)=0,$ where we sum over repeated indices.",
"After some additional algebra to evaluate the action of both derivative operators, one gets relations between moments and additional mixed quantities $\\sum _{\\begin{array}{c}p_1+p_2+p_3 =k+1 \\\\ p_1,p_2,p_3 \\ge 0\\end{array}} \\mathbb {E}\\bigl (\\mathrm {Tr}(S_2^{p_1})\\mathrm {Tr}(S_2^{p_2}) \\mathrm {Tr}(S_2^{p_2})\\bigr )+\\frac{(k+1)(k+2)}{2}\\mathbb {E}\\bigl (\\mathrm {Tr}(S_2^{k+1})\\bigr ) \\\\-N\\sum _{\\begin{array}{c}p_1+p_2=k+1 \\\\ p_1,p_2\\ge 0\\end{array}} \\Bigl [ \\mathbb {E}\\bigl ( \\mathrm {Tr}(S_2^{p_1})\\mathrm {Tr}(S_2^{p_2}X_2^*X_2)\\bigr ) +\\mathbb {E}\\bigl ( \\mathrm {Tr}(S_2^{p_1}X_1X_1^*)\\mathrm {Tr}(S_2^{p_2})\\bigr ) \\Bigr ] \\\\+N^2 \\mathbb {E}\\left( \\mathrm {Tr}(S_2^{k+2}) \\right)=0,$ where the first and second terms are obtained from the action of both derivatives operators on the monomial $\\bigl [X_1^{\\dagger }X_2^{\\dagger }X_2 S_2^{k}X_1X_1^{\\dagger }X_2^{\\dagger }\\bigr ]_{ac}$ .",
"The third term that involves mixed quantities is obtained by acting with one derivative operator on the monomial, while acting with the other derivative operator on the exponential factor.",
"The last term is obtained from the action of both derivative operator on the exponential factor.",
"These equations contain the mixed quantities already present in (REF ).",
"Thus we can use (REF ) to get rid of these terms in (REF ).",
"This leads to the equations on moments $\\sum _{\\begin{array}{c}p_1+p_2+p_3 =k+1\\\\ p_1,p_2,p_3 \\ge 0\\end{array}} \\mathbb {E}\\left( \\mathrm {Tr}(S_2^{p_1})\\mathrm {Tr}(S_2^{p_2}) \\mathrm {Tr}(S_2^{p_2})\\right) +\\frac{(k+1)(k+2)}{2}\\mathbb {E}\\left(\\mathrm {Tr}(S_2^{k+1})\\right)-N^2 \\mathbb {E}\\left(\\mathrm {Tr}(S_2^{k+2}) \\right)=0,$ which is trilinear in the traces of $S_2$ .",
"Notice that this family of equations extends to the value “$k=-1$ \" by replacing the monomial $\\bigl [X_1^{\\dagger }X_2^{\\dagger }X_2 S_2^{k}X_1X_1^{\\dagger }X_2^{\\dagger }\\bigr ]_{ac}$ by $\\bigl [X_1^{\\dagger }X_2^{\\dagger }\\bigr ]_{ac}$ .",
"Therefore we allow ourselves to set $k=k-1$ and to use our moments notation to get $\\sum _{\\begin{array}{c}p_1+p_2+p_3 =k\\\\ p_1,p_2,p_3 \\ge 0\\end{array}} m_{p_1,p_2,p_3}+\\frac{k(k+1)}{2}m_k -N^2 m_{k+1}=0.$ We then multiply the above equation by $\\frac{1}{x^{k+1}}$ and sum over $k\\ge 0$ in order to get an equation on the resolvents $\\sum _{k\\ge 0}\\sum _{\\begin{array}{c}p_1+p_2+p_3 =k\\\\ p_1,p_2,p_3 \\ge 0\\end{array}}\\frac{m_{p_1,p_2,p_3}}{x^{k+1}}+\\sum _{k\\ge 0}\\frac{k(k+1)}{2}\\frac{m_k}{x^{k+1}} -N^2 \\frac{m_{k+1}}{x^{k+1}}=0,$ which after a few manipulations rewrites $x^2\\overline{W}_3(x,x,x)+x\\partial _xW_1(x)+\\frac{1}{2}x^2\\partial _x^2W_1(x)-N^2xW_1(x)+N^3 =0.$ Note the interesting structural replacement of $\\overline{W}_2(x,x)$ appearing in (REF ) by $\\overline{W}_3(x,x,x)$ and the appearance of a derivative term.",
"Then we know from (REF ), () that $\\overline{W}_3(x,x,x)= N^3 W_{0,1}(x)^3+O(N)$ and $W_1(x)= N W_{0,1}(x)+O(1/N)$ .",
"Therefore we obtain the equation on $W_{0,1}(x)$ $x^2W_{0,1}(x)^3-xW_{0,1}(x)+1=0.$ This last equation relates to the equation satisfied by the generating function $G(u)$ of particular Fuss-Catalan numbers [29], [39], [45], $uG(u)^3-G(u)+1=0$ through the change of variables $W_{0,1}(x)=\\frac{1}{x}G(1/x)$ .",
"Consequently we have $W_{0,1}(x)=\\sum _{p\\ge 0} \\frac{C_p[3]}{x^{p+1}},$ where $C_p[D]$ are the Fuss-Catalan numbers of order $D$ , the usual Catalan numbers $C_p$ being the Fuss-Catalan numbers of order 2, that is $C_p=C_p[2]$ , and have the binomial coefficient form $C_p[D]=\\frac{1}{(D-1)p+1}\\binom{Dp}{p}.$ An explicit form of $W_{0,1}(x)$ can be written as follows.",
"First define $K_{\\pm }(u)=(\\sqrt{1+u}\\pm \\sqrt{u})^{1/3},$ then $G(u)$ writes $G(u)=\\frac{K_{+}\\left(-\\frac{27u}{4}\\right)-K_{-}\\left(-\\frac{27u}{4}\\right)}{\\sqrt{-3u}}.$ Finally one has $W_{0,1}(x)=\\frac{1}{x}G\\left(\\frac{1}{x}\\right).$ We study the solutions and the structure of (REF ) from a geometric perspective in the next sections.",
"Remark 4 Though in principle we need to first focus on the cut structure of $W_{0,1}$ to use the arguments that follow, we will in this remark content ourselves with a formal computation.",
"Starting from equation (REF ) we can also obtain a polynomial equation satisfied by the corresponding density by using the $\\delta , s$ operators along the cut.",
"Indeed with a similar method to that in Remark REF we have the equalities $&\\delta (x^2W_{0,1}(x)^3-xW_{0,1}(x)+1)=0\\\\&s(x^2W_{0,1}(x)^3-xW_{0,1}(x)+1)=0.$ This leads, using the same previously used notations, to the system $&\\frac{x^2}{4}(3u(x)^2+v(x)^2)-x=0\\\\&\\frac{x^2}{4}(u(x)^3+3v(x)^2u(x))-xu(x)+2=0$ which can be solved and leads to $\\rho _{0,1}(x)=\\frac{1}{2i\\pi }v(x)=\\frac{1}{2 \\pi }\\sqrt{\\frac{\\left(\\sqrt{81-12 x}+9\\right)^{2/3}}{2^{2/3} \\@root 3 \\of {3} x^{4/3}}+\\frac{2^{2/3}\\@root 3 \\of {3}}{\\left(\\left(\\sqrt{81-12 x}+9\\right) x\\right)^{2/3}}-\\frac{2}{x}},$ which is supported on $(0,27/4]$ , see the plot of the distribution on Fig.",
"REF .",
"Notice that this result can also be obtained by computing the free multiplicative product of two Marc̆enko-Pastur distribution of parameters $c_{1,2}=1$ .",
"A functional form equivalent to (REF ) is given in [44].",
"Figure: Plot of the eigenvalue density of the matrix S 2 S_2 in the large NN regime.Equation (REF ) possesses a $\\frac{1}{N}$ expansion.",
"This expansion results in a set of relations between $W_{g,1}(x)$ , $W_{g^{\\prime },2}(x,x)$ and $W_{g^{\\prime \\prime },3}(x,x,x)$ .",
"Indeed we have $0=x^2\\Bigl [\\frac{1}{N}\\sum _{g\\ge 0}N^{-2g}W_{g,3}(x,x,x)+3N\\sum _{g_1,g_2\\ge 0}N^{-2(g_1+g_2)}W_{g_1,1}(x)W_{g_2,2}(x,x) \\\\+N^3\\sum _{g_1,g_2,g_3\\ge 0}N^{-2(g_1+g_2+g_3)}W_{g_1,1}(x)W_{g_2,1}(x)W_{g_3,1}(x)\\Bigr ]\\\\+xN\\sum _{g\\ge 0} N^{-2g}\\partial _x W_{1,g}(x)+\\frac{N}{2}x^2\\sum _{g\\ge 0}N^{-2g}\\partial _x^2 W_1(x)-N^3x\\sum _{g\\ge 0}N^{-2g}W_{g,1}(x)+N^3.$ By collecting the coefficient of $N^{3-2g}$ , we obtain the following tower of equations $0=x^2\\left(W_{g-2,3}(x,x,x)+3\\sum _{g_1+g_2=g-1}W_{g_1,1}(x)W_{g_2,2}(x,x)+\\sum _{g_1+g_2+g_3=g}W_{g_1,1}(x)W_{g_2,1}(x)W_{g_3,1}(x)\\right)\\\\+x\\partial _xW_{g-1,1}(x)+\\frac{x^2}{2} \\partial _x^2W_{g-1,1}(x)-xW_{g,1}(x)+P_{g,1}(x),$ where we have $P_{g,1}(x)=\\delta _{g,0}$ .",
"In particular, the coefficient of $N^3$ of equation (REF ) produces equation (REF ).",
"The coefficient of $N$ produces an equation on the next-to-leading order $W_{1,1}(x)$ also involving $W_{0,1}(x)$ and $W_{0,2}(x,x)$ $3x^2W_{0,1}(x)W_{0,2}(x,x)+3x^2W_{0,1}(x)^2W_{1,1}(x)+x\\partial _xW_{0,1}(x)+\\frac{x^2}{2} \\partial _x^2W_{0,1}(x)-xW_{1,1}(x)=0.$ More generally, the coefficient of $N^{3-2g}$ for a fixed value of $g$ produces the equation for $W_{g,1}(x)$ in terms of the functions $W_{g^{\\prime },n^{\\prime }}$ such that $2-2g-1<2-2g^{\\prime }-n^{\\prime }$ and $n^{\\prime }\\le 3$ ."
],
[
"Equation for $W_2(x_1,x_2)$",
"In this section we use Schwinger-Dyson equation techniques to obtain a loop equation for $W_2(x_1,x_2)$ .",
"We start with slightly different identities that involve an additional trace insertion $\\mathrm {Tr}(S_2^q)$ .",
"This allows us to access relations between more general moments.",
"Schwinger-Dyson equations and loop equation for $W_2(x_1,x_2)$ and $W_{0,2}(x_1,x_2)$ .",
"Consider the vanishing integrals of total derivatives $&\\int \\mathrm {d}X_1\\mathrm {d}X_1^{\\dagger } \\mathrm {d}X_2\\mathrm {d}X_2^{\\dagger }\\frac{\\partial }{\\partial X_{1,ab}^{\\dagger }}\\left(\\bigl [X_1^{\\dagger }X_2^{\\dagger }X_2 S_2^{k+1}\\bigr ]_{ab} \\mathrm {Tr}(S_2^q)e^{-N\\mathrm {Tr}(X_1X_1^{\\dagger })}e^{-N\\mathrm {Tr}(X_2X_2^{\\dagger })}\\right) =0 \\\\&\\int \\mathrm {d}X_1\\mathrm {d}X_1^{\\dagger } \\mathrm {d}X_2\\mathrm {d}X_2^{\\dagger }\\frac{\\partial }{\\partial X_{2,ab}^{\\dagger }}\\left(\\bigl [S_2^{k+1}X_1X_1^{\\dagger }X_2^{\\dagger }\\bigr ]_{ab} \\mathrm {Tr}(S_2^q)e^{-N\\mathrm {Tr}(X_1X_1^{\\dagger })}e^{-N\\mathrm {Tr}(X_2X_2^{\\dagger })}\\right) =0,$ and the higher derivative one $\\int \\mathrm {d}X_1\\mathrm {d}X_1^{\\dagger } \\mathrm {d}X_2\\mathrm {d}X_2^{\\dagger }\\frac{\\partial }{\\partial X_{1,ab}^{\\dagger }}\\frac{\\partial }{\\partial X_{2,bc}^{\\dagger }}\\left( \\bigl [X_1^{\\dagger }X_2^{\\dagger }X_2 S_2^{k}X_1X_1^{\\dagger }X_2^{\\dagger }\\bigr ]_{ac}\\mathrm {Tr}(S_2^q)e^{-N\\mathrm {Tr}(X_1X_1^{\\dagger })}e^{-N\\mathrm {Tr}(X_2X_2^{\\dagger })}\\right)=0,$ where repeated indices are summed.",
"After evaluating explicitly the derivatives, the two first equations (REF ) and () lead to $&\\sum _{\\begin{array}{c}p_1+p_2=k+1 \\\\ \\lbrace p_i\\ge 0\\rbrace \\end{array}}\\mathbb {E}\\left(\\mathrm {Tr}(S_2^{p_1})\\mathrm {Tr}(S_2^{p_2}X_2^{\\dagger }X_2)\\mathrm {Tr}(S_2^q) \\right)+q\\mathbb {E}\\left( \\mathrm {Tr}(S_2^{k+q+1}X_2^{\\dagger }X_2)\\right)-N\\mathbb {E}\\left( \\mathrm {Tr}(S_2^{k+2})\\mathrm {Tr}(S_2^q)\\right)=0 \\\\&\\sum _{\\begin{array}{c}p_1+p_2=k+1 \\\\ \\lbrace p_i\\ge 0\\rbrace \\end{array}}\\mathbb {E}\\left(\\mathrm {Tr}(S_2^{p_1}X_1X_1^{\\dagger }) \\mathrm {Tr}(S_2^{p_2})\\mathrm {Tr}(S_2^q)\\right)+q\\mathbb {E}\\left(\\mathrm {Tr}(S_2^{k+q+1}X_1X_1^{\\dagger }) \\right)-N\\mathbb {E}\\left(\\mathrm {Tr}(S_2^{k+2})\\mathrm {Tr}(S_2^q) \\right)=0,$ where the first term of both equations (REF ) and () is obtained from the action of the derivative operator on the non-traced monomial.",
"The second term is obtained via the action of the derivative operator on the traced monomial term $\\mathrm {Tr}(S_2^q)$ .",
"The third term comes from the action of the derivative operator on the exponential factor.",
"These two equations involve mixed terms and cannot be written solely in terms of the moments of $S_2$ .",
"Meanwhile, the higher derivative equation (REF ) leads to $\\sum _{\\begin{array}{c}p_1+p_2+p_3=k+1 \\\\ \\lbrace p_i\\ge 0\\rbrace \\end{array}}\\mathbb {E}\\left(\\mathrm {Tr}(S_2^{p_1})\\mathrm {Tr}(S_2^{p_2})\\mathrm {Tr}(S_2^{p_3})\\mathrm {Tr}(S_2^q) \\right)+\\frac{(k+1)(k+2)}{2}\\mathbb {E}\\left( \\mathrm {Tr}(S_2^{k+1})\\mathrm {Tr}(S_2^q) \\right) \\\\- N\\sum _{\\begin{array}{c}p_1+p_2=k+1 \\\\ \\lbrace p_i\\ge 0\\rbrace \\end{array}}\\left[ \\mathbb {E}\\left(\\mathrm {Tr}(S_2^{p_1})\\mathrm {Tr}(S_2^{p_2}X_2^{\\dagger }X_2)\\mathrm {Tr}(S_2^q)\\right) + \\mathbb {E}\\left( \\mathrm {Tr}(S_2^{p_1}X_1X_1^{\\dagger })\\mathrm {Tr}(S_2^{p_2})\\mathrm {Tr}(S_2^q) \\right) \\right] \\\\+N^2\\mathbb {E}\\left(\\mathrm {Tr}(S_2^{k+2})\\mathrm {Tr}(S_2^q) \\right)+2\\sum _{\\begin{array}{c}p_1,p_2\\ge 0\\\\ p_1+p_2=k+1\\end{array}}q\\mathbb {E}\\left(\\mathrm {Tr}(S_2^{p_1})\\mathrm {Tr}(S_2^{p_2+q}) \\right)+\\sum _{n=1}^q q \\mathbb {E}\\left( \\mathrm {Tr}(S_2^{k+1+n})\\mathrm {Tr}(S_2^{n})\\right) \\\\- Nq\\left[ \\mathbb {E}\\left( \\mathrm {Tr}(S_2^{q+k+1}X_2^{\\dagger }X_2) \\right) + \\mathbb {E}\\left( \\mathrm {Tr}(S_2^{q+k+1}X_1X_1^{\\dagger })\\right) \\right] =0,$ where the two first terms come from the action of both derivatives operators on the non-traced monomial.",
"Each term of the second line comes from the action of one of the derivative on the exponential factor and of the other on the non-traced monomial.",
"The first term of the third line of (REF ) comes from the action of both derivatives on the exponential factor.",
"The second term of the third line is obtained as a sum of the action of the $X_1^{\\dagger }$ (resp.",
"$X_2^{\\dagger }$ ) derivative on the non-traced monomial and the action of the $X_2^{\\dagger }$ (resp.",
"$X_1^{\\dagger }$ ) derivative on the traced monomial $\\mathrm {Tr}(S_2^q)$ .",
"The last term of the third line is obtained from the action of both derivative operators on the traced monomial.",
"Finally the two terms of the fourth line of (REF ) are obtained by the action of $\\partial _{X_{1,ab}^{\\dagger }}$ (resp.",
"$\\partial _{X_{2,bc}^{\\dagger }}$ ) on the traced monomial and $\\partial _{X_{2,bc}^{\\dagger }}$ (resp.",
"$\\partial _{X_{1,ab}^{\\dagger }}$ ) on the exponential factor.",
"Combining equations (REF ), () and (REF ), rewriting some of the sums in a nicer way and using our moments notation we obtain $\\sum _{\\begin{array}{c}p_1+p_2+p_3=k+1 \\\\ \\lbrace p_i\\ge 0\\rbrace \\end{array}}m_{p_1,p_2,p_3,q} +\\frac{(k+1)(k+2)}{2}m_{k+1,q} - N^2 m_{k+2,q}+\\sum _{\\begin{array}{c}p_1,p_2\\ge 0\\\\ p_1+p_2=k+1\\end{array}}qm_{p_1,p_2+q}\\\\+\\sum _{\\begin{array}{c}p_1,p_2 \\ge 0 \\\\ p_1+p_2= k+q+1\\end{array}}q m_{p_1,p_2} =0.$ After performing the shift $k\\rightarrow k-1$ in (REF ), we multiply (REF ) by $\\frac{1}{x_1^{k+1}x_2^{q+1}}$ , and sum over $k, q\\ge 0$ .",
"Doing so we obtain the equation $0=&\\overline{W}_4(x_1,x_1,x_1,x_2)+\\frac{1}{x_1}\\partial _{x_1}\\overline{W}_2(x_1,x_2)+\\frac{1}{2}\\partial _{x_1}^2\\overline{W}_2(x_1,x_2) - \\frac{N^2}{x_1}\\overline{W}_2(x_1,x_2)-N^2A_2(x_1,x_2) \\\\&+\\frac{1}{x_1^2}\\partial _{x_2}\\left( x_1x_2\\frac{\\overline{W}_2(x_1,x_1)-\\overline{W}_2(x_1,x_2)}{x_1-x_2} \\right)+\\frac{1}{x_1^2}\\partial _{x_2}\\left( \\frac{x_1x_2\\overline{W}_2(x_1,x_1)-x_2^2\\overline{W}_2(x_2,x_2)}{x_1-x_2}\\right).$ with $A_2(x_1,x_2)=-\\frac{N}{x_1^2}W_1(x_2)$ .",
"We re-express this equation in terms of the connected resolvents to obtain $W_4(x_1,x_1,x_1,x_2)+3W_1(x_1)W_3(x_1,x_1,x_2)+3W_2(x_1,x_2)W_2(x_1,x_1)+3W_1(x_1)W_1(x_1)W_2(x_1,x_2) \\\\+\\frac{1}{x_1}\\partial _{x_1} W_2(x_1,x_2)+\\frac{1}{2}\\partial _{x_1}^2 W_2(x_1,x_2) - \\frac{N^2}{x_1} W_2(x_1,x_2)+\\frac{1}{x_1^2}\\partial _{x_2}\\left(x_1x_2\\frac{W_2(x_1,x_1)-W_2(x_1,x_2)}{x_1-x_2}\\right)\\\\+\\frac{1}{x_1^2}\\partial _{x_2}\\left(\\frac{x_1x_2W_2(x_1,x_1)-x_2^2W_2(x_2,x_2)}{x_1-x_2}\\right)+\\frac{1}{x_1^2}\\partial _{x_2}\\left(x_1x_2\\frac{W_1(x_1)W_1(x_1)-W_1(x_1)W_1(x_2)}{x_1-x_2}\\right) \\\\+\\frac{1}{x_1^2}\\partial _{x_2}\\left(\\frac{x_1x_2W_1(x_1)W_1(x_1)-x_2^2W_1(x_2)W_1(x_2)}{x_1-x_2}\\right)=0,$ where we used the fact that the terms factoring in front of $W_1(x_2)$ form the first loop equation (REF ).",
"From this equation we can get an equation on $W_{0,2}$ by inserting the $1/N$ expansion of the resolvents appearing in (REF ) and collecting the coefficients of $N^2$ .",
"This equation involves only already computed quantities and can be re-expressed as $\\frac{1}{x_1}\\left(3x_1W_{0,1}(x_1)^2-1 \\right) W_{0,2}(x_1,x_2)+\\frac{1}{x_1^2}\\partial _{x_2}\\left(x_1 x_2\\frac{W_{0,1}(x_1)W_{0,1}(x_1)-W_{0,1}(x_1)W_{0,1}(x_2)}{x_1-x_2} \\right) \\nonumber \\\\+\\frac{1}{x_1^2}\\partial _{x_2}\\left( \\frac{x_1x_2W_{0,1}(x_1)W_{0,1}(x_1)-x_2^2W_{0,1}(x_2)W_{0,1}(x_2)}{x_1-x_2}\\right)=0.$ First few relations between $c^{[0]}_{k}, c^{[0]}_{k_1,k_2}$ .",
"One can extract relations between the $c^{[0]}_{k}, c^{[0]}_{k_1,k_2}$ from equation (REF ).",
"These relations are obtained by expanding the equation at $x_1, x_2 = \\infty $ .",
"The first few examples are $&3 c^{[0]}_0 c^{[0]}_1-c^{[0]}_{1,1}=0,\\\\&2 (c^{[0]}_1)^2+6 c^{[0]}_0 c^{[0]}_2-c^{[0]}_{1,2}=0,\\\\&6 c^{[0]}_1 c^{[0]}_2+9 c^{[0]}_0 c^{[0]}_3-c^{[0]}_{1,3}=0.$ These relations allow to obtain the $c^{[0]}_{k_1,k_2}$ recursively knowing that $c^{[0]}_0,\\ c^{[0]}_1=1$ .",
"We can check these first few relations combinatorially.",
"For illustrative purposes we display the combinatorial maps interpretation of $3 c^{[0]}_0 c^{[0]}_1-c^{[0]}_{1,1}=0$ $3\\, \\left(\\raisebox {-3mm}{\\includegraphics [scale=0.65]{1st-term-comb-interpret.pdf}}\\right)\\quad - \\quad \\left(\\raisebox {-4mm}{\\includegraphics [scale=0.65]{2nd-term-comb-int-1.pdf}}\\quad + \\quad \\raisebox {-3mm}{\\includegraphics [scale=0.65]{3rd-term-comb-int-1.pdf}} \\quad + \\quad \\raisebox {-4mm}{\\includegraphics [scale=0.65]{4th-term-comb-int-1.pdf}} \\quad \\right)\\, =0.$ More generally, one has $0=3\\sum _{p_1+p_2+p_3=k-3}c^{[0]}_{p_1} c^{[0]}_{p_2}c^{[0]}_{p_3+1,q} - c^{[0]}_{k-1,q}+\\sum _{m=0}^{k+q-2}q\\ c^{[0]}_{k+q-m-2}c^{[0]}_{m}+\\sum _{m=0}^{k-2}q\\ c^{[0]}_{k-m-2}c^{[0]}_{m+q}.$"
],
[
"General loop equations",
"In this section we describe the general loop equations for $W_n(x_1,\\ldots , x_n)$ .",
"Because of the use of higher derivatives for Schwinger-Dyson equations, the case of $W_3(x_1,x_2,x_3)$ is still special compared to the cases $W_{n<3}$ .",
"We thus give the corresponding Schwinger-Dyson equations in details before stating the corresponding loop equations.",
"For the $W_{n>3}$ cases, the situation is very similar to the $W_{3}$ case.",
"Therefore we refrain from presenting the detailed derivation, and only state the corresponding loop equations.",
"Loop and Schwinger-Dyson equations for $W_3(x_1,x_2,x_3)$ .",
"We have to consider the equalities, $\\hspace{-11.38109pt}&\\int \\mathrm {d}X_1\\mathrm {d}X_1^{\\dagger } \\mathrm {d}X_2\\mathrm {d}X_2^{\\dagger }\\partial _{X_{1,ab}^{\\dagger }}\\left(\\bigl [X_1^{\\dagger }X_2^{\\dagger }X_2 S_2^{k+1}\\bigr ]_{ab} \\mathrm {Tr}(S_2^{q_1})\\mathrm {Tr}(S_2^{q_2})e^{-N\\mathrm {Tr}(X_1X_1^{\\dagger })}e^{-N\\mathrm {Tr}(X_2X_2^{\\dagger })}\\right) =0 \\\\\\hspace{-11.38109pt}&\\int \\mathrm {d}X_1\\mathrm {d}X_1^{\\dagger } \\mathrm {d}X_2\\mathrm {d}X_2^{\\dagger }\\partial _{X_{2,ab}^{\\dagger }}\\left(\\bigl [S_2^{k+1}X_1X_1^{\\dagger }X_2^{\\dagger }\\bigr ]_{ab} \\mathrm {Tr}(S_2^{q_1})\\mathrm {Tr}(S_2^{q_2})e^{-N\\mathrm {Tr}(X_1X_1^{\\dagger })}e^{-N\\mathrm {Tr}(X_2X_2^{\\dagger })}\\right) =0\\\\\\hspace{-11.38109pt}&\\int \\mathrm {d}X_1\\mathrm {d}X_1^{\\dagger } \\mathrm {d}X_2\\mathrm {d}X_2^{\\dagger }\\partial _{X_{1,ab}^{\\dagger }}\\partial _{X_{2,bc}^{\\dagger }}\\left( \\bigl [X_1^{\\dagger }X_2^{\\dagger }X_2 S_2^{k}X_1X_1^{\\dagger }X_2^{\\dagger }\\bigr ]_{ac}\\mathrm {Tr}(S_2^{q_1})\\mathrm {Tr}(S_2^{q_2})e^{-N\\left(\\mathrm {Tr}(X_1X_1^{\\dagger })-\\mathrm {Tr}(X_2X_2^{\\dagger })\\right)}\\right)=0.$ The inspection of these Schwinger-Dyson equations reveals that the only type of terms that we have not already faced are obtained when both derivatives $\\partial _{X_{1,ab}^{\\dagger }}$ , $\\partial _{X_{2,bc}^{\\dagger }}$ distribute over the two traced monomials $\\mathrm {Tr}(S_2^{q_1})$ , $\\mathrm {Tr}(S_2^{q_2})$ .",
"The distributed action of derivatives on the traced monomial leads to the term $2q_1q_2\\mathbb {E}\\left(\\mathrm {Tr}\\left(S_2^{q_1+q_2+k+1}\\right)\\right)=2q_1q_2m_{k+q_1+q_2}.$ The generating function of this term appearing in the corresponding loop equation will be $\\sum _{k,q_1,q_2\\ge 0}\\frac{2q_1q_2m_{k+q_1+q_2}}{x_1^{k+1}x_2^{q_1+1}x_3^{q_2+1}}= \\\\\\frac{2}{x_1}\\frac{\\partial ^2}{\\partial x_2\\partial x_3}\\left(\\frac{(x_2-x_3)x_1x_2x_3W_1(x_1)-(x_1-x_3)x_1x_2x_3W_1(x_2)+(x_1-x_2)x_1x_2x_3W_1(x_3)}{\\Delta (\\lbrace x_1,x_2,x_3\\rbrace )} \\right)$ where $\\Delta (\\lbrace x_1,x_2,x_3\\rbrace )=(x_3-x_2)(x_3-x_1)(x_2-x_1)$ is the Vandermonde determinant of the family of variables $\\lbrace x_1,x_2,x_3\\rbrace $ .",
"The remaining terms of the loop equations can be inferred by realizing that for all terms involved in either (REF ), (), (), one of the two traced monomials plays a spectator role for the action of the derivatives.",
"Consequently, one obtains the loop equation, $0=\\overline{W}_5(x_1,x_1,x_1,x_2,x_3)+\\frac{1}{x_1}\\partial _{x_1}\\overline{W}_3(x_1,x_2,x_3)+\\frac{1}{2}\\partial ^2_{x_1}\\overline{W}_3(x_1,x_2,x_3)-\\frac{N^2}{x_1}\\overline{W}_3(x_1,x_2,x_3)-N^2A_3(x_1,x_2,x_3)\\\\+\\frac{1}{x_1^2}\\partial _{x_2}\\left( x_1x_2\\frac{\\overline{W}_3(x_1,x_1,x_3)-\\overline{W}_3(x_1,x_2,x_3)}{x_1-x_2} \\right)+\\frac{1}{x_1^2}\\partial _{x_2}\\left( \\frac{x_1x_2\\overline{W}_3(x_1,x_1,x_3)-x_2^2\\overline{W}_3(x_2,x_2,x_3)}{x_1-x_2}\\right)\\\\+\\frac{1}{x_1^2}\\partial _{x_3}\\left( x_1x_3\\frac{\\overline{W}_3(x_1,x_1,x_2)-\\overline{W}_3(x_1,x_2,x_3)}{x_1-x_3} \\right)+\\frac{1}{x_1^2}\\partial _{x_3}\\left( \\frac{x_1x_3\\overline{W}_3(x_1,x_1,x_2)-x_3^2\\overline{W}_3(x_3,x_3,x_2)}{x_1-x_3}\\right)\\\\+\\frac{2}{x_1^3}\\frac{\\partial ^2}{\\partial x_2\\partial x_3}\\left(\\frac{(x_2-x_3)x_1x_2x_3W_1(x_1)-(x_1-x_3)x_1x_2x_3W_1(x_2)+(x_1-x_2)x_1x_2x_3W_1(x_3)}{\\Delta (\\lbrace x_1,x_2,x_3\\rbrace )} \\right),$ where we have set $A_3(x_1,x_2,x_3)=-\\frac{N}{x_1^2}\\overline{W}_2(x_2,x_3)$ .",
"We now introduce some notations in order to shorten expressions.",
"We denote $&\\tilde{\\mathcal {W}}_{n+2}(x_1,x_1,x_1;x_2,\\ldots ,x_n)=\\sum _{\\mu \\vdash [x_1,x_1,x_1]}\\sum _{\\bigsqcup _{i=1}^{|\\mu |}J_i=\\lbrace x_2,\\ldots ,x_n\\rbrace }\\prod _{\\mu _i\\in \\mu }W_{|\\mu _i|+|J_i|}(\\mu _i,J_i)\\\\&\\tilde{\\mathcal {W}}_{g,n+2}(x_1,x_1,x_1;x_2,\\ldots ,x_n)=\\sum _{\\mu \\vdash [x_1,x_1,x_1]}\\sum _{\\begin{array}{c}\\bigsqcup _{i=1}^{|\\mu |}J_i=\\lbrace x_2,\\ldots ,x_n\\rbrace \\\\\\sum _{i=1}^{|\\mu |}g_i=g+|\\mu |-2\\end{array}}\\prod _{\\mu _i\\in \\mu }W_{g_i,|\\mu _i|+|J_i|}(\\mu _i,J_i).$ The notation $\\mu \\vdash [x_1,x_1,x_1]$ needs to be explained.",
"The summation runs over the partitions $\\mu $ of the list $[x_1,x_1,x_1]$ in the following sense.",
"Firstly, in our notation the object $[x_a,x_b,x_c,\\ldots ]$ is a list of elements, that is an ordered multi-set.",
"More concretely the order of appearance of the elements in the list is important and so for example the instances $[x_1,x_2,x_1,x_1,x_4]$ , $[x_1,x_1,x_1,x_2,x_4]$ of lists are different (though they are the same multi-sets).",
"We now come to explain what we mean by partitions of lists.",
"A (denumerablewe will of course consider only the denumerable case since our lists are finite.)",
"list of elements can be represented as a set in the following way.",
"We send a list to the set of pairs $\\lbrace (\\textrm {element}, \\textrm {position in the list})\\rbrace $ .",
"For instance, the list $[x_1,x_2,x_1,x_1,x_4]\\mapsto \\lbrace (x_1,1),(x_2,2),(x_1,3),(x_1,4),(x_4,5)\\rbrace $ while the second list $[x_1,x_1,x_1,x_2,x_4]\\mapsto \\lbrace (x_1,1),(x_1,2),(x_1,3),(x_2,4),(x_4,5)\\rbrace $ which are indeed two different sets.",
"The partitions of the list $\\mu $ are the partitions of the corresponding set of pairs $(\\textrm {element}, \\textrm {position in the list})$ .",
"However, note that the elements of the partitions forget about the position in the list and thanks to the symmetry of the functions $W_n$ functions should be seen as subsets of the corresponding multi-set.",
"For instance, due to the fact that $\\mu $ is really a partition of a list, the partition $\\mu =\\lbrace \\lbrace x_1,x_1\\rbrace ,\\lbrace x_1\\rbrace \\rbrace $ with $\\mu _1=\\lbrace x_1,x_1\\rbrace , \\ \\mu _2=\\lbrace x_1\\rbrace $ of the list $[x_1,x_1,x_1]$ appears three times in the sum.",
"Some further notations are also required.",
"The sum over $\\bigsqcup _{i=1}^{|\\mu |}J_i=\\lbrace x_2,\\ldots ,x_n\\rbrace $ means that we sum over the decompositions into $|\\mu |$ (possibly empty) subsets $J_i$ of the set $\\lbrace x_2,\\ldots ,x_n\\rbrace $ .",
"For instance, in the case $n=3$ , one can consider the term indexed by the partition $\\mu =\\lbrace \\lbrace x_1,x_1\\rbrace ,\\lbrace x_1\\rbrace \\rbrace $ and the decomposition $J_1=\\emptyset , \\ J_2=\\lbrace x_2,x_3\\rbrace $ , which correspond to a term of the form $W_2(x_1,x_1)W_3(x_1,x_2,x_3)$ in the sum.",
"Note that these definitions are very similar to the ones appearing in [6].",
"We also introduce the notation $O_x=\\frac{1}{x_1}\\partial _{x_1}+\\frac{1}{2}\\partial _{x_1}^2.$ Using these notations the corresponding equation for connected resolvents writes $0=&\\tilde{\\mathcal {W}}_{5}(x_1,x_1,x_1;x_2,x_3)+O_xW_3(x_1,x_2,x_3)-\\frac{N^2}{x_1}W_3(x_1,x_2,x_3)\\nonumber \\\\&+\\frac{2}{x_1^3}\\frac{\\partial ^2}{\\partial x_2\\partial x_3}\\left(\\frac{(x_2-x_3)x_1x_2x_3W_1(x_1)-(x_1-x_3)x_1x_2x_3W_1(x_2)+(x_1-x_2)x_1x_2x_3W_1(x_3)}{\\Delta (\\lbrace x_1,x_2,x_3\\rbrace )} \\right)\\nonumber [1]\\\\&+\\frac{1}{x_1^2}\\partial _{x_2}\\left( x_1x_2\\left(\\sum _{\\begin{array}{c}J\\vdash [x_1,x_1,x_3]\\\\ J_i\\ne \\lbrace x_3\\rbrace , \\forall J_i\\end{array}}\\frac{\\prod _{J_i\\in J}W_{|J_i|}(J_i)}{x_1-x_2}- \\sum _{\\begin{array}{c}J\\vdash [x_1,x_2,x_3]\\\\ J_i\\ne \\lbrace x_3\\rbrace , \\forall J_i\\end{array}}\\frac{\\prod _{J_i\\in J}W_{|J_i|}(J_i)}{x_1-x_2}\\right) \\right)\\nonumber [2]\\\\&+\\frac{1}{x_1^2}\\partial _{x_2}\\left( x_1x_2\\sum _{\\begin{array}{c}J\\vdash [x_1,x_1,x_3]\\\\ J_i\\ne \\lbrace x_3\\rbrace , \\forall J_i\\end{array}}\\frac{\\prod _{J_i\\in J}W_{|J_i|}(J_i)}{x_1-x_2}- x_2^2\\sum _{\\begin{array}{c}J\\vdash [x_2,x_2,x_3]\\\\ J_i\\ne \\lbrace x_3\\rbrace , \\forall J_i\\end{array}}\\frac{\\prod _{J_i\\in J}W_{|J_i|}(J_i)}{x_1-x_2}\\right)\\nonumber [2]\\\\&+\\frac{1}{x_1^2}\\partial _{x_3}\\left( x_1x_3\\left(\\sum _{\\begin{array}{c}J\\vdash [x_1,x_1,x_2]\\\\ J_i\\ne \\lbrace x_2\\rbrace , \\forall J_i\\end{array}}\\frac{\\prod _{J_i\\in J}W_{|J_i|}(J_i)}{x_1-x_3}- \\sum _{\\begin{array}{c}J\\vdash [x_1,x_2,x_3]\\\\ J_i\\ne \\lbrace x_2\\rbrace , \\forall J_i\\end{array}}\\frac{\\prod _{J_i\\in J}W_{|J_i|}(J_i)}{x_1-x_3}\\right) \\right)\\nonumber [2]\\\\&+\\frac{1}{x_1^2}\\partial _{x_3}\\left( x_1x_3\\sum _{\\begin{array}{c}J\\vdash [x_1,x_1,x_2]\\\\ J_i\\ne \\lbrace x_2\\rbrace , \\forall J_i\\end{array}}\\frac{\\prod _{J_i\\in J}W_{|J_i|}(J_i)}{x_1-x_3}- x_3^2\\sum _{\\begin{array}{c}J\\vdash [x_3,x_3,x_2]\\\\ J_i\\ne \\lbrace x_2\\rbrace , \\forall J_i\\end{array}}\\frac{\\prod _{J_i\\in J}W_{|J_i|}(J_i)}{x_1-x_3}\\right).\\\\$ We can now extract the corresponding equation of order $g$ (that is the coefficient of $N^{-1-2g}$ in the expansion of (REF )).",
"The corresponding family of equations on $W_{g,3}$ can then be solved recursively provided that we know the $W_{g^{\\prime },n^{\\prime }}$ of lower orders, $0=\\tilde{\\mathcal {W}}_{g,5}(x_1,x_1,x_1;x_2,x_3)+O_xW_{g-1,3}(x_1,x_2,x_3)-\\frac{1}{x_1}W_{g,3}(x_1,x_2,x_3)\\\\+\\frac{2}{x_1^3}\\frac{\\partial ^2}{\\partial x_2\\partial x_3}\\left(\\frac{(x_2-x_3)x_1x_2x_3W_{g,1}(x_1)-(x_1-x_3)x_1x_2x_3W_{g,1}(x_2)+(x_1-x_2)x_1x_2x_3W_{g,1}(x_3)}{\\Delta (\\lbrace x_1,x_2,x_3\\rbrace )} \\right)[1]\\\\+\\frac{1}{x_1^2}\\partial _{x_2}\\left( x_1x_2\\left(\\sum _{\\begin{array}{c}J\\vdash [x_1,x_1,x_3]\\\\ J_i\\ne \\lbrace x_3\\rbrace , \\forall J_i\\\\g=\\sum _i g_i +4 -|J|\\end{array}}\\frac{\\prod _{J_i\\in J}W_{g_i,|J_i|}(J_i)}{x_1-x_2}- \\sum _{\\begin{array}{c}J\\vdash [x_1,x_2,x_3]\\\\ J_i\\ne \\lbrace x_3\\rbrace , \\forall J_i\\\\g=\\sum _i g_i +4 -|J|\\end{array}}\\frac{\\prod _{J_i\\in J}W_{g_i,|J_i|}(J_i)}{x_1-x_2}\\right) \\right)[2]\\\\+\\frac{1}{x_1^2}\\partial _{x_2}\\left( x_1x_2\\sum _{\\begin{array}{c}J\\vdash [x_1,x_1,x_3]\\\\ J_i\\ne \\lbrace x_3\\rbrace , \\forall J_i\\\\g=\\sum _i g_i +4 -|J|\\end{array}}\\frac{\\prod _{J_i\\in J}W_{g_i,|J_i|}(J_i)}{x_1-x_2}- x_2^2\\sum _{\\begin{array}{c}J\\vdash [x_2,x_2,x_3]\\\\ J_i\\ne \\lbrace x_3\\rbrace , \\forall J_i\\\\g=\\sum _i g_i +4 -|J|\\end{array}}\\frac{\\prod _{J_i\\in J}W_{g_i,|J_i|}(J_i)}{x_1-x_2}\\right)[2]\\\\+\\frac{1}{x_1^2}\\partial _{x_3}\\left( x_1x_3\\left(\\sum _{\\begin{array}{c}J\\vdash [x_1,x_1,x_2]\\\\ J_i\\ne \\lbrace x_2\\rbrace , \\forall J_i\\\\g=\\sum _i g_i +4 -|J|\\end{array}}\\frac{\\prod _{J_i\\in J}W_{g_i,|J_i|}(J_i)}{x_1-x_3}- \\sum _{\\begin{array}{c}J\\vdash [x_1,x_2,x_3]\\\\ J_i\\ne \\lbrace x_2\\rbrace , \\forall J_i\\\\g=\\sum _i g_i +4 -|J|\\end{array}}\\frac{\\prod _{J_i\\in J}W_{g_i,|J_i|}(J_i)}{x_1-x_3}\\right) \\right)[2]\\\\+\\frac{1}{x_1^2}\\partial _{x_3}\\left( x_1x_3\\sum _{\\begin{array}{c}J\\vdash [x_1,x_1,x_2]\\\\ J_i\\ne \\lbrace x_2\\rbrace , \\forall J_i\\\\g=\\sum _i g_i +4 -|J|\\end{array}}\\frac{\\prod _{J_i\\in J}W_{g_i,|J_i|}(J_i)}{x_1-x_3}- x_3^2\\sum _{\\begin{array}{c}J\\vdash [x_3,x_3,x_2]\\\\ J_i\\ne \\lbrace x_2\\rbrace , \\forall J_i\\\\g=\\sum _i g_i +4 -|J|\\end{array}}\\frac{\\prod _{J_i\\in J}W_{g_i,|J_i|}(J_i)}{x_1-x_3}\\right).$ We now state in full generality the loop equations.",
"General loop equations.",
"We obtain the higher order loop equations in full generality by starting with Schwinger-Dyson equalities of the same type than (REF ), (), (), but we now insert more traces of monomials of the matrix $S_2$ .",
"Doing so we obtain more relations between moments, and those relations can be translated into relations involving $W_n$ with higher values of $n$ .",
"As before, this first set of relations cannot be used to compute the $W_n$ as it does not close.",
"To solve this problem we perform the $1/N$ expansion which leads to a closed set of equations on $W_{g,n}$ .",
"We display both the equations on $W_n$ and the equations on $W_{g,n}$ for $(g,n)$ such that $2g-2+n>0$ .",
"With $I_{ij}=\\lbrace x_1,\\ldots ,x_n\\rbrace \\backslash \\lbrace x_i,x_j\\rbrace $ , $0=\\tilde{\\mathcal {W}}_{n+2}(x_1,x_1,x_1;x_2,\\ldots , x_n)+O_xW_{n}(x_1,\\ldots ,x_n)-\\frac{N^2}{x_1}W_{n}(x_1,\\ldots ,x_n)[1]\\\\+\\frac{2}{x_1^3}\\sum _{\\begin{array}{c}2\\le i<j\\le n\\end{array}}\\frac{\\partial ^2}{\\partial x_i\\partial x_j}\\left( \\frac{(x_i-x_j)x_1x_ix_jW_{n-2}(I_{ij})-(x_1-x_j)x_1x_ix_jW_{n-2}(I_{1j})+(x_1-x_i)x_1x_ix_jW_{n-2}(I_{1i})}{\\Delta (\\lbrace x_1,x_i,x_j\\rbrace )}\\right)[2]\\\\+\\frac{1}{x_1^2}\\sum _{i\\in [\\!",
"[ 2,n]\\!",
"]}\\partial _{x_i}\\left( x_1x_i\\left(\\sum _{\\begin{array}{c}J\\vdash \\lbrace x_1,x_1\\rbrace \\\\ \\bigsqcup _{k=1}^{|J|}K_k=\\lbrace x_2,\\ldots ,x_n\\rbrace \\backslash \\lbrace x_i\\rbrace \\end{array}}\\hspace{-34.1433pt}\\frac{\\prod _{J_l\\in J}W_{|J_l|+|K_l|}(J_l,K_l)}{x_1-x_i}\\, - \\hspace{-17.07164pt}\\sum _{\\begin{array}{c}J\\vdash \\lbrace x_1,x_i\\rbrace \\\\ \\bigsqcup _{k=1}^{|J|}K_k=\\lbrace x_2,\\ldots ,x_n\\rbrace \\backslash \\lbrace x_i\\rbrace \\end{array}}\\hspace{-34.1433pt}\\frac{\\prod _{J_l\\in J}W_{|J_l|+|K_l|}(J_l,K_l)}{x_1-x_i}\\right) \\right)[2]\\\\+\\frac{1}{x_1^2}\\sum _{i\\in [\\!",
"[ 2,n]\\!",
"]}\\partial _{x_i}\\left( x_1x_i\\hspace{-17.07164pt}\\sum _{\\begin{array}{c}J\\vdash \\lbrace x_1,x_1\\rbrace \\\\ \\bigsqcup _{k=1}^{|J|}K_k=\\lbrace x_2,\\ldots ,x_n\\rbrace \\backslash \\lbrace x_i\\rbrace \\end{array}}\\hspace{-34.1433pt}\\frac{\\prod _{J_l\\in J}W_{|J_l|+|K_l|}(J_l,K_l)}{x_1-x_i}\\, - \\hspace{5.69054pt}x_i^2\\hspace{-34.1433pt}\\sum _{\\begin{array}{c}J\\vdash \\lbrace x_i,x_i\\rbrace \\\\ \\bigsqcup _{k=1}^{|J|}K_k=\\lbrace x_2,\\ldots ,x_n\\rbrace \\backslash \\lbrace x_i\\rbrace \\end{array}}\\hspace{-34.1433pt}\\frac{\\prod _{J_l\\in J}W_{|J_l|+|K_l|}(J_l,K_l)}{x_1-x_i}\\right).$ For the equations on $W_{g,n}$ , write $0=\\tilde{\\mathcal {W}}_{g,n+2}(x_1,x_1,x_1;x_2,\\ldots , x_n)+O_xW_{g-1,n}(x_1,\\ldots ,x_n)-\\frac{1}{x_1}W_{g,n}(x_1,\\ldots ,x_n)[1]\\\\+\\frac{2}{x_1^3}\\sum _{\\begin{array}{c}2\\le i<j\\le n\\end{array}}\\frac{\\partial ^2}{\\partial x_i\\partial x_j}\\left( \\frac{(x_i-x_j)x_1x_ix_jW_{g,n-2}(I_{ij})-(x_1-x_j)x_1x_ix_jW_{g,n-2}(I_{1j})+(x_1-x_i)x_1x_ix_jW_{g,n-2}(I_{1i})}{\\Delta (\\lbrace x_1,x_i,x_j\\rbrace )}\\right)[2]\\\\+\\frac{1}{x_1^2}\\sum _{i\\in [\\!",
"[ 2,n]\\!",
"]}\\partial _{x_i}\\left( x_1x_i\\left(\\sum _{\\begin{array}{c}J\\vdash \\lbrace x_1,x_1\\rbrace \\\\ \\bigsqcup _{k=1}^{|J|}K_k=\\lbrace x_2,\\ldots ,x_n\\rbrace \\backslash \\lbrace x_i\\rbrace \\\\g=\\sum _lg_l -|J|+2\\end{array}}\\hspace{-34.1433pt}\\frac{\\prod _{J_l\\in J}W_{g_l,|J_l|+|K_l|}(J_l,K_l)}{x_1-x_i}\\, - \\hspace{-17.07164pt}\\sum _{\\begin{array}{c}J\\vdash \\lbrace x_1,x_i\\rbrace \\\\ \\bigsqcup _{k=1}^{|J|}K_k=\\lbrace x_2,\\ldots ,x_n\\rbrace \\backslash \\lbrace x_i\\rbrace \\\\g=\\sum _lg_l -|J|+2\\end{array}}\\hspace{-34.1433pt}\\frac{\\prod _{J_l\\in J}W_{g_l,|J_l|+|K_l|}(J_l,K_l)}{x_1-x_i}\\right) \\right)[2]\\\\+\\frac{1}{x_1^2}\\sum _{i\\in [\\!",
"[ 2,n]\\!",
"]}\\partial _{x_i}\\left( x_1x_i\\hspace{-17.07164pt}\\sum _{\\begin{array}{c}J\\vdash \\lbrace x_1,x_1\\rbrace \\\\ \\bigsqcup _{k=1}^{|J|}K_k=\\lbrace x_2,\\ldots ,x_n\\rbrace \\backslash \\lbrace x_i\\rbrace \\\\g=\\sum _lg_l -|J|+2\\end{array}}\\hspace{-34.1433pt}\\frac{\\prod _{J_l\\in J}W_{g_l,|J_l|+|K_l|}(J_l,K_l)}{x_1-x_i}\\, - \\hspace{5.69054pt}x_i^2\\hspace{-34.1433pt}\\sum _{\\begin{array}{c}J\\vdash \\lbrace x_i,x_i\\rbrace \\\\ \\bigsqcup _{k=1}^{|J|}K_k=\\lbrace x_2,\\ldots ,x_n\\rbrace \\backslash \\lbrace x_i\\rbrace \\\\g=\\sum _lg_l -|J|+2\\end{array}}\\hspace{-34.1433pt}\\frac{\\prod _{J_l\\in J}W_{g_l,|J_l|+|K_l|}(J_l,K_l)}{x_1-x_i}\\right).$ Using the family of equations (REF ) one can recursively compute any $W_{g,n}$ knowing the initial conditions $W_{0,1}(x)$ and $W_{0,2}(x_1,x_2)$ .",
"Moreover, starting from these equations it should be possible to obtain a topological recursion like formula.",
"Such a recursion formula certainly looks like the Bouchard-Eynard topological recursion formula introduced in [9], [6].",
"Establishing such a formula strongly depends on the analytic properties of the $W_{g,n}$ as well as the geometric information contained in $W_{0,1}$ and $W_{0,2}$ .",
"Thus in the next section we try to make explicit some of these properties.",
"We first focus on the geometry underlying the equation satisfied by $W_{0,1}$ , and then describe the analytic properties of the higher order terms, by: 1. doing explicit computations and 2. studying the structure of the loop equations.",
"A more detailed and systematic study of the analytical properties of the loop equations is postponed to further work on the product of $p$ rectangular Ginibre matrices."
],
[
"Spectral curve geometry",
"Before computing the first few solutions of the loop equations, we focus on studying the equation (REF ) on $W_{0,1}$ .",
"Indeed, this equation defines an affine algebraic curve $\\mathcal {C}$ , called the spectral curve, where by affine algebraic curve we mean the locus of zero in $(x,y)\\in \\hat{\\mathbb {C}}^2 = \\left(\\mathbb {C}\\cup \\lbrace \\infty \\rbrace \\right)^2$ of the polynomial $P(x,y)=x^2y^3-xy+1.$ This set of zeros of $P$ in $\\mathbb {C}^2$ is generically a (complex) codimension 1 subset of $\\mathbb {C}^2$ .",
"In particular it can be given the structure of a Riemann surface.",
"Computing the solutions $W_{0,1}(x)$ of (REF ) gives a parametrization of the curve away from the ramification points.",
"One of the goals of this section is to introduce a global, nicer parametrization called rational parametrization of the curve.",
"Using this parametrization allows us to simplify the resulting expressions of the solutions.",
"Indeed in the original $x$ variables, the solutions of (REF ) are multi-valued.",
"However one can fix that by promoting these solutions to meromorphic functions on the full affine curve defined by equation (REF ), the curve being the Riemann surface of $W_{0,1}(x)$ ."
],
[
"Basic properties of the curve",
"There are two finite ramification points in the $x$ -plane, one at $(x_{r_1},y_{r_1})=(27/4,2/9)$ , which is a simple ramification point and one at $(x_{r_2},y_{r_2})=(0,\\infty )$ which is a double ramification point.",
"There is also one ramification point at infinity $x_{r_\\infty }=\\infty $ which is a simple ramification point.",
"These ramifications are found from the condition that $P(x,y)=0$ and $\\partial _yP(x,y)=0$ .",
"We display the ramification profile in Fig.",
"REF .",
"Figure: Ramification profile of the curve 𝒞\\mathcal {C}.",
"We use colors to indicate permutations of sheets around ramification points.The cut structure is readily described in [27].",
"It is pictured in Fig.",
"REF , where the lowest sheet of the figure corresponds to the physical sheet that is corresponding to the solution analytic at infinity, whose coefficients of the Laurent expansion are the moments of $S_2$ .",
"The other two sheets correspond to the two other solutions of (REF ) that are not analytic at infinity.",
"Indeed they have a simple ramification point at infinity.",
"From the Fig.",
"REF we can infer that the monodromy group is generated by the transposition $\\tau _1 =(12)$ (obtained by going around $x_{r_1}$ in the physical sheet) and $\\tau _2=(132)$ (going around $x_{r_2}$ ).",
"These permutations are represented using colors on Fig.",
"REF .",
"The genus of the curve $\\mathcal {C}$ can be obtained by considering the Newton polygon of the curve.",
"The number of interior lattice points of the polygon drawn on Fig.",
"REF corresponds to the generic genus of the curve, that is the genus of the curve for generic enough coefficients of the polynomial $P$ .",
"However by fine tuning the coefficients of the polynomial one could in principle obtain a curve with smaller genus.",
"The generic genus is the maximal genus the curve can have.",
"In our case, $P(x,y)=x^2y^3-xy+1$ , the number of lattice points in the Newton polygon is zero, thus the genus of the curve is zero.",
"Since the genus of the curve is zero, there exists a rational parametrization.",
"That is there exists two rational functions $&x:\\hat{\\mathbb {C}} \\rightarrow \\hat{\\mathbb {C}}\\\\&y:\\hat{\\mathbb {C}} \\rightarrow \\hat{\\mathbb {C}},$ such that $x(z)^2y(z)^3-x(z)y(z)+1=0, \\quad \\forall z \\in \\hat{\\mathbb {C}}.$ These two functions can be found by solving the following system on the coefficients of $Q_x(z),Q_y(z)$ and $P_x(z),P_y(z)$ , $&Q_x(z) x(z)=P_x(z)\\\\&Q_y(z) y(z)=P_y(z)\\\\&x(z)^2y(z)^3-x(z)y(z)+1=0,$ where $Q_x(z),Q_y(z)$ and $P_x(z),P_y(z)$ are set to be polynomials of degree high enough for a solution to exist.",
"Then one obtains explicitly one possible parametrization $x(z)=\\frac{P_x(z)}{Q_x(z)}=\\frac{z^3}{1+z}, \\quad y(z)=\\frac{P_y(z)}{Q_y(z)}=-\\frac{1+z}{z^2}.$ Note that from this point of view, $y(z)$ is the analytic continuation of $W_{0,1}(x(z))$ .",
"The function $x$ can be seen as a cover $x:\\mathcal {C}\\rightarrow \\hat{\\mathbb {C}}$ of generic degree 3 (that is there are generically three values of $z$ corresponding to the same value of $x$ ).",
"As such, the zeroes of $\\mathrm {d}x$ corrrespond to the ramifications point of the cover.",
"One can then check that $\\mathrm {d}x=0$ at $z_{r_1}=0$ and $z_{r_2}=-3/2$ , corresponding to the values $x(0)=0$ and $x(-3/2)=27/4$ .",
"One also notices that the zero of $\\mathrm {d}x$ at $z=0$ is a double zero, thus confirming the fact that $x_{r_1}$ is a double ramification point.",
"Finally since $x=27/4$ is a simple ramification point, there is another pre-image of $27/4$ in $z$ variable, that is we have $x(3)=27/4$ .",
"This leads to the ramification profile shown on Fig.REF .",
"Figure: Cut structure of W 0,1 W_{0,1}."
],
[
"Computation of $w_{0,1}$ and {{formula:33cb87df-c6dc-408a-83d9-9cd28604610c}}",
"Using this parametrization we compute the functions $w_{g,n}(z_1,\\ldots ,z_n)=W_{g,n}(x(z_1),\\ldots , x(z_n))\\prod _{i=1}^n x^{\\prime }(z_i)+\\frac{\\delta _{g,0}\\delta _{n,2}x^{\\prime }(z_1)x^{\\prime }(z_2)}{(x(z_1)-x(z_2))^2}.$ We also denote $\\tilde{w}_{0,2}(z_1,z_2)=W_{0,2}(x(z_1),x(z_2))x^{\\prime }(z_1)x^{\\prime }(z_2)$ .",
"$w_{g,n}$ functions are meromorphic functions on $\\mathcal {C}$ , as such they are rational functions of their variables $z_i$ .",
"Consequently, they are much easier to manipulate than $W_{g,n}$ and their analytic properties are more transparent.",
"For $w_{0,1}(z)$ we already know that $y(z) = W_{0,1}(x(z))$ , thus $w_{0,1}(z)=y(z)x^{\\prime }(z)=-\\frac{2z+3}{1+z}.$ The original functions $W_{g,n}$ can be recovered using the inverse function $z(x)=-xW_{0,1}(x)=_{\\infty }-1-\\frac{1}{x}-\\frac{3}{x^2}-\\frac{12}{x^3}-\\frac{55}{x^4}+O\\left(\\frac{1}{x^5}\\right).$ Indeed one has, $&W_{g,n}(x_1,x_2,\\ldots ,x_n)=\\frac{w_{g,n}(z_1,z_2,\\ldots , z_n)}{x^{\\prime }(z_1)x^{\\prime }(z_2)\\ldots x^{\\prime }(z_n)}\\Bigr \\vert _{z_i=z(x_i)} \\textrm { for } (g,n)\\ne (0,2), \\\\&W_{0,2}(x_1,x_2)=\\frac{\\tilde{w}_{0,2}(z_1,z_2)}{x^{\\prime }(z_1)x^{\\prime }(z_2)}\\Bigr \\vert _{z_1=z(x_1), z_2=z(x_2)}.$ Note also that the corresponding coefficients of the expansion of $W_{g,n}$ at infinity, that is the $c^{[g]}_{k_1,\\ldots , k_n}$ , can be obtained by computing residues $c^{[g]}_{k_1,\\ldots , k_n}=\\underset{\\lbrace x_i\\rightarrow \\infty \\rbrace }{\\textrm {Res}}x_1^{k_1}\\ldots x_n^{k_n}W_{g,n}(x_1,x_2,\\ldots ,x_n)= \\underset{\\lbrace z_i\\rightarrow -1\\rbrace }{\\textrm {Res}}x(z_1)^{k_1}\\ldots x(z_n)^{k_n}w_{g,n}(x(z_1),x(z_2),\\ldots ,x(z_n)).$ It is also true that the residue in $z$ variables can equivalently be computed at infinity.",
"The passage from the $W_{g,n}$ to the $w_{g,n}$ functions takes into account the Jacobian of the change of variables.",
"For future convenience, we define $\\sigma (z)=\\frac{1}{x(z)}(1-3x(z)y(z)^2),$ where $\\sigma $ relates to $\\partial _yP$ since $\\sigma (z)=\\frac{1}{x(z)^2}\\partial _yP(x(z),y(z))$ .",
"So in particular $\\sigma $ vanishes at the ramification point $(x_{r_1},y_{r_1})=(27/4,2/9)$ and $x(z)^2\\sigma (z)$ has a zero of order 2 at $(x_{r_2},y_{r_2})=(0,\\infty )$ .",
"Figure: Newton polygon for the affine curve x 2 y 3 -xy+1=0x^2y^3-xy+1=0.",
"The number of ℕ 2 \\mathbb {N}^2 lattice points inside the polygon gives the generic genus of the curve.",
"Here there is no points inside the polygon so that the generic genus is zero, which implies that the genus is zero.Expression of $\\tilde{w}_{0,2}$ .",
"We have after multiplying (REF ) by $x^{\\prime }(z_1)x^{\\prime }(z_2)$ and performing a few additional manipulations $\\sigma (z_1) \\tilde{w}_{0,2}(z_1,z_2)=\\frac{x^{\\prime }(z_1)}{x(z_1)^2}\\partial _{z_2}\\left(x(z_1) x(z_2)\\frac{y(z_1)^2-y(z_1)y(z_2)}{x(z_1)-x(z_2)} \\right) \\\\+\\frac{x^{\\prime }(z_1)}{x(z_1)^2}\\partial _{z_2}\\left( \\frac{x(z_1)x(z_2)y(z_1)^2-x(z_2)^2y(z_2)^2}{x(z_1)-x(z_2)}\\right).$ From this equation $\\tilde{w}_{0,2}(z_1,z_2)$ can be computed in the variables $z_1,z_2$ , so that one obtains $\\tilde{w}_{0,2}(z_1,z_2)=\\frac{z_2^2 z_1^2+2 (z_2 z_1^2+ z_2^2 z_1) + z_1^2 +z_2^2 +4 z_2 z_1}{(z_2 z_1^2+z_2^2z_1+z_1^2+z_2^2+z_2 z_1)^2}.$ From this expression we can recover the limiting cumulants of the product of traces, $c^{[0]}_{i,j}=\\underset{z_1, z_2\\rightarrow \\infty }{\\textrm {Res}}\\, x(z_1)^ix(z_2)^j\\tilde{w}_{0,2}(z_1,z_2).$ We provide the reader with the first few orders on Table REF .",
"These numbers can be obtained easily via symbolic computation softwares.",
"Remark 5 Using a table of coefficients $c_{i,j}^{[0]}$ for $i,j$ running from 1 to 20 it is possible to make an experimental guess for the explicit form of these coefficients.",
"This is $c_{i,j}^{[0]}=\\frac{2 i j }{3(i+j)}\\binom{3i}{i}\\binom{3j}{j}.$ In particular we have checked that these numbers satisfy the recurrence equation (REF ) for the first few orders.",
"It would be interesting to prove or disprove this guess via, for instance, combinatorial means.",
"Table: Table of the first few cumulants c i,j [0] =lim N→∞ 𝔼 Tr (S 2 i ) Tr (S 2 j )-1 N 2 𝔼 Tr ( S 2 i )𝔼 Tr ( S 2 j )c^{[0]}_{i,j}=\\lim _{N\\rightarrow \\infty }\\mathbb {E}\\left(\\mathrm {Tr}(S_2^i)\\mathrm {Tr}(S_2^j)\\right)-\\frac{1}{N^2}\\mathbb {E}\\bigl (\\mathrm {Tr}(S_2^i)\\bigr )\\mathbb {E}\\bigl (\\mathrm {Tr}(S_2^j)\\bigr ).Universality for $w_{0,2}$ .",
"In this paragraph we explain in detail and a posterioriSince they can already easily be inferred from the explicit result of equation (REF ).",
"the analytic properties of $\\tilde{w}_{0,2}$ and $w_{0,2}$ .",
"We first argue that $\\tilde{w}_{0,2}$ does not have poles at the ramification points that is $z=-3/2,0$ .",
"We then consider the situation when $x(z_1)\\rightarrow x(z_2)$ .",
"First starting from the above remark that $x(z)^2\\sigma (z)=\\partial _yP(x(z),y(z))$ , we know that $x(z)^2\\sigma (z)$ has a double zero at $z=0$ and a simple zero at $z=-3/2$ , which makes it a source of poles as this factor appears in the denominator in front of the two terms of (REF ), see below $\\tilde{w}_{0,2}(z_1,z_2)=\\frac{x^{\\prime }(z_1)}{x(z_1)^2\\sigma (z_1)}\\partial _{z_2}\\left(x(z_1) x(z_2)\\frac{y(z_1)^2-y(z_1)y(z_2)}{x(z_1)-x(z_2)} \\right) \\\\+\\frac{x^{\\prime }(z_1)}{x(z_1)^2\\sigma (z_1)}\\partial _{z_2}\\left( \\frac{x(z_1)x(z_2)y(z_1)^2-x(z_2)^2y(z_2)^2}{x(z_1)-x(z_2)}\\right).$ We start by focusing on poles at the simple ramification point $z=-3/2$ .",
"We remind ourselves that $\\mathrm {d}x$ vanishes at the ramification points, and so $x^{\\prime }(z)$ has a simple zero at $z=-3/2$ .",
"Therefore $\\frac{x^{\\prime }(z_1)}{x(z_1)^2\\sigma (z_1)}$ is holomorphic at $z_1=-3/2$ .",
"Moreover, $x(z_1)$ and $y(z_1)$ are holomorphic at $z_1=-3/2$ .",
"As a consequence $\\tilde{w}_{0,2}$ is holomorphic at $z=-3/2$ in both $z_1$ and $z_2$ (thanks to the symmetry $z_1 \\leftrightarrow z_2$ ).",
"We now come back to the ratio $\\frac{x^{\\prime }(z_1)}{x(z_1)^2\\sigma (z_1)}$ for $z_1=0$ .",
"A similar argument is valid at $z_1=0$ .",
"Indeed $x^{\\prime }(z_1)$ has a double zero at $z_1=0$ and this cancels the double zero of $x(z_1)^2\\sigma (z_1)$ at $z_1=0$ .",
"In fact one can explicitly compute the ratio and find $\\frac{x^{\\prime }(z_1)}{x(z_1)^2\\sigma (z_1)}=\\frac{1}{1+z_1}$ which confirms our argument.",
"$x(z)$ is holomorphic at $z=0$ , but $y(z)$ is not, indeed it has a double pole at $z=0$ .",
"So the terms $x(z_1)x(z_2)y(z_1)^2$ could bring a simple pole at $z_1=0$ .",
"However, using the fact that $\\tilde{w}_{0,2}(z_1,z_2)$ is symmetric in its arguments, if such a simple pole exists at $z_1=0$ then one should have a simple pole at $z_2=0$ .",
"Using the fact that $x(z_2)$ has a third order zero at $z_2=0$ , and $y(z_2)$ has a double pole at $z_2=0$ one can show that $\\tilde{w}_{0,2}(z_1,z_2)$ is holomorphic at $z_2=0$ , therefore the apparent singularity at $z_1=0$ is a removable singularity.",
"Consequently, we have just shown that $\\tilde{w}_{0,2}(z_1,z_2)$ is holomorphic at the ramification points $z=-3/2, 0$ in both its variables.",
"Other possible singularities may occur at the singularities of $x(z)$ which possesses a simple pole at $z=-1$ and when $x(z_1)\\rightarrow x(z_2)$ .",
"First note that $\\frac{y(z_1)^2-y(z_1)y(z_2)}{x(z_1)-x(z_2)},$ has a double zero when $z_1\\rightarrow -1$ , thus $\\frac{x^{\\prime }(z_1)}{x(z_1)^2\\sigma (z_1)}\\partial _{z_2}\\left(x(z_1) x(z_2)\\frac{y(z_1)^2-y(z_1)y(z_2)}{x(z_1)-x(z_2)} \\right)$ is holomorphic when $z_1\\rightarrow -1$ since $\\frac{x(z_1)x^{\\prime }(z_1)}{x(z_1)^2\\sigma (z_1)}$ has a double pole at $z_1=-1$ .",
"A similar argument applies to the term $\\frac{x^{\\prime }(z_1)}{x(z_1)^2\\sigma (z_1)}\\partial _{z_2}\\left( \\frac{x(z_1)x(z_2)y(z_1)^2-x(z_2)^2y(z_2)^2}{x(z_1)-x(z_2)}\\right),$ thus showing that $\\tilde{w}_{0,2}(z_1,z_2)$ is holomorphic at $z_1=-1$ , and by symmetry at $z_2=-1$ .",
"We are now left with the situation $x(z_1)\\rightarrow x(z_2)$ .",
"A first possibility is $z_1\\rightarrow z_2$ .",
"In this case both ratios $\\frac{y(z_1)^2-y(z_1)y(z_2)}{x(z_1)-x(z_2)}, \\quad \\frac{x(z_1)x(z_2)y(z_1)^2-x(z_2)^2y(z_2)^2}{x(z_1)-x(z_2)},$ are holomorphic since the denominators and numerators have simultaneous simple zeroes.",
"So $\\tilde{w}_{0,2}(z_1, z_2)$ is holomorphic when $z_1 \\rightarrow z_2$ .",
"However, since $x(z)$ is a covering of degree three, there exists two (not globally defined) functions, $d_1(z), d_2(z)$ that leaves $x$ invariant, that is $x\\circ d_i=x, \\, i\\in \\lbrace 1,2\\rbrace $ .",
"These functions are the (non-trivial) solutions of the equation $\\frac{d(z)^3}{1+d(z)}=\\frac{z^3}{1+z}.$ This leads to the expressions $&d_1(z)=-\\frac{1}{2}\\frac{z^2+z+z\\sqrt{(z-3) (1+z)}}{1+z}, \\\\&d_2(z)=-\\frac{1}{2}\\frac{z^2+z-z\\sqrt{(z-3) (1+z)}}{1+z}.$ One can check that $x(d_1(z))=x(d_2(z))=x(z)$ .",
"In order to understand the pole structure of $\\tilde{w}_{0,2}(z_1, z_2)$ , one also needs to know how does $y(z)$ changes when composed with one of the $d_i$ .",
"One has the simple identities for $i\\in \\lbrace 1,2\\rbrace $ $y(d_i(z))=\\frac{d_i(z)}{z}y(z).$ Using these identities, one expects poles when $z_1 \\rightarrow d_{1,2}(z_2)$ .",
"Indeed, in this limit the numerators of (REF ) does not have zeroes anymore, while the denominators have simple zeroes.",
"Thus $\\tilde{w}_{0,2}(z_1, z_2)$ should have double poles when $z_1 \\rightarrow d_{1,2}(z_2)$ .",
"This is indeed what we find by requiring that the denominator of (REF ) vanishes.",
"Remark 6 The functions $d_i$ have interesting properties.",
"Indeed they permute the sheets of the covering $x:\\mathcal {C}\\rightarrow \\hat{\\mathbb {C}}$ .",
"Their behavior in a small neighborhood around a ramification point relates to the local deck transformation group of the cover.",
"Let us first focus on the double ramification point $z=0$ .",
"It is a fixed point of both $d_1$ and $d_2$ and around $z=0$ , we have $d_1(z)\\sim _0 e^{-\\frac{2i\\pi }{3}} z$ and $d_2(z)\\sim _0 e^{\\frac{2i\\pi }{3}} z$ thus they are inverse of each other locally, and generate the cyclic group $\\mathbb {Z}_3$ .",
"This cyclic group is the group generated by the permutation of the sheets $\\tau _2=(132)$ .",
"This group is the local deck transformation group around the ramification point at $z=0$ .",
"We now consider the behavior of $d_1, d_2$ at $z=-3/2$ .",
"In this case, only $d_1$ fixes $z=-3/2$ , while $d_2(-3/2)=3,\\, d_2(3)=-3/2$ , that is $d_2$ exchanges the ramification point with the point above it (see Fig.",
"REF ).",
"Note however that one has $d_1(3)=d_2(3)=-3/2$ as the two solutions $d_1,d_2$ of equation (REF ) merge at $z=3$ (as they also do at $z=1$ ).",
"This merging has the following interpretation.",
"At $z=-3/2$ two of the three sheets of the covering coincide.",
"Therefore, there remains effectively only two sheets to be permuted, that is why $d_1$ fixes $z=-3/2$ while $d_2$ permutes $z=-3/2$ with $z=3$ .",
"The action of the local deck transformation group at $z=-3/2$ relates to the action of $d_1$ in a small neighborhood of $z=-3/2$ .",
"Since $d_1(-3/2+\\epsilon )-d_1(-3/2)\\sim _{0}-\\epsilon $ , $d_1$ locally generates the cyclic group $\\mathbb {Z}_2$ corresponding to the group generated by the permutation $\\tau _1=(12)$ .",
"Similar arguments can be used to describe the local deck transformation group at the ramification point $z=\\infty $ .",
"We now come to the universality statement.",
"Indeed, we expect that a slightly different object than $\\tilde{w}_{0,2}(z_1,z_2)$ takes a universal form.",
"This is the reason for the shift introduced in (REF ).",
"The statement is that $w_{0,2}(z_1,z_2)$ should have a universal form, that is it should be the unique meromorphic function on the sphere with a double pole of order 2 on the diagonal with coefficient 1 and otherwise regular.",
"Indeed if we compute $w_{0,2}(z_1,z_2)$ we obtain $w_{0,2}(z_1,z_2)=\\tilde{w}_{0,2}(z_1,z_2)+\\frac{x^{\\prime }(z_1)x^{\\prime }(z_2)}{(x(z_1)-x(z_2))^2} = \\frac{1}{(z_1-z_2)^2}.$ We find exactly the expected universal form for a genus zero spectral curve.",
"Comment on probabilistic interpretation of $W_{0,1}(x)$ , $W_{1,1}(x)$ and $W_{0,2}(x_1,x_2)$ .",
"As stated earlier, $W_1(x)$ is the Stieltjes transform of the eigenvalues density of the matrix $S_2$ , that is $W_1(x)=\\int _{-\\infty }^{\\infty }\\mathrm {du} \\frac{\\rho _{1}(u)}{x-u}.$ In particular in the large $N$ limit we have that $W_{0,1}(x)=\\int _{-\\infty }^{\\infty }\\mathrm {du} \\frac{\\rho _{0,1}(u)}{x-u},$ and the computation of $W_{0,1}(x)$ uniquely determines $\\rho _{0,1}(x)$ .",
"The same property is also true for the exact density, i.e.",
"$W_1(x)$ uniquely determines $\\rho _1(x)$ .",
"This can be traced back to the Carlemann condition [4].",
"Indeed the Stieltjes transform $W_1(x)$ , (resp.",
"$W_{0,1}(x)$ ) contains the information on the whole moment sequence of $\\rho _1(x)$ (resp.",
"$\\rho _{0,1}(x)$ ).",
"The sequence of moments of both distributions can be shown to satisfy the Carlemann condition, and thus one expects that the knowledge of the Stieltjes transform is sufficient to reconstruct the densities $\\rho _1(x)$ , $\\rho _{0,1}(x)$ .",
"However it is known [25] that in general the truncation of the $1/N$ expansion of the resolvent does not determine a unique truncated density.",
"Indeed, there exists, a priori, multiple densities truncated at order $p$ , $\\rho ^{(p)}_1(x)=\\sum _{g\\ge 0}^pN^{-2g}\\rho _{g,1}(x)$ with the same truncated resolvent $\\sum _{g\\ge 0}^{p}N^{-2g}W_{g,1}(x) = \\int _{-\\infty }^{\\infty }\\mathrm {du} \\frac{\\rho ^{(p)}(u)}{x-u}.$ That is the computation of the corrections to $W_{0,1}(x)$ only determines Stieltjes class of densitiesThough this is not a rigorous justification, one can look at the truncated Carlemann criterion, for instance in the GUE case, and see that the Carlemann criterion is indeed not satisfied order-by-order in $1/N$ .",
"Only the large $N$ and the exact criterion are satisfied., often referred to as a smoothed density.",
"This is sufficient however to compute the corrections to the average $\\mathbb {E}(\\phi (x))$ where $\\phi (x)$ is any function analytic on the support of $\\rho _{0,1}(x)$ .",
"In particular, our later computation of the first few corrections to the large $N$ resolvent does not determine corrections $\\rho _{1,1}(x), \\rho _{2,1}(x),\\ldots $ The probabilistic interpretation of $W_{0,2}$ goes as follows.",
"$W_{2}$ is the Stieltjes transform of the connected part of the eigenvalue correlation function $W_{2}(x_1,x_2)=\\int _{-\\infty }^{\\infty }\\mathrm {d}u \\mathrm {d}v \\frac{\\rho _2(u,v)}{(x_1-u)(x_2-v)},$ and $\\rho _2(x_1,x_2)=\\mathbb {E}\\left(\\sum _{i=1}^N\\delta (x_1-\\lambda _i)\\sum _{j=1}^N\\delta (x_2-\\lambda _j)\\right)- \\rho _1(x_1)\\rho _1(x_2),$ where the $\\lambda _i$ are the eigenvalues of the matrix $S_2$ .",
"In the large $N$ limit, the centered random vector whose components are the traces of successive powers of the matrix $S_2$ , $\\left( \\mathrm {Tr}(S_2^i)-\\mathbb {E}(\\mathrm {Tr}(S_2^i))\\right)_{i=1}^k$ converges to a normal random vector of zero mean and variance $\\textrm {Var}_{m,n}$ $\\textrm {Var}_{m,n}=c^{[0]}_{m,n}= \\underset{z_1\\rightarrow -1}{\\textrm {Res}}\\underset{z_2\\rightarrow -1}{\\textrm {Res}}x(z_1)^m x(z_2)^n w_{0,2}(z_1,z_2),$ where the normality of this centered random vector at large $N$ follows from the fact that $W_n(x_1,\\ldots ,x_n)=O(1/N^{n-2})$ , that is the higher cumulants of the limiting distribution of the family $\\lbrace \\mathrm {Tr}(S_2^i)\\rbrace $ vanish at large $N$ .",
"This statement extends to the large $N$ limit of any linear statistics $A$ of the eigenvalues of the form $A=\\sum _{i=1}^N a(\\lambda _i),$ where $a$ is a sufficiently smooth function (analytic for instance), as we have $\\textrm {Var}(A)=\\oint _{\\Gamma }\\oint _{\\Gamma }\\frac{dx_1 dx_2}{(2i\\pi )^2} a(x_1)a(x_2) W_{0,2}(x_1,x_2),$ with $\\Gamma $ a contour encircling the cut $(0,27/4]$ of $W_{0,1}(x)$ ."
],
[
"Computation of $w_{1,1}$ and higher correlation functions.",
"From these data one can access the first correction to the resolvent which allows in turn to access a first correction to the large $N$ density.",
"The equation for $w_{1,1}(z)$ can be easily obtained from the equation (REF ) on $W_{1,1}(x)$ .",
"It reads $w_{1,1}(z)=\\frac{3x(z)^2}{x^{\\prime }(z)\\partial _yP(x(z),y(z))}y(z)\\tilde{w}_{0,2}(z,z) + \\frac{x(z)^2}{\\partial _yP(x(z),y(z))}\\left(\\partial _z y(z)-\\frac{x^{\\prime \\prime }(z)}{2x^{\\prime }(z)^2}\\partial _zy(z)+\\frac{1}{2x^{\\prime }(z)}\\partial ^2_z y(z) \\right).$ This leads to the result of the next paragraph.",
"Expression of $w_{1,1}(z)$ and analytic properties of (REF ).",
"We obtain, $w_{1,1}(z)=\\frac{z^4+7 z^3+21 z^2+24 z+9}{z^2 (2 z+3)^4}.$ We notice that the poles are located at $z=0$ and $z=-3/2$ , which are the zeroes of $\\mathrm {d}x$ .",
"However, starting from (REF ) one can only infer that the poles of $w_{1,1}(z)$ can be located at $z=0, -3/2, -1$ .",
"Indeed, one can easily obtain from the analytic properties of $x(z), y(z)$ and $\\tilde{w}_{0,2}(z,z)$ that the first term of the right hand side of (REF ) can have poles only at $z=0, -3/2$ , and rule out singularities at $z=-1, \\infty $ .",
"However when considering the derivatives term, that is the second term of equation (REF ), one can not rule out poles at $z=-1$ .",
"The explicit computation shows that the coefficient of these poles is zero.",
"Remark 7 Note that we can also produce a guess for the coefficients $c^{[1]}_n$ .",
"We need however to prove our first guess of Remark REF for $c_{i,j}^{[0]}$ to be able to prove this guess using the Schwinger-Dyson equations.",
"We provide our guess for purely informative purposes, $c^{[1]}_n=\\frac{(n-1)^2n}{6(3n-1)}\\binom{3n}{n}.$ Expression for higher correlations.",
"Using the loop equations (REF ) we can compute any $n$ -point resolvents recursively at any order.",
"We illustrate this claim by providing the first few resolvents of higher order.",
"One point case.",
"$&w_{0,1}(z)=-\\frac{2 z+3}{z+1}\\\\ &w_{1,1}(z)=\\frac{z^4+7 z^3+21 z^2+24 z+9}{z^2 (2 z+3)^4}\\\\ &w_{2,1}(z)=\\frac{9 z^9+153 z^8+1284 z^7+4227 z^6+7626 z^5+9246 z^4+8280 z^3+5220 z^2+1971 z+324}{z^3 (2 z+3)^{10}}.$ Two points case.",
"$&\\tilde{w}_{0,2}(z_1,z_2)=\\frac{z_2^2 z_1^2+2 (z_2 z_1^2+ z_2^2 z_1) + z_1^2 +z_2^2 +4 z_2 z_1}{(z_2 z_1^2+z_2^2z_1+z_1^2+z_2^2+z_2 z_1)^2}\\\\&w_{1,2}(z_1,z_2)=\\frac{pol(z_1,z_2)}{z_1^2 \\left(2 z_1+3\\right){}^6 z_2^2 \\left(2 z_2+3\\right){}^6},$ with $pol(z_1,z_2)$ a symmetric polynomial of $z_1,z_2$ of degree 12, $pol(z_1,z_2)=128 z_2^6 z_1^6+1280 z_2^5 z_1^6+6144 z_2^4 z_1^6+12288 z_2^3 z_1^6+12480 z_2^2 z_1^6+6912 z_2 z_1^6+1728 z_1^6+1280 z_2^6 z_1^5+12800 z_2^5 z_1^5\\\\+55680 z_2^4 z_1^5+108672 z_2^3 z_1^5+111168z_2^2 z_1^5+62208 z_2 z_1^5+15552 z_1^5+6144 z_2^6 z_1^4+55680 z_2^5 z_1^4+215352 z_2^4 z_1^4+405000 z_2^3 z_1^4\\\\+414234 z_2^2 z_1^4+233280 z_2 z_1^4+58320 z_1^4+12288 z_2^6 z_1^3+108672z_2^5 z_1^3+405000 z_2^4 z_1^3+768312 z_2^3 z_1^3+809838 z_2^2 z_1^3+466560 z_2 z_1^3\\\\+116640 z_1^3+12480 z_2^6 z_1^2+111168 z_2^5 z_1^2+414234 z_2^4 z_1^2+809838 z_2^3 z_1^2+888165 z_2^2z_1^2+524880 z_2 z_1^2+131220 z_1^2+6912 z_2^6 z_1\\\\+62208 z_2^5 z_1+233280 z_2^4 z_1+466560 z_2^3 z_1+524880 z_2^2 z_1+314928 z_2 z_1+78732 z_1+1728 z_2^6+15552 z_2^5+58320 z_2^4\\\\+116640z_2^3+131220 z_2^2+78732 z_2+19683.$ Three points case.",
"$w_{0,3}(z_1,z_2,z_3)=\\frac{24}{\\left(2 z_1+3\\right){}^2 \\left(2 z_2+3\\right){}^2 \\left(2 z_3+3\\right){}^2}.$ For all these computed $w_{g,n}$ , $(g,n)\\ne (0,1), (0,2)$ the poles are located at $z=0$ and $z=-3/2$ .",
"Therefore we can expect that the poles of $w_{g,n}$ , for $2g-2+n>0$ , are always located at $z=0$ and $z=-3/2$ , however this remains to be proven.",
"Remark 8 The computed $w_{g,n}$ are rational functions of the $z_i$ .",
"We notice that the numerator of these rational functions seems to be a polynomial with positive integer coefficients.",
"If this property is true for every $w_{g,n}$ , it would be interesting to understand if these positive integers have an enumerative (combinatorics or geometry) meaning."
],
[
"Conclusion",
"In this first paper on loop equations for matrix product ensembles, we have shown how to obtain loop equations for any resolvents for a random matrix defined as a product of two square complex Ginibre matrices without resorting to an eigenvalues or singular values reformulation of the problem.",
"Indeed, the eigenvalues reformulation is yet to access these observable quantities.",
"We used these loop equations to compute several terms of the expansion of the any resolvents $W_n$ .",
"In particular we accessed $W_{0,2}$ , giving us information on the fluctuations of linear statistics, as well as the first correction $W_{1,1}$ to $W_{0,1}$ .",
"We expect a similar technique to apply to the more general case of the product of $p\\ge 2$ rectangular Ginibre (complex or real) as well as to some other product ensembles, for instance the ensembles introduced in [26] that are closely related to the Hermite Muttalib-Borodin ensemble.",
"Several questions are suggested by this work.",
"The most straightforward one concerns the establishment of a topological recursion formula for the $w_{g,n}$ .",
"In the present case this topological recursion formula is certainly similar to the one devised in [9], [6] by Bouchard and al.",
"and Bouchard and Eynard.",
"We postpone the construction of such formula to further works.",
"Another interesting question oriented towards enumerative geometry concerns the application of the same technical means to the matrix model introduced by Ambjørn and Chekhov in [1], [2] which generates hypergeometric Hurwitz numbers.",
"In these works the spectral curve is obtained, however this is done via a matrix-chain approach that requires $p-1$ of the $p$ matrices to be invertible, thus ruling out the fully general case of rectangular matrices.",
"We hope this fully general case can be tackled using our higher derivatives technique.",
"Yet another related question is the following.",
"Free probability provides us with tools to determine the equation satisfied by the large $N$ limit of the resolvent of a product of matrices knowing the large $N$ limit of the resolvents of the members of the product.",
"These tools have been generalized to some extent to the 2-point resolvent in the works of Collins and al.",
"[15] in order to more systematically access the fluctuations of linear statistics.",
"One question is then the following.",
"Can we devise similar tools that would allow to construct the full set of loop equations for a product matrix knowing the loop equations satisfied by the member of the product (or, more realistically, the large $N$ sector of the loop equations)?",
"Finally, the loop equations can be interpreted as Tutte equations [24], [48], [49].",
"The loop equations described in this paper can also be interpreted combinatorially, and it would be interesting to understand the more general case of maps with an arbitrary number of black vertices in such a combinatorial setting.",
"Moreover, one would also like to understand if it is possible to merge two sets of Tutte equations together for two independent sets of maps with one type of edge in order to obtain Tutte equations for maps with two types of edges.",
"The combinatorial interpretation of the free multiplicative convolution described in [20] may be a useful starting point."
]
] | 1906.04390 |
[
[
"Exponential lower bound for Berge-Ramsey problems"
],
[
"Abstract We give an exponential lower bound for Berge-Ramsey problems."
],
[
"pdfpagemode=UseNone Gerbner and Palmer [6], generalizing the definition of hypergraph cycles due to Berge, introduced the following notion.",
"A hypergraph $H$ contains a Berge copy of a graph $G$ , if there are injections $\\Psi _1: V(G)\\rightarrow V(H)$ and $\\Psi _2: E(G) \\rightarrow E(H)$ such that for every edge $uv\\in E(G)$ the containment $\\Psi _1(u),\\Psi _1(v)\\in \\Psi _2(uv)$ holds, i.e., each graph edge can be mapped into a distinct hyperedge containing it to create a copy of $G$ .",
"If $|E(H)|=|E(G)|$ , then we say that $H$ is a Berge-$G$, and we denote such hypergraphs by $\\mathcal {B}G$ .",
"The study of Ramsey problems for such hypergraphs started independently in 2018 by three groups of authors [2], [5], [7].",
"Denote by $R_r(\\mathcal {B}G; c)$ the size of the smallest $N$ such that no matter how we $c$ -color the $r$ -edges of $K_N^r$ , the complete $r$ -uniform hypergraph, we can always find a monochromatic $\\mathcal {B}G$ .",
"In [2] $R_r(\\mathcal {B}K_n; c)$ was studied for $n=3,4$ .",
"In [5] it was conjectured that $R_r(\\mathcal {B}K_n; c)$ is bounded by a polynomial of $n$ (depending on $r$ and $c$ ), and they showed that $R_r(\\mathcal {B}K_n; c)=n$ if $r>2c$ and $R_r(\\mathcal {B}K_n; c)=n+1$ if $r=2c$ , while $R_3(\\mathcal {B}K_n; 2)< 2n$ (also proved in [7]).",
"In [7] a superlinear lower bound was shown for $r=c=3$ and for every other $r$ for large enough $c$ .",
"This was improved in [4] to $R_{r}(\\mathcal {B}K_n; c)=\\Omega (n^{d})$ if $c>(d-1)\\binom{r}{2}$ and $R_r(\\mathcal {B}K_n; c)=\\Omega (n^{1+1/(r-2)}/\\log n)$ .",
"We further improve these to disprove the conjecture of [5].",
"Theorem $R_{r}(\\mathcal {B}K_n; c)> (1+\\frac{1}{r^2})^{n-1}$ if $c>\\binom{r}{2}$ .",
"It is enough to prove the statement for $c=\\binom{r}{2}+1$ .",
"For $r=2$ this reduces to the classical Ramsey's theorem, so we can assume $r\\ge 3$ .",
"We can also suppose $n\\ge \\binom{r}{2}+1=c$ , or the lower bound becomes trivial.",
"Suppose $N\\le (1+\\frac{1}{r^2})^{n-1}$ .",
"Assign randomly (uniformly and independently) a forbidden color to every pair of vertices in $K_N^r$ .",
"Color the $r$ -edges of $K_N^r$ arbitrarily, respecting the following rule: if $\\lbrace u,v\\rbrace \\subset E$ , then the color of $E$ cannot be the forbidden color of $\\lbrace u,v\\rbrace $ .",
"Since $c>\\binom{r}{2}$ , this leaves at least one choice for each edge.",
"Following the classic proof of the lower bound of the Ramsey's theorem, now we calculate the probability of having a monochromatic $\\mathcal {B}K_n$ .",
"The chance of a monochromatic $\\mathcal {B}K_n$ on a fixed set of $n$ vertices for a fixed color is at most $(\\frac{c-1}{c})^{\\binom{n}{2}}$ , as the fixed color cannot be the forbidden one on any of the pairs of vertices.",
"Thus the expected number of monochromatic $\\mathcal {B}K_n$ 's is at most $c\\binom{N}{n}(\\frac{c-1}{c})^{\\binom{n}{2}}$ .",
"If this quantity is less than 1, then we know that a suitable coloring exists.",
"Since $c\\le n\\le n!$ , it is enough to show that $N< (\\frac{c}{c-1})^{\\frac{n-1}{2}}$ , but this is true using $c=\\binom{r}{2}+1$ and $r\\ge 3$ .",
"As was brought to my attention by an anonymous referee, my construction for $r=3$ and $c=4$ is essentially the same as the one used in the proof of Theorem 1(ii) in [3] for a different problem, the 4-color Ramsey number of the so-called hedgehog.",
"A hedgehog with body of order $n$ is a 3-uniform hypergraph on $n+\\binom{n}{2}$ vertices such that $n$ vertices form its body, and any pair of vertices from its body are contained in exactly one hyperedge, whose third vertex is one of the other $\\binom{n}{2}$ vertices, a different one for each hypderedge.",
"It is easy to see that such a hypergraph is a Berge copy of $K_n$ , and while their result, an exponential lower bound for the 4-color Ramsey number of the hedgehog, does not directly imply mine, their construction is such that it also avoids a monochromatic $\\mathcal {B}K_n$ .",
"It is an interesting problem to determine how $R_{r}(\\mathcal {B}K_n; c)$ behaves if $c\\le \\binom{r}{2}$ .",
"The first open case is $r=c=3$ , just like for hedgehogs."
]
] | 1906.04288 |
[
[
"Recovery of the electron-phonon interaction function in superconducting\n tantalum ballistic contacts"
],
[
"Abstract The experimentally observed nonlinearities of the current-voltage characteristics (CVCs) of tantalum-based point homo- and hetero- contacts in both normal and superconducting states related to electron-phonon interaction (EPI) were analyzed.",
"It was taken into account that additional nonlinearity of CVCs arising upon contact transition to the superconducting state (superconducting spectral component) is formed not only near the constriction in the region roughly equal to the contact diameter (as is the case for the normal state, and as predicted theoretically for the superconducting state), but also in a markedly larger region that is about the size of the coherence length.",
"In this case, a considerable role in the formation of this superconducting component is played by nonequilibrium phonons with low group velocity, which account for the experimentally observed sharpening of the phonon peaks in the EPI spectra (the second derivatives of the CVCs) during the superconducting transition of the contacts, instead of the theoretically expected peak broadening (spreading), and for the increase in the superconducting contribution to the point contact spectrum in the low and medium energy regions.",
"The high-energy part of the EPI spectrum changes much less significantly during the superconducting transition, which is attributable to suppression of the excess contact current by nonequilibrium quasi-particles.",
"A detailed procedure was proposed for the recovery of the EPI spectral function from the point contact spectrum contribution (the second derivative of the CVC) that arises during the superconducting transition of one or both contacting metals."
],
[
"INTRODUCTION",
"The usage of the nonlinearity of the current-voltage characteristics (CVC) of ballistic point contacts in the normal state to restore the electron-phonon interaction (EPI) function in metals is well known.",
"Several hundred publications are available on this subject and two integrating monographs have been published [1], [2].",
"Electron duplication takes place for these contacts in the on-state, namely, the electrons are split into two groups for which the energy difference between the occupied and unoccupied states on the Fermi surface is eV .",
"In other words, an electron that has traveled through the constriction from the opposite contact bank differs in energy from electrons within the present bank by exactly the applied voltage value.",
"At any point of the trajectory, the electron may lose excess energy by emitting a nonequilibrium phonon with the energy eV.",
"Since the contact is ballistic, the average energy relaxation length is much greater than the contact size.",
"Thus, the greater part of nonequilibrium phonon emissions would occur away from the constriction, within the contact banks.",
"If after scattering, the electron goes back through the constriction to the initial bank of the point contact from which it flew out with an additional energy, the contact resistance would increase.",
"These processes are referred to as backscattering.",
"These processes provide the basis for the Yanson EPI spectroscopy where the second derivative of the contact CVC directly reflects the structure of the EPI function.",
"It is obvious that, since during excess energy electron scattering via emission of nonequilibrium phonons, in the isotropic case, the direction of electron movement can change in an arbitrary way, then, due to geometric considerations, the backscattering processes are efficient only in a volume relatively proximate to the constriction.",
"If the contact is shaped like an orifice of diameter d in a thin wall, this volume is approximately equal to the volume of a sphere with the same diameter .",
"Hence, most of the electron scattering on nonequilibrium phonons does not participate in the formation of the EPI spectrum in Yanson spectroscopy.",
"The transition to the superconducting state brings about an additional CVC nonlinearity caused by the suppression (decrease) of the excess current (CVC difference between the superconducting and normal states of the contact metal at the same voltage) upon reabsorption of nonequilibrium phonons by Andreev-reflected electrons (hereinafter referred to as Andreev electrons, for brevity).",
"This nonlinearity can also be used for the recovery of the EPI function.",
"The appropriate theories that allow this to be done for S-c-N and S-c-S contacts (S is superconductor, N is normal metal, c is constriction) appeared in 1983 , ; however, practical attempts to recover the EPI function from the superconducting characteristics of these contacts were made relatively recently , , .",
"For tantalum-based contacts, the previously used procedure for EPI function recovery required further development, taking account of the ratio between the contact size and the coherence length.",
"The approaches considered in this article may prove to be fairly important in the analysis of experimental data of point contact EPI spectroscopy in superconductors."
],
[
"THEORETICAL GROUNDS",
"Reabsorption processes, in other words, Andreev electron reflections on nonequilibrium phonons, are not subject to geometrical constraints typical of backscattering processes; any reflection process is effective.",
"These processes can take place in the volume where nonequilibrium phonons and Andreev electrons coexist simultaneously.",
"Since the conversion of Andreev electrons to Cooper pairs occurs at the reduced coherence length $\\zeta $ ($\\tfrac{1}{\\zeta }=\\tfrac{1}{{{\\xi }_{0}}}+\\tfrac{1}{{{l}_{i}}}$ , ${\\xi }_{0}$ is the superconducting coherence length, $l_i$ is the length of scattering on impurities), then the volume, in the isotropic case, is equal to the volume of a sphere (or hemisphere for S-c-N point contacts) with the radius ${\\zeta }$ .",
"However, for ballistic contacts with great coherence and elastic relaxation lengths appearing in the theory, these processes are fairly probable in approximately the same volume as backscattering.",
"Here the volume restriction is caused by fast decrease in the concentrations of both nonequilibrium phonons and Andreev electrons moving away from the constriction.",
"High current density in the vicinity of the orifice provides also their high concentration, which rapidly declines as the current spreads.",
"Therefore, at large distances r from the constriction, the contact can be considered to be a point source of phonons, with their density decreasing as ${\\sim {\\ }1}/{{{r}^{2}}}\\;$ .",
"Since the minimum size in which the superconducting energy gap can vary coincides with the length $\\zeta \\gg {d}$ (d is the contact diameter), these scattering processes do not change the gap in the near-contact region, and suppression of the excess current is due to a minor decrease in the quantity of Andreev electrons.",
"The Khlus and Omel'yanchuk theory , of inelastic point contact EPI spectroscopy in the S-c-S and S-c-N point contacts was developed for ballistic contacts, i.e., for those contacts that obey the condition $d\\ll \\zeta $ , $\\upsilon _F/ \\omega _D$ ($\\upsilon _F$ is the Fermi velocity, ${{\\upsilon }_{F}}/{{\\omega }_{D}}\\sim {{l}_{\\varepsilon }}$ , where $l_{\\varepsilon }$ is the energy free path at the Debye energy $\\hbar \\omega _D$ ).",
"For S-c-S contacts, it was found that $\\frac{d{{I}_{exc}}}{dV}(V)\\!",
"=-\\frac{64}{3R}\\left( \\frac{\\Delta L}{\\hbar \\bar{v}} \\right)\\!",
"{{\\left[ {{G}^{N}}(\\omega )+\\frac{1}{4}{{G}^{S}}(\\omega ) \\right]}_{\\omega ={eV}/{\\hbar }\\;}}$ Here R is the contact resistance, L is a function that is fairly sophisticated for arbitrary arguments, $\\bar{v}$ is the velocity of electrons averaged over the Fermi surface, $G^{N}(\\omega )$ is the point contact (PC) EPI function, which is the same as that of point contacts in the normal state according to the Kulik-Omel'yanchuk-Shekhter (KOS) theory [2], $G^{S}(\\omega )$ is the superconducting PC EPI function differing from $G^{N}(\\omega )$ by a form factor, $\\Delta $ is the superconducting energy gap.",
"Furthermore, unlike the normal form factor, which is responsible for the backscattering contribution to the current, in the case of the superconducting form factor appearing in $G^{S}(\\omega )$ , the contribution to the current is made by electron-phonon collisions associated with Andreev reflection-like processes in the region of the contact, i.e., by of the conversion of quasi-electron excitations to quasi-holes.",
"A similar expression was obtained for the S-c-N contact : $\\begin{matrix}\\frac{1}{R(V)}-{{\\left( \\frac{1}{R(V)} \\right)}_{\\Delta =0}}= \\\\=-\\frac{32}{3R}\\times \\frac{d\\Delta }{h}\\times \\left[ \\frac{1}{v_{F}^{(1)}}\\times {{G}_{1}}\\left( \\omega \\right)+\\frac{1}{v_{F}^{(2)}}\\times {{G}_{2}}\\left( \\omega \\right) \\right] \\\\\\end{matrix}$ Here $G_{i}(\\omega )$ (i = 1, 2) is the EPI function of metals that form the point contact.",
"The relative phonon contribution to the excess current at $eV\\sim {{\\omega }_{D}}$ is about ${d\\cdot {{\\omega }_{D}}}/{{{v}_{F}}}$ , i.e., it is small when the condition $d\\ll {{{v}_{F}}}/{{{\\omega }_{D}}}$ holds.",
"This small contribution is very important to ensuring that equal changes in the nonequilibrium phonon flux density for different biases at the contact cause equal changes in the excess current.",
"Meanwhile, when suppression of the excess current is pronounced, this relationship is violated.",
"For this type of contacts, recovery of the EPI function would require a correction of the useful signal amplitude in the region of decreasing excess current .",
"For ballistic contacts, the current spreading results in Andreev electrons being spread over a larger space in which the concentration of nonequilibrium phonons is low; therefore, the relative decline of excess current for these contacts is minor.",
"The nonlinear deviation of the point contact resistance from $R_{0}=R$ at $V=0$ caused by scattering of nonequilibrium phonons on Andreev electrons is several times smaller than that caused by backscattering processes.",
"As a result, if $eV\\gg \\Delta $ , the point contact spectrum changes insignificantly upon transition to the superconducting state.",
"Khlus described this transformation of the second derivative of the CVC for the ballistic S-c-N point contact: $\\begin{matrix}\\frac{1}{R}\\cdot \\frac{dR}{dV}=\\frac{16ed}{3\\pi }\\cdot \\sum \\limits _{i=1,2}{\\frac{1}{v_{F}^{(i)}}\\cdot \\int \\limits _{0}^{\\infty }{\\frac{d\\omega }{\\Delta }\\cdot S\\left( \\frac{\\omega -eV}{\\Delta } \\right){{G}_{i}}(\\omega )}}.\\end{matrix}$ where $G_i(\\omega )$ are the EPI functions of the normal and superconducting metals that form the heterocontact, $S(x)$ is the smearing factor: $S(x)=\\theta (x-1)\\frac{2{{\\left( x-\\sqrt{{{x}^{2}}-1} \\right)}^{2}}}{\\sqrt{{{x}^{2}}-1}},$ here $\\theta (y)$ is the Heaviside theta function.",
"Upon the superconducting transition, the spectrum is additionally smeared and at $T\\rightarrow {0}$ , the resolution is determined by the value of $\\Delta $ .",
"In view of the relationship between the CVC derivative and the PC EPI function, it follows from expression (REF ) that $\\tilde{g}_{pc}^{S}(eV)=\\int \\limits _{0}^{\\infty }{\\frac{d\\omega }{\\Delta }S\\left( \\frac{\\omega -eV}{\\Delta } \\right)g_{pc}^{N}(\\omega )}.$ Thus, with the point contact spectra of the heterocontact in the normal and superconducting states at hand, one can compare the results of calculations and experiments.",
"In addition, as shown by Bobrov et al.",
", $g_{pc}^{S}(eV)=\\frac{1}{\\Delta }\\int \\limits _{0}^{eV}{\\left[ \\tilde{g}_{pc}^{S}(\\omega )-g_{pc}^{N}(\\omega ) \\right]d\\omega }$ Equation (REF ) obviously follows from the fact that the first derivative of the excess current is proportional to the EPI current.",
"It can be seen that the function $\\tilde{g}_{pc}^{S}(eV)$ reflects the transformation of the spectrum as the contact goes to the superconducting state.",
"It is proportional to the CVC second derivative, shifted to lower energy by a value of about $\\Delta $ , additionally broadened and, as a result, it has a slightly lower intensity.",
"The function $g_{pc}^{S}(eV)$ is proportional to the first derivative of excess current and does not contain additional broadening; according to calculations, its shift to lower energy is approximately two times less than that of $\\tilde{g}_{pc}^{S}(eV)$ ."
],
[
"COMPARISON OF THE THEORETICAL CONCLUSIONS\nWITH THE EXPERIMENT. TIN-BASED POINT CONTACTS",
"As an example, first, we will consider tin-based point contacts, which show good agreement between the theory and the experiment .",
"Figure REF Figure: (a) EPI spectra of the Sn-Cu point contact in the normal and superconductingstates.",
"B ˜ S {{\\tilde{B}}_{S}} and B N B_N are the background curves for the superconductingand normal spectra, respectively.",
"The superconductivity is suppressed by amagnetic field.",
"(b) Difference between the superconducting and normal spectraand the assumed shape of the background curve.",
"(c) The difference curve(after background subtraction).",
"(d) Point contact EPI functions recovered fromthe normal and superconducting states and by integration of the differencecurve.",
"For the convenience of comparison, the curve g pc S g_{pc}^{S} is aligned in amplitude with the curve g pc N g_{pc}^{N}.presents the spectra of a Sn-Cu point contact in the normal and superconducting states.",
"The difference curve and the background curves, $B_N$ , $B_S$ , and ${{\\tilde{B}}_{S}}$ , are also given.",
"It turned out that background is inherent in not only the Yanson point contact spectroscopy.",
"The background BN is clearly manifested as a nonzero second derivative of the CVC at displacements greater than those corresponding to the boundary frequency of the phonon spectrum, whereas the EPI function goes to zero.",
"The background is determined most often by the self-consistent iteration procedure [1].",
"It was found that mere integration of the difference curve ${{({{d}^{2}}V/d{{I}^{2}})}_{S}}-{{({{d}^{2}}V/d{{I}^{2}})}_{N}}$ is insufficient for obtaining the curve proportional to the EPI function $g_{pc}^{S}(eV)$ , since apart from the effect related to the gap at low biases, the second derivatives also differ in the background level beyond the boundary of the phonon spectrum.",
"This brings about the necessity to subtract the background curve $B_S$ from the difference curve before integration.",
"Apart from the obvious condition of being equal to zero at energies exceeding the maximum phonon spectrum frequency, the curve obtained after background subtraction should obey the rule of sums: the total areas under the curves above and below the abscissa axis must be equal.",
"The rule of sums follows from the fact that integration of the obtained curve gives a curve with a zero background.",
"It is clear that these criteria can be met for a multitude of different background curves; however, their variations, provided they are monotonic, do not induce considerable changes in the shapes and positions of phonon effects of the EPI function being recovered.",
"In order to verify the theory predictions, it is necessary to follow the transformation of the spectrum upon transition from the normal to superconducting state without the background component ${{\\tilde{B}}_{S}}$ , in order to get a curve proportional to $\\tilde{g}_{pc}^{S}(\\omega )$ .",
"In the absence of a normal state spectrum and difference curve with subtracted backgrounds, this is difficult to perform; however, when these curves are available, this problem is easily solved: let $\\tilde{S}$ be the spectrum in the superconducting state with background subtracted.",
"Then ${{\\tilde{B}}_{S}}=S-\\tilde{S}=S-[(N-{{B}_{N}})+(S-N-{{B}_{S}})]={{B}_{N}}+{{B}_{S}}$ .",
"The latter is even somewhat more convenient, as this allows for an easier approximation of the missing part of the curve in the small bias region near the gap effect.",
"For the convenience of comparison, the curve $g_{pc}^{S}$ in Fig.",
"[Fig1]1(d) is aligned in amplitude with the curve $g_{pc}^{N}$ .",
"As can be seen from comparison of Figs.",
"[Fig1]1(a) and [Fig1]1(d), the theory predictions are in good agreement with the experiment.",
"Upon transition to the superconducting state, the $\\tilde{g}_{pc}^{S}(\\omega )$ curves smear, decrease in the amplitude, and shift to lower energies by a value approximately equal to the gap.",
"The shapes of experimental and theoretical ${g}_{pc}^{S}$ curves differ in the high-energy region: the experimental curve is markedly more intense.",
"Apparently, this is attributable to increasing concentration of nonequilibrium phonons at the contact periphery caused by the decrease in the energy relaxation length of electrons close to the Debye energy."
],
[
"COMPARISON OF THE CHARACTERISTIC\nPARAMETERS FOR TIN AND TANTALUM",
"It turns out that even the complete correspondence of the point contact parameters to the requirements of the theory does not ensure that the experimental curves follow the theoretical model.",
"For example, as it was ascertained for ballistic point contacts based on tantalum , , , the superconducting transition is accompanied by a radical change in the shape, intensity, and positions of phonon effects.",
"Furthermore, differences from theory predictions are observed for both hetero- and homocontacts.",
"Instead of the expected broadening of the phonon peaks, they become strongly sharpened.",
"As a consequence, the amplitude of these peaks substantially increases.",
"In addition, the soft phonon mode, which is manifested in the normal state as a shoulder, is converted to a sharp peak as the contacts switch to the superconducting state.",
"This is manifested most vividly for relatively low-resistance point contacts, but even for high-resistance contacts, in which the ballistic condition is satisfied rather strictly, the deviations from theory predictions are quite large.",
"In order to understand what could be responsible for these crucial differences in the behaviors of tantalum- and tin-based point contacts, compare their characteristic parameters summarized in [Table 1]Table I.",
"Table: Characteristic parameters of tantalum- and tin-based point contacts.Since the Table presents the energy relaxation length at the Debye energies, $l_{\\varepsilon }^{D}$ , and the deviations in the tantalum spectra from the theory predictions already start at small biases, it appears reasonable to compare these lengths over the whole range of biases.",
"The inelastic mean free path of electrons for an arbitrary contact voltage can be estimated from the formula $\\frac{1}{{{l}_{\\varepsilon }}\\left( eV \\right)}=\\frac{2\\pi }{\\hbar {{v}_{F}}}\\int \\limits _{0}^{eV}{d\\omega \\,g(\\omega )},$ where $g(\\omega )$ is the EPI function.",
"Figure REF Figure: (a) Point contact EPI functions for tin and tantalum used to estimate theenergy relaxation rate vs. the contact bias.",
"(b) Energy relaxation lengths for tantalumin relation to tin vs. the bias.shows the point contact EPI functions for Ta and Sn [1] that were used to estimate the energy relaxation lengths.",
"The estimates were made using the electron parameters presented in the Table.",
"As can be seen in Fig.REF , the relaxation length is greater in tantalum than in tin and is in the 4-7 meV range, which is close to the soft mode energy in tantalum.",
"At other energies, $l_\\varepsilon $ is smaller in tantalum than in tin; it is approximately four times smaller at the Debye energies.",
"Considering these estimates, one can conclude that the reduced coherence length $\\zeta $ is the key factor responsible for the deviation from the theory`s predictions.",
"As follows from the [Table 1]Table, this value for tin coincides with the coherence length, $\\zeta \\approx \\xi _0$ , in view of the very large elastic scattering length $l_i$ , while in tantalum, $\\zeta \\sim $ 43 nm, i.e., the volume of the sphere limited by the reduced coherence length is 100 times smaller in tantalum than in tin.",
"For these reasons, and because of the lower Fermi velocity of electrons in tantalum, the concentration of Andreev electrons in this volume increases compared with that for tin.",
"Therefore, reabsorption of nonequilibrium phonons by Andreev electrons starts to play a noticeable role not only in the volume roughly equal to the contact diameter, as in tin, but also in the volume with the characteristic size of the reduced coherence length.",
"In other words, the excess current decreases not only because of decreasing quantity of the Andreev electrons within a volume about equal to the contact diameter, but also because of suppression of the gap in the contact region with a volume about equal to the coherence length."
],
[
"NONEQUILIBRIUM EFFECTS IN TANTALUM-BASED\nPOINT CONTACTS",
"Before considering specific features of the formation mechanism of phonon peaks in the tantalum-based superconducting contacts, we will pay attention to other nonlinear effects manifested in the CVCs of these contacts, which are also associated with the small coherence length.",
"These effects are not spectral and their position in the energy axis depends on both the point contact resistance and the temperature and magnetic field strength.",
"In the second derivative of the CVC, the low-temperature effect occurs immediately after the gap effect as a narrow and sharp surge and is greater in intensity.",
"This corresponds to a stepwise decrease in the excess current caused by the abrupt decrease in the superconducting gap in the region adjoining the contact.",
"This phenomenon is associated with the attained critical density of nonequilibrium quasi-particles in the near-contact region and is considered in detail in Refs., , .",
"Since the contacts investigated here are ballistic, most of electrons lose excess energy within the point contact banks, when scattered on nonequilibrium phonons.",
"In the superconducting state these electrons that have lost the excess energy are accumulated above the gap.",
"With increasing contact bias, the quantity of these nonequilibrium quasi-particles increases; when some critical concentration is reached, a part of the superconductor adjoining the constriction abruptly switches to a new, nonequilibrium state with a partly suppressed gap.",
"It is obvious that the smaller the space in which this switching takes place, the more rapidly the required concentration is reached.",
"The minimum size of this region cannot be smaller than the reduced coherence length $\\zeta $ .",
"The position of the nonequilibrium effect at a specified temperature for contacts with different resistance corresponds to the same injection power, being approximately 0.4 $\\mu $ W at 2K.",
"As the temperature or magnetic field increases, the relaxation rate of these quasi-particles above the gap increases; therefore, for attaining the critical concentration, it is necessary to increase the injection power.",
"This shifts the effect to higher voltages.",
"This rules out the interpretation of these effects as being due to the degradation of superconductivity caused by heating or suppression by the magnetic field.",
"Since the stepwise decrease in the excess current upon transition to the nonequilibrium state is insignificant, the change in the gap accompanied by the transition is also moderate.",
"Thus, the influence of the nonequilibrium effect on the superconducting spectral contribution is also small, except that it can be located at biases corresponding to some phonon effect and thus hamper observation of the phonon effect.",
"However, as follows from experimental results, the suppression of the excess current is not limited to the stepwise section resulting from the phase transition to a new nonequilibrium state.",
"On further increase of the voltage applied to the contact and, hence, increase in the injection power, the suppression of the excess current is rather smooth and is not accompanied by any indications in the CVC derivatives.",
"Depending on the resistance, this suppression may be rather small for high-resistance contacts and fairly large (several-fold larger) for low-resistance contacts.",
"In any case, this suppression has a pronounced influence on the generation of EPI-related effects."
],
[
"EFFECT OF THE NEAR-CONTACT REGION ON THE\nFORMATION OF THE SPECTRUM",
"Now we will consider the mechanism of formation of phonon effects related to the excess current.",
"Conventionally speaking, they represent a superposition of contributions from two spatially different regions: a contribution from the region about equal to point contact diameter, which corresponds to the theoretical model, and a contribution of the near-contact region about equal to the coherence length $\\zeta $ .",
"In principle, the mechanism of formation of phonon effects is the same in both regions and is related to reabsorption of nonequilibrium phonons by Andreev electrons.",
"However, since the second region is much larger than the contact diameter, here a considerable role is played by differences between the group velocities of nonequilibrium phonons generated by electrons with the excess energy $eV$ .",
"Since phonons with energies corresponding to the maximum density of states have the lowest group velocities, $\\partial \\omega /\\partial q=0$ , these phonons would leave this region more slowly and would thus be accumulated in a higher concentration.",
"Since the relative concentrations of nonequilibrium phonons and Andreev electrons determine the value of the negative contribution to the excess current, the greatest contribution to the spectrum will occur at the maximum density of phonon states.",
"This assumption is supported by sharpening of the phonon peaks.",
"In the region approximately equal to the contact diameter, because of small volume, slow phonons do not have time to concentrate, and their specific contribution to the nonlinearity would be the same as that of phonons with higher group velocity.",
"In terms of this model, it is clear that for selection of phonons with low group velocities, fast phonons must be free to leave the near-contact volume, i.e., the flight of phonons should correspond to the ballistic regime.",
"Thus, the ratio of contributions of the near-contact and remote regions would influence the degree of sharpening of phonon effects.",
"Evidently, the increase in the contact size would lead to increasing proportion of the latter contribution and, hence, to increasing degree of sharpening of phonon effects.",
"As shown below, this assumption is perfectly confirmed by experimental results."
],
[
"TA-CU HETEROCONTACTS",
"Unlike the spectra of Sn-Cu heterocontacts, the spectra of Ta-Cu heterocontacts do not show the contribution of copper .",
"In the spectra of heterocontacts of transition d-metals with Cu, Ag, and Au, the contribution of the latter is not manifested.",
"To begin with, consider a high-resistance (209 ohm) Ta-Cu heterocontact (see Fig.REF ).",
"Figure: (a) EPI spectra of the Ta-Cu point contact in the normal and superconductingstates.",
"T=1.4KT=1.4~K, R 0 R_0=210 ohm; the initial segment of the superconductingcurve containing the gap and nonequilibrium effects is scaled down by afactor of 100, B ˜ S {{\\tilde{B}}_{S}} and B N B_N are the background curves for the superconductingand normal spectra, respectively.",
"(b) The difference between the superconductingand normal spectra, and the assumed shape of the background curve.",
"(c)Difference curve (after background subtraction), scaling curve M, and the differencecurve multiplied by the scaling curve.",
"(d) Point contact EPI functionsrecovered from the spectra for normal and superconducting states and from thesuperconducting contribution to the spectrum by integration of the corrected differencecurve (c).",
"For convenience of comparison, the curve g pc S g_{pc}^{S} is aligned inamplitude with the curve g ˜ pc S \\tilde{g}_{pc}^{S}.",
"The scale is the same in all panels of the Figure.Formally, it satisfies all requirements of the theory: its diameter is $\\approx $ 2.2 nm, whereas the elastic scattering length in the tantalum bank of the contact at the liquid helium temperature is $\\approx $ 84 nm, the energy relaxation length at the Debye energy is approximately the same ($\\approx $ 90 nm), and the reduced coherence length in tantalum is $\\zeta \\approx $ 43 nm; in other words, all of the lengths are more than an order of magnitude greater than the contact diameter.",
"Nevertheless, the transformation of the spectrum upon transition to the superconducting state differs crucially from predictions of the theory for ballistic S-c-N point contacts.",
"Pay attention to the region of phonon energies.",
"First, in the superconducting state, there is no spectral smearing and no related decrease in the intensity.",
"Conversely, the phonon modes are markedly sharpened and their amplitude increases.",
"Since the resistance of this contact is very high, the above-noted nonequilibrium effect is located in a rather high-energy region and coincides with the soft phonon mode; therefore, it is impossible to follow the transformation of this mode upon the superconducting transition.",
"Regarding higher frequency regions of the spectrum, they can be examined most conveniently in Fig.",
"[Fig3]3(d), which shows the curves after background subtraction.",
"A comparison of $g^N_{pc}$ and $\\tilde{g}_{PC}^{S}$ demonstrates that upon transition to the superconducting state, the amplitude of the first phonon peak increases to the highest extent, while the growth of the high-energy peak is much less pronounced.",
"Therefore, it is impossible to perform a correct recovery of the EPI function from the superconducting contribution to the spectrum, after the same background subtraction procedure as was done for tin, by integration of the S-N-$B_S$ curve, because the rule of sums is not satisfied in this case.",
"This inconsistency was resolved by making a correction of the shape of the S-N-$B_S$ curve, which was done by multiplying it by the scaling curve M [see Fig.[Fig3]3(c)].",
"This empirical curve does not change the difference curve in the low-frequency region and increases its amplitude in the high-frequency region.",
"The result of recovery before and after the correction can be observed in Fig.REF .",
"Figure: A demonstration of how violating the rule of sums impacts the shape ofthe recovered EPI function.",
"EPI function with the background (1) recoveredwithout accounting for the correction of the suppressed high-frequency part ofthe S-N-B S B_S curve [see Fig.",
"[Fig3]3(c)] and without the background (2) recoveredfrom the corrected curve M(S-N-B S B_S).",
"The scale of the curves is the same asthat in Fig..The decrease in the amplitude of the superconducting contribution to the spectrum in the high-energy region is caused by suppression of the excess current in the contact by nonequilibrium quasi-particles mentioned above in the discussion dealing with nonspectral nonequilibrium effects in superconducting curves.",
"Figure: Differential resistance of the point contact whose characteristics areshown in Fig.",
"in the normal (N) and superconducting (S) states and excesscurrent vs. the bias.",
"The excess current panel shows the scaling curve M (seeFig.)",
"with designated excess current and voltage values corresponding to thestep position in the scaling curve.Figure REF presents the differential resistances of the point contact in the normal and superconducting states, the dependence of the excess current on the voltage, and the scaling curve for correction of the amplitude of the superconducting contribution to the spectrum [the same as in Fig.[Fig3]3(c)].",
"The vertical and horizontal segments refer to the voltage and excess current corresponding to the onset of suppression of the amplitude of the superconducting contribution to the spectrum.",
"Figure: (a) EPI spectra of the Ta-Cu point contact in the normal and superconductingstates.",
"T=1.7KT=1.7 K, R 0 R_0=73 ohm; the initial section of the superconductingcurve containing the gap and nonequilibrium effects is scaled down by a factor of100, B ˜ S {{\\tilde{B}}_{S}} and B N B_N are the background curves for the superconducting and normalspectra, respectively.",
"(b) The difference between the superconducting andnormal spectra and the assumed shape of the background curve.",
"(c) Differencecurve (after background subtraction), scaling curve M, and the difference curvemultiplied by the scaling curve.",
"(d) Point contact EPI functions recovered fromthe spectra for normal and superconducting states and from the superconductingcontribution to the spectrum by integration of the corrected difference curve (c).For the convenience of comparison, the curve g pc S g_{pc}^{S} is aligned in amplitude withthe curve g ˜ pc S \\tilde{g}_{pc}^{S}.",
"The scale is the same in all panels of the Figure.Let us consider the influence of the decrease in the point contact resistance.",
"Figure REF shows the spectra of the point contact with 73 ohm resistance.",
"In this case, the nonequilibrium effect already occurs at about $\\sim $ 5mV and coincides with the energy of the tantalum soft phonon mode by only an edge.",
"The shape of this mode in the superconducting state basically differs from that in the normal state and is manifested as a peak rather than a shoulder; the phonon peak at about 11.3 mV is also markedly sharper than that in the case of the 209 ohm contact.",
"If one refers to Fig.",
"[Fig6]6(d) , in the lowenergy part of the spectrum, attention is attracted by a much more pronounced increase in the peak amplitude upon transition to the superconducting state in comparison with the previous point contact; this supports the above idea that the contribution of the peripheral areas to the whole superconducting contribution increases with increasing point contact diameter.",
"However, the superconducting contribution for the high-energy part of the spectrum is markedly lower in this case than for the previous contact.",
"Figure: The first derivatives of the CVCs for a point contact whose characteristicsare shown in Fig.",
"in the normal (N) and superconducting (S) states andexcess current vs. the bias.",
"The excess current panel shows the scaling curveM (see Fig.)",
"with designated excess current and voltage values correspondingto the start of the ascending part of the scaling curve.According to Fig.REF , the suppression of the excess current with increasing contact bias is much greater in this case than in the former case, which is responsible for the observed result.",
"Owing to these factors together, the scaling curve for correction of the high-frequency part of the superconducting contribution is approximately twice greater for this contact than that for the contact considered above.",
"Figure: (a) EPI spectra of the Ta-Cu point contact in the normal and superconductingstates; N: T=4.6KT=4.6~K; S: T=1.7KT=1.7~K, R 0 R_0=16 ohm, the initial dashedsegment of the superconducting curve containing the gap and the nonequilibriumeffects is scaled down by a factor of 100, B N B_N is the background curve forthe normal spectrum.",
"(b) The difference between the superconducting andnormal spectra and the assumed shape of the background curve.",
"(c) Differencecurve (after background subtraction), scaling curve M, and the difference curvemultiplied by the scaling curve.",
"(d) Point contact EPI functions recovered fromthe spectra in the normal state and from the superconducting contribution to thespectrum by integrating the corrected difference curve (c).",
"For the convenienceof comparison, the curve g pc S g^S_{pc} is aligned in amplitude with the curve g pc N g^N_{pc} .",
"Thescale is the same in all panels of the Figure.Figure REF shows the EPI spectra for the Ta-Cu point contact (one of the lowest-resistance point contacts involved in the study).",
"When its resistance is 16 ohm, the diameter is close to 8.5 nm, which is almost 4 times greater than that of the previous point contact (the volume is almost 60 times greater).",
"Here the differences from the first two point contacts are much more pronounced.",
"For this contact, the nonequilibrium effect is located almost immediately after the gap effect at a voltage slightly above 2 mV.",
"This opens up the possibility to follow the fine structure of the initial section of the phonon spectrum of tantalum upon the transition to the superconducting state, because in the normal state, it is possible to follow only a gradual ascent of the spectrum in this energy range.",
"Figure: Second derivatives of the CVCs of the 16 ohm Ta-Cu point contact(see Fig.)",
"after background subtraction in the normal (N) and superconducting(S) states.",
"The scale of the curves is the same as that in Fig..Figure REF depicts the relevant EPI spectra in the superconducting and normal states after background subtraction.",
"For this contact in the superconducting state, the peak sharpening in the low-frequency region is much more pronounced than in the previous cases.",
"The spectral amplitude also greatly increases; meanwhile these changes in the high-energy part of the spectrum are much weaker.",
"In the difference curve [Fig.",
"[Fig8]8(c)], the high-energy part is barely seen against the initial part before the correction.",
"Figure: Differential resistance of the Ta-Cu point contact whose characteristicsare shown in Fig.",
"in the normal (N) and superconducting (S) states andexcess current vs. the bias.",
"The excess current panel shows the scaling curveM (see Fig.)",
"with designated excess current and voltage values correspondingto the start of the ascending part of the scaling curve.Figure REF shows the plots for differential resistances of this contact in the normal and superconducting states, excess current as a function of voltage, and the correction scaling curve.",
"As follows from the Figure, the scaling curve amplitude is approximately 6 times higher than that of the previous contact, which is evidently attributable to two factors mentioned above: a pronounced increase in the amplitude of the low-energy part of the superconducting contribution, and a more pronounced suppression of the highenergy part caused by decreasing excess current."
],
[
"Ta-Ta HOMOCONTACTS",
"The trends observed for heterocontacts are generally reproduced for homocontacts, although differences are also present: the spectra of homocontacts usually exhibit two nonequilibrium effects associated with attainment of the critical density of nonequilibrium quasi-particles in each bank.",
"Figure REF Figure: (a) EPI spectra of the Ta-Ta point contact in the normal and superconductingstates.",
"N: T=4.6T=4.6~K; S: T=2.0KT=2.0~K, R 0 R_0=64 ohm, the initial segments ofthe superconducting curve containing the gap and nonequilibrium effects arescaled down by factors of 300 and 10, respectively; B ˜ S \\tilde{B}_{S} and B N B_N are the backgroundcurves for the normal and superconducting spectra, respectively.",
"(b) Thedifference between the superconducting and normal spectra and the assumedshape of the background curve.",
"(c) Difference curve (after background subtraction),scaling curve M, and the difference curve multiplied by the scaling curve.",
"(d) Point contact EPI functions recovered from the spectra for normal and superconductingstates and from the superconducting contribution to the spectrum byintegration of the corrected difference curve (c).",
"The scale is the same in allpanels of the Figure.presents the EPI spectra of the 64 ohm Ta homocontact in the normal and superconducting states.",
"Since the temperature during the measurements in the superconducting state proved to be somewhat higher for this contact than for heterocontacts, and because of the presence of two nonequilibrium effects, all low-energy region down to less than 10 meV energies was inaccessible for observation of the behavior of the phonon effects.",
"Therefore, the initial section of the EPI function recovered from the superconducting contribution to the spectrum was approximated by a parabola.",
"Figure: Differential resistances of the Ta-Ta point contact whose characteristicsare shown in Fig.",
"in the normal (N) and superconducting (S) states andexcess current vs. the bias.",
"The excess current panel shows the scaling curveM (see Fig.)",
"with designated excess current and voltage values correspondingto the start of the ascending part of the scaling curve.Figure REF shows the differential resistances of the point contact in the normal and superconducting states, the energy dependence of the excess current, and the scaling curve.",
"As can be seen in Fig.",
"[Fig11]11(d), the sharpening and the intensity growth of the first peak upon the superconducting transition are markedly less pronounced in this case than for heterocontacts with a comparable resistance.",
"Regarding the superconducting contribution to the spectrum [Fig.",
"[Fig11]11(b)], its high-energy part is comparable in intensity to the low-energy one.",
"Therefore, the correction scaling curve M is relatively small in amplitude as compared with that of the heterocontacts.",
"Figure: a) EPI spectra of the Ta-Ta point contact in the normal and superconductingstates; N: T=4.6KT=4.6 K; S: T=2.55KT=2.55 K, R 0 R_0=17 ohm; the initial segments ofthe superconducting curve corresponding to the critical current and gap andnonequilibrium effects are scaled down by factors of 1000 and 100, respectively,B ˜ S \\tilde{B}_S and B N B_N are the background curves for the normal and superconductingspectra.",
"(b) The difference between the superconducting and normal spectraand the assumed shape of the background curve.",
"(c) Difference curve (afterbackground subtraction), scaling curve M, and the difference curve multiplied bythe scaling curve.",
"(d) Point contact EPI functions recovered from the spectra fornormal and superconducting states and from the superconducting contributionto the spectrum by integrating the corrected difference curve (c).",
"For the convenienceof comparison, the curve g pc S g^S_{pc} is aligned in amplitude with the curve g ˜ p S c\\tilde{g}^S_pc.The scale is the same in all panels of the Figure.Figure: Differential resistance of the Ta-Ta point contact whose characteristicsare shown in Fig.",
"in the normal (N) and superconducting (S) states andexcess current vs. the bias.",
"The excess current panel shows the scaling curveM (see Fig.)",
"with designated excess current and voltage values correspondingto the start of the ascending part of the scaling curve.Characteristics for the markedly lower-resistance homocontact (17 ohm) are shown in Figs.",
"REF and REF .",
"For this contact, the nonequilibrium effect was found to occur at less than 5 mV energy; therefore, the transformation of the soft mode during the superconducting transition of the contact was accessible for observation.",
"Like for heterocontacts, it is converted into a peak.",
"As follows from Fig.",
"[Fig13]13(d), pronounced sharpening and an amplitude increase of the first peak are also observed for this contact.",
"Regarding the high-energy part of the spectrum, the changes induced by the superconducting transition are moderate.",
"Nevertheless, a comparison with the proportion of the high-energy part of the superconducting contribution for the heterocontact with virtually the same resistance [Fig.",
"[Fig8]8(c)] demonstrates that this proportion is quite comparable in intensity with that of the low-energy part; therefore, the correction scaling curve is an order of magnitude smaller."
],
[
"RESULTS AND DISCUSSION",
"The conducted study of the ballistic point contacts based on tantalum demonstrated that all deviations from the predictions of the theoretical model are caused by involving a region that is about the size of the coherence length, which is adjacent to the constriction, in the formation of the EPI spectra upon the transition of the contact to the superconducting state.",
"Apart from other differences, these contacts show a pronounced dependence of the superconducting contribution to the spectrum on the contact bias and on the resistance.",
"This dependence, caused by partial suppression of the excess current by nonequilibrium quasi-particles, complicates the recovery of the EPI function from this contribution and requires a correction of its shape to ensure the fulfilment of the rule of sums, after subtraction of the superconducting background.",
"The search for the shape and intensity of the correction scaling curve was performed separately for each of the considered point contacts.",
"As has already been noted, the scaling curve intensity varies over wide limits, which is reflected in Fig.[Fig15]15(a).",
"However, the shapes of these curves were similar.",
"Figure [Fig15]15(b) shows all scaling curves M reduced to the same amplitude.",
"Moreover, the same correction curve proved to be applicable, with only the amplitude being varied.",
"For example, the curves for contacts 4 and 5 have the same shape, but differ in the amplitude approximately 3.7-fold.",
"Attention is attracted by the step present in all correction curves at similar energies with approximately the same amplitude on a reduced scale.",
"The curves for the excess current as a function of the bias show no peculiar effects at these energies, with the magnitude of the excess current smoothly descending.",
"By analogy with appearance of a step along the dependence of the excess current on the bias, associated with the attainment of the critical concentration of nonequilibrium quasi particles above the gap in the near-contact region, one can assume that there exists some threshold concentration of Andreev electrons determined by the excess current, below which the efficiency of reabsorption of nonequilibrium phonons sharply decreases, resulting in a sharp decrease in the superconducting contribution to the spectrum.",
"Further decreases in the concentration of Andreev electrons with increasing contact bias lead to a smooth decline of the superconducting contribution to the spectrum, which is reflected in the shape of the correction scaling curves.",
"This assumption is supported by the comparison of homo- and heterocontacts.",
"When the resistances are similar, the amplitude of the correction curves is much smaller for homocontacts, which is due to the doubled excess current.",
"However, as can be seen in Fig.REF , there are no obvious differences between the shapes of the correction curves for homo- and heterocontacts."
],
[
"BRIEF CONCLUSIONS",
"1.",
"The tantalum-based ballistic point contacts in the superconducting state show deviations from the predictions of theoretical models due to the involvement of a near-contact region with a size that is about equal to the reduced coherence length $\\zeta $ , in the formation of EPI spectra.",
"2.",
"The relative value of the superconducting contribution to the EPI spectrum depends on the contact resistance and increases with increasing contact diameter.",
"3.",
"The superconducting contribution decreases with increasing contact bias due to the suppression of excess current by the nonequilibrium quasi-particles, which leads to violation of the formulated rule of sums.",
"4.",
"Although the EPI functions are proportional to the first derivative of the excess current, the presence of the superconducting background and the need to correct the amplitude of the superconducting spectral contribution caused by violation of the rule of sums, bring about the necessity to recover the EPI functions from the CVC second derivatives.",
"5.",
"The procedure of recovering the EPI spectral function described in detail can be used to analyze the characteristics of ballistic point contacts based on a broad range of superconductors."
],
[
"ACKNOWLEDGMENTS",
"This study was supported by the National Academy of Sciences of Ukraine as a part of project FTs 4-19.",
"In conclusion, the author would like to express gratitude to A.V.",
"Khotkevich for numerous tips, comments, and discussions."
]
] | 1906.04380 |
[
[
"Secure Software-Defined Networking Based on Blockchain"
],
[
"Abstract Software-Defined Networking (SDN) separates the network control plane and data plane, which provides a network-wide view with centralized control (in the control plane) and programmable network configuration for data plane injected by SDN applications (in the application plane).",
"With these features, a number of drawbacks of the traditional network architectures such as static configuration, non-scalability and low efficiency can be effectively avoided.",
"However, SDN also brings with it some new security challenges, such as single-point failure of the control plane, malicious flows from applications, exposed network-wide resources and a vulnerable channel between the control plane and the data plane.",
"In this paper, we design a monolithic security mechanism for SDN based on Blockchain.",
"Our mechanism decentralizes the control plane to overcome single-point failure while maintaining a network-wide view.",
"The mechanism also guarantees the authenticity, traceability, and accountability of application flows, and hence secures the programmable configuration.",
"Moreover, the mechanism provides a fine-grained access control of network-wide resources and a secure controller-switch channel to further protect resources and communication in SDN."
],
[
"Motivation",
"In recent years, Software-Defined Networking (SDN) has received great attention from both the research community and the industry.",
"For example, Google has already implemented an SDN architecture, Google B4, on its data centers [1].",
"By separating the network control and the data plane, SDN overcomes several limitations of the traditional networks such as static configuration, non-scalability and low efficiency.",
"Due to the logical centralization of the control plane and the programmability of the configuration for the data plane, SDN provides a global view of network resources that enhances the performance and flexibility of the underlying networks.",
"Figure: The overview of attacks on SDN architectureHowever, applying SDN into networks also introduces some new security issues, which has been stressed in recent years among SDN researchers [2], [3], [4], [5].",
"The overview of attacks on SDN architecture is shown as Fig.",
"REF .",
"Most of these security issues exist in the application plane and the control plane.",
"On one hand, without authentication, applications may inject malicious configurations into network devices at will, which could reduce network availability, reliability and even lead to a network breakdown.",
"Note that application flows are network configurations sent by applications and are managed by controllers who install network configurations into switches.",
"Loss of traceability and accountability of application flows may cause trouble for network debugging.",
"In SDN, tracing and auditing application flows and network states can help monitor and replay network states or debug a broken-down network, and network behaviour information also can be applied to recognize attack patterns [6].",
"On the other hand, the control plane also exposes network-wide resources to all applications that will open a door for malicious applications.",
"Moreover, adopting a single controller may result in a single-point failure which can become an attractive target for DoS attacks.",
"Finally, the lack of authenticated controller-switch communication channel could lead to more severe security threats when the configuration-complex TLS protocol (Transport Plane Security, TLS) is not adopted.",
"Adversaries can launch man-in-the-middle attack and eavesdropping attack by seizing all packages between controllers and switches [7], [8].",
"In addition, malicious switches can also launch spoofing attacks by faking identities (e.g., IP addresses) which may lead to DoS/DDoS attacks [9], [10].",
"In order to address the above security problems, many proposals have been made in recent years [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28].",
"These proposals each targets at a specific security issue, but to the best of our knowledge, no attemps have been made which address all the common security issues simultaneously with a monolithic architecture.",
"Specifically, existing works that provide application flow authentication or flow secure constraint are to extend an individual secure module on a controller rather than a monolithic secure module in a multi-controllers environment.",
"Additionally, existing role-based access control schemes on network-wide resources are not fine-grained enough.",
"Moreover, optional TLS protocol and other cryptography-based authentication protocols presented in [27], [28] require multiple interactions (also called multiple pass) to build a controller-switch communication channel.",
"In a network with the physically decentralized control plane, simply combing those existing schemes fails to effectively solve all the common security issues simultaneously, because all secure modules must work seamlessly among multiple controllers."
],
[
"Contributions",
"In the paper, we present a Blockchain-based monolithic secure mechanism to effectively address multiple common security issues in SDNs.",
"In particular, the paper makes the following contributions.",
"$\\bullet $ Our mechanism decentralizes the control plane into multiple controllers while maintaining consensus among all controllers on network-wide resources.",
"$\\bullet $ To overcome the weakness of lacking traceability and accountability of application flows, all flows and network behaviours are recorded on Blockchain so that the network states can be easily replayed for auditing and debugging.",
"$\\bullet $ By assembling a lightweight and practical Attribute-Based Encryption (ABE) scheme, the access permissions of each application on network resources are defined and enforced to avoid resource abuse.",
"$\\bullet $ The effective authentication protocol HMQV (one-pass) is combined with Blockchain to protect the communication channel between the controller and switch against active attacks."
],
[
"Related Work",
"A variety of security measures directed against various security threats among different planes of SDN architecture have been proposed.",
"In terms of application plane, FRESCO[29] is a security development framework compatible with OpenFlow for SDN applications.",
"Cognition[11] was proposed to enforce the security of applications by defining cognitive functions.",
"OrchSec[12], an architecture considering the advantages of network-visibility and centralized control provided by SDN, was introduced to develop security applications.",
"FortNOX[13] extended NOX controller to provide security constraints on flow rules and an role-based authentication scheme for SDN applications.",
"FSL[14] presented a security authentication framework for flow-based network policies.",
"Similarly, Son et al.",
"proposed Flover[15] and Khurshid et al.",
"presented VeriFlow[16] to verify dynamic flow policies.",
"With the requirements to audit and track network process, Hadigol et al.",
"[17] studied network event debugger enabling network manager to track the root cause of a network bug.",
"OpenSAFE[18] was proposed to support security auditing and Flow Examination to analyze network traffic and filter network package.",
"On the control plane, a lot of frameworks with a decentralized control plane for OpenFlow were presented.",
"HyperFlow[19] was built on a distributed file system to realize network event distribution among multiple controllers.",
"Onix[20] implemented a physically distributed but logically centralized control platform to avoid threats brought from a single controller.",
"Also, SDN control frameworks such as ONOS[30], DISCO[21], yanc [22], PANE[23], and Flee[24] supporting the distributed network logic.",
"In order to secure network-wide resources, some security schemes [25], [26]were devoted to provide access control mechanism to protect resources from unconcerned SDN applications.",
"As for controller-switch channel, Transport Plane Security (TLS) was adopted on OpenFlow specification.",
"However, it became optional due to the insufferable drawbacks of TLS.",
"Apart from authenticated controller-switches communication, those were excellent security measures and systems, such as FRESCO [31], FloodGuard [32], AVANT-GUARD [33], FLOWGUARD [34], SE-Floodlight[35], SoftFirewall[36], CPRecovery [37] and so on [38]."
],
[
"Organization",
"The rest of the paper is organized as follows.",
"In Section , we provide a quick overview on Blockchain, Attribute-Based Encryption, and the HOMQV protocol.",
"Then, we describe our security requirements on OpenFlow/SDN in Section .",
"In Section , we present the design of the Blockchain-based monolithic module.",
"We analyze security issues of the construction in Section and in the Section we present a prototype implementation of our mechanism.",
"Lastly, we conclude the paper in Section ."
],
[
"Preliminaries",
"In this section, we give a brief introduction about Blockchain, Attribute-based encryption and HOMQV protocol.",
"Blockchain is originated from bitcoin and becomes an emerging technology as a decentralized, sharing, immutable database [39], [40], [41], [42], [43], [44].",
"Data in Blockchain is stored into blocks which are maintained as a chain.",
"Each block of Blockchain contains a timestamp and the reference, i.e., the hash of a previous block.",
"Blockchain is maintained in a peer-to-peer network.",
"The majority of Blockchain network nodes run a consensus protocol to achieve an agreement to generate a new block.",
"Meanwhile, the data, also called transactions, in the new block are also confirmed due to its consensus protocol.",
"Consensus protocols in the Blockchain setting can be implemented by several different agreement methods, such as POW-based (Proof of Work), BFT-based (Byzantine fault-tolerant) and POS-based (Proof of Stake).",
"We call them Blockchain protocols.",
"In the paper, we focus on the BFT-based Blockchain protocol [45].",
"This kind of protocol promises instant consensus[46].",
"In the meantime, Blockchain based on the kind of protocol can balance scalability and performance well, among which scalability means the number of participants and performance includes throughput and latency.",
"Vukolic et al.",
"[47] showed BFT-based Blockchains behave excellently in network performance and guarantee instant consensus.",
"It also presented that BFT-based Blockchains possess the excellent ability to support the large capability of clients.",
"By applying BFT-based Blockchains to our module, controllers in SDN play the roles as clients of the Blockchain, which demonstrates excellent network scalability of controllers.",
"On the other hand, there are two ways to write data on the Blockchain: transaction and smart contract.",
"Smart contract is a program that can automatically execute the partial and total operations pre-defined in the contract and output values as evidences supporting to be verified on Blockchain.",
"Smart contract usually provides an outer interactive interface and the interaction can be verified based on the cryptography so that smart contract is executed in strict accordance with the predefined logic.",
"In our context, we build security protocols by smart contract that can be automatically executed when the predefined conditions are triggered on the Blockchain.",
"Owing to the immutable recorded transactions and results of executed protocols, Blockchain supports network to trace any record linked with a specific time point.",
"Eventually, Blockchain enables the network-wide data to share some valuable features such as reliability, non-repudiation, traceability and auditability which are adapted to our security requirements.",
"Thus, this paper is interested in BFT-based Blockchains adapted to our security goals to construct a Blockchain-based secure module which strengthens the security in SDNs.",
"Attribute-based encryption (ABE) contributes to a fine-grained access control for encrypted data.",
"In an ABE system, a kind of encrypted resources are labeled with a set of descriptive attributes and a specific access structure which are associated with a private key of an access-user.",
"It determines the kind of encrypted resources that can be decrypted by the access-user with the private key satisfying the pre-defined access structure[48], [49], [50], [51], [52], [53].",
"We employ a lightweight ABE scheme which possesses execution efficiency and low communication costs[50] to achieve our secure and efficient requirements for access control.",
"As for the framework of ABE, four algorithms are introduced as follows and they will be used as black boxes.",
"$(PK,MK)$${\\sf =Setup}(\\kappa )$ : Taking input of the security parameter $\\kappa $ , this algorithm outputs the public parameters $PK$ and a master key $MK$ .",
"$E$${\\sf =Encryption}(m, attr, PK)$ : This is a randomized algorithm that takes as input a message $m$ , a set of attributes $attr$ , and the public parameters $PK$ .",
"It outputs the ciphertext $E$ .",
"$D$${\\sf =KeyGeneration}(\\textbf {A}, MK, PK)$ : This is a randomized algorithm that takes as input an access structure $\\textbf {A}$ , the master key $MK$ and the public parameters $PK$ .",
"It outputs a decryption key $D$ .",
"$M$${\\sf =Decryption}(E,D)$ : Taking as input the ciphertext $E$ that was encrypted under the set $attr$ of attributes, the decryption key $D$ for access control structure $\\textbf {A}$ and the public parameters $PK$ , it outputs the message $M$ if $attr \\in \\textbf {A}$ .",
"One-pass HMQV protocol (HOMQV) is a high-performance securely authenticated protocol which combines security, flexibility and efficiency [54].",
"Its security has been proved in [55].",
"Specifically, it uses a cyclic group $G$ of prime order $q$ and is generated by a given generator $g$ .",
"In the initial step, there are two communication parties Alice ($ID_{Alice}$ ) and Bob ($ID_{Bob}$ ) with the long-term keys $A=g^a$ , $B=g^b$ , respectively.",
"Before a key-exchange protocol, Bob first checks the key $A$ sent by Alice that $A\\in G^{\\prime }$ (if not it aborts).",
"Then Bob randomly chooses $y\\in _{R} Z_q$ , computes $Y=g^y$ and sends it to Alice.",
"Bob also computes a session key $H(\\sigma ,ID_{Alice},ID_{Bob},Y)$ where $\\sigma =A^{(y+eb)}$ and $e=H^{\\prime }(Y,ID_{Alice})$ .",
"When receiving $Y$ and $ID_{Bob}$ , Alice checks $Y$ and Bob's public key in $G^{\\prime }$ (if not it aborts) and then computes the session key $H(\\sigma ^{\\prime },ID_{Bob},ID_{Alice},Y)$ where $\\sigma ^{\\prime }=(YB^e)^a$ .",
"Finally, Alice and Bob share the same session key because of $\\sigma =\\sigma ^{\\prime }$ and the triple $(ID_{Alice},ID_{Bob},Y)$ is regarded as the session id.",
"However, applying the basic HOMQV protocol to controller-switch communication, controllers as receivers fail to resist reply attack since the protocol only has one pass.",
"In addition, it needs a certificate authority to update the long-term keys of two parties in the basic protocol.",
"Fortunately, the two security issues can be overcome by using the Blockchain, which enables a security-strengthen protocol applied into the authenticated controller-switch communication."
],
[
"Security Requirements",
"In this section, we list security requirements which should be achieved.",
"Application flows authentication.",
"The nature of high programmability and configurability on network devices of OpenFlow/SDN forces us to pay more security attention on the application plane.",
"Applications (e.g., traffic engineering) provide a variety of management services for network by configuring application flows versus new threats for network.",
"For example, a malicious or compromised application may inject malicious application flows into network devices thereby leading to a dramatic consequence.",
"Therefore, to authenticate application flows from legitimate applications and verify the authenticity of application flows are significant for the configurable OpenFlow/SDN.",
"Application flows tracing and accounting.",
"Traceability and accountability for application flows can assist operators to troubleshoot network once a network device breaks down or suffers from abnormal network behaviors.",
"On the other hand, for the scenarios of flow arbitration, a flow arbitration system with the duty to arbitrate conflict flows, needs to identify which flows are sent by which applications[56] where traceability and accountability for application flows are urgent to be provided.",
"Network behaviours auditing.",
"An audit system provides periodic auditing for network behaviours (network events associated with resulting network states), which helps to enforce the stability and robust the security of OpenFlow/SDN-based network.",
"Analyzing from auditing results by linking network events with respective network states in current time, operators can make adjustment of the network management and enhance network performance next time.",
"Furthermore, relying on the audit system, attack pattern recognition also can be supported to resist future network attacks.",
"Secure access control on network-wide resources.",
"It faces some potential threats that network-wide resources on the control plane are exposed to all applications.",
"For instance, Hartman et al.",
"[35] presented a kind of network security applications serving for firewall or intrusion detection that can access network resources of the firewall.",
"A malicious application may abuse the resources by utilizing the instance of the kind application to bypass the firewall.",
"Consequently, it is necessary to construct a secure access control mechanism which is customized to applications according to their categories and the network scope they are supposed to contribute to.",
"Decentralization of control plane.",
"A single controller is not feasible.",
"Obviously, a single-point failure may occur and the scalability to expand network is limited because a single controller should ensure the endurance capacity for network flows from various applications and manage a large number of devices.",
"On the other hand, a distributed control plane can improve the flexibility and resilience of network, e.g., each controller is responsible for a network slice with a certain number of devices.",
"In the meantime, we also require a distributed control plane to sustain logically centralized network view which is one of important features in SDN.",
"There are distributed controllers including Onix[20], HyperFlow[57], HP VAN SDN[10], ONOS[30], DISCO[21], yanc[22], PANE[23], and Fleet[24], but among which few of them can maintain a consistent view of the global network resources.",
"Controller-switch communication authentication.",
"The communication channel between controllers and switches suffers from some active attacks in SDN networks, such as man-in-the-middle attack, reply attack and spoofing attack.",
"Actually, the implementation of OpenFlow originally defined TLS[58] for the controller-switch communication.",
"However, it made TLS adoption in option on its latter versions due to the high complexity of configuration and high communication cost of TLS[59], [15].",
"This results in lots of security threats such as malicious flow insertion and flow modification when a controller installs a flow to a switch[7].",
"Thus, any device connected to a controller under an lightweight authentication protocol is inevitable."
],
[
"Concrete Design",
"In this section, we give a concrete design of the Blockchain-based monolithic secure mechanism as Fig.",
"REF , and introduce it from four aspects: Blockchain layer, entities building, transactions building and protocols building.",
"Additionally, we represent cryptographic primitives an asymmetric encryption algorithm and a digital signature algorithm as $\\textsf {AE}$ and $\\textsf {DS}$ , respectively.",
"$\\textsf {AE}$ algorithm is claimed by $\\textsf {(KeyGen, Enc, Dec)}$ , the key generation, encryption and decryption algorithms respectively and $\\textsf {DS}$ algorithm is defined by $\\textsf {(KeyGen, Sig, Ver)}$ , the key generation, signature and verification algorithms respectively.",
"A pair of public key and private key in the algorithms are represented as $PK$ and $SK$ ."
],
[
"Blockchain layer",
"Blockchain is used as a packaged and underlying component.",
"It provides functionalities of resource-recording and resource-sharing among multiple controllers on the control plane.",
"The functionality of resource-recording represents that Blockchain can be used to record network resources of each controller.",
"The functionality of resource-sharing demonstrates all recorded resources (mainly network events) are shared among all controllers, thereby maintaining the same network view.",
"We utilize the existing stable Blockchain platform to implement our requirements rather than building a new Blockchain.",
"The original consensus protocol of the applied Blockchain is not changed and it guarantees the reliability of our new network architecture.",
"The reason is that many Blockchain-based applications, in order to obtain new required functionalities, create a new Blockchain production (such as utilizing a variant of consensus protocol), but there exist some potential threats, e.g., making chain fork and resulting in a disastrous loss.",
"Thus, the mechanism applies a worth examining stable Blockchain as the underlying layer of the Blockchain layer.",
"Blockchain is used to write down all application flows and network events associating with the respective network states where those data are represented as raw transactions.",
"In addition to transactions, we build smart contract to implement security protocols which further satisfy the security requirements (e.g., to alert the failure of a controller in time).",
"The timestamping and trustworthy features of Blockchain enable the real-time reliability of all recorded application flows and all time-series of the network-wide views during the running process in the arbitrary time.",
"On the other hand, multiple controllers who are regraded as clients to participate in the underlying Blockchain undertake to record network data (from the application plane and from the device plane) as raw transactions into Blockchain.",
"The motivation of controllers to manage network, to an extend, keeps the liveness of the underlying Blockchain.",
"Meanwhile, all applications are obliged to provide network flows (or policies) through the control plane for network devices (e.g., OpenFlow/SDN switches) on the data plane.",
"OpenFlow/SDN switches also intend to work in network by sending messages (or events) to controllers or providing its resources to controllers.",
"For example, a switch will request its registered controller when it receives a coming package but fails to forward it.",
"Those network motivations of participating entities enable the underlying Blockchain to be applied significantly.",
"However, the question on selecting which kind of consensus protocol that the trustworthy underlying Blockchain is based on should be carefully considered.",
"We desire to gain confirmed transactions (network resources) without canceling on the Blockchain where there does not exist any fork or appear forks with overwhelming probability and to maintain consistency of those transactions (network resources) among all controllers.",
"We expect the underlying Blockchain layer to reach consensus under a negligible consensus latency and without the presence of the temporary forks.",
"The two requirements for the underlying Blockchain imply the property of consensus finality, which is proposed by Vukolic[47].",
"Consensus finality property is defined that once a valid block was appended to the Blockchain at some point in time, the block never was abandoned from the blockchian.",
"[47] also claimed and proved that any BFT-based Blockchain can satisfy the property of consensus finality, wherein it also supports the excellent network performance and thousands of clients, that is significant to apply to SDN.",
"[60] indicated that there are practical systems (e.g., Ripple network2 [61] or OpenBlockchain3 [62]) implementing the transformation from the eventually consistent POW consensus to the instantly consistent BFT consensus.",
"Moreover, the transaction processing ability per second is also concerned.",
"It implies throughput capacity of the Blockchain can perform, which determines the network performance of our module built on the Blockchain (e.g., throughput capacity of the control plane on network events which are network-wide resources).",
"Fortunately, BFT-based Blockchain protocols enjoy excellent performance on throughput[47].",
"Therefore, a kind of Blockchains based on BFT protocols such as Ripple network2 [61] are adopted to implement the Blockchain layer in new architecture of SDN."
],
[
"Entities building",
"In our context, we define entities who are actively participating in SDN including applications (i.e., $APP$ ), controllers (i.e., $CON$ ) and switches (i.e., $SWITCH$ ).",
"Formally, we give expressions to describe the entities in the form of multi-tuples as follows: $APP & =(\\textrm {ID_{app}}, \\textrm {PK_{app}}, \\textrm {SK_{app}}, \\textrm {category}) \\\\CON & =(\\textrm {ID_{contr}}, \\textrm {PK_{contr}}, \\textrm {SK_{contr}}, \\textrm {Slice})\\\\SWITCH & =(\\textrm {ID_{switch}}, \\textrm {PK_{switch}}, \\textrm {SK_{switch}}, \\textrm {Slice})$ Specifically, a unique identifier $ID$ is used to represent the identity of an entity.",
"An application can use the unique application package name as its identity $ID_{app}$ .",
"For controllers and switches, they use their unique IP addresses or Media Access Control(MAC) addresses, $ID_{contr}$ and $ID_{switch}$ respectively.",
"A switch works out a pre-setting cryptography puzzle to enable an effective IP address used to register the network.",
"This method limits the ability of switch to control IP addresses used in the network, which is resistant to spoofing attack.",
"The tuple $Slice$ represents the network slice a controller or a switch belongs to (The network of SDN composes of several network slices).",
"Additionally, we employ an asymmetric key generation algorithm, $(\\textrm {$ PK$}, \\textrm {$ SK$}) \\leftarrow \\textsf {KeyGen}$ , to create a pair of public key and private key for an entity.",
"According to the autonomous key-selection mechanism used in the context of Blockchain, each entity generates its keys as it desires by the $\\textsf {KeyGen}$ algorithm.",
"The tuple named $category$ in application expression is defined according to use cases among most of SDN applications.",
"We also define application flows from the application entities and network events created by the switch entities.",
"The entities building flows and creating events meanwhile are responsible to generate identities for them respectively.",
"$flow & =(\\textrm {ID_{flow}}, \\textrm {content}, \\textrm {PK_{app}}, \\textrm {ID_{contr}},\\textrm {ID_{switch}}) \\\\event & =(\\textrm {ID_{event}}, \\textrm {event}, \\textrm {PK_{switch}}, \\textrm {ID_{contr}}, \\textrm {ID_{switch}})$ The defined tuples for flows and events also indicate where they come from and where they contribute to.",
"When a flow is sent by an App, the App attaches the flow with its signature for the flow $Sign_{flow}$ by its private key $\\textrm {Sign_{flow}} &= \\textrm {\\textsf {DS.Sig}(SK_{app},ID_{flow}||ID_{contr}||content)})$ and the same process is necessary for an event."
],
[
"Transactions building",
"From the time when SDN entities participate in network and during the period they act in network, their all active histories are built into meta-data of transactions on the Blockchain.",
"The data are classified into three classes: transactions of registered entities, transactions of application flows and transactions of network events.",
"Transactions of registered entities that not only effectively indicate the existences of entities in SDN but also the connected relationships among them.",
"At first, each controller managing SDN provides register information which is described in a built entity.",
"And then they are generated into transactions $T_{contr}$ , $T_{contr} &=(\\textrm {ID_{T_{contr}}}, \\textrm {ID_{contr}}, \\textrm {PK_{contr}}, \\textrm {Slice}, \\textrm {\\textsf {DS.Sig}(SK_{contr},}\\\\&\\textrm {ID_{contr}||Slice)})$ in which $ID_{contr}$ concatenating $Slice$ can be verified by algorithm $\\textsf {DS.Ver}$$(PK_{contr}, \\textsf {DS.Sig}$$(SK_{contr}, ID_{contr}||Slice))$ that $ID_{contr}$ in the network slice $Slice$ is linked with $PK_{contr}$ and has been registered on the Blockchain layer.",
"An application connects with a controller to provide network flows for switches which connect with the controller.",
"Thus, the entity information of the application and the relationship information representing that the application connects with the controller will be recorded as $T_{app}$ and $T_{app-contr}$ .",
"$T_{app} &=(\\textrm {ID_{T_{app}}}, \\textrm {ID_{app}}, \\textrm {PK_{app}}, \\textrm {category}, \\textrm {ID_{contr}}, \\textrm {Sign_{app}})\\\\\\textrm {Sign_{app}} &= \\textrm {\\textsf {DS.Sig}(SK_{app}}, \\textrm {ID_{app}||category||ID_{contr})}\\\\T_{app-contr} &=(\\textrm {ID_{T_{app-contr}}}, \\textrm {ID_{T_{app}}}, \\textrm {ID_{T_{contr}}})$ From transactions $T_{app}$ , only the legitimate application owning the public key can succeed to verify $\\textsf {DS.Ver}$$(PK_{app}$ , $\\textsf {DS.Sig}$$(SK_{app},ID_{app}||category||ID_{contr}))$ the respective signature of the identity be accepted by the controller.",
"Additionally, transactions $T_{switch}$ and $T_{contr-switch}$ are generated once the controller-switch communication is built.",
"$&T_{switch}=(\\textrm {ID_{T_{switch}}}, \\textrm {ID_{switch}}, \\textrm {PK_{switch}}, \\textrm {Slice},\\textrm {ID_{contr}}, \\textrm {Com})\\\\&T_{contr-switch}=(\\textrm {ID_{T_{contr-switch}}}, \\textrm {ID_{T_{contr}}}, \\textrm {ID_{T_{switch}}})$ Note that a controller and a switch build an authenticated communication by using HOMQV protocol which has two security problems.",
"The two problems are that the origin HOMQV protocol fails to against reply attack and the sender of it needs to update a long-term key with a third party (the long-term key is the key of a switch used to construct a session key with a targeted controller).",
"We emphasize that the two security issues of the protocol can be overcome with the help of Blockchain.",
"The transaction $T_{switch}$ contains the information ($ID_{switch}$ and $PK_{switch}$ ) of a switch who launches a communicated request to a targeted controller following the HOMQV protocol.",
"Supposed that a compromised switch launches reply attack to a controller by frequently sending its information.",
"By auditing $T_{switch}$ on the Blockchain, a controller can refuse a replied request when the replied identity ($ID_{switch}$ and $PK_{switch}$ ) included in the $T_{switch}$ .",
"It is natural to understand because the Blockchain is regarded as a timestamping database and meanwhile stores communication histories among two parties of the protocol.",
"On the other hand, in the transaction $T_{switch}$ , the tuple $Com$ $Com= \\textrm {\\textsf {AE.Enc}(PK_{contr}, nonce, \\textsf {DS.Sig}(SK_{switch}, nonce))}$ of it is a commitment in encryption by the public key of a controller this switch connects with, which is helpful for the controller to confirm the identity of the switch when its long-term key has been updated.",
"The commitment includes a nonce the switch selects when the first connection is built.",
"Specifically, when the same but key-updated switch reconnects with the controller, the controller will audit the transaction $T_{switch}$ on the Blockchain.",
"It extracts the encrypted tuple from the $T_{switch}$ and makes an identity-verification challenge for the switch.",
"If the switch could answer the correct nonce that the controller verifies whether the encrypted value, with its public key of the nonce is equal to the value extracted from the logged connection transaction.",
"If it does, the verification is effective and the controller continues to share a new session key using its updated key.",
"An transaction of an application flow, represented by $T_{flow-afore}$ and $T_{flow-after}$ , includes the identity of the flow, the flow content, the identifier of the flow-stemming application, the identifier of the targeted controller and a signature signed by the flow-stemming application with its private key.",
"$T_{flow-afore}$ and $T_{flow-after}$ are on behalf of the transactions that some application flow is injected into the network by a specific application, passes by a related controller and ultimately is installed on a specific switch.",
"$T_{flow-afore}$ records the process from some specific application to a specific controller and the other one $T_{flow-after}$ demonstrates its trace from the controller to a specific device.",
"$T_{flow-afore} &=(\\textrm {ID_{T_{flow-afore}}}, \\textrm {ID_{flow}}, \\textrm {ID_{contr}}, \\textrm {PK_{app}},\\textrm {content},\\\\&\\textrm {Sign_{flow}})\\\\\\textrm {Sign_{flow}} &= \\textrm {\\textsf {DS.Sig}(SK_{app},ID_{flow}||ID_{contr}||content)})\\\\T_{flow-after} &=(\\textrm {ID_{T_{flow-after}}}, \\textrm {ID_{flow}}, \\textrm {ID_{contr}}, \\textrm {ID_{switch}}, \\textrm {state})\\\\T_{flow} &=(\\textrm {ID_{T_{flow}}}, \\textrm {ID_{T_{flow-afore}}}, \\textrm {ID_{T_{flow-after}}})$ Before generating $T_{flow-afore}$ , controllers need to authenticate all application flows.",
"A malicious flow will be filtered because controllers will audit the application creating the flow and verify its identity with the transaction $T_{app}$ .",
"The transaction $T_{flow-after}$ is recorded until the flow is installed into the flow table of a switch and the transaction contains the respective resource states $state$ sent by the switch where its description is omitted here because this process is similar with the process to generate events by switches as following.",
"Eventually, the last kind of transactions, that is, transactions of network events dedicate the network events provided by switches when the authenticated communication links have been built.",
"The transactions $T_{event}$ mainly are the dynamic states which are triggered by the respective application events (including in application flows) and additionally Pack_in messages.",
"$T_{event} &=(\\textrm {ID_{T_{event}}}, \\textrm {ID_{event}}, \\textrm {PK_{switch}}, \\textrm {ID_{contr}}, \\textrm {ID_{switch}}, \\textrm {event})$ Note that an authenticated communication link enables two parties to share a session key and any message among them is protected with the sharing session key.",
"However, messages (network events) included in a generated transaction $T_{event}$ on the Blockchain are in the form of non-encryption.",
"The messages are thought being authenticated because they come from a trustworthy controller-switch communication following the HOMQV protocol."
],
[
"Protocols building",
"Relying on the timestamping network records on the Blockchain, we defined necessary protocols to enforce security.",
"In the protocols, a lightweight and efficient ABE scheme and an authentication protocol HOMQV are implemented.",
"With the help of the defined protocols, our security goals are achieved.",
"We explain how to realize the security goals based on the Blockchain and introduce how to combine the Blockchain with ABE scheme and HOMQV protocol.",
"Security enhancement on the control plane We define protocols of detection and authentication for the newly coming application flows based on the existing transactions on the Blockchain.",
"At first, AuthFlowProtocol is implemented to enable the authentication of application flows.",
"In the protocol a controller first, by $IsRightFlow( )$ verifies the identity of an application creating the flow based on the transactions $T_{app}$ , $T_{app-contr}$ , $T_{contr}$ and $T_{contr-switch}$ .",
"Then, it uses the public key of the application to verify the flow content, by $verifyFlow(PK_{app})$ when the first step is valid.",
"[h] AuthFlowProtocol: To check whether the flow are related to the registered application and controller with the records $T_{app}$ , $T_{app-contr}$ , $T_{contr}$ and $T_{contr-switch}$ on Blockchain.",
"If the relationship records exist, the protocol continues and uses $PK_{app}$ to verify the signature.",
"Then, if the signature is signed by the application and the content of the sent flow does not be modified.",
"procedure call FlowReplyDetectionProtocol $IsRightFlow()$ $\\equiv $ true $verifyFlow(PK_{app})$ $\\equiv $ true $generateT_{flow-afore}(flow)$ end if end if end procedure Similarly, FlowReplyDetectionProtocol is used to detect the replied flows by auditing the identifiers of flows based on the transactions $T_{flow-afore}$ .",
"It is called by AuthFlowProtocol as its sub-protocol before executing the protocol logic to verify the identity of a flow application.",
"When the flow is installed in a switch by some controller and the associating network states are record, the transaction $T_{flow-after}$ is generated.",
"FlowReplyDetectionProtocol: To check $ID_{flow}$ and determine if it has existed.",
"If it has existed, the protocol uses $PK_{app}$ to gain $T_{app}$ .",
"If $T_{app}$ exists, the protocol decreases the reputation value of the application as punishment and returns false.",
"If it is a new flow, return true.",
"procedure $checkID_{flow}()$ $\\equiv $ true $getT_{app}(PK_{app})$ $\\equiv $ true $reduceReputation(T_{app}.ID_{app})$ return false end if end if return true end procedure Following the same mechanism as the protocol, the protocol AuditNetworkProtocol provides traceability of network behaviours by auditing the transactions $T_{flow\\_afore}$ , $T_{flow-after}$ and $T_{event}$ .",
"In particular, it enables the traceability of application flows when it has been injected into network and traces a cause of a network event.",
"It works because those logged network data (transactions) not only record the sent network flows among SDN entities but also demonstrate all network behaviour which has arose.",
"Within a running network, if the network suffers abnormal attacks, the attack processes also are logged as transactions.",
"On the other hand, with those logged transactions of attack trajectories, the future attacks lunched on the network can be recognized that is attacks pattern recognition.",
"The functionality of the protocol is configured flexibly by network managers according to the need of troubleshooting network.",
"As for the requirement to support notifications for the applications after a process of flows-arbitration is finished, we define the protocol FlowArbitrationLossNotifyProtocol.",
"By the protocol, an application lacking arbitration would gain a notification.",
"It audits the network records that demonstrate which one of conflicted flows targeted at the same switch is adopted.",
"Specifically, it checks the latest record that flow is generated into a transaction $T_{flow-after}$ and $T_{flow-afore}$ .",
"After finishing the process to arbitrate conflicted flows, the controller sends a notification to the application generating the arbitrated flow.",
"FlowArbitrationLossNotifyProtocol: Send a notification to an application which is out of arbitration.",
"procedure $getNewBlock( )$ $ID_{flow}$ $\\leftarrow $ $auditT_{flow-after}( )$ $ID_{T_{flow-afore}}$ $\\leftarrow $ $auditT_{flow-afore}(ID_{flow})$ $PK_{app}$ $\\leftarrow $ $auditT_{flow-afore}(ID_{T_{flow-afore}})$ $ID_{app}$ $\\leftarrow $ $auditT_{app}(PK_{app})$ $NotifySDN\\_APP(ID_{app})$ end procedure In order to ensure the real-time stable response for switches, controllers that the switches link with in SDN need to keep active.",
"Thus, ControllerFailedNotifyProtocol is defined to notify a switch when its directly connected controller has failed.",
"It depends on an assumption that, if some controller never participates in or becomes off-line, all transactions on the latest several blocks would not demonstrate any network behaviour of the controller.",
"That is, by auditing the transactions of the latest several blocks, the controller are identified being failed to a certain extent.",
"Based on the idea, we define the protocol which is executed automatically once some controller is alive-loss in a period of time on the Blockchain.",
"The protocol needs to periodically check whether all controllers are active by reading records of network behaviours with the latest block being created.",
"ControllerFailedNotifyProtocol: It is responsible to send a notification to the switches connecting with a controller when the controller is failed.",
"procedure $(T_{event}, T_{flow\\_after}) \\leftarrow getLastSixBlocks( )$ $auditAliveOfController$ ($T_{event}, T_{flow\\_after}$ ) $\\equiv $ null continue end if $auditAliveOfController$ ($T_{event}, T_{flow\\_after}$ ) $\\lnot \\equiv $ null $[ID_{T_{contr}}] \\leftarrow getFailedControllers( )$ $ID_{T_{contr}}$ in $[ID_{T_{contr}}]$ $[ID_{T_{switch}}]$ $\\leftarrow $ $auditT_{contr-switch}(ID_{T_{contr}})$ $[ID_{switch}] \\leftarrow getSwitches([ID_{T_{switch}}])$ $ID_{switch}$ in $[ID_{switch}]$ $NotifySwitch(ID_{switch})$ end for end for end if end procedure To stress it once again, we adopt multiple controllers on the control plane while a consistent network view of resources can be maintained using Blockchain as a sharing resource channel.",
"Since all controllers record all network events and collect network resources from devices connecting with it, the network-wide resources are public when the underlying Blockchain announces a new block.",
"Note that the lastly transactions are viewed valid and accepted consistently under the consensus mechanism of the underlying Blockchain, specifically, the aforementioned BFT-based Blockchain.",
"Therefore, depending on the creation of the reliable new block following the underlying consensus protocol, all controllers achieve consensus on the whole network view.",
"Figure: The access control on the network-wide topology resourcesSecure assess control on network-wide resources AccessControlProtocol: It provides attribute-based access control on network-wide resources with for SDN Apps.",
"procedure ${\\sf (PK,MK)}$ $\\leftarrow $ ABE.${\\sf Setup}$ () $getLatestTransactions$ ( ) $[T_{app}, T_{app-contr}]$ $\\leftarrow $ $getTappAndTapp\\_contr$ ( ) $T_{app}$ in $[T_{app}]$ $T_{app}$ in $[T_{app-contr}]$ $[T_{contr}]$ $\\leftarrow $ $getTapp\\_contr$ ( ) end for $[Attributes]$ $\\leftarrow $ $buildAttrForApp$ ($T_{app}, [T_{contr}]$ ) end for ${\\sf E}$ $\\leftarrow $ ABE.${\\sf Encryption}$ (${\\sf TD}$ , Attributes,${\\sf PK}$ ) ${\\sf D}$ $\\leftarrow $ ABE.${\\sf KeyGeneration}$ (${\\sf AC}$ ,${\\sf PK}$ , ${\\sf MK}$ ) ${\\sf M}$ $\\leftarrow $ ABE.${\\sf Decryption}$ (${\\sf D}$ ,${\\sf E}$ ) end procedure AuditAuthenRequestProtocol: It helps to resist replay attacks of connection requests from a switch.",
"If the switch has been connected, it sends three parameters(PK, Cipher, ID_of_request); otherwise, it sends two parameters(PK, Cipher).",
"$checkNumOfParam( )$ $\\equiv $ 2 $[T_{switch}]$ $\\leftarrow $ $getT_{switch}$ ( ) $checkResult$ $\\leftarrow $ $checkIsReplyRequest$ ($PK_{switch}$ , $C_{AE.Enc}(ID_{switch})$ , $[T_{switch}]$ ) $checkResult$ $\\equiv $ $false$ $T_{switch}$ $\\leftarrow $ $buildTransForSwitch$ ($PK_{switch}$ , $C_{AE.Enc}(ID_{switch})$ , $ID_{contr}$ ) $ID_{T_{switch}}$ $\\leftarrow $ $getFromT_{switch}$ ($T_{switch}$ ) $executeHOMQV( )$ end if $checkResult$ $\\equiv $ $true$ $reduceReputation(PK_{switch})$ return false end if end if $checkNumOfParam( )$ $\\equiv $ 3 Call SwitchChallengeProtocol SwitchChallengeProtocol $\\equiv $ $true$ $executeHOMQV( )$ end if return true end procedure SwitchChallengeProtocol: An honest key-updated switch who is intent to rebuild a connection needs to resend a new cipher $Com_{new} = \\textsf {AE.Enc}(PK_{contr}, nonce_{new}, \\textsf {DS.Sig}(SK_{switch}, nonce_{new}))$ which contains a new nonce and this nonce is required to be equal the nonce which is sent by the switch in the last authenticated connection.",
"procedure //get the last tuple of Tswitch, that is a cipher.",
"$[T_{switch}]$ $\\leftarrow $ $getT_{switch}$ ( ) $Com$ $\\leftarrow $ $T_{switch}.tuple[T_{switch}.length-1]$ $Signature$ $\\leftarrow $ $AE.Dec_{SK_{contr}}$ ($Com$ ) $Signature_{new}$ $\\leftarrow $ $AE.Dec_{SK_{contr}}$ ($Com_{new}$ ) $nonce_{new}$ $\\equiv $ $nonce$ and $\\textsf {DS.Ver}(PK_{switch}, Signature))$ $\\equiv $ $\\textsf {DS.Ver}(PK_{switch}, Signature_{new}))$ rerun true end if end if return false end procedure We apply ABE scheme to achieve secure access control on the network-wide resources.",
"Controllers manage the resources with fine-grained access by encrypting each resource with a set of related attributes.",
"Each individual SDN App keeps a private key associated with an access structure (${\\sf AC}$ ) consisting of a set of attributes and their relations (AND/OR).",
"Each attribute set is composed of App identities, App functionalities plus the relationships between the Apps and the controllers.",
"For example, a monitoring App is assigned with an attribute set that includes its identity, its function (i.e., monitoring) and two controller identities it has connected ($Attributes$ =$\\lbrace ID_{app}, ``monitoring\", ID_{contr_{1}}, ID_{contr_{2}}\\rbrace $ ).",
"In particular, [63] concludes application functionalities of majority of SDN Apps can be classified into the following five functions: traffic engineering; mobility and wireless; measurement and monitoring; security and dependability and data center networking.",
"After being encrypted, different kinds of network resources are only public to the proper Apps whose access structure related to its private key is satisfied with the set of attributes used to encrypt the kind of resources.",
"For example, as shown in Fig.",
"REF , the network resources are network-wide topology diagrams (${\\sf TD}$ ) which are maintained by controllers.",
"By utilizing the fine-grained access control, we encrypt different topology diagrams.",
"Different Apps access the diagrams (topology of some devices rather than all devices) they can decrypt with their private key.",
"Specifically, an example of the access control made on the network resources of topology diagrams is defined by the protocol AccessControlProtocol.",
"In the example, we take a traffic engineering App ${\\sf app1}$ obtaining topology resources of devices as shown in Fig.",
"REF .",
"We assume ${\\sf app1}$ has registered the controller ${\\sf contr1}$ and ${\\sf contr1}$ manages switches in the ${\\sf slice1}$ and ${\\sf slice2}$ , which means ${\\sf app1}$ can access the topology diagrams in the ${\\sf slice1}$ and ${\\sf slice2}$ .",
"Note that the relationships among ${\\sf app1}$ , ${\\sf contr1}$ and switches are recorded as the registered transactions that can be read by ${\\sf contr1}$ on the Blockchain.",
"In a word, we need to make an access policy for an App with the network functionality of traffic engineering and undertaking to provide services for switches in the ${\\sf slice1}$ and ${\\sf slice2}$ .",
"On the other hand, an access structure according to a set of attributes owning to ${\\sf app1}$ (i.e., $Attribute$ = $\\lbrace ID_{app1}, ``traffic\\ engineering\", ID_{contr1}\\rbrace $ ) is constructed by ${\\sf contr1}$ .",
"Then, ${\\sf contr1}$ executes ABE.${\\sf KeyGeneration}$ algorithm with the access structure to generate a private key ${\\sf D}$ for ${\\sf app1}$ .",
"Lastly, ${\\sf app1}$ uses ${\\sf D}$ to execute ABE.${\\sf Decryption}$ algorithm and obtain the topology diagrams.",
"Authenticated controller-switch communication The authenticated controller-switch communication is implemented by utilizing HOMQV protocol.",
"Meanwhile, in the context of Blockchain, we define two protocols to enhance the aforementioned security issues of HOMQV protocol.",
"On one hand, if being employed directly, HOMQV protocol fails to guarantee a controller resists replay attack launched by a switch who is ready to connect.",
"AuditAuthenRequestProtocol is defined to overcome effectively the issue.",
"On the other hand, a long term key which is used to key exchange to share a session key, could be self-updated as the switch pleases.",
"At that time, a switch needs to rebuild authenticated communication with the controller it connects last time.",
"The protocol SwitchChallengeProtocol is defined that a key-updated switch proves its existing connection with a controller in SDN and refreshes its connection with the controller."
],
[
"Security Analysis",
"In this section, we discuss security issues in our mechanism: authentication for application flows, replay attack detection for flows, notification of failed controllers for switches, secure access control for network-wide resources and authentication for controller-switch connection.",
"We first give fivefold secure knowledge which are guaranteed by utilized components in our mechanism.",
"1).",
"The underlying Blockchain layer is health.",
"Note that our mechanism applies a worth examining stable Blockchain as the underlying layer of the Control plane [61], [62].",
"Thus, that is reasonable for us to believe the underlying Blockchain layer is health, in which its record data are immutable and never abandoned.",
"2).",
"The lightweight ABE scheme is provably secure.",
"The security of the lightweight ABE scheme is provably secure in the attribute based selective-set model based on the ECDDH assumption, which is demonstrated in the work [50].",
"3).",
"The HOMQV protocol is a secure one-pass key-exchange protocol in the random oracle model and under the Gap-Diffie-Hellman (GDH) assumption.",
"The work [55] provides a formal analysis of the protocol's security.",
"Specifically, it assuming the hardness of Diffie-Hellman problem, proves the HOMQV protocol is secure which guarantees sender's forward secrecy and resilience to compromise of ephemeral data.",
"4).",
"The utilized asymmetric encryption algorithm is provably secure.",
"Our mechanism uses the classic public key cryptosystem [64] presented by Cramer et al.",
"which is provably secure against adaptive chosen ciphertext attack under standard intractability assumptions.",
"5).",
"The utilized digital signature algorithm is Strongly Existential Unforgeability.",
"The public-key signature algorithm such as Schnorr scheme [65] satisfies the security notion that an adversary could not output a new message-signature pair ($m^*$ , $\\sigma ^*$ ) with a totally different $\\sigma ^*$ even if the adversary has queried signatures on message $m^*$ .",
"Figure: Transaction auditing graph.",
"Note that a circle in dashed line represents a starting point in an auditing process, and directed edges in green lines, blue lines and purple lines are respectively related to the auditing process for authentication for application flows, replay attack detection for flows and notification of failed controllers for switches.Based on the aforementioned secure knowledge, we present our security analyses with the help of Fig.",
"REF .",
"Authentication for application flows.",
"The protocol AuthFlowProtocol authenticates the identity of an application flow by identifying whether the flow comes from a legitimate App which has registered the network and connects with some controller which manages switches in a network slice.",
"A flow (including its identity and the content) is signed by an App with its secret key and then verified by using the public key of the App.",
"The network only accepts legitimate flows but abandons any abnormal flow which is failed to be verified.",
"$IsRightFlow()$ in the protocol determines whether the App creating the flow is legitimate based on the transactions $T_{app}$ , $T_{app-contr}$ , $T_{contr}$ and $T_{contr-switch}$ .",
"Then, it uses the public key of the application to verify the flow signature to determine whether the content flow is modified by $verifyFlow(PK_{app})$ .",
"This process is shown by green circles and green directed edges in Fig.",
"REF .",
"It starts from $PK_{app}$ in the $flow$ and locates the transaction $T_{app}$ this $PK_{app}$ exists.",
"With $ID_{contr}$ , it indexes the transaction $T_{contr}$ and then via the relationship transaction $T_{contr-switch}$ the transaction $T_{switch}$ is located.",
"If the process above goes through, it means that the flow comes from the legitimate App and if the transaction in any step of this process does not exist, the flow is rejected.",
"Then, the signature $Sign_{flow}$ is verified by $PK_{app}$ of the App by using the verification algorithm of digital signature algorithm $\\textsf {DS.Ver}$$(PK_{app}, Sign_{flow})$ .",
"Replay attack detection for application flows.",
"The protocol FlowReplyDetectionProtocol detects replayed flows based on the logged records $T_{flow-afore}$ on the Blockchain.",
"When a newly coming flow is received, the identity of the flow is detected by auditing the transaction $T_{flow-afore}$ as shown in blue circles and blue directed edge.",
"The flow is accepted if the flow has never been sent.",
"Otherwise, the flow is rejected and the App sending this flow is punished by locating the transaction $T_{app}$ .",
"Notification of failed controllers for switches.",
"The protocol ControllerFailedNotifyProtocol can notify the switches when the controllers being connected break down.",
"By auditing the records of network behaviors $T_{flow-after}$ and $T_{event}$ within the latest 6 blocks, controllers $ID_{contr}$ without any active response can be found out.",
"Then, based on the relationship transaction $T_{contr-swith}$ , switches $ID_{switch}$ connecting with the failed controllers would be notified as shown in purple circles and purple directed edges.",
"On the other hand, the rest of two security issues are analyzed as follows.",
"Secure access control on network-wide resources.",
"The protocol AccessControlProtocol implemented by ABE scheme in [50] enables an App to access the respective resources when the attribute set of the App satisfies the access structure related to the encrypted resources.",
"Based on the acquired security knowledge that this ABE is provably secure, the access control mechanism is secure.",
"Authentication for controller-switch connection.",
"The protocol AuditAuthenRequestProtocol based on HOMQV protocol which has been proved security implements the authenticated communication between controllers and switches.",
"With an extra protocol SwitchChallengeProtocol, two existing security issues of the origin HOMQV protocol are worked out.",
"The two protocols implemented on the Blockchain provide secure authentication enhancement of controllers and switches."
],
[
"Proof-of-concept Implementation",
"As Floodlight [35] project puts the world's largest SDN ecosystem into practice[66], we decide to build our mechanism on the Floodlight project and illustrate the utility of our security enhancement.",
"On the other hand, we build Blockchain environment based on Hyperledger Fabric $V1.0$[67] which is an open project of Blockchain.",
"As shown in Fig.",
"REF , we demonstrate a schematic of our architecture prototype and compare it with original Floodlight architecture.",
"Focusing on the security goal we intend to achieve, we append our required Blockchain providers to the corresponding Floodlight application modules, in which Blockchain providers are implemented surrounding Floodlight application modules by applying programming method of Aspect Oriented Programming.",
"The Blockchain providers, TopologyBlockchainProvider and LinkBlockchainProvider are attached to the primary modules TopologyManager and LinkDiscovery respectively.",
"TopologyBlockchainProvider collects topology resources of network via TopologyManager while LinkBlockchainProvider monitors the status of links in network.",
"In the meantime, they are responsible to communicate with the defined security protocol AccessControlProtocol on Blockchain so that a customized access control mechanism for applications is provided.",
"The provider named ForwardBlockchainProvider undertakes to monitor network packages which are forwarded among devices.",
"It collects package information and forwarding pathes that are prepare for flow transaction and event transaction generation.",
"DeviceBlockchainProvider tracks information of network devices, which is used to generate entity transactions.",
"FlowBlockchainProvider is used to catch the newly flows via StaticFlowPusher, which is prepare for FlowTransactionGenerator.",
"Between the communication of module-appended Floodlight project and Blockchain, we construct an implemented project decoupling from the Floodlight project which includes four modules.",
"The four modules are EntityTransactionGenerator, FlowTransactionGenerator, EventTransactionGenerator and TopologyTransactionGenerator which undertake to encapsulate network data and generate transactions in the context of Blockchain.",
"In addition, the security protocols mentioned in section REF are implemented by smart contract which is validated and secure on Blockchain.",
"In order to connect with Blockchain, this exists an interface like Web3j devoted to build defined transactions and security protocols into Blockchain.",
"Note that Ethereum[68] which is another open source Blockchain project offers a library called Web3j for a variety of Jave application integrating with Ethereum.",
"For Hyperledger Blockchain, the third-party library, as Web3j for Ethereum, also is expected to be used to integrate our Java application based on Floodlight.",
"By calling the third-party library as middle interface between Floodlight project and Blockchain project, the communication with secure protocols on Blockchain can be achieved."
],
[
"Conclusion",
"SDN has become an emerging technology to enhance network performance.",
"With its extensive adoption, some security issues of SDN are exposed and imperative to be studied.",
"In the paper, we present a Blockchain-based monolithic secure mechanism for SDN.",
"By utilizing Blockchain to record all network flows and events and to implement secure protocols with smart contracts, the presented secure mechanism overcomes the common security issues in SDN.",
"In particular, the decentralized control plane tackles the problem of single-point failure and improves network scalability; application flows can be authenticated, tracked and accounted; network-wide resources are protected with access control scheme and authenticated communication channels are ensured between controllers and switches.",
"At last, the security analysis and an implementation prototype of our mechanism demonstrate the effectiveness of security improvement for SDN."
]
] | 1906.04342 |
[
[
"Rate-Splitting Unifying SDMA, OMA, NOMA, and Multicasting in MISO\n Broadcast Channel: A Simple Two-User Rate Analysis"
],
[
"Abstract Considering a two-user multi-antenna Broadcast Channel, this paper shows that linearly precoded Rate-Splitting (RS) with Successive Interference Cancellation (SIC) receivers is a flexible framework for non-orthogonal transmission that generalizes, and subsumes as special cases, four seemingly different strategies, namely Space Division Multiple Access (SDMA) based on linear precoding, Orthogonal Multiple Access (OMA), Non- Orthogonal Multiple Access (NOMA) based on linearly precoded superposition coding with SIC, and physical-layer multicasting.",
"The paper studies the sum-rate and shows analytically how RS unifies, outperforms, and specializes to SDMA, OMA, NOMA, and multicasting as a function of the disparity of the channel strengths and the angle between the user channel directions."
],
[
"Introduction",
"Linearly precoded Rate-Splitting (RS) with Successive Interference Cancellation (SIC) receivers has recently appeared as a powerful non-orthogonal transmission and robust interference management strategy for multi-antenna wireless networks [1].",
"Though originally introduced for the two-user Single-Input Single-Output Interference Channel (IC) in [2], RS has become an underpinning communication-theoretic strategy to tackle modern interference-related problems and has recently been successfully investigated in several Multiple-Input Single-Output (MISO) Broadcast Channel (BC) settings, namely, unicast-only transmission with perfect Channel State Information at the Transmitter (CSIT) [3], [4] and imperfect CSIT [5], [6], [7], [8], [9], [10], [11], [12], [13], (multigroup) multicast-only transmission [14], as well as superimposed unicast and multicast transmission [15].",
"Results highlight that RS provides significant benefits in terms of spectral efficiency [3], [6], [7], [9], [13], [14], [15], energy efficiency [4], robustness [8], and CSI feedback overhead reduction [6], [12] over conventional strategies used in LTE-A/5G that rely on fully treating interference as noise (e.g.",
"conventional multi-user linear precoding and Space Division Multiple Access - SDMA) or fully decoding interference (e.g.",
"power-domain Non-Orthogonal Multiple Access - NOMA [16]).",
"The key behind realizing those benefits is the ability of RS, through splitting messages into common and private parts, to partially decode interference and partially treat interference as noise.",
"Additionally, RS is an enabler for powerful multiple access designs that subsumes SDMA and NOMA as special cases and outperforms them both for a wide range of network loads (underloaded/overloaded regimes) and user deployments (for diverse channel directions/strengths and CSIT qualities) [3].",
"In this work, we build upon this last observation and show considering a simple two-user MISO BC with perfect CSIT that RS is a flexible framework for non-orthogonal transmission that generalizes, and subsumes as special cases, four seemingly completely different strategies, namely SDMA based on linear precoding, Orthogonal Multiple Access (OMA) where a resource is fully taken up by a single user, power-domain NOMA based on linearly precoded superposition coding with SIC, and physical-layer multicasting.",
"This is the first paper to show analytically how RS unifies, outperforms, and specializes to SDMA, OMA, NOMA, and multicasting as a function of the disparity of the user channel strengths and the angle between the user channel directions.",
"To that end, the paper differs from, and nicely complements, past works that analytically studied the rate performance of RS with imperfect CSIT [6], [9], [12] or looked at RS from an optimization perspective [3], [7], [8].",
"Notation: $|.|$ and $\\left\\Vert .\\right\\Vert $ refer to the absolute value of a scalar and the $l_2$ -norm of a vector.",
"$\\mathbf {I}$ is the identity matrix.",
"$\\mathbf {a}^{H}$ denotes the Hermitian transpose of vector $\\mathbf {a}$ .",
"I.i.d.",
"stands for independent and identically distributed.",
"$\\mathcal {CN}(0,\\sigma ^2)$ denotes the Circularly Symmetric Complex Gaussian distribution with zero mean and variance $\\sigma ^2$ .",
"$\\sim $ stands for “distributed as”."
],
[
"System Model: Rate-Splitting Architecture",
"We consider a MISO BC consisting of one transmitter with $n_t$ antennas and two single-antenna users.",
"As per Fig.",
"REF , the architecture relies on rate-splitting of two messages $W_1$ and $W_2$ intended for user-1 and user-2, respectively.",
"To that end, the message $W_k$ of user-$k$ is split into a common part $W_{\\mathrm {c},k}$ and a private part $W_{\\mathrm {p},k}$ .",
"The common parts $W_{\\mathrm {c},1}, W_{\\mathrm {c},2}$ of both users are combined into the common message $W_{\\mathrm {c}}$ , which is encoded into the common stream $s_{\\mathrm {c}}$ using a codebook shared by both users.",
"Hence, $s_{\\mathrm {c}}$ is a common stream required to be decoded by both users, and contains parts of the messages $W_1$ and $W_2$ intended for user-1 and user-2, respectively.",
"The private parts $W_{\\mathrm {p},1}$ and $W_{\\mathrm {p},2}$ , respectively containing the remaining parts of the messages $W_1$ and $W_2$ , are independently encoded into the private stream $s_1$ for user-1 and $s_2$ for user-2.",
"Out of the two messages $W_1$ and $W_2$ , three streams $s_{\\mathrm {c}}$ , $s_1$ , and $s_2$ are therefore created.",
"The streams are linearly precoded such that the transmit signal is given by $\\mathbf {x}=\\mathbf {p}_{\\mathrm {c}} s_{\\mathrm {c}}+\\mathbf {p}_1 s_1+\\mathbf {p}_2 s_2.$ Defining $\\mathbf {s}=[s_{\\mathrm {c}},s_1,s_2]^T$ and assuming that $\\mathbb {E}[\\mathbf {s}\\mathbf {s}^H]=\\mathbf {I}$ , the average transmit power constraint is written as $P_{\\mathrm {c}}+P_{1}+P_{2}\\le P$ where $P_{\\mathrm {c}}=\\left\\Vert \\mathbf {p}_{\\mathrm {c}}\\right\\Vert ^2$ and $P_{k}=\\left\\Vert \\mathbf {p}_{k}\\right\\Vert ^2$ with $k=1,2$ .",
"We refer to $\\mathbf {h}_k$ as the channel vector of user-$k$ , such that the signal received at user-$k$ can be written as $y_k=\\mathbf {h}_k^H \\mathbf {x}+n_k, \\hspace{14.22636pt}k=1,2,$ where $n_k\\sim \\mathcal {CN}(0,1)$ is Additive White Gaussian Noise (AWGN).",
"We further write the channel vectors as the product of their norm and direction as $\\mathbf {h}_k=\\left\\Vert \\mathbf {h}_k\\right\\Vert \\bar{\\mathbf {h}}_k$ , and assume without loss of generality $\\left\\Vert \\mathbf {h}_1\\right\\Vert \\ge \\left\\Vert \\mathbf {h}_2\\right\\Vert $ .",
"We also assume perfect CSI at the transmitter and the receivers.",
"Figure: Two-user system architecture with rate-splitting.At each user-$k$ , the common stream $s_{\\mathrm {c}}$ is first decoded into $\\widehat{W}_{\\mathrm {c}}$ by treating the interference from the private streams as noise.",
"Using SIC, $\\widehat{W}_{\\mathrm {c}}$ is re-encoded, precoded, and subtracted from the received signal, such that user-$k$ can decode its private stream $s_k$ into $\\widehat{W}_{\\mathrm {p},k}$ by treating the remaining interference from the other private stream as noise.",
"User-$k$ reconstructs the original message by extracting $\\widehat{W}_{\\mathrm {c},k}$ from $\\widehat{W}_{\\mathrm {c}}$ , and combining $\\widehat{W}_{\\mathrm {c},k}$ with $\\widehat{W}_{\\mathrm {p},k}$ into $\\widehat{W}_{k}$ .",
"Assuming Gaussian signalling and ideal SIC, the rate of the common stream is given by $R_{\\mathrm {c}}=\\min \\left(\\log _2\\left(1+\\frac{\\left|\\mathbf {h}_1^H \\mathbf {p}_{\\mathrm {c}}\\right|^2}{1+\\left|\\mathbf {h}_1^H \\mathbf {p}_{1}\\right|^2+\\left|\\mathbf {h}_1^H \\mathbf {p}_{2}\\right|^2}\\right),\\right.\\\\\\left.\\log _2\\left(1+\\frac{\\left|\\mathbf {h}_2^H \\mathbf {p}_{\\mathrm {c}}\\right|^2}{1+\\left|\\mathbf {h}_2^H \\mathbf {p}_{1}\\right|^2+\\left|\\mathbf {h}_2^H \\mathbf {p}_{2}\\right|^2}\\right)\\right),$ and the rates of the two private streams are obtained as $R_{k}=\\log _2\\left(1+\\frac{\\left|\\mathbf {h}_k^H \\mathbf {p}_{k}\\right|^2}{1+\\left|\\mathbf {h}_k^H \\mathbf {p}_{j}\\right|^2}\\right), k \\ne j.$ The rate of user-$k$ is given by $R_k+R_{\\mathrm {c},k}$ where $R_{\\mathrm {c},k}$ is the rate of the common part of the $k$ th user’s message, i.e., $W_{\\mathrm {c},k}$ , and it satisfies $R_{\\mathrm {c},1}+R_{\\mathrm {c},2}=R_{\\mathrm {c}}$ .",
"The sum-rate is therefore simply written as $R_{\\mathrm {s}}=\\sum _{k=1,2} R_k+R_{\\mathrm {c},k}=R_{\\mathrm {c}}+R_{1}+R_{2}$ .",
"By adjusting the message split and the power allocation to the common stream and the private streams, RS enables the decoding of part of the interference (thanks to the presence of the common stream) and treating the remaining part (the private stream of the other user) as noise.",
"Therefore, the introduced RS architecture allows the exploration of a wide range of strategies.",
"Among those strategies, there are four extreme cases, namely, SDMA, NOMA, OMA, and physical-layer multicasting.",
"Indeed, SDMA is obtained by allocating no power to the common stream ($P_{\\mathrm {c}}=0$ ) such that $W_k$ is encoded directly into $s_k$ .",
"No interference is decoded at the receiver using the common message, and the interference between $s_1$ and $s_2$ is fully treated as noise.",
"NOMA is obtained by encoding $W_2$ entirely into $s_{\\mathrm {c}}$ (i.e., $W_{\\mathrm {c}}=W_2$ ) and $W_1$ into $s_{1}$ , and turning off $s_2$ ($P_{2}=0$ ).",
"In this way, user-1 fully decodes the interference created by the message of user-2.",
"OMA is a sub-strategy of SDMA and NOMA and is obtained when only user-1 (with the stronger channel gain) is scheduled ($P_{\\mathrm {c}}=0, P_2=0$ ).",
"Multicasting is obtained by combining and encoding both $W_1$ and $W_2$ into $s_{\\mathrm {c}}$ , and turning off $s_1$ and $s_2$ ($P_1=0, P_2=0$ ).",
"The mapping of the messages to the streams is further illustrated in Fig.",
"REF .",
"Remark 1 Recall that the maximum number of interference-free streams (also called Degrees-of-Freedom DoF) in a two-user MISO BC is equal to 2.",
"From the above system model, both SDMA and RS can achieve such a DoF by precoding $s_1$ and $s_2$ using zero-forcing (ZF).",
"On the other hand, OMA, NOMA, and multicasting can achieve at most a DoF of 1 (irrespectively of how the precoders and power allocation are optimized), which leads to a rate loss at high Signal-to-Noise Ratio (SNR) in general multi-antenna settings, as already highlighted in [14], [3].",
"Figure: Mapping of messages to streams."
],
[
"Sum-Rate Analysis",
"Our objective is to derive tractable and insightful sum-rate expressions to illustrate the flexibility of RS in unifying SDMA, OMA, NOMA, and multicasting.",
"To that end, we do not optimize the precoding directions jointly with the power allocation as in [7], [3] but rather fix the precoding directions using ZF for the private streams, and adjust the power allocation among all the streamsSimulations in Section show that the conclusions drawn with the simple precoders also hold with the numerically optimized precoders of [7], [3]..",
"This leads to $\\left|\\mathbf {h}_2^H \\mathbf {p}_{1}\\right|=0$ , $\\left|\\mathbf {h}_1^H \\mathbf {p}_{2}\\right|=0$ , and $\\left|\\mathbf {h}_k^H \\mathbf {p}_{k}\\right|^2=\\left\\Vert \\mathbf {h}_k\\right\\Vert ^2 \\rho P_k$ , $k=1,2$ , where $\\rho =1-\\left|\\bar{\\mathbf {h}}_1^H\\bar{\\mathbf {h}}_2\\right|^2$ ($\\rho =0$ corresponds to aligned channels and $\\rho =1$ to orthogonal channels).",
"The precoder of the common stream is then to be designed such that $\\max _{\\mathbf {p}_{\\mathrm {c}}} \\min \\left(\\frac{\\left|\\mathbf {h}_1^H \\mathbf {p}_{\\mathrm {c}}\\right|^2}{1+\\left|\\mathbf {h}_1^H \\mathbf {p}_{1}\\right|^2},\\frac{\\left|\\mathbf {h}_2^H \\mathbf {p}_{\\mathrm {c}}\\right|^2}{1+\\left|\\mathbf {h}_2^H \\mathbf {p}_{2}\\right|^2}\\right).$ Defining $\\gamma _k^2=1+\\left|\\mathbf {h}_k^H \\mathbf {p}_{k}\\right|^2=1+\\left\\Vert \\mathbf {h}_k\\right\\Vert ^2 \\rho P_k$ , $k=1,2$ , and $\\tilde{\\mathbf {h}}_k=\\mathbf {h}_k/\\gamma _k$ , the problem is re-written as $\\max _{\\mathbf {p}_{\\mathrm {c}}} \\min \\left(\\big |\\tilde{\\mathbf {h}}_1^H \\mathbf {p}_{\\mathrm {c}}\\big |^2,\\big |\\tilde{\\mathbf {h}}_2^H \\mathbf {p}_{\\mathrm {c}}\\big |^2\\right).$ Following [17], the solution of (REF ) is $\\mathbf {p}_{\\mathrm {c}}=\\sqrt{P_{\\mathrm {c}}}\\mathbf {f}_{\\mathrm {c}}$ with the precoder direction $\\mathbf {f}_{\\mathrm {c}}$ ($\\left\\Vert \\mathbf {f}_{\\mathrm {c}}\\right\\Vert ^2=1$ ) given by $\\mathbf {f}_{\\mathrm {c}}=\\frac{1}{\\sqrt{\\lambda }}\\left(\\mu _1 \\tilde{\\mathbf {h}}_1 + \\mu _2 \\tilde{\\mathbf {h}}_2 e^{-j \\angle \\alpha _{12}}\\right),$ where $\\lambda =\\frac{\\alpha _{11} \\alpha _{22}-\\left|\\alpha _{12}\\right|^2}{\\alpha _{11} + \\alpha _{22}- 2 \\left|\\alpha _{12}\\right|},$ $\\left[\\begin{array}{c}\\mu _1 \\\\ \\mu _2\\end{array}\\right] =\\frac{1}{\\alpha _{11} + \\alpha _{22}- 2 \\left|\\alpha _{12}\\right|}\\left[\\begin{array}{c} \\alpha _{22}- \\left|\\alpha _{12}\\right| \\\\ \\alpha _{11}- \\left|\\alpha _{12}\\right|\\end{array}\\right],$ $\\left[\\begin{array}{cc}\\alpha _{11} & \\alpha _{12} \\\\ \\alpha _{12}^* & \\alpha _{22}\\end{array}\\right]=\\left[\\begin{array}{c} \\tilde{\\mathbf {h}}_1^H \\\\ \\tilde{\\mathbf {h}}_2^H \\end{array}\\right]\\left[\\begin{array}{cc}\\tilde{\\mathbf {h}}_1 & \\tilde{\\mathbf {h}}_2\\end{array}\\right].$"
],
[
"Sum-Rate at Finite SNR",
"The sum-rate with the above precoder designs can be written as $R_{\\mathrm {s}}=R_{\\mathrm {c}}+\\log _2\\left(\\gamma _1^2\\right)+\\log _2\\left(\\gamma _2^2\\right)$ , where $R_{\\mathrm {c}}\\!=\\!\\min \\!\\big (\\log _2\\big (1\\!+\\!\\big |\\tilde{\\mathbf {h}}_1^H \\mathbf {p}_{\\mathrm {c}}\\big |^2\\big ),\\log _2\\big (1\\!+\\!\\big |\\tilde{\\mathbf {h}}_2^H \\mathbf {p}_{\\mathrm {c}}\\big |^2\\big )\\big )$ .",
"With $\\mathbf {p}_{\\mathrm {c}}$ as per (REF ), following [17], $\\big |\\tilde{\\mathbf {h}}_1^H \\mathbf {p}_{\\mathrm {c}}\\big |\\!=\\!\\big |\\tilde{\\mathbf {h}}_2^H \\mathbf {p}_{\\mathrm {c}}\\big |$ , and we can write $R_{\\mathrm {c}}=\\log _2\\big (1+\\big |\\tilde{\\mathbf {h}}_2^H \\mathbf {p}_{\\mathrm {c}}\\big |^2\\big )$ , and the sum-rate simply as $R_{\\mathrm {s}}=\\log _2\\left(\\gamma _1^2\\right)+\\log _2\\left(\\gamma _2^2+\\big |\\mathbf {h}_2^H \\mathbf {p}_{\\mathrm {c}}\\big |^2\\right).$ Consider a fraction $t$ of the total transmit power $P$ is allocated to the private streams such that $P_1+P_2=tP$ and the remaining power $P_{\\mathrm {c}}=\\left(1-t\\right)P$ is allocated to the common stream.",
"For a given $t$ , the optimal values of $P_1$ and $P_2$ , maximizing the sum-rate of the private streams, are given by the Water-Filling (WF) solution $P_k=\\max \\left(\\mu -\\frac{1}{\\left\\Vert \\mathbf {h}_k\\right\\Vert ^2 \\rho },0\\right), \\hspace{8.5359pt} k=1,2, $ with the water level $\\mu $ chosen such that $P_1\\!+\\!P_2\\!=\\!tP$ , and set as $\\mu \\!=\\!\\frac{tP}{2}\\!+\\!\\frac{1}{2\\rho }\\left[\\frac{1}{\\left\\Vert \\mathbf {h}_1\\right\\Vert ^2}\\!+\\!\\frac{1}{\\left\\Vert \\mathbf {h}_2\\right\\Vert ^2}\\right]$ in the sequel.",
"Let us also introduce $\\Gamma \\!=\\!\\frac{1}{\\rho }\\left[\\frac{1}{\\left\\Vert \\mathbf {h}_2\\right\\Vert ^2}\\!-\\!\\frac{1}{\\left\\Vert \\mathbf {h}_1\\right\\Vert ^2}\\right]$ , which is a function of two main parameters: $\\rho $ reflecting the angle between the user channel directions, and $\\frac{1}{\\left\\Vert \\mathbf {h}_2\\right\\Vert ^2}-\\frac{1}{\\left\\Vert \\mathbf {h}_1\\right\\Vert ^2}$ reflecting the disparity of the channel strengths.",
"We can then identify two main regimes.",
"If $\\mu \\le \\frac{1}{\\left\\Vert \\mathbf {h}_2\\right\\Vert ^2 \\rho }$ , i.e., $tP \\le \\Gamma $ , we set $P_2=0$ and $P_1=tP$ according to (REF ), and RS specializes to multicasting for $t=0$ , NOMA for $0<t<1$ , and OMA for $t=1$ .",
"In this regime, $t$ needs to be adjusted so as to identify the best strategy among OMA, NOMA, and multicasting, and therefore efficiently allocate power across the common stream $s_{\\mathrm {c}}$ and the private stream $s_1$ ."
],
[
"RS/SDMA Regime",
"If $\\mu > \\frac{1}{\\left\\Vert \\mathbf {h}_2\\right\\Vert ^2 \\rho }$ , i.e.",
"$tP > \\Gamma $ , the WF solution (REF ) leads to $P_1=\\mu -\\frac{1}{\\left\\Vert \\mathbf {h}_1\\right\\Vert ^2\\rho }=\\frac{tP}{2}+\\frac{\\Gamma }{2}>0$ and $P_2=\\mu -\\frac{1}{\\left\\Vert \\mathbf {h}_2\\right\\Vert ^2\\rho }=\\frac{tP}{2}-\\frac{\\Gamma }{2}>0$ .",
"RS specializes to SDMA whenever $t$ is set to 1, but does not specializes to any other known scheme for $0<t<1$ .",
"In this regime, $t$ needs to be adjusted, as explained in the sequel, so as to allocate the power efficiently across the common stream and the two private streams.",
"Substituting the expressions of $P_k$ and $\\gamma _k^2$ , $k=1,2$ , into (REF ), we can write $R_{\\mathrm {s}}=\\log _2\\left(ac+\\left(ad+bc\\right)t+bdt^2\\right),$ where $b=\\frac{\\left\\Vert \\mathbf {h}_1\\right\\Vert ^2\\rho P}{2}$ , $a=1+\\frac{\\Gamma }{P}b$ , $d=\\frac{\\left\\Vert \\mathbf {h}_2\\right\\Vert ^2\\rho P}{2}-|\\mathbf {h}_2^H\\mathbf {f}_c|^2P$ , and $c=1-\\frac{\\Gamma }{P}d+|\\mathbf {h}_2^H\\mathbf {f}_c|^2(P-\\Gamma )$ .",
"The value of $t$ that maximizes $R_{\\mathrm {s}}$ is the solution of $\\frac{\\partial R_s}{\\partial t}=0$ , which is written as $t=-\\frac{a}{2b}-\\frac{c}{2d}$ .",
"Since $t\\le 1$ , the optimal value $t^{\\star }$ is given in closed form by (REF ) at the top of the next page.",
"For $t^{\\star }<1$ , RS yields a non-zero sum-rate enhancement over SDMA.",
"Table: NO_CAPTIONRemark 2 It is important to note that the solution $t=-\\frac{a}{2b}-\\frac{c}{2d}$ holds because the coefficients $a$ , $b$ , $c$ , $d$ are not functions of $t$ .",
"This could appear surprising since $c$ and $d$ are functions of $\\mathbf {f}_c$ , which, according to (REF ), is a function of $P_1$ and $P_2$ and therefore of $t$ .",
"However, interestingly, in the regime where $P_1>0$ and $P_2>0$ , we can show that $\\mathbf {f}_c$ is not a function of $t$ .",
"Making use of $P_1=\\frac{tP}{2}+\\frac{\\Gamma }{2}$ and $P_2=\\frac{tP}{2}-\\frac{\\Gamma }{2}$ , we can write $\\gamma _k^2=1+\\left\\Vert \\mathbf {h}_k\\right\\Vert ^2\\rho P_k=\\frac{f(t)}{\\left\\Vert \\mathbf {h}_j\\right\\Vert ^2}$ , $k,j=1,2$ and $k \\ne j$ , with $f(t)=\\frac{\\left\\Vert \\mathbf {h}_1\\right\\Vert ^2+\\left\\Vert \\mathbf {h}_2\\right\\Vert ^2+{\\left\\Vert \\mathbf {h}_1\\right\\Vert ^2\\left\\Vert \\mathbf {h}_2\\right\\Vert ^2\\rho P}t}{2}$ .",
"We then obtain $\\begin{aligned}&\\max _{\\mathbf {f}_{\\mathrm {c}}} \\min \\left(\\big |\\tilde{\\mathbf {h}}_1^H \\mathbf {f}_{\\mathrm {c}}\\big |^2,\\big |\\tilde{\\mathbf {h}}_2^H \\mathbf {f}_{\\mathrm {c}}\\big |^2\\right)\\\\\\Leftrightarrow & \\max _{\\mathbf {f}_c} \\min \\left(\\gamma _2^2{\\big |\\mathbf {h}_1^H\\mathbf {f}_c\\big |^2},{\\gamma _1^2\\big |\\mathbf {h}_2^H\\mathbf {f}_c\\big |^2}\\right)\\\\\\Leftrightarrow &\\max _{\\mathbf {f}_c} \\min \\left(f(t){\\big |\\bar{\\mathbf {h}}_1^H\\mathbf {f}_c\\big |^2},{f(t)\\big |\\bar{\\mathbf {h}}_2^H\\mathbf {f}_c\\big |^2}\\right)\\\\\\Leftrightarrow &\\max _{\\mathbf {f}_c} \\min \\left({\\big |\\bar{\\mathbf {h}}_1^H\\mathbf {f}_c\\big |^2},{\\big |\\bar{\\mathbf {h}}_2^H\\mathbf {f}_c\\big |^2}\\right),\\end{aligned}$ which reveals that $\\mathbf {f}_{\\mathrm {c}}$ is not a function of $t$ and the channel strength disparity, but only of the channel directions.",
"At high SNR, considering $0 < t \\le 1$ and $\\rho >0$ , the solution in (REF ) allocates power uniformly across the two private streams as $P_1=P_2=\\frac{tP}{2}>0$ .",
"Hence, only RS and SDMA are suitable strategies at high SNR.",
"The sum-rate in (REF ) can then be written as $R_{\\mathrm {s}}\\stackrel{P\\nearrow }{=}\\log _2\\big (\\left\\Vert \\mathbf {h}_1\\right\\Vert ^2 \\!",
"\\rho \\big )+2\\log _2\\left(P\\right)+\\log _2\\left(e t^2+f t\\right)$ with $e=\\frac{\\left\\Vert \\mathbf {h}_2\\right\\Vert ^2 \\rho }{4}\\!-\\!\\frac{\\left|\\mathbf {h}_2^H \\mathbf {f}_{\\mathrm {c}}\\right|^2}{2}$ , $f=\\frac{\\left|\\mathbf {h}_2^H \\mathbf {f}_{\\mathrm {c}}\\right|^2}{2}$ .",
"Not surprisingly, a DoF of 2 is achieved in (REF ).",
"More interesting is the fact that RS brings a constant sum-rate enhancement over SDMA.",
"Indeed, the value of $t$ that maximizes (REF ) is given by $t^{\\star }=\\min \\left(\\frac{-f}{2e},1\\right)=\\min \\left(\\frac{\\big |\\bar{\\mathbf {h}}_2^H \\mathbf {f}_{\\mathrm {c}}\\big |^2}{2\\big |\\bar{\\mathbf {h}}_2^H \\mathbf {f}_{\\mathrm {c}}\\big |^2-\\rho },1\\right),$ which coincides with (REF ) when $P\\rightarrow \\infty $ , and leads to a high SNR non-zero (whenever $0<t^{\\star }<1$ ) sum-rate gap between RS and SDMA ($t=1$ ) given by $\\Delta R_{\\mathrm {s}}=\\left.R_{\\mathrm {s}}\\right|_{t^{\\star }}-\\left.R_{\\mathrm {s}}\\right|_{t=1}=\\log _2\\left(\\frac{\\big |\\bar{\\mathbf {h}}_2^H \\mathbf {f}_{\\mathrm {c}}\\big |^4}{\\rho \\left(2\\big |\\bar{\\mathbf {h}}_2^H \\mathbf {f}_{\\mathrm {c}}\\big |^2-\\rho \\right)}\\right).",
"$ $t^{\\star }$ increases and $\\Delta R_{\\mathrm {s}}$ decreases as $\\rho $ increases, and both are not a function of the channel strengths.",
"The sum-rate gap between RS and NOMA/OMA/multicasting grows unbounded as $P\\!\\rightarrow \\!\\infty $ due to the difference in DoF (Remark 1)."
],
[
"Discussions",
"We can draw several insights from the above analysis.",
"First, for given $t$ , $\\rho $ , $\\left\\Vert \\mathbf {h}_1\\right\\Vert ^2$ , and $\\left\\Vert \\mathbf {h}_2\\right\\Vert ^2$ , as $P$ increases, the SNRs of the private streams increase, while the Signal-to-Interference-plus-Noise Ratio (SINR) of the common stream ultimately saturates (interference limited regime).",
"This suggests that the common message can only provide a constant rate improvement at high SNR, while the two private streams provide the DoF of 2.",
"Second, the quantity $\\rho $ is present in the SNRs of both private streams and has the effect of increasing/decreasing the SNRs of those two streams.",
"A lower $\\rho $ indicates that both private streams effectively operate at a lower SNR.",
"According to (REF ), for a given $t$ , a low $\\rho $ favors power allocation to a single private stream (NOMA/OMA/Multicasting regime) over a wider range of $P$ , and also leads to a smaller interference power (and therefore a higher rate) for the common stream.",
"A higher $\\rho $ leads to a higher effective SNR and therefore a better capability to support two private streams (RS/SDMA regime).",
"Third, as the disparity of channel strengths increases, the WF solution allocates a larger amount of power to the stronger user (user-1) over a wider range of $P$ (for a given $t$ ).",
"Beyond a certain disparity, for given $t$ , $P$ , and $\\rho $ , $P_2$ is turned off and RS specializes to NOMA/OMA."
],
[
"Evaluations",
"In this section, we first illustrate the above analysis and the preferred regions for the operation of NOMA, OMA, SDMA, and RS.",
"We assume $n_t=2$ , and channel vectors given by $\\mathbf {h}_1=1/\\sqrt{2}\\:[1,1]^H$ and $\\mathbf {h}_2=\\gamma /\\sqrt{2}\\:[1,e^{j \\theta }]^H$ .",
"Assuming the precoding strategies in Section and the WF power allocation (REF ), the colors in Fig.",
"REF (a) and (b) illustrate the optimum value (obtained from exhaustive search whenever not available in closed form) of $t$ that maximizes the sum-rate and the corresponding preferred communication strategy (RS, SDMA, NOMA, OMA) as a function of $\\rho =1-\\left|\\bar{\\mathbf {h}}_1^H\\bar{\\mathbf {h}}_2\\right|^2$ (ranging from 0 to 1) and $\\gamma _{\\mathrm {dB}}=20\\log _{10}(\\gamma )$ (ranging from 0 to -20dB), i.e., user-1 and user-2 have a long-term SNR of 20dB and $0 \\mathrm {dB}\\le 20 \\mathrm {dB}+\\gamma _{\\mathrm {dB}}\\le 20 \\mathrm {dB}$ , respectively.",
"Recall that SDMA is characterized by $t=1,P_1>0,P_2>0$ , NOMA by $0<t<1,P_1>0,P_2\\!=\\!0$ , OMA by $t\\!=\\!1,P_1\\!=\\!P,P_2\\!=\\!0$ , and multicast by $t\\!=\\!0,P_1\\!=\\!0,P_2\\!=\\!0$ .",
"For all other regimes, RS does not specialize to any other well-established scheme and is simply referred to as RS.",
"We observe that NOMA is preferred for deployments with small $\\rho $ , i.e., closely aligned users, and small $\\gamma $ , SDMA is preferred whenever $\\rho $ is sufficiently large, i.e., semi-orthogonal users, and RS bridges those two extremes.",
"OMA is preferred whenever $\\gamma $ is very small.",
"Recall that Fig.",
"REF is obtained for $P=100$ W. In Fig.",
"REF , we assess the evolution of the regions as a function of $P$ for $P=10$ W and $P=1000$ W (where the long term SNR is 10 dB and 30 dB, respectively).",
"As $P$ increases, RS becomes the dominant strategy for most deployment conditions.",
"Fig.",
"REF shows the relative sum-rate gain [%] of RS over dynamic switching between SDMA and NOMA, defined as $\\frac{R_{\\mathrm {s}}^{\\mathrm {RS}}-\\max (R_{\\mathrm {s}}^{\\mathrm {SDMA}},R_{\\mathrm {s}}^{\\mathrm {NOMA}})}{\\max (R_{\\mathrm {s}}^{\\mathrm {SDMA}},R_{\\mathrm {s}}^{\\mathrm {NOMA}})}\\!\\times \\!100$ , for $P\\!=\\!10,100,1000$ W and the precoders from Section .",
"RS provides explicit gains over dynamic switching for medium values of $\\rho $ .",
"The values in brackets indicate the relative sum-rate gains over SDMA and NOMA, respectively, i.e., $\\big (\\frac{R_{\\mathrm {s}}^{\\mathrm {RS}}-R_{\\mathrm {s}}^{\\mathrm {SDMA}}}{R_{\\mathrm {s}}^{\\mathrm {SDMA}}}\\!\\times \\!",
"100,\\frac{R_{\\mathrm {s}}^{\\mathrm {RS}}-R_{\\mathrm {s}}^{\\mathrm {NOMA}}}{R_{\\mathrm {s}}^{\\mathrm {NOMA}}}\\!\\times \\!",
"100\\big )$ .",
"Large gains over SDMA are observed for low to medium values of $\\rho $ , and over NOMA for medium to large values of $\\rho $ at low SNR and for all values of $\\rho $ and $\\gamma _{\\mathrm {dB}}$ at higher SNR.",
"Values $(0,0)$ indicate that OMA is the preferred strategy, and that RS, SDMA, and NOMA all specialize to OMA.",
"Fig.",
"REF is similar to Fig.",
"REF but now the Weighted Minimum Mean Square Error (WMMSE) precoding optimization framework for RS developed in [7], [3] is adopted.",
"Such framework optimizes all precoders ($\\mathbf {p}_{\\mathrm {c}}, \\mathbf {p}_1, \\mathbf {p}_2$ ) jointly with the power allocations so as to maximize the weighted sum-rate $\\sum _{k=1,2} u_k \\left(R_k+R_{\\mathrm {c},k}\\right)$ .",
"In those evaluations, the convergence tolerance of the WMMSE algorithm is set to $\\epsilon \\!=\\!10^{-3}$ [3].",
"When allocating equal weights or higher weights to the user with the stronger channel (namely user-1), NOMA has no benefit over SDMA.",
"When a higher weight is given to the weaker user (user-2), NOMA is able to outperform SDMA.",
"RS on the other hand always provides the same or better performance than both SDMA and NOMA for all weights, $\\rho $ , and $\\gamma _{\\mathrm {dB}}$ .",
"Though the precoders of Section are simple and not optimal, the insights obtained from the analysis and Fig.",
"REF are inline with those obtained from Fig.",
"REF .",
"Hence, irrespectively of the precoding strategies, i.e., simple or optimized, RS unifies and outperforms SDMA, OMA, NOMA, and multicasting.",
"We now change the channel model and assume i.i.d.",
"Rayleigh fading, i.e., the entries of $\\mathbf {h}_1$ and $\\mathbf {h}_2$ are $\\mathcal {CN}(0,1/n_t)$ and $\\mathcal {CN}(0,\\gamma ^2/n_t)$ .",
"We generate 10000 channel realizations.",
"Making use of the precoders in Section , we identify the preferred (i.e., sum-rate maximizing) strategy for each channel realization.",
"Fig.",
"REF displays the percentage a given strategy is the preferred option as a function of $P$ and $\\gamma _{\\mathrm {dB}}$ for $n_t=2$ .",
"OMA is preferred for low $P$ and low $\\gamma _{\\mathrm {dB}}$ , and RS becomes the preferred option as $P$ and/or $\\gamma _{\\mathrm {dB}}$ increase.",
"At high SNR, RS is the preferred option for about 75% of the channel realizations and SDMA for the remaining 25%.",
"Results with $n_t=4$ (not reproduced here due to the space constraint) show that NOMA almost disappears from the set of preferred strategies, and SDMA becomes more dominant (for about 60% of the channel realizations and RS for the remaining 40%).",
"This is natural since, as $n_t$ increases, the likelihood to experience large $\\rho $ increases, and $t^{\\star }$ has a higher chance of being equal to 1."
],
[
"Conclusions",
"RS unifies SDMA, OMA, NOMA, and multicasting under a single approach and provides a powerful framework for the design and optimization of non-orthogonal transmission, multiple access, and interference management strategies.",
"Thanks to its versatility, RS has the potential to tackle challenges of modern communication systems and is a gold mine of research problems for academia and industry, spanning fundamental limits, optimization, PHY and MAC layers, and standardization."
]
] | 1906.04474 |
[
[
"Likelihood-free approximate Gibbs sampling"
],
[
"Abstract Likelihood-free methods such as approximate Bayesian computation (ABC) have extended the reach of statistical inference to problems with computationally intractable likelihoods.",
"Such approaches perform well for small-to-moderate dimensional problems, but suffer a curse of dimensionality in the number of model parameters.",
"We introduce a likelihood-free approximate Gibbs sampler that naturally circumvents the dimensionality issue by focusing on lower-dimensional conditional distributions.",
"These distributions are estimated by flexible regression models either before the sampler is run, or adaptively during sampler implementation.",
"As a result, and in comparison to Metropolis-Hastings based approaches, we are able to fit substantially more challenging statistical models than would otherwise be possible.",
"We demonstrate the sampler's performance via two simulated examples, and a real analysis of Airbnb rental prices using a intractable high-dimensional multivariate non-linear state space model containing 13,140 parameters, which presents a real challenge to standard ABC techniques."
],
[
"1 Likelihood-free methods such as approximate Bayesian computation (ABC) have extended the reach of statistical inference to problems with computationally intractable likelihoods.",
"Such approaches perform well for small-to-moderate dimensional problems, but suffer a curse of dimensionality in the number of model parameters.",
"We introduce a likelihood-free approximate Gibbs sampler that naturally circumvents the dimensionality issue by focusing on lower-dimensional conditional distributions.",
"These distributions are estimated by flexible regression models either before the sampler is run, or adaptively during sampler implementation.",
"As a result, and in comparison to Metropolis-Hastings based approaches, we are able to fit substantially more challenging statistical models than would otherwise be possible.",
"We demonstrate the sampler's performance via two simulated examples, and a real analysis of Airbnb rental prices using a intractable high-dimensional multivariate non-linear state space model containing 13,140 parameters, which presents a real challenge to standard ABC techniques.",
"Key words: Approximate Bayesian computation; Gibbs sampler; State space models."
],
[
"Introduction",
"Likelihood-free methods refer to procedures that perform likelihood-based statistical inference, but without direct evaluation of the likelihood function.",
"This is attractive when the likelihood function is computationally prohibitive to evaluate due to dataset size or model complexity, or when the likelihood function is only known through a data generation process.",
"Some classes of likelihood-free methods include pseudo-marginal methods [6], [2], indirect inference Gourieroux1993 and approximate Bayesian computation handbook.",
"In particular, approximate Bayesian computation (ABC) methods form an approximation to the computationally intractable posterior distribution by firstly sampling parameter vectors from the prior, and conditional on these, generating synthetic datasets under the model.",
"The parameter vectors are then weighted by how well a vector of summary statistics of the synthetic datasets matches the same summary statistics of the observed data.",
"ABC methods have seen extensive application and development over the past 15 years.",
"See e.g.",
"handbook for a contemporary overview of this area.",
"However, ABC methods have mostly been limited to analyses with moderate numbers of parameters ($<50$ ) due to the inherent curse-of-dimensionality of matching larger numbers of summary statistics, in what may be viewed as a high-dimensional kernel density estimation problem [8].",
"For a fixed computational budget, the quality of the ABC posterior approximation deteriorates rapidly as the number of summary statistics (which is driven by the number of model parameters) increases nott+ofs17.",
"A number of techniques for extending ABC methods to higher dimensional models have been developed.",
"Post-processing techniques aim to reduce the approximation error by adjusting samples drawn from the ABC posterior approximation in a beneficial manner.",
"These include regression-adjustments Beaumont2002,blum+f10,blum+nps13, marginal adjustment Nott2012, and recalibration rodrigues+ps17,Prangle2013.",
"However, by their nature post-processing techniques are a means to improve an existing analysis rather than a principled approach to extend ABC methods to higher dimensions.",
"In addition, evidence is emerging that some of these procedures, in particular regression-adjustment, perform less well than is generally believed marin+rprr16,frazier+rr17.",
"Alternative model-based approximations to the intractable posterior have been developed, including Gaussian copula models Li2017, Gaussian mixture models bonassi+yw11, regression density estimation fan+ns13, Gaussian processes gutmann+c16, Bayesian indirect inference Drovandi2015,drovandi+mr17, variational Bayes tran+nk17 and synthetic likelihoods wood10,ong+ntsd16.",
"Each of these alternative models have appealing properties, although none of them fully address the high-dimensional ABC problem.",
"One technique that has some promise in helping extend ABC methods to higher dimensions is likelihood (or posterior) factorisation.",
"When the likelihood can be factorised into lower dimensional components, lower dimensional comparisons of summary statistics can be made, thereby side-stepping the curse of dimensionality to some extent.",
"This has been explored within hierarchical models by Bazin2010, within an expectation-propagation scheme by barthelme+c14, for discretely observed Markov models by white+kp15, and within the copula-ABC approach of Li2017.",
"However, such a factorisation is only available for particularly structured models (although see Li2017).",
"Other approaches include rephrasing summary statistic matching as a rare event problem prangle+ek16, and using local Bayesian optimisation techniques for high-dimensional intractable models meeds+w15,gutmann+c16.",
"In one particular take on posterior factorisation, Kousathanas2016 developed an ABC Markov chain Monte Carlo (MCMC) algorithm which only updates one parameter per iteration, so that the new candidate can be accepted or rejected based on a small subset of the summary statistics.",
"This approach can increase MCMC acceptance rates, although it is limited by the need to generate a synthetic dataset at each algorithm iteration, which may be computationally prohibitive if used for expensive simulators.",
"It also requires the identification of conditionally sufficient statistics for each parameter.",
"In this article we introduce a likelihood-free approximate Gibbs sampler that targets the high-dimensional posterior indirectly by approximating its full conditional distributions.",
"Low-dimensional regression-based models are constructed for each of these conditional distributions using synthetic (simulated) parameter value and summary statistic pairs, which then permit approximate Gibbs update steps.",
"In contrast to Kousathanas2016, synthetic datasets are not generated during each sampler iteration, thereby providing efficiencies for expensive simulator models, and only require sufficient synthetic datasets to adequately construct the full conditional models (e.g. fan+ns13).",
"Construction of the approximate conditional distributions can exploit known structures of the high-dimensional posterior, where available, to considerably reduce computational overheads.",
"The models themselves can also be constructed in localised or global forms.",
"In Section we introduce the method for constructing regression-based conditional distributions and for implementing the likelihood-free approximate Gibbs sampler, and discuss possible sampler variants.",
"In Section , we explore the performance of the algorithm under various sampler and model settings, and provide a real data analysis of an Airbnb dataset using an intractable state space model with 13,140 parameters in Section .",
"Section concludes with a discussion."
],
[
"Likelihood-free approximate Gibbs sampler",
"Suppose that $\\theta =(\\theta _1,\\ldots ,\\theta _D)^\\top $ is a $D$ -dimensional parameter vector, with associated prior distribution $\\pi (\\theta )$ , and a computationally intractable model for data $p(X|\\theta )$ .",
"Given the observed data, ${X_\\mathrm {obs}}$ , interest lies in the posterior distribution $\\pi (\\theta |{X_\\mathrm {obs}}) \\propto p({X_\\mathrm {obs}}|\\theta ) \\pi (\\theta )$ .",
"The ABC approximation is given by $\\pi _\\mathrm {ABC}(\\theta |{s_\\mathrm {obs}}) \\propto \\pi (\\theta )\\int K_h(\\Vert S(X)-{s_\\mathrm {obs}}\\Vert )p(X|\\theta )dX,$ where $s=S(X)$ is a vector of summary statistics, ${s_\\mathrm {obs}}=S({X_\\mathrm {obs}})$ and $K_h(u)=K(u/h)/h$ is a smoothing kernel with bandwidth parameter $h>0$ .",
"If the summary statistics $s$ are sufficient then the approximation error can be made arbitrarily small by taking $h \\rightarrow 0$ as in this case $\\pi _\\mathrm {ABC}(\\theta |{s_\\mathrm {obs}})$ will converge to the posterior distribution $\\pi (\\theta |{X_\\mathrm {obs}})$ .",
"Otherwise, for non-sufficient $s$ and $h>0$ the approximation is given as (REF ).",
"See e.g.",
"sisson+fb17 for further discussion on this approximation.",
"A simple procedure to draw samples from $\\pi _\\mathrm {ABC}(\\theta |{s_\\mathrm {obs}})$ is given in Algorithm .",
"More sophisticated algorithms are available (e.g. sisson+f18).",
"[tb] A simple importance sampling ABC algorithm Inputs: [noitemsep] An observed dataset ${X_\\mathrm {obs}}$ .",
"A prior $\\pi (\\theta )$ and intractable generative model $p(X|\\theta )$ .",
"An observed vector of summary statistics ${s_\\mathrm {obs}}=S({X_\\mathrm {obs}})$ .",
"A smoothing kernel $K_h(u)$ with scale parameter $h>0$ .",
"A positive integer $N$ defining the number of ABC samples.",
"Data simulation and weighting: For $i=1, \\ldots , N$ : Generate $\\theta ^{(i)} \\sim \\pi (\\theta )$ from the prior.",
"Generate $X^{(i)} \\sim p(X|\\theta ^{(i)})$ from the model.",
"Compute the summary statistics $s^{(i)}=S(X^{(i)})$ .",
"Compute the sample weight $w^{(i)} \\propto K_h(\\Vert s^{(i)}-{s_\\mathrm {obs}}\\Vert )$ .",
"Output: A set of weighted samples $\\lbrace (\\theta ^{(i)}, w^{(i)})\\rbrace _{i=1}^N$ from $\\pi _{ABC}(\\theta |{s_\\mathrm {obs}})$ .",
"Regression-adjustment post-processing methods Beaumont2002,blum+f10,blum+nps13 are commonly used to mitigate the effect of $h>0$ in (REF ) by fitting regression models of the form $\\theta _{d}|S \\sim f(\\theta _{d} | \\beta ^+_d, S)$ , for $d=1, \\ldots , D$ , based on the weighted samples $\\lbrace (\\theta ^{(i)},s^{(i)},w^{(i)})\\rbrace _{i=1}^N$ , that are as close as possible to the corresponding intractable marginal distributions $\\pi (\\theta _{d} | S)$ in the region of ${s_\\mathrm {obs}}$ .",
"For example, in the local linear approach of Beaumont2002 the fitted models are of the form $\\theta _d^{(i)} = \\alpha _d + \\beta _d^\\top (s^{(i)}-{s_\\mathrm {obs}}) + \\epsilon _d^{(i)},$ for $i=1,\\ldots ,N$ and $d=1,\\ldots ,D$ , where $\\alpha _d\\in \\mathbb {R}$ , $\\beta _d\\in \\mathbb {R}^q$ , $q$ is the length of the vector of summary statistics $s$ , and $\\epsilon _d^{(i)}\\sim N(0,\\sigma ^2_d)$ .",
"Here $\\beta ^+_d=(\\alpha _d,\\beta _d,\\sigma ^2_d)^\\top $ is the full vector of unknown regression parameters for model $d$ .",
"Regression-adjustment would then modify each $\\theta _d^{(i)}$ to reduce the discrepancy between $s^{(i)}$ and ${s_\\mathrm {obs}}$ via $\\theta ^{*(i)}_d=\\hat{\\beta }_d^\\top {s_\\mathrm {obs}}+ (\\theta _d^{(i)}-\\hat{\\beta }_d^\\top s^{(i)})$ where $\\hat{\\beta }_d$ denotes an estimated (e.g.",
"least squares) value of $\\beta _d$ .",
"To construct the likelihood-free approximate Gibbs sampler we similarly build regression models, but in this case we construct regression models of the form $\\theta _d|(S,\\theta _{-d}) \\sim f(\\theta _d|\\beta ^+_d,g_d(S,\\theta _{-d}))$ , where $\\theta _{-d}$ is the vector $\\theta $ but excluding $\\theta _d$ , so that $f(\\theta _d|\\beta ^+_d,g_d({s_\\mathrm {obs}},\\theta _{-d}))$ is as close as possible to the true conditional distribution $\\pi (\\theta _d|{s_\\mathrm {obs}},\\theta _{-d})$ of $\\pi (\\theta |{s_\\mathrm {obs}})$ .",
"The functions $g_d(S,\\theta _{-d})$ indicate the function of $\\bf {S}$ and $\\theta _{-d}$ used in the regression model to determine the conditional distribution of $\\theta _d$ , such as e.g.",
"main effects or interactions.",
"Clearly the appropriate dependent variables will vary with $d$ , but will typically be relatively low dimensional (see the analyses in Section for a guide on how these may be selected).",
"The approximate Gibbs sampler will then cycle through each of these conditional distributions in turn, drawing $\\theta _d\\sim f(\\theta _d|\\hat{\\beta }^+_d,{s_\\mathrm {obs}},\\theta _{-d})$ for $d=1,\\ldots ,D$ , conditioning on $s={s_\\mathrm {obs}}$ .",
"If $f(\\theta _d|\\hat{\\beta }^+_d,{s_\\mathrm {obs}},\\theta _{-d})=\\pi (\\theta _d|{s_\\mathrm {obs}},\\theta _{-d})$ then the resulting Gibbs sampler will exactly target $\\pi (\\theta |{s_\\mathrm {obs}})$ .",
"Otherwise, the resulting sampler will be an approximation (discussed further below).",
"This procedure is outlined in Algorithm .",
"[tb] Likelihood-free approximate Gibbs sampling (localised models) Inputs: [noitemsep] An observed dataset ${X_\\mathrm {obs}}$ .",
"A prior $\\pi (\\theta )$ and intractable generative model $p(X|\\theta )$ .",
"A sampling distribution $b(\\theta )$ describing a region of high posterior density.",
"An observed vector of summary statistics ${s_\\mathrm {obs}}=S({X_\\mathrm {obs}})$ .",
"A smoothing kernel $K_h(u)$ with scale parameter $h>0$ .",
"A positive integer $N$ defining the number of ABC samples.",
"A positive integer $M$ defining the number of Gibbs sampler iterations.",
"A collection of regression models $f(\\theta _{d} | \\beta ^+_d, g_d(S, \\theta _{-d}))$ to approximate each full conditional distribution $\\pi (\\theta _d|{s_\\mathrm {obs}},\\theta _{-d})$ for $d=1,\\ldots ,D$ .",
"Data simulation: For $i=1, \\ldots , N$ : Generate $\\theta ^{(i)} \\sim b(\\theta )$ from some suitable distribution $b(\\theta )$ .",
"Generate $X^{(i)} \\sim p(X|\\theta ^{(i)})$ from the model.",
"Compute the summary statistics $s^{(i)}=S(X^{(i)})$ .",
"Approximate Gibbs sampling: Initialise $\\tilde{\\theta }^{(0)}=(\\tilde{\\theta }_1^{(0)},\\ldots ,\\tilde{\\theta }_D^{(0)})^\\top $ .",
"For $m=1,\\ldots ,M$ : 11 For $d=1,\\ldots ,D$ : Denote by $\\theta ^\\star _{-d}=(\\tilde{\\theta }_1^{(m)},\\ldots ,\\tilde{\\theta }^{(m)}_{d-1},\\tilde{\\theta }^{(m-1)}_{d+1},\\ldots ,\\tilde{\\theta }^{(m-1)}_{D})^\\top $ the vector containing the most recently updated values of $\\tilde{\\theta }^{(\\cdot )}_j$ , $j\\ne d$ .",
"Set the regression weights $w_d^{(i)}=K_h(\\Vert g_d(s^{(i)}, \\theta ^{(i)}_{-d})-g_d({s_\\mathrm {obs}}, \\theta ^\\star _{-d})\\Vert )\\pi (\\theta )/b(\\theta )$ for $i=1,\\ldots ,N$ .",
"Fit a suitable regression model $\\theta _{d}|(S, \\theta _{-d}) \\sim f(\\theta _{d} | \\beta ^+_d, g_d(S, \\theta _{-d}))$ using the weighted samples $\\lbrace (\\theta ^{(i)},s^{(i)},w_d^{(i)})\\rbrace _{i=1}^N$ , so that $f(\\theta _{d} | \\hat{\\beta }^+_d, g_d({s_\\mathrm {obs}}, \\theta ^\\star _{-d}))$ locally approximates the full conditional distribution $\\pi (\\theta _d|{s_\\mathrm {obs}},\\theta ^\\star _{-d})$ .",
"Gibbs update: sample $\\tilde{\\theta }^{(m)}_d|({s_\\mathrm {obs}}, \\theta ^\\star _{-d}) \\sim f (\\theta _d | \\hat{\\beta }^+_d, g_d({s_\\mathrm {obs}}, \\theta ^\\star _{-d}) )$ .",
"Output: Realised Gibbs sampler output $\\tilde{\\theta }^{(0)},\\ldots ,\\tilde{\\theta }^{(M)}$ with target distribution $\\approx \\pi (\\theta |{s_\\mathrm {obs}})$ .",
"The algorithm begins similarly to many ABC algorithms, by drawing samples $\\lbrace (\\theta ^{(i)},s^{(i)})\\rbrace _{i=1}^N$ from the predictive distribution $(\\theta ^{(i)},X^{(i)})\\sim p(X|\\theta )b(\\theta )$ and computing $s^{(i)}=S(X^{(i)})$ .",
"In most standard ABC algorithms $b(\\theta )$ is the prior distribution $\\pi (\\theta )$ or an importance sampling distribution.",
"Then, a standard Gibbs sampler procedure is implemented by sampling each parameter in turn from an approximation to its full conditional distribution $\\theta ^{(m)}_d|({s_\\mathrm {obs}}, \\theta _{-d}) \\sim f (\\theta _d | \\hat{\\beta }^+_d, g_d({s_\\mathrm {obs}}, \\theta _{-d}) )$ .",
"These approximations are fitted using the pool of weighted samples $\\lbrace (\\theta ^{(i)},s^{(i)},w_d^{(i)})\\rbrace _{i=1}^N$ , where the weights $w_d^{(i)}\\propto K_h(\\Vert (s^{(i)}, \\theta ^{(i)}_{-d})-({s_\\mathrm {obs}}, \\theta ^\\star _{-d})\\Vert )\\pi (\\theta )/b(\\theta )$ ensure that higher importance is given to those samples which more closely match both the observed data ${s_\\mathrm {obs}}$ and the conditioned values of the parameters $\\theta _{-d}=\\theta _{-d}^\\star $ .",
"Clearly it is important that consideration be given to appropriate scaling of summary statistics and parameter values within the distance measure $\\Vert \\cdot \\Vert $ to avoid one or other dominating the comparison.",
"Note that it is only required that the full conditionals are estimated well in regions of high posterior density, rather than over the entirety of the support of $\\theta $ .",
"In this manner, the importance density $b(\\theta )$ can be chosen to place $\\theta ^{(i)}$ samples in regions where the conditional distributions need to be well approximated, which may be a much smaller region than specified by the prior $\\pi (\\theta )$ (e.g. fan+ns13).",
"One such strategy was successfully adopted by Fearnhead2012 who specified $b(\\theta )$ as proportional to the prior $\\pi (\\theta )$ but restricted to a region of high posterior density as identified by a pilot simulation.",
"Any appropriate regression technique can be used to construct the models $f (\\theta _d | \\beta ^+_d, g_d({s_\\mathrm {obs}}, \\theta _{-d}) )$ such as non-parametric models, GLMs, neural networks, semi-parametric models, lasso etc.",
"There are two possible ways to draw samples from each conditional regression model (step 2.2.4 in Algorithm ).",
"The first is when a parametric error distribution has been assumed, in which case a new sample may be drawn directly from the fitted distribution.",
"For example, if the regression model is specified such that $\\theta _d\\sim N(\\hat{\\mu },\\hat{\\sigma }^2)$ for specified $\\hat{\\mu }$ and $\\hat{\\sigma }^2$ , then a new value of $\\theta _d$ may be drawn directly from $N(\\hat{\\mu },\\hat{\\sigma }^2)$ .",
"Alternatively, when a parametric error distribution is not assumed, the (weighted) distribution of empirical residuals $r^{(i)}_d = \\theta ^{(i)}_d-\\hat{\\mu }$ can be constructed as $R^N_d(r)=\\sum _{i=1}^Nw_d^{*(i)}\\delta _{r_d^{(i)}}(r)$ where $w_d^{*(i)}=w_d^{(i)}/\\sum _{j=1}^Nw_d^{(j)}$ , and $\\delta _Z(z)$ is the Dirac measure, defined as $\\delta _Z(z)=1$ if $z\\in Z$ and $\\delta _Z(z)=0$ otherwise.",
"A new value of $\\theta _d$ is then given by $\\theta _d=\\hat{\\mu } + r$ where $r\\sim R^N_d(r)$ .",
"[tb] Likelihood-free approximate Gibbs sampling (global models) [Changes from Algorithm 2.]",
"Approximate Gibbs sampling: Initialise $\\tilde{\\theta }^{(0)}=(\\tilde{\\theta }_1^{(0)},\\ldots ,\\tilde{\\theta }_D^{(0)})^\\top $ .",
"Compute the sample weights $w^{(i)} \\propto K_h(\\Vert s^{(i)}-{s_\\mathrm {obs}}\\Vert )\\pi (\\theta )/b(\\theta )$ , for $i=1,\\ldots N$ .",
"For $d=1,\\ldots ,D$ : Fit a suitable regression model $\\theta _{d}|(S, \\theta _{-d}) \\sim f(\\theta _{d} | \\beta ^+_d, g_d(S, \\theta _{-d}))$ using the weighted samples $\\lbrace (\\theta ^{(i)},s^{(i)},w^{(i)})\\rbrace _{i=1}^N$ , so that $f(\\theta _{d} | \\hat{\\beta }^+_d, g_d({s_\\mathrm {obs}}, \\theta _{-d}))$ locally approximates the full conditional distribution $\\pi (\\theta _d|{s_\\mathrm {obs}},\\theta _{-d})$ .",
"For $m=1,\\ldots ,M$ : 11 For $d=1,\\ldots ,D$ : Denote by $\\theta ^*_{-d}=(\\tilde{\\theta }_1^{(m)},\\ldots ,\\tilde{\\theta }^{(m)}_{d-1},\\tilde{\\theta }^{(m-1)}_{d+1},\\ldots ,\\tilde{\\theta }^{(m-1)}_{D})^\\top $ the vector containing the most recently updated values of $\\tilde{\\theta }^{(\\cdot )}_j$ , $j\\ne d$ .",
"Gibbs update: sample $\\tilde{\\theta }^{(m)}_d|({s_\\mathrm {obs}}, \\theta ^\\star _{-d}) \\sim f (\\theta _d | \\hat{\\beta }^+_d, g_d({s_\\mathrm {obs}}, \\theta ^\\star _{-d}) )$ .",
"The computational overheads in Algorithm are in the initial data simulation stage (steps 1.1–1.3) which is standard in many ABC algorithms, and in the fitting of a separate regression model for each parameter $\\theta _d$ in each stage of the Gibbs sampler (steps 2.2.2–2.2.3).",
"For the latter, while it can be computationally cheap to fit any one regression model, repeating this $MD$ times during sampler implementation can clearly raise the computational burden.",
"There are two approaches that can reduce these costs, which can be implemented either separately or concurrently.",
"In certain cases, the model $p(\\theta |{s_\\mathrm {obs}})$ will have a structure such that several of the model parameters will have exactly the same form of full conditional distribution $\\pi (\\theta _d|{s_\\mathrm {obs}},\\theta _{-d})$ .",
"One such example is a hierarchical model (see Section REF ) where $x_{dj}\\sim p(x|\\theta _d)$ for $j=1,\\ldots ,n_d$ , and $\\theta _1,\\ldots ,\\theta _{D-1}\\sim N(\\theta _D,\\sigma ^2)$ .",
"Here the form of $\\pi (\\theta _d|{s_\\mathrm {obs}},\\theta _{-d})$ is identical for $d=1,\\ldots ,D-1$ .",
"Accordingly the regression model $f(\\theta _{d} | \\beta ^+_d, g_d(S, \\theta _{-d}))$ can be fitted by pooling the weighted samples $\\lbrace (\\theta ^{(i)},s^{(i)},w_d^{(i)})\\rbrace _{i=1}^N$ for $d=1,\\ldots ,D-1$ (each using different sub-elements of the vectors), thereby allowing computational savings in allowing the value of $N$ to be reduced.",
"Further, in the case where the conditional independence graph structure of the posterior is known (again, consider the hierarchical model), then the choice of which elements of $\\theta _{-d}$ should be included within the regression function $g_d(S,\\theta _{-d})$ is immediately specified as the neighbours of $\\theta _d$ on the conditional independence graph, and this does not then require independent elicitation.",
"Finally, in well-structured models, some parameters may be conditionally independent of all intractable nodes in the graph.",
"In such cases the corresponding true conditional distribution can be directly derived, instead of approximated by a regression model (see Section ).",
"A second approach is to choose the regression model $f(\\theta _{d} | \\beta ^+_d, g_d(S, \\theta _{-d}))$ sufficiently flexibly so that not only is it a good approximation of $\\pi (\\theta _{d} |{s_\\mathrm {obs}}, \\theta _{-d})$ when $\\theta _{-d}$ is fixed at a particular value, $\\theta ^\\star _{-d}$ , within the Gibbs sampler, but that the regression model holds globally for any $\\theta _{-d}$ .",
"Within Algorithm , the approximation of $\\pi (\\theta _{d} |{s_\\mathrm {obs}}, \\theta ^\\star _{-d})$ with $\\theta _{-d}=\\theta ^\\star _{-d}$ is achieved by weighting the $\\lbrace (\\theta ^{(i)},s^{(i)})\\rbrace _{i=1}^N$ samples in the region of $\\theta ^\\star _{-d}$ according to step 2.2.2.",
"If the regression model $f(\\theta _{d} | \\beta ^+_d, g_d(S, \\theta _{-d}))$ was a good approximation of $\\pi (\\theta _{d} |{s_\\mathrm {obs}}, \\theta _{-d})$ for any value of $\\theta _{-d}$ (in the region of high posterior density), then the $\\theta ^\\star _{-d}$ specific weighting of step 2.2.2 can be removed, all samples weighted as $w^{(i)}\\propto K_h(|s^{(i)}-{s_\\mathrm {obs}}\\Vert )\\pi (\\theta )/b(\\theta )$ , thereby localising on summary statistics only, and the regression models fitted once only, prior to implementing the Gibbs sampler.",
"This global model likelihood-free approximate Gibbs sampler is described in Algorithm .",
"Clearly the computational overheads of Algorithm are substantially lower than for the localised model version.",
"However, the localised version may be expected to be more accurate in practice, precisely due to the localised approximation of the full conditional distributions, and the difficulty in deriving sufficiently accurate global regression models.",
"In certain circumstances it can be seen that the likelihood-free approximate Gibbs sampler will exactly target the true partial posterior $\\pi (\\theta |{s_\\mathrm {obs}})$ .",
"In the case where the true conditional distributions $\\pi (\\theta _d|{s_\\mathrm {obs}},\\theta _{-d})$ are nested within the family of distributions described by $f(\\theta _{d} | \\beta ^+_d, g_d(S, \\theta _{-d}))$ , then as $N\\rightarrow \\infty $ , which in turn allows $h\\rightarrow 0$ , then $f(\\theta _{d} | \\hat{\\beta }^+_d, g_d(S, \\theta _{-d}))\\rightarrow \\pi (\\theta _d|{s_\\mathrm {obs}},\\theta _{-d})$ due to the law of large numbers ($N\\rightarrow \\infty $ ) and $h\\rightarrow 0$ eliminating the usual local ABC approximation error.",
"In this case, then Algorithms and will be exact.",
"In any other cases, $f(\\theta _{d} | \\hat{\\beta }^+_d, g_d(S, \\theta _{-d}))$ will be an approximation of $\\pi (\\theta _d|{s_\\mathrm {obs}},\\theta _{-d})$ .",
"This can be either a strong or weak approximation, whereby under a strong approximation $f(\\theta _{d} | \\beta ^+_d, g_d(S, \\theta _{-d}))$ can exactly describe $\\pi (\\theta _d|{s_\\mathrm {obs}},\\theta _{-d})$ but where $\\hat{\\beta }^+_d$ has not converged to $\\beta ^+_d$ (i.e.",
"finite $N$ ).",
"In this case, the likelihood-free approximate Gibbs sampler comes under the noisy Monte Carlo framework of alquier+feb16.",
"Under a weak approximation, $\\pi (\\theta _d|{s_\\mathrm {obs}},\\theta _{-d})$ is not nested within the family $f(\\theta _{d} | \\beta ^+_d, g_d(S, \\theta _{-d}))$ , and so $f(\\theta _{d} | \\hat{\\beta }^+_d, g_d(S, \\theta _{-d}))$ represents the closest approximation to $\\pi (\\theta _d|{s_\\mathrm {obs}},\\theta _{-d})$ available within the regression model's functional constraints.",
"This latter (weak) approximation can be arbitrarily good or poor.",
"When the fitted regression models only approximate the true posterior conditionals, then these may be incompatible in the sense that the set of approximate conditional distributions may not imply a joint distribution that is unique or even exists.",
"This is equally a criticism of the ABC-MCMC sampler of Kousathanas2016 as it is of the likelihood-free approximate Gibbs sampler, unless for the former it can be guaranteed that the subset of summary statistics used to update $\\theta _d$ in an ABC Metropolis-Hastings update step is sufficient for the full conditional distribution.",
"See e.g.",
"Arnold1999 for a book-length treatment of conditional specification of statistical models.",
"Incompatible conditional distributions are commonly encountered in the area of multivariate imputation by chained equations (MICE) also known as fully conditional specification (FCS), which is specifically designed for incomplete data problems [58].",
"In the simplified case of multivariate conditional distributions within exponential families, Arnold1999 found that determining appropriate constraints on the model parameters to ensure a valid joint density was often unattainable.",
"However, other authors have expressed uncertainty on the effects of incompatibility, and simulation studies have suggested that the problem may not be serious in practice Buuren2011,buuren+bgr06,drechsler+r08.",
"chen+ip15 have investigated the behaviour of the Gibbs sampler when the conditional distributions are potentially incompatible.",
"However, in a more general study of parameterisation within Bayesian modelling, gelman04 embraces the opportunities for inference based on inconsistent conditional distributions as a new class of models, motivated by computational and analytical convenience in order to bypass the limitations of joint models."
],
[
"Simulation studies",
"We examine the performance of the likelihood-free approximate Gibbs sampler in two simulation studies: a Gaussian mixture model using global regression models, and in a simple hierarchical model with both local and global regression models."
],
[
"A Gaussian mixture model",
"We consider the $D$ -dimensional Gaussian mixture model of Nott2012 where $ p(s|\\theta ) = \\sum _{b_1=0}^1 \\ldots \\sum _{b_D=0}^1 \\left[ \\prod _{i=1}^D \\omega ^{1-b_i} (1-\\omega )^{b_i} \\right] \\phi _D(s|\\mu (b, \\theta ), \\Sigma ),$ where $\\phi _D(x|a, B)$ denotes the multivariate Gaussian density with mean $a$ and covariance $B$ evaluated at $x$ , $\\omega \\in [0, 1]$ is a mixture weight, $\\mu (b, \\theta ) = ((1-2b_1)\\theta _1, \\ldots , (1-2b_D)\\theta _D)^\\top $ , $b=(b_1, \\ldots , b_D)^\\top $ with $b_i \\in \\lbrace 0, 1\\rbrace $ , and $\\Sigma =[\\Sigma _{ij}]$ is such that $\\Sigma _{ii}=1$ and $\\Sigma _{ij}=\\rho $ for $i \\ne j$ .",
"For illustration we consider the $D=2$ dimensional case, with ${s_\\mathrm {obs}}=(5/2, 5/2)^\\top $ , fix $\\omega =0.3$ and $\\rho =0.7$ as known constants and specify $\\pi (\\theta _d)$ as $U(-20, 40)$ for $d=1,2$ .",
"In this setting, the full conditional distributions for $\\theta _1$ and $b_1$ are given by $\\theta _1|(\\theta _2, b, s) & \\sim & N(\\mu _{\\theta _1}, \\sqrt{1-\\rho ^2})I(-20<\\theta _1<40),\\nonumber \\\\ \\mu _{\\theta _1}& =& s_1 -\\rho s_2 + \\rho \\theta _2 -2 s_1b_1 + 2\\rho s_2 b_1 -2\\rho b_1 \\theta _2 -2\\rho \\theta _2 b_2 + 4\\rho b_1 b_2 \\theta _2 \\\\b_1|(\\theta , b_2, s) & \\sim & \\text{Bernoulli}(L(p_{b_1})),\\nonumber \\\\p_{b_1}& =& \\ln \\left(\\frac{1-\\omega }{\\omega }\\right) - \\frac{2}{1-\\rho ^2} s_1\\theta _1 + \\frac{2\\rho }{1-\\rho ^2} s_2\\theta _1 - \\frac{2\\rho }{1-\\rho ^2} \\theta _1 \\theta _2 + \\frac{4\\rho }{1-\\rho ^2} b_2 \\theta _1 \\theta _2,\\nonumber $ where $L(x)=1/(1+\\exp (-x))$ denotes the logistic function.",
"The full conditional distributions for $\\theta _2$ and $b_2$ may be obtained by switching the indices in the above.",
"For this simple model we construct global regression models (Algorithm ).",
"We generate $N=1,000,000$ samples from the prior predictive distribution (i.e.",
"with $b(\\theta )=\\pi (\\theta )$ ) and specify $K_h(u)$ as the uniform kernel ($h=\\infty $ ).",
"As an illustration, we first naively attempt to approximate the full conditional distribution of $\\theta _1$ by a main-effects only (excluding $b$ ) Gaussian regression model $\\theta _1|(\\theta _2, b, s) \\sim N(\\beta _0 + \\beta _1 s_1 + \\beta _2 s_2 + \\beta _3 \\theta _2, \\sigma ^2)$ .",
"The resulting MLEs were $\\hat{\\beta } = (8.76, -0.31, 0, 0)^\\top $ (s.e.",
"$=(0.019,0.001,0.001,0.001)$ ) and $\\hat{\\sigma }=16.16$ , which suggests that $\\theta _1$ is conditionally independent of $s_2$ and $\\theta _2$ .",
"This can clearly be seen to be incorrect based on a simple graphical exploration of the synthetic samples.",
"This is a clear warning of the need to consider sufficiently flexible regression models, with interaction effects (as discussed in Nott2012 and as is evident in the form of $\\mu _{\\theta _1}$ ).",
"Instead, we specify the regression mean with all main effects and interactions and, because the number of samples $N$ is large, the resulting MLEs of $\\beta $ (and $\\sigma ^2$ ) matched the true values in (REF ) up to at least one decimal place (not shown).",
"Figure REF shows a kernel density estimate (KDE) of the differences between the fitted and true conditional mean values ($\\hat{\\mu }^{(i)}_{\\theta _1}-\\mu ^{(i)}_{\\theta _1}$ ) for each of the $N$ data points used in the regression.",
"In most cases, the absolute difference was less than $0.05$ .",
"Figure REF shows a KDE of the empirical residuals and the true $N(0, \\sqrt{1-\\rho ^2})$ error density.",
"The similarity suggests that in sampling from the regression model, randomly choosing a residual is essentially equivalent to sampling from the true Gaussian error distribution.",
"Given that we are fitting a regression model in the same family as the true conditional distribution, we have a strong approximation of $\\theta _1|(\\theta _2,b,s)$ (as defined in Section ) in this case.",
"Figure: Assessing the quality of the regression approximation.",
"(a) A kernel density estimate (KDE) of the differences between the fitted and true conditional mean values μ ^ θ 1 (i) -μ θ 1 (i) \\hat{\\mu }^{(i)}_{\\theta _1}-\\mu ^{(i)}_{\\theta _1}.",
"(b) The true N(0,1-ρ 2 )N(0, \\sqrt{1-\\rho ^2}) error density and the KDE of the fitted regression residuals.",
"(c) The fitted and the true conditional distribution p(b 1 =1|θ 1 ,s 1 =s 2 =2.5,b 2 =0,θ 2 =-2.5)p(b_1=1|\\theta _1, s_1=s_2=2.5, b_2=0, \\theta _2=-2.5).",
"(d) True versus estimated probability of changing the the state of the cluster indicator variable b 1 b_1.In a similar manner, we naturally model the conditional distribution of $b_1|(\\theta ,b_2,s)$ as a Bernoulli GLM with logistic link function, and all possible conditional main effects and interactions.",
"Figure REF examines the quality of this approximation by presenting the cdf's of the fitted and the true probabilities of $p(b_1=1|\\theta _1, s_1=s_2=2.5, b_2=0, \\theta _2=-2.5)$ .",
"The distributions are very similar, though still distinguishable.",
"An explanation for this is that for most of the $N$ samples, the conditional probability of $b_1$ is either (numerically) 0 or 1.",
"In other words, only the samples such that $\\theta $ is close to the origin are informative for the regression parameters.",
"This regression model is again a strong approximation to the true conditional distribution.",
"Figure: Likelihood-free approximate Gibbs sampler output.",
"(a) Sample path of the first 250 iterations of (θ 1 ,θ 2 )(\\theta _1,\\theta _2), with the values of (b 1 ,b 2 )(b_1,b_2) indicated by coloured points.",
"(b) Posterior density estimates (shading) based on 20,000 sampler iterations, and true posterior density contours.Figure REF illustrates the output of $M=20,000$ iterations of the resulting likelihood-free approximate Gibbs sampler, when initialised at $(\\theta _1, \\theta _2, b_1, b_2)=(0,-10,1,0)$ .",
"The sampler moves around the parameter space well, and visually appears to target the true posterior distribution.",
"During sampler implementation the true and estimated probabilities of switching the value of $b_1$ were recorded, and are illustrated in Figure REF .",
"Only a small proportion of the $M$ probabilities are larger than 0.2 (due to the form of the posterior), but on the whole the estimated probabilities are generally accurate, with a few exceptions.",
"For this example, the estimated conditional distributions (and associated switching probabilities of $b_1$ ) will approximate their true counterparts arbitrarily well as $N$ gets large, essentially due to the simple form of the true posterior distribution.",
"A better mixing approximate Gibbs sampler could also have been constructed for this posterior distribution, using a 4-level multinomial regression for the full conditional of $b|(\\theta ,s)$ and a bivariate Gaussian regression model for $\\theta |(b,s)$ ."
],
[
"A simple hierarchical model",
"We now compare the performance of a collection of approximate Gibbs sampler implementations for estimating a Gaussian hierarchical model with the ABC-MCMC method (ABC-PaSS; ABC with Parameter Specific Statistics) method introduced by Kousathanas2016, and the exact Gibbs sampler.",
"Hierarchical methods have been previously considered in the likelihood-free framework by e.g.",
"Bazin2010 and rodrigues+ns16.",
"The Gaussian hierarchical model, with parameters $\\theta =(\\mu _1, \\ldots , \\mu _U, \\mu , \\tau _{\\mu },\\tau _x)^\\top $ , is defined as $ X_{u\\ell } & \\sim N(\\mu _u, \\tau ^{-1}_x) \\\\\\mu _u & \\sim N(\\mu , \\tau ^{-1}_\\mu ) \\\\\\tau _x & \\sim \\text{Gamma}(\\alpha _x, \\nu _x) \\\\\\tau _\\mu & \\sim \\text{Gamma}(\\alpha _{\\mu }, \\nu _{\\mu }) \\\\\\mu & \\sim N(0, 1),$ latent] (mu) $\\mu $ ; latent, below=of mu, yshift=.25cm] (muu) $\\mu _u$ ; latent, right=1 of muu] (taumu) $\\tau _\\mu $ ; obs, below=of muu, yshift=.25cm] (x) $X_{u\\ell }$ ; latent, right=of x, xshift=0cm] (taux) $\\tau _x$ ; [inner sep=0.3cm,xshift=0cm,yshift=0cm] plate1 (x) $\\ell =1, \\ldots , L$; [inner sep=0.3cm, xshift=-0cm, yshift=0cm] plate2 (muu) (plate1) $u=1, \\ldots , U$ a; mu, taumu muu ; muu, taux x ; where $X_{u\\ell }$ denotes the $\\ell $ -th observation in group $u$ , for $\\ell =1, \\ldots , L$ and $u=1, \\ldots , U$ .",
"The model is tractable, allowing direct comparison between the exact and approximate posteriors.",
"The full conditional distributions and prior specification for this model are given in Table REF .",
"Table: Prior and full conditional distributions of the Gaussian hierarchical model.",
"Non-estimated conditionals (×\\times ) use the full conditional distribution within the Gibbs/ABC-PaSS samplers.The structure of this model may be exploited to simplify sampler computations in three meaningful ways, as discussed in Section .",
"First, $\\pi (\\mu _u|{s_\\mathrm {obs}}, \\theta _{-u})$ is identical for $u=1,\\ldots ,U$ , so these distributions only need to be approximated for one group.",
"Second, the nodes which should be included within the regression function $g_d(S,\\theta _{-d})$ are easily identified from the graph.",
"Third, it is only necessary to approximate the full conditional distribution of parameters that are conditionally dependent on intractable quantities.",
"In the following we only update $\\mu _u$ and $\\tau _x$ using approximate likelihood-free methods, and use the full conditional distributions for $\\mu $ and $\\tau _\\mu $ (Table REF ).",
"We compare the exact Gibbs sampler with three different approximation strategies (each using the same $g_d(\\cdot )$ functions): (a) Simple global: The conditional models are approximated by global linear regression models (Algorithm ); (b) Simple local: The conditional models have the same linear form as the simple global approach, but fits are localised at each Gibbs iteration (Algorithm ); (c) Flexible global: The conditional models are globally approximated (Algorithm ) by non-linear conditional heteroscedastic feed-forward multilayer artificial neural network models (blum+f10).",
"The distribution of each unit mean $\\mu _1, \\ldots , \\mu _U$ depends on the data exclusively through the corresponding unit-specific summary statistics $S_u=( \\overline{X}_u, \\hat{\\tau }_u)^\\top $ (e.g.",
"Bazin2010), where $\\overline{X}_{u}$ and $\\hat{\\tau }_u$ are the sample mean and precision of the data in group $u$ , respectively, and therefore we take $g_{u}(S, \\theta _{-u})=(1, \\mu , \\tau _\\mu , \\tau _x, S_u^\\top )^\\top $ .",
"Recall (Table REF ) that the conditional mean $\\mathbb {E}(\\mu _u|\\ldots )$ is a non-linear function of the covariates, and the conditional variance $\\mathbb {V}(\\mu _u|\\ldots )$ is not constant throughout the covariate space.",
"Consequently, for the linear and non-linear model approaches, we approximate the true conditional distribution by $\\mu _u | (S, \\theta _{-u}) &\\sim & N \\left((1, \\mu , \\tau _\\mu , \\tau _x, S_u^\\top )^\\top \\beta _{\\mu _u}, V_{\\mu _u} \\right),\\\\\\mu _u | (S, \\theta _{-u}) &=& m(g_{\\mu _u}(\\cdot )) + \\sigma (g_{\\mu _u}(\\cdot )) \\times \\zeta ,$ respectively, where $\\zeta $ is a random variable with mean zero and fixed variance.",
"The conditional expectation is estimated as $\\hat{m}(g_{\\mu _u}(\\cdot ))$ with a neural network using the R function h2o.deeplearning H2o2018 with default model settings.",
"The variance term $\\sigma (g_{\\mu _u}(\\cdot ))$ is similarly estimated by a gamma neural network fitted over the squared residuals $r^2=({\\mu _u} - \\hat{m}(g_{\\mu _u}(\\cdot )))^2$ .",
"An approximate sample from the full conditional distribution is then ${\\mu _u^*}=\\hat{m}(g_{\\mu _u}(\\cdot )) + \\hat{\\sigma }(g_{\\mu _u}(\\cdot )) \\times \\frac{{\\mu _u}^{(i)} - \\hat{m}^{(i)}}{\\hat{\\sigma }^{(i)}},$ where $i$ is randomly selected from $1, \\ldots , N$ (step 2.2.4 in Algorithm ).",
"For the full conditional distribution of $\\tau _x$ , after discarding uninformative nodes, we defined $g_{\\tau _x}(1, \\mu _1, \\ldots , \\mu _U, S_\\tau )=(1, \\overline{\\mu }, \\hat{\\tau }_{\\mu _u}, S_\\tau ^\\top )^\\top ,\\qquad S_\\tau =(\\overline{\\overline{X}}, \\tau _{\\overline{X}}, \\overline{\\hat{\\tau }}, \\tau _{\\hat{\\tau }})^\\top $ where the symmetric summary statistics are $\\overline{\\overline{X}} = \\frac{1}{U}\\sum _{u=1}^U \\overline{X}_{u}$ , $\\overline{\\hat{\\tau }} = \\frac{1}{U}\\sum _{u=1}^U \\hat{\\tau }_u$ , $\\tau _{\\overline{X}}=\\left[\\frac{1}{U-1} \\sum _{u=1}^U \\left(\\overline{X}_{u} - \\overline{\\overline{X}}\\right)^2 \\right]^{-1}\\qquad \\mbox{and}\\qquad \\tau _{\\hat{\\tau }}=\\left[\\frac{1}{U-1} \\sum _{u=1}^U \\left(\\hat{\\tau }_u - \\overline{\\hat{\\tau }} \\right)^2 \\right]^{-1}.\\\\$ The covariate vector $(\\mu _1, \\ldots , \\mu _U)$ was also summarised by its mean and precision, $\\overline{\\mu }$ and $\\hat{\\tau }_{\\mu _u}$ $\\overline{\\mu } = \\frac{1}{U}\\sum _{u=1}^U \\mu _u \\quad \\text{and} \\quad \\hat{\\tau }_{\\mu _u}=\\left[\\frac{1}{U-1} \\sum _{u=1}^U \\left(\\mu _u - \\overline{\\mu } \\right)^2 \\right]^{-1}.\\\\$ Sampling from the full conditional distribution of $\\tau _x$ is achieved following the same procedure as for $\\mu _u$ , except that we use a gamma (rather than Gaussian) neural network model for the non-linear mean function.",
"The essential idea behind the ABC-PaSS method Kousathanas2016 is to use approximately conditionally sufficient summary statistics within low-dimensional conditional Metropolis-Hastings updates.",
"To conduct a fair comparison with approximate Gibbs sampling, to update $\\tau _x$ and $\\mu _u$ , at each iteration we draw proposals from their known (in this case) full conditional distributions.",
"This favourably gives ABC-PaSS the best possible proposal distribution, and so allows the comparison between algorithms to focus on the form of the update mechanism.",
"The summary statistics used for each parameter update are the same as for the approximate Gibbs samplers ($S_\\tau $ and $S_u$ for $\\tau _x$ and $\\mu _u$ respectively).",
"Generating $S_u$ only requires simulating data from group $u$ .",
"The updates for $\\mu $ and $\\tau _\\mu $ are performed using Gibbs updates, as before.",
"We consider a single `iteration' of the ABC-PaSS algorithm to update each model parameter in turn.",
"We generate $L=10$ observations from $U=10$ groups with $\\mu =0$ , $\\tau _{\\mu }=\\tau _x=1$ and $\\alpha _\\mu =\\nu _\\mu =\\alpha _x=\\nu _x=1$ .",
"We simulate $M=10,000$ iterations from each sampler.",
"For the approximate Gibbs samplers, we first generated $N=10,000$ synthetic datasets from the prior predictive distribution.",
"For the global models we chose $K_h$ to be uniform, with $h$ determined to select the closest $5,000$ samples (in terms of Euclidean distance) to the observed symmetric summary statistics.",
"For the local model, for each localised regression model we kept the closest 10% of the $5,000$ samples.",
"For the kernels $K_h$ in the Metropolis-Hastings updates of the ABC-PaSS algorithm we set $h=0.5, 2$ for $\\mu _u$ and $\\tau _x$ respectively.",
"Each simulation was replicated a total of 500 times.",
"Figure: Performance of the exact Gibbs sampler, approximate Gibbs samplers (shades of blue) and the modified ABC-PaSS sampler (red) for the simple hierarchical model.",
"(a) Mean relative MSE and observed (90%90\\%) coverage for τ x \\tau _x and μ u \\mu _u based on 500 replications.",
"(b) Log-scale boxplot of sampler implementation times.",
"(c, d) Estimated marginal posteriors for μ 1 \\mu _1 and τ x \\tau _x for a single simulation.",
"(e, f) Typical Markov chain (only the last 5000 iterations, for clarity) sample paths for τ x \\tau _x for the ABC-PaSS and the flexible-global approximate Gibbs sampler.Figure REF shows the relative (mean) MSE and observed coverage $90\\%$ credibility intervals for $\\tau _x$ and $\\mu _u$ with respect to the exact Gibbs sampler.",
"ABC-PaSS performed significantly worse than the other samplers.",
"For the approximate Gibbs samplers the simple global model performed well for $\\mu _u$ , but was clearly worse for $\\tau _x$ , when compared to the other model specifications which performed relatively well.",
"Panel REF illustrates the time taken to run each sampler.",
"The exact Gibbs sampler takes less than 1s to complete, while the simple global approach and ABC-PaSS take less than 20s on average.",
"The remaining methods had comparable times ($\\sim $ 100s).",
"These times are broken down in Table REF .",
"The ABC-PaSS algorithm does not fit regression models and dataset generation is performed within the MCMC sampler.",
"For this example, generating 10,000 synthetic datasets took only 6.58 seconds.",
"The flexible-global approach required the fit of four Deep Learning regression models (modelling both mean and variance of $\\tau _x$ and $\\mu _u$ ), each computationally expensive.",
"Whereas the simple-local approach required 20,000 regression model fits, making each sampler iteration 3.5 times slower than the flexible-global strategy.",
"This simulation suggests that in applications where synthetic sampling is an expensive operation rodrigues+ps17, ABC-PaSS will be largely inefficient.",
"In comparison, the approximate Gibbs samplers make more efficient and repeated use of each synthetic sample, within each sampler iteration.",
"In practice the optimal approach will be determined by balancing the cost of synthetic dataset simulation and the required number of MCMC samples.",
"Table: Mean time (seconds) and number of synthetic dataset generations for each sampler, based on 500 replicate chains.",
"Figures for synthetic samples and regression fits are for operations before the sampler is run, excluding those indexed by **, which are performed within the sampler.Figures REF and REF show the estimated marginal posterior densities for $\\mu _1$ and $\\tau _x$ .",
"All approximate Gibbs implementations reasonably estimate the true density (black line), but the performance of ABC-PaSS is clearly poor.",
"For this algorithm, by setting the $K_h$ kernel scale parameter to $h=0.5,2$ for $\\mu _u$ and $\\tau _x$ we achieved Metropolis-Hastings acceptance rates of 20% and 18% respectively.",
"Lowering $h$ could improve the accuracy of this algorithm, however the sampler acceptance rates would fall further, and already the chain is experiencing poor mixing (the `sticking' phenomenon; sisson+ft07) in the tail of the distribution (Figure REF ).",
"In contrast, mixing for the approximate Gibbs sampler is excellent (Figure REF ).",
"Of the approximate Gibbs samplers, the simple-global approach performs least well for $\\tau _x$ – this is hardly surprising given the large differences between the exact conditional distributions and the simple regression models.",
"However, localising the regressions (light blue line) at each stage of the Gibbs sampler produces a major improvement in the quality of the approximation.",
"The same applies when the chosen regression models are flexible enough to accommodate non-linearities, interactions and heteroscedasticity (dark blue line)."
],
[
"A state space model of ",
"We analyse a time series dataset containing Airbnb property rental prices in the city of Seattle, WA, USA in 2016.",
"The dataset, available at kaggle.com, consists of 928,151 entries, each corresponding to an available listed space (property, room, etc) at a given date.",
"The price distribution of these data on each day is non-Gaussian even after transformation.",
"Hence we use the more flexible $g$ -and-$k$ distribution [30], [49], which has an intractable density function, but a tractable quantile function $Q(q|A, B, g, k) = A + B \\left[1 + c\\frac{1-\\exp \\lbrace -gz(q)\\rbrace }{1+\\exp \\lbrace -gz(q)\\rbrace } \\right] (1+z(q)^2)^k z(q),$ for $B>0$ and $k>-0.5$ (with $c=0.8$ ), where $z(q)$ denotes the $q$ -th quantile of the standard Gaussian distribution.",
"As a simple 4-parameter univariate model with an intractable density, this distribution has gained popularity in the ABC literature [20], [23], [42].",
"Figure REF shows $L$ -moments estimates of each $g$ -and-$k$ parameter Peters2016 for each day in the Airbnb dataset.",
"Each parameter exhibits a dynamic level with a weekly seasonal effect, and a sudden shift induced by the start and end of the extended summer season (1st April to 31st September), as well as additional stochastic variation potentially depending on other factors.",
"The series are also dependent with e.g.",
"a strong negative correlation between scale ($B$ ) and kurtosis ($k$ ).",
"Figure: L-moments gg-and-kk distribution parameter estimates (A,B,g,kA, B, g, k) for each day in the Airbnb dataset (Peters et al.",
"2016).We construct the following intractable non-linear state space model: $\\text{Observation distribution:} && y_t & \\sim g\\mbox{-and-}k(\\beta _t),\\quad t=1, \\ldots , T \\\\\\text{Link function:} && h(\\beta _t) & = \\lambda _t = F^\\top _t \\theta _t \\\\\\text{System equation:} && (\\theta _t|\\theta _{t-1}) & = G_t \\theta _{t-1} + w_t, \\quad w_t \\sim N(\\mathbf {0}, W_t) \\\\\\text{Prior distribution:} && \\theta _0 & \\sim N(m_0, C_0),$ where $y_t$ denotes the vector of (log) prices observed at time $t$ , $F_t$ is a known $p \\times 4$ design matrix that maps the state vector $\\theta _t$ to the linear predictor $\\lambda _t=(\\lambda _{1,t},\\ldots ,\\lambda _{4,t})^\\top $ , $G_t$ is a known $p \\times p$ evolution matrix that dictates the system's dynamics, $W_t$ is a possibly unknown covariance matrix, and $\\beta _t = (\\lambda _{1, t}, \\exp (\\lambda _{2, t}), \\lambda _{3, t}, \\exp (\\lambda _{4, t})-0.5)^\\top =(A, B, g, k)^\\top _t$ represents the $g$ -and-$k$ distribution parameters.",
"The link function $h(\\cdot )$ ensures that $\\beta _t$ respects the constraints imposed by the observation distribution.",
"We assume that given $\\theta _t$ , the observations $y_t$ are independent and identically distributed.",
"The sequence of errors $w_t$ are also assumed to be independent.",
"Specification of $F_t$ and $G_t$ is provided in Appendix A.1.",
"For this analysis we set $m_0 = \\bf 0$ and $C_0 = 10^{7} \\bf I$ , where $\\bf 0$ is a vector of zeros and $\\bf I$ is the identity matrix, and $W_t = W= \\text{diag}(1/\\tau _1, \\ldots , 1/\\tau _p)$ , with $\\tau _i \\sim \\text{Gamma}(\\alpha =10^{-10}, \\nu =10^{-10})$ , for $i=1, \\ldots , p$ .",
"State space models provide a flexible and well-structured framework to probabilistically describe an extensive array of applied problems West1997,Petris2010.",
"West1985 introduced dynamic generalised linear models, which relaxed the linearity and Gaussian assumptions, allowing the observations to follow other members of the exponential family.",
"Other works have focused on specific observation distributions, such as the Beta da-Silva2011 and the Dirichlet da-Silva2013.",
"Computational hurdles have limited the use of intractable dynamic models such as the one considered here, but increasing efforts to tackle this issue are being made Jasra2012,Dean2014,Martin2014,Calvet2012,Yildirim2013,Picchini2016,Martin2016.",
"Our approach extends the method given by Peters2016.",
"Writing $\\Theta =(\\theta _0, \\ldots , \\theta _T)$ , the joint distribution factorises as $ p(\\Theta , W, y_1, \\ldots , y_T) = p(\\theta _0) p({W}) \\prod _{t=1}^T\\left[ p(\\theta _t|\\theta _{t-1}, W) p(y_t|\\lambda _t) \\right].$ The data $y_t$ only depend on the system state through $\\lambda _t$ , so the full conditional distribution for $\\theta _t$ can be conveniently factorised as $p(\\theta _t|\\cdot ) & =p(\\theta _t | \\theta _{t-1}, \\theta _{t+1}, W, \\lambda _t) p(\\lambda _t | \\theta _{t-1}, \\theta _{t+1}, W, y_t).$ One can sample from this distribution in two stages: $\\lambda _t^* \\sim p(\\lambda _t | \\cdot )$ and then $\\theta _t^* \\sim p(\\theta _t | \\lambda _t^*, \\cdot )$ .",
"All full conditional distributions are tractable (see Appendix A.2) apart from $p(\\lambda _t | \\cdot )$ .",
"To approximate the linear predictor's conditional distribution, $p(\\lambda _t | \\theta _{t-1}, \\theta _{t+1}, W, y_t)$ , we reduce the dimension of the conditioning set by replacing the observed data $y_t$ by the summary statistic $s_t=g(\\hat{\\beta _t})$ , where $\\hat{\\beta _t}$ is the L-moments estimator of $\\beta _t$ given $y_t$ and $g(\\cdot )$ is the link function defined above.",
"While not fully sufficient, these statistics are highly informative and nearly unbiased for all sample sizes and parameters Peters2016.",
"It is useful to recognise that $p(\\lambda _t | \\theta _{t-1}, \\theta _{t+1}, W, s_t) = p(\\lambda _t | \\phi _t, s_t)$ , where $\\phi _t = (f_t, q_t, n_t)$ , and where $f_t = F^\\top _t a_t,$ $q_t = F^\\top _t R_t F_t,$ and $n_t$ is the sample size at time $t$ .",
"As this structure is valid throughout the evolution period, the time label can be effectively dropped, which reduces the problem to approximating the distribution of a 4-dimensional vector, $\\lambda $ , conditional on 13 variables ($q_t$ is a diagonal matrix).",
"Without loss of generality, we write $ (\\lambda | \\phi , s) = \\mu _\\lambda + \\Sigma _\\lambda ^{1/2} \\epsilon _\\lambda ,$ where $\\mu _\\lambda $ and $\\Sigma _\\lambda ^{1/2}$ , as functions of $\\phi $ and $s$ , respectively denote the mean and the (Cholesky) square root of the covariance of $(\\lambda | \\phi , s)$ .",
"$\\epsilon _\\lambda $ follows an unknown standardised distribution (that may also depend on $\\phi $ and $s$ ).",
"Even without knowledge of the distribution of $\\epsilon _\\lambda $ , given the moments of the joint vector, $\\begin{pmatrix}\\left.\\begin{array}{c}\\lambda \\\\s\\end{array}\\right|& \\hspace{-5.69046pt} \\phi \\end{pmatrix}\\sim \\begin{bmatrix}\\begin{pmatrix}f\\\\f\\end{pmatrix},& \\Omega _{\\phi } =\\begin{pmatrix}\\Omega _{11} & \\Omega _{12} \\\\\\Omega _{21} & \\Omega _{22}\\end{pmatrix}\\end{bmatrix},$ Linear Bayes (Hartigan1969,Goldstein1976; and Nott2012 in an ABC context) can be employed to give the estimators $\\hat{\\mu }_\\lambda = f+ \\Omega _{12} \\Omega _{22}^{-1} (s- f) \\quad \\mbox{and}\\quad \\hat{\\Sigma }_\\lambda = \\Omega _{11} - \\Omega _{12} \\Omega _{22}^{-1} \\Omega _{21}.$ To draw an approximate sample from $p(\\lambda _t | \\theta _{t-1}, \\theta _{t+1}, W, y_t)$ within the Gibbs sampler we a) estimate the covariance matrix $\\Omega _{\\phi }$ , b) compute the conditional moments in (REF ), c) draw an approximate sample for $\\epsilon _\\lambda $ , and d) plug-in the obtained values into (REF ).",
"To build the regression models we generate $N=5000$ samples of $\\phi $ uniformly on a hypercube that roughly covers the region that might be visited during the Gibbs run: the means $f$ have the same range as observed in $\\lbrace s_{obs,t}\\rbrace $ , the diagonal elements of $q$ are in the interval $(0, 10^{-5})$ , and $n$ spans the observed sample sizes.",
"See e.g.",
"Fearnhead2012, fan+ns13 for other strategies.",
"For each sample $\\phi ^{(i)}=(f^{(i)}, q^{(i)}, n^{(i)})$ , $i=1, \\ldots , N$ , we draw $(\\lambda , s)^{(i)} \\sim (\\lambda , s| \\phi ^{(i)}) = p(s| \\lambda , n^{(i)}) p(\\lambda | f^{(i)}, q^{(i)})$ .",
"Recall that $(\\lambda ,s)$ only depends on $t$ through $\\phi $ , so only a small single days' data needs to be generated.",
"For each step $m=1,\\ldots ,M$ in the approximate Gibbs sampler and for each $t=1,\\ldots ,T$ , conditional on the current value of $\\phi ^{*}_t$ we estimate $\\Omega _{\\phi ^{*}_{t}} = \\text{V}\\begin{pmatrix}\\left.\\begin{array}{c}\\lambda \\\\s\\end{array}\\right|& \\hspace{-5.69046pt} \\phi ^{*}_{t}\\end{pmatrix}\\approx \\int \\text{V}\\begin{pmatrix}\\left.\\begin{array}{c}\\lambda - f\\\\s- f\\end{array}\\right|& \\hspace{-5.69046pt} q, n\\end{pmatrix}K_h(\\Vert \\phi -\\phi ^{*}_{t}\\Vert ) p(\\phi ) d\\phi $ by computing the kernel-weighted sample covariance matrix over the centered samples $(\\lambda ^{(i)} - f^{(i)}, s^{(i)} - f^{(i)})$ , $i=1, \\ldots , N$ .",
"We used the Epanechnikov kernel $K_h$ , with bandwidth chosen such that the closest 2000 samples had non-zero weight.",
"For each $\\hat{\\Omega }_{\\phi ^{*}_t}$ we then compute $\\hat{\\mu }_{\\lambda _t}^{*}$ and $\\hat{\\Sigma }_{\\lambda _t}^{*}$ from (REF ).",
"The empirical residuals are then given by $\\epsilon _{\\lambda }^{i, *} = (\\hat{\\Sigma }_{\\lambda _t}^{*})^{-1/2} (\\lambda ^{(i)} - \\hat{\\mu }_{\\lambda _t}^{*})$ , $i=1,\\ldots ,N$ .",
"Finally, an approximate sample from the full conditional distribution $p(\\lambda _t | \\phi ^{*}_{t}, s_t)$ is obtained by $\\lambda _t^{**} = \\hat{\\mu }_{\\lambda _t}^{*} + (\\hat{\\Sigma }_{\\lambda _t}^{*})^{1/2} \\epsilon _\\lambda ^{k, *} \\sim \\int p(\\lambda _t | \\phi , s_t) K_h(\\Vert \\phi -\\phi ^{*}_t\\Vert ) p(\\phi ) d\\phi ,$ where the index $k$ is drawn from $(1, \\ldots , N)$ with probability $\\propto K_h(\\Vert \\phi ^{(k)}-\\phi ^{*}_t\\Vert )$ .",
"Figure REF shows some of the estimated model components of $A_t$ .",
"In Figure REF , the deseasonalised posterior estimates (original scale) are plotted over the L-moment estimates, revealing the overall shape of the (location of the) price changes over the course of the year, with higher prices in the summer months.",
"There is a clearly noticeable step change in prices for the duration of the high season.",
"The season effect parameters are estimated to be effectively constant throughout the high season, with $\\hat{\\theta }_{9, t}^{[1]} \\approx 0.024$ for all such $t$ .",
"That is, prices are expected to uniformly increase by about $\\exp (0.024)-1=2.4\\%$ during the high season.",
"Figure: Estimated components ofA t A_t.",
"(a)Posterior mean (red line) of the deseasonalised parameter exp(θ 1,t [1] +θ 3,t [1] δ(t))\\exp (\\theta _{1, t}^{[1]} + \\theta _{3, t}^{[1]} \\delta (t)), with 95%95\\% HPD intervals (shading) and L-moments estimates (grey lines); (b) Associated estimates of exp(A t )\\exp (A_t) (dots); (c) Estimated seasonal effect of the linear predictor λ 1,t \\lambda _{1,t} given the posterior mean for θ 3,t [1] \\theta _{3, t}^{[1]}; (d) Residual plot for A t A_t, showing the differences s obs 1,t -λ ^ 1,t {s_\\mathrm {obs}}_{1, t} - \\hat{\\lambda }_{1, t}.",
"Panels (e), (f): sampler trace plots for AA at time t=1t=1 and its average summer effect θ 9,t [1] \\theta _{9,t}^{[1]}.The points in Figure REF are the estimated location parameter means when including the estimated seasonality (Figure REF ), for example, showing an average price increase of around $5.6\\%$ from Thursdays to Fridays.",
"The residual plot (Figure REF ) exhibits a slight lack-of-fit, suggesting some kind of annual sinusoidal modelling is required.",
"The highest residual was observed on Valentine's weekend when, perhaps, there may be an increase in demand from couples.",
"The lowest residual was on the first day of the high season: Friday, April 1st.",
"These results were based on $M=1$ million approximate Gibbs sampler iterations, retaining every 20th sample, and then discarding the first 25,000 iterations as burn-in.",
"The sampler was initialised from estimates obtained by fitting a simple state space model (that assumes each series in Figure REF follows an independent dynamic linear model, with pre-specified matrices $W^{[i]}$ ) by Kalman smoothing.",
"There are 13,140 unknown parameters in the model, and assessing chain convergence is not trivial.",
"Trace plots of the location parameter $A$ at time $t=1$ and its average summer effect ($\\theta _{9,t}^{[1]}$ ) are displayed in Figure REF e,f.",
"It would be extremely challenging for regular ABC methods to handle a model of this size and complexity.",
"However, computationally this analysis was still expensive – it took almost 10 days to generate the 1 million Gibbs sampler iterations in R on a HP device with an Intel Core i7-4790 CPU (3.6GHz) with 16 GB of RAM.",
"In addition, Gibbs samplers result in slowly mixing chains when performing low dimensional parameter block updates, although this low dimensionality is exactly the feature required for ABC methods to function well.",
"In this analysis use of Linear Bayes allowed us to model the vector $\\lambda $ jointly, rather than separately for each of its elements.",
"This accounts for its full correlation structure and naturally handles heteroscedasticity.",
"With separate univariate regressions, one would have to accommodate possible interaction terms and model the variance explicitly."
],
[
"Discussion",
"Because it suffers from the curse of dimensionality, ABC performs most effectively for lower dimensional models with lower dimensional summary statistics.",
"In order to consider more complex and higher-dimensional models, such as the 13140 parameter dynamic model considered in Section , this dimensionality must be structurally lowered.",
"This is achieved with the likelihood-free approximate Gibbs sampler.",
"As the full conditional distributions are approximated by regression models, this approach can substantially outperform related Metropolis-Hastings based samplers (e.g. Kousathanas2016).",
"We considered various strategies for constructing the regression models.",
"Localising bespoke regression models at each iteration of the approximate Gibbs sampler can approximate the true conditional distributions more accurately than global regression models that are fitted once, which ultimately leads to lower posterior approximation errors.",
"However, they are correspondingly more expensive to implement.",
"Similarly, simple regression models are faster to fit than more sophisticated models, at the price of greater approximation.",
"The simulations in Section REF demonstrated that non-linear deep learning models substantially improved the posterior estimates.",
"Similar to the Metropolis-Hastings ABC-MCMC algorithm of Kousathanas2016, the likelihood-free approximate Gibbs sampler embraces the spirit of Bayesian modelling with potentially inconsistent conditional distributions, as advocated by gelman04.",
"This potential inconsistency can be greatly diminished if the fitted regression models are sufficiently flexible so that they can approximate the true conditional distributions arbitrarily well.",
"Whether this is possible or not is model and regression model specific.",
"Very recent work by clarte+rrs19 provides interesting theoretical insights on the conditions under which this will be possible in the ABC context.",
"One possible drawback of the likelihood-free Gibbs sampler is that it trades off the greater accuracy of lower-dimensional ABC models for slower mixing Markov chains, particularly in more complex models, due to the Gibbs updates.",
"However, this is a genuine tradeoff, and for some problems these tools are potentially the only feasible option.",
"."
],
[
"Acknowledgements",
"GSR is funded by the CAPES Foundation via the Science Without Borders program (BEX 0974/13-7).",
"DJN is supported by a Singapore Ministry of Education Academic Research Fund Tier 1 grant (R-155-000-189-114).",
"SAS is supported by the Australia Research Council through the Discovery Project Scheme (FT170100079), and the Australian Centre of Excellence for Mathematical and Statistical Frontiers (ACEMS, CE140100049).",
"The authors are grateful to Wilson Ye Chen and Gareth W. Peters for generously providing the code used to compute the L-moment estimate of parameters of the $g$ -and-$k$ distribution.",
"Each $g$ -and-$k$ parameter $\\beta _{t}^{[1]}$ (with $\\beta _t=(\\beta _{t}^{[1]},\\ldots ,\\beta _{t}^{[4]})^\\top $ ), $i=1, \\ldots , 4$ , is defined by its own system parameters, $\\theta _{t}^{[i]}$ , and the matrices $F_{t}^{[i]} = (E_2, E_6, \\delta (t))^\\top $ and $G_{t}^{[i]} =G^{[i]} =\\begin{pmatrix}J_2 & {\\bf 0}_{2 \\times 6} & {\\bf 0}_{2 \\times 1} \\\\{\\bf 0}_{6 \\times 2} & P_6 & {\\bf 0}_{6 \\times 1} \\\\{\\bf 0}_{1 \\times 2} & {\\bf 0}_{1 \\times 6} & 1\\end{pmatrix},\\quad \\text{where } J_2 =\\begin{pmatrix}1 & 1 \\\\0 & 1 \\\\\\end{pmatrix},\\quad P_6 =\\begin{pmatrix}-{\\bf 1}_{1 \\times 5} & -1 \\\\{\\bf I}_5 & {\\bf 0}_{5 \\times 1} \\\\\\end{pmatrix},$ $E_n=(1, 0, \\ldots , 0)$ is an $n$ -dimensional vector, $\\delta (t)$ is an indicator function that takes value 1 if $t$ is in the summer season and 0 otherwise, and ${\\bf 1}$ denotes a matrix of ones.",
"$J_2$ , which is a Jordan block, implies a local-linear trend for the latent level $\\theta _{1, t}^{[i]}$ .",
"$P_6$ is a permutation matrix that models the weekly seasonal effect, which impacts the series though $\\theta _{3, t}^{[i]}$ .",
"The summer-effect is described by $\\theta _{9, t}^{[i]}$ .",
"The model () becomes fully specified by setting $F_t = F_t^{[i]} \\otimes {\\bf I_4}, \\quad G_t = G^{[i]} \\otimes {\\bf I_4} \\quad \\text{and} \\quad \\theta _t=(\\theta _t^{[1]}, \\ldots , \\theta _t^{[4]}),$ where $\\otimes $ is the Kronecker product.",
"This specification imposes those features perceived to drive the Airbnb data, however alternative models could be adopted.",
"For more details on how to specify the matrix of a dynamic model, see e.g. Petris2009.",
"The full conditional distribution (FCD) of the system's initial state $\\theta _0$ is $p(\\theta _0|\\cdot ) \\sim N(a_0, \\Sigma _0),$ where $\\Sigma _0 = (G^\\top _{1} W^{-1} G_{1} + C_0^{-1})^{-1}$ and $a_0=\\Sigma _0 (C_0^{-1} m_0 + G^\\top _{1} W^{-1} \\theta _1)$ .",
"To facilitate sampling the system's state $\\theta _T$ , we augment the parameter space to keep track of the parameter $\\theta _{T+1}$ , with FCD given by $p(\\theta _{T+1}|\\cdot ) \\sim N(G_{T+1} \\theta _T, W)$ .",
"The FCD of the error's precisions $\\tau _i$ are given by $p(\\tau _i|\\cdot ) \\sim \\text{Gamma}\\left( \\alpha +\\frac{T+1}{2}, \\nu + \\frac{\\sum _{t=1}^{T+1} w_{ti}^2}{2} \\right),$ where $w_t = \\theta _t - G_t \\theta _{t-1}$ represents the system innovation at time $t$ .",
"For the system state $\\theta _t$ , the model equations imply that $\\begin{pmatrix}\\left.\\begin{array}{c}\\theta _t \\\\\\lambda _t\\end{array}\\right|& \\hspace{-5.69046pt} \\theta _{t-1}, \\theta _{t+1}, W\\end{pmatrix}\\sim \\text{N}\\begin{bmatrix}\\begin{pmatrix}a_t \\\\f_t\\end{pmatrix},&\\begin{pmatrix}R_t & R_t F_t \\\\F^\\top _t R_t & q_t\\end{pmatrix}\\end{bmatrix},$ where $f_t = F^\\top _t a_t,$ $q_t = F^\\top _t R_t F_t,$ $a_t = R_t (W^{-1} G_t \\theta _{t-1} + G^\\top _{t+1} W^{-1} \\theta _{t+1}),$ and $R_t = (G^\\top _{t+1} W^{-1} G_{t+1} + W^{-1})^{-1}.$ It then follows from the conditional properties of the multivariate normal distribution that $p(\\theta _t | \\theta _{t-1}, \\theta _{t+1}, W, \\lambda _t) = N(\\mu _t, \\Sigma _t),$ where $\\mu _t=a_t + R_t F_t q_t^{-1} (\\lambda _t - f_t)$ and $\\Sigma _t=R_t - R_t F_t q_t^{-1} F^\\top _t R_t.$"
]
] | 1906.04347 |
[
[
"A physically-consistent, flexible and efficient strategy to convert\n local boundary conditions into nonlocal volume constraints"
],
[
"Abstract Nonlocal models provide exceptional simulation fidelity for a broad spectrum of scientific and engineering applications.",
"However, wider deployment of nonlocal models is hindered by several modeling and numerical challenges.",
"Among those, we focus on the nontrivial prescription of nonlocal boundary conditions, or volume constraints, that must be provided on a layer surrounding the domain where the nonlocal equations are posed.",
"The challenge arises from the fact that, in general, data are provided on surfaces (as opposed to volumes) in the form of force or pressure data.",
"In this paper we introduce an efficient, flexible and physically consistent technique for an automatic conversion of surface (local) data into volumetric data that does not have any constraints on the geometry of the domain and on the regularity of the nonlocal solution and that is not tied to any discretization.",
"We show that our formulation is well-posed and that the limit of the nonlocal solution, as the nonlocality vanishes, is the local solution corresponding to the available surface data.",
"Quadratic convergence rates are proved for the strong energy and L-2 convergence.",
"We illustrate the theory with one dimensional numerical tests whose results provide the ground work for realistic simulations."
],
[
"Introduction and motivation",
"Nonlocal models employ integral rather than differential operators which allows them to relax the regularity constraints of partial differential equations (PDEs) and to capture effects arising from long-range forces at the microscale and mesoscale, not accounted for by PDEs.",
"Consequently, nonlocal models provide exceptional simulation fidelity for a broad spectrum of applications such as fracture mechanics [17], [18], [26], anomalous subsurface transport [5], [24], [25], phase transitions [4], [8], [14], image processing [1], [15], [16], [19], multiscale and multiphysics systems [2], [3], MHD [23], and stochastic processes [6], [9], [20], [22].",
"The main difference between PDE models and the nonlocal models we consider is that, in the former case, interactions between two domains only occur due to contact, whereas in the latter case, interactions can occur at a distance.",
"In this work, for simplicity of the exposition and without loss of generality (see Remark REF ), we consider the nonlocal counterpart of elliptic differential operators.",
"In its simplest form, the action of a nonlocal diffusion operator on a scalar function $u:{\\mathbb {R}^n}\\rightarrow \\mathbb {R}$ is given by $\\mathcal {L}u(\\mathbf {x})=C\\int _{\\mathbb {R}^d}\\big (u(\\mathbf {y})-u(\\mathbf {x})\\big ) \\,\\gamma (\\mathbf {x},\\mathbf {y})\\,d\\mathbf {y}\\qquad \\mathbf {x}\\in {\\mathbb {R}^d},$ where the kernel function $\\gamma $ , usually with bounded support, is related to the specific application and determines the smoothing properties of $\\mathcal {L}$ .",
"The integral form above allows us to catch long-range interactions so that every point in a domain interacts with a neighborhood of points.",
"Also, such form reduces the regularity requirements for the solution, which is able to describe discontinuous (for e.g.",
"fracture mechanics) or anomalous (for e.g.",
"subsurface dispersion) behaviors.",
"However, the increased accuracy of nonlocal models comes at a price: several modeling and numerical challenges arise.",
"These include the nontrivial prescription of “nonlocal” boundary conditions, the often prohibitively expensive numerical solution and the definition of model parameters (such as $\\gamma $ ), often unknown or subject to uncertainty.",
"All these (open) problems can hinder wider deployment of nonlocal models and are the subject of current research in the fast-growing nonlocal community.",
"In this work we focus on the first challenge.",
"Because of nonlocal interactions, when solving a nonlocal problem in a bounded domain, the prescription of classical boundary conditions does not guarantee the well-posedness of the equations [11]; in fact, in general, nonlocal boundary conditions, or, more properly, volume constraints, must be defined on a layer surrounding the domain.",
"However, it is often the case that such information is not available, whereas it is easy to measure surface (local) data.",
"Consequently, one of the biggest challenges to be addressed before nonlocal models can be widely applied in realistic contexts is the conversion of local boundary conditions, defined on surfaces, into volume constraints, defined on volumes.",
"Previous attempts to tackle the conversion are either too expensive (solving an optimization problem) or too restrictive (requiring conditions on geometry or dimensionality).",
"The first approach is an optimization-based coupling method that mimics generalized overlapping domain-decomposition formulations [10].",
"The main idea is to decompose the domain into a local and nonlocal subdomains where the former is placed in a neighborhood of the part of the boundary where only surface data are available.",
"This choice allows both the local and nonlocal problems to be well-posed and circumvents the prescription of volume constraints when not available.",
"On the other hand, this method requires the solution of a nonlocal minimization problem whose algorithm may require several computation of the nonlocal solutions, dramatically increasing the computational effort.",
"Paper [7] is the first that interprets the nonlocal Neumann boundary condition as a body force acting on the boundary layer of the domain, where $L^1$ convergence of nonlocal solutions to the corresponding local ones is shown.",
"Later in [28], a careful modification of the body force in a one dimensional setting is found that leads to a second order uniform convergence of solutions as the nonlocal interaction vanishes.",
"The second order convergence result is then extended to two dimensions in [31], where the curvature of the computation domain plays an important role in the definition of the modified body force.",
"Recently, [13] achieves the second order uniform convergence in one dimension with another approach.",
"To the best of our knowledge, no work has yet discussed second order nonlocal approximations to the local Neumann boundary value problems in space dimension higher than two.",
"The complexity of geometric bodies to be dealt with in high dimension is an obvious hindrance.",
"We propose a computationally cheap, flexible and physically consistent method for an efficient conversion that has no constraints on dimensionality, geometry, regularity of the nonlocal solution and that is not tied to any discretization.",
"Our main and most promising approach consists of three simple steps.",
"A Solution of a computationally cheap local model using available surface data.",
"B Derivation, from A, of forces corresponding to the local solution in the thick nonlocal layer.",
"C Solution of the nonlocal model using the forces derived in B.",
"Note that the forces computed in B are equivalent to nonlocal Neumann data, which is used in C as volume constraint for the solution of the nonlocal problem.",
"Also note that local and nonlocal problems are completely uncoupled; this feature becomes very powerful when dealing with large scale problems (as it is often the case in engineering applications); in fact, local and nonlocal solvers can be used as black boxes and the overall cost of the proposed method is the same of a nonlocal problem, for given nonlocal boundary data.",
"This is due to the fact that the cost of solving the local problem is negligible compared to the one of the nonlocal problem.",
"Note that the uncoupling of local and nonlocal equations allows for completely independent discretizations of the local and nonlocal equationsAs an example, one can use a mesh-free discretization for the nonlocal models and a mesh-based one for the local model..",
"In fact, application of the nonlocal operator to the discretized local solution in step B only requires projection of the latter onto the nonlocal discretization space.",
"Furthermore, this approach is such that the nonlocal solution computed in C reduces to the solution computed in A, as the nonlocal interactions vanish, with a quadratic rate of convergence for both the (nonlocal) energy and $L^2$ norms with respect to the characteristic interaction length.",
"A few considerations are in order.",
"Even though we do not require additional regularity of the nonlocal solution, we do assume that the given surface data is such that the corresponding local problem computed in A is well-posed (for, e.g., the classical Poisson equation square integrability over the boundary of the force/pressure data is enough to guarantee the existence and uniqueness of the local solution).",
"We also mention that in the analysis of the asymptotic behavior of the nonlocal solution for vanishing nonlocality we assume that the local solution belongs to $C^4$ .",
"However, this additional regularity is not required in practice.",
"We expect the proposed strategy to advance the state of the art for predictive nonlocal modeling by providing an efficient in-demand tool that will impact a broad class of applications and unlock the full potential of nonlocal models.",
"Note that we also introduce an alternative, more straightforward, strategy that has exactly the same properties of the approach described in A–C, but delivers solutions whose behavior is closer to the local one.",
"The paper is organized as follows.",
"In the following section we introduce the notation and recall relevant results of the nonlocal vector calculus, a theory developed in the last decade by Du et al.",
"[12] that allows one to study nonlocal diffusion problems in a very similar way as PDEs by framing nonlocal equations in a variational setting.",
"In Section we introduce two alternative strategies to the conversion problem, discuss their properties, and provide a qualitative comparison.",
"In Section we study the convergence to the local limit of the nonlocal solution for the most promising strategy and show quadratic strong convergence in both the nonlocal energy norm and $L^2$ norm.",
"In Section we illustrate the theoretical results in a one-dimensional setting."
],
[
"Preliminaries",
"In this section we introduce the nonlocal vector calculus and recall results relevant to this paper.",
"Let ${\\Omega }$ be a bounded open domain in ${\\mathbb {R}^d}$ , $d=1,2,3$ , with Lipschitz-continuous boundary $\\partial {\\Omega }$ and ${\\alpha }(\\mathbf {x},\\mathbf {y}) \\colon {\\mathbb {R}^d}\\times {\\mathbb {R}^d}\\rightarrow {\\mathbb {R}^d}$ be an antisymmetric function, i.e.",
"${\\alpha }(\\mathbf {y},\\mathbf {x})=-{\\alpha }(\\mathbf {x},\\mathbf {y})$ .",
"For the functions $u(\\mathbf {x})\\colon {\\mathbb {R}^d}\\rightarrow \\mathbb {R}$ and ${\\nu }(\\mathbf {x},\\mathbf {y})\\colon {\\mathbb {R}^d}\\times {\\mathbb {R}^d}\\rightarrow {\\mathbb {R}^d}$ we define the nonlocal divergence $\\mathcal {D}\\colon {\\mathbb {R}^d}\\rightarrow \\mathbb {R}$ of ${\\nu }(\\mathbf {x},\\mathbf {y})$ as $\\mathcal {D}\\big ({\\nu }\\big )(\\mathbf {x}) := \\int _{{\\mathbb {R}^d}} \\big ({\\nu }(\\mathbf {x},\\mathbf {y})+{\\nu }(\\mathbf {y},\\mathbf {x})\\big )\\cdot {\\alpha }(\\mathbf {x},\\mathbf {y})\\,d\\mathbf {y}\\qquad \\mathbf {x}\\in {\\mathbb {R}^d}$ and the nonlocal gradient $\\mathcal {G}\\colon {\\mathbb {R}^d}\\times {\\mathbb {R}^d}\\rightarrow {\\mathbb {R}^d}$ of $u(\\mathbf {x})$ as $\\mathcal {G}\\big (u\\big )(\\mathbf {x},\\mathbf {y}) := \\big (u(\\mathbf {y})-u(\\mathbf {x})\\big ) {\\alpha }(\\mathbf {x},\\mathbf {y}) \\qquad \\mathbf {x},\\mathbf {y}\\in {\\mathbb {R}^d}.$ It is shown in [12] that the adjoint $\\mathcal {D}^*=-\\mathcal {G}$ .",
"Next, we define the nonlocal diffusion $\\mathcal {L}\\colon {\\mathbb {R}^d}\\rightarrow \\mathbb {R}$ of $u(\\mathbf {x})$ as a composition of the nonlocal divergence and gradient operators, i.e.",
"$\\mathcal {L}u(\\mathbf {x}) := \\mathcal {D}\\big (\\mathcal {G}u\\big )(\\mathbf {x}) =2\\int _{\\mathbb {R}^d}\\big (u(\\mathbf {y})-u(\\mathbf {x})\\big ) \\,\\gamma (\\mathbf {x},\\mathbf {y})\\,d\\mathbf {y}\\qquad \\mathbf {x}\\in {\\mathbb {R}^d},$ where $\\gamma (\\mathbf {x},\\mathbf {y}):={\\alpha }(\\mathbf {x},\\mathbf {y})\\cdot {\\alpha }(\\mathbf {x},\\mathbf {y})$ is a non-negative symmetric kernelThere are more general representations of the nonlocal diffusion operator, these are associated with nonsymmetric and not necessarily positive kernel functions.",
"In such cases $\\mathcal {L}$ may define a model for non-symmetric diffusion phenomena, we mention e.g.",
"nonsymmetric jump processes [9]..",
"Note that this is the same operator introduced in Section .",
"We define the interaction domain of an open bounded region ${\\Omega }\\in {\\mathbb {R}^d}$ as ${\\Omega _I}= \\lbrace \\mathbf {y}\\in {\\mathbb {R}^d}\\setminus {\\Omega }: \\; \\gamma (\\mathbf {x},\\mathbf {y})\\ne 0, \\; \\mathbf {x}\\in {\\Omega }\\rbrace ,$ and set ${\\overline{\\Omega }}={\\Omega }\\cup {\\Omega _I}$ .",
"This domain contains all points outside of ${\\Omega }$ that interact with points inside of ${\\Omega }$ ; as such, ${\\Omega _I}$ is the volume where nonlocal boundary conditions must be prescribed to guarantee the well-posedness of nonlocal euqations (see Section REF ).",
"We make the following assumptions: for $\\mathbf {x}\\in {\\Omega }$ $\\left\\lbrace \\begin{array}{ll}\\gamma (\\mathbf {x},\\mathbf {y}) > 0 \\quad &\\forall \\, \\mathbf {y}\\in B_\\varepsilon (\\mathbf {x})\\\\[2mm]\\gamma (\\mathbf {x},\\mathbf {y}) = 0 \\quad &\\forall \\, \\mathbf {y}\\in {{\\overline{\\Omega }}} \\setminus B_\\varepsilon (\\mathbf {x}),\\end{array}\\right.$ where $B_\\varepsilon (\\mathbf {x}) = \\lbrace \\mathbf {y}\\in {{\\overline{\\Omega }}}: \\; \\Vert \\mathbf {x}-\\mathbf {y}\\Vert <\\varepsilon ,\\; \\mathbf {x}\\in {\\Omega }\\rbrace $ and $\\varepsilon $ is the interaction radius or horizon.",
"For such kernels the interaction domain is a layer of thickness $\\varepsilon $ that surrounds ${\\Omega }$ , i.e.",
"${\\Omega _I}= \\lbrace \\mathbf {y}\\in {\\mathbb {R}^d}\\setminus {\\Omega }: \\; \\Vert \\mathbf {y}-\\mathbf {x}\\Vert <\\varepsilon , \\;\\mathbf {x}\\in {\\Omega }\\rbrace .$ We refer to Figure REF (left) for an illustration of a two-dimensional domain, the support of $\\gamma $ and the induced interaction domain.",
"Figure: Left: the domain Ω{\\Omega }, the support of γ\\gamma at a point 𝐱∈Ω\\mathbf {x}\\in {\\Omega }, B δ (𝐱)B_\\delta (\\mathbf {x}), and the induced interaction domain Ω I {\\Omega _I}.",
"Right: two-dimensional configuration.",
"Here, Ω N ∪Ω D =Ω I {\\Omega _N}\\cup {\\Omega _D}={\\Omega _I}, Ω∪Ω I =Ω ¯{\\Omega \\cup \\Omega _I}={\\overline{\\Omega }} and Γ N ∪Γ D =Γ\\Gamma _N\\cup \\Gamma _D=\\Gamma .Corresponding to the divergence operator $\\mathcal {D}({\\nu })$ we introduce a nonlocal interaction operator $\\mathcal {N}({\\nu })(\\mathbf {x})=-\\int _{\\overline{\\Omega }}\\left({\\nu }(\\mathbf {x},\\mathbf {y})+{\\nu }(\\mathbf {y},\\mathbf {x})\\right){\\alpha }(\\mathbf {x},\\mathbf {y})\\,d\\mathbf {y}\\qquad \\mathbf {x}\\in {\\Omega _I}.$ The integral $\\int _{\\Omega _I}\\mathcal {N}({\\nu })\\,d\\mathbf {x}$ generalizes the notion of a flux $\\int _{\\partial {\\Omega }}{\\bf q}\\cdot {\\bf n}\\,dA$ through the boundary of a domain, with $\\mathcal {N}({\\nu })$ playing the role of a flux density ${\\bf q}\\cdot {\\bf n}$ .",
"The key difference between (REF ) and a conventional flux is that in the former the flux is a volume integral, whereas in the latter it is a boundary integral.",
"Nonetheless, the nonlocal divergence and interaction operators satisfy a nonlocal Gauss theorem $\\int _{\\Omega }\\mathcal {D}({\\nu })\\,d\\mathbf {x}=\\int _{\\Omega _I}\\mathcal {N}({\\nu })\\,d\\mathbf {x}$ .",
"We refer to [12] for additional nonlocal vector calculus results, including generalized nonlocal Green's identities.",
"We respectively introduce the nonlocal energy semi-norm, nonlocal energy space, and nonlocal volume-constrained energy space $\\begin{array}{ll}& |||v|||^2 := \\displaystyle \\frac{1}{2}\\int _{{\\overline{\\Omega }}}\\int _{{{\\overline{\\Omega }}}}(\\mathcal {G}v)^2\\,d\\mathbf {y}\\, d\\mathbf {x}\\\\ [5mm]& V({\\overline{\\Omega }}) := \\left\\lbrace v \\in L^2({\\overline{\\Omega }}) \\,\\,:\\,\\, |||v|||_{{\\overline{\\Omega }}} < \\infty \\right\\rbrace \\\\[3mm]& V_c({\\overline{\\Omega }}) := \\left\\lbrace v\\in V({{\\overline{\\Omega }}}) \\,\\,:\\,\\, v=0\\;{\\rm on}\\;{\\Omega _D}\\right\\rbrace \\;\\;\\hbox{ for ${\\Omega _D}\\subseteq {\\Omega _I}$.",
"}\\end{array}$ We also define the volume-trace space $\\widetilde{V}_c({\\overline{\\Omega }}):=\\lbrace v|_{\\Omega _D}: \\,v\\in V({\\overline{\\Omega }})\\rbrace $ and the dual spaces $V^{\\prime }({\\overline{\\Omega }})$ and $V^{\\prime }_c({\\overline{\\Omega }})$ with respect to $L^2$ -duality pairings.",
"We consider kernels such that the corresponding energy norm satisfies a Poincaré-like inequality, i.e.",
"$\\Vert v\\Vert _{0,{{\\overline{\\Omega }}}}\\le C_{pn}|||v|||$ for all $v\\in V_c({\\overline{\\Omega }})$ , where $C_{pn}$ is referred to as the nonlocal Poincaré constant.",
"Kernels satisfying this property can be found in [11]; for such kernelsThe nonlocal Poincaré inequality holds for an even more general class of properly scaled, non-increasing, kernel functions, see [21]., in [21], it is shown that the Poincaré constant is independent of $\\varepsilon $ if $\\varepsilon \\in (0, \\varepsilon _0]$ with a certain fixed number $\\varepsilon _0$ .",
"A popular example is the class of integrable kernelsSpecifically, we are referring to kernels for which there exist positive constants $\\gamma _1$ and $\\gamma _2$ such that $\\gamma _1\\le \\int _{{\\overline{\\Omega }}\\cap B_\\varepsilon (\\mathbf {x})} \\gamma (\\mathbf {x},\\mathbf {y})\\,d\\mathbf {y}$ and $\\int _{\\overline{\\Omega }}\\gamma ^2(\\mathbf {x},\\mathbf {y})\\,d\\mathbf {y}\\le \\gamma _2^2$ for all $\\mathbf {x}\\in {\\Omega }$ .",
"for which $V({\\overline{\\Omega }})$ and $V_c({\\overline{\\Omega }})$ are equivalent to $L^2({{\\overline{\\Omega }}})$ and $L^2_c({\\overline{\\Omega }})$ ; in this case, the operator $\\mathcal {L}$ is such that $\\mathcal {L}:L^2({\\overline{\\Omega }})\\rightarrow L^2({\\overline{\\Omega }})$ [11]."
],
[
"Volume-constrained nonlocal diffusion problems",
"We refer to the simplified configuration in Figure REF (right); here we let $\\Gamma =\\partial {\\overline{\\Omega }}$ , ${\\Omega _I}={\\Omega _N}\\cup {\\Omega _D}$ such that ${\\Omega _N}\\cap {\\Omega _D}=\\emptyset $ and $\\Gamma =\\Gamma _N\\cup \\Gamma _D$ such that $\\Gamma _N\\cap \\Gamma _D=\\emptyset $ .",
"For $s\\in V^{\\prime }_c({\\overline{\\Omega }})$ , $g_n\\in V^{\\prime }({\\Omega _N})$ and $v_n\\in \\widetilde{V}_c({\\overline{\\Omega }})$ , we want to solve $\\left\\lbrace \\begin{array}{ll}-\\displaystyle \\mathcal {L}{u_n}= s & \\mathbf {x}\\in {\\Omega }\\\\[3mm]-\\displaystyle \\mathcal {N}(\\mathcal {G}{u_n}) = {g_n}& \\mathbf {x}\\in {\\Omega _N}\\\\[3mm]{u_n}= v_n & \\mathbf {x}\\in {\\Omega _D},\\end{array}\\right.$ where (REF )$_2$ and (REF )$_3$ are the nonlocal counterpart of a Neumann and Dirichlet boundary conditions, referred to as Neumann and Dirichlet volume constraints, respectively.",
"More specifically, by composition of the nonlocal interaction and gradient operators we have that (REF )$_2$ corresponds to $-\\displaystyle \\mathcal {N}(\\mathcal {G}{u_n})(\\mathbf {x}) = \\int _{\\Omega \\cup \\Omega _I}({u_n}(\\mathbf {x})-{u_n}(\\mathbf {y}))\\gamma (\\mathbf {x},\\mathbf {y})\\,d\\mathbf {y}= {g_n}\\quad \\forall \\,\\mathbf {x}\\in {\\Omega _N}.$ As for local equations, the weak form of (REF ) is obtained by multiplying both sides by a test function $z\\in V_c$ and integrating over ${\\Omega }$ , i.e.",
"$\\displaystyle -\\int _{\\Omega }\\mathcal {L}{u_n}z\\,d\\mathbf {x}= \\int _{\\Omega }sz\\,d\\mathbf {x}\\quad \\forall \\, z\\in V_c({\\overline{\\Omega }}).$ Using nonlocal integration by parts [12] and the Neumann constraint, (REF ) is equivalent to $\\begin{aligned}\\displaystyle \\int _{\\overline{\\Omega }}& \\int _{\\overline{\\Omega }}\\mathcal {G}{u_n}\\mathcal {G}z \\,d\\mathbf {y}\\,d\\mathbf {x}= -\\int _{\\Omega _N}\\mathcal {N}(\\mathcal {G}{u_n}) z\\,d\\mathbf {x}+ \\int _{\\Omega }sz\\,d\\mathbf {x}\\quad \\Rightarrow \\\\[3mm]\\displaystyle \\int _{\\overline{\\Omega }}&\\int _{\\overline{\\Omega }}({u_n}(\\mathbf {x})-{u_n}(\\mathbf {y}))(z(\\mathbf {x})-z(\\mathbf {y}))\\gamma (\\mathbf {x},\\mathbf {y})\\,d\\mathbf {y}\\,d\\mathbf {x}=\\int _{\\Omega _N}{g_n}z\\,d\\mathbf {x}+ \\int _{\\Omega }sz\\,d\\mathbf {x}\\quad \\Rightarrow \\\\[3mm]a&(u,z) = F(z),\\end{aligned}$ where the bilinear form and the linear functional are defined as $a(u,z)=\\langle u,z\\rangle _{V_c}$ and $F(v)=\\int _{\\Omega _N}{g_n}z\\,d\\mathbf {x}+ \\int _{\\Omega }sz\\,d\\mathbf {x}$ .",
"It can be easily shown [11] that for every $\\gamma (\\cdot ,\\cdot )$ satisfying the Poincaré inequality $a(\\cdot ,\\cdot )$ is coercive and continuous in $V_c({\\overline{\\Omega }})\\times V_c({\\overline{\\Omega }})$ and that $F(\\cdot )$ is continuous in $V_c({\\overline{\\Omega }})$ .",
"Thus, by the Lax-Milgram theorem problem (REF ) is well-posed."
],
[
"Proposed strategies",
"In engineering applications it is often the case that data are only available on the boundary $\\Gamma $ and not in ${\\Omega _I}$ ; in particular, most of the times, we are given force or pressure data (i.e.",
"a local Neumann boundary condition) on parts of $\\Gamma $ .",
"As shown in [11] and as recalled above, this is not enough for the well-posedness of problem (REF ).",
"We make the following assumptions.",
"A1 The kernel function $\\gamma $ is such that the limit of the nonlocal diffusion operator is the classical Laplacian, i.e.",
"$\\mathcal {L}w(\\mathbf {x})\\rightarrow \\Delta w(\\mathbf {x}), \\quad {\\rm as} \\quad \\varepsilon \\rightarrow 0.$ This is obtained by scaling $\\gamma $ using some appropriate constant proportional to a power of $\\varepsilon $ .",
"A2 There exists a local (differential) operator that approximates well enough the nonlocal one when the solution does not feature a nonlocal behavior, i.e.",
"does not exhibit irregularities.",
"Because of assumption (REF ) in A1, we use the classical Laplacian $\\Delta $ as the approximation of $\\mathcal {L}$ in (REF ).",
"We refer to this model as the surrogate local model.",
"A3 Only the following data are available: 1.",
"$g_l\\in L^2(\\Gamma _N)$ : local Neumann boundary data on $\\Gamma _N$ ; 2.",
"$v_n\\in \\widetilde{V}_c({\\overline{\\Omega }})$ on ${\\Omega _D}$ : nonlocal Dirichlet data; 3.",
"$s\\in V_c^{\\prime }({\\overline{\\Omega }})$ : forcing term over ${\\overline{\\Omega }}$ .",
"Once again, these do not guarantee existence and uniqueness of a nonlocal solution.",
"Remark 3.1 We point out that our strategy is readily applicable to a much broader class of nonlocal operators as long as A2 holds.",
"As an example, this approach could be applied to a linear nonlocal elasticity model (specifically the linear peridynamic solid model [27]) for which the corresponding surrogate local model is the classical Navier-Cauchy equation of linear elasticity, as the latter is the local limit of the former.",
"Our goal is to design a strategy to automatically convert $g_l$ into a nonlocal volume constraint (either of Neumann or Dirichlet type) on ${\\Omega _N}$ .",
"In the next sections we introduce two conversion approaches and present qualitative comparison results.",
"Note that the conversion problem is an ill-posed inverse problem as there exists an infinite number of nonlocal data corresponding to $g_l$ for which the associated nonlocal problem is well-posed.",
"However, among all possible choices, we look for a strategy such that the corresponding nonlocal solution, say ${\\widetilde{u}_n}$ , satisfies ${\\widetilde{u}_n}\\rightarrow u_l \\;\\; {\\rm as} \\;\\; \\varepsilon \\rightarrow 0 \\quad {\\rm in} \\;\\; V({\\overline{\\Omega }}) \\;\\; {\\rm and} \\;\\; L^2({\\overline{\\Omega }}),$ where $u_l$ is the solution of the following (surrogate) Poisson equation $\\left\\lbrace \\begin{array}{ll}-\\Delta {u_l}= s & \\mathbf {x}\\in {\\overline{\\Omega }}\\\\[3mm]-\\nabla {u_l}\\cdot {\\bf n} = g_l & \\mathbf {x}\\in \\Gamma _N \\\\[3mm]{u_l}= v_n & \\mathbf {x}\\in \\Gamma _D,\\end{array}\\right.$ i.e.",
"the solution of the local problem with boundary data as in A2.",
"Here, by prescribing the Dirichlet condition on $\\Gamma _D$ we are assuming that $v_n|_{\\Gamma _D}$ exists and is such that $v_n|_{\\Gamma _D}\\in H^\\frac{1}{2}(\\Gamma _D)$Note that, even though this is a regularity requirement (not desirable in nonlocal contexts), we are not assuming $v_n\\in H^1({\\Omega _N})$ , but only that $v_n$ has a well-defined trace on $\\Gamma _D$ .."
],
[
"Neumann strategy",
"This is our main and most promising strategy.",
"The key idea is to use the available data in A3 to solve the surrogate problem in ${\\overline{\\Omega }}$ and utilize the local solution ${u_l}$ to compute the corresponding force, say ${\\widetilde{g}_n}$ , over ${\\Omega _N}$ .",
"It is clear from the right hand side in (REF ) that the nonlocal Neumann data is indeed a forcing term acting on ${\\Omega _N}$ ; thus, ${\\widetilde{g}_n}$ will be used as an approximation of ${g_n}$ to solve (REF ).",
"We proceed step by step.",
"1N Solve the surrogate local problem (REF ).",
"2N Compute the forces on ${\\Omega _N}$ associated with ${u_l}$ .",
"This is achieved by applying the nonlocal Neumann operator $\\mathcal {N}(\\mathcal {G}\\,\\cdot )$ to ${u_l}$ , i.e.",
"$-\\mathcal {N}(\\mathcal {G}{u_l})(\\mathbf {x})={\\widetilde{g}_n}(\\mathbf {x})$ , for $\\mathbf {x}\\in {\\Omega _N}$ .",
"This represents an approximation of the nonlocal Neumann data ${g_n}$ .",
"Note that, for the same reasons as for the operator $\\mathcal {L}$ , the Neumann operator $\\mathcal {N}(\\mathcal {G}\\,\\cdot )$ also maps $V$ into $V^{\\prime }$ .",
"This implies that ${\\widetilde{g}_n}\\in V^{\\prime }({\\Omega _N}).$ 3N Compute an approximation of the nonlocal solution ${u_n}$ , say ${\\widetilde{u}_n}$ , using ${\\widetilde{g}_n}$ as Neumann data, i.e.",
"solve $\\left\\lbrace \\begin{array}{ll}-\\displaystyle \\mathcal {L}{\\widetilde{u}_n}= s & \\mathbf {x}\\in {\\Omega }\\\\[3mm]-\\displaystyle \\mathcal {N}(\\mathcal {G}{\\widetilde{u}_n}) = {\\widetilde{g}_n}& \\mathbf {x}\\in {\\Omega _N}\\\\[3mm]{\\widetilde{u}_n}= v_n & \\mathbf {x}\\in {\\Omega _D}.\\end{array}\\right.$ Because of (REF ), problem (REF ) is well-posed."
],
[
"Dirichlet strategy",
"We present an alternative, and more straightforward, that approach consists in using $u_l$ computed as in 1N as Dirichlet volume constraint for the nonlocal problem in ${\\Omega _N}$ .",
"Thus, we have the following procedure.",
"1D Solve the surrogate local problem (REF ).",
"2D Solve the following nonlocal problem: $\\left\\lbrace \\begin{array}{ll}-\\displaystyle \\mathcal {L}\\widetilde{u}_{n,D} = s & \\mathbf {x}\\in {\\Omega }\\\\[3mm]\\widetilde{u}_{n,D} = u_l & \\mathbf {x}\\in {\\Omega _N}\\\\[3mm]\\widetilde{u}_{n,D} = v_n & \\mathbf {x}\\in {\\Omega _D}.\\end{array}\\right.$ Because of its regularity, $u_l\\in \\widetilde{V}({\\overline{\\Omega }})$ and, thus, problem (REF ) is well-posed.",
"This approach cleary delivers a solution that is unable to catch nonlocal behaviors in a neighborhood of the Neumann boundary.",
"This effect is less strong in the previous approach because, instead of prescribing a local constraint on the solution itself, the Neumann approach only prescribes an equivalence of forces allowing the solution to feature a nonlocal behavior.",
"In other words, the locality constraint is weaker.",
"This is confirmed by one-dimensional numerical results.",
"We consider ${\\Omega }=(0,1)$ , ${\\overline{\\Omega }}=(-\\varepsilon ,1+\\varepsilon )$ , and ${\\Omega _N}=(-\\varepsilon ,0)$ .",
"We test both homogeneous and non-homogeneous Neumann conditions; specifically, we consider the following problem settings.",
"A $s=-12x^2-6/5\\varepsilon ^2$ , $g_l=-4\\varepsilon ^3$ and $v_n=x^4$ ; B $s=-12x^2-6/5\\varepsilon ^2$ , $g_l=2/5\\varepsilon ^2(8-13\\varepsilon )$ and $v_n = x^4+2x+3/5\\varepsilon ^2(x^2+2x-3-4\\varepsilon -\\varepsilon ^2)$ , where dependence of the data on $\\varepsilon $ is only for testing purposes.",
"We do not specify discretization details as they are not relevant for now.",
"In Figure REF we report ${\\widetilde{u}_n}$ , $\\widetilde{u}_{n,D}$ and ${u_l}$ for A (left) and B (right) in a region around the Neumann boundary.",
"Results show that in both cases the solutions obtained with the Neumann and Dirichlet approaches are significantly different in the zoomed area; in fact, while $\\widetilde{u}_{n,D}$ is, by construction, on top of $u_l$ , ${\\widetilde{u}_n}$ only reproduces its normal derivative.",
"Figure: Comparison of solutions obtained with Neumann (u ˜ n {\\widetilde{u}_n}) and Dirichlet (u ˜ n,D \\widetilde{u}_{n,D}) strategies for case A (left) and B (right) around the Neumann boundary.Note that when the data are such that local and nonlocal models are equivalentFor the operators under considerations, we have equivalence for polynomials up to the third order, see numerical experiments in Section for an illustration., the two approaches coincide and we have that $\\widetilde{u}_{n,D}={\\widetilde{u}_n}={u_l}$ .",
"This is confirmed by numerical experiments in Section ."
],
[
"Convergence to the local limit",
"In this section we study the limiting behavior of the solution as the nonlocal interactions vanish, i.e.",
"as $\\varepsilon \\rightarrow 0$ .",
"We introduce the errors $e_E = |||{\\widetilde{u}_n}-{u_l}|||\\quad {\\rm and} \\quad e_0 = \\Vert {u_n}-{\\widetilde{u}_n}\\Vert _{0,{\\overline{\\Omega }}}.$ The following proposition provides a bound for $e_E$ for the Neumann approach.",
"Theorem 4.1 Let $\\varepsilon _0\\in (0,\\infty )$ and $\\mathcal {U}_l:=\\lbrace {u_l}\\in C^4({\\overline{\\Omega }}): {u_l}\\hbox{ solves (\\ref {eq:nonlocal-Neumann}) for }\\varepsilon \\in (0,\\varepsilon _0]\\rbrace $ be a family of solutions of (REF ).",
"Then, for all ${u_l}\\in \\mathcal {U}_l$ $e_E \\le C \\varepsilon ^2 \\Vert D^{(4)}{u_l}\\Vert _{\\infty ,{\\overline{\\Omega }}}\\,,$ where $C$ is a positive constant independent of $\\varepsilon $ and ${u_l}$ and $D^{(4)}$ indicates the 4-th derivative operator.",
"Recall that, by definition, ${\\widetilde{u}_n}$ and ${u_l}$ satisfy $\\left\\lbrace \\begin{array}{ll}-\\displaystyle \\mathcal {L}{\\widetilde{u}_n}= s = -\\Delta {u_l}& \\mathbf {x}\\in {\\Omega }\\\\[3mm]-\\displaystyle \\mathcal {N}({\\widetilde{u}_n}) = {\\widetilde{g}_n}= -\\displaystyle \\mathcal {N}({u_l}) & \\mathbf {x}\\in {\\Omega _N}\\\\[3mm]{\\widetilde{u}_n}= v_n & \\mathbf {x}\\in {\\Omega _D}.\\end{array}\\right.$ We introduce the following nonlocal auxiliary problem for the local solution ${u_l}$ : $\\left\\lbrace \\begin{array}{ll}-\\displaystyle \\mathcal {L}{u_l}= {s_l}= -\\int _{\\overline{\\Omega }}({u_l}(\\mathbf {y})-{u_l}(\\mathbf {x}))\\gamma (\\mathbf {x},\\mathbf {y})\\,d\\mathbf {y}& \\mathbf {x}\\in {\\Omega }\\\\[3mm]-\\displaystyle \\mathcal {N}({u_l}) = {\\widetilde{g}_n}& \\mathbf {x}\\in {\\Omega _N}\\\\[3mm]{u_l}= v_n & \\mathbf {x}\\in {\\Omega _D}.\\end{array}\\right.$ In order to estimate $e_E$ we first consider the point-wise difference $s(\\mathbf {x})\\!-\\!",
"{s_l}(\\mathbf {x})$ .",
"By the Taylor's theorem $|s(\\mathbf {x})-{s_l}(\\mathbf {x})| = \\left|\\int _{\\overline{\\Omega }}({u_l}(\\mathbf {y})-{u_l}(\\mathbf {x}))\\gamma (\\mathbf {x},\\mathbf {y})\\,d\\mathbf {y}- \\Delta {u_l}\\right|\\le \\widetilde{C} \\varepsilon ^2 |D^{(4)}{u_l}|_{\\infty ,{\\overline{\\Omega }}}\\,,$ where $\\widetilde{C}$ is a positive constant independent of $\\varepsilon $ and ${u_l}$ and $D^{(4)}$ indicates the 4-th derivative operator.",
"Next, we consider the weak forms of (REF ) and (REF ) for the same test function $z\\in V_c$ ; we have $\\int _{\\overline{\\Omega }}\\int _{\\overline{\\Omega }}({\\widetilde{u}_n}(\\mathbf {x})-{\\widetilde{u}_n}(\\mathbf {y}))(z(\\mathbf {x})-z(\\mathbf {y}))\\gamma (\\mathbf {x},\\mathbf {y})\\,d\\mathbf {y}\\,d\\mathbf {x}=\\int _{\\Omega _N}{\\widetilde{g}_n}\\,z\\,d\\mathbf {x}+ \\int _{\\Omega }s\\,z\\,d\\mathbf {x},$ $\\int _{\\overline{\\Omega }}\\int _{\\overline{\\Omega }}({u_l}(\\mathbf {x})-{u_l}(\\mathbf {y}))(z(\\mathbf {x})-z(\\mathbf {y}))\\gamma (\\mathbf {x},\\mathbf {y})\\,d\\mathbf {y}\\,d\\mathbf {x}=\\int _{\\Omega _N}{g_n}\\,z\\,d\\mathbf {x}+ \\int _{\\Omega }{s_l}\\,z\\,d\\mathbf {x}.$ Subtraction, yields $\\begin{aligned}\\int _{\\overline{\\Omega }}\\int _{\\overline{\\Omega }}({\\widetilde{u}_n}(\\mathbf {x})-{u_l}(\\mathbf {x})-{\\widetilde{u}_n}(\\mathbf {y})+{u_l}(\\mathbf {y}))(z(\\mathbf {x})-z(\\mathbf {y}))\\gamma (\\mathbf {x},\\mathbf {y})\\,d\\mathbf {y}\\,d\\mathbf {x}= \\int _{\\Omega }(s-{s_l})\\,z\\,d\\mathbf {x}.\\end{aligned}$ By taking $z={\\widetilde{u}_n}-{u_l}\\in V_c$ , we have $|||{\\widetilde{u}_n}-{u_l}|||^2 \\le \\int _{\\Omega }(s-{s_l})\\,({\\widetilde{u}_n}-{u_l})\\,d\\mathbf {x}\\le \\Vert s-{s_l}\\Vert _{0,{\\Omega }} \\Vert {\\widetilde{u}_n}-{u_l}\\Vert _{0,{\\Omega }} \\le C \\varepsilon ^2 \\Vert D^{(4)}{u_l}\\Vert _{\\infty ,{\\overline{\\Omega }}} |||{\\widetilde{u}_n}-{u_l}|||,$ where we omitted the higher order terms because not relevant and where the last inequality follows from the Poincaré inequality.",
"Then, the thesis follows by dividing both sides by $|||{\\widetilde{u}_n}-{u_l}|||$ .",
"Remark 4.1 Theorem REF implies that the convergence rate for the $L^2$ norm of the difference between local and nonlocal solutions, i.e.",
"$e_0$ , is at least quadratic.",
"In fact, by the Poincaré inequality, we have $e_0 = \\Vert {\\widetilde{u}_n}-{u_l}\\Vert _{0,{\\overline{\\Omega }}} \\le C_{n,p} |||{\\widetilde{u}_n}-{u_l}||| \\le \\widehat{C} \\varepsilon ^2 \\Vert D^{(4)}{u_l}\\Vert _{\\infty ,{\\overline{\\Omega }}}.$ Remark 4.2 A simple modification of the proof of Theorem REF yields the same convergence result for the Dirichlet strategySimply extend the Dirichlet condition to the whole interaction domain and disregard the term on ${\\Omega _N}$ in the weak forms..",
"The same convergence rate is inherited by the $L^2$ norm, as described in Remark REF .",
"Remark 4.3 Theorem REF implies that when the data $g_l$ , $s$ and $v_n$ are smooth enough to have $\\mathcal {L}{u_l}=\\Delta {u_l}$ , then ${\\widetilde{u}_n}={u_l}$ .",
"We use this observation to conduct a consistency test for the proposed conversion method."
],
[
"Numerical tests",
"With the purpose of illustrating the theoretical results, in this section we report the results of one-dimensional numerical tests.",
"Even though preliminary, these results are promising and provide the ground work for realistic simulations.",
"We consider the one-dimensional configuration in Figure REF ; we let $a=0$ , $b=1$ , $\\Gamma _N=\\lbrace x=-\\varepsilon \\rbrace $ , $\\Gamma _D=\\lbrace x=1+\\varepsilon \\rbrace $ , and $\\gamma (x,y)=\\dfrac{3}{\\varepsilon ^3}\\,\\mathcal {X}(|x-y|<\\varepsilon ).$ This integrable kernel is such that $\\mathcal {L}w \\rightarrow \\Delta w$ as $\\varepsilon \\rightarrow 0$ .",
"In all our tests we discretize the nonlocal equation with the finite element method (FEM) and utilize piecewise linear finite elements.",
"The domain ${\\overline{\\Omega }}$ is partitioned in intervals of the same size $h$ .",
"We denote the FEM solutions by ${\\widetilde{u}_n}^h$ and $\\widetilde{u}_{n,D}^h$ and introduce the discrete counterparts of the $e_E$ and $e_{0}$ , i.e.",
"$e_{E,h}=|||w^h-{u_l}|||\\quad {\\rm and} \\quad e_{0,h}=\\Vert w^h-{u_l}\\Vert ,$ where $w^h$ is either ${\\widetilde{u}_n}^h$ or $\\widetilde{u}_{n,D}^h$ .",
"We test both the consistency and the convergence to local limits.",
"Figure: One-dimensional configuration.Remark 5.1 As mentioned in the introduction, our conversion method is not tied to any discretization; in fact, both mesh-free and mesh-based methods can be employed.",
"FEM is, in general, quite expensive for large scale nonlocal simulations, but affordable in a one-simensional setting.",
"An advantage of using the piecewise linear FEM is the asymptotic compatibility, a property studied in [29], [30] on the robustness of numerical schemes under change of $\\varepsilon $ .",
"Remark 5.2 Note that since we use manufactured solutions for which the local solution can be computed explicitly, we do not approximate the local problem."
],
[
"Consistency",
"We consider local solutions ${u_l}$ such that $\\mathcal {L}{u_l}=\\Delta {u_l}$ .",
"According to Remark REF and to the discussion in Section REF the approximate nonlocal solutions ${\\widetilde{u}_n}$ and $\\widetilde{u}_{n,D}$ are such that ${\\widetilde{u}_n}=\\widetilde{u}_{n,D}={u_l}$ .",
"Thus, we consider the following problem settings: A ${u_l}=x$ , $g_l=1$ , $v_l=1+\\varepsilon $ and $s=0$ ; B ${u_l}=x^3$ , $g_l=3\\varepsilon ^2$ , $v_l=(1+\\varepsilon )^3$ and $s=-6x$ .",
"Note that for both A and B we have that $s=-\\mathcal {L}u_l=-\\Delta u_l$ .",
"For the sake of comparison and to illustrate our theory we consider both the Neumann approach described in Section REF and the Dirichlet approach described in Section REF .",
"As mentioned above, we expect the two approaches to be equivalent when the local and nonlocal operators are equivalent.",
"Additionally, in case A we expect ${\\widetilde{u}_n}=\\widetilde{u}_{n,D}={u_l}$ and the FEM solution to be $\\epsilon $ -machine accurate because the exact solution belongs to the space of discretized solutions; in case B we also expect ${\\widetilde{u}_n}=\\widetilde{u}_{n,D}={u_l}$ and $e_{E,h}$ to be independent of $\\varepsilon $ .",
"Numerical tests confirm that for both Neumann and Dirichlet approaches in case A $e_{E,h}=\\epsilon $ and in case B, $e_{E,h}\\cong 9$ e-5 for a grid of size $h=2^{-6}$ and for several values of $\\varepsilon $ for both strategies.",
"In Figure REF we report illustrations of the numerical solutions for both tests cases: ${\\widetilde{u}_n}$ , $\\widetilde{u}_{nD}$ , and $u_l$ are superimposed.",
"Figure: Nonlocal solutions obtained with the Neumann and Dirichlet strategies and local solution for linear (left) and cubic (right) tests.",
"Up to discretization error, the solutions coincide."
],
[
"Convergence to local limits",
"We perform numerical tests on the convergence of ${\\widetilde{u}_n}$ and $\\widetilde{u}_{n,D}$ to the local limit.",
"We consider the data $g_l=2+5\\varepsilon ^4$ , $v_n=x(2+x^4)$ , and $s=-20x^3$ ; the corresponding local solution is given by ${u_l}=x(2+x^4)$ .",
"With the purpose of “hiding” the discretization error we compute the nonlocal solution on a very fine grid, i.e.",
"$h=2^{-12}$ ; for decreasing values of $\\varepsilon $ we report results in Tables REF and REF for the Neumann and Dirichlet strategies respectively.",
"The observed rates for $e_{E,h}$ and $e_{0,h}$ are in alignment with Theorem REF and Remark REF .",
"Next, for simoultaneously decreasing values of $\\varepsilon $ , we test the asymptotic compatibility [29], [30] of our scheme; results are reported in Tables REF and REF for the Neumann and Dirichlet strategies respectively.",
"Also in this case, we have a second order convergence rate.",
"Note that we consider pairs $(h,\\varepsilon )=(\\varepsilon ^2,\\varepsilon )$ ; this choice is motivated by the fact that, for piecewise linear finite element approximations, a linear dependence between $h$ and $\\varepsilon $ would compromise the convergence rate of the energy norm due to the influence of the discretization error on the local-limit error.",
"The choice of $h$ , makes the discretization error negligible so that the only contribution to the errors is given by the interaction length.",
"As a confirmation, in Table REF we report the same results for the pairs $(h,\\varepsilon )=(\\varepsilon /4,\\varepsilon )$ , we consider the Neumann approach only.",
"While the convergence of $e_{0,h}$ is still quadratic, the convergence rate of $e_{E,h}$ asymptotically deteriorates (an additional pair with respect to previous tables is added to show deterioration).",
"Finally, note that, as expected, the errors obtained with the Dirichlet approach are lower that those obtained with the Neumann.",
"Table: Neumann approach: energy and L 2 L^2 norm of the difference between local and discretized nonlocal solution for h=2 -12 h=2^{-12} and decreasing values of ε\\varepsilon .Table: Dirichlet approach: energy and L 2 L^2 norm of the difference between local and discretized nonlocal solution for h=2 -12 h=2^{-12} and decreasing values of ε\\varepsilon .Table: Neumann approach: energy and L 2 L^2 norm of the difference between local and discretized nonlocal solution for simultaneously decreasing values of ε\\varepsilon and hh such that (h,ε)=(ε 2 ,ε)(h,\\varepsilon )=(\\varepsilon ^2,\\varepsilon ).Table: Dirichlet approach: energy and L 2 L^2 norm of the difference between local and discretized nonlocal solution for simultaneously decreasing values of ε\\varepsilon and hh such that (h,ε)=(ε 2 ,ε)(h,\\varepsilon )=(\\varepsilon ^2,\\varepsilon ).Table: Neumann approach: energy and L 2 L^2 norm of the difference between local and discretized nonlocal solution for simultaneously decreasing values of ε\\varepsilon and hh such that (h,ε)=(ε/4,ε)(h,\\varepsilon )=(\\varepsilon /4,\\varepsilon )."
],
[
"Conclusion",
"We introduced a flexible, physically consistent and efficient strategy for the conversion of surface local data into volumetric data in the context of nonlocal modeling and simulation.",
"Our technique does not have regularity constraints on the nonlocal solution, it can be applied in any dimension, and converges to the solution of the corresponding local problem as the nonlocality vanishes.",
"More specifically, we achieve second order convergence of the energy norm as the nonlocal interaction vanish in any dimension and only requiring the local solution to belong to $C^4$ (which can be obtained when the boundary of the domain, the boundary data and the source term are smooth enough).",
"Furthermore, even if numerical results are only in one dimension, the implementation of this approach in two and three dimensions is straightforward and only requires PDE and nonlocal solvers that can be used as black boxes, i.e.",
"the proposed method does not require any implementation effort.",
"Also, the computational cost is the same as the one required by a single nonlocal simulation."
],
[
"Acknowledgments",
"Marta D'Elia was supported by Sandia National Laboratories (SNL), SNL is a multimission laboratory managed and operated by National Technology and Engineering Solutions of Sandia, LLC., a wholly owned subsidiary of Honeywell International, Inc., for the U.S. Department of Energys National Nuclear Security Administration contract number DE-NA0003525.",
"Specifically, this work was supported through the Sandia National Laboratories Laboratory-directed Research and Development (LDRD) program.",
"This paper describes objective technical results and analysis.",
"Any subjective views or opinions that might be expressed in the paper do not necessarily represent the views of the U.S. Department of Energy or the United States Government.",
"SAND Number: SAND2019-6287.",
"The research of Xiaochuan Tian is supported in part by the U.S. NSF grant DMS-1819233.",
"Yue Yu is supported by the U.S. NSF grant DMS-1620434 and the Lehigh faculty research grant.",
"The authors would like to thank Dr. D. Littlewood (Sandia Natioanl Laboratories, NM) for useful discussions and insights."
]
] | 1906.04259 |
[
[
"Shadow of a rotating squashed Kaluza-Klein black hole"
],
[
"Abstract We study the shadow of a rotating squashed Kaluza-Klein (KK) black hole and the shadow is found to possess distinct properties from those of usual rotating black holes.",
"It is shown that the shadow for a rotating squashed KK black hole is heavily influenced by the specific angular momentum of photon from the fifth dimension.",
"Especially, as the parameters lie in a certain special range, there is no any shadow for a black hole, which does not emerge for the usual black holes.",
"In the case where the black hole shadow exists, the shadow shape is a perfect black disk and its radius decreases with the rotation parameter of the black hole.",
"Moreover, the change of the shadow radius with extra dimension parameter also depends on the rotation parameter of black hole.",
"Finally, with the latest observation data, we estimate the angular radius of the shadow for the supermassive black hole Sgr $A^{*}$ at the centre of the Milky Way galaxy and the supermassive black hole in $M87$."
],
[
"Introduction",
"Recently, the Event Horizon Telescope team showcased the first image of the supermassive black hole in the center of the giant elliptical galaxy M87 [1], [2], [3], [4], [5], [6].",
"This event is terribly exciting because it confirms once again that there exists exactly black hole in our Universe.",
"Moreover, the information carried by the image is of great benefit to the understanding of the black hole shadow and the matter accretion process.",
"Black hole shadow is a two-dimensional dark region in the observer's sky, where light rays from the source fall into an event horizon.",
"It is well known that the shape and size of shadow depend on the black hole parameters [7], [8], which implies that the shadow could be regarded as a potential tool to identify black holes.",
"Thus, the shadows of black holes with different parameters have been studied recently in various theories of gravity [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41], [42], [43], [44].",
"For example, the shadow is a perfect rounded silhouette for a Schwarzschild black hole, but it becomes a “D\"-shaped silhouette for a fast rotating black hole due to dragging effect [7], [8].",
"Moreover, the cusp silhouettes of shadows with small eye lashes are found in the spacetime of Kerr black hole with Proca hair [9] and the Konoplya-Zhidenko rotating non-Kerr black hole [10].",
"The self-similar fractal structures appear in the black hole shadows when the equations of photon motion are not variable separable in the background spacetimes due to the chaotic motion of photon [11], [12], [13], [14], [15], [16], [17], [18].",
"Recently, the shadows have been investigated for high-dimensional black holes including the Schwarzschild-Tangherlini black hole [45], the five-dimensional rotating Myers-Perry black hole [46] and the five-dimensional rotating Einstein-Maxwell-Chern-Simons black hole [47], and the rotating black hole in Randall-Sundrum theories [48].",
"These studies show that the extra dimension imprints in the black hole shadows.",
"Here, we focus on the rotating squashed KK black hole, which is a kind of interesting Kaluza-Klein type metrics with the special topology and asymptotical structure [49].",
"This family of black holes have squashed $S^3$ horizons.",
"In the vicinity of horizon, the black hole has a structure like a five-dimensional black hole, but it behaves as the four-dimensional black holes with a constant twisted $S^1$ fiber in the far region.",
"In these squashed KK black holes, the size of compactified extra dimension can be adjusted by the parameter $r_{\\infty }$ .",
"Recent investigations showed that the information of the size of the extra dimension for a KK black hole with squashed horizons imprints in its spectrum of Hawking radiation [50], [51], quasinormal frequencies [52], [53], precession of a gyroscope in a circular orbit [54] and strong gravitational lensing [55], which could open a possible window to observe extra dimensions in the future.",
"In the present work we study the shadow of a rotating squashed KK black hole and probe the features of the black hole shadow caused by the fifth dimension.",
"The paper is organized as follows.",
"In Sec.",
"$II$ , we introduce briefly the rotating squashed Kaluza-Klein black hole spacetime and the corresponding null geodesics.",
"In Sec.",
"$III$ , we study the features in the shadow of a rotating squashed Kaluza-Klein black hole and find that the shadow is heavily influenced by the specific angular momentum of photon from the fifth dimension.",
"Finally, we present a summary and some discussions in the last section."
],
[
"The rotating squashed Kaluza-Klein black hole spacetime and null geodesics ",
"The neutral rotating squashed KK black hole is a vacuum axisymmetric solution of Einstein field equation, which can be obtained by applying the squashing transformation technique to a five-dimensional Kerr black hole with two equal angular momenta [49].",
"The metric of this rotating squashed KK black hole can be expressed as $d s^{2}=-d \\tilde{t}^{\\ 2}+\\frac{\\Sigma _{0}}{\\Delta _{0}} k(r)^{2} dr^{2}+\\frac{r^{2}+a^{2}}{4}\\left[k(r)(\\sigma _{1}^{2}+\\sigma _{2}^{2})+\\sigma _{3}^{2}\\right]+\\frac{M}{r^{2}+a^{2}}(d \\tilde{t}-\\frac{a}{2} \\sigma _{3})^{2},$ with $\\Sigma _{0} &=&r^{2}(r^{2}+a^{2}), \\\\\\Delta _{0} &=&(r^{2}-r_{+}^{2})(r^{2}-r_{-}^{2}), \\\\k(r)&=&\\frac{(r_{\\infty }^{2}-r_{+}^{2})(r_{\\infty }^{2}-r_{-}^{2})}{(r_{\\infty }^{2}-r^{2})^{2}},$ and $\\sigma _{1} &=&-\\sin \\tilde{\\psi } d \\theta +\\cos \\tilde{\\psi } \\sin \\theta d \\phi ,\\\\ \\sigma _{2} &=&\\cos \\tilde{\\psi } d \\theta +\\sin \\tilde{\\psi } \\sin \\theta d \\phi , \\\\\\sigma _{3} &=&d \\tilde{\\psi }+\\cos \\theta d \\phi , $ where the angular coordinates satisfy $0<\\theta <\\pi , 0<\\phi <2 \\pi $ and $ 0<\\tilde{\\psi }<4 \\pi $ .",
"The quantity $r_\\infty $ corresponds to the spatial infinity and the polar coordinate $r$ runs in the range $\\ 0<r<r_\\infty $ .",
"The parameters $M$ and $a$ are associated with the mass and angular momenta of the black hole, respectively.",
"The outer and inner horizons of the black hole are located at $r_{+}$ and $r_{-}$ , which are functions of $M$ and $a$ , i.e., $r^2_{\\pm }=M-2a^2\\pm \\sqrt{M^2-4a^2M}$ .",
"The squashed parameter $k(r_{+})$ deforms the shape of black hole horizon.",
"Introducing a radial coordinate [49] $\\rho =\\tilde{\\rho }_{0} \\frac{r^{2}}{r_{\\infty }^{2}-r^{2}},$ with $\\tilde{\\rho }_{0}^{2}=\\frac{(r_{\\infty }^{2}+a^{2})[(r_{\\infty }^{2}+a^{2})^{2}-M r_{\\infty }^{2}]}{4 r_{\\infty }^{4}}.$ One can find that the metric (REF ) can be rewritten as $d s^{2}=-d \\tilde{t}^{\\ 2}+U d \\rho ^{2}+R^{2}(\\sigma _{1}^{2}+\\sigma _{2}^{2})+W^{2} \\sigma _{3}^{ 2}+V(d \\tilde{t}-\\frac{a}{2} \\sigma _{3})^{2},$ with $\\begin{aligned}K^{2}=\\frac{\\rho +\\tilde{\\rho }_{0}}{\\rho +\\frac{a^{2}}{r_{\\infty }^{2}+a^{2}} \\tilde{\\rho }_{0}},\\;\\;\\;\\;\\;\\;\\;& V=\\frac{M}{r_{\\infty }^{2}+a^{2}} K^{2}, \\;\\;\\;\\;\\;\\;\\;\\quad W^{2}=\\frac{r_{\\infty }^{2}+a^{2}}{4 K^{2}}, \\\\ R^{2}=\\frac{\\left(\\rho +\\tilde{\\rho }_{0}\\right)^{2}}{K^{2}},\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;& U=\\left(\\frac{r_{\\infty }^{2}}{r_{\\infty }^{2}+a^{2}}\\right)^{2} \\frac{\\tilde{\\rho }_{0}^{2}}{W^{2}-\\frac{r_{\\infty }^{2}}{4} \\frac{\\rho }{\\rho +\\tilde{\\rho }_{0}} V}.\\end{aligned}$ As the rotation parameter $a$ vanishes, the metric (REF ) reduces to that of a five-dimensional Schwarzschild black hole with squashed horizon.",
"When $r_{\\infty }\\rightarrow \\infty $ ($k(r)\\rightarrow 1$ ), the squashing effect disappears and then the metric of the usual five-dimensional Kerr black hole with two equal angular momenta is recovered in this limit.",
"Adopting to the coordinates transformation [49] $\\tilde{t}=h\\ t, \\quad \\quad \\quad \\tilde{\\psi }=\\psi -j\\ t, $ with $h=\\sqrt{\\frac{\\left(r_{\\infty }^{2}+a^{2}\\right)^{2}-M r_{\\infty }^{2}}{\\left(r_{\\infty }^{2}+a^{2}\\right)^{2}+M a^{2}}}, \\quad \\quad j=\\frac{2 M a}{\\left(r_{\\infty }^{2}+a^{2}\\right)^{2}+M a^{2}}, $ it is found that the cross-term between $d \\tilde{t}$ and $\\sigma _{3}$ in the asymptotic form (i.e., $\\rho \\rightarrow \\infty $ ) of the metric (REF ) vanishes, which means that the asymptotic topology of the spacetime (REF ) has the same form as the Schwarzschild squashed KK black hole spacetime.",
"From the Komar mass $M_k$ for the black hole (REF ) [55] $\\begin{aligned}M_{k} &=\\frac{M \\pi }{2 G_{5}} \\frac{(r_{\\infty }^{2}+a^{2})^{2}-M a^{2}}{\\sqrt{(r_{\\infty }^{2}+a^{2})^{2}+M a^{2}} \\sqrt{(r_{\\infty }^{2}+a^{2})^{2}-M r_{\\infty }^{2}}} \\\\&=\\frac{M}{4 G_{4}} \\frac{(r_{\\infty }^{2}+a^{2})^{2}-M a^{2}}{(r_{\\infty }^{2}+a^{2})^{2}+M a^{2}} \\frac{\\sqrt{r_{\\infty }^{2}+a^{2}}}{\\sqrt{(r_{\\infty }^{2}+a^{2})^{2}-M r_{\\infty }^{2}}}.\\end{aligned}$ we obtain the relationship between the five-dimensional and four-dimensional gravitational constants $G_{5}=2 \\pi r_{\\infty }^{\\prime } G_{4}, $ with $r_{\\infty }^{\\prime }=\\sqrt{\\frac{\\left(r_{\\infty }^{2}+a^{2}\\right)^{2}+M a^{2}}{r_{\\infty }^{2}+a^{2}}}.",
"$ Obviously, the expression of $r_{\\infty }^{\\prime }$ is more complicated than that of $r_{\\infty }$ .",
"However, the geometric interpretation for $r_{\\infty }^{\\prime }$ is clearer than that of $r_{\\infty }$ in the rotating squashed KK black hole spacetime (REF ) [49], which implies that the parameter $r_{\\infty }^{\\prime }$ is better than $r_{\\infty }$ for the compactified dimension.",
"As the rotation parameter $a$ disappears, we find that $r_{\\infty }^{\\prime }$ reduces to $r_{\\infty }$ and then the relationship (REF ) tends to the usual form in the Schwarzschild squashed KK black hole spacetime, i.e., $G_{5}=2\\pi r_{\\infty } G_{4}$ .",
"As in Ref.",
"[54], one can introduce a quantity $\\rho _{M}$ , which is related to Komar mass $M_{k}$ by $\\rho _{M}\\equiv 2 G_{4} M_{k}=\\frac{M}{2} \\frac{(r_{\\infty }^{2}+a^{2})^{2}-M a^{2}}{(r_{\\infty }^{2}+a^{2})^{2}+M a^{2}} \\frac{\\sqrt{r_{\\infty }^{2}+a^{2}}}{\\sqrt{(r_{\\infty }^{2}+a^{2})^{2}-M r_{\\infty }^{2}}}.$ For a sake of simplifying the calculation, we make use of the following transformations $r^{\\prime 2}=\\frac{(r_{\\infty }^{2}+a^{2})^{2}+M a^{2}}{r_{\\infty }^{4}+r^{2} a^{2}} r^{2},\\quad \\quad \\quad \\rho _{0}^{2}=\\frac{r_{\\infty }^{\\prime 2}-M}{4},\\quad \\quad \\quad b=\\sqrt{\\frac{M}{r_{\\infty }^{2}+a^{2}}} \\ a,$ and then find that the radial coordinate (REF ) and the quantity $\\rho _M$ can be rewritten as $\\rho &=&\\rho _{0} \\frac{r^{\\prime 2}}{r_{\\infty }^{\\prime 2}-r^{\\prime 2}},\\quad \\quad \\quad \\nonumber \\\\ \\rho _{M}&=&\\frac{\\rho _{0} M}{r_{\\infty }^{\\prime 2}-M} \\frac{(r_{\\infty }^{2}+a^{2})^{2}-M a^{2}}{(r_{\\infty }^{2}+a^{2})^{2}+M a^{2}}=\\frac{M \\rho _{0}}{r_{\\infty }^{\\prime 2}-M}(1-\\frac{2 b^{2}}{r_{\\infty }^{\\prime 2}}).$ With these transformations, one can find that the metric (REF ) can be given in a new form [55] $d s^{2}=-A(\\rho ) d t^{2}+B(\\rho ) d \\rho ^{2}+C(\\rho )(d \\theta ^{2}+\\sin ^{2} \\theta d \\phi ^{2})+D(\\rho )(d \\psi +\\cos \\theta d \\phi )^{2}-2 H(\\rho ) d t(d \\psi +\\cos \\theta d \\phi ), $ where $\\begin{aligned} A(\\rho ) &=\\frac{4-\\mathcal {V}(a j-2)^{2}-4 j^{2} \\mathcal {W}^{2}}{4 h^{2}}, \\quad \\quad \\quad B(\\rho )=\\mathcal {U}(\\rho ), \\quad \\quad \\quad C(\\rho )=\\mathcal {R}^{2}(\\rho ), \\\\D(\\rho ) &=\\frac{a^{2} \\mathcal {V}+4 \\mathcal {W}^{2}}{4}, \\quad \\quad \\quad \\quad H(\\rho )=\\frac{2 a \\mathcal {V}-(a^{2} \\mathcal {V}+4 \\mathcal {W}^{2}) j}{4 h}, \\end{aligned} $ with $\\begin{array}{c}{\\mathcal {K}^{2}=\\frac{\\rho +\\frac{r_{\\infty }^{2}+a^{2}}{r_{\\infty }^{2}} \\rho _{0}}{\\rho +\\frac{a^{2}}{r_{\\infty }^{2}} \\rho _{0}}, \\quad \\quad \\quad \\mathcal {V}=\\frac{M}{r_{\\infty }^{2}+a^{2}} \\mathcal {K}^{2}, \\quad \\quad \\quad \\mathcal {W}^{2}=\\frac{r_{\\infty }^{2}+a^{2}}{4 \\mathcal {K}^{2}}}, \\\\ {\\mathcal {R}^{2}(\\rho )=(\\rho +\\frac{a^{2}}{r_{\\infty }^{2}} \\rho _{0})(\\rho +\\frac{r_{\\infty }^{2}+a^{2}}{r_{\\infty }^{2}} \\rho _{0}), \\quad \\quad \\quad \\quad \\mathcal {U}(\\rho )=\\frac{\\rho _{0}^{2}}{\\mathcal {W}^{2}-\\frac{r_{\\infty }^{2}}{4} \\frac{\\rho }{\\mathcal {K} \\mathcal {R}} \\mathcal {V}}}.\\end{array} $ There are only three independent parameters among the parameters $\\rho _{0}$ , $M$ , $a$ , $r_{\\infty }$ , $r_{\\infty }^{\\prime }$ , $\\rho _{M}$ and $b$ .",
"Here, we select the parameters $\\rho _0$ , $\\rho _M $ and $\\ b$ as the independent parameters and the others are related to them by $\\begin{aligned}r_{\\infty }^{2}\\ = \\ &2 \\rho _{0}(\\rho _{M}+\\rho _{0})-b^{2}+\\sqrt{b^{4}-4 b^{2} \\rho _{0}(\\rho _{0}-\\rho _{M})+4 \\rho _{0}^{2}(\\rho _{M}+\\rho _{0})^{2}},\\\\&-\\frac{b^{2}(4\\rho _{0}^{2}-b^{2})}{2\\rho _{0}(\\rho _{M}-\\rho _{0})+b^{2}+\\sqrt{b^{4}-4 b^{2} \\rho _{0}(\\rho _{0}-\\rho _{M})+4 \\rho _{0}^{2}(\\rho _{M}+\\rho _{0})^{2}}},\\\\r_{\\infty }^{\\prime 2}\\ = \\ &b^{2}+2 \\rho _{0} \\rho _{M}+2 \\rho _{0}^{2}+\\sqrt{b^{4}-4 b^{2} \\rho _{0}(\\rho _{0}-\\rho _{M})+4 \\rho _{0}^{2}(\\rho _{M}+\\rho _{0})^{2}}, \\\\M\\ =\\ &b^{2}+2 \\rho _{0} \\rho _{M}-2 \\rho _{0}^{2}+\\sqrt{b^{4}-4 b^{2} \\rho _{0}(\\rho _{0}-\\rho _{M})+4 \\rho _{0}^{2}(\\rho _{M}+\\rho _{0})^{2}}, \\\\a\\ =\\ &b\\left[1+\\frac{4 \\rho _{0}^{2}-b^{2}}{2 \\rho _{0}(\\rho _{M}-\\rho _{0})+b^{2}+\\sqrt{b^{4}-4 b^{2} \\rho _{0}(\\rho _{0}-\\rho _{M})+4 \\rho _{0}^{2}(\\rho _{M}+\\rho _{0})^{2}}}\\right] ^{\\frac{1}{2}}.\\end{aligned}$ With these quantities, all of coefficients in the metric (REF ) can be expressed as functions of the parameters $\\rho _0$ , $\\rho _M $ and $\\ b$ .",
"With this functions, we can further study the shadow of the rotating squashed KK black hole (REF ).",
"In the background of the rotating squashed KK black hole (REF ), the Lagrange density of a photon propagation along null geodesics can be expressed as $\\begin{aligned}\\mathcal {L}=\\frac{1}{2}g_{\\mu \\nu }\\dot{x}^{\\mu }\\dot{x}^{\\nu },\\end{aligned}$ where a dot represents a derivative with respect to affine parameter $\\lambda $ along the geodesics.",
"Since all of metric functions $g_{\\mu \\nu }$ are independent of the coordinates $t$ , $\\phi $ and $\\psi $ , there are three conserved quantities for the photon propagation $\\begin{aligned}E&=-p_{t}=-g_{tt} \\dot{t}-g_{t\\phi } \\dot{\\phi }-g_{t \\psi } \\dot{\\psi },\\\\L_{\\phi }&=p_{\\phi }=g_{t\\phi }\\dot{t}+g_{\\phi \\phi }\\dot{\\phi }+g_{\\phi \\psi } \\dot{\\psi },\\\\L_{\\psi }&=p_{\\psi }=g_{t\\psi }\\dot{t}+g_{\\phi \\psi }\\dot{\\phi }+g_{\\psi \\psi }\\dot{\\psi },\\end{aligned}$ where $E$ is energy of the photon.",
"$L_{\\phi }$ and $L_{\\psi }$ denote its angular momentum in the $\\phi $ and $\\psi $ directions, respectively.",
"With these conserved quantities, the null geodesics for a photon can be further simplified as $\\begin{aligned}&\\dot{t}=\\frac{D(\\rho )E-H(\\rho )L_{\\psi }}{A(\\rho )D(\\rho )+H(\\rho )^{2}},\\quad \\quad \\quad \\quad \\dot{\\phi }=\\frac{L_{\\phi }-\\cos {\\theta } L_{\\psi }}{\\sin ^{2}{\\theta } \\ C(\\rho )},\\\\&\\dot{\\psi }=\\frac{H(\\rho )E+A(\\rho )L_{\\psi }}{A(\\rho )D(\\rho )+H(\\rho )^{2}}-\\frac{(L_{\\phi }-L_{\\psi }\\cos {\\theta })\\cos {\\theta }}{\\sin ^{2}{\\theta } \\ C(\\rho )},\\end{aligned}$ and $\\begin{aligned}&C(\\rho )^2\\ \\dot{\\rho }^{2}=R(\\rho )=\\frac{C(\\rho )^2}{B(\\rho )}\\bigg [\\frac{D(\\rho )E^{2}-2H(\\rho )EL_{\\psi }-A(\\rho )L_{\\psi }^{2}}{A(\\rho )D(\\rho )+H(\\rho )^{2}}-\\frac{L_{\\phi }^{2}+Q}{C(\\rho )}\\bigg ],\\\\&C(\\rho )^{2}\\ \\dot{\\theta }^{2}=\\Theta (\\theta )=-\\frac{(L_{\\phi }-\\cos {\\theta } L_{\\psi })^2}{\\sin ^{2}{\\theta }}+L_{\\phi }^{2}+Q,\\end{aligned}$ where $Q$ is the Carter constant.",
"From the above equation, we find that the $\\theta -$ component equation is independent of the rotation parameter and it is exactly identical to the equation in the static squashed KK black hole spacetime, which would yield some special properties of the shadow of the black hole (REF ).",
"The circular orbits satisfy $\\begin{aligned}R(\\rho )=0,\\qquad \\qquad R^{\\prime }(\\rho )=0.\\end{aligned}$ It is well known that the unstable circular orbits are very important to determine the boundary of the shadow casted by a black hole."
],
[
"Shadow of a rotating squashed Kaluza-Klein black hole",
"We assume that the observer is located in the spatial infinity in the rotating squashed KK black hole spacetime.",
"As $\\rho \\rightarrow \\infty $ , the metric (REF ) becomes $d s^{2}&=&-d t^{2}+ d \\rho ^{2}+\\rho ^2(d \\theta ^{2}+\\sin ^{2} \\theta d \\phi ^{2})+\\frac{r^{\\prime 2}_{\\infty }}{4}(d \\psi +\\cos \\theta d \\phi )^{2},$ which describes a four-dimensional Minkowski spacetime with a constant twisted $S_1$ fiber.",
"Setting $dw=\\frac{r^{\\prime }_{\\infty }}{2}(d \\psi +\\cos \\theta d \\phi )$ , and making the coordinate transformation $x=\\rho \\sin \\theta \\cos \\phi ,\\quad \\quad \\quad \\quad y=\\rho \\cos \\theta , \\quad \\quad \\quad \\quad z=\\rho \\sin \\theta \\sin \\phi ,$ one can find that the above metric can be rewritten as $d s^{2}&=&-d t^{2}+d x^{2}+dy^2+dz^2+ dw^{2},$ which has a form of five-dimensional Minkowski metric.",
"Thus, in the rotating squashed KK black hole spactime, the observer basis at the spatial infinity $\\left\\lbrace e_{\\hat{t}}, e_{\\hat{x}}, e_{\\hat{y}}, e_{\\hat{z}}, e_{\\hat{w}}\\right\\rbrace $ can be expanded as a form in the coordinate basis $\\left\\lbrace \\partial _{t}, \\partial _{\\rho }, \\partial _{\\theta }, \\partial _{\\phi }, \\partial _{\\psi }\\right\\rbrace $ [56], [57] $e_{\\hat{\\mu }}=e_{\\hat{\\mu }}^{\\nu } \\partial _{\\nu }.$ Here $e^{\\nu }_{\\hat{\\mu }}$ is the transform matrix which satisfies $g_{\\mu \\nu }e^{\\mu }_{\\hat{\\alpha }}e^{\\nu }_{\\hat{\\beta }}=\\eta _{\\hat{\\alpha }\\hat{\\beta }}$ .",
"$\\eta _{\\hat{\\alpha }\\hat{\\beta }}$ is the five-dimensional Minkowski metric.",
"In general, it is not unique for the transformation (REF ) which satisfies both the spatial rotations and Lorentz boosts.",
"For a rotating squashed KK black hole spacetime (REF ), one can choice a convenient decomposition associated with a reference frame with zero axial angular momentum in relation to spatial infinity [56], [57] $e_{\\hat{\\mu }}^{\\nu }=\\left( \\begin{array}{ccccc}{\\zeta } & {0} & {0} & {\\lambda } & {\\chi } \\\\{0} & {A^{\\rho }} & {0} & {0} & {0} \\\\{0} & {0} & {A^{\\theta }} & {0} & {0} \\\\{0} & {0} & {0} & {A^{\\phi }} & {\\varepsilon } \\\\{0} & {0} & {0} & {0} & {A^{\\psi }}\\end{array}\\right),$ where $\\zeta $ , $\\lambda $ , $\\chi $ , $\\varepsilon $ , $A^r$ , $A^{\\theta }$ , $A^{\\phi }$ and $\\ A^{\\psi }$ are real coefficients.",
"According to the Minkowski normalization $e_{\\hat{\\mu }}e^{\\hat{\\nu }}=\\delta _{\\hat{\\mu }}^{\\;\\hat{\\nu }},$ one can obtain $\\begin{aligned}&A^{\\rho }=\\frac{1}{\\sqrt{g_{\\rho \\rho }}},\\quad \\quad \\quad A^{\\theta }=\\frac{1}{\\sqrt{g_{\\theta \\theta }}},\\quad \\quad \\quad A^{\\phi }=\\sqrt{\\frac{g_{\\psi \\psi }}{g_{\\phi \\phi }g_{\\psi \\psi }-g^2_{\\phi \\psi }}},\\\\&A^{\\psi }=\\frac{1}{\\sqrt{g_{\\psi \\psi }}},\\quad \\quad \\quad \\varepsilon =-\\frac{g_{\\phi \\psi }}{g_{\\psi \\psi }}\\sqrt{\\frac{g_{\\psi \\psi }}{g_{\\phi \\phi }g_{\\psi \\psi }-g^2_{\\phi \\psi }}},\\\\&\\zeta =\\frac{\\sqrt{g_{\\phi \\psi }^{2}-g_{\\phi \\phi } g_{\\psi \\psi }}}{\\sqrt{g_{t t} g_{\\phi \\phi } g_{\\psi \\psi }+2 g_{t \\phi } g_{t \\psi } g_{\\phi \\psi }-g_{t \\phi }^{2} g_{\\psi \\psi }-g_{t \\psi }^{2} g_{\\phi \\phi }-g_{\\phi \\psi }^{2} g_{t t}}},\\\\&\\lambda =\\frac{g_{t\\phi } g_{\\psi \\psi }-g_{t\\psi } g_{\\phi \\psi }}{\\sqrt{g_{\\phi \\psi }^{2}-g_{\\phi \\phi } g_{\\psi \\psi }} \\sqrt{g_{t t} g_{\\phi \\phi } g_{\\psi \\psi }+2 g_{t \\phi } g_{t \\psi } g_{\\phi \\psi }-g_{t \\phi }^{2} g_{\\psi \\psi }-g_{t \\psi }^{2} g_{\\phi \\phi }-g_{\\phi \\psi }^{2} g_{t t}}},\\\\&\\chi =\\frac{g_{t\\psi } g_{\\phi \\phi }-g_{t\\phi } g_{\\phi \\psi }}{\\sqrt{g_{\\phi \\psi }^{2}-g_{\\phi \\phi } g_{\\psi \\psi }} \\sqrt{g_{t t} g_{\\phi \\phi } g_{\\psi \\psi }+2 g_{t \\phi } g_{t \\psi } g_{\\phi \\psi }-g_{t \\phi }^{2} g_{\\psi \\psi }-g_{t \\psi }^{2} g_{\\phi \\phi }-g_{\\phi \\psi }^{2} g_{t t}}}.\\end{aligned}$ Therefore, the locally measured five-momentum $p^{\\hat{\\mu }}$ of a photon is computed by the projection of its five-momentum $p^{\\mu }$ into $e_{\\hat{\\mu }}$ $\\begin{aligned} p^{\\hat{t}} &=\\zeta E-\\lambda p_{\\phi }-\\chi p_{\\psi }, \\quad \\quad \\quad \\quad p^{\\hat{x}} =\\frac{1}{\\sqrt{g_{\\rho \\rho }}}p_{\\rho }, \\\\p^{\\hat{y}} &=\\frac{1}{\\sqrt{g_{\\theta \\theta }}}p_{\\theta },\\quad \\quad \\quad \\quad p^{\\hat{z}} =\\sqrt{\\frac{g_{\\psi \\psi }}{g_{\\phi \\phi }g_{\\psi \\psi }-g^2_{\\phi \\psi }}}\\bigg (p_{\\phi }-\\frac{g_{\\phi \\psi }}{g_{\\psi \\psi }}p_{\\psi }\\bigg ) ,\\quad \\quad \\quad \\quad p^{\\hat{w}} =\\frac{1}{\\sqrt{g_{\\psi \\psi }}} p_{\\psi }.\\end{aligned}$ The four-vector $\\vec{p}$ is the photon's linear momentum with components $\\ p_{\\hat{x}}$ , $p_{\\hat{y}} $ , $p_{\\hat{z}} $ and $p_{\\hat{w}}$ in the orthonormal basis $\\left\\lbrace e_{\\hat{x}}, e_{\\hat{y}}, e_{\\hat{z}}, e_{\\hat{w}}\\right\\rbrace $ , $\\begin{aligned}\\vec{p}&=p^{\\hat{x}} e_{\\hat{x}}+p^{\\hat{y}} e_{\\hat{y}}+p^{\\hat{z}}e_{\\hat{z}}+p^{\\hat{w}} e_{\\hat{w}}.\\end{aligned}$ Combing the transformation (REF ) with the geometry of the photon's linear momentum, we have $p^{\\hat{x}} =\\tilde{p} \\cos \\alpha \\cos \\beta , \\quad \\quad \\quad p^{\\hat{y}} =\\tilde{p} \\sin \\alpha ,\\quad \\quad \\quad p^{\\hat{z}} =\\tilde{p} \\cos \\alpha \\sin \\beta ,\\quad \\quad \\quad p^{\\hat{w}}=\\sqrt{|\\vec{p}|^2-\\tilde{p}^2}.$ where $\\tilde{p}=\\sqrt{(p^{\\hat{x}})^2+(p^{\\hat{y}})^2+(p^{\\hat{z}})^2}$ .",
"In general, there should be three celestial coordinates of photon's image in the observer's sky in the five-dimensional background spacetime.",
"Since in the spatial infinity the rotating squashed KK black hole spacetime has a structure of four-dimensional Minkowski spacetime with a constant twisted $S_1$ fiber, the fifth dimension is compacted for the observer in the position far from black hole in this case.",
"This allows us to adopt the usual celestial coordinates $(x,y)$ in the four-dimensional case to describe the position of photon's image in the observer's sky because the celestial coordinates $(x,y)$ could be observed really by our astronomical experiments.",
"Moreover, it is very convenient for us to compare our results with those obtained in the usual four-dimensional black hole spacetimes.",
"The celestial coordinates $(x,y)$ for the photon's image in observer's sky for a rotating squashed KK black hole (REF ) are $x &=&-\\lim _{\\rho _{obs} \\rightarrow \\infty } \\rho _{obs}\\tan \\beta =-\\lim _{\\rho _{obs} \\rightarrow \\infty }\\rho _{obs} \\frac{p^{\\hat{z}}}{p^{\\hat{x}}}=-\\frac{\\xi _{\\phi }-\\xi _{\\psi }\\cos \\theta _0}{\\sin {\\theta _0}}\\frac{r^{\\prime }_{\\infty }}{\\sqrt{r^{\\prime 2}_{\\infty }-4\\xi ^2_{\\psi }}},\\nonumber \\\\y &=&\\lim _{\\rho _{obs} \\rightarrow \\infty } \\rho _{obs}\\frac{\\tan \\alpha }{\\cos {\\beta }}=\\lim _{\\rho _{obs} \\rightarrow \\infty } \\rho _{obs} \\frac{p^{\\hat{y}}}{p^{\\hat{x}}}= \\frac{r^{\\prime }_{\\infty }}{\\sqrt{r^{\\prime 2}_{\\infty }-4\\xi ^2_{\\psi }}} \\sqrt{\\eta +\\xi _{\\phi }^2-\\bigg (\\frac{\\xi _{\\phi }-\\xi _{\\psi }\\cos \\theta _0}{\\sin {\\theta _0}}\\bigg )^2}.$ Here $\\xi _{\\phi }\\equiv L_{\\phi }/E$ , $\\xi _{\\psi }\\equiv L_{\\psi }/E$ , $\\eta \\equiv Q/E^2$ , and $\\rho _{obs}$ is the distance between black hole and the observer.",
"$\\theta _{0}$ is the inclination angle of observer.",
"From Eq.",
"(REF ), it is easy to obtain $x^{2}+y^{2}=\\frac{r^{\\prime 2}_{\\infty }(\\eta +\\xi _{\\phi }^2)}{r^{\\prime 2}_{\\infty }-4\\xi ^2_{\\psi }},$ which implies that the shape of the shadow is a perfect black disk for a rotating squashed KK black hole (REF ).",
"It is different from that of a usual rotating black hole.",
"This special property of shadow of a rotating squashed KK black hole could be explained by a fact that the $\\theta $ - component equation is independent of the rotation parameter in this special black hole spacetime.",
"Moreover, we find that the radius of the image in the observer's sky caused by the photon with $\\xi _{\\psi }$ is $R_s=\\frac{r^{\\prime }_{\\infty }\\sqrt{\\eta +\\xi _{\\phi }^2}}{\\sqrt{r^{\\prime 2}_{\\infty }-4\\xi ^2_{\\psi }}}=\\frac{r^{\\prime }_{\\infty }}{\\sqrt{r^{\\prime 2}_{\\infty }-4\\xi ^2_{\\psi }}}\\sqrt{\\frac{C(\\rho _{ps})[D(\\rho _{ps})-2H(\\rho _{ps})\\xi _{\\psi }-A(\\rho _{ps})\\xi ^2_{\\psi }]}{A(\\rho _{ps})D(\\rho _{ps})+H(\\rho _{ps})^{2}}},$ Figure: Changes of R s R_s with ξ ψ \\xi _{\\psi } for the rotating squashed KK black hole with different ρ 0 \\rho _0 and bb.",
"Here we set ρ M =1\\rho _M=1.where $\\rho _{ps}$ is the photon sphere radius of photon with the specific angular momentum $\\xi _{\\psi }$ .",
"Obviously, the radius $R_s$ is a function of parameters $\\xi _{\\psi }$ , $\\rho _0$ , $\\rho _M$ and $b$ .",
"Thus, for the black hole with fixed parameters $\\rho _0$ , $\\rho _M$ and $b$ , the quantity $R_s$ has different value for the photon with different $\\xi _{\\psi }$ .",
"This means that the radius of the black hole shadow is determined by the minimum value of $R_s$ , i.e., $R_{BH}|_{(\\rho _0,\\rho _M, b)}=\\text{Minimum} [R_s(\\rho _0, \\rho _M, b,\\xi _{\\psi })]|_{(\\rho _0,\\rho _M, b)}.$ Figure: ξ ψ =0.96\\xi _{\\psi }=0.96Obviously, it is heavily influenced by the specific angular momentum $\\xi _{\\psi }$ of the photon.",
"From Eq.",
"(REF ), one can find that $\\xi _{\\psi }$ should be limited in the range $\\xi _{\\psi }\\in (-\\frac{r^{\\prime }_{\\infty }}{2},\\frac{r^{\\prime }_{\\infty }}{2})$ .",
"Here, in Fig.REF , we present the changes of $R_s$ with $\\xi _{\\psi }$ for different black hole parameters in the rotating squashed KK black hole spacetime, which indicates that the specific angular momentum of photon $\\xi _{\\psi }$ affects sharply the black hole shadow.",
"For the fixed $b=0$ , $\\rho _0=0.3$ , we find that $R_s$ has a minimum value $R_{s_{\\text{min}}}=2.8426$ as $\\xi _{\\psi }=0$ , which means that there exists black hole shadow with its radius $R_{BH}=2.8426$ in this case.",
"However, for the case with $b=0.4$ and $\\rho _0=0.6$ , we obtain the minimum value $R_{s_{\\text{min}}}=0$ corresponding $\\xi _{\\psi }=0.996112$ , which is in the range $\\xi _{\\psi }\\in (-\\frac{r^{\\prime }_{\\infty }}{2},\\frac{r^{\\prime }_{\\infty }}{2})$ .",
"The similar behavior also appears in the case with $b=0.7$ and $\\rho _0=0.8$ .",
"The special behavior $R_{s_{\\text{min}}}=0$ means that there is no black shadow for a black hole in these cases, which is novel since it is impossible to appear in the usual black hole spacetimes.",
"Figure: ξ ψ =0.997\\xi _{\\psi }=0.997The mathematical reason is that the factor in the right side of Eq.",
"(REF ) $D(\\rho _{ps})-2H(\\rho _{ps})\\xi _{\\psi }-A(\\rho _{ps})\\xi ^2_{\\psi }=0$ in this case.",
"Here, we also present the propagation of photon with different $\\xi _{\\psi }$ in the rotating squashed KK black hole spacetime with the parameters ($\\rho _0=0.6$ , $b=0.2$ ) in Fig.",
"REF and ($\\rho _0=0.6$ ,$b=0.4$ ) in Fig.",
"REF , respectively.",
"In Fig.",
"REF , we find that the rotating squashed KK black hole with parameters $\\rho _0=0.6$ and $b=0.2$ can capture the photons with various values of $\\xi _{\\psi }$ as they approach the black hole, which means that black hole shadow exists in this case.",
"However, for the black hole with parameters $\\rho _0=0.6$ and $b=0.4$ , we find from Fig.",
"REF that for $\\xi _{\\psi }\\ge 0.9 $ , the impact parameter of the photon captured by black hole first increases and then decreases with increase of $\\xi _{\\psi }$ , which is consistent with the change of $R_s$ in Fig.",
"REF (c).",
"Especially, as $\\xi _{\\psi }\\in (\\xi _{\\psi _c}, \\frac{r^{\\prime }_{\\infty }}{2})$ , where $\\xi _{\\psi _c}$ is the positive root of the equation $D(\\rho _{ps})-2H(\\rho _{ps})\\xi _{\\psi }-A(\\rho _{ps})\\xi ^2_{\\psi }=0$ , we find that the photons near black hole change their propagation direction and then become far away from the black hole.",
"This implies that these photons can not captured by black hole so that they could reach the observer, which yields that there is no shadow in this case.",
"Therefore, the specific angular momentum $\\xi _{\\psi }$ of photon from the fifth dimension plays an important role in the formation of no shadow for a rotating squashed KK black hole.",
"The phenomenon of black hole without black shadow would vanish if there exists the further constraint on the specific angular momentum $\\xi _{\\psi }$ of photon from the fifth dimension.",
"Figure: The parameter regions of existence and nonexistence of black hole shadow for a rotating squashed KK black hole.",
"The black hole shadow exists only in the region II and there is no shadow in the region IIII.",
"In the region IIIIII, there is no horizon and the metric () does not describe geometry of a black hole.",
"Here we set ρ M =1\\rho _M=1.In Fig.",
"REF , we present the parameter regions of existence and nonexistence of black hole shadow for a rotating squashed KK black hole.",
"The black hole shadow exists only in the region $I$ and there is no black shadow in the region $II$ .",
"In the region $III$ , there is no horizon and the metric (REF ) does not describe geometry of a black hole.",
"Figure: Dependence of the radius of the shadow on the rotation parameter bb and the scale of transition ρ 0 \\rho _0 for a rotating squashed KK black hole.",
"The green dashed line denote the boundary between existence and nonexistence of black hole shadow.In Fig.",
"REF , we present the dependence of the radius of the shadow on the rotation parameter $b$ and the scale of transition $\\rho _0$ in the case of the spacetime parameters lie in the region $I$ in which there exists black hole shadow for a rotating squashed KK black hole.",
"From Fig.REF , for the case with existence of black hole shadow, one can find that the shadow radius $R_{BH}$ decreases with the rotation parameter $b$ of black hole, which is similar to those in the usual rotating black holes.",
"With the increase of extra dimension parameter $\\rho _0$ , the radius $R_{BH}$ increases monotonically in the case of $b=0$ .",
"In the rotating cases with the smaller $b$ , we find that $R_{BH}$ first decreases and then increases with the parameter $\\rho _0$ .",
"For the cases with the larger $b$ , $R_{BH}$ decreases monotonically with the parameter $\\rho _0$ .",
"Finally, we make use of the metric (REF ) and estimate the angular radius of the black hole shadow by using the observable $R_{BH}$ as $\\theta _{BH}=R_{BH}\\mathcal {M}/D_O$ , where $D_O$ is the distance between the observer and the black hole.",
"For an arbitrary black hole of mass $\\mathcal {M}$ and distance $D_O$ from the observer, the angular radius can be expressed as $\\theta _{BH}=9.87098\\times 10^{-6} R_{BH}(\\mathcal {M}/M_{\\odot })(1kpc/DO)\\mu as$ [27], [32] with the mass of the Sun $M_{\\odot }$ .",
"In Table (I), we present the angular radius of the black hole by the metric (REF ) for the supermassive black hole Sgr $A^{*}$ located at the Galactic center and the supermassive black hole in $M87$ , respectively.",
"Here we use the mass $\\mathcal {M} = 4.3\\times 10^{6} M_{\\odot }$ and the observer distance $D_O = 8.3 kpc$ for the black hole Sgr $A^{*}$[59], and $\\mathcal {M} = 6.5\\times 10^{9} M_{\\odot }$ and $D_O=16.8 Mpc$ for the black hole in the $M87$ [6].",
"Table: The numerical estimation for the angular radius of the black shadow for the supermassive black hole Sgr A* in our Galaxy and the black hole in M87M87 by using the metric of a rotating squashed KK black hole.The latest observation indicates that the angular diameter of $M87$ black hole is $42\\pm 3\\mu as$ [6].",
"Combining it with the data in Table (REF ), one can find that there is a room for the theoretical model of such a rotating squashed KK black hole."
],
[
"Summary and Discussion",
"We have studied the shadow of a rotating squashed Kaluza-Klein black hole and find that the shadow possesses some novel properties differed from those of other rotating black holes.",
"Firstly, the shadow shape is a perfect black disk for the black hole in the allowed parameter regions of $\\rho _0$ and rotation parameter $b$ .",
"It is different from that of the usual Kerr rotating black hole where the shape gradually changes from disk to “D\"-shape with the increase of rotation parameter.",
"The circular silhouette of the shadow for the rotating squashed KK black hole is caused by the spacetime property that there are two equal rotational parameters in this special black hole spacetime, which also leads to that the $\\theta $ - component equation is independent of the rotation parameter.",
"Moreover, since the radius $R_s$ of the image in the observer's sky caused by the photon falling into black hole horizon depends on the specific angular momentum $\\xi _{\\psi }$ of photon, only the minimum value of $R_s$ is the radius of the black hole shadow, which yields that the shadow for a rotating squashed KK black hole is heavily influenced by the specific angular momentum $\\xi _{\\psi }$ of photon.",
"Especially, as the black hole parameters lie in a certain special range, we find that there is no shadow for a black hole since the minimum value $R_{s_{\\text{min}}}=0$ in these special cases, which is novel since it does not emerge in the usual black hole spacetimes.",
"It must be noted that the black hole without shadow is not caused by that light rays can penetrate the black hole, but by that the photons near black hole with some special range of $\\xi _{\\psi }$ change their propagation direction and then become far away from the black hole.",
"Therefore, the specific angular momentum $\\xi _{\\psi }$ of photon from the fifth dimension plays an important role in the formation of no black shadow for a rotating squashed KK black hole.",
"The phenomenon of black hole without black shadow would vanish if there exists the further constraint on the specific angular momentum $\\xi _{\\psi }$ of photon from the fifth dimension.",
"In the case where black hole shadow exists, we find that the radius of the black hole shadow $R_{BH}$ decreases with the rotation parameter $b$ of black hole, which is similar to those in the usual rotating black holes.",
"With the increase of extra dimension parameter $\\rho _0$ , the radius of the black hole shadow $R_{BH}$ increases monotonically in the case with $b=0$ .",
"In the rotating cases with the smaller $b$ , we find that $R_{BH}$ first decreases and then increases with the parameter $\\rho _0$ .",
"For the cases with the larger $b$ , $R_{BH}$ decreases monotonically with the parameter $\\rho _0$ .",
"Finally, we make use of the metric (REF ) and estimate the angular radius of the black hole shadow by the observation data from the supermassive black hole Sgr $A^{*}$ located at the Galactic center and the supermassive black hole in $M87$ , which implies that there is a room for the theoretical model of such a rotating squashed KK black hole."
],
[
"Acknowledgments",
"This work was partially supported by the National Natural Science Foundation of China under Grant No.",
"11875026, the Scientific Research Fund of Hunan Provincial Education Department Grant No.",
"17A124.",
"J. Jing's work was partially supported by the National Natural Science Foundation of China under Grant No.",
"11875025."
]
] | 1906.04456 |
[
[
"The Formation of Bimodal Dust Species in Nova Ejecta I: Chemical\n Conditions"
],
[
"Abstract The formation of bimodal dust species (that is, both of silicate and amorphous carbon dust grains are observed in a nova eruption) in nova ejecta is still debated.",
"Using the Modules for Experiments in Stellar Astrophysics code and considering the effects of WD's mass, mass-accretion rate and the chemical profiles of WD which are described by new parameter --- mixing depth on the chemical abundances of nova ejecta, we investigate the possibility that bimodal dust species are produced in a nova eruption.",
"We find that $C/O$ (the ratio of carbon number density to oxygen number density) of nova ejecta is affected by the mixing depth.",
"For the model with a small mixing depth, the $C/O$ of nova ejecta can evolute from lager than 1.0 to smaller than 1.0 in a whole eruption, which provides the chemical condition for the formation of bimodal dust species."
],
[
"Introduction",
"It is well known, dust is the most important ingredient of the interstellar medium.",
"It is mainly produced by the stellar winds of asymptotic giant branch (AGB) stars and supernova (SN) ejecta.",
"Recently, [38] suggested that the dust produced by common-envelope (CE) ejecta is not negligible [65], [66].",
"[67] investigated the evolution of interstellar dust and stardust in the solar neighbourhood, and found that the fraction of stardust produced by SNe is about 15% and it is about 85% for AGB stars.",
"[38] compared the dust masses produced by CE ejecta with those produced by AGB stars for the solar metallicity, and found that the dust produced in CE ejecta may be quite significant and could even dominate under certain circumstances.",
"However, the former greatly depends on the input parameters[38].",
"Based on the popular view of point, due to the high binding energy of CO, the dust species produced by these sources depend on the abundance ratio of the carbon to the oxygen ($C/O$ ) in their environment.",
"For example, M-type ($C/O<1$ ) AGB star can produce silicate dust grains (olivine, pyroxene or quartz)[16], while amorphous carbon dust grains (graphite, diamond or silicon carbide) originate from C-type ($C/O>1$ ) star[14].",
"Of course, this expectation results from an assumption that the CO abundance reaches its saturation value.",
"However, under some environment (such as SN, nova), there is a strong radiation field which can compromise CO molecule and reduce the criticality of the C/O.",
"[47] investigated the chemical evolution of nova ejecta, and found that amorphous carbon dust grains can be produced in an oxygen rich environment because of neutral reactions in a shielded region.",
"Similarly, dust has been observed in some nova eruptions [19], [17].",
"Although the dust produced by nova is less than 4% of that produced by AGB stars[10], dust formation in the nova ejecta is very interesting.",
"About 50% of novae can produce dust [24].",
"Surprisingly, in some novae, both of silicate and amorphous carbon dust grains (that is, the bimodal dust species) are observed in a nova eruption [18], [13].",
"Typically, the silicate grains are observed after amorphous carbon grains are detected during nova eruptions.",
"Due to no infrared echoes from the pre-existing silicate dust observed in nova V1280 Sco, [53] considered that the silicate grains were newly produced during nova eruptions.",
"The bimodal dust species should be produced in a nova ejecta.",
"There is still debating about the origin of the bimodal dust species.",
"[53] suggested that the amorphous carbon grains form in the nova ejecta, but they were not sure that the silicate grains were produced either in the expanding nova ejecta or in the interaction zone of the nova ejecta and the oxygen-rich circumstellar medium.",
"[9] considered that CO is destroyed by the high energy particles accelerated by the shock, and then the bimodal dust species can form.",
"For the former, V1280 Sco is a classical nova.",
"This means that the companion of white dwarf (WD) is a main-sequence star or dwarf and its matter is transferred via Roche lobe.",
"The circumstellar medium does not originate from the companion but from the pre-nova ejecta.",
"Its composition should be similar with that of the nova ejecta.",
"For the latter, as [9] estimated, the effects of non-thermal decomposition are very uncertain, so play a less significant role.",
"Under the assumption of saturated CO abundance, the bimodal dust species mean that there should be two different chemical environments which is noted by $C/O<1$ and $C/O>1$ in a nova ejecta.",
"The chemical abundances in the nova ejecta become the key to understand the formation of the bimodal dust species.",
"However, it is well known that they strongly depend on the model of nova eruption.",
"The nova eruption is a thermonuclear runaway (TNR) occurring on the surface of accreting WD in a close binary.",
"It has been about four decades since [59] first used a nuclear reaction network to calculate the TNR of a nova.",
"The nova models have been investigated by many literatures [49], [29], [63], [37], [40], [39], [7], [8].",
"See [56], [31], [57] and [27] for the recent reviews.",
"In these theoretical models, the chemical abundances in nova ejecta are mainly determined by the TNR and the mixing from the WDs.",
"Unfortunately, our knowledge of the mixing is extremely limited.",
"In general, in the 1D model, the range of the mixing is between 25% and 75%[29], [58].",
"Recent multidimensional (2D and 3D) models have showed that the Kelvin-Helmholtz hydrodynamic instabilities can dredge up material from the underlying WD and enrich the accreted envelope with (outer)core material[23], [4], [6].",
"In addition, most of models assume that WD has an uniform chemical composition.",
"However, we have known that WDs with different masses have different chemical compositions, and the chemical abundances around the surface of a WD are deeply varietal with a depth increase.",
"In the present paper, we investigate the effects of the chemical profile in WD and mixing depth on the chemical abundances in nova ejecta, and discuss whether there is an environment for the formation of bimodal dust species in nova ejecta.",
"In §2, we give theoretical models about nova.",
"The main results are in §3.",
"The conclusions are given in §4."
],
[
"Nova Models",
"In nova models, the WD mass, the composition of the accreted matter, the mass-accretion rate, convection, and the mixing prescription play an important role.",
"The WDs with different masses are produced by stars with different masses, and they have undergone different nucleosynthesis.",
"Therefore, their chemical compositions are different.",
"We use Modules for Experiments in Stellar Evolution (MESA, [rev.",
"10108]; [44], [45], [46]) to create WD models which include the CO WDs with masses of 0.6 and 1.0 $\\rm M_\\odot $ , and ONe WDs with masses of 1.2 and 1.3 $\\rm M_\\odot $ , respectively.",
"Usually, if WDs are produced via single star model, there are some unburnt He- and H-rich layers above the CO cores or unburnt C-rich layers above ONe cores [28].",
"However, most of WDs in nova involve binary interaction including Roche lobe mass transfer and CE evolution before they form [64], [36].",
"[20] showed that the binary interaction can greatly affect the WD masses and their chemical compositions.",
"These WDs probably are stripped H-rich layers, or even He-rich layers.",
"In [7], the H-rich and He-rich layers of WDs removed artificially.",
"The Roche lobe mass transfer or CE evolution usually involve a donor with H-rich envelope which is finally transferred to its companion or is ejected[12], [43].",
"Therefore, we remove H-rich layers artificially when WDs form in this work.",
"Figure: The profiles of the chemical abundances for He, C, O, Ne and Mg around thesurface of WDs with different masses.",
"Every WD is showed in a sub-figure.",
"The WD masses andspecies are given in the bottom region.Figure REF shows the profiles of the chemical abundances for He, C, O, Ne and Mg around the surface of WDs with different masses.",
"Obviously, the profiles of the chemical abundances for WDs with different masses are different greatly.",
"The main reason is that WDs with different masses originate from main-sequence stars with different masses and they undergo different nuclear reactions.",
"In nova model, the mixing degree of the accreted matter with the matter of WD is crucial parameter.",
"In fact, during the progress of the TNR or the accretion, the mixing may occur.",
"There are several mechanisms for such mixing: [48] assumed the mixing by a diffusion layer[33], [15]; [11] considered the mixing by shear instability due to differential rotation [42], [55], [34]; [51] proposed the mixing by gravity waves[2], [3], [1]; [62] suggested the mixing by convective overshoot[54], [21], [22], [23].",
"These mechanisms are put forward in the framework of 1D or 2D simulations and have shortcomings of themselves[35].",
"Considered that such convective mixing can only be simulated in the framework of three dimensions, [5] carried a 3D nuclear-hydrodynamic simulation for the mixing, and found that Kelvin-Helmholtz instabilities can naturally result in self-enrichment of the accreted envelopes with material from the underlying WD[4], [6].",
"Their results are consistent with observations.",
"Following [7] and [52], we use 1D nova model in test suit of MESA to simulate the nova eruption.",
"The main input parameters are listed as follows: (I) The mixing depth Obviously, based on Figure REF , the compositions of TNR material will be different when the mixing occurs in different depth from the surface of WD.",
"Recently, in order to investigate the C-rich dust in CO nova outbursts, [28] also considered the chemical profiles for the outer WD layers which are characterized by different C and O content .",
"In this work, we introduce a free parameter, mixing depth ($\\delta =\\frac{M_{\\rm mix}}{M_{\\rm WD}}$ ), which is the ratio of the mixed mass of WD to the total mass.",
"In order to discuss the effect of the parameter $\\delta $ , we take it as 0.001, 0.01, 0.05 and 0.1 in different simulations.",
"(II)The nuclear network The element abundances of the accreted matter are similar with these of the Sun.",
"Because the temperature during the TNR can reach up to $2-4 \\times 10^8$ K, the nuclei as heavy as Ar and Ca may be synthesized.",
"In our model, we select 52 isotopes from $^1$ H to $^{41}$ Ca.",
"These isotopes refer to 386 nuclear reactions from pp chains, CNO cycle to Ca burning (such as $^{41}$ Ca($n,\\alpha $ )$^{38}$ Ar, $^{41}$ Ca($p,\\alpha $ )$^{38}$ K), and so on.",
"(III)The mass-loss rate The mass loss occurs when the luminosity of WD during the TNR closes to Eddington luminosity.",
"According to [7], the mass-loss rate is given by $\\dot{M}=-2\\eta _{\\rm Edd}\\frac{L-L_{\\rm Edd}}{v_{\\rm esc}^2},$ where $v_{\\rm esc}=\\sqrt{2{\\rm G}M/R}$ is the escape velocity, $L$ and $L_{\\rm Edd}=4\\pi {\\rm G c}M/\\kappa $ are the luminosity of nova and Eddington luminosity, respectively.",
"Here, ${\\rm G}$ and ${\\rm c}$ are the gravitational constant and light velocity, $M$ and $R$ are the WD's mass and radius, respectively.",
"The $\\kappa $ is the Rosseland mean opacity.",
"The parameter $\\eta _{\\rm Edd}$ is set to 1, which simply assumes that the radiative energy of $L-L_{\\rm Edd}$ is completely used to eject matter around the surface of WD.",
"(IV)The mass-accretion rate and the core temperature It is widely accepted that the nova eruption is affected by not only the WD's mass and the chemical abundances, but also the mass-accretion rate and the core temperature of the WD.",
"Here, we take different mass-accretion rates ($1\\times 10^{-7}$ , $1\\times 10^{-9}$ and $1\\times 10^{-11} {\\rm M}_\\odot $ yr$^{-1}$ ) in different simulations.",
"The effects of the core temperature of WD on nova have been discussed by many literatures [60], [63], [30].",
"In general, a cooler WD can produce stronger nova outburst.",
"In this work, we do not consider its effect, and take a constant core temperature of $10^7$ K."
],
[
"Results and Discussions",
"We simulate 48 models for nova eruption by combining 3 parameters (4 WD masses, 3 mass-accretion rates and 4 mixing depths).",
"Tables 1—4 in appendix show all models and results."
],
[
"Parameter Effects",
"In our models, the matter accreted is hydrogen-rich.",
"Therefore, the energy released during TNR mainly originates from hydrogen burning.",
"Due to very high temperature ($>10^8$ K) in the reaction zone, the CNO cycle is the main way for hydrogen burning, which is showed in Figure REF for typical models.",
"Simultaneously, WD masses, the mass-accretion rates and the mixing depth parameter ($\\delta $ ) have great effects on TNRs.",
"With similar previous studies [49], [63], the lower mass-accretion rate of a WD is, the stronger TNR is, and a higher WD's mass is, the shorter and the stronger nova eruption is.",
"A large mixing depth can trigger an earlier TNR because it can provide more C and O elements for the hydrogen burning in CNO cycle.",
"Therefore, in these models, the hydrogen mass burned is less than that with a small mixing depth and the maximum temperature during TNR is also lower (See Tables 1—4 in Appendix).",
"Figure: The energy released by different nuclear reactions.",
"The left and rightpanels give these produced by the p-p chain and CNO cycle, respectively.The up and down two panels are for models with 1.0 M ⊙ \\rm M_\\odot CO WDs and 1.3 M ⊙ \\rm M_\\odot ONe WDs, respectively.The thin and thick lines represent the results of the models withδ=0.001\\delta =0.001 and 0.1, respectively.",
"The solid, dashed and dotted linesshow the models with the mass-accretion rates of 1×10 -7 1\\times 10^{-7},1×10 -9 1\\times 10^{-9} and 1×10 -11 M ⊙ 1\\times 10^{-11} \\rm M_\\odot yr -1 ^{-1},respectively.Figure REF shows the evolution of luminosity during a whole nova eruption.",
"In our models, the duration of a nova eruption greatly depends on the mixing depth and WD mass, while it is weakly affected by the mass-accretion rates.",
"It increases from about 100 days to about 700 days when $\\delta $ increases from 0.001 to 0.1.",
"On the observations, a nova eruption can last several weeks or many months, even serval years.",
"Therefore, we are not able to constraint the value of mixing depth parameter.",
"In short, besides of the core temperature, the theoretical simulation of nova eruptions in this work greatly depends on the uncertain three parameters: the WD mass, the mass-accretion rate and the mixing depth.",
"Figure: The evolution of luminosity during a whole nova eruption in different models.The up and down two panels are for models with 1.0 M ⊙ \\rm M_\\odot CO WDs and 1.3 M ⊙ \\rm M_\\odot ONe WDs, respectively.The left panels are for the model with different δ\\delta (δ=0.001\\delta =0.001 and 0.1) but witha fixed mass-accretion rate (1×10 -9 M ⊙ 1\\times 10^{-9} \\rm M_\\odot yr -1 ^{-1}),while the right panels are for the model with different mass-accretion rates(1×10 -9 1\\times 10^{-9} and 1×10 -11 M ⊙ 1\\times 10^{-11} \\rm M_\\odot yr -1 ^{-1})but a fixed δ\\delta (δ=0.001\\delta =0.001)."
],
[
"Element Abundances in the Ejecta",
"Our models assume that the WD accretes solar composition material.",
"After the mixing of the accreted matter with the matter of WD, some matter is ejected during nova eruptions.",
"Therefore, the chemical abundances of ejecta also are influenced by the mass-accretion rate, the mixing depth and WD's type.",
"Usually, TNR can trigger nucleosynthesis up to the charge number $\\sim $ 20[26].",
"Considering that H, He, C, N, O, Mg, Al, Si and S may affect the formation of dust in nova ejecta and these elements can be compared with those in [29], we show the average chemical abundances of these elements in the ejecta in Figure REF .",
"Compared with the mass-accretion rate (Comparing the left two panels with the right two panels in Figure REF ), the mixing depth and the WD's type have greater effects on the chemical abundances of the ejecta.",
"[29] assumed that CO WD is composed of 49.5% of C, 49.5% of O and 1% of Ne, and the initial composition of ONe WD comes from C burning nucleosynthesis calculations from [50].",
"In our work, with the enhance of the mixing depth, more C, O or Ne are involved into TNRs.",
"Therefore, the results of the CO WD model with large mixing depth ($\\delta $ =0.1) are closed to those of CO3 model in [29] although some input parameters are different.",
"However, the results of ONe WD in two works are different because of the different chemical abundances of ONe WD.",
"The variations of the element abundances from the accreted matter to the mixed matter result from the mixing, and are determined by the chemical profiles of WD and the mixing depth.",
"The variations from the mixed matter to the ejecta are triggered by the TNR.",
"Therefore, as Figure REF and Tables 1—4 show, for a low-mass CO WD, the mixing can change the abundances of C, O and Mg while the TNR only varies the abundances of elements lighter than O element because the maximum temperature during nova eruptions hardly gets to $\\sim 2.0\\times 10^8$ K; for a high-mass ONe WD, the elements lighter than Ca will be involved in some nucleosynthesises.",
"These results are consist with those of [29].",
"Figure: The differences of chemical abundances for elements which can affect the formation ofdust in nova ejecta in models with different mass-accretion rates (M ˙=10 -9 \\dot{M}=10^{-9} and 10 -11 M ⊙ 10^{-11} \\rm M_\\odot yr -1 ^{-1})and different mixing depths (δ=0.001\\delta =0.001 and 0.1).The black and green cycles represent the average chemical abundancesof the ejecta in the models with different mixing depths (δ=0.001\\delta =0.001 and 0.1) during a whole eruption, respectively.The red cycles in up and down pannel represent the results of CO3 and ONe5 models from , respectively.Due to very high binding energy of CO, the species of dust greatly depend on $C/O$ in the ejecta.",
"Figure REF gives the $C/O$ evolutions in nova ejecta and the amount of mass ejected for different models.",
"Obviously, the $C/O$ of ejecta is larger than 1.0 at the beginning of the nova eruption, while then quickly becomes smaller than 1.0 due to the nucleosynthesises of TNR.",
"As the bottom two panels of Figure REF shows, about less than 10% of mass ejected is carbon-rich.",
"It is possible that the amorphous carbon dust is firstly formed, and soon the silicate dust is formed in the nova ejecta, which is similar with the observations found by [18], [13], [53].",
"Figure: Similar with Figure , but for the evolutions of C/OC/O in nova ejecta and themass ejected (ΔM e \\Delta M_{\\rm e}).Figure REF shows the evolution of the convective and the mixing regions during nova eruptions in different models.",
"Before the outburst, the convection produced by the accretion occurs in a very thin layer under the WD surface, and the mixing triggered by the thermohaline mechanism always exists in the thick core region.",
"During the outburst, the envelope accumulated on the WD surface rapidly expands up to several hundred times of the WD radius, and the convection still occurs in the bottom of the envelope.",
"Based on the right two panels of Figure REF , compared with WD mass, the mass involved in the convection region is insignificant, that is, the mixing between the matter accreted and the WD's matter during the outburst process in our models mainly occurs in a thin layer on the bottom of the envelope accumulated.",
"Figure: The evolution of the convective and the mixing regions during nova eruptions in different models.The beginning of the eruption is represented by t=0t=0.The left two panels show the convective and the mixing regions along the coordinate of WD's radius, while the righttwo panels give those along the coordinate of WD's mass.",
"The convective region lies between the black and red solid lines,and the thermohaline mixing region is located between the black and red dashed lines.",
"The input parameters of modelsare given in the blanks.Figure REF shows the $C/O$ evolutions for all nova models simulated in the present paper.",
"For the nova models with 0.6 $\\rm M_\\odot $ CO WD, $C/O$ of the ejecta is always larger than 1.0 except the models with very low mass-accretion rate ($1\\times 10^{-11} \\rm M_\\odot $ yr$^{-1}$ ) and small mixing depth ($\\delta =0.001$ , and 0.01).",
"For the nova models with 1.0 $\\rm M_\\odot $ CO WD, $C/O$ can evolve from larger than 1.0 to lower than 1.0 in all models, while the $C/O$ in the nova models with 1.2 $\\rm M_\\odot $ ONe WD can do so only in the models with small mixing depth ($\\delta =0.001$ , and 0.01).",
"The main reason is that the mixing in these models with large mixing depth ($\\delta =0.05$ , and 0.1) can result in a very low $C/O$ because the O abundance of the WD from the surface to the inside quickly rises over C abundances (See Figure REF ).",
"This reason is suitable to all models with 1.3 $\\rm M_\\odot $ ONe WD.",
"More interestingly, in the models with very low mass-accretion rate ($1\\times 10^{-11}\\rm M_\\odot $ yr$^{-1}$ ), the TNR deletes amounts of O element so that $C/O$ in the ejecta changes from smaller than 1.0 to higher than 1.0.",
"This means that the silicate dust may be produced at first, and the amorphous carbon dust forms after that.",
"Figure: The C/OC/O evolutions on the surface of WDs during nova eruptions.The masses of WDs and the values of parameter δ\\delta are given in the top zone of everypanel.",
"The solid, dashed and dash-dotted lines represent the models with the mass-accretionrates of 1×10 -7 1\\times 10^{-7}, 1×10 -9 1\\times 10^{-9} and 1×10 -11 M ⊙ 1\\times 10^{-11} \\rm M_\\odot yr -1 ^{-1},respectively.The bimodal dust species have been observed in following six novae: V1370 Aql, V842 Cen, QV Vul, V2676 Oph, V1280 Sco and V1065 Cent [61], [25], [53], [32].",
"Here, the masses of CO WDs in V1280 Sco, V842 Cen and V2676 Oph have been observationally estimated.",
"They are $\\le 0.6$ , 0.88 and 0.6 (by slow evolution of light curves) or 1.1 (by nucleosynthesis) $\\rm M_\\odot $ , respectively [53], [41], [32].",
"These observations have a weak constrain on our model.",
"In addition, based on [63], the different core temperatures of WDs can result in an uncertainty up to a factor of about 10.",
"Considering that the input parameters (the mixing depth, the WD's mass and the mass-accretion rate) in this work have lead to a large scatter of our results, the present paper does not discuss its effects.",
"Simultaneously, we have not conducted 3D simulations in this work since they are extremely time- consuming and 1D simulations are accurate enough for our goals."
],
[
"Conclusions",
"In order to discuss the possibility for the formation of bimodal dust species, we use MESA to investigate the chemical abundances of nova ejecta.",
"Having considered the chemical profiles of WD, the new input parameter, mixing depth, is introduced to describe the mixing zone when TNR occurs.",
"The effects of WD mass, mass-accretion rate and mixing depth on the nova eruption are studied.",
"The effects of the first two parameters ( WD mass and mass-accretion rate) are similar with the previous work, that is, the lower mass-accretion rate of a WD is, the stronger the TNR is, and the higher WD's mass is, the shorter and the stronger nova eruption is.",
"For new parameter—the mixing depth, we find that a large mixing depth can trigger an earlier TNR because it can provide more C and O elements for the hydrogen burning in CNO cycle.",
"Therefore, in these models, the hydrogen mass burned is less than that with a small mixing depth and the maximum temperature during TNR is also lower.",
"We focus on the $C/O$ evolution during a whole nova ejecta, and find that it greatly depends on the mixing depth.",
"This means that the chemical profiles of WD greatly affect nova eruption.",
"For the models of CO or ONe WDs with a small mixing depth ($\\delta =0.001$ ), the $C/O$ of nova ejecta may be larger than 1.0 at the beginning of the nova eruption, and then quickly becomes smaller than 1.0 due to the nucleosynthesises of TNR.",
"However, for a large mixing depth ($\\delta \\ge 0.05$ ), the $C/O$ for ONe WD is always smaller than 1.0, and even for model of ONe WD with low mass-accretion ($10^{-11}\\rm M_\\odot $ yr$^{-1}$ ), the $C/O$ of the ejecta changes from smaller than 1.0 to higher than 1.0.",
"Considering the bimodal dust species have been observed in CO and ONe WDs, we suggest that the mixing depth should be a very small value.",
"This means that the mixing only occurs in a thin layer close to the surface of WD."
],
[
"Acknowledgments",
"This work received the generous support of the National Natural Science Foundation of China, project Nos.",
"11763007, 11863005, 11803026 and 11503008.",
"We would also like to express our gratitude to the Tianshan Youth Project of Xinjiang No.2017Q014."
],
[
"Input Parameters and Yields of Nova Models",
"Tables 1—4 show the input parameters for all models and results including the envelope's mass ($M_{\\rm en}$ ) before TNR occurs, the mass ejected ($M_{\\rm ej}$ ) during nova eruption, and the maximum temperature ($T_{\\rm max}$ ) of TNR during eruption.",
"The chemical abundances (mass fraction) of isotopes from $^1$ H to $^{40}$ Ca are given.",
"Table: Input parameters for the models with 0.6M ⊙ 0.6 \\rm M_\\odot CO WD and results.",
"ΔM en \\Delta M_{\\rm en} and ΔM ej \\Delta M_{\\rm ej} are the envelope's massbefore TNR occurs and the mass ejected during nova eruptions in unit of 10 -5 M ⊙ 10^{-5}\\rm M_\\odot .",
"T max T_{\\rm max} isthe maximum temperature (in unit of 10 8 K10^8 \\rm K) of TNR during nova eruption .",
"i X^{\\rm i}X represents the yields (mass fraction)of isotope XX in the nova ejecta.Table: Similar with Table but for models with 1.0 M ⊙ \\rm M_\\odot CO WD.Table: Similar with Table but for models with 1.2 M ⊙ \\rm M_\\odot ONe WD£¬and ΔM en \\Delta M_{\\rm en} and ΔM ej \\Delta M_{\\rm ej} are in unit of 10 -6 M ⊙ 10^{-6}\\rm M_\\odot .Table: Similar with Table but for models with 1.3 M ⊙ \\rm M_\\odot ONe WD£¬and ΔM en \\Delta M_{\\rm en} and ΔM ej \\Delta M_{\\rm ej} are in unit of 10 -6 M ⊙ 10^{-6}\\rm M_\\odot ."
]
] | 1906.04369 |
[
[
"Some Features of Semiclassical Chiral Transport in Rotating Frames"
],
[
"Abstract Semiclassical chiral kinetic theories in the presence of electromagnetic fields as well as vorticity can be constructed by means of some different relativistic or nonrelativistic approaches.",
"To cover the noninertial features of rotating frames one can start from the modified quantum kinetic equation of Wigner function in Minkowski spacetime.",
"It provides a relativistic chiral transport equation whose nonrelativistic limit yields a consistent three-dimensional kinetic theory which does not depend explicitly on spatial coordinates.",
"Recently a chiral transport equation in curved spacetime has been proposed and its nonrelativistic limit in rotating coordinates was considered in the absence of electromagnetic fields.",
"We show that the modified theory can be extended to curved spacetime.",
"The related particle current density and chiral transport equation for an inertial observer in the rotating frame are derived.",
"A novel three-dimensional chiral kinetic transport equation is established by inspecting the nonrelativistic limit of the curved spacetime approach in the rotating frame for a comoving observer in the presence of electromagnetic fields.",
"It explicitly depends on spatial coordinates.",
"We prove that it is consistent with the chiral anomaly, chiral magnetic and vortical effects."
],
[
"Introduction",
"In the presence of external electromagnetic fields the charged, massless Dirac particles which can be right- or left-handed, exhibit unusual features like the chiral magnetic effect [1], [2], [3] and the chiral separation effect [4], [5].",
"These manifest themselves in heavy-ion collisions [1], [2].",
"There are also some similar phenomena due to rotation of coordinate frame which affect dynamical behavior of chiral particles: The chiral vortical effect [6], [7], [8], [9], [10] and the local polarization effect [11], [12], [13].",
"Dynamical features of chiral particles have been investigated mostly within the semiclassical kinetic theories which offer an intuitive understanding of their collective phenomena.",
"Kinetic theories can be studied either in a manifestly covariant way in Minkowski spacetime or within the nonrelativistic approach where the physical content is apparent.",
"Three-dimensional (3D) semiclassical chiral transport theories based on [14], [15], are usually inspected by making allowance for only the external electromagnetic fields or the rotation of coordinate frame.",
"The first 3D semiclassical kinetic theory of chiral particles which takes into account the external electromagnetic fields as well as the rotation of the coordinate frame and consistent with the chiral anomaly was constructed in [16].",
"It generates the chiral anomalous effects correctly as it should be.",
"A relativistic chiral kinetic theory (CKT) can be defined [17], [18], [19] starting from the quantum kinetic equation (QKE) obeyed by the Wigner function [20], [21].",
"However, it does not take into account all noninertial effects like the Coriolis force.",
"To overcome this shortcoming QKE was modified by means of some frame dependent terms in [22] .",
"To have better insights into the transport properties of particles, it is convenient to consider the 3D CKT generated by the relativistic chiral transport equation (CTE).",
"However, this limit is challenging, the nonrelativistic CKT should be consistent with chiral anomaly and anomalous currents.",
"The modified approach [22] gives rise to a consistent 3D CKT which does not explicitly depend on the spatial coordinates $\\mathbf {x},$ in the presence of both electrodynamic fields and vorticity.",
"It is preferable to work with a CTE which does not explicitly depend on the local positions of particles but global features like the angular velocity $\\mathbf {\\omega }$ of the rotating frame.",
"Nonetheless, explicit rotation velocity dependence is pertinent to 3D transport equations.",
"In fact, the 3D CTE established in [16] depends explicitly on $\\mathbf {u} =\\mathbf {x} \\times \\mathbf {\\omega }.$ This dependence is crucial in obtaining the continuity relation: If the rotation velocity dependent terms are suspended, there will appear some undesired terms in the Liouville equation.",
"Relativistic chiral kinetic theories which we deal with are the ones provided by the quantum kinetic equation without referring explicitly to the equilibrium distribution functions.",
"In the approach of [23], [24] the solution of quantum kinetic equation derived in [13] was employed to define a CKT.",
"Yet another methods of furnishing chiral transport equations were presented in [25], [26].",
"Recently CKT was studied in curved spacetime where the 3D limit in rotating coordinates has been exposed only for vanishing electromagnetic fields [27].",
"Considering QKE in curved spacetime [28] effectively means to modify it in a frame dependent manner.",
"In fact the Coriolis force has been acquired by choosing the metric adequate to deal with rotating coordinates [27].",
"The nonrelativistic limit in the absence of electromagnetic fields was discussed for two different choices of four-velocities corresponding to inertial and rotating observers.",
"For the latter choice the CTE which they obtained is the same with the one derived in [16] at $\\mathbf {x}\\approx 0.$ We would like to understand how the formulations established in [16] and [22] are related to the curved space formulation of chiral particles in rotating coordinates.",
"We will first show that the modified QKE approach can be extended to curved spacetime.",
"We study it in rotating coordinates by choosing adequately the metric tensor.",
"The modification is relevant only for an inertial observer as far as the centrifugal force is ignored.",
"We calculate the particle current density for an inertial observer in terms of the equilibrium distribution function resulting from two different choices of fluid four-velocities.",
"Then the CKE in the absence of electromagnetic fields is obtained.",
"The phase space measure and first time derivatives of phase space variables which we acquire, differ from the ones presented in [27].",
"In fact, CKT which we acquire is partially in accord with the one established in [16].",
"We then study the 3D limit of four-dimensional CTE for a rotating observer in the presence of electromagnetic fields.",
"The CKT which follows possesses $\\mathbf {x}$ dependent terms and resembles the one established in [16].",
"We show that it is consistent with the chiral anomaly, the chiral vortical and magnetic effects.",
"Hence it constitutes a new 3D CTE.",
"In Sec.",
"II we briefly review the QKE and the existing modifications which take into account noninertial properties of rotating coordinates.",
"The modified kinetic equation in curved space is presented in Sec.",
"III.",
"The particle current density and the resulting 3D CKT are obtained for an inertial observer.",
"Sec.",
"IV is devoted to CKT for a rotating observer in the presence of external electromagnetic fields.",
"We obtain a novel 3D CKT which gives rise to the continuity equation and anomalous chiral effects correctly.",
"In the last section the results acquired and future directions are discussed."
],
[
"Wigner Function Formalisms",
"Chiral theories which we consider are based on the quantum kinetic equation $\\gamma ^\\mu \\left(p_\\mu + \\frac{i\\hbar }{2} \\mathfrak {D}^{{\\scriptscriptstyle {(I)}}}_\\mu \\right) W(x,p) = 0.$ $(x_\\mu ,p_\\mu )$ are the eight-dimensional phase space variables.",
"The superscript $I=(O,M,C),$ standing for original, modified and curved, indicates the different choices of derivative terms.",
"As it will explicitly be given shortly, their differences lie in how they treat the inertial properties of coordinate frame.",
"$W(x,p)$ is the Wigner function for spin-$1/2$ fermions which can be decomposed in terms of the Clifford algebra generators whose coefficients are the scalar, pseudoscalar, vector, axial-vector and tensor fields.",
"We are interested in the chiral vector fields given by the vector and axial-vector field components ${\\cal V}_\\mu $ and ${\\cal A}_\\mu $ as ${\\cal J}^\\mu _{\\scriptscriptstyle { \\chi }}= \\frac{1}{2} ({\\cal V}^\\mu + \\chi {\\cal A}^\\mu ).",
"\\nonumber $ $\\chi =\\pm 1,$ indicates the right- and left-handed fermions.",
"We take into consideration only the right-handed chiral vector field to expose our results, $ {\\cal J}^\\mu \\equiv {\\cal J}_1^\\mu .$ The equations which they obey decouple from the other components, leading to $p_\\mu {\\cal J}^\\mu & = & 0, \\\\\\hbar \\epsilon ^{\\mu \\nu \\alpha \\rho } \\mathfrak {D}^{{\\scriptscriptstyle {(I)}}}_\\alpha {\\cal J}_\\rho &=& - 2 (p^\\mu {\\cal J}^{ \\nu } - p^\\nu {\\cal J}^{\\mu }) ,\\\\\\mathfrak {D}^{{\\scriptscriptstyle {(I)}}}_\\mu {\\cal J}^{ \\mu }& = & 0.$ The original QKE [20], [21] was derived starting from the Dirac equation in Minkowski spacetime yielding the derivative terms $\\mathfrak {D}^{{\\scriptscriptstyle {(O)}}}_\\mu \\equiv \\nabla _\\mu =\\partial _\\mu - F_{\\mu \\nu } \\partial ^\\nu _p .",
"$ The derivatives with respect to the phase space variables denoted $ \\partial _\\mu \\equiv \\partial / \\partial x^\\mu ,\\ \\partial ^\\mu _p \\equiv \\partial / \\partial p_\\mu .$ We set $c=k=1$ as well as $Q =1 $ which is the electric charge of chiral particle coupled to the external electromagnetic fields described by $F_{\\mu \\nu }.$ For the sake of simplicity we deal with the electromagnetic field strength satisfying $\\partial _\\rho F_{\\mu \\nu }=0.$ In spite of the fact that the original QKE is covariant, it does not explicitly depend on the inertial properties of reference frame.",
"The fluid vorticity or equivalently the angular velocity of the frame appears in its solution [13].",
"Therefore, the CTE designated by (REF ) does not lead to the Coriolis force.",
"To surmount this disadvantage in [22] the derivative terms of the original QKE is modified by means of the four-velocity $n^\\mu $ of the frame satisfying $n_\\mu n^\\mu =1,$ as $\\mathfrak {D}^{{\\scriptscriptstyle {(M)}}}_\\mu \\equiv \\tilde{\\nabla }_\\mu ^{\\scriptscriptstyle {(n)}}= \\nabla _\\mu +\\left[ \\partial _\\nu n^\\alpha p_\\alpha n_\\mu -\\partial _\\mu n^\\alpha p_\\alpha n_\\nu \\right] \\partial ^\\nu _p .",
"$ We introduced the modification guided by the circulation tensor which provides the noninertial properties of fluids.",
"This approach revealed to be essential in obtaining a formulation of 3D CTE which is not explicitly dependent of the spatial coordinates $\\mathbf {x}.$ In fact, it is the unique $\\mathbf {x}$ independent 3D CKT consistent with the chiral anomaly when both the external electromagnetic fields and fluid vorticity are present.",
"Hence, as far as these properties are considered the modification terms seem to be unique.",
"This formalism possesses some similarities with the effective field theory approach [8] but it is not possible to introduce a gauge field which generates the proposed modification of QKE.",
"On the other hand frame dependent terms can also be incorporated in the original QKE by extending it to curved spacetime.",
"This has been achieved by means of “horizontal lift of the derivative operator in the cotangent bundle\" [28], [27] yielding $\\mathfrak {D}^{{\\scriptscriptstyle {(C)}}}_\\mu \\equiv \\Delta _{\\mu }= \\partial _\\mu +\\left[\\Gamma ^\\lambda _{\\mu \\nu }p_\\lambda - F_{\\mu \\nu } \\right] \\partial ^\\nu _p .",
"$ $\\Gamma ^\\lambda _{\\mu \\nu }$ denote the Christoffel symbols.",
"We suppressed spin connection because it does not show up in (REF )-().",
"We are concerned with the semiclassical approximation where the Wigner function is expanded in Planck constant and the terms up to the first order are retained.",
"In course of obtaining CKT one first solves (REF ) and ().",
"For $\\nabla _\\mu ,$ they were solved in [17], [18], [19].",
"As it has been demonstrated in [27], this solution can be extended to the curved spacetime by substituting $\\nabla _\\mu $ with $\\Delta _\\mu ,$ giving ${\\cal J}^{\\mu } = p^\\mu f \\delta (p^2) + \\hbar \\tilde{F}^{\\mu \\nu } p_\\nu f \\delta ^\\prime (p^2) + \\hbar S_{\\scriptscriptstyle {(n)}}^{\\mu \\nu } ({\\Delta }_{\\nu } f) \\delta (p^2) ,$ where $ \\delta ^\\prime (p^2) = d\\delta (p^2)/dp^2.$ One also expands the distribution function in $\\hbar $ and retain the first order: $f \\equiv f^{0} +\\hbar f^{1} .",
"$ The spin tensor $S^{\\mu \\nu }_{\\scriptscriptstyle {(n)}}=\\frac{\\epsilon ^{\\mu \\nu \\rho \\sigma } p_\\rho n_\\sigma }{ 2 n \\cdot p} ,$ and the dual electromagnetic field strength $\\tilde{F}^{\\mu \\nu } =\\epsilon ^{\\mu \\nu \\alpha \\rho }F_{\\alpha \\rho }/2,$ are introduced.",
"By making use of (REF ) in the remaining equation () one attains the relativistic CKT in curved spacetime: $\\delta \\Big ( p^2 + \\frac{ \\hbar n_\\alpha \\tilde{F}^{\\alpha \\beta } p_\\beta }{n \\cdot p} \\Big )\\Bigg [ p \\cdot \\Delta + \\frac{ \\hbar n_\\mu \\tilde{F}^{\\mu \\nu } \\Delta _\\nu }{n \\cdot p} +\\hbar \\Delta _\\mu S_{\\scriptscriptstyle {(n)}}^{\\mu \\nu }\\Delta _\\nu \\Big ]f=0.",
"$ We would like to examine the behavior of massless Dirac particles in rotating coordinates.",
"Metric components of the coordinate frame which rotates with the constant angular velocity $\\mathbf {\\omega }$ were given in [29], [30] as $g_{00}=1-\\mathbf {u}^2,\\ \\ g_{0i}=u^i,\\ \\ g_{ij}=-\\delta _{ij}.$ We deal with nonrelativistic rotations, thus we let the rotation velocity $u^i= \\epsilon ^{ijk}x^j\\omega ^k$ be much smaller than the speed of light: $|\\mathbf {u}| \\ll 1.$ $\\epsilon ^{ijk}$ is totally antisymmetric and $\\epsilon ^{123}=1.$ The related Christoffel symbols can be seen to be $\\Gamma ^{j}_{0i}=\\Gamma ^{j}_{i0}=\\epsilon ^{ijk} \\omega ^k, \\ \\ \\Gamma ^i_{00}= \\epsilon ^{ijk} u^j \\omega ^k,\\ \\ \\Gamma ^0_{\\mu \\nu }= \\Gamma ^i_{jk} =0.$ It yields a vanishing Riemann tensor.",
"Following [27] we consider two different observers: The inertial observer with the four-velocity $u_{\\mu }= (1,0,0,0),\\ \\ u^\\mu =(1,u^i),$ and the rotating observer with the four-velocity $v^\\mu =\\frac{1}{\\sqrt{g_{00}}} \\delta ^\\mu _0,\\ \\ v_{ \\mu } =(\\sqrt{g_{00}}, \\frac{1}{\\sqrt{g_{00} }} u^i ).$ The 3D CKT established in [16] embraces the Coriolis force as well as the centrifugal force.",
"However, when one studies the 3D theories generated by the 4D CKTs the centrifugal terms are ignored.",
"Also here we do not take into account those effects which means that $O({\\mathbf {u}}^2)$ terms and their derivatives with respect to spatial coordinates assumed to be vanishing.",
"In accord with this assumption we adopt the approximation $O(u^iu^j) \\approx 0,\\ O(u^i\\omega ^j) \\approx 0.",
"$ It is worth mentioning that this approximation is consistent with the fact that $|\\mathbf {u}| \\ll 1.$"
],
[
"Modified Kinetic Equation in Curved spacetime",
"The modified theory can be extended to the curved spacetime by substituting $\\mathfrak {D}^{{\\scriptscriptstyle {(I)}}}_\\mu $ in (REF ) with ${\\Delta }_{{\\scriptscriptstyle {(n)}}\\mu }= \\partial _\\mu +\\left[\\Gamma ^\\lambda _{\\mu \\nu }p_\\lambda - F_{\\mu \\nu } + \\partial _\\mu n^\\alpha p_\\alpha n_\\nu - \\partial _\\nu n^\\alpha p_\\alpha n_\\mu \\right] \\partial ^\\nu _p .$ The signs of modification terms are altered with respect to (REF ) as it is convenient for considering $p_\\mu $ as the momentum variables.",
"The semiclassical solution of (REF ) and () can be attained as ${\\cal J}_{\\scriptscriptstyle {(n)}}^{\\mu } &=& \\left(1-\\frac{\\hbar }{n \\cdot p} S_{\\scriptscriptstyle {(n)}}^{\\sigma \\rho } g_{\\rho \\nu }( {\\partial }_{ \\sigma }n^\\nu ) \\right) p^\\mu f \\delta (p^2) \\nonumber \\\\&&+ \\hbar \\tilde{F}^{\\mu \\nu } p_\\nu f \\delta ^\\prime (p^2) - \\hbar \\epsilon ^{\\mu \\nu \\alpha \\rho } p_\\nu (\\partial _\\alpha n^{\\beta } ) p_\\beta n_\\rho f \\delta ^\\prime (p^2) \\nonumber \\\\&&+ \\hbar S^{\\mu \\nu }_{\\scriptscriptstyle {(n)}}({\\Delta }_{{\\scriptscriptstyle {(n)}}\\nu } f^{0}) \\delta (p^2).$ We deal with the semiclassical approximation, so that in (REF ) the first order part of the distribution function $f^1,$ is arbitrary.",
"In fact, we used this freedom to write the second term in accord with the Minkowski spacetime formulation [22].",
"It is worth noting that to determine the distribution function completely one should solve the remaining equation ().",
"We first would like to discuss 3D currents resulting from (REF ) in rotating coordinates by choosing $n_\\mu $ appropriately.",
"Because of ignoring the centrifugal effects we deal with $\\mathbf {u}$ satisfying (REF ).",
"Then, the unique possibility is to choose $n_\\mu =u_\\mu ,$ because for $n_\\mu =v_\\mu $ the modification terms in (REF ) vanish.",
"The zeroth component of (REF ) yields particle number density and the spatial components are used to define the chiral particle current density as $j^i_{{\\scriptscriptstyle {(u)}}} =\\int \\frac{d^4p}{4\\pi ^3 \\hbar ^{3}} {\\cal J}^{i}_{{\\scriptscriptstyle {(u)}}} .$ We would like to emphasize the fact that terms which are at most linear in $\\hbar $ are kept, obviously up to the $\\hbar ^{-3}$ factor which is due to the definition of momentum space volume.",
"Its partial integration reads $j_{{\\scriptscriptstyle {(u)}}}^{\\mu } &=& \\int \\frac{d^4p}{4\\pi ^3 \\hbar ^{3}} \\Big [ p^\\mu f - \\hbar \\partial _p^0\\left(\\tilde{F}^{\\mu \\nu } p_\\nu f/2p_0 \\right) \\nonumber \\\\&&+\\hbar \\epsilon ^{\\mu \\nu \\alpha \\rho } \\partial _p^0\\left(p_\\nu (\\partial _\\alpha u^{\\beta } ) p_\\beta u_\\rho f/2p_0 \\right) \\nonumber \\\\&&-\\frac{\\hbar }{u \\cdot p} p^\\mu S_{\\scriptscriptstyle {(u)}}^{\\lambda \\rho } g_{\\rho \\nu }\\left( {\\partial }_{ \\lambda }u^\\nu \\right) f^0+ \\hbar S^{\\mu \\nu }_{\\scriptscriptstyle {(u)}}({\\Delta }_{{\\scriptscriptstyle {(u)}}\\nu } f^{0}) \\Big ] \\delta (p^2).",
"\\nonumber $ We adopt the equilibrium distribution function for rotating coordinates as it is given in [31], [27]: $f_{eq}(x,p)= \\left[1 +e^{ \\left(p\\cdot U-\\mu +\\frac{\\hbar }{2} S^{\\mu \\nu }_{\\scriptscriptstyle {(n)}}\\partial _\\mu U_\\nu \\right)/T}\\right]^{-1}.",
"$ $U_\\mu $ is the four-velocity of fluid and $n_\\mu =u_\\mu .$ One can first perform the $p_0$ integral by solving the mass-shell condition $p^2=0$ as $p_0=\\pm |\\mathbf {p}|-p_iu^i ,$ where $|\\mathbf {p}|=\\sqrt{p_ip_i}.$ Then we write $\\delta (p^2) =\\frac{\\theta (p_0) \\delta (p_0- |\\mathbf {p}|+p_iu^i)}{ 2|\\mathbf {p}|} +\\frac{\\theta (-p_0) \\delta (p_0+ |\\mathbf {p}|+p_iu^i)}{ 2|\\mathbf {p}|} .$ We only display the positive part explicitly.",
"After integrating over $p_0,$ in the vicinity of $\\mathbf {x} \\approx 0$ the current density can be written as $j^i_{\\scriptscriptstyle {(u)}}&= &\\int \\frac{d^3p}{(2\\pi \\hbar )^3}\\Big [-p_i +\\frac{\\hbar }{2 |\\mathbf {p}|} \\omega ^i -\\frac{3\\hbar }{2|\\mathbf {p}|^3}\\omega ^j p_j p_i\\nonumber \\\\& &+\\frac{\\hbar }{2|\\mathbf {p}|}\\epsilon ^{ijk}F_{jk}\\frac{\\partial }{\\partial |\\mathbf {p}|}+ \\frac{\\hbar }{2|\\mathbf {p}|^3}\\epsilon ^{ijk}p_jF_{0k}\\nonumber \\\\& &+ \\frac{\\hbar }{2 |\\mathbf {p}|^2}\\epsilon ^{ijk} p_j \\frac{\\partial }{\\partial x_k}\\Big ] f_{eq}|_{p_0=|\\mathbf {p}|-p_iu^i } .$ Observe that $p_i/2 |\\mathbf {p}|^3$ is the Berry curvature.",
"There are two possibilities of choosing the fluid four-velocity in the equilibrium distribution (REF ).",
"For $U_\\mu =u_\\mu ,$ contributions arising from the second and third terms in (REF ) cancel each other, so that the chiral vortical effect is not generated.",
"(REF ) only leads to the magnetic current $\\mathbf {j}_{\\scriptscriptstyle {(u)}}^{\\scriptscriptstyle { B}}=\\frac{\\mu }{4\\pi ^2 \\hbar ^2} \\mathbf {B},$ where $\\frac{1}{2}\\epsilon ^{ijk}F_{jk}=-B^i.$ It provides the chiral magnetic effect.",
"The other possibility is to set $U_\\mu =v_\\mu ,$ which yields $\\mathbf {j}_{\\scriptscriptstyle {(u)}}=\\mathbf {j}_{\\scriptscriptstyle {(u)}}^{\\scriptscriptstyle { B}}+ \\mathbf {j}_{\\scriptscriptstyle {(u)}}^\\omega ,$ where $\\mathbf {j}_{\\scriptscriptstyle {(u)}}^\\omega =\\left(\\frac{\\mu ^2}{2}+\\frac{T^2\\pi ^2}{6}\\right)\\frac{\\mathbf {\\omega }}{2\\pi ^2\\hbar ^2},$ describes chiral vortical effect correctly.",
"Obviously, to work out the momentum integrals both particle and antiparticle contributions should be taken into account.",
"Although angular velocity dependence of the chiral current obtained in [27] differs from (REF ), the results regarding chiral vortical effect are consistent.",
"Let us switch off the external electromagnetic fields and plug (REF ) into the remaining Eq.",
"().",
"For the sake of simplicity we set terms which are second order in angular velocity into zero and work in the vicinity of $\\mathbf {x} \\approx 0.$ After integrating over $p_0$ we obtain $\\left(\\sqrt{\\kappa }\\frac{\\partial }{\\partial t} +\\sqrt{\\kappa }\\dot{x}^i\\frac{\\partial }{\\partial x^i}+ \\sqrt{\\kappa } \\dot{p}_i \\frac{\\partial }{\\partial p_i} \\right)f(t,\\mathbf {x},\\mathbf {p})=0,$ where $f(t,\\mathbf {x},\\mathbf {p})\\equiv f_{eq}(x,p_i,p_0=|\\mathbf {p}|-p_iu^i).$ The phase space measure and the first time derivatives of phase space variables are $&\\sqrt{\\kappa } = 1&+\\frac{p_i\\omega ^i}{|\\mathbf {p}|^2}, \\\\&\\sqrt{\\kappa } \\dot{x}^i = &- \\mathrm {v}^i- \\frac{\\hbar \\hat{ p}_j\\omega ^j \\hat{ p}_i }{|\\mathbf {p}|} , \\\\&\\sqrt{\\kappa } \\dot{p}_i = & 2|\\mathbf {p}|\\epsilon ^{ijk}\\mathrm {v}^j\\omega ^k .",
"$ We introduced the “canonical velocity\" $\\mathrm {v}^i=\\hat{ p}_i -\\frac{\\hbar \\omega ^i}{2|\\mathbf {p}|} +\\frac{\\hbar \\hat{ p}_j\\omega ^j \\hat{ p}_i }{2|\\mathbf {p}|} .$ (REF ) and () are in accord with the CKT derived in [16].",
"Let us compare (REF )-() with the CKT obtained for inertial fluid in [27] where phase space measure is 1 and $\\ \\dot{x}^i =-\\hat{ p}_i +u^i ,\\ \\dot{p}_i =\\epsilon ^{ijk} p_j\\omega ^k .$ First of all, some versions of CKT are related by phase space coordinate transformations [32], seemingly this is not the case for the formalisms which we compare.",
"In the latter approach phase space measure and $\\dot{{\\mathbf {x}}}$ at $\\mathbf {x} \\approx 0, $ do not possess angular velocity dependence and in contrary to () Coriolis force with the factor of 2 is generated if one does not suppress the $\\mathbf {x}$ dependence.",
"In fact, the latter formalism is classical but the one derived here possesses quantum corrections.",
"This fact will be important especially when one considers collisions.",
"Moreover, in Minkowski spacetime the modification of QKE is necessary to obtain a Coriolis like term and a satisfactory $\\mathbf {x}$ dependent 3D CKT in the presence of electromagnetic fields."
],
[
"3D CKT in the Presence of Electromagnetic Fields",
"In this section we let the particle 4-velocity be $U_\\mu =v_\\mu $ and consider the 4D CKT in the comoving frame $n_\\mu =v_\\mu ,$ provided by (REF ) as $\\delta ( p^2 + \\frac{\\hbar v_\\alpha \\tilde{F}^{\\alpha \\beta } p_\\beta }{v \\cdot p})\\lbrace p \\cdot \\Delta + \\frac{ \\hbar v_\\mu \\tilde{F}^{\\mu \\nu } \\Delta _\\nu }{v \\cdot p} +\\hbar \\Delta _\\mu S^{\\mu \\nu }_{\\scriptscriptstyle {(v)}}\\Delta _\\nu \\rbrace f=0.",
"$ In [27] this transport equation was studied in the absence of electromagnetic fields.",
"They showed that working with $p^\\mu =g^{\\mu \\nu }p_\\nu ,$ suits well in obtaining the 3D CKT which preserves the symmetry between magnetic field and angular velocity.",
"For this choice of momentum variables (REF ) is expressed as $\\Delta _{\\mu }= \\partial _\\mu -\\left[\\Gamma ^\\nu _{\\mu \\lambda }p^\\lambda + F_{\\mu \\lambda } g^{\\lambda \\nu } \\right] \\partial _{p\\,\\nu }.$ We would like to study (REF ) in the presence of the external electric and magnetic fields defined by $F_{0i}=(\\mathbf {E} -\\mathbf {u}\\times \\mathbf {B})^i$ and $\\frac{1}{2}\\epsilon ^{ijk}F_{jk}=-B^i.$ Note that in a frame rotating with the angular velocity $\\mathbf {\\omega },$ Maxwell equations are given in terms of $\\mathbf {B}$ and $\\mathbf {E}^\\prime =\\mathbf {E} -\\mathbf {u}\\times \\mathbf {B}.$ To derive the 3D CKT we would like to integrate (REF ) over $p^0.$ To this aim we need to solve $p^2 + \\hbar \\frac{v_\\mu \\tilde{F}^{{\\scriptscriptstyle {\\mu \\nu }}} p_\\nu }{v \\cdot p}=0,$ for $p^0$ in terms of $p^i.$ By adopting the notation of [27] let us introduce $\\mathbf {q} =(p^1,p^2,p^3).$ The mass-shell condition can be solved as $p^0=\\pm \\frac{|\\mathbf {q}|}{\\sqrt{g_{00}}}\\sqrt{1+\\frac{(\\mathbf {q}\\cdot \\mathbf {u})^2}{\\mathbf {q}^2}+\\hbar \\frac{F_{\\mu \\nu } S^{\\mu \\nu }_{\\scriptscriptstyle {(v)}}}{\\mathbf {q}^2}} - \\frac{\\mathbf {q}\\cdot \\mathbf {u}}{g_{00}} .$ Centrifugal terms are not taken into account, hence the rotation velocity satisfies (REF ).",
"Moreover, we deal with weak external fields, so that we set $O(E^2) \\approx 0,$ and let the angular velocity and the magnetic field be in the same direction: $\\mathbf {\\omega }\\times \\mathbf {B}=0.$ Therefore, in the semiclassical approximation, (REF ) yields $p^0=\\epsilon _q^+=|\\mathbf {q}| - \\mathbf {q}\\cdot \\mathbf {u} -\\hbar |\\mathbf {q}|\\mathbf {b} \\cdot \\mathbf {B}^\\prime ,$ where $\\mathbf {b}=\\mathbf {q}/2|\\mathbf {q}|^3$ is the Berry curvature and $\\mathbf {B}^\\prime =\\mathbf {B}-\\mathbf {E}\\times \\mathbf {u} $ is the external magnetic field observed in the coordinate frame moving with the velocity $\\mathbf {\\omega }\\times \\mathbf {x},$ given by the Lorentz transformation with the Lorentz factor $\\gamma =1/\\sqrt{1-\\mathbf {u}^2} \\approx 1.$ Observe that this is the mass-shell condition which we should employ in performing $p^0$ integrals.",
"As we will see the effective dispersion relation arising in the 3D CKT is independent of the electric field.",
"We only exhibit the particle solution, although antiparticles are essential to perform momentum integrals.",
"The positive part of the delta function becomes $\\delta ^+ (p^2 + \\hbar \\frac{v_\\mu \\tilde{F}^{{\\scriptscriptstyle {\\mu \\nu }}} p_\\nu }{v \\cdot p}) =\\frac{ \\delta (p^0- \\epsilon _q^+)}{ 2|\\mathbf {q}|}\\left[1- 2\\hbar \\mathbf {b} \\cdot \\left( \\mathbf {E}\\times \\mathbf {u} -\\mathbf {B} \\right)\\right] .$ Now, by integrating (REF ) over $p^0,$ one acquires $\\left( \\sqrt{\\eta } \\frac{\\partial }{\\partial t} + \\sqrt{\\eta } \\dot{{\\mathbf {x}}} \\cdot \\frac{\\partial }{\\partial \\mathbf {x}} + \\sqrt{\\eta } \\dot{\\mathbf {q}} \\cdot \\frac{\\partial }{\\partial \\mathbf {q}}+I_0 \\frac{\\partial }{\\partial p^0}\\right) f|_{p^0=\\epsilon _q^+}=0.",
"$ The phase space measure and the first time derivatives of phase space variables are $&\\sqrt{\\eta } =1+ & \\ \\hbar \\mathbf {b}\\cdot (\\mathbf {B} + 2|\\mathbf {q}|\\mathbf {\\omega } )- \\mathbf {\\nu }\\cdot \\mathbf {u} , \\\\&\\sqrt{\\eta } \\cdot \\dot{{\\mathbf {x}}}=&{\\mathbf {\\nu }} + \\mathbf {E}^\\prime \\times \\mathbf {b}+ \\hat{\\mathbf {q}} \\cdot \\mathbf {b}(\\mathbf {B}+ 2 |\\mathbf {q}| \\mathbf {\\omega }) \\nonumber \\\\&&+ 2 \\hat{\\mathbf {q}} \\ \\mathbf {b} \\cdot [ \\mathbf {u} \\times \\mathbf {E}], \\\\&\\sqrt{\\eta } \\cdot \\dot{{\\mathbf {q}}}=& \\mathbf {E}^\\prime + {\\mathbf {\\nu }} \\times (\\mathbf {B} + 2{\\cal E} \\mathbf {\\omega }) + \\hbar \\mathbf {E} \\cdot (\\mathbf {B} + |\\mathbf {q}| \\mathbf {\\omega }) \\mathbf {b} \\nonumber \\\\&&+\\hbar |\\mathbf {q}| \\mathbf {b} \\cdot \\mathbf {\\omega }\\mathbf {E}-[ \\mathbf {u} \\times \\mathbf {E}] \\times {\\mathbf {\\nu }} \\nonumber \\\\&& +\\hbar \\hat{\\mathbf {q}}\\cdot \\mathbf {b} \\mathbf {E}\\cdot \\mathbf {B}\\mathbf {u}+ 2 \\mathbf {E}\\cdot (\\hat{\\mathbf {q}} \\times \\mathbf {u}) \\mathbf {b} \\times \\mathbf {B}.",
"$ We introduced the “canonical velocity\" ${\\mathbf {\\nu }}=\\partial {\\cal E}/\\partial \\mathbf {q}$ with ${\\cal E} =q - \\hbar q^2 ( \\mathbf {b}\\cdot \\mathbf {\\omega }) - \\hbar q ( \\mathbf {b}\\cdot \\mathbf {B}).",
"\\nonumber $ Observe that it is the semiclassical dispersion relation of a right-handed Weyl particle subject to the external electromagnetic fields in rotating coordinates [16].",
"When electromagnetic fields are present the coefficient of $\\partial f/\\partial p^0$ in (REF ) does not vanish.",
"Its integral over $p^0$ turns out to be $&I_0=&\\hat{\\mathbf {q}}\\cdot \\mathbf {E}^\\prime \\left( 1+ 2 \\hbar \\mathbf {b} \\cdot \\mathbf {B}+\\hbar | \\mathbf {q} | \\mathbf {b} \\cdot \\mathbf {\\omega }\\right)+\\hbar \\mathbf {b}\\cdot \\mathbf {q} \\mathbf {E} \\cdot \\mathbf {\\omega }\\nonumber \\\\&&-\\mathbf {E} \\cdot \\mathbf {u} - (1+ 2 \\hbar \\mathbf {b} \\cdot \\mathbf {B}) \\hat{\\mathbf {q}} \\cdot (\\mathbf {B} \\times \\mathbf {u})-\\hbar \\mathbf {b} \\cdot \\mathbf {u} \\mathbf {E} \\cdot \\mathbf {B} .",
"\\nonumber $ However, one can show that it can be expressed as $I_0= \\sqrt{\\eta }\\dot{\\mathbf {q}}\\cdot (\\partial {\\epsilon _q^+} /\\partial \\mathbf {q}).$ Hence, when we integrate (REF ) over $p^0,$ the coefficients of $\\partial f/\\partial p^\\mu ,$ namely the last two terms of (REF ), lead to $& &\\sqrt{\\eta }\\dot{\\mathbf {q}} \\cdot \\left[ \\frac{\\partial f(x, p) }{\\partial \\mathbf {q}}\\right]_{p^0= {\\epsilon _q^+}} +\\sqrt{\\eta }\\dot{\\mathbf {q}}\\cdot \\frac{\\partial {\\epsilon _q^+}}{\\partial \\mathbf {q}} \\left[\\frac{\\partial f( x,q) }{\\partial p^0}\\right]_{p^0= {\\epsilon _q^+}} \\nonumber \\\\& & = \\sqrt{\\eta }\\dot{\\mathbf {q}}\\ \\frac{\\partial f(t,\\mathbf {x}, {\\epsilon _q^+}, \\mathbf {q}) }{\\partial \\mathbf {q}}.",
"\\nonumber $ Therefore, we establish the 3D CTE as $\\left( \\sqrt{\\eta } \\frac{\\partial }{\\partial t} + \\sqrt{\\eta } \\dot{{\\mathbf {x}}} \\cdot \\frac{\\partial }{\\partial \\mathbf {x}} + \\sqrt{\\eta } \\dot{\\mathbf {q}} \\cdot \\frac{\\partial }{\\partial \\mathbf {q}}\\right) f (t,\\mathbf {x},\\mathbf {q})=0, $ where $f (t,\\mathbf {x},\\mathbf {q}) \\equiv f(t,\\mathbf {x}, p^0=\\epsilon _q^+, \\mathbf {q}).$ By making use of (REF )-() and the Maxwell equations in rotating coordinates one can show that the Liouville equation satisfied by the measure is $\\frac{\\partial }{\\partial t}\\sqrt{\\eta } + \\frac{\\partial }{\\partial \\mathbf {x}} \\left(\\sqrt{\\eta } \\dot{{\\mathbf {x}}}\\right)+ \\frac{\\partial }{\\partial \\mathbf {q}}\\left(\\sqrt{\\eta } \\dot{\\mathbf {q}}\\right)= \\left(2\\pi \\delta (\\mathbf {q}) +2 \\hat{\\mathbf {b}}\\cdot \\mathbf {u} \\right) \\mathbf {E}\\cdot \\mathbf {B} .$ In 3D the chiral particle number and current densities are defined as $n & = & \\int [dq] \\sqrt{\\eta } f, \\\\\\mathbf {j} & = & \\int [dq]\\sqrt{\\eta } \\cdot \\dot{\\mathbf {x}} f + \\mathbf {j}_{{\\scriptscriptstyle { M}}} , $ where $[dq] =d^3q/(2\\pi \\hbar )^3$ and $\\mathbf {j}_{{\\scriptscriptstyle { M}}}=\\mathbf {\\nabla }\\times \\int [dq] \\hbar {\\cal E} \\mathbf {b} f ,$ is the magnetization current [14], [33], [34], [17].",
"By making use of (REF ) and (REF ), one can show that the 4-divergence of the 4-current $(n,\\mathbf {j})$ yields the continuity equation with source: $\\frac{\\partial n}{\\partial t} + \\mathbf {\\nabla } \\cdot \\mathbf {j} = \\frac{ \\mathbf {{ E}}\\cdot \\mathbf {B}}{(2\\pi \\hbar )^2} f|_{\\mathbf {q} =0} .$ Note that on the right-hand side only $ f^0$ appears.",
"This continuity equation is consistent with the chiral anomaly.",
"Let us now focus on the currents which are proportional to $\\mathbf {B}$ and $\\mathbf {\\omega },$ by setting $\\mathbf {E}=0.$ For $p^0=\\epsilon _q^+,$ the equilibrium distribution function (REF ) with $n_\\mu =U_\\mu =v_\\mu ,$ becomes $f(t,\\mathbf {x}, \\mathbf {p})=\\frac{1}{e^{({\\cal E}-\\mu )/T}+1}.$ By plugging it into () and employing () one attains the chiral magnetic and vortical effects correctly: $\\mathbf {j} = \\frac{ \\left(3\\mu ^2+T^2\\pi ^2\\right)}{12\\pi ^2\\hbar ^2} \\mathbf {\\omega }+\\frac{\\mu }{4\\pi ^2 \\hbar ^2} \\mathbf {B}.$ To calculate the integrals we have taken into account both particle and antiparticle contributions.",
"We conclude that (REF )-(REF ) describe a consistent 3D CKT."
],
[
"Discussions",
"In Minkowski spacetime the modified chiral QKE generates a consistent 3D CKT which does not explicitly depend on spatial coordinates.",
"In Sec.",
"we studied its curved spacetime formulation by ignoring centrifugal force terms imposing the conditions (REF ).",
"The modification terms survive for the inertial observer: $n_\\mu =u_\\mu .$ We derived the current density at $\\mathbf {x} \\approx 0$ and show that the chiral vortical effect is generated for the particle (fluid) velocity $U_\\mu =v_\\mu .$ For $U_\\mu =u_\\mu $ the current density does not lead to any angular momentum dependent term.",
"This is not surprising because the equilibrium distribution function of rotating fluid in Minkowski spacetime [31] is consistent only with the former choice.",
"We derived the transport equation in the absence of electromagnetic fields as in (REF ).",
"It furnishes the phase space measure and velocities which in part coincide with the ones obtained in [16] at $\\mathbf {x} \\approx 0.$ Hence, we can conclude that the modified QKE in curved spacetime is needed to acquire the phase space measure and the first time derivatives of phase space variables correctly for the inertial observer.",
"In Sec.",
"the novel CTE (REF ) is established for $n_\\mu =v_\\mu .$ It is similar to the CKT obtained directly in 3D [16].",
"The difference is mainly in the explicitly $\\mathbf {x}$ -dependent terms of (REF )-().",
"The 3D CKT of [16] was constructed starting from the scalar and vector fields which can be associated with Coriolis and centrifugal forces experienced by a massive particle.",
"One can examine if a similar approach in 3D exists which can be associated with the new 3D CKT.",
"Another open question is how to incorporate the centrifugal force in the 3D CKT starting with the 4D curved spacetime formulation of chiral particles.",
"It is a hard task, because one should not only keep terms at the order of $\\mathbf {u}^2$ but also, at least initially, the $\\mathbf {u}^3$ terms whose derivatives with respect to spatial coordinates are at the order of $\\mathbf {u}^2.$ This would also clarify if an underlying 3D construction mentioned above exists.",
"We only presented transport equations without collisions.",
"The explicitly $\\mathbf {x}$ -dependent CKT of [16] has been extended to cover collisions by adopting the relaxation time method [35].",
"This allowed us to study nonlinear transport properties of chiral plasma.",
"The novel formalism constructed here can be studied in a similar manner.",
"Collisions can be introduced by means of the relaxation time formalism and the particle current densities provided by them can be calculated.",
"Once this is done one can compare the particle currents generated by these CKTs.",
"This would serve as a testing ground for deciding which CKT suits better with the observable effects.",
"We would like to thank Xu-Guang Huang for the illuminative correspondence on their work.",
"This work is supported by the Scientific and Technological Research Council of Turkey (TÜBİTAK) Grant No.",
"117F328."
]
] | 1906.04504 |
[
[
"$f$-wave superfluidity from repulsive interaction in Rydberg-dressed\n Fermi gas"
],
[
"Abstract Interacting Fermi gas provides an ideal model system to understand unconventional pairing and intertwined orders relevant to a large class of quantum materials.",
"Rydberg-dressed Fermi gas is a recent experimental system where the sign, strength, and range of the interaction can be controlled.",
"The interaction in momentum space has a negative minimum at $q_c$ inversely proportional to the characteristic length-scale in real space, the soft-core radius $r_c$.",
"We show theoretically that single-component (spinless) Rydberg-dressed Fermi gas in two dimensions has a rich phase diagram with novel superfluid and density wave orders due to the interplay of the Fermi momentum $p_F$, interaction range $r_c$, and interaction strength $u_0$.",
"For repulsive bare interactions $u_0>0$, the dominant instability is $f$-wave superfluid for $p_Fr_c\\lesssim 2$, and density wave for $p_Fr_c\\gtrsim 4$.",
"The $f$-wave pairing in this repulsive Fermi gas is reminiscent of the conventional Kohn-Luttinger mechanism, but has a much higher $T_c$.",
"For attractive bare interactions $u_0<0$, the leading instability is $p$-wave pairing.",
"The phase diagram is obtained from functional renormalization group that treats all competing many-body instabilities in the particle-particle and particle-hole channels on equal footing."
],
[
"Introduction",
"Understanding the many-body instabilities and symmetry breaking in strongly interacting fermions in two-dimension (2D) holds the key to several long-standing problems in condensed matter physics.",
"One example is the precise mechanism by which unconventional superconductivity with various pairing symmetries emerges from repulsive interactions, in materials ranging from cuprate [1], ruthenate [2], and pnictide [3] superconductors.",
"These and other correlated quantum materials typically display intertwined vestigial orders, e.g.",
"in the so-called pseudogap region where charge density waves, pairing, and other fluctuations compete.",
"Recently, ultracold Fermi gases [4], [5] of atoms and molecules have become a promising experimental platform to tackle some of these open problems by realizing Hamiltonians such as the Fermi-Hubbard model [6], [7], [8] with tunable interactions [9].",
"This offers opportunity to deepen our understanding of the “pairing glue\" in repulsively interacting systems, and shed light on the complex interplay of quantum fluctuations in distinct channels for simple and highly controlled Hamiltonians.",
"In this paper, we show theoretically that Rydberg-dressed Fermi gas of alkali atoms with tunable long-range interactions gives rise to not only $p$ -wave topological superfluids for attractive bare interactions, but also $f$ -wave superfluid with high transition temperatures stemming from repulsive bare interactions.",
"Rydberg atoms and Rydberg-dressed atoms haven long been recognized for their potential in quantum simulation and quantum information [10], [11], [12], [13], [14].",
"Recent experiments have successfully demonstrated a panoply of two-body interactions in cold gases of Rydberg-dressed alkali atoms [15], [16], [17], [18], [19], [20], [21].",
"In Rydberg dressing, the ground state atom (say $n_0S$ ) is weakly coupled to a Rydberg state (say $nS$ or $nD$ ) with large principal number $n$ by off-resonant light with Rabi frequency $\\Omega $ and detuning $\\Delta $ .",
"The coupling can be achieved for example via a two-photon process involving an intermediate state $n_1 P$ to yield longer coherence times [22].",
"The huge dipole moments of the Rydberg states lead to strong interactions that exceed the natural van der Waals interaction by a factor that scales with powers of $n$ [12], [13].",
"The interaction between two Rydberg-dressed atoms takes the following form [22]: $V(\\mathbf {r}) = \\frac{u_0}{r^6+r_c^6}.$ Here $r=|\\mathbf {r}|$ is the inter-particle distance, $u_0=(\\Omega /2\\Delta )^4C_6$ is the interaction strength, $C_6$ is the van der Waals coefficient, and $r_c=|C_6/2\\hbar \\Delta |^{1/6}$ is the soft-core radius and the characteristic scale for the interaction range.",
"As shown in Fig.",
"REF , $V({\\bf r})$ has a step-like soft-core for $r\\lesssim r_c$ before decaying to a van der Waals tail at long distances.",
"Both $u_0$ and $r_c$ can be tuned experimentally via $\\Omega $ and $\\Delta $ [22].",
"Moreover, by choosing proper Rydberg states (e.g.",
"$nS$ versus $nD$ for $^6$ Li with $n>30$ [23]) $C_6$ and $u_0$ can be made either repulsive or attractive.",
"By choosing proper $n$ , $\\Delta $ and $\\Omega $ , atom loss can be reduced to achieve a sufficiently long life time to observe many-body phenomena [22], [18], [20], [24].",
"Previous theoretical studies have explored the novel many-body phenomena associated with interaction Eq.",
"(REF ) in bosonic [22], [25], [26], [27], [28], [29], [30], [31] and fermionic gases [24] including the prediction of topological superfluids [23] and topological density waves [32].",
"Here we consider single-component Rydberg Fermi gases confined in 2D [33], where mean-field and random phase approximation (RPA) become unreliable due to enhanced quantum fluctuations.",
"Our goal is to set up a theory to systematically describe the competing many-body phases of 2D Rydberg-dressed Fermi gas by treating them on equal footing beyond the weak-coupling regime and RPA.",
"We achieve this by solving the functional renormalization group flow equations for the fermionic interaction vertices.",
"The resulting phase diagram (Fig.",
"REF ) is much richer than the RPA prediction [33] and reveals an unexpected $f$ -wave phase.",
"The paper is organized as follows.",
"In Sec.",
"we introduce many-body phases of Rydberg-dressed Fermi gas within mean-field from the standard Cooper instability analysis and Random Phase Approximation.",
"In Sec.",
"we present the numerical implementation of Functional Renormalization Group to this problem and in Sec.",
"we show many-body phases beyond mean field calculation which manifest intertwined quantum fluctuations in pairing and density-wave channels.",
"In Sec.",
", we summarize our study and implications of our findings for future experimental developments in ultracold gases."
],
[
"Rydberg-dressed Fermi gas",
"We first highlight the unique properties of Rydberg-dressed Fermi gas by comparing it with other well-known Fermi systems with long-range interactions such as the electron gas and dipolar Fermi gas.",
"Correlations in electron liquid are characterized by a single dimensionless parameter $r_s$ , the ratio of Coulomb interaction energy to kinetic energy.",
"In the high density limit $r_s\\ll 1$ , the system is weakly interacting while in the low density limit $r_s\\gg 1$ , Wigner crystal is formed.",
"The intermediate correlated regime with $r_s\\sim 1$ can only be described by various approximations [34].",
"Similarly, dipolar Fermi gas also has a power-law interaction that lacks a scale, so a parameter analogous to $r_s$ can be introduced which varies monotonically with the density [35].",
"The situation is different in Rydberg-dressed Fermi gas with interaction given by Eq.",
"(REF ).",
"From the inter-particle spacing $1/\\sqrt{2\\pi n}$ and the Fermi energy $\\epsilon _F=2\\pi n/m$ (we put $\\hbar =1$ and $k_B=1$ ) in terms of areal density $n$ , one finds that the ratio of interaction energy to kinetic energy scales as $n^2/[1+(2\\pi r_c^2)^3 n^3]$ , which varies non-monotonically with $n$ unlike electron liquid due to $r_c$ (Fig.",
"REF a inset).",
"Distinctive feature of the interaction $V(\\mathbf {r})$ is revealed by its Fourier transform in 2D [33], $V(\\mathbf {q}) = g G\\left({q^6r_c^6}/{6^6}\\right),\\;\\;\\; g=\\pi u_0/3r_c^4,$ where $\\mathbf {q}$ is the momentum, $q=|\\mathbf {q}|$ , $g$ is the coupling strength and $G$ is the Meijer G-function In Mathematica, this Meijer G-function is called with MeijerG[{{},{}},{{0,1/3,2/3,2/3},{0,1/3}},$z^6/6^6$ ] where $z=qr_c$ ..",
"The function $V(\\mathbf {q})$ , plotted in Fig.",
"REF b, develops a negative minimum at $q=q_c\\sim 4.82/r_c$ .",
"This is the momentum space manifestation of the step-like interaction potential Eq.",
"(REF ).",
"These unique behaviors are the main culprits of its rich phase diagram.",
"Figure: Single-component Fermi gas with Rydberg-dressed interactions in 2D.",
"(a) The interaction potential Eq.",
"()shows a step-like soft core of radius r c r_c and a long-range tail.",
"(Inset) Ratio of the interaction to kinetic energy varies non-monotonically with density.",
"(b) The Rydberg-dressedinteraction Eq.",
"() in momentum spaceattains a negative minimum at q c ∼4.82/r c q_c\\sim 4.82/r_c.",
"(c) For attractiveinteractions, the critical temperatures in different angular momentumℓ\\ell channels (in arbitrary units) from the solution of the Cooperproblem.",
"The leading instability is pp-wave, ℓ=±1\\ell =\\pm 1.",
"Maximum T c T_cis around p F r c ≈2p_Fr_c\\approx 2.",
"(d) For repulsive interactions, random phaseapproximation points to a density-wave order.",
"False color (shading) shows theordering wave vector of density modulations.Starting from the free Fermi gas, increasing the interaction $g$ may lead to a diverging susceptibility and drive the Fermi liquid into a symmetry-broken phase.",
"We first give a qualitative discussion of potential ordered phases using standard methods to orient our numerical FRG results later.",
"For attractive interactions, $u_0<0$ , an arbitrarily small $g$ is sufficient to drive the Cooper instability.",
"By decomposing $V(\\mathbf {q}=\\mathbf {p}_F-\\mathbf {p}_F^{\\prime })$ into angular momentum channels, $V(2p_F\\sin \\frac{\\theta }{2})=\\sum _\\ell V_\\ell e^{i\\ell \\theta }$ where $\\theta $ is the angle between $\\mathbf {p}_{F}$ and $\\mathbf {p}_F^{\\prime }$ , one finds different channels decouple and the critical temperature of the $\\ell $ -th channel $T_c(\\ell ) \\sim e^{-1/N_0V_\\ell }$ [37] with $N_0=m/2\\pi $ being the density of states.",
"Thus the leading instability comes from the channel with the largest $V_\\ell $ (hence the largest $T_c$ ).",
"Fig.",
"REF c illustrates $T_c(\\ell )$ as a function of $r_c$ for fixed $p_F$ .",
"It is apparent that the dominant instability is in the $\\ell =\\pm 1$ channel, i.e., $p$ -wave pairing.",
"Its $T_c$ develops a dome structure and reaches maximum around $p_Fr_c\\approx 2$ .",
"For large $r_c$ , higher angular momentum channels start to compete with the $\\ell =\\pm 1$ channel.",
"For repulsive bare interactions, $u_0>0$ , a sufficiently strong interaction $g$ can induce an instability toward the formation of (charge) density waves.",
"This has been shown recently [33] for 2D Rydberg-dressed Fermi gas using random phase approximation (RPA) which sums over a geometric series of “bubble diagrams\" to yield the static dielectric function, $\\epsilon (\\mathbf {q})=1-V(\\mathbf {q})\\chi _0(\\mathbf {q})$ where the Linhard function $\\chi _0(\\mathbf {q})=-N_0[ 1- \\Theta (q-2k_F) \\sqrt{q^2-4k_F^2}/q]$ .",
"The onset of density wave instability is signaled by $\\epsilon (\\mathbf {q})=0$ at some wave vector $q=q_{ins}$ , i.e.",
"the softening of particle-hole excitations.",
"Within RPA, $q_{ins}$ always coincides with $q_c$ , and the resulting phase diagram is shown in Fig.",
"REF d. While these standard considerations capture the $p$ -wave pairing and density wave order, they fail to describe the physics of intertwined scattering between particle-particle and particle-hole channels.",
"We show below that this missing ingredient exhibits significant effects, leading to the emergence of a robust $f$ -wave superfluid in the repulsive regime.",
"For a detailed comparison between RPA and FRG see Ref.",
"[38]."
],
[
"Numerical Implementation of Functional Renormalization Group",
"Functional renormalization group (FRG) is a powerful technique that can accurately predict the many-body instabilities of strongly interacting fermions [39].",
"It implements Wilson's renormalization group for interacting fermions in a formally exact manner by flowing the generating functional of the many-body system $\\Gamma $ as a sliding momentum scale $\\Lambda $ is varied.",
"Starting from the bare interaction $V(\\mathbf {q})$ at a chosen ultraviolet scale $\\Lambda _{UV}$ , higher energy fluctuations are successively integrated out to yield the self-energy $\\Sigma $ and effective interaction vertex $\\Gamma $ at a lower scale $\\Lambda <\\Lambda _{UV}$ .",
"As $\\Lambda $ is lowered toward a very small value $\\Lambda _{IR}$ , divergences in the channel coupling matrices and susceptibilities point to the development of long-range order.",
"Its advantage is that all ordering tendencies are treated unbiasedly with full momentum resolution.",
"The main draw back is its numerical complexity: at each RG step, millions of running couplings have to be retained.",
"FRG has been applied to dipolar Fermi gas [38], [40] and extensively benchmarked against different techniques [41], [42], [43], [41], [44].",
"For more details about the formalism, see reviews [39] and [45].",
"Note that our system is a continuum Fermi gas, not a lattice system extensively studied and reviewed in [39].",
"The central task of FRG is to solve the coupled flow equations for self-energy $\\Sigma _{1^{\\prime },1}$ and two-particle vertex $\\Gamma _{1^{\\prime },2^{\\prime };1,2}$ [39]: $\\partial _\\Lambda \\Sigma _{1^{\\prime },1} &= -\\sum _{2} S_{2} \\Gamma _{1^{\\prime },2;1,2},\\nonumber \\\\\\partial _\\Lambda \\Gamma _{1^{\\prime },2^{\\prime };1,2} &= \\sum _{3,4} \\Pi _{3,4}\\big [\\frac{1}{2} \\Gamma _{1^{\\prime },2^{\\prime };3,4} \\Gamma _{3,4;1,2}-\\Gamma _{1,^{\\prime }4;1,3}\\Gamma _{3,2^{\\prime };4,2}\\nonumber \\\\&+\\Gamma _{2^{\\prime },4;1,3}\\Gamma _{3,1^{\\prime };4,2}\\big ],$ Here the short-hand notation $1\\equiv (\\omega _1,\\mathbf {p}_1)$ , $1,2$ ($1^{\\prime },2^{\\prime }$ ) label the incoming (outgoing) legs of the four-fermion vertex $\\Gamma $ , and the sum stands for integration over frequency and momentum, $\\Sigma \\rightarrow \\int d\\omega d^2\\mathbf {p}/(2\\pi )^3$ .",
"Diagrammatically, the first term in Eq.",
"(REF ) is the BCS diagram in the particle-particle channel, and the second and third terms are known as the ZS and ZS' diagram in the particle-hole channel [46].",
"The polarization bubble $\\Pi _{3,4} = G_{3} S_{4} + S_{3} G_{4}$ contains the product of two scale-dependent Green functions defined by $G_{\\omega ,\\mathbf {p}} =\\frac{\\Theta (|\\xi _\\mathbf {p}|-\\Lambda ) }{i\\omega -\\xi _\\mathbf {p}-\\Sigma _{\\omega ,\\mathbf {p}} } ,\\quad \\quad S_{\\omega ,\\mathbf {p}} =\\frac{\\delta (|\\xi _\\mathbf {p}|-\\Lambda ) }{i\\omega -\\xi _\\mathbf {p}-\\Sigma _{\\omega ,\\mathbf {p}} }.$ Note that $G$ , $S$ , $\\Sigma $ and $\\Gamma $ all depend on the sliding scale $\\Lambda $ , we suppressed their $\\Lambda $ -dependence in equations above for brevity.",
"Figure: Phase diagram of Rydberg-dressed spinless Fermi gas in 2D based onFRG.",
"Tuning the interaction range r c r_c and interaction strength ggyields Fermi Liquid (FL), pp-wave superfluid (p-SF), ff-wave superfluid(f-SF), and density-wave (DW).",
"False color (shading) indicates thecritical scaleΛ c \\Lambda _c of the instability where brighter (darker) regions havehigher (lower) T c T_c.",
"Panels labelled with 𝒫 1 \\mathcal {P}_1,𝒫 2 \\mathcal {P}_2 and 𝒫 3 \\mathcal {P}_3 show the details of renormalizationflow and vertex function for points marked with white diamonds on thephase diagram.",
"The leading eigenvalues for a few channels (see legends)are shown on the left.",
"The maps of vertex functionΓ(𝐩 F1 ' ,𝐩 F2 ' ,𝐩 F1 )\\Gamma (\\mathbf {p}_{F1}^{\\prime },\\mathbf {p}_{F2}^{\\prime },\\mathbf {p}_{F1}) are shown on the right for fixed𝐩 F1 =(-p F ,0)\\mathbf {p}_{F1}=(-p_F,0).",
"Superfluid (density wave) order displays diagonal(horizontal and vertical) correlations.Several well-justified approximations are used to make the flow equations computationally tractable.",
"To identify leading instabilities, the self-energy can be safely dropped, and the frequency dependence of $\\Gamma $ can be neglected [39].",
"As a result, the frequency integral of the fermion loops in Eq.",
"(REF ) can be performed analytically.",
"Furthermore, we retain the most relevant dependence of $\\Gamma $ on $\\mathbf {p}$ by projecting all off-shell momenta radially onto the Fermi surface [39].",
"Then, $\\Gamma $ is reduced to $\\Gamma _{1^{\\prime },2^{\\prime };1,2}\\rightarrow \\Gamma (\\mathbf {p}_{F1}^{\\prime },\\mathbf {p}_{F2}^{\\prime },\\mathbf {p}_{F1})$ where the last momentum variable is dropped because it is fixed by conservation, and the subscript in $\\mathbf {p}_F$ indicates radial projection onto the Fermi surface.",
"The initial condition for $\\Gamma $ at the ultraviolet scale $\\Lambda _{UV}$ is given by the antisymmetrized bare interaction $V(\\mathbf {q})$ , $\\Gamma (\\mathbf {p}_{F1}^{\\prime },\\mathbf {p}_{F2}^{\\prime },\\mathbf {p}_{F1})\\big |_{\\Lambda _{UV}}\\equiv \\frac{1}{2}[V(\\mathbf {p}_{F1}^{\\prime }-\\mathbf {p}_{F1})-V(\\mathbf {p}_{F2}^{\\prime }-\\mathbf {p}_{F1})].$ We solve the flow equation by the Euler method on a logarithmic grid of $\\Lambda $ consisting of $10^3$ RG steps going from $\\Lambda _{UV}=0.99E_F$ down to $\\Lambda _{IR}=10^{-3}E_F$ .",
"Each $\\mathbf {p}_{F}$ is discretized on an angular grid with up to hundreds of patches on the Fermi surface To speed up the calculation, the FRG algorithm is adapted to run parallel on Graphic Processing Units.. We monitor the flow of $\\Gamma (\\mathbf {p}_{F1}^{\\prime },\\mathbf {p}_{F2}^{\\prime },\\mathbf {p}_{F1})$ which contains hundreds of millions of running coupling constants.",
"When the absolute value of a running coupling constant in $\\Gamma $ exceeds a threshold, e.g.",
"$50E_F$ , signaling an imminent divergence, we terminate the flow, record the critical scale $\\Lambda _c$ , and analyze the vertex to diagnose the instability.",
"If the flow continues smoothly down to $\\Lambda _{IR}$ , we conclude the Fermi liquid is stable down to exponentially small temperatures.",
"Scanning the parameter space $(g,r_c)$ gives the phase diagram, whereas $\\Lambda _c$ provides a rough estimate of the $T_c$ of each ordered phase.",
"Two complementary methods are employed to identify the leading instability from the large, complex data set of $\\Gamma $ .",
"First, we plot $\\Gamma (\\mathbf {p}_{F1}^{\\prime },\\mathbf {p}_{F2}^{\\prime },\\mathbf {p}_{F1})$ at $\\Lambda _c$ against the angular directions of $\\mathbf {p}_{F1}^{\\prime }$ and $\\mathbf {p}_{F2}^{\\prime }$ for fixed $\\mathbf {p}_{F1}=(-p_F,0)$ This is done without loss of generality due to the rotational invariance.",
"to reveal the dominant correlations between particles on the Fermi surface.",
"The color map (Fig.",
"REF , lower right columns) shows diagonal structures ($\\mathbf {p}_{F1}^{\\prime }=-\\mathbf {p}_{F2}^{\\prime }$ ) for pairing instability, and horizontal-vertical structures (scattering $\\mathbf {p}_{F1}\\rightarrow \\mathbf {p}_{F1}^{\\prime }$ with momentum transfer close to 0 or $2p_F$ ) for density waves [49], [45].",
"This method directly exposes the pairing symmetry through the number of nodes along the diagonal structures: a $p$ -wave phase has one node, and an $f$ -wave phase has three nodes, etc.",
"In the second method, we construct the channel matrices from $\\Gamma $ , e.g.",
"$V_{BCS}(\\mathbf {p}^{\\prime },\\mathbf {p})=\\Gamma (\\mathbf {p}^{\\prime },-\\mathbf {p}^{\\prime },\\mathbf {p})$ for the pairing channel, and $V^\\mathbf {q}_{DW}(\\mathbf {p}^{\\prime },\\mathbf {p})=\\Gamma (\\mathbf {p}+\\mathbf {q}/2,\\mathbf {p}^{\\prime }-\\mathbf {q}/2,\\mathbf {p}-\\mathbf {q}/2)$ for the density wave channel.",
"Different values of $\\mathbf {q}$ , e.g.",
"$\\mathbf {q}_i=(q_i,0)$ with $q_i\\in \\lbrace 0.05p_F,0.5p_F,p_F,2p_F\\rbrace $ for $i\\in \\lbrace 1,...,4\\rbrace $ respectively, are compared (see DW$_i$ in Fig.",
"REF , left column).",
"The channel matrices are then diagonalized and their the most negative eigenvalues are monitored.",
"This method provides a clear picture of the competition among the channels.",
"The eigenvector of the leading divergence exposes the orbital symmetry, e.g.",
"$p$ - or $f$ -wave, of the incipient order."
],
[
"Phase diagram from FRG",
"The resulting phase diagram is summarized in the top panel of Fig.",
"REF .",
"In addition to the Fermi liquid, three ordered phases are clearly identified.",
"Here the filled circles mark the phase boundary, the color indicates the critical scales $\\Lambda _c$ which is proportional to $T_c$ [39], and the dash lines are guide for the eye and they roughly enclose the regions where $\\Lambda _c$ is higher than the numerical IR scale $\\Lambda _{IR}$ .",
"For attractive interactions $g<0$ , e.g.",
"at the point $\\mathcal {P}_1$ , the leading eigenvalues are from $V_{BCS}$ and doubly degenerate with $p$ -wave symmetry.",
"The vertex map also reveals diagonal structures with single node (Fig.",
"REF ), confirming a $p$ -wave superfluid phase.",
"While the FRG here cannot directly access the wavefunction of the broken symmetry phase, mean field argument favors a $p_x+ip_y$ ground state because it is fully gapped and has the most condensation energy.",
"Thus Rydberg-dressed Fermi gas is a promising system to realize the $p_x+ip_y$ topological superfluid.",
"Our analysis suggests that the optimal $T_c$ is around $p_Fr_c\\sim 2$ and $T_c$ increases with $|u_0|$ .",
"For repulsive interactions $g>0$ , which channel gives the leading instability depends intricately on the competition between $p_F$ and $r_c$ .",
"(a) First, FRG reveals a density wave phase for $p_Fr_c \\gtrsim 4$ , in broad agreement with RPA.",
"For example, at point $\\mathcal {P}_3$ , the most diverging eigenvalue comes from $V_{DW}$ , and the vertex map shows clear horizontal-vertical structures (Fig.",
"REF ).",
"Note the separations between the horizontal/vertical lines, and relatedly the ordering wave vector, depend on $r_c$ .",
"(b) For $p_Fr_c\\lesssim 4$ , however, the dominant instability comes from the BCS channel despite that the bare interaction is purely repulsive in real space.",
"In particular, for small $p_Fr_c\\lesssim 2$ , such as the point $\\mathcal {P}_2$ in Fig.",
"REF , the pairing symmetry can be unambiguously identified to be $f$ -wave: the vertex map has three nodes, the most diverging eigenvalues of $V_{BCS}$ are doubly degenerate, and their eigenvectors follow the form $e^{\\pm i 3\\theta }$ .",
"This $f$ -wave superfluid is the most striking result from FRG.",
"(c) For $p_Fr_c$ roughly between 2 and 4, sandwiched between the density wave and $f$ -wave superfluid, lies a region where the superfluid paring channel strongly intertwines with the density wave channel.",
"While the leading divergence is still superfluid, it is no longer pure $f$ -wave, and it becomes increasingly degenerate with a subleading density wave order.",
"This hints at a coexistence of superfluid and density wave.",
"To determine the phase boundary, we trace the evolution of $\\Lambda _c$ along a few vertical cuts in the phase diagram, and use the kinks in $\\Lambda _c$ as indications for the transition between the density wave and superfluid phase, or a change in pairing symmetry within the superfluid (see inset, top panel of Fig.",
"REF ).",
"We have checked the phase boundary (filled circles) determined this way is consistent with the eigenvalue flow and vertex map.",
"Cooper pairing can occur in repulsive Fermi liquids via the Kohn-Luttinger (KL) mechanism through the renormalization of fermion vertex by the particle-hole fluctuations.",
"Even for featureless bare interactions $V(\\mathbf {q})=U>0$ , the effective interaction $V_{\\ell }$ in angular momentum channel $\\ell $ can become attractive due to over-screening by the remaining fermions [50].",
"In 2D, the KL mechanism becomes effective at higher orders of perturbation theory, e.g.",
"$U^3$ , and the leading pairing channel is believed to be $p$ -wave [51].",
"Here, the effective interaction is also strongly renormalized from the bare interaction by particle-hole fluctuations.",
"We have checked that turning off the ZS and ZS' channels eliminates superfluid order on the repulsive side.",
"However, our system exhibits $f$ -wave pairing with a significant critical temperature in contrast to usual KL mechanism with exponentially small $T_c$ .",
"This is because the Rydberg-dressed interaction already contains a “pairing seed\": $V(\\mathbf {q})$ develops a negative minimum in momentum space for $q=q_c$ unlike the featureless interaction $U$ .",
"Among all the scattering processes $(\\mathbf {p}_{F},-\\mathbf {p}_{F})\\rightarrow (\\mathbf {p}^{\\prime }_F,-\\mathbf {p}^{\\prime }_F)$ , those with $q=|\\mathbf {p}^{\\prime }_F-\\mathbf {p}_F|\\sim q_c$ favor pairing.",
"It follows that pairing on the repulsive side occurs most likely when the Fermi surface has a proper size, roughly $2p_F\\sim q_c$ , in broad agreement with the FRG phase diagram.",
"These considerations based on the bare interaction and BCS approach, however, are insufficient to explain the $f$ -wave superfluid revealed only by FRG, which accurately accounts the interference between the particle-particle and particle-hole channels.",
"The pairing seed and over screening conspire to give rise to a robust $f$ -wave superfluid with significant $T_c$ ."
],
[
"Conclusion",
"We developed an unbiased numerical technique based on FRG to obtain the phase diagram for the new system of Rydberg-dressed Fermi gas to guide future experiment.",
"We found an $f$ -wave superfluid with unexpectedly high $T_c$ driven by repulsive interactions beyond the conventional Kohn-Luttinger paradigm.",
"The physical mechanism behind the $T_c$ enhancement is traced back to the negative minimum in the bare interaction, as well as the renormalization of the effective interaction by particle-hole fluctuations.",
"These results contribute to our understanding of unconventional pairing from repulsive interactions, and more generally, competing many-body instabilities of fermions with long-range interactions.",
"Our analysis may be used for optimizing $T_c$ by engineering effective interactions using schemes similar to Rydberg dressing.",
"Our FRG approach can also be applied to illuminate the rich interplay of competing density wave and pairing fluctuations in solid state correlated quantum materials.",
"Note that f-wave pairing has been previously discussed in the context of fermions on the p-orbital bands [52], [53].",
"This work is supported by NSF Grant No.",
"PHY-1707484, AFOSR Grant No.",
"FA9550-16-1-0006 (A.K.",
"and E.Z.",
"), ARO Grant No.",
"W911NF-11-1-0230, and MURI-ARO Grant No.",
"W911NF-17-1-0323 (A.K.).",
"X.L.",
"acknowledges support by National Program on Key Basic Research Project of China under Grant No.",
"2017YFA0304204 and National Natural Science Foundation of China under Grants No.",
"11774067."
]
] | 1906.04235 |
[
[
"Recognizing License Plates in Real-Time"
],
[
"Abstract License plate detection and recognition (LPDR) is of growing importance for enabling intelligent transportation and ensuring the security and safety of the cities.",
"However, LPDR faces a big challenge in a practical environment.",
"The license plates can have extremely diverse sizes, fonts and colors, and the plate images are usually of poor quality caused by skewed capturing angles, uneven lighting, occlusion, and blurring.",
"In applications such as surveillance, it often requires fast processing.",
"To enable real-time and accurate license plate recognition, in this work, we propose a set of techniques: 1) a contour reconstruction method along with edge-detection to quickly detect the candidate plates; 2) a simple zero-one-alternation scheme to effectively remove the fake top and bottom borders around plates to facilitate more accurate segmentation of characters on plates; 3) a set of techniques to augment the training data, incorporate SIFT features into the CNN network, and exploit transfer learning to obtain the initial parameters for more effective training; and 4) a two-phase verification procedure to determine the correct plate at low cost, a statistical filtering in the plate detection stage to quickly remove unwanted candidates, and the accurate CR results after the CR process to perform further plate verification without additional processing.",
"We implement a complete LPDR system based on our algorithms.",
"The experimental results demonstrate that our system can accurately recognize license plate in real-time.",
"Additionally, it works robustly under various levels of illumination and noise, and in the presence of car movement.",
"Compared to peer schemes, our system is not only among the most accurate ones but is also the fastest, and can be easily applied to other scenarios."
],
[
"Introduction",
"License plate detection and recognition (LPDR) of vehicles has been playing an increasingly important role in building intelligent societies and cities.",
"LPDR can be exploited to enforce the security of communities, enable road safety, and facilitate the collection of payment for parking or road toll [8].",
"Although there are many recent efforts on LPDR [26], [25], [27], [2], existing systems generally cannot recognize the plate at high accuracy while also completing LPDR in real-time.",
"LPDR performance suffers when there is no advanced hardware to capture high-quality images or the recognition is needed for a plate attached to a fast moving car [8].",
"The difficulty lies in the extreme diversity of character patterns.",
"Characters on the plates can be in different sizes, fonts, and colors depending on their states and nations, distorted by the viewpoint of camera, and captured with low-quality images due to lighting, shadows, occlusion, or blurring.",
"LPDR is made even harder if there is a requirement for real-time processing.",
"The highly complicated backgrounds introduce additional challenges, and often lead to false alarms in plate detection.",
"Some example backgrounds include the general texts on shop boards, text-like patterns on the car windows, as well as guardrails and bricks on the road whose textures are similar to license plates.",
"A complete LPDR process is typically composed of two phases: license plate detection (LPD) and license plate recognition (LPR).",
"LPD aims to identify and localize a license plate, and generate a bounding box around the plate.",
"Usually, a verification process is also needed to guarantee that the plate is correctly detected.",
"In LPR, plates are firstly segmented into several characters, and then each is recognized by a recognizer.",
"The aim of this work is to achieve LPDR at high accuracy in real-time, and it is critical to have LPD and LPR both efficient.",
"To reduce the running-time in the LPD procedure, we apply the most efficient edge-based algorithm.",
"Despite its quick processing speed, this method is sensitive to noise, and cannot deal with broken edges and remove irrelevant edges properly.",
"The segmentation of characters is difficult if there exist faked picture borders.",
"Being able to learn mid and high level features from the training data, convolutional neural networks (CNNs) have shown good recognition performances in many vision tasks, such as image classification and object detection [11].",
"Although CNN appears to be promising in recognizing general characters, it is often difficult to obtain enough data to well train CNN for LPDR, given the diversity of plate types, the varying quality of images captured during car mobility, and the complex backgrounds around plates.",
"This will cause a CNN to overfit easily.",
"Finally, existing LPDR systems often take separate and independent algorithms for the candidate verification [16], [21], which introduces additional computation cost.",
"To address these issues, we implement a complete LPDR system to enable real-time and accurate license plate detection and recognition, with the structure shown in Fig.",
"REF .",
"The contributions of our work are as follows: We propose a contour reconstruction scheme to quickly localize the plate in the presence of incomplete or distorted plate contours in a practical environment.",
"We refine the license plate candidates with a simple zero-one-alternation method to effectively remove the irrelevant edges and noise.",
"We introduce a few strategies in our design to overcome the overfitting problem for more accurate character recognition, including data augmentation, incorporation of SIFT features to the network, and transfer learning to determine the initial training parameters for the network.",
"We propose a two-phase verification method to determine the correct plate: 1) We first apply a statistical filter in the LPD stage to effectively remove the wrong plates from the candidate set to reduce the processing overhead for further processing in a LPDR system; and 2) We take advantage of our accurate character recognizer to help verify the candidates after the LPR process.",
"Without need of extra technique and time to verify the plate, the second phase is equivalent to shortening the system pipeline, which further improves the efficiency of our LPDR system.",
"Figure: The LPDR system structure.The paper is structured as follows: we introduce the related work in Section .",
"We introduce the LPD and LPR problems, and our methods and CNN model to address the issues in Section and Section .",
"The experimental results are presented and analyzed in Section .",
"Finally, we conclude the paper and introduce our future work in Section ."
],
[
"Related Work",
"We briefly introduce the previous work which targets for different phases of the LPDR system."
],
[
"License Plate Detection",
"For the license plate detection (LPD), previous studies [3], [5] generally try to capture certain morphological, color or textural features of a license plate.",
"They are either computationally expensive and thus not suitable for real-time systems, or very easy to be affected by the color change in plates.",
"Hough transform methods [13] assume that license plates are defined by lines around them, and require a large memory space and considerable amount of computing time.",
"Histogram-based approaches [17] do not work properly on the images with big noise or license plates tilted.",
"Learning methods with sliding window [2], [26] suffer from high computational cost with their applying the classifier to a sequence of rectangles within an image.",
"In addition, not all objects are box-shaped, and representations may be polluted by features not belonging to the object.",
"Edge-based approaches are the simplest and fastest [1], [9], and are employed in our work.",
"However, existing methods are sensitive to unwanted edges often appearing close to the license plates, which may lead to a wrong detection.",
"We propose a simple method which exploits contour reconstruction together with statistical filtering to quickly and accurately detect the license plate.",
"Different from the use of global search such as with sliding windows, our design follows the nature of human visual detection, where an attention is driven to certain locations by low-level features of images, such as contours, edges, and texts, rather than uniformly to all locations."
],
[
"License Plate Recognition",
"The License plate recognition (LPR) step consists of three parts: preprocessing, segmentation, and character recognition.",
"In preprocessing, skew correction and image refinement are often applied to deskew the image and remove the unnecessary borders and noises.",
"Then a projection-based method is applied for segmentation [1], which often does not work well when there are redundant borders and other edges around the license plate, especially when the faked borders appear at the top or bottom part of the plate.",
"Based on the character features of license plates, we propose a zero-one-alternation method to correctly remove the unwanted borders for more accurate segmentation.",
"Character Recognition (CR) in a general context has been widely studied.",
"The template matching method [26] is simple and straightforward, but is vulnerable to font, rotation, noise, and thickness changes.",
"SVMs [4] and shallow BP neural networks [12] are also popular, but they are not good enough to get the most important information from the characters.",
"CNN-based methods have been proven to be very efficient in image recognition and classification with their ability of learning richer and higher level representations of features [11], [24], [14], [23], [22], [6].",
"They are invariant under small shifts, distortions, and noise.",
"Despite the potentials, CNNs need a huge amount of training data to avoid “overfitting”, while it is often very hard to get a big training set in real-world applications.",
"With the large variety of license plates and their differences in varying environment conditions, getting enough data for training becomes even harder.",
"This will significantly compromise the CR performance.",
"To improve the CR accuracy, we propose various schemes for data augmentation, aggregate SIFT features, and exploit transfer learning."
],
[
"License Plate Detection",
"License plate detection (LPD) is the first critical step of the LPDR system.",
"Our LPD consists of two major procedures: candidate plate detection and candidate plate verification.",
"Candidate plate is defined as the area that potentially contains a license plate.",
"To more effectively detect the candidate image region of the plate, we propose a set of schemes based on edge detection, so that the scheme can better work under different backgrounds, motion blur, light conditions, and tilt angles."
],
[
"Candidate Plate Detection",
"With the edge detection, some of the edges may form closed contours, and one may be around the license plate.",
"A license plate is generally bounded by a rectangle area which contains some texts.",
"However, in a practical environment, the contour of the license plate may not be a rectangle.",
"It may be broken or tilted (skewed).",
"In order to improve the detection accuracy, we first propose to reconstruct the edges to form complete rectangular candidate contours that possibly contain the plate.",
"To speed up the detection process, we further propose a set of schemes to reduce the candidate set.",
"As an intuitive way of finding the location of a license plate, one can first get rectangular contours after the edge detection, and then select the one with the right width, height, and convex area.",
"This method suffers when the contours are broken or greatly tilted.",
"It also cannot be applied to detect the license plates with a wide range of height, width, and area.",
"To better deal with the plate candidates with broken or highly tilted contours, we propose to reconstruct a complete contour around the plate based on the extreme points found on the edge map.",
"As shown in Fig.",
"REF , if we have the top-most point $A$ , the right-most point $B$ , the bottom-most point $C$ , and the left-most point $D$ , we can reconstruct the candidate contour by taking the rectangle $A^\\prime $$B^\\prime $$C^\\prime $$D^\\prime $ , where $x_{A^\\prime } = x_D, y_{A^\\prime } = y_A, x_{B^\\prime } = x_B, y_{B^\\prime } = y_A, x_{C^\\prime } = x_B, y_{C^\\prime } = y_C, x_{D^\\prime } = x_D, y_{D^\\prime } = y_C.$ With this procedure, no matter the contour is broken, tilted, or concave, we can always reconstruct a proper rectangle around it.",
"For example in Fig.2(c), a broken contour with only two borders is still able to be reconstructed.",
"Although simple, our performance studies indicate that this process helps to significantly improve the accuracy of LPD.",
"Figure: Candidates of different scenarios and the corresponding reconstruction results: (a) a normal candidate.",
"(b) A tilted candidate.",
"(c) A tilted broken candidate"
],
[
"Coarse Candidate Plate Verification",
"After the detection process, there may exist multiple candidate plates.",
"To reduce the later processing overhead, we will filter out the wrong candidates and keep only the ones most likely to be correct.",
"We propose a Statistical Filtering method in this section.",
"Before presenting our scheme, we first introduce a vectorization process to translate a plate image containing alphanumeric characters to a “pixel vector” by summing up the “pixel matrix” along its columns and then normalizing it by 255: $\\textbf {v} = \\frac{\\sum _{i}\\textbf {M}_{ij} }{255}.$ We observe that the pixel vector exhibits certain statistical regularities distinguishable from the background.",
"In the plots of the plate candidates and their 1-D pixel vectors on Fig.",
"REF , we can see that a true candidate has some distinctive statistical regularities different from the false ones.",
"For example, a true license plate generally has a sequence of “peaks” and “valleys”, and the number of peaks is larger than most of the false ones.",
"Definition 1 A value is considered to be a “peak” if it is a local maximum of the pixel vector and the other minima points on its left and right are smaller than it by a threshold.",
"A similar definition can be applied to “valley”.",
"In the Fig.",
"REF , suppose the threshold is 3, then there are 8 peaks on the correct plate, while there are only 2 peaks on the wrong plate.",
"A candidate with more than 6 peaks (the number of characters in the plate) will be considered as a potential plate and passed to the subsequent procedures.",
"The others will be eliminated.",
"Figure: Candidates and their normalized pixel vectors."
],
[
"License Plate Recognition",
"License plate recognition (LPR) consists of three steps: pre-processing, segmentation, and character recognition (CR).",
"Pre-processing procedures such as skew correction and image binarization are commonly taken first to prepare an image for the subsequent steps.",
"A refining procedure is then applied to remove the unnecessary parts (e.g., the bordering regions) from the binary image, so it will ideally look like that in Fig.4(b).",
"Finally, with the binary image in Fig.4(b), a pixel vector can be built, as shown by the plot in Fig.4(e).",
"The segmentation is performed along the red dotted lines passing through the valleys of the 1-D vector.",
"This segmentation method is called vertical projection.",
"The segmented letters are shown in Fig.4(f).",
"With the existence of noisy borders around the plate, the segmentation step is often difficult.",
"Although the left and right borders can be removed through the literature method [25], existing methods often fail to remove the top and bottom borders in the presence of distracting image contents.",
"This may lead the segmentation error.",
"We propose a simple zero-one-alternation (ZOA) method which can effectively and quickly remove the top and bottom borders.",
"After the characters are segmented from the license plate, a robust CR method that can work in different situations is followed to better recognize the segmented characters.",
"Figure: Plate preprocessing and segmentation illustration: (a) The binary plate.",
"(b) The refined binary image after ZOA processing.",
"(c) Normalized pixel vector for (a).",
"(d) Number of zero-one-alternations along rows.",
"(e) Normalized pixel vector for (b).",
"(f) The segmented characters."
],
[
"Refinement with Zero-One-Alternation",
"A binary image after the pre-processing has two values, 1 (255) or 0.",
"To more effectively remove the noisy top and bottom borders outside the characters of the license plate, we propose a novel scheme called zero-one-alternation (ZOA).",
"Definition 2 Along each row of the binary image matrix, the number of changes from 0 to 1 or from 1 to 0 is the number of zero-one-alternations (ZOAs).",
"By scanning row-by-row from the top to the bottom of the binary image, if we can observe similar number of ZOAs (more than $\\alpha $ ) from row $i$ until row $i+n$ , with $n \\ge \\beta $ , it is very likely that the characters are contained within a region of $n$ rows.",
"In Fig.4(d), this region is from point 1 to point 2.",
"$\\alpha $ and $\\beta $ are the only parameters need to be predefined.",
"$\\alpha $ and $\\beta $ can be set by the smallest number of alphanumeric characters and the lowest height of the characters in pixels in all the license plates.",
"In this work, we set $\\alpha =6$ and $\\beta =8$ .",
"Note that these two are heuristic parameters, but should work well in most cases.",
"After removing the top and bottom borders using ZOA, we get Fig.4(b).",
"Without this step, the plate might be mistakenly recognized as “7E67Q9” instead of “7F6709”.",
"To remove the left and right borders, we further apply a commonly used method in [25].",
"The segmentation can be efficiently performed over the properly trimmed image through the vertical projection introduced earlier."
],
[
"Character Recognition",
"The last key phase of a LPDR system is to recognize the segmented characters of the license plate.",
"It is necessary and important to propose a robust CR method that can work in different situations.",
"Due to the large variety of license plates and their images differ in varying environment conditions, there is no specific dataset for plate character recognition.",
"Thus, we create a dataset by ourselves, as introduced in Section REF .",
"However, the training data available are far from enough.",
"This will significantly compromise the CR performance.",
"In our created dataset, we have 36 classes and 3,000 examples in total.",
"So on average less than 100 training data are available for each class, where a typical classification problem requires more than 1,000 training data for each class.",
"Thus, the character recognition for license plates is challenging.",
"With very limited training data, the learning faces the problems of over-fitting, where the trained model does not apply well to new data.",
"To enable higher quality CR for LPDR system, we will exploit the use of data augmentation, a newly designed SIFT-CNN model, and the transfer learning."
],
[
"Data Augmentation",
"In order to make full use of the limited training samples, we consider two methods for data augmentation.",
"In the first method, we augment data via a number of random affine transforms to avoid the duplication of data in the training set.",
"We gradually increase the amount of augmentation till we have a low testing error.",
"This helps to alleviate the overfitting problem and make our model more easily applied to new scenarios.",
"A big challenge of performing accurate CR for license plate is that a plate image is often subject to different illumination and noise.",
"In the second method, we vary the illumination of our training data and inject Gaussian noise into our model so that it performs more robustly under different image quality.",
"Our augmentation parameters are summarized in table REF .",
"Table: Data augmentation setup"
],
[
"Incorporating SIFT feature vector into CNN",
"CNN requires massive training data to work well in different scenarios.",
"As an another scheme, scale-invariant feature transform (SIFT) method [15] can extract image descriptors for recognition without requiring a large amount of data.",
"SIFT searches an image at multiple scales and positions to look for the regions that have the maxima or minima with high contrast, called keypoints.",
"The image gradients can be tracked with a histogram and represented as a descriptor to characterize the appearance of a keypoint.",
"To describe a keypoint, the region around the keypoint is divided into $4 \\times 4$ subregions, within each gaussian derivatives are computed in 8 orientation planes.",
"So a 128-dimension descriptor vector is formed for each keypoint, and this procedure is repeated for all keypoints to obtain a set of vectors for the image.",
"SIFT descriptors are invariant to translations, rotations, and scaling of the image, and robust to moderate transformations and illumination variations.",
"Despite these benefits, SIFT filters are fixed and cannot vary for a different source of data, which compromises its performance.",
"To take advantage of both schemes while avoiding their limitations, we propose a new SIFT-CNN model which integrates the vector extracted by SIFT with the feature vector of CNN to increase the accuracy of character recognition.",
"Our method is inspired by a bag-of-words (BoW) representation where image features are treated as words.",
"In BoW, detection and description of image features are first applied, followed by assigning feature descriptors to a set of predetermined clusters (like vocabulary).",
"Finally, a vector called a BoW is used to track the number of occurrences of words in the vocabulary.",
"Fig.",
"REF shows our SIFT-CNN model, where a SIFT branch is added to extract the image feature vector as follows: Using SIFT method to extract the keypoint descriptors of the image.",
"Assigning each keypoint descriptor to a set of clusters whose centers are determined using K-means based on the descriptors of training data.",
"Constructing the feature vector $\\mathbf {v}$ for each image to track the number of keypoints assigned to each cluster, and normalizing $\\mathbf {v}$ by $\\frac{\\mathbf {v}}{\\vert \\mathbf {v} \\vert _{l_2}}$ .",
"Concatenating the feature vector $\\mathbf {v}$ with the feature vector extracted in the fully-connected layer of CNN to form a final feature vector.",
"We set the number of clusters to 256, so that the length of the feature vector is the same as that of the CNN feature vector.",
"Then the concatenated feature vector has the size 512, and will be used to classify the image.",
"Figure: Network architectures for CR problem.",
"Left: traditional CNN-8 model as a reference.",
"Right: CNN-8 incorporated with SIFT (SIFT-CNN) model."
],
[
"Efficient Learning across Applications",
"Many machine learning methods work well only under a common assumption, the training and testing data are taken from the same feature space and follow the same distribution [18].",
"When the distribution changes, most statistical models need to be rebuilt using newly collected training data.",
"It would be helpful to exploit an emerging technique called the transfer learning, which extracts the knowledge from one or more source tasks and applies the knowledge to a target task.",
"For a small training dataset, without enough knowledge on the data, randomly initializing a neural network can make the training result worse since it is easily to get stuck in local minima.",
"However, if a neural network can start from an already trained feature extractor, it can “borrow” some knowledge from a task already learnt to the new task.",
"Despite the potential, it is crucial and difficult to determine the right knowledge sources and the amount of knowledge to use for a new learning task.",
"If too few are transferred, we may not get enough knowledge from source tasks, while transferring too many parameters will not only lead to redundancy but may also compromise the training performance if many parameters are irrelevant.",
"To further improve the quality of character recognition, we apply transferring learning to the training of our CNN model.",
"Our preliminary studies indicate that features learnt from the low-level and mid-level of a neural network are more effective for transfer learning.",
"We “borrow” the parameters of a SIFT-CNN model, trained on the dataset ICDAR [10].",
"The dataset contains more than 62 classes, which include 26 upper-case letters, 26 lower-case letters, 10 digits classes and other classes.",
"Our studies indicate that this dataset is excellent in extracting low and mid level features of the document letters.",
"Initially, we attempted to apply the model trained with this data set directly to recognizing the characters on the license plates.",
"However, the CR accuracy is only 84%, since the license plate characters are quite different from the document letters in fonts, types and classes.",
"Instead, as shown in Fig.",
"REF , we train a new model based on license plate data.",
"We “borrow” the convolutional layers $1-4$ of the trained SIFT-CNN model, and rebuild two convolutional layers $CA-CB$ and two fully-connected layers $FCA-FCB$ with the number of neurons being 512 and 36 respectively."
],
[
"Hybrid CNN model",
"For recognizing characters on license plates, we apply Hybrid CNN which exploits a SIFT-CNN model with transfer learning.",
"We train our model using stochastic gradient descent (SGD) algorithm with an annealed learning rate.",
"We test our model with different number of transferred layers and fine-tuned layers, and we find four transferred layers and four fine-tuned layers can achieve the best result.",
"Thus, we fix the 4 transferred layers and only fine-tune the 4 rebuilt layers using the target training data created for this work.",
"The training process stops when there is no improvement in performance for 5 epochs.",
"In the experiment section, we will show that the transfer learning together with fine-tuning can improve the recognition performance, especially when the training data are very limited.",
"If we fine-tune all layers, it will not only take a very long time, but may also break the good feature filters already built by the original model.",
"We only fine-tune the last two convolutional layers and the two fully-connected layers.",
"The features extracted from ICDAR dataset by the low-level and mid-level convolutional layers are very general, and can be easily applied to the classification of our images.",
"The higher convolutional layers trained with the ICDAR dataset, however, do not fit well in our system and cannot provide satisfying CR results.",
"Figure: Hybrid CNN model.",
"Note: SC-Special Characters."
],
[
"Fine Candidate Verification with Voting",
"After the statistical filtering in the LPD phase, we might still have more than one candidate plate, usually two or three.",
"To determine which one is correct, all the remaining candidates should go through a second verification phase.",
"We propose a“voting” method taking advantage of the results from the CR process for more accurate verification without need of additional computing time and resources.",
"As every character is recognized independently, a candidate is more likely to be a correct one if every character is more likely to be correctly recognized.",
"So every character is “voting” for the candidate they belong to.",
"The probability of the character $i$ to be the right label $y_i$ is $P_i(y=y_i|\\textbf {X})$ , as shown in table REF .",
"The probability of the candidate to be the right plate is $\\hat{P}=\\prod _{i=1}^{n} P_{i}$ , where $n$ is the number of segmented characters.",
"The one with a higher probability will be selected as the license plate.",
"From table REF , we can tell that the first candidate is much more likely to be the right candidate than the second one since the first one has a higher probability $\\hat{P}$ .",
"Once the CR process is completed, the verification is followed to choose the right candidate and the final result is achieved.",
"Table: Illustration of “voting” on a real & a false candidate."
],
[
"Experiments",
"We compare the performance of our system with several state-of-the-art LPDR schemes, and we emphasize the key similarities and differences.",
"To demonstrate the accuracy and efficiency of our proposed LPDR scheme, we conduct a set of experiments over natural car images taken in different environments and nations.",
"Our experiments are run on a dual-core 2.7 GHz Intel i5 machine, and our algorithms are realized in C++ due to its efficiency."
],
[
"Dataset",
"We test our system on two datasets.",
"The first one is the AOLP benchmark dataset [8] with three subsets: access control (AC), law enforcement (LE), and road patrol (RP).",
"AC refers to the cases that a vehicle passes a fixed passage with a low speed or full stop.",
"The 681 images were captured under different illuminations and weather conditions, with the resolution of each image at $240 \\times 320$ .",
"LE refers to the cases that a vehicle violates traffic laws and is captured by roadside cameras.",
"The 757 image samples were collected with the image resolution at $240 \\times 320$ or $480 \\times 640$ .",
"RP refers to the cases that a camera is installed on a patrolling vehicle, and the 611 images were taken with arbitrary viewpoints and distances, with the resolution of each image at $240 \\times 320$ .",
"The second dataset, LongIsland, contains 300 samples of car images we collect ourselves by driving on local roads of Long Island.",
"We drove with the average speed of 30 mph on a cloudy day.",
"The resolution of each image is $480 \\times 640$ .",
"The dataset consists of car images of different backgrounds, motion blurring, skew angles, capturing distances, and sizes.",
"The images in the dataset are labelled manually.",
"Two alphanumeric dataset are used in the CR phase.",
"The ICDAR dataset [10] consists of about 12,000 samples.",
"It includes classes for 10 digit numbers, 26 classes for upper-case characters, 26 classes for lower-case characters, and many other special symbols.",
"We use the 6548 samples of 10 digits and 26 upper-case characters to train our CNN models.",
"We create the second dataset ourselves by cropping the car and motorcycle license plates of different fonts, nations, and states from the Internet.",
"It consists of 3,000 training samples of 36 different classes, which include 26 upper-case letters and 10 digits.",
"The two datasets are merged together and then divided into three subsets - training subset ($\\frac{5}{7}$ of the samples in the merged dataset), validation subset ($\\frac{1}{7}$ of the merged dataset), and test subset ($\\frac{1}{7}$ of the merged dataset).",
"The training subset is used for model training, the validation subset for validating our models and enforcing early-stopping of training when the validation accuracy does not have perceivable improvement over time, and the test subset for testing the performance of our models.",
"When training our Hybrid CNN model, we divide the two datasets in the same way.",
"However, we first train our model on ICDAR dataset to get the initial parameters, then fine-tune the parameters with the plate character dataset we created."
],
[
"Evaluation Metrics and LDPR Results",
"To measure the processing speed of our system, we calculate the average number of images processed by our system in one second, or frames per second (FPS).",
"The performance of plate detection is evaluated using the precision and recall rate [10], two most widely used evaluation criteria for detecting the general texts in natural images.",
"Precision is defined as the ratio of the correctly detected license plates and the total number of detected plates.",
"Recall is the ratio of the correctly detected license plates and the total number of groundtruth.",
"A license plate is correctly detected if it is totally enclosed by a bounding box, and the Intersection over Union (IoU), which is area of overlap divided by area of union, is greater than 0.5.",
"We evaluate the plate recognition performance with recognition rate, which is defined as the number of correctly recognized license plates divided by the total number of correctly detected plates.",
"Note that a correctly recognized license plate means all the characters on the plate are recognized correctly.",
"Some of the results of our system are shown in Fig.",
"REF .",
"We will evaluate the detailed performance on LPD and LPR in Section REF and REF respectively, and show the impact of motion on the LPDR performance in Section REF ."
],
[
"Performance of Plate Detection",
"Our system exploits Contour Reconstruction and Hybrid CNN for plate detection and recognition, and we call it CR-HybridCNN.",
"Similar abbreviations are given to the other systems likewise.",
"We compare the performance of plate detection of our system with four other methods proposed within the recent three years and showed to have good LPD and LPR performance in their papers.",
"The detection performance of all methods are evaluated with the AOLP dataset with the results shown in Table REF .",
"Based on the metrics of evaluation, our proposed system is the most efficient one, and it can process 90 frames per second for images in all three subsets.",
"While it is slightly less accurate than EC-LDA, the processing speed is about 20 times larger.",
"Next we introduce each reference LPDR method and show the difference of our design.",
"Edge-clustering and Linear Discriminant Analysis (EC-LDA) [8] uses an Expectation-Maximization (EM) edge clustering algorithm to extract regions with dense sets of edges which have shapes similar to plates.",
"The clusters with their edge densities larger than a predefined threshold and edges of plate-like shapes will be considered as plate candidates.",
"As the system needs to predefine eight threshold values and parameters such as the number of clusters and the penalty weight, it prohibits the system from working well in other scenarios.",
"In addition, the clustering algorithm is computationally expensive with its needs for a large number of iterations to obtain a good result.",
"Our system replaces all of these computationally expensive part with only one simple reconstruction method, which only needs two general thresholds without other parameters.",
"Thus our algorithm has its accuracy comparable to EC-LDA, but can work much faster system and better in new scenarios than EC-LDA.",
"Extremal Regions and RBM (ER-RBM) [7] applies morphological transformations with several rounds of close and open operations to do coarse detection and produce the candidate license plates.",
"Then Extremal Regions method is used to detect the license plates at the fine level.",
"Each stage of this complex pipeline must be precisely tuned independently, so the system is slow, and it takes more than 0.2 seconds on average to process an image.",
"Similar to ER-RBM, our system also tries to first coarsely detect license plates and then detect the final results by comparing several candidates.",
"However, our system uses the simple statistical filtering method to complete the coarse detection quickly, and reduces candidates to a small number (usually fewer than three) in one round rather than several rounds of transformations.",
"In addition, in the fine detection stage, we exploit the results from character recognition to vote for the final result without additional processing.",
"Thus our processing speed is more than 10 times faster.",
"Other Real-Time LPDR Systems Many research efforts in LPDR focus on speeding up the detection pipeline.",
"However, after testing on our own machine, only KNN-SVM [20] and Color with Edge Detection and KNN (CE-KNN) [19] can run in real-time (with 30 FPS or better).",
"However, their detection accuracy is sacrificed.",
"KNN-SVM combines edge detection with morphological transformation for plate detection.",
"CE-KNN uses color detection and edge detection to process the image separately, and then uses the information from the two to locate the final license plate.",
"Both of the two methods need to limit the size of the candidates to be within a small range, so that the final candidate within the range is considered as the plate.",
"If the sizes of license plates vary over a large range, the system's performance will suffer.",
"Our license plate detector is general and does not depend on the application scenarios, dataset, and sizes of license plates.",
"Table: Comparing the detection accuracy and speed of plate detectors.",
"Precision, recall, and frames per second (FPS) are used as evaluation metrics.",
"Our system has the fastest detector while still ensuring its detection precision and recall rate comparable with the most accurate system EC-LDA, with its detection precision and recall less than 2% lower on average."
],
[
"Performance of Plate Recognition",
"In EC-LDA, local binary pattern (LBP) features are extracted and classified using a two-layer LDA classifier.",
"The training samples are randomly selected from the three subsets and processed by the EC-LDA system.",
"This random selection of training data cannot prevent the use of same data for testing, which may compromise the effectiveness in performance evaluation.",
"In ER-RBM, a restricted Boltzmann machine is used as the classifier.",
"The results from experiments of the paper show that RBM performs better than SVM in classifying plate characters.",
"In KNN-SVM, KNN is used as the initial step to classify all datasets and then multi-class SVM is performed only over the smaller dataset with similar characters.",
"It performs better than traditional SVM classifiers.",
"CE-KNN proposes to use normalization, image thinning, and feature extraction as pre-processing.",
"The extracted features include the slope of stroke, the amplitude of inflection point, and the depth of profile.",
"Then the features are used to train the classifier.",
"In our system, we use deep convolutional neural networks, which have been tested to work well in many computer vision tasks [11].",
"We also apply several data augmentation techniques to increase training data, and exploit the transfer learning in our SIFT-CNN model to first classify general upper-case letters and digits.",
"The designed network is fine-tuned with training over additional plate characters in the dataset created by our own."
],
[
"Effectiveness of the Hybrid-CNN Model",
"The character recognition model plays an important role in plate recognition.",
"The Hybrid-CNN model used in our CR part is built on an eight-layer CNN model, with the incorporation of data augmentation and SIFT feature vectors, and the application of transfer learning to set up the initial parameters for training.",
"We denote the original model without extra techniques as “CNN”, the model using data augmentation as “Aug-CNN”, and the model with data augmentation and SIFT vectors as “SIFT-CNN”, based on which, “Hybrid-CNN” is created with the addition of transfer learning.",
"As shown in Table REF , with use of these three techniques, the classification accuracy improves from 0.848 to 0.892, 0.918, and 0.964, respectively.",
"Table: Classification performance of different CNN models on 36 characters."
],
[
"Comparison with other Schemes on LPR",
"The recognition results of all methods on AOLP dataset are presented in Table REF .",
"Compared to other schemes studied, our system achieves the highest FPS for all of the three subsets.",
"It has the highest recognition rate on the Subset AC.",
"For other two subsets, its recognition rates are only slightly lower, with less than 1% below the EC-LDA system.",
"The results demonstrate that our method can perform well on different datasets.",
"Table: Recognition rates (RR) and frames per second (FPS) of different license plate recognizers.",
"Our system is the fastest one while also achieving the recognition rate comparable with the most accurate system EC-LDA."
],
[
"License Plate Detection and Recognition with Motion",
"It's easier to have a good performance on a fixed dataset with all the parameters well tuned.",
"However, both the detection and recognition become harder when cars are on move.",
"We further compare our system with others using the LongIsland dataset, with images captured during driving.",
"Table REF shows that our system achieves the highest accuracy and FPS for both plate detection and recognition, which demonstrates its robustness.",
"EC-LDA has high detection precision, recall, and recognition rate on the AOLP dataset, but its performance drops off considerably when applied to the LongIsland dataset.",
"Part of the drop comes from the difficulty of detecting and recognizing license plates with motion blur.",
"The difficulty of fine-tuning the large number of parameters also affects its performance.",
"A small parameter shift in one step may compromise the ones finely tuned in the previous steps.",
"Its use of training data from the same dataset is another factor that prevents it from working well in other scenarios.",
"Table: Quantitative results on LongIsland dataset.",
"Our system has the fastest detection and recognition speed, and is also more accurate than other systems.",
"Note: RR-Recognition Rate."
],
[
"Conclusion",
"We have introduced a license plate detection and recognition (LPDR) system and demonstrated its accuracy and efficiency under different conditions.",
"The accuracy and efficiency are achieved with a set of schemes we propose: an efficient license plate detection algorithm with the support of contour reconstruction, a refinement method based on zero-one-alternation to effectively remove the unwanted boarders for more accurate character segmentation, an efficient hybrid-CNN model along with various techniques to overcome the overfitting problem, and two-phase verification to determine the correct plate at low cost.",
"In our future work, we plan to apply our algorithms to LPDR in videos and make the pipeline shorter, which can be done by removing segmentation part in the plate recognition step to further improve the performance of the system."
]
] | 1906.04376 |
[
[
"The binary millisecond pulsar PSR J1023+0038 -- II. Optical spectroscopy"
],
[
"Abstract We present time-resolved optical spectroscopy of the `redback' binary millisecond pulsar system PSR J1023+0038 during both its radio pulsar (2009) and accretion disc states (2014 and 2016).",
"We provide observational evidence for the companion star being heated during the disc-state.",
"We observe a spectral type change along the orbit, from G5 to F6 at the secondary star's superior and inferior conjunction, respectively, and find that the corresponding irradiating luminosity can be powered by the high energy accretion luminosity or the spin-down luminosity of the neutron star.",
"We determine the secondary star's radial velocity semi-amplitude from the metallic (primarily Fe and Ca) and Halpha absorption lines during these different states.",
"The metallic and Halpha radial velocity semi-amplitude determined from the 2009 pulsar-state observations allows us to constrain the secondary star's true radial velocity K_2=276.3+/-5.6 km/s and the binary mass ratio q=0.137+/-0.003.",
"By comparing the observed metallic and Halpha absorption-line radial velocity semi-amplitudes with model predictions, we can explain the observed semi-amplitude changes during the pulsar-state and during the pulsar/disc-state transition as being due to different amounts of heating and the presence of an accretion disc, respectively."
],
[
"Introduction",
"Radio millisecond pulsars (MSP) are fast rotating magnetic neutron stars, that were spun up via the transfer of angular momentum from the companion star to the neutron star in a low mass X-ray binary (LMXB) , .",
"With the discovery of the first “transitional” MSP (tMSP) PSR J1023+0038 , the connection between rotation-powered MSPs and accretion-powered LMXBs was established.",
"The MSPs M28I and XSS J12270–4859 have also switched back and forth between a rotation-powered radio pulsar state (“pulsar-state”) and an accretion-powered state which shows X-ray pulsations and accretion disc signatures (“disc-state”).",
"PSR J1023+0038 was discovered by as part of the “Faint Images of the Radio Sky at Twenty Centimeters” (FIRST) survey.",
"It was initially classified as a cataclysmic variable in 2001 because the optical counterpart to the radio source displayed short time-scale flickering and a blue optical spectrum with double-peaked emission lines, associated with an accretion disc .",
"Optical photometry taken in 2003 and 2004 did not show the rapid, large flickering events seen in the 2001 light curve, suggesting that the system had changed state.",
"Only a repetitive 4.75476(2) h single-humped modulation was observed .",
"Indeed confirmed this state transition because the optical spectrum taken in 2003 was dominated by strong absorption features and lacked the prominent emission lines observed in the 2001 spectrum.",
"They performed a time-resolved optical spectroscopic and photometric study of PSR J1023+0038, in the what we know now to be the 'pulsar-state'.",
"They found the companion star to be a late-type G5 star with an absorption-line radial velocity semi-amplitude 268$\\pm $ 3 km s$^{-1}$ modulated on an orbital period $P_{\\rm orb}$ =4.754 h. The optical light curves taken in 2004 revealed a single-humped modulation, explained in terms of an X-ray heated companion star in a system with an orbital inclination angle of $i\\sim 55^{\\circ }$ .",
"Combining the photometric and radial velocity studies led to the conclusion that the system was not a cataclysmic variable with a white dwarf, but instead an X-ray binary harbouring a neutron star.",
"Indeed, the X-ray binary scenario also explained the 2004 X-ray observations which were dominated by a hard X-ray power-law component .",
"In 2007 the system was detected as a radio pulsar with a spin period of 1.69 ms , confirming that the primary was indeed a neutron star and that PSR J1023+0038 is a low-mass X-ray binary.",
"This was the first direct evidence of an MSP in a binary system transitioning between two distinct states, and gave significant support to the “recycled” scenario for the origin of MSPs.",
"Between 2008 and 2012 PSR J1023+0038 was consistently observed as an eclipsing radio millisecond pulsar, where the radio eclipses are attributed to ionized material being forced off the companion star by the pulsar wind, as seen in other similar MSPs called \"black-widows\" .",
"In 2013 June, after about 10 years in the pulsar-state, PSR J1023+0038 transitioned back to the disc-state, as witnessed by the increased X-ray and gamma-ray flux (Fig.",
"REF ) and the disappearance of radio pulsations.",
"PSR J1023+0038 was detected as a radio pulsar in 2013 June 15, but eight days later it was not detected in the radio .",
"After the transition, the system brightened by $\\sim $ 1 mag in the optical and showed several broad double-peaked emission lines , .",
"There was also a disappearance of radio pulsations and an increase in the X-ray and gamma-ray luminosities by a factor of $\\sim $ 20 and $\\sim $ 10, respectively , , , .",
"All these factors provided compelling evidence that an accretion disc had re-formed, indicating that PSR J1023+0038 had switched back from a radio millisecond pulsar (pulsar-state) to a low-mass X-ray binary (disc-state).",
"In we presented the results of our optical photometric campaign, which revealed unprecedented fast variability.",
"Here we present the results of our optical spectroscopic campaign of PSR J1023+0038 in the disc-state, including those summarized in , as well as archival optical spectroscopy taken during the pulsar-state.",
"Previous spectroscopic studies do not cover full orbital cycles, have poor spectral resolution or do not compare the different states , , .",
"We perform a spectral type, radial velocity curve analysis and compute Doppler maps of the H$\\alpha $ emission-line in both the pulsar- and disc-state.",
"Finally, we compare the radial velocity semi-amplitude with model predictions, which account for the effects of heating and an accretion disc.",
"Figure: Normalized phase-averagedACAM spectrum of PSR J1023+0038 in the disc-state.The main emission lines are identified (vertical lines).Table: Summary of optical spectroscopic observations of PSR J1023+0038.We observed PSR J1023+0038 between November 2013 and April 2014 with the William Herschel Telescope (WHT), at the Roque de los Muchachos Observatory on La Palma, in order to obtain low-resolution optical spectra of the system in the disc state .",
"We used the double-armed Intermediate dispersion Spectrograph and Imaging System (ISIS) optical spectrometer with the R600B and the R600R gratings as well as the Auxiliary-port CAMera (ACAM) with the 400 lines/mm transmission volume phase holographic (VPH) grating.",
"These observations were taken about 6 months after PSR J1023+0038 transitioned from the radio pulsar to the disc state around 2013 June 30 , .",
"The images were first de-biased, and flat-fielded using standard procedures within irafiraf is distributed by the National Optical Astronomy Observatory, which is operated by the Association of Universities for Research in Astronomy, Inc., under cooperative agreement with the National Science Foundation.",
"http://iraf.noao.edu/.",
"After subtracting the bias level, the images were divided by a median sky flat field that was normalised by fitting high order spline functions to remove the detector specific spectral response.",
"The 1-D spectra were then extracted using optimal extraction as implemented in pamela , which is part of the starlinkhttp://starlink.eao.hawaii.edu/starlink software.",
"The wavelength calibration was done using CuNe/CuAr arc lamp spectra taken before, after and between science spectra, extracted from the same region as the closest target spectrum in time.",
"Sets of about 20 to 50 line centroid positions, depending on the instrument, were fitted with a 4–6 order polynomial to obtain the dispersion relation, which was interpolated linearly in time to account for instrumental flexure when calibrating the spectra.",
"The wavelength calibration was subsequently checked and refined using the sky emission lines, which were also used to measure the spectral resolution (full width at half-maximum, FWHM, of the sky lines).",
"The exposure time, wavelength range, dispersion and resolution are shown in Table REF ."
],
[
"Archival data",
"We downloaded the Isaac Newton Group archival data of PSR J1023+0038 carried out on 2009 April 12 with ISIS on the WHT (P.I.",
"Jonker).",
"ISIS was fitted with the H2400B and the R1200R gratings in the blue and red arm, respectively and a 0.82 arcsec slit width was used.",
"The spectra were extracted using the same procedure as the WHT 2014 data.",
"The exposure time, wavelength range, dispersion and resolution are shown in Table REF .",
"We also downloaded the ESO Very Large Telescope (VLT) archival data of PSR J1023+0038 taken on 2016 January 30 under programme 096.D-0808 (P.I.",
"Bassa) with the X-SHOOTER medium resolution (R= 4000 to 7000) spectrograph .",
"A slit width of 1.0 arcsec in the UVB arm (2989–5560 Å) and 0.9 arcsec in the VIS arm (5337–10200 Å) and NIR arms (0.994–2.478 $\\mu $ m) was used.",
"The UVB and VIS arm CCDs were binned by a factor of 2 and the exposure times were 360 s in the UVB arm, 372 s in the VIS arm and 400 s in the NIR arm.",
"A total of 42 spectra were taken in nodding mode in each arm.",
"The data reduction pipeline was used to optimally extract and calibrate the spectra .",
"The mean dispersion was 10 km s$^{-1}$ and the spectral resolution measured from the sky lines was 40 km s$^{-1}$ .",
"In this paper we only use the UVB and VIS arm spectra, since no telluric stars were observed.",
"The exposure time, wavelength range, dispersion and resolution are shown in Table REF ."
],
[
"Average spectrum",
"In Fig.",
"REF we show the disc-state normalized phase-averaged WHT ACAM 2014 spectrum of PSR J1023+0038.",
"Broad Balmer and He lines emission lines are clearly seen, which also show a double-horned profile related to the presence of an accretion disc.",
"Figure: The cross-correlation radial velocity analysis and the optimal subtractionto determine the spectral type of the secondary star.",
"The results areshown using all orbital phases.",
"As a function of template star spectraltype, we show the values of K 2 K_{\\rm 2} and determined from a circularorbit fit to the radial velocities, and the values for v rot sin 𝑖v_{\\rm rot}\\,\\rm sin\\,{\\it i}, f star f_{\\rm star} and χ 2 \\chi ^2 obtained from the optimal subtraction.",
"The left,middle and right columns are the results for the ISIS 2009 (pulsar-state),ISIS 2014 (disc-state) and X-SHOOTER 2016 (disc-state) spectra,respectively.Figure: Left: The heliocentric radial-velocity curve (filled circles) of thesecondary star in PSR J1023+0038, determined using the metallic (primarily Fe andCa) lines.",
"We show the 2004 pulsar-state , ISIS 2009pulsar-state, ISIS 2014 disc-state and the X-SHOOTER 2016 disc-state radialvelocities.",
"Right: The heliocentric radial-velocity curve (filled circles)of the secondary star in PSR J1023+0038, determined using the residual Hα\\alpha absorption-line.",
"We show the ISIS 2009 pulsar-state and the X-SHOOTER 2016disc-state radial velocities.",
"The solid line shows an eccentric orbit fitto the data and the red crosses show the residual after subtracting thefit.",
"The data have been folded on the orbital ephemeris and are showntwice for clarity.Figure: Left: the X-SHOOTER 2016 spectrum taken in the disc-state,the ISIS 2014 spectrum taken in the disc-state and the ISIS 2009 spectrumtaken in the pulsar-state.",
"(The gap in the X-SHOOTER 2016 data is due to anartifact in the reduction pipeline).",
"Right: the corresponding optimallybroadened template star spectrum.",
"The spectra have beenoffset for clarity.Figure: The results of the optimal subtraction to determine the spectral type ofthe secondary star using spectra taken around phase 0.0 and 0.5.",
"We findthe spectral type to be ∼\\sim G5 and ∼\\sim F5, around phase 0.0 and 0.5,respectively.Table: Results of the fit to the radial velocity data using the metallicand Hα\\alpha absorption-line, taken at different epochs.",
"Note that in each case,spectra between orbital phase 0.0 and 1.0 have been used."
],
[
"Analysis",
"In order to measure the absorption-line radial velocity curve and determine the spectral type of the companion star in PSR J1023+0038, we use cross-correlation and optimal subtraction techniques, respectively .",
"All analysis was performed with the mollyhttp://deneb.astro.warwick.ac.uk/phsaap/software/ package.",
"For a given template star we optimally subtract the variance-weighted Doppler-shifted target spectrum from the rotationally broadened template star spectrum.",
"We perform a $\\chi ^2$ test on the residuals of the subtraction, where the optimal value for the spectral type is obtained by minimizing $\\chi ^2$ .",
"Since both procedures require template star spectra, we compiled a library of high signal-to-noise A to K star template spectra with luminosity class V from the UVES Paranal Observatory Project (UVESPOP) archive .",
"We normalize the spectra by determining a low-order spline fit to the continuum level and then divide the spectra by the fitted continuum.",
"We degrade the template star spectra to match the same spectral resolution and heliocentric velocity scale as the target and interpolate all the spectra onto a logarithmic wavelength scale with the pixel size set by the target spectra."
],
[
"Radial velocity curve",
"Firstly, we measure the projected orbital velocity of the companion star in PSR J1023+0038 by cross-correlating the spectra of the target with the template star of interest.",
"We normalize the spectra similar as above and interpolate the spectra onto a logarithmic wavelength scale set by the target spectra.",
"As the absorption lines in the target spectrum are rotationally broadened, the template star is broadened before performing the cross-correlation analysis, using the broadening estimated by .",
"We use the wavelength range 5500–6900 Å which contains numerous weak metallic absorption lines of Fe, Ca and Mg, excluding the H and He emission lines and interstellar features.",
"We perform a least-squares fit of the radial velocities versus time using a circular orbit.",
"The orbital period $P_{\\rm orb}$ was fixed at the pulsar timing value $P_{\\rm orb}$ =0.1980963569(3) d .",
"$T_{\\rm 0}$ was constrained to lie near the middle of the time span over which the observations were taken, and defines orbital phase 0 as inferior conjunction of the secondary star (see results in Table REF ).",
"determine the radial velocities of PSR J1023+0038 in 2004 in the pulsar-state by cross-correlating spectra of PSR J1023+0038 with template stars.",
"The wavelength range and spectral type template stars they use in the cross-correlation covers the wavelength range and template stars star used here.",
"Here we use their radial velocities and perform the same least-squares fit as above."
],
[
"Spectral type",
"Using the orbital parameters derived from the radial velocity fits above, we Doppler-correct and average the target spectra to the reference frame of the template star of interest, over a defined orbital phase range.",
"In principle, the absorption features arising from the companion star should be relatively sharp because the Doppler shifts due to orbital motion have been corrected.",
"However, the absorption lines are still broadened, due to the instrumental resolution of the spectrograph, the rotation of the secondary star and by changes in the orbital velocity during individual exposures.",
"The broadening function that affects the target's absorption lines consists of the convolution of the instrumental profile with the rotational broadening profile of the secondary star.",
"There is also a further smearing due to the orbital velocity shift of the secondary star during a given exposure $\\rm \\Delta t$ at a given orbital phase, $\\rm \\Delta \\rm V_{\\rm smear} \\sim | \\frac{\\partial V_{\\rm 2}}{\\partial t}|\\Delta \\rm t_\\phi $ .",
"The orbital smearing is maximal at the conjunction phases and for an exposure time of 600 s it is $\\sim $ 60 km s$^{-1}$ .",
"Therefore, it is necessary to correct for smearing because the smearing is comparable to the rotational broadening .",
"For each target exposure, we first smear the template spectrum by convolution with a rectangular profile with an amount corresponding to the target's radial velocity and exposure time, and then compute the average template star spectrum.",
"To determine the secondary star's spectral type we compare a broadened version of the average, smeared template star spectrum with the target spectrum.",
"We broaden the template star from 50 to 100 km s$^{-1}$ by convolution with the rotational profile.",
"We subtract a constant, $f_{\\rm star}$ , representing the fraction of light from the template star, multiplied by a smeared, rotationally broadened version of the template star.",
"We eliminate long-scale trends in the residual spectrum by applying a high-pass Gaussian filter of FWHM=400 km s$^{-1}$ .",
"We use a linear limb-darkening coefficient of 0.62 [1] appropriate for 6500 Å and a mid-GV star.",
"We perform a $\\chi ^2$ test on the high-pass filtered residuals of the subtraction, where the optimal values of $v_{\\rm rot}\\,\\rm sin\\,{\\it i}$ and $f_{\\rm star}$ were obtained by minimizing $\\chi ^2$ .",
"For the optimal subtraction procedure we use the wavelength range 6000–6900 Å which is common to all data sets, excluding the emission lines.",
"We also exclude the region 6250–6350 Å because it is contaminated by the weak Doppler smeared diffuse interstellar bands at 6283 Å and the 6300 Å [OI] sky emission-line.",
"Figure: Trailed spectra and Doppler maps of the Hα\\alpha emission-line atdifferent epochs.",
"From left to right, WHT ISIS 2009, VLT X-SHOOTER 2016and WHT ISIS 2014.",
"From top to bottom; trailed spectra, computed spectraand Doppler maps.White and black show absorption and emission, respectively.The ISIS 2009 pulsar-state spectra clearly show thenarrow Hα\\alpha absorption-line from the secondary star (white S-wave), which can alsobe seen in the VLT X-SHOOTER 2016 and WHT ISIS 2014 disc-state spectra.The Roche lobe of the secondary star is also plotted for qq=0.141 andK 2 K_{\\rm 2}=270 km s -1 ^{-1}.",
"The circles show the gas stream and the blue cross,the position of the neutron star.Figure: Isolating the narrow Hα\\alpha narrow absorption component arising from thesecondary star in PSR J1023+0038.",
"We show the average Hα\\alpha profile in therest frame of the system's centre-of-mass (top) and the average spectrum of thesecondary star without the accretion disc profile, showing the narrowHα\\alpha absorption component (bottom).The spectra have been offset for clarity.Figure: (a) An example of the diagnostic diagram for the Hα\\alpha emission lineof the X-SHOOTER 2016 The panels show the fitted sine-wave parameters areshown as a function of the Gaussian separation Δv\\rm \\Delta v. (b) Theoptimal result of the double-Gaussian technique to measure the Hα\\alpha (black) and Hβ\\beta (blue) emission-line radial velocity curve.",
"Theresulting best-fit sinusoidal model is shown (solid line).",
"We show theresults for the X-SHOOTER 2016 (top) and ISIS 2014 (bottom) disc-statespectra."
],
[
"Metallic absorption lines",
"In Fig.",
"REF we show the results using the ISIS 2009, ISIS 2014 and X-SHOOTER 2016 spectra of PSR J1023+0038 between orbital phase 0.0 and 1.0, for template stars in the F to K spectral range.",
"The $\\chi ^2$ versus spectral type obtained from the optimal subtraction has a minimum at G2$\\pm $ 2, F8$\\pm $ 2 and G0$\\pm $ 2 for the ISIS 2009 pulsar-state, ISIS 2014 disc-state and X-SHOOTER 2016 disc-state, respectively.",
"The uncertainties in the spectral type were estimated from a parabolic fit to the minimum.",
"For each data set, the result of an circular orbit fit to the radial velocities obtained using the optimal spectral type is shown in Table REF .",
"For the optimal spectral type, we obtain metallic absorption-line radial velocity semi-amplitudes of $K_{\\rm 2,Z}$ =282.2$\\pm $ 3.2 km s$^{-1}$ , 274.9$\\pm $ 2.1 km s$^{-1}$ and 270.0$\\pm $ 1.7 km s$^{-1}$ , respectively.",
"The 1$\\sigma $ errors are quoted where the error bars have been rescaled so that the reduced $\\chi ^2$ of the optimal subtraction fit is 1.",
"Fig.",
"REF (left) we show the radial velocity curves obtained from the ISIS 2009, ISIS 2014 and X-SHOOTER 2016 spectra folded on the ephemeris determined from the optimal spectral type.",
"The value for $T_0$ determined independently for each dataset is consistent with the calculated $T_0$ adopting an ephemeris with constant orbital period.",
"To determine the secondary star's rotational broadening the spectral resolution of the data should be less than the combined effects of the instrumental resolution plus orbital smearing.",
"The ISIS 2014 spectra has the worst resolution and hence cannot be used to determine the secondary star's rotational broadening.",
"The instrumental spectral resolution is 24, 70 and 40 km s$^{-1}$ for the ISIS 2009, 2014 and X-SHOOTER 2016 spectra, respectively.",
"The maximum smearing is $\\sim $ 60 km s$^{-1}$ for the ISIS 2009 and 2014 spectra, and $\\sim $ 35 km s$^{-1}$ for the X-SHOOTER 2016 spectra.",
"For the ISIS 2009 and X-SHOOTER 2016 spectra we obtain a $v_{\\rm rot}\\,\\rm sin\\,{\\it i}$ of 77.7$\\pm $ 2.7 km s$^{-1}$ and 74.9$\\pm $ 0.9 km s$^{-1}$ , respectively.",
"The 1$\\sigma $ errors are quoted where the error bars have been rescaled so that the reduced $\\chi ^2$ of the optimal subtraction fit is 1.",
"However, the ISIS 2014 data with the worst resolution, cannot resolve the star's rotational broadening and hence the $v_{\\rm rot}\\,\\rm sin\\,{\\it i}$ we obtain is biased towards the blended lines.",
"In Fig.",
"REF we show the Doppler-averaged spectrum of the ISIS 2004, ISIS 2009 and X-SHOOTER 2016 data taken as well as the optimal rotationally broadened template star for each data set.",
"As one can see, the ISIS 2009 spectrum shows deeper absorption lines compared to the template star, indeed for any spectral type $f_{\\rm star}$ is greater than 1, suggesting that the secondary star is peculiar (see Section )."
],
[
"Effects of irradiation",
"Although the binary has a circular orbit, the observed radial-velocity curve of the secondary star can appear non-circular.",
"Heating effects can distort the radial-velocity curve leading to an apparent eccentric orbit , .",
"We therefore fit the X-SHOOTER 2016 radial-velocity curve with an eccentric orbit to determine the effects of heating.",
"We perform a least-squares fit of the radial velocities versus time using an eccentric orbit of the form, $V_{\\rm 2}= \\gamma + K_{\\rm 2}[\\cos (\\theta +\\omega ) + e\\cos \\omega ],$ where $\\gamma $ is the systemic velocity of the binary, $\\theta $ is the true anomaly that varies with time, $e$ is the eccentricity of the orbit, $\\omega $ is the periastron angle and $K_{\\rm 2}$ is the radial velocity semi-amplitude .",
"The value for $\\theta (t)$ is given by $\\tan [\\theta (t)/2] = \\sqrt{\\frac{1+e}{1-e}} \\tan (E/2)$ , where $E$ is the eccentric anomaly, obtained by numerically solving the Kepler equation $E-e \\sin E = \\frac{2\\pi }{P}(t-T_{\\rm 0})$ , where $T_{\\rm 0}$ is the time of periastron passage and $P_{\\rm orb}$ is the orbital period of the system.",
"For the X-SHOOTER 2016 spectra, the G0 template star gives a fit with $e$ =0.017$\\pm $ 0.002, which is significantly better than a circular orbit at the 99.4 per cent level.",
"To see if the spectral type and thus effective temperature is different on the heated and non-heated hemisphere of the secondary star, we repeat the spectra type determination analysis in Section by averaging the target spectra around orbital phase $\\phi $ =0.00$\\pm $ 0.15 and orbital phase $\\phi $ =0.50$\\pm $ 0.15.",
"For the X-SHOOTER 2016 spectra, with high signal-to-noise, we find that the secondary star's spectral type around phase 0.0 and 0.5 is significantly different, $\\sim $ G5 and $\\sim $ F6, respectively (see Fig.",
"REF )."
],
[
"H$\\alpha $ absorption-line",
"A trailed spectrogram of the ISIS 2009 pulsar-state data clearly shows narrow H$\\alpha $ in absorption (see Fig.",
"REF ), which arises from the secondary star.",
"For the disc-state spectra, the accretion double-peaked emission-line profile contaminates the narrow H$\\alpha $ absorption-line.",
"Despite this, narrow H$\\alpha $ absorption is clearly seen in the data taken with the highest spectral resolution.",
"It is clearly observed in the X-SHOOTER 2016 spectra and to lesser extent in the ISIS 2014 spectra, which has a poorer spectral resolution.",
"To determine the radial velocities of the H$\\alpha $ absorption-line, we cross-correlate each individual target spectrum with a rotationally broadened H$\\alpha $ absorption-line profile, using only the region around H$\\alpha $ .",
"We then perform a least-squares fit of the radial velocities versus orbital phase using an eccentric orbit and determine the radial velocity semi-amplitude $K_{\\rm 2,H\\alpha }$ (see Section ).",
"For the ISIS 2014 and X-SHOOTER 2016 disc-state spectra, before we perform the cross-correlation, we isolate the narrow H$\\alpha $ absorption-line by subtracting the average accretion disc profile.",
"The narrow H$\\alpha $ absorption-line is not clearly resolved in the ISIS 2014 spectra, mainly due to the spectral resolution of the data, and so we only analyse the X-SHOOTER 2016 spectra.",
"In Fig.",
"REF (right) we show the radial velocity curves obtained from the ISIS 2009 and X-SHOOTER 2016 spectra, where we measure $K_{\\rm 2,H\\alpha }$ =270.4$\\pm $ 5.4 and 293.6$\\pm $ 4.2 km s$^{-1}$ , respectively (the 1$\\sigma $ errors are quoted where the error bars have been rescaled so that the reduced $\\chi ^2$ of the fit is 1).",
"The ISIS 2009 radial velocity curve is consistent with a circular orbit, however, the X-SHOOTER 2016 radial velocity curve is eccentric.",
"It is not clear if this eccentricity is real or if it is due to the uncertainty in disentangling the narrow H$\\alpha $ absorption-line modulation from the double-peaked disc emission-line profile.",
"In Fig.",
"REF the average Doppler-corrected H$\\alpha $ absorption-line spectrum of the X-SHOOTER 2016 and ISIS 2014 spectra in the rest frame of the secondary star are shown."
],
[
"Doppler maps",
"Doppler tomography is used to deduce the accretion structures in binary systems .",
"The method inverts phase-resolved spectra into an equivalent image of brightness distribution in velocity space .",
"It is able to separate various sources of emission, such as from the secondary star or accretion disc, in velocity space.",
"Some of the basic assumptions of Doppler tomography are that velocity vectors rotate with the binary, motion is parallel to the orbital plane and the flux from any point is constant in time.",
"Violations of these assumptions do not imply that Doppler tomography cannot be performed, but care must be taken when interpreting the Doppler maps.",
"To compute the Doppler maps of PSR J1023+0038 we used the normalised continuum subtracted spectra and the Python/C++ maximum entropy Doppler tomography code https://github.com/trmrsh/trm-doppler developed by T. Marsh.",
"We model both absorption and emission-line modulated components.",
"In Fig.",
"REF we show the trailed spectra and Doppler maps of the H$\\alpha $ emission-line seen in the 2014 and 2016 disc-state.",
"We also compute the Doppler map of the ISIS 2009 pulsar-state spectra, which shows the narrow H$\\alpha $ absorption-line arising from the secondary star.",
"We do not compute the Doppler maps of the H$\\beta $ and other lines because the ISIS 2009 and 2014 spectra do not have sufficient signal-to-noise for a comparison with the X-SHOOOTER spectra.",
"The disc-state trailed spectra show the characteristic double-peaked tramlines, a signature of a Keplerian accretion disc, that show up as a constant emission ring-like structure of high intensity in the Doppler maps.",
"The trailed spectra also show a narrow absorption, which is in phase with the secondary star, and a narrow emission-line feature which is shifted by $\\sim $ -0.1 phase with respect to the secondary star, which arises from the accretion disc.",
"Indeed, the crossing of this emission-line feature with the H$\\alpha $ absorption-line feature at orbital phase $\\sim $ 0.7 seems to be responsible for the larger deviations observed in the H$\\alpha $ velocities (see Fig.",
"REF ).The relative strength of the blue and red emission-line peaks vary over the orbital period.",
"and results in an axisymmetric Doppler map with enhanced emission between orbital phase 0.25 and 0.50."
],
[
"The neutron star's radial velocity",
"The emission lines generated from a uniform symmetric accretion disc around a neutron star will have the same velocity shift as the neutron star.",
"The centroids of the lines will produce a sinusoidal phase-dependent modulation corresponding to the orbital motion of the neutron star.",
"The addition of other sources of phase-dependent modulation, such as from a bright spot, results in a radial velocity curve that is offset in phase with respect to the true motion of the neutron star.",
"Therefore, in order to exclude these contributions, one examines the wings of the emission-line profile corresponding to material orbiting at small disc radii from the neutron star, where the contamination by other sources are assumed to be minimal.",
"To measure the radial velocities of the Balmer emission lines we use the well established double-Gaussian technique of .",
"This technique allows one to extract radial velocity curves from the wings of the emission-line profile, which are expected to follow the motion of the neutron star.",
"By convolving the emission-line with double-Gaussian function with varying separation the radial velocity curve is determined.",
"We apply this technique to the high spectral resolution ISIS 2014 and the X-SHOOTER 2016 spectra, where we can clearly see the double-peaked emission-line profile from the accretion disc.",
"We use a Gaussian width of 600 km s$^{-1}$ and vary the separation $\\rm \\Delta v$ from 600 to 3600 km s$^{-1}$ in steps of 200 km s$^{-1}$ .",
"At each Gaussian separation the resulting radial velocity curve was fitted with a sine function of the form $V = \\gamma - K\\sin [2\\pi (\\phi -\\phi _0)]$ where $V$ is the radial velocity, $K$ the velocity semi-amplitude, $\\gamma $ the systemic velocity, $\\phi $ the orbital phase, and $\\phi _0$ is the phase at superior conjunction of the neutron star.",
"The results of the radial velocity analysis as well as the fractional error in the amplitude ($\\sigma K/K$ ), which is a function of the Gaussian separation, are inspected in the form of a diagnostic diagram .",
"By plotting such a diagram it is possible to select the value of $K$ that most closely represents the actual $K_{\\rm 1}$ .",
"The point beyond which the noise in the continuum begins to dominate the signal from the emission-line wings corresponds to a sharp increase in $\\sigma K/K$ .",
"Usually $K_{\\rm 1}$ obtained from a diagnostic diagram is the value corresponding to when $\\sigma K/K$ is at a minimum (see Fig.",
"REF ).",
"For the H$\\alpha $ emission-line we obtain $ K_{\\rm 1}$ =72$\\pm $ 5 km s$^{-1}$ and $\\phi _0$ =0.05$\\pm $ 0.01, whereas for the H$\\beta $ emission-line we obtain $K_{\\rm 1}$ =135$\\pm $ 9 km s$^{-1}$ and $\\phi _0$ =0.05$\\pm $ 0.01 for the ISIS 2014 spectra (see Fig.",
"REF ).",
"For the X-SHOOTER 2016 spectra we obtain $K_{\\rm 1}$ =73$\\pm $ 3 km s$^{-1}$ and $\\phi _0$ =0.028$\\pm $ 0.005 using the H$\\alpha $ emission-line and $K_{\\rm 1}$ =118$\\pm $ 6 km s$^{-1}$ and $\\phi _0$ =0.050$\\pm $ 0.007 using the H$\\beta $ emission-line.",
"Note that the H$\\alpha $ and H$\\beta $ $K_{\\rm 1}$ values for the 2014 and 2016 spectra are consistent with each other.",
"The projected semi-major axis of the pulsar orbit measured from radio timing observations, $x_{\\rm 1}= a_{\\rm 1}\\sin \\,i$ =0.343356 light-seconds allows one to determine the radial velocity semi-amplitude of the neutron star $K_{\\rm 1} = 2\\pi c x_{\\rm 1} /P_{\\rm orb}$ = 38 km s$^{-1}$ (where $a$ is the binary separation and $c$ is the speed of light).",
"The fact that we obtain $K_{\\rm 1}$ values (see Section REF ) that are different for the H$\\alpha $ and H$\\beta $ emission lines wings with relatively large phase offsets, which do not agree with the expected value derived from radio timing, means that the wings of the emission-line profiles do not follow the motion of the neutron star.",
"Indeed, it has been shown that bright-spot non-Keplerian motion can induce a measured $K_{\\rm 1}$ larger than the true value , .",
"The motion of the emission lines reflect the region where that emission lines are excited, further out in the accretion disc."
],
[
"Doppler maps",
"The X-ray and optical flaring mode-switching behaviour of PSR J1023+0038 in the disc-state has been extensively studied by a number of authors , , , .",
"The optical and X-ray passive and active mode-switching behaviour have transition time-scales of <20 s, and it has been suggested that they are due to fast of a propeller, where matter in the disk in-flow is propelled away by the rapidly rotating neutron star magnetosphere .",
"were able to isolate the passive and active-state spectra from time resolved H$\\alpha $ spectroscopy taken in 2017.",
"Narrow H$\\alpha $ absorption-line feature can be clearly seen in the active-state spectra, which is not present in the passive-state spectra.",
"They also found evidence for a lack of H$\\alpha $ emission around orbital phase 0.25–0.50 in the active-state and interpret it as matter being ejected from the system via the propeller effect .",
"Our 2014 and 2016 observations also shows narrow H$\\alpha $ absorption-line feature but in contrast to we observe excess H$\\alpha $ emission around orbital phase 0.25–0.50.",
"Note that use the orbital ephemeris given in , which agrees with our ephemeris (see Table REF ).",
"Figure: Theoretical [A84,A41] values measured from the accretion disc Hα\\alpha emission-line profile for each [α,R\\alpha ,R] pair .",
"The redand black lines represent constant α\\alpha and RR,respectively.",
"Our observed Hα\\alpha value is shown as the solid filledblue circle.Figure: A comparison of the PSR J1023+0038 pulsar- and disc-state spectrum with syntheticspectra.",
"From top to bottom: the 2016 disc-state, 2009 pulsar-statespectrum and the spectrum of a rotationally-broadened G5V star.We also over-plot the rotationally broadened synthetic spectra of a 5600 Klogg\\log \\,g=4.5 star with Z = --1.0 (blue dashed line), ++0.0 (red solid line)and ++1.0 (yellow dot-dashed line).",
"The observed spectra have been shiftedto the rest frame of the synthetic spectrum."
],
[
"Accretion disc size",
"One can estimate the inner and outer accretion disc radii in accreting MSPs by measuring the emission-line wing and peak separations, respectively.",
"The Keplerian velocity of an annulus of gas in Keplerian motion at radius $r$ around a neutron star of mass $M_{\\rm 1}$ orbiting at an inclination angle $i$ is given by $V_{\\rm kep} = \\sqrt{ \\frac{ G M_{\\rm 1} }{ r } } \\sin i.$ For a double-peaked emission-line profile from an accretion disc, the Keplerian velocity of the outer disc radius ($r_{\\rm out}$ ) corresponds to half the peak-to-peak separation, while the Keplerian velocity at the inner accretion disc radius ($r_{\\rm in}$ ) corresponds to half the wing separation (full width at zero intensity).",
"We measure velocities using spectra in which the wings and peaks of the accretion disc emission-line are not affected by bright-spot emission or absorption from the secondary star, which is around orbital phase 0.0.",
"We use the average 2016 X-SHOOTER disc-spectra taken around orbital phase 0.0 and measure the H$\\alpha $ half peak-to-peak separation and half wing separation to be 367$\\pm $ 6 km s$^{-1}$ and $\\sim $ 2000 km s$^{-1}$ , respectively.",
"Note that the wing separation is difficult to establish because of the uncertainty in where the line wings end and the continuum begins and possible contamination from other absorption and disc emission lines.",
"We estimate the ratio $R=r_{\\rm in}/r_{\\rm out}$ to be $\\sim $ 0.03.",
"However, it should be noted that there is clear evidence from studies of cataclysmic variable that the for the outer disc is sub-Keplerian and so $R$ is underestimated.",
"Accounting for this $\\sim $ 20 per cent effect, gives $R\\sim $ 0.04 We can also estimate $R$ using the method developed by .",
"Smak's method assumes an axially symmetric accretion disc in Keplerian motion and a power-law flux distribution $f \\propto r^{-\\alpha }$ for the disc emission.",
"He found a relationship between the pairs [A84,A41] and [$\\alpha ,R$ ], where parameters A84 = $\\log W_{\\rm 0.4} - \\log W_{\\rm 0.8}$ and A41 = $\\log W_{\\rm 0.1} - \\log W_{\\rm 0.4}$ can be measured from the emission-line profile ($W_{\\rm 0.8}$ , $W_{\\rm 0.4}$ and $W_{\\rm 0.1}$ are the emission-line widths at the fractions 0.8, 0.4 and 0.1 of the peak height above the continuum, respectively).",
"We apply Smak's method to the average 2016 X-SHOOTER H$\\alpha $ disc-state spectrum taken around orbital phase 0.0.",
"In Fig.",
"REF we show the theoretical [A84,A41] pairs for a given [$\\alpha ,R$ ] as well as our observed values, [A84,A41] = [0.19,0.22], which corresponds to $R\\sim $ 0.075 and $\\alpha \\sim $ 2.0.",
"Assuming that the accretion disc radius does not exceed the maximum disc radius determined by tidal interactions with the secondary star, approximated by $r_{\\rm tidal}/a$ = 0.6 /(1 + q) $\\sim $ 0.94 $\\rm R_{}$ ($q$ =0.141, $M_{\\rm 1}$ +$M_{\\rm 2}$ =2.0 $\\rm M_{}$ and $a$ is the binary separation; ), and $R\\sim $ 0.040–0.075, we estimate $r_{\\rm in}\\sim $ 0.04–0.07 $\\rm R_{} \\sim $ 26,000–50,000 km.",
"This is the location of the inner disc radius that produces the bulk of the H$\\alpha $ emission which lies further out from from the inner edge of the accretion disc $\\sim 150$ km , .",
"also estimate $r_{\\rm in}$ by modelling the 2001 disc-state continuum spectrum with a simple accretion disc model.",
"They obtained $r_{\\rm in}$ $\\sim $ 14,000 km.",
"which is about a factor of 2 closer to the neutron star compared to what we measure from our 2016 disc-state spectrum.",
"However, it should be noted that the 2001 spectrum was taken just before PSR J1023+0038 switched to the pulsar-state sometime between 2001–2003 , whereas our 2016 spectrum was just after PSR J1023+0038 transitioned into the disc-state in 2013 .",
"This suggests that during the extended period of accretion in the disc-state, the region responsible for the H$\\alpha $ emission drifts inwards towards the neutron star, before transitioning to the pulsar-state."
],
[
"A peculiar secondary star",
"In Fig.",
"REF we show the variance-weighted Doppler-averaged spectrum of PSR J1023+0038 taken in 2009 and 2016, as well a G5V template star artificially broadened by 75 km s$^{-1}$ for comparison.",
"As one can see, the pulsar-state 2009 spectrum shows deeper absorption lines compared to the F8 or G5V star, in particular the Ca i lines at 6439.075 Å, 6493.781 Å and Ca i 6462.567 Å, and the $\\sim $ 6495 Å blend of Ca i 6493.781 Å, Fe i 6494.980 Å and Ba ii 6496.897 Å.",
"This can also be seen in the value for $f_{\\rm star}$ obtained in Section , which are all greater than 1 for all spectral type.",
"A full chemical analysis is beyond the scope of this paper, but we compare the observed PSR J1023+0038 spectrum with synthetic high-resolution spectra from the PHOENIX library .",
"We show the synthetic spectrum for a 5600 K $\\log \\,g$ =4.5 star with a metallicity Z of $-$ 1.0, 0.0 and $+$ 1.0.",
"We find that the 2016 disc-state spectrum of PSR J1023+0038 and the G5V spectrum are reasonably well described by the Z=0.0 synthetic spectrum.",
"However, the 2009 pulsar-state spectrum is better described by the Z=$+$ 1.0 synthetic spectrum.",
"Fe and Ca seems to be over-abundant in the atmosphere of the secondary star, and so makes the secondary star in PSR J1023+0038 peculiar.",
"In general the features in the 2009 pulsar-state spectrum are $\\sim $ 5 times stronger than in the 2016 disc-state spectrum."
],
[
"Spectral type changes along the orbit",
"The light curves taken when PSR J1023+0038 was in the pulsar-state , show an asymmetric single-humped modulation, which is expected from the combined effects of the tidally-locked secondary star's ellipsoidal modulation and the high-energy emission from the pulsar wind heating of the inner face of the secondary star.",
"The light curve observed in the disc-state is very similar in shape and amplitude , suggesting that the effect of heating on the secondary star is the same when the system is in the pulsar- or disc-state and that the main source of irradiation of the secondary star is the high-energy emission from the pulsar relativistic wind.",
"The asymmetric maximum suggests that the heating does not come directly from the isotropic pulsar wind, but is due to non-thermal X-ray emission produced by the intra-binary shock between the pulsar and secondary star's wind .",
"Given that heating with a bolometric irradiating luminosity of $10^{34}$ erg s$^{-1}$ has a pronounced effect on the phase-resolved $V$ -band light curves in the pulsar-state, we expect to see similar effects on the phase-resolved spectroscopy.",
"Indeed, the 2016 disc-state spectra show evidence for a spectral type change across the orbit, from G5 at phase 0.0 to F6 at phase 0.5, corresponding to a temperature change of 5650 K to 6340 K, respectively .",
"This is also supported by the significant eccentric fit to the metallic absorption-line radial velocity curve (see Table REF ), where irradiation distorts the radial velocity curve of the secondary star, as the centre-of-light of the absorption lines changes with orbital phase.",
"Significant eccentric orbits are not detected in the 2009 pulsar-state and 2014 disc-state spectra, because of the lower quality and spectral resolution of the 2009 and 2014 spectra compared to the 2016 disc-state spectra.",
"The moderate inclination angle of 54$^\\circ $ of the system implies that the contrast between the observed spectral type at different orbital phases is reduced, which means that the observer always sees the heated inner face of the secondary star and thus the true spectral type of the secondary star is cooler.",
"Simultaneous modelling of the optical light- and radial-velocity curves during the pulsar-state should reveal the true spectral type of the secondary star , Figure: The observed secondary star's metallic (top) and Hα\\alpha (bottom)absorption-line radial velocity semi-amplitude using optical spectra takenin 2004 and 2009 in the pulsar-state (filled red circles) and in 2014 and2016 in the disc-state (blue crosses), respectively (seeTable ).",
"The vertical band marks the time when thesystem transitioned from a LMXB to a MSP around 2000-2001 and from a MSP to a LMXB in 2013 ."
],
[
"The secondary star's radial velocity curve",
"The absorption lines arising from the secondary star are Doppler shifted as the star moves in its orbit.",
"Therefore, the radial velocity at a given orbital phase depends on the observed centre-of-light of the lines that are used to determine the radial velocities.",
"Heating shifts the centre-of-light of the secondary, weighted by the strength of the absorption lines, from the centre-of-mass of the star.",
"This results in a distortion of the radial velocity curve leading to an apparent elliptical/eccentric orbit with a spurious radial velocity semi-amplitude , .",
"Variable amounts of heating will produce different radial velocity semi-amplitudes.",
"In Fig.",
"REF we show the different values for $K_{\\rm 2,Z}$ and $K_{\\rm 2,H\\alpha }$ determined between 2004 and 2016, when the system was either in the pulsar- (P) or disc- (D) state.",
"The value for $K_{\\rm 2,Z}^{\\rm P}$ determined in 2004 and 2009 in the pulsar-state are different at the 2.8$\\sigma $ level, and so we can assume that there is no significant change in $K_{\\rm 2,Z}$ The value for $K_{\\rm 2,Z}^{\\rm D}$ determined from the 2014 and 2016 spectra, when the system was in the disc-state are similar at the 1.8$\\sigma $ level, which is not surprising given that the X-ray and $\\gamma $ -ray flux are also very similar (see Fig.",
"REF ).",
"During the 2009 to 2014 transition the change in $K_{\\rm 2,Z}$ is only significant at the 1.9$\\sigma $ level.",
"However, the 2009 to 2016 transition the change in $K_{\\rm 2,Z}$ is significant at the 3.4$\\sigma $ level which can be explained by the accretion disc shadowing (see Section REF ).",
"Figure: (a) We show the apparent radial velocity curve deviation from a circular orbit forthe pulsar-state (top panel) and disc-state (bottom panel).",
"In each panelwe plot the radial velocity deviation using the metallic absorption linesin the 6000–6500Å spectral range (solid blue) and the Balmerabsorption line (dashed red).",
"We assume a T eff T_{\\rm eff}=5660 K secondarystar, qq=0.14, ii=43 ∘ ^\\circ , K 2 K_{\\rm 2}=270 km s -1 ^{-1} and L X =10 34 L_{\\rm X}=10^{34} erg s -1 ^{-1}.",
"For the pulsar-state we assume f Roche f_{\\rm Roche}=0.83, withno accretion disc and for the disc-state we assume f Roche f_{\\rm Roche}=1 andan accretion disc that extends to the tidal radius with an opening angleof 20 ∘ ^\\circ .",
"On the right we show projected maps of the observedmetallic and Balmer line-strength distribution on the secondary star atorbital phase 0.75 using xrbcurve , .",
"(b) We show the simulated absorption-line profile in the pulsar-state (redsolid line) and disc-state (blue dashed line), observed at orbital phase0.25.",
"The same model parameters as above are used.",
"Note the change in theline profile shape."
],
[
"The pulsar-state radial velocity curve",
"We can use our X-ray binary model xrbcurve , to determine the line flux distribution and hence radial velocity curve arising from the secondary star in PSR J1023+0038, for a system in the pulsar- and disc-state, where one may expect different amounts of heating.",
"An important parameter that determines the shape and amplitude of the radial velocity curve is the effects of heating and how the line-strength versus temperature relation used in determining the radial velocity.",
"We assume the ”deep heating\" approximation which means that each element radiates as predicted by a model atmosphere for a single star.",
"The spectral type of the secondary stars in MSPs and X-ray binaries are typically later than F and their spectra contain metal absorption lines.",
"The strongest absorption metallic lines in the red part of the optical spectrum are the Fe ı, Ca i and Ba ii lines (see Fig.",
"REF ).",
"Using UVESPOP spectra we find that the line strength versus temperature relation of the metallic absorption lines in the 6000–6500 Å spectral range decreases with increasing temperature.",
"Similarly, for the H$\\alpha $ absorption-line (Balmer), we find that the line strength increases with increasing temperature , .",
"Therefore, we can compute the secondary star's metallic ($K_{\\rm 2,Z}$ ) or Balmer ($K_{\\rm 2,H}$ ) absorption-line radial velocity semi-amplitude.",
"Equations REF to are the conditions we find due the effects of irradiation and/or accretion disc on the metallic and Balmer line radial velocity curves semi-amplitudes.",
"$K_{\\rm 2,Z}^{\\rm P} > K_{\\rm 2,H}^{\\rm P} \\\\K_{\\rm 2,Z}^{\\rm D} > K_{\\rm 2,H}^{\\rm D} \\\\K_{\\rm 2,Z}^{\\rm D} < K_{\\rm 2,Z}^{\\rm P} \\\\K_{\\rm 2,H}^{\\rm D} > K_{\\rm 2,H}^{\\rm P} \\\\K_{\\rm 2,Z}^{\\rm D} > K_{\\rm 2,H}^{\\rm D} $ In the pulsar-state, high energy relativistic particles produced by the pulsar wind are mostly likely responsible for heating the secondary star's inner face.",
"The hot inner face thus produces less metallic line flux compared to the non-heated face of the star, and so the centre-of-light is shifted towards the non-heated side of the star, resulting in a spuriously high radial velocity semi-amplitude; $K_{\\rm 2,Z}^{\\rm P} > K_{\\rm 2}$ .",
"In Fig.",
"REF we show the metallic and Balmer absorption-line radial velocity curve deviations.",
"The model predicts that the metallic radial velocity semi-amplitude is greater than the Balmer radial velocity semi-amplitude in both the pulsar- and disc-states (see Equation REF ).",
"In contrast, the hot inner face produces more Balmer flux on the heated face compared to the non-heated face of the star, and so the centre-of-light is shifted towards the heated side of the star, resulting in a spuriously low radial velocity semi-amplitude; $K_{\\rm 2,H}^{\\rm P}< K_{\\rm 2}$ .",
"Our ISIS 2009 observations in the pulsar-state imply 270.4 $<K_{\\rm 2}<$ 282.2 km s$^{-1}$ (or $K_{\\rm 2}$ = 276.3$\\pm $ 5.6 km s$^{-1}$ )."
],
[
"The pulsar- to disc-state radial velocity",
"The pulsar- to disc-state transition involves an increase in the gamma-ray and X-ray flux , resulting in a change in the metallic and Balmer line radial velocity semi-amplitude because of the effects of the accretion disc shadowing.",
"In Fig.",
"REF we show the effects the accretion disc has on the shape of the metallic absorption-line profile.",
"We show the simulated line profile in the pulsar- and disc-state observed at orbital phase 0.25.",
"The elements of area near the inner Lagrangian point contribute flux to the negative wing of the line profile.",
"We assume a bolometric irradiating luminosity of 10$^{34}$ erg s$^{-1}$ which is the same in the pulsar- and disc-state.",
"In the pulsar-state elements of area on the hot irradiated inner face of the star produce less metallic line flux (because of the line-strength versus temperature relation) compared to the non-heated face of the star, and so the centre-of-light of the line profile (blue solid line) is shifted towards the non-heated side of the star.",
"The appearance of an accretion disc shadows the elements area near the inner Lagrangian point (negative wing of the line profile at phase 0.25), which intrinsically have lower temperatures because of gravity darkening, and so contribute more flux than in the pulsar-state.",
"The line profile is skewed and the effect of the disc shadowing is to shift the centre-of-light of the line profile towards the inner Lagrangian point, resulting in a lower radial velocity semi-amplitude compared to in the pulsar-state (see Equation ).",
"For the H$\\alpha $ absorption-line, the opposite line-strength versus temperature relation implies that the centre-of-light of the absorption-line profile is shifted away from the inner Lagrangian point.",
"This results in a higher semi-amplitude compared to when there is no accretion disc (see Equation ).",
"Finally the accretion disc shadowing also gives the condition in Equation .",
"We find that the 2009 pulsar-state data satisfies equation REF and the metallic and Balmer radial velocities determined from the 2009 to 2014/2016 transition satisfies equation and , respectively.",
"However, the 2016 disc-state data does not satisfy equation or , but this could be due to problems in disentangling the narrow H$\\alpha $ absorption-line modulation from the double-peaked disc emission-line profile.",
"Therefore the observed changes in the $K_{\\rm 2,Z}$ and $K_{\\rm 2,H}$ in the pulsar- and disc-state can be explained to some extent by the presence of an accretion disc."
],
[
"The Roche lobe filling factor in the pulsar-state",
"In contrast to the disc-state, in the pulsar-state no accretion disc is present and hence the secondary star does not fully fill its Roche lobe.",
"Indeed, observations of the MSP secondary stars in the pulsar-state show that they underfill their Roche lobes; the Roche lobe filling factor is in the range $f_{\\rm Roche}$ =0.80–0.95 , .",
"Given that the star's rotational broadening depends on its radius, the star's radius is smaller in the pulsar-state compared to in the disc-state.",
"It can be shown that the star's equivalent volume radius depends on the binary mass ratio and Roche lobe filling factor, as well as its radial velocity semi-amplitude .",
"Using equation (2) in we can calculate the change in the rotational broadening for different values of $f_{\\rm Roche}$ ($q$ =0.14, see Section REF ) and compare it to the observed change in $v_{\\rm rot}\\,\\rm sin\\,{\\it i}$ .",
"Given the observed X-ray/gamma-ray luminosities, the effects of irradiation in the determination of $v_{\\rm rot}\\,\\rm sin\\,{\\it i}$ in the two states are similar and so appropriately cancel out.",
"In Section we determine the secondary star's rotational broadening in the pulsar-state to be larger than in the disc-state (see Table REF ).",
"However, it should be noted that the pulsar-state $v_{\\rm rot}\\,\\rm sin\\,{\\it i}$ measurement has a large error.",
"The ratio of pulsar-state to disc-state $v_{\\rm rot}\\,\\rm sin\\,{\\it i}$ value is 1.037$\\pm $ 0.038.",
"Using the 3$\\sigma $ lower limits, we find a change in $v_{\\rm rot}\\,\\rm sin\\,{\\it i}$ of $>$ 0.923, which corresponds to a Roche lobe filling factor of $f_{\\rm Roche}$ >0.78, agreeing with what was obtained by by modelling the $V$ -band light curve obtained by in the pulsar-state."
],
[
"Irradiating luminosity",
"In the pulsar- and or disc-state, the relativistic pulsar wind and and/or X-ray/gamma-ray emission from the accretion disc are the dominant sources of irradiation.",
"We can compute the irradiating luminosity temperature difference of hemispheres of the secondary star and compare it to the observed X-ray and gamma-ray luminosities, as well as to the pulsar's spin down luminosity ($L_{\\rm sd}$ ) and determine the source of the driving mechanism.",
"The “irradiation temperature” is given by $T_{\\rm irr}^4 =T_{\\rm day}^4 -T_{\\rm night}^4$ , where $T_{\\rm night}$ and $T_{\\rm day}$ are the temperatures observed at orbital phase 0.0 and 0.5, respectively.",
"The irradiating luminosity is then $L_{\\rm irr}=4\\pi a^2\\sigma T_{\\rm irr}^4$ where $a$ is the orbital separation, which is related to $L_{\\rm sd}$ via the efficiency of irradiation parameter; $\\eta _{\\rm irr}=L_{\\rm irr}/L_{\\rm sd}$ .",
"Using $T_{\\rm night}$ =5650 K and $T_{\\rm day}$ =6340 K we find $T_{\\rm irr}$ =4943 K and $L_{\\rm irr}=6.5\\times 10^{33}$ erg s$^{-1}$ , which implies $\\eta _{\\rm irr}$ =14 percent.",
"For PSR J1023+0038 in the disc-state, the observed X-ray (0.5–10 keV) luminosity is $2\\times 10^{33}$ erg s$^{-1}$ , the gamma-ray (0.2–20 GeV) luminosity is $6\\times 10^{33}$ erg s$^{-1}$ and the pulsar's spin-down luminosity (corrected for the Shklovskii effect) is $L_{\\rm sd}=4.8\\times 10^{34}$ erg s$^{-1}$ .",
"The irradiation efficiency of 14 percent we derive is similar to what is observed in other MSPs .",
"Indeed, the energetics suggest that the pulsar's the relativistic wind, powered by the rotational spin down of the neutron star can drive the observed heating mechanism in the disc-state.",
"However, in PSR J1023+0038 the observed X-ray and gamma-ray (most likely due to accretion) luminosities are also sufficient to provide the observed irradiating luminosity."
],
[
"The system parameters",
" determined the binary mass ratio of $q$ =0.141$\\pm $ 0.002 ($q$ =$K_{\\rm 1}$ /$K_{\\rm 2}$ ) by measuring the radial velocity semi-amplitude of the neutron star determined from radio timing $K_{\\rm 1}$ =38 km s$^{-1}$ and the secondary star's radial velocity semi-amplitude 268$\\pm $ 4 km s$^{-1}$ determined by .",
"We update this value using our $K_{\\rm 2}$ value determined in Section REF to obtain $q$ =0.137$\\pm $ 0.003.",
"We can also determine the mass ratio from the fact that the Roche lobe filling star's rotational broadening radius depends only $q$ and $K_{\\rm 2}$ in the disc-state , .",
"Using $v_{\\rm rot}\\,\\rm sin\\,{\\it i}$ =74.9$\\pm $ 0.9 km s$^{-1}$ and our $K_{\\rm 2}$ value we obtain $q$ =0.20$\\pm $ 0.03.",
"It should be noted that this value is uncertain because the limb-darkening coefficient used in the determination of $v_{\\rm rot}\\,\\rm sin\\,{\\it i}$ leads to an overestimation of secondary star's rotational broadening.",
"Also, irradiation leads to an underestimation of $v_{\\rm rot}\\,\\rm sin\\,{\\it i}$ .",
"Despite this, the value for $q$ is consistent, at the 2-$\\sigma $ level, with the revised value determined earlier.",
"From the mass function equation, one can use $K_{\\rm 2}$ , $q$ , the orbital period and the binary inclination $i$ to determine the masses of the compact object and secondary star ($M_{\\rm 2}$ ).",
"Using our values for $K_{\\rm 2}$ = 276.3$\\pm $ 5.6 km s$^{-1}$ and $q$ =0.137$\\pm $ 0.003 we obtain $M_{\\rm 1} = \\frac{0.56 \\pm 0.03}{\\sin ^3\\,i} M_{\\odot }, \\hspace{42.67912pt}M_{\\rm 2} = \\frac{0.077 \\pm 0.005}{\\sin ^3\\,i} M_{\\odot }$ For a neutron star with a mass in the range 1.4 to 3 $\\rm M_{}$ , corresponding to the canonical and maximum theoretical mass of a neutron star , respectively, implies $i$ between 47 and 35$^\\circ $ , respectively.",
"modelled the multi-colour ($B$ , $V$ and $I$ ) optical lightcurves of PSR J1023+0038 in the 2009 pulsar-state with a fully Roche-lobe filling irradiated secondary star model.",
"They noted that assuming that if the secondary fills its Roche lobe its mass can be inferred from the distance ($D_{\\rm kpc}$ ), because the distance determines the temperature and size of the star.",
"If the star underfills its Roche lobe then $D_{\\rm kpc} < 2.20 (M_{\\rm 2}/M_{\\odot } )^{1/3}$ .",
"used this relation with $K_{\\rm 2}$ measured by , $K_{\\rm 1}$ from radio timing and the distance $D_{\\rm kpc}$ of 1.37 kpc from the known radio parallax to determine $M_{\\rm 1}$ .",
"We use the same method but with our revised values for $K_{\\rm 2}$ and $q$ to obtain $M_{\\rm 1}>1.76\\pm $ 0.16 $\\rm M_{}$ , $M_{\\rm 2}>0.24\\pm $ 0.02 $\\rm M_{}$ and $i$ <43$\\pm $ 2$^\\circ $ .",
"The mass estimates are lower limits because the secondary star in the pulsar-state can substantially underfills its Roche lobe.",
"The $V$ -band optical lightcurve and a radial velocity curve presented in was also modelled by .",
"Allowing for the secondary star's Roche lobe filling factor to vary, they find that the secondary star underfills its Roche lobe and $i\\sim $ 54$\\pm $ 5$^\\circ $ .",
"This inclination angle is much higher than what we estimate above and gives an unusually low mass for the neutron star; $M_{\\rm 1}=1.1\\pm $ 0.2 $\\rm M_{}$ .",
"Given that we expect relatively massive ($\\sim $ 1.8 $\\rm M_{}$ ) neutron stars to reside in MSPs, we suspect their determination of $i$ is biased.",
"Simultaneous modelling of multi-band lightcurves with high resolution spectroscopy is needed.",
"This will allow one to determine effective temperature of the secondary, its true radial velocity semi-amplitude, inclination angle and hence the mass of the neutron star."
],
[
"Conclusions",
"We present time-resolved optical spectroscopy of the binary millisecond pulsar PSR J1023+0038 during its 2009 radio-powered pulsar-state and during its accretion-powered disc-states in 2014 and 2016.",
"Below we list the main results of this paper.",
"We provide observational support for the companion star being heated during the disc-state.",
"We observe a spectral type change along the orbit, from $\\sim $ G5 to $\\sim $ F6 at the secondary star's superior and inferior conjunction, respectively, which correspond to the \"day\" and \"night\" side temperatures of the secondary star.",
"We find that the irradiating luminosity can be powered by the spin down luminosity of the neutron star, as is the case on many other MSPs, or by the accretion luminosity of the accretion disc.",
"We determine the secondary star's radial velocity semi-amplitude from the metallic (primarily Fe and Ca) and H$\\alpha $ absorption lines during the pulsar- and different disc-states.",
"We find that the observed changes in the metallic radial velocity semi-amplitude is only significant (at the 3.4$\\sigma $ level) for the 2009 pulsar-state to 2016 disc-state transition, which can be explained by the accretion disc shadowing.",
"The metallic and H$\\alpha $ radial velocity semi-amplitude determined from the 2009 pulsar-state observations allows us to constrain the secondary star's true radial velocity $K_{\\rm 2}$ =276.3$\\pm $ 5.6 km s$^{-1}$ and the binary mass ratio $q$ =0.137$\\pm $ 0.003.",
"Doppler maps of the disc-state spectra show characteristic double-peaked emission-line profile arising from the accretion disc, a narrow absorption, which is in phase with the secondary star, and a narrow emission-line feature which is shifted by $\\sim $ -0.1 phase with respect to the secondary star, which arises from the accretion disc.",
"From the average emission-line profile of the accretion disc in 2016, we place constraints on the inner to outer disc radii ratio, 0.04–0.075.",
"By comparing the observed metallic and H$\\alpha $ absorption-line radial velocity semi-amplitudes with model predictions, we can explain the observed semi-amplitude changes during the different pulsar states and during the pulsar/disc-state transition as being due to different amounts of heating and the presence of an accretion disc, respectively.",
"We thank Tom Marsh for the use of his molly, pamela and Doppler tomography programs.",
"We acknowledge the use of data from the UVES Paranal Observatory Project (ESO DDT Program ID 266.D-5655).",
"M.L.",
"is supported by EU's Horizon 2020 programme through a Marie Sklodowska-Curie Fellowship (grant No.",
"702638).",
"Based on observations made with the WHT telescope operated by the Instituto de Astrofísica de Canarias in the Spanish Observatories of el Roque de los Muchachos (La Palma).",
"Based on observations made with ESO Telescopes at the La Silla Paranal Observatory under ESO programme 096.D-0808.",
"This paper makes use of data obtained from the Isaac Newton Group Archive which is maintained as part of the CASU Astronomical Data Centre at the Institute of Astronomy, Cambridge.",
"The Starlink software is currently supported by the East Asian Observatory.",
"Facilities: WHT (ISIS), WHT (ACAM), VLT (X-SHOOTER)"
]
] | 1906.04524 |
[
[
"Subspace Attack: Exploiting Promising Subspaces for Query-Efficient\n Black-box Attacks"
],
[
"Abstract Unlike the white-box counterparts that are widely studied and readily accessible, adversarial examples in black-box settings are generally more Herculean on account of the difficulty of estimating gradients.",
"Many methods achieve the task by issuing numerous queries to target classification systems, which makes the whole procedure costly and suspicious to the systems.",
"In this paper, we aim at reducing the query complexity of black-box attacks in this category.",
"We propose to exploit gradients of a few reference models which arguably span some promising search subspaces.",
"Experimental results show that, in comparison with the state-of-the-arts, our method can gain up to 2x and 4x reductions in the requisite mean and medium numbers of queries with much lower failure rates even if the reference models are trained on a small and inadequate dataset disjoint to the one for training the victim model.",
"Code and models for reproducing our results will be made publicly available."
],
[
"Introduction",
"Deep neural networks (DNNs) have been demonstrated to be vulnerable to adversarial examples [37] that are typically formed by perturbing benign examples with an intention to cause misclassifications.",
"*The first two authors contributed equally to this work.",
"According to the amount of information that is exposed and possible to be leveraged, an intelligent adversary shall adopt different categories of attacks.",
"Getting access to critical information (e.g., the architecture and learned parameters) about a target DNN, the adversaries generally prefer white-box attacks [37], [7], [24], [2], [23].",
"After a few rounds of forward and backward passes, such attacks are capable of generating images that are perceptually indistinguishable to the benign ones but would successfully trick the target DNN into making incorrect classifications.",
"Whereas, so long as little information is exposed, the adversaries will have to adopt black-box attacks [28], [22], [3], [25], [13], [26], [38], [14], [8] instead.",
"In general, black-box attacks require no more information than the confidence score from a target and thus the threat model is more realistic in practice.",
"Over the past few years, remarkable progress has been made in this regard.",
"While initial efforts reveal the transferability of adversarial examples and devote to learning substitute models [28], [22], recent methods focus more on gradient estimation accomplished via zeroth-order optimizations [3], [25], [13], [26], [38], [14].",
"By issuing classification queries to the target (a.k.a., victim model), these methods learn to approach its actual gradient w.r.t.",
"any input, so as to perform adversarial attacks just like in the white-box setting.",
"Despite many practical merits, high query complexity is virtually inevitable for computing sensible estimations of input-gradients in some methods, making their procedures costly and probably suspicious to the classification system.",
"Following this line of research, we aim at reducing the query complexity of the black-box attacks.",
"We discover in this paper that, it is possible that the gradient estimations and zeroth-order optimizations can be performed in subspaces with much lower dimensions than one may suspect, and a principled way of spanning such subspaces is considered by utilizing “prior gradients” of a few reference models as heuristic search directions.",
"Our method, for the first time, bridges the gap between transfer-based attacks and the query-based ones.",
"Powered by the developed mechanism, we are capable of trading the attack failure rate in favor of the query efficiency reasonably well.",
"Experimental results show that our method can gain significant reductions in the requisite numbers of queries with much lower failure rates, in comparison with previous state-of-the-arts.",
"We show that it is possible to obtain the reference models with a small training set disjoint to the one for training CIFAR-10/ImageNet targets."
],
[
"Related Work",
"One common and crucial ingredient utilized in most white-box attacks is the model gradient w.r.t the input.",
"In practical scenarios, however, the adversaries may not be able to acquire detailed architecture or learned parameters of a model, preventing them from adopting gradient-based algorithms directly.",
"One initial way to overcome this challenge is to exploit transferability [37].",
"Ever since the adversarial phenomenon was discovered [37], [7], it has been presented that adversarial examples crafted on one DNN model can probably fool another, even if they have different architectures.",
"Taking advantage of the transferability, Papernot et al.",
"[27], [28] propose to construct a dataset which is labeled by querying the victim model, and train a substitute model as surrogate to mount black-box attacks.",
"Thereafter, Liu et al.",
"[22] study such transfer-based attacks over large networks on ImageNet [32], and propose to attack an ensemble of models for improved performance.",
"Despite the simplicity, attacks function solely on the transferability suffer from high failure rates.",
"An alternative way of mounting black-box attacks is to perform gradient estimation.",
"Suppose that the prediction probabilities (i.e., the confidence scores) of the victim model is available, methods in this category resort to zeroth-order optimizations.",
"For example, Chen et al.",
"[3] propose to accomplish this task using pixel-by-pixel finite differences, while Ilyas et al.",
"[13] suggest to apply a variant of natural evolution strategies (NES) [33].",
"With the input-gradients appropriately estimated, they proceed as if in a white-box setting.",
"In practice, the two are combined with the C&W white-box attack [2] and PGD [23], respectively.",
"Though effective, owing to the high dimensionality of natural images, these initial efforts based on accurate gradient estimation generally require (tens of) thousands of queries to succeed on the victim model, which is very costly in both money and time.",
"Towards reducing the query complexity, Tu et al.",
"[38] and Ilyas et al.",
"[14] further introduce an auto-encoding and a bandit mechanisms respectively that incorporate spatial and temporal priors.",
"Similarly, Bhagoji et al.",
"[26] show the effectiveness of random grouping and principal components analysis in achieving the goal.",
"In extreme scenarios where only final decisions of the victim model are exposed, adversarial attacks can still be performed [1], [4].",
"Such black-box attacks are in general discrepant from the score-based attacks, and we restrict our attention to the latter in this paper.",
"As have been briefly reviewed, methods in this threat model can be divided into two categories, i.e., the transfer-based attacks (which are also known as the oracle-based attacks) and query-based attacks.",
"Our method, probably for the first time, bridges the gap between them and therefore inherits the advantages from both sides.",
"It differs from existing transfer-based attacks in a sense that it takes gradients of reference models as heuristic search directions for finite difference gradient estimation, and benefit from the heuristics, it is far more (query-)efficient than the latest query-based attacks."
],
[
"Motivations",
"Let us consider attacks on an image classification system.",
"Formally, the black-box attacks of our interest attempt to perturb an input $\\mathbf {x}\\in \\mathbb {R}^n$ and trick a victim model $f:\\mathbb {R}^n\\rightarrow \\mathbb {R}^k$ to give an incorrect prediction $\\operatornamewithlimits{arg\\,max}_i f(\\mathbf {x})_i\\ne y$ about its label $y$ .",
"While, on account of the high dimensionality of input images, it is difficult to estimate gradient and perform black-box attacks within a few queries, we echo a recent claim that the limitation can be reasonably ameliorated by exploiting prior knowledge properly [14].",
"In this section, we will shed light on the motivations of our method.",
"Figure: Black-box attack in low-dimensional random subspaces."
],
[
"Attack in Linear Subspaces?",
"Natural images are high-dimensional and spatially over-redundant, which means not all the pixels (or combinations of pixels) are predictive of the image-level labels.",
"A classification model offers its predictions typically through mining discriminative components and suppressing irrelevant variations from raw images [19].",
"One reasonable hypothesis worth exploring in this spirit is that, it is probably less effective to perturb an image on some specific pixels (or along certain directions) when attacking a black-box model.",
"From a geometric point of view, that said, the problem probably has a lower intrinsic dimension than $n$ , just like many other ones [20].",
"To verify this, we try estimating gradients and mounting attacks on low-dimensional subspaces for images, which is bootstrapped by generating $m<n$ random basis vectors $\\mathbf {u}_0, \\ldots \\mathbf {u}_{m-1}$ sequentially on condition of each being orthogonal to the prior ones.",
"We utilize the bandit optimization advocated in a recent paper [14] for gradient estimation, and adopt the same iterative attack (i.e., PGD) as in it.",
"Recall that the bandit mechanism updates its estimation $\\mathbf {g}_t$ at each step by a scaled search direction: $\\mathbf {\\Delta }_t = \\frac{l(\\mathbf {g}_t + \\delta \\mathbf {u}_t^{\\prime })-l(\\mathbf {g}_t - \\delta \\mathbf {u}_t^{\\prime })}{\\delta } \\mathbf {u}_t^{\\prime },$ in which $\\mathbf {u}_t^{\\prime }$ is the search direction sampled from a Gaussian distribution, $\\delta >0$ is a step size that regulates the directional estimation, and $l(\\cdot )$ calculates the inner product between its normalized input and the precise model gradient.",
"The mechanism queries a victim model twice at each step of the optimization procedure for calculating $\\mathbf {\\Delta }_t$ , after which a PGD step based on the current estimation is applied.",
"Interested readers can check the insightful paper [14] for more details.",
"In this experiment, once the basis $\\lbrace \\mathbf {u}_0, \\ldots \\mathbf {u}_{m-1}\\rbrace $ is established for a given image, they are fixed over the whole optimization procedure that occurs on the $m$ -dimensional subspace instead of the original $n$ -dimensional one.",
"More specifically, the search direction $\\mathbf {u}_t^{\\prime }$ is yielded by combining the generated basis vectors with Gaussian coefficients, i.e., $\\mathbf {u}_t^{\\prime }=\\sum _i \\alpha _i\\mathbf {u}_i$ and $\\alpha _i\\sim \\mathcal {N}(0,1)$ .",
"We are interested in how the value of $m$ affects the failure rate and the requisite number of queries of successful attacks.",
"By sampling 1,000 images from the CIFAR-10 test set, we craft untargeted adversarial examples for a black-box wide residual network (WRN) [41] with an upper limit of 2,000 queries for efficiency reasons.",
"As depicted in Figure REF , after $m>500$ , all three concerned metrics (i.e., failure rate, mean and median query counts) barely change.",
"Moreover, at $m=2000$ , the failure rate already approaches $\\sim $ 10%, which is comparable to the result gained when the same optimization is applied in the original image space which has $n=3072$ dimensions.",
"See the red dotted line in Figure REF for this baseline.",
"Similar phenomenon can be observed on other models using other attacks as well, which evidences that the problem may indeed have a lower dimension than one may suspect and it complements the study of the intrinsic dimensionality of training landscape of DNNs in a prior work [20]."
],
[
"Prior Gradients as Basis Vectors? ",
"Since the requisite number of queries at $m=2000$ is already high in Figure REF , we know that the random basis vectors boost the state-of-the-art only to some limited extent.",
"Yet, it inspires us to explore more principled subspace bases for query-efficient attacks.",
"To achieve this goal, we start from revisiting and analyzing the transfer-based attacks.",
"We know from prior works that even adversarial examples crafted using some single-step attacks like the fast gradient (sign) [18] can transfer [28], [22], hence one can hypothesize that the gradients of some “substitute” models are more helpful in spanning the search subspaces with reduced dimensionalities.",
"A simple yet plausible way of getting these gradients involved is to use them directly as basis vectors.",
"Note that unlike the transfer-based attacks in which these models totally substitute for the victim when crafting adversarial examples, our study merely considers their gradients as priors.",
"We refer to such models and gradients as reference models and prior gradients respectively throughout this paper for clarity.",
"More importantly, we can further let these basis vectors be adaptive when applying an iterative attack (e.g., the basic iterative method [18] and PGD [23]), simply by recalculating the prior gradients (w.r.t the current inputs which may be candidate adversarial examples) at each step.",
"Different zeroth-order optimization algorithms can be readily involved in the established subspaces.",
"For simplicity, we will stick with the described bandit optimization in the sequel of this paper and we leave the exploration on other algorithms like the coordinate-wise finite differences [3] and NES [13] to future works.",
"An experiment is similarly conducted to compare attacks in the gradient-spanned subspacesGranted, the prior gradients are almost surely linearly independent and thus can be regarded as basis vectors.",
"and the random ones, in which the WRN is still regarded as the victim model.",
"We compare mounting black-box attacks on different subspaces spanned by the (adaptive) prior gradients and randomly generated vectors as described before.",
"Figure REF summarizes our main results.",
"As in Figure REF , we illustrate the attack failure rates in Figure REF .",
"Apparently, the prior gradients are much more promising than its random counterparts when spanning search subspaces.",
"For more insights, we project normalized WRN gradients onto the two sorts of subspaces and further compare the mean squared residuals of projection under different circumstances in Figure REF .",
"It can be seen that the gradient-spanned subspaces indeed align better with the precise WRN gradients, and over misalignments between the search subspaces and precise model gradients lead to high failure rates.",
"Figure: Comparison of (a) the failure rates when attacking WRN, and (b) mean squared residuals of projecting the precise gradient onto subspaces spanned by random directions or prior gradients.",
"We collect nine models as candidates to obtain the prior gradients: AlexNet , VGG-11/13/16/19 , and ResNet-20/32/44/56 .",
"We add prior gradients corresponding to models from deep to shallow one by one to the basis set."
],
[
"Our Subspace Attack",
"As introduced in the previous section, we reckon that it is promising to apply the gradient of some reference models to span the search subspace for mounting black-box attacks.",
"However, there remain some challenges in doing so.",
"First, it should be computationally and memory intensive to load all the reference models and calculate their input-gradients as basis vectors.",
"Second, it is likely that an “universal” adversarial example for a victim model is still far away from such subspaces, which means mounting attacks solely on them may lead to high failure rate as encountered in the transfer-based attacks.",
"We will discuss the issues and present our solutions in this section.",
"We codename our method subspace attack and summarize it in Algorithm , in which the involved hyper-parameters will be carefully explained in Section .",
"[h] Subspace Attack [1] Input: a benign example $\\mathbf {x}\\in \\mathbb {R}^n$ , its label $y$ , a set of $m$ reference models $\\lbrace f_0,\\ldots ,f_{m-1}\\rbrace $ , a chosen attack objective function $\\mathcal {L}(\\cdot , \\cdot )$ , and the victim model from which the output of $f$ can be inferred.",
"Output: an adversarial example $\\mathbf {x}_{\\mathrm {adv}}$ fulfills $||\\mathbf {x}_{\\mathrm {adv}} - \\mathbf {x}||_\\infty \\le \\epsilon $ .",
"Initialize the adversarial example to be crafted $\\mathbf {x}_{\\mathrm {adv}}\\leftarrow \\mathbf {x}$ .",
"Initialize the gradient to be estimated $\\mathbf {g}\\leftarrow \\mathbf {0}$ .",
"Initialize the drop-out/layer ratio $p$ .",
"not successful Choose a reference model whose index is $i$ uniformly at random Calculate a prior gradient with drop-out/layer ratio $p$ as $\\mathbf {u}\\leftarrow \\frac{\\partial \\mathcal {L}(f_i(\\mathbf {x}_{\\mathrm {adv}}, p), y)}{\\partial \\mathbf {x}_{\\mathrm {adv}}}$ $\\mathbf {g}_+\\leftarrow \\mathbf {g}+\\tau \\mathbf {u},\\quad \\mathbf {g}_-\\leftarrow \\mathbf {g}-\\tau \\mathbf {u}$ $\\mathbf {g}_+^{\\prime }\\leftarrow \\mathbf {g}_+/\\Vert \\mathbf {g}_+\\Vert _2,\\quad \\mathbf {g}_-^{\\prime }\\leftarrow \\mathbf {g}_- /\\Vert \\mathbf {g}_-\\Vert _2$ $\\mathbf {\\Delta }_t\\leftarrow \\frac{\\mathcal {L}(f(\\mathbf {x}_\\mathrm {adv}+\\delta \\mathbf {g}_+^{\\prime }), y)-\\mathcal {L}(f(\\mathbf {x}_\\mathrm {adv}+\\delta \\mathbf {g}_-^{\\prime }), y)}{\\tau \\delta }\\mathbf {u}$ $\\mathbf {g}\\leftarrow \\mathbf {g}+\\eta _{\\mathbf {g}}\\mathbf {\\Delta }_t$ $\\mathbf {x}_{\\mathrm {adv}}\\leftarrow \\mathbf {x}_{\\mathrm {adv}} + \\eta \\cdot \\mathrm {sign}(\\mathbf {g})$ $\\mathbf {x}_{\\mathrm {adv}}\\leftarrow \\mathrm {Clip}(\\mathbf {x}_{\\mathrm {adv}}, \\mathbf {x}-\\epsilon , \\mathbf {x}+\\epsilon )$ $\\mathbf {x}_{\\mathrm {adv}}\\leftarrow \\mathrm {Clip}(\\mathbf {x}_{\\mathrm {adv}}, 0, 1)$ Update the drop-out/layer ratio $p$ following our policy return $\\mathbf {x}_{\\mathrm {adv}}$"
],
[
"Coordinate Descent for Efficiency",
"If one of the prior gradients happens to be well-aligned with the gradient of the victim model, then “an adaptive” one-dimensional subspace suffices to mount the attack.",
"Nevertheless, we found that it is normally not the case, and increasing the number of reference models and prior gradients facilitates the attack, which can be partially explained by the fact that they are nearly orthogonal to each other in high-dimensional spaces [22].",
"Definitely, it is computationally and memory intensive to calculate the input-gradients of a collection of reference models at each step of the optimization.",
"Given a set of basis vectors, off-the-shelf optimization procedures for black-box attacks either estimate the optimal coefficients for all vectors before update [3] or give one optimal scaling factor overall [14].",
"For any of them, the whole procedure is somewhat analogous to a gradient descent whose update directions do not necessarily align with single basis vectors.",
"It is thus natural to make an effort based on coordinate descent [39], which operates along coordinate directions (i.e., basis vectors) to seek the optimum of an objective, for better efficiency.",
"In general, the algorithm selects a single coordinate direction or a block of coordinate directions to proceed iteratively.",
"That said, we may only need to calculate one or several prior gradients at each step before update and the complexity of our method is significantly reduced.",
"Experimental results in Section show that one single prior gradient suffices."
],
[
"Drop-out/layer for Exploration",
"As suggested in Figure REF , one way of guaranteeing a low failure rate in our method is to collect adequate reference models.",
"However, it is usually troublesome in practice, if not infeasible.",
"Suppose that we have collected a few reference models which might not be adequate, and we aim to reduce the failure rate whatsoever.",
"Remind that the main reason of high failure rates is the imperfect alignment between our search subspaces and the precise gradients (cf., Figure REF ), however, it seems unclear how to explore other possible search directions without training more reference models.",
"One may simply try adding some random vectors to the basis set for better alignment and higher subspace-dimensions, although they bare the ineffectiveness as discussed in Section and we also found in experiments that this strategy does not help much.",
"Our solution to resolve this issue is inspired by the dropout [35] and “droplayer” (a.k.a., stochastic depth) [12] techniques.",
"Drop-out/layer, originally serve as regularization techniques, randomly drop a subset of hidden units or residual blocks (if exist) from DNNs during training.",
"Their successes indicate that a portion of the features can provide reasonable predictions and thus meaningful input-gradients, which implies the possibility of using drop-out/layer invoked gradients to enrich our search priors We examine the generated input-gradients in this manner and found that most of them are still independent.. By temporarily removing hidden units or residual blocks, we can acquire a spectrum of prior gradients from each reference model.",
"In experiments, we append dropout to all convolutional/fully-connect layer (except the final one), and we further drop residual blocks out in ResNet reference models."
],
[
"Experiments",
"In this section, we will testify the effectiveness of our subspace attack by comparing it with the state-of-the-arts in terms of the failure rate and the number of queries (of successful attacks).",
"We consider both untargeted and targeted $\\ell _\\infty $ attacks on CIFAR-10 [16] and ImageNet [32].",
"All our experiments are conducted on a GTX 1080 Ti GPU with PyTorch [29].",
"Our main results for untargeted attacks are summarized in Table REF , and the results for targeted attacks are reported in the supplementary material.",
"Table: Performance of different black-box attacks with ℓ ∞ \\ell _\\infty constraint under untargeted setting.",
"The maximum perturbation is ϵ=8/255\\epsilon =8/255 for CIFAR-10, and ϵ=0.05\\epsilon =0.05 for ImageNet.",
"A recent paper also reports its result on WRN similarly, which achieves a failure rate of 1.0% with 7680 queries.",
"PyramidNet* in the table indicates PyramidNet+ShakeDrop+AutoAugment ."
],
[
"Evaluation Metrics and Settings.",
"As in prior works [13], [26], [14], we adopt the failure rate and the number of queries to evaluate the performance of attacks using originally correctly classified images.",
"For untargeted settings, an attack is considered successful if the model prediction is different from the ground-truth, while for the targeted settings, it is considered successful only if the victim model is tricked into predicting the target class.",
"We observe that the number of queries changes dramatically between different images, thus we report both the mean and median number of queries of successful attacks to gain a clearer understanding of the query complexity.",
"Following prior works, we scale the input images to $[0, 1]$ , and set the maximum $\\ell _\\infty $ perturbation to $\\epsilon =8/255$ for CIFAR-10 and $\\epsilon =0.05$ for ImageNet.",
"We limit to query victim models for at most 10,000 times in the untargeted experiments and 50,000 times in the targeted experiments, as the latter task is more difficult and requires more queries.",
"In all experiments, we invoke PGD [23] to maximize the hinge logit-diff adversarial loss from Carlini and Wagner [2].",
"The PGD step size is set to $1/255$ for CIFAR-10 and $0.01$ for ImageNet.",
"At the end of each iteration, we clip the candidate adversarial examples back to $[0, 1]$ to make sure they are still valid images.",
"We initialize the drop-out/layer ratio as $0.05$ and increase it by $0.01$ at the end of each iteration until it reaches $0.5$ throughout our experiments.",
"Other hyper-parameters like the OCO learning rate $\\eta _{\\mathbf {g}}$ and the finite-difference step sizes (i.e., $\\delta ,\\tau $ ) are set following the paper [14].",
"We mostly compare our method with NES [13] and Bandits-TD [14], and their official implementations are directly used.",
"We apply all the attacks on the same set of clean images and victim models for fair comparison.",
"For Bandits-TD on ImageNet, we craft adversarial examples on a resolution of $50\\times 50$ and upscale them according to specific requests from the victim models (i.e., $299\\times 299$ for Inception-v3, $331\\times 331$ for PNAS-Net, and $224\\times 224$ for SENet) before query, just as described in the paper [14].",
"We do not perform such rescaling on CIFAR-10 since no performance gain is observed."
],
[
"Victim and Reference Models.",
"On CIFAR-10, we consider three victim models: (a) a WRN [41] with 28 layers and 10 times width expansion Pre-trained model: https://github.com/bearpaw/pytorch-classification, which yields 4.03% error rate on the test set; (b) a model obtained via neural architecture search named GDAS [6] Pre-trained model: https://github.com/D-X-Y/GDAS, which has a significantly different architecture than our AlexNet and VGGNet reference models and shows 2.81% test error rate; (c) a 272-layer PyramidNet+Shakedrop model [9], [40] trained using AutoAugment [5] with only 1.56% test error rate, Unlike the other two models that are available online, this one is trained using scripts from: https://github.com/tensorflow/models/tree/master/research/autoaugment which is the published state-of-the-art on CIFAR-10 to the best of our knowledge.",
"As for reference models, we simply adopt the AlexNet and VGG-11/13/16/19 architectures with batch normalizations [15].",
"To evaluate in a more data-independent scenario, we choose an auxiliary dataset (containing only 2,000 images) called CIFAR-10.1 [30] to train the reference models from scratch.",
"We also consider three victim models on ImageNet: (a) an Inception-v3 [36] which is commonly chosen [13], [14], [4], [38] with 22.7% top-1 error rate on the official validation set; (b) a PNAS-Net-5-Large model [21] whose architecture is obtained through neural architecture search, with a top-1 error rate of 17.26%; (c) an SENet-154 model [11] with a top-1 error rate of 18.68% Pre-trained models: https://github.com/Cadene/pretrained-models.pytorch.",
"We adopt ResNet-18/34/50 as reference architectures, and we gather 30,000+45,000 images from an auxiliary dataset [31] and the ImageNet validation set to train them from scratch.",
"The clean images for attacks are sampled from the remaining 5,000 ImageNet official validation images and hence being unseen to both the victim and reference models."
],
[
"Comparison with The State-of-the-arts",
"In this section we compare the performance of our subspace attack with previous state-of-the-art methods on CIFAR-10 and ImageNet under untargeted settings.",
"On CIFAR-10, we randomly select 1,000 images from its official test set, and mount all attacks on these images.",
"Table REF summarizes our main results, in which the fifth to seventh columns compare the mean query counts, median query counts and failure rates.",
"On all three victim models, our method significantly outperforms NES and Bandits-TD in both query efficiency and success rates.",
"By using our method, we are able to reduce the mean query counts by a factor of 1.5 to 2.1 times and the median query counts by 2.1 to 4.4 times comparing with Bandits-TD which incorporates both time and spatial priors [14].",
"The PyramidNet+ShakeDop+AutoAugment [5] model, which shows the lowest test error rate on CIFAR-10, also exhibits the best robustness under all considered black-box attacks.",
"More interestingly, even if the victim model is GDAS, whose architecture is designed by running neural architecture search and thus being drastically different from that of the reference models, our prior gradients can still span promising subspaces for attacks.",
"To the best of our knowledge, we are the first to attack PyramidNet+ShakeDrop+AutoAugment which is a published state-of-the-art and GDAS which has a searched architecture in the black-box setting.",
"For ImageNet, we also randomly sample 1,000 images from the ImageNet validation set for evaluation.",
"Similar to the results on CIFAR-10, the results on ImageNet also evidence that our method outperforms the state-of-the-arts by large margins.",
"Moreover, since the applied reference models are generally more “old-fashioned” and computationally efficient than the victim models that are lately invented, our method introduces little overhead to the baseline optimization algorithm."
],
[
"Dropout Ratios and Training Scales",
"We are interested in how the dropout ratio would affect our attack performance.",
"To figure it out, we set an upper limit of the common dropout ratio $p$ to 0.0, 0.2, 0.5 respectively to observe how the query complexity and the failure rate vary when attacking the WRN victim model.",
"With the AlexNet and VGGNet reference models trained on CIFAR-10.1 [30], we see from the bottom of Table REF that more dropout indicates lower failure rate, verifying that exploration via dropout well amends the misalignments between our subspaces and the victim model gradients.",
"Table: Impact of the dropout ratio and training scale on CIFAR-10.",
"The victim model is WRN.It might also be intriguing to evaluate how the performance of our method varies with the scale of training set for yielding reference models.",
"We attempt to evaluate it empirically by training AlexNet and VGGNets from scratch using different numbers of training images.",
"More specifically, we enlarge our training set by further using the CIFAR-10 official training and test images, excluding the 1,000 images for mounting attacks of course.",
"In addition to the CIFAR-10.1 dataset as used, we try two larger sets: (a) the official CIFAR-10 training set which consists of 50,000 images; In this special setting the reference models and the victim model share the same training data.",
"(b) a set built by augmenting CIFAR-10.1 with 8,000 CIFAR-10 test images, whose overall size is 2,000+8,000=10,000.",
"It can be seen from Table REF that by training reference models with 8,000 more images, the query counts could be cut by over 2$\\times $ without dropout, and the failure rate decreases as well.",
"We believe that the performance gain is powered by better generalization ability of the reference models.",
"In a special scenario where the reference and the victim models share the same training set, our method requires only 59 queries on average to succeed on 98.6% of the testing images without dropout.",
"The performance of our method with dropout is also evaluated on the basis of these reference models, and we can see that dropout is capable of reducing the failure rates significantly regardless of the reference training set.",
"While for the query complexity, we may observe that more powerful reference models generally require less exploration governed by dropout to achieve efficient queries."
],
[
"Choice of Reference Models and Prior Gradients",
"We investigate the impact of number and architecture of reference models for our method by evaluating our attack using different reference model sets, and report the performance in Table REF .",
"As in previous experiments, reference models are trained on CIFAR-10.1, and the maximum dropout ratio is set to 0.5.",
"We see that increasing the number of reference models indeed facilitates the attack in both query efficiency and success rates, just like in the exploratory experiment where dropout is absent.",
"We also compare using “gradient descent” and “coordinate descent” empirically.",
"On CIFAR-10 we choose the same five reference models as previously reported, and at each iteration we compute all five prior gradients and search in the complete subspace.",
"We combine all the prior gradients with Gaussian coefficients to provide a search direction in it.",
"Experimental results demonstrate that with significantly increased run-time, both the query counts and failure rates barely change (mean/median queries: 389/62, failure rate: 0.3%), verifying that our coordinate-descent-flavored policy achieves a sensible trade-off between efficiency and effectiveness."
],
[
"Conclusion",
"While impressive results have been gained, state-of-the-art black-box attacks usually require a large number of queries to trick a victim classification system, making the process costly and suspicious to the system.",
"In this paper, we propose the subspace attack method, which reduces the query complexity by restricting the search directions of gradient estimation in promising subspaces spanned by input-gradients of a few reference models.",
"We suggest to adopt a coordinate-descent-flavored optimization and drop-out/layer to address some potential issues in our method and trade off the query complexity and failure rate.",
"Extensive experimental results on CIFAR-10 and ImageNet evidence that our method outperforms the state-of-the-arts by large margins, even if the reference models are trained on a small and inadequate dataset disjoint to the one for training the victim models.",
"We also evaluate the effectiveness of our method on some winning models (e.g., PyramidNet+ShakeDrop+AutoAugment [5] and SENet [11]) on these datasets and models whose architectures are designed by running neural architecture search (e.g., GDAS [6] and PNAS [21]).",
"[pages=1]supp.pdf [pages=2]supp.pdf [pages=3]supp.pdf"
]
] | 1906.04392 |
[
[
"Anderson-Bernoulli Localization on the 3D lattice and discrete unique\n continuation principle"
],
[
"Abstract We consider the Anderson model with Bernoulli potential on the 3D lattice, and prove localization of eigenfunctions corresponding to eigenvalues near zero, the lower boundary of the spectrum.",
"We follow the framework by Bourgain-Kenig and Ding-Smart, and our main contribution is a 3D discrete unique continuation, which says that any eigenfunction of the harmonic operator with bounded potential cannot be too small on a significant fractional portion of all the points.",
"Its proof relies on geometric arguments about the 3D lattice."
],
[
"Main result and background",
"In the 3D Anderson-Bernoulli model on the lattice, we consider the random Schrödinger operator $H:=-\\Delta +\\delta V$ , acting on the space $\\ell ^2(\\mathbb {Z}^3)$ .",
"Here $\\delta > 0$ is the disorder strength, $\\Delta $ is the discrete Laplacian: $\\Delta u(a) =- 6 u(a)+ \\sum _{b \\in \\mathbb {Z}^3, |a-b|=1}u(b) ,\\; \\forall u \\in \\ell ^2(\\mathbb {Z}^3), a \\in \\mathbb {Z}^3,$ and $V:\\mathbb {Z}^3 \\rightarrow \\lbrace 0,1\\rbrace $ is the Bernoulli random potential; i.e.",
"for each $a \\in \\mathbb {Z}^3$ , $V(a)=1$ with probability $\\frac{1}{2}$ independently.",
"Here and throughout this paper, $|\\cdot |$ denotes the Euclidean norm.",
"Our main result is as follows.",
"Theorem 1.1 There exists $\\lambda _{*}>0$ , depending on $\\delta $ , such that almost surely the following holds.",
"For any function $u : \\mathbb {Z}^3 \\rightarrow \\mathbb {R}$ and $\\lambda \\in [0,\\lambda _{*}]$ , if $H u=\\lambda u$ and $\\inf _{n \\ge 0} \\sup _{a \\in \\mathbb {Z}^3} (1+|a|)^{-n}|u(a)|< \\infty $ , we have $\\inf _{t>0} \\sup _{a \\in \\mathbb {Z}^3} \\exp (t|a|) |u(a)|<\\infty $ .",
"In literature, this phenomenon is sometimes called “Anderson localization” (near the edge of the spectrum).",
"It also implies that $H$ has pure point spectrum in $[0, \\lambda _{*}]$ (see e.g.",
"[17]).",
"Note that this is related to but different from “dynamical localization” (see e.g.",
"discussions in [2]).",
"The Anderson models are widely used to describe spectral and transport properties of disordered media, such as moving quantum mechanical particles, or electrons in a metal with impurities.",
"The mathematical study of their localization phenomena can be traced back to the 1980s (see e.g.",
"[18]), and since then there have been many results in models on both discrete and continuous spaces.",
"In most early works, some regularity conditions on the distribution of the random potential are needed.",
"In [14], Fröhlich and Spencer used a multi-scale analysis argument to show that if $\\lbrace V(a):a\\in \\mathbb {Z}^d\\rbrace $ are i.i.d.",
"bounded random variables with bounded probability density, then the resolvent decays exponentially when $\\delta $ is large enough or energy is sufficiently small.",
"Then in [13], together with Martinelli and Scoppola, they proved Anderson localization under the same condition.",
"This result was strengthened later by [9], where the same results were proved under the condition that the distribution of $\\lbrace V(a):a\\in \\mathbb {Z}^d\\rbrace $ are i.i.d., bounded, and Hölder continuous.",
"It remains an interesting problem to remove these regularity conditions.",
"As described at the beginning of [11], when using the Anderson models to study alloy type materials, it is natural to expect the random potential to take only finitely many values.",
"A particular case is where the random potential are i.i.d.",
"Bernoulli variables.",
"For the particular case of $d=1$ , in the above mentioned paper [9] the authors proved that for the discrete model on $\\mathbb {Z}$ , Anderson localization holds for the full spectrum when the i.i.d.",
"random potential is non-degenerate and has some finite moment.",
"This includes the Bernoulli case.",
"In [3] a new proof is given for the case where the random potential has bounded support.",
"In [11], the continuous model on $\\mathbb {R}$ was studied, and Anderson localization was proved for the full spectrum when the i.i.d.",
"random potential is non-degenerate and has bounded support.",
"For higher dimensions, a breakthrough was then made by Bourgain and Kenig.",
"In [4], they studied the continuous model $\\mathbb {R}^d$ , for $d \\ge 2$ , and proved Anderson-Bernoulli localization near the bottom of the spectrum.",
"An important ingredient is the unique continuation principle in $\\mathbb {R}^{d}$ , i.e.",
"[4].",
"It roughly says that, if $u:\\mathbb {R}^d \\rightarrow \\mathbb {R}$ satisfies $\\Delta u= V u$ for some bounded $V$ on $\\mathbb {R}^{d}$ , and $u$ is also bounded, then $u$ can not be too small on any ball with positive radius.",
"Using this unique continuation principle together with the Sperner lemma, they proved a Wegner estimate, which was used to prove the exponential decay of the resolvent.",
"In doing this, many aspects of the usual multi-scale analysis framework were adapted; and in particular, they introduced the idea of “free sites”.",
"See [6] for some more discussions.",
"Later, Germinet and Klein [15] incorporated the new ideas of [4] and proved localization (in a strong form) near the bottom of the spectrum in the continuous model, for any non-degenerate potential with bounded support.",
"The Anderson-Bernoulli localization on lattices in higher dimensions remained open.",
"There were efforts toward this goal by relaxing the condition that $V$ only takes two values (see [16]).",
"Recently, the work of Ding and Smart [10] proved Anderson-Bernoulli localization near the edge of the spectrum on the 2D lattice.",
"As discussed in [4], the approach there cannot be directly applied to the lattice model, due to the lack of a discrete version of the unique continuation principle.",
"A crucial difference between the lattice $\\mathbb {Z}^d$ and $\\mathbb {R}^d$ is that one could construct a function $u:\\mathbb {Z}^d \\rightarrow \\mathbb {R}$ , such that $\\Delta u= V u$ holds for some bounded $V$ , but $u$ is supported on a lower dimensional set (see Remark REF below for an example on 3D lattice).",
"Hence, a suitable “discrete unique continuation principle” in $\\mathbb {Z}^d$ would state that, if a function $u$ satisfies $-\\Delta u + V u=0$ in a finite (hyper)cube, then $u$ can not be too small (compared to its value at the origin) on a substantial portion of the (hyper)cube.",
"In [10], a randomized version of the discrete unique continuation principle on $\\mathbb {Z}^2$ was proved.",
"The proof was inspired by [5], where unique continuation principle was proved for harmonic functions (i.e.",
"$V=0$ ) on $\\mathbb {Z}^2$ .",
"An important observation exploited in [5] is that the harmonic function has a polynomial structure.",
"More recently, following this line, [19] studied the 2D lattice model with $\\frac{1}{2}$ -Bernoulli potential and large disorder, and localization was proved outside finitely many small intervals.",
"Our Theorem REF in this paper settles the Anderson-Bernoulli localization near the edge of the spectrum on the 3D lattice.",
"Our proof follows the framework of [4] and [10].",
"Our main contribution is the proof of a 3D discrete unique continuation principle.",
"Unlike the 2D case, where some randomness is required, in 3D our discrete unique continuation principle is deterministic, and allows the potential $V$ to be an arbitrary bounded function.",
"It is also robust, in the sense that certain “sparse set” can be removed and the result still holds; and this makes it stand for the multi-scale analysis framework (see Theorem REF below).",
"The most innovative part of our proof is to explore the geometry of the 3D lattice.",
"Let us also mention that Anderson localization is not expected through the whole spectrum in $\\mathbb {Z}^3$ , when the potential is small.",
"There might be a localization-delocalization transition.",
"To be more precise, it is conjectured that there exists $\\delta _0>0$ such that, for any $\\delta <\\delta _0$ , $-\\Delta +\\delta V$ has purely absolutely continuous spectrum in some spectrum range (see e.g.",
"[22]).",
"Localization and delocalization phenomenons are also studied for other models, see e.g.",
"[2] and [1] for regular tree graphs and expander graphs, and see [7], [8], [26] and [23], [24] for random band matrices."
],
[
"An outline of the proof of the 3D discrete unique continuation principle",
"In this subsection we explain the most important ideas in the proof of the 3D discrete unique continuation principle.",
"The formal statement of the 3D discrete unique continuation principle is Theorem REF below.",
"It is stated to fit the framework of [4] and [10].",
"To make a clear outline, we state a simplified version here.",
"Definition 1.2 For any $a \\in \\mathbb {Z}^3$ , and $r \\in \\mathbb {R}_+$ , the set $a + \\left([-r, r] \\cap \\mathbb {Z}\\right)^3$ is called a cube, or $2r$ -cube, and we denote it by $Q_r(a)$ .",
"Particularly, we also denote $Q_r:= Q_r(\\mathbf {0})$ .",
"Theorem 1.3 There exists constant $p>\\frac{3}{2}$ such that the following holds.",
"For each $K > 0$ , there is $C_{1}>0$ , such that for any large enough $n\\in \\mathbb {Z}_+$ , and functions $u, V: \\mathbb {Z}^3 \\rightarrow \\mathbb {R}$ with $\\Delta u=Vu$ in $Q_n$ and $\\Vert V \\Vert _{\\infty } \\le K $ , we have that $ \\left| \\left\\lbrace a \\in Q_{n} : |u(a)| \\ge \\exp (-C_{1} n ) |u(\\mathbf {0})| \\right\\rbrace \\right| \\ge n^{p}.$ Remark 1.4 The power of $\\frac{3}{2}$ should not be optimal.",
"We state it this way because it is precisely what we need (in the proof of Lemma REF below).",
"To prove Theorem REF , we first prove a version with a loose control on the magnitude of the function but with a two-dimensional support.",
"It is a simplified version of Theorem REF below.",
"Theorem 1.5 For each $K > 0$ , there is $C_{2}$ depending only on $K$ , such that for any $n \\in \\mathbb {Z}_+$ and functions $u, V: \\mathbb {Z}^3 \\rightarrow \\mathbb {R}$ with $\\Delta u=Vu$ in $Q_n$ and $\\Vert V \\Vert _{\\infty } \\le K $ , we have that $ \\left| \\left\\lbrace a \\in Q_{n} : |u(a)| \\ge \\exp (-C_{2} n^{3}) |u(\\mathbf {0})| \\right\\rbrace \\right| \\ge C_3 n^2(\\log _2 n)^{-1}.$ Here $C_3$ is a universal constant.",
"Remark 1.6 The power of $n^2$ can not be improved.",
"Consider the case where $V\\equiv 0$ , and $u:(x,y,z)\\mapsto (-1)^{x}\\exp (s z) {1}_{x=y}$ , where $s \\in \\mathbb {R}_+$ is the constant satisfying $\\exp (s)+\\exp (-s)=6$ .",
"One can check that $\\Delta u_{0}\\equiv 0$ , while $|\\lbrace a \\in Q_n:u_{0}(a) \\ne 0\\rbrace |=|\\lbrace (x,y,z) \\in Q_{n}:x=y\\rbrace |=(2n+1)^2$ .",
"To prove Theorem REF , we find many disjoint translations of $Q_{n^{1/3}}$ inside $Q_n$ , and use Theorem REF on each of these translations.",
"This is made precise by Theorem REF in Section .",
"The foundation of the arguments there is the “cone property”, given in Section , which says that from any point in $\\mathbb {Z}^3$ , we can find a chain of points, where $|u|$ decays at most exponentially.",
"Such property is also used in other parts of the paper.",
"The proof of Theorem REF is based on geometric arguments on $\\mathbb {Z}^3$ .",
"We consider four collections of planes in $\\mathbb {R}^3$ .",
"Definition 1.7 Let $\\mathbf {e}_1:=(1, 0, 0)$ , $\\mathbf {e}_2 := (0,1,0)$ , and $\\mathbf {e}_3 := (0,0,1)$ to be the standard basis of $\\mathbb {R}^3$ , and denote $\\lambda _1 := \\mathbf {e}_1+\\mathbf {e}_2+\\mathbf {e}_3$ , $\\lambda _2 := -\\mathbf {e}_1+\\mathbf {e}_2+\\mathbf {e}_3$ , $\\lambda _3 := \\mathbf {e}_1-\\mathbf {e}_2+\\mathbf {e}_3$ , $\\lambda _4:= -\\mathbf {e}_1-\\mathbf {e}_2+\\mathbf {e}_3$ .",
"For any $k \\in \\mathbb {Z}$ , and $\\tau \\in \\lbrace 1, 2, 3, 4\\rbrace $ , denote $\\mathcal {P}_{\\tau , k} := \\left\\lbrace a \\in \\mathbb {R}^3: a\\cdot \\lambda _{\\tau } = k \\right\\rbrace $ .",
"We note that the intersection of $\\mathbb {Z}^3$ with each of these planes is a 2D triangular lattice.",
"Besides, there is a family of regular tetrahedrons in $\\mathbb {R}^3$ , whose four faces are orthogonal to $\\lambda _1, \\lambda _2, \\lambda _3, \\lambda _4$ , respectively.",
"Using these tetrahedrons, we construct some polyhedrons $\\mathfrak {P} \\subset \\mathbb {R}^3$ , called pyramid.",
"For each of these pyramid $\\mathfrak {P}$ , the boundary $\\partial \\mathfrak {P}$ consists of subsets of some of the planes $\\mathcal {P}_{\\tau , k}$ (where $\\tau \\in \\lbrace 1, 2, 3, 4\\rbrace $ and $k \\in \\mathbb {Z}$ ).",
"See Figure REF for an illustration.",
"Using these observations, we lower bound $\\left|\\left\\lbrace a \\in Q_{n} : |u(a)| \\ge \\exp (-C_{2} n^{3}) |u(\\mathbf {0})| \\right\\rbrace \\cap \\partial \\mathfrak {P}\\right|$ .",
"To be more precise, we define such 2D triangular lattice as follows.",
"Definition 1.8 In $\\mathbb {R}^2$ , denote $\\xi :=(-1, 0)$ and $\\eta :=\\left(\\frac{1}{2}, \\frac{\\sqrt{3}}{2}\\right)$ .",
"Define the triangular lattice as $\\Lambda :=\\lbrace s\\xi + t\\eta : s,t \\in \\mathbb {Z}\\rbrace $ .",
"For $a \\in \\Lambda $ and $n\\in \\mathbb {Z}_{\\ge 0}$ , denote $T_{a;n}:=\\left\\lbrace a+s\\xi +t\\eta : t,s\\in \\mathbb {Z},-n\\le t \\le 2n, t-n\\le s \\le n \\right\\rbrace .$ Then $T_{a;n}$ is an equilateral triangle of lattice points with center $a$ , such that on each side there are $3n+1$ lattice points.",
"Now we state the bound we need.",
"Theorem 1.9 There exist constants $C_{4} >5$ and $\\epsilon _{1} >0$ such that the following is true.",
"For any $n\\in \\mathbb {Z}_+$ and any function $u: T_{\\mathbf {0};n} \\rightarrow \\mathbb {R}$ , if $|u ( a ) +u ( a - \\xi ) +u ( a + \\eta )| <C_{4}^{-n} |u (\\mathbf {0} ) |$ for any $a \\in T_{\\mathbf {0};\\left\\lfloor \\frac{n}{2}\\right\\rfloor }$ , then $ \\left| \\lbrace a \\in T_{\\mathbf {0};n} : | u ( a) | >C_{4}^{-n} |u ( \\mathbf {0} ) | \\rbrace \\right| >\\epsilon _{1} n^{2}.$ This theorem can be seen as a triangular version of [5].",
"Our proof is also similar to the arguments there, using the fact that the function $u$ has an approximate polynomial structure."
],
[
"Organization of remaining text",
"In Section , we state and prove the “cone properties”.",
"In Section , we introduce our discrete unique continuation (Theorem REF ), and explain how to prove the resolvent estimate (Theorem REF ) from it, by adapting the framework from [4] and [10].",
"The next three sections are devoted to the proof of our discrete unique continuation (Theorem REF ): in Section we prove the estimates on triangular lattice, i.e.",
"Theorem REF and its corollaries, using arguments similar to those in [5]; in Section , we state and prove Theorem REF (a stronger version of Theorem REF ) by constructing pyramids and using Theorem REF ; finally, in Section we do induction on scales, and deduce Theorem REF from Theorem REF .",
"We have three appendices.",
"In Appendix we state some auxiliary results from [10] that are used in the general framework.",
"Appendix is devoted to the base case of the multi-scale analysis in the general framework.",
"In Appendix we give some details on deducing Anderson localization (Theorem REF ) from decay of the resolvent (Theorem REF ), following existing arguments (from [4], [6], [15]).",
"The authors thank Professor Jian Ding, the advisor of Linjun Li, for introducing this problem to them, explaining the idea of “free sites” from [4], reading early versions of this paper, and providing very helpful suggestions on formulating the text.",
"The authors thank Professor Charles Smart for explaining the ideas in the proof of [10] (i.e.",
"Lemma REF ).",
"The authors also thank anonymous referees for reading this paper carefully, and for their valuable feedbacks which led to many improvements in the text."
],
[
"Cone properties",
"In this section we state and prove the “cone properties”, which are widely used throughout the rest of this paper.",
"Definition 2.1 For each $a \\in \\mathbb {Z}^3$ , and $\\tau \\in \\left\\lbrace 1,2,3\\right\\rbrace $ , denote the cone $\\mathcal {C}_a^{\\tau } := \\left\\lbrace b \\in \\mathbb {Z}^3: |(b-a) \\cdot \\mathbf {e}_{\\tau }| \\ge \\sum _{\\tau ^{\\prime } \\in \\left\\lbrace 1,2,3\\right\\rbrace \\setminus \\left\\lbrace \\tau \\right\\rbrace }|(b-a) \\cdot \\mathbf {e}_{\\tau ^{\\prime }}| \\right\\rbrace .$ For each $k \\in \\mathbb {Z}$ , let $\\mathcal {C}_a^{\\tau }(k) := \\mathcal {C}_a^{\\tau } \\cap \\left\\lbrace b \\in \\mathbb {Z}^3: (b-a) \\cdot \\mathbf {e}_{\\tau } = k \\right\\rbrace $ be a section of the cone.",
"We also denote $\\mathcal {C}:=\\mathcal {C}_{\\mathbf {0}}^{3}$ , for simplicity of notations.",
"First, we have the “local cone property”.",
"Lemma 2.2 For any $u:\\mathbb {Z}^3 \\rightarrow \\mathbb {R}$ , $a \\in \\mathbb {Z}^3$ , and $v\\in \\left\\lbrace \\pm \\mathbf {e}_1, \\pm \\mathbf {e}_2, \\pm \\mathbf {e}_3 \\right\\rbrace $ , if $|\\Delta u(a+v)| \\le K|u(a+v)|$ , we have $ \\max _{b \\in a + v + \\left\\lbrace \\mathbf {0}, \\pm \\mathbf {e}_1, \\pm \\mathbf {e}_2, \\pm \\mathbf {e}_3 \\right\\rbrace \\setminus \\left\\lbrace a\\right\\rbrace }|u(b)| \\ge (K+11)^{-1}|u(a)|.$ Without loss of generality we assume that $v = \\mathbf {e}_1$ .",
"We have $|u(a)|\\le (6+K)|u(a+\\mathbf {e}_1)| + |u(a+2\\mathbf {e}_1)| + |u(a+\\mathbf {e}_1-\\mathbf {e}_2)| + |u(a+\\mathbf {e}_1+\\mathbf {e}_2)| \\\\ + |u(a+\\mathbf {e}_1+\\mathbf {e}_3)| + |u(a+\\mathbf {e}_1-\\mathbf {e}_3)|\\le (K+11) \\max _{b \\in a + \\mathbf {e}_1 + \\left\\lbrace \\mathbf {0}, \\pm \\mathbf {e}_1, \\pm \\mathbf {e}_2, \\pm \\mathbf {e}_3 \\right\\rbrace \\setminus \\left\\lbrace a\\right\\rbrace } |u(b)|,$ and our conclusion follows.",
"With Lemma REF , we can inductively construct an oriented “chain” from $\\mathbf {0}$ to the boundary of a cube, and inside a cone.",
"Lemma 2.3 Let $K \\in \\mathbb {R}_+$ , and $u, V: \\mathbb {Z}^3 \\rightarrow \\mathbb {R}$ , such that $\\Vert V\\Vert _{\\infty } \\le K$ , and $\\Delta u = Vu$ in $Q_n$ for some $n \\in \\mathbb {Z}_+$ .",
"For any $a \\in Q_{n-2}$ , $\\tau \\in \\left\\lbrace 1,2,3\\right\\rbrace $ , $\\iota \\in \\left\\lbrace 1, -1\\right\\rbrace $ , and $k \\in \\mathbb {Z}_{\\ge 0}$ , if $\\mathcal {C}_a^{\\tau }(\\iota k) \\subset Q_n$ , then there exists $w \\in \\mathbb {Z}_{\\ge 0}$ , and a sequence of points $a = a_0, a_1, \\cdots , a_w \\in \\mathcal {C}_a^{\\tau }\\cap Q_{n}$ , such that for any $1 \\le i \\le w$ , we have $a_i - a_{i-1} \\in ( \\iota \\mathbf {e}_{\\tau } + \\left\\lbrace \\mathbf {0},\\pm \\mathbf {e}_1, \\pm \\mathbf {e}_2, \\pm \\mathbf {e}_3\\right\\rbrace ) \\setminus \\left\\lbrace \\mathbf {0}\\right\\rbrace $ , $|u(a_i)| \\ge (K+11)^{-1}|u(a_{i-1})|$ ; and $(a_w-a) \\cdot (\\iota \\mathbf {e}_{\\tau }) \\in \\left\\lbrace k-1, k \\right\\rbrace $ .",
"We prove the case where $\\iota = 1$ , and the other case follows the same arguments.",
"We define the sequence inductively.",
"Let $a_0 := a$ .",
"Suppose we have $a_i \\in \\mathcal {C}_a^{\\tau }$ , with $0 \\le (a_i-a) \\cdot \\mathbf {e}_{\\tau } < k-1$ .",
"Then $a_i + \\mathbf {e}_{\\tau } + \\left\\lbrace \\mathbf {0}, \\pm \\mathbf {e}_1, \\pm \\mathbf {e}_2, \\pm \\mathbf {e}_3\\right\\rbrace \\subset Q_n$ .",
"Let $a_{i+1}:= \\operatorname{argmax}_{b \\in a_i + \\mathbf {e}_{\\tau } + \\left\\lbrace \\mathbf {0}, \\pm \\mathbf {e}_1, \\pm \\mathbf {e}_2, \\pm \\mathbf {e}_3\\right\\rbrace \\setminus \\left\\lbrace a_i\\right\\rbrace } |u(b)|.$ Then we have that $a_{i+1} - a_{i} \\in \\mathbf {e}_{\\tau } + \\left\\lbrace \\mathbf {0}, \\pm \\mathbf {e}_1, \\pm \\mathbf {e}_2, \\pm \\mathbf {e}_3\\right\\rbrace \\setminus \\left\\lbrace \\mathbf {0}\\right\\rbrace $ , $0 \\le (a_{i+1}-a) \\cdot \\mathbf {e}_{\\tau } \\le k$ , and $a_{i+1} \\in \\mathcal {C}_a^{\\tau }$ .",
"By Lemma REF , we also have that $|u(a_{i+1})| \\ge (K+11)^{-1}|u(a_i)|$ .",
"This process will terminate when $(a_i-a) \\cdot \\mathbf {e}_{\\tau } \\ge k-1$ for some $i\\in \\mathbb {Z}_{\\ge 0}$ .",
"Then we let $w=i$ ; and from the construction we know that $(a_i-a) \\cdot \\mathbf {e}_{\\tau } \\in \\left\\lbrace k-1, k\\right\\rbrace $ .",
"Thus we get the desired sequence of lattice points.",
"We also have a Dirichlet boundary version, whose proof is similar.",
"Lemma 2.4 Take any $n \\in \\mathbb {Z}_+$ , $K\\in \\mathbb {R}_+$ , and $u, V:Q_n \\rightarrow \\mathbb {R}$ , such that $\\Vert V\\Vert _{\\infty } \\le K$ and $\\Delta u = Vu$ with Dirichlet boundary condition.",
"For any $a \\in Q_{n}$ , $\\tau \\in \\left\\lbrace 1,2,3\\right\\rbrace $ , $\\iota \\in \\left\\lbrace 1, -1\\right\\rbrace $ , and $k \\in \\mathbb {Z}_{\\ge 0}$ , if $\\mathcal {C}_a^{\\tau }(\\iota k) \\cap Q_n \\ne \\emptyset $ , then the result of Lemma REF still holds.",
"In particular, we have $a_w \\in (\\mathcal {C}_a^{\\tau }(\\iota (k-1))\\cup \\mathcal {C}_{a}^{\\tau }(\\iota k))\\cap Q_{n}$ and $|u(a_{w})|\\ge (K+11)^{-k}|u(a)|$ .",
"Again we only prove the case where $\\iota = 1$ , and define the sequence inductively.",
"The only difference is that, given some $a_i \\in \\mathcal {C}_a^{\\tau }$ , if $0\\le (a_i-a) \\cdot \\mathbf {e}_{\\tau } < k-1$ , now we let $a_{i+1}:= \\operatorname{argmax}_{b \\in (a_i + \\mathbf {e}_{\\tau } + \\left\\lbrace \\mathbf {0}, \\pm \\mathbf {e}_1, \\pm \\mathbf {e}_2, \\pm \\mathbf {e}_3\\right\\rbrace \\setminus \\left\\lbrace a_i\\right\\rbrace ) \\cap Q_{n}} |u(b)|.$ By the Dirichlet boundary condition, we still have that $a_{i+1} - a_{i} \\in \\mathbf {e}_{\\tau } + \\left\\lbrace \\mathbf {0}, \\pm \\mathbf {e}_1, \\pm \\mathbf {e}_2, \\pm \\mathbf {e}_3\\right\\rbrace \\setminus \\left\\lbrace \\mathbf {0}\\right\\rbrace $ , $0\\le (a_{i+1}-a) \\cdot \\mathbf {e}_{\\tau } \\le k$ , $a_{i+1} \\in \\mathcal {C}_a^{\\tau } \\cap Q_{n}$ , and $|u(a_{i+1})| \\ge (K+11)^{-1}|u(a_i)|$ ."
],
[
"General framework",
"This section is about the framework, based on the arguments in [10].",
"We formally state the discrete unique continuation principle (Theorem REF ), and explain how to deduce Theorem REF from it.",
"For some results from [10] that are used in this section, we record them in Appendix for easy reference purpose.",
"As in [10], these arguments essentially work for any i.i.d.",
"potential $V$ that is bounded and nontrivial.",
"For simplicity we only study the $\\frac{1}{2}$ -Bernoulli case with disorder strength $\\delta = 1$ .",
"Borrowing the formalism from [4] and [10], we allow $V$ to take values in the interval $[0, 1]$ , for the purpose of controlling the number of eigenvalues in proving the Wegner estimate (in the proof of Claim REF below).",
"In other words, we study the operator $H=-\\Delta +V$ , where $V$ takes value in the space $[0, 1]^{\\mathbb {Z}^3}$ , equipped with the usual Borel sigma-algebra, and the distribution is given by the product of the $\\frac{1}{2}$ -Bernoulli measure (which is supported on $\\lbrace 0, 1\\rbrace ^{\\mathbb {Z}^3}$ ).",
"We let $\\operatorname{sp}(H)$ be the spectrum of $H$ , then it is well known that, almost surely $\\operatorname{sp}(H)=[0,13]$ (see, e.g.",
"[2]).",
"For any cube $Q \\subset \\mathbb {Z}^3$ , let $P_{Q}:\\ell ^2(\\mathbb {Z}^3) \\rightarrow \\ell ^2(Q)$ be the projection operator onto cube $Q$ , i.e.",
"$P_{Q}u=u | _{Q}$ .",
"Define $H_{Q}:=P_{Q} H P_{Q}^{\\dag }$ , where $P_{Q}^{\\dag }$ is the adjoint of $P_Q$ .",
"Then $H_{Q}:\\ell ^{2}(Q)\\rightarrow \\ell ^{2}(Q)$ is the restriction of $H$ on $Q$ with Dirichlet boundary condition.",
"Throughout this section, by “dyadic”, we mean a number being an integer power of 2.",
"The following result on decay of the resolvent is a 3D version of Theorem [10], and it directly implies Theorem REF .",
"Theorem 3.1 There exist $\\kappa _{0}>0$ , $0<\\lambda _{*}<1$ and $L_{*}>1$ such that $\\mathbb {P}\\left[ \\left|(H_{Q_L}-\\lambda )^{-1}(a,b)\\right| \\le \\exp \\left(L^{1-\\lambda _{*}}-\\lambda _{*} |a-b|\\right), \\;\\forall a,b \\in Q_{L}\\right] \\ge 1-L^{-\\kappa _{0}}$ for any $\\lambda \\in [0,\\lambda _{*}]$ and dyadic scale $L \\ge L_{*}$ .",
"From Theorem REF , the arguments in [4] prove Anderson localization in $[0,\\lambda _{*}]$ (Theorem REF ).",
"See Appendix for the details.",
"To prove Theorem REF , we will prove a 3D analog of [10], i.e.",
"Theorem REF below.",
"Except for replacing all 2D objects by 3D objects, the essential differences are: We need more information on the the frozen sites defined in [10], rather than only knowing they're “$\\eta _k$ -regular” (see [10]).",
"We need a 3D Wegner estimate, an analog of [10].",
"We now set up some geometric notations.",
"Definition 3.2 For any sets $A,B \\subset \\mathbb {R}^3$ , let $\\operatorname{dist}(A,B):=\\inf _{a \\in A,b \\in B}|a-b|,$ and $\\operatorname{diam}(A):=\\sup _{a,b \\in A}|a-b|.$ If $A=\\lbrace b \\in \\mathbb {R}^3:|a-b| < r\\rbrace $ , for some $r>0$ and $a \\in \\mathbb {R}^3$ , we call $A$ a (open) ball and denote its radius as $\\operatorname{radi}(A):=r$ .",
"The following definitions are used to describe the frozen sites, and are stronger than being “$\\eta _k$ -regular” in [10].",
"Definition 3.3 Let $d \\in \\mathbb {Z}_{\\ge 0}$ , $N \\in \\mathbb {Z}_+$ , and $C,\\varepsilon >0$ , $l \\ge 1$ .",
"A set $Z \\subset \\mathbb {R}^3$ is called $(N,l,\\varepsilon )$ -scattered if $Z=\\bigcup _{j \\in \\mathbb {Z}_+,1 \\le t \\le N}Z^{(j,t)}$ is a union of open balls such that, for each $j \\in \\mathbb {Z}_+$ and $t\\in \\lbrace 1,\\cdots ,N\\rbrace $ , $\\operatorname{radi}(Z^{(j,t)})=l$ ; for any $j \\ne j^{\\prime } \\in \\mathbb {Z}_+$ and $t\\in \\lbrace 1,2,\\cdots ,N\\rbrace $ , $\\operatorname{dist}(Z^{(j,t)},Z^{(j^{\\prime },t)}) \\ge l^{1+\\varepsilon }$ .",
"A set $Z \\subset \\mathbb {R}^3$ is called $C$ -unitscattered, if we can write $Z=\\bigcup _{j\\in \\mathbb {Z}_+} Z^{(j)}$ , where each $Z^{(j)} \\subset \\mathbb {R}^3$ is an open unit ball with center in $\\mathbb {Z}^{3}$ and $\\forall j\\ne j^{\\prime } \\in \\mathbb {Z}_+, \\operatorname{dist}(Z^{(j)},Z^{(j^{\\prime })})\\ge C.$ Let $l_{1}, \\cdots l_d > 1$ , we say that the vector $\\vec{l}=(l_1,l_2,\\cdots ,l_d)$ is $\\varepsilon $ -geometric if for each $2 \\le i \\le d$ , we have $l_{i-1}^{1+2\\varepsilon } \\le l_i$ .",
"Given a vector of positive reals $\\vec{l}=(l_1,l_2,\\cdots ,l_d)$ , a set $E\\subset \\mathbb {R}^3$ is called an $(N,\\vec{l},C,\\varepsilon )$ -graded set if there exist sets $E_{0}, \\cdots , E_d \\subset \\mathbb {R}^3$ , such that $E=\\bigcup _{i=0}^{d}E_i$ and the following holds: $\\vec{l}$ is $\\varepsilon $ -geometric, $E_0$ is a $C$ -unitscattered set, for any $1 \\le i\\le d$ , $E_i$ is an $(N,l_i,\\varepsilon )$ -scattered set.",
"For each $1 \\le i \\le d$ , we say that $l_i$ is the $i$ -th scale length of $E$ .",
"In particular, $l_1$ is called the first scale length.",
"We also denote $l_0 := 1$ .",
"Let $A \\subset \\mathbb {R}^3$ , and $E$ be an $(N,\\vec{l},C,\\varepsilon )$ -graded set and $\\overline{C},\\overline{\\varepsilon }>0$ .",
"Then $E$ is said to be $(\\overline{C},\\overline{\\varepsilon })$ -normal in $A$ , if $E_0\\cap A \\ne \\emptyset $ implies $\\overline{C}\\le \\operatorname{diam}(A)$ , and $E_i\\cap A \\ne \\emptyset $ implies $l_i \\le \\operatorname{diam}(A)^{1-\\frac{\\overline{\\varepsilon }}{2}}$ for any $i\\in \\lbrace 1,\\cdots ,d\\rbrace $ .",
"In [10], a 2D Wegner estimate [10] is proved and used in the multi-scale analysis.",
"We will prove the 3D Wegner estimate based on our 3D discrete unique continuation, and we need to accommodate the frozen sites which emerge from the multi-scale analysis.",
"For this we refine Theorem REF as follows.",
"Theorem 3.4 There exists a constant $p>\\frac{3}{2}$ , such that for any $N \\in \\mathbb {Z}_+$ , $K \\in \\mathbb {R}_+$ , and small enough $\\varepsilon \\in \\mathbb {R}_+$ , there exist $C_{\\varepsilon ,K},C_{\\varepsilon ,N}>0$ to make the following statement hold.",
"Take $n \\in \\mathbb {Z}_+$ with $n>C_{\\varepsilon ,N}^{4}$ and functions $u, V: \\mathbb {Z}^3 \\rightarrow \\mathbb {R}$ satisfying $\\Delta u=Vu,$ and $\\Vert V \\Vert _{\\infty } \\le K $ in $Q_n$ .",
"Let $\\vec{l}$ be a vector of positive reals, and $E\\subset \\mathbb {Z}^3$ be an $(N,\\vec{l},\\varepsilon ^{-1},\\varepsilon )$ -graded set with the first scale length $l_1>C_{\\varepsilon ,N}$ and be $(1,\\varepsilon )$ -normal in $Q_n$ .",
"Then we have that $ \\left| \\left\\lbrace a \\in Q_{n}\\setminus E : |u(a)| \\ge \\exp (-C_{\\varepsilon ,K} n ) |u(\\mathbf {0})| \\right\\rbrace \\right| \\ge n^{p}.$ Assuming Theorem REF , we can prove the 3D Wegner estimate.",
"For simplicity of notations, for any $A\\subset \\mathbb {Z}^{3}$ , we denote $V_{A}:=V|_{A}$ , the restriction of the potential function $V$ on $A$ .",
"Lemma 3.5 (3D Wegner estimate) There exists $\\varepsilon _0>0$ such that, if $\\varepsilon > \\delta >0$ , $\\varepsilon $ is small enough, and $\\overline{\\lambda } \\in \\operatorname{sp}(H)=[0,13]$ , $N_1 \\ge 1$ is an integer and $\\vec{l}$ is a vector of positive reals, $L_0>\\cdots >L_5\\ge C_{\\varepsilon ,\\delta ,N_{1}}$ with $L_{j}^{1-2\\delta }\\ge L_{j+1}\\ge L_{j}^{1-\\frac{1}{2}\\varepsilon }$ for $j=0,1,2,3,4$ , where $C_{\\varepsilon ,\\delta ,N_{1}}$ is a (large enough) constant, and $L_0$ , $L_3$ are dyadic, $Q \\subset \\mathbb {Z}^3$ and $Q$ is an $L_0$ -cube, $Q^{\\prime }_1,Q^{\\prime }_2,\\cdots ,Q^{\\prime }_{N_1} \\subset Q$ , and $Q_k^{\\prime }$ is an $L_3$ -cube for each $k=1,2,\\cdots ,N_{1}$ (we call them “defects”), $G \\subset \\bigcup _{k=1}^{N_{1}}Q^{\\prime }_{k}$ with $0<|G|<L_{0}^{\\delta }$ , $E$ is a $(1000 N_{1},\\vec{l},\\varepsilon ^{-1},\\varepsilon )$ -graded set with the first scale length $l_1 \\ge C_{\\varepsilon ,\\delta ,N_{1}}$ and ${V}:E\\cap Q \\rightarrow \\lbrace 0,1\\rbrace $ , for any $L_3$ -cube $Q^{\\prime }\\subset Q\\setminus \\bigcup _{k=1}^{N_1}Q^{\\prime }_{k}$ , $E$ is $(1,\\varepsilon )$ -normal in $Q^{\\prime }$ , for any $V:\\mathbb {Z}^3\\rightarrow [0,1]$ with $V_{E\\cap Q}={V}$ , $|\\lambda -\\overline{\\lambda }|\\le \\exp (-L_5)$ and $H_{Q} u=\\lambda u$ , we have $\\exp (L_4)\\Vert u\\Vert _{\\ell ^2(Q\\setminus \\bigcup _{k}Q^{\\prime }_{k})} \\le \\Vert u\\Vert _{\\ell ^{2}(Q)}\\le (1+L_{0}^{-\\delta })\\Vert u\\Vert _{\\ell ^2(G)}.$ Then $\\mathbb {P}\\left[\\Vert (H_{Q}-\\overline{\\lambda })^{-1}\\Vert \\le \\exp (L_1)\\big | \\; V_{E\\cap Q}={V}\\right]\\ge 1-L_{0}^{C\\varepsilon -\\varepsilon _0},$ where $C$ is a universal constant, and $\\Vert \\cdot \\Vert $ denotes the operator norm.",
"The proof is similar to that of [10], after changing 2D notations to corresponding 3D notations.",
"The major difference is in Claim REF and REF (corresponding to [10]), where Theorem REF is used.",
"This is also the reason why we need the constant $p>\\frac{3}{2}$ in Theorem REF .",
"[Proof of Lemma REF ] Let $\\varepsilon _{0}<p-\\frac{3}{2}$ where $p > \\frac{3}{2}$ is the constant in Theorem REF .",
"In this proof, we will use $c, C$ to denote small and large universal constants.",
"We let $\\lambda _1 \\ge \\lambda _2 \\ge \\cdots \\ge \\lambda _{(L_{0}+1)^{3}}$ be the eigenvalues of $H_{Q}$ .",
"For each $1 \\le k \\le (L_0+1)^3$ , choose eigenfunctions $u_k$ such that $\\Vert u_k\\Vert _{\\ell ^{2}(Q)}=1$ and $H_{Q} u_{k}= \\lambda _{k} u_{k}$ .",
"We may think of $\\lambda _{k}$ and $u_{k}$ as deterministic functions of the potential $V_{Q}\\in [0,1]^{Q}$ .",
"Let $E^{\\prime }=\\left(\\bigcup _{k=1}^{N_1}Q^{\\prime }_{k}\\right) \\cup (E\\cap Q)$ , then for any event $\\mathcal {E}$ , $\\mathbb {P}\\left[\\mathcal {E}\\big |\\; V_{E\\cap Q}={V}\\right]=2^{-|E^{\\prime }\\setminus E|}\\sum _{{V}^{\\prime }:E^{\\prime }\\rightarrow \\lbrace 0,1\\rbrace , {V}^{\\prime }|_{E\\cap Q}={V}}\\mathbb {P}\\left[\\mathcal {E}\\big |\\;V_{E^{\\prime }}={V}^{\\prime }\\right].$ By the simple fact that the average is bounded from above by the maximum, we only need to prove $\\mathbb {P}\\left[\\Vert (H_{Q}-\\overline{\\lambda })^{-1}\\Vert > \\exp (L_1)\\big | \\; V_{E^{\\prime }}={V}^{\\prime }\\right]\\le L_{0}^{C\\varepsilon -\\varepsilon _0},$ for any ${V}^{\\prime }:E^{\\prime }\\rightarrow \\lbrace 0,1\\rbrace $ with ${V}^{\\prime }|_{E\\cap Q}={V}$ .",
"Claim 3.6 There is a constant $C_{N_1}$ such that the following is true.",
"Suppose $u$ satisfies $H_{Q} u= \\lambda u$ for some $\\lambda \\in [0,13]$ .",
"Then there is $a^{\\prime }\\in \\mathbb {Z}^3$ , such that $Q_{\\frac{L_{3}}{2}}(a^{\\prime }) \\subset Q \\setminus \\bigcup _{k}Q^{\\prime }_{k}$ , and $|u(a^{\\prime })| \\ge \\exp (-C_{N_1} L_{3}) \\Vert u\\Vert _{\\ell ^{\\infty }(Q)}.$ Without loss of generality, we assume $Q=Q_{\\frac{L_0}{2}}(\\mathbf {0})$ .",
"Take $a_0 \\in Q$ such that $|u(a_{0})|=\\Vert u\\Vert _{\\ell ^{\\infty }(Q)}$ .",
"We assume without loss of generality that $a_0 \\cdot \\mathbf {e}_{\\tau }\\le 0$ , for each $\\tau \\in \\lbrace 1, 2, 3\\rbrace $ .",
"Since each $Q^{\\prime }_{k}$ is an $L_3$ -cube, by the Pigeonhole principle, there is $x^{\\prime }_{0} \\in [a_0 \\cdot \\mathbf {e}_1+100 N_{1} L_3, a_0 \\cdot \\mathbf {e}_1+200 N_{1} L_3]$ , such that $\\lbrace b \\in Q: b \\cdot \\mathbf {e}_1 \\in [x^{\\prime }_{0}-16L_{3},x^{\\prime }_{0}+16L_{3}]\\rbrace \\cap \\bigcup _{k=1}^{N_1}Q^{\\prime }_{k} =\\emptyset .$ Now we iteratively apply the cone property Lemma REF with $K=13$ .",
"Recall the notations of cones from Definition REF , and note that $(K+11)<\\exp (5)$ .",
"We find $a_{1}\\in (\\mathcal {C}_{a_0}^{1}(x^{\\prime }_{0}-a_0 \\cdot \\mathbf {e}_1)\\cup \\mathcal {C}_{a_0}^{1}(x^{\\prime }_{0}-a_0 \\cdot \\mathbf {e}_1+1)) \\cap Q$ with $|u(a_{1})| \\ge \\exp (-1000 N_{1} L_{3}) |u(a_{0})|,$ and $a_{2} \\in (\\mathcal {C}_{a_1}^{2}(4L_{3})\\cup \\mathcal {C}_{a_1}^{2}(4L_{3}+1)) \\cap Q$ with $|u(a_{2})| \\ge \\exp (-(1000 N_{1}+20) L_{3}) |u(a_{0})|,$ and $a_{3} \\in (\\mathcal {C}_{a_2}^{3}(2L_{3})\\cup \\mathcal {C}_{a_2}^{3}(2L_{3}+1)) \\cap Q$ with $|u(a_{3})| \\ge \\exp (-(1000 N_{1}+30) L_{3}) |u(a_{0})|.$ By (REF ), we have $|a_{1} \\cdot \\mathbf {e}_1-x^{\\prime }_{0}|\\le 1$ and $-\\frac{L_0}{2}\\le a_{1} \\cdot \\mathbf {e}_{\\tau }\\le 200 N_{1} L_{3}+1$ for $\\tau =2,3$ .",
"Then $|a_{2} \\cdot \\mathbf {e}_1-x^{\\prime }_{0}|\\le 4L_{3}+2$ , and $-\\frac{L_0}{2}+4L_{3} \\le a_2 \\cdot \\mathbf {e}_2 \\le (200N_{1}+4)L_{3}+2$ , and $-\\frac{L_0}{2} \\le a_2 \\cdot \\mathbf {e}_3 \\le (200N_{1}+4)L_{3}+2$ .",
"Finally, we have $|a_3 \\cdot \\mathbf {e}_1-x^{\\prime }_{0}|\\le 6L_{3}+3$ , and $-\\frac{L_0}{2}+2L_{3}-1 \\le a_3 \\cdot \\mathbf {e}_2 \\le (200N_{1}+6)L_{3}+3$ , and $-\\frac{L_0}{2}+2L_{3} \\le a_3 \\cdot \\mathbf {e}_3 \\le (200N_{1}+6)L_{3}+3$ .",
"This implies $Q_{\\frac{L_{3}}{2}}(a_{3}) \\subset Q\\setminus \\bigcup _{k=1}^{N_1}Q^{\\prime }_{k}$ and the claim follows by letting $a^{\\prime }=a_{3}$ and $C_{N_1}=1000N_{1}+30$ .",
"Claim 3.7 For any $\\lambda \\in [0,13]$ , $H_{Q} u= \\lambda u$ implies $\\left|\\left\\lbrace a\\in Q:|u(a)| \\ge \\exp \\left(-\\frac{L_2}{4} \\right) \\Vert u\\Vert _{\\ell ^{2}(Q)} \\right\\rbrace \\setminus E^{\\prime } \\right| \\ge \\left(\\frac{L_{3}}{2}\\right)^{p} .$ By applying Claim REF to $u$ , we can find a cube $Q_{\\frac{L_{3}}{2}}(a^{\\prime }) \\subset Q \\setminus \\bigcup _{k}Q^{\\prime }_{k}$ for some $a^{\\prime } \\in \\mathbb {Z}^3$ , such that $|u(a^{\\prime })| \\ge \\exp (-C_{N_1} L_{3}) \\Vert u\\Vert _{\\ell ^{\\infty }(Q)}\\ge \\exp (-C_{N_1} L_{3})(L_{0}+1)^{-\\frac{3}{2}} \\Vert u\\Vert _{\\ell ^{2}(Q)}$ .",
"By Condition 8, $E$ is $(1,\\varepsilon )$ -normal in $Q_{\\frac{L_{3}}{2}}(a^{\\prime })$ .",
"Applying Theorem REF to cube $Q_{\\frac{L_{3}}{2}}(a^{\\prime })$ with graded set $E$ , function $u$ , and $K=13$ , and letting $\\frac{1}{4}C_{\\varepsilon ,\\delta ,N_1}^{2\\delta }>C_{N_1}+C_{\\varepsilon ,K}$ where $C_{\\varepsilon ,K}$ is the constant in Theorem REF , the claim follows.",
"Claim 3.8 Let $s_{i}=\\exp (-L_{1}+(L_2-L_4+C)i)$ for each $i \\in \\mathbb {Z}$ .",
"For $1 \\le k_1 \\le k_2 \\le (L_{0}+1)^{3}$ and $0 \\le \\ell \\le C L_{0}^{\\delta }$ , we have $\\mathbb {P}\\left[\\mathcal {E}_{k_1,k_2,\\ell } \\big |\\; V_{E^{\\prime }} = {V}^{\\prime } \\right] \\le CL_{0}^{\\frac{3}{2}}L_{3}^{-p}$ where $\\mathcal {E}_{k_1,k_2,\\ell }$ denotes the event $|\\lambda _{k_1}-\\overline{\\lambda }|,|\\lambda _{k_2}-\\overline{\\lambda }|< s_{\\ell },\\; |\\lambda _{k_1 -1}-\\overline{\\lambda }|,|\\lambda _{k_2+1}-\\overline{\\lambda }|\\ge s_{\\ell +1}.$ For $i=0,1$ , we let $\\mathcal {E}_{k_1,k_2,\\ell ,i}$ denote the event $\\mathcal {E}_{k_1,k_2,\\ell }\\ \\cap \\ \\left\\lbrace \\left|\\left\\lbrace a\\in Q:|u_{k_1}(a)| \\ge \\exp \\left(-\\frac{L_2}{4} \\right), V(a)=i \\right\\rbrace \\setminus E^{\\prime } \\right| \\ge \\frac{L_{3}^{p}}{8} \\right\\rbrace \\cap \\lbrace V_{E^{\\prime }}={V}^{\\prime }\\rbrace .$ Since we are under the event $V_{E^{\\prime }}={V}^{\\prime }$ , we can view $\\mathcal {E}_{k_1,k_2,\\ell ,0}$ and $\\mathcal {E}_{k_1,k_2,\\ell ,1}$ as subsets of $\\lbrace 0,1\\rbrace ^{Q\\setminus E^{\\prime }}$ .",
"Observe that $\\mathcal {E}_{k_1,k_2,\\ell }\\cap \\lbrace V_{E^{\\prime }}={V}^{\\prime }\\rbrace \\subset \\mathcal {E}_{k_1,k_2,\\ell ,0} \\cup \\mathcal {E}_{k_1,k_2,\\ell ,1}$ by Claim REF .",
"Fix $i \\in \\lbrace 0, 1\\rbrace $ .",
"For each $\\omega \\in \\mathcal {E}_{k_1,k_2,\\ell ,i}$ , we denote $S_1(\\omega ):=\\lbrace a \\in Q \\setminus E^{\\prime }:\\omega (a)=1-i\\rbrace ,$ and $S_2(\\omega ):=\\left\\lbrace a \\in Q\\setminus E^{\\prime }: \\omega (a)=i, |u_{k_1}(a)|\\ge \\exp \\left(- \\frac{L_2}{4} \\right)\\right\\rbrace .$ By definition of $\\mathcal {E}_{k_1,k_2,\\ell ,i}$ , we have $|S_2(\\omega )|\\ge \\frac{L_{3}^{p}}{8}$ .",
"For each $\\omega \\in \\mathcal {E}_{k_1,k_2,\\ell ,i}$ , $a\\in S_{2}(\\omega )$ , we define $\\omega ^{a}$ as $\\omega ^{a}(a):=1-\\omega (a),\\; \\omega ^{a}(a^{\\prime }):=\\omega (a^{\\prime }),\\; \\forall a^{\\prime } \\in Q\\setminus E^{\\prime }, a^{\\prime }\\ne a.$ We claim that $\\omega ^{a} \\notin \\mathcal {E}_{k_1,k_2,\\ell ,i}$ .",
"In the case where $i=0$ , because of Condition 9 and $a \\notin \\bigcup _{k}Q^{\\prime }_{k}$ , we have $\\sum _{|\\lambda _{k}-\\overline{\\lambda }| < \\exp (-L_{5})}u_{k}(a)^{2}<\\exp (-c L_{4})$ .",
"Now we apply Lemma REF to $H_{Q}-\\overline{\\lambda }+s_{\\ell }$ with $r_1=2s_{\\ell }$ , $r_2=s_{\\ell +1}$ , $r_3=\\exp (-\\frac{1}{2} L_2)$ , $r_4=\\exp (-c L_4)$ and $r_5=\\exp (-L_5)$ .",
"Then $\\lambda _{k_1}$ moves out of interval $(\\overline{\\lambda }-s_{\\ell },\\overline{\\lambda }+s_{\\ell })$ when $\\omega (a)$ is changed from 0 to 1.",
"Thus we have $\\omega ^{a} \\notin \\mathcal {E}_{k_1,k_2,\\ell ,0}$ .",
"The case where $i=1$ is similar.",
"From this, we know that for any two $\\omega ,\\omega ^{\\prime } \\in \\mathcal {E}_{k_1,k_2,\\ell ,i}$ , $S_1(\\omega ) \\subset S_1(\\omega ^{\\prime })$ implies $S_1(\\omega ^{\\prime })\\cap S_{2}(\\omega )=\\emptyset $ .",
"Since $|Q\\setminus E^{\\prime }| \\le (L_0+1)^{3}-(L_{3}+1)^{3} \\le L_{0}^3$ , we can apply Theorem REF with set $\\lbrace S_{1}(\\omega ):\\omega \\in \\mathcal {E}_{k_1,k_2,\\ell ,i}\\rbrace $ and $\\rho =\\frac{1}{8} L_{0}^{-3}L_{3}^{p}$ , and we conclude that $\\mathbb {P}[\\mathcal {E}_{k_1,k_2,\\ell ,i}| \\; V_{E^{\\prime }} = {V}^{\\prime }] \\le C L_{0}^{\\frac{3}{2}} L_{3}^{-p}$ .",
"Claim 3.9 There is a set $K \\subset \\lbrace 1,2,\\cdots ,(L_{0}+1)^{3}\\rbrace $ depending only on $E^{\\prime }$ and ${V}^{\\prime }$ , such that $|K|\\le CL_{0}^{\\delta }$ and $\\lbrace \\Vert (H_{Q}-\\overline{\\lambda })^{-1}\\Vert >\\exp (L_{1})\\rbrace \\cap \\lbrace V_{E^{\\prime }} ={V}^{\\prime }\\rbrace \\subset \\bigcup _{\\begin{array}{c}k_1,k_2 \\in K\\\\ k_{1}\\le k_{2}\\end{array}}\\bigcup _{0\\le \\ell \\le CL_{0}^{\\delta }} \\mathcal {E}_{k_1,k_2,\\ell }.$ Conditioning on $V_{E^{\\prime }}={V}^{\\prime }$ , we view $\\lambda _k$ and $u_k$ as functions on $[0,1]^{Q\\setminus E^{\\prime }}$ .",
"Let $1\\le k_{1}<\\cdots <k_{m}\\le (L_{0}+1)^{3}$ be all indices $k_{i}$ such that there is at least one $\\omega \\in [0,1]^{Q\\setminus E^{\\prime }}$ with $|\\lambda _{k_{i}}(w)-\\overline{\\lambda }|\\le \\exp (-L_{2})$ .",
"To prove the claim, it suffices to prove that $m\\le C L_{0}^{\\delta }$ .",
"Indeed, then we can always find an $0\\le \\ell \\le m$ such that the annulus $\\left[\\overline{\\lambda }-s_{\\ell +1},\\overline{\\lambda }+s_{\\ell +1}\\right]\\setminus \\left[\\overline{\\lambda }-s_{\\ell },\\overline{\\lambda }+s_{\\ell }\\right]$ contains no eigenvalue of $H_{Q}$ .",
"Since $\\bigcup _{k} Q^{\\prime }_{k}\\subset E^{\\prime }$ , Condition 9 implies that for any $\\omega \\in [0,1]^{Q\\setminus E^{\\prime }}$ with $|\\lambda _{k_{i}}(\\omega )-\\overline{\\lambda }|\\le \\exp (-L_{5})$ , we have $\\Vert u_{k_{i}}(\\omega )\\Vert _{\\ell ^{\\infty }(Q\\setminus E^{\\prime })}\\le \\exp (-L_{4})$ .",
"In particular, if there is $\\omega _{0}\\in \\left[0,1\\right]^{Q\\setminus E^{\\prime }}$ such that $|\\lambda _{k_{i}}(\\omega _{0})-\\overline{\\lambda }|\\le \\exp (-L_{2})$ , then by eigenvalue variation, $|\\lambda _{k_{i}}(\\omega ) -\\overline{\\lambda }|\\le \\exp (-L_{4})$ holds for all $\\omega \\in \\lbrace 0,1\\rbrace ^{Q\\setminus E^{\\prime }}$ .",
"Indeed, let $\\omega _{t}=(1-t)\\omega _{0}+t\\omega $ for $t\\in [0,1]$ .",
"We compute $\\begin{split}|\\lambda _{k_{i}}(\\omega _{t})-\\overline{\\lambda }|\\le & |\\lambda _{k_{i}}(\\omega _{0})-\\overline{\\lambda }| +\\int _{0}^{t} \\Vert u_{k_{i}}(\\omega _{s})\\Vert ^{2}_{\\ell ^{2}(Q\\setminus E^{\\prime })} ds\\\\\\le & \\exp (-L_{2})+\\int _{0}^{t} |Q| \\exp (-2L_{4})+{1}_{|\\lambda _{k_{i}}(\\omega _{s})-\\overline{\\lambda }|\\ge \\exp (-L_{5})} ds\\\\\\le & \\exp (-L_{4})+{1}_{\\max _{0\\le s\\le t}|\\lambda _{k_{i}}(\\omega _{s})-\\overline{\\lambda }|\\ge \\exp (-L_{5})}\\end{split}$ and conclude by continuity.",
"By (REF ) and Condition 9, for all $\\omega \\in \\lbrace 0,1\\rbrace ^{Q\\setminus E^{\\prime }}$ , we have $1=\\Vert u_{k_{i}}(\\omega )\\Vert _{\\ell ^{2}(Q)}\\ge \\Vert u_{k_{i}}(\\omega )\\Vert _{\\ell ^{2}(G)}\\ge 1-C L_{0}^{-\\delta }$ .",
"In particular, we have $|\\langle u_{k_{i}}(\\omega ), u_{k_{j}}(\\omega ) \\rangle _{\\ell ^{2}(G)}-{1}_{i=j}| \\le CL_{0}^{-\\delta }\\le (5|G|)^{-\\frac{1}{2}}$ .",
"By Lemma REF we have that $m\\le C|G|\\le CL_{0}^{\\delta }$ .",
"Finally, $\\mathbb {P}[\\Vert (H_{Q}-\\overline{\\lambda })^{-1}\\Vert > \\exp (L_1)| \\; V_{E^{\\prime }} = {V}^{\\prime }] \\le \\sum _{k_1,k_2 \\in K}\\sum _{1 \\le \\ell \\le CL_{0}^{\\delta }} \\mathbb {P}[\\mathcal {E}_{k_1,k_2,\\ell }|\\; V_{E^{\\prime }} = {V}^{\\prime }]$ and thus $\\mathbb {P}[\\Vert (H_{Q}-\\overline{\\lambda })^{-1}\\Vert > \\exp (L_1)| \\; V_{E^{\\prime }} = {V}^{\\prime }]\\le CL_{0}^{\\frac{3}{2}+3\\delta }L_{3}^{-p}\\le L_{0}^{C\\varepsilon -\\varepsilon _{0}},$ so our conclusion follows.",
"We now prove Theorem REF by a multi-scale analysis argument.",
"In the remaining part of this section, by “dyadic cube”, we mean a cube $Q_{2^{n}}(a)$ for some $a \\in 2^{n-1}\\mathbb {Z}^3$ and $n \\in \\mathbb {Z}_{+}$ .",
"For each $k,m\\in \\mathbb {Z}_{+}$ and each $2k$ -cube $Q$ , we denote by $m Q$ the $2 m k$ -cube with the same center as $Q$ .",
"Theorem 3.10 (Multi-scale Analysis) There exists $\\kappa > 0$ , such that for any $\\varepsilon _{*}>0$ , there are $\\varepsilon _{*}>\\varepsilon >\\nu >\\delta >0$ , $M,N\\in \\mathbb {Z}_+$ , dyadic scales $L_k$ , for $k\\in \\mathbb {Z}_{\\ge 0}$ , with $\\left\\lfloor \\log _{2}L_{k+1}^{1-6 \\varepsilon } \\right\\rfloor =\\log _{2} L_{k}$ , decay rates $1 \\ge m_{k} \\ge L_{k}^{-\\delta }$ for $k \\in \\mathbb {Z}_{\\ge 0}$ , such that for any $0 \\le \\overline{\\lambda } \\le \\exp (-L_{M}^{\\delta })$ , we have random sets $\\mathcal {O}_{k} \\subset \\mathbb {R}^3$ for $k\\in \\mathbb {Z}_{\\ge 0}$ with $\\mathcal {O}_{k} \\subset \\mathcal {O}_{k+1}$ (depending on the Bernoulli potential $V$ ), and the following six statements hold for any $k \\in \\mathbb {Z}_{\\ge 0}$ : When $k \\le M$ , $\\mathcal {O}_{k} \\cap \\mathbb {Z}^{3}= \\left\\lceil \\varepsilon ^{-1} \\right\\rceil \\mathbb {Z}^{3}$ .",
"When $k\\ge M+1$ , $\\mathcal {O}_{k}$ is an $(N,\\vec{l},(2\\varepsilon )^{-1},2\\varepsilon )$ -graded random set with $\\vec{l}=(L_{M+1}^{1-2\\varepsilon },L_{M+2}^{1-2\\varepsilon },\\cdots ,L_{k}^{1-2\\varepsilon })$ .",
"For any $L_k$ -cube $Q$ , the set $\\mathcal {O}_k$ is $(1,2\\varepsilon )$ -normal in $Q$ .",
"For any $i\\in \\mathbb {Z}_{\\ge 0}$ and any dyadic $2^iL_k$ -cube $Q$ , the set $\\mathcal {O}_{k}\\cap Q$ is $V_{\\mathcal {O}_{k-1}\\cap 3Q}$ -measurable.",
"For any dyadic $L_k$ -cube $Q$ , it is called good (otherwise bad), if for any potential $V^{\\prime }: \\mathbb {Z}^{3}\\rightarrow [0,1]$ with $V^{\\prime }_{\\mathcal {O}_k\\cap Q}=V_{\\mathcal {O}_k\\cap Q}$ , we have $|(H^{\\prime }_{Q}-\\overline{\\lambda })^{-1}(x,y)|\\le \\exp (L_{k}^{1-\\varepsilon }-m_{k}|x-y|),\\; \\forall x,y \\in Q.$ Here $H^{\\prime }_Q$ is the restriction of $-\\Delta +V^{\\prime }$ on $Q$ with Dirichlet boundary condition.",
"Then $Q$ is good with probability at least $1-L_{k}^{-\\kappa }$ .",
"$m_{k} = m_{k-1}-L_{k-1}^{-\\nu }$ when $k \\ge M+1$ .",
"Throughout the proof, we use $c,C$ to denote small and large universal constants.",
"Let $\\kappa $ be any number with $0<\\kappa <\\varepsilon _{0}$ , where $\\varepsilon _{0}$ is from Lemma REF .",
"Let small reals $\\varepsilon ,\\delta ,\\nu $ satisfy Condition 1 and to be determined.",
"Let $M \\in \\mathbb {Z}_{+}$ satisfy $\\frac{3}{5}\\delta <(1-6\\varepsilon )^{M}<\\frac{4}{5}\\delta $ ; such $M$ must exist as long as $\\varepsilon <\\frac{1}{24}$ .",
"Leave $N$ to be determined, and let $L_0$ be large enough with $L_{0} \\ge \\max \\left\\lbrace C_{\\delta ,\\varepsilon }, C_{\\varepsilon ,\\delta ,N}\\right\\rbrace $ , where $C_{\\delta ,\\varepsilon }$ is the constant in Proposition REF and $C_{\\varepsilon ,\\delta ,N}$ is the constant in Lemma REF (with $N_1=N$ ).",
"For $k>0$ , let $L_{k}$ be dyadic numbers satisfying Condition 3.",
"Fix $\\overline{\\lambda }\\in \\left[0, \\exp (-L_{M}^{\\delta })\\right]$ .",
"When $k=0,1,\\cdots ,M$ , let $\\mathcal {O}_{k}=\\bigcup _{a \\in \\left\\lceil \\varepsilon ^{-1} \\right\\rceil \\mathbb {Z}^3} o_{a}$ , where $o_{a}$ is the open unit ball centered at $a$ .",
"Then Statement 1, 3, 4 hold.",
"Let $m_{k}:=L_{k}^{-\\delta }$ .",
"Proposition REF implies Statement 5 for $k=1,2,\\cdots ,M$ .",
"We now prove by induction for $k>M$ .",
"Assume that Statement 1 to 6 hold for all $k^{\\prime }<k$ .",
"For any $0< k^{\\prime } < k$ , by Lemma REF , any bad dyadic $L_{k^{\\prime }}$ -cube $Q$ must contain a bad $L_{k^{\\prime }-1}$ -cube.",
"For any $0 < i \\le k$ , and a bad $L_{k-i}$ -cube $Q^{\\prime }\\subset Q$ , we call $Q^{\\prime }$ a hereditary bad $L_{k-i}$ -subcube of $Q$, if there exists a sequence $Q^{\\prime }=\\overline{Q}_i \\subset \\overline{Q}_{i-1} \\subset \\cdots \\subset \\overline{Q}_1 \\subset Q$ , where for each $j=1, \\cdots , i$ , $\\overline{Q}_j$ is a bad $L_{k-j}$ -cube.",
"We also call such sequence $\\lbrace \\overline{Q}_{j}\\rbrace _{1\\le j \\le i}$ a hereditary bad chain of length $i$ .",
"Note that the set of hereditary bad chains of $Q$ is $V_{\\mathcal {O}_{k-1}\\cap Q}$ -measurable.",
"Claim 3.11 When $\\varepsilon $ is small enough, there exists $N^{\\prime }$ depending on $M,\\kappa ,\\delta ,\\varepsilon $ , such that, for any dyadic $L_k$ -cube $Q$ , $\\mathbb {P}\\left[\\text{$Q$ has no more than $N^{\\prime }$ hereditary bad chain of length $M$}\\right] \\ge 1-L_{k}^{-10}.$ Writing $N^{\\prime }=(N^{\\prime \\prime })^{M}$ , we have $\\begin{split}& \\mathbb {P}[\\text{$Q$ has more than $N^{\\prime }$ hereditary bad chain of length $M$}]\\\\\\le & \\sum _{\\begin{array}{c}Q^{\\prime }\\subset Q\\\\\\text{$Q^{\\prime }$ is a dyadic $L_{j}$-cube}\\\\ k-M<j\\le k\\end{array}} \\mathbb {P}[\\text{$Q^{\\prime }$ contains more than $N^{\\prime \\prime }$ bad $L_{j-1}$-subcubes}].\\end{split}$ We can use inductive hypothesis to bound this by $\\begin{split}& \\sum _{\\begin{array}{c}Q^{\\prime }\\subset Q\\\\\\text{$Q^{\\prime }$ is a dyadic $L_{j}$-cube}\\\\ k-M<j\\le k\\end{array}} \\left(\\frac{L_{j}}{L_{j-1}}\\right)^{CN^{\\prime \\prime }} (L_{j-1}^{-\\kappa })^{cN^{\\prime \\prime }}\\le \\sum _{k-M<j\\le k} \\left(\\frac{L_{k}}{L_{j}}\\right)^{C} \\left(\\frac{L_{j}}{L_{j-1}}\\right)^{CN^{\\prime \\prime }} (L_{j-1}^{-\\kappa })^{cN^{\\prime \\prime }}\\\\\\le & C M L_{k}^{C} \\max _{k-M<j\\le k}L_{j-1}^{(C\\varepsilon -c\\kappa )N^{\\prime \\prime }}\\le C M L_{k}^{C} (L_{k}^{(C\\varepsilon -c\\kappa )N^{\\prime \\prime }}+L_{k}^{(C\\varepsilon -c\\kappa )\\delta N^{\\prime \\prime }}).\\end{split}$ Here we used that $L_{k-M} > L_{k}^{\\frac{\\delta }{2}}$ in the last inequality.",
"The claim follows by taking $\\varepsilon $ sufficiently small (depending on $\\kappa $ ) and $N^{\\prime \\prime }$ large enough (depending on $M,\\kappa ,\\delta ,\\varepsilon $ ).",
"Now we let $N:=1000N^{\\prime }$ .",
"We call a dyadic $L_{k}$ -cube $Q$ ready if $Q$ has no more than $N^{\\prime }$ hereditary bad chain of length $M$ .",
"The event that $Q$ is ready is $V_{\\mathcal {O}_{k-1} \\cap Q}$ -measurable.",
"Suppose $Q$ is an $L_k$ -cube and is ready.",
"Let $Q^{\\prime \\prime \\prime }_{1},\\cdots ,Q^{\\prime \\prime \\prime }_{N^{\\prime }}\\subset Q$ be a complete list of all hereditary bad $L_{k-M}$ -subcubes of $Q$ .",
"Let $Q^{\\prime \\prime }_{1},\\cdots ,Q^{\\prime \\prime }_{N^{\\prime }} \\subset Q$ be the corresponding bad $L_{k-1}$ -cubes, such that $Q^{\\prime \\prime \\prime }_{i} \\subset Q^{\\prime \\prime }_{i}$ for each $i=1,2,\\cdots ,N^{\\prime }$ .",
"These cubes are chosen in a way such that $\\lbrace Q^{\\prime \\prime }_{1},\\cdots ,Q^{\\prime \\prime }_{N^{\\prime }}\\rbrace $ contains all the bad $L_{k-1}$ -cubes in $Q$ .",
"By Lemma REF , we can choose a dyadic scale $L^{\\prime }$ satisfying $L_{k}^{1-3\\varepsilon } \\le L^{\\prime } \\le L_{k}^{1-2\\varepsilon }$ and disjoint $L^{\\prime }$ -cubes $Q^{\\prime }_{1},\\cdots ,Q^{\\prime }_{N^{\\prime }}\\subset Q$ such that, for every $Q^{\\prime \\prime }_{i}$ , there is a $Q^{\\prime }_{j}$ such that $Q^{\\prime \\prime }_{i}\\subset Q^{\\prime }_{j}$ and $\\operatorname{dist}(Q^{\\prime \\prime }_{i},Q \\setminus Q^{\\prime }_{j})\\ge \\frac{L^{\\prime }}{8}$ .",
"For each $i=1,2,\\cdots ,N^{\\prime }$ , we let $O_{Q,i}$ be the ball in $\\mathbb {R}^3$ , with the same center as $Q^{\\prime }_{i}$ and with radius $L_{k}^{1-2\\varepsilon }$ .",
"We can choose $O_{Q,i},Q^{\\prime \\prime }_{i},Q^{\\prime \\prime \\prime }_{i}$ in a $V_{\\mathcal {O}_{k-1}\\cap Q}$ -measurable way.",
"Now we let $\\mathcal {O}_{k}$ be the union of $\\mathcal {O}_{k-1}$ and balls $O_{Q,1},\\cdots ,O_{Q,N^{\\prime }}$ , for each ready $L_{k}$ -cube $Q$ ; i.e.",
"$\\mathcal {O}_{k}:=\\mathcal {O}_{k-1} \\cup \\left(\\bigcup _{\\text{$Q$ is an $L_k$-cube and is ready}}\\left( \\bigcup _{i=1}^{N^{\\prime }} O_{Q,i} \\right)\\right),$ and let $m_{k}=m_{k-1}-L_{k-1}^{-\\nu }$ .",
"From induction hypothesis we have $m_k\\ge L_{k-1}^{-\\delta }-L_{k-1}^{-\\nu }\\ge L_k^{-\\delta }$ .",
"We now verify Statement 2 to 6.",
"First note that Statement 4 and 6 hold for $k$ by the above construction.",
"Claim 3.12 Statement 2 and 3 hold for $k$ .",
"From (REF ), we let $\\tilde{\\mathcal {O}}_{k^{\\prime }}:=\\bigcup _{\\text{$Q$ is an $L_{k^{\\prime }}$-cube and is ready}} \\bigcup _{i=1}^{N^{\\prime }} O_{Q,i}$ for $k^{\\prime }>M$ .",
"Then we have that $\\mathcal {O}_{k}=\\mathcal {O}_{M} \\cup \\left(\\bigcup _{k^{\\prime }=M+1}^{k}\\tilde{\\mathcal {O}}_{k^{\\prime }}\\right)$ , and we claim that $\\mathcal {O}_{M}$ is $(2\\varepsilon )^{-1}$ -unitscattered, $\\tilde{\\mathcal {O}}_{k^{\\prime }}$ is an $(N,L_{k^{\\prime }}^{1-2\\varepsilon },2\\varepsilon )$ -scattered set for each $k^{\\prime }>M$ .",
"By these two claims, Statement 2 holds by Condition 3.",
"Now we check these two claims.",
"For the first one, just note that $\\mathcal {O}_{M}=\\bigcup _{a \\in \\left\\lceil \\varepsilon ^{-1} \\right\\rceil \\mathbb {Z}^3} o_{a}$ , then use Definition REF .",
"For the second one, when $k^{\\prime }>M$ the set $\\tilde{\\mathcal {O}}_{k^{\\prime }}$ is the union of $N^{\\prime }$ balls $O_{Q,1},O_{Q,2},\\cdots ,O_{Q,N^{\\prime }}$ for each ready $L_{k^{\\prime }}$ -cube $Q$ , and each ball $O_{Q,i}$ has radius $L_{k^{\\prime }}^{1-2\\varepsilon }$ .",
"Denote the collection of dyadic $L_{k^{\\prime }}$ -cubes by $\\mathcal {Q}_{k^{\\prime }}:=\\left\\lbrace Q_{\\frac{L_{k^{\\prime }}}{2}}(a):a \\in \\frac{L_{k^{\\prime }}}{4} \\mathbb {Z}^3\\right\\rbrace $ .",
"We can divide $\\mathcal {Q}_{k^{\\prime }}$ into at most 1000 subsets $\\mathcal {Q}_{k^{\\prime }}=\\bigcup _{t=1}^{1000}\\mathcal {Q}^{(t)}_{k^{\\prime }}$ , such that any two $L_{k^{\\prime }}$ -cubes in the same subset have distance larger than $L_{k^{\\prime }}$ , i.e.",
"$\\text{$\\operatorname{dist}(Q, Q^{\\prime }) \\ge L_{k^{\\prime }}$ for any $t \\in \\left\\lbrace 1,2,\\cdots ,1000\\right\\rbrace $ and any $Q\\ne Q^{\\prime } \\in \\mathcal {Q}^{(t)}_{k^{\\prime }}$.",
"}$ For each $1 \\le t \\le 1000$ and $1 \\le j \\le N^{\\prime }$ , let $\\mathfrak {O}^{(t,j)}_{k^{\\prime }}=\\left\\lbrace O_{Q,j}:\\text{$Q$ is ready and $Q \\in \\mathcal {Q}^{(t)}_{k^{\\prime }}$}\\right\\rbrace $ .",
"Then for any two $O \\ne O^{\\prime } \\in \\mathfrak {O}^{(t,j)}_{k^{\\prime }}$ , by (REF ), we have $\\operatorname{dist}(O, O^{\\prime }) \\ge L_{k^{\\prime }}-2L_{k^{\\prime }}^{1-2\\varepsilon }\\ge L_{k^{\\prime }}^{1-4\\varepsilon ^2}=(\\operatorname{radi}(O))^{1+2\\varepsilon }= (\\operatorname{radi}(O^{\\prime }))^{1+2\\varepsilon }.$ From Definition REF , we have that $\\tilde{\\mathcal {O}}_{k^{\\prime }}=\\bigcup _{1 \\le t \\le 1000,1 \\le j \\le N^{\\prime }} \\left(\\bigcup \\mathfrak {O}^{(t,j)}_{k^{\\prime }}\\right)$ is an $(N,L_{k^{\\prime }}^{1-2\\varepsilon },2\\varepsilon )$ -scattered set since $N=1000 N^{\\prime }$ .",
"Thus the second claim holds.",
"Finally, since $\\operatorname{radi}(O_{Q,i})=L_{k^{\\prime }}^{1-2\\varepsilon }< \\operatorname{diam}(Q)^{1-\\varepsilon }$ for any ready $L_{k^{\\prime }}$ -cube $Q$ and $1\\le i \\le N^{\\prime }$ , we have that $\\mathcal {O}_{k}$ is $(1,2\\varepsilon )$ -normal in any $L_{k}$ -cube.",
"Hence Statement 3 holds.",
"Now it remains to check Statement 5 for $k$ .",
"Claim 3.13 If $Q$ is an $L_{k}$ -cube and $Q$ is ready, then for any $1 \\le i \\le N^{\\prime }$ , we have $\\exp (c L_{k-1}^{1-\\delta }) \\Vert u\\Vert _{\\ell ^{\\infty }\\left(Q^{\\prime }_{i}\\setminus \\bigcup _{j=1}^{N^{\\prime }} Q^{\\prime \\prime }_{j}\\right)} \\le \\Vert u\\Vert _{\\ell ^{2}(Q^{\\prime }_{i})} \\le (1+\\exp (-c L_{k-M}^{1-\\delta }))\\Vert u\\Vert _{\\ell ^{2}\\left(Q^{\\prime }_{i}\\cap \\bigcup _{j=1}^{N^{\\prime }}Q^{\\prime \\prime \\prime }_{j}\\right)},$ for any $\\lambda \\in \\mathbb {R}$ with $|\\lambda -\\overline{\\lambda }|\\le \\exp (-2 L_{k-1}^{1-\\varepsilon })$ , and any $u:Q^{\\prime }_{i}\\rightarrow \\mathbb {R}$ with $H_{Q^{\\prime }_{i}}u=\\lambda u$ .",
"If $a\\in Q^{\\prime }_{i}\\setminus \\bigcup _{j=1}^{N^{\\prime }}Q^{\\prime \\prime \\prime }_{j}$ , then there is a $j^{\\prime }=1,\\cdots ,M$ and a good $L_{k-j^{\\prime }}$ -cube $Q^{\\prime \\prime }\\subset Q^{\\prime }_{i}$ with $a\\in Q^{\\prime \\prime }$ and $\\operatorname{dist}(a,Q_{i}^{\\prime }\\setminus Q^{\\prime \\prime })\\ge \\frac{1}{8}L_{k-j^{\\prime }}$ .",
"Moreover, if $a\\in Q^{\\prime }_{i}\\setminus \\bigcup _{j=1}^{N^{\\prime }} Q^{\\prime \\prime }_{j}$ , then we can take $j^{\\prime }=1$ .",
"By the definition of good and Lemma REF , $|u(a)|\\le 2\\exp \\left(L_{k-j^{\\prime }}^{1-\\varepsilon } -\\frac{1}{8}m_{k-j^{\\prime }}L_{k-j^{\\prime }}\\right) \\Vert u\\Vert _{\\ell ^{1}(Q^{\\prime }_{i})}\\le \\exp (-c L_{k-j^{\\prime }}^{1-\\delta })\\Vert u\\Vert _{\\ell ^{2}(Q^{\\prime }_{i})}.$ In particular, we see that $\\Vert u\\Vert _{\\ell ^{\\infty }\\left(Q^{\\prime }_{i}\\setminus \\bigcup _{j=1}^{N^{\\prime }} Q^{\\prime \\prime }_{j}\\right)}\\le \\exp (-c L_{k-1}^{1-\\delta }) \\Vert u\\Vert _{\\ell ^{2}(Q^{\\prime }_{i})}$ and $\\Vert u\\Vert _{\\ell ^{\\infty }(Q^{\\prime }_{i}\\setminus \\bigcup _{j=1}^{N^{\\prime }}Q^{\\prime \\prime \\prime }_{j})}\\le \\exp (-c L_{k-M}^{1-\\delta })\\Vert u\\Vert _{\\ell ^{2}(Q^{\\prime }_{i})}.$ These together imply the claim.",
"Claim 3.14 If $Q$ is an $L_{k}$ -cube, and for any $1 \\le i \\le N^{\\prime }$ , $\\mathcal {E}_{i}(Q)$ denotes the event that $\\text{Q is ready and $\\Vert (H_{Q^{\\prime }_{i}}-\\overline{\\lambda })^{-1}\\Vert \\le \\exp (L_{k}^{1-4\\varepsilon })$},$ then $\\mathbb {P}[\\mathcal {E}_{i}(Q)]\\ge 1-L_{k}^{C\\varepsilon -\\varepsilon _{0}}$ .",
"Recall that the event where $Q$ is ready is $V_{\\mathcal {O}_{k-1} \\cap Q}$ -measurable, and subcubes $Q^{\\prime }_{i}$ 's are also $V_{\\mathcal {O}_{k-1} \\cap Q}$ -measurable.",
"Assuming $\\varepsilon >5\\delta $ , we apply Lemma REF with $2\\varepsilon >\\delta >0$ , $N_1=N^{\\prime }$ , and to the cube $Q^{\\prime }_{i}$ with scales $L^{\\prime } \\ge L_{k}^{1-4\\varepsilon } \\ge L_{k}^{1-5\\varepsilon } \\ge L_{k-1} \\ge L_{k-1}^{1-2\\delta } \\ge 2 L_{k-1}^{1-\\varepsilon }$ (recall that $L^{\\prime }$ is the scale chosen above satisfying (REF )), defects $\\left\\lbrace Q^{\\prime \\prime }_{j}:Q^{\\prime \\prime }_{j} \\subset Q^{\\prime }_{i}\\right\\rbrace $ , $G=\\bigcup _{1\\le j \\le N^{\\prime }:Q^{\\prime \\prime \\prime }_{j} \\subset Q^{\\prime }_{i}}Q^{\\prime \\prime \\prime }_{j}$ , and $E=\\mathcal {O}_{k-1}$ .",
"Note that $L_{k}^{\\frac{\\delta }{2}} < L_{k-M} < L_{k}^{\\frac{9\\delta }{10}}$ .",
"Condition 9 of Lemma REF is given by Claim REF .",
"By Claim REF this claim follows.",
"Claim 3.15 If $Q$ is an $L_{k}$ -cube and $\\mathcal {E}_{1}(Q),\\cdots ,\\mathcal {E}_{N^{\\prime }}(Q)$ hold, then $Q$ is good.",
"We apply Lemma REF to the cube $Q$ with small parameters $\\varepsilon >\\nu >0$ , scales $L_{k}\\ge L_{k}^{1-\\varepsilon }\\ge L^{\\prime } \\ge L_{k}^{1-3\\varepsilon }\\ge L_{k}^{1-4\\varepsilon }\\ge L_{k-1}\\ge L_{k-1}^{1-\\varepsilon }$ , and defects $Q^{\\prime }_{1},\\cdots ,Q^{\\prime }_{N^{\\prime }}$ .",
"We conclude that $|(H_{Q}-\\overline{\\lambda })^{-1}(a,b)|\\le \\exp (L_{k}^{1-\\varepsilon }-m_{k}|a-b|).$ Since $Q^{\\prime }_{i} \\subset \\mathcal {O}_{k}$ when $Q$ is ready, the events $\\mathcal {E}_{i}(Q)$ are $V_{\\mathcal {O}_{k}\\cap Q}$ -measurable, thus $Q$ is good.",
"By combining Claim REF , Claim REF , and letting $C\\varepsilon <\\varepsilon _{0}-\\kappa $ , we have that Statement 5 holds for $k$ .",
"Thus the induction principle proves the theorem.",
"[Proof of Theorem REF ] Apply Theorem REF with any $\\varepsilon _{*}<\\frac{\\kappa }{100}$ , then there are $\\left\\lbrace L_{k}\\right\\rbrace _{k \\in \\mathbb {Z}_{\\ge 0}}$ , $\\left\\lbrace m_{k}\\right\\rbrace _{k \\in \\mathbb {Z}_{\\ge 0}}$ , $\\varepsilon $ , $\\delta $ , $\\nu $ , $N$ and $M$ such that the statements of Theorem REF hold.",
"Let $k_*\\in \\mathbb {Z}_+$ be large enough with $k_*\\ge M+2$ and let $L_{*}= L_{k_*}$ .",
"Fix dyadic scale $L\\ge L_{*}$ , and let $k$ be the maximal integer such that $L \\ge L_{k+1}$ .",
"Then $L_{k}^{1+6\\varepsilon } \\le L_{k+1} \\le L < L_{k+2} \\le L_{k}^{1+15\\varepsilon }$ .",
"Denote $\\mathcal {Q}:=\\left\\lbrace Q:\\text{$Q$ is a dyadic $L_{k}$-cube and $Q \\cap Q_{L}\\ne \\emptyset $}\\right\\rbrace .$ Then $Q_{L} \\subset \\bigcup _{Q\\in \\mathcal {Q}}Q$ and $|\\mathcal {Q}|\\le 1000\\left(\\frac{L}{L_{k}}\\right)^{3} \\le L_{k}^{100\\varepsilon }\\le L_{k}^{100 \\varepsilon _{*}}$ .",
"By elementary observations, for any $a \\in Q_{L}$ , there is a $Q\\in \\mathcal {Q}$ such that $a \\in Q$ and $\\operatorname{dist}(a,Q_{L}\\setminus Q) \\ge \\frac{1}{8}L_{k}$ .",
"Fix $\\lambda \\in [0,\\exp (-L_{M}^{\\delta })]$ .",
"For each $Q\\in \\mathcal {Q}$ , define $A_{Q}$ to be the following event: $\\text{$|(H_{Q}-\\lambda )^{-1}(a,b)|\\le \\exp (L_{k}^{1-\\varepsilon }-m_{k}|a-b|)$ for each $a,b \\in Q$}.$ By Lemma REF , $\\bigcap _{Q \\in \\mathcal {Q}} A_{Q}$ implies $|(H_{Q_L}-\\lambda )^{-1}(a,b)|\\le \\exp (L^{1-\\varepsilon }-m|a-b|), \\forall a,b \\in Q_L,$ where $m=m_{k}-L_{k}^{-\\delta }$ .",
"Note that for $k\\ge k_*-1\\ge M+1$ we have $m=m_{k}-L_{k}^{-\\delta } \\ge L_{k_*-2}^{-\\delta }-L_{k_*-2}^{-\\nu }-\\cdots -L_{k-1}^{-\\nu }-L_{k}^{-\\delta }>\\delta _{0}$ for some $\\delta _{0}>0$ independent of $k$ .",
"Here the inequalities are by Condition 4 and Statement 6 in Theorem REF , and the fact that $L_{k}$ increases super-exponentially and $k_*$ is large enough.",
"By Theorem $\\ref {thm:multiscale}$ , for each $Q \\in \\mathcal {Q}$ we have $\\mathbb {P}[A_{Q}]\\ge 1-L_{k}^{-\\kappa }.$ Thus $\\mathbb {P}\\left[\\bigcap _{Q \\in \\mathcal {Q}} A_{Q}\\right] \\ge 1-|\\mathcal {Q}|L_{k}^{-\\kappa } \\ge 1-L_{k}^{-\\kappa + 100\\varepsilon _{*}}.$ Hence our theorem follows by letting $\\kappa _{0}=\\frac{\\kappa - 100\\varepsilon _{*}}{1+15\\varepsilon }$ and $\\lambda _{*}=\\min \\left\\lbrace \\delta _{0},\\exp (-L_{M}^{\\delta }),\\varepsilon \\right\\rbrace $ ."
],
[
"Polynomial arguments on triangular lattice",
"The goal of this section is to prove Theorem REF , which is a triangular lattice version of [5].",
"Our proof closely follows that in [5], which employs the polynomial structure of $u$ and the Remez inequality, and a Vitalli covering argument."
],
[
"Notations and basic bounds",
"Before starting the proof, recall Definition REF for some basic geometric objects.",
"Here we need more notations for geometric patterns in $\\Lambda $ .",
"Definition 4.1 We denote $\\gamma :=\\xi + \\eta =\\left(-\\frac{1}{2}, \\frac{\\sqrt{3}}{2}\\right)$ .",
"For each $b=s\\xi +t\\eta \\in \\Lambda $ , we denote $\\xi (b):=s$ and $\\eta (b):=t$ .",
"For $a \\in \\Lambda $ and $m\\in \\mathbb {Z}_{\\ge 0}$ , we denote the $\\xi $ -edge, $\\eta $ -edge, and $\\gamma $ -edge of $T_{a;m}$ to be the sets $\\begin{split}&\\left\\lbrace a-m\\eta +s\\xi :-2m\\le s \\le m\\right\\rbrace \\cap \\Lambda ,\\\\&\\left\\lbrace a+m\\xi +t\\eta :-m \\le t \\le 2m\\right\\rbrace \\cap \\Lambda ,\\\\&\\left\\lbrace a-m\\xi +s \\xi + s\\eta :-m\\le s \\le 2m\\right\\rbrace \\cap \\Lambda \\end{split}$ respectively, each containing $3m+1$ points.",
"In this section, an edge of $T_{a;m}$ means one of its $\\xi $ -edge, $\\eta $ -edge and $\\gamma $ -edge.",
"For $a \\in \\Lambda $ and $m,\\ell \\in \\mathbb {Z}_{\\ge 0}$ , denote $P_{a;m,\\ell }:=\\left\\lbrace a+s\\xi +t\\eta :-\\ell \\le t \\le 0, -m+t \\le s \\le 0 \\right\\rbrace \\cap \\Lambda $ , a trapezoid of lattice points.",
"Especially, when $\\ell =0$ , $P_{a;m,\\ell }=\\left\\lbrace a+s\\xi :0 \\le s \\le m\\right\\rbrace $ is a segment parallel to $\\xi $ .",
"The $ \\emph {lower edge}$ of $P_{a;m,\\ell }$ is defined to be the set $P_{a-\\ell \\eta ;m+\\ell ,0}$ , and the $ \\emph {upper edge}$ of $P_{a;m,\\ell }$ is defined to be the set $P_{a;m,0}$ .",
"The $\\emph {left leg}$ of $P_{a;m,\\ell }$ is the set $\\left\\lbrace a+t \\eta :-\\ell \\le t \\le 0\\right\\rbrace \\cap \\Lambda $ , and the $\\emph {right leg}$ of $P_{a;m,\\ell }$ is the set $\\left\\lbrace a-m\\xi -t\\gamma : 0 \\le t \\le \\ell \\right\\rbrace \\cap \\Lambda $ .",
"See Figure REF for an illustration of $T_{a;m}$ and $P_{a;m,\\ell }$ .",
"Figure: T a;m T_{a;m} is the set of lattice points in the triangle region; P a;m,ℓ P_{a;m,\\ell } is the set of lattice points in the trapezoid region.The following lemma can be proved using a straight forward induction.",
"Lemma 4.2 Let $R, S \\in \\mathbb {R}_+$ , $a \\in \\Lambda $ , and $m \\in \\mathbb {Z}_+$ .",
"Suppose $u:\\Lambda \\rightarrow \\mathbb {R}$ satisfies $|u ( b ) +u ( b-\\xi ) +u ( b+\\eta )| \\le R$ for any $b\\in T_{a;m}$ with $\\eta (b)-\\xi (b)<m$ , and $|u| \\le S$ on one of three edges of $T_{a;m}$ .",
"Then $|u(b)| \\le 2^{3m}S+(2^{3m}-1)R$ for each $b \\in T_{a;m}$ .",
"By symmetry, we only need to prove the result when $|u| \\le S$ on the $\\xi $ -edge of $T_{a;m}$ .",
"Without loss of generality we also assume that $a=\\mathbf {0}$ .",
"We claim that for each $k=0, 1, \\cdots , 3m$ , $|u(b)| \\le 2^{k}S+(2^{k}-1)R$ for any $b \\in T_{\\mathbf {0};m}$ with $\\eta (b)=k-m$ .",
"We prove this claim by induction on $k$ .",
"The base case of $k=0$ holds by the assumptions.",
"We suppose that the statement is true for $0, 1, \\cdots , k$ .",
"For any $b \\in T_{\\mathbf {0};m}$ with $\\eta (b)=k-m$ and $\\xi (b)>k-2m$ , we have $b,b-\\xi \\in T_{\\mathbf {0};m}$ and $\\eta ( b ) = \\eta (b-\\xi ) = k-m$ .",
"By (REF ) and the induction hypothesis, $|u(b+\\eta )|\\le |u(b)|+|u(b-\\xi )| +R \\le 2(2^{k}S+(2^k-1)R)+R=2^{k+1}S+(2^{k+1}-1)R.$ Then our claim holds by induction, and the lemma follows from our claim."
],
[
"Key lemmas via polynomial arguments",
"In this subsection we prove two key results, Lemma REF and REF below, which are analogous to [5] and [5], respectively.",
"We will use the Remez inequality [21].",
"More precisely, we will use the following discrete version as stated and proved in [5].",
"Lemma 4.3 ([5]) Let $d, \\ell \\in \\mathbb {Z}_{+}$ , and $p$ be a polynomial with degree no more than $d$ .",
"For $M\\in \\mathbb {R}_+$ , suppose that $|p| \\le M$ on at least $d+\\ell $ integer points on a closed interval $I$ , then on $I$ we have $|p| \\le \\left(\\frac{4|I|}{\\ell }\\right)^d M.$ Now we prove the following bound of $|u|$ in a trapezoid, given that $|u|$ is small on the upper edge and on a substantial fraction of the lower edge of the trapezoid.",
"Lemma 4.4 Let $R, K \\in \\mathbb {R}_+$ , $\\ell ,m \\in \\mathbb {Z}_+$ with $\\ell \\le \\frac{m}{10}$ , and $a\\in \\Lambda $ .",
"There is a universal constant $C_5 > 1$ (independent of $a,m,\\ell ,K,R$ ), such that the following is true.",
"Suppose $u:P_{a;m,\\ell } \\rightarrow \\mathbb {R}$ is a function satisfying that: (REF ) holds for any $b\\in P_{a-\\eta ;m,\\ell -1}$ , $|u| \\le K$ on the upper edge of $P_{a;m,\\ell }$ , $|u| \\le K$ for at least half of the points in the lower edge of $P_{a;m,\\ell }$ .",
"Then $|u| \\le C_5^{\\ell +m}(K+R)$ in $P_{a;m,\\ell }$ .",
"We assume without loss of generality that $a=\\mathbf {0}$ .",
"We first claim that there is a function $v:P_{\\mathbf {0};m,\\ell }\\rightarrow \\mathbb {R}$ satisfying the following four conditions: $v=0$ on $\\left\\lbrace -t\\eta : 1 \\le t \\le \\ell \\right\\rbrace $ .",
"$v=u$ on $P_{\\mathbf {0};m,0}$ .",
"For each point $b \\in P_{-\\eta ;m,\\ell -1}$ , $v(b)+v(b-\\xi )+v(b+\\eta )=u(b)+u(b-\\xi )+u(b+\\eta ).$ $\\Vert v\\Vert _{\\infty } \\le 4^{\\ell +m}(K+R)$ .",
"We construct the function $v$ by first defining it on $\\left\\lbrace -t\\eta : 0 \\le t \\le \\ell \\right\\rbrace $ and $P_{\\mathbf {0};m,0}$ , then iterating (REF ) line by line.",
"More precisely, for $-m \\le s\\le 0$ , we let $v(s\\xi )=u(s\\xi )$ .",
"For each $t = -1, -2 \\cdots , -\\ell $ , we first define $v(t \\eta )=0$ , then define $v((s-1)\\xi +t\\eta ):=-v(s\\xi +t\\eta )-v(s\\xi +(t+1)\\eta )+u (s\\xi +t\\eta ) +u ((s-1)\\xi +t\\eta ) +u (s\\xi +(t+1)\\eta )$ for all $-m+t+1 \\le s\\le 0$ .",
"Then we have defined $v(s\\xi +t\\eta )$ for $-\\ell \\le t \\le 0$ and $-m+t \\le s \\le 0 $ .",
"By our construction, $v$ satisfies Condition 1 to 3.",
"Now we prove $v$ satisfies Condition 4.",
"First, (REF ) implies that $|v(b)+v(b-\\xi )+v(b+\\eta )| \\le R$ for any $b\\in P_{-\\eta ;m,\\ell -1}$ .",
"Using this and $|v|\\le K$ on $P_{\\mathbf {0};m,0}$ , by an induction similar to that in the construction of $v$ , we can prove that $|v(-s\\xi -t\\eta )| \\le 2^{s+t}K+(2^{s+t}-1)R$ for each $0 \\le t \\le \\ell $ and $0 \\le s\\le m+t $ .",
"In particular, $|v| \\le (K+R)4^{\\ell +m}$ on any point in trapezoid $P_{\\mathbf {0};m,\\ell }$ , and $v$ satisfies Condition 4.",
"Let $w:=u-v$ , then $w=0$ on $P_{\\mathbf {0};m,0}$ and $w(b)+w(b-\\eta )+w(b-\\gamma )=0$ for each $b \\in P_{\\mathbf {0};m,\\ell -1} $ .",
"Also, $|w| \\le (K+R)4^{\\ell +m}+K \\le (K+R)5^{\\ell +m}$ on at least half of points in the lower edge of $P_{\\mathbf {0};m,\\ell }$ .",
"Since $\\ell \\le \\frac{m}{10}$ , we have $\\left| \\left\\lbrace 0 \\le s \\le m+\\ell :|w(-s\\xi -\\ell \\eta )| \\le (K+R)5^{\\ell +m}\\right\\rbrace \\right|\\ge \\frac{m+\\ell }{2} \\ge 5\\ell .$ We claim that for each $0 \\le t \\le \\ell $ , if we denote $g_{t}(s)=(-1)^{s}w(-s\\xi -t\\eta ), \\; \\forall 0 \\le s \\le m+t,s\\in \\mathbb {Z},$ then $g_t$ is a polynomial of degree at most $t$ .",
"We prove the claim by induction on $t$ .",
"For $t=0$ , this is true since $w=0$ on the upper edge of $P_{\\mathbf {0};m, \\ell }$ .",
"Suppose the statement is true for $t$ , then since $g_{t+1}(s)-g_{t+1}(s+1) =(-1)^{s}w(-s\\xi -(t+1)\\eta )-(-1)^{s-1}w((-s-1)\\xi -(t+1)\\eta )\\\\=-(-1)^{s}w(-s\\xi -t\\eta )=-g_{t}(s),$ for all $0 \\le s \\le m+t, s\\in \\mathbb {Z}$ , we have that $g_{t+1}$ is a polynomial of degree at most $t+1$ .",
"Hence our claim holds.",
"In particular, $g_{\\ell }(s) = (-1)^s w(-s\\xi -\\ell \\eta )$ is a polynomial of degree at most $\\ell $ .",
"Hence by (REF ) and Lemma REF , there exists a constant $C>0$ such that $|w(-s\\xi -\\ell \\eta )| \\le 5^{\\ell +m}C^{\\ell }(K+R)$ for each $0 \\le s \\le m+\\ell $ .",
"Thus on the lower edge of $P_{\\mathbf {0};m,\\ell }$ , $|u| \\le |w|+|v| \\le 5^{\\ell +m}C^{\\ell }(K+R)+4^{\\ell +m}(K+R) \\le (5 C +4)^{\\ell +m}(K+R),$ Finally, by an inductive argument similar to the proof of Lemma REF , and letting $C_5=10 C+8$ , we get $|u| \\le 2^{\\ell }(5C+4)^{\\ell +m}(K+R)+(2^{\\ell }-1)R\\le C_5^{\\ell +m}(K+R)$ in $P_{\\mathbf {0};m,\\ell }$ .",
"Our next lemma is obtained by applying Lemma REF repeatedly.",
"Lemma 4.5 Let $m,\\ell \\in \\mathbb {Z}_+$ with $\\ell \\le m \\le 2\\ell $ , $K, R \\in \\mathbb {R}_+$ , and $a \\in \\Lambda $ .",
"Let $u:P_{a;m,\\ell }\\rightarrow \\mathbb {R}$ be a function satisfying (REF ) for each $b\\in P_{a-\\eta ;m,\\ell -1}$ .",
"If $|u| \\le K$ on $P_{a;m,0}$ and $ \\left|\\left\\lbrace b\\in P_{a;m,\\ell }:|u(b)|>K\\right\\rbrace \\right| \\le \\frac{1}{10^5}m\\ell $ , then $|u| \\le (K+R)C_{6}^{\\ell }$ in $P_{a;m, \\left\\lfloor \\frac{\\ell }{2} \\right\\rfloor }$ , where $C_6>1$ is a universal constant.",
"If $\\ell \\le 120$ , then the result holds trivially since $\\frac{1}{10^5}m\\ell \\le \\frac{2}{10^5}\\ell ^2 <1$ .",
"From now on we assume that $\\ell \\ge 120$ , and let $C_6 = C_5^{1000}$ where $C_5$ is the constant in Lemma REF .",
"For each $k = 0,1,\\cdots ,29$ , we choose an $l_{k} \\in \\left\\lbrace \\left\\lfloor \\frac{2k}{60}\\ell \\right\\rfloor , \\left\\lfloor \\frac{2k}{60}\\ell \\right\\rfloor +1, \\cdots ,\\left\\lfloor \\frac{2k+1}{60} \\ell \\right\\rfloor -1\\right\\rbrace $ such that $\\left|\\left\\lbrace b:|u(b)| \\le K\\right\\rbrace \\cap P_{a-l_{k}\\eta ;m+l_{k},0}\\right| \\ge \\frac{1}{2}(m+l_k).$ Such $l_{k}$ must exist, since otherwise, $\\left| \\left\\lbrace b\\in P_{a;m,\\ell }:|u(b)|> K\\right\\rbrace \\right| > \\frac{1}{2}\\cdot \\frac{1}{60}m\\ell > \\frac{1}{10^5}m\\ell ,$ which contradicts with an assumption in the statement of this lemma.",
"In particular, we can take $l_{0}=0$ .",
"From the definition, we have $l_{k+1}-l_{k} \\le \\frac{1}{20}\\ell \\le \\frac{1}{20}m$ and $l_{k+1}-l_{k} \\ge \\frac{1}{60}\\ell \\ge \\frac{1}{120}m$ .",
"For each $k=0,1,\\cdots ,28$ , let $P_k=P_{a-l_{k}\\eta ;m+l_{k},l_{k+1}-l_{k}}$ , then we claim that $|u| \\le C_{6}^{l_{k+1}}(K+R)$ on $P_k$ .",
"We prove this claim by induction on $k$ .",
"For $k=0$ , we use Lemma REF for $P_{a;m,l_{1}}$ to get $|u| \\le (K+R)C_5^{l_{1}+m} \\le (K+R)C_5^{121l_{1}} \\le (K+R)C_{6}^{l_{1}}$ in $P_0=P_{a;m,l_{1}}$ .",
"Suppose the statement holds for $k$ , then $|u| \\le (K+R)C_{6}^{l_{k+1}}$ in $P_{a-l_{k+1}\\eta ;m+l_{k+1},0}$ which is the upper edge of $P_{k+1}$ .",
"We use Lemma REF again for $P_{k+1}$ , and get $|u| \\le (K+R)C_{6}^{l_{k+2}}$ in $P_{k+1}$ .",
"Thus the claim follows.",
"Since $l_{29} \\ge \\frac{29}{30}\\ell -1 \\ge \\left\\lfloor \\frac{1}{2}\\ell \\right\\rfloor +1$ when $\\ell \\ge 120$ , we have $P_{a;m,\\left\\lfloor \\frac{\\ell }{2} \\right\\rfloor }\\subset \\bigcup _{k=0}^{28}P_k$ .",
"Then the lemma is implied by this claim."
],
[
"Proof of Theorem ",
"In this subsection we finish the proof of Theorem REF .",
"The key step is a triangular analogue of [5] (Lemma REF below); then we finish using a Vitalli covering argument.",
"[Proof of Theorem REF ] Let $\\epsilon _1 = \\frac{1}{10^{18}}$ , and $C_4=6C_6>6$ where $C_6$ is the constant in Lemma REF .",
"We note that now Theorem REF holds trivially when $n<10^9$ , so below we assume that $n\\ge 10^9$ .",
"We argue by contradiction, i.e.",
"we assume that $ \\left|\\left\\lbrace b\\in T_{\\mathbf {0};n} : | u (b) | >K \\right\\rbrace \\right| \\le \\epsilon _{1} n^2,$ where we take $K=C_{4}^{-n}|u(\\mathbf {0})|$ .",
"We first define a notion of triangles on which $|u|$ is “suitably bounded”.",
"For this, we let $R=C_{4}^{-n}|u(\\mathbf {0})|$ as well, and we define a triangle $T_{a;m}\\subset T_{\\mathbf {0};\\left\\lfloor \\frac{n}{2}\\right\\rfloor }$ as being good if $m$ is even and $|u| \\le (K+R) \\left(\\frac{C_{4}}{3}\\right)^{3m}$ on any point in $T_{a;m}$ .",
"We choose points $a_i \\in T_{\\mathbf {0};\\left\\lfloor \\frac{n}{20} \\right\\rfloor }$ for $1 \\le i \\le \\left\\lfloor \\frac{n^2}{10^6} \\right\\rfloor $ , such that each $T_{a_i,2} \\subset T_{\\mathbf {0};\\left\\lfloor \\frac{n}{20} \\right\\rfloor }$ , and $T_{a_i,2}\\cap T_{a_j,2}=\\emptyset $ for any $i \\ne j$ .",
"Denote $S:=\\left\\lbrace T_{a_i,2}:1 \\le i \\le \\left\\lfloor \\frac{n^2}{10^6} \\right\\rfloor \\right\\rbrace $ .",
"By (REF ), for at least half of the triangles in $S$ , $|u| \\le K$ on each of them.",
"Hence, there are at least $\\frac{n^2}{10^{7}}$ good triangles in $S$ .",
"Denote $Q=\\left\\lbrace a_{i}:1 \\le i \\le \\left\\lfloor \\frac{n^2}{10^6} \\right\\rfloor ,\\text{ $T_{a_{i},2}$ is good}\\right\\rbrace .$ For any $a \\in Q$ , let $l_{a}=\\max \\left\\lbrace l\\in \\mathbb {Z}_{+}:\\text{$T_{a,l}$ is good and $T_{a,l}\\subset T_{\\mathbf {0};\\left\\lfloor \\frac{n}{2}\\right\\rfloor }$}\\right\\rbrace $ .",
"Denote $X_a=T_{a;l_{a}}$ for each $a\\in Q$ .",
"If there exists $a \\in Q$ with $l_a \\ge \\frac{n}{30}$ , then this maximal triangle contains $\\mathbf {0}$ , and $|u(\\mathbf {0})| \\le \\left(\\frac{C_4}{3}\\right)^{3l_{a}}(K+R) \\le \\left(\\frac{C_4}{3}\\right)^{n}(K+R) < |u(\\mathbf {0})|$ , which is impossible.",
"Hence $l_a \\le \\frac{n}{30}$ for any $a\\in Q$ .",
"For any $a \\in Q$ , denote $Y_a := T_{a;4l_a}$ .",
"Then $Y_{a} \\subset T_{\\mathbf {0};\\left\\lfloor \\frac{n}{2}\\right\\rfloor }$ .",
"We need the following result on good triangles.",
"Lemma 4.6 For any $m\\in \\mathbb {Z}_+$ and $a\\in \\Lambda $ the following is true.",
"Let $T_1=T_{a;2m}$ , $T_2=T_{a;5m}$ and $T_3=T_{a;8m}$ (see Figure REF for an illustration).",
"If $T_3\\subset T_{\\mathbf {0};\\left\\lfloor \\frac{n}{2}\\right\\rfloor }$ , and $\\left|\\left\\lbrace b \\in T_3:|u(b)|>K\\right\\rbrace \\right|\\le \\frac{m^2}{10^6}$ , and $T_{1}$ is good, then $T_2$ is also good.",
"We assume this result for now and continue our proof of Theorem REF .",
"We have that $\\left|\\left\\lbrace b \\in Y_{a}:|u(b)|>K\\right\\rbrace \\right| \\ge \\frac{l_{a}^2}{10^7},\\;\\;\\forall a \\in Q,$ since otherwise, by Lemma REF with $T_1=X_a$ and $T_3=Y_a$ , there is a good triangle strictly containing $X_a$ and this contradicts with the maximal property of $X_a$ .",
"Finally we apply Vitalli's covering theorem to the collection of triangles $\\left\\lbrace Y_{a}:a \\in Q\\right\\rbrace $ .",
"We can find a subset $\\tilde{Q}\\subset Q$ such that $\\left|\\bigcup _{a \\in \\tilde{Q}}Y_{a}\\right| \\ge \\frac{1}{16}|\\bigcup _{a \\in {Q}}Y_{a}|$ , and $Y_a \\cap Y_{a^{\\prime }}=\\emptyset $ for any $a\\ne a^{\\prime } \\in \\tilde{Q}$ .",
"Hence $\\left| \\left\\lbrace a \\in T_{\\mathbf {0};\\left\\lfloor \\frac{n}{2}\\right\\rfloor }: |u(a)| >K \\right\\rbrace \\right| \\ge \\frac{1}{10^7}\\left|\\bigcup _{a \\in \\tilde{Q}}Y_{a}\\right| > \\frac{1}{10^9} \\left|\\bigcup _{a \\in {Q}}Y_{a}\\right|.$ Since $Q \\subset \\bigcup _{a \\in {Q}}Y_{a}$ , we have $\\left|\\bigcup _{a \\in {Q}}Y_{a}\\right| \\ge |Q| > \\frac{n^2}{10^{7}}$ , so $\\left| \\left\\lbrace a \\in T_{\\mathbf {0};\\left\\lfloor \\frac{n}{2}\\right\\rfloor }: |u(a)| >K \\right\\rbrace \\right| > \\frac{1}{10^{9}}\\cdot \\frac{n^2}{10^{7}} = \\frac{n^2}{10^{16}}$ .",
"This contradicts with our assumption (REF ) since $\\epsilon _{1}=\\frac{1}{10^{18}}$ .",
"Figure: The thick lines indicate edges of T 1 T_1, T 2 T_2, and T 3 T_3.",
"The blue segment indicates L 1 L_1 and the red segment indicates L 2 L_{2}.",
"The yellow region indicates P 1 ' P^{\\prime }_{1} and the union of yellow region and green region indicates P 1 P_{1}.It remains to prove Lemma REF .",
"[Proof of Lemma REF ] We first note that $u$ satisfies (REF ) for any $b\\in T_{\\mathbf {0};\\left\\lfloor \\frac{n}{2}\\right\\rfloor }$ .",
"Without loss of generality, we assume $a=\\mathbf {0}$ .",
"Define $F:\\Lambda \\rightarrow \\Lambda $ to be the counterclockwise rotation around $\\mathbf {0}$ by $\\frac{2\\pi }{3}$ , i.e.",
"$F(s_1\\xi +t_1\\eta )=(t_1-s_1)\\xi -s_1\\eta $ for any $s_1,t_1 \\in \\mathbb {Z}$ .",
"We first consider the trapezoid $P_1:=P_{2m\\xi -2m\\eta ;6m,6m}$ .",
"The upper edge of $P_1$ is exactly the $\\xi $ -edge of $T_1$ and the lower edge of $P_1$ is contained in the $\\xi $ -edge of $T_3$ .",
"Denote $P^{\\prime }_1:=P_{2m\\xi -2m\\eta ;6m,3m}$ , $K_1:=(K+R)(2C_6)^{6m}$ and $K_2:=(K_1+R)C_{6}^{6m}$ .",
"Then $|u| \\le K_1$ in $T_1$ since $T_1$ is good.",
"In particular, $|u| \\le K_1$ on the upper edge of $P_1$ .",
"We also have $ \\left| \\left\\lbrace b \\in P_1:|u(b)|>K\\right\\rbrace \\right| \\le \\frac{36}{10^5}m^2$ , by $P_1\\subset T_3$ and the assumption of this lemma.",
"Thus by Lemma REF , we deduce that $|u| \\le K_2$ in $P^{\\prime }_1$ .",
"Let $P_2:=F(P_1)$ and $P_3:=F^{-1}(P_1)$ .",
"A symmetric argument for $P_2$ and $P_3$ implies that $|u| \\le K_2$ also holds in $P^{\\prime }_2:=F(P^{\\prime }_1)$ and $P^{\\prime }_3:=F^{-1}(P^{\\prime }_1)$ .",
"Consider the triangles $T^{\\prime }_1:=T_{3m\\xi +6m\\eta ;2m}$ , $T^{\\prime }_2:=T_{3m\\xi -3m\\eta ; 2m}$ and $T^{\\prime }_3:=T_{-6m\\xi -3m\\eta ;2m}$ (see Figure REF ).",
"We have $T^{\\prime }_2=F(T^{\\prime }_1)$ and $T^{\\prime }_3=F^{-1}(T^{\\prime }_1)$ .",
"We claim that $|u| \\le (K_2+R)2^{6m}$ in $\\bigcup _{i=1,2,3}T^{\\prime }_i$ .",
"By symmetry, we only need to prove the claim in $T^{\\prime }_1$ .",
"Denote $L_1 := \\left\\lbrace s\\xi +4m\\eta : -m \\le s \\le 2m \\right\\rbrace $ and $L_2 := \\left\\lbrace s\\xi +4m\\eta : 2m \\le s \\le 5m \\right\\rbrace $ .",
"Note that the $\\xi $ -edge of triangle $T^{\\prime }_1$ is the set of points $\\left\\lbrace s\\xi +4m\\eta : -m \\le s \\le 5m \\right\\rbrace = L_1 \\cup L_2.$ Since $F^{-1}(L_1)=\\left\\lbrace -4m\\xi +(s-4m)\\eta :-m \\le s \\le 2m \\right\\rbrace \\subset P^{\\prime }_1,$ and $F(L_2)=\\left\\lbrace (4m+t)\\xi +t\\eta : -5m \\le t \\le -2m \\right\\rbrace \\subset P^{\\prime }_1,$ we have $L_1 \\subset F(P^{\\prime }_1)=P^{\\prime }_2$ and $L_2 \\subset F^{-1}(P^{\\prime }_1)=P^{\\prime }_3$ .",
"Hence $|u| \\le K_2$ on $L_1 \\cup L_2$ , i.e.",
"the $\\xi $ -edge of $T^{\\prime }_1$ .",
"By Lemma REF , $|u| \\le (K_2+R)2^{6m}$ in $T^{\\prime }_1$ , and our claim holds.",
"Since $\\left(\\bigcup _{i=1,2,3} T^{\\prime }_i\\right) \\cup \\left(\\bigcup _{i=1,2,3} P^{\\prime }_i\\right) \\cup T_1 = T_2$ , we have $|u| \\le (K_2+R)2^{6m}$ in $T_2$ .",
"We also have that $2^{6m}(K_2+R) = 2^{12m}C_{6}^{12m}K+ (2^{12m}C_{6}^{12m}+2^{6m}C_{6}^{6m}+2^{6m})R \\le \\left(\\frac{C_4}{3}\\right)^{15m}(K+R),$ so $T_2$ is good.",
"To apply Theorem REF to prove Theorem REF in the next section, we actually need the following two corollaries.",
"Corollary 4.7 Let $a \\in \\Lambda $ , and $m, \\ell \\in \\mathbb {Z}_{\\ge 0}$ with $m \\ge 2\\ell $ .",
"Take any nonempty $L\\subset \\left\\lbrace a-t\\xi :t \\in \\mathbb {Z}, \\ell \\le t \\le m-\\ell \\right\\rbrace ,$ and function $u: P_{a;m,\\ell } \\rightarrow \\mathbb {R}$ such that $|u ( b ) +u ( b-\\xi ) +u ( b+\\eta )| \\le C_4^{-2\\ell }\\min _{c\\in L} |u(c)|,$ for any $b$ with $\\left\\lbrace b,b-\\xi , b+\\eta \\right\\rbrace \\subset P_{a;m,\\ell }$ .",
"Then $\\left| \\left\\lbrace b \\in P_{a;m,\\ell } : |u(b)| \\ge C_{4} ^{-2\\ell }\\min _{c\\in L} |u(c)| \\right\\rbrace \\right| \\ge \\epsilon _2 (\\ell +1)^2$ whenever $L$ contains at least one element; and $\\left| \\left\\lbrace b \\in P_{a;m,\\ell } : |u(b)| \\ge C_{4} ^{-2\\ell }\\min _{c\\in L} |u(c)| \\right\\rbrace \\right| \\ge \\epsilon _2 (m+2) (\\ell +1)$ if $m \\ge 2\\ell +2$ and $L= \\left\\lbrace a-t\\xi :t \\in \\mathbb {Z}, \\ell +1 \\le t \\le m-\\ell -1 \\right\\rbrace $ .",
"Here $\\epsilon _2$ is a universal constant.",
"If $\\ell \\le 10^9$ then the conclusion holds trivially by taking $\\epsilon _2$ small enough.",
"From now on we assume $\\ell >10^9$ .",
"We denote $P:=P_{a;m,\\ell }$ , for simplicity of notations.",
"Without loss of generality, we assume that $\\min _{c\\in L} |u(c)| = 1$ .",
"First we prove $\\left| \\left\\lbrace b \\in P : |u(b)| \\ge C_{4} ^{-2\\ell } \\right\\rbrace \\right| \\ge \\frac{\\epsilon _1 (\\ell +1)^2}{100},$ which implies (REF ).",
"We take $a^{\\prime }\\in L$ .",
"By (REF ), for any $b \\in P_{a-\\xi ;m-2,\\ell -2}$ and $0<k_1<\\ell $ , if $|u(b)| \\ge C_4 ^{-k_1}$ , then $|u ( b-\\eta )| \\ge C_4 ^{-k_1-1}$ or $|u ( b-\\gamma )| \\ge C_4 ^{-k_1-1} $ .",
"Thus we can inductively pick $a_1=a^{\\prime },a_2, \\cdots ,a_{\\left\\lfloor \\frac{\\ell }{3}\\right\\rfloor } \\in P$ , such that for each $i=1,2, \\cdots , \\left\\lfloor \\frac{\\ell }{3}\\right\\rfloor $ , $|u(a_i)| \\ge C_4 ^{-i+1}$ , and $a_i=a^{\\prime }-s_i\\xi -i\\eta $ with $s_i - s_{i-1} \\in \\left\\lbrace 0, 1\\right\\rbrace $ for each $2 \\le i \\le \\left\\lfloor \\frac{\\ell }{3}\\right\\rfloor $ .",
"In particular, we have $\\left|u\\left(a_{\\left\\lfloor \\frac{\\ell }{3}\\right\\rfloor }\\right)\\right| \\ge C_4 ^{-\\ell }$ .",
"Denote $T^{\\prime }:=T_{a_{\\left\\lfloor \\frac{\\ell }{3}\\right\\rfloor };2\\left\\lfloor \\frac{\\ell }{18}\\right\\rfloor }$ .",
"Then $T^{\\prime } \\subset P$ , and we can apply Theorem REF in $T^{\\prime }$ with $n=2\\left\\lfloor \\frac{\\ell }{18}\\right\\rfloor $ , thus $(\\ref {eq:trapep1})$ follows.",
"For the case where $L= \\left\\lbrace a-t\\xi :t \\in \\mathbb {Z}, \\ell +1 \\le t \\le m-\\ell -1 \\right\\rbrace $ , we prove $\\left| \\left\\lbrace b \\in P : |u(b)| \\ge C_{4} ^{-2\\ell } \\right\\rbrace \\right| \\ge \\epsilon _1 \\left(\\frac{(m+2)(\\ell +1)}{800}-\\frac{(\\ell +1)^2}{100}\\right).$ When $m \\le 8\\ell $ , (REF ) is trivial.",
"From now on we assume that $m > 8\\ell $ .",
"Denote $l := \\left\\lceil \\frac{m-2\\ell -1}{4\\ell } \\right\\rceil -1$ .",
"We take $b_1:=a-(\\ell +1)\\xi $ .",
"Let $b_i:=b_1-4\\ell (i-1)\\xi $ where $i=2,\\cdots ,l$ .",
"For each $1 \\le i \\le l$ , consider the trapezoid $P_i:=P_{b_i;2\\ell ,\\ell }$ .",
"We note that these trapezoids are disjoint, and $P_i \\subset P$ for each $1 \\le i \\le l$ (see Figure REF for an illustration).",
"We apply the same arguments in the proof of (REF ), with $P$ substituted by each $P_i$ , and we get $\\left| \\left\\lbrace b \\in P_i : |u(b)| \\ge C_{4} ^{-2\\ell } \\right\\rbrace \\right| \\ge \\frac{\\epsilon _1 (\\ell +1)^2}{100},$ for each $1 \\le i \\le l$ .",
"By summing over all $i$ we get (REF ).",
"Finally, we can deduce (REF ) from $(\\ref {eq:trapep1})$ and $(\\ref {eq:trapep2})$ .",
"Figure: An illustration of P i P_i's.",
"The thick line indicates LL.For the next corollary, we set up notations for reversed trapezoids.",
"Definition 4.8 For any $a\\in \\Lambda $ , $m,\\ell \\in \\mathbb {Z}_{\\ge 0}$ with $\\ell \\le m$ , we denote $P^{r}_{a;m,\\ell }:=\\left\\lbrace a-t\\xi -s\\eta :s \\le t \\le m, 0 \\le s \\le \\ell \\right\\rbrace \\cap \\Lambda ,$ which is also a trapezoid, but its orientation is different from that of $P_{a;m,\\ell }$ (see Figure REF for an illustration).",
"We also denote $\\left\\lbrace a-t\\xi :0 \\le t \\le m\\right\\rbrace \\cap \\Lambda $ to be the upper edge of $P^{r}_{a;m,\\ell }$ .",
"Corollary 4.9 Let $a \\in \\Lambda $ , and $m, \\ell \\in \\mathbb {Z}_{\\ge 0}$ with $m \\ge \\ell $ .",
"Let $L$ be a nonempty subset of the upper edge of $P^{r}_{a;m,\\ell }$ .",
"Figure: An illustration of Corollary : P a;m,ℓ r P^{r}_{a;m,\\ell } is the set of lattice points in the region surrounded by black lines.P a ' +ℓ 5+1ξ;2ℓ 5+2,ℓ 5 P_{a^{\\prime }+\\left(\\left\\lfloor \\frac{\\ell }{5}\\right\\rfloor +1\\right)\\xi ;2\\left\\lfloor \\frac{\\ell }{5}\\right\\rfloor +2,\\left\\lfloor \\frac{\\ell }{5}\\right\\rfloor } is the blue region, andP a-ℓ 5+2ξ;m-2ℓ 5-4,ℓ 5 P_{a-\\left(\\left\\lfloor \\frac{\\ell }{5}\\right\\rfloor +2\\right)\\xi ;m-2\\left\\lfloor \\frac{\\ell }{5}\\right\\rfloor -4,\\left\\lfloor \\frac{\\ell }{5}\\right\\rfloor }is the union of the blue and red regions.Take a function $u: P^{r}_{a;m,\\ell } \\rightarrow \\mathbb {R}$ such that $|u ( b ) +u ( b-\\xi ) +u ( b+\\eta )| \\le C_4^{-2\\ell }\\min _{c\\in L} |u(c)|,$ for any $b$ with $\\left\\lbrace b,b-\\xi ,b+\\eta \\right\\rbrace \\subset P^{r}_{a;m,\\ell }$ .",
"Then $\\left| \\left\\lbrace b \\in P^{r}_{a;m,\\ell } : |u(b)| \\ge C_{4} ^{-2\\ell }\\min _{c\\in L} |u(c)| \\right\\rbrace \\right| \\ge \\epsilon _3 (\\ell +1)^2,$ if $L=\\left\\lbrace a-\\left\\lfloor \\frac{m}{2}\\right\\rfloor \\xi \\right\\rbrace $ or $L=\\left\\lbrace a-\\left\\lceil \\frac{m}{2}\\right\\rceil \\xi \\right\\rbrace $ .",
"And $\\left| \\left\\lbrace b \\in P^{r}_{a;m,\\ell } : |u(b)| \\ge C_{4} ^{-2\\ell }\\min _{c\\in L} |u(c)| \\right\\rbrace \\right| \\ge \\epsilon _3 (m+2)(\\ell +1),$ if $L=\\left\\lbrace a-t\\xi :t\\in \\mathbb {Z}, 1 \\le t \\le m-1\\right\\rbrace $ .",
"Here $\\epsilon _3$ is a universal constant.",
"If $m \\le 10^9$ , then the conclusion holds trivially by taking $\\epsilon _3$ small enough.",
"From now on we assume that $m>10^9$ .",
"If $L=\\left\\lbrace a-\\left\\lfloor \\frac{m}{2}\\right\\rfloor \\xi \\right\\rbrace $ or $L=\\left\\lbrace a-\\left\\lceil \\frac{m}{2}\\right\\rceil \\xi \\right\\rbrace $ , let $a^{\\prime }=a-\\left\\lfloor \\frac{m}{2}\\right\\rfloor \\xi $ or $a^{\\prime }=a-\\left\\lceil \\frac{m}{2}\\right\\rceil \\xi $ respectively.",
"Consider $P_{a^{\\prime }+\\left(\\left\\lfloor \\frac{\\ell }{5}\\right\\rfloor +1\\right)\\xi ;2\\left\\lfloor \\frac{\\ell }{5}\\right\\rfloor +2,\\left\\lfloor \\frac{\\ell }{5}\\right\\rfloor }\\subset P^{r}_{a;m,\\ell }$ (blue region in Figure REF ).",
"Using Corollary REF for this trapezoid, we get (REF ).",
"If $L=\\left\\lbrace a-t\\xi :t\\in \\mathbb {Z}, 1 \\le t \\le m-1\\right\\rbrace $ , consider $P_{a-\\left(\\left\\lfloor \\frac{\\ell }{5}\\right\\rfloor +2\\right)\\xi ;m-2\\left\\lfloor \\frac{\\ell }{5}\\right\\rfloor -4,\\left\\lfloor \\frac{\\ell }{5}\\right\\rfloor } \\subset P^{r}_{a;m,\\ell }$ (union of blue and red regions in Figure REF ).",
"Using Corollary REF for this trapezoid, we get (REF )."
],
[
"Geometric substructure on 3D lattice",
"In this section we state and prove the following stronger version of Theorem REF which incorporates a graded set (which is defined in Definition REF ).",
"Theorem 5.1 For any $K\\in \\mathbb {R}_+$ , $N\\in \\mathbb {Z}_+$ , and small enough $\\varepsilon \\in \\mathbb {R}_+$ , we can find large $C_{2} \\in \\mathbb {R}_+$ depending only on $K$ and $C_{\\varepsilon ,N} \\in \\mathbb {R}_+$ depending only on $\\varepsilon , N$ , such that the following statement is true.",
"Take integer $n>C_{\\varepsilon ,N}$ and functions $u, V: \\mathbb {Z}^3 \\rightarrow \\mathbb {R}$ , satisfying $\\Delta u = V u$ in $Q_{n}$ and $\\Vert V\\Vert _{\\infty } \\le K$ .",
"Let $\\vec{l}$ be a vector of positive reals, and $E \\subset \\mathbb {Z}^3$ be any $(N, \\vec{l},\\varepsilon ^{-1}, \\varepsilon )$ -graded set, with the first scale length $l_1 > C_{\\varepsilon , N}$ .",
"If $E$ is $(1,2\\varepsilon )$ -normal in $Q_{n}$ , then we have that $ \\left| \\left\\lbrace a \\in Q_{n} : |u(a)| \\ge \\exp (-C_{2} n^{3}) |u(\\mathbf {0})| \\right\\rbrace \\setminus E \\right| \\ge C_3 n^2(\\log _2 n)^{-1}.$ Here $C_3$ is a universal constant.",
"The first result we need is based on the “cone property” of the function $u$ , as discussed in Section .",
"We remind the reader of the notations $\\mathbf {e}_\\tau $ , for $\\tau =1,2,3$ ; and $\\lambda _\\tau $ , $\\mathcal {P}_{\\tau ,k}$ , for $\\tau \\in \\left\\lbrace 1,2,3,4\\right\\rbrace $ and $k\\in \\mathbb {Z}$ , from Definition REF ; and the cones from Definition REF .",
"Proposition 5.2 Let $K \\in \\mathbb {R}_+$ , $n \\in \\mathbb {Z}_+$ , and $u, V$ satisfy (REF ) in $Q_n$ , with $\\Vert V\\Vert _{\\infty }\\le K$ .",
"Then there exists $\\tau \\in \\left\\lbrace 1, 2, 3, 4\\right\\rbrace $ , such that for any $0 \\le i \\le \\frac{n}{10}$ there is $a_i \\in \\left(\\mathcal {P}_{\\tau , i}\\cup \\mathcal {P}_{\\tau , i+1}\\right) \\cap \\mathcal {C}\\cap Q_{\\frac{n}{10} + 1}$ with $|u(a_i)| \\ge (K+11)^{-n}|u(\\mathbf {0})|$ .",
"We can assume that $n\\ge 10$ since otherwise this proposition holds obviously.",
"We argue by contradiction.",
"Denote $\\Upsilon :=\\left\\lbrace b \\in Q_{n}:|u(b)|\\ge (K+11)^{-n}|u(\\mathbf {0})|\\right\\rbrace $ .",
"If the statement is not true, then for each $\\tau \\in \\left\\lbrace 1, 2, 3, 4\\right\\rbrace $ , there is $i_{\\tau } \\in \\left[0, \\frac{n}{10}\\right]$ , such that $ \\left(\\mathcal {P}_{\\tau , i_\\tau }\\cup \\mathcal {P}_{\\tau , i_\\tau +1}\\right) \\cap \\mathcal {C}\\cap \\Upsilon \\cap Q_{\\frac{n}{10}+1}= \\emptyset .$ Define $B_{in}:=\\bigcap _{\\tau =1}^4\\left\\lbrace a \\in \\mathcal {C}: a\\cdot \\lambda _{\\tau } < i_{\\tau } \\right\\rbrace $ , $B_{bd}:=\\bigcap _{\\tau =1}^4\\left\\lbrace a \\in \\mathcal {C}: a\\cdot \\lambda _{\\tau } \\le i_{\\tau } + 1 \\right\\rbrace \\setminus B_{in}$ , $B_{out}:= \\mathcal {C}\\setminus \\left( B_{in}\\cup B_{bd} \\right)$ .",
"Then for any $a \\in B_{in}$ and $b \\in B_{out}$ , we have $\\Vert a - b \\Vert _{1} \\ge 3$ .",
"Since $i_1, i_2, i_3, i_4 \\le \\frac{n}{10}$ , we have that $B_{bd} \\subset \\mathcal {C}\\cap \\left\\lbrace a \\in \\mathbb {Z}^3: |a \\cdot \\mathbf {e}_1| + |a\\cdot \\mathbf {e}_2| + a \\cdot \\mathbf {e}_3 \\le \\frac{n}{10} + 1 \\right\\rbrace \\subset Q_{\\frac{n}{10} + 1}.$ Then the condition (REF ) implies that $\\Upsilon \\cap B_{bd} = \\emptyset $ .",
"We now apply Lemma REF to starting point $a_0 = \\mathbf {0}$ , in the $\\mathbf {e}_3$ direction, and $k = n$ .",
"Let $\\mathbf {0} = a_0, a_1, \\cdots , a_w \\in \\mathcal {C}\\cap \\mathbb {Z}^3$ be the chain.",
"Then $a_0 \\in B_{in}$ , and $a_w \\cdot \\mathbf {e}_3 \\ge n-1$ , which implies that $a_w \\in B_{out}$ (since otherwise, $a_w\\cdot \\mathbf {e}_3 = \\frac{1}{4}\\sum _{\\tau =1}^4 a_w\\cdot \\lambda _\\tau \\le \\frac{1}{4}\\sum _{\\tau =1}^4 i_\\tau +1\\le \\frac{n}{10}+1$ ).",
"Thus $a_w\\ne a_0$ and $w\\ge 1$ .",
"Since $|u(a_i)|\\ge (K+11)^{-1} |u(a_{i-1})|$ for each $i=1, \\cdots , w$ , we also have that each $a_i \\in \\Upsilon $ .",
"As $\\Upsilon \\cap B_{bd} = \\emptyset $ , we can find $1 \\le i \\le w$ , such that $a_{i-1} \\in B_{in}$ and $a_i \\in B_{out}$ .",
"This implies that $\\Vert a_{i-1} - a_i \\Vert _1 \\ge 3$ , which contradicts with the construction of the chain from Lemma REF .",
"Proposition 5.3 For any $K\\in \\mathbb {R}_+$ , $N\\in \\mathbb {Z}_+$ , and small enough $\\varepsilon >0$ , we can find $C_7, C_{\\varepsilon ,N}\\in \\mathbb {R}_+$ , where $C_7$ depends only on $K$ and $C_{\\varepsilon ,N}$ depends only on $\\varepsilon , N$ , such that following statement is true.",
"Take integer $n>C_{\\varepsilon ,N}$ , and let functions $u, V$ satisfy (REF ) in $Q_n$ , and $\\Vert V\\Vert _{\\infty }\\le K$ .",
"Let $\\vec{l}$ be a vector of positive reals, and $E$ be an $(N,\\vec{l},\\varepsilon ^{-1},\\varepsilon )$ -graded set with the first scale length $l_1>C_{\\varepsilon ,N}$ , and be $(1,2\\varepsilon )$ -normal in $Q_n$ .",
"For any $\\tau \\in \\left\\lbrace 1,2,3,4\\right\\rbrace $ , $k\\in \\mathbb {Z}$ , $0 \\le k \\le \\frac{n}{10}$ , and $a_0 \\in \\mathcal {P}_{\\tau , k} \\cap Q_{\\frac{n}{4}}$ , there exists $h \\in \\mathbb {Z}_+$ , such that $ \\left|\\left\\lbrace a \\in Q_n \\cap \\bigcup _{i=0}^h \\mathcal {P}_{\\tau , k + i}: |u(a)| \\ge \\exp (- C_{7} n^{3})|u(a_0)|\\right\\rbrace \\setminus E \\right| > C_{8} hn(\\log _2(n))^{-1} .$ Here $C_8$ is a universal constant.",
"In Section REF , Theorem REF is proved by applying Proposition REF to each point $a_i$ obtained from Proposition REF .",
"The next two subsections are devoted to the proof of Proposition REF .",
"We will work with $\\tau =1$ only, and the cases where $\\tau =2,3,4$ follow the same arguments.",
"Assuming the result does not hold, we can find many “gaps”, i.e.",
"intervals that do not intersect the set $\\lbrace |u(a)|:a \\in Q_n \\setminus E, a \\cdot \\lambda _{1} \\ge k \\rbrace $ .",
"These gaps will allow us to construct geometric objects on $\\mathbb {Z}^3$ .",
"We first find many “pyramids” in $\\left\\lbrace a \\in Q_n: a \\cdot \\lambda _{1} \\ge k \\right\\rbrace $ (see Lemma REF ), then we prove Proposition REF assuming a lower bound on the number of desired points in each “pyramid” (Proposition REF ).",
"In Section REF we prove Proposition REF , by studying “faces” of each “pyramid”, and using corollaries of Theorem REF ."
],
[
"Decomposition into pyramids",
"In this subsection we define pyramids in $Q_n$ , and in the next subsection we study the structure of each of these pyramids.",
"We need some further geometric objects in $\\mathbb {R}^3$ .",
"Definition 5.4 For simplicity of notations we denote $\\overline{\\lambda }_2=\\lambda _2=-\\mathbf {e}_1+\\mathbf {e}_2+\\mathbf {e}_3$ , $\\overline{\\lambda }_3=\\lambda _3=\\mathbf {e}_1-\\mathbf {e}_2+\\mathbf {e}_3$ , and $\\overline{\\lambda }_4=-\\lambda _4=\\mathbf {e}_1+\\mathbf {e}_2-\\mathbf {e}_3$ .",
"Then $\\lambda _1\\cdot \\overline{\\lambda }_2=\\lambda _1\\cdot \\overline{\\lambda }_3=\\lambda _1\\cdot \\overline{\\lambda }_4=1$ , and $\\overline{\\lambda }_2\\cdot \\overline{\\lambda }_3=\\overline{\\lambda }_2\\cdot \\overline{\\lambda }_4=\\overline{\\lambda }_3\\cdot \\overline{\\lambda }_4=-1$ .",
"For any $a \\in \\mathbb {R}^3$ , $r \\in \\mathbb {Z}_+$ , denote $\\mathbf {t}_r(a)=a+r\\mathbf {e}_1+r\\mathbf {e}_2+2r\\mathbf {e}_3$ .",
"Then $\\mathbf {t}_r(a)\\cdot \\overline{\\lambda }_2 = a\\cdot \\overline{\\lambda }_2+2r$ , $\\mathbf {t}_r(a)\\cdot \\overline{\\lambda }_3 = a\\cdot \\overline{\\lambda }_3+2r$ , and $\\mathbf {t}_r(a)\\cdot \\overline{\\lambda }_4 = a\\cdot \\overline{\\lambda }_4$ .",
"Denote $\\mathcal {T}_{a, r} := \\left\\lbrace b \\in \\mathcal {P}_{1, a \\cdot \\lambda _1}:b \\cdot \\overline{\\lambda }_\\tau \\le \\mathbf {t}_r(a) \\cdot \\overline{\\lambda }_\\tau , \\forall \\tau \\in \\lbrace 2,3,4\\rbrace \\right\\rbrace ,$ and let $\\mathring{\\mathcal {T}}_{a, r}$ be the interior of $\\mathcal {T}_{a, r}$ in $\\mathcal {P}_{1, a \\cdot \\lambda _1}$ .",
"Respectively, $\\mathring{\\mathcal {T}}_{a, r}$ and $\\mathcal {T}_{a, r}$ are the open and closed equilateral triangles with side length $2\\sqrt{2}r$ in the plane $\\mathcal {P}_{1, a \\cdot \\lambda _1}$ , and $a$ is the midpoint of one side.",
"When $a \\in \\mathbb {Z}^3$ , there are $2r+1$ lattice points on each side of $\\mathcal {T}_{a, r}$ .",
"We also take $\\mathfrak {T}_{a,r}:= \\left\\lbrace b \\in \\mathbb {R}^3: b\\cdot \\lambda _1 \\ge a \\cdot \\lambda _1, \\; b \\cdot \\overline{\\lambda }_\\tau \\le \\mathbf {t}_r(a) \\cdot \\overline{\\lambda }_\\tau , \\forall \\tau \\in \\lbrace 2,3,4\\rbrace \\right\\rbrace ,$ which is a (closed) regular tetrahedron, with four faces orthogonal to $\\lambda _1, \\lambda _2, \\lambda _3, \\lambda _4$ respectively.",
"The point $\\mathbf {t}_r(a)$ is a vertex of $\\mathfrak {T}_{a,r}$ , and $\\mathcal {T}_{a,r}$ is the face orthogonal to $\\lambda _1$ .",
"(See Figure REF for an illustration) For any $k \\in \\mathbb {Z}$ , denote $\\pi _k(a)$ to be the orthogonal projection of $a$ onto $\\mathcal {P}_{1, k}$ .",
"The purpose of the following lemma is to find some triangles ($\\mathcal {T}_{a_i,r_i}$ for $a_i$ , $r_i$ in Lemma REF ) in $\\mathcal {P}_{1,k}\\cup \\mathcal {P}_{1,k+1}$ , and these triangles will be basements of pyramids to be constructed in the proof of Proposition REF .",
"Lemma 5.5 Let $N \\in \\mathbb {Z}_{+}$ , and $\\varepsilon >0$ and be small enough, then there exists $C_{\\varepsilon ,N}>0$ such that the following statement is true.",
"Suppose we have a function $u:\\mathbb {Z}^3\\rightarrow \\mathbb {R}$ , $n, k \\in \\mathbb {Z}$ , $n>C_{\\varepsilon ,N}$ , $k \\in \\mathbb {Z}\\cap \\left[0, \\frac{n}{10}\\right)$ , $a_{0} \\in \\mathcal {P}_{1, k} \\cap Q_{\\frac{n}{4}}$ , a vector of positive reals $\\vec{l}$ , and an $(N,\\vec{l},\\varepsilon ^{-1},\\varepsilon )$ -graded set $E$ with the first scale length $l_1>C_{\\varepsilon ,N}$ , and $E$ being $(1,2 \\varepsilon )$ -normal in $Q_{n}$ , $D \\in \\mathbb {R}_+$ , and $0 < g_1, \\cdots , g_{100n} < |u(a_0)|$ , such that $g_i \\le g_{i+1}\\exp (-Dn)$ for each $1 \\le i \\le 100n - 1$ .",
"Then we can find $m \\in \\mathbb {Z}_+$ , $r_1, r_2 \\cdots , r_m \\in \\mathbb {Z}\\cap \\left[0, \\frac{n}{32}\\right)$ , $a_1, a_2, \\cdots , a_m \\in \\left(\\mathcal {P}_{1, k}\\cup \\mathcal {P}_{1, k+1} \\right) \\cap Q_{\\frac{n}{2}}$ and $s_{1},s_{2},\\cdots ,s_{m} \\in \\left\\lbrace 1,2,\\cdots , 100n\\right\\rbrace $ , satisfying the following conditions: $\\sum _{i=1}^m (r_i+1) \\ge \\frac{n}{100}$ .",
"for each $1 \\le i \\le m$ , we have $|u(a_{i})| \\ge \\exp (Dn)g_{s_i}$ , and $|u(b)|<g_{s_i}$ for any $b \\in (\\mathring{\\mathcal {T}}_{\\pi _{k}(a_{i}),r_{i}}\\cup \\mathring{\\mathcal {T}}_{\\pi _{k+1}(a_{i}),r_{i}}) \\cap \\mathbb {Z}^3$ .",
"for any point $a \\in \\mathcal {P}_{1, k}$ , we have $a \\in \\mathcal {T}_{\\pi _k(a_i), r_i}$ for at most two $1 \\le i \\le m$ .",
"$E$ is $(\\varepsilon ^{-\\frac{1}{2}},\\varepsilon )$ -normal in $\\mathfrak {T}_{a_i, r_i}$ for each $1 \\le i \\le m$ .",
"Denote $R:= \\left\\lbrace a \\in (\\mathcal {P}_{1,k} \\cup \\mathcal {P}_{1,k+1}) \\cap Q_{\\frac{n}{2}} : |u(a)| \\ge \\exp (Dn)g_1 \\right\\rbrace $ .",
"For each $a \\in R$ , denote $I(a):=\\max \\left\\lbrace i \\in \\left\\lbrace 1, \\cdots , 100n\\right\\rbrace : |u(a)|\\ge \\exp (Dn) g_i\\right\\rbrace ,$ and we let $r(a)$ be the largest integer, such that $0 \\le r(a) < \\frac{n}{32}$ , and $|u(b)| \\le g_{I(a)},\\; \\forall b\\in \\left(\\mathring{\\mathcal {T}}_{\\pi _k(a), r(a)} \\cup \\mathring{\\mathcal {T}}_{\\pi _{k+1}(a), r(a)}\\right)\\cap \\mathbb {Z}^3.$ Suppose $\\vec{l}=(l_1,l_2,\\cdots ,l_d)$ .",
"We write $E=\\bigcup _{i=0}^{d}E_{i}$ where $E_i$ is a $(N,l_i,\\varepsilon )$ -scattered set for $0<i\\le d$ , and $E_0$ is a $\\varepsilon ^{-1}$ -unitscattered set.",
"We write $E_{i}=\\bigcup _{t=1}^N\\bigcup _{j\\in \\mathbb {Z}_+}E_{i}^{(j,t)}$ , where each $E_{i}^{(j,t)}$ is an open ball, and $\\operatorname{dist}(E_i^{(j,t)}, E_i^{(j^{\\prime },t)})\\ge l_i^{1+\\varepsilon }$ , $\\forall j\\ne j^{\\prime } \\in \\mathbb {Z}_+$ .",
"We also write $E_{0}=\\bigcup _{j \\in \\mathbb {Z}_{+}} o_{j}$ where each $o_{j}$ is an open unit ball, such that $\\forall j\\ne j^{\\prime } \\in \\mathbb {Z}_+$ we have $\\operatorname{dist}(o_{j},o_{j^{\\prime }})\\ge \\varepsilon ^{-1}$ .",
"If $r(a) \\ge \\frac{n}{100}$ for any $a \\in R$ , then Condition 1 to 3 hold by letting $m=1$ , $a_1=a$ , $r_1=r(a)$ and $s_1=I(a)$ .",
"Now we show that Condition 4 also holds (when $C_{\\varepsilon ,N}$ is large enough).",
"Since $E$ is $(1,2\\varepsilon )$ -normal in $Q_{n}$ , $l_{i}<4n^{1-\\varepsilon },$ whenever $E_{i} \\cap Q_{n}\\ne \\emptyset $ .",
"Then since $n>C_{\\varepsilon ,N}$ , by taking $C_{\\varepsilon ,N}$ large enough we have $n>300\\varepsilon ^{-\\frac{1}{2}}$ , and $l_{i}<4n^{1-\\varepsilon }< r(a)^{1-\\frac{\\varepsilon }{2}}.$ Thus $E$ is $(\\varepsilon ^{-\\frac{1}{2}},\\varepsilon )$ -normal in $\\mathfrak {T}_{a_1, r_1}$ .",
"From now on, we assume $r(a) < \\frac{n}{100}$ for each $a \\in R$ .",
"We also assume that $n > 100$ by letting $C_{\\varepsilon , N} > 100$ .",
"For each $0<i\\le d$ , $1 \\le t \\le N$ , and $j\\in \\mathbb {Z}_+$ , denote $B_{i}^{(j,t)}$ to be the open ball with radius $l_{i}^{1+\\frac{2}{3}\\varepsilon }$ and the same center as $E_{i}^{(j,t)}$ .",
"Let $\\tilde{B}_{i}^{(j,t)}:=B_{i}^{(j,t)} \\cap \\mathcal {P}_{1, k}$ , which is either a 2D open ball on the plane $\\mathcal {P}_{1, k}$ , or $\\emptyset $ .",
"For each $j \\in \\mathbb {Z}_{+}$ , let $B_{j}$ be the open ball with radius $\\varepsilon ^{-\\frac{2}{3}}$ and has the same center as $o_{j}$ .",
"Denote $\\tilde{B}_{j}:=B_{j} \\cap \\mathcal {P}_{1,k}$ .",
"We define a graph $G$ as follows.",
"The set of vertices of $G$ is $V(G):=\\left\\lbrace \\mathcal {T}_{\\pi _{k}(a),r(a)+1}:a \\in R\\right\\rbrace \\\\\\cup \\left\\lbrace \\tilde{B}_{i}^{(j,t)}: 1 \\le i\\le d, 1 \\le t\\le N, j\\in \\mathbb {Z}_+, \\tilde{B}_{i}^{(j,t)} \\ne \\emptyset \\right\\rbrace \\cup \\left\\lbrace \\tilde{B}_{j}:j \\in \\mathbb {Z}_{+}, \\tilde{B}_{j} \\ne \\emptyset \\right\\rbrace .$ For any $v_1, v_2 \\in V(G)$ , there is an edge connecting $v_1,v_2$ if and only if $v_1 \\cap v_2\\ne \\emptyset $ .",
"Claim 5.6 There is $a_{\\infty } \\in R$ , such that $\\mathcal {T}_{\\pi _{k}(a_0),r(a_0)+1}$ and $\\mathcal {T}_{\\pi _{k}(a_{\\infty }),r(a_{\\infty })+1}$ are in the same connected component in $G$ , and $\\left(\\mathcal {T}_{\\pi _{k}(a_{\\infty }),r(a_{\\infty })+1} \\cup \\mathcal {T}_{\\pi _{k+1}(a_{\\infty }),r(a_{\\infty })+1}\\right) \\cap \\mathbb {Z}^3 \\lnot \\subset Q_{\\frac{n}{2}}$ .",
"We let $b_0 : = a_0$ .",
"For any $i \\in \\mathbb {Z}_{\\ge 0}$ , if $b_i \\in R$ , we choose $b_{i+1} \\in \\mathbb {Z}^3 \\cap \\left(\\mathring{\\mathcal {T}}_{\\pi _k(b_i), r(b_i)+1} \\cup \\mathring{\\mathcal {T}}_{\\pi _{k+1}(b_i), r(b_i)+1}\\right)\\setminus \\left(\\mathring{\\mathcal {T}}_{\\pi _k(b_i), r(b_i)} \\cup \\mathring{\\mathcal {T}}_{\\pi _{k+1}(b_i), r(b_i)}\\right),$ with the largest $|u(b_{i+1})|$ (choose any one if not unique).",
"As $b_{i+1} \\in \\mathbb {Z}^3 \\cap \\left(\\mathring{\\mathcal {T}}_{\\pi _k(b_i), r(b_i)+1} \\cup \\mathring{\\mathcal {T}}_{\\pi _{k+1}(b_i), r(b_i)+1}\\right)$ , we have that $ b_{i+1} \\cdot (-\\mathbf {e}_1-\\mathbf {e}_2+2\\mathbf {e}_3) \\ge b_{i} \\cdot (-\\mathbf {e}_1-\\mathbf {e}_2+2\\mathbf {e}_3) + 1.$ By the definition of $r(b_i)$ , we have that $|u(b_{i+1})| \\ge g_{I(b_i)} \\ge \\exp (Dn) g_{I(b_i)-1}$ , thus $I(b_{i+1}) \\ge I(b_i)-1$ .",
"The construction terminates when we get some $q \\in \\mathbb {Z}_+$ such that $b_q \\notin R$ .",
"We let $a_{\\infty } := b_{q-1}$ , and we show that it satisfies all the conditions.",
"From the construction, for each $i=0,\\cdots , q-1$ we have that $\\pi _k(b_{i+1}) \\in \\mathring{\\mathcal {T}}_{\\pi _k(b_i), r(b_i)+1}$ , so there is an edge in $G$ connecting $\\mathcal {T}_{\\pi _{k}(b_i),r(b_i)+1}$ and $\\mathcal {T}_{\\pi _{k}(b_{i+1}),r(b_{i+1})+1}$ .",
"This implies that $\\mathcal {T}_{\\pi _{k}(b_0),r(b_0)+1}$ and $\\mathcal {T}_{\\pi _{k}(b_{q-1}),r(b_{q-1})+1}$ are in the same connected component in $G$ .",
"If $\\left(\\mathcal {T}_{\\pi _{k}(b_{q-1}),r(b_{q-1})+1} \\cup \\mathcal {T}_{\\pi _{k+1}(b_{q-1}),r(b_{q-1})+1}\\right) \\cap \\mathbb {Z}^3 \\subset Q_{\\frac{n}{2}}$ , we have $b_q \\in Q_{\\frac{n}{2}}$ .",
"By (REF ) we have that $b_{q} \\cdot (-\\mathbf {e}_1-\\mathbf {e}_2+2\\mathbf {e}_3) \\ge b_{0} \\cdot (-\\mathbf {e}_1-\\mathbf {e}_2+2\\mathbf {e}_3) + q$ .",
"Since $b_0, b_q \\in Q_{\\frac{n}{2}}$ , we have $q \\le 4n$ .",
"This means that $I(b_q) \\ge I(b_0) - q \\ge 100n-4n > 1$ .",
"Then we have that $b_q \\in R$ , which contradicts with its construction.",
"This means that $a_{\\infty }=b_{q-1}$ satisfies all the conditions stated in the claim.",
"We define a weight on the graph $G$ , by letting each vertex in $\\left\\lbrace \\mathcal {T}_{\\pi _{k}(a),r(a)+1}:a \\in R\\right\\rbrace $ (which are triangles) have weight 2, and each other vertex (which are balls) have weight 1.",
"The weights are defined this way for the purpose of proving Condition 4.",
"We then take a path $\\gamma _{path} =\\left\\lbrace v_1, v_2, \\cdots , v_{p}\\right\\rbrace $ such that $\\pi _{k}(a_{0})\\in v_1$ and $\\pi _{k}(a_{\\infty }) \\in v_{p}$ , and has the least total weight (among all such paths).",
"Then all these vertices are mutually different.",
"For each $i=1,2,\\cdots ,p-1$ there is an edge connecting $v_{i}$ and $v_{i+1}$ , and these are all the edges in the subgraph induced by these vertices.",
"Note that each $v_i$ is either a ball or a triangle in $\\mathcal {P}_{1. k}$ .",
"See Figure REF for an illustration.",
"Figure: The path γ path \\gamma _{path}Suppose all the triangles in $\\gamma _{path}$ are $\\left\\lbrace \\mathcal {T}_{a_{i},r(a_{i})+1}:1 \\le i \\le m\\right\\rbrace $ .",
"Let $r_{i}:=r(a_{i})$ and $s_i:=I(a_i)$ .",
"We claim that these $a_{i}$ , $r_{i}$ and $s_i$ for $1 \\le i \\le m$ satisfy all the conditions.",
"Condition 2 follows from the definition of $r_i=r(a_i)$ .",
"As $\\gamma _{path}$ is a least weighted path, we have that $v_{i^{\\prime }} \\cap v_{i^{\\prime \\prime }}=\\emptyset $ whenever $|i^{\\prime }-i^{\\prime \\prime }|>1$ , thus Condition 3 follows as well.",
"We next verify Condition 1.",
"For this, we need to show that in the path, triangles constitute a substantial fraction.",
"This is incorporated in Claims REF and REF below.",
"Denote $\\ell _{i}:=\\operatorname{diam}(v_{i})$ , for each $1 \\le i \\le p$ .",
"As $r(a_{\\infty }) < \\frac{n}{100}$ , we have $a_{\\infty } \\notin Q_{\\frac{n}{2}-\\frac{n}{20}}$ ; also note that $a_0 \\in Q_{\\frac{n}{4}}$ , so we have $\\ell _{total}:=\\sum _{i=1}^{p} \\ell _{i}\\ge \\operatorname{dist}(Q_{\\frac{n}{4}},\\mathbb {Z}^3 \\setminus Q_{\\frac{n}{2}-\\frac{n}{20}}) \\ge \\frac{n}{5}.$ For each $1 \\le i \\le d$ and $1 \\le t \\le N$ , denote $\\mathcal {V}_{i,t}:=\\left\\lbrace v \\in \\gamma _{path}:\\exists j\\in \\mathbb {Z}_+, v=\\tilde{B}_{i}^{(j,t)}\\right\\rbrace $ .",
"Claim 5.7 If $\\mathcal {V}_{i,t} \\ne \\emptyset $ , then $\\sum _{i^{\\prime } : v_{i^{\\prime }} \\in \\mathcal {V}_{i,t}} \\ell _{i^{\\prime }} \\le \\ell _{total} l_{i}^{-\\frac{\\varepsilon }{4}}$ , provided that $\\varepsilon $ is small enough and $C_{\\varepsilon ,N}$ is large enough.",
"Since $\\mathcal {V}_{i,t} \\ne \\emptyset $ and $E$ is $(1,2\\varepsilon )$ -normal in $Q_{n}$ , we have $C_{\\varepsilon ,N} \\le l_{i} \\le n^{1-\\varepsilon }$ .",
"Case 1: $|\\mathcal {V}_{i,t}| = 1$ .",
"Suppose $\\lbrace v_{i^{\\prime }}\\rbrace = \\mathcal {V}_{i,t}$ .",
"Then by (REF ), when $C_{\\varepsilon ,N}$ is large enough we have $\\ell _{i^{\\prime }} \\le 2l_{i}^{1+\\frac{2}{3}\\varepsilon } \\le \\frac{nl_{i}^{-\\frac{\\varepsilon }{4}}}{5} \\le \\ell _{total} l_{i}^{-\\frac{\\varepsilon }{4}}.$ Case 2: $|\\mathcal {V}_{i,t}| > 1$ .",
"Write $\\mathcal {V}_{i,t}=\\left\\lbrace v_{i_1},v_{i_2},\\cdots ,v_{i_q}\\right\\rbrace $ , where $1 \\le i_1 < i_2 < \\cdots < i_q \\le p$ , and $q \\ge 2$ .",
"For each $w \\in \\left\\lbrace 1,2,\\cdots ,q-1\\right\\rbrace $ , consider the part of $\\gamma _{path}$ between $v_{i_{w}}$ and $v_{i_{w+1}}$ .",
"By letting $C_{\\varepsilon ,N}$ large enough we have $\\sum _{i^{\\prime }=i_w}^{i_{w+1}} \\ell _{i^{\\prime }} \\ge \\operatorname{dist}(v_{i_{w}},v_{i_{w+1}}) \\ge l_{i}^{1+\\varepsilon }-2l_{i}^{1+\\frac{2}{3}\\varepsilon }\\ge 2(\\ell _{i_{w}}+\\ell _{i_{w+1}}) l_{i}^{\\frac{\\varepsilon }{4} }.$ Summing (REF ) through all $w \\in \\left\\lbrace 1,2,\\cdots ,q-1\\right\\rbrace $ , we get $\\ell _{total} \\ge \\frac{1}{2}\\sum _{w \\in \\left\\lbrace 1,2,\\cdots ,q-1\\right\\rbrace } \\sum _{i^{\\prime }=i_w}^{i_{w+1}} \\ell _{i^{\\prime }}\\ge \\left(\\sum _{v_{i^{\\prime }} \\in \\mathcal {V}_{i,t}} \\ell _{i^{\\prime }}\\right) l_{i}^{\\frac{\\varepsilon }{4}}.$ Then the claim follows as well.",
"Let $\\mathcal {V}_{0}:=\\left\\lbrace v_{i^{\\prime }} \\in \\gamma _{path}:\\exists j\\in \\mathbb {Z}_{+}, v_{i^{\\prime }}=\\tilde{B}_{j}\\right\\rbrace $ .",
"Claim 5.8 If $\\mathcal {V}_{0} \\ne \\emptyset $ , then $\\sum _{v_{i^{\\prime }} \\in \\mathcal {V}_{0}} \\ell _{i^{\\prime }} \\le \\varepsilon ^{\\frac{1}{4}} \\ell _{total}$ , provided that $\\varepsilon $ is small enough and $C_{\\varepsilon ,N}$ is large enough.",
"This is by the same arguments as the proof of Claim REF .",
"From Claim REF and Claim REF , by making $\\varepsilon $ small and $C_{\\varepsilon ,N}$ large enough, from $l_1>C_{\\varepsilon ,N}$ and $l_{i+1}\\ge l_{i}^{1+2\\varepsilon }$ , we have $\\sum _{i^{\\prime }:\\text{$v_{i^{\\prime }}$ is a 2D ball}} \\ell _{i^{\\prime }}=\\sum _{v_{i^{\\prime }} \\in \\mathcal {V}_{0}} \\ell _{i^{\\prime }} +\\sum _{1 \\le i \\le d,1 \\le t\\le N}\\sum _{v_{i^{\\prime }} \\in \\mathcal {V}_{i,t}} \\ell _{i^{\\prime }} \\le \\varepsilon ^{\\frac{1}{4}} \\ell _{total} + N \\ell _{total} \\sum _{i=1}^{\\infty } l_{i}^{-\\frac{\\varepsilon }{4}} \\le \\frac{\\ell _{total}}{100} .$ Now we have that $\\sum _{i=1}^m (r_i+1) \\ge (2\\sqrt{2})^{-1} \\sum _{i^{\\prime }:v_{i^{\\prime }}\\text{\\;is\\;a\\;triangle}} \\ell _{i^{\\prime }} \\ge (2\\sqrt{2})^{-1}\\frac{99}{100} \\ell _{total} > \\frac{n}{100},$ where the last inequality is due to (REF ).",
"Then Condition 1 follows.",
"It remains to check Condition 4.",
"We prove by contradiction.",
"Suppose for some $1 \\le i^{\\prime } \\le m$ , $E$ is not $(\\varepsilon ^{-\\frac{1}{2}},\\varepsilon )$ -normal in $\\mathfrak {T}_{a_{i^{\\prime }},r_{i^{\\prime }}}$ .",
"There are only two cases: Case 1: There exists $1\\le i\\le d$ and $E_{i}^{(j,t)}$ , such that $E_{i}^{(j,t)} \\cap \\mathfrak {T}_{a_{i^{\\prime }},r_{i^{\\prime }}} \\ne \\emptyset $ and $l_{i} > \\operatorname{diam}(\\mathfrak {T}_{a_{i^{\\prime }},r_{i^{\\prime }}})^{1-\\frac{\\varepsilon }{2}}.$ Recall that $B_{i}^{(j,t)}$ is the ball with radius $l_{i}^{1+\\frac{2}{3}\\varepsilon }$ and the same center as $E_{i}^{(j,t)}$ .",
"By (REF ) and letting $C_{\\varepsilon ,N}$ large enough, we have $\\operatorname{radi}(B_{i}^{(j,t)})-l_i=l_{i}^{1+\\frac{2}{3}\\varepsilon }-l_{i} >\\operatorname{diam}(\\mathfrak {T}_{a_{i^{\\prime }},r_{i^{\\prime }}}) + 3.$ This implies that $\\mathcal {T}_{\\pi _{k}(a_{i^{\\prime }}),r_{i^{\\prime }}+1} \\subset B_{i}^{(j,t)}$ and $\\mathcal {T}_{\\pi _{k}(a_{i^{\\prime }}),r_{i^{\\prime }}+1} \\subset \\tilde{B}_{i}^{(j,t)}$ .",
"If we substitute $\\mathcal {T}_{\\pi _{k}(a_{i^{\\prime }}),r_{i^{\\prime }}+1}$ by $\\tilde{B}_{i}^{(j,t)}$ in the path $\\gamma _{path}$ , then the new path has lower weight than $\\gamma _{path}$ .",
"This contradicts with the fact that $\\gamma _{path}$ is a least weight path.",
"Case 2: $E_{0} \\cap \\mathfrak {T}_{a_{i^{\\prime }},r_{i^{\\prime }}} \\ne \\emptyset $ and $\\varepsilon ^{-\\frac{1}{2}} > \\operatorname{diam}(\\mathfrak {T}_{a_{i^{\\prime }},r_{i^{\\prime }}})$ .",
"Then $\\mathcal {T}_{\\pi _{k}(a_{i^{\\prime }}),r_{i^{\\prime }}+1} \\subset B_{j}$ and $\\mathcal {T}_{\\pi _{k}(a_{i^{\\prime }}),r_{i^{\\prime }}+1} \\subset \\tilde{B}_{j}$ for some $j \\in \\mathbb {Z}_{+}$ , since $\\operatorname{radi}(B_{j})-1=\\varepsilon ^{-\\frac{2}{3}}-1>\\varepsilon ^{-\\frac{1}{2}}+3 > \\operatorname{diam}(\\mathfrak {T}_{a_{i^{\\prime }},r_{i^{\\prime }}}) + 3$ .",
"By the same reason as Case 1, we reach a contradiction.",
"Thus Condition 4 holds and the conclusion follows.",
"Now we work on each tetrahedron $\\mathfrak {T}_{a_i, r_i}$ .",
"We will construct a pyramid in each of them, and show that on the boundary of the pyramid, the number of points $b$ such that $b\\notin E$ , $|u(b)|\\ge \\exp (-C_{2}n^{3})$ , is at least in the order of $r_i^2+1$ .",
"Figure: An illustration of the constructions in Definition and .",
"The colored triangles are 𝒯 a,r \\mathcal {T}_{a,r} and 𝒯 a,r,b \\mathcal {T}_{a,r,b}.We start by defining a family of regular tetrahedrons.",
"Recall that in Definition REF , we have defined the tetrahedron $\\mathfrak {T}_{a,r}$ with one face being $\\mathcal {T}_{a,r}$ .",
"Definition 5.9 Let $a\\in \\mathbb {Z}^3$ , $r \\in \\mathbb {Z}_+$ .",
"For each $b \\in \\mathfrak {T}_{a,r} \\cap \\mathbb {Z}^3$ , we define a regular tetrahedron $\\mathfrak {T}_{a,r,b}$ characterized by the following conditions.",
"Its four faces are orthogonal to $\\lambda _1, \\overline{\\lambda }_2, \\overline{\\lambda }_3, \\overline{\\lambda }_4$ respectively.",
"For $\\tau \\in \\left\\lbrace 2,3,4\\right\\rbrace $ , we consider the distances between the faces of $\\mathfrak {T}_{a,r}$ and $\\mathfrak {T}_{a,r,b}$ that are orthogonal to $\\overline{\\lambda }_{\\tau }$ , and they are the same for each $\\tau $ .",
"The point $b$ is at the boundary of the face orthogonal to $\\lambda _1$ .",
"Formally, we denote $F_{a,r,b}:= \\max \\left\\lbrace F : b\\cdot \\overline{\\lambda }_\\tau \\le \\mathbf {t}_r(a) \\cdot \\overline{\\lambda }_\\tau - F,\\; \\forall \\tau \\in \\lbrace 2,3,4\\rbrace \\right\\rbrace .$ Then $F_{a,r,b}\\ge 0$ since $b \\in \\mathfrak {T}_{a,r}$ , and $\\frac{F_{a,r,b}}{\\sqrt{3}}$ would be the distance between the faces of $\\mathfrak {T}_{a,r}$ and $\\mathfrak {T}_{a,r,b}$ that are orthogonal to $\\overline{\\lambda }_{\\tau }$ , for each $\\tau \\in \\left\\lbrace 2,3,4\\right\\rbrace $ .",
"Define $\\mathfrak {T}_{a,r,b}:= \\left\\lbrace c \\in \\mathbb {R}^3: c\\cdot \\lambda _1 \\ge b \\cdot \\lambda _1, \\;b\\cdot \\overline{\\lambda }_\\tau \\le \\mathbf {t}_r(a) \\cdot \\overline{\\lambda }_\\tau - F_{a,r,b},\\; \\forall \\tau \\in \\lbrace 2,3,4\\rbrace \\right\\rbrace ,$ and let $\\mathring{\\mathfrak {T}}_{a,r,b}$ be the interior of $\\mathfrak {T}_{a,r,b}$ .",
"We denote $\\mathcal {T}_{a,r,b}:= \\mathfrak {T}_{a, r, b} \\cap \\mathcal {P}_{1, b\\cdot \\lambda _1}$ to be the face of $\\mathfrak {T}_{a, r, b}$ orthogonal to $\\lambda _1$ , and we denote its three edges as $\\mathcal {L}_{a,r,b,\\tau } :=\\left\\lbrace c\\in \\mathcal {T}_{a,r,b}: c\\cdot \\overline{\\lambda }_\\tau = \\mathbf {t}_r(a)\\cdot \\overline{\\lambda }_\\tau - F_{a, r,b}\\right\\rbrace ,\\;\\forall \\tau \\in \\lbrace 2,3,4\\rbrace .$ Then $b$ is on one of these three edges.",
"We denote the three vertices by $\\mathbf {v}_{a,r,b,\\tau }:= \\bigcap _{\\tau ^{\\prime } \\in \\left\\lbrace 2, 3, 4\\right\\rbrace \\setminus \\left\\lbrace \\tau \\right\\rbrace }\\mathcal {L}_{a,r,b,\\tau ^{\\prime }},\\; \\tau \\in \\left\\lbrace 2,3,4\\right\\rbrace ,$ or equivalently, $\\mathbf {v}_{a,r,b,\\tau }$ is the unique point characterized by $\\mathbf {v}_{a,r,b,\\tau } \\cdot \\lambda _1=b\\cdot \\lambda _1$ , and $\\mathbf {v}_{a,r,b,\\tau } \\cdot \\overline{\\lambda }_{\\tau ^{\\prime }}=\\mathbf {t}_r(a)\\cdot \\overline{\\lambda }_{\\tau ^{\\prime }}-F_{a,r,b}$ for $\\tau ^{\\prime }\\in \\lbrace 2,3,4\\rbrace \\setminus \\lbrace \\tau \\rbrace $ .",
"As $b\\cdot \\lambda _1$ and each $\\mathbf {t}_r(a)\\cdot \\overline{\\lambda }_{\\tau ^{\\prime }}-F_{a,r,b}$ are integers and have the same parity, we have $\\mathbf {v}_{a,r,b,\\tau }\\in \\mathbb {Z}^3$ .",
"We also denote the interior of these three edges by $\\mathring{\\mathcal {L}}_{a,r,b,\\tau } := \\mathcal {L}_{a,r,b,\\tau } \\setminus \\left\\lbrace \\mathbf {v}_{a,r,b,2},\\mathbf {v}_{a,r,b,3},\\mathbf {v}_{a,r,b,4}\\right\\rbrace , \\;\\tau \\in \\left\\lbrace 2,3,4\\right\\rbrace .$ We now define the pyramid using these tetrahedrons.",
"Definition 5.10 Take any $a\\in \\mathbb {Z}^3$ , $r \\in \\mathbb {Z}_+$ .",
"For any $b \\in \\mathfrak {T}_{a,r}\\bigcap \\mathbb {Z}^3$ let $\\mathring{\\mathfrak {H}}_{a,r,b} := \\left\\lbrace c \\in \\mathbb {R}^3:c \\cdot \\lambda _1 > b \\cdot \\lambda _1\\right\\rbrace \\setminus \\mathfrak {T}_{a,r,b},$ which is an open half space minus a regular tetrahedron.",
"Let $\\mathfrak {H}_{a,r,b}$ be the closure of $\\mathring{\\mathfrak {H}}_{a,r,b}$ .",
"Let $\\Gamma \\subset \\mathbb {Z}^3$ , such that $a\\in \\Gamma $ and $\\mathring{\\mathcal {T}}_{a, r} \\cap \\Gamma = \\emptyset $ .",
"We consider the collection of sets $\\left\\lbrace \\mathfrak {H}_{a, r, b}\\right\\rbrace _{b \\in \\mathfrak {T}_{a,r} \\cap \\Gamma }$ .",
"They form a partially ordered set (POSET) by inclusion of sets.",
"We take all the maximal elements in $\\left\\lbrace \\mathfrak {H}_{a, r, b}\\right\\rbrace _{b \\in \\mathfrak {T}_{a,r} \\cap \\Gamma }$ , and denote them as $\\mathfrak {H}_{a, r, b_1}, \\cdots , \\mathfrak {H}_{a, r, b_m}$ .",
"In particular $\\mathfrak {H}_{a, r, a} = \\mathfrak {H}_{a, r}$ is maximal since $\\mathring{\\mathcal {T}}_{a, r} \\cap \\Gamma = \\emptyset $ , so we can assume that $b_1 = a$ .",
"(For each $2 \\le i \\le m$ , the choice of each $b_i \\in \\mathfrak {T}_{a,r} \\cap \\Gamma $ may not be unique, but always gives the same $\\mathfrak {H}_{a, r, b_i}$ .)",
"We note that since each $\\mathfrak {H}_{a, r, b_i}$ is maximal, all the numbers $b_i \\cdot \\lambda _1$ for $1 \\le i \\le m$ must be mutually different, so we can assume that $b_1 \\cdot \\lambda _1 < \\cdots < b_m \\cdot \\lambda _1$ .",
"The pyramid is defined as $\\mathfrak {P}_{a, r, \\Gamma } :=\\mathfrak {T}_{a,r,b_m}\\cup \\bigcup _{i=1}^{m-1} \\left(\\mathfrak {T}_{a,r,b_i}\\cap \\left\\lbrace c \\in \\mathbb {R}^3:c \\cdot \\lambda _1 \\le b_{i+1} \\cdot \\lambda _1\\right\\rbrace \\right),$ and we let $\\mathring{\\mathfrak {P}}_{a, r, \\Gamma }$ be the interior of $\\mathfrak {P}_{a, r, \\Gamma }$ .",
"Note that in this definition, $\\mathfrak {P}_{a, 0, \\Gamma }:=\\left\\lbrace a\\right\\rbrace $ .",
"Finally, let $\\partial \\mathfrak {P}_{a, r, \\Gamma }:=\\mathfrak {P}_{a, r, \\Gamma } \\setminus (\\mathring{\\mathfrak {P}}_{a, r, \\Gamma }\\cup \\mathring{\\mathcal {T}}_{a,r})$ be the boundary of the pyramid (without the interior of its basement).",
"See Figure REF for an example of pyramid.",
"In words, we construct the pyramid $\\mathfrak {P}_{a,r,\\Gamma }$ by stacking together some “truncated” regular tetrahedrons $\\mathfrak {T}_{a,r,b}$ , for $b \\in \\Gamma $ , so that $\\mathfrak {P}_{a,r,\\Gamma }$ intersects $\\Gamma $ only at its boundary.",
"Indeed, for each $b\\in \\mathfrak {T}_{a,r}\\cap \\Gamma $ we have $b \\in \\mathfrak {H}_{a,r,b}$ , and $\\mathring{\\mathfrak {P}}_{a, r, \\Gamma } \\cap \\mathfrak {H}_{a,r,b} = \\emptyset $ .",
"Our key step towards proving Proposition REF is the following estimate about points on the boundary of a pyramid.",
"Figure: Pyramid 𝔓 a,r,Γ \\mathfrak {P}_{a,r,\\Gamma }, where Γ\\Gamma is the collection of red points.Proposition 5.11 There exists a constant $C_{9}$ , such that for any $K\\in \\mathbb {R}_+$ , $N \\in \\mathbb {Z}_+$ , and any small enough $\\varepsilon \\in \\mathbb {R}_+$ , there are small $C_{10} \\in \\mathbb {R}_+$ depending only on $K$ and large $C_{\\varepsilon , N} \\in \\mathbb {R}_+$ depending only on $\\varepsilon , N$ , such that the following statement holds.",
"Take any $g\\in \\mathbb {R}_+$ , $n, r\\in \\mathbb {Z}_+$ with $0 \\le r<\\frac{n}{32}$ , and functions $u, V$ satisfying $\\Delta u = Vu$ in $Q_n$ and $\\Vert V\\Vert _{\\infty }\\le K$ .",
"Suppose we have that $\\Gamma :=\\left\\lbrace b \\in Q_{n}: |u(b)|\\ge \\exp (3 C_{10} n) g\\right\\rbrace $ , and $a \\in \\Gamma \\cap Q_{\\frac{n}{2}}$ ; $|u(b)| < g$ for each $b \\in \\mathring{\\mathcal {T}}_{a,r} \\cap \\mathbb {Z}^3$ , and either $|u(b)| < g$ for each $b \\in \\mathring{\\mathcal {T}}_{a-\\frac{\\lambda _1}{3}, r} \\cap \\mathbb {Z}^3$ or $|u(b)| < g$ for each $b \\in \\mathring{\\mathcal {T}}_{a+\\frac{\\lambda _1}{3} ,r} \\cap \\mathbb {Z}^3$ ; $\\vec{l}$ is a vector of positive reals, $E$ is an $(N,\\vec{l},\\varepsilon ^{-1},\\varepsilon )$ -graded set; in addition, the first scale length of $E$ is $l_1>C_{\\varepsilon ,N}$ , and $E$ is $(\\varepsilon ^{-\\frac{1}{2}},\\varepsilon )$ -normal in $\\mathfrak {T}_{a, r}$ ; for each $b \\in Q_{n}$ with $b \\cdot \\lambda _1 \\ge a \\cdot \\lambda _1$ , $g \\le |u(b)| \\le \\exp (3C_{10} n) g$ implies $b \\in E$ .",
"Then $\\left| \\left\\lbrace b \\in \\partial \\mathfrak {P}_{a,r,\\Gamma }\\cap \\mathbb {Z}^3: |u(b)|\\ge \\exp (C_{10}n)g \\right\\rbrace \\setminus E \\right| \\ge C_{9} (r^2 +1).$ The proof of Proposition REF is left for the next subsection.",
"We now finish the proof of Proposition REF assuming it.",
"[Proof of Proposition REF ] The idea is to first apply Lemma REF to find some triangles $\\mathcal {T}_{a_i,r_i}$ in $\\mathcal {P}_{1,k}\\cup \\mathcal {P}_{1,k+1}$ , and build pyramids using these triangles as basements, then apply Proposition REF to lower bound the number of desired points on the boundary of each pyramid and finally sum them up.",
"For the parameters, we take $C_{7}=\\max \\left\\lbrace 6C_{10},\\log (K+11)\\right\\rbrace $ where $C_{10}$ is the constant in Proposition REF .",
"We leave $C_8$ to be determined.",
"We require that $\\varepsilon $ is small as required by Lemma REF and Proposition REF ; and for each such $\\varepsilon $ we let $C_{\\varepsilon , N}$ be large enough as required by Lemma REF and Proposition REF .",
"Without loss of generality, we assume $\\tau =1$ .",
"We can also assume $n > 100$ , by letting $C_{\\varepsilon , N} > 100$ .",
"Denote $\\Upsilon :=\\left\\lbrace a \\in Q_{n}: |u(a)| \\ge \\exp (- C_{7} n^{3})|u(a_0)|, a \\cdot \\lambda _1 \\ge k \\right\\rbrace \\setminus E.$ If $|\\Upsilon | \\ge n^2$ , the conclusion follows by letting $h = 3n$ and $C_{8}<\\frac{1}{3}$ .",
"Now we assume that $|\\Upsilon | < n^2$ .",
"The interval $[\\exp (-C_{7} n^3)|u(a_0)|, |u(a_0)|)$ is the union of $2n^2$ disjoint intervals, which are $\\left[\\exp \\left(-\\frac{C_{7}(i+1)n}{2}\\right)|u(a_0)|,\\exp \\left(-\\frac{C_{7}in}{2}\\right)|u(a_0)|\\right), \\; i = 0, \\cdots , 2n^2-1.$ By the Pigeonhole principle, at least $n^2$ of these intervals do not intersect the set $\\left\\lbrace |u(a)| : a \\in \\Upsilon \\right\\rbrace $ ; i.e., we can find $\\exp (-C_{7} n^3)|u(a_0)| \\le g_1, \\cdots , g_{n^2} \\le |u(a_0)|$ , such that $g_i \\le g_{i+1}\\exp \\left(-\\frac{C_{7}n}{2}\\right)$ , for each $1 \\le i \\le n^2-1$ , and $\\left\\lbrace a \\in Q_n: |u(a)| \\in \\bigcup _{i=1}^{n^2} \\left[g_i, g_i \\exp \\left(\\frac{C_{7} n}{2}\\right)\\right), a \\cdot \\lambda _1 \\ge k \\right\\rbrace \\subset E .$ We remark that actually we just need $g_1,\\cdots , g_{100n}$ to apply Lemma REF , rather than $n^2$ numbers; but we cannot get a better quantitative lower bound for $|u|$ by optimizing this, since applying the Pigeonhole principle to $2n^2$ parts or $n^2+100n$ parts does not make any essential difference.",
"As we assume that $a_0 \\in \\mathcal {P}_{1, k} \\cap Q_{\\frac{n}{4}}$ and $0 \\le k \\le \\frac{n}{10}$ , we can apply Lemma REF with $D=\\frac{C_{7}}{2}$ .",
"Then we can find some $a_1, \\cdots , a_m$ , $r_1, \\cdots , r_m$ and $g_{s_1}, \\cdots , g_{s_m}$ , satisfying the conditions there.",
"In particular, we have $|u(a_{i})| \\ge g_{s_i} \\exp \\left(\\frac{C_{7}n}{2}\\right) > \\exp (-C_{7} n^3)|u(a_0)|$ , for each $1\\le i \\le m$ .",
"If $m > n$ , we can just take $h=2$ , and (REF ) holds by taking $C_{8}$ small.",
"Now assume that $m\\le n$ .",
"We argue by contradiction, assuming that (REF ) does not hold.",
"As $C_{7} \\ge 6 C_{10}$ , we can apply Proposition REF to $a=a_i$ , $r=r_{i}$ and $g=g_{s_i}$ for each $i=1,2,\\cdots ,m$ , and get that $\\left|\\Upsilon \\cap \\mathfrak {T}_{a_i, r_i}\\right|\\ge \\left| \\left\\lbrace b \\in \\mathfrak {T}_{a_i,r_i}\\cap \\mathbb {Z}^3: |u(b)|\\ge \\exp (C_{10}n)g_{s_i} \\right\\rbrace \\setminus E \\right| \\ge C_{9} (r_{i}^2 +1).$ As we have assumed that $(\\ref {eq:lvh})$ does not hold, for each $h \\in \\mathbb {Z}_+$ , $C_{9}\\sum _{i=1}^m {1}_{h > 4r_i}(r_i^2+1)\\le \\sum _{i=1}^m {1}_{h > 4r_i} \\left|\\Upsilon \\cap \\mathfrak {T}_{a_i, r_i}\\right|\\le 2\\left| \\left(\\bigcup _{i=0}^h \\mathcal {P}_{1, k + i} \\right) \\cap \\Upsilon \\right| \\le 2C_{8}hn(\\log _2 n)^{-1}$ where the second inequality is due to the fact that any point is contained in at most two tetrahedrons $\\mathfrak {T}_{a_i, r_i}$ , by Conclusion 3 in Lemma REF .",
"Take $l := \\left\\lfloor \\log _2 n\\right\\rfloor - 5$ .",
"For each $0 \\le j \\le l$ , let $M_j=|\\left\\lbrace i:1 \\le i \\le m, 2^j \\le r_i +1 <2^{j+1}\\right\\rbrace |$ .",
"Then we have that $ \\sum _{j=0}^{l}2^j M_j \\ge \\frac{1}{2}\\sum _{i=1}^m (r_i +1) \\ge \\frac{n}{200} ,$ by Lemma REF .",
"For each $0 \\le s \\le l$ , by taking $h = 2^{s+3}$ in equation (REF ) we get $ C_{9}\\sum _{j=0}^{s} 2^{2j} M_j \\le C_{8}2^{s+4}n (l+5)^{-1} .$ Multiplying both sides of (REF ) by $2^{-s}$ and summing over all $s\\in \\mathbb {Z}_{\\ge 0}$ , we get $ \\sum _{j=0}^{l}2^j M_j\\le \\sum _{s=0}^l\\sum _{j=0}^{s} 2^{2j-s} M_j \\le \\sum _{s=0}^l 2^4 C_{8}(C_{9})^{-1} n (l+5)^{-1} < 2^4 C_{8}(C_{9})^{-1} n.$ This contradicts with (REF ) whenever $C_8 < (200\\cdot 2^4)^{-1}C_9$ ."
],
[
"Multi-layer structure of the pyramid and estimates on the boundary",
"The purpose of this subsection is to prove Proposition REF .",
"We first show that, under slightly different conditions, there are many points in $\\Gamma $ on the boundary of a pyramid without removing the graded set.",
"Proposition 5.12 There exists a constant $C_{9}^{\\prime }$ , so that for any $K \\in \\mathbb {R}_+$ , there is $C_{10} >K+11$ , relying only on $K$ , and the following is true.",
"Take any $g\\in \\mathbb {R}_+$ , $n,r\\in \\mathbb {Z}$ with $0 \\le r<\\frac{n}{32}$ , and functions $u, V$ satisfying $\\Delta u = Vu$ in $Q_n$ and $\\Vert V\\Vert _{\\infty }\\le K$ .",
"Suppose we have $\\Gamma :=\\left\\lbrace b \\in Q_{n}: |u(b)|\\ge \\exp (3 C_{10} n) g\\right\\rbrace $ , and $a \\in \\Gamma \\cap Q_{\\frac{n}{2}}$ ; $|u(b)| < g$ for each $b \\in \\mathring{\\mathcal {T}}_{a,r} \\cap \\mathbb {Z}^3$ , and either $|u(b)| < g$ for each $b \\in \\mathring{\\mathcal {T}}_{a-\\frac{\\lambda _1}{3} ,r} \\cap \\mathbb {Z}^3$ or for each $b \\in \\mathring{\\mathcal {T}}_{a+\\frac{\\lambda _1}{3} ,r} \\cap \\mathbb {Z}^3$ ; $h:= \\max \\lbrace a \\cdot \\lambda _1\\rbrace \\cup \\left\\lbrace b\\cdot \\lambda _1: b \\in \\mathring{\\mathfrak {P}}_{a,r,\\Gamma } \\cap \\mathbb {Z}^3, |\\mathcal {L}_{a,r,b,2}\\cap \\mathbb {Z}^3| \\ge \\frac{r}{4} \\right\\rbrace $ , and $|u(b)| \\le \\exp (C_{10} n)g$ for each $b \\in \\mathring{\\mathfrak {P}}_{a,r,\\Gamma } \\cap \\mathbb {Z}^3$ with $b \\cdot \\lambda _1 \\le h$ .",
"Then $ \\left| \\left\\lbrace b \\in \\partial \\mathfrak {P}_{a,r,\\Gamma }: |u(b)|\\ge \\exp (C_{10}n)g \\right\\rbrace \\right| \\ge C_{9}^{\\prime } (r^2+1) .$ To prove Proposition REF , we analyze the structure of the pyramid boundary $\\partial \\mathfrak {P}_{a,r,\\Gamma }$ .",
"Specifically, we study faces of it and estimate the number of lattice points $b$ with $|u(b)|\\ge \\exp (C_{10}n)g$ on each face.",
"For some of the faces, we can show that the number of such points is proportional to the area of the face.",
"This is by observing that the lattice $\\mathbb {Z}^3$ restricted to the face is a triangular lattice, and then using results from Section .",
"Finally we sum up the points on all the faces and get the conclusion.",
"[Proof of Proposition REF ] We can assume that $r>100$ , since otherwise the statement holds by taking $C_{9}^{\\prime }<10^{-5}$ .",
"We take $a=b_1,\\cdots ,b_m$ from the definition of $\\mathfrak {P}_{a,r,\\Gamma }$ .",
"As $\\mathfrak {H}_{a, r, b_1}, \\cdots , \\mathfrak {H}_{a, r, b_m}$ are all the maximal elements in $\\left\\lbrace \\mathfrak {H}_{a, r, b}\\right\\rbrace _{b \\in \\mathfrak {T}_{a,r} \\cap \\Gamma }$ , we have that $\\bigcup _{b \\in \\mathfrak {T}_{a,r} \\cap \\Gamma } \\mathfrak {H}_{a, r, b}=\\bigcup _{i=1}^m \\mathfrak {H}_{a, r, b_i} .$ We can also characterize $\\mathring{\\mathfrak {P}}_{a,r,\\Gamma }$ as the half space $\\lbrace b\\in \\mathbb {R}^3: b\\cdot \\lambda _1>a\\cdot \\lambda _1\\rbrace $ minus $\\bigcup _{i=1}^m \\mathfrak {H}_{a,r,b_i}$ .",
"For each $s \\in \\mathbb {Z}$ , we take $m_s \\in \\left\\lbrace 1, \\cdots , m\\right\\rbrace $ to be the maximum number such that $b_{m_s} \\cdot \\lambda _1 \\le s$ .",
"We first study the faces of $\\partial \\mathfrak {P}_{a,r,\\Gamma }$ that are orthogonal to $\\lambda _1$ .",
"For $2\\le i \\le m$ , we denote $\\mathring{\\mathcal {R}}_{i}:=\\mathring{\\mathfrak {T}}_{a,r,b_{i-1}}\\cap \\mathcal {P}_{1, b_i \\cdot \\lambda _1}=\\left\\lbrace b\\in \\mathcal {P}_{1,b_i\\cdot \\lambda _1}: b\\cdot \\overline{\\lambda }_\\tau < \\mathbf {t}_r(a)\\cdot \\overline{\\lambda }_\\tau - F_{a,r,b_{i-1}},\\;\\forall \\tau \\in \\lbrace 2,3,4\\rbrace \\right\\rbrace .$ Let $\\mathcal {R}_i$ be the closure of $\\mathring{\\mathcal {R}}_i$ , then $\\mathcal {R}_i \\supset \\mathcal {T}_{a,r,b_i}$ and it has the same center as $\\mathcal {T}_{a,r,b_i}$ .",
"We denote the side length of $\\mathcal {R}_i$ to be $\\theta _i$ .",
"Note that the three vertices of $\\mathcal {R}_i$ are in $\\frac{1}{2}\\mathbb {Z}^3$ , so $\\frac{\\theta _i}{\\sqrt{2}}\\in \\frac{1}{2}\\mathbb {Z}_{\\ge 0}$ .",
"Further, for each $1 \\le i \\le m+1$ , we denote the side length of $\\mathcal {T}_{a,r,b_i}$ to be $\\vartheta _i$ .",
"Note that the vertices of $\\mathcal {T}_{a,r,b_i}$ are $\\mathbf {v}_{a,r,b_i,\\tau }$ , for $\\tau \\in \\lbrace 2,3,4\\rbrace $ , and each of them is in $\\mathbb {Z}^3$ .",
"Thus we have $\\frac{\\vartheta _i}{\\sqrt{2}} = |\\mathcal {L}_{a,r,b_i,2}\\cap \\mathbb {Z}^3|-1 \\in \\mathbb {Z}_{\\ge 0}$ .",
"We also obviously have that $2\\sqrt{2}r = \\vartheta _1 > \\theta _2 > \\vartheta _2 > \\cdots > \\theta _m > \\vartheta _m\\ge 0$ .",
"For simplicity of notations, we also denote $b_{m+1}:= \\operatorname{argmax}_{b \\in \\mathfrak {P}_{a,r,\\Gamma }} b \\cdot \\lambda _1$ , and $\\theta _{m+1}=\\vartheta _{m+1}=0$ .",
"The following results will be useful in analyzing the face $\\mathcal {R}_i$ , for $1 \\le i \\le m_{h+1}$ .",
"Claim 5.13 For any $2\\le i \\le m_{h+1}$ and $b \\in \\mathring{\\mathcal {R}}_i \\cap \\mathbb {Z}^3$ , if $b+\\mathbf {e}_1-\\mathbf {e}_3, b+\\mathbf {e}_2-\\mathbf {e}_3\\in \\mathring{\\mathcal {R}}_i$ , then we have $ |u(c)| < \\exp (C_{10}n)g,\\;\\forall c \\in \\left\\lbrace b-\\mathbf {e}_3, b-\\mathbf {e}_1-\\mathbf {e}_3, b-\\mathbf {e}_2-\\mathbf {e}_3, b-2\\mathbf {e}_3\\right\\rbrace .$ Claim 5.14 If $C_{10} > K+11$ , then for each $2 \\le i \\le m_h$ , there exists $\\tau _i \\in \\left\\lbrace 2,3,4\\right\\rbrace $ , such that $b_i \\in \\mathcal {L}_{a,r,b_i,\\tau _i}$ , and $|u(b)|\\ge \\exp (2C_{10}n)g$ , $\\forall b \\in \\mathring{\\mathcal {L}}_{a,r,b_i,\\tau _i} \\cap \\mathbb {Z}^3$ .",
"We continue our proof assuming these claims.",
"Fix $2 \\le i \\le m_{h+1}$ .",
"For any $b \\in \\mathring{\\mathcal {R}}_i \\cap \\mathbb {Z}^3$ with $b+\\mathbf {e}_1-\\mathbf {e}_3, b+\\mathbf {e}_2-\\mathbf {e}_3\\in \\mathring{\\mathcal {R}}_i$ , since $\\Delta u(b-\\mathbf {e}_3)=(V u)(b-\\mathbf {e}_3)$ , and $|V(b-\\mathbf {e}_3)| \\le K$ , by Claim REF we have $\\left|u(b)+u(b+\\mathbf {e}_1-\\mathbf {e}_3)+u(b+\\mathbf {e}_2-\\mathbf {e}_3) \\right| \\\\\\le (K+6)\\max _{c \\in \\left\\lbrace b-\\mathbf {e}_3, b-\\mathbf {e}_1-\\mathbf {e}_3, b-\\mathbf {e}_2-\\mathbf {e}_3, b-2\\mathbf {e}_3\\right\\rbrace } |u(c)|\\le (K+6)\\exp (C_{10} n)g.$ We take $C_{10} > 2\\ln (C_4(K+6))$ where $C_{4}$ is the constant in Theorem REF .",
"Then if $i \\le m_h$ , using Claim REF and $b_i\\in \\Gamma $ , we have $ \\left|u(b)+u(b+\\mathbf {e}_1-\\mathbf {e}_3)+u(b+\\mathbf {e}_2-\\mathbf {e}_3) \\right|<C_4^{-2n}\\min _{c \\in \\left(\\mathring{\\mathcal {L}}_{a,r,b_i,\\tau _i}\\cap \\mathbb {Z}^3\\right) \\cup \\left\\lbrace b_i\\right\\rbrace } |u(c)|,$ where $\\tau _i \\in \\left\\lbrace 2,3,4\\right\\rbrace $ is given by Claim REF .",
"If $m_{h}< m_{h+1}$ and $i = m_{h+1}$ , as $b_i\\in \\Gamma $ we have $ \\left|u(b)+u(b+\\mathbf {e}_1-\\mathbf {e}_3)+u(b+\\mathbf {e}_2-\\mathbf {e}_3) \\right|<C_4^{-2n} |u(b_i)|.$ Without loss of generality, we assume that $\\tau _i = 2$ in the former case, and $b_i \\in \\mathcal {L}_{a,r,b_{m_{h+1}},2}$ in the later case.",
"We consider the following trapezoid in $\\mathcal {R}_i$ : $\\mathring{\\mathcal {W}}_i:= \\left\\lbrace b \\in \\mathring{\\mathcal {R}}_i: b \\cdot \\overline{\\lambda }_2 \\ge b_i\\cdot \\overline{\\lambda }_2 \\right\\rbrace ,$ and let $\\mathcal {W}_i$ be the closure of $\\mathring{\\mathcal {W}}_i$ .",
"See Figure REF for an illustration of $\\mathcal {W}_i$ .",
"Then $\\mathring{\\mathcal {W}}_i\\cap \\mathbb {Z}^3$ can be treated as $P_{\\mathbf {0}; \\frac{\\vartheta _i}{\\sqrt{2}} + 2\\left\\lceil \\frac{\\theta _i - \\vartheta _i}{3\\sqrt{2}} \\right\\rceil -2, \\left\\lceil \\frac{\\theta _i - \\vartheta _i}{3\\sqrt{2}} \\right\\rceil -1}$ (see Definition REF ).",
"We apply Corollary REF to $\\mathring{\\mathcal {W}}_i\\cap \\mathbb {Z}^3$ , with $L = \\mathring{\\mathcal {L}}_{a,r,b_i,2} \\cap \\mathbb {Z}^3$ if $\\vartheta _i \\ge 2\\sqrt{2}$ (thus $\\mathring{\\mathcal {L}}_{a,r,b_i,2} \\cap \\mathbb {Z}^3$ is not empty) and $i \\le m_{h}$ , and with $L = \\left\\lbrace b_i\\right\\rbrace $ otherwise.",
"If $i\\le m_h$ , we have $\\frac{\\epsilon _2(\\theta _{i} - \\vartheta _{i})^2}{(3\\sqrt{2})^2}\\ge \\frac{\\epsilon _2(\\theta _i+2\\vartheta _i)(\\theta _i-\\vartheta _i)}{5\\cdot 3\\sqrt{2}\\cdot 3\\sqrt{2}}$ when $\\vartheta _i =\\sqrt{2}$ or 0, since $\\theta _{i} - \\vartheta _{i}\\ge \\frac{\\sqrt{2}}{2}$ .",
"Thus we always have $\\left| \\left\\lbrace b\\in \\mathring{\\mathcal {W}}_i\\cap \\mathbb {Z}^3: |u(b)| \\ge C_4^{-\\frac{2(\\theta _i - \\vartheta _i)}{3\\sqrt{2}}} \\min _{c \\in \\left(\\mathring{\\mathcal {L}}_{a,r,b_i,2}\\cap \\mathbb {Z}^3\\right) \\cup \\left\\lbrace b_i\\right\\rbrace } |u(c)| \\right\\rbrace \\right|\\ge \\frac{\\epsilon _2(\\theta _i+2\\vartheta _i)(\\theta _i-\\vartheta _i)}{5\\cdot 3\\sqrt{2}\\cdot 3\\sqrt{2}}.$ Since $\\frac{\\theta _i - \\vartheta _i}{3\\sqrt{2}} < n$ , and $C_4^{-2n} \\min _{c \\in \\left(\\mathring{\\mathcal {L}}_{a,r,b_i,2}\\cap \\mathbb {Z}^3\\right)\\cup \\left\\lbrace b_i\\right\\rbrace } |u(c)| \\ge \\exp (C_{10} n)g$ by Claim REF , we have $ \\left|\\left\\lbrace b \\in \\mathring{\\mathcal {W}}_i\\cap \\mathbb {Z}^3: |u(b)|\\ge \\exp (C_{10}n)g \\right\\rbrace \\right|\\ge \\frac{\\epsilon _2(\\theta _i+2\\vartheta _i)(\\theta _i-\\vartheta _i)}{5\\cdot 3\\sqrt{2}\\cdot 3\\sqrt{2}}\\ge \\frac{\\epsilon _2\\theta _i(\\theta _i-\\vartheta _i)}{5\\cdot 3\\sqrt{2}\\cdot 3\\sqrt{2}}.$ If $m_{h}<m_{h+1}$ and $i=m_{h+1}$ , we have $\\left| \\left\\lbrace b\\in \\mathring{\\mathcal {W}}_i\\cap \\mathbb {Z}^3: |u(b)| \\ge C_4^{-\\frac{2(\\theta _i - \\vartheta _i)}{3\\sqrt{2}}} |u(b_{i})| \\right\\rbrace \\right| \\ge \\frac{\\epsilon _2(\\theta _{i} - \\vartheta _{i})^2}{(3\\sqrt{2})^2}.$ Since $\\frac{\\theta _i - \\vartheta _i}{3\\sqrt{2}} < n$ , and $C_4^{-2n}|u(b_{i})| \\ge \\exp (C_{10} n)g$ , we have $ \\left|\\left\\lbrace b \\in \\mathring{\\mathcal {W}}_{i}\\cap \\mathbb {Z}^3: |u(b)|\\ge \\exp (C_{10}n)g \\right\\rbrace \\right|\\ge \\frac{\\epsilon _2(\\theta _{i} - \\vartheta _{i})^2}{(3\\sqrt{2})^2}.$ For the cases where $\\tau _i = 3, 4$ , $i\\le m_h$ , and the cases where $m_{h}<m_{h+1}=i$ and $b_{i} \\in \\mathcal {L}_{a,r,b_{m_{h+1}},3}$ or $\\mathcal {L}_{a,r,b_{m_{h+1}},4}$ , we can argue similarly and get (REF ) and (REF ) as well.",
"We then study other faces of $\\mathfrak {P}_{a,r,\\Gamma }$ .",
"Again we fix $2 \\le i \\le m_h$ , and assume that $\\tau _i = 2$ , for $\\tau _i$ given by Claim REF .",
"We define $\\hat{\\mathcal {S}}_i := \\left\\lbrace b \\in \\mathcal {P}_{2, b_i\\cdot \\lambda _2}:b \\cdot \\overline{\\lambda }_\\tau < \\mathbf {t}_r(a)\\cdot \\overline{\\lambda }_\\tau -F_{a,r,b_i}, \\forall \\tau \\in \\lbrace 3,4\\rbrace ,\\;b_i \\cdot \\lambda _1 \\le b \\cdot \\lambda _1 < b_{i+1} \\cdot \\lambda _1 \\right\\rbrace .$ Let $\\mathring{\\mathcal {S}}_i:=\\left\\lbrace b \\in \\hat{\\mathcal {S}}_i : b\\cdot \\lambda _1 < h+1\\right\\rbrace $ , and $\\mathcal {S}_i$ be the closure of $\\mathring{\\mathcal {S}}_i$ .",
"Then $\\mathcal {S}_i\\subset \\mathcal {P}_{2,\\lambda _2\\cdot b_i}$ and is a trapezoid.",
"It is a face of $\\partial \\mathfrak {P}_{a, r, \\Gamma }$ , for $2\\le i<m_h$ , and for $i=m_h$ when $m_{h+1}>m_h$ ; and it is part of a face of $\\partial \\mathfrak {P}_{a, r, \\Gamma }$ for $i=m_h$ when $m_{h+1}=m_h$ .",
"See Figure REF for an illustration.",
"Figure: Faces 𝒮 i \\mathcal {S}_i, 𝒲 i \\mathcal {W}_i, and 𝒮 i-1 \\mathcal {S}_{i-1}, in the pyramid boundary ∂𝔓 a,r,Γ \\partial \\mathfrak {P}_{a,r,\\Gamma }.The yellow triangle is the intersection of 𝔓 a,r,Γ \\mathfrak {P}_{a,r,\\Gamma } with the plane b·λ 1 =h+1b\\cdot \\lambda _1=h+1, and the blue lines are ℒ a,r,b i ,τ i \\mathcal {L}_{a,r,b_{i},\\tau _{i}} and ℒ a,r,b i-1 ,τ i-1 \\mathcal {L}_{a,r,b_{i-1},\\tau _{i-1}}.Claim 5.15 For $b \\in \\mathring{\\mathcal {S}}_i\\cap \\mathbb {Z}^3$ , if $b + \\mathbf {e}_1 + \\mathbf {e}_2, b + \\mathbf {e}_1 + \\mathbf {e}_3 \\in \\mathring{\\mathcal {S}}_i$ , then $ |u(c)| < \\exp (C_{10}n)g,\\;\\forall c \\in \\left\\lbrace b + \\mathbf {e}_1, b + \\mathbf {e}_1 - \\mathbf {e}_2, b + \\mathbf {e}_1 - \\mathbf {e}_3, b + 2\\mathbf {e}_1\\right\\rbrace .$ We leave the proof of this claim for later as well.",
"By Claim REF , and arguing as for (REF ) above, we conclude that $\\forall b \\in \\mathring{\\mathcal {S}}_i\\cap \\mathbb {Z}^3$ with $b + \\mathbf {e}_1 +\\mathbf {e}_2, b + \\mathbf {e}_1+\\mathbf {e}_3 \\in \\mathring{\\mathcal {S}}_i$ , $|u(b) + u(b+\\mathbf {e}_1+\\mathbf {e}_2)+u(b+\\mathbf {e}_1+\\mathbf {e}_3)| < C_4^{-2n}\\min _{c\\in \\left(\\mathring{\\mathcal {L}}_{a,r,b_i,2}\\cap \\mathbb {Z}^3\\right)\\cup \\lbrace b_i\\rbrace }|u(c)|.$ Let's first assume that $\\mathring{\\mathcal {S}}_i \\cap \\mathbb {Z}^3 \\ne \\emptyset $ .",
"Then we have $\\mathring{\\mathcal {L}}_{a,r,b_i,2} \\cap \\mathbb {Z}^3 \\ne \\emptyset $ , and $\\vartheta _{i} \\ge 2\\sqrt{2}$ .",
"If $i < m_{h+1}$ , then $b_{i+1} \\cdot \\lambda _1 \\le h+1$ , so we treat $\\mathring{\\mathcal {S}}_i \\cap \\mathbb {Z}^3$ as $P^r_{\\mathbf {0}; \\frac{\\vartheta _i}{\\sqrt{2}}-2, \\left\\lceil \\frac{\\vartheta _i-\\theta _{i+1}}{\\sqrt{2}}\\right\\rceil -1}$ (from Definition REF ), and $\\mathcal {L}_{a,r,b_i,2} \\cap \\mathbb {Z}^3$ is its upper edge.",
"If $i=m_h=m_{h+1} \\ge 2$ , then $b_{i+1}\\cdot \\lambda _1 > h+1$ , and we treat $\\mathring{\\mathcal {S}}_{i} \\cap \\mathbb {Z}^3$ as $P^r_{\\mathbf {0}; \\frac{\\vartheta _{i}}{\\sqrt{2}}-2, \\left\\lceil \\frac{\\vartheta _{i}-\\theta _{i+1}}{\\sqrt{2}} - \\frac{b_{i+1}\\cdot \\lambda _1 - (h+1)}{2} \\right\\rceil -1}$ .",
"We apply Corollary REF to the trapezoid, with $L = \\mathring{\\mathcal {L}}_{a,r,b_i,2} \\cap \\mathbb {Z}^3$ if it is not empty; similar to the study of $\\mathcal {W}_i$ , we conclude that $ \\left| \\left\\lbrace b \\in \\mathcal {S}_i \\cap \\mathbb {Z}^3: |u(b)|>\\exp (C_{10}n)g \\right\\rbrace \\right| > \\frac{\\epsilon _3\\vartheta _i (\\vartheta _i - \\theta _{i+1})}{\\sqrt{2} \\cdot \\sqrt{2}},$ if $2 \\le i < m_{h+1}$ , and $ \\left| \\left\\lbrace b \\in \\mathcal {S}_{i} \\cap \\mathbb {Z}^3: |u(b)|>\\exp (C_{10}n)g \\right\\rbrace \\right|> \\frac{\\epsilon _3 \\vartheta _{i}}{\\sqrt{2} } \\left(\\frac{\\vartheta _{i}-\\theta _{i+1}}{\\sqrt{2}} - \\frac{b_{i+1}\\cdot \\lambda _1 - (h+1)}{2} \\right),$ if $i = m_h = m_{h+1} \\ge 2$ .",
"In the case where $\\mathring{\\mathcal {S}}_i \\cap \\mathbb {Z}^3 = \\emptyset $ , we have $\\vartheta _{i} \\le \\sqrt{2}$ , and these inequalities still hold, since $b_i\\in \\mathcal {S}_i \\cap \\mathbb {Z}^3$ and $|u(b_i)|>\\exp (C_{10}n)g$ .",
"When $\\tau _i = 3,4$ , we can define $\\mathcal {S}_i$ analogously, and obtain (REF ) and (REF ) as well.",
"In addition, we consider $\\hat{\\mathcal {S}}_1 := \\left\\lbrace b \\in \\mathcal {P}_{4, a\\cdot \\lambda _4 }:b \\cdot \\overline{\\lambda }_\\tau < \\mathbf {t}_r(a)\\cdot \\overline{\\lambda }_2,\\;\\forall \\tau \\in \\lbrace 2,3\\rbrace ,\\;a \\cdot \\lambda _1 \\le b \\cdot \\lambda _1 < b_{2} \\cdot \\lambda _1 \\right\\rbrace ,$ and let $\\mathring{\\mathcal {S}}_1:=\\left\\lbrace b \\in \\hat{\\mathcal {S}}_1 : b\\cdot \\lambda _1 < h+1\\right\\rbrace $ , and $\\mathcal {S}_1$ be the closure of $\\mathring{\\mathcal {S}}_1$ .",
"We treat $\\mathcal {S}_1$ differently (from $\\mathcal {S}_i$ for $2\\le i \\le m_h$ ) because Claim REF cannot be extended to $i=1$ .",
"Also note that by taking $\\mathcal {S}_1\\subset \\mathcal {P}_{4,a\\cdot \\lambda _4}$ , $\\mathcal {S}_1$ is defined as (possibly part of) the face in $\\partial \\mathfrak {P}_{a,r,\\Gamma }$ that contains $a=b_1$ .",
"Using arguments similar to those for $\\mathcal {S}_i$ , $2 \\le i \\le m_h$ , we treat $\\mathring{\\mathcal {S}}_1 \\cap \\mathbb {Z}^3$ as $P^r_{\\mathbf {0};\\frac{\\vartheta _1}{\\sqrt{2}}-2, \\left\\lceil \\frac{\\vartheta _1 - \\theta _2}{\\sqrt{2}}\\right\\rceil -1}$ if $m_{h+1} > 1$ , and as $P^r_{\\mathbf {0};\\frac{\\vartheta _1}{\\sqrt{2}}-2, \\left\\lceil \\frac{\\vartheta _1 - \\theta _2}{\\sqrt{2}} - \\frac{b_{2}\\cdot \\lambda _1 - (h+1)}{2} \\right\\rceil -1}$ if $m_{h+1}=1$ .",
"Then we apply Corollary REF to it with $L = \\left\\lbrace a \\right\\rbrace $ .",
"We conclude that $ \\left| \\left\\lbrace b \\in \\mathcal {S}_{1} \\cap \\mathbb {Z}^3: |u(b)|>\\exp (C_{10}n)g \\right\\rbrace \\right|>{\\left\\lbrace \\begin{array}{ll}\\frac{\\epsilon _3(\\vartheta _1 - \\theta _2)^2}{(\\sqrt{2})^2}, & \\; m_{h+1} > 1,\\\\\\epsilon _3 \\left(\\frac{\\vartheta _1-\\theta _{2}}{\\sqrt{2}} - \\frac{b_{2}\\cdot \\lambda _1 - (h+1)}{2} \\right)^2 & \\; m_{h+1} = 1.\\end{array}\\right.",
"}$ We now put together the bounds we've obtained so far, for all $\\mathcal {S}_i$ and $\\mathcal {W}_i$ that are contained in $\\lbrace b\\in \\mathbb {R}^3: b\\cdot \\lambda _1 \\le h+1\\rbrace $ .",
"Case 1: $m_h = m_{h+1}$ .",
"In this case we consider $\\mathcal {S}_i$ for $1 \\le i \\le m_h$ and $\\mathcal {W}_i$ for $2 \\le i \\le m_h$ .",
"We first show that $h\\ne a \\cdot \\lambda _1$ .",
"For this we argue by contradiction.",
"Assume the contrary, i.e.",
"$h= a \\cdot \\lambda _1$ .",
"From the definition of $h$ we have that $|\\mathcal {L}_{a,r,c,2}\\cap \\mathbb {Z}^3|< \\frac{r}{4}$ , for any $c \\in \\mathring{\\mathfrak {P}}_{a,r,\\Gamma } \\cap \\mathbb {Z}^3$ with $c\\cdot \\lambda _1 = h+1= a \\cdot \\lambda _1+1$ .",
"As we assumed that $r>100$ , we must have $b_2 \\cdot \\lambda _1 = a \\cdot \\lambda _1+1 = h+1$ , and this implies $m_{h+1}=2$ .",
"However, by $h= a \\cdot \\lambda _1$ we have $m_h=1$ .",
"This contradicts with the assumption that $m_h = m_{h+1}$ .",
"We next show that $ \\frac{\\theta _{m_h+1}}{\\sqrt{2}}+ \\frac{b_{m_h+1}\\cdot \\lambda _1 - (h+1)}{2} < \\frac{r}{2}.$ By the definition of $h$ and $h\\ne a\\cdot \\lambda _1$ , we can find $c \\in \\mathring{\\mathfrak {P}}_{a,r,\\Gamma } \\cap \\mathbb {Z}^3$ with $c\\cdot \\lambda _1 = h$ and $|\\mathcal {L}_{a,r,c,2}\\cap \\mathbb {Z}^3|\\ge \\frac{r}{4}$ .",
"Since we assumed that $r > 100$ , using $m_{h}=m_{h+1}$ we have $\\mathring{\\mathfrak {P}}_{a,r,\\Gamma } \\cap \\mathcal {P}_{1, h+1} \\cap \\mathbb {Z}^3 \\ne \\emptyset $ .",
"This implies that $b_{m_h+1} \\cdot \\lambda _1 = b_{m_{h+1}+1} \\cdot \\lambda _1 > h+1$ (since otherwise, by the definiton of $m_{h+1}$ , we must have $m_{h+1} = m$ and $b_{m+1}\\cdot \\lambda _1 \\le h+1$ , implying $\\mathring{\\mathfrak {P}}_{a,r,\\Gamma } \\cap \\mathcal {P}_{1, h+1} = \\emptyset $ ).",
"Also note that $b_{m_h} \\cdot \\lambda _1 \\le h$ , so we can find $b \\in \\mathbb {Z}^3$ , and $b$ in the closure of $\\hat{\\mathcal {S}}_{m_h}$ , such that $b \\cdot \\lambda _1 = h+1$ or $h+2$ .",
"As $|\\mathcal {L}_{a,r,b,2}\\cap \\mathbb {Z}^3| = \\frac{\\theta _{m_h+1}}{\\sqrt{2}} + 1 + \\frac{(b_{m_h+1}-b)\\cdot \\lambda _1 }{2}$ , we have $|\\mathcal {L}_{a,r,b,2}\\cap \\mathbb {Z}^3| \\ge \\frac{\\theta _{m_h+1}}{\\sqrt{2}}+ \\frac{b_{m_h+1}\\cdot \\lambda _1 - (h+1)}{2}$ .",
"On the other hand, using $|\\mathcal {L}_{a,r,c,2}\\cap \\mathbb {Z}^3|\\ge \\frac{r}{4}$ and $r > 100$ again, we have $\\mathring{\\mathfrak {P}}_{a,r,\\Gamma } \\cap \\mathcal {P}_{1, b\\cdot \\lambda _1} \\cap \\mathbb {Z}^3 \\ne \\emptyset $ .",
"By the maximum property of $h$ , for any $b^{\\prime } \\in \\mathring{\\mathfrak {P}}_{a,r,\\Gamma } \\cap \\mathcal {P}_{1, b\\cdot \\lambda _1} \\cap \\mathbb {Z}^3$ , we have $|\\mathcal {L}_{a,r,b^{\\prime },2}\\cap \\mathbb {Z}^3|< \\frac{r}{4}$ .",
"Then $|\\mathcal {L}_{a,r,b,2}\\cap \\mathbb {Z}^3| < \\frac{r}{4}+3 < \\frac{r}{2}$ , and (REF ) follows.",
"If $m_h = m_{h+1} = 1$ , by (REF ) we have that $ \\left| \\left\\lbrace b \\in \\mathfrak {P}_{a,r,\\Gamma } \\cap \\mathbb {Z}^3: |u(b)|>\\exp (C_{10}n)g \\right\\rbrace \\right|>\\epsilon _3 \\left(\\frac{\\vartheta _1-\\theta _{2}}{\\sqrt{2}} - \\frac{b_{2}\\cdot \\lambda _1 - (h+1)}{2} \\right)^2\\\\>\\epsilon _3\\left(2r - \\frac{r}{2}\\right)^2 > \\epsilon _3 r^2,$ where we use (REF ) and the fact that $\\vartheta _1 = 2\\sqrt{2}r$ .",
"If $m_h = m_{h+1}> 1$ , we note that for all $2 \\le i \\le m_{h}$ , these $\\mathcal {W}_i$ are mutually disjoint; and for all $1 \\le i \\le m_h$ , these $\\mathcal {S}_i$ are mutually disjoint.",
"By (REF ),(REF ),(REF ),(REF ) and taking a small enough $\\epsilon _4>0$ , we have that $ \\begin{split}&\\left| \\left\\lbrace b \\in \\mathfrak {P}_{a,r,\\Gamma } \\cap \\mathbb {Z}^3: |u(b)|>\\exp (C_{10}n)g \\right\\rbrace \\right|\\\\>&\\epsilon _4\\left(\\frac{(\\vartheta _1 - \\theta _2)^2}{2}+ \\sum _{i=2}^{m_h} \\theta _i(\\theta _i - \\vartheta _{i}) + \\vartheta _i(\\vartheta _i - \\theta _{i+1})- \\vartheta _{m_h} \\frac{b_{m_h+1}\\cdot \\lambda _1 - (h+1)}{\\sqrt{2}}\\right) \\\\=&\\epsilon _4\\Bigg (\\frac{\\vartheta _1^2}{4}+\\frac{(\\vartheta _1 - 2\\theta _2)^2}{4}+ \\frac{\\sum _{i=2}^{m_h} (\\theta _i - \\vartheta _i)^2 + \\sum _{i=2}^{m_h-1} (\\vartheta _i - \\theta _{i+1})^2}{2}\\\\&+ \\frac{\\left(\\vartheta _{m_h} - \\theta _{m_h+1} - \\frac{b_{m_h+1}\\cdot \\lambda _1 - (h+1)}{\\sqrt{2}} \\right)^2}{2}-\\frac{\\left(\\theta _{m_h+1} + \\frac{b_{m_h+1}\\cdot \\lambda _1 - (h+1)}{\\sqrt{2}} \\right)^2}{2}\\Bigg )\\\\\\ge &\\epsilon _4\\left(\\frac{\\vartheta _1^2}{4}-\\frac{\\left(\\theta _{m_h+1} + \\frac{b_{m_h+1}\\cdot \\lambda _1 - (h+1)}{\\sqrt{2}} \\right)^2}{2}\\right) .\\end{split}$ Using (REF ), we get $ \\left| \\left\\lbrace b \\in \\mathfrak {P}_{a,r,\\Gamma } \\cap \\mathbb {Z}^3: |u(b)|>\\exp (C_{10}n)g \\right\\rbrace \\right|>\\epsilon _4\\left(2r^2-\\frac{r^2}{4}\\right)> \\epsilon _4 r^2.$ Case 2: $m_h < m_{h+1}$ .",
"In this case we consider $\\mathcal {S}_i$ for $1 \\le i \\le m_h$ and $\\mathcal {W}_i$ for $2 \\le i \\le m_h+1=m_{h+1}$ .",
"Similar to the first case, by (REF ),(REF ),(REF ),(REF ) and taking a small enough $\\epsilon _5>0$ , we have $ \\begin{split}&\\left| \\left\\lbrace b \\in \\mathfrak {P}_{a,r,\\Gamma } \\cap \\mathbb {Z}^3: |u(b)|>\\exp (C_{10}n)g \\right\\rbrace \\right|\\\\\\ge &\\epsilon _5\\left(\\frac{(\\vartheta _1 - \\theta _2)^2}{2}+ \\sum _{i=2}^{m_h} \\theta _i(\\theta _i - \\vartheta _{i}) + \\vartheta _i(\\vartheta _i - \\theta _{i+1})+(\\theta _{m_{h+1}} - \\vartheta _{m_{h+1}})^2\\right) \\\\= &\\epsilon _5\\left(\\frac{\\vartheta _1^2}{4}+\\frac{(\\vartheta _1 - 2\\theta _2)^2}{4}+ \\sum _{i=2}^{m_h}\\frac{(\\theta _i - \\vartheta _i)^2 + (\\vartheta _i - \\theta _{i+1})^2}{2}+ \\frac{(\\theta _{m_{h+1}} - 2\\vartheta _{m_{h+1}})^2}{2}-\\vartheta _{m_{h+1}}^2\\right) \\\\\\ge &\\epsilon _5\\left(\\frac{\\vartheta _1^2}{4}-\\vartheta _{m_{h+1}}^2\\right) .\\end{split}$ We now show that $\\vartheta _{m_{h+1}} < r$ .",
"Since $m_{h+1} > m_h$ , we have $b_{m_{h+1}} \\cdot \\lambda _1 = h+1$ .",
"If $\\vartheta _{m_{h+1}} \\ge r$ , then $|\\mathcal {L}_{a,r,b_{m_{h+1}},2}\\cap \\mathbb {Z}^3| \\ge \\frac{r}{\\sqrt{2}} + 1$ , and we can find $b \\in \\mathring{\\mathfrak {P}}_{a,r,\\Gamma } \\cap \\mathcal {P}_{1, h+1}$ , such that $|\\mathcal {L}_{a,r,b,2}\\cap \\mathbb {Z}^3| \\ge \\frac{r}{\\sqrt{2}} - 2 > \\frac{r}{4}$ .",
"This contradicts with the definition of $h$ .",
"With $\\vartheta _{m_{h+1}} < r$ , and note that $2\\sqrt{2}r = \\vartheta _1 \\ge \\theta _2$ , we have $ \\left| \\left\\lbrace b \\in \\mathfrak {P}_{a,r,\\Gamma } \\cap \\mathbb {Z}^3: |u(b)|>\\exp (C_{10}n)g \\right\\rbrace \\right|>\\epsilon _5\\left(2r^2-r^2\\right)= \\epsilon _5 r^2.$ By taking $C_{9}^{\\prime }$ small enough, we get (REF ) by each of (REF ), (REF ), and (REF ).",
"Figure: An illustration of points in Claim .The red point (b-𝐞 3 b-\\mathbf {e}_3) is in 𝒫 1,b i ·λ 1 -1 \\mathcal {P}_{1,b_i\\cdot \\lambda _1-1} and the blue points are in 𝒫 1,b i ·λ 1 -2 \\mathcal {P}_{1,b_i\\cdot \\lambda _1-2}.The point cc is among the red and blue points and is in 𝔓 ˚ a,r,Γ ∪𝒯 ˚ a,r ∪𝒯 ˚ a-λ 1 3,r \\mathring{\\mathfrak {P}}_{a,r,\\Gamma }\\cup \\mathring{\\mathcal {T}}_{a,r}\\cup \\mathring{\\mathcal {T}}_{a-\\frac{\\lambda _1}{3},r}.It remains to prove Claim REF , REF , and REF .",
"[Proof of Claim REF ] Take any $c \\in \\left\\lbrace b-\\mathbf {e}_3, b-\\mathbf {e}_1-\\mathbf {e}_3, b-\\mathbf {e}_1-\\mathbf {e}_3, b-2\\mathbf {e}_3\\right\\rbrace $ .",
"Since $c \\cdot \\overline{\\lambda }_{2} \\le b \\cdot \\overline{\\lambda }_2, c \\cdot \\overline{\\lambda }_{3} \\le b \\cdot \\overline{\\lambda }_3$ , and $c \\cdot \\overline{\\lambda }_{4} \\le (b+\\mathbf {e}_1-\\mathbf {e}_3) \\cdot \\overline{\\lambda }_4$ , and $b, b+\\mathbf {e}_1-\\mathbf {e}_3 \\in \\mathring{\\mathcal {R}}_i \\subset \\mathring{\\mathfrak {T}}_{a,r}$ , we have that $ c\\cdot \\overline{\\lambda }_\\tau < \\mathbf {t}_r(a)\\cdot \\overline{\\lambda }_\\tau - F_{a,r,b_{i-1}} \\le \\mathbf {t}_r(a)\\cdot \\overline{\\lambda }_\\tau ,\\;\\forall \\tau \\in \\lbrace 2,3,4\\rbrace .$ We first consider the case where $c \\notin \\mathring{\\mathfrak {T}}_{a,r}$ .",
"Then we have that $a \\cdot \\lambda _1 \\ge c\\cdot \\lambda _1 \\ge b \\cdot \\lambda _1 - 2 = b_i \\cdot \\lambda _1 - 2 \\ge a \\cdot \\lambda _1 - 1$ , where the last inequality is due to $b_i \\in \\mathring{\\mathfrak {T}}_{a,r}$ .",
"If $c \\cdot \\lambda _1 = a \\cdot \\lambda _1$ , we have $c \\in \\mathring{\\mathcal {T}}_{a,r}$ by (REF ); and by the second condition of Proposition REF we have that $|u(c)| < g$ .",
"If $c \\cdot \\lambda _1 = a \\cdot \\lambda _1-1$ , we have $c \\in \\mathring{\\mathcal {T}}_{a-\\frac{\\lambda _1}{3}, r}$ by (REF ).",
"As $b_i \\cdot \\lambda _1 > a\\cdot \\lambda _1$ , and $b_i \\cdot \\lambda _1 = b \\cdot \\lambda _1 \\le c \\cdot \\lambda _1 +2$ , we have that $b_i \\cdot \\lambda _1 = a\\cdot \\lambda _1 + 1$ , thus $b_i \\in \\mathring{\\mathcal {T}}_{a+\\frac{\\lambda _1}{3}, r}$ .",
"Since $|u(b_i)|\\ge \\exp (3C_{10} n)g$ , by the second condition of Proposition REF we have $|u(c)| < g$ .",
"Now we assume that $c \\in \\mathring{\\mathfrak {T}}_{a,r}$ .",
"For any $j$ , if $i\\le j \\le m$ , as $c\\cdot \\lambda _1 < b_i \\cdot \\lambda _1$ , we have that $c\\cdot \\lambda _1 < b_j \\cdot \\lambda _1$ , and thus $c\\notin \\mathfrak {H}_{a,r,b_j}$ .",
"If $1 \\le j \\le i-1$ , we have $b_j\\cdot \\lambda _1 \\le b_{i-1}\\cdot \\lambda _1$ , so $F_{a,r,b_j} \\le F_{a,r,b_{i-1}}$ (since otherwise $\\mathfrak {H}_{a,r,b_{i-1}}$ is not maximal).",
"By (REF ) we have that $c\\cdot \\overline{\\lambda }_\\tau < \\mathbf {t}_r(a)\\cdot \\overline{\\lambda }_\\tau - F_{a,r,b_{j}},\\;\\forall \\tau \\in \\lbrace 2,3,4\\rbrace ,$ thus $c\\notin \\mathfrak {H}_{a,r,b_j}$ .",
"Then by the definition of $\\mathring{\\mathfrak {P}}_{a,r,\\Gamma }$ , we have that $c \\in \\mathring{\\mathfrak {P}}_{a,r,\\Gamma }$ .",
"As $c \\cdot \\lambda _1 \\le b_{i} \\cdot \\lambda _1 - 1\\le b_{m_{h+1}} \\cdot \\lambda _1 - 1 \\le h$ , we have $|u(c)| \\le \\exp (C_{10}n)g$ by the third condition of Proposition REF .",
"Claim REF can be proved in a similar way.",
"Figure: An illustration of points in Claim .The red point (b+𝐞 1 b+\\mathbf {e}_1) is in 𝒫 2,b i ·λ 2 -1 \\mathcal {P}_{2,b_i\\cdot \\lambda _2-1} and the blue points are in 𝒫 2,b i ·λ 2 -2 \\mathcal {P}_{2,b_i\\cdot \\lambda _2-2}.The point cc is among the red and blue points and is in 𝔓 ˚ a,r,Γ ∪𝒯 ˚ a,r \\mathring{\\mathfrak {P}}_{a,r,\\Gamma }\\cup \\mathring{\\mathcal {T}}_{a,r}.",
"[Proof of Claim REF ] We take $c \\in \\left\\lbrace b + \\mathbf {e}_1, b + \\mathbf {e}_1 - \\mathbf {e}_2, b + \\mathbf {e}_1 - \\mathbf {e}_3, b + 2\\mathbf {e}_1 \\right\\rbrace $ , then $c \\cdot \\overline{\\lambda }_2 < b \\cdot \\overline{\\lambda }_2 = b_i \\cdot \\overline{\\lambda }_2$ , and $c \\cdot \\overline{\\lambda }_\\tau \\le b \\cdot \\overline{\\lambda }_\\tau + 2$ for $\\tau \\in \\lbrace 3,4\\rbrace $ .",
"Since $b, b+\\mathbf {e}_1+\\mathbf {e}_2, b+\\mathbf {e}_1+\\mathbf {e}_3 \\in \\mathring{\\mathcal {S}}_i$ , we have that $b \\cdot \\overline{\\lambda }_3 + 2 = (b+\\mathbf {e}_1+\\mathbf {e}_3) \\cdot \\overline{\\lambda }_3 < \\mathbf {t}_r(a)\\cdot \\overline{\\lambda }_3 - F_{a,r,b_{i}}$ , and $b \\cdot \\overline{\\lambda }_4 + 2 = (b+\\mathbf {e}_1+\\mathbf {e}_2) \\cdot \\overline{\\lambda }_4 < \\mathbf {t}_r(a)\\cdot \\overline{\\lambda }_4 + F_{a,r,b_{i}}$ ; then $ c\\cdot \\overline{\\lambda }_\\tau < \\mathbf {t}_r(a)\\cdot \\overline{\\lambda }_\\tau - F_{a,r,b_{i}} \\le \\mathbf {t}_r(a)\\cdot \\overline{\\lambda }_\\tau ,\\;\\forall \\tau \\in \\lbrace 2,3,4\\rbrace ,$ We claim that $c\\notin \\mathfrak {H}_{a,r,b_j}$ for any $1 \\le j \\le m$ : for $j > i$ , note that $b + \\mathbf {e}_1 + \\mathbf {e}_2 \\in \\mathring{\\mathcal {S}}_i$ , so $c\\cdot \\lambda _1 \\le b \\cdot \\lambda _1 + 2 = (b+\\mathbf {e}_1+\\mathbf {e}_2) \\cdot \\lambda _1 < b_{i+1} \\cdot \\lambda _1 $ ; for $j \\le i$ , this is implied by (REF ).",
"Thus $c \\in \\mathring{\\mathfrak {P}}_{a,r,\\Gamma }\\cup \\mathring{\\mathcal {T}}_{a,r}$ , since we also have $c \\cdot \\lambda _1 \\ge b \\cdot \\lambda _1 \\ge b_i \\cdot \\lambda _1 \\ge a \\cdot \\lambda _1$ .",
"If $c\\in \\mathring{\\mathcal {T}}_{a,r}$ , by the second condition of Proposition REF , we have $|u(c)|\\le g<\\exp (C_{10} n)g$ .",
"If $c \\in \\mathring{\\mathfrak {P}}_{a,r,\\Gamma }$ , using the fact that $b + \\mathbf {e}_1 + \\mathbf {e}_2 \\in \\mathring{\\mathcal {S}}_i$ again, we have $c\\cdot \\lambda _1 \\le b\\cdot \\lambda _1+2 = (b + \\mathbf {e}_1 + \\mathbf {e}_2)\\cdot \\lambda _1 \\le h$ , and this implies that $|u(c)|\\le \\exp (C_{10} n)g$ by the third condition of Proposition REF .",
"Lastly, we prove Claim REF , using Claim REF above and the local cone property (from Section ).",
"[Proof of Claim REF ] Throughout this proof, we assume that $\\left(\\bigcup _{\\tau \\in \\left\\lbrace 2,3,4\\right\\rbrace }\\mathring{\\mathcal {L}}_{a,r,b_i,\\tau }\\right) \\cap \\mathbb {Z}^3 \\ne \\emptyset $ .",
"We first show that we can find point $b \\in \\left(\\bigcup _{\\tau \\in \\left\\lbrace 2,3,4\\right\\rbrace }\\mathring{\\mathcal {L}}_{a,r,b_i,\\tau }\\right) \\cap \\mathbb {Z}^3$ , such that $|u(b)| \\ge (K+11)^{-1}\\exp (3C_{10}n)g.$ This is obviously true if $b_i \\in \\bigcup _{\\tau \\in \\left\\lbrace 2,3,4\\right\\rbrace }\\mathring{\\mathcal {L}}_{a,r,b_i,\\tau }$ ; otherwise, by symmetry we assume that $b_i = \\mathbf {v}_{a,r,b_i,4}$ .",
"By Lemma REF , $\\max _{c \\in \\left\\lbrace b_i - \\mathbf {e}_3, b_i -\\mathbf {e}_3+\\mathbf {e}_1, b_i -\\mathbf {e}_3+\\mathbf {e}_2, b_i -\\mathbf {e}_3-\\mathbf {e}_1, b_i -\\mathbf {e}_3-\\mathbf {e}_2, b_i -2\\mathbf {e}_3\\right\\rbrace } |u(c)| \\ge (K+11)^{-1}\\exp (3C_{10}n)g.$ As $b_i, b_i - \\mathbf {e}_3 + \\mathbf {e}_1, b_i - \\mathbf {e}_3 + \\mathbf {e}_2 \\in \\mathring{\\mathcal {R}}_i$ , by Claim REF , we have $\\max _{c \\in \\left\\lbrace b_i -\\mathbf {e}_3+\\mathbf {e}_1, b_i -\\mathbf {e}_3+\\mathbf {e}_2\\right\\rbrace } |u(c)| \\ge (K+11)^{-1}\\exp (3C_{10}n)g.$ Note that $b_i -\\mathbf {e}_3+\\mathbf {e}_1, b_i -\\mathbf {e}_3+\\mathbf {e}_2 \\in \\bigcup _{\\tau \\in \\left\\lbrace 2,3,4\\right\\rbrace }\\mathring{\\mathcal {L}}_{a,r,b_i,\\tau }$ , so we can choose $b \\in \\left\\lbrace b_i -\\mathbf {e}_3+\\mathbf {e}_1, b_i -\\mathbf {e}_3+\\mathbf {e}_2\\right\\rbrace $ and the condition is satisfied.",
"Now by symmetry we assume that there is $b \\in \\mathring{\\mathcal {L}}_{a,r,b_i,4}\\cap \\mathbb {Z}^3$ so that $|u(b)| \\ge (K+11)^{-1}\\exp (3C_{10}n)g.$ We prove that, for any $b^{\\prime } \\in \\mathring{\\mathcal {L}}_{a,r,b_i,4}\\cap \\mathbb {Z}^3$ , we have $|u(b^{\\prime })| \\ge \\exp (2C_{10}n)g$ .",
"We argue by contradiction, and assume that there is $b^{\\prime } \\in \\mathring{\\mathcal {L}}_{a,r,b_i,4}\\cap \\mathbb {Z}^3$ so that $|u(b^{\\prime })|<\\exp (2C_{10}n)g$ .",
"Without loss of generality, we also assume that $b^{\\prime } \\cdot \\mathbf {e}_1 < b \\cdot \\mathbf {e}_1$ .",
"Consider the sequence of points in $\\mathring{\\mathcal {L}}_{a,r,b_i,4}\\cap \\mathbb {Z}^3$ between $b$ and $b^{\\prime }$ .",
"We iterate this sequence from $b$ to $b^{\\prime }$ , by adding $-\\mathbf {e}_1+\\mathbf {e}_2$ at each step.",
"We let $c$ be the first one such that $|u(c-\\mathbf {e}_1+\\mathbf {e}_2)| < (K+11)^{-1}|u(c)|$ .",
"The existence of such $c$ is ensured by that $|u(b^{\\prime })|<(K+11)\\exp (-C_{10}n)|u(b)|$ , $|\\mathring{\\mathcal {L}}_{a,r,b_i,4}\\cap \\mathbb {Z}^3| < 2r < \\frac{n}{16}$ , and $C_{10}>K+11$ .",
"For such $c$ we also have $c, c-\\mathbf {e}_1+\\mathbf {e}_2\\in \\mathring{\\mathcal {L}}_{a,r,b_i,4}\\cap \\mathbb {Z}^3$ , and $|u(c)| \\ge (K+11)^{-1-2r}\\exp (3C_{10}n)g > \\exp \\left(\\frac{5C_{10}n}{2}\\right)g$ .",
"Since $c, c-\\mathbf {e}_1+\\mathbf {e}_2, c-\\mathbf {e}_1+\\mathbf {e}_3 \\in \\mathring{\\mathcal {R}}_i$ , by Claim REF we have $|u(c^{\\prime })| < \\exp (C_{10}n)g < (K+11)^{-1}|u(c)|,\\;\\forall c^{\\prime } \\in \\left\\lbrace c-\\mathbf {e}_1, c-\\mathbf {e}_1-\\mathbf {e}_3, c-\\mathbf {e}_1-\\mathbf {e}_2, c-2\\mathbf {e}_1\\right\\rbrace .$ For $c-\\mathbf {e}_1+\\mathbf {e}_3$ , as $c-\\mathbf {e}_1+\\mathbf {e}_3 \\in \\mathring{\\mathfrak {P}}_{a,r,\\Gamma }$ and $(c-\\mathbf {e}_1+\\mathbf {e}_3) \\cdot \\lambda _1 = c\\cdot \\lambda _1 \\le h$ , we have $|u(c-\\mathbf {e}_1+\\mathbf {e}_3)| \\le \\exp (C_{10}n)g< (K+11)^{-1}|u(c)|$ by the third condition of Proposition REF .",
"Then we get a contradiction with Lemma REF .",
"The next step is to control the points in a graded set $E$ .",
"Proposition 5.16 For $C_{9}^{\\prime }$ from Proposition REF , any small enough $\\varepsilon >0$ , and any $N \\in \\mathbb {Z}_+$ , there exists $C_{\\varepsilon ,N}>0$ such that the following is true.",
"Let $n \\in \\mathbb {Z}_+$ , $r \\in \\mathbb {Z}$ , $0 \\le r < \\frac{n}{32}$ .",
"Let $\\Gamma \\subset Q_{n}$ , $a \\in \\Gamma \\cap Q_{\\frac{n}{2}}$ such that $\\mathring{\\mathcal {T}}_{a, r} \\cap \\Gamma = \\emptyset $ .",
"Suppose that $\\vec{l}$ is a vector of positive reals, and $E$ is an $(N,\\vec{l},\\varepsilon ^{-1},\\varepsilon )$ -graded set with the first scale length $l_1>C_{\\varepsilon ,N}$ .",
"If $E$ is $(\\varepsilon ^{-\\frac{1}{2}},\\varepsilon )$ -normal in $\\mathfrak {T}_{a,r}$ , then $\\left|E\\cap \\partial \\mathfrak {P}_{a, r, \\Gamma }\\cap \\mathbb {Z}^3\\right|\\le \\frac{C^{\\prime }_{9}}{2}(r^2+1).$ If $r<\\frac{1}{10\\sqrt{\\varepsilon }}$ , since $E$ is $(\\varepsilon ^{-\\frac{1}{2}},\\varepsilon )$ -normal in $\\mathfrak {T}_{a,r}$ , we have $E \\cap \\mathfrak {T}_{a,r}=\\emptyset $ when $C_{\\varepsilon ,N}$ is large, and our conclusion holds.",
"From now on, we assume that $r\\ge \\frac{1}{10\\sqrt{\\varepsilon }}$ .",
"Denote $\\pi :=\\pi _{a \\cdot \\lambda _1}$ for the simplicity of notations.",
"Evidently, for any two $b_1,b_2 \\in \\partial \\mathfrak {P}_{a, r, \\Gamma }$ , $\\frac{1}{10} |b_1-b_2| \\le |\\pi (b_1)-\\pi (b_2)| \\le |b_1-b_2|.$ Suppose $\\vec{l}=(l_1,\\cdots ,l_d)$ , where $l_{i+1} \\ge l_{i}^{1+2\\varepsilon }$ for each $1\\le i\\le d-1$ .",
"Write $E=\\bigcup _{i=0}^{d}E_{i}$ , where $E_{0}$ is a $\\varepsilon ^{-1}$ -unitscattered set and $E_{i}$ is an $(N,l_{i},\\varepsilon )$ -scattered set.",
"It suffices to prove that there exists a universal constant $C$ such that for each $1 \\le i \\le d$ , $ \\left|E_{i} \\cap \\partial \\mathfrak {P}_{a, r, \\Gamma } \\cap \\mathbb {Z}^3\\right| \\le C N l_{i}^{-\\varepsilon } r^{2},$ and $\\left|E_{0} \\cap \\partial \\mathfrak {P}_{a, r, \\Gamma } \\cap \\mathbb {Z}^3\\right| \\le C \\varepsilon ^{2} r^{2}.$ Then with (REF ) and (REF ) we can take $\\varepsilon $ small enough, such that $C\\varepsilon ^{2}<\\frac{C^{\\prime }_{9}}{4}$ ; and take $C_{\\varepsilon ,N}$ large enough, such that $\\sum _{i=1}^{\\infty } C N l_{i}^{-\\varepsilon } \\le \\sum _{i=1}^{\\infty } C N l_{1}^{-\\varepsilon (1+2\\varepsilon )^{i-1}} \\le \\frac{C^{\\prime }_{9}}{4}.$ Thus we get (REF ).",
"We first prove (REF ).",
"As in Definition REF , for each $1 \\le i \\le d$ , we write $E_{i}=\\bigcup _{j \\in \\mathbb {Z}_{+}, 1 \\le t \\le N} E^{(j,t)}_{i}$ where each $E^{(j,t)}_{i}$ is a open ball with radius $l_i$ , and $\\operatorname{dist}(E^{(j,t)}_i,E^{(j^{\\prime },t)}_i) \\ge l_i^{1+\\varepsilon }$ for each $j\\ne j^{\\prime }$ .",
"Claim 5.17 For any $1 \\le i \\le d$ , we have $\\left|\\left\\lbrace (j,t):E_{i}^{(j,t)} \\cap \\partial \\mathfrak {P}_{a, r, \\Gamma } \\ne \\emptyset \\right\\rbrace \\right| < C N l_{i}^{-2-\\varepsilon } r^{2}$ , where $C$ is a universal constant.",
"The proof is via a simple packing argument.",
"Assume that $E_i\\cap \\mathfrak {T}_{a,r}\\ne \\emptyset $ (since otherwise the claim obviously holds).",
"Denote $\\tilde{\\mathcal {T}}_{a,r}$ to be the closed equilateral triangle in $\\mathcal {P}_{1,a\\cdot \\lambda _1}$ , such that it has the same center and orientation as $\\mathcal {T}_{a,r}$ , and its side length is $100 r$ .",
"For any $j, t$ , let $B^{(j,t)}_{i}$ be the open ball with radius $l_{i}^{1+\\frac{\\varepsilon }{2}}$ and with the same center as $E^{(j,t)}_{i}$ .",
"Since $E$ is $(\\varepsilon ^{-\\frac{1}{2}},\\varepsilon )$ -normal in $\\mathfrak {T}_{a,r}$ , we have $\\operatorname{diam}(B^{(j,t)}_{i}) \\le 10 r^{1-\\frac{\\varepsilon ^2}{4}}$ .",
"Suppose $E_{i}^{(j,t)} \\cap \\partial \\mathfrak {P}_{a, r, \\Gamma } \\ne \\emptyset $ , we then have $\\pi (B^{(j,t)}_{i}) \\subset \\tilde{\\mathcal {T}}_{a,r}$ .",
"In addition, if for some $j^{\\prime }\\ne j$ we have $E_{i}^{(j^{\\prime },t)} \\cap \\partial \\mathfrak {P}_{a, r, \\Gamma } \\ne \\emptyset $ as well, by $\\operatorname{dist}(E^{(j,t)},E^{(j^{\\prime },t)}) \\ge l_i^{1+\\varepsilon }$ and (REF ), we have that (when $C_{\\varepsilon , N}$ is large enough) $\\pi (B^{(j,t)}_{i}) \\cap \\pi (B^{(j^{\\prime },t)}_{i})=\\emptyset $ .",
"Thus for any $t$ , $\\left|\\left\\lbrace j:E_{i}^{(j,t)} \\cap \\partial \\mathfrak {P}_{a, r, \\Gamma } \\ne \\emptyset \\right\\rbrace \\right| l_{i}^{2+\\varepsilon } < \\operatorname{Area}(\\tilde{\\mathcal {T}}_{a,r}),$ since $\\operatorname{Area}(\\pi (B_{i}^{(j,t)}))>l_{i}^{2+\\varepsilon }$ for any $j, t$ .",
"Our claim follows by observing that $\\operatorname{Area}(\\tilde{\\mathcal {T}}_{a,r}) \\le C r^{2}$ .",
"Claim 5.18 There exists some universal constant $C$ such that for any $j \\in \\mathbb {Z}_{+}$ , $t\\in \\left\\lbrace 1,2,\\cdots ,N\\right\\rbrace $ and $i\\in \\left\\lbrace 1,2,\\cdots ,d\\right\\rbrace $ , $\\left|E^{(j,t)}_{i} \\cap \\partial \\mathfrak {P}_{a, r, \\Gamma }\\cap \\mathbb {Z}^3\\right|\\le C l_{i}^{2}$ .",
"By (REF ), $\\pi $ is a injection from $\\partial \\mathfrak {P}_{a, r, \\Gamma }$ , so we only need to show $\\left|\\pi (E_{i}^{(j,t)})\\cap \\pi (\\mathbb {Z}^3)\\right| \\le C l_{i}^2.$ We note that $\\pi (\\mathbb {Z}^3)$ is a triangular lattice on $\\mathcal {P}_{1, a\\cdot \\lambda _1}$ , with constant lattice length $\\frac{\\sqrt{6}}{3}$ and $\\pi (E_{i}^{(j,t)})$ is a 2D ball with radius at least $C_{\\varepsilon ,N}$ .",
"Assuming $C_{\\varepsilon ,N} > 10$ , we have $\\left|\\pi (E_{i}^{(j,t)})\\cap \\pi (\\mathbb {Z}^3)\\right| \\le 10 \\operatorname{Area}(\\pi (E_{i}^{(j,t)}))$ and our claim follows.",
"Now by Claim REF , $\\left|E_{i} \\cap \\partial \\mathfrak {P}_{a, r, \\Gamma }\\cap \\mathbb {Z}^3\\right| \\le \\sum _{j,t}\\left|E^{(j,t)}_{i} \\cap \\partial \\mathfrak {P}_{a, r, \\Gamma }\\cap \\mathbb {Z}^3\\right|\\le \\sum _{j,t}\\left|\\left\\lbrace (j,t):E_{i}^{(j,t)} \\cap \\partial \\mathfrak {P}_{a, r, \\Gamma } \\ne \\emptyset \\right\\rbrace \\right| C l_{i}^{2}.$ Then by Claim REF , we get (REF ).",
"As for (REF ), since by (REF ) $\\pi $ is a injection on $\\partial \\mathfrak {P}_{a, r, \\Gamma }$ , we only need to show $\\left|\\pi \\left(E_{0} \\cap \\partial \\mathfrak {P}_{a, r, \\Gamma }\\cap \\mathbb {Z}^3\\right)\\right|\\le C \\varepsilon ^{2} r^{2}$ for some universal constant $C$ .",
"By (REF ) and the fact that $E_{0}$ is $\\varepsilon ^{-1}$ -unitscattered, we have $|\\pi (b)-\\pi (b^{\\prime })| \\ge \\frac{\\varepsilon ^{-1}}{10}$ for any $b \\ne b^{\\prime } \\in E_{0}\\cap \\partial \\mathfrak {P}_{a, r, \\Gamma }\\cap \\mathbb {Z}^3$ (since $b$ and $b^{\\prime }$ are centers of different unit balls in $E_0$ ).",
"Thus (REF ) follows from $\\operatorname{Area}(\\pi (\\mathfrak {P}_{a, r, \\Gamma })) < 100 r^{2}$ .",
"[Proof of Proposition REF ] We assume that $r > 1000$ , since otherwise the statement holds by taking $C_{9}$ small enough.",
"To apply Proposition REF , we need to check its third condition.",
"We argue by contradiction, and assume that there exists $b \\in \\mathring{\\mathfrak {P}}_{a,r,\\Gamma } \\cap \\mathbb {Z}^3$ with $b\\cdot \\lambda _1 \\le h$ , and $|u(b)|>\\exp (C_{10}n)g$ .",
"Consider the triangle $\\mathcal {P}_{1, b \\cdot \\lambda _1} \\cap \\mathring{\\mathfrak {P}}_{a,r,\\Gamma }$ , and write it as $\\lbrace c \\in \\mathcal {P}_{1, b \\cdot \\lambda _1} : c\\cdot \\overline{\\lambda }_\\tau < \\mathbf {t}_r(a) \\cdot \\overline{\\lambda }_\\tau - F^{\\prime }, \\forall \\tau = 2,3,4\\rbrace $ for some $F^{\\prime }\\ge 0$ .",
"From the definition of $h$ , the its side length is at least $\\sqrt{2}\\left(\\frac{r}{4}-1\\right)$ .",
"Consider the three sets $ \\left\\lbrace c \\in \\mathcal {P}_{1, b \\cdot \\lambda _1} : c\\cdot \\overline{\\lambda }_\\tau > \\mathbf {t}_r(a) \\cdot \\overline{\\lambda }_\\tau - F^{\\prime } - \\frac{r}{10} \\right\\rbrace $ where $\\tau \\in \\lbrace 2,3,4\\rbrace $ (see Figure REF ).",
"The intersection of all three of them is empty, so by symmetry, we can assume that $b$ is not in the first one, i.e.",
"$b \\cdot \\overline{\\lambda }_2 \\le \\mathbf {t}_r(a) \\cdot \\overline{\\lambda }_2 - F^{\\prime } - \\frac{r}{10}.$ Figure: The three green areas are given by () and do not have common intersection, so b=c 0 ∈𝒫 1,b·λ 1 ∩𝔓 ˚ a,r,Γ b=c_0 \\in \\mathcal {P}_{1,b\\cdot \\lambda _1}\\cap \\mathring{\\mathfrak {P}}_{a,r,\\Gamma } is outside one of them, and we can construct a path in 𝔓 ˚ a,r,Γ \\mathring{\\mathfrak {P}}_{a,r,\\Gamma } from it by using the cone property.Now we apply Lemma REF , starting from $b$ and in the $-\\mathbf {e}_1$ direction.",
"Since $r < \\frac{n}{32}$ and $a \\in Q_{\\frac{n}{2}}$ , we can find a sequence of points $b=c_0, c_1, \\cdots , c_r$ , such that for any $1 \\le i \\le r$ , we have $|u(c_i)| \\ge (K+11)^{-1}|u(c_{i-1})|$ , and $c_i - c_{i-1} \\in \\left\\lbrace -\\mathbf {e}_1, -\\mathbf {e}_1 + \\mathbf {e}_2, -\\mathbf {e}_1+\\mathbf {e}_3, -\\mathbf {e}_1-\\mathbf {e}_2, -\\mathbf {e}_1-\\mathbf {e}_3, -2\\mathbf {e}_1 \\right\\rbrace $ .",
"Then we have that $c_i \\cdot \\overline{\\lambda }_2 \\le c_{i-1} \\cdot \\overline{\\lambda }_2 + 2$ , $c_i \\cdot \\overline{\\lambda }_3 \\le c_{i-1} \\cdot \\overline{\\lambda }_3$ and $c_i \\cdot \\overline{\\lambda }_4 \\le c_{i-1} \\cdot \\overline{\\lambda }_4$ .",
"This means that for $1 \\le i \\le \\frac{r}{30}$ , $ \\begin{split}&c_i \\cdot \\overline{\\lambda }_2 \\le b\\cdot \\overline{\\lambda }_2 + \\frac{r}{15} < \\mathbf {t}_r(a) \\cdot \\overline{\\lambda }_2 - F^{\\prime }, \\\\&c_i \\cdot \\overline{\\lambda }_\\tau \\le b \\cdot \\overline{\\lambda }_\\tau \\le \\mathbf {t}_r(a) \\cdot \\overline{\\lambda }_\\tau - F^{\\prime },\\;\\forall \\tau \\in \\lbrace 3,4\\rbrace .\\end{split}$ Also, for $i \\le \\frac{r}{30}$ , we have $ |u(c_i)| \\ge (K+11)^{-\\frac{r}{30}}|u(c_{0})| > \\exp \\left(\\frac{C_{10}n}{2}\\right)g,$ when $C_{10}>K+11$ .",
"Since $c_{i-1} \\lambda _1 - 2 \\le c_i \\cdot \\lambda _1 \\le c_{i-1} \\lambda _1$ , by the second condition of Proposition REF , we have that $a \\cdot \\lambda _1 < c_i \\cdot \\lambda _1\\le b \\cdot \\lambda _1$ for each $1 \\le i \\le \\frac{r}{30}$ .",
"With (REF ) this implies that $c_i \\in \\mathring{\\mathfrak {P}}_{a,r,\\Gamma }$ for each $1 \\le i \\le \\frac{r}{30}$ .",
"See Figure REF for an illustration.",
"By the definition of the pyramid $\\mathfrak {P}_{a,r,\\Gamma }$ , for $0 \\le i \\le \\frac{r}{30}$ we have that $c_i \\notin \\Gamma $ , thus $c_i \\in E$ by (REF ) and the fourth condition of Proposition REF .",
"For $l \\in \\mathbb {R}_+$ with $1 \\le l < (2\\sqrt{2}r)^{1-\\frac{\\varepsilon }{2}}$ , and any $(1,l,\\varepsilon )$ -scattered set $Z$ , the number of balls in $Z$ that intersect $\\left\\lbrace c_i\\right\\rbrace _{i=1}^{\\left\\lfloor \\frac{r}{30} \\right\\rfloor }$ is at most $2\\left\\lfloor \\frac{r}{30} \\right\\rfloor l^{-1-\\varepsilon }+1$ .",
"This is because, otherwise, there must exist $1\\le i_1 < i_2 \\le \\left\\lfloor \\frac{r}{30} \\right\\rfloor $ , such that $|i_1-i_2|< \\frac{l^{1+\\varepsilon }}{2}$ , and $c_{i_1}$ and $c_{i_2}$ are contained in different balls.",
"By construction the distance between $c_{i_1}$ and $c_{i_2}$ is at most $2|i_1-i_2|$ , and this contradicts with the fact that $Z$ is $(1,l,\\varepsilon )$ -scattered.",
"For each ball in $Z$ , it contains at most $2l$ points in $\\left\\lbrace c_i\\right\\rbrace _{i=1}^{\\left\\lfloor \\frac{r}{30} \\right\\rfloor }$ .",
"This is because $c_i\\cdot \\mathbf {e}_1\\le c_{i-1}\\cdot \\mathbf {e}_1-1$ for $1<i\\le \\left\\lfloor \\frac{r}{30} \\right\\rfloor $ , and the diameter of each ball is $2l$ .",
"Thus we have $ \\left| Z \\cap \\left\\lbrace c_i\\right\\rbrace _{i=1}^{\\left\\lfloor \\frac{r}{30} \\right\\rfloor } \\right| \\le 2l\\cdot \\left( 2\\left\\lfloor \\frac{r}{30} \\right\\rfloor l^{-1-\\varepsilon }+1\\right) < r l^{-\\varepsilon } + 2l.$ Similarly, for any $\\varepsilon ^{-1}$ -unitscattered set $Z$ , we have $ \\left| Z \\cap \\left\\lbrace c_i\\right\\rbrace _{i=1}^{\\left\\lfloor \\frac{r}{30} \\right\\rfloor } \\right| <r\\varepsilon +2.$ For the set $E$ which is $(\\varepsilon ^{-\\frac{1}{2}},\\varepsilon )$ -normal in $\\mathfrak {P}_{a,r,\\Gamma }$ , using (REF ) and (REF ) we have $ \\left| E \\cap \\left\\lbrace c_i\\right\\rbrace _{i=1}^{\\left\\lfloor \\frac{r}{30} \\right\\rfloor } \\right|<r\\varepsilon +2+\\sum _{1 \\le i \\le d: l_i < (2\\sqrt{2}r)^{1-\\frac{\\varepsilon }{2}}}Nr l^{-\\varepsilon }_i + 2Nl_i$ We have that $Nr\\sum _{i=1}^{d} l_i^{-\\varepsilon }\\le Nr\\sum _{i=1}^{\\infty } l_1^{-\\varepsilon (1+2\\varepsilon )^{i-1}}<Nr\\sum _{i=1}^{\\infty } C_{\\varepsilon , N}^{-\\varepsilon (1+2\\varepsilon )^{i-1}}<Nr\\sum _{i=1}^{\\infty } C_{\\varepsilon , N}^{-\\varepsilon } C_{\\varepsilon , N}^{-2(i-1)\\varepsilon ^2}=\\frac{NrC_{\\varepsilon , N}^{-\\varepsilon }}{1-C_{\\varepsilon , N}^{-2\\varepsilon ^2}},$ and when $C_{\\varepsilon , N}$ is large enough this is less than $\\frac{r}{100}$ .",
"Also, when $(2\\sqrt{2}r)^{1-\\frac{\\varepsilon }{2}} > C_{\\varepsilon , N}>100$ , and $\\varepsilon < \\frac{1}{200}$ , we have $ \\sum _{1 \\le i \\le d: l_i < (2\\sqrt{2}r)^{1-\\frac{\\varepsilon }{2}}}2Nl_i< 2\\left(\\frac{\\log \\left(\\frac{\\log (2\\sqrt{2}r)}{\\log (C_{\\varepsilon , N})}\\right)}{\\log (1+2\\varepsilon )} + 1 \\right)N(2\\sqrt{2}r)^{1-\\frac{\\varepsilon }{2}}<\\frac{4\\log (\\log (2\\sqrt{2}r))}{\\varepsilon }N(2\\sqrt{2}r)^{1-\\frac{\\varepsilon }{2}},$ where the first inequality is due to that there are at most $\\left\\lceil \\frac{\\log \\left(\\frac{\\log (2\\sqrt{2}r)}{\\log (C_{\\varepsilon , N})}\\right)}{\\log (1+2\\varepsilon )} \\right\\rceil $ terms in the summation, and each is at most $2N(2\\sqrt{2}r)^{1-\\frac{\\varepsilon }{2}}$ .",
"We further have that (REF ) is less than $\\frac{r}{100}$ when $C_{\\varepsilon , N}$ is large enough.",
"When $(2\\sqrt{2}r)^{1-\\frac{\\varepsilon }{2}} \\le C_{\\varepsilon , N}$ , the left hand side of (REF ) is zero.",
"Thus the left hand side of (REF ) is less than $\\frac{3r}{100}+2<\\frac{r}{30}$ when $\\varepsilon < \\frac{1}{100}$ and $C_{\\varepsilon , N}$ is large enough.",
"This contradicts with the fact that $c_i \\in E$ for each $0 \\le i \\le \\frac{r}{30}$ .",
"Finally, the conclusion follows from Proposition REF and REF , by taking $C_{9}= \\frac{1}{2}C^{\\prime }_{9}$ and the same $C_{10}$ as in Proposition REF ."
],
[
"Proof of Theorem ",
"In this subsection we assemble results in previous subsections together and finish the proof of Theorem REF .",
"[Proof of Theorem REF ] By taking $C_{\\varepsilon , N}$ large we can assume that $n>100$ .",
"We prove the result for $C_{3}=\\frac{1}{60}C_{8}$ and $C_{2}=\\max \\left\\lbrace 2C_{7}, 2\\log (K+11) \\right\\rbrace $ , where $C_{8},C_{7}$ are the constants in Proposition REF .",
"We let $\\varepsilon $ be small enough, and $C_{\\varepsilon ,N}$ be the same as required by Proposition REF .",
"By Proposition REF , there exists $\\tau \\in \\left\\lbrace 1,2,3,4\\right\\rbrace $ , and $a_i \\in \\left(\\mathcal {P}_{\\tau , i}\\cup \\mathcal {P}_{\\tau , i+1}\\right) \\cap \\mathcal {C}\\cap Q_{\\frac{n}{10} + 1}$ for $i=0,1,\\cdots ,\\left\\lfloor \\frac{n}{10} \\right\\rfloor - 1$ , such that $|u(a_{i})|\\ge (K+11)^{-n}|u(\\mathbf {0})|$ .",
"For each $i=0,1,\\cdots ,\\left\\lfloor \\frac{n}{10} \\right\\rfloor -1$ , we apply Proposition REF to $a_i$ , and find $h_i \\in \\mathbb {Z}_+$ , such that $\\left| \\left\\lbrace a \\in Q_n \\cap \\bigcup _{j=0}^{h_i} \\mathcal {P}_{\\tau , a_i\\cdot \\lambda _1+j}: |u(a)| \\ge \\exp (-C_{7}n^3)|u(a_i)| \\ge \\exp (-C_{2}n^3)|u(\\mathbf {0})| \\right\\rbrace \\setminus E \\right| \\\\ > C_{8}h_i n (\\log _2(n))^{-1}.$ Now for some $m\\in \\mathbb {Z}_{\\ge 0}$ , we define a sequence of nonnegative integers $i_1 < \\cdots < i_m$ inductively.",
"Let $i_1 := 0$ .",
"Given $i_k$ , if $a_{i_k} \\cdot \\lambda _{\\tau } + h_k + 1 \\le \\left\\lfloor \\frac{n}{10} \\right\\rfloor -1$ , we let $i_{k+1} := a_{i_k} \\cdot \\lambda _{\\tau } + h_{i_k} + 1$ ; otherwise, let $m=k$ and the process terminates.",
"Obviously, the sets $\\left\\lbrace a \\in Q_n \\cap \\bigcup _{j=0}^{h_{i_k}} \\mathcal {P}_{\\tau , a_{i_k}\\cdot \\lambda _1+j} : |u(a)| \\ge \\exp (-C_{2}n^3)|u(\\mathbf {0})| \\right\\rbrace \\setminus E$ for $k=1, \\cdots , m$ are mutually disjoint.",
"Besides, we have that $a_{i_1}\\cdot \\lambda _{\\tau } \\le 1$ and $a_{i_m}\\cdot \\lambda _{\\tau } + h_{i_m} \\ge \\left\\lfloor \\frac{n}{10} \\right\\rfloor -1$ ; and for each $1 \\le k < m$ , $a_{i_{k+1}} \\cdot \\lambda _{\\tau } - a_{i_k} \\cdot \\lambda _{\\tau } \\le h_{i_k} + 2$ .",
"This implies that $\\sum _{j=1}^m (h_{i_k}+2) \\ge \\left\\lfloor \\frac{n}{10} \\right\\rfloor -2$ , thus $\\sum _{j=1}^m h_{i_k} > \\frac{n}{60}$ , and $\\left| \\left\\lbrace a \\in Q_{n} : |u(a)| \\ge \\exp (-C_{2} n^{3}) |u(\\mathbf {0})| \\right\\rbrace \\setminus E \\right| \\ge C_{8}\\left(\\sum _{k=1}^m h_{i_k}\\right) n (\\log _2(n))^{-1} \\\\ > C_3n^2(\\log _2(n))^{-1}$ which is (REF )."
],
[
"Recursive construction: proof of discrete unique continuation",
"We deduce Theorem REF from Theorem REF in this section.",
"The key step is the following result.",
"Theorem 6.1 There exist universal constants $\\beta $ and $\\alpha >\\frac{5}{4}$ such that for any positive integers $m \\le n$ and any positive real $K$ , the following is true.",
"For any $u,V:\\mathbb {Z}^3 \\rightarrow \\mathbb {R}$ such that $\\Delta u=V u$ in $Q_{n}$ and $\\Vert V \\Vert _{\\infty } \\le K$ , we can find a subset $\\Theta \\subset Q_n$ with $|\\Theta | \\ge \\beta \\left(\\frac{n}{m}\\right)^{\\alpha } $ , such that $|u(b)| \\ge ( K+11 )^{-12n} | u ( \\mathbf {0} ) |$ for each $b \\in \\Theta $ .",
"$Q_m(b) \\cap Q_m(b^{\\prime })=\\emptyset $ for $b,b^{\\prime } \\in \\Theta $ , $b \\ne b^{\\prime }$ .",
"$Q_m(b)\\subset Q_n $ for each $b \\in \\Theta $ .",
"The proof of Theorem REF is based on the cone property, i.e.",
"Lemma REF , and induction on $\\frac{n}{m}$ .",
"We first set up some notations.",
"Definition 6.2 A set $B \\subset \\mathbb {Z}^3$ is called a cuboid if there are integers $t_{\\tau }\\le k_{\\tau }$ , for $\\tau =1,2,3$ , such that $B=\\left\\lbrace b \\in \\mathbb {Z}^3: t_{\\tau } \\le b\\cdot \\mathbf {e}_{\\tau } \\le k_{\\tau },\\tau =1,2,3 \\right\\rbrace .$ We denote $p^+(B):=k_1$ , $p^-(B):=t_1$ , and $q^+(B):=k_2$ , $q^-(B):=t_2$ .",
"A cuboid is called even if $t_{\\tau }, k_{\\tau }$ are even for each $\\tau = 1,2,3$ .",
"[Proof of Theorem REF ] Without loss of generality we assume that $u ( \\mathbf {0} ) =1$ .",
"Take $\\alpha =1.251>\\frac{5}{4}$ , and leave $\\beta $ to be determined.",
"We denote $f_{m} ( x ) = \\beta (\\frac{x}{m}) ^{\\alpha }$ for $x>0$ .",
"Then we have the following two inequalities: $4\\cdot 4^{-\\alpha } +4\\cdot 8^{-\\alpha } >1, \\;6\\cdot 4^{-\\alpha } >1.$ This implies that there exists universal $N_0>10^8$ such that, for any positive integers $m,n$ with $n>N_0 m$ and any real $\\beta >0$ , we have $ 4f_{m}\\left(\\frac{n}{4}-3\\right)+4f_{m}\\left(\\frac{n}{8}-2\\right)> f_{m}(n+7)$ and $ 4f_{m}\\left(\\frac{n}{4}-3\\right)+2f_{m}\\left(\\frac{n}{4}-2\\right)> f_{m}(n+7).$ We let $\\beta =(N_{0}+7)^{-\\alpha }$ , and fix $m \\in \\mathbb {Z}_+$ .",
"We need to show that, when $n \\ge m$ , there is $\\Theta \\subset Q_n$ , such that $|\\Theta | \\ge f_{m}(n)$ , and $\\Theta $ satisfies the three conditions in the statement.",
"For this, we do induction on $n$ .",
"First, it holds trivially when $m \\le n \\le N_0 m+7$ by the choice of $\\beta $ .",
"For simplicity of notations below, we only work on $n$ that divides 8.",
"At each step, we take some $n>N_0 m\\ge 10^8 m$ with $\\frac{n}{8}\\in \\mathbb {Z}$ and suppose our conclusion holds for all smaller $n$ .",
"Then we show that we can find a subset $\\Theta \\subset Q_{n}$ with $|\\Theta | \\ge f_m(n+7) $ , such that the conditions in the statement are satisfied.",
"Thus the conclusion holds for $n,n+1,\\cdots , n+7$ .",
"By Lemma REF , and using the notations in Definition REF , we pick $a_1 \\in \\mathcal {C}^{3}_{\\mathbf {0}} ( \\frac{n}{2} ) \\cup \\mathcal {C}_{\\mathbf {0}}^{3} ( \\frac{n}{2} +1 )$ and $a_2 \\in \\mathcal {C}^{3}_{\\mathbf {0}} ( - \\frac{n}{2}) \\cup \\mathcal {C}_{\\mathbf {0}}^{3} ( - \\frac{n}{2} -1 ) $ such that $|u(a_1)|, | u ( a_2 ) | \\ge (K+11 )^{-n}$ .",
"For simplicity of notations, we denote $Q^1$ as the even cuboid such that $Q_{ \\frac{n}{2} -2} ( a_1 )\\subset Q^1\\subset Q_{ \\frac{n}{2} -1} ( a_1 )$ ; and $Q^2$ as the even cuboid such that $Q_{\\frac{n}{2}-2}(a_2) \\subset Q^2 \\subset Q_{\\frac{n}{2}-1}(a_2)$ .",
"Then we use Lemma REF again to pick $\\begin{split}&a_{11} \\in \\mathcal {C}_{a_1}^{3}\\left( \\frac{n}{4} -1\\right)\\cup \\mathcal {C}_{a_1}^{3}\\left( \\frac{n}{4} \\right),\\\\&a_{12}\\in \\mathcal {C}_{a_1}^{3}\\left(- \\frac{n}{4} +1\\right) \\cup \\mathcal {C}_{a_1}^{3}\\left(- \\frac{n}{4} \\right),\\\\&a_{21} \\in \\mathcal {C}_{a_2}^{3}\\left( \\frac{n}{4} -1\\right)\\cup \\mathcal {C}_{a_2}^{3}\\left( \\frac{n}{4} \\right),\\\\&a_{22}\\in \\mathcal {C}_{a_2}^{3}\\left(- \\frac{n}{4} +1\\right) \\cup \\mathcal {C}_{a_2}^{3}\\left(- \\frac{n}{4} \\right),\\end{split}$ such that $|u(a_{11})|, |u(a_{12})|,|u(a_{21})|,|u(a_{22})|\\ge ( K+11 )^{-2n}$ .",
"For $i,j\\in \\left\\lbrace 1,2\\right\\rbrace $ , let $Q^{ij}$ be an even cuboid such that $Q_{\\frac{n}{4}-3}(a_{ij})\\subset Q^{ij}\\subset Q_{\\frac{n}{4}-2}(a_{ij})$ .",
"Comparing the coordinates of $a_{ij}$ 's, we see $Q^{ij}$ 's are pairwise disjoint.",
"By inductive hypothesis, we can find $4f (\\frac{n}{4}-3)$ points in $Q^{11}\\cup Q^{12}\\cup Q^{21}\\cup Q^{22}$ , such that for each $b$ among them, $| u(b) | \\ge ( K+11)^{-2n} ( K+11 )^{-12 (\\frac{n}{4}-3) } \\ge ( K+11 )^{-12n}$ and all $Q_m(b)$ are mutually disjoint, and contained in $Q^{11} \\cup Q^{12}\\cup Q^{21} \\cup Q^{22}$ .",
"Let $B$ be the minimal cuboid containing $Q^1\\cup Q^2$ , $B_1$ be the minimal cuboid containing $Q^{11}\\cup Q^{12}$ , and $B_2$ be the minimal cuboid containing $Q^{21}\\cup Q^{22}$ .",
"Let $g^{(r)}:=p^+(Q_n)-p^+(B)$ , $g^{(l)}:=p^-(B)-p^-(Q_n)$ , $g^{(r)}_{1}:=p^+(Q^1)-p^+(B_1)$ , $g^{(l)}_{1}:=p^-(B_1)-p^-(Q^1)$ , $g^{(r)}_{2}:=p^+(Q^2)-p^+(B_2)$ and $g^{(l)}_{2}:=p^-(B_2)-p^-(Q^2)$ .",
"Similarly, in the $\\mathbf {e}_2$ -direction, let $h^{(u)}:=q^+(Q_n)-q^+(B)$ , $h^{(d)}:=q^-(B)-q^-(Q_n)$ , $h^{(u)}_{1}:=q^+(Q^1)-q^+(B_1)$ , $h^{(d)}_{1}:=q^-(B_1)-q^-(Q^1)$ , $h^{(u)}_{2}:=q^+(Q^2)-q^+(B_2)$ and $h^{(d)}_{2}:=q^-(B_2)-q^-(Q^2)$ .",
"See Figure REF for an illustration of these definitions.",
"Figure: The projection onto the 𝐞 1 𝐞 2 \\mathbf {e}_1\\mathbf {e}_2 plane.From the above definitions, $g^{(r)}+g^{(l)}+h^{(u)}+h^{(d)}=4n-(p^+(B)-p^-(B))-(q^+(B)-q^-(B)).$ Observe that $(p^+(B)-p^-(B))+(q^+(B)-q^-(B))\\le |(a_1-a_2)\\cdot \\mathbf {e}_1|+|(a_1-a_2)\\cdot \\mathbf {e}_2|+4\\left( \\frac{n}{2} -1 \\right).$ As $a_1 \\in \\mathcal {C}^{3}_{\\mathbf {0}} ( \\frac{n}{2} ) \\cup \\mathcal {C}_{\\mathbf {0}}^{3} ( \\frac{n}{2} +1 ) $ , we have $|a_1\\cdot \\mathbf {e}_1|+|a_1\\cdot \\mathbf {e}_2|\\le |a_1\\cdot \\mathbf {e}_3| \\le \\frac{n}{2}+1$ ; and similarly, we have $|a_2\\cdot \\mathbf {e}_1|+|a_2\\cdot \\mathbf {e}_2|\\le \\frac{n}{2}+1$ .",
"Using these and (REF ), and triangle inequality, we have $(p^+(B)-p^-(B))+(q^+(B)-q^-(B)) \\le 3n-2.$ Thus with (REF ) we have $g^{(r)}+g^{(l)}+h^{(u)}+h^{(d)} \\ge n+2.$ The same argument applying to smaller cubes $Q^{1}$ and $Q^{2}$ , we have $g^{(r)}_{1}+g^{(l)}_{1}+h^{(u)}_{1}+h^{(d)}_{1} \\ge \\frac{n}{2}+2$ and $g^{(r)}_{2}+g^{(l)}_{2}+h^{(u)}_{2}+h^{(d)}_{2} \\ge \\frac{n}{2}+2.$ Summing them together we get $g^{(r)}+g^{(l)}+g^{(r)}_{1}+g^{(l)}_{1}+g^{(r)}_{2}+g^{(l)}_{2}+h^{(u)}+h^{(d)}+h_{1}^{(u)}+h^{(d)}_{1}+h^{(u)}_{2}+h^{(d)}_{2} \\ge 2n+6.$ As these $g$ 's and $h$ 's are exchangeable, we assume without loss of generality that $g^{(r)}+g^{(l)}+g^{(r)}_{1}+g^{(l)}_{1}+g^{(r)}_{2}+g^{(l)}_{2} \\ge n + 3.$ By symmetry, we assume without loss of generality that $a_1 \\cdot \\mathbf {e}_1 \\le a_2 \\cdot \\mathbf {e}_1$ ; consequently $p^-(Q^1)\\le p^-(Q^2)$ .",
"We discuss two possible cases.",
"Case 1: $p^+(B_2) \\le p^+(Q^1)$ or $p^-(B_1) \\ge p^-(Q^2)$ .",
"By symmetry again, it suffices to consider the scenario for $p^+(B_2) \\le p^+(Q^1)$ .",
"See Figure REF for an illustration.",
"Figure: The projection onto the 𝐞 1 𝐞 2 \\mathbf {e}_1\\mathbf {e}_2 plane in Case 1.Consider cuboids $\\begin{split}& U_l:=\\left\\lbrace b \\in \\mathbb {Z}^3: |b\\cdot \\mathbf {e}_2|, |b\\cdot \\mathbf {e}_3| \\le n-1, -n+1 \\le b\\cdot \\mathbf {e}_1 \\le p^-(Q^1)-1\\right\\rbrace , \\\\& U_r:=\\left\\lbrace b \\in \\mathbb {Z}^3: |b\\cdot \\mathbf {e}_2|, |b\\cdot \\mathbf {e}_3| \\le n-1, p^+(Q^1)+1 \\le b\\cdot \\mathbf {e}_1 \\le n-1\\right\\rbrace .\\end{split}$ Then $U_l,U_r,B_1,B_2$ are mutually disjoint, since $p^+(B_2) \\le p^+(Q^1)$ and $p^-(Q^1) \\le p^-(Q^2)$ .",
"Now we use Lemma REF to pick points $\\begin{split}& c_1 \\in \\mathcal {C}^{1}_{\\mathbf {0}} \\left( \\frac{1}{2}(p^-(Q^1) -n) \\right) \\cup \\mathcal {C}_{\\mathbf {0}}^{1} \\left( \\frac{1}{2}(p^-(Q^1) -n) +1 \\right) ,\\\\& c_2 \\in \\mathcal {C}^{1}_{\\mathbf {0}} \\left( \\frac{1}{2}(p^+(Q^1) +n) \\right) \\cup \\mathcal {C}_{\\mathbf {0}}^{1} \\left( \\frac{1}{2}(p^+(Q^1) +n) +1\\right) ,\\end{split}$ such that $|u(c_1)|, |u(c_2)| \\ge (K+11)^{-n}$ .",
"Denote $I_1:=\\frac{p^-(Q^1)+n}{2}-2, I_2:=\\frac{n-p^+(Q^1)}{2}-2$ .",
"Then $I_1, I_2 \\le \\frac{n}{2}$ .",
"We also have $(p^-(Q^1)+n)+(n-p^+(Q^1)) = 2n+p^-(Q^1)-p^+(Q^1)\\ge n+2,$ so $I_1+I_2 \\ge \\frac{n}{2}-3.$ We use inductive hypothesis on $Q_{I_1}(c_1) \\subset U_l$ , if $I_1 > m$ ; and on $Q_{I_2}(c_2) \\subset U_r$ , if $I_2 > m$ .",
"Note that $U_l, U_r, B_1, B_2$ are mutually disjoint.",
"Thus we get $f_{m}(I_1){1}_{I_1 > m} + f_m(I_2){1}_{I_2 > m}$ points in $\\mathbb {Z}^3$ , such that for each point $b$ among them, $|u(b)| \\ge (K+11)^{-n}(K+11)^{-12\\cdot \\frac{n}{2}} \\ge (K+11)^{-12n}$ , $Q_m(b)\\cap Q_m(b^{\\prime })=\\emptyset $ for another $b^{\\prime }\\ne b$ among them, $Q_m(b)\\subset Q_n\\setminus (Q^{11}\\cup Q^{12}\\cup Q^{21}\\cup Q^{22})$ .",
"We now show that $ f_{m}(I_1){1}_{I_1 > m} + f_m(I_2){1}_{I_2 > m} \\ge 2f_{m}\\left(\\frac{n}{4}-2\\right).$ If $I_1, I_2>m$ , (REF ) follows by convexity and monotonicity of the function $f_{m}$ , and (REF ).",
"If $I_1\\le m$ , by (REF ) and the assumption that $n>N_0 m \\ge 10^8 m$ , we have $I_2\\ge \\frac{n}{2}-3-m > 10^7 m$ .",
"Then by monotonicity of $f_m$ we have $f_m(I_2){1}_{I_2 > m} = f_m(I_2) \\ge f_m\\left(\\frac{n}{2}-3-m\\right) \\ge 2f_{m}\\left(\\frac{n}{4}-2\\right)$ , which implies (REF ).",
"The case when $I_2\\le m$ is symmetric.",
"Now together with the $4f_{m}\\left(\\frac{n}{4}-3\\right)$ points we found in $Q^{11}\\cup Q^{12}\\cup Q^{21}\\cup Q^{22}$ , we have a set of at least $4f_{m}\\left(\\frac{n}{4}-3\\right)+2f_{m}\\left(\\frac{n}{4}-2\\right)$ points in $Q_n$ , satisfying all the three conditions.",
"Case 2: $p^+(B_2) > p^+(Q^1)$ and $p^-(B_1) < p^-(Q^2)$ .",
"See Figure REF for an illustration.",
"Figure: The projection onto the 𝐞 1 𝐞 2 \\mathbf {e}_1\\mathbf {e}_2 plane in Case 2.Denote $\\begin{split}& U_1:=\\left\\lbrace b \\in \\mathbb {Z}^3: |b\\cdot \\mathbf {e}_2|, |b\\cdot \\mathbf {e}_3| \\le n-1, -n+1 \\le b\\cdot \\mathbf {e}_1 \\le p^-(B_1)-1\\right\\rbrace , \\\\& U_2:=\\left\\lbrace b \\in \\mathbb {Z}^3: |b\\cdot \\mathbf {e}_2|, |b\\cdot \\mathbf {e}_3| \\le n-1, p^+(B_2)+1 \\le b\\cdot \\mathbf {e}_1 \\le n-1\\right\\rbrace , \\\\& U_3:=\\left\\lbrace b \\in \\mathbb {Z}^3: |b\\cdot \\mathbf {e}_2| \\le n-1, 1 \\le b\\cdot \\mathbf {e}_3 \\le n-1, p^+(B_1)+1 \\le b\\cdot \\mathbf {e}_1 \\le p^+(Q^1)-1\\right\\rbrace , \\\\& U_4:=\\left\\lbrace b \\in \\mathbb {Z}^3: |b\\cdot \\mathbf {e}_2| \\le n-1, -n+1 \\le b\\cdot \\mathbf {e}_3 \\le -1, p^-(Q^2)+1 \\le b\\cdot \\mathbf {e}_1 \\le p^-(B_2)-1\\right\\rbrace .\\end{split}$ We note that $U_1$ , $U_2$ , $U_3$ , $U_4$ , $B_1$ and $B_2$ are mutually disjoint.",
"We use Lemma REF to pick the following points: $\\begin{split}& c_1 \\in \\mathcal {C}^{1}_{\\mathbf {0}} \\left( \\frac{1}{2}\\left(p^-\\left(B_1\\right) -n\\right) \\right) \\cup \\mathcal {C}_{\\mathbf {0}}^{1} \\left( \\frac{1}{2}\\left(p^-\\left(B_1\\right) -n\\right) +1 \\right) ,\\\\& c_2 \\in \\mathcal {C}^{1}_{\\mathbf {0}} \\left( \\frac{1}{2}\\left(p^+\\left(B_2\\right) +n\\right) \\right) \\cup \\mathcal {C}_{\\mathbf {0}}^{1} \\left( \\frac{1}{2}\\left(p^+\\left(B_2\\right) +n\\right) +1 \\right),\\\\& c_3 \\in \\mathcal {C}^{1}_{a_1} \\left( \\frac{1}{2}\\left(p^+\\left(B_1\\right) + p^+\\left(Q^1\\right)\\right) -a_1 \\cdot \\mathbf {e}_1 \\right) \\cup \\mathcal {C}_{a_1}^{1} \\left( \\frac{1}{2}\\left(p^+\\left(B_1\\right) + p^+\\left(Q^1\\right)\\right) -a_1 \\cdot \\mathbf {e}_1 +1 \\right),\\\\& c_4 \\in \\mathcal {C}^{1}_{a_2} \\left( \\frac{1}{2}\\left(p^-\\left(B_2\\right) + p^-\\left(Q^2\\right)\\right) -a_2 \\cdot \\mathbf {e}_1 \\right) \\cup \\mathcal {C}_{a_2}^{1} \\left( \\frac{1}{2}\\left(p^-\\left(B_2\\right) + p^-\\left(Q^2\\right)\\right) -a_2 \\cdot \\mathbf {e}_1 +1 \\right),\\end{split}$ such that $|u(c_i)| \\ge (K+11)^{-3n}$ for each $i=1,2,3,4$ .",
"Denote $J_1:=\\frac{p^-(B_1)+n}{2} -2$ , $J_2:=\\frac{n-p^+(B_2)}{2} -2$ , $J_3:=\\frac{p^+(Q^1)-p^+(B_1)}{2} -2$ , and $J_4:=\\frac{p^-(B_2)-p^-(Q^2)}{2} -2$ .",
"For each $i=1,2,3,4$ , if $J_i > m$ , we use inductive hypothesis on $Q_{J_i}(c_i) \\subset U_i$ (note that $Q_{J_i}(c_i)$ is disjoint from $Q_{11}$ , so $J_i \\le \\frac{3n}{4}$ ).",
"As the sets $B_1$ , $B_2$ , $U_1$ , $U_2$ , $U_3$ and $U_4$ are mutually disjoint, we can find $\\sum _{i=1}^4 f_{m}(J_i){1}_{J_i>m}$ points in $\\bigcup _{i=1}^4 U_i$ , such that for each point $b$ among them, $|u(b)|\\ge (K+11)^{-3n}(K+11)^{-12 \\cdot \\frac{3n}{4}}=(K+11)^{-12n}$ , $Q_m(b)\\cap Q_m(b^{\\prime })=\\emptyset $ for another $b^{\\prime }\\ne b$ among them, $Q_m(b)\\subset Q_n\\setminus (Q^{11}\\cup Q^{12}\\cup Q^{21}\\cup Q^{22})$ .",
"By (REF ), we have $(p^-(B_1)+n)+(n-p^+(B_2))+(p^+(Q^1)-p^+(B_1))+(p^-(B_2)-p^-(Q^2))\\\\=g^{(r)}+g^{(l)}+g^{(r)}_{1}+g^{(l)}_{1}+g^{(r)}_{2}+g^{(l)}_{2} \\ge n+3,$ thus $J_1+J_3+J_3+J_4 \\ge \\frac{n}{2}-7$ .",
"Similar to (REF ) above, by monotonicity and convexity of $f_{m}$ , and $n>N_0 m \\ge 10^8 m$ , we have $\\sum _{i=1}^4 f_{m}(J_i){1}_{J_i>m} \\ge 4f_{m}\\left(\\frac{n}{8}-2\\right).$ This implies that, together with the $4f_{m}\\left(\\frac{n}{4}-3\\right)$ points we found in $Q^{11}\\cup Q^{12}\\cup Q^{21}\\cup Q^{22}$ , we have a set of at least $4f_{m}\\left(\\frac{n}{4}-3\\right)+4f_{m}\\left(\\frac{n}{8}-2\\right)$ points in $Q_n$ , satisfying all the three conditions.",
"In conclusion, by (REF ) and (REF ), in each case, we can always find a $\\Theta \\subset Q_n$ satisfying the three conditions, with $|\\Theta | \\ge f_{m}(n+7)$ .",
"Thus Theorem REF follows from the principle of induction.",
"Now we prove Theorem REF .",
"[Proof of Theorem REF ] Let $p:=\\frac{1}{3}\\alpha + \\frac{13}{12}$ , then $p>\\frac{3}{2}$ since $\\alpha >\\frac{5}{4}$ .",
"Without loss of generality, we assume that $u(\\mathbf {0})=1$ .",
"Suppose $\\vec{l}=(l_1,l_2,\\cdots ,l_d)$ .",
"Since $E$ is $(N,\\vec{l},\\varepsilon ^{-1},\\varepsilon )$ -graded, we can write $E=\\bigcup _{i=0}^{d}E_{i}$ where $E_i$ is an $(N,l_i,\\varepsilon )$ -scattered set for $i>0$ and $E_0$ is a $\\varepsilon ^{-1}$ -unitscattered set.",
"We also write $E_{i}=\\bigcup _{j \\in \\mathbb {Z}_+,1 \\le t \\le N}E_{i}^{(j,t)}$ , where each $E_{i}^{(j,t)}$ is an open ball with radius $l_{i}$ and $\\operatorname{dist}(E_{i}^{(j,t)},E_{i}^{(j^{\\prime },t)}) \\ge l_{i}^{1+\\varepsilon }$ whenever $j \\ne j^{\\prime }$ .",
"We assume without loss of generality that $l_{d} \\le 4n^{1-\\frac{\\varepsilon }{2}}$ .",
"Otherwise, since $E$ is $(1,\\varepsilon )$ -normal in $Q_{n}$ , we can replace $E$ by $E_{0} \\cup \\left(\\bigcup _{l_{i} \\le 4n^{1-\\frac{\\varepsilon }{2}} } E_{i}\\right)$ .",
"Let $n_{k}:=\\left\\lfloor l_{d-k} \\right\\rfloor $ for $k=0,1,\\cdots ,d$ .",
"Claim 6.3 We can assume there is $M \\in \\mathbb {Z}_{+}$ such that $n^{\\frac{1}{3}(1-4\\varepsilon )}+1 \\le n_{M} \\le n^{\\frac{1}{3}}$ .",
"Suppose there is no such $M\\in \\mathbb {Z}_{+}$ , we then add a level of empty set with scale length equal $n^{\\frac{1}{3}(1-2\\varepsilon )}$ .",
"More specifically, let $k$ be the largest nonnegative integer satisfying $l_{k}\\le n^{\\frac{1}{3}(1-4\\varepsilon )}$ , then $l_{k+1} > n^{\\frac{1}{3}}$ .",
"We let $l^{\\prime }_{i}=l_{i}$ and $E^{\\prime }_{i}=E_{i}$ for each $0 \\le i \\le k$ .",
"Let $l^{\\prime }_{k+1}=n^{\\frac{1}{3}(1-2\\varepsilon )}$ and $E^{\\prime }_{k+1}$ be any $(N,l^{\\prime }_{k},\\varepsilon )$ -scattered set that is disjoint from $Q_n$ .",
"Let $l^{\\prime }_{i}=l_{i-1}$ and $E^{\\prime }_{i}=E_{i-1}$ for $i \\ge k+2$ .",
"Then for each $1\\le i \\le d+1$ , we have $(l_{i-1}^{\\prime })^{1+2\\varepsilon } \\le l_i^{\\prime }$ , and $E^{\\prime }_{i}$ is $(N,l^{\\prime }_{i},\\varepsilon )$ -scattered.",
"Also, as $n > C_{\\varepsilon , N}^4$ we still have $l_1^{\\prime } > C_{\\varepsilon , N}$ .",
"Evidently, by replacing $E$ with $\\bigcup _{i=0}^{d+1} E^{\\prime }_{i}$ , our claim holds with $M=k+1$ .",
"Now we inductively construct subsets $\\Theta _{k} \\subset Q_n$ for $k=0,1,\\cdots ,M$ , such that the following conditions hold.",
"$|\\Theta _{k}| \\ge \\left(\\frac{\\beta }{2}\\right)^{2k+2} \\left(\\frac{n}{n_{k}}\\right)^{\\alpha }$ .",
"For any $a \\in \\Theta _{k}$ , we have $|u(a)| \\ge (K+11)^{-24(k+1) n}$ .",
"For any $a,a^{\\prime } \\in \\Theta _{k}$ with $a \\ne a^{\\prime }$ , we have $Q_{n_{k}}(a) \\cap Q_{n_{k}}(a^{\\prime })=\\emptyset $ .",
"For any $a \\in \\Theta _{k}$ , we have $Q_{n_{k}}(a) \\subset Q_n$ .",
"When $k>0$ , for any $a \\in \\Theta _{k}$ , there exists $a^{\\prime } \\in \\Theta _{k-1}$ such that $Q_{n_{k}}(a) \\subset Q_{n_{k-1}}(a^{\\prime })$ .",
"For any $a \\in \\Theta _{k}$ and $d-k\\le i\\le d$ , we have $E_{i} \\cap Q_{n_{k}}(a) = \\emptyset $ .",
"Let $n^{\\prime }_{0}:=\\min \\left\\lbrace \\left\\lfloor \\frac{1}{4}n_0^{1+\\varepsilon } \\right\\rfloor , n\\right\\rbrace $ .",
"By using Theorem REF for $m=n^{\\prime }_{0}$ , we get a subset $\\Theta ^{\\prime }_{0} \\subset Q_n$ such that $|\\Theta ^{\\prime }_{0}| \\ge \\beta \\left(\\frac{n}{n^{\\prime }_{0}}\\right)^{\\alpha }$ and $\\Theta ^{\\prime }_{0}$ satisfies Condition 1 to 3 in Theorem REF .",
"For each fixed $t \\in \\left\\lbrace 1,2,\\cdots ,N\\right\\rbrace $ and $j\\ne j^{\\prime } \\in \\mathbb {Z}_+$ , by definition we have $\\operatorname{dist}(E_{d}^{(j,t)},E_{d}^{(j^{\\prime },t)}) \\ge 4n^{\\prime }_{0}$ .",
"This implies $\\left|\\left\\lbrace (j,t):E_{d}^{(j,t)} \\cap Q_{n^{\\prime }_{0}}(a) \\ne \\emptyset \\right\\rbrace \\right| \\le N,$ for each $a \\in \\Theta ^{\\prime }_{0}$ .",
"For each $a \\in \\Theta ^{\\prime }_{0}$ , by using Theorem REF for $Q_{n^{\\prime }_{0}}(a)$ and $m=n_{0}$ , we get a subset $\\Theta ^{(a)}_{0} \\subset Q_{n^{\\prime }_{0}}(a)$ such that $|\\Theta ^{(a)}_{0}| \\ge \\beta (\\frac{n^{\\prime }_{0}}{n_{0}}) ^{\\alpha }$ and $\\Theta ^{(a)}_{0}$ satisfies Condition 1 to 3 in Theorem REF .",
"For each $j, t$ we have $\\left|\\left\\lbrace b \\in \\Theta ^{(a)}_{0}:Q_{n_{0}}(b) \\cap E_{d}^{(j, t)} \\ne \\emptyset \\right\\rbrace \\right| \\le 100$ .",
"This is because for each $b \\in \\Theta ^{(a)}_{0}$ with $Q_{n_{0}}(b) \\cap E_{d}^{(j, t)} \\ne \\emptyset $ , the cube $Q_{n_{0}}(b)$ is contained in the closed ball of radius $2\\sqrt{3}n_0+l_d<(2\\sqrt{3}+1)n_0+1$ with the same center as $E_{d}^{(j, t)}$ .",
"As we have $Q_{n_{0}}(b) \\cap Q_{n_{0}}(b^{\\prime })=\\emptyset $ for $b \\ne b^{\\prime } \\in \\Theta ^{(a)}_{0}$ , the number of such $b \\in \\Theta ^{(a)}_{0}$ is at most $\\frac{(2(2\\sqrt{3}+1)n_0+2)^3}{(2n_0+1)^3}<100$ .",
"Thus by (REF ), we have $\\left|\\left\\lbrace b \\in \\Theta ^{(a)}_{0}:Q_{n_{0}}(b) \\cap E_{d} \\ne \\emptyset \\right\\rbrace \\right| \\le 100 N.$ Let $\\tilde{\\Theta }^{(a)}_{0}:=\\Theta ^{(a)}_{0} \\setminus \\left\\lbrace b \\in \\Theta ^{(a)}_{0}:Q_{n_{0}}(b) \\cap E_{d} \\ne \\emptyset \\right\\rbrace $ for each $a \\in \\Theta ^{\\prime }_{0}$ , and $\\Theta _{0}=\\bigcup _{a \\in \\Theta ^{\\prime }_{0}} \\tilde{\\Theta }^{(a)}_{0}$ .",
"Now we check the conditions.",
"Condition 6 is from the definition, and Condition 5 automatically holds since $k=0$ .",
"Condition 2 to 4 hold by the conditions in Theorem REF .",
"For Condition 1, recall that $l_{d} \\ge l_{1} \\ge C_{\\varepsilon ,N}$ , and $l_d\\le 4n^{1-\\frac{\\varepsilon }{2}}$ .",
"By letting $C_{\\varepsilon ,N}$ large enough we have $n^{\\prime }_{0}> l_{d} ^{1+\\frac{\\varepsilon }{2}}$ , and then $\\frac{1}{2} \\beta ( \\frac{n^{\\prime }_{0}}{n_{0}}) ^{\\alpha } > \\frac{1}{2}\\beta l_d^{\\frac{1}{2} \\alpha \\varepsilon } \\ge \\frac{1}{2}\\beta C_{\\varepsilon ,N}^{\\frac{1}{2} \\alpha \\varepsilon } > 100N$ .",
"Thus for each $a \\in \\Theta ^{\\prime }_{0}$ we have $|\\tilde{\\Theta }^{(a)}_{0}|\\ge |\\Theta ^{(a)}_{0}|-100N \\ge \\frac{1}{2} \\beta ( \\frac{n^{\\prime }_{0}}{n_{0}}) ^{\\alpha }$ .",
"This implies that $|\\Theta _{0}| = \\sum _{a \\in \\Theta ^{\\prime }_{0}} |\\tilde{\\Theta }^{(a)}_{0}| \\ge \\left(\\frac{1}{2} \\beta \\left(\\frac{n^{\\prime }_{0}}{n_{0}}\\right) ^{\\alpha }\\right)\\left(\\beta \\left(\\frac{n}{n^{\\prime }_{0}}\\right) ^{\\alpha }\\right) > \\left(\\frac{\\beta }{2}\\right)^{2} \\left(\\frac{n}{n_{0}}\\right)^{\\alpha }.$ Suppose we have constructed $\\Theta _{k}$ , for some $0\\le k < M$ , we proceed to construct $\\Theta _{k+1}$ .",
"Note that as $l_{d-k-1}^{1+2\\varepsilon } \\le l_{d-k}$ , we have $n_{k} \\ge n_{k+1}^{1+2\\varepsilon } -1$ .",
"Let $n^{\\prime }_{k+1}=\\left\\lfloor \\frac{1}{4} n_{k+1}^{1+\\varepsilon }\\right\\rfloor $ .",
"Take an arbitrary $a_{0} \\in \\Theta _{k}$ , use Theorem REF for $Q_{n_{k}}(a_0)$ with $m=n^{\\prime }_{k+1}$ , we get a subset $\\Theta ^{\\prime (a_0)}_{k+1} \\subset Q_{n_{k}}(a_0)$ such that $|\\Theta ^{\\prime (a_0)}_{k+1}| \\ge \\beta \\left(\\frac{n_k}{n^{\\prime }_{k+1}}\\right)^{\\alpha }$ and $\\Theta ^{\\prime (a_0)}_{k+1}$ satisfies Condition 1 to 3 in Theorem REF .",
"For each fixed $t \\in \\left\\lbrace 1,2,\\cdots ,N\\right\\rbrace $ and $j \\ne j^{\\prime } \\in \\mathbb {Z}_+$ , by definition we have $\\operatorname{dist}(E_{d-k-1}^{(j,t)},E_{d-k-1}^{(j^{\\prime },t)}) \\ge 4n^{\\prime }_{k+1}$ .",
"This implies, for each $a \\in \\Theta ^{\\prime (a_0)}_{k+1}$ , $\\left|\\left\\lbrace (j,t):E_{d-k-1}^{(j,t)} \\cap Q_{n^{\\prime }_{k+1}}(a) \\ne \\emptyset \\right\\rbrace \\right| \\le N.$ For each $a \\in \\Theta ^{\\prime (a_0)}_{k+1}$ , by using Theorem REF for $Q_{n^{\\prime }_{k+1}}(a)$ and $m=n_{k+1}$ , we get a subset $\\Theta ^{(a)}_{k+1} \\subset Q_{n^{\\prime }_{k+1}}(a)$ such that $|\\Theta ^{(a)}_{k+1}| \\ge \\beta \\left(\\frac{n^{\\prime }_{k+1}}{n_{k+1}}\\right)^{\\alpha }$ and $\\Theta ^{(a)}_{k+1}$ satisfies Condition 1 to 3 in Theorem REF .",
"By (REF ), $\\left|\\left\\lbrace b \\in \\Theta ^{(a)}_{k+1}:Q_{n_{k+1}}(b) \\cap E_{d-k-1} \\ne \\emptyset \\right\\rbrace \\right| \\le 100 N.$ Let $\\tilde{\\Theta }^{(a)}_{k+1}:=\\Theta ^{(a)}_{k+1} \\setminus \\left\\lbrace b \\in \\Theta ^{(a)}_{k+1}:Q_{n_{k+1}}(b) \\cap E_{d-k-1} \\ne \\emptyset \\right\\rbrace $ .",
"Then $|\\tilde{\\Theta }^{(a)}_{k+1}|\\ge |\\Theta ^{(a)}_{k+1}|-100N \\ge \\frac{1}{2} \\beta \\left(\\frac{n^{\\prime }_{k+1}}{n_{k+1}}\\right)^{\\alpha }$ , when $C_{\\varepsilon , N}$ is large enough; and for each $b \\in \\tilde{\\Theta }^{(a)}_{k+1}$ , $Q_{n_{k+1}}(b) \\cap E_{i} \\ne \\emptyset $ implies $i \\le d-k-2$ .",
"Then $\\left|\\bigcup _{a \\in \\Theta ^{\\prime (a_0)}_{k+1}} \\tilde{\\Theta }^{(a)}_{k+1}\\right| = \\sum _{a \\in \\Theta ^{\\prime (a_{0})}_{k+1}} |\\tilde{\\Theta }^{(a)}_{k+1}| \\ge \\left(\\frac{\\beta }{2}\\right)^{2} \\left(\\frac{n_{k}}{n_{k+1}}\\right)^{\\alpha }.$ Now let $\\Theta _{k+1}:=\\bigcup _{a_{0} \\in \\Theta _{k}} \\bigcup _{a \\in \\Theta ^{\\prime (a_0)}_{k+1}} \\tilde{\\Theta }^{(a)}_{k+1}$ .",
"Then Condition 2 to 6 hold for $k+1$ obviously.",
"As for Condition 1, $|\\Theta _{k+1}| = \\sum _{a_0 \\in \\Theta _{k}} \\left|\\bigcup _{a \\in \\Theta ^{\\prime (a_0)}_{k+1}} \\tilde{\\Theta }^{(a)}_{k+1}\\right| \\ge |\\Theta _{k}| \\left(\\frac{\\beta }{2}\\right)^{2} \\left(\\frac{n_{k}}{n_{k+1}}\\right)^{\\alpha } \\ge \\left(\\frac{\\beta }{2}\\right)^{2k+4} \\left(\\frac{n}{n_{k+1}}\\right)^{\\alpha },$ where the second inequality is true since Condition 1 holds for $k$ .",
"Inductively, we have constructed $\\Theta _{M}$ such that $|\\Theta _{M}| \\ge \\left(\\frac{\\beta }{2}\\right)^{2M+2} \\left(\\frac{n}{n_{M}}\\right)^{\\alpha }$ .",
"For any $a \\in \\Theta _{M}$ , we have $|u(a)| \\ge (K+11)^{-24 (M+1) n}$ .",
"For any $a,a^{\\prime } \\in \\Theta _{M}$ with $a \\ne a^{\\prime }$ , we have $Q_{n_{M}}(a) \\cap Q_{n_{M}}(a^{\\prime })=\\emptyset $ .",
"For any $a \\in \\Theta _{M}$ , we have $Q_{n_{M}}(a) \\subset Q_n$ .",
"For any $a \\in \\Theta _{M}$ and $d-M\\le i\\le d$ , we have $E_{i} \\cap Q_{n_{M}}(a) = \\emptyset $ .",
"As $l_{d-k-1}^{1+2\\varepsilon } \\le l_{d-k}$ for each $0 \\le k < M$ , we have $n_{M} \\le l_d^{\\left(\\frac{1}{1+2\\varepsilon }\\right)^{M}}\\le n^{\\left(\\frac{1}{1+2\\varepsilon }\\right)^{M}}$ .",
"Note that $n_{M} > n^{\\frac{1}{3}(1-4\\varepsilon )}$ , thus $\\left(\\frac{1}{1+2\\varepsilon }\\right)^{M} \\ge \\frac{1}{3}(1-4\\varepsilon )$ .",
"From this we have $M < 2 \\varepsilon ^{-1} .$ Since $l_{d-M-1}^{1+2\\varepsilon } \\le l_{d-M}$ and $l_{d-M}\\ge l_1 \\ge C_{\\varepsilon , N}$ we have $l_{d-M-1}< n_M^{1-\\varepsilon }$ when $C_{\\varepsilon , N}$ is large enough.",
"Then for each $a \\in \\Theta _{M}$ , by Condition 5 we have that $E$ is $(1,2\\varepsilon )$ -normal in $Q_{n_{M}}(a)$ .",
"For any $a \\in \\Theta _{M}$ , we apply Theorem REF to $Q_{n_{M}}(a)$ , then $\\left|\\left\\lbrace b\\in Q_{n_M}(a): |u(b)| \\ge (K+11)^{-24 (M+1) n} \\exp (-C_{2}n^{3}_{M})\\right\\rbrace \\setminus E\\right| \\ge C_{3} \\frac{n_{M}^{2}}{\\log (n_{M})}.$ Let $C_{\\varepsilon ,K}=C_{2}+96 \\log (K+11) \\varepsilon ^{-1}$ .",
"From (REF ), (REF ) and $n^{\\frac{1}{3}(1-4\\varepsilon )}<n_{M}<n^{\\frac{1}{3}}$ , we have $\\left|\\left\\lbrace b\\in Q_{n_M}(a): |u(b)| \\ge \\exp (-C_{\\varepsilon ,K}n )\\right\\rbrace \\setminus E\\right| \\ge C_{3}\\frac{n_{M}^{2}}{\\log (n_{M})}.$ Since $Q_{n_{M}}(a) \\cap Q_{n_{M}}(a^{\\prime })=\\emptyset $ when $a \\ne a^{\\prime } \\in \\Theta _{M}$ , in total we have $\\left|\\left\\lbrace b\\in Q_{n}: |u(b)| \\ge \\exp (-C_{\\varepsilon ,K}n )\\right\\rbrace \\setminus E\\right| \\ge C_{3}\\frac{n_{M}^{2}}{\\log (n_{M})}|\\Theta _{M}|\\\\\\ge C_{3} \\left(\\frac{\\beta }{2}\\right)^{2M+2} n^{\\frac{2}{3}(1-4\\varepsilon )+\\frac{2}{3}\\alpha }(\\log (n_{M}))^{-1} \\ge n^{p},$ where the last inequality holds by taking $\\varepsilon $ small enough, and then $C_{\\varepsilon ,N}$ large enough (recall that we require $n>C_{\\varepsilon ,N}^{4}$ ).",
"tocsectionReferences Auxiliary lemmas for the framework In our general framework several results from [10] are used, and some of them are also used in Appendix below as well.",
"For the convenience of readers we record them here.",
"There are a couple of results from linear algebra.",
"The first of them is an estimate on the number of almost orthonormal vectors, which appears in [25] as well as [10].",
"Lemma 7.1 ([25][10]) Assume $v_{1},\\cdots ,v_{m} \\in \\mathbb {R}^{n}$ such that $|v_{i}\\cdot v_{j}-{1}_{i=j}|\\le (5n)^{-\\frac{1}{2}}$ , then $m\\le \\frac{5-\\sqrt{5}}{2}n$ .",
"The second one is about the variation of eigenvalues.",
"Lemma 7.2 ([10]) Suppose the real symmetric $n\\times n$ matrix $A$ has eigenvalues $\\lambda _{1}\\ge \\cdots \\ge \\lambda _{n}\\in \\mathbb {R}$ with orthonormal eigenbasis $v_{1},\\cdots ,v_{n}\\in \\mathbb {R}^{n}$ .",
"If $1\\le i\\le j\\le n$ , $1\\le k\\le n$ $0<r_{1}<r_{2}<r_{3}<r_{4}<r_{5}<1$ $r_{1}\\le c\\min \\lbrace r_{3}r_{5},r_{2}r_{3}/r_{4}\\rbrace $ where $c>0$ is a universal constant $0<\\lambda _{j}\\le \\lambda _{i}<r_{1}<r_{2}<\\lambda _{i-1}$ $v^{2}_{j,k}\\ge r_{3}$ $\\sum _{r_{2}<\\lambda _{\\ell }<r_{5}} v_{\\ell ,k}^{2}\\le r_{4}$ then the $i$ -th largest eigenvalue $\\lambda ^{\\prime }_{i}$ (counting with multiplicity) of $A+e_{k} e_{k}^{\\dag }$ is at least $r_{1}$ , where $e_{k}$ is the $k$ -th standard basis element and $e_{k}^{\\dag }$ is its transpose.",
"We then state the generalized Sperner's theorem, used in the proof of our 3D Wegner estimate (Lemma REF ).",
"Theorem 7.3 ([10]) Suppose $\\rho \\in (0,1]$ , and $\\mathcal {A}$ is a set of subsets of $\\lbrace 1,\\cdots ,n\\rbrace $ satisfying the following.",
"For every $A\\in \\mathcal {A}$ , there is a set $B(A)\\subset \\lbrace 1,\\cdots ,n\\rbrace \\setminus A$ such that $|B(A)|\\ge \\rho (n-|A|)$ , and $A^{\\prime }\\cap B(A)=\\emptyset $ for any $A\\subset A^{\\prime }\\in \\mathcal {A}$ .",
"Then $|\\mathcal {A}|\\le 2^{n}n^{-\\frac{1}{2}}\\rho ^{-1}.$ For the next several results, in [10] they are stated and proved in the 2D lattice setting, but the proofs work, essentially verbatim, in the 3D setting.",
"The following covering lemma is used in the multi-scale analysis.",
"Recall that by “dyadic” we mean an integer power of 2.",
"Lemma 7.4 ([10]) There is a constant $C>1$ such that following holds.",
"Suppose $K\\ge 1$ is an integer, $\\alpha \\ge C^{K}$ is a dyadic scale, $L_{0}\\ge \\alpha L_{1}\\ge L_{1}\\ge \\alpha L_{2}\\ge L_{2}$ are dyadic scales, $Q\\subset \\mathbb {Z}^{3}$ is an $L_{0}$ -cube, and $Q^{\\prime \\prime }_{1},\\cdots Q^{\\prime \\prime }_{K}\\subset Q$ are $L_{2}$ -cubes.",
"Then there is a dyadic scale $L_{3}\\in [L_{1},\\alpha L_{1}]$ and disjoint $L_{3}$ -cubes $Q^{\\prime }_{1},\\cdots ,Q^{\\prime }_{K}\\subset Q$ , such that for each $Q^{\\prime \\prime }_{k}$ there is $Q^{\\prime }_{j}$ with $Q^{\\prime \\prime }_{k}\\subset Q^{\\prime }_{j}$ and $\\operatorname{dist}(Q^{\\prime \\prime }_{k},Q\\setminus Q^{\\prime }_{j})\\ge \\frac{1}{8}L_{3}$ .",
"We need the following continuity of resolvent estimate.",
"It is stated in a slightly different way from [10], so we add a proof here.",
"Lemma 7.5 ([10]) If for $\\lambda \\in \\mathbb {R}$ , $\\alpha >\\beta >0$ , and a cube $Q\\subset \\mathbb {Z}^{3}$ , we have $|(H_{Q}-\\lambda )^{-1}(a,b)|\\le \\exp (\\alpha -\\beta |a-b|) \\text{ for $a,b\\in Q$},$ then for $\\lambda ^{\\prime }$ with $|\\lambda ^{\\prime }-\\lambda |\\le \\frac{1}{2} |Q|^{-1} \\exp (-\\alpha )$ , we have $ |(H_{Q}-\\lambda ^{\\prime })^{-1}(a,b)|\\le 2\\exp (\\alpha -\\beta |a-b|) \\text{ for $a,b\\in Q$}.$ We first prove (REF ) assuming $\\lambda ^{\\prime }$ is not an eigenvalue of $H_Q$ .",
"By resolvent identity we have, $(H_{Q}-\\lambda ^{\\prime })^{-1}=(H_{Q}-\\lambda )^{-1} + (H_{Q}-\\lambda ^{\\prime })^{-1}(\\lambda ^{\\prime }-\\lambda )(H_{Q}-\\lambda )^{-1}.$ Let $\\gamma =\\max _{a,b\\in Q} \\exp (\\beta |a-b|-\\alpha ) |(H_{Q}-\\lambda ^{\\prime })^{-1}(a,b)|$ .",
"Then for any $a,b\\in Q$ , $\\begin{split} &|(H_{Q}-\\lambda ^{\\prime })^{-1}(a,b)|\\\\ \\le &|(H_{Q}-\\lambda )^{-1}(a,b)|+|\\lambda ^{\\prime }-\\lambda |\\sum _{c\\in Q}|(H_{Q}-\\lambda ^{\\prime })^{-1}(a,c)||(H_{Q}-\\lambda )^{-1}(c,b)|\\\\ \\le &\\exp (\\alpha -\\beta |a-b|)+|\\lambda ^{\\prime }-\\lambda |\\sum _{c \\in Q} \\exp (\\alpha -\\beta |a-c|) \\exp (\\alpha -\\beta |c-b|) \\gamma \\\\ \\le &\\exp (\\alpha -\\beta |a-b|)+|\\lambda ^{\\prime }-\\lambda ||Q|\\exp (2\\alpha -\\beta |a-b|) \\gamma \\\\ \\le &\\exp (\\alpha -\\beta |a-b|)+\\frac{1}{2}\\exp (\\alpha -\\beta |a-b|) \\gamma .\\end{split}$ This implies $\\gamma \\le 1+\\frac{1}{2}\\gamma $ and thus $\\gamma \\le 2$ and (REF ) follows.",
"Now we can deduce that $|\\det (H_{Q}-\\lambda ^{\\prime })^{-1}|$ is uniformly bounded for $\\lambda ^{\\prime }$ that is not an eigenvalue of $H_Q$ and satisfies $|\\lambda ^{\\prime }-\\lambda |\\le \\frac{1}{2} |Q|^{-1} \\exp (-\\alpha )$ .",
"By continuity of the determinant (as a function of $\\lambda ^{\\prime }$ ), we conclude that $H_{Q}$ has no eigenvalue in $\\left[\\lambda -\\frac{1}{2} |Q|^{-1} \\exp (-\\alpha ),\\lambda +\\frac{1}{2} |Q|^{-1} \\exp (-\\alpha )\\right]$ .",
"Thus our conclusion follows.",
"We also need the following result to deduce exponential decay of the resolvent in a cube from the decay of the resolvent in subcubes.",
"Lemma 7.6 ([10]) Suppose $\\varepsilon >\\delta >0$ are small, $K\\ge 1$ is an integer and $\\lambda \\in [0,13]$ , $L_{0}\\ge \\cdots \\ge L_{6}$ are large enough (depending on $\\varepsilon ,\\delta ,K$ ) with $L_{k}^{1-\\varepsilon }\\ge L_{k+1}$ , $1\\ge m\\ge 2L_{5}^{-\\delta }$ represents the exponential decay rate, $Q\\subset \\mathbb {Z}^{3}$ is an $L_{0}$ -cube, $Q^{\\prime }_{1},\\cdots ,Q^{\\prime }_{K}\\subset Q$ are disjoint $L_{2}$ -cubes with $\\Vert (H_{Q^{\\prime }_{k}}-\\lambda )^{-1}\\Vert \\le \\exp (L_{4})$ , for all $a\\in Q$ , one of the following holds there is $Q^{\\prime }_{k}$ with $a\\in Q^{\\prime }_{k}$ and $\\operatorname{dist}(a,Q\\setminus Q^{\\prime }_{k})\\ge \\frac{1}{8}L_{2}$ there is an $L_{5}$ -cube $Q^{\\prime \\prime }\\subset Q$ such that $a\\in Q^{\\prime \\prime }$ , $\\operatorname{dist}(a,Q\\setminus Q^{\\prime \\prime })\\ge \\frac{1}{8}L_{5}$ , and $|(H_{Q^{\\prime \\prime }}-\\lambda )^{-1}(b,b^{\\prime })|\\le \\exp (L_{6}-m|b-b^{\\prime }|)$ for $b,b^{\\prime }\\in Q^{\\prime \\prime }$ .",
"Then $|(H_{Q}-\\lambda )^{-1}(a,a^{\\prime })|\\le \\exp (L_{1}-\\tilde{m}|a-a^{\\prime }|)$ for $a,a^{\\prime }\\in Q$ where $\\tilde{m}=m-L_{5}^{-\\delta }$ .",
"The principal eigenvalue This appendix sets up the base case in the induction proof of Theorem REF .",
"We follow [10], and generalize their result to higher dimensions.",
"We take $d \\in \\mathbb {Z}$ , $d>2$ , and denote $Q_{n}:=\\left\\lbrace a \\in \\mathbb {Z}^d:\\Vert a\\Vert _{\\infty } \\le n\\right\\rbrace $ instead.",
"Theorem 8.1 Let $\\overline{V}:Q_{n}\\rightarrow \\left[0,1\\right]$ be any potential function, and $R>0$ large enough, such that for any $a \\in Q_n$ , there exists $b\\in Q_n$ with $\\overline{V}(b)=1$ and $|a-b|<R$ .",
"Let $\\overline{H}:\\ell ^2(Q_n) \\rightarrow \\ell ^2(Q_n)$ , $\\overline{H}=-\\Delta +\\overline{V}$ , with Dirichlet boundary condition.",
"Then its principal eigenvalue is no less than $CR^{-d}$ , where $C$ is a constant depending only on $d$ .",
"Let $\\lambda _0$ denote the principal eigenvalue, then by e.g.",
"[12] we have $\\lambda _{0}=\\sup _{u:Q_{n} \\rightarrow \\mathbb {R}_{+}} \\min _{Q_{n}} \\frac{\\overline{H}u}{u}.$ Hence we lower bound $\\lambda _0$ by constructing a function $u$ .",
"Let $\\tilde{G}:\\mathbb {Z}^d \\rightarrow \\mathbb {R}$ be the lattice Green's function; i.e.",
"for any $a \\in \\mathbb {Z}^d$ , $\\tilde{G}(a)$ is the expected number of times that a (discrete time) simple random walk starting at $\\mathbf {0}$ gets to $a$ .",
"Let $G:=\\tilde{G}/2d$ .",
"Then $G$ is the only function such that $-\\Delta G= \\delta _{\\mathbf {0}}$ (where $\\delta _{\\mathbf {0}}(\\mathbf {0})=1$ and $\\delta _{\\mathbf {0}}(a)=0$ for $a \\ne \\mathbf {0}$ ), and $0 \\le G(a) \\le G(\\mathbf {0})$ for any $a\\in \\mathbb {Z}^d$ .",
"In addition, for any $a \\in \\mathbb {Z}^d$ with $a \\ne \\mathbf {0}$ , by e.g.",
"[20] we have $G(a)=\\frac{C_{d}}{|a|^{d-2}}+O\\left(\\frac{1}{|a|^{d}}\\right),$ where $C_d$ is a constant depending only on $d$ .",
"Hence $\\frac{4 C_{d}}{5 |a|^{d-2}} \\le G(a) \\le \\frac{3 C_{d}}{2 |a|^{d-2}}$ when $|a|$ is large enough.",
"We define $u:\\mathbb {Z}^d \\rightarrow \\mathbb {R}_+$ as $u(a):=1 + G(\\mathbf {0}) - G(a) -\\varepsilon _{d} R^{-d} |a|^2, \\; \\forall a \\in \\mathbb {Z}^d,$ where $\\varepsilon _{d} > 0$ is a small enough constant depending on $d$ .",
"Then $ -\\Delta u =-\\delta _{\\mathbf {0}}+2d\\varepsilon _{d} R^{-d},$ and for any $a \\in \\mathbb {Z}^d$ with $|a|<3R$ , we have $0 < u(a) \\le 1+G(\\mathbf {0}) $ .",
"Assume that $R$ is large enough.",
"For any $a$ with $2R<|a|<3R$ , we have $u(a) \\ge 1+G(\\mathbf {0})-\\frac{3C_{d}}{2 (2R)^{d-2}}-9 \\varepsilon _{d} R^{-d+2}$ ; and for any $a$ with $|a|<R$ , $u(a) \\le 1+G(\\mathbf {0})-\\frac{4 C_{d}}{5 R^{d-2}} \\le 1+G(\\mathbf {0})-\\frac{3 C_{d}}{2 (2R)^{d-2}}-9\\varepsilon _{d} R^{-d+2} $ , as long as $\\varepsilon _{d}<\\frac{C_{d}}{180}$ (also note that here we have $d>2$ ).",
"Thus $ \\min _{2R<|a|<3R} u(a) \\ge \\max _{|a|<R} u(a)$ Now we define $u_0 : Q_n \\rightarrow \\mathbb {R}_+$ , as $u_{0}(a):=\\min _{|a-b|<3R,\\overline{V}(b)=1} u(a-b),\\; \\forall a \\in Q_n$ .",
"Pick an arbitrary $a^{\\prime } \\in Q_n$ , by (REF ) there is $b^{\\prime }$ with $|a^{\\prime }-b^{\\prime }| \\le 2R$ such that $u_{0}(a^{\\prime })=u(a^{\\prime }-b^{\\prime })$ and $\\overline{V}(b^{\\prime })=1$ .",
"For any $a^{\\prime \\prime } \\in Q_{n}$ with $|a^{\\prime \\prime }-a^{\\prime }|=1$ , since $|a^{\\prime \\prime }-b^{\\prime }| \\le 2R+1 <3R$ , we have $u_{0}(a^{\\prime \\prime })=\\min _{|a^{\\prime \\prime }-b|<3R,\\overline{V}(b)=1} u(a^{\\prime \\prime }-b) \\le u(a^{\\prime \\prime }-b^{\\prime }).$ Thus by (REF ), and Dirichlet boundary condition, $ \\begin{split}\\overline{H} u_{0} (a^{\\prime })= &2d u_{0}(a^{\\prime })-\\sum _{a^{\\prime \\prime }\\in Q_{n}, |a^{\\prime }-a^{\\prime \\prime }|=1} u_{0}(a^{\\prime \\prime }) +\\overline{V}(a^{\\prime })u_{0}(a^{\\prime })\\\\\\ge &2d u(a^{\\prime }-b^{\\prime })-\\sum _{a^{\\prime \\prime }\\in Q_{n}, |a^{\\prime }-a^{\\prime \\prime }|=1} u(a^{\\prime \\prime }-b^{\\prime }) +\\overline{V}(a^{\\prime })u(a^{\\prime }-b^{\\prime })\\\\\\ge &-\\Delta u(a^{\\prime }-b^{\\prime }) +\\overline{V}(a^{\\prime })u(a^{\\prime }-b^{\\prime })\\\\= &-\\delta _{\\mathbf {0}}(a^{\\prime }-b^{\\prime })+ 2d\\varepsilon _{d}R^{-d} +\\overline{V}(a^{\\prime })u(a^{\\prime }-b^{\\prime }) \\\\\\ge &2d\\varepsilon _{d}R^{-d}.\\end{split}$ Since $a^{\\prime }$ is arbitrary and $0<u_{0}(a^{\\prime })\\le 1+G(\\mathbf {0})$ , by (REF ) and letting $C=\\frac{2d \\varepsilon _{d}}{1+G(\\mathbf {0})}$ , we have $\\lambda _{0} \\ge C R^{-d}$ .",
"Remark 8.2 The exponent in $R^{-d}$ is optimal.",
"Consider a potential $\\overline{V}$ such that $\\overline{V}(a)=1$ only if $a \\in \\lceil R \\rceil \\mathbb {Z}^{d} \\cap Q_{n}$ and $\\overline{V}(a)=0$ otherwise.",
"In this case we have that $\\lambda _{0} \\le 8d R^{-d}+4d n^{-1}$ .",
"To see this, consider the test function $\\phi (a)=1-\\overline{V}(a)$ for $a \\in Q_{n}$ and use the variational principle $\\lambda _{0} \\le \\frac{\\langle \\phi , \\overline{H} \\phi \\rangle }{\\Vert \\phi \\Vert _{2}^{2}}$ .",
"Corollary 8.3 Let $\\overline{H}$ , $C$ be defined as in Theorem REF .",
"Let $0\\le \\lambda <\\frac{CR^{-d}}{2} $ .",
"Then $\\Vert (\\overline{H}-\\lambda )^{-1}\\Vert \\le \\frac{2R^{d}}{C}$ and $|(\\overline{H}-\\lambda )^{-1}(a,b)| \\le \\frac{2R^{d}}{C} \\exp \\left(-\\frac{C R^{-d}}{8d+2} |a-b|\\right)$ for any $a,b \\in Q_{n}$ .",
"As the principal eigenvalue of $\\overline{H}$ is no less than $C R^{-d}$ , we have $\\Vert (\\overline{H}-\\lambda )^{-1}\\Vert \\le \\frac{2R^{d}}{C} $ .",
"Let $T:= I - \\frac{1}{4 d +1} (\\overline{H}-\\lambda ) $ .",
"Since any eigenvalue of $\\overline{H}$ is in $\\left[C R^{-d},4d+1\\right]$ , the eigenvalues of $T$ are in $\\left[0, 1 - \\frac{C}{8d+2} R^{-d}\\right]$ , so $\\Vert T\\Vert \\le 1 - \\frac{C}{8d+2} R^{-d}$ .",
"Note that for each $i>0$ and $a,b \\in Q_{n}$ , $T^{i}(a,b)=0$ if $|a-b|>i$ .",
"Then we have $|(\\overline{H}-\\lambda )^{-1}(a,b)|=(4d+1)^{-1}|(I-T)^{-1}(a,b)|\\le (4d+1)^{-1} \\sum _{i \\ge 0}|T^{i}(a,b)|\\\\= (4d+1)^{-1} \\sum _{i \\ge |a-b|}|T^{i}(a,b)|\\le (4d+1)^{-1} \\sum _{i \\ge |a-b|} \\Vert T\\Vert ^{i}\\le \\frac{2R^{d}}{C} \\exp \\left(-\\frac{C R^{-d}}{8d+2} |a-b|\\right),$ so the corollary follows.",
"Finally, we have the following result, which implies the base case in the induction proof of Theorem REF .",
"Proposition 8.4 Let $d=3$ , and $V$ be the Bernoulli potential, i.e.",
"$\\mathbb {P}(V(a)=0)=\\mathbb {P}(V(a)=1)=\\frac{1}{2}$ for each $a \\in \\mathbb {Z}^3$ independently.",
"For any $0<\\delta <\\frac{1}{10}$ and $\\varepsilon >0$ , there exists $C_{\\delta ,\\varepsilon }$ such that for any $n>C_{\\delta ,\\varepsilon }$ and $0\\le \\lambda < \\frac{Cn^{-\\frac{3 \\delta }{10}}}{2}$ , with probability at least $1-n^{-1}$ the following is true.",
"Take any $V^{\\prime }:\\mathbb {Z}^3\\rightarrow [0,1]$ such that $V^{\\prime }_{Q_{n} \\cap \\lceil \\varepsilon ^{-1}\\rceil \\mathbb {Z}^{3}} = V_{Q_{n} \\cap \\lceil \\varepsilon ^{-1}\\rceil \\mathbb {Z}^{3}}$ .",
"Let $H^{\\prime }_{Q_n}$ be the restriction of $-\\Delta +V^{\\prime }$ on $Q_n$ with Dirichlet boundary condition.",
"Then we have $\\Vert (H^{\\prime }_{Q_{n}}-\\lambda )^{-1}\\Vert \\le \\exp (n^{2\\delta }),$ and $ \\text{$|(H^{\\prime }_{Q_{n}}-\\lambda )^{-1}(a,b)| \\le n^{2\\delta } \\exp (-n^{-\\delta }|a-b|)$ for any $a,b \\in Q_{n}$}.$ Let $R:=n^{\\frac{\\delta }{10}}$ , and let $A$ denote the following event: $\\forall a \\in Q_{n}, \\exists b \\in Q_{n}\\cap \\lceil \\varepsilon ^{-1}\\rceil \\mathbb {Z}^{3}, \\; \\text{s.t.}",
"\\; |a-b| \\le R, V(b)=1.$ Then $A$ only depends on $V_{Q_{n} \\cap \\lceil \\varepsilon ^{-1}\\rceil \\mathbb {Z}^{3}}$ .",
"Using Corollary REF with $d=3$ , we have that (REF ) and (REF ) hold under the event $A$ , when $n$ is large enough.",
"Finally, since there are $(2n+1)^{3}$ points in $Q_{n}$ , and inside each ball of radius $R$ , there are at least $\\frac{1}{8}n^{\\frac{3\\delta }{10}} \\varepsilon ^{3}$ points in $\\lceil \\varepsilon ^{-1}\\rceil \\mathbb {Z}^{3} \\cap Q_{n}$ , we have $\\mathbb {P}(A^{c}) \\le (2n+1)^{3} 2^{- \\frac{1}{8} n^{\\frac{3\\delta }{10}} \\varepsilon ^{3}} \\le n^{-1}$ , when $n$ is large enough.",
"Deducing Anderson localization from the resolvent estimate The arguments in this appendix originally come from [4] (see also [15] and [6]).",
"These previous works are about the continuous space model.",
"For completeness and for the reader's convenience, we adapt the arguments for the lattice model, thus deducing Theorem REF from Theorem REF .",
"As in Section , in this appendix, by “dyadic” we mean an integer power of 2, and by “dyadic cube”, we mean a cube $Q_{2^{n}}(a)$ for some $a \\in 2^{n-1}\\mathbb {Z}^3$ and $n \\in \\mathbb {Z}_{+}$ .",
"For any $k \\in \\mathbb {Z}_{+}$ , we define $\\Omega _{k}:=\\lbrace u:\\mathbb {Z}^{3}\\rightarrow \\mathbb {R}: |u(a)|\\le k (1+|a|)^{k},\\;\\; \\forall a\\in \\mathbb {Z}^{3},\\;\\; \\text{and}\\;\\; u(\\mathbf {0})=1\\rbrace .$ Since the law of $H$ is invariant under translation, to prove Theorem REF , it suffices to show that for any $k\\in \\mathbb {Z}_{+}$ , almost surely $\\inf _{t>0} \\sup _{a \\in \\mathbb {Z}^3} \\exp (t|a|) |u(a)|<\\infty ,$ for any $u\\in \\Omega _{k}$ and $\\lambda \\in [0, \\lambda _*]$ with $H u=\\lambda u$ .",
"Denote $\\mathcal {I}=(0,\\lambda _{*})$ .",
"We first see that it suffices to prove (REF ) for any $u\\in \\Omega _{k}$ and $\\lambda \\in \\mathcal {I}$ with $H u=\\lambda u$ , by applying the following lemma to $\\lambda =0$ and $\\lambda =\\lambda _{*}$ .",
"Lemma 9.1 Suppose $\\lambda \\in [0,\\lambda _{*}]$ and $k\\in \\mathbb {Z}_{+}$ .",
"Then almost surely, there is no $u\\in \\Omega _k$ with $H u =\\lambda u$ .",
"Let $L_{i}=2^{i}$ for $i\\in \\mathbb {Z}_{+}$ .",
"By Theorem REF and the Borel-Cantelli lemma, almost surely, there exists $i^{\\prime }>0$ , such that for any $i>i^{\\prime }$ , $\\left|(H_{Q_{L_{i}}}-\\lambda )^{-1}(a,b)\\right| \\le \\exp \\left(L_{i}^{1-\\lambda _{*}}-\\lambda _{*} |a-b|\\right), \\;\\forall a,b \\in Q_{L_{i}}.$ Assume there exists $u\\in \\Omega _k$ with $H u=\\lambda u$ .",
"For each large enough $i$ we have $|u(\\mathbf {0})|=\\left|\\sum _{\\begin{array}{c}a\\in Q_{L_{i}},a^{\\prime }\\in \\mathbb {Z}^{3}\\setminus Q_{L_{i}}\\\\|a-a^{\\prime }|=1\\end{array}} (H_{Q_{L_{i}}}-\\lambda )^{-1}(\\mathbf {0},a) u(a^{\\prime })\\right|\\le 6\\cdot (2L_i +1)^2\\exp \\left(-\\frac{\\lambda _{*}L_{i}}{2}\\right)k(1+\\sqrt{3}L_{i})^{k}$ which converges to zero as $i\\rightarrow \\infty $ .",
"Thus $u(\\mathbf {0})=0$ , which contradicts with the fact that $u\\in \\Omega _{k}$ .",
"Let us fix $k\\in \\mathbb {Z}_{+}$ and denote by $\\sigma _k(H)$ the set of all $\\lambda \\in \\mathcal {I}$ , such that $Hu=\\lambda u$ for some $u \\in \\Omega _k$ .",
"For each $L\\in \\mathbb {Z}_{+}$ , denote by $\\sigma (H_{Q_{L}})$ the set of eigenvalues of $H_{Q_{L}}$ .",
"The first key step is to prove that for any large enough $L$ , with high probability, the distance between any $\\lambda \\in \\sigma _k(H)$ and $\\sigma (H_{Q_{L}})$ is small, exponentially in $L$ .",
"Proposition 9.2 There exist $\\kappa ^{\\prime },c_1 >0$ such that for any dyadic $L$ large enough, we can find a $V_{Q_{L}}$ -measurable event $\\mathcal {E}_{wloc}^{(L)}$ , such that $\\mathbb {P}\\left[\\mathcal {E}_{wloc}^{(L)}\\right] \\ge 1-L^{-\\kappa ^{\\prime }},$ and under the event $\\mathcal {E}_{wloc}^{(L)}$ , we have $\\operatorname{dist}(\\lambda , \\sigma (H_{Q_{L}})\\cap \\mathcal {I}) \\le \\exp (-c_1 L)$ for any $\\lambda \\in \\sigma _k(H)\\cap \\left[\\exp (-c_1\\sqrt{L}),\\lambda _{*}-\\exp (-c_1\\sqrt{L})\\right].$ The next key step is to strengthen Proposition REF so that each $\\lambda \\in \\sigma _k(H)$ is not only exponentially close to $\\sigma (H_{Q_{L}})$ , but also exponentially close to a finite subset $S\\subset \\sigma (H_{Q_{L}})$ with $|S|<L^{\\delta ^{\\prime }}$ for arbitrarily small $\\delta ^{\\prime }$ .",
"Proposition 9.3 For any $\\delta ^{\\prime }>0$ , there exist $\\kappa ^{\\prime \\prime },c_2 >0$ such that for each dyadic $L$ large enough (depending on $\\delta ^{\\prime }$ ), we can find a $V_{Q_{L}}$ -measurable event $\\mathcal {E}_{sloc}^{(L)}$ with $\\mathbb {P}\\left[\\mathcal {E}_{sloc}^{(L)}\\right] \\ge 1-L^{-\\kappa ^{\\prime \\prime }},$ and under the event $\\mathcal {E}_{sloc}^{(L)}$ , there exists a finite set $S\\subset \\sigma (H_{Q_L})\\cap \\mathcal {I}$ with $|S|<L^{\\delta ^{\\prime }}$ such that $\\operatorname{dist}(\\lambda , S) \\le \\exp (-c_2 L)$ for any $\\lambda \\in \\sigma _k(H) \\cap \\left[\\exp (-L^{c_2}),\\lambda _{*}-\\exp (-L^{c_2}) \\right]$ .",
"Proposition REF and REF are discrete versions of [6] and [6] respectively.",
"See also [15].",
"Now we leave the proofs of these two propositions to the next two subsections, and prove localization assuming them.",
"[Proof of Theorem REF ] We apply Proposition REF with $\\delta ^{\\prime }<\\kappa _{0}$ where $\\kappa _{0}$ is the constant in Theorem REF .",
"Take large enough dyadic $L$ , and consider the annulus $A_{L}=Q_{5L}\\setminus Q_{2L}$ .",
"We cover $A_L$ by $2L$ -cubes $\\lbrace Q^{(j)}:1\\le j\\le 1000\\rbrace $ that are disjoint with $Q_{L}$ , such that for each $a\\in A_{L}$ there is $1\\le j\\le 1000$ with $a\\in Q^{(j)}$ and $\\operatorname{dist}(a,\\mathbb {Z}^{3}\\setminus Q^{(j)})\\ge \\frac{1}{8}L$ .",
"Apply Theorem REF to each of $Q^{(j)}$ 's and to each energy $\\lambda \\in S\\subset \\sigma (H_{Q_L})\\cap \\mathcal {I}$ , we have $\\mathbb {P}\\left[\\mathcal {E}^{(L)}_{ann}\\big |\\; \\mathcal {E}_{sloc}^{(L)}\\right]\\ge 1-1000L^{\\delta ^{\\prime }-\\kappa _{0}}$ where $\\mathcal {E}^{(L)}_{ann}$ denotes the event: $\\left|(H_{Q^{(j)}}-\\lambda )^{-1}(a,b)\\right| \\le \\exp \\left(L^{1-\\lambda _{*}}-\\lambda _{*} |a-b|\\right), \\;\\forall 1\\le j\\le 1000,\\;\\forall a,b \\in Q^{(j)},\\;\\text{and }\\forall \\lambda \\in S.$ Then by Proposition REF we have $ \\mathbb {P}\\left[\\mathcal {E}^{(L)}_{ann}\\cap \\mathcal {E}_{sloc}^{(L)}\\right]\\ge (1-L^{-\\kappa ^{\\prime \\prime }})(1-1000L^{\\delta ^{\\prime }-\\kappa _{0}})\\ge 1-L^{-\\kappa ^{\\prime \\prime \\prime }},$ for some constant $\\kappa ^{\\prime \\prime \\prime }>0$ and large enough $L$ .",
"Under the event $\\mathcal {E}^{(L)}_{ann}\\cap \\mathcal {E}_{sloc}^{(L)}$ , we take any $u\\in \\Omega _{k}$ with $H u=\\lambda u$ and $\\lambda \\in [\\exp (-L^{c_2}),\\lambda _{*}-\\exp (-L^{c_2})]$ , and $\\lambda ^{\\prime }\\in S$ with $|\\lambda -\\lambda ^{\\prime }|<\\exp (-c_2 L)$ .",
"Thus using Lemma REF , we have $ \\begin{split}\\Vert u\\Vert _{\\ell ^{\\infty }(A_{L})}&\\le 2\\exp \\left(L^{1-\\lambda _{*}}-\\frac{1}{8}\\lambda _{*} L\\right) \\Vert u\\Vert _{\\ell ^{1}(Q_{6L})}\\\\&\\le 2\\exp \\left(L^{1-\\lambda _{*}}-\\frac{1}{8}\\lambda _{*} L\\right)k(6\\sqrt{3}L+1)^{k} (12L+1)^{3} \\le \\exp (-c^{\\prime } L)\\end{split}$ for some constant $c^{\\prime }<\\frac{\\lambda _{*}}{8}$ and large enough $L$ .",
"Now we consider the event $\\mathcal {E}_{loc}=\\bigcup _{i^{\\prime }\\ge 0} \\bigcap _{i\\ge i^{\\prime }} (\\mathcal {E}^{(2^{i})}_{ann}\\cap \\mathcal {E}_{sloc}^{(2^{i})}).$ We have $\\mathbb {P}[\\mathcal {E}_{loc}]=1$ by (REF ).",
"Note that for any $\\lambda \\in \\mathcal {I}$ , we have $\\lambda \\in [\\exp (-L^{c_2 }),\\lambda _{*}-\\exp (-L^{c_2})]$ for large enough $L$ .",
"We also have that $\\bigcup _{i\\ge i^{\\prime }}A_{2^{i}}=\\mathbb {Z}^{3}\\setminus Q_{2^{i^{\\prime }+1}}$ for any $i^{\\prime }\\in \\mathbb {Z}_{+}$ .",
"By (REF ) we have that (REF ) holds under the event $\\mathcal {E}_{loc}$ .",
"Then localization is proved.",
"The first spectral reduction For simplicity of notations, for any $\\lambda \\in \\mathbb {R}$ , dyadic scale $L$ , and $a\\in \\mathbb {Z}^3$ , we say $Q_{L}(a)$ is $\\lambda $ -good if $\\left|(H_{Q_L(a)}-\\lambda )^{-1}(b,b^{\\prime })\\right| \\le \\exp \\left(L^{1-\\lambda _{*}}-\\lambda _{*} |b-b^{\\prime }|\\right), \\;\\forall b,b^{\\prime } \\in Q_{L}(a).$ Otherwise, we call it $\\lambda $ -bad.",
"By Theorem REF , for any large enough dyadic scale $L$ and $\\lambda \\in [0, \\lambda _{*}]$ , we have $\\mathbb {P}[\\text{$Q_{L}(a)$ is $\\lambda $-bad}]\\le L^{-\\kappa _{0}}.$ [Proof of Proposition REF ] Throughout the proof, we use $C$ to denote large universal constants.",
"For a dyadic scale $L$ , we construct a graph $G_{L}$ whose vertices are all the dyadic $2L$ -cubes.",
"The edges are given as follows: for any $a\\ne a^{\\prime }\\in \\frac{L}{2}\\mathbb {Z}^{3}$ , there is an edge connecting $Q_{L}(a)$ and $Q_{L}(a^{\\prime })$ if and only if $Q_{L}(a)\\cap Q_{L}(a^{\\prime })\\ne \\emptyset $ .",
"Fix large dyadic scale $L$ .",
"Take the dyadic scale $L_{0}\\in \\left\\lbrace \\sqrt{L},\\sqrt{2L}\\right\\rbrace $ .",
"For any $\\lambda \\in \\mathcal {I}$ , denote by $\\mathcal {E}^{\\lambda }_{per}$ the event that there is a path of $\\lambda $ -bad $2L_0$ -cubes $\\overline{Q}_{1},\\cdots ,\\overline{Q}_{m}$ in $G_{L_{0}}$ such that $\\text{$\\overline{Q}_{1}\\cap Q_{\\frac{L}{2}}\\ne \\emptyset $ and $\\overline{Q}_{m}\\cap Q_{L}=\\emptyset $}.$ Under the event $\\mathcal {E}^{\\lambda }_{per}$ , suppose that $\\Gamma _{0}=(\\overline{Q}_{1},\\cdots ,\\overline{Q}_{m})$ is such a path with the shortest length.",
"Since $\\operatorname{dist}(Q_{\\frac{L}{2}},\\mathbb {Z}^{3}\\setminus Q_{L})\\ge \\frac{L}{2}$ , we have $m\\ge \\frac{L}{4\\sqrt{3}L_{0}}$ .",
"By definition of dyadic cubes and that $\\Gamma _{0}$ has the shortest length, there are at least $\\frac{m}{1000}$ disjoint $\\lambda $ -bad cubes in $\\Gamma _{0}$ .",
"Hence, $\\mathbb {P}[\\mathcal {E}^{\\lambda }_{per}] \\le \\sum _{m\\ge \\frac{L}{4\\sqrt{3}L_{0}}} C L^{3} 1000^{m}(L_{0}^{-\\kappa _{0}})^{\\frac{m}{1000}}\\le 2C L^{3} (1000 L_{0}^{-\\frac{\\kappa _{0}}{1000}})^{\\frac{L}{4\\sqrt{3}L_{0}}}\\le L_{0}^{-c^{\\prime }L_{0}}$ for some $c^{\\prime }>0$ .",
"Here the first inequality is by (REF ), and counting the total number of $G_{L_0}$ paths with length $m$ and one end intersecting $Q_{\\frac{L}{2}}$ .",
"Claim 9.4 Under the event $(\\mathcal {E}^{\\lambda }_{per})^{c}$ , any $\\lambda ^{\\prime }\\in \\sigma _{k}(H)$ with $|\\lambda ^{\\prime }-\\lambda |\\le \\exp (- L^{1-\\frac{\\lambda _{*}}{2}}_{0})$ satisfies $\\operatorname{dist}(\\lambda ^{\\prime },\\sigma (H_{Q_{\\frac{3}{2}L}}))\\le \\exp (-\\epsilon ^{\\prime } L_{0})$ for a universal constant $\\epsilon ^{\\prime }>0$ .",
"Denote the set of all the $\\lambda $ -bad $L_{0}$ -cubes contained in $Q_{\\frac{3}{2}L}$ by $\\mathcal {S}$ .",
"We consider $\\mathbb {Z}^{3}$ as a graph with edges between nearest neighbors.",
"Consider the set $S_{0}:=(\\bigcup \\mathcal {S})\\cup Q_{\\frac{L}{2}}\\subset Q_{\\frac{3}{2}L}$ .",
"Let $S_{1}$ be the maximal connected component of $S_{0}$ which contains $Q_{\\frac{L}{2}}$ .",
"Then $(\\mathcal {E}^{\\lambda }_{per})^{c}$ implies $S_{1}\\subset Q_{L+2L_{0}}$ .",
"Denote $\\partial ^{-}S_{1}=\\lbrace a\\in S_{1}:\\text{$|a-a^{\\prime }|=1$ for some $a^{\\prime }\\in \\mathbb {Z}^{3}\\setminus S_{1}$}\\rbrace ,$ and $\\partial ^{+}S_{1}=\\lbrace a\\in \\mathbb {Z}^{3}\\setminus S_{1}:\\text{$|a-a^{\\prime }|=1$ for some $a^{\\prime }\\in S_{1}$}\\rbrace .$ Assume $\\lambda ^{\\prime }$ satisfies the hypothesis in the claim, then there is $u\\in \\Omega _{k}$ such that $H u= \\lambda ^{\\prime } u$ .",
"For any $a^{\\prime }\\in \\partial ^{-}S_{1}\\cup \\partial ^{+}S_{1}$ , there is a dyadic $L_{0}$ -cube $Q^{\\prime }$ such that $a^{\\prime }\\in Q^{\\prime }$ and $\\operatorname{dist}(a^{\\prime },\\mathbb {Z}^{3}\\setminus Q^{\\prime })\\ge \\frac{1}{8}L_{0}$ .",
"By maximality of $S_{1}$ , we have $Q^{\\prime }$ is $\\lambda $ -good.",
"Thus by Lemma REF , $\\begin{split}|u(a^{\\prime })|\\le &2\\exp (L_{0}^{1-\\lambda _{*}}-\\frac{1}{8}\\lambda _{*} L_{0})\\Vert u\\Vert _{\\ell ^{1}(Q_{L+4L_{0}})}\\\\\\le &2\\exp (L_{0}^{1-\\lambda _{*}}-\\frac{1}{8}\\lambda _{*} L_{0})(2L+8L_{0}+1)^{3}k(\\sqrt{3}L+4\\sqrt{3}L_{0}+1)^{k}\\\\\\le &\\exp (-\\frac{1}{10}\\lambda _{*} L_{0})\\end{split}$ for large enough $L_{0}$ .",
"Let $u_{*}:Q_{\\frac{3}{2}L}\\rightarrow \\mathbb {R}$ be defined by $u_{*}=u$ on $S_{1}$ and $u_{*}=0$ on $Q_{\\frac{3}{2}L}\\setminus S_{1}$ .",
"Then $(H_{Q_{\\frac{3}{2}L}}-\\lambda ^{\\prime }) u_{*}(a)={\\left\\lbrace \\begin{array}{ll}0 & \\text{if\\;} a\\in Q_{\\frac{3}{2}L}\\setminus (\\partial ^{-}S_{1}\\cup \\partial ^{+}S_{1}),\\\\\\sum _{|a^{\\prime }-a|=1,a^{\\prime }\\in \\partial ^{+}S_{1}} u(a^{\\prime }) & \\text{if\\;} a\\in \\partial ^{-}S_{1},\\\\-\\sum _{|a^{\\prime }-a|=1,a^{\\prime }\\in \\partial ^{-}S_{1}} u(a^{\\prime }) &\\text{if\\;} a\\in \\partial ^{+}S_{1}.\\end{array}\\right.",
"}$ By (REF ), we have $\\Vert (H_{Q_{\\frac{3}{2}L}}-\\lambda ^{\\prime }) u_{*}\\Vert _{\\ell ^{2}(Q_{\\frac{3}{2}L})}\\le 6(3L+1)^{\\frac{3}{2}}\\exp (-\\frac{1}{10}\\lambda _{*} L_{0})\\le \\exp (-\\epsilon ^{\\prime }L_{0})\\Vert u_{*}\\Vert _{\\ell ^{2}(Q_{\\frac{3}{2}L})}$ for large enough $L$ .",
"Here, we used $\\Vert u_{*}\\Vert _{\\ell ^{2}(Q_{\\frac{3}{2}L})}\\ge 1$ since $\\mathbf {0}\\in S_{1}$ and $u(\\mathbf {0})=1$ .",
"By expanding $u_{*}$ into a linear combination of eigenvectors of $H_{Q_{\\frac{3}{2}L}}$ , (REF ) guarantees that there is an eigenvalue $\\lambda _{0}$ of $H_{Q_{\\frac{3}{2}L}}$ such that $|\\lambda ^{\\prime }-\\lambda _{0}|\\le \\exp (-\\epsilon ^{\\prime }L_{0})$ .",
"Our claim follows.",
"Denote $\\lambda ^{(h)}=h\\exp (-L_{0})$ for $h\\in \\mathbb {Z}_{+}$ and let $\\mathcal {E}^{0}_{trap}=\\bigcap _{\\lambda ^{(h)}\\in \\mathcal {I}} (\\mathcal {E}^{\\lambda ^{(h)}}_{per})^{c}.$ Then by (REF ), $\\mathbb {P}[\\mathcal {E}^{0}_{trap}]\\ge 1-\\lambda _{*}\\exp (L_{0})L^{-c^{\\prime } L_{0}}_{0}\\ge 1-L^{-10}$ for large $L$ .",
"Claim 9.5 Under the event $\\mathcal {E}^{0}_{trap}$ , any $\\lambda \\in [0, \\lambda _{*}]\\cap \\sigma _{k}(H)$ satisfies $\\operatorname{dist}(\\lambda ,\\sigma (H_{Q_{\\frac{3}{2}L}}))\\le \\exp (-\\epsilon ^{\\prime } L_{0}).$ For any $\\lambda \\in [0, \\lambda _{*}]$ , there exists an $h\\in \\mathbb {Z}_{+}$ such that $\\lambda ^{(h)}\\in \\mathcal {I}$ and $|\\lambda -\\lambda ^{(h)}|\\le \\exp (-L_{0}^{1-\\frac{\\lambda _{*}}{2}})$ .",
"Our claim follows from Claim REF .",
"Let $q$ be the smallest positive integer such that $2^{\\frac{1}{q}}-1<\\frac{\\lambda _{*}}{2}$ and let $\\tau =2^{\\frac{1}{q}}-1$ .",
"Define $\\tilde{L}_{1}=L_{0}^{1+\\tau }$ and $\\tilde{L}_{i+1}=\\tilde{L}_{i}^{1+\\tau }$ for $i=1,2,\\cdots ,q-1$ .",
"Then $L\\le \\tilde{L}_{q}=L_{0}^{2}\\le 2L$ .",
"Let $L_i$ be the (unique) dyadic scale such that $L_i\\in [\\tilde{L}_i, 2\\tilde{L}_i)$ for each $i=1,\\cdots ,q$ .",
"Let $M_{i}=\\frac{3}{2}L+C^{\\prime }\\sum _{1\\le j\\le i}L_{j}$ for each $i=1,\\cdots ,q$ and $M_{0}=\\frac{3}{2}L$ .",
"Here $C^{\\prime }$ is a large constant to be determined.",
"Then $M_{i}\\le \\frac{3}{2}L + 4C^{\\prime } i L\\le \\left(\\frac{3}{2}+4C^{\\prime } q\\right)L$ for each $0\\le i\\le q$ .",
"In addition, we denote $M_{q+1}=2^wL$ where $w$ is the smallest integer with $2^w>3+8C^{\\prime }q$ , and let $L_{q+1}=L_q$ .",
"For any $\\lambda \\in \\mathcal {I}$ and any $j\\in \\lbrace 1,\\cdots ,q+1\\rbrace $ , denote by $\\mathcal {E}^{\\lambda ,j}_{per}$ the following event: there exists a path of $\\lambda $ -bad $2L_j$ -cubes in $G_{L_{j}}$ , say $\\overline{Q}_{1},\\cdots ,\\overline{Q}_{m}$ , such that $\\begin{split}&\\overline{Q}_{i}\\subset Q_{M_{j}}\\setminus Q_{M_{j-1}}, \\; \\forall i\\in \\lbrace 1,\\cdots ,m\\rbrace ,\\\\&\\overline{Q}_{1}\\cap Q_{M_{j-1}+10L_{j}}\\ne \\emptyset ,\\\\&\\overline{Q}_{m}\\cap Q_{M_{j}-10L_{j}}\\ne \\emptyset .\\end{split}$ Under the event $\\mathcal {E}^{\\lambda ,j}_{per}$ , suppose that $\\Gamma _{0}=(\\overline{Q}_{1},\\cdots ,\\overline{Q}_{m})$ in $G_{L_{j}}$ is such a path with the shortest length.",
"Since $\\operatorname{dist}(Q_{M_{j-1}+10L_{j}},\\mathbb {Z}^{3}\\setminus Q_{M_{j}-10L_{j}})\\ge (C^{\\prime }-20) L_{j}$ , we have $m\\ge \\frac{C^{\\prime }}{4}$ when $C^{\\prime }$ is large enough.",
"By definition of dyadic cubes and that $\\Gamma _{0}$ has the shortest length, there are at least $\\frac{m}{1000}$ disjoint $\\lambda $ -bad cubes in $\\Gamma _{0}$ .",
"Hence, $\\mathbb {P}[\\mathcal {E}^{\\lambda ,j}_{per}] \\le \\sum _{m\\ge \\frac{C^{\\prime }}{4}}C (C^{\\prime } L)^{3} 1000^{m}(L_{j}^{-\\kappa _{0}})^{\\frac{m}{1000}}\\le 2C (C^{\\prime } L)^{3} (1000 L_{j}^{-\\frac{\\kappa _{0}}{1000}})^{\\frac{C^{\\prime }}{4}}\\le L^{-10}.$ Here the first inequality is by (REF ) and counting the number of paths in $G_{L_j}$ with length $m$ and one end intersecting $Q_{M_{j-1}+10L_j}$ , and the last inequality is by taking $C^{\\prime }$ large enough.",
"By adapting the proof of Claim REF we can get the following result.",
"Claim 9.6 Under the event $(\\mathcal {E}^{\\lambda ,j}_{per})^{c}$ , any $\\lambda ^{\\prime }\\in \\sigma _{k}(H)$ with $|\\lambda ^{\\prime }-\\lambda |\\le \\exp (- L_{j}^{1-\\frac{\\lambda _{*}}{2}})$ satisfies $\\operatorname{dist}(\\lambda ^{\\prime },\\sigma (H_{Q_{M_{j}}} ))\\le \\exp (-\\epsilon ^{\\prime \\prime } L_{j})$ for a universal constant $\\epsilon ^{\\prime \\prime }>0$ .",
"Note that, given $\\lambda \\in \\mathcal {I}$ , the event $\\mathcal {E}^{\\lambda ,j}_{per}$ is $V_{Q_{M_{j}}\\setminus Q_{M_{j-1}}}$ -measurable.",
"Hence, the event $\\mathcal {E}^{j}_{trap}:=\\left(\\bigcup _{\\lambda \\in \\sigma (H_{Q_{M_{j-1}}})\\cap \\mathcal {I}} \\mathcal {E}^{\\lambda ,j}_{per}\\right)^{c}$ satisfies $\\mathbb {P}[\\mathcal {E}^{j}_{trap}| V_{Q_{M_{j-1}}}]\\ge 1-(M_{j-1}+1)^{3}L^{-10} \\ge 1-L^{-6}$ by (REF ) and (REF ) for large enough $L$ .",
"For each $0\\le j\\le q+1$ , $\\mathcal {E}^{j}_{trap}$ is $V_{Q_{M_{j}}}$ -measurable, thus the event $\\mathcal {E}_{trap}:=\\bigcap _{0\\le j\\le q+1}\\mathcal {E}^{j}_{trap}$ is $V_{Q_{M_{q+1}}}$ -measurable.",
"By (REF ) and (REF ), we have $\\mathbb {P}[\\mathcal {E}_{trap}]\\ge 1- (q+2) L^{-6}\\ge 1-L^{-5}.$ Claim 9.7 Under the event $\\mathcal {E}_{trap}$ , any $\\lambda \\in [\\exp (-\\epsilon ^{\\prime \\prime \\prime } L_{0}/2),\\lambda _{*}-\\exp (-\\epsilon ^{\\prime \\prime \\prime } L_{0}/2)]\\cap \\sigma _{k}(H)$ satisfies $\\operatorname{dist}(\\lambda ,\\sigma (H_{Q_{M_{q+1}}}))\\le \\exp (-\\epsilon ^{\\prime \\prime \\prime } L)$ for some $\\epsilon ^{\\prime \\prime \\prime }>0$ .",
"Let $\\epsilon ^{\\prime \\prime \\prime }=\\min \\lbrace \\epsilon ^{\\prime },\\epsilon ^{\\prime \\prime }\\rbrace $ .",
"Let $\\lambda \\in [\\exp (-\\epsilon ^{\\prime \\prime \\prime } L_{0}/2),\\lambda _{*}-\\exp (-\\epsilon ^{\\prime \\prime \\prime } L_{0}/2)]\\cap \\sigma _{k}(H)$ .",
"We inductively prove that, $\\operatorname{dist}(\\lambda ,\\sigma (H_{Q_{L_{j}}}))\\le \\exp (-\\epsilon ^{\\prime \\prime \\prime } L_{j})$ for any $0\\le j\\le q+1$ .",
"Thus, in particular, we have $\\operatorname{dist}(\\lambda ,\\sigma (H_{Q_{M_{q+1}}}))\\le \\exp (-\\epsilon ^{\\prime \\prime \\prime } L_{q+1})\\le \\exp (-\\epsilon ^{\\prime \\prime \\prime } L),$ and the claim follows.",
"For the case $j=0$ , by Claim REF , $\\operatorname{dist}(\\lambda ,\\sigma (H_{Q_{M_{0}}}))\\le \\exp (-\\epsilon ^{\\prime } L_{0})$ .",
"Assume the conclusion holds for some $j<q+1$ , then $|\\lambda -\\lambda _{0}|\\le \\exp (-\\epsilon ^{\\prime \\prime \\prime } L_{j})$ for some $\\lambda _{0}\\in \\sigma (H_{Q_{M_{j}}})$ .",
"As $\\lambda \\in [\\exp (-\\epsilon ^{\\prime \\prime \\prime } L_{0}/2),\\lambda _{*}-\\exp (-\\epsilon ^{\\prime \\prime \\prime } L_{0}/2)]$ , we must have $\\lambda _0 \\in \\mathcal {I}$ .",
"Since $\\tau <\\frac{\\lambda _{*}}{2}$ , for $L$ large enough we have $\\epsilon ^{\\prime \\prime \\prime }L_{j} > L_{j+1}^{1-\\frac{\\lambda _{*}}{2}}$ and $|\\lambda -\\lambda _{0}|\\le \\exp (- L_{j+1}^{1-\\frac{\\lambda _{*}}{2}})$ .",
"Thus Claim REF implies $\\operatorname{dist}(\\lambda ,\\sigma (H_{Q_{M_{j+1}}}))\\le \\exp (-\\epsilon ^{\\prime \\prime } L_{j+1})$ .",
"Finally, since $M_{q+1}=2^wL$ and $w$ is a constant, the proposition follows from Claim REF and (REF ).",
"The second spectral reduction For any positive integers $L^{\\prime \\prime }>L^{\\prime }$ , we denote the annulus $A_{L^{\\prime \\prime },L^{\\prime }}=Q_{L^{\\prime \\prime }}\\setminus Q_{L^{\\prime }}$ .",
"Take any $\\delta >0$ .",
"For $\\lambda \\in \\mathcal {I}$ and $L^{\\prime \\prime }>2L^{\\prime }$ , let $\\mathcal {E}_{L^{\\prime \\prime },L^{\\prime }}^{(\\lambda )}$ denote the following event: there exists a subset $G^{(\\lambda )}_{L^{\\prime \\prime },L^{\\prime }}\\subset A_{L^{\\prime \\prime },L^{\\prime }}$ with $|G^{(\\lambda )}_{L^{\\prime \\prime },L^{\\prime }}|\\le (L^{\\prime })^{\\frac{\\delta }{2}}$ such that, for any $a\\in A_{L^{\\prime \\prime },2L^{\\prime }}\\setminus G^{(\\lambda )}_{L^{\\prime \\prime },L^{\\prime }}$ , there is a $\\lambda $ -good cube $Q_{L^{\\prime \\prime \\prime }}(b)\\subset A_{L^{\\prime \\prime },L^{\\prime }}$ such that $\\operatorname{dist}(a,Q_{L^{\\prime \\prime }}\\setminus Q_{L^{\\prime \\prime \\prime }}(b))\\ge \\frac{1}{8}L^{\\prime \\prime \\prime }$ , and $(L^{\\prime })^{\\frac{\\delta }{10}}\\le L^{\\prime \\prime \\prime }\\le L^{\\prime }$ .",
"Note that, $\\mathcal {E}^{(\\lambda )}_{L^{\\prime \\prime },L^{\\prime }}$ is $V_{A_{L^{\\prime \\prime },L^{\\prime }}}$ -measurable.",
"Lemma 9.8 Let $\\varepsilon , \\delta >0$ be small enough.",
"Suppose $L^{\\prime },L^{\\prime \\prime }$ are dyadic, satisfying $(L^{\\prime })^{1+\\frac{1}{2}\\varepsilon }<L^{\\prime \\prime }<(L^{\\prime })^{1+\\varepsilon }$ , and $L^{\\prime }$ is large enough (depending on $\\varepsilon , \\delta $ ).",
"Then for any $\\lambda \\in \\mathcal {I}$ we have $\\mathbb {P}[\\mathcal {E}_{L^{\\prime \\prime },L^{\\prime }}^{(\\lambda )}]\\ge 1-(L^{\\prime })^{-10}$ .",
"Let $\\tilde{L}^{(0)}=L^{\\prime }$ , $\\tilde{L}^{(i+1)}=(\\tilde{L}^{(i)})^{1-\\varepsilon }$ , and $L^{(i)}$ be the (unique) dyadic scale with $L^{(i)}\\in [\\tilde{L}^{(i)}, 2\\tilde{L}^{(i)})$ , for $i\\in \\mathbb {Z}_{\\ge 0}$ .",
"Let $M^{\\prime }\\in \\mathbb {Z}_{+}$ such that $\\frac{1}{10}\\delta <(1-\\varepsilon )^{M^{\\prime }}<\\frac{1}{6}\\delta $ .",
"For any dyadic $2L^{(M^{\\prime })}$ -cube $Q\\subset A_{L^{\\prime \\prime },L^{\\prime }}$ , we call it hereditary bad if there are $\\lambda $ -bad dyadic cubes $Q^{(0)},\\cdots ,Q^{(M^{\\prime })}=Q$ such that, $Q^{(i+1)}\\subset Q^{(i)}\\subset A_{L^{\\prime \\prime },L^{\\prime }}$ for each $0\\le i\\le M^{\\prime }-1$ and $Q^{(i)}$ is a dyadic $2L^{(i)}$ -cube.",
"By (REF ), and the same arguments in the proof of Claim REF , the following is true.",
"For small enough $\\varepsilon $ , there exists $N\\in \\mathbb {Z}_{+}$ depending on $\\varepsilon ,\\delta $ , such that with probability at least $1-(L^{\\prime })^{-10}$ , $|\\lbrace Q\\subset A_{L^{\\prime \\prime },L^{\\prime }}:\\text{$Q$ is a hereditary bad $2L^{(M^{\\prime })}$-cube}\\rbrace |<N.$ Let $G^{(\\lambda )}_{L^{\\prime \\prime },L^{\\prime }}=\\bigcup \\lbrace Q\\subset A_{L^{\\prime \\prime },L^{\\prime }}:\\text{$Q$ is a hereditary bad $2L^{(M^{\\prime })}$-cube}\\rbrace $ .",
"Then (REF ) implies $|G^{(\\lambda )}_{L^{\\prime \\prime },L^{\\prime }}|\\le N (2L^{(M^{\\prime })}+1)^{3}\\le (L^{\\prime })^{\\frac{\\delta }{2}}$ for large enough $L^{\\prime }$ .",
"For each $a\\in A_{L^{\\prime \\prime },2L^{\\prime }}\\setminus G^{(\\lambda )}_{L^{\\prime \\prime },L^{\\prime }}$ , there is $0\\le i^{\\prime }\\le M^{\\prime }$ and a $\\lambda $ -good cube $Q_{L^{(i^{\\prime })}}(b)\\subset A_{L^{\\prime \\prime },L^{\\prime }}$ such that $\\operatorname{dist}(a,Q_{L^{\\prime \\prime }}\\setminus Q_{L^{(i^{\\prime })}}(b))\\ge \\frac{1}{8}L^{(i^{\\prime })}$ .",
"Since $(L^{\\prime })^{\\frac{\\delta }{10}}\\le L^{(i^{\\prime })}\\le L^{\\prime }$ , our claim follows.",
"For any large enough dyadic scales $L^{\\prime },L^{\\prime \\prime }$ with $(L^{\\prime })^{1+\\frac{1}{2}\\varepsilon }<L^{\\prime \\prime }<(L^{\\prime })^{1+\\varepsilon }$ , we denote $\\mathcal {E}_{L^{\\prime \\prime },L^{\\prime }}^{supp}=\\bigcap _{\\lambda \\in \\sigma (H_{Q_{L^{\\prime }}})\\cap \\mathcal {I}}\\mathcal {E}_{L^{\\prime \\prime },L^{\\prime }}^{(\\lambda )}$ .",
"Then by Lemma REF , as each $\\mathcal {E}_{L^{\\prime \\prime },L^{\\prime }}^{(\\lambda )}$ is $V_{A_{L^{\\prime \\prime },L^{\\prime }}}$ -measurable, we have $\\mathbb {P}[\\mathcal {E}_{L^{\\prime \\prime },L^{\\prime }}^{supp}]\\ge 1-(L^{\\prime })^{-6}.$ [Proof of Proposition REF ] In this proof we let $\\varepsilon >0$ be a small universal constant, and $\\delta >0$ be a number depending on $\\delta ^{\\prime }$ .",
"Both of them are to be determined.",
"Now we fix dyadic scale $L$ large enough (depending on $\\epsilon ,\\delta $ and thus depending on $\\delta ^{\\prime }$ ).",
"Let $\\tilde{L}_{0}=L$ , $\\tilde{L}_{i+1}=\\tilde{L}^{1-\\frac{3}{4}\\varepsilon }_{i}$ , and $L_{i}$ be the (unique) dyadic scale with $L_{i}\\in [\\tilde{L}_{i}, 2\\tilde{L}_{i})$ , for $i\\in \\mathbb {Z}_{\\ge 0}$ .",
"Pick $M\\in \\mathbb {Z}_{+}$ such that $\\frac{1}{10}\\delta <(1-\\frac{3}{4}\\varepsilon )^{M}< \\frac{1}{6}\\delta $ .",
"Write $\\overline{L_{i}}=\\frac{1}{16} L_{i}$ for $0\\le i\\le M$ and let $\\mathcal {E}^{supp}=\\bigcap _{0\\le i\\le M-1} \\mathcal {E}^{supp}_{L_{i},\\overline{L_{i+1}}}.$ Then by (REF ), $\\mathbb {P}[\\mathcal {E}^{supp}]\\ge 1-M \\left(\\frac{L_{M}}{16}\\right)^{-6}\\ge 1-L^{-\\frac{\\delta }{2}}$ as $L$ is large enough.",
"For $0\\le i\\le M$ , denote by $\\Theta _{i}$ the set of eigenvalues $\\lambda \\in \\sigma (H_{Q_{L_{i}}})$ such that, $\\lambda \\in [(M-i+1)\\exp (-L^{\\frac{\\delta }{20}}),\\lambda _{*}-(M-i+1)\\exp (-L^{\\frac{\\delta }{20}})],$ and $\\operatorname{dist}(\\lambda ,\\sigma (H_{Q_{\\overline{L_{j}}}})),\\operatorname{dist}(\\lambda ,\\sigma (H_{Q_{L_{j}}})) \\le 2^{i}\\exp (-c^{\\prime } L_{j}) \\quad \\forall j\\in \\lbrace i,i+1,\\cdots ,M\\rbrace .$ Here the constant $c^{\\prime }=\\frac{c_1}{20}$ where $c_1$ is the constant from Proposition REF .",
"Claim 9.9 Under the event $\\mathcal {E}^{supp}$ , for any $1\\le i\\le M$ and $\\lambda \\in \\Theta _{i}$ , there exists $G^{(i-1)}\\subset Q_{L_{i-1}}$ with $10\\le |G^{(i-1)}|\\le L^{\\frac{2}{3}\\delta }$ such that the following holds.",
"For any $\\lambda ^{\\prime }\\in \\sigma (H_{Q_{L_{i-1}}})$ and $u\\in \\ell ^{2}(Q_{L_{i-1}})$ with $|\\lambda -\\lambda ^{\\prime }|\\le 2^{i-1}\\exp (-c^{\\prime } L_{i})$ and $H_{Q_{L_{i-1}}} u=\\lambda ^{\\prime } u$ , we have $\\Vert u\\Vert _{\\ell ^{2}(G^{(i-1)})}\\ge (1-|G^{(i-1)}|^{-2})\\Vert u\\Vert _{\\ell ^{2}(Q_{L_{i-1}})}$ .",
"Since $\\lambda \\in \\Theta _{i}$ , there are $\\lambda ^{(j)}\\in \\sigma (H_{Q_{\\overline{L_{j}}}})$ such that $|\\lambda -\\lambda ^{(j)}|\\le 2^{i}\\exp (-c^{\\prime } L_{j})$ for each $i\\le j\\le M$ .",
"Let $G^{(i-1)}_{*}=\\bigcup _{i-1\\le j\\le M-1} G^{(\\lambda ^{(j+1)})}_{L_{j},\\overline{L_{j+1}}}.$ Then $|G^{(i-1)}_{*}|\\le M L^{\\frac{\\delta }{2}}$ .",
"Suppose $\\lambda ^{\\prime }$ and $u$ satisfy the hypothesis.",
"Then $|\\lambda ^{\\prime }-\\lambda ^{(j)}|\\le |\\lambda ^{\\prime }-\\lambda |+|\\lambda -\\lambda ^{(j)}|\\le 2^{i-1}\\exp (-c^{\\prime } L_{i})+2^{i}\\exp (-c^{\\prime } L_{j})\\le 2^{M+1}\\exp (-c^{\\prime } L_{j})$ for each $i\\le j\\le M$ .",
"Denote $L^{\\prime }_{j}=\\frac{1}{2}L_{j}$ for each $i\\le j\\le M-1$ and $L^{\\prime }_{i-1}=L_{i-1}$ .",
"Pick an arbitrary $a\\in Q_{L_{i-1}}\\setminus Q_{L_{M}}$ , there exists $j^{\\prime }\\in \\lbrace i-1,\\cdots ,M-1\\rbrace $ such that $a\\in A_{L^{\\prime }_{j^{\\prime }},2\\overline{L_{j^{\\prime }+1}}}$ .",
"If $a\\notin G^{(i-1)}_{*}$ , by definition of $G^{\\lambda ^{(j^{\\prime }+1)}}_{L_{j^{\\prime }},\\overline{L_{j^{\\prime }+1}}}$ , there exists a $\\lambda ^{(j^{\\prime }+1)}$ -good cube $Q_{L^{\\prime \\prime \\prime }}(b)$ such that $\\overline{L_{j^{\\prime }+1}}\\ge L^{\\prime \\prime \\prime }\\ge \\overline{L_{j^{\\prime }+1}}^{\\frac{\\delta }{10}}\\ge L^{\\frac{\\delta ^{2}}{100}}$ , and $\\operatorname{dist}(a,Q_{L_{j^{\\prime }}}\\setminus Q_{L^{\\prime \\prime \\prime }}(b))\\ge \\frac{1}{8}L^{\\prime \\prime \\prime }$ .",
"Then since $a\\in Q_{L^{\\prime }_{j^{\\prime }}}$ , we have $\\operatorname{dist}(a,Q_{L_{i-1}}\\setminus Q_{L^{\\prime \\prime \\prime }}(b))\\ge \\frac{1}{8}L^{\\prime \\prime \\prime }$ .",
"We also have that $|\\lambda ^{\\prime }-\\lambda ^{(j^{\\prime }+1)}|\\le 2^{M+1}\\exp (-c^{\\prime } L_{j^{\\prime }+1})\\le 2^{M+1}\\exp (-16 c^{\\prime } L^{\\prime \\prime \\prime }).$ Then by Claim REF we have, $|u(a)|\\le 2\\exp \\left((L^{\\prime \\prime \\prime })^{1-\\lambda _{*}}-\\frac{1}{8}\\lambda _{*}L^{\\prime \\prime \\prime }\\right) \\Vert u\\Vert _{\\ell ^{1}(Q_{L_{i-1}})}\\le L^{-10} \\Vert u\\Vert _{\\ell ^{2}(Q_{L_{i-1}})}.$ Hence, by letting $G^{(i-1)}=G^{(i-1)}_{*}\\cup Q_{L_{M}}$ , we have $10\\le |G^{(i-1)}|\\le |G^{(i-1)}_{*}|+ |Q_{L_{M}}|\\le ML^{\\frac{\\delta }{2}}+100L^{\\frac{\\delta }{2}}\\le L^{\\frac{2}{3}\\delta }$ , and $\\Vert u\\Vert _{\\ell ^{2}(G^{(i-1)})}\\ge \\big (1- (2L_{i-1}+1)^{3} L^{-20}\\big )^{\\frac{1}{2}} \\Vert u\\Vert _{\\ell ^{2}(Q_{L_{i-1}})} \\ge (1-|G^{(i-1)}|^{-2}) \\Vert u\\Vert _{\\ell ^{2}(Q_{L_{i-1}})}.$ Thus our claim follows.",
"Claim 9.10 Under the event $\\mathcal {E}^{supp}$ , for any $1\\le i\\le M$ and $\\lambda \\in \\Theta _{i}$ , we have $|\\lbrace \\lambda ^{\\prime }\\in \\sigma (H_{Q_{L_{i-1}}}):|\\lambda -\\lambda ^{\\prime }|\\le 2^{i-1}\\exp (-c^{\\prime } L_{i})\\rbrace |\\le 2L^{\\frac{2}{3}\\delta }.$ Let $\\lambda _{1},\\cdots ,\\lambda _{p} \\in \\sigma (H_{Q_{L_{i-1}}})$ be all the eigenvalues (counting with multiplicity) in the interval $[\\lambda -2^{i-1}\\exp (-c^{\\prime } L_{i}),\\lambda +2^{i-1}\\exp (-c^{\\prime } L_{i})].$ Let $u_{1},\\cdots ,u_{p}$ be the corresponding (mutually orthogonal) eigenvectors with $H_{Q_{L_{i-1}}}u_{s}=\\lambda _{s} u_{s}$ and $\\Vert u_{s}\\Vert _{\\ell ^{2}(Q_{L_{i-1}})}=1$ for $1\\le s\\le p$ .",
"By Claim REF , $\\Vert u_{s}\\Vert _{\\ell ^{2}(G^{(i-1)})}\\ge 1-|G^{(i-1)}|^{-2}$ for $1\\le s\\le p$ .",
"Thus we have $|\\langle u_{s_{1}},u_{s_{2}} \\rangle _{\\ell ^{2}(G^{(i-1)})}-{1}_{s_{1}=s_{2}}|\\le 2|G^{(i-1)}|^{-2}$ for $1\\le s_{1},s_{2}\\le p$ .",
"By Lemma REF , we have $p\\le 2|G^{(i-1)}|\\le 2L^{\\frac{2}{3}\\delta }$ .",
"Claim 9.11 We have $|\\Theta _{0}|\\le L^{M \\delta }$ under the event $\\mathcal {E}^{supp}$ .",
"Suppose $\\mathcal {E}^{supp}$ holds.",
"For each $1\\le i\\le M$ and $\\lambda \\in \\Theta _{i-1}$ , there are $\\lambda ^{(j)}\\in \\sigma (H_{Q_{L_{j}}})$ and $\\overline{\\lambda ^{(j)}}\\in \\sigma (H_{Q_{\\overline{L_{j}}}})$ with $|\\lambda -\\lambda ^{(j)}|, |\\lambda -\\overline{\\lambda ^{(j)}}|\\le 2^{i-1}\\exp (-c^{\\prime } L_{j})$ , for $i\\le j\\le M$ .",
"In particular, $|\\lambda -\\lambda ^{(i)}|\\le 2^{i-1}\\exp (-c^{\\prime } L_{i})$ .",
"Thus $|\\lambda ^{(i)}-\\lambda ^{(j)}|\\le 2^{i-1}(\\exp (-c^{\\prime } L_{j})+\\exp (-c^{\\prime } L_{i}))\\le 2^{i}\\exp (-c^{\\prime } L_{j})$ and similarly $|\\lambda ^{(i)}-\\overline{\\lambda ^{(j)}}|\\le 2^{i}\\exp (-c^{\\prime } L_{j})$ for $i\\le j\\le M$ .",
"Moreover, $\\lambda \\in \\Theta _{i-1}$ implies that $\\lambda \\in [(M-i+2)\\exp (-L^{\\frac{\\delta }{20}}),\\lambda _{*}-(M-i+2)\\exp (-L^{\\frac{\\delta }{20}})]$ and thus $\\lambda ^{(i)}\\in [(M-i+1)\\exp (-L^{\\frac{\\delta }{20}}),\\lambda _{*}-(M-i+1)\\exp (-L^{\\frac{\\delta }{20}})].$ These imply $\\lambda ^{(i)}\\in \\Theta _{i}$ .",
"Hence, we have $\\Theta _{i-1}\\subset \\lbrace \\lambda \\in \\sigma (H_{Q_{L_{i-1}}}):\\operatorname{dist}(\\lambda ,\\Theta _{i})\\le 2^{i-1}\\exp (-c^{\\prime } L_{i})\\rbrace .$ Together with Claim REF , we have $|\\Theta _{i-1}|\\le 2L^{\\frac{2}{3}\\delta } |\\Theta _{i}|$ for $1\\le i \\le M$ .",
"Since $|\\Theta _{M}|\\le |\\sigma (H_{Q_{L_{M}}})|\\le 10L_{M}^{3}\\le 100 L^{\\frac{\\delta }{2}}$ , we have $|\\Theta _{0}|\\le 100 L^{\\frac{\\delta }{2}}\\cdot 2^{M} L^{\\frac{2}{3} M \\delta } \\le L^{M\\delta }$ .",
"Now we denote $\\mathcal {E}_{sloc}^{(L)}=\\mathcal {E}^{supp}\\cap \\bigcap _{0\\le i\\le M} \\mathcal {E}^{(L_{i})}_{wloc}\\cap \\mathcal {E}_{wloc}^{(\\overline{L_{i}})}$ .",
"By Proposition REF and (REF ), $\\mathbb {P}[ \\mathcal {E}_{sloc}^{(L)}]\\ge 1- L^{-\\frac{\\delta }{2}} -2(M+1) L^{-\\frac{1}{20}\\kappa ^{\\prime }\\delta }\\ge 1-L^{-\\kappa ^{\\prime \\prime }}$ for some small $\\kappa ^{\\prime \\prime }>0$ depending on $\\delta ,M$ .",
"Take $c_2=\\min \\lbrace \\frac{\\delta }{30}, c^{\\prime }\\rbrace $ .",
"Under the event $\\mathcal {E}_{sloc}^{(L)}$ , for any $\\lambda \\in \\sigma _{k}(H)\\cap [\\exp (-L^{c_2}),\\lambda _{*}-\\exp (-L^{c_2})],$ we claim that $\\operatorname{dist}(\\lambda ,\\Theta _{0})\\le \\exp (-c_2 L).$ To see this, by definition of $\\mathcal {E}^{(L_{i})}_{wloc}$ and $ \\mathcal {E}_{wloc}^{(\\overline{L_{i}})}$ , (REF ) implies $\\operatorname{dist}(\\lambda ,\\sigma (H_{Q_{L_{i}}}))\\le \\exp (-c_1 L_{i})$ and $\\operatorname{dist}(\\lambda ,\\sigma (H_{Q_{\\overline{L_{i}}}})) \\le \\exp (-\\frac{c_1}{16} L_{i})$ for each $0\\le i\\le M$ .",
"In particular, there is $\\lambda _{0}\\in \\sigma (H_{Q_{L}})$ such that $|\\lambda -\\lambda _{0}|\\le \\exp (-c_1 L)$ .",
"Since $c^{\\prime }=\\frac{c_1}{20}$ , we have $\\lambda _{0}\\in \\left[(M+1)\\exp (-L^{\\frac{\\delta }{20}}),\\lambda _{*}-(M+1)\\exp (-L^{\\frac{\\delta }{20}})\\right]$ by (REF ), and also $\\begin{split}&\\operatorname{dist}(\\lambda _{0},\\sigma (H_{Q_{L_{i}}}))\\le |\\lambda -\\lambda _{0}|+\\operatorname{dist}(\\lambda ,\\sigma (H_{Q_{L_{i}}}))\\le \\exp (-c_1 L)+\\exp (-c_1 L_{i})\\le \\exp (-c^{\\prime } L_{i}),\\\\&\\operatorname{dist}(\\lambda _{0},\\sigma (H_{Q_{\\overline{L_{i}}}}))\\le |\\lambda -\\lambda _{0}|+\\operatorname{dist}(\\lambda ,\\sigma (H_{Q_{\\overline{L_{i}}}}))\\le \\exp (-c_1 L)+\\exp (-\\frac{c_1}{16} L_{i})\\le \\exp (-c^{\\prime } L_{i}),\\end{split}$ for $0\\le i\\le M$ .",
"Hence $\\lambda _{0}\\in \\Theta _{0}$ and (REF ) follows.",
"Finally, observe that $|\\Theta _{0}|\\le L^{M \\delta }\\le L^{\\frac{\\log (\\frac{1}{10}\\delta )}{\\log (1-\\frac{3}{4}\\varepsilon )} \\delta }\\le L^{\\delta ^{\\prime }}$ by taking $\\delta $ small enough (depending on $\\delta ^{\\prime }$ ), the proposition follows by letting $S=\\Theta _{0}$ ."
],
[
"Auxiliary lemmas for the framework",
"In our general framework several results from [10] are used, and some of them are also used in Appendix below as well.",
"For the convenience of readers we record them here.",
"There are a couple of results from linear algebra.",
"The first of them is an estimate on the number of almost orthonormal vectors, which appears in [25] as well as [10].",
"Lemma 7.1 ([25][10]) Assume $v_{1},\\cdots ,v_{m} \\in \\mathbb {R}^{n}$ such that $|v_{i}\\cdot v_{j}-{1}_{i=j}|\\le (5n)^{-\\frac{1}{2}}$ , then $m\\le \\frac{5-\\sqrt{5}}{2}n$ .",
"The second one is about the variation of eigenvalues.",
"Lemma 7.2 ([10]) Suppose the real symmetric $n\\times n$ matrix $A$ has eigenvalues $\\lambda _{1}\\ge \\cdots \\ge \\lambda _{n}\\in \\mathbb {R}$ with orthonormal eigenbasis $v_{1},\\cdots ,v_{n}\\in \\mathbb {R}^{n}$ .",
"If $1\\le i\\le j\\le n$ , $1\\le k\\le n$ $0<r_{1}<r_{2}<r_{3}<r_{4}<r_{5}<1$ $r_{1}\\le c\\min \\lbrace r_{3}r_{5},r_{2}r_{3}/r_{4}\\rbrace $ where $c>0$ is a universal constant $0<\\lambda _{j}\\le \\lambda _{i}<r_{1}<r_{2}<\\lambda _{i-1}$ $v^{2}_{j,k}\\ge r_{3}$ $\\sum _{r_{2}<\\lambda _{\\ell }<r_{5}} v_{\\ell ,k}^{2}\\le r_{4}$ then the $i$ -th largest eigenvalue $\\lambda ^{\\prime }_{i}$ (counting with multiplicity) of $A+e_{k} e_{k}^{\\dag }$ is at least $r_{1}$ , where $e_{k}$ is the $k$ -th standard basis element and $e_{k}^{\\dag }$ is its transpose.",
"We then state the generalized Sperner's theorem, used in the proof of our 3D Wegner estimate (Lemma REF ).",
"Theorem 7.3 ([10]) Suppose $\\rho \\in (0,1]$ , and $\\mathcal {A}$ is a set of subsets of $\\lbrace 1,\\cdots ,n\\rbrace $ satisfying the following.",
"For every $A\\in \\mathcal {A}$ , there is a set $B(A)\\subset \\lbrace 1,\\cdots ,n\\rbrace \\setminus A$ such that $|B(A)|\\ge \\rho (n-|A|)$ , and $A^{\\prime }\\cap B(A)=\\emptyset $ for any $A\\subset A^{\\prime }\\in \\mathcal {A}$ .",
"Then $|\\mathcal {A}|\\le 2^{n}n^{-\\frac{1}{2}}\\rho ^{-1}.$ For the next several results, in [10] they are stated and proved in the 2D lattice setting, but the proofs work, essentially verbatim, in the 3D setting.",
"The following covering lemma is used in the multi-scale analysis.",
"Recall that by “dyadic” we mean an integer power of 2.",
"Lemma 7.4 ([10]) There is a constant $C>1$ such that following holds.",
"Suppose $K\\ge 1$ is an integer, $\\alpha \\ge C^{K}$ is a dyadic scale, $L_{0}\\ge \\alpha L_{1}\\ge L_{1}\\ge \\alpha L_{2}\\ge L_{2}$ are dyadic scales, $Q\\subset \\mathbb {Z}^{3}$ is an $L_{0}$ -cube, and $Q^{\\prime \\prime }_{1},\\cdots Q^{\\prime \\prime }_{K}\\subset Q$ are $L_{2}$ -cubes.",
"Then there is a dyadic scale $L_{3}\\in [L_{1},\\alpha L_{1}]$ and disjoint $L_{3}$ -cubes $Q^{\\prime }_{1},\\cdots ,Q^{\\prime }_{K}\\subset Q$ , such that for each $Q^{\\prime \\prime }_{k}$ there is $Q^{\\prime }_{j}$ with $Q^{\\prime \\prime }_{k}\\subset Q^{\\prime }_{j}$ and $\\operatorname{dist}(Q^{\\prime \\prime }_{k},Q\\setminus Q^{\\prime }_{j})\\ge \\frac{1}{8}L_{3}$ .",
"We need the following continuity of resolvent estimate.",
"It is stated in a slightly different way from [10], so we add a proof here.",
"Lemma 7.5 ([10]) If for $\\lambda \\in \\mathbb {R}$ , $\\alpha >\\beta >0$ , and a cube $Q\\subset \\mathbb {Z}^{3}$ , we have $|(H_{Q}-\\lambda )^{-1}(a,b)|\\le \\exp (\\alpha -\\beta |a-b|) \\text{ for $a,b\\in Q$},$ then for $\\lambda ^{\\prime }$ with $|\\lambda ^{\\prime }-\\lambda |\\le \\frac{1}{2} |Q|^{-1} \\exp (-\\alpha )$ , we have $ |(H_{Q}-\\lambda ^{\\prime })^{-1}(a,b)|\\le 2\\exp (\\alpha -\\beta |a-b|) \\text{ for $a,b\\in Q$}.$ We first prove (REF ) assuming $\\lambda ^{\\prime }$ is not an eigenvalue of $H_Q$ .",
"By resolvent identity we have, $(H_{Q}-\\lambda ^{\\prime })^{-1}=(H_{Q}-\\lambda )^{-1} + (H_{Q}-\\lambda ^{\\prime })^{-1}(\\lambda ^{\\prime }-\\lambda )(H_{Q}-\\lambda )^{-1}.$ Let $\\gamma =\\max _{a,b\\in Q} \\exp (\\beta |a-b|-\\alpha ) |(H_{Q}-\\lambda ^{\\prime })^{-1}(a,b)|$ .",
"Then for any $a,b\\in Q$ , $\\begin{split} &|(H_{Q}-\\lambda ^{\\prime })^{-1}(a,b)|\\\\ \\le &|(H_{Q}-\\lambda )^{-1}(a,b)|+|\\lambda ^{\\prime }-\\lambda |\\sum _{c\\in Q}|(H_{Q}-\\lambda ^{\\prime })^{-1}(a,c)||(H_{Q}-\\lambda )^{-1}(c,b)|\\\\ \\le &\\exp (\\alpha -\\beta |a-b|)+|\\lambda ^{\\prime }-\\lambda |\\sum _{c \\in Q} \\exp (\\alpha -\\beta |a-c|) \\exp (\\alpha -\\beta |c-b|) \\gamma \\\\ \\le &\\exp (\\alpha -\\beta |a-b|)+|\\lambda ^{\\prime }-\\lambda ||Q|\\exp (2\\alpha -\\beta |a-b|) \\gamma \\\\ \\le &\\exp (\\alpha -\\beta |a-b|)+\\frac{1}{2}\\exp (\\alpha -\\beta |a-b|) \\gamma .\\end{split}$ This implies $\\gamma \\le 1+\\frac{1}{2}\\gamma $ and thus $\\gamma \\le 2$ and (REF ) follows.",
"Now we can deduce that $|\\det (H_{Q}-\\lambda ^{\\prime })^{-1}|$ is uniformly bounded for $\\lambda ^{\\prime }$ that is not an eigenvalue of $H_Q$ and satisfies $|\\lambda ^{\\prime }-\\lambda |\\le \\frac{1}{2} |Q|^{-1} \\exp (-\\alpha )$ .",
"By continuity of the determinant (as a function of $\\lambda ^{\\prime }$ ), we conclude that $H_{Q}$ has no eigenvalue in $\\left[\\lambda -\\frac{1}{2} |Q|^{-1} \\exp (-\\alpha ),\\lambda +\\frac{1}{2} |Q|^{-1} \\exp (-\\alpha )\\right]$ .",
"Thus our conclusion follows.",
"We also need the following result to deduce exponential decay of the resolvent in a cube from the decay of the resolvent in subcubes.",
"Lemma 7.6 ([10]) Suppose $\\varepsilon >\\delta >0$ are small, $K\\ge 1$ is an integer and $\\lambda \\in [0,13]$ , $L_{0}\\ge \\cdots \\ge L_{6}$ are large enough (depending on $\\varepsilon ,\\delta ,K$ ) with $L_{k}^{1-\\varepsilon }\\ge L_{k+1}$ , $1\\ge m\\ge 2L_{5}^{-\\delta }$ represents the exponential decay rate, $Q\\subset \\mathbb {Z}^{3}$ is an $L_{0}$ -cube, $Q^{\\prime }_{1},\\cdots ,Q^{\\prime }_{K}\\subset Q$ are disjoint $L_{2}$ -cubes with $\\Vert (H_{Q^{\\prime }_{k}}-\\lambda )^{-1}\\Vert \\le \\exp (L_{4})$ , for all $a\\in Q$ , one of the following holds there is $Q^{\\prime }_{k}$ with $a\\in Q^{\\prime }_{k}$ and $\\operatorname{dist}(a,Q\\setminus Q^{\\prime }_{k})\\ge \\frac{1}{8}L_{2}$ there is an $L_{5}$ -cube $Q^{\\prime \\prime }\\subset Q$ such that $a\\in Q^{\\prime \\prime }$ , $\\operatorname{dist}(a,Q\\setminus Q^{\\prime \\prime })\\ge \\frac{1}{8}L_{5}$ , and $|(H_{Q^{\\prime \\prime }}-\\lambda )^{-1}(b,b^{\\prime })|\\le \\exp (L_{6}-m|b-b^{\\prime }|)$ for $b,b^{\\prime }\\in Q^{\\prime \\prime }$ .",
"Then $|(H_{Q}-\\lambda )^{-1}(a,a^{\\prime })|\\le \\exp (L_{1}-\\tilde{m}|a-a^{\\prime }|)$ for $a,a^{\\prime }\\in Q$ where $\\tilde{m}=m-L_{5}^{-\\delta }$ ."
],
[
"The principal eigenvalue ",
"This appendix sets up the base case in the induction proof of Theorem REF .",
"We follow [10], and generalize their result to higher dimensions.",
"We take $d \\in \\mathbb {Z}$ , $d>2$ , and denote $Q_{n}:=\\left\\lbrace a \\in \\mathbb {Z}^d:\\Vert a\\Vert _{\\infty } \\le n\\right\\rbrace $ instead.",
"Theorem 8.1 Let $\\overline{V}:Q_{n}\\rightarrow \\left[0,1\\right]$ be any potential function, and $R>0$ large enough, such that for any $a \\in Q_n$ , there exists $b\\in Q_n$ with $\\overline{V}(b)=1$ and $|a-b|<R$ .",
"Let $\\overline{H}:\\ell ^2(Q_n) \\rightarrow \\ell ^2(Q_n)$ , $\\overline{H}=-\\Delta +\\overline{V}$ , with Dirichlet boundary condition.",
"Then its principal eigenvalue is no less than $CR^{-d}$ , where $C$ is a constant depending only on $d$ .",
"Let $\\lambda _0$ denote the principal eigenvalue, then by e.g.",
"[12] we have $\\lambda _{0}=\\sup _{u:Q_{n} \\rightarrow \\mathbb {R}_{+}} \\min _{Q_{n}} \\frac{\\overline{H}u}{u}.$ Hence we lower bound $\\lambda _0$ by constructing a function $u$ .",
"Let $\\tilde{G}:\\mathbb {Z}^d \\rightarrow \\mathbb {R}$ be the lattice Green's function; i.e.",
"for any $a \\in \\mathbb {Z}^d$ , $\\tilde{G}(a)$ is the expected number of times that a (discrete time) simple random walk starting at $\\mathbf {0}$ gets to $a$ .",
"Let $G:=\\tilde{G}/2d$ .",
"Then $G$ is the only function such that $-\\Delta G= \\delta _{\\mathbf {0}}$ (where $\\delta _{\\mathbf {0}}(\\mathbf {0})=1$ and $\\delta _{\\mathbf {0}}(a)=0$ for $a \\ne \\mathbf {0}$ ), and $0 \\le G(a) \\le G(\\mathbf {0})$ for any $a\\in \\mathbb {Z}^d$ .",
"In addition, for any $a \\in \\mathbb {Z}^d$ with $a \\ne \\mathbf {0}$ , by e.g.",
"[20] we have $G(a)=\\frac{C_{d}}{|a|^{d-2}}+O\\left(\\frac{1}{|a|^{d}}\\right),$ where $C_d$ is a constant depending only on $d$ .",
"Hence $\\frac{4 C_{d}}{5 |a|^{d-2}} \\le G(a) \\le \\frac{3 C_{d}}{2 |a|^{d-2}}$ when $|a|$ is large enough.",
"We define $u:\\mathbb {Z}^d \\rightarrow \\mathbb {R}_+$ as $u(a):=1 + G(\\mathbf {0}) - G(a) -\\varepsilon _{d} R^{-d} |a|^2, \\; \\forall a \\in \\mathbb {Z}^d,$ where $\\varepsilon _{d} > 0$ is a small enough constant depending on $d$ .",
"Then $ -\\Delta u =-\\delta _{\\mathbf {0}}+2d\\varepsilon _{d} R^{-d},$ and for any $a \\in \\mathbb {Z}^d$ with $|a|<3R$ , we have $0 < u(a) \\le 1+G(\\mathbf {0}) $ .",
"Assume that $R$ is large enough.",
"For any $a$ with $2R<|a|<3R$ , we have $u(a) \\ge 1+G(\\mathbf {0})-\\frac{3C_{d}}{2 (2R)^{d-2}}-9 \\varepsilon _{d} R^{-d+2}$ ; and for any $a$ with $|a|<R$ , $u(a) \\le 1+G(\\mathbf {0})-\\frac{4 C_{d}}{5 R^{d-2}} \\le 1+G(\\mathbf {0})-\\frac{3 C_{d}}{2 (2R)^{d-2}}-9\\varepsilon _{d} R^{-d+2} $ , as long as $\\varepsilon _{d}<\\frac{C_{d}}{180}$ (also note that here we have $d>2$ ).",
"Thus $ \\min _{2R<|a|<3R} u(a) \\ge \\max _{|a|<R} u(a)$ Now we define $u_0 : Q_n \\rightarrow \\mathbb {R}_+$ , as $u_{0}(a):=\\min _{|a-b|<3R,\\overline{V}(b)=1} u(a-b),\\; \\forall a \\in Q_n$ .",
"Pick an arbitrary $a^{\\prime } \\in Q_n$ , by (REF ) there is $b^{\\prime }$ with $|a^{\\prime }-b^{\\prime }| \\le 2R$ such that $u_{0}(a^{\\prime })=u(a^{\\prime }-b^{\\prime })$ and $\\overline{V}(b^{\\prime })=1$ .",
"For any $a^{\\prime \\prime } \\in Q_{n}$ with $|a^{\\prime \\prime }-a^{\\prime }|=1$ , since $|a^{\\prime \\prime }-b^{\\prime }| \\le 2R+1 <3R$ , we have $u_{0}(a^{\\prime \\prime })=\\min _{|a^{\\prime \\prime }-b|<3R,\\overline{V}(b)=1} u(a^{\\prime \\prime }-b) \\le u(a^{\\prime \\prime }-b^{\\prime }).$ Thus by (REF ), and Dirichlet boundary condition, $ \\begin{split}\\overline{H} u_{0} (a^{\\prime })= &2d u_{0}(a^{\\prime })-\\sum _{a^{\\prime \\prime }\\in Q_{n}, |a^{\\prime }-a^{\\prime \\prime }|=1} u_{0}(a^{\\prime \\prime }) +\\overline{V}(a^{\\prime })u_{0}(a^{\\prime })\\\\\\ge &2d u(a^{\\prime }-b^{\\prime })-\\sum _{a^{\\prime \\prime }\\in Q_{n}, |a^{\\prime }-a^{\\prime \\prime }|=1} u(a^{\\prime \\prime }-b^{\\prime }) +\\overline{V}(a^{\\prime })u(a^{\\prime }-b^{\\prime })\\\\\\ge &-\\Delta u(a^{\\prime }-b^{\\prime }) +\\overline{V}(a^{\\prime })u(a^{\\prime }-b^{\\prime })\\\\= &-\\delta _{\\mathbf {0}}(a^{\\prime }-b^{\\prime })+ 2d\\varepsilon _{d}R^{-d} +\\overline{V}(a^{\\prime })u(a^{\\prime }-b^{\\prime }) \\\\\\ge &2d\\varepsilon _{d}R^{-d}.\\end{split}$ Since $a^{\\prime }$ is arbitrary and $0<u_{0}(a^{\\prime })\\le 1+G(\\mathbf {0})$ , by (REF ) and letting $C=\\frac{2d \\varepsilon _{d}}{1+G(\\mathbf {0})}$ , we have $\\lambda _{0} \\ge C R^{-d}$ .",
"Remark 8.2 The exponent in $R^{-d}$ is optimal.",
"Consider a potential $\\overline{V}$ such that $\\overline{V}(a)=1$ only if $a \\in \\lceil R \\rceil \\mathbb {Z}^{d} \\cap Q_{n}$ and $\\overline{V}(a)=0$ otherwise.",
"In this case we have that $\\lambda _{0} \\le 8d R^{-d}+4d n^{-1}$ .",
"To see this, consider the test function $\\phi (a)=1-\\overline{V}(a)$ for $a \\in Q_{n}$ and use the variational principle $\\lambda _{0} \\le \\frac{\\langle \\phi , \\overline{H} \\phi \\rangle }{\\Vert \\phi \\Vert _{2}^{2}}$ .",
"Corollary 8.3 Let $\\overline{H}$ , $C$ be defined as in Theorem REF .",
"Let $0\\le \\lambda <\\frac{CR^{-d}}{2} $ .",
"Then $\\Vert (\\overline{H}-\\lambda )^{-1}\\Vert \\le \\frac{2R^{d}}{C}$ and $|(\\overline{H}-\\lambda )^{-1}(a,b)| \\le \\frac{2R^{d}}{C} \\exp \\left(-\\frac{C R^{-d}}{8d+2} |a-b|\\right)$ for any $a,b \\in Q_{n}$ .",
"As the principal eigenvalue of $\\overline{H}$ is no less than $C R^{-d}$ , we have $\\Vert (\\overline{H}-\\lambda )^{-1}\\Vert \\le \\frac{2R^{d}}{C} $ .",
"Let $T:= I - \\frac{1}{4 d +1} (\\overline{H}-\\lambda ) $ .",
"Since any eigenvalue of $\\overline{H}$ is in $\\left[C R^{-d},4d+1\\right]$ , the eigenvalues of $T$ are in $\\left[0, 1 - \\frac{C}{8d+2} R^{-d}\\right]$ , so $\\Vert T\\Vert \\le 1 - \\frac{C}{8d+2} R^{-d}$ .",
"Note that for each $i>0$ and $a,b \\in Q_{n}$ , $T^{i}(a,b)=0$ if $|a-b|>i$ .",
"Then we have $|(\\overline{H}-\\lambda )^{-1}(a,b)|=(4d+1)^{-1}|(I-T)^{-1}(a,b)|\\le (4d+1)^{-1} \\sum _{i \\ge 0}|T^{i}(a,b)|\\\\= (4d+1)^{-1} \\sum _{i \\ge |a-b|}|T^{i}(a,b)|\\le (4d+1)^{-1} \\sum _{i \\ge |a-b|} \\Vert T\\Vert ^{i}\\le \\frac{2R^{d}}{C} \\exp \\left(-\\frac{C R^{-d}}{8d+2} |a-b|\\right),$ so the corollary follows.",
"Finally, we have the following result, which implies the base case in the induction proof of Theorem REF .",
"Proposition 8.4 Let $d=3$ , and $V$ be the Bernoulli potential, i.e.",
"$\\mathbb {P}(V(a)=0)=\\mathbb {P}(V(a)=1)=\\frac{1}{2}$ for each $a \\in \\mathbb {Z}^3$ independently.",
"For any $0<\\delta <\\frac{1}{10}$ and $\\varepsilon >0$ , there exists $C_{\\delta ,\\varepsilon }$ such that for any $n>C_{\\delta ,\\varepsilon }$ and $0\\le \\lambda < \\frac{Cn^{-\\frac{3 \\delta }{10}}}{2}$ , with probability at least $1-n^{-1}$ the following is true.",
"Take any $V^{\\prime }:\\mathbb {Z}^3\\rightarrow [0,1]$ such that $V^{\\prime }_{Q_{n} \\cap \\lceil \\varepsilon ^{-1}\\rceil \\mathbb {Z}^{3}} = V_{Q_{n} \\cap \\lceil \\varepsilon ^{-1}\\rceil \\mathbb {Z}^{3}}$ .",
"Let $H^{\\prime }_{Q_n}$ be the restriction of $-\\Delta +V^{\\prime }$ on $Q_n$ with Dirichlet boundary condition.",
"Then we have $\\Vert (H^{\\prime }_{Q_{n}}-\\lambda )^{-1}\\Vert \\le \\exp (n^{2\\delta }),$ and $ \\text{$|(H^{\\prime }_{Q_{n}}-\\lambda )^{-1}(a,b)| \\le n^{2\\delta } \\exp (-n^{-\\delta }|a-b|)$ for any $a,b \\in Q_{n}$}.$ Let $R:=n^{\\frac{\\delta }{10}}$ , and let $A$ denote the following event: $\\forall a \\in Q_{n}, \\exists b \\in Q_{n}\\cap \\lceil \\varepsilon ^{-1}\\rceil \\mathbb {Z}^{3}, \\; \\text{s.t.}",
"\\; |a-b| \\le R, V(b)=1.$ Then $A$ only depends on $V_{Q_{n} \\cap \\lceil \\varepsilon ^{-1}\\rceil \\mathbb {Z}^{3}}$ .",
"Using Corollary REF with $d=3$ , we have that (REF ) and (REF ) hold under the event $A$ , when $n$ is large enough.",
"Finally, since there are $(2n+1)^{3}$ points in $Q_{n}$ , and inside each ball of radius $R$ , there are at least $\\frac{1}{8}n^{\\frac{3\\delta }{10}} \\varepsilon ^{3}$ points in $\\lceil \\varepsilon ^{-1}\\rceil \\mathbb {Z}^{3} \\cap Q_{n}$ , we have $\\mathbb {P}(A^{c}) \\le (2n+1)^{3} 2^{- \\frac{1}{8} n^{\\frac{3\\delta }{10}} \\varepsilon ^{3}} \\le n^{-1}$ , when $n$ is large enough."
],
[
"Deducing Anderson localization from the resolvent estimate",
"The arguments in this appendix originally come from [4] (see also [15] and [6]).",
"These previous works are about the continuous space model.",
"For completeness and for the reader's convenience, we adapt the arguments for the lattice model, thus deducing Theorem REF from Theorem REF .",
"As in Section , in this appendix, by “dyadic” we mean an integer power of 2, and by “dyadic cube”, we mean a cube $Q_{2^{n}}(a)$ for some $a \\in 2^{n-1}\\mathbb {Z}^3$ and $n \\in \\mathbb {Z}_{+}$ .",
"For any $k \\in \\mathbb {Z}_{+}$ , we define $\\Omega _{k}:=\\lbrace u:\\mathbb {Z}^{3}\\rightarrow \\mathbb {R}: |u(a)|\\le k (1+|a|)^{k},\\;\\; \\forall a\\in \\mathbb {Z}^{3},\\;\\; \\text{and}\\;\\; u(\\mathbf {0})=1\\rbrace .$ Since the law of $H$ is invariant under translation, to prove Theorem REF , it suffices to show that for any $k\\in \\mathbb {Z}_{+}$ , almost surely $\\inf _{t>0} \\sup _{a \\in \\mathbb {Z}^3} \\exp (t|a|) |u(a)|<\\infty ,$ for any $u\\in \\Omega _{k}$ and $\\lambda \\in [0, \\lambda _*]$ with $H u=\\lambda u$ .",
"Denote $\\mathcal {I}=(0,\\lambda _{*})$ .",
"We first see that it suffices to prove (REF ) for any $u\\in \\Omega _{k}$ and $\\lambda \\in \\mathcal {I}$ with $H u=\\lambda u$ , by applying the following lemma to $\\lambda =0$ and $\\lambda =\\lambda _{*}$ .",
"Lemma 9.1 Suppose $\\lambda \\in [0,\\lambda _{*}]$ and $k\\in \\mathbb {Z}_{+}$ .",
"Then almost surely, there is no $u\\in \\Omega _k$ with $H u =\\lambda u$ .",
"Let $L_{i}=2^{i}$ for $i\\in \\mathbb {Z}_{+}$ .",
"By Theorem REF and the Borel-Cantelli lemma, almost surely, there exists $i^{\\prime }>0$ , such that for any $i>i^{\\prime }$ , $\\left|(H_{Q_{L_{i}}}-\\lambda )^{-1}(a,b)\\right| \\le \\exp \\left(L_{i}^{1-\\lambda _{*}}-\\lambda _{*} |a-b|\\right), \\;\\forall a,b \\in Q_{L_{i}}.$ Assume there exists $u\\in \\Omega _k$ with $H u=\\lambda u$ .",
"For each large enough $i$ we have $|u(\\mathbf {0})|=\\left|\\sum _{\\begin{array}{c}a\\in Q_{L_{i}},a^{\\prime }\\in \\mathbb {Z}^{3}\\setminus Q_{L_{i}}\\\\|a-a^{\\prime }|=1\\end{array}} (H_{Q_{L_{i}}}-\\lambda )^{-1}(\\mathbf {0},a) u(a^{\\prime })\\right|\\le 6\\cdot (2L_i +1)^2\\exp \\left(-\\frac{\\lambda _{*}L_{i}}{2}\\right)k(1+\\sqrt{3}L_{i})^{k}$ which converges to zero as $i\\rightarrow \\infty $ .",
"Thus $u(\\mathbf {0})=0$ , which contradicts with the fact that $u\\in \\Omega _{k}$ .",
"Let us fix $k\\in \\mathbb {Z}_{+}$ and denote by $\\sigma _k(H)$ the set of all $\\lambda \\in \\mathcal {I}$ , such that $Hu=\\lambda u$ for some $u \\in \\Omega _k$ .",
"For each $L\\in \\mathbb {Z}_{+}$ , denote by $\\sigma (H_{Q_{L}})$ the set of eigenvalues of $H_{Q_{L}}$ .",
"The first key step is to prove that for any large enough $L$ , with high probability, the distance between any $\\lambda \\in \\sigma _k(H)$ and $\\sigma (H_{Q_{L}})$ is small, exponentially in $L$ .",
"Proposition 9.2 There exist $\\kappa ^{\\prime },c_1 >0$ such that for any dyadic $L$ large enough, we can find a $V_{Q_{L}}$ -measurable event $\\mathcal {E}_{wloc}^{(L)}$ , such that $\\mathbb {P}\\left[\\mathcal {E}_{wloc}^{(L)}\\right] \\ge 1-L^{-\\kappa ^{\\prime }},$ and under the event $\\mathcal {E}_{wloc}^{(L)}$ , we have $\\operatorname{dist}(\\lambda , \\sigma (H_{Q_{L}})\\cap \\mathcal {I}) \\le \\exp (-c_1 L)$ for any $\\lambda \\in \\sigma _k(H)\\cap \\left[\\exp (-c_1\\sqrt{L}),\\lambda _{*}-\\exp (-c_1\\sqrt{L})\\right].$ The next key step is to strengthen Proposition REF so that each $\\lambda \\in \\sigma _k(H)$ is not only exponentially close to $\\sigma (H_{Q_{L}})$ , but also exponentially close to a finite subset $S\\subset \\sigma (H_{Q_{L}})$ with $|S|<L^{\\delta ^{\\prime }}$ for arbitrarily small $\\delta ^{\\prime }$ .",
"Proposition 9.3 For any $\\delta ^{\\prime }>0$ , there exist $\\kappa ^{\\prime \\prime },c_2 >0$ such that for each dyadic $L$ large enough (depending on $\\delta ^{\\prime }$ ), we can find a $V_{Q_{L}}$ -measurable event $\\mathcal {E}_{sloc}^{(L)}$ with $\\mathbb {P}\\left[\\mathcal {E}_{sloc}^{(L)}\\right] \\ge 1-L^{-\\kappa ^{\\prime \\prime }},$ and under the event $\\mathcal {E}_{sloc}^{(L)}$ , there exists a finite set $S\\subset \\sigma (H_{Q_L})\\cap \\mathcal {I}$ with $|S|<L^{\\delta ^{\\prime }}$ such that $\\operatorname{dist}(\\lambda , S) \\le \\exp (-c_2 L)$ for any $\\lambda \\in \\sigma _k(H) \\cap \\left[\\exp (-L^{c_2}),\\lambda _{*}-\\exp (-L^{c_2}) \\right]$ .",
"Proposition REF and REF are discrete versions of [6] and [6] respectively.",
"See also [15].",
"Now we leave the proofs of these two propositions to the next two subsections, and prove localization assuming them.",
"[Proof of Theorem REF ] We apply Proposition REF with $\\delta ^{\\prime }<\\kappa _{0}$ where $\\kappa _{0}$ is the constant in Theorem REF .",
"Take large enough dyadic $L$ , and consider the annulus $A_{L}=Q_{5L}\\setminus Q_{2L}$ .",
"We cover $A_L$ by $2L$ -cubes $\\lbrace Q^{(j)}:1\\le j\\le 1000\\rbrace $ that are disjoint with $Q_{L}$ , such that for each $a\\in A_{L}$ there is $1\\le j\\le 1000$ with $a\\in Q^{(j)}$ and $\\operatorname{dist}(a,\\mathbb {Z}^{3}\\setminus Q^{(j)})\\ge \\frac{1}{8}L$ .",
"Apply Theorem REF to each of $Q^{(j)}$ 's and to each energy $\\lambda \\in S\\subset \\sigma (H_{Q_L})\\cap \\mathcal {I}$ , we have $\\mathbb {P}\\left[\\mathcal {E}^{(L)}_{ann}\\big |\\; \\mathcal {E}_{sloc}^{(L)}\\right]\\ge 1-1000L^{\\delta ^{\\prime }-\\kappa _{0}}$ where $\\mathcal {E}^{(L)}_{ann}$ denotes the event: $\\left|(H_{Q^{(j)}}-\\lambda )^{-1}(a,b)\\right| \\le \\exp \\left(L^{1-\\lambda _{*}}-\\lambda _{*} |a-b|\\right), \\;\\forall 1\\le j\\le 1000,\\;\\forall a,b \\in Q^{(j)},\\;\\text{and }\\forall \\lambda \\in S.$ Then by Proposition REF we have $ \\mathbb {P}\\left[\\mathcal {E}^{(L)}_{ann}\\cap \\mathcal {E}_{sloc}^{(L)}\\right]\\ge (1-L^{-\\kappa ^{\\prime \\prime }})(1-1000L^{\\delta ^{\\prime }-\\kappa _{0}})\\ge 1-L^{-\\kappa ^{\\prime \\prime \\prime }},$ for some constant $\\kappa ^{\\prime \\prime \\prime }>0$ and large enough $L$ .",
"Under the event $\\mathcal {E}^{(L)}_{ann}\\cap \\mathcal {E}_{sloc}^{(L)}$ , we take any $u\\in \\Omega _{k}$ with $H u=\\lambda u$ and $\\lambda \\in [\\exp (-L^{c_2}),\\lambda _{*}-\\exp (-L^{c_2})]$ , and $\\lambda ^{\\prime }\\in S$ with $|\\lambda -\\lambda ^{\\prime }|<\\exp (-c_2 L)$ .",
"Thus using Lemma REF , we have $ \\begin{split}\\Vert u\\Vert _{\\ell ^{\\infty }(A_{L})}&\\le 2\\exp \\left(L^{1-\\lambda _{*}}-\\frac{1}{8}\\lambda _{*} L\\right) \\Vert u\\Vert _{\\ell ^{1}(Q_{6L})}\\\\&\\le 2\\exp \\left(L^{1-\\lambda _{*}}-\\frac{1}{8}\\lambda _{*} L\\right)k(6\\sqrt{3}L+1)^{k} (12L+1)^{3} \\le \\exp (-c^{\\prime } L)\\end{split}$ for some constant $c^{\\prime }<\\frac{\\lambda _{*}}{8}$ and large enough $L$ .",
"Now we consider the event $\\mathcal {E}_{loc}=\\bigcup _{i^{\\prime }\\ge 0} \\bigcap _{i\\ge i^{\\prime }} (\\mathcal {E}^{(2^{i})}_{ann}\\cap \\mathcal {E}_{sloc}^{(2^{i})}).$ We have $\\mathbb {P}[\\mathcal {E}_{loc}]=1$ by (REF ).",
"Note that for any $\\lambda \\in \\mathcal {I}$ , we have $\\lambda \\in [\\exp (-L^{c_2 }),\\lambda _{*}-\\exp (-L^{c_2})]$ for large enough $L$ .",
"We also have that $\\bigcup _{i\\ge i^{\\prime }}A_{2^{i}}=\\mathbb {Z}^{3}\\setminus Q_{2^{i^{\\prime }+1}}$ for any $i^{\\prime }\\in \\mathbb {Z}_{+}$ .",
"By (REF ) we have that (REF ) holds under the event $\\mathcal {E}_{loc}$ .",
"Then localization is proved."
],
[
"The first spectral reduction",
"For simplicity of notations, for any $\\lambda \\in \\mathbb {R}$ , dyadic scale $L$ , and $a\\in \\mathbb {Z}^3$ , we say $Q_{L}(a)$ is $\\lambda $ -good if $\\left|(H_{Q_L(a)}-\\lambda )^{-1}(b,b^{\\prime })\\right| \\le \\exp \\left(L^{1-\\lambda _{*}}-\\lambda _{*} |b-b^{\\prime }|\\right), \\;\\forall b,b^{\\prime } \\in Q_{L}(a).$ Otherwise, we call it $\\lambda $ -bad.",
"By Theorem REF , for any large enough dyadic scale $L$ and $\\lambda \\in [0, \\lambda _{*}]$ , we have $\\mathbb {P}[\\text{$Q_{L}(a)$ is $\\lambda $-bad}]\\le L^{-\\kappa _{0}}.$ [Proof of Proposition REF ] Throughout the proof, we use $C$ to denote large universal constants.",
"For a dyadic scale $L$ , we construct a graph $G_{L}$ whose vertices are all the dyadic $2L$ -cubes.",
"The edges are given as follows: for any $a\\ne a^{\\prime }\\in \\frac{L}{2}\\mathbb {Z}^{3}$ , there is an edge connecting $Q_{L}(a)$ and $Q_{L}(a^{\\prime })$ if and only if $Q_{L}(a)\\cap Q_{L}(a^{\\prime })\\ne \\emptyset $ .",
"Fix large dyadic scale $L$ .",
"Take the dyadic scale $L_{0}\\in \\left\\lbrace \\sqrt{L},\\sqrt{2L}\\right\\rbrace $ .",
"For any $\\lambda \\in \\mathcal {I}$ , denote by $\\mathcal {E}^{\\lambda }_{per}$ the event that there is a path of $\\lambda $ -bad $2L_0$ -cubes $\\overline{Q}_{1},\\cdots ,\\overline{Q}_{m}$ in $G_{L_{0}}$ such that $\\text{$\\overline{Q}_{1}\\cap Q_{\\frac{L}{2}}\\ne \\emptyset $ and $\\overline{Q}_{m}\\cap Q_{L}=\\emptyset $}.$ Under the event $\\mathcal {E}^{\\lambda }_{per}$ , suppose that $\\Gamma _{0}=(\\overline{Q}_{1},\\cdots ,\\overline{Q}_{m})$ is such a path with the shortest length.",
"Since $\\operatorname{dist}(Q_{\\frac{L}{2}},\\mathbb {Z}^{3}\\setminus Q_{L})\\ge \\frac{L}{2}$ , we have $m\\ge \\frac{L}{4\\sqrt{3}L_{0}}$ .",
"By definition of dyadic cubes and that $\\Gamma _{0}$ has the shortest length, there are at least $\\frac{m}{1000}$ disjoint $\\lambda $ -bad cubes in $\\Gamma _{0}$ .",
"Hence, $\\mathbb {P}[\\mathcal {E}^{\\lambda }_{per}] \\le \\sum _{m\\ge \\frac{L}{4\\sqrt{3}L_{0}}} C L^{3} 1000^{m}(L_{0}^{-\\kappa _{0}})^{\\frac{m}{1000}}\\le 2C L^{3} (1000 L_{0}^{-\\frac{\\kappa _{0}}{1000}})^{\\frac{L}{4\\sqrt{3}L_{0}}}\\le L_{0}^{-c^{\\prime }L_{0}}$ for some $c^{\\prime }>0$ .",
"Here the first inequality is by (REF ), and counting the total number of $G_{L_0}$ paths with length $m$ and one end intersecting $Q_{\\frac{L}{2}}$ .",
"Claim 9.4 Under the event $(\\mathcal {E}^{\\lambda }_{per})^{c}$ , any $\\lambda ^{\\prime }\\in \\sigma _{k}(H)$ with $|\\lambda ^{\\prime }-\\lambda |\\le \\exp (- L^{1-\\frac{\\lambda _{*}}{2}}_{0})$ satisfies $\\operatorname{dist}(\\lambda ^{\\prime },\\sigma (H_{Q_{\\frac{3}{2}L}}))\\le \\exp (-\\epsilon ^{\\prime } L_{0})$ for a universal constant $\\epsilon ^{\\prime }>0$ .",
"Denote the set of all the $\\lambda $ -bad $L_{0}$ -cubes contained in $Q_{\\frac{3}{2}L}$ by $\\mathcal {S}$ .",
"We consider $\\mathbb {Z}^{3}$ as a graph with edges between nearest neighbors.",
"Consider the set $S_{0}:=(\\bigcup \\mathcal {S})\\cup Q_{\\frac{L}{2}}\\subset Q_{\\frac{3}{2}L}$ .",
"Let $S_{1}$ be the maximal connected component of $S_{0}$ which contains $Q_{\\frac{L}{2}}$ .",
"Then $(\\mathcal {E}^{\\lambda }_{per})^{c}$ implies $S_{1}\\subset Q_{L+2L_{0}}$ .",
"Denote $\\partial ^{-}S_{1}=\\lbrace a\\in S_{1}:\\text{$|a-a^{\\prime }|=1$ for some $a^{\\prime }\\in \\mathbb {Z}^{3}\\setminus S_{1}$}\\rbrace ,$ and $\\partial ^{+}S_{1}=\\lbrace a\\in \\mathbb {Z}^{3}\\setminus S_{1}:\\text{$|a-a^{\\prime }|=1$ for some $a^{\\prime }\\in S_{1}$}\\rbrace .$ Assume $\\lambda ^{\\prime }$ satisfies the hypothesis in the claim, then there is $u\\in \\Omega _{k}$ such that $H u= \\lambda ^{\\prime } u$ .",
"For any $a^{\\prime }\\in \\partial ^{-}S_{1}\\cup \\partial ^{+}S_{1}$ , there is a dyadic $L_{0}$ -cube $Q^{\\prime }$ such that $a^{\\prime }\\in Q^{\\prime }$ and $\\operatorname{dist}(a^{\\prime },\\mathbb {Z}^{3}\\setminus Q^{\\prime })\\ge \\frac{1}{8}L_{0}$ .",
"By maximality of $S_{1}$ , we have $Q^{\\prime }$ is $\\lambda $ -good.",
"Thus by Lemma REF , $\\begin{split}|u(a^{\\prime })|\\le &2\\exp (L_{0}^{1-\\lambda _{*}}-\\frac{1}{8}\\lambda _{*} L_{0})\\Vert u\\Vert _{\\ell ^{1}(Q_{L+4L_{0}})}\\\\\\le &2\\exp (L_{0}^{1-\\lambda _{*}}-\\frac{1}{8}\\lambda _{*} L_{0})(2L+8L_{0}+1)^{3}k(\\sqrt{3}L+4\\sqrt{3}L_{0}+1)^{k}\\\\\\le &\\exp (-\\frac{1}{10}\\lambda _{*} L_{0})\\end{split}$ for large enough $L_{0}$ .",
"Let $u_{*}:Q_{\\frac{3}{2}L}\\rightarrow \\mathbb {R}$ be defined by $u_{*}=u$ on $S_{1}$ and $u_{*}=0$ on $Q_{\\frac{3}{2}L}\\setminus S_{1}$ .",
"Then $(H_{Q_{\\frac{3}{2}L}}-\\lambda ^{\\prime }) u_{*}(a)={\\left\\lbrace \\begin{array}{ll}0 & \\text{if\\;} a\\in Q_{\\frac{3}{2}L}\\setminus (\\partial ^{-}S_{1}\\cup \\partial ^{+}S_{1}),\\\\\\sum _{|a^{\\prime }-a|=1,a^{\\prime }\\in \\partial ^{+}S_{1}} u(a^{\\prime }) & \\text{if\\;} a\\in \\partial ^{-}S_{1},\\\\-\\sum _{|a^{\\prime }-a|=1,a^{\\prime }\\in \\partial ^{-}S_{1}} u(a^{\\prime }) &\\text{if\\;} a\\in \\partial ^{+}S_{1}.\\end{array}\\right.",
"}$ By (REF ), we have $\\Vert (H_{Q_{\\frac{3}{2}L}}-\\lambda ^{\\prime }) u_{*}\\Vert _{\\ell ^{2}(Q_{\\frac{3}{2}L})}\\le 6(3L+1)^{\\frac{3}{2}}\\exp (-\\frac{1}{10}\\lambda _{*} L_{0})\\le \\exp (-\\epsilon ^{\\prime }L_{0})\\Vert u_{*}\\Vert _{\\ell ^{2}(Q_{\\frac{3}{2}L})}$ for large enough $L$ .",
"Here, we used $\\Vert u_{*}\\Vert _{\\ell ^{2}(Q_{\\frac{3}{2}L})}\\ge 1$ since $\\mathbf {0}\\in S_{1}$ and $u(\\mathbf {0})=1$ .",
"By expanding $u_{*}$ into a linear combination of eigenvectors of $H_{Q_{\\frac{3}{2}L}}$ , (REF ) guarantees that there is an eigenvalue $\\lambda _{0}$ of $H_{Q_{\\frac{3}{2}L}}$ such that $|\\lambda ^{\\prime }-\\lambda _{0}|\\le \\exp (-\\epsilon ^{\\prime }L_{0})$ .",
"Our claim follows.",
"Denote $\\lambda ^{(h)}=h\\exp (-L_{0})$ for $h\\in \\mathbb {Z}_{+}$ and let $\\mathcal {E}^{0}_{trap}=\\bigcap _{\\lambda ^{(h)}\\in \\mathcal {I}} (\\mathcal {E}^{\\lambda ^{(h)}}_{per})^{c}.$ Then by (REF ), $\\mathbb {P}[\\mathcal {E}^{0}_{trap}]\\ge 1-\\lambda _{*}\\exp (L_{0})L^{-c^{\\prime } L_{0}}_{0}\\ge 1-L^{-10}$ for large $L$ .",
"Claim 9.5 Under the event $\\mathcal {E}^{0}_{trap}$ , any $\\lambda \\in [0, \\lambda _{*}]\\cap \\sigma _{k}(H)$ satisfies $\\operatorname{dist}(\\lambda ,\\sigma (H_{Q_{\\frac{3}{2}L}}))\\le \\exp (-\\epsilon ^{\\prime } L_{0}).$ For any $\\lambda \\in [0, \\lambda _{*}]$ , there exists an $h\\in \\mathbb {Z}_{+}$ such that $\\lambda ^{(h)}\\in \\mathcal {I}$ and $|\\lambda -\\lambda ^{(h)}|\\le \\exp (-L_{0}^{1-\\frac{\\lambda _{*}}{2}})$ .",
"Our claim follows from Claim REF .",
"Let $q$ be the smallest positive integer such that $2^{\\frac{1}{q}}-1<\\frac{\\lambda _{*}}{2}$ and let $\\tau =2^{\\frac{1}{q}}-1$ .",
"Define $\\tilde{L}_{1}=L_{0}^{1+\\tau }$ and $\\tilde{L}_{i+1}=\\tilde{L}_{i}^{1+\\tau }$ for $i=1,2,\\cdots ,q-1$ .",
"Then $L\\le \\tilde{L}_{q}=L_{0}^{2}\\le 2L$ .",
"Let $L_i$ be the (unique) dyadic scale such that $L_i\\in [\\tilde{L}_i, 2\\tilde{L}_i)$ for each $i=1,\\cdots ,q$ .",
"Let $M_{i}=\\frac{3}{2}L+C^{\\prime }\\sum _{1\\le j\\le i}L_{j}$ for each $i=1,\\cdots ,q$ and $M_{0}=\\frac{3}{2}L$ .",
"Here $C^{\\prime }$ is a large constant to be determined.",
"Then $M_{i}\\le \\frac{3}{2}L + 4C^{\\prime } i L\\le \\left(\\frac{3}{2}+4C^{\\prime } q\\right)L$ for each $0\\le i\\le q$ .",
"In addition, we denote $M_{q+1}=2^wL$ where $w$ is the smallest integer with $2^w>3+8C^{\\prime }q$ , and let $L_{q+1}=L_q$ .",
"For any $\\lambda \\in \\mathcal {I}$ and any $j\\in \\lbrace 1,\\cdots ,q+1\\rbrace $ , denote by $\\mathcal {E}^{\\lambda ,j}_{per}$ the following event: there exists a path of $\\lambda $ -bad $2L_j$ -cubes in $G_{L_{j}}$ , say $\\overline{Q}_{1},\\cdots ,\\overline{Q}_{m}$ , such that $\\begin{split}&\\overline{Q}_{i}\\subset Q_{M_{j}}\\setminus Q_{M_{j-1}}, \\; \\forall i\\in \\lbrace 1,\\cdots ,m\\rbrace ,\\\\&\\overline{Q}_{1}\\cap Q_{M_{j-1}+10L_{j}}\\ne \\emptyset ,\\\\&\\overline{Q}_{m}\\cap Q_{M_{j}-10L_{j}}\\ne \\emptyset .\\end{split}$ Under the event $\\mathcal {E}^{\\lambda ,j}_{per}$ , suppose that $\\Gamma _{0}=(\\overline{Q}_{1},\\cdots ,\\overline{Q}_{m})$ in $G_{L_{j}}$ is such a path with the shortest length.",
"Since $\\operatorname{dist}(Q_{M_{j-1}+10L_{j}},\\mathbb {Z}^{3}\\setminus Q_{M_{j}-10L_{j}})\\ge (C^{\\prime }-20) L_{j}$ , we have $m\\ge \\frac{C^{\\prime }}{4}$ when $C^{\\prime }$ is large enough.",
"By definition of dyadic cubes and that $\\Gamma _{0}$ has the shortest length, there are at least $\\frac{m}{1000}$ disjoint $\\lambda $ -bad cubes in $\\Gamma _{0}$ .",
"Hence, $\\mathbb {P}[\\mathcal {E}^{\\lambda ,j}_{per}] \\le \\sum _{m\\ge \\frac{C^{\\prime }}{4}}C (C^{\\prime } L)^{3} 1000^{m}(L_{j}^{-\\kappa _{0}})^{\\frac{m}{1000}}\\le 2C (C^{\\prime } L)^{3} (1000 L_{j}^{-\\frac{\\kappa _{0}}{1000}})^{\\frac{C^{\\prime }}{4}}\\le L^{-10}.$ Here the first inequality is by (REF ) and counting the number of paths in $G_{L_j}$ with length $m$ and one end intersecting $Q_{M_{j-1}+10L_j}$ , and the last inequality is by taking $C^{\\prime }$ large enough.",
"By adapting the proof of Claim REF we can get the following result.",
"Claim 9.6 Under the event $(\\mathcal {E}^{\\lambda ,j}_{per})^{c}$ , any $\\lambda ^{\\prime }\\in \\sigma _{k}(H)$ with $|\\lambda ^{\\prime }-\\lambda |\\le \\exp (- L_{j}^{1-\\frac{\\lambda _{*}}{2}})$ satisfies $\\operatorname{dist}(\\lambda ^{\\prime },\\sigma (H_{Q_{M_{j}}} ))\\le \\exp (-\\epsilon ^{\\prime \\prime } L_{j})$ for a universal constant $\\epsilon ^{\\prime \\prime }>0$ .",
"Note that, given $\\lambda \\in \\mathcal {I}$ , the event $\\mathcal {E}^{\\lambda ,j}_{per}$ is $V_{Q_{M_{j}}\\setminus Q_{M_{j-1}}}$ -measurable.",
"Hence, the event $\\mathcal {E}^{j}_{trap}:=\\left(\\bigcup _{\\lambda \\in \\sigma (H_{Q_{M_{j-1}}})\\cap \\mathcal {I}} \\mathcal {E}^{\\lambda ,j}_{per}\\right)^{c}$ satisfies $\\mathbb {P}[\\mathcal {E}^{j}_{trap}| V_{Q_{M_{j-1}}}]\\ge 1-(M_{j-1}+1)^{3}L^{-10} \\ge 1-L^{-6}$ by (REF ) and (REF ) for large enough $L$ .",
"For each $0\\le j\\le q+1$ , $\\mathcal {E}^{j}_{trap}$ is $V_{Q_{M_{j}}}$ -measurable, thus the event $\\mathcal {E}_{trap}:=\\bigcap _{0\\le j\\le q+1}\\mathcal {E}^{j}_{trap}$ is $V_{Q_{M_{q+1}}}$ -measurable.",
"By (REF ) and (REF ), we have $\\mathbb {P}[\\mathcal {E}_{trap}]\\ge 1- (q+2) L^{-6}\\ge 1-L^{-5}.$ Claim 9.7 Under the event $\\mathcal {E}_{trap}$ , any $\\lambda \\in [\\exp (-\\epsilon ^{\\prime \\prime \\prime } L_{0}/2),\\lambda _{*}-\\exp (-\\epsilon ^{\\prime \\prime \\prime } L_{0}/2)]\\cap \\sigma _{k}(H)$ satisfies $\\operatorname{dist}(\\lambda ,\\sigma (H_{Q_{M_{q+1}}}))\\le \\exp (-\\epsilon ^{\\prime \\prime \\prime } L)$ for some $\\epsilon ^{\\prime \\prime \\prime }>0$ .",
"Let $\\epsilon ^{\\prime \\prime \\prime }=\\min \\lbrace \\epsilon ^{\\prime },\\epsilon ^{\\prime \\prime }\\rbrace $ .",
"Let $\\lambda \\in [\\exp (-\\epsilon ^{\\prime \\prime \\prime } L_{0}/2),\\lambda _{*}-\\exp (-\\epsilon ^{\\prime \\prime \\prime } L_{0}/2)]\\cap \\sigma _{k}(H)$ .",
"We inductively prove that, $\\operatorname{dist}(\\lambda ,\\sigma (H_{Q_{L_{j}}}))\\le \\exp (-\\epsilon ^{\\prime \\prime \\prime } L_{j})$ for any $0\\le j\\le q+1$ .",
"Thus, in particular, we have $\\operatorname{dist}(\\lambda ,\\sigma (H_{Q_{M_{q+1}}}))\\le \\exp (-\\epsilon ^{\\prime \\prime \\prime } L_{q+1})\\le \\exp (-\\epsilon ^{\\prime \\prime \\prime } L),$ and the claim follows.",
"For the case $j=0$ , by Claim REF , $\\operatorname{dist}(\\lambda ,\\sigma (H_{Q_{M_{0}}}))\\le \\exp (-\\epsilon ^{\\prime } L_{0})$ .",
"Assume the conclusion holds for some $j<q+1$ , then $|\\lambda -\\lambda _{0}|\\le \\exp (-\\epsilon ^{\\prime \\prime \\prime } L_{j})$ for some $\\lambda _{0}\\in \\sigma (H_{Q_{M_{j}}})$ .",
"As $\\lambda \\in [\\exp (-\\epsilon ^{\\prime \\prime \\prime } L_{0}/2),\\lambda _{*}-\\exp (-\\epsilon ^{\\prime \\prime \\prime } L_{0}/2)]$ , we must have $\\lambda _0 \\in \\mathcal {I}$ .",
"Since $\\tau <\\frac{\\lambda _{*}}{2}$ , for $L$ large enough we have $\\epsilon ^{\\prime \\prime \\prime }L_{j} > L_{j+1}^{1-\\frac{\\lambda _{*}}{2}}$ and $|\\lambda -\\lambda _{0}|\\le \\exp (- L_{j+1}^{1-\\frac{\\lambda _{*}}{2}})$ .",
"Thus Claim REF implies $\\operatorname{dist}(\\lambda ,\\sigma (H_{Q_{M_{j+1}}}))\\le \\exp (-\\epsilon ^{\\prime \\prime } L_{j+1})$ .",
"Finally, since $M_{q+1}=2^wL$ and $w$ is a constant, the proposition follows from Claim REF and (REF )."
],
[
"The second spectral reduction",
"For any positive integers $L^{\\prime \\prime }>L^{\\prime }$ , we denote the annulus $A_{L^{\\prime \\prime },L^{\\prime }}=Q_{L^{\\prime \\prime }}\\setminus Q_{L^{\\prime }}$ .",
"Take any $\\delta >0$ .",
"For $\\lambda \\in \\mathcal {I}$ and $L^{\\prime \\prime }>2L^{\\prime }$ , let $\\mathcal {E}_{L^{\\prime \\prime },L^{\\prime }}^{(\\lambda )}$ denote the following event: there exists a subset $G^{(\\lambda )}_{L^{\\prime \\prime },L^{\\prime }}\\subset A_{L^{\\prime \\prime },L^{\\prime }}$ with $|G^{(\\lambda )}_{L^{\\prime \\prime },L^{\\prime }}|\\le (L^{\\prime })^{\\frac{\\delta }{2}}$ such that, for any $a\\in A_{L^{\\prime \\prime },2L^{\\prime }}\\setminus G^{(\\lambda )}_{L^{\\prime \\prime },L^{\\prime }}$ , there is a $\\lambda $ -good cube $Q_{L^{\\prime \\prime \\prime }}(b)\\subset A_{L^{\\prime \\prime },L^{\\prime }}$ such that $\\operatorname{dist}(a,Q_{L^{\\prime \\prime }}\\setminus Q_{L^{\\prime \\prime \\prime }}(b))\\ge \\frac{1}{8}L^{\\prime \\prime \\prime }$ , and $(L^{\\prime })^{\\frac{\\delta }{10}}\\le L^{\\prime \\prime \\prime }\\le L^{\\prime }$ .",
"Note that, $\\mathcal {E}^{(\\lambda )}_{L^{\\prime \\prime },L^{\\prime }}$ is $V_{A_{L^{\\prime \\prime },L^{\\prime }}}$ -measurable.",
"Lemma 9.8 Let $\\varepsilon , \\delta >0$ be small enough.",
"Suppose $L^{\\prime },L^{\\prime \\prime }$ are dyadic, satisfying $(L^{\\prime })^{1+\\frac{1}{2}\\varepsilon }<L^{\\prime \\prime }<(L^{\\prime })^{1+\\varepsilon }$ , and $L^{\\prime }$ is large enough (depending on $\\varepsilon , \\delta $ ).",
"Then for any $\\lambda \\in \\mathcal {I}$ we have $\\mathbb {P}[\\mathcal {E}_{L^{\\prime \\prime },L^{\\prime }}^{(\\lambda )}]\\ge 1-(L^{\\prime })^{-10}$ .",
"Let $\\tilde{L}^{(0)}=L^{\\prime }$ , $\\tilde{L}^{(i+1)}=(\\tilde{L}^{(i)})^{1-\\varepsilon }$ , and $L^{(i)}$ be the (unique) dyadic scale with $L^{(i)}\\in [\\tilde{L}^{(i)}, 2\\tilde{L}^{(i)})$ , for $i\\in \\mathbb {Z}_{\\ge 0}$ .",
"Let $M^{\\prime }\\in \\mathbb {Z}_{+}$ such that $\\frac{1}{10}\\delta <(1-\\varepsilon )^{M^{\\prime }}<\\frac{1}{6}\\delta $ .",
"For any dyadic $2L^{(M^{\\prime })}$ -cube $Q\\subset A_{L^{\\prime \\prime },L^{\\prime }}$ , we call it hereditary bad if there are $\\lambda $ -bad dyadic cubes $Q^{(0)},\\cdots ,Q^{(M^{\\prime })}=Q$ such that, $Q^{(i+1)}\\subset Q^{(i)}\\subset A_{L^{\\prime \\prime },L^{\\prime }}$ for each $0\\le i\\le M^{\\prime }-1$ and $Q^{(i)}$ is a dyadic $2L^{(i)}$ -cube.",
"By (REF ), and the same arguments in the proof of Claim REF , the following is true.",
"For small enough $\\varepsilon $ , there exists $N\\in \\mathbb {Z}_{+}$ depending on $\\varepsilon ,\\delta $ , such that with probability at least $1-(L^{\\prime })^{-10}$ , $|\\lbrace Q\\subset A_{L^{\\prime \\prime },L^{\\prime }}:\\text{$Q$ is a hereditary bad $2L^{(M^{\\prime })}$-cube}\\rbrace |<N.$ Let $G^{(\\lambda )}_{L^{\\prime \\prime },L^{\\prime }}=\\bigcup \\lbrace Q\\subset A_{L^{\\prime \\prime },L^{\\prime }}:\\text{$Q$ is a hereditary bad $2L^{(M^{\\prime })}$-cube}\\rbrace $ .",
"Then (REF ) implies $|G^{(\\lambda )}_{L^{\\prime \\prime },L^{\\prime }}|\\le N (2L^{(M^{\\prime })}+1)^{3}\\le (L^{\\prime })^{\\frac{\\delta }{2}}$ for large enough $L^{\\prime }$ .",
"For each $a\\in A_{L^{\\prime \\prime },2L^{\\prime }}\\setminus G^{(\\lambda )}_{L^{\\prime \\prime },L^{\\prime }}$ , there is $0\\le i^{\\prime }\\le M^{\\prime }$ and a $\\lambda $ -good cube $Q_{L^{(i^{\\prime })}}(b)\\subset A_{L^{\\prime \\prime },L^{\\prime }}$ such that $\\operatorname{dist}(a,Q_{L^{\\prime \\prime }}\\setminus Q_{L^{(i^{\\prime })}}(b))\\ge \\frac{1}{8}L^{(i^{\\prime })}$ .",
"Since $(L^{\\prime })^{\\frac{\\delta }{10}}\\le L^{(i^{\\prime })}\\le L^{\\prime }$ , our claim follows.",
"For any large enough dyadic scales $L^{\\prime },L^{\\prime \\prime }$ with $(L^{\\prime })^{1+\\frac{1}{2}\\varepsilon }<L^{\\prime \\prime }<(L^{\\prime })^{1+\\varepsilon }$ , we denote $\\mathcal {E}_{L^{\\prime \\prime },L^{\\prime }}^{supp}=\\bigcap _{\\lambda \\in \\sigma (H_{Q_{L^{\\prime }}})\\cap \\mathcal {I}}\\mathcal {E}_{L^{\\prime \\prime },L^{\\prime }}^{(\\lambda )}$ .",
"Then by Lemma REF , as each $\\mathcal {E}_{L^{\\prime \\prime },L^{\\prime }}^{(\\lambda )}$ is $V_{A_{L^{\\prime \\prime },L^{\\prime }}}$ -measurable, we have $\\mathbb {P}[\\mathcal {E}_{L^{\\prime \\prime },L^{\\prime }}^{supp}]\\ge 1-(L^{\\prime })^{-6}.$ [Proof of Proposition REF ] In this proof we let $\\varepsilon >0$ be a small universal constant, and $\\delta >0$ be a number depending on $\\delta ^{\\prime }$ .",
"Both of them are to be determined.",
"Now we fix dyadic scale $L$ large enough (depending on $\\epsilon ,\\delta $ and thus depending on $\\delta ^{\\prime }$ ).",
"Let $\\tilde{L}_{0}=L$ , $\\tilde{L}_{i+1}=\\tilde{L}^{1-\\frac{3}{4}\\varepsilon }_{i}$ , and $L_{i}$ be the (unique) dyadic scale with $L_{i}\\in [\\tilde{L}_{i}, 2\\tilde{L}_{i})$ , for $i\\in \\mathbb {Z}_{\\ge 0}$ .",
"Pick $M\\in \\mathbb {Z}_{+}$ such that $\\frac{1}{10}\\delta <(1-\\frac{3}{4}\\varepsilon )^{M}< \\frac{1}{6}\\delta $ .",
"Write $\\overline{L_{i}}=\\frac{1}{16} L_{i}$ for $0\\le i\\le M$ and let $\\mathcal {E}^{supp}=\\bigcap _{0\\le i\\le M-1} \\mathcal {E}^{supp}_{L_{i},\\overline{L_{i+1}}}.$ Then by (REF ), $\\mathbb {P}[\\mathcal {E}^{supp}]\\ge 1-M \\left(\\frac{L_{M}}{16}\\right)^{-6}\\ge 1-L^{-\\frac{\\delta }{2}}$ as $L$ is large enough.",
"For $0\\le i\\le M$ , denote by $\\Theta _{i}$ the set of eigenvalues $\\lambda \\in \\sigma (H_{Q_{L_{i}}})$ such that, $\\lambda \\in [(M-i+1)\\exp (-L^{\\frac{\\delta }{20}}),\\lambda _{*}-(M-i+1)\\exp (-L^{\\frac{\\delta }{20}})],$ and $\\operatorname{dist}(\\lambda ,\\sigma (H_{Q_{\\overline{L_{j}}}})),\\operatorname{dist}(\\lambda ,\\sigma (H_{Q_{L_{j}}})) \\le 2^{i}\\exp (-c^{\\prime } L_{j}) \\quad \\forall j\\in \\lbrace i,i+1,\\cdots ,M\\rbrace .$ Here the constant $c^{\\prime }=\\frac{c_1}{20}$ where $c_1$ is the constant from Proposition REF .",
"Claim 9.9 Under the event $\\mathcal {E}^{supp}$ , for any $1\\le i\\le M$ and $\\lambda \\in \\Theta _{i}$ , there exists $G^{(i-1)}\\subset Q_{L_{i-1}}$ with $10\\le |G^{(i-1)}|\\le L^{\\frac{2}{3}\\delta }$ such that the following holds.",
"For any $\\lambda ^{\\prime }\\in \\sigma (H_{Q_{L_{i-1}}})$ and $u\\in \\ell ^{2}(Q_{L_{i-1}})$ with $|\\lambda -\\lambda ^{\\prime }|\\le 2^{i-1}\\exp (-c^{\\prime } L_{i})$ and $H_{Q_{L_{i-1}}} u=\\lambda ^{\\prime } u$ , we have $\\Vert u\\Vert _{\\ell ^{2}(G^{(i-1)})}\\ge (1-|G^{(i-1)}|^{-2})\\Vert u\\Vert _{\\ell ^{2}(Q_{L_{i-1}})}$ .",
"Since $\\lambda \\in \\Theta _{i}$ , there are $\\lambda ^{(j)}\\in \\sigma (H_{Q_{\\overline{L_{j}}}})$ such that $|\\lambda -\\lambda ^{(j)}|\\le 2^{i}\\exp (-c^{\\prime } L_{j})$ for each $i\\le j\\le M$ .",
"Let $G^{(i-1)}_{*}=\\bigcup _{i-1\\le j\\le M-1} G^{(\\lambda ^{(j+1)})}_{L_{j},\\overline{L_{j+1}}}.$ Then $|G^{(i-1)}_{*}|\\le M L^{\\frac{\\delta }{2}}$ .",
"Suppose $\\lambda ^{\\prime }$ and $u$ satisfy the hypothesis.",
"Then $|\\lambda ^{\\prime }-\\lambda ^{(j)}|\\le |\\lambda ^{\\prime }-\\lambda |+|\\lambda -\\lambda ^{(j)}|\\le 2^{i-1}\\exp (-c^{\\prime } L_{i})+2^{i}\\exp (-c^{\\prime } L_{j})\\le 2^{M+1}\\exp (-c^{\\prime } L_{j})$ for each $i\\le j\\le M$ .",
"Denote $L^{\\prime }_{j}=\\frac{1}{2}L_{j}$ for each $i\\le j\\le M-1$ and $L^{\\prime }_{i-1}=L_{i-1}$ .",
"Pick an arbitrary $a\\in Q_{L_{i-1}}\\setminus Q_{L_{M}}$ , there exists $j^{\\prime }\\in \\lbrace i-1,\\cdots ,M-1\\rbrace $ such that $a\\in A_{L^{\\prime }_{j^{\\prime }},2\\overline{L_{j^{\\prime }+1}}}$ .",
"If $a\\notin G^{(i-1)}_{*}$ , by definition of $G^{\\lambda ^{(j^{\\prime }+1)}}_{L_{j^{\\prime }},\\overline{L_{j^{\\prime }+1}}}$ , there exists a $\\lambda ^{(j^{\\prime }+1)}$ -good cube $Q_{L^{\\prime \\prime \\prime }}(b)$ such that $\\overline{L_{j^{\\prime }+1}}\\ge L^{\\prime \\prime \\prime }\\ge \\overline{L_{j^{\\prime }+1}}^{\\frac{\\delta }{10}}\\ge L^{\\frac{\\delta ^{2}}{100}}$ , and $\\operatorname{dist}(a,Q_{L_{j^{\\prime }}}\\setminus Q_{L^{\\prime \\prime \\prime }}(b))\\ge \\frac{1}{8}L^{\\prime \\prime \\prime }$ .",
"Then since $a\\in Q_{L^{\\prime }_{j^{\\prime }}}$ , we have $\\operatorname{dist}(a,Q_{L_{i-1}}\\setminus Q_{L^{\\prime \\prime \\prime }}(b))\\ge \\frac{1}{8}L^{\\prime \\prime \\prime }$ .",
"We also have that $|\\lambda ^{\\prime }-\\lambda ^{(j^{\\prime }+1)}|\\le 2^{M+1}\\exp (-c^{\\prime } L_{j^{\\prime }+1})\\le 2^{M+1}\\exp (-16 c^{\\prime } L^{\\prime \\prime \\prime }).$ Then by Claim REF we have, $|u(a)|\\le 2\\exp \\left((L^{\\prime \\prime \\prime })^{1-\\lambda _{*}}-\\frac{1}{8}\\lambda _{*}L^{\\prime \\prime \\prime }\\right) \\Vert u\\Vert _{\\ell ^{1}(Q_{L_{i-1}})}\\le L^{-10} \\Vert u\\Vert _{\\ell ^{2}(Q_{L_{i-1}})}.$ Hence, by letting $G^{(i-1)}=G^{(i-1)}_{*}\\cup Q_{L_{M}}$ , we have $10\\le |G^{(i-1)}|\\le |G^{(i-1)}_{*}|+ |Q_{L_{M}}|\\le ML^{\\frac{\\delta }{2}}+100L^{\\frac{\\delta }{2}}\\le L^{\\frac{2}{3}\\delta }$ , and $\\Vert u\\Vert _{\\ell ^{2}(G^{(i-1)})}\\ge \\big (1- (2L_{i-1}+1)^{3} L^{-20}\\big )^{\\frac{1}{2}} \\Vert u\\Vert _{\\ell ^{2}(Q_{L_{i-1}})} \\ge (1-|G^{(i-1)}|^{-2}) \\Vert u\\Vert _{\\ell ^{2}(Q_{L_{i-1}})}.$ Thus our claim follows.",
"Claim 9.10 Under the event $\\mathcal {E}^{supp}$ , for any $1\\le i\\le M$ and $\\lambda \\in \\Theta _{i}$ , we have $|\\lbrace \\lambda ^{\\prime }\\in \\sigma (H_{Q_{L_{i-1}}}):|\\lambda -\\lambda ^{\\prime }|\\le 2^{i-1}\\exp (-c^{\\prime } L_{i})\\rbrace |\\le 2L^{\\frac{2}{3}\\delta }.$ Let $\\lambda _{1},\\cdots ,\\lambda _{p} \\in \\sigma (H_{Q_{L_{i-1}}})$ be all the eigenvalues (counting with multiplicity) in the interval $[\\lambda -2^{i-1}\\exp (-c^{\\prime } L_{i}),\\lambda +2^{i-1}\\exp (-c^{\\prime } L_{i})].$ Let $u_{1},\\cdots ,u_{p}$ be the corresponding (mutually orthogonal) eigenvectors with $H_{Q_{L_{i-1}}}u_{s}=\\lambda _{s} u_{s}$ and $\\Vert u_{s}\\Vert _{\\ell ^{2}(Q_{L_{i-1}})}=1$ for $1\\le s\\le p$ .",
"By Claim REF , $\\Vert u_{s}\\Vert _{\\ell ^{2}(G^{(i-1)})}\\ge 1-|G^{(i-1)}|^{-2}$ for $1\\le s\\le p$ .",
"Thus we have $|\\langle u_{s_{1}},u_{s_{2}} \\rangle _{\\ell ^{2}(G^{(i-1)})}-{1}_{s_{1}=s_{2}}|\\le 2|G^{(i-1)}|^{-2}$ for $1\\le s_{1},s_{2}\\le p$ .",
"By Lemma REF , we have $p\\le 2|G^{(i-1)}|\\le 2L^{\\frac{2}{3}\\delta }$ .",
"Claim 9.11 We have $|\\Theta _{0}|\\le L^{M \\delta }$ under the event $\\mathcal {E}^{supp}$ .",
"Suppose $\\mathcal {E}^{supp}$ holds.",
"For each $1\\le i\\le M$ and $\\lambda \\in \\Theta _{i-1}$ , there are $\\lambda ^{(j)}\\in \\sigma (H_{Q_{L_{j}}})$ and $\\overline{\\lambda ^{(j)}}\\in \\sigma (H_{Q_{\\overline{L_{j}}}})$ with $|\\lambda -\\lambda ^{(j)}|, |\\lambda -\\overline{\\lambda ^{(j)}}|\\le 2^{i-1}\\exp (-c^{\\prime } L_{j})$ , for $i\\le j\\le M$ .",
"In particular, $|\\lambda -\\lambda ^{(i)}|\\le 2^{i-1}\\exp (-c^{\\prime } L_{i})$ .",
"Thus $|\\lambda ^{(i)}-\\lambda ^{(j)}|\\le 2^{i-1}(\\exp (-c^{\\prime } L_{j})+\\exp (-c^{\\prime } L_{i}))\\le 2^{i}\\exp (-c^{\\prime } L_{j})$ and similarly $|\\lambda ^{(i)}-\\overline{\\lambda ^{(j)}}|\\le 2^{i}\\exp (-c^{\\prime } L_{j})$ for $i\\le j\\le M$ .",
"Moreover, $\\lambda \\in \\Theta _{i-1}$ implies that $\\lambda \\in [(M-i+2)\\exp (-L^{\\frac{\\delta }{20}}),\\lambda _{*}-(M-i+2)\\exp (-L^{\\frac{\\delta }{20}})]$ and thus $\\lambda ^{(i)}\\in [(M-i+1)\\exp (-L^{\\frac{\\delta }{20}}),\\lambda _{*}-(M-i+1)\\exp (-L^{\\frac{\\delta }{20}})].$ These imply $\\lambda ^{(i)}\\in \\Theta _{i}$ .",
"Hence, we have $\\Theta _{i-1}\\subset \\lbrace \\lambda \\in \\sigma (H_{Q_{L_{i-1}}}):\\operatorname{dist}(\\lambda ,\\Theta _{i})\\le 2^{i-1}\\exp (-c^{\\prime } L_{i})\\rbrace .$ Together with Claim REF , we have $|\\Theta _{i-1}|\\le 2L^{\\frac{2}{3}\\delta } |\\Theta _{i}|$ for $1\\le i \\le M$ .",
"Since $|\\Theta _{M}|\\le |\\sigma (H_{Q_{L_{M}}})|\\le 10L_{M}^{3}\\le 100 L^{\\frac{\\delta }{2}}$ , we have $|\\Theta _{0}|\\le 100 L^{\\frac{\\delta }{2}}\\cdot 2^{M} L^{\\frac{2}{3} M \\delta } \\le L^{M\\delta }$ .",
"Now we denote $\\mathcal {E}_{sloc}^{(L)}=\\mathcal {E}^{supp}\\cap \\bigcap _{0\\le i\\le M} \\mathcal {E}^{(L_{i})}_{wloc}\\cap \\mathcal {E}_{wloc}^{(\\overline{L_{i}})}$ .",
"By Proposition REF and (REF ), $\\mathbb {P}[ \\mathcal {E}_{sloc}^{(L)}]\\ge 1- L^{-\\frac{\\delta }{2}} -2(M+1) L^{-\\frac{1}{20}\\kappa ^{\\prime }\\delta }\\ge 1-L^{-\\kappa ^{\\prime \\prime }}$ for some small $\\kappa ^{\\prime \\prime }>0$ depending on $\\delta ,M$ .",
"Take $c_2=\\min \\lbrace \\frac{\\delta }{30}, c^{\\prime }\\rbrace $ .",
"Under the event $\\mathcal {E}_{sloc}^{(L)}$ , for any $\\lambda \\in \\sigma _{k}(H)\\cap [\\exp (-L^{c_2}),\\lambda _{*}-\\exp (-L^{c_2})],$ we claim that $\\operatorname{dist}(\\lambda ,\\Theta _{0})\\le \\exp (-c_2 L).$ To see this, by definition of $\\mathcal {E}^{(L_{i})}_{wloc}$ and $ \\mathcal {E}_{wloc}^{(\\overline{L_{i}})}$ , (REF ) implies $\\operatorname{dist}(\\lambda ,\\sigma (H_{Q_{L_{i}}}))\\le \\exp (-c_1 L_{i})$ and $\\operatorname{dist}(\\lambda ,\\sigma (H_{Q_{\\overline{L_{i}}}})) \\le \\exp (-\\frac{c_1}{16} L_{i})$ for each $0\\le i\\le M$ .",
"In particular, there is $\\lambda _{0}\\in \\sigma (H_{Q_{L}})$ such that $|\\lambda -\\lambda _{0}|\\le \\exp (-c_1 L)$ .",
"Since $c^{\\prime }=\\frac{c_1}{20}$ , we have $\\lambda _{0}\\in \\left[(M+1)\\exp (-L^{\\frac{\\delta }{20}}),\\lambda _{*}-(M+1)\\exp (-L^{\\frac{\\delta }{20}})\\right]$ by (REF ), and also $\\begin{split}&\\operatorname{dist}(\\lambda _{0},\\sigma (H_{Q_{L_{i}}}))\\le |\\lambda -\\lambda _{0}|+\\operatorname{dist}(\\lambda ,\\sigma (H_{Q_{L_{i}}}))\\le \\exp (-c_1 L)+\\exp (-c_1 L_{i})\\le \\exp (-c^{\\prime } L_{i}),\\\\&\\operatorname{dist}(\\lambda _{0},\\sigma (H_{Q_{\\overline{L_{i}}}}))\\le |\\lambda -\\lambda _{0}|+\\operatorname{dist}(\\lambda ,\\sigma (H_{Q_{\\overline{L_{i}}}}))\\le \\exp (-c_1 L)+\\exp (-\\frac{c_1}{16} L_{i})\\le \\exp (-c^{\\prime } L_{i}),\\end{split}$ for $0\\le i\\le M$ .",
"Hence $\\lambda _{0}\\in \\Theta _{0}$ and (REF ) follows.",
"Finally, observe that $|\\Theta _{0}|\\le L^{M \\delta }\\le L^{\\frac{\\log (\\frac{1}{10}\\delta )}{\\log (1-\\frac{3}{4}\\varepsilon )} \\delta }\\le L^{\\delta ^{\\prime }}$ by taking $\\delta $ small enough (depending on $\\delta ^{\\prime }$ ), the proposition follows by letting $S=\\Theta _{0}$ ."
]
] | 1906.04350 |
[
[
"Compact inverse categories"
],
[
"Abstract The Ehresmann-Schein-Nambooripad theorem gives a structure theorem for inverse monoids: they are inductive groupoids.",
"A particularly nice case due to Jarek is that commutative inverse monoids become semilattices of abelian groups.",
"It has also been categorified by DeWolf-Pronk to a structure theorem for inverse categories as locally complete inductive groupoids.",
"We show that in the case of compact inverse categories, this takes the particularly nice form of a semilattice of compact groupoids.",
"Moreover, one-object compact inverse categories are exactly commutative inverse monoids.",
"Compact groupoids, in turn, are determined in particularly simple terms of 3-cocycles by Baez-Lauda."
],
[
"Introduction",
"Inverse monoids model partial symmetry [24], and arise naturally in many combinatorial constructions [8].",
"The easiest example of an inverse monoid is perhaps a group.",
"There is a structure theorem for inverse monoids, due to Ehresmann-Schein-Nambooripad [9], [10], [27], [26], that exhibits them as inductive groupoids.",
"The latter are groupoids internal to the category of partially ordered sets with certain extra requirements.",
"By a result of Jarek [19], the inductive groupoids corresponding to commutative inverse monoids can equivalently be described as semilattices of abelian groups.",
"A natural typed version of an inverse monoid is an inverse category [22], [6].",
"This notion can for example model partial reversible functional programs [12].",
"The easiest example of an inverse category is perhaps a groupoid.",
"DeWolf-Pronk have generalised the ESN theorem to inverse categories, exhibiting them as locally complete inductive groupoids.",
"This paper investigates `the commutative case', thus fitting in the bottom right cell of Figure REF .",
"Figure: Overview of structure theorems for inverse categories.However, let us emphasise two ways in which Figure REF is overly simplified.",
"First, the term `commutative case' is misleading: we mean considering compact inverse categories.",
"More precisely, we prove that compact inverse categories correspond to semilattices of compact groupoids.",
"Compact inverse categories are only commutative in that their endohomset of scalars is always commutative.",
"In particular, the categorical composition of the compact inverse category can be as noncommutative as you like.",
"We expect that the tensor product also need not be symmetric.",
"But compact categories are interesting in their own right: they model quantum entanglement [17]; they model linear logic [29]; and they naturally extend traced monoidal categories modelling feedback [20].",
"Second, our result is not a straightforward special case of DeWolf-Pronk [7], nor of Jarek [19], but instead rather a common categorification.",
"We prove that one-object compact inverse categories are exactly commutative inverse monoids.",
"Semilattices of groupoids are a purely categorical notion, whereas ordered groupoids have more ad hoc aspects.",
"Compact groupoids are also known as 2-groups or crossed modules, and have fairly rigid structure themselves, due to work by Baez and Lauda [5].",
"We take advantage of this fact to ultimately show that there is a (weak) 2-equivalence of (weak) 2-categories of compact inverse categories, and semilattices of 3-cocycles.",
"Section starts by recalling the ESN structure theorem for inverse monoids, and its special commutative case due to Jarek in a language that the rest of the paper will follow.",
"Section discusses the generalisation of the ESN theorem to inverse categories due to DeWolf and Pronk, and its relation to semilattices of groupoids.",
"Section is the heart of the paper, and considers additional structure on inverse categories that was hidden for inverse monoids.",
"It shows that the construction works for compact inverse categories, and argues that this is the right generalisation of inverse monoids in this sense.",
"After all this theory, Section lists examples.",
"We have chosen to treat examples after theory; that way they can illustrate not just compact inverse categories, but also the construction of the structure theorem itself.",
"Section then moves to a 2-categorical perspective, to connect to the structure theorem for compact groupoids due to Baez and Lauda.",
"Finally, Section discusses the many questions left open and raised in the paper."
],
[
"Inverse monoids",
"An inverse monoid is a monoid where every element $x$ has a unique element $x^\\dag $ satisfying $x=xx^\\dag x$ and $x^\\dag =x^\\dag x x^\\dag $ [24].",
"Equivalently, the monoid carries an involution $\\dag $ such that $x=xx^\\dag x$ and $xx^\\dag yy^\\dag =yy^\\dag y xx^\\dag $ for all elements $x$ and $y$ .",
"Inverse monoids and involution-respecting homomorphisms form a category $\\mathbf {InvMon}$ , and commutative inverse monoids form a full subcategory $\\mathbf {cInvMon}$ .",
"This section recalls structure theorems for inverse monoids.",
"In general they correspond to inductive groupoids by the Ehresmann-Schein-Nambooripad theorem [9], [10], [26], [27], that we now recall.",
"Definition 1 A (bounded meet-)semilattice is a partially ordered set with a greatest element $\\top $ , in which any two elements $s$ and $t$ have a greatest lower bound $s \\wedge t$ .",
"A morphism of semilattices is a function $f$ satisfying $f(\\top )=\\top $ and $f(s \\wedge t) = f(s) \\wedge f(t)$ .",
"We regard a semilattice as a category by letting elements be objects and having a unique morphism $s \\rightarrow t$ when $s \\le t$ , that is, when $s \\wedge t = s$ .",
"We will disregard size issues altogether; either by restricting to small categories throughout the article, or by allowing semilattices (and monoids) that are large – the only place it seems to matter is Lemma REF below.",
"Recall that a groupoid is a category whose every morphism is invertible.",
"Definition 2 An ordered groupoid is a groupoid internal to the category of partially ordered sets and monotone functions, together with a choice of restriction $(f|A) \\colon A \\rightarrow B$ for each $f \\colon A^{\\prime } \\rightarrow B$ and $A \\le A^{\\prime }$ satisfying $(f|A)\\le f$ .",
"Explicitly, the sets $G_0$ and $G_1$ of objects and arrows are partially ordered, and the functions $\\begin{tikzpicture}[xscale=2.5,font=\\small ]\\node (0) at (0,0){G_0};\\node (1) at (1,0){G_1};\\node (2) at (2,0){G_2};[->] (0) to node[above=-.5mm]{id} (1);[->] (1) to[out=-135,in=-45] node[below=-1mm]{cod} (0);[->] (1) to[out=135,in=45] node[above]{dom} (0);[->] (2) to node[above]{comp} (1);[->] (1) to[out=60,in=0] +(0,.7) node[above]{inv} to[out=180,in=120] (1);\\end{tikzpicture}$ are all monotone, where $G_2=\\lbrace (g,f) \\in G_1^2 \\mid \\mathrm {dom}(g)=\\mathrm {cod}(f) \\rbrace $ is ordered by $(g,f) \\le (g^{\\prime },f^{\\prime })$ when $g \\le g^{\\prime }$ and $f \\le f^{\\prime }$ .",
"An inductive groupoid is an ordered groupoid whose partially ordered set of objects forms a semilattice.",
"A morphism of ordered groupoids is a functor $F$ that is monotone in morphisms, that is, $F(f)\\le F(g)$ when $f \\le g$ .",
"Inductive groupoids and their morphisms form a category $\\mathbf {IndGpd}$ .",
"Theorem 3 There is an equivalence $\\mathbf {InvMon} \\simeq \\mathbf {IndGpd}$ .",
"See [24] or [7] for details.",
"An inverse monoid $M$ turns into an inductive groupoid as follows.",
"Objects are idempotents $ss^\\dag =s \\in M$ .",
"Every element of $M$ is a morphism $x \\colon x^\\dag x \\rightarrow xx^\\dag $ .",
"The identity on $s$ is $s$ itself, and composition is given by multiplication in $M$ .",
"Inverses are given by $x^{-1}=x^\\dag $ .",
"The order $x \\le y$ holds when $x=yx^\\dag x$ .",
"The restriction of $x \\colon x^\\dag x \\rightarrow xx^\\dag $ to $s^\\dag s = s \\le x^\\dag x$ is $xs$ .",
"Observe from the proof of the previous theorem that commutative inverse monoids correspond to inductive groupoids where every morphism is an endomorphism.",
"Moreover, the endohomsets are abelian groups.",
"Hence commutative inverse monoids correspond to a semilattice of abelian groups.",
"Definition 4 A semilattice over a subcategory $\\mathbf {V}$ of $\\mathbf {Cat}$ is a functor $F \\colon \\mathbf {S}^\\textrm {\\rm op}\\rightarrow \\mathbf {V}$ where $\\mathbf {S}$ is a semilattice and all categories $F(s)$ have the same objects.",
"A morphism of semilattices $F \\rightarrow F^{\\prime }$ over $\\mathbf {V}$ is a morphism of semilattices $\\varphi \\colon \\mathbf {S} \\rightarrow \\mathbf {S^{\\prime }}$ together with a natural transformation $\\theta \\colon F \\Rightarrow F^{\\prime } \\circ \\varphi $ .",
"Write $\\mathbf {SLat}[\\mathbf {V}]$ for the category of semilattices over $\\mathbf {V}$ and their morphisms.",
"The ordinary category of semilattices can be recovered by choosing $\\mathbf {V}$ to be the category containing as its single object the terminal category $\\mathbf {1}$ .",
"In the commutative case, the ESN theorem simplifies, as worked out by Jarek [19].",
"The following formulation chooses $\\mathbf {V}=\\mathbf {Ab}$ , regarding an abelian group as a one-object category.",
"Theorem 5 If $M$ is a commutative inverse monoid, then $\\mathbf {S} = \\lbrace s \\in M \\mid ss^\\dag =s \\rbrace , \\qquad s \\wedge t = st, \\qquad \\top =1,$ is a semilattice, and for each $s \\in \\mathbf {S}$ , $F(s) = \\lbrace x \\in M \\mid xx^\\dag =s \\rbrace $ is an abelian group with multiplication inherited from $M$ and unit $s$ , giving a semilattice of abelian groups $F \\colon \\mathbf {S} \\rightarrow \\mathbf {Ab}$ by $F(s \\le t)(x) \\rightarrow sx$ .",
"If $F \\colon \\mathbf {S} \\rightarrow \\mathbf {Ab}$ is a semilattice of abelian groups, then $M = \\coprod _{s \\in \\mathbf {S}} F(s)$ is a commutative inverse monoid under $xy & = F(s \\wedge t \\le s)(x) \\cdot F(s \\wedge t \\le t)(y) && \\text{ if }x \\in F(s),\\ y \\in F(t), \\\\x^\\dag & = x^{-1} \\in F(s) && \\text{ if }x \\in F(s), \\\\1 & = 1 \\in F(\\top ).$ This gives an equivalence $\\mathbf {cInvMon} \\simeq \\mathbf {SLat}[\\mathbf {Ab}]$ .",
"First, let $M$ be an inverse monoid.",
"To see that $\\mathbf {S}$ is a semilattice, it suffices to show that it is a commutative idempotent monoid.",
"Commutativity is inherited from $M$ , and idempotence follows from the fact that $M$ is an inverse monoid: $(xx^\\dag )^2=xx^\\dag xx^\\dag =xx^\\dag $ .",
"Next we verify that each $F(s)$ is an abelian group.",
"It is closed under multiplication: if $x,y \\in F(s)$ , then $(xy)(xy)^\\dag = xx^\\dag y^\\dag y = ss^\\dag =s$ so also $xy \\in F(s)$ .",
"It has $s$ as a unit: if $x \\in F(s)$ , then $sx=xx^\\dag x=x$ .",
"The inverse of $x \\in F(s)$ is given by $x^\\dag $ , because $xx^\\dag =s$ by definition.",
"Furthermore, the diagram $F$ is functorial: clearly $F(s \\le t) \\circ F(r \\le s) (x)=Rx=F(r \\le t)(x)$ , and $F(s \\le s)(x)=sx=xx^\\dag x=x$ .",
"It is also well-defined: if $s \\le t$ and $x \\in F(t)$ , then $sx(sx)^\\dag =sxx^\\dag s^\\dag = sts^\\dag = ss^\\dag =s$ so $sx \\in F(t)$ .",
"Now let $F \\in \\mathbf {SLat}[\\mathbf {Ab}]$ .",
"Then $1 \\in F(\\top )$ acts as a unit in $M$ : if $x \\in F(s)$ then $x 1 = F(s \\le s)(x) \\cdot F(s \\le \\top )(1) = x \\cdot 1 = x \\in F(s)$ .",
"The multiplication is clearly associative and commutative, so $M$ is an abelian monoid.",
"It is an inverse monoid because $xx^\\dag x = xx^{-1}x = x$ is computed within $F(s)$ .",
"Next we move to morphisms.",
"Given a morphism $f \\colon M \\rightarrow M^{\\prime }$ of commutative inverse monoids, define a morphism $F \\rightarrow F^{\\prime }$ of their associated semilattices of abelian groups as follows: $\\varphi \\colon \\mathbf {S} \\rightarrow \\mathbf {S^{\\prime }}$ is just $\\varphi (s)=f(s)$ , and $\\theta _s \\colon F(s) \\rightarrow F^{\\prime }(f(s))$ is just $\\theta _s(x)=f(x)$ .",
"This is clearly functorial $\\mathbf {cInvMon} \\rightarrow \\mathbf {SLat}[\\mathbf {Ab}]$ .",
"Conversely, given a morphism $(\\varphi ,\\theta ) \\colon F \\rightarrow F^{\\prime }$ of semilattices of abelian groups, define a homomorphism $M \\rightarrow M^{\\prime }$ of their associated commutative inverse monoids by $F(s) \\ni x \\mapsto \\theta _s(x) \\in F(\\varphi (s))$ .",
"This is clearly functorial $\\mathbf {SLat}[\\mathbf {Ab}] \\rightarrow \\mathbf {cInvMon}$ .",
"Finally, turning a commutative inverse monoid $M$ into a semilattice of abelian groups and that in turn into a commutative inverse monoid ends up with the exact same monoid $M$ .",
"A semilattice of abelian groups $F \\colon \\mathbf {S} \\rightarrow \\mathbf {Ab}$ gets mapped to the inverse monoid $\\coprod _s F(s)$ , which in turn gets mapped to the following semilattice of abelian groups $G \\colon \\mathbf {T} \\rightarrow \\mathbf {Ab}$ .",
"The semilattice $\\mathbf {T}$ is given by $\\lbrace t \\in F(s) \\mid s \\in \\mathbf {S}, t=tt^\\dag \\rbrace =\\lbrace t \\in F(s) \\mid s \\in \\mathbf {S}, t=tt^{-1}=1\\rbrace =\\lbrace 1 \\in F(s) \\mid s \\in \\mathbf {S}\\rbrace $ ; clearly $s \\mapsto 1 \\in F(s)$ is an isomorphism $\\varphi \\colon \\mathbf {S} \\rightarrow \\mathbf {T}$ .",
"The abelian group $G(\\varphi (s))$ is given by $\\lbrace x \\mid xx^\\dag =s \\rbrace = \\lbrace x \\mid 1=xx^{-1}=s\\rbrace = \\lbrace x \\in F(s)\\rbrace $ ; clearly $x \\mapsto x$ is a natural isomorphism $\\theta _s \\colon F(s) \\rightarrow G(\\varphi (s))$ .",
"Thus $G \\simeq F$ , and the two functors implement an equivalence."
],
[
"Inverse categories",
"This section extends the previous one to a typed setting.",
"A dagger category is a category with a contravariant involution $\\dag $ that acts as the identity on objects.",
"A dagger functor is a functor between dagger categories satisfying $F(f^\\dag )=F(f)^\\dag $ .",
"An inverse category is a dagger category where $f = f f^\\dag f$ and $ff^\\dag gg^\\dag = gg^\\dag ff^\\dag $ for any pair of morphisms $f$ and $g$ with the same domain [6].",
"Equivalently, it is a category where every morphism $f\\colon A \\rightarrow B$ allows a unique morphism $f^\\dag \\colon B \\rightarrow A$ satisfying $f=ff^\\dag f$ and $f^\\dag =f^\\dag f f^\\dag $ ; thus every functor between inverse categories is in fact a dagger functor.",
"Inverse categories and (dagger) functors form a category $\\mathbf {InvCat}$ , and groupoids and functors form a full subcategory $\\mathbf {Gpd}$ .",
"The ESN theorem extends to inverse categories, as worked out by DeWolf and Pronk [7].",
"Definition 6 A locally complete inductive groupoid is an ordered groupoid with a partition of the semilattice $G_0$ of objects into semilattices $\\lbrace M_i\\rbrace $ such that two objects are comparable if and only if they are in the same semilattice $M_i$ .",
"Locally complete inductive groupoids form a subcategory $\\mathbf {lcIndGpd}$ of $\\mathbf {IndGpd}$ of those functors that preserve greatest lower bounds of objects.",
"Theorem 7 There is an equivalence $\\mathbf {InvCat}\\simeq \\mathbf {lcIndGpd}$ .",
"See [7] for details.",
"An inverse category $\\mathbf {C}$ turns into a locally complete inductive groupoid as follows.",
"Objects are idempotents $ff^\\dag $ for some endomorphism $f \\colon A \\rightarrow A$ in $\\mathbf {C}$ .",
"These partition into the semilattices of idempotents on a fixed object $A$ .",
"Every morphism $f \\colon A \\rightarrow B$ of $\\mathbf {C}$ becomes a morphism $f^\\dag f \\rightarrow ff^\\dag $ .",
"The identity on $ff^\\dag $ is $ff^\\dag $ itself, and composition is inherited from $\\mathbf {C}$ .",
"Inverses are given by $f^{-1}=f^\\dag $ .",
"The order $f \\le g$ holds when $f=gf^\\dag f$ ; clearly two identity morphisms are comparable exactly when they endomorphisms on the same object.",
"The restriction of $f \\colon f^\\dag f \\rightarrow ff^\\dag $ to $s^\\dag s=s \\le f^\\dag f$ is $fs$ .",
"Lemma 8 If $F \\colon \\mathbf {S}^\\textrm {\\rm op}\\rightarrow \\mathbf {Gpd}$ is a semilattice of groupoids, there is a well-defined inverse category $\\mathbf {C}$ with the same objects as $F(\\top )$ and morphisms $\\mathbf {C}(A,B)=\\coprod _{s \\in \\mathbf {S}} F(s)\\big ( A, B \\big )\\text{.",
"}$ If $(\\varphi ,\\theta )$ is a morphism $F \\rightarrow F^{\\prime }$ of semilattices of groupoids, then there is a dagger functor $\\mathbf {C} \\rightarrow \\mathbf {C^{\\prime }}$ between their associated categories, given by $A \\mapsto \\theta _\\top (A)$ on objects and $F(s) \\ni f \\mapsto \\theta _s(f) \\in F^{\\prime }(\\varphi (s))$ on morphisms.",
"This gives a functor $\\mathbf {SLat}[\\mathbf {Gpd}] \\rightarrow \\mathbf {InvCat}$ .",
"The composition of $f \\in F(s)(A,B)$ and $g \\in F(t)(A,B)$ is given by $F(s \\wedge t \\le t)(g) \\circ F(s \\wedge t \\le s)(f) \\in F(s \\wedge t)(A,C)$ ; this is clearly associative.",
"The identity on $A$ is given by $\\mathrm {id}_{A} \\in F(\\top )(A,A)$ : if $f \\in F(s)(A,B)$ , then $f \\circ \\mathrm {id}_{A} = F(s \\wedge \\top \\le \\top )(\\mathrm {id}_{A}) \\circ F(s \\wedge \\top \\le s)(f) = \\mathrm {id}_{}\\circ F(s \\le s)(f)=f$ .",
"The dagger of $f \\in F(s)(A,B)$ is given by $f^{-1} \\in F(s)(B,A)$ ; this clearly is an inverse category.",
"Combining Theorem REF and Lemma REF , we see that a semilattice of groupoids $F \\colon \\mathbf {S}^\\textrm {\\rm op}\\rightarrow \\mathbf {Gpd}$ gives rise to a locally complete inductive groupoid $\\mathbf {G}$ where: objects are $\\coprod _{A \\in F(\\top )} \\coprod _{s \\in \\mathbf {S}} \\lbrace f^\\dag f \\mid f \\in F(s)(A,A) \\rbrace $ ; there is an arrow $(f^\\dag f)_{A,s} \\rightarrow (ff^\\dag )_{B,s}$ for each $f \\in F(s)(A,B)$ ; the composition of $f \\in F(s)(A,B)$ and $g \\in F(t)(B,C)$ is computed as $F(s \\wedge t \\le t)(g) \\circ F(s \\wedge t \\le s)(f)$ .",
"Not every locally complete inductive groupoid comes from a semilattice of groupoids in this way.",
"Instead, locally complete inductive groupoids correspond to certain functors $\\mathbf {S}^\\textrm {\\rm op}\\rightarrow \\mathbf {Gpd}$ where $\\mathbf {S}$ may be a disjoint union of several semilattices; a `multi-semilattice' of groupoids.",
"Notice that the objects of $\\mathbf {G}$ are doubly-indexed: once by an object of the category $F(\\top )$ , and once by an element of the semilattice $\\mathbf {S}$ .",
"Locally complete inductive groupoids and semilattices of groupoids have different ways of bookkeeping the same data, each emphasising one of these two indices.",
"In the remainder of the paper, we will prefer to work with semilattices of groupoids rather than the more general locally complete inductive groupoids for two reasons.",
"First, the extra structure we will consider does not require `multi-semilattices', but instead is uniform enough so semilattices suffice.",
"Second, semilattices of groupoids form a purely categorical concept, whereas ordered groupoids require extra conditions on groupoids internal to the category of partially ordered sets that are somewhat ad hoc.",
"For example, this perspective will later enable us to remove the restriction that all groupoids in a semilattice of groupoids must have the same objects; see Lemma REF below."
],
[
"Compact inverse categories",
"There is another way to categorify inverse monoids, that takes advantage of a degree of commutativity.",
"Instead of moving from inverse monoids to inverse categories, in this section we move to compact inverse categories.",
"The presence of the tensor product means that the latter specialise to commutative inverse monoids in the one-object case.",
"By a compact inverse category we mean an inverse category that is also a compact dagger category under the same dagger [17].",
"Here, a dagger category is compact when it is symmetric monoidal, $(f \\otimes g)^\\dag = f^\\dag \\otimes g^\\dag $ for all morphisms $f$ and $g$ , all coherence isomorphisms are inverted by their own daggers, and every object $A$ allows an object $A^*$ and a morphism $\\eta _A \\colon I \\rightarrow A^* \\otimes A$ satisfying $\\mathrm {id}_{A} =\\lambda _A \\circ (\\varepsilon \\otimes \\mathrm {id}_{A}) \\circ \\alpha \\circ (\\mathrm {id}_{A} \\otimes \\eta ) \\circ \\rho _A^{-1}$ for $\\varepsilon =\\sigma \\circ \\eta ^\\dag $ where $\\sigma $ is the swap map.",
"Let us first show that compact inverse categories indeed generalise commutative inverse monoids, because the property of compactness is hidden in the one-object case.",
"Proposition 9 One-object compact (dagger/inverse) categories are exactly commutative (involutive/inverse) monoids.",
"Let $M$ be a commutative monoid.",
"Regard it as a one-object monoidal category.",
"The one object is the tensor unit, and in any monoidal category, the tensor unit $I$ is its own dual $I^*=I$ , since $\\eta =\\lambda _I^{-1}$ and $\\varepsilon =\\rho _I$ satisfy (REF ) by coherence [17].",
"If the monoid is involutive/inverse, then the category is clearly dagger/inverse.",
"Conversely, a one-object (dagger) category is clearly an (involutive) monoid.",
"If the category is monoidal, then the monoid is necessarily that of scalars $I \\rightarrow I$ , where tensor and composition coincide and are commutative [1].",
"We now set out to generalise Theorem REF to compact inverse categories $\\mathbf {C}$ .",
"They have the right modicum of commutativity to take advantage of Lemma REF : the monoid $\\mathbf {C}(I,I)$ of scalars is always commutative, any morphism $f \\colon A \\rightarrow B$ can be multiplied with a scalar $s \\colon I \\rightarrow I$ to give $s \\bullet f = \\lambda \\circ (s \\otimes f) \\circ \\lambda ^{-1}$ , and any endomorphism $f \\colon A \\rightarrow A$ has a trace $\\operatorname{Tr}(f) = \\varepsilon \\circ (f \\otimes \\mathrm {id}_{A^*}) \\circ \\sigma \\circ \\eta \\colon I \\rightarrow I$ .",
"Furthermore, any morphism $f \\colon A \\rightarrow B$ has a dual $f^* = (\\mathrm {id}_{A^*} \\otimes \\varepsilon _B) \\circ (\\mathrm {id}_{A^*} \\otimes f \\otimes \\mathrm {id}_{B^*}) \\circ (\\eta _A \\otimes \\mathrm {id}_{B^*}) \\colon B^* \\rightarrow A^*$ , satisfying $\\operatorname{Tr}(f^*)=\\operatorname{Tr}(f)^*$ when $A=B$ .",
"We will write $\\operatorname{tr}(f)$ instead of $\\operatorname{Tr}(f)^*$ .",
"The form of the following lemma resembles the categorical no-cloning theorem [2], and is the heart of the matter.",
"Lemma 10 In a compact inverse category, any endomorphism $f$ equals $\\operatorname{tr}(f) \\bullet \\mathrm {id}_{}$ .",
"Let $f \\colon A \\rightarrow A$ be an endomorphism.",
"Compactness provides $\\eta \\colon I \\rightarrow A^* \\otimes A$ and $\\varepsilon \\colon A \\otimes A^* \\rightarrow I$ satisfying the snake equations.",
"In terms of $g=\\varepsilon \\otimes \\mathrm {id}_{A}$ and $h=\\mathrm {id}_{A} \\otimes \\eta ^\\dag = \\mathrm {id}_{A} \\otimes (\\varepsilon \\circ \\sigma )$ , and suppressing coherence isomorphisms, these equations read $gh^\\dag = \\mathrm {id}_{A} = hg^\\dag $ .",
"It follows that $hh^\\dag = gh^\\dag hh^\\dag = gh^\\dag = \\mathrm {id}_{A}\\text{,}\\\\g^\\dag h = g^\\dag g h^\\dag h = h^\\dag h g^\\dag g = h^\\dag g\\text{.",
"}$ Therefore $g = hh^\\dag g = hg^\\dag h = h$ , and so $f= g \\circ (\\mathrm {id}_{A} \\otimes f^* \\otimes \\mathrm {id}_{A}) \\circ h^\\dag = h \\circ (\\mathrm {id}_{A} \\otimes f^* \\otimes \\mathrm {id}_{A}) \\circ h^\\dag = \\operatorname{Tr}(f^*) \\bullet \\mathrm {id}_{A}\\text{.",
"}$ Proposition 11 A compact dagger category is a compact inverse category if and only if every morphism $f$ satisfies $f=\\operatorname{tr}(f f^\\dag ) \\bullet f$ .",
"Suppose we're given a compact inverse category.",
"By Lemma REF , the endomorphism $f f^\\dag $ equals $\\operatorname{tr}(f f^\\dag f f^\\dag ) \\bullet \\mathrm {id}_{}= \\operatorname{tr}(f f^\\dag ) \\bullet \\mathrm {id}_{}$ .",
"Hence $f=\\operatorname{tr}(f f^\\dag ) \\bullet f$ .",
"Conversely, suppose given a compact dagger category in which every morphism satisfies $f=\\operatorname{tr}(f f^\\dag ) \\bullet f$ .",
"We will prove that this is a restriction category with $\\bar{f} = \\operatorname{tr}(f f^\\dag ) \\bullet \\mathrm {id}_{}$ , by verifying the four axioms [6].",
"First, $f \\bar{f} = \\operatorname{tr}(f f^\\dag ) \\bullet f = f$ .",
"Second, $\\bar{f} \\bar{g} = \\operatorname{tr}(f f^\\dag ) \\bullet \\operatorname{tr}(g g^\\dag ) \\bullet \\mathrm {id}_{}= \\bar{g} \\bar{f}$ if $\\operatorname{dom}(f)=\\operatorname{dom}(g)$ .",
"Third, $& \\operatorname{tr}(ff^\\dag )^\\dag \\circ \\operatorname{tr}(ff^\\dag ) \\\\& = (\\varepsilon \\otimes \\varepsilon ) \\circ (\\sigma \\otimes \\mathrm {id}_{}) \\circ (ff^\\dag \\otimes \\mathrm {id}_{}\\otimes ff^\\dag \\otimes \\mathrm {id}_{}) \\circ (\\mathrm {id}_{}\\otimes \\sigma )\\circ (\\eta \\otimes \\eta ) \\\\& = \\varepsilon \\circ (ff^\\dag ff^\\dag \\otimes \\mathrm {id}_{}) \\circ \\sigma \\circ \\eta \\\\& = \\varepsilon \\circ (ff^\\dag \\otimes \\mathrm {id}_{}) \\circ \\sigma \\circ \\eta \\\\ & = \\operatorname{tr}(ff^\\dag ) $ by Lemma REF .",
"Therefore, for $\\operatorname{dom}(f)=\\operatorname{dom}(g)$ : $\\overline{g \\bar{f}}& = \\overline{\\operatorname{tr}(f f^\\dag ) \\bullet g} \\\\& = \\operatorname{tr}\\big [ \\operatorname{tr}(f f^\\dag )^\\dag \\bullet \\operatorname{tr}(f f^\\dag ) \\bullet g g^\\dag \\big ] \\bullet \\mathrm {id}_{}\\\\& = \\operatorname{tr}(f f^\\dag )^\\dag \\bullet \\operatorname{tr}(f f^\\dag ) \\bullet \\operatorname{tr}(g g^\\dag ) \\bullet \\mathrm {id}_{}\\\\& = \\operatorname{tr}(f f^\\dag ) \\bullet \\operatorname{tr}(g g^\\dag ) \\bullet \\mathrm {id}_{}\\\\& = \\bar{g} \\bar{f}.$ Fourth, $\\bar{g} f= \\operatorname{tr}(g g^\\dag ) \\bullet f= \\operatorname{tr}(g g^\\dag ) \\bullet \\operatorname{tr}(f f^\\dag ) \\bullet f$ , and $f \\overline{g f}= \\operatorname{tr}(g f f^\\dag g^\\dag ) \\bullet f$ .",
"The two are equal by a similar computation as (REF ).",
"Finally, taking $g=f^\\dag $ shows that $\\bar{f}=\\operatorname{tr}(f f^\\dag ) \\bullet \\mathrm {id}_{}= g f$ by Lemma REF and similarly $\\bar{g}=f g$ .",
"Therefore the category is compact inverse [6].",
"Next we build up to generalise Theorem REF , starting with the replacement for abelian groups.",
"A compact groupoid is a compact dagger category where any morphism $f$ is inverted by $f^\\dag $ .",
"Lemma 12 Compact groupoids are precisely compact inverse categories with invertible scalars.",
"Let $\\mathbf {C}$ be a compact inverse category with invertible scalars.",
"By Lemma REF , all endomorphisms are invertible.",
"Let $f \\colon A \\rightarrow B$ be any morphism.",
"Then $f f^\\dag $ is an isomorphism, and so $f$ is (split) monic.",
"Because $f=ff^\\dag f$ , it follows that $ff^\\dag = \\mathrm {id}_{B}$ .",
"Similarly $f^\\dag f$ is an isomorphism, so $f$ is (split) epic, whence $f^\\dag f=\\mathrm {id}_{A}$ .",
"Thus $f$ is invertible.",
"We can now show that any compact inverse category is a semilattice of compact groupoids.",
"Write $\\mathbf {CptInvCat}$ for the category of compact inverse categories and (strong) monoidal dagger functors, and $\\mathbf {CptGpd}$ for the full subcategory of compact groupoids and (strong) monoidal functors.",
"Proposition 13 If $\\mathbf {C}$ is a compact inverse category, then $\\mathbf {S} = \\lbrace s \\in \\mathbf {C}(I,I) \\mid ss^\\dag =s \\rbrace , \\qquad s \\wedge t = st, \\qquad \\top = \\mathrm {id}_{I},$ is a semilattice, and for each $s \\in \\mathbf {S}$ , there is a compact groupoid $F(s)$ with the same objects as $\\mathbf {C}$ and morphisms $F(s)(A,B) = \\lbrace f \\in \\mathbf {C}(A,B) \\mid \\operatorname{tr}(ff^\\dag )=s \\rbrace \\text{,}$ giving a semilattice $F \\colon \\mathbf {S}^\\textrm {\\rm op}\\rightarrow \\mathbf {CptGpd}$ of compact groupoids $F(s\\le t)(f) \\mapsto s \\bullet f$ .",
"The assignment $\\mathbf {C} \\mapsto F$ extends to a functor $\\mathbf {CptInvCat} \\rightarrow \\mathbf {SLat}[\\mathbf {CptGpd}]$ by sending a morphism $G \\colon \\mathbf {C} \\rightarrow \\mathbf {C^{\\prime }}$ to $\\varphi (s)=\\psi _0^{-1} \\circ G(s)\\circ \\psi _0, \\qquad \\theta _s(A)=G(A), \\qquad \\theta _s(f)=G(f)\\text{,}$ where $\\psi _0 \\colon I^{\\prime } \\rightarrow G(I)$ is the structure isomorphism.",
"First, $\\mathbf {S}$ is a commutative idempotent monoid by definition.",
"Next, we verify that $F(s)$ is a compact groupoid.",
"Composition is well-defined: if $f \\colon A \\rightarrow B$ and $g \\colon B \\rightarrow C$ satisfy $\\operatorname{tr}(ff^\\dag )=s=\\operatorname{tr}(gg^\\dag )$ , then by Lemma REF and linearity and cyclicity of trace: $\\operatorname{tr}\\big ((gf)(gf)^\\dag \\big )& = \\operatorname{tr}(g^\\dag g ff^\\dag ) \\\\& = \\operatorname{tr}\\big [ (\\operatorname{tr}(g^\\dag g) \\bullet \\mathrm {id}_{B}) \\circ (\\operatorname{tr}(ff^\\dag ) \\bullet \\mathrm {id}_{B}) \\big ] \\\\& = \\operatorname{tr}(g^\\dag g) \\bullet \\operatorname{tr}(ff^\\dag ) \\bullet \\operatorname{tr}(\\mathrm {id}_{B}) \\\\& = \\operatorname{tr}(ff^\\dag ) \\bullet \\operatorname{tr}(\\mathrm {id}_{B}) \\\\& = \\operatorname{tr}\\big [ \\mathrm {id}_{B} \\circ (\\operatorname{tr}(ff^\\dag ) \\bullet \\mathrm {id}_{B})] \\\\& = \\operatorname{tr}(\\mathrm {id}_{B} \\circ ff^\\dag ) \\\\& = \\operatorname{tr}(ff^\\dag ) \\\\& = s\\text{.",
"}$ It is clear that $s \\bullet \\mathrm {id}_{A}$ play the role of identities in $F(s)$ .",
"The category $F(s)$ is monoidal, because if $\\operatorname{tr}(ff^\\dag ) = s = \\operatorname{tr}(gg^\\dag )$ , then $\\operatorname{tr}((f \\otimes g) (f \\otimes g)^\\dag ) = \\operatorname{tr}(ff^\\dag \\otimes gg^\\dag ) = \\operatorname{tr}(ff^\\dag ) \\operatorname{tr}(gg^\\dag ) = s$ .",
"It also inherits the dagger from $\\mathbf {C}$ : if $\\operatorname{tr}(ff^\\dag )=s$ , then also $\\operatorname{tr}(f^\\dag f)=\\operatorname{tr}(ff^\\dag )=s$ .",
"Consequently, $F(s)$ inherits the property of being an inverse category from $\\mathbf {C}$ .",
"Moreover, $F(s)$ is a compact dagger category: the units and counits are given by $s \\bullet \\eta _A$ and $s \\bullet \\varepsilon _A$ .",
"Finally, scalars $x \\in F(s)(I,I)$ are those scalars $x \\in \\mathbf {C}(I,I)$ satisfying $x^\\dag x=s$ , and form an abelian group with inverse $x^\\dag $ and unit $s$ : for $xs=xx^\\dag x=x$ ; if $x^\\dag x=s=y^\\dag y$ then $(xy)^\\dag (xy)=x^\\dag x y^\\dag y = s^\\dag s = s$ ; and $xx^\\dag =s$ .",
"Lemma REF therefore makes $F(s)$ a compact groupoid.",
"Notice that $F$ is a well-defined functor: if $s \\le t$ and $\\operatorname{tr}(ff^\\dag )=t$ , then $st=t$ , so $\\operatorname{tr}((sf)(sf)^\\dag ) = s s^\\dag \\operatorname{tr}(ff^\\dag ) = st=s$ .",
"Now consider morphisms.",
"If $\\mathbf {G} \\colon \\mathbf {C} \\rightarrow \\mathbf {C^{\\prime }}$ is a monoidal dagger functor, say with structure isomorphisms $\\psi _0 \\colon I^{\\prime } \\rightarrow G(I)$ and $\\psi _{A,B} \\colon G(A) \\otimes ^{\\prime } G(B) \\rightarrow G(A \\otimes B)$ , then it is easy to see that $\\varphi $ is a semilattice homomorphism, and that $\\theta _s$ is a well-defined monoidal dagger functor that is moreover natural in $s$ , because monoidal functors preserve dual objects and hence traces.",
"Finally, it is clear that the assignment $G \\mapsto (\\varphi ,f)$ is functorial.",
"Notice that $\\mathbf {S}$ contains all dimension scalars $\\dim (A)=\\operatorname{tr}(\\mathrm {id}_{A})$ .",
"Lemma 14 If $F \\colon \\mathbf {S} \\rightarrow \\mathbf {CptGpd}$ is a semilattice of compact groupoids, then the category $\\mathbf {C}$ of Lemma REF is a compact inverse category, and this gives a functor $\\mathbf {SLat}[\\mathbf {CptGpd}] \\rightarrow \\mathbf {CptInvCat}$ .",
"Define the tensor product on objects on $\\mathbf {C}$ as in $F(\\top )$ , and set the tensor unit $I$ in $\\mathbf {C}$ to be that of $F(\\top )$ .",
"The fact that $F(s \\le \\top )$ are monoidal functors gives structure isomorphisms $\\psi _s \\colon A \\otimes _s B \\rightarrow A \\otimes B$ , where we write $\\otimes _s$ for the tensor product in $F(s)$ , and $\\psi \\colon I_s \\rightarrow I$ , where we write $I_s$ for the tensor unit in $F(s)$ .",
"Define the tensor product of $f \\in F(s)(A,B)$ and $g \\in F(t)(C,D)$ to be $\\psi _{s\\wedge t} \\circ \\big ( F(s \\wedge t \\le s)(f) \\otimes _{s \\wedge t} F(s \\wedge t \\le t)(g) \\big ) \\circ \\psi _{s\\wedge t}^{-1}$ in $F(s \\wedge t)\\big ( A \\otimes C, B \\otimes D \\big )$ .",
"Taking coherence isomorphisms and dual objects as in $F(\\top )$ , a tedious but straightforward calculation proves that the triangle and pentagon axioms are satisfied, that the snake equations are satisfied, and that $\\mathbf {C}$ is a compact inverse category.",
"An even more tedious but still straightforward calculation shows that the functor induced by a morphism of semilattices of compact groupoids is monoidal.",
"Theorem 15 The functors of Proposition REF and Lemma REF implement an equivalence $\\mathbf {CptInvCat} \\simeq \\mathbf {SLat}[\\mathbf {CptGpd}]$ .",
"Starting with a compact inverse category $\\mathbf {C}$ , turning it into a semilattice of compact groupoids $F$ , and turning that into compact inverse category again, results in the exact same compact inverse category $\\mathbf {C}$ .",
"For example, the old homset $\\mathbf {C}(A,B)$ equals the new homset $\\coprod _{s \\in \\mathbf {C}(I,I) \\mid ss^\\dag =s} \\lbrace f \\in \\mathbf {C}(A,B) \\mid \\operatorname{tr}(ff^\\dag )=s \\rbrace $ because any morphism $f$ in $\\mathbf {C}$ is of the form $s \\bullet f$ for some scalar $ss^\\dag =s=\\operatorname{tr}(ff^\\dag )$ by Proposition REF .",
"Similarly, the new tensor product of $f \\in F(s)(A,B)$ and $g \\in F(t)(C,D)$ is $& \\psi _{s \\wedge t} \\circ \\big ( F(s \\wedge t \\le s)(f) \\otimes F(s \\wedge t \\le s)(g) \\big ) \\circ \\psi _{s \\wedge t}^{-1} \\\\& = \\psi _{s \\wedge t} \\circ (stf \\otimes stg) \\circ \\psi _{s \\wedge t}^{-1} \\\\& = \\psi _{s \\wedge t} \\circ (st \\bullet (f \\otimes g)) \\circ \\psi _{s \\wedge t}^{-1} \\\\& = (st \\bullet (f \\otimes g)) \\circ \\psi _{s \\wedge t} \\circ \\psi _{s \\wedge t}^{-1} \\\\& = (s \\bullet f)\\otimes (t \\bullet g) \\\\& = f \\otimes g\\text{,}$ again by Proposition REF , and because the natural isomorphism $\\psi $ cooperates with unitors and hence scalar multiplication, and so equals the old tensor product.",
"Now start with a semilattice of compact groupoids $F \\colon \\mathbf {S}^\\textrm {\\rm op}\\rightarrow \\mathbf {CptGpd}$ .",
"Lemma REF turns it into a compact inverse category $\\mathbf {C}$ , which in turn becomes the following semilattice of compact groupoids $G \\colon \\mathbf {T}^\\textrm {\\rm op}\\rightarrow \\mathbf {CptGpd}$ .",
"The semilattice $\\mathbf {T}$ is $\\coprod _{s \\in \\mathbf {S}} \\lbrace t \\in F(s)(I,I) \\mid tt^\\dag =t\\rbrace = \\coprod _{s \\in \\mathbf {S}} \\lbrace \\mathrm {id}_{I} \\in F(s)(I,I) \\rbrace $ because each $F(s)$ is a groupoid, so $s \\mapsto \\mathrm {id}_{I} \\in F(s)(I,I)$ is a semilattice isomorphism $\\varphi \\colon \\mathbf {S} \\rightarrow \\mathbf {T}$ .",
"The construction of Proposition REF gives $G(\\varphi (s))$ the same objects as $F(\\top )$ .",
"Morphisms $A \\rightarrow B$ in $G(\\varphi (s))$ are $f\\colon A \\rightarrow B$ in $F(t)(A,B)$ for some $t \\in \\mathbf {S}$ satisfying $\\varphi (s)=\\operatorname{tr}(ff^\\dag )$ .",
"Because $F(t)$ is a groupoid, $s$ must be $t$ , so $G(\\varphi (s))$ and $F(t)$ have the exact same homsets and identities, and we may take $\\theta $ to be the identity functor.",
"Going through the construction of $G$ shows that $\\theta $ is in fact a monoidal dagger functor."
],
[
"examples",
"This section lists examples of compact inverse categories $\\mathbf {C}$ .",
"For each example we will indicate how Proposition REF works by writing $\\mathbf {C}_0$ for the semilattice $\\mathbf {S}$ and $\\mathbf {C}_s$ for the compact groupoid $F(s)$ .",
"Example 16 (The fundamental compact groupoid) Any topological space $X$ with a fixed chosen point $x \\in X$ gives rise to a compact groupoid $\\mathbf {C}$ : The objects of $\\mathbf {C}$ are paths from $x_0$ to $x_0$ , more precisely, continuous functions $f \\colon [0,1] \\rightarrow X$ with $f(0)=f(1)=x$ .",
"The arrows $f \\rightarrow g$ are homotopy classes of paths, more precisely, continuous functions $h \\colon [0,1]^2 \\rightarrow X$ such that $h(s,0)=f(s)$ , $h(s,1)=g(s)$ , and $h(0,t)=h(1,t)=x_0$ , where $h$ and $h^{\\prime }$ are identified when there is a continuous function $H \\colon [0,1]^3 \\rightarrow X$ with $H(s,t,0)=h(s,t)$ , $H(s,t,1)=h^{\\prime }(s,t)$ , $H(s,0,u)=f(s)$ , $H(s,1,u)=g(s)$ , and $H(0,t,u)=H(1,t,u)=x_0$ .",
"The tensor product of objects is composition of paths according to some fixed reparametrisation, the tensor unit is the constant path.",
"Reparametrisation leads to associators and unitors.",
"Dual objects are given by reversal of paths.",
"The dagger is given by reversal of homotopies.",
"The unit $\\eta _f$ is the “birth of a double loop”, a homotopy that “grows” from the constant path to the path $f^\\dag \\circ f$ by travelling progressively further along $f$ before travelling back along $f^\\dag $ .",
"The counit $\\varepsilon _f$ is the “contraction of a double loop”, a homotopy that “shrinks” from the path $f^\\dag \\circ f$ to the constant path.",
"In this case $\\mathbf {C}_0$ is a one-element semilattice, and $\\mathbf {C}_s=\\mathbf {C}$ is already a groupoid.",
"Example 17 Any abelian group $\\mathbf {C}$ , considered as a discrete monoidal category, is a compact groupoid.",
"In this case $\\mathbf {C}_0$ is a one-element semilattice, and $\\mathbf {C}_s=\\mathbf {C}$ is already a groupoid.",
"Lemma 18 If $\\mathbf {C}$ is a compact (dagger/inverse) category, and $S$ a family of (dagger) idempotents, then $\\mathrm {Split}_S(\\mathbf {C})$ is again (dagger/inverse) compact.",
"In terms of Theorem REF , $\\mathrm {Split}_S(\\mathbf {C})_0 \\simeq \\mathbf {C}_0$ , and $\\mathrm {Split}_S(\\mathbf {C})_s = \\mathrm {Split}_{S_s}(\\mathbf {C}_s)$ , where $S_s = \\lbrace p \\in S \\mid \\operatorname{tr}(p)=s \\rbrace $ .",
"Let $p \\colon A \\rightarrow A$ be in $S$ .",
"Define $\\eta _p = (p^* \\otimes p) \\circ \\eta _A \\colon \\mathrm {id}_{I} \\rightarrow p \\otimes p^*$ and $\\varepsilon _p = \\varepsilon _A \\circ (p \\otimes p^*) \\colon p^* \\otimes p \\rightarrow \\mathrm {id}_{I}$ ; these are well-defined morphisms in $\\mathrm {Split}_S(\\mathbf {C})$ .",
"Then indeed the snake equations hold: $p = (\\varepsilon _A \\otimes p) \\circ (p \\otimes p^* \\otimes p) \\circ (p \\otimes \\eta _A) = (\\varepsilon _p \\otimes p) \\circ (p \\otimes \\eta _p)$ .",
"If $\\mathbf {C}$ has a dagger, then so does $\\mathrm {Split}_S(\\mathbf {C})$ , and $\\eta _p = (\\varepsilon _p \\circ \\sigma )^\\dag $ .",
"Example 19 If $\\mathbf {C}$ and $\\mathbf {D}$ are compact inverse categories, then so is $\\mathbf {C} \\times \\mathbf {D}$ .",
"In this case $(\\mathbf {C} \\times \\mathbf {D})_0 \\simeq \\mathbf {C}_0 \\times \\mathbf {D}_0$ , and $(\\mathbf {C} \\times \\mathbf {D})_{(s,t)} = \\mathbf {C}_s \\times \\mathbf {D}_t$ .",
"If $\\mathbf {C}$ and $\\mathbf {D}$ are compact groupoids, then so is $\\mathbf {C} \\times \\mathbf {D}$ .",
"Example 20 If $\\mathbf {C}$ is a compact inverse category, and $\\mathbf {G}$ is a groupoid, then $[\\mathbf {G},\\mathbf {C}]_\\dag $ , the category of functors $F \\colon \\mathbf {G} \\rightarrow \\mathbf {C}$ satisfying $F(f^{-1})=F(f)^\\dag $ and natural transformations, is again a compact inverse category.",
"In this case $([\\mathbf {G},\\mathbf {C}]_\\dag )_0 \\simeq \\mathbf {C}_0$ , and $([\\mathbf {G},\\mathbf {C}]_\\dag )_s$ has as morphisms natural transformations whose every component is in $\\mathbf {C}_s$ .",
"If $\\alpha \\colon F \\Rightarrow G$ is a natural transformation, its dagger is given by $(\\alpha ^\\dag )_A = (\\alpha _A)^\\dag \\colon G(A) \\rightarrow F(A)$ ; naturality of $\\alpha ^\\dag $ follows from naturality of $\\alpha $ together with the conditions $F(f)^\\dag =F(f^{-1})$ and $G(f)^\\dag =G(f^{-1})$ .",
"This makes $[\\mathbf {G},\\mathbf {C}]_\\dag $ into a dagger category.",
"It inherits the property $\\alpha =\\alpha \\alpha ^\\dag \\alpha $ componentwise from $\\mathbf {C}$ , and is therefore an inverse category.",
"The tensor product of objects is given by $(F \\otimes G)(A) = F(A) \\otimes G(A)$ , and on morphisms by $(F \\otimes G)(f) = F(f) \\otimes G(f)$ .",
"The tensor unit is the functor that is constantly $I$ .",
"Because the coherence isomorphisms in $\\mathbf {C}$ are unitary, this makes $[\\mathbf {G},\\mathbf {C}]_\\dag $ into a well-defined dagger symmetric monoidal category.",
"Finally, the dual object of $F \\colon \\mathbf {G} \\rightarrow \\mathbf {C}$ is given by $F^*(A)=F(A)^*$ and $F^*(f)=F(f)_*$ .",
"The unit $\\eta _F \\colon I \\Rightarrow F^* \\otimes F$ is given by $(\\eta _F)_A = \\eta _{F(A)}$ , and the counit by $(\\varepsilon _F)_A=\\varepsilon _{F(A)}$ .",
"These are natural because any morphism $f \\colon A \\rightarrow B$ in $\\mathbf {G}$ satisfies $ff^\\dag = \\mathrm {id}_{A}$ , whence $(F(f)_* \\otimes F(f)) \\circ \\eta _{F(A)} = (\\mathrm {id}_{B^*} \\otimes f) \\circ (\\mathrm {id}_{B^*} \\otimes f^\\dag ) \\circ \\eta _{F(B)} = \\eta _{F(B)}$ .",
"This makes $[\\mathbf {G},\\mathbf {C}]_\\dag $ a compact inverse category."
],
[
"Compact groupoids",
"This section moves to a 2-categorical perspective, to connect to a characterisation of compact groupoids.",
"A compact groupoid is the same thing as a coherent 2-group [5].",
"It is also known as a crossed module.",
"Compact groupoids are classified by two abelian groups $G$ and $H$ and an element of the third cohomology group of $G$ with coefficients in $H$ , as worked out by Baez and Lauda [5].",
"The following proposition makes this more precise.",
"In the nonsymmetric case, $G$ need not be abelian, and there is an additional action of $G$ on $H$ .",
"Proposition 21 A compact groupoid $G$ is, up to equivalence, defined by the following data: the (abelian) group $G$ of isomorphism classes of objects of $\\mathbf {C}$ , under $\\otimes $ , with unit $I$ , and inverse given by dual objects; the abelian group $H$ of scalars $\\mathbf {C}(I,I)$ under composition with unit $\\mathrm {id}_{I}$ and inverse $\\dag $ ; the conjugation action $G \\times H \\rightarrow H$ that takes $(A,s)$ to $\\operatorname{tr}(A \\otimes s)=s$ ; the 3-cocycle $G \\times G \\times G \\rightarrow H$ that takes $(A,B,C)$ to $\\operatorname{Tr}(\\alpha _{A,B,C})$ .",
"The above data form the objects of a (weak) 2-category $\\mathbf {Cocycle}$ , with 1- and 2-cells as in [5].",
"See [5].",
"The trick is the following.",
"First, we may assume that $\\mathbf {C}$ is skeletal.",
"Then, we may adjust the tensor product such that all unitors and units and counits (but not the associators!)",
"are identities.",
"The pentagon equation ensures that the trace of the associator is in fact a 3-cocycle.",
"The proof of Theorem REF is the only place where we have used that in a semilattice $F$ of categories all $F(s)$ must have the same objects.",
"It was needed because if the functor $\\theta _s$ is to be an isomorphism, it must give a bijection between the objects of $F(s)$ and $F(\\top )$ .",
"We now move to a (weak) 2-categorical perspective to remove this restriction.",
"Definition 22 Redefine the category $\\mathbf {SLat}[\\mathbf {V}]$ of Definition REF to become a (weak) 2-category as follows: 0-cells are functors $F \\colon \\mathbf {S}^\\textrm {\\rm op}\\rightarrow \\mathbf {Cat}$ for some semilattice $\\mathbf {S}$ ; 1-cells $F \\rightarrow F^{\\prime }$ consist of a morphism $\\varphi \\colon \\mathbf {S} \\rightarrow \\mathbf {S^{\\prime }}$ of semilattices and a natural transformation $\\theta \\colon F \\Rightarrow F^{\\prime } \\circ \\varphi $ ; 2-cells $(\\varphi ,\\theta ) \\rightarrow (\\varphi ^{\\prime },\\theta ^{\\prime })$ exist when $\\varphi \\le \\varphi ^{\\prime }$ and then are natural transformations $\\gamma \\colon \\theta \\Rrightarrow \\theta ^{\\prime } \\circ (\\mathrm {id}_{}* (\\varphi \\le \\varphi ^{\\prime }))$ .",
"Composition is by pasting.",
"$\\begin{tikzpicture}\\node (J) at (0,1) {\\mathbf {S}^\\textrm {\\rm op}};\\node (J^{\\prime }) at (0,-1) {\\mathbf {S^{\\prime }}^\\textrm {\\rm op}};\\node (C) at (5,0) {\\mathbf {V}};[->] (J) to[out=0,in=90,looseness=.5] node[above]{F} (C);[->] (J^{\\prime }) to[out=0,in=-90,looseness=.5] node[below]{F^{\\prime }} (C);[->] (J) to[out=-120,in=120] node[left]{\\varphi } (J^{\\prime });[->] (J) to[out=-60,in=60] node[right]{\\varphi ^{\\prime }} (J^{\\prime });\\node at (0,0) {\\le };[twocell] (3,.5) to[out=-60,in=60] node[right]{\\theta ^{\\prime }} (3,-.5);[twocell] (2,.5) to[out=-120,in=120] node[left]{\\theta } (2,-.5);[threecell] (2.2,0) to node[above]{\\gamma } (2.8,0);\\end{tikzpicture}$ Write $\\mathbf {SLat}_=[\\mathbf {CptGpd}]$ for the full sub-2-category where all categories $F(s)$ have the same objects.",
"To be precise, in $\\mathbf {SLat}[\\mathbf {CptGpd}]$ , 2-cells $\\gamma $ are modifications: for each $s \\in \\mathbf {S}$ and $A \\in F^{\\prime }(\\varphi (s))$ , there is a morphism $\\gamma _{s,A} \\colon \\theta _s(A) \\rightarrow \\theta ^{\\prime }_s\\big (F^{\\prime }\\big (\\varphi (s) \\le \\varphi ^{\\prime }(s)\\big )(A)\\big )$ that is natural in $s$ as well as $A$ .",
"Lemma 23 There is a (weak) 2-equivalence $\\mathbf {SLat}[\\mathbf {CptGpd}] \\simeq \\mathbf {SLat}_=[\\mathbf {CptGpd}]$ .",
"First, observe that two 0-cells $F,G \\colon \\mathbf {S}^\\textrm {\\rm op}\\rightarrow \\mathbf {CptGpd}$ are equivalent in $\\mathbf {SLat}[\\mathbf {CptGpd}]$ exactly when there is a natural monoidal equivalence $F(s) \\simeq G(s)$ .",
"Therefore, it suffices to construct, for each $F$ , such a $G$ such that each $G(s)$ has the same objects.",
"Let $\\kappa _s$ be the cardinality of the objects of $F(s)$ , and let $\\kappa $ be the maximum of all $\\kappa _s$ .",
"Define $G(s)$ to be equal to $F(s)$ , except that we add $\\kappa $ isomorphic copies of the tensor unit $I$ .",
"There is an obvious monoidal structure on $G(s)$ , and by construction there is a monoidal equivalence $F(s) \\simeq G(s)$ , so that $G(s)$ is automatically a compact groupoid.",
"We may furthermore relabel the objects of $G(s)$ to be ordinal numbers, so that all $G(j)$ have the same objects.",
"Theorem 24 There is a (weak) 2-equivalence $\\mathbf {CptInvCat} \\simeq \\mathbf {SLat}[\\mathbf {Cocycle}]$ , where $\\mathbf {CptInvCat}$ has natural transformations as 2-cells.",
"The (weak) 2-equivalence $\\mathbf {CptGpd}\\simeq \\mathbf {Cocycle}$ of [5] induces a (weak) 2-equivalence $\\mathbf {SLat}[\\mathbf {CptGpd}] \\simeq \\mathbf {SLat}[\\mathbf {Cocycle}]$ by postcomposition.",
"Combine this with the equivalence $\\mathbf {SLat}_=[\\mathbf {CptGpd}] \\simeq \\mathbf {SLat}[\\mathbf {CptGpd}]$ of Lemma REF and the equivalence $\\mathbf {CptInvCat} \\simeq \\mathbf {SLat}_=[\\mathbf {CptGpd}]$ of Theorem REF ; the latter still holds after the change of Definition REF ."
],
[
"Concluding remarks",
"We conclude by discussing the many questions left open and raised in this paper.",
"First, one could investigate generalising the results in this paper from categories to semicategories.",
"Second, one could investigate generalising the results in this paper from compact categories to monoidal categories where every object has a dual."
],
[
"Traced inverse categories",
"Inverse categories provide semantics for reversible programs, but higher-order aspects of reversible programming remain unclear.",
"Compact categories are closed and hence provide semantics for higher-order programming.",
"Theorem REF shows that compact inverse categories are, in a sense, degenerate.",
"But one of the most interesting aspects of higher-order programming, tail recursion, doesn't need compact categories for semantics, and can already be modeled in traced monoidal categories.",
"(But see also [21].)",
"Now every traced monoidal category can be monoidally embedded in a compact category [20].",
"One can prove that there exists a left dagger biadjoint to the forgetful functor from dagger compact categories to dagger traced categories.",
"There is also a left adjoint to the forgetful functor from compact inverse categories to compact dagger categories, but the latter is not faithful.",
"Hence there is a left dagger biadjoint to the forgetful functor from compact inverse categories to traced inverse categories, but its unit does not embed any traced inverse category into a compact inverse category.",
"Therefore Theorem REF does not show that all traced inverse categories degenerate.",
"Indeed, the category $\\mathbf {PInj}$ of sets and injections is the universal inverse category [22], and is also traced [18], [14], but it fails Lemma REF , irrespective of which tensor product it carries, as the swap map on the two element set is not a scalar multiple of the identity.",
"That leaves a valid question: what do traced inverse categories look like?"
],
[
"Idempotents",
"A subunit in a monoidal category $\\mathbf {C}$ is a subobject $r \\colon R \\rightarrowtail I$ for which $r \\otimes \\mathrm {id}_{R}$ is invertible [11]; they form a semilattice $\\mathrm {ISub}(\\mathbf {C})$ .",
"The following lemma shows that in compact inverse categories, up to splitting idempotents, the semilattice $\\mathbf {C}_0$ is precisely that of subunits.",
"See also [28] for structure theorems of inverse categories in which all idempotents split.",
"Lemma 25 Let $\\mathbf {C}$ be a compact inverse category.",
"A map $r \\colon R \\rightarrow I$ is a subunit if and only if $r^\\dag r = \\mathrm {id}_{}$ .",
"Any subunit $r$ induces an element $rr^\\dag $ of $\\mathbf {C}_0$ .",
"If idempotents split in $\\mathbf {C}$ , any element $\\mathbf {C}_0$ is $rr^\\dag $ for a unique subunit $r$ ; this gives a isomorphism between the semilattices $\\mathbf {C}_0$ and $\\mathrm {ISub}(\\mathbf {C})$ .",
"For (a), first notice that if $r \\colon R \\rightarrow I$ is monic, then because $r=rr^\\dag r$ in fact $r$ is an isometry, that is, $r^\\dag r = \\mathrm {id}_{}$ .",
"We will show that for isometries $r$ , the condition that $r \\otimes \\mathrm {id}_{R}$ is invertible holds automatically, with the inverse being $r^\\dag \\otimes \\mathrm {id}_{R}$ .",
"It suffices to show that $(r \\otimes \\mathrm {id}_{R})(r^\\dag \\otimes \\mathrm {id}_{R})=\\mathrm {id}_{I \\otimes R}$ .",
"But $\\mathrm {id}_{I \\otimes R} = \\mathrm {id}_{I} \\otimes (r^\\dag r) = \\mathrm {id}_{I} (r^\\dag (r r^\\dag ) r) = (r r^\\dag ) \\otimes (r^\\dag r) = (r r^\\dag ) \\otimes \\mathrm {id}_{R}\\text{.",
"}$ Thus the subunits are precisely the (subobjects represented by) isometries.",
"Part (b) is obvious: if $r$ is an isometry, then $s=rr^\\dag \\colon I \\rightarrow I$ satisfies $s=ss^\\dag $ .",
"Part (c) follows from [6], as does the fact that the maps of (b) and (c) are each other's inverses.",
"It is easy to see that both maps preserve the order structure using [11].",
"Now there are two ways to `localise' $\\mathbf {C}$ to $r \\in \\mathrm {ISub}(\\mathbf {C})$ .",
"The localisation $\\mathbf {C}\\big |_r$ according to [11] has objects $A$ such that $r \\otimes \\mathrm {id}_{A}$ is invertible, and all morphisms between those objects.",
"The localisation $\\mathbf {C}_{rr^\\dag }$ above has all objects, but only those morphisms $f$ satisfying $\\operatorname{tr}(ff^\\dag )=rr^\\dag $ .",
"These two localisations are different.",
"The former localises with respect to the tensor product, whereas the latter localises with respect to composition.",
"Generally, taking semilattices of categories is a completion procedure.",
"Does it generalise to (weak) 2-categories?",
"If so, the above may be the special cases of a single object and of unique 2-cells, and could form a higher-categorical analogue of the Eckmann-Hilton argument in the Baez-Dolan stabilisation hypothesis [4].",
"Is there a relationship with [15]?"
],
[
"Internal descriptions",
"Groupoids are precisely special dagger Frobenius algebras in the category $\\mathbf {Rel}$ of sets and relations [16].",
"Compact groupoids are precisely special dagger Frobenius algebras in the category $\\mathbf {Rel}(\\mathbf {Gp})$ of relations over the regular category of groups, see [13].",
"Can inverse categories similarly be described as certain monoids in a category of relations?"
],
[
"Bratteli diagrams and C*-algebras",
"Describing compact inverse categories through a diagram of groupoids resembles describing an AF C*-algebra as a diagram of finite-dimensional C*-algebras [3].",
"It is very fruitful to work with this so-called Bratteli diagram directly rather than with the C*-algebra itself.",
"More generally, inverse semigroups are a popular way to generate C*-algebras [8], as it is easier to work with the inverse semigroup directly, and moreover this captures many important classes of C*-algebras (see e.g.",
"[30]): AF C*-algebras, graph C*-algebras, tiling C*-algebras, self-similar group C*-algebras, subshift C*-algebras, C*-algebras of ample étale groupoids, and C*-algebras of Boolean dynamical systems.",
"There is also a multiply-typed version building a C*-algebra from a so-called higher rank graph [23].",
"Can one similarly generate a C*-algebra from a compact inverse category, and is there a relationship to these other constructions?",
"A first step might be to extend [25] to possibly infinite categories by adding a norm."
]
] | 1906.04248 |
[
[
"Crystal Volumes and Monopole Dynamics"
],
[
"Abstract The low velocity dynamic of a doubly periodic monopole, also called a monopole wall or monowall for short, is described by geodesic motion on its moduli space.",
"This moduli space is hyperkaehler and non-compact.",
"We establish a relation between the Kaehler potential of this moduli space and the volume of a region in Euclidean three-space cut out by a plane arrangement associated with each monowall."
],
[
"Introduction",
"A monopole wall or a monowall is a BPS monopole on $\\mathbb {R}\\times S^1\\times S^1.$ The latter space is endowed with coordinates $(x,\\theta ,\\varphi )$ and the product Euclidean metric $dx^2+d\\theta ^2+d\\varphi ^2$ with respective circle radii $r_\\theta $ and $r_\\varphi $ , i.e.",
"$\\theta \\sim \\theta +2\\pi r_\\theta $ and $\\varphi \\sim \\varphi +2\\pi r_\\varphi .$ In detail, a monowall is a Hermitian bundle $E\\rightarrow \\mathbb {R}\\times S^1\\times S^1$ of rank $h$ with a pair $(A,\\Phi )$ consisting of a connection $A$ on $E$ and a Higgs field $\\Phi $ , which is an endomorphism of $E,$ satisfying the Bogomolny equation $*D_A\\Phi =-F_A.$ Here $F_A$ is the curvature of the connection (so in a local trivialization the curvature two-form is $F_A=dA+A\\wedge A$ where $A$ is the connection one-form), $*$ is the Hodge star operator, and $D_A$ the covariant differential.",
"We impose the same asymptotic condition as in [7], namely that the eigenvalues of the Higgs field grow at most linearly, having the form $\\Phi &=\\frac{\\mathrm {i}}{2\\pi r_\\theta r_\\varphi } \\mathrm {diag} (Q^\\iota _\\pm x + M^\\iota _\\pm ) + O(|x|^{-1}),$ as $x\\rightarrow \\pm \\infty ,$ and the connection one-form has the form $\\mathcal {A}=\\frac{-\\mathrm {i}}{2\\pi r_\\theta r_\\varphi } \\mathrm {diag}\\left(Q^\\iota _\\pm \\frac{\\theta d\\varphi -\\varphi d\\theta }{2} + r_\\varphi \\chi ^{\\theta ,\\iota }_\\pm d\\theta + r_\\theta \\chi ^{\\varphi ,\\iota }_\\pm d\\varphi \\right)+O(|x|^{-1}).$ Here $\\iota =1,2,\\ldots ,h.$ See [7] for the detailed discussion of charges $Q^\\iota _\\pm \\in \\mathbb {Q}$ , consistency conditions, and field asymptotics.",
"As argued in [7], it is natural to enlarge the scope of our problem by allowing for Dirac-type monopole singularities at some points $p_1^+, \\ldots , p_{v_+}^+$ and $p_1^-, \\ldots , p_{v_-}^-$ in $\\mathbb {R}\\times S^1\\times S^1.$ At these points the Higgs field has (up to unitary gauge transformation) a prescribed singularity $\\Phi =\\mathrm {i}\\begin{pmatrix}\\frac{\\pm 1}{2|\\vec{x}-\\vec{p}_\\sigma ^\\pm |} &0_{1\\times (n-1)}\\\\0_{(n-1)\\times 1}&0_{(n-1)\\times (n-1)}\\end{pmatrix}+O(|\\vec{x}-\\vec{p}_\\sigma ^\\pm |^0).$ The first study of monopole walls, that we are aware of, was undertaken by Ki-Myeong Lee in [14], where the deformation theory of monopole walls with arbitrary compact simple Lie gauge group was studied.",
"The numerical study by Richard Ward of an $SU(2)$ monowall appeared in [22] and [23].",
"The spectral curve was used in [7] to study the deformation theory of $U(h)$ monowalls.",
"Hamanaka et al [12] used monowall scattering to compute the moduli space asymptotic metric for $U(2)$ monowalls.",
"The interior of these moduli spaces was probed by Maldonado and Ward in [17] via special geodesics.",
"A systematic description of the asymptotic region of the monowall moduli space and classification of the monowall moduli spaces of real dimension four appeared in [3].",
"For a general $U(N)$ monopole, the asymptotic moduli space metric in the regime of widely separated constituents was found in [5].",
"Monowalls relate to a number of significant problems involving non-abelian Hodge theory [16], mirror symmetry [21], Calabi-Yau moduli spaces and quantum gauge theories in five dimensions [3], [1], and integrable systems [19]."
],
[
"Spectral Data of a Monowall",
"The Bogomolny equation (REF ) implies the compatibility of the following linear system $\\left\\lbrace \\begin{array}{l}(D_{\\varphi }+ \\mathrm {i}\\Phi ) V(x,\\theta ,\\varphi )=0,\\\\(D_x+\\mathrm {i}D_{\\theta }) V(x,\\theta ,\\varphi )=0.\\end{array}\\right.$ Here $D_x=D_{\\frac{\\partial }{\\partial x}}$ is the covariant derivative along the $x$ -direction, etc.",
"It follows that the holonomy $W(s):=V(x,\\theta ,2\\pi r_\\varphi )V(x,\\theta ,0)^{-1}\\in GL(h,\\mathbb {C})$ around the $\\varphi $ -direction has eigenvalues that are meromorphic in the complex coordinate $s:=\\exp \\frac{x+\\mathrm {i}\\theta }{r_\\theta }\\in \\mathbb {C}^*.$ This motivates introducing the holomorphic spectral curve $\\mathbb {S}_\\varphi =\\left\\lbrace (s,t)\\, |\\, \\mathrm {det}\\,(t-W(s))=0\\right\\rbrace \\subset \\mathbb {C}^*\\times \\mathbb {C}^*.$ Moreover, the asymptotic conditions (REF ) and prescribed Dirac singularity conditions (REF ) ensure that this spectral curve is algebraic, given by $P(s,t)=0$ , with the spectral polynomial $P(s,t)=Q(s)\\,\\mathrm {det}\\big (t-W(s)\\big )=\\sum _{(m,n)\\in \\mathcal {N}} C_{m,n} s^m t^n.$ Here $Q(s)$ is the lowest degree common multiple of the denominators of the rational functions $q_j(s)$ appearing as coefficients of the characteristic polynomial $\\mathrm {det}(t-W(s))=t^n+q_1(s)t^{n-1}+q_2(s)t^{n-2}+\\ldots +q_n(s).$ This defines $P(s,t)$ up to an overall constant nonzero factor.",
"This ambiguity can be fixed, if desired, by imposing the dictionary order on the vertices $(m,n)\\in \\mathcal {N}$ and requiring the coefficient $C_{m_0,n_0}$ for the minimal vertex $(m_0,n_0)$ to be one.",
"The Newton polygon $\\mathcal {N}$ is the minimal convex hull of all the points $(m,n)\\in \\mathbb {Z}\\times \\mathbb {Z}$ for which $C_{m,n}\\ne 0$ .",
"The height of $\\mathcal {N}$ is equal to the monopole bundle rank $h.$ Note that our preferential treatment of the $\\varphi $ coordinate leading to the definition of the spectral curve was somewhat arbitrary.",
"One can instead consider the modified holonomy around the $\\theta $ direction and obtain a different spectral curve $\\mathbb {S}_\\theta ,$ now covering the $\\mathbb {C}^*$ factor with coordinate $s^{\\prime }=\\exp \\frac{x-\\mathrm {i}\\varphi }{r_{\\varphi }}$ .",
"Let Per$(\\mathcal {N})$ denote the set of integer perimeter points of $\\mathcal {N}$ and let Int$(\\mathcal {N})$ denote the set of its integer interior points.",
"As demonstrated in [7], the Newton polygon is entirely determined by the charge values $Q^\\iota _\\pm $ and the numbers of singularities $v_+$ and $v_-,$ while the perimeter coefficients $C_{m,n}$ with $(m,n)\\in \\mathrm {Per}(\\mathcal {N})$ are determined by the constants $M^\\iota _\\pm \\in \\mathbb {R}$ and $\\chi _\\pm ^{\\varphi ,\\iota }\\in [0,2\\pi )$ appearing in the asymptotic conditions (and by the $s$ -coordinates of the Dirac singularities $p_\\sigma ^{x,\\pm }+\\mathrm {i}p_\\sigma ^{\\theta ,\\pm }$ ).",
"See [7] for details.",
"The interior coefficients, on the other hand, are some of the moduli (parameterizing the $L^2$ deformations) of the monopole solution, thus producing a family $\\mathcal {B}_\\mathcal {N}$ of curves (with fixed perimeter coefficients).",
"In fact, the total number of real moduli of a monowall is equal to four times the number of internal points: $4\\times |\\mathrm {Int}(\\mathcal {N})|$ and the moduli space is the universal Jacobian fibration of this family $\\mathcal {B}_\\mathcal {N}.$ We shall focus on the region in the moduli space with large $C_{m,n}$ and large differences between them (as specified in Section using the secondary fan).",
"The generic curve $\\mathbb {S}_\\varphi $ for a family $\\mathcal {B}_\\mathcal {N}$ is a punctured Riemann surface of genus $|\\mathrm {Int}(\\mathcal {N})|$ with $|\\mathrm {Per}(\\mathcal {N})|$ punctures.",
"Since, for any given monowall, $\\mathbb {S}_\\varphi $ is a curve of eigenvalues it (generically) comes equipped with a Hermitian eigen-line bundle $\\mathcal {L}\\rightarrow \\mathbb {S}_\\varphi $ with a flat connection $\\nabla $ .",
"The triplet $(\\mathbb {S}_\\varphi ,\\mathcal {L},\\nabla )$ is the spectral data encoding the monowall solution $(A,\\Phi ),$ up to a gauge transformation, with its parameters and moduli correspondence as follows: The holonomy of $\\nabla $ around each puncture is valued in $U(1)$ and is determined by the asymptotic conditions.",
"This is how the $|\\mathrm {Per}(\\mathcal {N})|$ triplets of parameters $(M^\\iota _\\pm ,\\chi _\\pm ^{\\theta ,\\iota },\\chi _\\pm ^{\\varphi ,\\iota })$ of the boundary conditions translate to the spectral data [7]: $M^\\iota _\\pm +\\mathrm {i}\\chi _\\pm ^{\\theta ,\\iota }$ determine the position of each puncture, while $\\chi _\\pm ^{\\varphi ,\\iota }$ determines the $\\nabla $ holonomy around it.",
"Viewing a (generic) curve $\\mathbb {S}_\\varphi $ as a sphere with $|\\mathrm {Int}(\\mathcal {N})|$ handles, one can associate each handle to an internal point $(m,n)$ of $\\mathcal {N}$ and choose a symplectic homology basis $\\lbrace A_f,B_{f^{\\prime }} | A_f\\cap B_{f^{\\prime }}=\\delta _{ff^{\\prime }}, A_f\\cap A_{f^{\\prime }}=0=B_f\\cap B_{f^{\\prime }}\\rbrace $ of the compactified Riemann surface $\\overline{\\mathbb {S}}_\\varphi $ with each pair $(A_f,B_f)=(A_{m,n},B_{m,n})$ associated to the $f=(m,n)$ -th handle.",
"Thus, each internal point $f=(m,n)\\in \\mathcal {N}$ has four moduli associated to it: two real moduli $R_f$ and $\\Theta _f$ in $C_{m,n}=\\exp \\frac{R_f+\\mathrm {i}\\Theta _f}{r_\\theta }$ and two moduli $\\Phi _f\\sim \\Phi _f+2\\pi r_\\varphi $ and $T_f\\sim T_f+2\\pi $ specifying the holonomies $e^{\\mathrm {i}\\frac{\\Phi _f}{r_\\varphi }}$ and $e^{\\mathrm {i}T_f}$ of $\\nabla $ around the cycles $A_f$ and $B_f,$ respectively.",
"Let us emphasize an important distinction between parameters and moduli.",
"Variations of moduli correspond to $L^2$ deformations of the solution $(A,\\Phi )$ of the Bogomolny equation (REF ), while variations of parameters result in deformations of the solution that are not square integrable.",
"Physically, moduli correspond to all directions in the space of (gauge equivalence classes of) solutions that have finite mass, while the parameters are the remaining transverse coordinates.",
"As a result, a monowall can slowly evolve in time with moduli changing, while all parameters will have to remain fixed, since their time evolution would require infinite energy.",
"In other words the space of all monowalls with the given Newton polygon $\\mathcal {N}$ is fibered over the parameter space.",
"The base is parameterized by the $3|\\mathrm {Per}(\\mathcal {N})|$ parameters and the fiber is what we call the moduli space.",
"The coordinates on the moduli space are the $4|\\mathrm {Int}(\\mathcal {N})|$ moduli.",
"The $L^2$ norm on the tangent space of pairs $(A,\\Phi )$ induces the metric on each moduli space."
],
[
"The Crystal",
"Given a monowall and its spectral polynomial with coefficients $C_{m,n}$ , set $R_{m,n}:=r_\\theta \\ln |C_{m,n}|$ and consider the set of planes $\\lbrace (x,y,z)\\, |\\, z=m x + n y + R_{m,n}\\rbrace \\subset \\mathbb {R}^3.$ Let us call the convex domain above all of these planes the cut crystal: ${cut}=\\lbrace (x,y,z)\\, |\\, z\\ge m x + n y + R_{m,n},\\ \\forall (m,n)\\in \\mathcal {N}\\rbrace .$ Its surface is the graph of the function $M(x,y)=\\max _{(m,n)\\in \\mathcal {N}} \\lbrace m x + n y + R_{m,n} \\rbrace .$ The shape of the cut crystal depends on the moduli (and parameters) and we shall be interested in how its volume changes with the change in the moduli $R_{m,n}.$ Since the cut crystal has infinite volume, to keep track of these changes, let us also consider the domain above all of the perimeter planes only: ${0}=\\lbrace (x,y,z)\\, |\\, z\\ge m x + n y +R_{m,n},\\ \\forall (m,n)\\in \\mathrm {Per}(\\mathcal {N})\\rbrace .$ Call it the the blocked crystal.",
"Its surface is the graph of the function $m(x,y)=\\max _{(m,n)\\in \\mathrm {Per}(\\mathcal {N})} \\lbrace m x + n y + R_{m,n} \\rbrace .$ It is completely determined by the asymptotic conditions and is independent of the moduli, and it satisfies $m(x,y)\\le M(x,y).$ Thus, clearly, ${cut}\\subseteq {0}$ and the planes corresponding to the interior points of $\\mathcal {N}$ cut ${cut}$ out of the blocked crystal 0.",
"We call the volume of the difference of the two crystals 0 and ${cut}$ the cut volume $\\mathcal {V}(R_f):=\\mathrm {Vol}\\,(0\\setminus {cut})=\\mathrm {Vol}\\lbrace (x,y,z)\\, |\\, m(x,y)\\le z\\le M(x,y)\\rbrace .$ It is a function of $|\\mathcal {N}|$ variables $R_f$ , one for each integer point of $\\mathcal {N}.$ Intuitively, for large moduli a monowall would split into subwalls, as demonstrated in [6].",
"As argued in Section , the subwall positions are well approximated by the $x-$ positions of the vertices of this cut crystal.",
"It was conjectured in [3] that the Kähler potential of a monowall moduli space is related to this cut volume (REF ).",
"This paper refines this conjecture and proves it.",
"This is based on the asymptotic metric found in [5], obtained by analyzing subwall dynamic interactions via the Gibbons-Manton approach [9], reviewed in Sections .",
"This metric approximates the metric on the moduli space end with exponential accuracy.",
"The Kähler potential of this asymptotic metric is presented in Section .",
"This Kähler potential, in turn, is the Generalized Legendre Transform (GLT) of Lindström and Roček [15], [11] of the function $G.$ The main result of this paper is that the GLT function $G$ encoding the asymptotic monowall metric equals the cut volume: $G=\\mathcal {V}.$ The exact meaning and the proof of this relation are spelled out in Section .",
"It can be summarized as follows: in the regime of far separated subwalls the monowall Kähler potential is the Generalized Legendre Transform of the cut volume."
],
[
"Subwall Positions and Spectral Curve Branch Points",
"As monowall moduli increase, the monowall splits into subwalls.",
"Let us explore the dependence of these subwall positions on the moduli."
],
[
"The Secondary Fan and the Monowall Spine",
"There is significant information about the monowall contained in the cut crystal.",
"Its surface consists of faces (each face contained in one of the planes (REF ) and thus each has an associated integer point $f=(m,n)\\in \\mathcal {N}$ ), edges at which these faces meet, and vertices.",
"The projection of the cut crystal edges and vertices on the $(x,y)-$ plane is a graph, that we call the spine, as illustrated in Figure REF .",
"From this description the spine is dual to a regular subdivision [8] of the Newton polygon $\\mathcal {N},$ in which the two integer points of $\\mathcal {N}$ are connected by an edge if and only if the corresponding faces of the cut crystal meet at a crystal edge.",
"Each spine edge is normal to the correspoinding edge of the subdivision of $\\mathcal {N}.$ Figure: Newton polygon 𝒩\\mathcal {N} with colored integer points and two examples of its regular triangulations (a) and (b).The corresponding spines in black and their color-coded faces (c) and (d), with each face corresponding to an integer point of 𝒩\\mathcal {N} in (a) and (b), respectively.Figure: Two examples of the tent functions (a) and (b) and their corresponding plane arrangements (c) and (d) for the Newton polynomialF(s,t)=blue5s+cyanAst-orange5s 2 t+red20t 2 +grayBst 2 +violet20s 2 t 2 -darkgreen5t 3 F(s,t)= {blue}{5 s} + {cyan}{A s t} - {orange}{5 s^2 t} + {red}{20 t^2} + {gray}{B s t^2} + {violet}{20 s^2 t^2} - {darkgreen}{5 t^3} with(A,B)=(120,27)(A,B)=(120,27) (left) and (A,B)=(20,90)(A,B)=(20,90) (right).A regular subdivision is defined in the following way.",
"Consider a real valued function $l(m,n)$ on the integer points of $\\mathcal {N}$ and the convex hull in $\\mathbb {R}^3$ of the set of downward rays $\\lbrace (m,n,z)\\, |\\, (m,n)\\in \\mathcal {N}, z\\le l(m,n)\\rbrace .$ The part of this hull's surface that is not vertical is a graph of a concave function over the interior of $\\mathcal {N}$ in $\\mathbb {R}^2.$ Let us call it the tent function.",
"It is piecewise linear, with corners at (some of the points) $(m,n,l(m,n))$ .",
"The edges of this surface project onto $\\mathcal {N}$ giving a regular subdivision of $\\mathcal {N}.$ (Note, in some of the literature, e.g.",
"in [8] itself, such subdivisions are called coherent subdivisions instead of regular subdivisions.)",
"In our case, choosing $l(m,n)=R_{m,n}=r_\\theta \\ln |C_{m,n}|$ results in a subdivision dual to the monowall spine.",
"Also, note that the resulting tent function is the negative of the Legendre transform of the cut crystal surface function $M(x,y)$ of Eq.",
"(REF ).",
"As a result, the space $\\mathbb {R}^{|\\mathcal {N}|}$ with coordinates $R_f=R_{m,n}$ is subdivided into cones labelled by regular subdivisions of $\\mathcal {N}.$ These cones form the secondary fan $F(\\mathcal {N})$ of $\\mathcal {N}.$ Moving to infinity within a given cone results in the monowall splitting into subwalls of certain types, determined by the elements of that corresponding subdivision.",
"Each polygon appearing in the subdivision corresponds to a subwall.",
"The secondary fan $F(\\mathcal {N})$ is encoded in the secondary polytope $\\Sigma (\\mathcal {N}).$ In fact, the two are dual to each other: each ray of $F(\\mathcal {N})$ is normal to a face of $\\Sigma (\\mathcal {N})$ and two rays are connected by a wedge if the corresponding faces of $\\Sigma (\\mathcal {N})$ share an edge.",
"The $i-$ th coordinate of the vertex of $\\Sigma (N)$ can be read off from its corresponding regular triangulation as the total area of the triangles for which the $i-$ th integer point of $\\mathcal {N}$ is a vertex.",
"See [8] for many fascinating details.",
"There is a partial order on all regular subdivisions given by refinement.",
"The maximally refined subdivisions are the regular triangulations with each triangle of areaAs in [8], we normalize the area of a basic triangle with vertices $(0,0),(1,0),$ and $(0,1)$ to be one, instead of a half.",
"1.",
"This is the case we are most interested in here, as it corresponds to the monowall maximally split into elementary subwalls.",
"Each regular triangulation labels a cone (of maximal dimension) in the secondary fan (see [8]), with other regular subdivisions labelling its lower-dimensional cones.",
"According to [3], the secondary fan is in the space $\\mathbb {R}^{|\\mathcal {N}|}$ with coordinates $R_{m,n},$ which include both moduli and parameters.",
"A generic direction lies in the interior of a single cone of the secondary fan and corresponds to some regular triangulation.",
"Fixing all parameters gives a slice $\\mathbb {R}^{|\\mathrm {Int}(\\mathcal {N})|}$ of $\\mathbb {R}^{|\\mathcal {N}|}=\\mathbb {R}^{|\\mathrm {Per}(\\mathcal {N})|}\\times \\mathbb {R}^{|\\mathrm {Int}(\\mathcal {N})|}$ .",
"The intersection of this slice with the secondary fan divides this slice into regions, some compact and some noncompact.",
"The `down-facing' cones of the secondary fan correspond to triangulations not involving any internal points of $\\mathcal {N}$ as triangle vertices.",
"(These form the associahedral face of the secondary polytope, its largest face.)",
"The maximally refined triangulations described above correspond to `upward-facing' cones.",
"It is these latter that correspond to asymptotic directions in the monowall moduli space (the noncompact regions of the secondary cone subdivision of an $\\mathbb {R}^{|\\mathrm {Int}(\\mathcal {N})|}$ slice).",
"Such, regular triangulations describe generic asymptotics of a monowall moduli space.",
"To summarize, for a regular triangulation there is the following correspondence illustrated in Figure REF : each face $f$ of the spineA spine face is the projection of a face of the cut crystal.",
"corresponds to an integer point $(m,n)$ in the Newton polygon $\\mathcal {N},$ each edge of a spine is an interface between faces $f_1$ and $f_2$ and it is orthogonal to the edge of the triangulation connecting the two corresponding integer points $(m_1,n_1)$ and $(m_2,n_2)$ of $\\mathcal {N},$ and each vertex $a$ of the spine corresponds to a triangle $\\Delta _a$ of the triangulation Triang$(\\mathcal {N})$ of $\\mathcal {N}.$ Clearly, point 2. above can be stated as: the spine edge connecting vertices $a$ and $b$ is orthogonal to an edge of the triangulation of the Newton polygon that is shared by the triangles $\\Delta _a$ and $\\Delta _b.$"
],
[
"Subwall Positions",
"Let us now explore the generic asymptotic region in the monowall moduli space by fixing a regular maximal triangulation and moving along a ray in the corresponding upward facing cone of the secondary fan.",
"Each triangle of this triangulation corresponds to a vertex of the spine.",
"We claim that (up to a constant, moduli independent, shift) the $x-$ position of this vertex is the position of the corresponding subwall into which the monowall splits.",
"To be exact, we understand the position of the subwall to be the point of (partial) gauge symmetry restoration, i.e.",
"the branch point of the spectral curve $\\mathbb {S}_\\varphi .$"
],
[
"Spine Vertex",
"Consider a triangle $\\Delta _a$ of the regular maximal triangulation.",
"Say its vertices are $(m_1,n_1),(m_2,n_2),$ and $(m_3,n_3)$ ordered counterclockwise.",
"A crystal vertex corresponding to that triangle is positioned at the point $(x_a,y_a,z_a)$ satisfying $m_1 x_a+n_1 y_a+R_{m_1,n_1}&=z_a,\\nonumber \\\\m_2 x_a+n_2 y_a+R_{m_2,n_2}&=z_a,\\\\m_3 x_a+n_3 y_a+R_{m_3,n_3}&=z_a,\\nonumber $ with solution $\\left({\\begin{matrix}x_a\\\\y_a\\\\z_a\\end{matrix}}\\right)=\\frac{-1}{\\left|{\\begin{matrix}m_1-m_3 & m_2-m_3\\\\n_1-n_3 & n_2-n_3\\end{matrix}}\\right|}\\left({\\begin{matrix}n_3-n_2&n_1-n_3&n_2-n_1\\\\m_2-m_3&m_3-m_1&m_1-m_2\\\\m_2n_3-m_3n_2&m_3n_1-m_1n_3&m_1n_2-m_2n_1\\end{matrix}}\\right)\\left({\\begin{matrix}R_{m_1,n_1}\\\\R_{m_2,n_2}\\\\R_{m_3,n_3}\\end{matrix}}\\right).$ Since the triangulation is maximal and the points $(m_i,n_i)$ are numbered counterclockwise, the triangle has minimal area, thus the denominator in (REF ) is $+ 1$ and the $x-$ position of the spine vertex is $x_a=(n_2-n_3)R_{m_1,n_1}+(n_3-n_1) R_{m_2,n_2}+(n_1-n_2)R_{m_3,n_3}.$ To simplify the notation, let $R_j=R_{m_j,n_j}$ and $\\delta n_{ij}=n_i-n_j$ and same for other quantities, then $x_a&=\\delta n_{23}R_1+\\delta n_{31}R_2+\\delta n_{12}R_3\\\\&=-n_1\\delta R_{23}-n_2\\delta R_{31}-n_3\\delta R_{12}\\\\&=\\delta n_{23}\\delta R_{12}-\\delta n_{12}\\delta R_{23}.$"
],
[
"Spectral Curve Branch Points",
"The holonomy of $D_\\varphi +\\mathrm {i}\\Phi $ breaks the $U(n)$ gauge symmetry, and when the gauge symmetry is maximally broken to $U(1)^h$ the Bogomolny equation for the resulting $U(1)^h$ fields is Abelian, implying that the $U(1)^h$ Higgs field is harmonic.",
"Thus, at large distances the Higgs field is linear.",
"Therefore, it is exactly the regions where the gauge symmetry is at least partially restored that can be viewed as the sources of electromagnetic fields.",
"This argument (at least in the limit of large separation of all subwalls) associates the magnetic charge to the regions where some eigenvalues of the holonomy coincide.",
"In other words, the monowall consists of subwalls positioned at the branch points of the spectral curve.",
"These subwalls carry magnetic $U(1)^h$ charges and the magnetic field is constant between them.",
"Our immediate task is finding the locations of the branch points, in particular, their $x-$ coordinates, with $x=r_\\theta \\ln |s|,$ and comparing them with the $x-$ locations of the spine vertices found above.",
"As moduli become large, so does the spectral curve.",
"To keep the whole curve in view, we rescale the coordinates accordingly.",
"To begin, let $s=\\exp (\\frac{1}{\\hbar }\\frac{a}{r_\\theta }+\\mathrm {i}\\frac{\\theta }{r_\\theta }), t=\\exp (\\frac{1}{\\hbar }\\frac{b}{r_\\theta }+\\mathrm {i}\\alpha ),$ and $C_{m,n}=\\exp (\\frac{1}{\\hbar }\\frac{l_{mn}}{r_\\theta }+\\mathrm {i}\\frac{\\Theta _{mn}}{r_\\theta }).$ We consider the relevant locations $a$ of the branch points as $\\hbar $ is sent to zero, which corresponds to the large moduli region.",
"From the basic Puiseux expansion, each branch point is governed by three relevant monomials of the spectral polynomial $P(s,t)=\\sum _{(m,n)\\in \\mathcal {N}}C_{m,n}s^m t^n$ (see [6] for details), corresponding to the vertices of some triangle $\\Delta $ of the given regular triangulation of $\\mathcal {N}$ , therefore we can focus on $C_{m_1,n_1}s^{m_1}t^{n_1}+C_{m_2,n_2}s^{m_2}t^{n_2}+C_{m_3,n_3}s^{m_3}t^{n_3}=0.$ The other terms are exponentially small ($\\sim e^{-K/\\hbar }$ with some $K>0$ ) in the moduli.",
"If needed, relabel the vertices so that $n_3\\le n_1,n_2.$ Let $N_j=n_j-n_3$ and $M_j=m_j-m_3,$ for $j=1,2.$ If needed, exchange the indices 1 and 2 to have the counterclockwise orientation, so that $\\begin{vmatrix}N_1&N_2\\\\ M_1&M_2\\end{vmatrix}=1.$ Now, the above equation reads $C_{m_1,n_1}s^{M_1}t^{N_1}+C_{m_2,n_2}s^{M_2}t^{N_2}+C_{m_3,n_3}=0.$ In the new variables $S:=s^{M_1}t^{N_1}$ and $T:=s^{M_2}t^{N_2},$ Eq.",
"(REF ) becomes $C_{m_1,n_1} S+C_{m_2,n_2} T+C_{m_3,n_3}=0,$ thus $T=-\\frac{C_{m_1,n_1}}{C_{m_2,n_2}}\\left(S+\\frac{C_{m_3,n_3}}{C_{m_1,n_1}}\\right)=A(S-\\alpha ),$ with $A= -\\frac{C_{m_1,n_1}}{C_{m_2,n_2}}$ and $\\alpha =-\\frac{C_{m_3,n_3}}{C_{m_1,n_1}}.$ In terms of $S$ and $T$ the original variables are $s&=S^{-N_2}T^{N_1}=A^{N_1}S^{-N_2}(S-\\alpha )^{N_1},\\\\t&=S^{M_2} T^{-M_1}=A^{N_1}S^{M_2}(S-\\alpha )^{-M_1}.$ Branching of $t$ as a function of $s$ can only occur at the branch points of $S(s),$ the solution of (REF ).",
"These occur at the roots of the discriminant of the polynomial $Q(S)=A^{N_1}(S-\\alpha )^{N_1}-s S^{N_2}.$ The discriminant is proportional to the resultant $R(Q,Q^{\\prime })$ , which we now compute.",
"Let us list some basic properties of the resultant (see e.g.",
"[20]) of a pair of polynomials: $R(f,g)&=(-1)^{\\deg f\\cdot \\deg g} R(g,f),\\\\R(gq+r,g)&=b^{\\deg (gq+r) -\\deg r}R(r,g),\\ \\text{where\\ } b \\text{ is the leading coefficient of \\ } g,\\\\R(f_1f_2,g)&=R(f_1,g)R(f_2,g),\\\\R(f,a)&=a^{\\deg f}=R(a,f),\\\\R(f,x-\\alpha )&=f(\\alpha ),\\\\R(x^n-\\alpha , x^m-\\beta )&=(-1)^m (\\alpha ^{m^{\\prime }}-\\beta ^{n^{\\prime }})^d,$ where $d=GCD(n,m)$ , $n=n^{\\prime } d$ and $m=m^{\\prime } d.$ Clearly, $Q^{\\prime }(S)&=N_1 A^{N_1}(S-\\alpha )^{N_1-1}-N_2 s S^{N_2-1},\\\\Q(S)&=A^{N_1}(S-\\alpha )^{N_1}-s S^{N_2}\\nonumber \\\\&=\\frac{S-\\alpha }{N_1}Q^{\\prime }(S)+s \\frac{N_2-N_1}{N_1} (S-\\frac{N_2}{N_2-N_1}\\alpha )S^{N_2-1}.$ And the resultant is $R(Q,Q^{\\prime })=(N_1 A^{N_1})^{N_1-N_2-1}R\\left(s \\frac{N_2-N_1}{N_1} S^{N_2-1}(S-\\frac{N_2}{N_2-N_1}\\alpha ),Q^{\\prime }\\right)\\\\=(N_1 A^{N_1})^{N_1-N_2-1}\\left(s \\frac{N_2-N_1}{N_1}\\right)^{N_2}R(S,Q^{\\prime })^{N_2-1}R\\left(S-\\frac{N_2}{N_2-N_1}\\alpha , Q^{\\prime }\\right) \\\\=(N_1 A^{N_1})^{N_1-N_2-1}\\left(s \\frac{N_2-N_1}{N_1}\\right)^{N_2}\\left(Q^{\\prime }(0)\\right)^{N_2-1}Q^{\\prime }\\left(\\frac{N_2}{N_2-N_1}\\alpha \\right) \\\\=(N_1 A^{N_1})^{N_1-N_2-1}\\left(s \\frac{N_2-N_1}{N_1}\\right)^{N_2}\\left( N_1 A^{N_1}(-\\alpha )^{N_1-1}\\right)^{N_2-1}\\\\\\left(N_1 A^{N_1}\\left(\\frac{N_1}{N_2-N_1}\\alpha \\right)^{N_1-1}-N_2 s \\left(\\frac{N_2}{N_2-N_1}\\alpha \\right)^{N_2-1}\\right).$ It vanishes at $s=(-1)^{N_2}\\left(\\frac{N_1}{N_2}\\right)^{N_1}\\frac{1}{(N_2-N_1)^{N_1-N_2}} \\frac{C_{m_1,n_1}^{N_2}}{C_{m_2,n_2}^{N_1}}C_{m_3,n_3}^{N_1-N_2}.$ This is the position of a branch point corresponding to the triangle $\\Delta $ .",
"In terms of the spatial position $x=r_\\theta \\ln |s|=a/\\hbar $ of this branch point of the spectral curve, we have $\\hbar x_\\Delta =a=\\left((n_2-n_3) l_{m_1,n_1}+(n_3-n_1) l_{m_2,n_2}+(n_1-n_2) l_{m_3,n_3}\\right)\\\\+\\hbar \\ln \\left(\\frac{N_1}{N_2}\\right)^{N_1}\\frac{1}{(N_2-N_1)^{N_1-N_2}},$ which matches the position of the vertex of the spine (REF ) up to $O(\\hbar ^0)$ terms.",
"The $\\theta =r_\\theta \\mathrm {Arg} (s)$ coordinate of the branch point is read off as the imaginary part of (REF ): $\\theta _\\Delta =\\left((n_2-n_3) \\Theta _{m_1,n_1}+(n_3-n_1) \\Theta _{m_2,n_2}+(n_1-n_2) \\Theta _{m_3,n_3}\\right).$ Thus, for large values of $R_{m,n}=r_\\theta \\ln |C_{m,n}|=\\frac{l_{m,n}}{\\hbar }$ the subwalls are positioned at the $x-$ locations of the vertices of the spine.",
"Moreover, the $x$ and $\\theta $ positions of the wall associated to the spine vertex $a$ (corresponding to the triangle $\\Delta _a$ of the triangulation) are expressed via the same relation $x_a&=\\sum _{f=1}^3 c_a^f R_f,&\\theta _a&=\\sum _{f=1}^3 c_a^f \\Theta _f,$ where the sum is over the three spine faces containing to the vertex $a.$ When these three faces are numbered counterclockwise, the coefficients $c_a^f$ are $c_a^1= (n_2-n_3), c_a^2= (n_3-n_1),$ and $c_a^3= (n_1-n_2).$"
],
[
"Subwall Charges and Inter-wall Fields",
"If the vertices of the spine indicate the subwall positions, the spine edges approximate the eigenvalues of the Higgs field between the walls (to exponential accuracy in distance to the nearest wall).",
"Away from all subwalls the $U(n)$ gauge symmetry is broken to $U(1)^h$ with each $U(1)$ factor associated to one $D_\\varphi +\\mathrm {i}\\Phi $ holonomy eigenvalue $t^\\iota $ .",
"We order these eigenvalues in decreasing order of $y^\\iota =r_\\theta \\ln |t^\\iota |$ so that $y^1\\ge y^2\\ge \\ldots \\ge y^n.$ Thus, each $y^\\iota $ is a continuous, piecewise linear function $\\mathbf {\\phi }^\\iota $ of $x$ with kinks at the spine vertices.",
"Away from the walls we have Higgs field $\\Phi =\\frac{\\mathrm {i}}{2\\pi r_\\theta r_\\varphi }\\mathrm {diag}(y^\\iota )+O(e^{- \\frac{d_{\\mathrm {wall}}}{\\Lambda }}),$ where $d_\\mathrm {wall}$ is the distance to the closest wall and the constant $\\Lambda $ is the characteristic wall width, computed in [5].",
"Since each spine edge, orthogonal to the $(m_{ff^{\\prime }},n_{ff^{\\prime }})=(m_{f}-m_{f^{\\prime }},n_{f}-n_{f^{\\prime }})$ edge of the triangulation of $\\mathcal {N}$ , corresponds to $|n_{ff^{\\prime }}|$ of the ordered eigenvalues, now all non-vertical edges of the spine are labelled by the factor indices $\\iota $ , as illustrated in Figure REF .",
"If we associate each index value to some distinct color, then the spine consists of continuous lines going left to right, each piecewise linear with kinks at the spine vertices.",
"Each colored line is a graph of a function $y^\\iota (x)$ .",
"It corresponds to an (approximate) Higgs diagonal value and the slope $S_e=-\\frac{n_e}{m_e}=-\\frac{n_{h(e)}-n_{t(e)}}{m_{h(e)}-m_{t(e)}}$ of the line is the magnetic field of the corresponding $U(1)$ factor.",
"The difference $g^\\iota =S^\\iota _\\mathrm {right}-S^\\iota _\\mathrm {left}$ in line slopes $S^\\iota _\\mathrm {right}$ and $S^\\iota _\\mathrm {left}$ across the wall is the magnetic charge in that $\\iota -$ th $U(1)$ factor of the wall corresponding to this spine vertex.",
"Next, we interpret the resulting fields as a superposition of individual subwall contributions and explore the subwall dynamics."
],
[
"Subwall Interactions",
"The moduli space of a monowall is of real dimension $4|\\mathrm {Int}(\\mathcal {N})|$ with half of the moduli being the coefficients $C_{m,n}=\\exp {\\frac{R_{m,n}+\\mathrm {i}\\Theta _{m,n}}{r_\\theta }},$ (for $(m,n)\\in \\mathrm {Int}\\,\\mathcal {N}$ ) of the spectral curve $\\mathbb {S}_\\varphi $ and the other half $(\\Phi _{m,n},T_{m,n})$ parameterizing the Hermitian eigen-line bundle with a flat connection over $\\mathbb {S}_\\varphi .$ We view this moduli space as a three-torus fibration over the $|\\mathrm {Int}(\\mathcal {N})|-$ dimensional space $\\mathbb {R}^{|\\mathrm {Int}(\\mathcal {N})|}$ with base coordinates $R_{m,n}=r_\\theta \\ln |C_{m,n}|, (m,n)\\in \\mathrm {Int}\\,\\mathcal {N}$ and the fiber coordinates $(\\Theta _{m,n},\\Phi _{m,n},T_{m,n}).$ The space of `long' moduli and parameters $R_{m,n}$ factors as a direct product $\\mathbb {R}^{|\\mathcal {N}|}=\\mathbb {R}^{|\\mathrm {Int}(\\mathcal {N})|}\\times \\mathbb {R}^{|\\mathrm {Per}(\\mathcal {N})|}$ of the space of all `long' moduli and of the space all `long' parameters.",
"As we discussed, this space $\\mathbb {R}^{|\\mathcal {N}|}$ contains the secondary fan whose maximal cones are indexed by the regular triangulations of the Newton polygon $\\mathcal {N}.$ The space of long moduli is obtained by fixing the values of the long parameters.",
"This is the base space of the moduli space fibered by tori.",
"It traverses this fan, and the fan subdivides it into polytopal regions, each region corresponding to a phase of the monowall.",
"The monowall in each phase, labelled by a triangulation Triang$(\\mathcal {N}),$ is well approximated (for sufficiently large moduli) by an array of subwalls.",
"And $a$ -th subwall corresponds to a triangle $\\Delta _a\\in \\mathrm {Triang}(\\mathcal {N})$ and it carries $h$ Abelian charges $(g^1_a,\\ldots , g^h_a)$ (defined by the slopes of the two sides of the triangle $\\Delta _a$ to which the graph of $y_\\iota (x)$ is associated).",
"Away from any subwall the Higgs field is essentially diagonal $\\Phi =\\frac{\\mathrm {i}}{2\\pi r_\\theta r_\\varphi }\\mathrm {diag}(y^\\iota )+O(e^{- \\frac{d_{\\mathrm {wall}}}{\\Lambda }})$ with $y^\\iota =\\frac{Q^\\iota _+ +Q^\\iota _-}{2} x+\\frac{M^\\iota _+ +M^\\iota _-}{2}+\\sum _{\\Delta \\in \\mathrm {Triang}(\\mathcal {N})} \\frac{g_\\Delta ^\\iota }{2} |x-x_\\Delta | +O(e^{- \\frac{d_{\\mathrm {wall}}}{\\Lambda }}).$ Here $Q^\\iota _+-Q^\\iota _-=\\sum _\\Delta g^\\iota _\\Delta $ and $M^\\iota _- - M^\\iota _+=\\sum _\\Delta g_\\Delta ^\\iota x_\\Delta ,$ and the subwalls' positions are $x_\\Delta =(n_2-n_3)R_{m_1,n_1}+(n_3-n_1)R_{m_2,n_2}+(n_1-n_2)R_{m_3,n_3}.$ Let us discuss the meaning of (REF ) in detail, neglecting the exponentially small terms from now on.",
"A single Abelian wall positioned at $(x_a,\\theta _a,\\varphi _a)$ produces fields $\\Phi =\\mathrm {diag}(\\Phi ^\\iota )$ and $A=\\mathrm {diag}(A^\\iota )$ with $\\Phi ^\\iota _a&=\\frac{\\mathrm {i}}{2\\pi r_\\theta r_\\varphi } \\frac{1}{2} g_a^\\iota |x-x_a|,\\\\A^\\iota _a&=\\frac{\\mathrm {i}}{2\\pi r_\\theta r_\\varphi } \\mathrm {sign}(x-x_a) \\frac{1}{2}g_a^\\iota \\frac{(\\varphi -\\varphi _a)d\\theta - (\\theta -\\theta _a)d\\varphi }{2}.$ The superposition of such fields has left-right symmetric asymptotics.",
"To accommodate general monowall charges, let $\\bar{Q}^\\iota &=\\frac{Q^\\iota _+ +Q^\\iota _-}{2}&&\\text{and}&\\bar{M}^\\iota &=\\frac{M^\\iota _+ +M^\\iota _-}{2},&$ so that the total fields are $\\Phi ^\\iota &= \\frac{i}{2 \\pi r_\\theta r_\\varphi } \\left( \\bar{Q}^\\iota x+ \\bar{M}^\\iota \\right)+\\sum _a \\Phi ^\\iota _a, \\\\A^\\iota &=\\frac{-\\mathrm {i}}{2 \\pi r_\\theta r_\\varphi } \\left( \\bar{Q}^\\iota \\frac{\\theta d\\varphi -\\varphi d\\theta }{2} + r_\\varphi \\bar{\\chi }_{\\theta }^\\iota d\\theta + r_\\theta \\bar{\\chi }_{\\varphi }^\\iota d\\varphi \\right)+\\sum _a A^\\iota _a.$ Now, any variation of the moduli produces some motion of the subwalls.",
"A moving charged wall produces Liénard-Wiechert potentials [5], instead of those of the static potentials of Eq.",
"(REF ).",
"In addition, each wall has an associated electromagnetic phase.",
"Time dependence of this phase produces an electric charge $q_a.$ Thus, each subwall (with varying moduli) becomes a dyonic moving wall with magnetic charges $g_a^\\iota $ , respective electric charges $q_a g_a^\\iota $ and velocity $\\vec{V}_a.$ We spell out the explicit expressions for these potentials next."
],
[
"Moving Dyonic Subwalls",
"To avoid superficial prefactors let $\\Phi ^\\iota =\\frac{\\mathrm {i}}{2\\pi r_\\theta r_\\varphi }\\mathbf {\\phi }^\\iota $ and $A^\\iota =\\frac{-\\mathrm {i}}{2\\pi r_\\theta r_\\vartheta } \\mathbf {a}^\\iota $ , so that for an Abelian monowall $*d\\mathbf {a}^\\iota =d\\mathbf {\\phi }^\\iota .$ An elementary static wall positioned at $x=0,\\theta =0$ produces $\\mathbf {\\phi }^\\iota (x)&=\\frac{1}{2} g^\\iota |x|,&\\mathbf {a}^\\iota (x)&=\\frac{1}{2} g^\\iota \\, \\eta _{{\\vec{x}}},$ where the one-form $\\eta _{\\vec{x}}$ satisfies $*d\\eta _{\\vec{x}}=d|x|,$ for example $\\eta _{\\vec{x}}={\\left\\lbrace \\begin{array}{ll} \\frac{\\theta d\\varphi - \\varphi d\\theta }{2},& \\text{for } x>0\\\\-\\frac{\\theta d\\varphi - \\varphi d\\theta }{2},& \\text{for } x<0\\end{array}\\right.",
"}.$ The superposition of such subwalls as in (REF ) produces the functions (REF ) read off from the spine with $y^\\iota =\\mathbf {\\phi }^\\iota $ .",
"A BPS dyon with electric charge $q$ and magnetic charge $e$ satisfies BPS equations $B=\\frac{e}{\\sqrt{e^2+q^2}}\\nabla \\Phi $ and $E=\\frac{q}{\\sqrt{e^2+q^2}}\\nabla \\Phi $ [4], so an elementary dyonic wall can be described by the pentuple $(\\mathbf {\\phi },\\mathbf {a}_0,\\mathbf {a},\\tilde{\\mathbf {a}}_0,\\tilde{\\mathbf {a}})$ consisting of the scalar Higgs field $\\mathbf {\\phi }$ , an electro-magnetic potential consisting of the time component function $\\mathbf {a}_0$ and a `vector' component one-form $\\mathbf {a},$ and dual electromagnetic potentials $\\tilde{\\mathbf {a}}_0$ (a function) and $\\tilde{\\mathbf {a}}$ (a one-form).",
"These are related by electromagnetic duality $+d\\mathbf {a}=B^\\flat &=\\tilde{E}^\\flat =-d\\tilde{\\mathbf {a}}_0-\\dot{\\tilde{\\mathbf {a}}},\\\\d\\mathbf {a}_0+\\dot{\\mathbf {a}}=-E^\\flat &=\\tilde{B}^\\flat =*d\\tilde{\\mathbf {a}}.$ Here $E^\\flat , B^\\flat $ are the one-forms metric dual to the electric and magnetic vector fields, and $\\tilde{E}^\\flat , \\tilde{B}^\\flat $ are in the same relation as the electro-magnetic dual fields $\\tilde{E}:=B$ and $\\tilde{B}:=-E.$ In these terms the fields of the $b-$ th static dyonic wall with magnetic charges $g_b^\\iota $ and electric charges $q_b g_b^\\iota $ positioned at $\\vec{x}=\\vec{x}_b$ are $\\mathbf {\\phi }^\\iota (x)&=\\frac{1}{2} g^\\iota _b \\sqrt{1+q_b^2} |x-x_b|,$ $\\mathbf {a}^\\iota (x)&=\\frac{1}{2} g_b^\\iota \\eta _{{\\vec{x}}-{\\vec{x}}_b},&\\mathbf {a}_0^\\iota (x)&=-q_b \\frac{1}{2} g_b^\\iota |x-x_b|,\\\\\\tilde{\\mathbf {a}}^\\iota (x)&=-q_b\\frac{1}{2} g_b^\\iota \\eta _{{\\vec{x}}-{\\vec{x}}_b},&\\tilde{\\mathbf {a}}_0^\\iota (x)&=- \\frac{1}{2} g_b^\\iota |x-x_b|.$ The Lorentz boost (accompanied by the proper time delay) produced the Liénard-Wiechert potential produced by the moving dyonic wall.",
"Our focus is on the dynamics of slowly moving walls, thus, we neglect terms higher than quadratic in the resulting Lagrangian.",
"In particular, the typical time delay terms of the form $\\sqrt{\\vec{x}^2-(\\vec{x} \\times \\vec{V})^2}$ can be safely replaced by $|x|.$ The resulting fields are ${\\mathbf {\\phi }}^\\iota _b(x)&= \\sqrt{1+q_b^2} \\frac{1}{2} g_b^\\iota |x-x_b|\\sqrt{1-\\vec{V}_b^2}\\nonumber \\\\&=\\frac{1}{2} g_b^\\iota |x-x_b|\\left(1+\\frac{q_b^2}{2}-\\frac{\\vec{V}^2}{2}\\right)+o(V_b^2,q_b^2),$ ${\\mathbf {a}}^\\iota _b(x)&=\\frac{1}{2} g_b^\\iota \\eta _{{\\vec{x}}-{\\vec{x}}_b}- \\frac{1}{2} q_b g_b^\\iota |x-x_b| \\vec{V}_b^\\flat +o(V_b^2,q_b^2),\\\\{\\mathbf {a}}_{0b}^\\iota (x)&=- \\frac{1}{2} q_b g_b^\\iota |x-x_b|+\\frac{1}{2} g_b^\\iota \\eta _{{\\vec{x}}-{\\vec{x}}_b}(\\vec{V}_b)+O(V_b^2,q_b^2),\\\\{\\tilde{\\mathbf {a}}}^\\iota _b(x)&=- \\frac{1}{2} q_b g_b^\\iota \\eta _{{\\vec{x}}-{\\vec{x}}_b} - \\frac{1}{2} g_b^\\iota |x-x_b| \\vec{V}_b^\\flat +O(V_b^2,q_b^2),\\\\{\\tilde{\\mathbf {a}}}_{0b}^\\iota (x)&=-\\frac{1}{2} g_b^\\iota |x-x_b|- \\frac{1}{2} q_b g_b^\\iota \\eta _{{\\vec{x}}-{\\vec{x}}_b}(\\vec{V}_b)+o(V_b^2,q_b^2).$ Here $\\vec{x}=(x,\\theta ,\\varphi ),$ $\\vec{V}^\\flat =V^x dx+V^\\theta d\\theta +V^\\varphi d\\varphi ,$ and $\\eta (\\vec{V})=\\eta _x V^x+\\eta _\\theta V^\\theta +\\eta _\\varphi V^\\varphi $ is the value of the one-form $\\eta $ on the vector $\\vec{V}.$ From now on we drop the higher order terms in $V$ and $q.$ A dyonic wall $a$ moves in the background of fields $(\\mathbf {\\phi },\\mathbf {a},\\mathbf {a}_0,\\tilde{\\mathbf {a}},\\tilde{\\mathbf {a}}_0)$ which are the sum of contributions of all other walls.",
"For example, (keeping up to quadratic terms in $V$ and $q$ ) the Higgs field that the $a$ -th wall experiences is $\\mathbf {\\phi }^\\iota (x_a)&=\\bar{Q}^\\iota x_a+\\bar{M}^\\iota +\\sum _{b} {\\mathbf {\\phi }}^\\iota _b(x_a)\\nonumber \\\\&=\\bar{Q}^\\iota x_a+\\bar{M}^\\iota +\\sum _{b} \\frac{1}{2} g_b^\\iota |x_a-x_b| \\left(1+\\frac{q_b^2}{2}-\\frac{\\vec{V}_b^2}{2}\\right).$ Similarly, $\\mathbf {a}^\\iota (x_a)&=\\bar{Q}^\\iota \\frac{\\theta _a d\\varphi -\\varphi _a d\\theta }{2}+ r_\\varphi \\bar{\\chi }^\\iota _\\theta d\\theta + r_\\theta \\bar{\\chi }^\\iota _\\varphi d\\varphi +\\sum _{b} \\hat{\\mathbf {a}}^\\iota _b(\\vec{x}_a), \\\\\\mathbf {a}^\\iota _0(x_a)&=\\sum _{b} \\hat{\\mathbf {a}}^\\iota _{0b}(\\vec{x}_a), \\\\\\tilde{\\mathbf {a}}^\\iota (x_a) &= \\sum _{b} \\hat{\\tilde{\\mathbf {a}}}^\\iota _b(\\vec{x}_a), \\\\\\tilde{\\mathbf {a}}^\\iota _0(x_a) &= - \\bar{Q}^\\iota x_a - \\bar{M}^\\iota + \\sum _{b} \\hat{\\tilde{\\mathbf {a}}}^\\iota _{0b}(\\vec{x}_a).$ Note, that since the fields produced by any given subwall itself vanish at its location, there are no self-interaction terms, and the sums above are extended over all walls.",
"The resulting relativistic Lagrangian $\\hat{L}_a$ governing the $a$ -th subwall dynamics is $\\hat{L}_a= \\sum _\\iota \\big \\lbrace -g_a^\\iota \\mathbf {\\phi }^\\iota (x_a) \\sqrt{1+q_a^2}\\sqrt{1-\\vec{V}_a^2}&+ q_a g_a^\\iota [\\mathbf {a}^\\iota (x_a)(\\vec{V}_a) - \\mathbf {a}_0^\\iota (x_a)]\\nonumber \\\\&+g_a^\\iota [\\tilde{\\mathbf {a}}^\\iota (x_a)(\\vec{V}_a)- \\tilde{\\mathbf {a}}_0^\\iota (x_a)]\\big \\rbrace ,$ with the background fields given by (REF –).",
"This Lagrangian $\\hat{L}_a$ governing the motion of one of the subwalls should be understood as the part of the effective Lagrangian $\\hat{L}$ of the whole monowall governing the motion of all subwalls.",
"In particular it is the part of $\\hat{L}$ that contains $\\vec{x}_a.$ Note, that the two subwall interaction is symmetric, e.g.",
"$g_a^\\iota \\sqrt{1+q_a^2}\\sqrt{1-\\vec{V}_a^2}\\mathbf {\\phi }^\\iota _b(x_a)=g_b^\\iota \\sqrt{1+q_b^2}\\sqrt{1-\\vec{V}_b^2}\\mathbf {\\phi }^\\iota _a(x_b).$ Thus, combining individual subwall Lagrangians $\\hat{L}_a$ into oneEach pairwise interaction contributes once.",
"(and keeping terms up to quadratic in velocities and electric charges): $ \\hat{L} = \\frac{1}{2} U^{ab} (\\vec{V}_a \\cdot \\vec{V}_b - q_a q_b)+ q_a W^{ab} (\\vec{V}_b) ,$ with implicit summation over the repeated indices $a$ and $b$ and $U^{aa}&= \\sum _{\\iota =1}^n g_a^\\iota \\left(\\bar{Q}^\\iota x_a+\\bar{M}^\\iota + \\frac{1}{2}\\sum _b g^\\iota _b|x_a-x_b|\\right),\\\\U^{ab}&=- \\frac{1}{2}\\sum _{\\iota =1}^n g_a^\\iota g_b^\\iota |x_a-x_b|, \\qquad \\text{for\\ } a\\ne b,$ and $W^{aa}&= \\sum _{\\iota =1}^n g_a^\\iota \\left(\\bar{Q}^\\iota \\frac{\\theta _a d\\varphi -\\varphi _a d\\theta }{2} + r_\\varphi \\bar{\\chi }_\\theta ^\\iota d\\theta + r_\\theta \\bar{\\chi }_\\varphi ^\\iota d\\varphi + \\frac{1}{2}\\sum _b g_b^\\iota \\eta _{\\vec{x}_a-\\vec{x}_b}\\right),\\\\W^{ab}&= - \\frac{1}{2}\\sum _{\\iota =1}^n g_a^\\iota g_b^\\iota \\eta _{\\vec{x}_a-\\vec{x}_b},\\qquad \\text{for\\ } a\\ne b.$"
],
[
"Positions",
"As we discussed, the motion of the subwalls is highly choreographed, since the subwalls' positions are dictated by the plane arrangement.",
"Via Eqs.",
"(REF ): $x_a=\\sum _{\\begin{array}{c}f \\\\ V(f)\\ni a\\end{array}}c_a^f R_f,$ where the sum is over the three faces $f$ that have $a$ as a their vertex and the coefficients are $c_a^f=n_{f^{\\prime }}-n_{f^{\\prime \\prime }}$ as in Sec.",
"REF .",
"In fact, the $\\theta $ -position of the wall is determined by the same relation $\\theta _a=\\sum _{\\begin{array}{c}f \\\\ V(f)\\ni a\\end{array}} c_a^f \\Theta _f.$ As mentioned in Sec.",
"REF , one can consider another spectral curve $\\mathbb {S}_\\theta $ .",
"Analysis of its branch points leads to the same $\\varphi $ subwall position relation $\\varphi _a=\\sum _{\\begin{array}{c}f \\\\ V(f)\\ni a\\end{array}} c_a^f \\Phi _f.$ Next, we focus on understanding the relations between the electric charges $q_a$ of the subwalls.",
"Namely, we shall now demonstrate that they also satisfy the same relation $q_a=\\sum _{\\begin{array}{c}f \\\\ V(f)\\ni a\\end{array}} c_a^f Q_f,$ for independent variables $Q_f$ , one for each internal spine face."
],
[
"Electric Charges",
"Let $V$ denote the set of spine vertices, $E$ – the set of spine edges, and $F$ – the set of spine faces.",
"We shall orient the edges rightwards (and up, if vertical).",
"For an edge $e\\in E$ let $h(e)\\in V$ denote its head and let $t(e)\\in V$ denote its tail.",
"Then the crystal vertex position $(x_a,y_a,z_a)$ is determined from the system of equations (REF ) $z_a&=m_f x_a+n_f y_a+R_f,$ satisfied for all faces $f\\in F$ for which $a\\in V$ is a vertex of $f$ : $a\\in V(f).$ Taking the difference of adjacent faces, one gets the spine vertex position $(x_a,y_a)$ from the equations $&&&&M_e&:=m_{h(e)}-m_{t(e)},\\nonumber \\\\M_e x_a+N_e y_a+L_e&=0,&&\\text{where}&N_e&:=n_{h(e)}-n_{t(e)},\\\\&&&&L_e&:=R_{h(e)}-R_{t(e)},\\nonumber $ for any edge $e\\in E$ beginning or ending at the vertex $a\\in V.$ Note that solutions $((x_a,y_a))_{a\\in V}$ of (REF ) are in one-to-one correspondence with solutions $((x_a,y_a,z_a))_{a\\in V}$ of (REF ).",
"Also, Eqs.",
"(REF ) describes a system $(V_3,E_2)$ of $V$ points $[x_a,y_a,1]$ in $\\mathbb {R}P^2$ and $E$ lines $\\lbrace [x,y,1] | M_e x+N_e y+L_e=0 \\rbrace $ in the same in $\\mathbb {R}P^2$ , such that each point has three lines passing through it (corresponding to three edges $e$ for which $a$ is a vertex) and each line has two points on it (corresponding to the two ends of $e\\in E$ ).",
"The reciprocal view of the dual $\\mathbb {R}P^2$ with coordinates $[M,N,L]$ gives the $(E_2,V_3)$ system of $E$ points $[M_e,N_e,L_e]$ and $V$ lines $\\lbrace [M,N,L] | x_a M + y_a N + L=0 \\rbrace $ such that each point has two lines through it and each line has three points on it.",
"Note, that the whole system is completely determined by the set of distinct points $(x_a,y_a)$ , since (using the first $(V_3,E_2)$ configuration) each line is determined by two points on it.",
"Consider triplets $(x_a,y_a,w_a)$ with $w_a$ the (coincident) eigenvalue of $\\mathbf {a}_0$ at the wall $a$ where this eigenvalue has a kink.",
"As earlier, we define $W_e:=w_{h(e)}-w_{t(e)}$ for each spine edge $e.$ Then, comparing the electric flux change across the subwall (LHS below) with the electric charge $q_ag_a^\\iota $ (RHS below), one has $\\frac{W_{e^{out}}}{X_{e^{out}}} - \\frac{W_{e^{in}}}{X_{e^{in}}}=q_a\\left(\\frac{Y_{e^{out}}}{X_{e^{out}}} - \\frac{Y_{e^{in}}}{X_{e^{in}}}\\right).$ This was our very definition of the electric charge $q_a g_a^\\iota $ .",
"This relation implies that $p_a:=\\frac{W_e-q_a Y_e}{X_e}$ is the same for any edge $e$ beginning or ending at $a$ .",
"This implies that $u_a:=x_a p_a+y_a q_a -w_a=x_b p_a + y_b q_a -w_b$ for any edge $\\overline{ab}\\in E.$ Which in turn is equivalent to $P_e x_a+Q_e y_a-U_e=0,$ for any $a\\in V$ and any $e\\in E$ beginning or ending at $a$ .",
"Note, that (REF ) is also a $(V_3,E_2)$ system.",
"In fact, since it has the same set of points $(x_a,y_a)_{a\\in V}$ it must be the same system of projective lines and points.",
"Thus, $M_e Q_e&=N_e P_e,&Q_e L_e&=- U_e N_e.$ We take the last equation as determining $U_e=-\\frac{L_e}{N_e} Q_e$ .",
"The first equation, on the other hand, reads $S_e:=-M_e q_a+N_e p_a=-M_e q_b+N_e p_b,$ for any edge $\\overline{ab}\\in E.$ Since, $M_e=m_{f_{left}}-m_{f_{right}}$ and $N_e=n_{f_{left}}-n_{f_{right}}$ , we have $\\sum _{e, h(e)=a} S_e-\\sum _{e,t(e)=a} S_e=0$ and therefore the function $\\lbrace S_e\\rbrace _e$ on edges is potential on the dual graph, in other words, there is a function $Q_f$ such that $S_e=Q_{f_{left}}-Q_{f_{right}}$ .",
"Here $f_{left}$ denotes the spine face to the left of the oriented spine edge $e$ , and $f_{right}$ denotes the one to its right.",
"Substituting this into Eq.",
"(REF ), $m_{f_{left}}q_a + n_{f_{left}}(-p_a) + Q_{f_{left}}=m_{f_{right}}q_a + n_{f_{right}}(-p_a) + Q_{f_{right}}=:r_a.$ We conclude that the set of triples $(q_a,-p_a,r_a)$ satisfies exactly the same system of equations as the triples $(x_a,y_a,w_a)$ with the role of $R_f$ played by $Q_f.$ Thus, $q_a$ are expressed via (REF ): $q_a&=\\delta n_{23}Q_1+\\delta n_{31}Q_2+\\delta n_{12}Q_3=-n_1\\delta Q_{23}-n_2\\delta Q_{31}-n_3\\delta Q_{12}\\nonumber \\\\&=\\delta n_{23}\\delta Q_{12}-\\delta n_{12}\\delta Q_{23},$ and $q_a=\\sum _{f, V(f)\\ni a} c_a^f Q_f.$"
],
[
"The Asymptotic Metric on the Moduli Space",
"Now we are ready to read off the asymptotic metric on the moduli space within each maximal cone of the secondary fan.",
"So far we can conclude that the effective Lagrangian (REF ), expressed in terms of the moduli $\\vec{X}_f=\\left({\\begin{matrix}R_{f}\\\\ \\Theta _{f}\\\\ \\Phi _{f}\\end{matrix}}\\right)$ and independent charges $Q_f$ , is $\\hat{L}=\\frac{1}{2} c_a^f U^{ab}c_b^{f^{\\prime }}(\\dot{\\vec{X}}_f\\cdot \\dot{\\vec{X}}_{f^{\\prime }} - Q_f Q_{f^{\\prime }})+ Q_f c_a^f W^{ab}(\\dot{\\vec{X}}_{f^{\\prime }})c_b^{f^{\\prime }}.$ To lighten our notation from here on we denote by $cUc=[(cUc)^{ff^{\\prime }}]$ the matrix with entries $(cUc)^{ff^{\\prime }}:=c_a^f U^{ab}c_b^{f^{\\prime }}$ and similarly for $cWc.$ The conserved charges $Q_f$ should be viewed as momenta associated with the electromagnetic phase moduli $T_f.$ In order to express the Lagrangian in terms of the moduli we perform the Legendre transform in $Q_f:$ $\\dot{T}_f&=\\frac{\\partial \\hat{L}}{\\partial Q_f}=-(cUc)^{ff^{\\prime }}Q_{f^{\\prime }} + (cWc)^{ff^{\\prime }}(\\dot{\\vec{X}}_{f^{\\prime }}),\\\\L&=\\hat{L}-Q_{f^{\\prime }}\\frac{\\partial \\hat{L}}{\\partial Q_{f^{\\prime }}}.$ This yields the effective Lagrangian: $L=\\frac{1}{2} (cUc)^{ff^{\\prime }}\\dot{\\vec{X}}_f\\cdot \\dot{\\vec{X}}_{f^{\\prime }}\\\\+\\frac{1}{2}\\Big (\\dot{T}_f - (cWc)^{f\\check{f}}(\\dot{\\vec{X}}_{\\check{f}})\\Big ) (cUc)^{-1}_{ff^{\\prime }} \\Big (\\dot{T}_{f^{\\prime }} - (cWc)^{f^{\\prime }\\hat{f}}(\\dot{\\vec{X}}_{\\hat{f}})\\Big ),$ which describes free motion of a point on a manifold with the Pedersen-Poon [18] type metric $g=(cUc)^{ff^{\\prime }}d{\\vec{X}}_f\\cdot d{\\vec{X}}_{f^{\\prime }}+\\Big (d{T}_f - (cWc)^{f}\\Big ) (cUc)^{-1}_{ff^{\\prime }} \\Big (d{T}_{f^{\\prime }} - (cWc)^{f^{\\prime }}\\Big ).$ This is the asymptotic metric on the moduli space of the monowall.",
"Its terms are written in terms of the $U$ and $W$ of Eqs.",
"(REF -REF ) and the coefficients $c_a^f$ appearing in Eq.",
"(REF ).",
"Let us emphasize that each generic ray in the moduli space lies in a cone labelled by a regular triangulation of the Newton polygon.",
"Thus, the end of the moduli space is divided into sectors, each with the corresponding asymptotic metric (REF ).",
"The triangulation determines both the coefficients $c_a^f$ and the order of the subwalls' positions $x_a.$"
],
[
"The Kähler Potential and the Generalized Legendre Transform",
"Consider approaching the infinity of the moduli space within some maximal cone of the secondary fan.",
"Such a cone is specified by a triangulation Triang$(\\mathcal {N})$ of the Newton polygon $\\mathcal {N}.$ As we now demonstrate, the Kähler potential $K$ of the asymptotic metric is encoded in a single function: $G(\\bar{M}^\\iota ;x_1,...,x_n) = \\sum \\limits _{\\iota =1}^h \\Bigg [ \\sum \\limits _{a\\in \\mathrm {Triang}(\\mathcal {N})} g_{a}^\\iota \\left( \\bar{M}^\\iota \\frac{x_a^2}{2} + \\bar{Q}^\\iota \\frac{x_a^3}{6} \\right) \\\\+ \\frac{1}{2} \\sum \\limits _{\\begin{array}{c}a,b \\\\ a>b\\end{array}} g_a^\\iota g_b^\\iota \\frac{(x_a-x_b)^3}{6} \\Bigg ].$ The relation is via the Generalized Legendre Transform of [15], [11] as follows.",
"Number the subwalls from left to right, so that $x_1<x_2<\\ldots <x_N,$ and introduce a Laurent polynomial in the auxiliary variable $\\zeta $ for each subwall $\\hat{\\eta }_a(\\zeta ) := \\frac{\\theta _a +\\mathrm {i}\\varphi _a}{2\\zeta }+ x_a-\\frac{\\theta _a -\\mathrm {i}\\varphi _a}{2}\\zeta ,$ and let $\\hat{\\mathcal {V}}^\\iota (\\zeta ) := (r_\\theta \\bar{\\chi }_{\\varphi }^\\iota - \\mathrm {i}r_\\varphi \\bar{\\chi }_\\theta ^\\iota )\\frac{1}{\\zeta } + \\bar{M}^\\iota - (r_\\theta \\bar{\\chi }_{\\varphi }^\\iota + \\mathrm {i}r_\\varphi \\bar{\\chi }_\\theta ^\\iota ) \\zeta .$ Note, that thanks to (REF – REF ) the polynomial coefficients $x_a,\\theta _a,\\varphi _a$ associated to the positions of each wall are functions of the respective moduli (and parameters) $R_{f},\\Theta _{f},\\Phi _{f}$ : $x_a&=c_a^f R_f,&\\theta _a&=c_a^f \\Theta _f,&\\varphi _a&=c_a^f \\Phi _f.&$ Consider the Generalized Legendre Transform of the following auxiliary function $F(R_f,\\Theta _f,\\Phi _f)=\\frac{-1}{2 \\pi \\mathrm {i}} \\oint _0 \\frac{d\\zeta }{\\zeta } {G}\\left(\\hat{\\mathcal {V}}; \\hat{\\eta }_1,...,\\hat{\\eta }_N\\right),$ of the parameters and of three quarters of the moduli.",
"Half of the complex moduli are $Z_f:=\\frac{\\Theta _f+\\mathrm {i}\\Phi _f}{2}.$ The contour integration above is over a counterclockwise oriented small circle around zero.",
"The remaining half of the moduli $U_f$ are related to the above coordinates by $U_f+\\bar{U}_f:=\\frac{\\partial F}{\\partial R_f}=F_{R_f}.$ Importantly, $F$ constructed this way is guaranteed to satisfy the Laplace type system of equations $(\\partial _{Z_f}\\partial _{\\bar{Z}_{f^{\\prime }}}+\\partial _{R_f}\\partial _{R_{f^{\\prime }}})F=0.$ The Kähler potential $K$ is the Legendre transform of $F:$ $K(Z_f,U_f)=F-\\sum _{f\\in \\mathrm {Int}(\\mathcal {N})} R_f (U_f+\\bar{U}_f),$ with $R_f$ on the right-hand side understood as functions of $Z_f$ and $U_f$ determined by $(\\ref {Eq:LegMom}).$ As usual for the Legendre transform $K_{U_f}=-R_f$ and $K_{Z_f}=F_{Z_f}.$ This gives $K_{U_f \\bar{U}_{f^{\\prime }}}=-[F_{RR}]^{-1}_{ff^{\\prime }}$ , which is the negative inverse of the matrix $F_{RR}=(F_{R_f R_{f^{\\prime }}}).$ Also $K_{U_f\\bar{Z}_{f^{\\prime }}}=[F_{RR}]^{-1}_{f\\hat{f}}F_{R_{\\hat{f}}\\bar{Z}_{f^{\\prime }}},$ as well as $K_{Z_f\\bar{Z}_{f^{\\prime }}}=-(F_{R_fR_{f^{\\prime }}}+F_{Z_fR_{\\hat{f}}}[F_{RR}]^{-1}_{\\hat{f}\\check{f}} F_{R_{\\check{f}}\\bar{Z}_{f^{\\prime }}}).$ The resulting metric $g^{GLT}=4(K_{Z_f\\bar{Z}_{f^{\\prime }}}dZ_f d\\bar{Z}_{f^{\\prime }}+K_{Z_f\\bar{U}_{f^{\\prime }}}dZ_f d\\bar{U}_{f^{\\prime }}+K_{U_f\\bar{Z}_{f^{\\prime }}}dU_f d\\bar{Z}_{f^{\\prime }}+K_{U_f\\bar{U}_{f^{\\prime }}}dU_f d\\bar{U}_{f^{\\prime }}),$ is directly expressed in terms of $F$ : $g^{GLT}= - 4dZ_f F_{R_f R_{f^{\\prime }}}d\\bar{Z}_{f^{\\prime }}- 4(dU_f - dZ_{\\hat{f}}F_{Z_{\\hat{f}}R_f})[F_{RR}]^{-1}_{ff^{\\prime }}(d\\bar{U}_{f^{\\prime }} - F_{R_{f^{\\prime }}\\bar{Z}_{\\check{f}}} d\\bar{Z}_{\\check{f}}),$ which in terms of the real moduli $R_f, \\Theta _f, \\Phi _f$ and $T_f:=2\\mathrm {Im}\\, U_f$ reads $g^{GLT}= - F_{R_f,R_{f^{\\prime }}}(dR_f dR_{f^{\\prime }}+d\\Theta _f d\\Theta _{f^{\\prime }}+d\\Phi _f d\\Phi _{f^{\\prime }})\\\\- (dT_f - W^f)[F_{RR}]^{-1}_{ff^{\\prime }}(dT_{f^{\\prime }} - W^{f^{\\prime }}).$ with the one-form $W^f=-\\mathrm {i}dZ_{\\hat{f}}F_{Z_{\\hat{f}}R_f}+\\mathrm {i}d\\bar{Z}_{\\hat{f}}F_{\\bar{Z}_{\\hat{f}}R_f}.$ The exact metric coefficients can be easily evaluated observing that $G_{\\hat{\\eta }_a\\hat{\\eta }_b}=\\sum _\\iota \\Bigg [\\delta _{ab} g_a^\\iota (\\hat{\\mathcal {V}}^\\iota +\\bar{Q}^\\iota \\hat{\\eta }_a+\\frac{1}{2} \\sum _{c, a>c} g_c^\\iota (\\hat{\\eta }_a-\\hat{\\eta }_c)- \\frac{1}{2} \\sum _{c, c>a} g_c^\\iota (\\hat{\\eta }_a-\\hat{\\eta }_c))\\\\- \\frac{1}{2} g_a^\\iota g_b^\\iota \\mathrm {sign}(a-b)(\\hat{\\eta }_a-\\hat{\\eta }_b)\\Bigg ],$ and by direct calculation $F_{R_f R_{f^{\\prime }}}=\\sum _{a,b}c_a^f c_b^{f^{\\prime }}\\frac{(-1)}{2\\pi \\mathrm {i}}\\oint \\frac{d\\zeta }{\\zeta } G_{\\hat{\\eta }_a\\hat{\\eta }_b}\\\\= - \\sum _a c_a^f c_a^{f^{\\prime }} g_a^\\iota \\left( \\bar{M}^\\iota +\\bar{Q}^j x_a+\\sum _c\\frac{1}{2} g_c^\\iota |x_a-x_c| \\right)+ \\sum _{a,b}\\frac{1}{2}c_a^f g_a^\\iota c_b^{f^{\\prime }} g_b^\\iota |x_a-x_b|,$ $F_{R_f Z_{f^{\\prime }}}=\\sum _{a,b}c_a^f c_b^{f^{\\prime }}\\frac{(-1)}{2\\pi \\mathrm {i}}\\oint \\frac{d\\zeta }{\\zeta } \\frac{1}{\\zeta } G_{\\hat{\\eta }_a\\hat{\\eta }_b}\\\\= \\sum _a c_a^f c_a^{f^{\\prime }}g_a^\\iota \\Big (r_\\theta \\bar{\\chi }_\\varphi +\\mathrm {i}r_\\varphi \\bar{\\chi }_\\theta + \\bar{Q}^\\iota \\frac{\\theta _a-\\mathrm {i}\\varphi _a}{2}+\\frac{1}{2}\\sum _c g_c^\\iota \\frac{\\theta _{ac}-\\mathrm {i}\\varphi _{ac}}{2}\\mathrm {sign}(x_a-x_c)\\Big )\\\\-\\sum _{a,b}\\frac{1}{2} c_a^f g_a^\\iota c_b^{f^{\\prime }}g_b^\\iota \\frac{\\theta _{ab}-\\mathrm {i}\\varphi _{ab}}{2} \\mathrm {sign}(x_a-x_b).$ Using $F_{R_f\\bar{Z}_{f^{\\prime }}}=\\sum _{a,b}c_a^f c_b^{f^{\\prime }}\\frac{1}{2\\pi \\mathrm {i}}\\oint \\frac{d\\zeta }{\\zeta } (-\\zeta )G_{\\hat{\\eta }_a\\hat{\\eta }_b}=\\overline{F_{R_f Z_{f^{\\prime }}}},$ one has $W^f=2 \\mathrm {Im}\\, dZ_{\\hat{f}}F_{Z_{\\hat{f}}R_f}=\\sum _{a}c_a^f g_a^\\iota \\left( r_\\theta \\bar{\\chi }_\\varphi d\\varphi _a+r_\\varphi \\bar{\\chi }_\\theta d\\theta _a+\\bar{Q}^\\iota \\frac{\\theta _a d\\varphi _a-\\varphi _a d\\theta _a}{2}\\right.\\\\+\\frac{1}{2}\\sum _c g_c^\\iota \\frac{\\theta _{ac} d\\varphi _a-\\varphi _{ac} d\\theta _a}{2}\\mathrm {sign}(x_a-x_c)\\\\\\left.-\\sum _{b}\\frac{1}{2}g_b^\\iota \\frac{\\theta _{ab} d\\varphi _b-\\varphi _{ab} d\\theta _b}{2}\\mathrm {sign}(x_a-x_b)\\right),$ which exactly matches Eqs.",
"(REF -REF ).",
"Thus, we directly verified that the resulting GLT metric (REF ) with (REF ) and (REF ) exactly matches the asymptotic metric (REF ) obtained from the subwall dynamics: $g^{GLT}= g.$"
],
[
"Cut Volume",
"We make use of the Lawrence formula [13] for the volume of a simple convex polytope $P=\\lbrace x\\in \\mathbb {R}^n\\, |\\, \\vec{a}_i\\cdot \\vec{x}\\le b_i, i=1,\\ldots ,m \\rbrace :$ $\\mathrm {Vol}(P)=\\sum _{\\vec{v}\\in \\mathrm {Vert}(P)} N_{\\vec{v}},$ which is a sum of signed volumes of simplices with $N_{\\vec{v}}=\\frac{1}{n!",
"}\\frac{(\\vec{c}\\cdot \\vec{v}+d)^n}{\\gamma _1\\gamma _2\\ldots \\gamma _n|\\det (a_{i_1},a_{i_2},\\ldots ,a_{i_n})|},$ the signed volume of the simplex with its apex at $\\vec{v}$ and its base in the base plane $\\vec{c}\\cdot \\vec{x}+d=0.$ Here $\\vec{v}$ is one of the vertices of $P$ with exactly $n$ of the planes $\\vec{a}_{i_1}\\cdot \\vec{x}=b_{i_1},\\ldots ,\\vec{a}_{i_n}\\cdot \\vec{x}=b_{i_n}$ passing through it, the corresponding simplex is cut out by these $n$ planes and the base plane $\\vec{c}\\cdot \\vec{v}+d\\ge 0$ with some fixed vector $\\vec{c}$ not normal to any of the polygon planes, the constants $\\gamma _1,\\ldots ,\\gamma _n$ are the coefficients in the decomposition $\\vec{c}=\\gamma _1 \\vec{a}_{i_1}+\\ldots +\\gamma _n \\vec{a}_{i_n}.$ Let us gain some appreciation of this formula (REF ) by proving it.",
"In dimension $n=3$ , let $\\vec{e}_1,\\vec{e}_2,\\vec{e}_3$ be the simplex edges emanating from its main vertex $\\vec{v}$ and ending on its base plane $\\vec{c}\\cdot \\vec{x}+d=0.$ Let $\\vec{b}$ be a point in this base plane and let $\\vec{v}_0=\\vec{v}-\\vec{b}$ be its height, i.e.",
"$\\vec{c}\\cdot \\vec{b}+d=0$ and $\\vec{c}\\cdot \\vec{v}+d=\\vec{c}\\cdot \\vec{v}_0.$ Clearly the symplex volume is $\\mathrm {Vol}_{\\vec{v}}=\\frac{1}{3!",
"}\\det (\\vec{e}_1,\\vec{e}_2,\\vec{e}_3).$ The corresponding vectors $\\vec{a}_1,\\vec{a}_2,\\vec{a}_3$ are normal to respective simplex faces, and thus each $a_i$ is proportional to the vector product $\\vec{e}_j\\times \\vec{e}_k$ of the two edges of that simplex face: $(\\vec{a}_1,\\vec{a}_2,\\vec{a}_3)=(\\vec{e}_2\\times \\vec{e}_3,\\vec{e}_3\\times \\vec{e}_1,\\vec{e}_1\\times \\vec{e}_2)\\left({\\begin{matrix}\\alpha _1&0&0\\\\0&\\alpha _2&0\\\\0&0&\\alpha _3\\end{matrix}}\\right).$ To lighten our notation let $Det=\\det (\\vec{e}_1,\\vec{e}_2,\\vec{e}_3).$ By construction $\\vec{c}=(\\vec{a}_1,\\vec{a}_2,\\vec{a}_3)\\gamma $ , thus $\\begin{pmatrix}\\gamma _1\\\\ \\gamma _2\\\\ \\gamma _3\\end{pmatrix}=(\\vec{a}_1,\\vec{a}_2,\\vec{a}_3)^{-1}\\vec{c}=\\left({\\begin{matrix}\\alpha _1^{-1}&0&0\\\\0&\\alpha _2^{-1}&0\\\\0&0&\\alpha _3^{-1}\\end{matrix}}\\right)\\frac{1}{Det}\\begin{pmatrix}\\vec{e}_1^{\\,T}\\\\\\vec{e}_2^{\\,T}\\\\\\vec{e}_3^{\\,T}\\end{pmatrix}\\vec{c},$ giving $\\gamma _j=\\frac{\\vec{e}_j\\cdot \\vec{c}}{\\alpha _j Det}.$ Noting that $(\\vec{e}_2\\times \\vec{e}_3,\\vec{e}_3\\times \\vec{e}_1,\\vec{e}_1\\times \\vec{e}_2)={Det}\\cdot \\begin{pmatrix}\\vec{e}_1^{\\,T}\\\\\\vec{e}_2^{\\,T}\\\\\\vec{e}_3^{\\,T}\\end{pmatrix}^{-1},$ we have the Lawrence formula take the form $N_{\\vec{v}}=\\frac{1}{3!",
"}\\frac{(\\vec{c}\\cdot \\vec{v}+d)^3}{\\gamma _1\\gamma _2\\gamma _3|\\det (\\vec{a}_{1},\\vec{a}_{2},\\vec{a}_{3})|}=\\frac{1}{3!",
"}\\frac{(\\vec{c}\\cdot \\vec{v}_0)^3}{\\frac{\\vec{c}\\cdot \\vec{e}_1}{\\alpha _1 Det}\\frac{\\vec{c}\\cdot \\vec{e}_2}{\\alpha _2 Det}\\frac{\\vec{c}\\cdot \\vec{e}_3}{\\alpha _3 Det}|\\alpha _1\\alpha _2\\alpha _3|\\frac{|Det|^3}{|\\det (\\vec{e}_{1},\\vec{e}_{2},\\vec{e}_{3})|}}\\\\=\\pm \\frac{1}{3!}",
"Det=\\pm \\mathrm {Vol}_{\\vec{v}}.$ We used $\\vec{c}\\cdot \\vec{v}_0=\\vec{c}\\cdot \\vec{e}_1=\\vec{c}\\cdot \\vec{e}_2=\\vec{c}\\cdot \\vec{e}_3,$ since $\\vec{c}$ is normal to the base plane continaining $\\vec{e}_i-\\vec{e}_j$ and $\\vec{v}_0-\\vec{e}_j.$ The signs in the Lawrence formula are chosen already so that the individual simplex volumes contribute with different signs and the polygon volume does not depend on the choice of the base plane $\\vec{c}\\cdot \\vec{x}+d=0.$ Let us choose a high horizontal plane $z=M$ for a very large value $M.$ The cut volume is the difference of the volume of the (convex) regularized blocked crystal $\\bar{\\mathcal {C}}_0=\\lbrace (x,y,z)\\,|\\, m_fx+n_fy+R_f\\le z\\le M, f\\in \\mathrm {Int}(\\mathcal {N})\\rbrace ,$ and the (convex) regularized cut crystal $\\bar{\\mathcal {C}}_{cut}=\\lbrace (x,y,z)\\,|\\, m_fx+n_fy+R_f\\le z\\le M, f\\in \\mathcal {N}\\rbrace .$ The Lawrence formula applies to both $\\bar{\\mathcal {C}}_0$ and $\\bar{\\mathcal {C}}_{cut}$ and thus the cut volume is $\\mathcal {V}=\\mathrm {Vol}(\\bar{\\mathcal {C}}_{0})-\\mathrm {Vol}(\\bar{\\mathcal {C}}_{cut})=\\sum _{\\begin{array}{c}(ppp)\\\\(ppt)\\end{array}}N_{\\vec{v}}-\\sum _{\\begin{array}{c}(ppi)\\\\(pii)\\\\(iii)\\\\(ppt)\\end{array}}N_{\\vec{v}}=\\sum _{\\begin{array}{c}(ppp)\\end{array}}N_{\\vec{v}}-\\sum _{\\begin{array}{c}(ppi)\\\\(pii)\\\\(iii)\\end{array}}N_{\\vec{v}},$ where the first sum is over the vertices $(ppp)$ at which three perimeter planes (i.e.",
"planes corresponding to the points in Per$(\\mathcal {N})$ ) meet or $(ppt)$ at which two perimeter and one top plane meet.",
"The last sum is over the vertices $(**i)$ involving an internal plane as well as the vertices $(ppt)$ involving the top plane.",
"The latter $(ppt)$ contributions cancel (as, indeed, the cut volume does not depend on the choice of the high top plane).",
"The remaining $(ppp)$ contributions are moduli independent, thus, up to a constant, the volume we are interested in is $\\bar{\\mathcal {V}}=-\\sum _{a\\in V}N_{\\vec{v}_a},$ with $\\vec{v}_a$ the apex of the cone cut out by $m_fx+n_fy-z\\le -R_f$ for three internal points $(m_f,n_f)\\in \\mathcal {N}$ forming the $\\Delta _{a}$ triangle of the triangulation.",
"In the Lawrence formula we choose $\\vec{c}=(1,0,0)^T$ and $d=0$ , and have $(a_{f_1},a_{f_2},a_{f_3})=\\left({\\begin{matrix} m_1&m_2&m_3\\\\n_1&n_2&n_3\\\\-1&-1&-1\\end{matrix}}\\right).$ Thus, $\\det (a_{f_1},a_{f_2},a_{f_3})=m_{31}n_{21}-m_{21}n_{31}=-\\delta _{123}$ , where $\\delta _{123}$ is the area of the triangleHere we use our conventions of the footnote on page REF , i.e.",
"$\\delta _{123}$ is twice the conventional triangle area.",
"$\\left((m_1,n_1),(m_2,n_2),(m_3,n_3)\\right).$ And the relevant factors read off from $(\\gamma _1,\\gamma _2,\\gamma _3)^T=(a_{f_1},a_{f_2},a_{f_3})^{-1}(1,0,0)^T$ are $\\gamma _1=n_{23}/\\delta _{123}, \\gamma _2=n_{31}/\\delta _{123}, \\gamma _3=n_{12}/\\delta _{123}$ .",
"The resulting volume formula is $\\bar{\\mathcal {V}}=-\\sum _{a\\in V}N_{\\vec{v}_a}=-\\sum _{a\\in V} \\frac{\\delta _{a}^2}{n_{12}n_{23}n_{31}} \\frac{x_a^3}{6},$ with the triangle $((m_i,n_i))_{i=1,2,3}$ positively oriented."
],
[
"GLT Function",
"The GLT function (REF ) is $\\begin{aligned}G&=\\frac{1}{12}\\sum _\\iota \\left(\\sum _a g_a^\\iota (6\\bar{M}^\\iota x_a^2+2\\bar{Q}^\\iota x_a^3)+\\sum _{a>b} g_a^\\iota g_b^\\iota (x_a-x_b)^3\\right)\\\\& =\\frac{1}{12}\\sum _\\iota \\left(\\sum _a 3 g_a^\\iota x_a^2(2\\bar{M}^\\iota + 2 \\bar{Q}^\\iota x_a)\\right.\\end{aligned} \\\\+3\\sum _{a>b} g_a^\\iota g_b^\\iota (x_a^3-x_a^2x_b)+3\\sum _{b>a} g_a^\\iota g_b^\\iota (x_b x_a^2-x_a^3)\\\\\\left.",
"-2\\sum _a g_a^\\iota x_a^3(2\\bar{Q}^\\iota +\\sum _{b | b<a} g_b^\\iota - \\sum _{b | b>a} g_b^\\iota )\\right).$ In terms of the Higgs field (REF ), this reads $G=\\sum _a\\sum _\\iota g_a^\\iota \\mathbf {\\phi }^\\iota (x_a)\\frac{x_a^2}{2}- \\sum _\\iota \\sum _a g_a^\\iota \\frac{x_a^3}{6}\\left(2\\bar{Q}^\\iota + \\sum _{b | b<a} g_b^\\iota - \\sum _{b | b>a} g_b^\\iota \\right).$ For any given subwall $a$ all $U(1)$ factors $\\iota $ which have a nonzero charge $g_a^\\iota $ have the same value $\\mathbf {\\phi }^\\iota (x_a)$ , while the charges themselves satisfy $\\sum _\\iota g_a^\\iota =0$ , sinceWe suppose for concreteness that $n_2>n_1>n_3.$ $n_{13}$ of the $U(1)$ factors have $g_a^\\iota =\\frac{m_{13}}{n_{13}}-\\frac{m_{23}}{n_{23}}$ and $n_{21}$ of the $U(1)$ factors have $g_a^\\iota =\\frac{m_{21}}{n_{21}}-\\frac{m_{23}}{n_{23}}:$ $\\pm \\sum _\\iota g_a^\\iota &=n_{13} \\left(\\frac{m_{13}}{n_{13}}-\\frac{m_{23}}{n_{23}} \\right)+n_{21} \\left(\\frac{m_{21}}{n_{21}}-\\frac{m_{23}}{n_{23}} \\right) \\\\ \\nonumber &=m_{13}+m_{21}-m_{23}=0.$ Thus, the first term in (REF ) vanishes.",
"If we let $S_a^\\iota $ denote the magnetic flux in the $\\iota $ -th $U(1)$ factor to the right of the $a$ -th subwall, then $g_a^\\iota =S_a^\\iota -S_{a-1}^\\iota $ and $\\bar{Q}^\\iota =(S_0^\\iota +S_N^\\iota )/2$ , as defined in (REF ).",
"Thus, the second term in (REF ) is a telescoping series: $\\sum _{b | b<a}g_a^\\iota =S_{a-1}^\\iota -S_0^\\iota $ and $\\sum _{b | b>a}g_a^\\iota =S_{N}^\\iota -S_a^\\iota $ , therefore, the last term becomes $ \\sum _{\\iota ,a} g_a^\\iota \\frac{x_a^3}{6} \\left( S_a^\\iota +S_{a-1}^\\iota \\right)=\\sum _{\\iota ,a} \\frac{x_a^3}{6}((S_a^\\iota )^2-(S_{a-1}^\\iota )^2).$ Summing over the $U(1)$ factors, $\\sum _\\iota \\left((S_a^\\iota )^2-(S_{a-1}^\\iota )^2 \\right) = n_{21}\\left(\\frac{m_{21}}{n_{21}}\\right)^2+n_{32}\\left( \\frac{m_{32}}{n_{32}}\\right)^2 - n_{31}\\left(\\frac{m_{31}}{n_{31}}\\right)^2=\\frac{(m_{21}n_{32}-n_{21}m_{32})^2}{n_{21}n_{32}n_{31}}.$ As a result $G=-\\sum _a \\frac{x_a^3}{6}\\frac{\\delta _{a}^2}{n_{12}n_{23}n_{31}}.$ Here $\\delta _a$ is twice the conventional area of the triangle $(m_i,n_i), i=1,2,3$ associated with $a$ -th subwall.",
"Comparing to (REF ) we conclude that the GLT function is equal to the cut volume: $G= \\mathcal {V}.$"
],
[
"Outlook",
"The effective dynamics of a monopole wall are given by the electromagnetic interaction of its constituents.",
"In the low speed approximation it produces the effective Lagrangian from which we read off the resulting asymptotic moduli space metric.",
"We proved that the Kähler potential of this metric is the Generalized Legendre Transform of the regularized crystal volume cut out by the plane arrangement.",
"The latter volume can be easily read off from the monopole charges, parameters, and moduli.",
"The remaining challenge is to find the Kähler potential for the whole moduli space.",
"With this goal in mind we now pose some questions and take the liberty of making some speculations.",
"There is a more refined volume function at hand that could capture some of the Kähler potential subleading asymptotic behavior.",
"Consider the Ronkin function $\\mathcal {R}^\\varphi _{C_{m,n}}(x,y)=\\frac{1}{(2\\pi \\mathrm {i})^2}\\oint \\limits _{\\begin{array}{c}|s|=\\exp {\\frac{x}{r_\\theta }}\\\\|t|=\\exp {\\frac{y}{r_\\theta }}\\end{array}}\\ln |P(s,t)| \\frac{ds}{s}\\frac{dt}{t}.$ It is linear outside of the amoeba $\\mathcal {A}:=\\lbrace (\\ln |s|,\\ln |t|) : P(s,t)=0 \\rbrace $ with $\\mathcal {R}^\\varphi _{C_{m,n}}(x,y)=mx+ny+\\tilde{R}_{m,n}.$ Note that as moduli approach infinity $\\tilde{R}_{m,n}\\rightarrow R_{m,n}.$ These planes lead to a function $\\tilde{m}(x,y):=\\max _{(m,n)\\in \\mathcal {N}}\\lbrace mx+ny+\\tilde{R}_{m,n}\\rbrace .$ The region above the graph of $\\mathcal {R}^\\varphi _{C_{m,n}}$ is the melted crystal.",
"One can consider the volume of the region $\\lbrace (x,y,z) : \\tilde{m}(x,y)<z<\\mathcal {R}^\\varphi _{C_{m,n}}(x,y)\\rbrace $ and use this melted volume $\\mathcal {V}_{melt}$ instead of the cut volume $\\mathcal {V}$ used in this paper.",
"For large moduli these two volumes $\\mathcal {V}_{melt}$ and $\\mathcal {V}$ are exponentially close to each other and thus produce the same asymptotic.",
"One might seek to combine the two Ronkin functions $\\mathcal {R}^\\varphi $ and $\\mathcal {R}^\\theta $ , for example, incorporating both $\\theta $ and $\\varphi $ spectral curves $\\mathbb {S}^\\theta $ and $\\mathbb {S}^\\varphi $ to encode the complete Kähler potential.",
"The relation between the two Legendre transforms that we used can be summarized in the following diagram: Table: NO_CAPTIONThis leads to a question: Is there a more direct relation between the tent function and the Kähler potential?",
"Is there a natural physical meaning of the Legendre transform of the Ronkin function in this context?",
"Let us conclude with a conjecture for the auxiliary function $G$ for the exact Kähler potential.",
"To begin, we define the Twistor Spectral curve $S^{\\mathrm {tw}}$ [2] via the Hitchin scattering problem [10].",
"The space of oriented lines in the covering space $\\mathbb {R}^3$ of the base space $\\mathbb {R}\\times S^1\\times S^1$ is the minitwistor space $T\\mathbb {P}^1$ .",
"Each line $\\ell $ is determined by the unit vector of its direction $\\hat{n}$ , (which determines the point with the complex coordinate $\\zeta $ on the Riemann sphere $\\mathbb {P}^1$ ) and the line's displacement from the origin (which is a point in the tangent plane at $\\hat{n}$ with coordinate $\\eta \\in T_\\zeta \\mathbb {P}^1=\\mathbb {C}\\cup \\lbrace \\infty \\rbrace $ ).",
"For each line, consider the scattering problem $(D_{\\hat{n}}+\\Phi )\\psi =0.$ For some lines this problem has an $L^2$ solutions.",
"These lines are called the spectral lines.",
"Each line in $\\mathbb {R}^3$ is a point in $T\\mathbb {P}^1$ and the set of all spectral lines forms a curve $S_0^{\\mathrm {tw}}$ in $T\\mathbb {P}^1.$ Since our initial problem is invariant under discrete shifts in the $\\theta $ and $\\varphi $ directions, the curve $S_0^{\\mathrm {tw}}$ descends to a curve $S^{\\mathrm {tw}}$ in the quotient space $\\mathcal {Z}:=T\\mathbb {P}^1/2\\pi ( r_\\theta \\hat{n}_\\theta \\mathbb {Z}\\oplus r_\\varphi \\hat{n}_\\varphi \\mathbb {Z}),$ which is the space of geodesics in $\\mathbb {R}\\times S^1\\times S^1.$ Let $\\lbrace \\eta _1(\\zeta ),\\ldots ,\\eta _n(\\zeta )\\rbrace $ be the local branches of this twistor spectral curve.",
"We conjecture that $G=\\mathcal {V}_{melt}\\left(\\frac{\\eta _1}{\\zeta },\\ldots ,\\frac{\\eta _n}{\\zeta }\\right)$ produces the exact Kähler potential.",
"The challenge in using such a relation is that even for the conventional monopoles in $\\mathbb {R}^3$ the twistor curve is notoriously difficult to find, as it should satisfy a complicated `triviality condition'.",
"In addition, for monowalls, the curve $S^{\\mathrm {tw}}$ is contained in the minitwistor space $\\mathcal {Z}$ that is non-Hausdorff, while its cover $S^{\\mathrm {tw}}_0\\subset T\\mathbb {P}^1$ is of infinite genus.",
"Some recent approaches, such as in [16], provide promising perspectives on this problem."
],
[
"Acknowledgments",
"SCh is grateful to the organizers of the 2019 workshop “Microlocal Methods in Analysis and Geometry” at CIRM–Luminy and to the Institute des Hautes Études Scientifiques, Bures-sur-Yvette where the final stages of this work were completed.",
"SCh received funding from the European Research Council under the European Union Horizon 2020 Framework Programme (h2020) through the ERC Starting Grant QUASIFT (QUantum Algebraic Structures In Field Theories) nr.",
"677368.",
"RC thanks the Marshall Foundation for her Dissertation Fellowship funding."
]
] | 1906.04454 |
[
[
"SALT: Subspace Alignment as an Auxiliary Learning Task for Domain\n Adaptation"
],
[
"Abstract Unsupervised domain adaptation aims to transfer and adapt knowledge learned from a labeled source domain to an unlabeled target domain.",
"Key components of unsupervised domain adaptation include: (a) maximizing performance on the target, and (b) aligning the source and target domains.",
"Traditionally, these tasks have either been considered as separate, or assumed to be implicitly addressed together with high-capacity feature extractors.",
"When considered separately, alignment is usually viewed as a problem of aligning data distributions, either through geometric approaches such as subspace alignment or through distributional alignment such as optimal transport.",
"This paper represents a hybrid approach, where we assume simplified data geometry in the form of subspaces, and consider alignment as an auxiliary task to the primary task of maximizing performance on the source.",
"The alignment is made rather simple by leveraging tractable data geometry in the form of subspaces.",
"We synergistically allow certain parameters derived from the closed-form auxiliary solution, to be affected by gradients from the primary task.",
"The proposed approach represents a unique fusion of geometric and model-based alignment with gradients from a data-driven primary task.",
"Our approach termed SALT, is a simple framework that achieves comparable or sometimes outperforms state-of-the-art on multiple standard benchmarks."
],
[
"Results on Office-Home Dataset ",
"While we reported aggregate statistics in Table 3 of the main paper for the Office-Home Dataset , here, in Table REF we report detailed performance across all pairs of DA tasks for this dataset.",
"From Table REF we observe that while SALT consistently outperforms baseline methods including the recent DeepJdot , it also achieves comparable performance to the highest reported – CDAN in all DA tasks.",
"Table: Classification accuracy on Office-Home dataset.",
"Best performance is shown in bold, and the second best in bold italic."
]
] | 1906.04338 |
[
[
"POSSIS: predicting spectra, light curves and polarization for\n multi-dimensional models of supernovae and kilonovae"
],
[
"Abstract We present POSSIS, a time-dependent three-dimensional Monte Carlo code for modelling radiation transport in supernovae and kilonovae.",
"The code incorporates wavelength- and time-dependent opacities and predicts viewing-angle dependent spectra, light curves and polarization for both idealized and hydrodynamical explosion models.",
"We apply the code to a kilonova model with two distinct ejecta components, one including lanthanide elements with relatively high opacities and the other devoid of lanthanides and characterized by lower opacities.",
"We find that a model with total ejecta mass $M_\\mathrm{ej}=0.04\\,M_\\odot$ and half-opening angle of the lanthanide-rich component $\\Phi=30^\\circ$ provides a good match to GW 170817 / AT 2017gfo for orientations near the polar axis (i.e.",
"for a system viewed close to face-on).",
"We then show how crucial is the use of self-consistent multi-dimensional models in place of combining one-dimensional models to infer important parameters as the ejecta masses.",
"We finally explore the impact of $M_\\mathrm{ej}$ and $\\Phi$ on the synthetic observables and highlight how the relatively fast computation times of POSSIS make it well-suited to perform parameter-space studies and extract key properties of supernovae and kilonovae.",
"Spectra calculated with POSSIS in this and future studies will be made publicly available."
],
[
"Introduction",
"The field of time-domain astronomy has witnessed a rapid growth in the past decade thanks to the advent of optical sky surveys, including but not limited to PanSTARRS [29], the Palomar Transient Factory (PTF, [37]), the All-sky Automated Survey for Supernovae (ASAS-SN, [48]), the Dark Energy Survey (DES, [13]), the Asteroid Terrestrial-impact Last Alert System (ATLAS, [55]) and the Zwicky Transient Facility (ZTF, [22]).",
"Nowadays, about five supernovae (SNe) are discovered every nightBased on statistics available at https://wis-tns.weizmann.ac.il/stats-maps for classified supernovae.",
"and this number is expected to increase significantly when the Large Synoptic Sky Survey [39], [27] comes online.",
"Current surveys are also well-suited [4], [21] to rapidly scan large regions of the sky to search for electromagnetic counterparts of gravitational-wave events and specifically kilonovae (KNe).",
"At the same time, the continuous improvement in computational resources has led to a rapid increase in the available hydrodynamical models for both SNe and KNe.",
"A progress in time-domain astronomy is therefore critically tied to connecting state-of-the-art explosion models with the wealth of available and future observations.",
"Among different techniques, a powerful approach to provide such connection is via radiative transfer calculations, which simulate the propagation of light through an external medium and study the interaction between radiation and matter via absorption and scattering processes.",
"This allows the prediction of synthetic observables – as light curves, spectra and polarization – that can then be compared to data to place constraints on models.",
"Over the past three decades, sophisticated radiative transfer codes have been developed and used to investigate both SNe and KNe [24], [7], [23], [57], [14], [31], [36], [6], [28], [52], [20], [60], [33], [44], [16].",
"These codes have lead to a better understanding of these phenomena and placed important constraints on the underlying physics.",
"However, simulations performed with some of these codes are typically computationally expensive and thus restricted to sampling only a few realizations of the full parameter space.",
"In addition, some codes work in one dimension and do not capture ejecta inhomogeneities and asymmetries and thus the corresponding viewing-angle dependence of the synthetic observables.",
"Here, we report on upgrades to the time-dependent multi-dimensional Monte Carlo radiative transfer code possis (POlarization Spectral Synthesis In Supernovae), originally developed as a test-code in [10].",
"Unlike other radiative transfer codes, possis does not solve the radiative transfer equation but rather requires opacities as input.",
"This assumption speeds up the calculation significantly and allows the undertaking of parameter-space studies to constrain key properties of the modelled system.",
"In addition, possis works in three dimensions and is thus well-suited to studying intrinsically asymmetric models and predict their observability at different viewing angles.",
"The paper is organized as follows.",
"We provide an outline of possis in Section , focussing particularly on the new features introduced to the code.",
"We then present a two-component KN model against which we test our code in Section .",
"We finally show and discuss synthetic observables (spectra, light curves and polarization) for this specific model in Section , before summarizing in Section .",
"Spectra computed in this and future works are made available at: https://mattiabulla.wixsite.com/personal/models."
],
[
"Outline of the code",
"Here we provide a summary of the Monte Carlo radiative transfer code possis and outline the new features introduced in this work.",
"possis was first presented as a test-code in [10] and used to model polarization of both SNe [26] and KNe [9] at individual time snapshots.",
"The main changes introduced in this work are the energy treatment and a temporal dependence in both opacities and ejecta properties, which then allow us to produce time-dependent spectra (flux and polarization) and broad-band light curves."
],
[
"Model grid",
"A three-dimensional Cartesian grid is given at some reference time $t_0$ , with velocity $v_i$ , density $\\rho _{i,0}$ and temperature $T_{i,0}$ provided for each grid cell $i$ .",
"In the case of KNe, the electron fraction $Y_{\\mathrm {e},i}$ is also given.",
"The code assumes homologous expansion, i.e.",
"the velocity $v_i$ in each cell is constant (free expansion) and the corresponding radial coordinate $r_i$ is given by $r_i = v_i t$ at any time $t$ .",
"The grid is expanded at each time-step $j$ , the density is scaled as $\\rho _{ij} = \\rho _{i,0}\\,\\bigg ( \\frac{t_j}{t_0} \\bigg )^{-3}$ according to homologous expansion while the temperature is scaled as $T_{ij} = T_{i,0}\\,\\bigg ( \\frac{t_j}{t_0} \\bigg )^{-\\alpha }$ with $\\alpha >0$ ."
],
[
"Opacities",
"possis can handle line opacity from bound-bound transitions ($\\kappa _\\mathrm {bb}$ ) and continuum opacity from either electron scattering ($\\kappa _\\mathrm {es}$ ), bound-free ($\\kappa _\\mathrm {bf}$ ) or free-free ($\\kappa _\\mathrm {ff}$ ) absorption.",
"Wavelength-dependent opacities can be given either at each time-step or at a reference time $t_\\mathrm {ref}$ together with a function $f_\\mathrm {opac}(t)$ describing their temporal evolution.",
"Two separate modes can be selected to treat bound-bound opacities.",
"The first mode (sob-mode) treats bound-bound opacities using the Sobolev approximation [51], in which photons interact with each line at a single frequency and thus at a specific location along their trajectory throughout the ejecta.",
"In the second mode (abs-mode), a polynomial fit to the bound-bound opacity is performed (see e.g.",
"[26]) and used together with the bound-free and free-free opacities as representative of a “pseudo-continuum” absorption component.",
"The former approach is well-suited to predict spectral features associated to individual line transitions, while the latter allows one to predict a featureless “pseudo-continuum” flux level."
],
[
"Creating photon packets",
"A number $N_\\mathrm {ph}$ of Monte Carlo quanta are created at any time-step.",
"Each of these quanta is assigned a location $\\textbf {x}$ and an initial direction $\\textbf {n}$ , energy $e$ , frequency $\\nu $ and normalized Stokes vector $\\textbf {s}=(1,q,u)$ As in [10] we neglect the Stokes parameter $V$ describing circular polarization..",
"Following [2] and [40], each Monte Carlo quantum is treated as a packet of identical and indivisible photons (hereafter referred to as packet).",
"As explained below, this implies that the same energy is assigned to all packets and that this energy is kept constant during all the interactions.",
"The location $\\textbf {x}$ is selected either on a pre-defined photospheric surface or according to the distribution of radioactive material, while the initial direction $\\textbf {n}$ is sampled assuming either isotropic emission or constant surface brightness.",
"As mentioned above, packets are treated as identical and carry the same amount of energy throughout the simulation.",
"The total energy from the relevant radioactive decay processes, $E_\\mathrm {tot}(t_j)$ , is then divided equally among all the packets $e(t_j) = \\frac{E_\\mathrm {tot}(t_j)\\,\\epsilon _\\mathrm {th}}{ N_\\mathrm {ph}}~~,$ where $\\epsilon _\\mathrm {th}$ is a thermalization efficiency, i.e.",
"we neglect $\\gamma $ -ray transport and assume that a fraction $\\epsilon _\\mathrm {th}$ of $E_\\mathrm {tot}(t_j)$ is deposited and made available for ultraviolet-optical-infrared radiation.",
"The initial frequency $\\nu $ is chosen by sampling the thermal emissivity $S(\\nu ) = B(\\nu ,T)\\,\\kappa _\\mathrm {tot}(\\nu )~~,$ where $\\kappa _\\mathrm {tot}(\\nu )$ is the total opacity and $B(\\nu ,T)$ is the Planck function at temperature $T$ .",
"Finally, packets are created unpolarized, i.e.",
"their normalized Stokes vector is set to $\\textbf {s} = (1,0,0)$ .",
"Figure: A meridional cross-section of the two-component kilonova model adopted in this study.",
"A “lanthanide-rich” component is distributed around the merger plane (with half-opening angle Φ\\Phi ) and characterized by high opacities from lanthanides (red region).",
"A “lanthanide-free” component is distributed at higher latitudes and characterized by lower opacities (blue region).",
"Synthetic observables are calculated for different viewing angles Θ obs \\Theta _\\mathrm {obs}."
],
[
"Propagating photon packets",
"Each packet is propagated throughout the ejecta until it interacts with matter.",
"The propagation of a packet is performed in the rest frame, while interactions are treated in the comoving frame.",
"This involves transforming properties like the direction of propagation and frequency from rest frame to comoving frame (and viceversa) every time an interaction with matter occurs (see [10] for details).",
"Which event occurs is chosen depending on the mode selected to treat bound-bound opacity (see Section REF ).",
"In the sob-mode, the procedure outlined in [10] is adopted to select whether a line or continuum interaction occurs.",
"In the abs-mode, instead, a continuum event is selected.",
"When a continuum interaction is selected in either modes, a random number $\\xi $ is drawn from a uniform distribution over the interval $[0,1)$ to determine the nature of the event.",
"Specifically, electron scattering is selected if $\\xi <\\frac{\\kappa _\\mathrm {es}}{\\kappa _\\mathrm {es}+\\kappa _{abs}}~~,$ where $\\kappa _\\mathrm {abs}=\\kappa _\\mathrm {bf}+\\kappa _\\mathrm {ff}$ in the sob-mode while $\\kappa _\\mathrm {abs}=\\kappa _\\mathrm {bb}+\\kappa _\\mathrm {bf}+\\kappa _\\mathrm {ff}$ in the abs-mode.",
"Continuum absorption is chosen otherwise.",
"Upon interaction, the properties of a packet are updated according to the specific event that occurred.",
"In the case of electron scattering, a new direction and Stokes vector are calculated according to the scattering angles randomly selected (see [10]) while the frequency of the packet is kept unchanged.",
"In the cases where bound-bound, bound-free or free-free opacity is selected, the packet is instead re-emitted isotropically, with no polarization and with a new frequency.",
"The latter is calculated using the “two-level atom” (TLA) approach described by [31], in which a packet can be re-emitted either at the same frequency or at a new frequency sampled from the thermal emissivity of the given cell (see equation REF ).",
"The probability of redistribution is controlled by the redistribution parameter $\\epsilon $ , which is set to $\\epsilon =0.9$ following [41].",
"The procedure described in this Section is repeated until the packet leaves the computational boundary.",
"Figure: Bound-bound line opacities κ bb \\kappa _\\mathrm {bb} adopted in this study for the lanthanide-free (blue) and lanthanide-rich (red) component.",
"Opacities are shown at three different epochs: 1.5 (solid), 5 (dashed) and 10 (dot-dashed) days after the merger.",
"Vertical lines show the range of opacities at 1 d and 0.2, 0.5 and 1 μ\\mu m spanned by models with Y e ≤0.25Y_\\mathrm {e}\\le 0.25 (lanthanide-rich, red) and Y e >0.25Y_\\mathrm {e}>0.25 (lanthanide-free, blue) from state-of-the-art calculations by ."
],
[
"Collecting photon packets",
"Two different approaches are used simultaneously by possis to predict synthetic observables: a direct counting technique (DCT) and an event-based technique (EBT).",
"In the former approach – typically adopted in Monte Carlo radiative transfer codes – packets escaping the computational boundary are collected in different angular bins according to their final directions $\\textbf {n}$ .",
"The resulting spectra are then computed as $\\begin{pmatrix} I \\\\ Q \\\\ U \\end{pmatrix} = \\sum \\frac{e}{\\Delta t~ \\Delta \\nu ~4\\pi r^2 }~\\textbf {s}_\\text{f} ~~,$ where $r$ is the distance between the observer and the system and the sum is performed over all the packets arriving to the observer with a final Stokes vector $\\textbf {s}_\\mathrm {f}$ in the time interval [$t-\\Delta t/2$ , $t-\\Delta t/2$ ] and frequency range [$\\nu -\\Delta \\nu /2$ , $\\nu -\\Delta \\nu /2$ ].",
"$I$ is used to calculate flux spectra and light curves, while all three Stokes parameters $I$ , $Q$ and $U$ are used to compute polarization spectra.",
"In the EBT, virtual packets are created every time a Monte Carlo packet interact with matter.",
"Virtual packets are then sent directly to $N_\\mathrm {obs}$ specific observer orientations defined at the start of the simulation, with energy, frequency and Stokes vector equal to those calculated for the real packets after the interaction (see Section REF ).",
"Virtual packets are weighted according to the probability of reaching the observer, which takes into account (i) the probability per unit solid angle $dP/d\\Omega |_\\text{EBT}$ of being scattered in the observer direction (see equation 16 of [10]) and (ii) the probability of reaching the computational boundary (and thus the observer) without further interaction, $e^{-\\tau _\\mathrm {esc}}$ (where $\\tau _\\mathrm {esc}$ is the optical depth to the boundary, see equation 17 of [10]).",
"To speed up the calculations, we follow [10] and neglect virtual packets with $\\tau _\\mathrm {esc}> \\tau ^\\mathrm {max}_\\mathrm {esc}=10$ .",
"Synthetic observables can then be calculated for the pre-defined $N_\\mathrm {obs}$ observer viewing angles.",
"In particular, spectra are computed as $\\begin{pmatrix} I \\\\ Q \\\\ U \\end{pmatrix} = \\sum \\frac{e}{\\Delta t~ \\Delta \\nu ~r^2 }~s_\\text{f} \\cdot \\bigg (\\frac{dP}{d\\Omega }\\bigg |_\\text{EBT} ~e^{-\\tau _\\text{esc}}\\bigg )~~.$ for each viewing angle.",
"Compared to the DCT, the EBT allows one to calculate synthetic observables with much smaller Monte Carlo noise levels and avoids the need to average contributions from different angles in the same angular bin [10]."
],
[
"A test model for kilonovae",
"As mentioned in Section , our radiative transfer code possis is well-suited to calculate synthetic observables for both SN and KN models.",
"In this study, however, we choose to test the code possis by computing spectra, light curves and polarization for the two-component KN model of [9].",
"Fig.",
"REF shows a meridional cross-section of the adopted ejecta morphology.",
"The model is axially symmetric and characterized by two distinct ejecta components: (i) a “lanthanide-rich” component distributed around the merger plane with half-opening angle $\\Phi $ and (ii) a “lanthanide-free” component distributed at higher latitudes.",
"Broadly speaking, these two components can be thought of as the dynamical ejecta and lanthanide-free post-merger ejecta (disk wind), respectively.",
"We adopt the main source of opacities in KNe, i.e.",
"electron scattering and bound-bound opacities.",
"We fix opacities at a reference time $t_\\mathrm {ref}=1.5$ d after the merger and use simple prescriptions for their time-evolution.",
"Choices of the opacities are guided by numerical simulations from [53], with bound-bound opacities treated in the abs-mode (see Section REF ).",
"As shown in Fig.",
"REF , we adopt a power-law dependence of bound-bound opacities on wavelength below 1 $\\mu $ m while we choose the same value of $\\kappa _\\mathrm {bb}$ at longer wavelengths.",
"Specifically, electron scattering opacities are taken as $\\kappa _\\mathrm {es}^\\mathrm {lf}=\\kappa _\\mathrm {es}^\\mathrm {lr}=0.01\\,\\bigg (\\frac{t}{t_\\mathrm {ref}}\\bigg )^{-\\gamma }~\\mathrm {cm}^2\\,\\mathrm {g}^{-1}$ while bound-bound opacities controlled by their value at 1 $\\mu $ m, which is allowed to vary as $\\kappa _\\mathrm {bb}^\\mathrm {lf}[1\\mu \\mathrm {m}]=5\\times 10^{-3}\\,\\bigg (\\frac{t}{t_\\mathrm {ref}}\\bigg )^{\\gamma }~\\mathrm {cm}^2\\,\\mathrm {g}^{-1}$ for the lanthanide-free component and as $\\kappa _\\mathrm {bb}^\\mathrm {lr}[1\\mu \\mathrm {m}]=1.0\\,\\bigg (\\frac{t}{t_\\mathrm {ref}}\\bigg )^{\\gamma }~\\mathrm {cm}^2\\,\\mathrm {g}^{-1}$ for the lanthanide-rich component.",
"Models with different choices of $\\gamma $ are calculated, but in this study we will focus on results with $\\gamma =1$ , a value that is found to give good fits to the AT 2017gfo data (see Section ).",
"We adopt a power-law density profile, i.e.",
"the density in each cell $i$ is initialized as $\\rho _{i,0} = A\\,r_i^{-\\beta }~~,$ where the power-law index is set to $\\beta =3$ (in line with predictions from hydrodynamical calculations, [25], [52]) and the scaling constant $A$ derived to give a desired ejecta mass $M_\\mathrm {ej}$ .",
"The temperature is assumed to be uniform throughout the ejecta, its initial value set to $T_{i,0}=5000$ K and the power-law index describing the temporal evolution (see equation REF ) fixed to $\\alpha =0.4$ .",
"The total energy $E_\\mathrm {tot}(t_j)$ is calculated from the nuclear-heating rates of [35] and a thermalization factor $\\epsilon _\\mathrm {th}=0.5$ is assumed.",
"Packets are created according to the distribution of radioactive materials and assuming isotropic emission.",
"Flux spectra and light curves presented in this work are extracted from simulations using $N_\\mathrm {ph}=10^6$ , while polarization spectra are from higher signal-to-noise calculations with $N_\\mathrm {ph}=2\\times 10^7$ .",
"Observables are computed between 0.5 and 15 d after the merger ($\\Delta t=0.5$ d) and in the wavelength range $0.1-2.3\\,\\mu $ m ($\\Delta \\lambda =0.022\\,\\mu $ m).",
"The EBT approach is adopted and $N_\\mathrm {obs}=11$ viewing angles are taken from pole ($\\Theta _\\mathrm {obs}=0$ ) to equator ($\\Theta _\\mathrm {obs}=\\pi /2$ ) equally-spaced in cosine, i.e.",
"$\\Delta (\\cos \\Theta )=0.1$ .",
"We will focus most of the discussion on a fiducial model with $M_\\mathrm {ej}=0.04\\,M_\\odot $ and $\\Phi =30^\\circ $ (denoted as nsns mej0.04 phi30) while we explore the impact of these two parameters on the light curves in Section REF .",
"The fiducial model is characterized by an ejecta mass of $M_\\mathrm {ej}^\\mathrm {lr}=0.016\\,M_\\odot $ in the lanthanide-rich component and an ejecta mass of $M_\\mathrm {ej}^\\mathrm {lf}=0.024\\,M_\\odot $ in the lanthanide-free component."
],
[
"Synthetic observables",
"Here, we present viewing-angle dependent synthetic observables calculated for the model described in Section .",
"We show spectral energy distributions (SEDs) in Section REF , broad-band light curves in Section REF and polarization spectra in Section REF ."
],
[
"Spectral energy distribution",
"SEDs in the first week after the merger are shown in Fig.",
"REF for the nsns mej0.04 phi30 model seen from two different orientations: one looking at the system face-on ($\\cos \\theta _\\mathrm {obs}=1$ , left panels) and one edge-on ($\\cos \\theta _\\mathrm {obs}=0$ , right panels).",
"At all wavelengths, SEDs are fainter when the system is viewed edge-on compared to face-on.",
"This is a direct consequence of the higher opacities (Section ) and then more severe line-blocking that packets experience trying to escape the ejecta through equatorial rather than polar regions.",
"Each panel of Fig.",
"REF shows the contribution to the total flux of packets coming from the two distinct components.",
"Packets travelling into the lanthanide-rich region are very likely to interact multiple times with lines and thus to be first absorbed and then re-emitted at longer wavelengths.",
"Hence, flux coming from the lanthanide-rich region emerges preferentially in the infrared.",
"In contrast, interactions with lines occur less frequently for packets travelling in the lanthanide-free component.",
"Hence, flux coming from the lanthanide-free region emerges preferentially in the optical, while the infrared re-processed flux is roughly an order of magnitude smaller compared to that from the lanthanide-rich region.",
"The re-processing mechanism described above is also time-dependent.",
"Packets interacting multiple times with lines typically take longer to diffuse out and to finally escape the ejecta.",
"This leads to a clear evolution from an SED peaking in the optical at early times (1 d after the merger) to an SED peaking in the infrared at later times (7 d after the merger).",
"For both viewing angles, this is highlighted by the relative increase of infrared compared to optical flux in the lanthanide-free component.",
"The predicted time-evolution accounts for the transition from a so-called “blue” KN to a “red” KN that was observed in AT 2017gfo [12], [46], [32], [49], [50].",
"Fig.",
"REF shows the sum of a one-component lanthanide-free model ($\\Phi =0^\\circ $ ) with a one-component lanthanide-rich model ($\\Phi =90^\\circ $ ), in the following referred to as the 1cLF+1cLR model.",
"Combinations of this sort have been reported in the literature to infer the presence of two ejecta components in GW 170817/AT 2017gfo and to extract their ejecta masses (e.g.",
"[30], [11], [34] and [45]).",
"Fig.",
"REF highlights how SEDs thus calculated are different from those computed with our self-consistent two-component model at different times.",
"At 1 d after the merger (upper panel), the 1cLF+1cLR model has nearly the same brightness as the face-on two-component model ($\\cos \\theta _\\mathrm {obs}=1$ ) in the optical, but it is a factor of $\\sim 2$ fainter in the infrared.",
"At later epochs (e.g.",
"7 d, lower panel) the difference is even stronger, with the 1cLF+1cLR model inconsistent with any viewing angle of the two-component model.",
"Based on this comparison, we argue against combining one-component models with different compositions to interpret KN data and infer key parameters as e.g.",
"ejecta masses $M_\\mathrm {ej}^\\mathrm {lf}$ and $M_\\mathrm {ej}^\\mathrm {lr}$ ."
],
[
"Broad-band light curves",
"Fig.",
"REF shows broad-band light curves predicted for the nsns mej0.04 phi30 model.",
"In particular, ugrizyJH light curves are shown for $N_\\mathrm {obs}=11$ viewing angles against data collected in the same bands for AT 2017gfo [3], [5], [11], [12], [15], [17], [32], [46], [50], [54], [56], [58], [59].",
"Owing to the difference in SEDs at different orientations (see Section REF ), the viewing-angle dependence of the light curves is also quite strong.",
"Specifically, an observer in the merger plane (system viewed edge-on, $\\cos \\theta _\\mathrm {obs}=0$ ) would see a KN $\\sim $ 1$-$ 1.5 mag fainter than an observer along the polar axis (system viewed face-on, $\\cos \\theta _\\mathrm {obs}=1$ ) depending on the specific filters.",
"The small panels in Fig.",
"REF highlight a viewing-angle dependence in the light-curve shape as well.",
"In particular, the magnitude difference between a face-on and edge-on KN tends to decrease with time.",
"This is a direct consequence of the different diffusion time-scales at different orientations, with photons escaping the ejecta near the equator interacting multiple times within the lanthanide-rich component and thus arriving to the observer later (see also discussion in Section REF ).",
"Because line opacities are higher at optical rather than infrared wavelengths, this effect is most evident in the gri filters, highlighting the importance of optical observations to constrain the inclination of future KN events.",
"After scaling the model fluxes to the distance inferred for AT 2017gfo [19], [38], we find a better agreement with data for viewing angles close to the polar axis (blue lines in Fig.",
"REF ).",
"This is consistent with previous findings suggesting that AT 2017gfo was observed at 15$^\\circ \\lesssim \\theta _\\mathrm {obs}\\lesssim 30^\\circ $ ($0.87\\lesssim \\cos \\theta _\\mathrm {obs}\\lesssim 0.97$ ) from the polar axis [1], [46], [56], [18], [42], [43].",
"The good agreement is especially true at bluer wavelengths (ugri).",
"For redder filters (zyJH), models are consistent with data in the first week after the merger whereas they tend to decline more slowly than observed at later epochs.",
"This discrepancy points to an incorrect assumption for the time-dependence of opacities in the near-infrared (see discussion in Section ).",
"The impact of the ejecta mass $M_\\mathrm {ej}$ on the predicted light curves is shown in the top panels of Fig.",
"REF for an observer looking at the system from the polar axis (face-on, $\\cos \\theta _\\mathrm {obs}=1$ ).",
"A larger $M_\\mathrm {ej}$ translates into a brighter KN in all filters following the increase in the amount of radioactive material (i.e.",
"energy budget).",
"At the same time, however, higher ejecta masses provide larger opacities to radiation.",
"As shown in Fig.",
"REF , this has two effects when moving to increasingly larger masses: the increase in brightness tends to plateau and the light curves tend to peak later (due to increasingly larger diffusion time-scales, see especially near-infrared bands).",
"The impact of the half-opening angle $\\Phi $ on the predicted light curves is shown in the bottom panels of Fig.",
"REF for an observer along the polar axis (face-on, $\\cos \\theta _\\mathrm {obs}=1$ ).",
"For the same total mass $M_\\mathrm {ej}$ , varying the $\\Phi $ value has the effect of changing the relative fraction of mass in one compared to the other ejecta component, i.e.",
"$M_\\mathrm {ej}^\\mathrm {lf}$ vs $M_\\mathrm {ej}^\\mathrm {lr}$ .",
"At bluer wavelengths, both components contribute to the spectrum (see Fig.",
"REF ).",
"Reducing $\\Phi $ leads to smaller opacities from the lanthanide-rich region and a larger flux contribution from the lanthanide-free component, effects which combine to give brighter KNe.",
"Redder wavelengths, instead, are dominated by flux coming from the lanthanide-rich region (Fig.",
"REF ).",
"Initially, reducing $\\Phi $ decreases the opacities from the lanthanide-rich component, thus leading to brighter KNe in the infrared.",
"This increase in brightness is seen when lowering $\\Phi $ from 75 to 30$^\\circ $ .",
"Reducing $\\Phi $ from 30 to 15$^\\circ $ , however, leads to a fainter KN in the infrared following a decrease in $M_\\mathrm {ej}^\\mathrm {lr}$ and thus in the energy budget from the lanthanide-rich region."
],
[
"Polarization",
"Polarization spectra at 1.5 d after the merger are shown in the bottom panel of Fig.",
"REF .",
"Predictions refer to an equatorial orientation ($\\cos \\theta _\\mathrm {obs}=0$ ), for which the polarization signal is expected to be maximized [9].",
"Moving the observer from the equator to the pole leads to a smaller polarization signal, with both $Q$ and $U$ consistent with zero when the system is viewed face-on ($\\cos \\theta _\\mathrm {obs}=1$ ) due to the axial symmetry of the adopted model.",
"Figure: Upper panel: relative importance of electron-scattering compared to bound-bound opacity (κ es /κ bb \\kappa _\\mathrm {es}/\\kappa _\\mathrm {bb}) at 1.5 d after the merger and at different wavelengths.",
"Opacities in the lanthanide-free component are shown in blue, while those from the lanthanide-rich component in red.",
"Lower panel: QQ (black) and UU (grey) polarization spectra at 1.5 ±\\pm 0.5 d after the merger (average of 3 time-bins).",
"Spectra as calculated from possis are shown with thin lines, while thick lines show a re-binned version to decrease the Monte Carlo noise (bin size = 4).Given the axial symmetry of the model, the $U$ Stokes parameter is consistent with zero at all wavelengths.",
"Following [8], deviations of $U$ from zero can thus be used as a proxy for Monte Carlo noise, which for the case of $N_\\mathrm {ph}=2\\times 10^7$ used in these simulations is $\\sigma = |U|\\lesssim 0.1$ per cent at all wavelengths.",
"A net polarization signal is instead predicted across the $Q$ Stokes parameter.",
"In line with what was found by [9], all the polarization signal is created in the lanthanide-free region as electron scattering is a sub-dominant source of opacity in the lanthanide-rich component at all wavelengths ($\\kappa _\\mathrm {es}/\\kappa _\\mathrm {bb}\\lesssim 0.01$ , see upper panel of Fig.",
"REF ).",
"The overall Stokes vectors coming from the lanthanide-free component are aligned in the horizontal direction, thus resulting in a negative $Q$ value (see also fig.",
"2 in [9]).",
"The wavelength-dependence of the signal can be readily understood from the relative importance of electron scattering over bound-bound opacity ($\\kappa _\\mathrm {es}/\\kappa _\\mathrm {bb}$ ) in different spectral regions (see upper panel of Fig.",
"REF ).",
"At wavelengths bluer than 0.5 $\\mu $ m, $\\kappa _\\mathrm {es}/\\kappa _\\mathrm {bb}\\lesssim 0.01$ and thus the depolarizing effect of line opacities leads to $Q\\sim 0$ .",
"Moving from 0.5 to 1 $\\mu $ m increases $\\kappa _\\mathrm {es}/\\kappa _\\mathrm {bb}$ from $\\sim $ 0.01 to 2 (see equations REF and REF ), with the effect of increasing $Q$ from zero to $-$ 0.6 per cent.",
"The same polarization level is finally predicted at all wavelengths larger than 1 $\\mu $ m, following the adopted choice of keeping the bound-bound opacity in the infrared fixed to the same value (see Section ).",
"The polarization signal drops very rapidly with time as a consequence of the fast increase of bound-bound opacity (i.e.",
"decrease of $\\kappa _\\mathrm {es}/\\kappa _\\mathrm {bb}$ , see Fig.",
"REF ).",
"$Q$ reaches values of $-0.2$ per cent in the infrared at 2.5 d after the merger and becomes negligible at later epochs.",
"This behaviour is in good agreement with what was found in [9], with differences in the absolute polarization levels due to the different choices for the opacities."
],
[
"Conclusions",
"In this study, we presented possis, a Monte Carlo radiative transfer code that is well-suited to predict viewing-angle dependent observables for multi-dimensional models of SNe and KNe.",
"Building on previous works [10], [9], we upgraded the code to incorporate an energy treatment of radiation and a time-dependence of both opacities and ejecta properties.",
"Thanks to these upgrades, possis can calculate (i) spectral energy distributions (SEDs) at different times, (ii) broad-band light curves and (iii) polarization spectra for SN and KN models.",
"We tested possis against the two-component KN model discussed in [9], in which the ejecta are characterized by a first component around the equatorial plane and rich in lanthanide elements and by a second component at polar regions and devoid of lanthanides.",
"We presented synthetic observables for different viewing angles and demonstrated the power of possis to constrain the system inclination through the comparison of predicted SEDs, light curves and polarization spectra with KN observations.",
"Given the relatively fast computation times ($\\sim $ hours on a single core for $N_\\mathrm {ph}=10^5$ and $N_\\mathrm {obs}=11$ ), possis using the abs-mode (see Section REF ) is well-suited to undertake parameter-space study to place constraints on key properties of SNe and KNe (e.g.",
"ejecta mass, temperature, angular extent of the two components).",
"Here, we presented a proof-of-concept of such parameter-space study by investigating the impact of the chosen ejecta mass and angular extent of the two components on the synthetic observables.",
"Although we focused on testing possis against a model with an idealized ejecta morphology, the code is completely flexible in terms of the input geometry.",
"This will allow us to explore the more complex ejecta structure produced by multi-dimensional hydrodynamical models, predicting viewing-angle dependent observables that can be used to interpret data and place constraints on models."
],
[
"Acknowledgements",
"I thank the anonymous referee for helping to improve the quality of the paper.",
"I am very grateful to S. A. Sim, A. Goobar and H. F. Stevance for useful comments and suggestions.",
"I acknowledge support from the G.R.E.A.T research environment funded by the Swedish National Science Foundation."
]
] | 1906.04205 |
[
[
"Intertemporal Community Detection in Human Mobility Networks"
],
[
"Abstract We introduce a community detection method that finds clusters in network time-series by introducing an algorithm that finds significantly interconnected nodes across time.",
"These connections are either increasing, decreasing, or constant over time.",
"Significance of nodal connectivity within a set is judged using the Weighted Configuration Null Model at each time-point, then a novel significance-testing scheme is used to assess connectivity at all time points and the direction of its time-trend.",
"We apply this method to bikeshare networks in New York City and Chicago and taxicab pickups and dropoffs in New York to find and illustrate patterns in human mobility in urban zones.",
"Results show stark geographical patterns in clusters that are growing and declining in relative usage across time and potentially elucidate latent economic or demographic trends."
],
[
"Introduction",
"[rgb]0,0,0Much research has been done in recent years in the analysis of real world networks.",
"One particular area of active interest is community detection.",
"Broadly, community detection is an unsupervised exploratory data analysis method that extracts subsets of vertices in a network that are more densely connected within the subset than between the subsets in a given network.",
"[rgb]0,0,0A majority of the research on community detection [rgb]0,0,0in networks has dealt with static networks [1].",
"However, many real-world networks exhibit dynamic properties, such as human mobility networks in urban systems.",
"These networks include commuting patterns over time [2], location based social networks [3], taxicab travel patterns [4] and cell phone call records [5].",
"Understanding the structures of these networks [rgb]0,0,0reveals underlying trends in human mobility and provides important information [rgb]0,0,0for the management of urban infrastructure.",
"There are many human mobility patterns [rgb]0,0,0that can be represented as networks with high temporal resolution because of the presence of origin and destination locations and time stamps associated with the trips.",
"[rgb]0,0,0 For example, bikeshare systems are rich and remarkably comprehensive in tracking mobility patterns within a city.",
"By 2019, over 2,000 cities have created bikeshare systems around the world.",
"In 2018, according to the National Association of City Transportation Officials, 36.5 million trips were completed in over 100 cities in the United [rgb]0,0,0States using these systems.",
"Many of these systems have stations where users can rent the bikes and deposit them at another station at the end of the trip.",
"These stations allow the system operator to track the precise origins and destinations of individual trips by time-of-day and day-of- week.",
"Another mode of travel in cities is by automobile, which trips can be modeled as networks.",
"In particular, taxicabs in cities are regulated and therefore location and time data of these cab pickup and dropoff locations are often reported to the regulators.",
"The increased usage of often less-regulated ridesharing services (Uber, Lyft etc.)",
"have reduced taxi trips in the last few years.",
"Much research has been done on network analyses [6], [7] [8], [9], but most do not fully take into account the dependencies induced by the network structures and temporal trends.",
"Many of these studies have also focused mostly on demand estimation [10], [11].",
"In this work, we develop a method to identify clusters of significantly connected nodes in a time-series of weighted networks.",
"Identification of such clusters [rgb]0,0,0allows us to understand the nature of geographical, economic and cultural relationships, when these networks are rooted in cities.",
"[rgb]0,0,0 Identifying trajectories of connectivity in clusters across time may reveal structural changes within the mobility patterns in the city.",
"[rgb]0,0,0 We develop an intertemporal community detection method to analyze the structure of long-term trends in time-series of networks to understand global and local trends.",
"In particular, we attempt to determine whether such trends are uniformly distributed across the networks, or whether certain communities exhibit countervailing trends in interconnectivity [rgb]0,0,0when compared with others.",
"We aim to identify and partition the nodes that potentially have driven this global trend, as well as the communities that exhibit locally specific trends.",
"[rgb]0,0,0 The objective of the community detection method in this study is to find groups of nodes that are consistently connected across time and [rgb]0,0,0exhibit increasing, decreasing, or stable trends in connectivity.",
"We use the assumptions of the weighted configuration model as posited in [12], [13] to scale and whiten the time-series of node-set connectivities.",
"In doing so, we remove much of the overall graph effects that represent network-wide signals at a given time-point, such as weather and other city-wide phenomena.",
"Though normalization removes much of the seasonality, some autoregressivity still [rgb]0,0,0persists: we ignore these effects but future work should account for such behavior.",
"Analysis of time-varying weighted graphs [rgb]0,0,0allows us to gain more insight [rgb]0,0,0into the nature of the city as a complex [rgb]0,0,0accumulation of micro-level spatial activity patterns.",
"While this method of intertemporal community detection is developed for data structured like mobility systems, it can be adapted for any type of time-series network data with registered nodes (such as inter county commuting patterns, internet traffic, etc.)"
],
[
"Data and Network Construction",
"We apply intertemporal community detection to data from two bikeshare systems and a taxicab trips.",
"[rgb]0,0,0Bikeshare trip data for Divvy (Chicago) and Citibike (New York) are publicly available on their respective websites [14], [15].",
"The two bikeshare systems provide [rgb]0,0,0contrasting cases.",
"Divvy ridership increased steadily between 2014-2016 from 2.7 to 3.6 million, but overall ridership declined slightly from 3.8 million trips in 2017 to $\\sim $ 3.6 million in 2018 [16].",
"The Citibike system, on the other hand, has consistently increased in usage from 14 million in 2016 to 16 million in 2017 and 18 million in 2018 [15].",
"The publicly available datasets include trip start and stop times for each trip between stations.",
"In our analyses, we focus on the time period between July 2016 and June 2018.",
"We omit all stations that were newly introduced or removed within this period.",
"547 nodes (7.4 million trips) in Chicago and 583 nodes (8.4 million trips) in New York remain in the dataset used for this study.",
"One common problem in bikeshare systems is the issue of supply-demand mismatch in ridership.",
"A station in a high-activity area of a large city is often empty or full at certain times of the day [17], [18], [19], [20], [21], [10].",
"A full or empty station prevents an otherwise possible trip.",
"Load rebalancing is a well-studied problem for bikeshare systems in order to solve the inefficiencies associated with queuing between bikes in stations with finite numbers of slots for bikes at each station.",
"Real-time data on station status [rgb]0,0,0rebalancing exist for New York and Chicago [22].",
"However, historical station inventory data is only available for New York City [23] and not Chicago.",
"Thus, for the New York bikeshare system, we find communities with and without demand adjustment (see section REF for details on the method).",
"The taxicab data for New York is from the Taxicab and Limosuine Commission [24].",
"We use data from January 2017 to 2019 because trips from the ridehailing apps (such as Uber, Jio) are only included since 2017 in the data.",
"263 zones cover all the five boroughs of New York, and the dataset includes over 453 million trips between these zones.",
"From these datasets, we construct the observed time-series of networks as $\\lbrace G_t \\rbrace _{1 \\le t \\le T}$ .",
"In all these datasets, we aggregate the trips between a pair of nodes for each week.",
"The weekly aggregation smooths the diurnal variations and keeps the time-series long enough for time-domain.",
"Thus, each time $t$ corresponds to a week, where $T$ is the total number of time periods.",
"The indicator $A_{uv,t}$ represents the presence of any trips at time $t$ between $u$ and $v$ .",
"We use the number of trips [rgb]0,0,0between two nodes per week $t$ as the edge weight $W_{uv,t} $ .",
"In network $G_t$ the degree of node $u$ is defined as $\\deg _{u,t} = \\sum _{v: v \\ne u} A_{uv,t} $ and strengths are defined as $S_{u,t} = \\sum _{v: v \\ne u} W_{uv,t} $ at each time-unit $t$ across total time $T$ [13].",
"We define the index set $[n]$ as the set of all nodes $u$ , which represent stations."
],
[
"Detecting Intertemporal Communities",
"In this section, we describe a method based on iterative testing of node-set connectivities to extract statistically significant communities across time [13], [12].",
"We use a similar approach but account for and classify the types of time dependency.",
"We posit that trends across time are generally increasing, decreasing or stable and account for these types of time dependence.",
"To this end, we adjust connectivities to time-decay and find trends using equivalence testing [25], [26].",
"We introduce a method to find clusters that are significantly connected across time and exhibit differing trends in connectivity by building on the iterative testing framework of Palowitch et al.([12]).",
"In that work, the weights on the edges incident on each node $u$ are modeled as $\\widehat{W}_{uv}=\\xi _{uv} \\bigg (\\frac{s_u s_v}{s_T} \\bigg ) \\big / \\bigg ( \\frac{d_u d_v}{d_T} \\bigg )$ where each $d_u$ represents the degree of node $u$ , where each $s_{u}$ represents the strength from node $u$ , and $s_T, d_T$ represents the global sum of strengths and degrees.",
"Communities are sets of vertices that have edges that are significantly interconnected within the set but not connected outside the set.",
"Prior work [12], [13] use the above null model to identify communities within a single graph.",
"Random variables $\\xi _{uv}$ with variance $\\kappa $ are constructed so as to satisfy the weighted configuration model [13], [12].",
"When $W_{uv,t}$ are structured in time-series, the variances $\\kappa _t$ of random variable $\\xi _{uv,t}$ summarize the overall variability of each network across time.",
"We utilize these time-varying characteristics of the networks to find intertemporal communities."
],
[
"Intertemporal Configuration Null Model",
"We extend the method used in prior work [12], [13] by developing an intertemporal null model wherein a baseline model is extracted from a time-series of registered networks and iteratively subjected to hypothesis tests for trends and local deviance.",
"We detect significant communities across the time-series of networks if the trend and variation components are significantly different from those of the baseline model.",
"These communities signal subsections of the network that are either strongly interconnected at either the beginning or end of the time-period, or consistently connected throughout the entire time period.",
"We define the intertemporal null model for a given node set $B$ (as in [12], [13]) to determine if it is significantly interconnected across all time-points according to the hypothesized trend.",
"We search for communities that are increasing if its nodes are significantly connected at time $t=1$ , but not necessarily significantly connected as $t$ is late.",
"decreasing if its nodes are significantly connected at time $t=T$ , but not necessarily significantly connected when $t$ is early.",
"[rgb]0,0,0stable (or neutral) if its nodes are significantly connected across all time points.",
"Within $B$ , we posit that a time-series of relative connectivity may be decomposed into trend and variation components.",
"Trend denotes the presence of a constant time-trend in the relative connectivity amongst nodes in set $B$ .",
"Variation denotes the aspects of the node-set connectivity that do not vary systematically across time."
],
[
"Null Model for Node-Set Connectivity",
"The estimate for each edge weight $W_{uv,t}$ at time $t$ is a simple extension of (REF ), which is the null model for a single graph.",
"$\\widehat{W}_{uv,t}={\\left\\lbrace \\begin{array}{ll}\\xi _{uv,t} (\\frac{s_{u,t} s_{v,t}}{s_{T,t}}) \\big / (\\frac{d_{u,t} d_{v,t}}{d_{T,t}}) & \\text{if } u \\ne v \\\\0 & \\text{if } u = v\\end{array}\\right.",
"}$ Each $\\widehat{W}_{uv,t}$ is an weighted edge on a random time-varying graph $\\mathcal {G}_t$ , where each $u$ has fixed degrees $d_{u,t}$ and strengths $s_{u,t}$ .",
"Each graph $G_t$ at time $t$ has total degrees $d_{T,t} = \\sum _v d_{v,t}$ and total strengths $s_{T,t} = \\sum _v s_{v,t}$ .",
"The analogous node-set connectivity $S(u,B, {G}_t) $ (as [12]), is $S(u,B, {G}_t) = \\sum _{v \\ne u, v \\in B} W_{uv, t}$ with means and variances $ \\mathbb {E}[S(u,B,\\mathcal {G}_t)] &= \\sum _{v \\in B} \\frac{s_{u,t} s_{v,t}}{ s_{T,t}} ; \\\\\\text{Var}( S(u,B ,\\mathcal {G}_t)) &= \\sum _{v \\in B} \\frac{ (\\frac{s_{u,t} s_{v,t}}{ s_{T,t} } )^2}{ \\frac{d_{u,t} d_{v,t}}{ d_{T,t} } } \\left( \\kappa _t - \\frac{d_{u,t} d_{v,t}}{d_{T,t}} + 1 \\right).$ detailed derivations of these values can be found in the text of [12].",
"To search for intertemporal clusters, we investigate the significance of connectivities of $B_t$ to $v_t$ across time $t$ .",
"The normalized connectivity score $Z_t(v,B)$ represents the sums of weights of a given node set $B$ in graphs $ G_t $ : $Z_t (v,B) = \\frac{ S(v,B, {G}_t) - \\mathbb {E}[ S (v,B, \\mathcal {G}_t) ] }{\\text{Var}( S(v,B, \\mathcal {G}_t)) } , \\quad { t = 1,...,T}$"
],
[
"Identifying Nodes that are Significantly Bordering Across Time ",
"We use iterative testing to identify nodes that are significantly connected to their neighbors through time.",
"Methods developed in previous literature [12], [13] have applied this method to a fixed graph $G$ .",
"We use the same method of deriving significance of the probability that $v$ is significantly connected to $u$ in set $B$ as in Palowitch et al.",
"([12]).",
"The proposed algorithm relies on an iterative procedure starting at iteration step $k=1$ , then repeated until the results do not change.",
"The objective is to find sets $B$ such that for each $v \\in B$ , $v$ is significantly connected to $u$ across all time points $T$ .",
"At a given step $k>1$ , for fixed time $t$ , for a set of nodes $B_{k,t}$ and a bordering node $u$ , the score of node-set connectivity is determined by (REF ): $S(u,B_{k,t}, {G}_t) = \\sum _{v \\ne u, v \\in B_{k,t}} W_{uv, t}.$ After the normalizing calculation (REF ) is performed, a p-value for each $v \\in B_{k,t}$ is then determined: $p(u,B_{k,t}, {G}_t ) = \\mathbb {P}( S(u,B_{k,t}, {G}_t) > S(u,B_{k,t}, \\mathcal {G}_t ) ).$ For each time point $t$ , the p-value $p(u,B_{k,t}, {G}_t )$ is then corrected for false-discovery rate correction as in [27].",
"The non-significant nodes are rejected and the a set of significant nodes is retained.",
"We describe additional steps to find significant nodes in the following sections REF - REF to account for time-decay in significant bordering nodes and describe the testing of trends in REF ."
],
[
"Time-Decay Adjusted False Discovery Rate Correction",
"To identify significantly interconnected nodes for a given time $t$ , a slightly augmented version of the Benjamini-Hochberg [28] procedure used in [12], [13] is used.",
"The only difference is that the FDR-adjusted p-value $p^*_u$ is multiplied by decay term $a_t$ , contingent on if the communities are hypothesized to be increasing, decreasing, or stable in connectivity over time.",
"For a fixed time $t$ , iteration step $k$ , and set $B_{k,t}$ , we find all the nodes that are significantly connected to $B_{k,t}$ across all time $t=1,...,T$ after calculating the p-value as in (REF ).",
"The output set at iteration $K$ and time $t$ is written as $M_k(B_k)$ , described in more detail in later sections in equation (REF ).",
"We define $a_t$ is an exponential decay term to adjust for the shifting time-window of significance.",
"It is defined as: $a_t :={\\left\\lbrace \\begin{array}{ll}\\left( 1 - \\exp \\left( - \\frac{ t-1 }{T} \\right) \\right) a_0^+ & \\text{if trend is increasing} \\\\\\left( \\exp \\left( - \\frac{ t-1 }{T}\\right) - a^-_0 \\right) \\bigg / (1-a^-_0) & \\text{ if trend is decreasing }\\\\1 & \\text{if trend is neutral.}\\end{array}\\right.",
"}$ The terms $ a_0^+ $ and $a_0^-$ are defined such that $a_t$ is 0 at time 1 and 1 at time $T$ if the trend is increasing, and 1 at time 1 and 0 at time $T$ if the trend is decreasing: $a_0^+ := 1 - \\exp \\left(-\\frac{T-1}{T} \\right) ; \\quad \\quad a_0^- := \\exp \\left(-\\frac{T-1}{T} \\right) .$ If the trend is posited to be increasing, then the algorithm allows more permissive selection of `significantly' bordering nodes when time $t$ is early, but is more penalizing when $t$ approaches $T$ .",
"When $t$ is 1, then $a_t$ is equal to zero.",
"In this case, all $p^*_u$ are zero and will automatically be counted as significant if $u$ borders $B_{k,t}$ .",
"When $t$ is $T$ , then $ a_t $ is 1, so the FDR correction is identical to BH.",
"The threshold for the maximum allowable p-value increases as $t$ decreases so that negligible connections (when $t$ is early) that become stronger (when $t$ is late) are deemed significant.",
"Conversely, when the trend is posited to be decreasing, the same kind of correction is made in reverse because the multiplier is subtracted by one.",
"Because the multiplier to the adjusted p-value is always less than 1, the procedure is always less conservative than the Benjamini-Hochberg method and lets nodes that otherwise would not be significant at a given time-period be deemed as“significant\" based on their potential to be significant given their trajectory.",
"If the trend is posited to be neutral, then we use the ordinary BH rejection procedure."
],
[
"Bonferroni Interval for Bordering Frequencies",
"The previous section REF details significance testing for the collections of nodes $ B_{k,t}$ at each time period.",
"To determine whether the collections of nodes are significantly connected to $u$ at all times, we apply a second testing step using Bonferroni Correction.",
"This correction is applied to the frequencies of nodes whose p-values have been deemed significant by the BH correction (in the previous section).",
"The product of Bonferroni confidence intervals is used to define the significance of the neighboring frequency at iteration step $k$ , for each $B_{k,t} $ across all time $t=1,...,T$ .",
"For a set $ B_{k,t}$ , we define $m_t( B_{k,t})$ as the set of nodes that are found to be `significantly bordering' described by Section REF : $m_t( B_{k,t}) = \\# \\lbrace \\text{$u$: $u$ is significantly bordering $ B_{k,t}$ at time } t \\rbrace .$ A large value of $m_t(B_{k,t})$ for all $t$ signifies a large collection of nodes that significantly border $B_{k,t}$ and results in a false discovery interval that is close to $T$ , and hence $v$ must border $B_{k,t}$ for nearly all time $T$ for it to be significant.",
"Conversely, if $m_t(B_{k,t})$ is small, then the required frequency for $v$ to be significant is not as high.",
"During this step, we assume away dependency between $B_{k,t}$ .",
"We define $FDI_{\\alpha ,k}$ to be the threshold for false discovery interval of all significantly adjacent nodes to node $B_{k,t}$ $ FDI_{\\alpha ,k} = \\prod _{t =1,..., T}\\left( 1 - \\frac{\\alpha }{m_t( B_{k,t})} \\right) \\cdot T $ where $ 1 - {\\alpha }/ {m_t( B_{k,t})} $ is the Bonferroni confidence level at each time point.",
"The product of these intervals cross all time multiplied by the total time $T$ gives the threshold of significantly bordering nodes across all time.",
"Figure: Example of set BB at times t=1,2t=1,2.",
"u 1 u_1 is significantly connected when t=1t=1, but not when t=2t=2.",
"So for arbitrary iteration step kk, let B k =BB_k = B, then m t (B k,t ){m_t( B_{k,t})} is m 1 (B 1,k )=B k ⋃{u 1 ,u 2 ,u 3 ,u 4 }{m_1(B_{1,k})}= B_k \\bigcup \\lbrace u_1, u_2, u_3, u_4 \\rbrace at t=1t=1 , but m 2 (B 2,k )=B k ⋃{u 2 ,u 3 ,u 4 }{m_2(B_{2,k})}= B_k \\bigcup \\lbrace u_2, u_3,u_4 \\rbrace at t=2t=2.Now we define the $B_k^0$ as the combined list of all the nodes in any $B_{k,t}$ : $B^0_k = \\bigcup _{t = 1,...,T} B_{k,t}.",
"$ For each $v \\in B^0_k$ , we define the bordering frequency $N_v (B_k)$ as the counts of $v$ which are significantly bordering $ B_{k,t}$ across all time $t$ .",
"A significant $N_v(B_k)$ suggests that $v$ is more frequently bordered across time than other nodes.",
"Each $v$ significantly borders all $ B_{k,t}$ if $ FDI_{\\alpha , k}< N_v( B_{k,t})$ that is, if $ B_{k,t}$ borders $v$ enough times across $t$ for it to be significant overall in the time-period $1,...,T $ [29].",
"Finally, we take the union of all nodes $v$ that satisfy the “significantly neighboring\" criteria (REF ) and denote the set $M_k(B_k)$ $ M_k(B_k) = \\bigcup _{v \\in B^0_k } \\lbrace v: FDI_{\\alpha , k} < N_v(B_k)\\rbrace .$ The resulting set $M_k(B_k) $ represents the nodes that are significantly connected across time, given the appropriate time-window adjustments.",
"We then check if the trends are actually as hypothesized."
],
[
"We define the sum of $Z_t (v,B) $ in (REF ) as $\\mathbf {Z}(B)$ to gauge the significance of the time-trend of a cluster.",
"$\\mathbf {Z}(B) &= \\biggl \\lbrace \\sum _{v \\in B} Z_t(v, B) \\biggr \\rbrace _ { 1 \\le t \\le T}\\\\& := \\mathbf {V}(B) + \\sum _{v \\in B }\\beta _{v, B} \\mathbf {t},$ Moreover, for a given community $B$ that is significantly connected across time $t = 1,..., T$ , we write the vector of node-set connectivity $ \\mathbf {Z}(B) $ as the sum of trend and variation components, where $\\beta _{ u,B} \\mathbf {t}$ represents the trend component which is linearly dependent on time and $\\mathbf {V}(B) $ represents the variation component that is stationary across time.",
"The previous sections describe discovery of node-sets that are significantly connected across time, this section details testing for their trends.",
"If the trends are posited to be positive or negative, then one-sided t-tests are used, respectively with null hypotheses $H_{0,+}: \\beta _{ v, B} \\le 0 $ and $H_{0,-}: \\beta _{ v, B} \\ge 0 $ .",
"More detail can be found in the appendix (section ).",
"If the the trend is positied to be negligible (stable), then the two sided test: $H_0:& \\beta _{v,B}\\ne 0 ; \\quad \\quad H_1: \\beta _{v,B} = 0$ is used.",
"The hypothesis is flipped (compared to the positive or negative tests) in order to test if the trend is equal to zero.",
"We invoke equivalence testing methods ([26]) to determine significance in relation to a pre-selected symmetric interval $[-U,U]$ about zero.",
"Given an set $B_k$ at iteration $k$ , we first find all nodes $v^*$ that are significantly bordering across time as described in Section REF and label these nodes as $M_k(B_k)$ as in (REF ).",
"We then assess the significance of the trends of each of the nodes $v \\in M_k(B_k)$ in relation to set $B_k$ .",
"Calculation of trend employs test statistic for node-set connectivity $S(u,B_k, {G}_t)$ : $ \\mathbf {Z}(v,B_k) = \\biggl \\lbrace \\frac{ S(v,B_k, {G}_t) - \\mathbb {E}[ S (v,B_k, \\mathcal {G}_t) ] }{\\text{Var}( S(v,B_k, \\mathcal {G}_t)) } \\biggr \\rbrace _ { 1 \\le t \\le T}.$ Using $B_k$ and $ M_k(B_k)$ , we then find the time trend $\\beta _{v,B_k}$ for each $v \\in M_k(B_k)$ .",
"We assume that intertemporal communities have trends that are increasing, decreasing, or neutral.",
"We use the equivalence testing method to assess trend significance [25], [26] .",
"Even if a trend is significant, its impact may be negligible and should be assumed to be “zero\".",
"A bounding energy barrier $U>0$ is chosen to control the size of the desired time-trends.",
"A positive $U$ is chosen as a lower bound for a positive trend, $-U$ is used as a upper bound for a negative trend.",
"A symmetric bounding interval of $[-U, U]$ about zero is used for a neutral trend.",
"Hypothesis tests are conducted for the time trend for set $B_k$ (at iteration $k$ ) and node $v$ .",
"Significances of trend $\\beta _{v,B}$ (assuming fixed $B:=B_k$ at iteration $k$ ) are calculated using the difference of the estimates with the upper bounds $U$ (if positive) and lower bound $-U$ (if negative).",
"T-tests for these differences $\\beta _{v,B} - U $ or $\\beta _{v,B} + U $ are then performed to assess significance while excluding very small trends.",
"To determine whether a node-set has a significantly negligible (neutral) trend, we utilize the approach outlined by Dixon et al.",
"[26] and use two one-sided tests to determine if $\\beta _{v,B}$ is significantly outside the interval $[-U,U]$ .",
"Details on the test statistics can be found in the appendix in REF for increasing and decreasing trends and REF for the neutral trend.",
"Before iterative testing in the general case for step $k$ , we first initialize according to section REF ."
],
[
"Iteration and Overlap Filtering Steps",
"After the procedures for selecting nodes that are both significant in connectivity (Section REF ) and trend $\\beta _{v, B} $ depending on the posited direction of trajectory (Section REF ), we derive p-values from the $t$ -statistic of the time-trend .",
"The nodes whose trends are significant after incorporating the FDR correction with significance level $\\alpha $ , are retained.",
"We update the set $B_{k+1}$ with the inclusion of the new nodes $v$ that are both significantly connected to $B_k$ across all time $t$ and have a significant trend according to the trend hypothesis.",
"The procedure is repeated until the set becomes stable such that $B_{k} = B_{k+1}$ for all candidate sets.",
"In all the applications used in this study, this process takes 3 to 5 iterations.",
"After stable sets are found from the iteration steps, they are filtered by their Jaccard overlaps [12], [13].",
"We use an overlap threshold of 0.50 to remove clusters with over 50% overlap; more details on this procedure can be found in prior work [12].",
"After filtering by Jaccard overlaps, communities of size 3 or less are removed, as dyadic or triadic relationships between nodes may be too localized to be meaningful in a larger scale."
],
[
"Effect of Normalizing Edges",
"Modeling network time-series using the weighted configuration model places edge-weights in a relative scale when they are normalized by their expectations and variances, which are functions of global $\\kappa _t$ .",
"Global $\\kappa _t$ is shown to be highly seasonal (fig.",
"REF ) in the Divvy system in Chicago but less so for the NYC taxicab and Citibike data.",
"The variances in the taxicab data [rgb]0,0,0experience a sudden increase in the middle of 2017 and [rgb]0,0,0thereafter consistently increase through time.",
"The high seasonality of $\\kappa _t$ in Chicago and the effects of its removal by normalization [rgb]0,0,0are apparent in figure REF .",
"Scaling edge weights is especially useful in time-series networks where seasonal effects dominate much of the variation (in the Divvy data) or the trend (in NYC taxicab data).",
"Figure: Global variance parameter κ t \\kappa _t from 2016 to 2018 for the Divvy system in Chicago (left), the Citibike system in New York City (center), and κ t \\kappa _t for NYC taxicab networks (right) from 2017 to 2018"
],
[
" Results",
"[rgb]0,0,0We report results for a range of values for tuning parameters $\\alpha $ and $U$ for observed demand $\\lbrace G_t\\rbrace _{1\\le t \\le T}$ .",
"In the Divvy Network, we fix $\\alpha $ at 0.05 and $U = 0.007$ as well as $0.009$ because these settings [rgb]0,0,0capture clusters of moderate sizes across all trend categories and also [rgb]0,0,0show distinct geographical divisions.",
"Under these tuning parameters, we [rgb]0,0,0find five clusters with decreasing connectivities over time and five clusters with increasing connectivities.",
"[rgb]0,0,0We find only one cluster with a stable trend at the 0.05 significance level.",
"There is a stark division in trends between the northern and southern parts of the city (fig.",
"REF ).",
"At the 5% significance level, clusters with significantly decreasing trends [rgb]0,0,0are mostly found in the southern and western parts of the city, while clusters with significantly increasing trends [rgb]0,0,0are mostly found in the northern and central parts of the city.",
"Interestingly, the decreasing clusters map to a nearly concentric outer ring around the central part of the city, while the increasing clusters stretch from the Loop northwards along the shore of Lake Michigan.",
"One stable cluster is located in the Loop.",
"Figure: Intertemporal communities of increasing or decreasing trends amongst Divvy stations in 2016-2018 under varying significance levels and bounding parameters UU using the network time-series {G t }\\lbrace G_t\\rbrace uncorrected for load-imbalance.",
"n B n_B represents the number of found communities and |B| ¯\\bar{|B|} represent the mean size of communities.It is useful to focus on one community to illustrate the effect of edge normalization (see section REF ).",
"In figure REF , while the raw edge weights show a stable trend, the normalization $\\mathbf {Z}(B)$ shows an increasing trend.",
"Thus, the collection of five stations in the Lincoln Park neighborhood in Chicago is classified as a cluster with an increasing time trend rather than a stable one.",
"Figure: top: Total trips in a community in networks G t G_t with increasing normalized connectivity over time comprising 5 stations around the Lincoln Park Neighborhood in Chicago.",
"bottom: Map of stations in BB.In New York City, many clusters are found in the Citibike system in networks of observed demand (raw counts of trips) $G_t$ (fig.",
"REF ).",
"We set $\\alpha $ to 0.05 and $U$ to 0.007 and 0.009 to allow direct comparison to the clusters in the Divvy system in Chicago.",
"When $U$ is set to 0.007, increasing and decreasing clusters are found throughout Manhattan and Brooklyn while stable clusters are mostly concentrated around Lower Manhattan.",
"When $U$ is set to 0.009, increasing and decreasing clusters decrease in size and number but stable clusters increase in size and geographical scale: All of the land-area in Manhattan and part of north Brooklyn is covered in these stable clusters.",
"Figure: Intertemporal Communities of increasing (↑\\uparrow ), decreasing (↓\\downarrow ), and stable (→\\rightarrow ) trends amongst stations in years 2016-2018 under varying significance levels and bounding parameters UU in the uncorrected networks G t G_t.",
"n B n_B represents the number of found communities and |B| ¯\\bar{|B|} represents the mean size of communities.The geographical domain that the taxicab network covers is much larger than the bikeshare network, which only spans Manhattan and Brooklyn.",
"[rgb]0,0,0In two settings of $U$ , clusters are decreasing in connectivity across much of the Bronx, Queens, and much of Brooklyn.",
"Clusters are consistently increasing in eastern parts of [rgb]0,0,0Queens.",
"One cluster appears to consistently link Staten Island to southern Brooklyn for both values of $U$ .",
"[rgb]0,0,0Clusters are stable around the denser parts of the city, as is the case in Upper Manhattan when $U$ is 0.01 and [rgb]0,0,0in Upper and Lower Manhattan, Central Brooklyn, and Astoria in Queens when $U$ is 0.02.",
"Figure: Intertemporal Communities of increasing, decreasing, and stable trends in taxicab networks amongst zones in years 2017-2018 in New York City under varying significance levels and bounding parameters UU.",
"n B n_B represents the number of found communities and |B| ¯\\bar{|B|} represents the mean size of communities rounded to the nearest integer.",
"[rgb]0,0,0"
],
[
"Effect of Demand Correction",
"We apply the intertemporal community detection algorithm [rgb]0,0,0to the demand-corrected (DC) time-series networks $ \\lbrace \\tilde{G}_t \\rbrace _{1\\le t \\le T}$ with weights $\\tilde{W}_{uv,t}$ in the Citibike system.",
"We use the same significance $\\alpha =0.05$ and set barrier $U$ to 0.007 and 0.009 as in observed trip networks in NYC and the Divvy system in Chicago.",
"The obtained communities retain similar geographical characteristics as those in uncorrected graphs, but with some key differences.",
"When $U$ is set at 0.007, the decreasing and increasing clusters in the demand-corrected networks are localized in approximately similar geographical regions as in non-corrected networks.",
"Increasing clusters are mostly located in Upper and Lower Manhattan as well as Southern Brooklyn.",
"Decreasing clusters are present in some small areas throughout Manhattan but pervasively cover swathes of northern Brooklyn around the Williamsburg region.",
"Stable clusters mostly span Midtown Manhattan but also extend to northern Manhattan and parts of Brooklyn.",
"When $U$ is increased to 0.009, the increasing and decreasing clusters shrink in size and number and the stable clusters expand.",
"Increasing clusters are more visibly located in Upper and Lower Manhattan (similar to the clusters in the graphs of observed demand) at the higher threshold.",
"Decreasing clusters are interspersed throughout the city but large coherent areas are more clearly located around northern Brooklyn, also as in the observed graphs $G_t$ .",
"The stable graphs, however, are much larger and cover much more ground in Lower Manhattan (fig.",
"REF ) .",
"Figure: Intertemporal Communities of increasing, decreasing, and neutral trends amongst Citibike stations in years 2016-2018 in New York City under varying significance levels and bounding parameters UU in the demand-corrected networks G ˜ t \\tilde{G}_t.",
"n B n_B represents the number of found communities and |B| ¯\\bar{|B|} represents the mean size of communities rounded to the nearest integer."
],
[
"Discussion",
"[rgb]0,0,0In the Citibike, Divvy, and NYC taxicab systems, we observe a trade-off between increasing or decreasing clusters and stable clusters [rgb]0,0,0depending on the choice of $U$ .",
"If $U$ is larger, then there is “more room\" for a trend to be classified as stable, but less so for increasing or decreasing trends.",
"Discovery of more increasing and decreasing clusters when $U$ is increased suggests that these clusters are increasing or decreasing in connectivity [rgb]0,0,0at different rates from the other clusters.",
"When $U$ is large, increasing and decreasing clusters vanish but more stable clusters persist.",
"The interaction between $\\alpha $ and $U$ is not entirely linear or monotonic.",
"Though a decrease in $\\alpha $ may correspond to an increase in $U$ , a lower $\\alpha $ implies that the nodes are more connected at each time-instance and does not necessarily mean that the trend is higher.",
"Figures REF and REF show that in both $G_t$ and $\\tilde{G}_t$ , clusters appear as $U$ becomes larger and $\\alpha $ stays the same.",
"Such behavior may be attributed to FDR correction.",
"A lower barrier $U$ may yield more significantly connected nodes but with weaker trends.",
"The sensitivity of community detection to the choice of parameter is an important issue [6].",
"We compare the extracted communities under different tuning parameters $U$ and $\\alpha $ .",
"In results from the observed network $ G_t $ in Chicago, the choices of $\\alpha $ and $U$ produce generally similar results over a range of values (fig.",
"REF ).",
"Shifting $U$ from $.007$ to $.009$ induces discovery of more increasing and decreasing clusters, but the bound is too tight for any significant sets [rgb]0,0,0to be found under the hypothesis tests in (REF ).",
"More increasing clusters are found in high-traffic areas surrounding cultural amenities such as the Adler Planetarium in the Loop when $U=0.005$ .",
"Several other clusters in the Near North Side and the Loop [rgb]0,0,0are found under this threshold that are not found under the settings in figure REF .",
"Stable clusters are mostly found also in the Loop and the Near North Side and share a lot of commonalities with increasing clusters at less stringent thresholds; we deduce they have a moderate increasing trend.",
"[rgb]0,0,0 Our analysis is exploratory in nature and only summarizes the trajectories of network structures in time but not their underlying causes.",
"While this work is focused on methodological aspects of temporal community detection, results suggest that it might be useful to think about the causal mechanisms that underlie the different types of clusters in the bikeshare networks.",
"The geographical patterns of the cluster map to different neighborhood characteristics.",
"[rgb]0,0,0 Chicago is an conventionally viewed as a monocentric city focused on the downtown [10][rgb]0,0,0, so it is unsurprising that stable clusters are found around the Loop.",
"The southwestern part of the city is comparatively less affluent.",
"Pilsen, in particular, is a predominantly Latino neighborhood, although its demographic composition is rapidly changing due to gentrification [30], [31].",
"Lincoln Park and Lakeview are known to be affluent and mostly white residential neighborhoods.",
"Increasing, stable, and decreasing trends map closely to these neighborhoods of differing socioeconomic characteristics and suggest latent factors undergirding the decreasing trends that are driving the overall slight decrease in trips from 2016 to 2018.",
"At the same fixed parameters for $U$ and $\\alpha $ , clusters in NYC are more numerous and less geographically spread out than Chicago, possibly because the city is much denser and more populous.",
"Moreover, the seasons are milder, which induces less variation in trends.",
"Figure REF shows that the global variance parameter $\\kappa _t$ [rgb]0,0,0 of the Divvy system is highly seasonal, [rgb]0,0,0unlike that of the Citibike system.",
"Several areas appear to be persistently decreasing in both demand-corrected and observed networks.",
"The easternmost group of four stations in Williamsburg is found in both thresholds of $U$ in demand-corrected and observed networks, suggesting a plausibly real relative decreasing trend in this neighborhood.",
"Citibike has much higher overall usage and thus a lower significance level $\\alpha $ (compared to Divvy) is used to locate the clusters that are significantly anomalously connected.",
"The Citibike system is globally consistently increasing as opposed to Divvy which is slightly decreasing from 2017 to 2018.",
"The edge-normalizing step of the community detection algorithm (section REF ) makes such station-specific adjustments affect the whole network, thereby affecting the entire system.",
"Regardless, similarities persist in clusters in both DC and uncorrected graphs.",
"The demand corrected (DC) networks $\\tilde{G}_t$ when $U$ is 0.007 and 0.009 yield similar decreasing clusters to [rgb]0,0,0those of the uncorrected networks $G_t$ .",
"Demand-adjustment makes a considerable difference in some clusters in the Citibike system.",
"Among decreasing clusters, most of the discovered communities in both uncorrected and corrected graphs are located in Brooklyn, with some scattered around Manhattan (figs.",
"REF , REF ).",
"However, clusters in general are smaller and less numerous in the DC case.",
"The opposite case is observed for increasing clusters: discovered communities in DC graphs $\\tilde{G}_t$ are much more geographically defined in Upper and Lower Manhattan and southern Brooklyn and notably much larger, more numerous, and overlapping.",
"Stable clusters are confined to Manhattan in the observed graph $G_t$ , but appear to form different shapes and extend to Brooklyn in DC graphs $\\tilde{G}_t$ .",
"Adjusting for demand-correction thus reveals stronger, more cohesive increasing trends [rgb]0,0,0within the ridership and suggests that observed trips do not adequately capture the latent increasing signals that are distorted by load imbalances from empty stations.",
"A common feature of decreasing clusters in both DC and non-DC networks across all choices of $U$ is the presence of large clusters around the Williamsburg neighborhood in north Brooklyn.",
"Small pockets of the neighborhood are clustered into increasing or neutral clusters, but decreasing clusters are dominant.",
"On the other hand, south Brooklyn has more of a mixture of trends.",
"Though certain regions in DC and non-DC graphs are grouped into decreasing clusters, the decreasing clusters in DC graphs are very large, encompassing nearly all of the southern part of the city, with some clusters extending to southern Manhattan.",
"Due to the fact that trips by taxicabs are, in general, longer than the trips by bike, the clusters that are found cover larger areas.",
"We choose $U$ to be 0.01 and 0.02 to maximize differentiation between clusters based on an assumed negligibility of trend.",
"Most increasing clusters generally look the same with one exception.",
"One of the decreasing clusters [rgb]0,0,0changes shape rather drastically as $U$ is [rgb]0,0,0increased from 0.01 to 0.02[rgb]0,0,0 and goes from covering nearly all of Brooklyn to covering northern Brooklyn and part of Queens.",
"These patterns mostly demarcate general regions and may illustrate decline in usage within these areas due to other transit options such as bikeshares in Brooklyn and the Bronx.",
"Increasing clusters are generally similar across choices of $U$ and highlight the corridor between Brooklyn and Staten Island, which may be illustrative of the lack of public transit between these regions.",
"The neutral clusters only demarcate a small region around Central Manhattan when $U$ is 0.01, but expand to cover most of Manhattan when $U$ is 0.02.",
"These patterns may be indicative of the consistent usage of taxicabs in the busiest parts of the city.",
"[rgb]0,0,0There are some similarities between clusters of taxicab trips and Citibike trips: stable clusters are found around Central Manhattan in both cases (DC and non-DC cases for Citibike), and southern Brooklyn is found to be increasing (DC case) just as in the taxicab network (though the latter case is connected to Staten Island) However, because the geographical scale of the taxicab dataset is much larger than that of Citibike, and signal different kinds of mobility and connectivity, the similarities in clustering geographies may not be reflective of similar underlying trends.",
"Results in Chicago and New York illustrate the similarities and differences between the two systems.",
"The Citibike system is larger than Divvy (18 million trips in 2018 vs 3.6 million) and is growing at a much faster rate.",
"As such, the significance thresholds for connectivity are stronger for Citibike and [rgb]0,0,0its communities represent more densely connected collections of nodes.",
"[rgb]0,0,0There are no large contiguous areas of consistent decreasing trends like in the ring of clusters surrounding Chicago's city center (fig.",
"REF ).",
"The decreasing clusters in Chicago may be indicative of specific geographical patterns that drive the global slight decrease in the Divvy system because the clusters are geographically coherent.",
"However, the decreasing clusters in NYC are distinctively countervailing with respect to to the overall increasing global trend, though certain clusters (like in Williamsburg) are persistent across several settings and parameters.",
"Because the proposed method is for exploratory purposes, these summarizing claims should be verified in a more rigorous way in future research.",
"Furthermore, the choice of $U$ varies by application.",
"We use an ad-hoc scheme to select $U$ whose resultant neutral clusters yield approximately the same amount of nodes as the increasing and decreasing clusters combined.",
"However, because most of the results we present include two different values of $U$ in order to show the differences in results due to adjusting the parameters , the results in this study may not strictly adhere to this criteria.",
"However, results from Fig.",
"REF , Fig.",
"REF and REF all approximately follow ths heuristic when $U=.009$ , though they may have different $\\alpha $ 's.",
"Different applications of intertemporal community detection may call for different criteria for tuning parameters.",
"For example, setting $U$ to be small so as to not allow discovery of any neutral clusters (i.e.",
"Fig.",
"REF , $U=.007$ ) may also be a suitable option.",
"In future work, more principled approaches for setting tuning parameters utilizing cross-validations may be investigated."
],
[
"Future Work",
"In both Chicago and New York City, there may be several explanations for the underlying signals that cause the clusters to decrease in connectivity.",
"Further work may examine what these signals are and how these signals may function.",
"One explanation may be that decreasing trends are symptoms of displacement, destabilizing steady ridership among long-term inhabitants in gentrifying neighborhoods.",
"Another may be differential rates of attention given to load rebalancing in stations in different neighborhoods with varying resources.",
"Causal analysis of these phenomena are outside the scope of this study, but our exploratory results are useful in initializing conversations about changes in mobility patterns [rgb]0,0,0within and between neighborhoods.",
"Future work may analyze the relationship between the discovered communities and factors such as new construction, bike lanes, weather, incomes, and demographic characteristics.",
"The methods devised in this study can be applied to a variety of data in network time-series format, particularly human mobility networks.",
"The method can be applied to bikeshare networks in other cities, or may be applied to other networks of transportation in urban systems.",
"Future work may elaborate on the theoretical properties of intertemporal community detection.",
"The null model described in section REF may also have further use in statistical inference or [rgb]0,0,0in forecasting future patterns.",
"Another extension [rgb]0,0,0would be to account directly for the spatiotemporal aspects of trips in the methodology.",
"Our work currently relies on historic station inventory data for the analysis of the Citibike system.",
"We do not have access to historical inventory data for Chicago and thus are not able to estimate demand.",
"Though similarities between corrected and non-corrected networks in the NYC bikeshare system shows that there may be some use in using only non-corrected data in Chicago, there are limitations in drawing conclusions for demarcations of functional mobility zones using only observed demand.",
"We propose a method in REF , but further estimation of demand without historical station inventory data should be explored in future work in conjunction with community detection."
],
[
"Conclusions",
"[rgb]0,0,0We proposed a novel method to cluster networks representing bikeshare systems that vary across time.",
"Our community detection method combines usage of a configuration null model with [rgb]0,0,0a trend model to describe the expected trajectory of the graph evolutions.",
"We use a significance-testing methodology to assess whether nodes are anomalously connected to each other within and across time-periods.",
"By using the proposed method, we are able to filter some of the system-wide seasonal effects and [rgb]0,0,0map geographically coherent communities of latent human mobility signals in the bikeshare stations in Chicago and New York and the taxicab network in New York.",
"The methods used in this paper may be applied to other situations where it is important to study the evolution of structures within networks."
],
[
"Corrections for Forgone Trips Due to Load Imbalance Given Load Rebalancing Data",
"We estimate the functionals $ \\mathbb {P}( \\tilde{E}_{u,Y}) $ by taking the average rate [rgb]0,0,0at which a station [rgb]0,0,0is empty i.e.",
"yields no available bikes.",
"$ \\mathbb {P}( \\tilde{E}_{u,Y}) $ are calculated as the ratio of the time-intervals that a station is empty to the total intervals during peak-times (i.e.",
"when users could plausibly check out or return bikes).",
"The ratio represents the probability of a station being empty when a user accesses it.",
"A high ratio signifies that the station is usually empty, and so it is more frequently load-imbalanced due to high usage, hence more weight should be proportionally accounted for to estimate the trips that could have been taken if the system was perfectly balanced.",
"Let $\\tilde{E}_{u,Y}$ be the event that a typical trip in year $Y$ from or to station $u$ is foregone owing to load imbalance and let $\\mathbb {P}( \\tilde{E}_{u,Y})$ be its associated probability.",
"$\\mathbb {P}( \\tilde{E}_{u,Y})$ is approximated as: $ \\mathbb {P}( \\tilde{E}_{u,Y})\\approx \\frac{ \\# \\lbrace \\text{intervals when } u \\text{ is empty in year } Y\\rbrace }{ \\# \\lbrace \\text{total intervals in station } u \\text{ in year } Y \\rbrace }.$ Observed demand $W_{uv,t}$ for each edge between stations $u,v$ during time-index $t$ (weeks in this analysis) are then converted to estimated demand $\\tilde{W}_{uv,t}$ as follows: $\\tilde{W}_{uv, t} = W_{uv, t } ( 1+ \\mathbb {P}( \\tilde{E}_{u,Y}) ) ( 1+ \\mathbb {P}( \\tilde{E}_{v,Y}) ), \\quad t \\in Y.$ We refer to the time-series of graphs comprised of these demand-corrected (DC) weights as $\\lbrace \\tilde{ G }_t \\rbrace _{ 1 \\le t \\le T}$ .",
"For this study, we assume that this probability is constant over the year.",
"Seasonal effects may be influential in this calculation but will be deferred to future research.",
"We assume that a full station induces a negligible impact on load imbalance compared to empty stations.",
"Each probability is calculated as the proportion of time-intervals that the station is empty.",
"We construct networks of estimated demand to correct for trips that could not have taken place due to full or empty stations and find communities within these networks to more accurately find communities of trip demand in a human mobility network [11], [32], [32]."
],
[
"Corrections for Forgone Trips Due to Load Imbalance Without Rebalancing Data",
"Though real-time data on station status (e.g.",
"number of open slots) exist and are available online [22], we do not have access to the historical load rebalancing data and as such we need to estimate the probability of foregone trips.",
"To determine the presence of these forgone trips induced by full or empty stations, we look for anomalous gaps in usage of stations on the days that it is heavily utilized.",
"We refer to these gaps due to forgone trips as load imbalance.",
"We describe a simple significance-testing based method that corrects the counts of trips between stations (edge weights) in each graph $G_{t}$ for week $t$ in each year $Y$ .",
"We have omitted the results of this analysis of the Divvy System in Chicago, though results from this study can be made available on request.",
"Let $\\tilde{E}_{u,Y}$ be the event that a typical trip in year $Y$ from or to station $u$ is foregone owing to load imbalance, we write $\\mathbb {P}( \\tilde{E}_{u,Y})$ as its associated probability.",
"For this paper, we assume that this probability is constant over the year.",
"Seasonal effects may be influential in this calculation but will be deferred to future research.",
"Sums-of-trips $W_{uv,t}$ , or observed demand, for each edge between stations $u,v$ during time-index $t$ (weeks in this analysis) are then converted to estimated demand $\\tilde{W}_{uv,t}$ as follows $\\tilde{W}_{uv, t} = W_{uv, t } ( 1+ \\mathbb {P}( \\tilde{E}_{u,Y}) ) ( 1+ \\mathbb {P}( \\tilde{E}_{v,Y}) ), \\quad t \\in Y$ We refer to the time-series of graphs comprised of these demand-corrected weights as $\\lbrace \\tilde{ G }_t \\rbrace _{ 1 \\le t \\le T}$ .",
"We now describe how to estimate the functionals $ \\mathbb {P}( \\tilde{E}_{u,Y}) $ ."
],
[
"Calculating Significant Gaps in Station Activity",
"A time interval for station $u$ is an interval between any two consecutive events (arrivals or departures).",
"We first formulate a methodology to judge if a time interval is anomalous or not.",
"We call such an unnaturally long time interval a gap.",
"Gaps may occur because of load imbalance or random events not related to load imbalance.",
"We posit that the probability of the occurrence of a foregone trip is: $ \\mathbb {P}( \\tilde{E}_{u,Y})\\approx \\frac{ \\# \\lbrace \\text{gaps in station } u \\text{ in year } Y \\text{ due to load imbalance} \\rbrace }{ \\# \\lbrace \\text{intervals between trips in station } u \\text{ in year } Y \\rbrace }.$ We assume that typical waiting times (in seconds) between consecutive events (start and end of trips) at a station $u$ on day $d$ , $w_{u,d}$ follows an exponential distribution with mean $\\delta _{u,d}$ [17].",
"Note that the cardinality of waiting times is equivalent to the strengths $S_{u,d}$ , or sum-of-trips, of station $u$ on day $d$ subtracted by 1.",
"We filter out the first and last 10% of trips that occurred during day $d$ are censored to filter out the longer gaps during the early and late times of the day, hence only restricting the times $s$ to non-dormant hours, so let $S^*_{u,d}-1$ represent the number of trips excluding the first and last $10\\%$ of trips.",
"We count the number of anomalies per day assuming that high-activity stations are rebalancing at least several times a day [19].",
"To determine anomalies in durations between activity, we first define waiting-times.",
"Let $\\theta _{1,u,d} < \\theta _{2,u,d} <...<\\theta _{S^*_u,u,d} $ denote the time points of consecutive activity on day $d$ at station $u$ after removing the upper and lower $10\\%$ of trip-times.",
"Let $\\mathcal {S}^*_{u,d}$ represent the collection of intervals $ \\lbrace [ \\theta _{i,u,d} , \\theta _{i-1,u,d} ] \\rbrace $ and let $w_{i,u,d} = \\theta _{i,u,d} - \\theta _{i-1,u,d} $ denote the length of these corresponding intervals.",
"We define the sample mean $\\bar{ \\delta }_{u,d}$ as $\\bar{ \\delta }_{u,d} = \\frac{1}{S^*_{u,d}-1} \\sum _{i = 1}^{S^*_{u,d}} w_{i,u,d}.$ Let $I_{u,d}$ be the number of time-intervals $w_{u,u,d} \\in \\mathcal {S}^*_{u,d}$ whose lengths are significantly greater than $ \\delta _{u,d} $ under significance level $\\alpha $ after being corrected by the Benjamini-Hochberg false-discovery rate rejection procedure [28].",
"This procedure will be described in the later section REF and will be used in the community detection algorithm.",
"Precisely: $I_{u,d} =\\# \\lbrace w_{i,u,d} : w_{i,u,d} > \\bar{ \\delta }_{u,d} \\text{ at } \\alpha , \\text{ FDR corrected across } w_{i,u,d} \\in \\mathcal {S}^*_{u,d} \\rbrace .$ $I_{u,d}$ represents the estimated number of gaps in waiting-times.",
"These values may represent gaps due to either load imbalance or typical events such as a break in usage during lunch, or an adverse weather event.",
"We assume that these typical events are different from load imbalance.",
"We do not have data on events that could have led to these gaps caused by typical events.",
"However, we can determine a summary measure of the gaps that occurred when the station is operating in excess, which we define as the condition when the number of trips is significantly greater than the number of slots in the stations.",
"We can also determine the total sum of the gaps that may be due to random, typical, conditions when the station is not operating in excess.",
"We posit that the difference of the gaps under these two conditions provides a reasonable approximation of the gaps owing to load imbalance."
],
[
"Finding Stations with Excess Demand",
"We define $C_{u,Y}$ as the carrying capacity, or number of slots, in a station $u$ in year $Y$ .",
"Typically, carrying capacities of stations are updated once per year.",
"If $C_{u,Y}$ of a station (in and outflows) are exceeded significantly at a given day $d$ by the total trips (daily strengths) $S_{u,d}$ , then we consider the possibility of a overfilled or empty station may influence the decisions of a potential user.",
"We define excess demand $D_{u^*,d}$ in stations $u^*$ where $\\lbrace u^*: S_{u^*,d} \\ge C_{u^*,Y}\\rbrace $ as: $D_{u^*,d} = ( S_{u^*,d} - C_{u^*,Y} ) \\sim \\text{Poi} (\\lambda _d)$ We assume that the counts of excess demand on day $d$ at station $u$ adheres to a Poisson distribution across all stations $u \\in [ n]$ on day $d$ .",
"Functionals related to the total number of trips between periods of times are conventionally modeled as Poisson [17].",
"Let $\\lambda _d$ be the typical network-level excess level of demand in day $d$ and let $\\bar{\\lambda }_d$ be its sample mean: $\\bar{\\lambda }_d = \\frac{1}{n} \\sum _{u = 1}^n D_{u,d}.$ To determine whether station $u$ is operating in excess on a given day $d$ in year $Y$ , we use the Benjamini-Hochberg false-discovery rate correction ( section REF ) to find the stations that are significantly over capacity on day $d$ .",
"We evaluate the p-value of excess demand $D_{u,d}$ at station $u$ by testing every $u \\in [n]$ on day $d$ against the sample mean $\\bar{\\lambda }_d$ under a Poisson distribution under fixed significance $\\alpha $ .",
"We introduce a binary random variable $Q_{u,d}$ to denote if a station is significantly in excess.",
"Let the value of $Q_{u,d}=1$ if $ D_{u,d}$ is judged to be significantly anomalous from $ \\bar{\\lambda }_d$ under significance level $\\alpha $ with false discovery rate correction across stations $u^*$ with excess demand above 0, otherwise, let $Q_{u,d}=0$ .",
"Note that $Q_{u,d}$ is zero for all $u$ such that $\\lbrace u: S_{u,d} < C_{u,Y}\\rbrace $ , but it is zero for some stations $u^*$ such that $\\lbrace u^*: S_{u^*,d} \\ge C_{u^*,Y}\\rbrace $ ."
],
[
"Estimating Foregone Trips",
"Gaps may be due to typical baseline events or to load imbalance.",
"On a given day, a station may be visited above or below its average rate of activity due to chance.",
"However, if the station significantly exceeds demand (number of trips far exceed the number of slots) on such a day, then there is more reason to believe that the gaps in waiting-times between usage are plausibly related to load imbalance.We approximate the gaps using methods described in the previous sections.",
"Let $\\hat{g}_{u,Y}^E$ denote the total approximated number of gaps in activity in station $u$ over year $Y$ on the days $d$ when the station is operating in excess (i.e.",
"$ Q_{u,d}$ ).",
"We assume that the indicator for station $u$ for a gap is independent of the fact that the station is over capacity on day $d$ .",
"The estimated counts of gaps when the station is operating in excess is expressed as: $\\hat{g}^E_{u, Y} & = \\sum _{d \\in Y } I_{u,d} Q_{u,d}$ Recall that $1-Q_{u,d}$ denotes the judgement by the FDR procedure of a non-anomalous demand on day $d$ .",
"Let $\\hat{g}_{u,Y}^{b}$ denote the sum of the number of gaps on days when the excess demand of station $u$ is not significantly anomalous with respect to $\\text{Poi} ( \\bar{\\lambda }_d) $ .",
"Here $ 1-Q_{u,d} =1$ representative of a typical day with baseline anomalies.",
"These counts are estimated as: $\\hat{g}_{u, Y}^b & = \\sum _{d \\in Y } I_{u,d} (1- Q_{u,d}) $ Here $ \\hat{g}_{u, Y}^b $ represents the natural number of anomalous gaps from the days not distorted by too much activity in a station that would give rise to full or empty stations.",
"In contrast, $g^E_{u,Y}$ represents an estimate of anomalous intervals (gaps) in stations owing to excess demand.",
"We assume load imbalance can only occur when there is excess demand, and gaps due to excess demand comprise baseline and baseline gaps.",
"We remove the baseline gaps from gaps owing to excess demand by subtracting $ \\hat{g}_{u, Y}^b $ from $g^E_{u,Y}$ to refine the estimate of gaps induced by load imbalance.",
"Because load imbalance can only decrease the efficiency of the system by reducing the number of trips, the demand-correction probability can only be increased and the numerator of (REF ) is: $ \\# \\lbrace \\text{gaps due to load imbalance in station } u \\text{ in year } Y \\rbrace \\approx \\big ( \\hat{g}^E_{u, Y}- \\hat{g}^b_{u, Y} \\big ) ^+$ The probability of a forgone trip (REF ) can be estimated by $ \\mathbb {P}( \\tilde{E}_{u,Y}) &\\approx \\frac{ \\big ( \\hat{g}^E_{u, Y}- \\hat{g}^b_{u, Y} \\big ) ^+}{ \\sum _{d \\in Y} (S^*_{u,d} - 1) }$ where the denominator, which represents the total number of time-intervals in all days across year $Y$ , can be represented by the sum of trips (daily strengths excluding first and last 10% of trips) of station $u$ in each day $d$ .",
"We use these probabilities to construct a demand-corrected time-series of graphs $\\lbrace \\tilde{G}_t\\rbrace _{1 \\le t \\le T}$ and find communities in these networks in addition to the uncorrected graphs."
],
[
"Testing for Increasing and Decreasing Trends among Node-Sets",
"For the time trend expressed w.r.t.",
"$t$ given a set $B$ , node $v$ , we test for hypotheses for trend about a symmetric interval $[-U,U]$ close to zero.",
"These hypotheses test for a null hypothesis of zero in equivalence testing.",
"The null hypotheses are written as follows: $&H_{0,+}: \\beta ^+_{ v, B} \\le U&H_{1, +}: \\beta ^+_{v, B} > U,\\\\&H_{0,-}: \\beta ^-_{ v, B} \\ge -U&H_{1, -}: \\beta ^-_{v, B} < -U.$ We calculate the significance of $\\beta _{v,B}$ using the difference of the estimates as well as the (pre-specified) upper and lower bounds of the trend.",
"In order to filter out the trends that are negligible, we perform a t-test for the regression statistic subtracted by the upper or lower bound $U$ , divided by the standard error of the estimate, $s(\\beta _{v,B})$ .",
"Defining such a bound allows us to exclude the very small but still significant trends and only find clusters that are increasing or decreasing with considerable magnitude.",
"$t_{\\text{upper}} (v,B) &= \\frac{ \\hat{\\beta }^+_{vB} - U}{s(\\beta ^+_{vB})} , \\quad \\quad t_{\\text{lower}}(v,B) = - \\frac{ \\hat{\\beta }^-_{vB} - (-U)}{s(\\beta ^-_{vB})}$ The corresponding p-values of $ t_{\\text{upper}} $ and $t_{\\text{lower}}$ , respectively, with significance $\\alpha /2$ (for one-sided tests) and with degrees of freedom $n - 2$ [rgb]0,0,0, represent the trend of connectivity of node $v$ in relation to set $B$ .",
"Typical of ordinary least squares, the degrees of freedom are discounted by the slope and intercept terms."
],
[
"Testing for Neutral Trends among Node-Sets",
"P-values of the similarity of neutral trends to $U$ are obtained by taking the maximum of the p-values associated with the t-statistics $ t_{\\text{neutral},a } $ and $ t_{\\text{neutral}, b} $ , respectively, with significance $\\alpha $ and degrees of freedom $n - 2$ .",
"To determine the t-statistic of a negligible trend, we utilize the approach outlined in [26].",
"To test for whether a trend is negligible, the typical hypothesis test for a regression coefficient is inverted and split instead into two one-sided tests.",
"$H_{0,a}: \\beta _{v,B}\\ \\ge U , \\quad \\quad & H_{1,a}: \\beta _{v,B} < U ,\\\\H_{0,b}: \\beta _{v,B} \\le -U , \\quad \\quad & H_{1,b}: \\beta _{v,B} > -U.$ Dixon et al.",
"([26]) used the following pair of t-statistics to test for these hypotheses: $t_{\\text{neutral}, a } &= \\frac{ \\hat{\\beta }_{uv} - (-U)}{ s(\\beta _{uv})} ;\\quad \\quad t_{\\text{neutral}, b } = \\frac{ U - \\hat{\\beta }_{uv}}{ s(\\beta _{uv})}$ and obtained the corresponding p-values for the probability of the alternative hypothesis by taking the maximum of the p-values associated with the t-statistics $ t_{\\text{neutral},a } $ and $ t_{\\text{neutral}, b} $ , respectively, with significance $\\alpha $ and with degrees of freedom $n - 2$ ."
],
[
"Initializing Time Trend of $\\xi _{uv, t}$",
"To initialize the iterative search procedure, all individual nodes $u \\in 1,...,n$ .",
"We calculate $M_0(u)$ for all $B_0(u)=u$ following the procedures from REF at iterative step $k=0$ .",
"Within $M_0(u)$ , we calculate each normalized $W_{uv,t } | A_{uv,t}$ by the following equation for all $v$ that are significantly connected to $u$ across all time $T$ : $Z_{t}(u,v) &= \\frac{ W_{uv,t } - \\mathbb {E}[W_{uv,t} | A_{uv,t}] }{ \\text{Var}(W_{uv,t} | A_{uv,t}) }$ where $\\mathbb {E}[W_{uv,t} | A_{uv,t}] = \\frac{ \\frac{s_{u,t} s_{v,t}}{ s_{T,t} }}{\\frac{ d_{u,t} d_{v,t} }{ d_{T,t}} }; \\quad \\text{Var}(W_{uv,t} | A_{uv,t}) =\\left( \\frac{ \\frac{s_{u,t} s_{v,t}}{ s_{T,t} }}{\\frac{ d_{u,t} d_{v,t} }{ d_{T,t}} }\\right) ^2 \\kappa _t$ Next, we find the linear trends of each $Z_{ t} (u,v)$ across time $t=1,...,T$ and take the nodes with trends that are either significantly positive or negative.",
"We write $\\mathbf {Z}(u,v) $ as the vectorized time series of $Z_{t}(u,v)$ .",
"The trend is calculated as the coefficient with time $t=1,...,T$ from ordinary least squares (OLS), between nodes $u$ and $v$ .",
"$ \\hat{\\beta }_{uv} $ is determined to be significantly increasing, decreasing, or stable (neutral) using the method described in the following section REF , but only using a single node $v$ in place of a set $B.$ If $ \\beta _{uv} $ is significant at the $\\alpha $ level (in OLS), then denote the nodes $v$ that are significantly connected and increasing or decreasing with initializing node $u$ as $v^{**}$ .",
"We construct an initializing set $B_1$ with these nodes $\\lbrace u, {v}^{**} \\rbrace $ for step $k=1$ ."
],
[
"Observed Trips in Citibike at $\\alpha =0.01$",
"Several clusters are found in New York city in networks of observed demand $G_t$ (fig.",
"REF ) when $\\alpha $ is set to 0.01 as the smallest significance threshold that would allow discovery of clusters in all trend-types .",
"When $U$ is set at 0.009, only decreasing clusters are found.",
"The three decreasing clusters are all located on Long Island in Brooklyn.",
"When $U$ is set at 0.012, two decreasing clusters disappear but many stable clusters are found.",
"The stable clusters are all located in Manhattan spanning several parts of the island.",
"The remaining decreasing cluster is located in Williamsburg.",
"Figure: Intertemporal Communities of increasing, decreasing, and neutral trends amongst stations in years 2016-2018 under varying significance levels and bounding parameters UU in the uncorrected networks G ˜ t \\tilde{G}_t.",
"n B n_B represents the number of found communities and |B| ¯\\bar{|B|} represents the mean size of communities.",
"[rgb]0,0,0 [rgb]0,0,0We apply the intertemporal community detection [rgb]0,0,0algorithm to the demand-corrected (DC) time-series of networks $ \\lbrace \\tilde{G}_t \\rbrace _{1\\le t \\le T}$ with weights $\\tilde{W}_{uv,t}$ in the Citibike system.",
"The obtained communities retain similar geographical characteristics as communities in uncorrected graphs, but with some differences.",
"When $U$ is set at 0.009, there are a few decreasing and increasing clusters in the demand-corrected networks but no stable clusters were found.",
"In addition to the clusters found in the non-corrected networks, two clusters in the Upper East Side of Manhattan are found to be decreasing.",
"The increasing clusters are both located in Central Brooklyn around the Clinton Hill neighborhood.",
"When $U$ is increased to 0.012, the increasing clusters vanish and only two decreasing clusters remain.",
"Figure: Intertemporal Communities of increasing, decreasing, and neutral trends amongst stations in years 2016-2018 in New York City under varying significance levels and bounding parameters UU in the demand-corrected networks G ˜ t \\tilde{G}_t.",
"n B n_B represents the number of found communities and |B| ¯\\bar{|B|} represents the mean size of communities rounded to the nearest integer.The authors thank the two referees for an in depth reading of the entire manuscript and detailed comments that lead to a significant improvement of the original submission.",
"The authors also thank Eric Hanss for providing helpful information about bikeshare systems, Hannah Loftus for helpful comments on Chicago geography, and Professor Eliza Rose for providing helpful comments on New York geography.",
"All the trip data are available on the Divvy website [14] (https://www.divvybikes.com/system-data) and Citibike [15].",
"Reloading data are available from OpenBUS [23].",
"The code to implement the methods described in this manuscript is available upon request."
]
] | 1906.04582 |
[
[
"Heterogeneous network approach to predict individuals' mental health"
],
[
"Abstract Depression and anxiety are critical public health issues affecting millions of people around the world.",
"To identify individuals who are vulnerable to depression and anxiety, predictive models have been built that typically utilize data from one source.",
"Unlike these traditional models, in this study, we leverage a rich heterogeneous data set from the University of Notre Dame's NetHealth study that collected individuals' (student participants') social interaction data via smartphones, health-related behavioral data via wearables (Fitbit), and trait data from surveys.",
"To integrate the different types of information, we model the NetHealth data as a heterogeneous information network (HIN).",
"Then, we redefine the problem of predicting individuals' mental health conditions (depression or anxiety) in a novel manner, as applying to our HIN a popular paradigm of a recommender system (RS), which is typically used to predict the preference that a person would give to an item (e.g., a movie or book).",
"In our case, the items are the individuals' different mental health states.",
"We evaluate four state-of-the-art RS approaches.",
"Also, we model the prediction of individuals' mental health as another problem type - that of node classification (NC) in our HIN, evaluating in the process four node features under logistic regression as a proof-of-concept classifier.",
"We find that our RS and NC network methods produce more accurate predictions than a logistic regression model using the same NetHealth data in the traditional non-network fashion as well as a random-approach.",
"Also, we find that the best of the considered RS approaches outperforms all considered NC approaches.",
"This is the first study to integrate smartphone, wearable sensor, and survey data in an HIN manner and use RS or NC on the HIN to predict individuals' mental health conditions."
],
[
"Introduction",
"Mental disorders such as depression and anxiety have been recognized as critical public health issues.",
"Mental disorders are one of the leading causes of both injury and disability for people around the world [78], [11].",
"According to the World Health Organization (WHO), in 2015, 322 million people were living with depression (4.4% of the global population) and 264 million people were living with anxiety (3.6% of the global population) [50].",
"Moreover, depression and anxiety are major contributors to suicides [50].",
"Although early interventions significantly reduce the risk of developing mental disorders, about two-thirds of people do not seek appropriate treatments due to a lack of awareness of their mental illness [6], [1].",
"One way to overcome this issue is to develop predictive models to enable individuals to recognize their risks of mental disorders.",
"Fortunately, smartphones, wearable sensors, and online social media provide a wealth of data relevant to individuals' mental health [22], [61], [14].",
"Using such data, health care providers could alert people who are at risk for depression and anxiety to get timely treatment.",
"In terms of developing models to predict mental health, existing studies can be categorized into three different groups according to the types of data that they use [46], [22]: 1) one group of studies rely on smartphone usage data [9], [54], [2], such as incoming and outgoing call frequency and duration; 2) another group of studies use wearable sensor data, such as physical activity, skin conductance, and heart rate [77], [52], [26]; and 3) the remaining group of studies use social media behavioral data, such as text or image content on social media platforms [81], [14], [1].",
"However, existing studies have several limitations.",
"First, among group 1 and group 2 studies, most are conducted on a limited number of individuals (fewer than 50) and a limited time period (less than one month) [46] except for one that collects smartphone data from 111 individuals for seven months [9].",
"Critically, the small number of individuals may not be representative of a larger population and thus the obtained results might not be generalizable.",
"In addition, data collected from a short time period may be affected by special events such as holidays and thus the obtained results might not be reliable.",
"Group 3 studies do not suffer from a limited number of individuals or a short time period because data is collected online, but the prediction performance of their models may be affected by low data quality resulting from fake accounts and noise in text and image contents [22], [81].",
"Second, all of the existing studies focused on a single type of data except for two [61], [9].",
"Models using a single type of data are often less accurate than models integrating multiple types of data [9], [61].",
"For example, it was shown that combining different types of data including information on individuals' personality traits, weather conditions, and smartphone data yields more accurate performance in predicting individuals' stress levels than individual data types [9].",
"Third, among the existing studies, only three [80], [40], [42] have utilized a network approach to model and analyze data by capturing relationships between entities.",
"Clearly, there is a shortage of network-based methods that could utilize relations between entities to make mental health predictions [81].",
"Developing new approaches of this type is essential because networks are powerful models of complex real-world systems, including social networks, and because social networks play an important role in individuals' mental health conditions [60], [68], [62], [44].",
"Moreover, networks allow for data integration in an elegant way and can be used to make predictions about individuals' health-related traits [31], [84].",
"For instance, a network approach was used to model multiple types of clinical data of patients in order to diagnose diseases [31].",
"To address the limitations of the existing studies, we propose the following contributions.",
"We leverage a rich data set from the NetHealth study to predict individuals' mental health conditions.",
"The NetHealth study collected smartphone data, wearable sensor (Fitbit) data, and individuals' trait data from surveys from approximately 700 undergraduate student participants at the University of Notre Dame during 2015 to 2019 [41], [55], [18].",
"The NetHealth data is more representative of a large population since it contains a larger number of individuals (student participants) than previous studies.",
"In addition, the NetHealth data covers a longer time period than the data from the existing studies and thus may be more reliable.",
"Moreover, the NetHealth study contains multiple types of data for the same set of individuals, which allows us to have a more comprehensive and thus hopefully more accurate understanding of the individuals' mental health conditions compared to the existing studies that only focused on a single type of data.",
"In this study, we aim to predict individuals' mental health conditions by integrating multiple types of data including individuals' social interactions, i.e., their SMS communications collected from smartphones, health-related behaviors collected via Fitbits, and individuals' traits (e.g., personalities) collected from surveys (Figure REF ).",
"We divide these data collected from the three sources (smartphones, Fitbits, and surveys) into five dimensions: individuals' social interactions (i.e., SMS communications), personality traits, social status, physical health, and well-being, with each of the last four dimensions consisting of several components (Figure REF ).",
"To predict individuals' mental health conditions, we first divide individuals into training and testing sets.",
"In the prediction task, we leverage all individuals' information as shown in the left box of Figure REF as well as the training individuals' mental health conditions to train a predictive model.",
"Then we use the trained model to predict the testing individuals' mental health conditions.",
"To integrate the different types of information, we model the NetHealth data as a heterogeneous information network (HIN), which is an effective tool to fuse information by considering multiple types of nodes and edges [65].",
"As a promising paradigm, HIN analysis has been applied to a variety of computational and applied tasks, such as recommendation systems in predicting movie ratings [63], node classification in predicting authors or venues in academic networks [16], clustering in visualization tasks [79], [73], link prediction in inferring future co-authorships [83], and network alignment in predicting protein function from biological data [27].",
"In our study, we apply HIN analysis to the task of mental health prediction.",
"Figures REF and REF show the network schema and visualization of our HIN, respectively.",
"Our constructed HIN consists of six node types (individual, personality traits, social status, physical health, well-being, and mental health).",
"For the individual node type, different nodes correspond to different student participants of the NetHealth study.",
"For each of the other five node types (personality traits, social status, etc.",
"), different nodes correspond to different personal characteristics of the given type, i.e., different values of the components listed under the given dimension in Figure REF .",
"For example, a node of the personality traits type represents some combination of an individual's agreeableness, conscientiousness, extroversion, neuroticism, and openness.",
"As an illustration, one combination, i.e., a node of the personality traits type, may be low agreeableness - low conscientiousness - low extroversion - low neuroticism - low openness, and another combination, i.e., another node of the personality traits type, may be high agreeableness - high conscientiousness - high extroversion - high neuroticism - high openness.",
"See Section REF for details.",
"Clearly, there are multiple nodes of each type because there are multiple possible combinations of personal characteristics of the given type, and each combination corresponds to one node of that type.",
"We connect the nodes of the network by forming six edge types.",
"We construct the individual - individual edge type by connecting two individuals if they communicate through SMS at least once during our study period.",
"We construct edges of the other five types by connecting the individual node type to each of the other five node types if the given individual has the given combination of personal characteristics of the given type.",
"See Section REF for details.",
"We construct the HIN in this way in order to be able to predict an individual's mental health state (likelihood of being depressed or anxious) by relying on the information about: 1) the individual's combinations of the four traits (personality traits, social status, physical health, and well-being), 2) the individual's position in their social network (i.e., which other individuals they are linked to), 3) combinations of the four traits of the individual's (direct or indirect) network neighbors, and 4) mental health states of the individual's (direct or indirect) network neighbors.",
"For example, we argue that if an individual is linked (directly or indirectly) to many other individuals with whom she/he shares many combinations of the four traits, and if these individuals are depressed or anxious, then the individual in question is also likely to be depressed or anxious.",
"So, our HIN offers an elegant and convenient yet important way to model influence between individuals.",
"More specifically, we have recognized that we can model the problem of mental health prediction from our HIN in a novel manner, as applying to the HIN a popular paradigm of a recommender system (RS).",
"RS is widely used to suggest a personalized list of items (e.g., movies, books, or new friends) to individuals based on their preferences to help them find the most relevant items [58].",
"In other words, for an individual $i$ , based on the history of $i$ 's behaviors (e.g., rating movies, liking contents on social media, or friendships with other individuals), an RS approach calculates a personalized ranking score on a set of new items (i.e., movies, contents, or friends, respectively) and suggests the top-ranked items to $i$ .",
"[58], [74].",
"In our case, items are the individuals' different mental health states, i.e., we predict what mental health state an individual is likely to have.",
"Unlike their homogeneous counterparts that consider a single type of individual-item interaction (i.e., edge), HIN-based RS approaches take advantage of multiple node and edge types [63].",
"Multiple types of HIN-based RS approaches exist.",
"One major approach category is based on matrix factorization.",
"For example, DEDICOM, one of the earliest such approaches, is able to capture correlations between different types of nodes through matrix factorization.",
"Such correlations are captured by including multiple edge types into the learning task at hand [4].",
"RESCAL can be considered as a mathematically relaxed version of DEDICOM and was shown to perform better in recommendation tasks [48].",
"DMF further used novel objective functions compared to DEDICOM and RESCAL to learn latent feature parameters, which led to additional improvement of the prediction performance in recommendation tasks [17].",
"Another major approach category incorporates into the recommendation process the notion of a metapath, which follows a specific sequence of multiple types of nodes and edges of a HIN to capture additional HIN information for recommendation.",
"For example, HeteRec learns metapath-based latent features (representations or embeddings) of nodes (individuals and items) through a Bayesian ranking optimization technique [82].",
"Similarly, HERec learns node features through metapath-based random walks, but those are then transformed by fused functions and integrated into a matrix factorization framework with the goal of deriving more informative node features for recommendation [63].",
"Unlike the above methods that cannot deal with weighted graphs (i.e., edges), SemRec aimed to account for edge weights, and it did so via weighted metapaths in RS to generate better recommendations [66].",
"Different from the many traditional RS methods that are based on matrix factorization, MCRec used a deep neural network with a co-attention mechanism to leverage the metapath-based context for recommendation [32].",
"Unlike the above methods that did not consider social (i.e., individual-individual) interaction data, DSR accounted for edges between individuals and used an extended social similarity regularization, which imposed constraints on both similar and dissimilar nodes in its RS framework [34].",
"Similarly, SimMF integrated both social interactions and attribution information of individuals and items to make recommendations; it did so via metapath similarity measures [64].",
"Of all HIN-based RS methods mentioned above, we apply four prominent or recent ones, namely DMF [17], HERec [63], RESCAL [48], and DEDICOM [4], to the HIN to predict edges between nodes of the individual type and nodes of the mental health type, denoted as target edges in our HIN (Figure REF ).",
"In addition, to evaluate the power of RS, we model the problem of mental health prediction in an alternative manner, as node classification (NC) in our HIN.",
"NC predicts labels of nodes based on their features that characterize positions of nodes in a network [8].",
"In our case, node labels are individuals' different mental health states.",
"In our study, we extract four prominent features of individuals from our HIN: graphlets [45], colored graphlets [27], DeepWalk [51], and Metapath2vec++ [16].",
"Then, we put network features into a logistic regression classifier to make predictions, which is a common approach in NC [24].",
"Note that we choose these four features for the following reasons.",
"The first two are based on homogeneous and heterogeneous graphlets, respectively, where graphlets are small subgraphs, i.e., Lego-like building blocks, of complex networks [45], [27].",
"The graphlet-based features, which intuitively capture how many subgraphs of each type a node participates in, were extensively demonstrated to be powerful measures of the network position of a node in numerous tasks [47].",
"The last two are prominent homogeneous and heterogeneous network embedding methods, respectively.",
"Network embedding has received significant attention in the last several years, owing to of its ability to automatically learn low-dimensional latent node features that likely preserve the original high-dimensional network structure [51], [16], [83], [67].",
"To evaluate the prediction performance of the four RS and four NC network methods, we compare them against a logistic regression classifier using the same NetHealth data in a non-network fashion [46], and against a random guess method [21].",
"To our knowledge, this is the first study to use smartphone, wearable sensor, and survey data in an HIN model and use RS or NC on the HIN to predict individuals' mental health conditions.",
"Our findings are as follows.",
"For both depression and anxiety prediction, among all network methods, DMF (an RS method) makes the most accurate predictions.",
"Compared to the random guess method, for both depression and anxiety, all RS and NC methods except for RESCAL and DEDICOM are significantly more accurate.",
"Compared to the non-network method, for depression, DMF and DeepWalk are significantly more accurate, graphlets, colored graphlets, and Metapath2vec++ are marginally more accurate, HERec is comparable, and RESCAL and DEDICOM are significantly less accurate; and for anxiety, DMF is significantly more accurate, DeepWalk is marginally more accurate, HERec, graphlets, colored graphlets, and Metapath2vec++ are marginally less accurate, and RESCAL and DEDICOM are significantly less accurate.",
"Our results indicate that (at the minimum) the best of the network methods outperforms the random guess method and the non-network method.",
"In addition, the best RS method outperforms all NC methods.",
"This confirms the power of networks and RS in particular in predicting mental health.",
"We explore whether the different types of methods (RS, NC, non-network) identify different sets of depressed/anxious individuals.",
"If this is true, we might be able to make more accurate predictions by combining the different method types.",
"Specifically, we study overlaps between depressed/anxious individuals correctly predicted by three representative methods: the most accurate RS method (DMF), the most accurate NC method (DeepWalk), and the non-network method.",
"For depression, we find that DMF's and DeepWalk's predictions significantly overlap.",
"Additionally, DeepWalk's and the non-network method's predictions also significantly overlap.",
"But DMF's and the non-network method's predictions do not significantly overlap.",
"For anxiety prediction, we observe similar results.",
"The reasons for these observations could be: 1) DMF and DeepWalk are both network methods and differ only in one aspect - they are different types of methods (RS versus NC); 2) DeepWalk and the non-network method both use the logistic regression to make predictions and differ only in one aspect - DeepWalk uses network features while the non-network method does not; and 3) DMF differs from the non-network method in two ways: the former is a network method and it does not use logistic regression to make predictions, while the latter is a non-network method that uses logistic regression.",
"In other words, it could be that the more similar the two approaches are in terms of their methodologies, the more similar their predictions.",
"Our results suggest that because of the non-significant overlap between DMF's and the non-network method's predictions, by combining the two methods, we might be able to make more accurate predictions.",
"Exploring this is beyond the scope of this paper and is the subject of our future work.",
"In our prediction approaches mentioned above, we have integrated all types of information about individuals, represented by the different edge types in the HIN, into our (RS and NC) network methods to make mental health predictions.",
"To study whether some edge types might be more informative (i.e., have more predictive power) than others, we investigate the effect of using all possible combinations of edge types (including all edge types) on the prediction performance.",
"For this analysis, we focus only on DMF as the most accurate of all analyzed methods in the above evaluations.",
"We consider a series of DMF instances that correspond to all possible combinations of all five non-target edge types from our HIN (the target edge type needs to be always included by default).",
"For example, on the one extreme, we consider each of the five non-target edge types alone, and on the other extreme, we consider all five non-target edge types combined (which has been the case in the above evaluations).",
"An example of an edge type combination in-between these two extremes is the combination of the individual - individual, individual - physical health, and individual - personality traits edge types.",
"We find that for depression, the combination of the individual - physical health and individual - well-being edge types is significantly more accurate than the rest of the combinations, including the combination of all edge types.",
"For anxiety, the combination of the individual - personality traits and individual - well-being edge types is marginally more accurate than six other edge type combinations, including the combination of all edge types, and is significantly more accurate than the rest of the combinations.",
"Our results indicate that we can make more accurate predictions using some subset of edge types than using all edge types.",
"Exploring the reasons behind this observation is beyond the scope of this paper and is the subject of our future work.",
"Figure: The summary of our data and the goal of our study.Figure: Network schema of the HIN that we construct from the NetHealth data.",
"Circles denote our six node types.",
"Connections between circles denote our six edge types.",
"The red line indicates the target (individual - mental health) edge type that we try to predict.Figure: Network visualization of the HIN constructed from the NetHealth data.",
"Node colors represent the six node types from the network schema in Figure .",
"The table below the visualization summarizes the number of nodes or edges of each type and the total number of nodes or edges across all types.",
"For detailed explanation behind the numbers of nodes and edges of the given type, see Section ."
],
[
"HIN construction",
"Specific data used in this paper.",
"The data used in this paper come from the NetHealth study, an institutional review board-approved effort that collected smartphone, Fitbit, and survey data from approximately 700 undergraduate student volunteers at the University of Notre Dame, who entered the study as freshmen in August 2015 [55], [18].",
"Smartphone data was collected via a monitoring application (CIMON) installed on individuals' (student participants') smartphones that periodically synchronizes SMS logs [30].",
"Notably, we did not collect information about the content of the messages.",
"Instead, we collected information on who communicates with whom and when.",
"Health behavioral data was collected through Fitbit Charge HR devices.",
"In this study, we consider two specific Fitbit metrics, individuals' step counts and their sleep duration.",
"We conducted periodic surveys that asked questions about individuals' demographics, personality traits, mental health, and other subjects.",
"From surveys, we extract 15 characteristics that can be divided into five categories: 1) personality traits: agreeableness, conscientiousness, extraversion, neuroticism, and openness, as measured by the Big-Five personality test [23]; 2) social status: gender, race, religion, and parents' educations; 3) physical health: sleep quality measured by the Pittsburgh Sleep Quality Index (PSQI) [10]; 4) well-being: body image, happiness, health, loneliness [12], and self-esteem [59]; and 5) mental health: depression measured by the Center for Epidemiological Studies Depression Scale (CES-D) test [38] and anxiety measured by the State-Trait Anxiety Inventory (STAI) test [70].",
"Selection of the study time period and pool of individuals.",
"In this paper, we select one year from August 2015 to August 2016 as the study period since our previous study showed that the majority of individuals are actively involved in the NetHealth study during this period but not during other periods [41].",
"This period covers 52 weeks, of which 31 are school weeks and 21 are break weeks.",
"In this study, we focus only on school weeks because our previous study showed that school weeks have meaningful social network structures, while break weeks do not [41].",
"The NetHealth data contain SMS logs of 615 iPhone users and 96 Android users.",
"In this study, we only consider iPhone data since we encountered problems with Android data consistency.",
"Finally, we keep only SMS data where both the sender and receiver are among the 615 iPhone users, i.e., the participants of our study.",
"In other words, we discard SMS data where the sender or receiver is not a participant of the NetHealth study (e.g., is a participant's family member or friend).",
"We use the following three criteria established in our previous study [41] to select individuals who have high-quality data: Students' first SMS activity date was before or on the start date of our considered study period and their last SMS activity was on or after the end date of our considered study period, i.e., students actively sent or received SMSs during the one-year study period.",
"Students have high-quality Fitbit data, i.e., their Fitbit data are valid.",
"By “valid”, we mean that a student wore the Fitbit at least 80 percent of the time in more than half of the days during the one-year study period.",
"Students completed the survey taken in July 2015, which was conducted before students entered the school, and the survey taken in January 2016, which was conducted during our study period.",
"The reasons why we need both surveys are discussed below.",
"274 out of the 615 individuals satisfy the three criteria and thus form the final pool of individuals considered in this study.",
"The HIN.",
"One of the key contributions of this study is to establish an HIN using the NetHealth data.",
"An HIN is a network consisting of multiple types of nodes and edges [73].",
"In this study, we construct an HIN consisting of six node types and six edge types (listed below).",
"Formally, the HIN can be represented as $G=(V,E)$ , where $V = \\bigcup _{i}V_i$ , $i\\in \\lbrace 1,2,3,4,5,6\\rbrace $ , i.e., $V$ is the union of all nodes over all six node types, and $E = \\bigcup _{j}E_ji$ , $j\\in \\lbrace 1,2,3,4,5,6\\rbrace $ i.e., $E$ is the union of all edges over all six edge types.",
"The node types include individual, personality traits, physical health, social status, well-being, and mental health, as follows (also, see Figures REF and REF ).",
"The individual node type represents students who participate in the NetHealth study.",
"There are 274 nodes of the individual node type.",
"The personality traits node type represents a combination of five personality characteristics including agreeableness, conscientiousness, extroversion, neuroticism, and openness, extracted from the survey taken in January 2016.",
"We use data from this survey because it is the only one conducted during our study period.",
"The distributions of personality characteristics scores are shown in Supplementary Figure S1.",
"We combine these five characteristics because they belong to the same type of the individuals' information, as also suggested by domain experts who have conducted the NetHealth study.",
"The individuals' personality characteristics scores vary from person to person.",
"For example, the individuals' agreeableness scores could be 2.1, 2.2, 3.4, 4.2, 4.8, etc.",
"Combining such absolute score values across the different personality characteristics and over all individuals would generate a huge number of possible combinations.",
"To address this in an elegant way, for each personality characteristic, we divide the individuals, i.e., their scores, into three groups, as is typically done [61]: the high-score group (the top 25 percent of the scores), the medium-score group (the middle 50 percent of the scores), and the low-score group (the lowest 25 percent of the scores).",
"Then, we form nodes of the personality traits type by combining the group labels of the five personality characteristics.",
"For example, a node of the personality traits type may be low agreeableness - medium conscientiousness - medium extroversion - medium neuroticism - high openness.",
"The number of all possible combinations of the five personality characteristics is $3^5=243$ .",
"Note that some possible combinations are not present in the NetHealth data.",
"So, the number of combinations that correspond to scores of at least one individual in our data, i.e., the number of nodes of the personality traits type, is 114.",
"The social status node type represents a combination of four social status characteristics including gender, race, religion, and parents' education, extracted from the survey taken in July 2015.",
"We use social status data from this survey because it is the only one containing the individuals' social status information.",
"(This is the reason why we need both surveys in the third criterion when choosing which pool of individuals to consider.)",
"The distributions of the 274 individuals' social status scores are shown in Supplementary Figure S2.",
"For example, a node of the social status type may be female - white - Catholic - both parents received bachelor's degrees.",
"Given two possible genders, five possible races, four possible religions, and three possible options for parents' education, the number of all possible combinations of the four social status characteristics is $2 \\times 5 \\times 4 \\times 3 = 120$ .",
"The number of combinations that correspond to scores of at least one individual in our data, i.e., the number of nodes of the social status type, is 55.",
"The physical health node type represents a combination of three physical health characteristics including sleep quality, average sleep duration, and average step counts.",
"We obtain the individuals' sleep quality scores from the survey taken in January 2016 and their average sleep duration and average step counts during the study period from the Fitbit data.",
"The distributions of the scores are shown in Supplementary Figure S3.",
"To combine sleep quality, average sleep duration, and average step counts, we divide scores of each physical health characteristic into three groups: the high-score group (the top 25 percent of the scores), the medium-score group (the middle 50 percent of the scores), and the low-score group (the lowest 25 percent of the scores).",
"Then, we form nodes of the physical health type by combining the group labels of the three physical health characteristics.",
"For example, a node of the physical health type may be low sleep quality - medium average sleep duration - high step counts.",
"The number of all possible combinations of the three physical health characteristics is $3^3=27$ .",
"The number of combinations that correspond to scores of at least one individual in our data, i.e., the number of nodes of the physical health type, is 27.",
"The well-being node type represents a combination of five well-being characteristics including body image, happiness, health, loneliness, and self-esteem, extracted from the survey taken in January 2016.",
"The distributions of the scores are shown in Supplementary Figure S4.",
"To combine the five well-being characteristics, we divide scores of each characteristic into three groups: the high-score group (the top 25 percent of the scores), the medium-score group (the middle 50 percent of the scores), and the low-score group (the lowest 25 percent of the scores).",
"Then, we form nodes of the well-being type by combining the group labels of the five well-being characteristics.",
"For example, a node of the well-being type may be low body image - medium happiness - medium health - medium loneliness - high self-esteem.",
"The number of all possible combinations of the five well-being characteristics is $3^5=243$ .",
"The number of combinations that correspond to scores of at least one individual in our data, i.e., the number of nodes of the well-being type, is 87.",
"The mental health node type represents a mental health condition: either depression or anxiety.",
"(Technically, we analyze each of depression and anxiety via its respective HIN.)",
"For depression, we use one node to denote having “depressed” condition, and another node to denote having “non-depressed” condition.",
"Similarly, for anxiety, we use one node to denote having “anxious” condition, and another node to denote having “non-anxious” condition.",
"The individuals' depression and anxiety scores are collected from the survey taken in January 2016.",
"We consider individuals whose depression scores are above 15 as depressed, which is suggested in the literature [36], [33], [60].",
"Out of the 274 individuals, 67 individuals (24.5%) are depressed according to this criterion.",
"We consider individuals whose anxiety scores are above 40 as anxious, which is suggested in the literature [25], [35], [13].",
"Out of the 274 individuals, 106 individuals (38.7%) are anxious according to this criterion.",
"The six edge types in our HIN include individual - individual, individual - personality traits, individual - social status, individual - physical health, individual - well-being, and individual - mental health, as follows (also, see Figures REF and REF ).",
"The individual - individual edge type is constructed by connecting two individuals if they have communicated through SMS at least once within our study period.",
"This results in 1354 edges between our 274 individuals.",
"We assign a weight to each edge, corresponding to the total number of SMSs exchanged between two given individuals within the study period.",
"The remaining five edge types are constructed by connecting the individual node type to the other five node types (personality traits, physical health, social status, well-being, and mental health).",
"Here, an edge is formed if the given individual has the given combination of personal characteristics of the given type.",
"For example, if an individual has low agreeableness - low conscientiousness - low extroversion- low neuroticism - low openness, we connect the individual to the node of the personality traits type representing such combination.",
"In our data, an individual can only have one combination of personal characteristics of a given node type (e.g., for the personality traits node type, if an individual is linked to a node corresponding to low agreeableness - low conscientiousness - low extroversion - low neuroticism - low openness, he/she cannot be linked to a node corresponding to high agreeableness - high conscientiousness - high extroversion - high neuroticism - high openness or to any other combination).",
"Therefore, an individual is connected to exactly one node of each of the five node types.",
"Thus, for each of the five edge types, the number of edges is the same as the number of the individuals, i.e., 274.",
"The individual - mental health edge type is considered as the target edge type since this is what we try to predict."
],
[
"Network methods that we adapt to the task of predicting mental health conditions",
"We model the problem of predicting individuals' mental health conditions as RS and NC.",
"RS methods.",
"RS is widely used to suggest a personalized list of items (e.g., movies, books, or new friends) to individuals based on their preferences, in order to help them find the most relevant items [58].",
"In this study, we use four RS methods: DMF [17], RESCAL [48], DEDICOM [4], and HERec [63].",
"The first three methods are of the multi-relational matrix factorization (MRMF) type [37].",
"In MRMF, the target edge type is to be predicted and the remaining types of edges as used as side information.",
"For example, if the task is to recommend movies to individuals, the individual - movie edge type is used as the target edge type and other types of edges such as movie - genre, movie - actor, etc.",
"are used as side information.",
"An MRMF model is trained using a proportion of target edges as well as side information, and then the model is used to predict the rest of the target edges.",
"MRMF computes latent features of all types of nodes, which are used to map individuals to items (the edge type to be predicted) that the individuals will prefer.",
"However, instead of using only one edge type, as in standard factorization schemes [29], MRMF allows for the creation of additional latent features based on other side information (other edge types).",
"By operating on latent features between a pair of nodes, we can obtain a value denoting the probability that the given node pair has an edge.",
"MRMF optimizes the loss function to minimize the difference between probability values of all node pairs (represented by a matrix) and the adjacency matrix representing real edges.",
"The three MRMF methods that we consider use different loss functions (that is, they differ in how they mathematically define the difference between probability values of all node pairs and the adjacency matrix) and different optimization methods (that is, they differ in how they minimize their respective loss functions) [75], [76].",
"We use all three MRMF-based RS approaches to predict target edges (i.e., edges of the individual - mental health type).",
"For more details on the three MRMF RS methods, see Supplementary Section S1.",
"As a different approach type, HERec works as follows.",
"First, it learns latent features of nodes from the HIN through metapath-based random walks.",
"For this step, to mimic as closely as possible the case studies in the HERec paper [63], i.e., to give HERec the best-case advantage, we consider all possible metapath types from the individual node type to the individual node type via each of the other five considered node types.",
"That is, we consider the following five types of metapaths: individual - well-being - individual, individual - personality traits - individual, individual - social status - individual, individual - physical health - individual, and individual - mental health - individual.",
"Second, to derive more effective (i.e., better-informed) node representations, HERec uses fused functions to transform the latent node features from the the previous step into a more suitable form for recommendations.",
"For this second step, we consider the personalized non-linear fusion function, which was shown to yield the best performance in the recommendation task among three functions considered in the HERec paper [63].",
"Third, HERec blends the transformed features into a matrix factorization framework to predict rating scores of individuals.",
"For this step, we model the problem of rating prediction as the task of predicting individual - mental health links (i.e., target edges).",
"Namely, we use HERec to predict rating scores of all individuals, sort the predicted scores from highest to lowest, and select a cutoff where individuals with scores above the cutoff will be predicted as depressed (anxious) and those with scores below the cutoff as non-depressed (non-anxious).",
"Note that HERec has the node feature dimension as input parameter.",
"The method uses the default value of 30 for this parameter.",
"To give HERec the best-case advantage, we evaluated additional values of this parameter, namely 10 and 20, i.e., lower-dimensional features than used in the HERec paper, because our data has fewer nodes than the data from the HERec study.",
"However, we found that reducing the dimension decreased HERec's perfrmance.",
"So, we ended up reporting results for the default setting, as this option yielded the best accuracy.",
"When we predict target edges of the individual - mental health type using the four RS methods discussed above, we randomly divide the individuals into 80% training individuals and 20% testing individuals.",
"This step is conducted using stratified sampling to ensure that the split data maintain the percentage of the individuals who are depressed (or anxious).",
"We use target edges of training individuals as well as the other types of edges to train an RS model, and then use the model to predict target edges of testing individuals.",
"In other words, we hide a proportion of target edges and use the remaining edges to train the model, and then use the model to predict the hidden target edges.",
"Formally, given the HIN $G=(V,E)$ , let $I \\in V$ and $M \\in V$ denote the sets of all individuals and all mental health conditions, respectively.",
"Then, the goal is to predict {$i_{test}$ –$m$ }, i.e., the target edges of the testing individuals, where $i_{test}\\in I$ and $m\\in M$ .",
"To predict an individual as depressed or non-depressed, we calculate probability values of the two target edges that connect the individual to the node representing “depressed” condition and the node representing “non-depressed” condition, respectively.",
"We predict an individual as anxious or non-anxious in the same manner.",
"Of the two target edges, we select the one with the higher probability value as the final prediction.",
"NC methods.",
"We model the problem of mental health prediction in an alternative manner, as NC that uses the structural information encoded in the network to predict nodes' labels.",
"For example, in a social network consisting of individuals with known labels being smoker or non-smoker and individuals with unknown labels, the NC method predicts the latter as smokers or non-smokers based on their network features, i.e., their network “relationships” with the former [8].",
"Network features characterize properties of nodes in a network, such as nodes' network positions and their neighborhood information.",
"In order to make mental health predictions using NC methods, our first step is to extract the individuals' network features from our constructed HIN.",
"We learn the individuals' network features based on all edge types except for the target edge type.",
"Instead, we use the target edge type to label the individuals according to their mental health states, for the purpose of NC training and testing (see below).",
"For example, if an individual connects to a node representing the “depressed” condition, we assign a “depressed” label to the individual.",
"We extract four features for the individuals: graphlets [45], colored graphlets [27], DeepWalk [51], and Metapath2vec++ [16].",
"Among these four, graphlets and DeepWalk learn features of nodes from a homogeneous network containing a single node type and a single edge type; in other words, when applied to our HIN, they ignore the different node and edge types and consider the network as homogeneous.",
"Colored graphlets and Metapath2vec++ learn features of nodes from a heterogeneous network consisting of multiple types of nodes and edges.",
"We use the two homogeneous network methods because 1) these methods are widely used to extract features of nodes from a network and 2) there is a limited number of methods that extract nodes' features from a heterogeneous network.",
"In more detail, graphlets are small connected non-isomorphic induced subgraphs of a network [45].",
"Based on the notion of graphlets, the graphlet degree vector (GDV) is a network feature that summarizes a node's extended network neighborhood.",
"In this study, we use both homogeneous graphlets [45] and heterogeneous graphlets [27] to calculate nodes' GDVs.",
"The DeepWalk method [51] uses random walks to generate node sequences in a homogeneous network.",
"The node sequences are put into a skip-gram language model to learn network features.",
"The Metapath2vec++ method [16] formalizes metapath-based random walks which are restricted to only transitions between certain types of nodes.",
"Then, Metapath2vec++ leverages a heterogeneous skip-gram model to learn nodes' features vectors from the metapath-based random walks.",
"For more details on the four NC methods, see Supplementary Section S1.",
"In NC, we use the same training and testing individuals that we have used in RS in order to compare the different types of network methods in an unbiased way.",
"After learning the individuals' network features, the second step in NC is to train a classifier (e.g., logistic regression, random forest, or support vector machine) based on training individuals' network features and node labels and make predictions on testing individuals' node labels.",
"Node labels in NC and target edges in RS are conceptually equivalent.",
"We model target edges as node labels in NC and this is why we do not include target edges when learning network features of individuals.",
"In this study, we use logistic regression as a proof-of-concept classifier, because this particular classifier is often used to predict individuals’ mental health conditions [46], [39], [56].",
"The output of a logistic regression classifier is a set of probability scores, one score per individual.",
"For each logistic regression classifier used in this study (one classifier per network feature), we choose a probability cutoff where individuals with probability scores above the cutoff value will be classified as depressed/anxious, and those with probability scores below the cutoff value will be classified as non-depressed/non-anxious.",
"For each logistic regression classifier used in this study, we choose a cutoff value such that the proportion of the individuals who are predicted as depressed/anxious is equal to the proportion of the individuals who are actually depressed/anxious, as is typically done [20].",
"RS and NC are mathematically different since they work in different ways as discussed above.",
"But ultimately, their inputs and outputs are the same, which makes them directly comparable.",
"Comparison of RS and NC with a non-network method and a random guess method.",
"To investigate the effectiveness of the RC and NC network methods, we compare the network methods to each other as well as to a non-network method [46] and a random guess method [21].",
"For the non-network method and the random guess method, we use the same training and testing individuals that we have used in RS and NC.",
"We implement the non-network method as follows, making it as fairly comparable to our RS and NC network methods as possible.",
"The non-network method is also a logistic regression classifier that utilizes the same NetHealth data as our network methods do but in a non-network fashion.",
"Specifically, the non-network method uses the 15 characteristics collected from the surveys (Section REF ), average step counts and average sleep duration collected from Fitbits, and numbers of SMS messages sent or received during the study period collected from smartphones.",
"For each of these data types, we divide individuals into three groups: the high-score group, the medium-score group, and the low-score group, which is the same approach that we have used when forming nodes of our HIN (Section REF ).",
"We do this to ensure that the step of dividing individuals to groups is consistent and is thus not a factor that can account for differences in the results.",
"The random guess method works as follows.",
"For example, in the depression prediction task, suppose that $x\\%$ of the individuals are depressed and $(1-x)\\%$ of the individuals are non-depressed.",
"To make predictions, the random guess method randomly chooses $x\\%$ of all individuals and predicts them as depressed, and it predicts the remaining individuals as non-depressed."
],
[
"Evaluation methodology",
"We use 5-fold cross-validation to evaluate the performance of the four RS and four NC methods, the non-network method, and the random guess method.",
"We divide all individuals into five equal-sized subsets, such that each subset contains the same proportion of depressed/anxious individuals as present in the considered pool of individuals.",
"We use one of the subsets as the testing set and the union of the other four subsets as the training set.",
"We repeat this process five times until every subset has served as the testing set.",
"We calculate the average and standard deviation of evaluation measures (defined below) over the five runs.",
"Moreover, we make a prediction about an individual when the individual is part of the testing set.",
"This way, we are able to predict mental health conditions for all individuals through the cross-validation process.",
"Given a prediction for an individual, there are four outcomes [19].",
"Taking depression as an example, a true positive (TP) represents an individual who is depressed and is also predicted as depressed.",
"A false negative (FN) represents an individual who is depressed but is predicted as non-depressed.",
"A true negative (TN) represents an individual who is non-depressed and is also predicted as non-depressed.",
"A false positive (FP) represents an individual who is non-depressed but is predicted as depressed.",
"In our study, we consider the following four evaluation measures: $Precision = {TP \\over (TP+FP)}$ : of all predictions, how many are correctly predicted as depressed.",
"$Recall = {TP \\over (TP+FN)}$ : of all depressed people, how many are correctly predicted as depressed.",
"$F1 \\,score = {2TP \\over (2TP+FP+FN)}$ : the harmonic mean of precision and recall.",
"$Accuracy = {(TP+TN) \\over (TP+TN+FP+FN)}$ : of all predictions, how many are correctly predicted (as either depressed or non-depressed).",
"To statistically compare the performance of the different methods when predicting the individuals' mental health conditions, we use the Wilcoxon signed-rank test, which is a non-parametric test used to compare paired samples to assess whether their distributions differ [57].",
"Since for each method we compare its prediction performance against the rest of the methods that we evaluate, we adjust p-values via false discovery rate (FDR) estimation to account for multiple test correction [49].",
"In this study, we use the adjusted p-value threshold of 0.05.",
"We explore whether the most accurate RS method (DMF), the most accurate NC method (DeepWalk), and the non-network method identify different sets of individuals as depressed/anxious.",
"If this is true, we might be able to make more accurate predictions by combining the different method types.",
"For each pair of the three methods, we examine whether sets of depressed/anxious individuals correctly predicted by the two methods are significantly overlapping through a hypergeometric test.",
"Suppose that $S$ is the set of all depressed individuals, $A$ is the set of depressed individuals correctly predicted by one of the two methods, $B$ is the set of depressed individuals correctly predicted by the other method, and $O$ is the overlap between $A$ and $B$ .",
"Then, the $p$ -value (i.e., the probability of obtaining the overlap of size |O| or greater) is: $P (X \\ge |O|) = 1 - \\sum _{i=0}^{|O|-1}\\frac{{S \\atopwithdelims ()i}{|S|-|A| \\atopwithdelims ()|B|-i}}{{|S| \\atopwithdelims ()|B|}}$ We say that the overlap is significant if its $p$ -value is $<$ 0.05."
],
[
"RESULTS",
"In this study, we investigate three research questions: Q1: How do RS, NC, non-network, and random approaches compare to each other in terms of accuracy of predicting individuals' mental health?",
"Q2: Do the most accurate RS method, the most accurate NC method, and the non-network method identify different sets of anxious/depressed individuals, i.e., are they complementary to each other?",
"Q3: What is the impact of using different types of information about the individuals, represented by different edge types in the HIN, on the performance of mental health prediction?",
"Q1: How do RS, NC, non-network, and random approaches compare to each other in terms of accuracy of predicting individuals' mental health?",
"Depression prediction.",
"Among all network methods that we evaluate, DMF, an RS method, is the most accurate and is significantly (p-value$<$ 0.05) more accurate than the rest of the network methods in terms of all evaluation measures (Figures REF , REF and Supplementary Figure S5).",
"Our results indicate that the most accurate RS method (DMF) outperforms all NC methods.",
"However, this observation could be due to the classifier used in NC.",
"Instead of the logistic regression classifier, using other classifiers such as random forest and support vector machine might improve the performance of NC.",
"Recall that the reason why we have used logistic regression is that this particular classifier is typically used in the field of mental health prediction [46], [39], [56].",
"When we compare the RS and NC network methods to the random guess method, we find that all network methods, except RESCAL and DEDICOM, are significantly (p-value$<$ 0.05) more accurate in terms of all evaluation measures (Figures REF , REF and Supplementary Figure S5).",
"The most accurate method that we evaluate, DMF, achieves gains of 115% in terms of precision, 150% in terms of recall, 131% in terms of F1 score, and 21% in terms of accuracy over the random guess method.",
"Note that when we compare an approach, say DMF, to the random guess method, we measure the gain (i.e., relative change) of DMF over the random guess method as $\\frac{Performance_{\\,DMF} - Performance_{\\,random}}{Performance_{\\,random}}$ .",
"For example, if $Performance_{\\,DMF}$ is 0.612 and $Performance_{\\,random}$ is 0.245, the gain is $\\frac{0.612 - 0.245}{0.245}= 1.5 = 150\\%$ .",
"RESCAL and DEDICOM do not accurately predict depression - RESCAL and DEDICOM show similar (i.e., not significantly different) values as the random guess method, in terms of all evaluation measures (Figures REF , REF and Supplementary Figure S5).",
"Note that the superiority of DMF over the other two RS methods (RESCAL and DEDICOM) is not surprising: DMF was already shown to perform better than RESCAL in recommendation tasks [17].",
"In turn, RESCAL was already shown to perform better than DEDICOM in recommendation tasks [48].",
"Therefore, we expected DMF to work the best of these three RS approaches.",
"However, it is surprising that in our task of predicting mental health, RESCAL and DEDICOM produce random-like results.",
"Also, it is at least somewhat surprising that DMF is superior than the fourth considered RS approach, HERec, given that the latter is a more recent approach than the former.",
"Examining why RESCAL and DEDICOM produce random-like results and why DMF is superior to HERec is non-trivial, given the heuristic-like nature of these methods in the recommendation task, without many if any theoretic guarantees.",
"As such, this is out of the scope of the current study.",
"Figure: Method performance when predicting depression and anxiety with respect to precision and recall.",
"Each column shows the prediction performance of the corresponding method averaged over five runs of the 5-fold cross-validation; the error bar represents the corresponding standard deviation.",
"The column corresponding to the random guess method is marked in red and its average precision or recall is shown by the red line.",
"The column corresponding to the non-network method is marked in blue and its average precision or recall is shown by the blue line.",
"RS denotes “recommender system”.",
"NC denotes “node classification”.",
"Analogous results for F1-score and accuracy are shown in Supplementary Figure S5 for depression and in Supplementary Figure S6 for anxiety.When we compare the network methods to the non-network method, we find that: 1) DMF and DeepWalk are significantly (p-value$<$ 0.05) more accurate in terms of all evaluation measures (Figures REF , REF and Supplementary Figure S5); 2) graphlets, colored graphlets, and Metapath2vec++ are marginally (i.e., not significantly) more accurate in terms of all evaluation measures (Figures REF , REF and Supplementary Figure S5); 3) HERec is comparable in terms of all evaluation measures (Figures REF , REF and Supplementary Figure S5); and 4) RESCAL and DEDICOM are significantly (p-value$<$ 0.05) less accurate in terms of all evaluation measures (Figures REF , REF and Supplementary Figure S5).",
"The most accurate method that we evaluate, DMF, achieves gains (as defined above) of 41% in terms of precision, 64% in terms of recall, 52% in terms of F1 score, and 11% in terms of accuracy over the non-network method.",
"In summary, our results show that for depression, the best RS method significantly outperforms all NC methods.",
"In addition, six out of the eight network methods significantly outperform the random guess method.",
"Moreover, both the best RS method and the best NC method significantly outperform the non-network method.",
"This confirms the power of network methods and RS in particular in predicting depression.",
"Anxiety prediction.",
"Among all network methods that we evaluate, DMF, an RS method, is the most accurate and is significantly (p-value$<$ 0.05) more accurate than the rest of the network methods in terms of all evaluation measures (Figures REF , REF and Supplementary Figure S6).",
"Our results indicate that the most accurate RS method (DMF) outperforms all NC methods.",
"When we compare the RS and NC network methods to the random guess method, we find that all network methods, except RESCAL and DEDICOM, are significantly (p-value$<$ 0.05) more accurate than the random guess method in terms of all evaluation measures (Figures REF , REF and Supplementary Figure S6).",
"The most accurate method that we evaluate, DMF, achieves gains of 72% in terms of precision, 101% in terms of recall, 86% in terms of F1 score, and 43% in terms of accuracy over the random guess method.",
"RESCAL and DEDICOM do not accurately predict anxiety - RESCAL and DEDICOM show similar (i.e., not significantly different) values as the random guess method, in terms of all evaluation measures (Figures REF , REF and Supplementary Figure S6).",
"When we compare the network methods to the non-network method, we find that: 1) DMF is significantly (p-value$<$ 0.05) more accurate in terms of all evaluation measures (Figures REF , REF and Supplementary Figure S6); 2) DeepWalk is marginally more accurate and HERec, graphlets, colored graphlets, and Metapath2vec++ are marginally less accurate in terms of all evaluation measures (Figures REF , REF and Supplementary Figure S6); and 3) RESCAL and DEDICOM are significantly (p-value$<$ 0.05) less accurate in terms of all evaluation measures (Figures REF , REF and Supplementary Figure S6).",
"The most accurate method that we evaluate, DMF, achieves gains of 20% in terms of precision, and 40% in terms of recall, 29% in terms of F1 score, and 16% in terms of accuracy over the non-network method.",
"In summary, the results for anxiety are similar to those for depression.",
"This confirms the power of network methods and RS in particular in predicting anxiety in addition to depression.",
"Q2: Do the most accurate RS method, the most accurate NC method, and the non-network method identify different sets of anxious/depressed individuals, i.e., are they complementary to each other?",
"To answer Q2, we examine the overlap of sets of depressed/anxious individuals correctly predicted by DMF, DeepWalk, and the non-network method.",
"We find that among the three methods, DMF identifies the largest number of depressed/anxious individuals (Figures REF and REF ), which is reflected by its highest prediction accuracy (Figure REF ).",
"Using DMF, we predict as depressed 41 out of all 67 actually depressed individuals and as anxious 77 out of all 106 actually anxious individuals.",
"The other two methods combined correctly predict additional 11 depressed individuals and 20 anxious individuals who are missed by DMF.",
"Most of the depressed/anxious individuals are correctly predicted by at least one of the three methods; only 15 out of all 67 depressed individuals and 9 out of all 106 anxious individuals are not correctly predicted by any of the three methods.",
"Figure: Sizes of overlaps between individuals correctly predicted as (a) depressed and (b) anxious by the best RS network method (DMF), the best NC network method (DeepWalk), and the non-network method.Results regarding the overlap between predictions of DMF, DeepWalk, and the non-network method are qualitatively similar for depression and anxiety (Figures REF and REF ).",
"Taking depression as an example, we find that the two sets of depressed individuals correctly predicted by DMF and DeepWalk overlap significantly (p-value$<$ 0.05) (Figure REF ).",
"This could be because DMF and DeepWalk are both network methods and differ only in one aspect - they are different types of methods (RS and NC).",
"Moreover, the two sets of depressed individuals correctly predicted by DeepWalk and the non-network method overlap significantly (p-value$<$ 0.05) (Figure REF ).",
"This could be because the two methods both use logistic regression to make predictions and differ only in one aspect - DeepWalk uses network features while the non-network method uses non-network features.",
"However, the two sets of depressed individuals correctly predicted by DMF and the non-network method do not significantly overlap (Figure REF ).",
"This could be because DMF differs from the non-network method in two ways: the former is a network method and it does not use logistic regression to make predictions, while the latter is a non-network method that uses logistic regression.",
"In other words, it could be that the more similar the two approaches are in terms of their methodologies, the more similar their predictions.",
"Since the two sets of depressed/anxious individuals correctly predicted by DMF and the non-network method do not significantly overlap, the combination of the two methods' ideas may be able to correctly predict more of the depressed/anxious individuals.",
"Thus, we could potentially use ensemble learning algorithms to combine DMF and the non-network method to achieve more accurate predictions [15], [53].",
"Q3: What is the impact of using different types of information about the individuals, represented by different edge types in the HIN, on the performance of mental health prediction?",
"To answer Q3, we train a series of instances of DMF, which is the most accurate method we have evaluated thus far (when considering all types of information about individuals).",
"Each instance is trained by using the target (i.e., individual - mental health) edge type and one or more of the other five non-target edge types.",
"We analyze all possible combinations of the five non-target edge types.",
"Each DMF instance uses one of the combinations.",
"In total, we train 31 DMF instances corresponding to 31 possible combinations of the five non-target edge types.",
"In this section, edge types are denoted as follows: I: individual - individual, P: individual - personality traits, S: individual - social status, F: individual - physical health (where “F” stands for mostly Fitbit-based physical health data), and W: individual - well-being.",
"Then, the different edge type combinations are denoted by the corresponding combinations of the I, P, S, F, and W acronyms.",
"Depression prediction.",
"We find that among all combinations, the FW combination—representing the combination of the individual - physical health and individual - well-being (W) edge types—is the most accurate in terms of all evaluation measures (Figure REF and Supplementary Figure S7).",
"In more detail, the FW combination is significantly (p-value$<$ 0.05) more accurate than the rest of combinations, including the combination of all five non-target edge types (“All”), in terms of all evaluation measures (Figure REF and Supplementary Figure S7).",
"Potential reasons why the FW combination is more accurate than the “All” combination are as follows.",
"First, DMF is an MRMF method (Section REF ).",
"MRMF typically contains a large number of parameters and may be prone to overfitting, meaning that MRMF may fit well on the training data but not predict well on the testing data [71].",
"The complexity (the number of parameters) of the “All” combination is higher than the complexity of the FW combination.",
"Thus, the higher number of parameters of the former may cause its overfitting, which in turn may cause its lower prediction performance.",
"Second, some edge types in the “All” combination may be less informative than the individual - physical health (F) and individual - well-being (W) edge types in the FW combination when predicting depression.",
"Using edge types that may be suboptimally informative may lower the prediction performance compared to using only edge types that are optimally informative.",
"In other words, it might not be surprising that using some subset of all data types might be more informative/accurate than using all data types.",
"Specifically, in our evaluation, since each of the F and W edge types alone is more accurate than any one of S, I, and P edge types alone (Figure REF ), it might not be surprising that the FW edge combination is more accurate than the All combination.",
"Importantly, it is the case that the other edge types alone, namely S and I (although not P), are performing significantly better than at random (Figure REF ), meaning that they do contain some predictive power.",
"So, it is the subject of our future work to understand how to significantly improve upon the FW combination while incorporating the S and I (and possibly even P) edge types, i.e., how to get a truly synergistic, multiplicative effect when integrating the different data types.",
"In Section , we discuss a possible direction towards achieving this goal.",
"Figure: Performance of DMF (the best network method) when using different combinations of data (i.e., edge types) to predict depression, with respect to precision.",
"In this figure, edge types are denoted as follows.",
"I: individual - individual, P: individual - personality traits, S: individual - social status, F: individual - physical health, W: individual - well-being.",
"“All” denotes the combination of all edge types, and it corresponds to the DMF method shown in Figure .",
"Each column shows the prediction performance of the corresponding edge type combination averaged over five runs in the 5-fold cross-validation; the error barrepresents the corresponding standard deviation.",
"Columns are sorted from left to right according to their heights from high to low, i.e., edge type combinations are sorted from left to right in decreasing order of their prediction performance.",
"Columns to the left of the vertical blue line are the six top performing combinations that we focus on in the text.",
"The horizontal red line shows the precision of the random guess method.Besides FW, there exist five other combinations that all have comparable (i.e., not significantly different) prediction performance to each other and are all significantly (p-value$<$ 0.05) more accurate than the rest of the combinations in terms of all evaluation measures (Figure REF and Supplementary Figure S7).",
"So, these five combinations can all be considered as the second best result, inferior only to the FW combination.",
"These combinations are FSW, “All” (the combination including all types of edges), IFSW, PFSW, and IPFW.",
"We find that all of these combinations contain the individual - well-being (W) and individual - physical health (F) edge types.",
"This agrees with the literature - well-being traits such as body image and self-esteem [72], [69], as well as physical health traits such as physical activity [28] and sleep [5], [3] are correlated with depression, which validates our HIN-based predictive framework.",
"Anxiety prediction.",
"For anxiety, we find that among all combinations, the PW combination—representing the combination of the individual - personality traits (P) and individual - well-being (W) edge types—is the most accurate in terms of all evaluation measures (Figure REF and Supplementary Figure S8).",
"Figure: Performance of DMF (the best network method) when using different combinations of data (i.e., edge types) to predict anxiety, with respect to precision.",
"In this figure, edge types are denoted as follows.",
"I: individual - individual, P: individual - personality traits, S: individual - social status, F: individual - physical health, W: individual - well-being.",
"“All” denotes the combination of all edge types, and it corresponds to the DMF method shown in Figure .",
"Each column shows the prediction performance of the corresponding edge type combination averaged over five runs in the 5-fold cross-validation; the error barrepresents the corresponding standard deviation.",
"Columns are sorted from left to right according to their heights from high to low, i.e., edge type combinations are sorted from left to right in decreasing order of their prediction performance.",
"Columns to the left of the vertical blue line are the seven top performing combinations that we focus on in the text.",
"The horizontal red line shows the precision of the random guess method.Unlike for depression, the best combination for anxiety, PW, is only marginally more accurate than six other combinations including FW, PFW, PSW, IPW, “All” (the combination of all types of edges), and IPFW.",
"In other words, the PW combination and the six other combination all have comparable (i.e., not significantly different) performance in terms of all evaluation measures.",
"Plus, all seven combinations are significantly (p-value$<$ 0.05) more accurate than the rest of the combinations in terms of all evaluation measures (Figure REF and Supplementary Figure S8).",
"Focusing on the seven best combinations, the individual - personality traits (P), individual - well-being (W), and individual - physical health (F) edge types are frequently included.",
"This finding agrees with the literature that personality traits [7], well-being traits such as self-esteem [69], and physical health traits such as physical activity [43] and sleep [3] are correlated with anxiety, which further validates our HIN-based predictive framework.",
"Unlike for depression, for anxiety, the All combination is one of the best-scoring combinations.",
"However, it is still not significantly better than the other six best-scoring combinations.",
"So, just like for depression, for anxiety, it is also the subject of our future work to understand how to get a multiplicative effect with our HIN data integrative framework.",
"The promise to improve is certainly there, especially because multiple individual edge types, namely P, W, and I (although not F and S) are all performing better than at random when used alone (Figure REF ), meaning that each of them has some predictive power.",
"Hence, it should be possible to use these three combined, possibly also with F and S, to get the IPW combination, possibly also the All combination, to perform significantly better than the other edge combinations.",
"Again, in Section , we propose a step in this direction.",
"An additional observation is that the FW combination, the most accurate combination in depression prediction, is the second-best combination in anxiety prediction.",
"This indicates that physical health (F) and well-being (W) are good predictors of both depression and anxiety.",
"On the other hand, the PW combination, the most accurate combination in anxiety prediction, is not among the best combinations in predicting depression.",
"This indicates that personality traits (P) may not be a key factor in predicting depression, while they are a good predictor of anxiety.",
"These results suggest that depression prediction and anxiety prediction have both similarities and differences.",
"This might be explained by a reasonably large overlap between the depressed individuals and the anxious individuals in our data.",
"Namely, of the 67 depressed individuals and 106 anxious individuals, 51 individuals are both depressed and anxious ($p$ -value$<$ 0.05).",
"So, this overlap could explain the similarities between the depression prediction and anxiety prediction results.",
"On the other hand, a number of individuals are depressed but not anxious and vice versa, which could explain the differences between the depression prediction and anxiety prediction results."
],
[
"Conclusion",
"In this paper, we integrate individuals' smartphone, wearable sensor, and survey data into an HIN and apply state-of-the-art RS and NC methods to the HIN to predict the individuals' mental health conditions.",
"Our results indicate that among all of the network methods, DMF, an RS method, is the best, i.e., RS is better than NC as evaluated in our study.",
"DMF outperforms the non-network method as well as the random guess method in terms of all evaluation measures.",
"This confirms the power of network-based analyses of NetHealth data and RS in particular in predicting mental health.",
"This study can be extended in several ways.",
"1) Because the NetHealth study has collected time series data, adding temporal information such as dynamic social interaction data [42] into our HIN could perhaps yield a truly multiplicative effect of data integration, i.e., lead to improvement compared to using the currently best-performing FW and PW data type combinations for depression and anxiety, respectively.",
"Note that doing this is non-trivial, as traditional RS and NC methods are designed for static HINs.",
"This is why we have not considered the data's temporal nature in the current study and why instead we plan to do so in our future work.",
"2) In this study, we focus on the task of mental health prediction as a proof-of-concept.",
"But our HIN framework could be generalized to predict any of the individuals’ traits available in the NetHealth data.",
"In other words, our framework could be generalized to predict any edge type and not just the individual - mental health edge type, as we do in the current study.",
"To do that, we would just need to treat a desired edge type as the target edge type and the rest of the edge types (including the individual - mental health edge type) as side information.",
"3) Our framework could also be generalized to new and larger data sets containing more participants and more types of data when such data sets become available.",
"To do that, we would just need to model the new data set as a new HIN that may contain a larger number of nodes and edges and more node and edge types than the HIN constructed in this study, and then apply RS and NC methods to the new HIN.",
"4) Because of the non-significant overlap between DMF’s and the non-network method’s predictions, ensemble learning methods could be developed to combine the two methods' ideas in order to further improve prediction performance compared to each of the methods individually.",
"Exploring this is beyond the scope of this paper and is the subject of our future work."
],
[
"Acknowledgement",
"This work was funded by the National Institutes of Health (NIH) 1R01HL117757, National Science Foundation (NSF) CAREER CCF-1452795, and Air Force Office of Scientific Research (AFOSR) YIP FA9550-16-1-0147 grants.",
"We thank the entire NetHealth team for their useful discussions during weekly meetings.",
"We especially thank the following team members: Rachael Purta for helping us access the Fitbit data, Afzal Hossain for help with the SMS data, and Louis Faust for assistance with understanding the Fitbit and survey data.",
"Importantly, we thank all NetHealth study individuals for generously volunteering and consenting to share their data, without whom our study would not have been possible."
]
] | 1906.04346 |
[
[
"On the LoRa Modulation for IoT: Waveform Properties and Spectral\n Analysis"
],
[
"Abstract An important modulation technique for Internet of Things (IoT) is the one proposed by the LoRa allianceTM.",
"In this paper we analyze the M-ary LoRa modulation in the time and frequency domains.",
"First, we provide the signal description in the time domain, and show that LoRa is a memoryless continuous phase modulation.",
"The cross-correlation between the transmitted waveforms is determined, proving that LoRa can be considered approximately an orthogonal modulation only for large M. Then, we investigate the spectral characteristics of the signal modulated by random data, obtaining a closed-form expression of the spectrum in terms of Fresnel functions.",
"Quite surprisingly, we found that LoRa has both continuous and discrete spectra, with the discrete spectrum containing exactly a fraction 1/M of the total signal power."
],
[
"Introduction",
"The most typical IoT scenario involves devices with limited energy, that need to be connected to the Internet via wireless links.",
"In this regard, lpwan aim to offer low data rate communication capabilities over ranges of several kilometers [1], [2], [3], [4].",
"Among the current communication systems, that proposed by the lora alliance (Low power long Range) [5] is one of the most promising, with an increasing number of IoT applications, including smart metering, smart grid, and data collection from WSN for environmental monitoring [6], [7], [8], [9], [10], [11].",
"Several works discuss the suitability of the LoRa communication system when the number of IoT devices increases [12], [13], [14], [15].",
"The modulation used by LoRa, related to Chirp Spread Spectrum, has been originally defined by its instantaneous frequency [16].",
"Few recent papers attempted to provide a description of the lora modulation in the time domain, but, as will be detailed below, they are not complaint with the original LoRa signal model.",
"The LoRa performance has been analyzed by simulation or by considering it as an orthogonal modulation [17], [18], [19].",
"On the other hand, the spectral characteristics of LoRa have not been addressed in the literature.",
"In this paper we provide a complete characterization of the LoRa modulated signal.",
"In particular, we start by developing a mathematical model for the modulated signal in the time domain.",
"The waveforms of this $M$ -ary modulation technique are not orthogonal, and the loss in performance with respect to an orthogonal modulation is quantified by studying their cross-correlation.",
"The characterization in the frequency domain is given in terms of the power spectrum, where both the continuous and discrete parts are derived.",
"The found analytical expressions are compared with the spectrum of LoRa obtained by experimental data.",
"The main contributions of this paper can be summarized as follows: we provide the analytical expression of the signal for the $M$ -ary LoRa chirp modulation in the time domain (both continuous-time and discrete-time); we derive the cross-correlation between the LoRa waveforms, and prove that the modulation is non-orthogonal; we prove that the waveforms are asymptotically orthogonal for increasingly large $M$ ; we derive explicit closed-form expressions of the continuous and discrete spectra of the LoRa signal in terms of the Fresnel functions; we prove that the power of the discrete spectrum is exactly a fraction $1/M$ of the overall signal power; we compare the analytical expression of the spectrum with experimental data from commercial LoRa devices; we show how the analytical expressions of the spectrum can be used to investigate the compliance of the LoRa modulation with the spectral masks regulating the out-of-band emissions and the power spectral density.",
"The provided time and spectral characterization of the LoRa signal is an analytical tool for the system design, as it allows suitable selection of the modulation parameters in order to fulfill the given requirements.",
"For example, our analysis clarifies how the spreading factor, maximum frequency deviation, and transmitted power determine the occupied bandwidth, shape of the power spectrum and its compliance with spectrum regulations, system spectral efficiency, total discrete spectrum power, maximum cross-correlation, and SNR penalty with respect to orthogonal modulations.",
"Throughout the manuscript, we define the indicator function $g_T(t)=1$ for $0\\le t <T$ and $g_T(t)=0$ elsewhere, and indicate as $u(t)$ the unit step function.",
"The Dirac's delta is indicated as $\\delta (x)$ , and its discrete version as $\\delta _m$ , with $\\delta _0=1$ , $\\delta _m=0 \\, \\forall m\\ne 0$ .",
"We also indicate with $C(x)\\triangleq \\int _{0}^{x} \\cos \\left(t^{2} {\\pi }/{2} \\right) \\, dt$ and $S(x) \\triangleq \\int _{0}^{x} \\sin \\left(t^{2} {\\pi }/{2}\\right) \\, dt\\,$ the Fresnel functions[20]."
],
[
"LoRa Signal Model",
"The LoRa frequency shift chirp spread spectrum modulation has been originally described in terms of the instantaneous frequency reported in [16].",
"It is an $M$ -ary digital modulation, where the $M$ possible waveforms at the output of the modulator are chirp modulated signals over the frequency interval $(f_0-B/2, f_0+B/2)$ with $M$ different initial frequencies.",
"The data modulated signal is usually preceded by synchronization waveforms, not considered here.",
"For the data, the instantaneous frequency is linearly increased, and then wrapped to $f_0-B/2$ when it reaches the maximum frequency $f_0+B/2$ , an operation that mathematically can be seen as a reduction modulo $B$ .",
"Having the instantaneous frequency sweeping over $B$ does not imply that the signal bandwidth is $B$ , as will be discussed in Section .",
"For LoRa the parameters are chosen such that $M=2^{\\text{SF}}$ with $\\text{SF}$ integer, and $B T_{s}=M$ , where $T_{s}$ is the symbol interval.",
"The bit-rate of the modulation is $R_b&=\\frac{1}{T_{s}} \\log _2 M = \\frac{\\text{SF}}{T_{s}}= B \\frac{\\text{SF}}{2^{\\text{SF}}}$ The ratio between the chip-rate $R_c=M/T_{s}=B$ and the bit-rate is thereforeIn spread-spectrum literature this is what is usually called spreading factor.",
"However, in the LoRa terminology $\\text{SF}$ is called the spreading factor.",
"$\\eta =\\frac{R_c}{R_b}=\\frac{B}{R_b}=\\frac{2^{\\text{SF}}}{\\text{SF}} \\,.$ Its reciprocal $1/\\eta $ can be seen as the modulation spectral efficiency in $\\text{bit/s/Hz}$ .",
"Some values of the spectral efficiency are reported in Table REF for $M$ ranging from $2^{3}$ to $2^{12}$ ."
],
[
"Continuous-time description",
"To describe mathematically the signal in the time domain, let us start for clarity by assuming that the frequency interval over which to linearly sweep the frequency is $[0, B]$ as depicted in Fig.",
"REF .",
"Figure: Example of the instantaneous frequency f(t;a)f(t;a) as a function of time for two different modulating symbols a 1 ,a 2 ∈{0,...,M-1}a_1, a_2 \\in \\lbrace 0, \\ldots , M-1\\rbrace .For the time interval $t \\in [0,T_{s}[$ and a symbol $a\\in \\lbrace 0, 1, \\ldots , M-1\\rbrace $ the instantaneous frequency in LoRa can thus be written as $f(t;a)&=a\\, \\frac{B}{M}+\\frac{B}{T_{s}}\\,t \\pmod {B} \\nonumber \\\\&= a\\, \\frac{B}{M}+\\frac{B}{T_{s}}\\,t-B\\, u\\left(t-\\tau _{a}\\right) && 0 \\le t < T_{s}$ where $a\\, {B}/{M}$ is the initial frequency which depends on the modulating symbol, and $\\tau _{a}=T_{s}\\left(1-\\frac{a}{M}\\right)$ is the time instant where, after a linear increase, the instantaneous frequency reaches the maximum; for the remaining part of the symbol interval the instantaneous frequency is still linearly increasing, but reduced modulo $B$ by subtracting $B$ .",
"Assuming the modulation starts at $t=0$ , from (REF ) the phase $\\phi (t;a)$ for $t\\in [0,T_{s}[$ is given by $\\phi (t;a) &\\triangleq 2\\pi \\int _{0}^{t} f(\\tau , a) \\, d\\tau \\nonumber \\\\&= 2\\,\\pi \\left[a\\, \\frac{B}{M} \\,t+\\frac{B}{2\\,T_{s}}\\,t^2-B\\,\\left(t-\\tau _{a}\\right)\\, u (t-\\tau _{a}) \\right] \\,.",
"$ Also, with the LoRa parameters we see from (REF ) that the product $B\\tau _{a}=M-a$ is an integer, and can therefore be omitted in the phase.",
"Note that a factor $1/2$ for the quadratic term is missing in the phase definitions reported in [17], [18], [19], making the instantaneous frequency of the signal not complaint with that of LoRa.",
"That difference also propagated in the discrete-time version of the signals used in [18], [19], so that even the time-discrete analysis made there is not applicable to the LoRa signal.",
"Property 1 The LoRa modulation is a memoryless continuous phase modulation with $\\phi (0;a)=\\phi (T_{s};a)$ .",
"The initial phase is $\\phi (0;a)=0$ .",
"The phase at the end of the symbol interval is $\\phi (T_{s};a)&=2\\,\\pi \\left[a\\, \\frac{B}{M} T_{s}+\\frac{B}{2}T_{s}-B \\left(T_{s}-\\tau _{a}\\right)\\, u (T_{s}-\\tau _{a}) \\right] \\nonumber \\\\&=2\\,\\pi \\left(a+\\frac{M}{2}-M\\, u (T_{s}-\\tau _{a})\\right) = 0 \\pmod {2\\pi } \\nonumber $ where the last equality is due to that $a+{M}/{2}-M u (T_{s}-\\tau _{a})$ is always an integer.",
"In other words, the initial and final phases are coincident, irrespectively on the symbol $a$ .",
"From this property we see that the LoRa modulation can be interpreted as a continuous phase memoryless modulation, where the transmitted waveform in each symbol interval depends only on the symbol in that interval, and not on previous or successive symbols.",
"This can be visualized through the phase diagram which tracks the evolution of the phase over time.",
"In Fig.",
"REF , the phase diagram for two consecutive LoRa modulated symbols is shown as a function of time.",
"It can be noted that each waveform starts and ends with the same phase.",
"Figure: The phase diagram as a function of time over two consecutive LoRa modulated symbols, indicated in blue and orange.The complex envelope of the modulated signal is $x(t;a)&=\\gamma \\exp \\left\\lbrace j\\, \\phi (t;a)\\right\\rbrace , && 0 \\le t < T_{s}$ where $\\gamma =\\sqrt{2\\, P_{\\text{s}}}$ accounts for the passband signal power $P_{\\text{s}}$ .",
"In the following we will assume $\\gamma =1$ unless otherwise stated.",
"By introducing a frequency shift $-B /2$ , the complex envelope centered at frequency zero for the interval $[0,T_{s}[$ is $\\!\\!\\!\\!x(t;a)&= \\exp \\left\\lbrace j 2 \\pi B t \\left[\\frac{a}{M}\\!-\\!\\frac{1}{2}\\!+\\!\\frac{B t}{2 M}\\!",
"-\\!",
"u\\left(t-\\frac{M-a}{B}\\right) \\right]\\right\\rbrace \\,.",
"$ Due to the memoryless nature of the modulation, the complex envelope of the LoRa signal can be written as $i(t)&=\\sum _n x(t- nT_{s};a_n) g_{T_{s}}(t-nT_{s})$ where $a_n$ is the symbol transmitted in the time interval $[nT_{s}, (n+1)T_{s}[$ .",
"We remark that, as this is a frequency modulated signal, we have $|i(t)|=1$ and the power of the signal $i(t)$ is one.",
"The passband modulated signal centered at $f_0$ is then $s(t)=\\Re \\left\\lbrace i(t) e^{j 2 \\pi f_0 t}\\right\\rbrace $ .",
"Table: Spectral efficiency, maximum cross-correlation, 99%99\\%-power bandwidth, total discrete spectrum power, and maximum SNR penalty.Property 2 The cross-correlation between the continuous time waveforms $x(t; \\ell )$ and $x(t; m)$ with $\\ell \\ne m$ is $C_{\\ell ,m}&=\\frac{1}{T_{s}} \\int _{0}^{T_{s}} x(t; \\ell ) x^*(t; m) dt= \\nonumber \\\\&=M \\frac{e^{j 2 \\pi \\ell (m-\\ell )/M}-e^{j 2 \\pi m (m-\\ell )/M}}{j 2 \\pi (M-|m-\\ell |) |m-\\ell | }$ and $C_{\\ell ,\\ell }=1$ .",
"It follows that the waveforms $x(t; \\ell )$ and $x(t; m)$ are orthogonal (i.e, $C_{\\ell ,m}=0$ ) only for $|m-\\ell |=2^{(p+\\text{SF})/2}$ with $p\\ge 0$ an odd (even) integer for odd (even) $\\text{SF}$ .",
"Moreover, since $\\Re \\left\\lbrace C_{\\ell ,m}\\right\\rbrace &=M \\frac{\\sin \\left(\\frac{2 \\pi \\ell (m-\\ell )}{M}\\right)-\\sin \\left(\\frac{2 \\pi m (m-\\ell )}{M}\\right)}{2 \\pi (M-|m-\\ell |) |m-\\ell | }$ we have that the passband waveforms $\\Re \\left\\lbrace x(t; \\ell ) e^{j 2 \\pi f_0 t}\\right\\rbrace $ and $\\Re \\left\\lbrace x(t; m) e^{j 2 \\pi f_0 t}\\right\\rbrace $ are orthogonal (i.e, $\\Re \\left\\lbrace C_{\\ell ,m}\\right\\rbrace =0$ ) only when $(m-\\ell )^2/M$ is an integer, or when $(m^2-\\ell ^2)/M -1/2$ is an integer.",
"We assume $f_0 \\gg B$ so that the passband waveforms are orthogonal when $\\Re \\left\\lbrace C_{\\ell ,m}\\right\\rbrace =0$ .",
"Also, the maximum cross-correlation can be upper bounded as $\\max _{\\ell \\ne m} \\left|\\Re \\left\\lbrace C_{\\ell ,m}\\right\\rbrace \\right| \\le \\max _{\\ell \\ne m} \\left|C_{\\ell ,m}\\right| \\le \\frac{1}{\\sqrt{2\\,M}-1}.$ Hence, the waveforms are asymptotically orthogonal for increasing $M$ : $\\lim _{M \\rightarrow \\infty } \\left| \\left\\lbrace C_{\\ell ,m}\\right\\rbrace \\right| = \\delta _{\\ell -m} \\,.$ The crosscorrelation between the continuous time waveforms $x(t; \\ell )$ and $x(t; m)$ with $\\ell \\ne m$ and $\\ell >m$ can be written as $C _ { \\ell , m }&=\\frac{ 1 }{T_{s}} \\int _ { 0 } ^ {T_{s}} e^{j 2 \\pi \\frac{B}{M}(\\ell -m)\\,t - B\\, t\\, \\left[u\\left(t-\\tau _{\\ell }\\right)-u\\left(t-\\tau _{m}\\right) \\right] }\\, d t \\nonumber \\\\&=\\underbrace{\\frac{ 1 }{T_{s}} \\int _ { 0 } ^ {T_{s}} e^{j 2 \\pi \\frac{B}{M}(\\ell -m)t} \\,dt}_{0} -{\\frac{ 1 }{T_{s}} \\int _ { \\tau _{\\ell } } ^ {\\tau _{m}} e^{j 2 \\pi \\frac{B}{M}(\\ell -m)t}\\, dt} \\nonumber \\\\&\\phantom{=}+ { \\frac{ 1 }{T_{s}} \\int _ { \\tau _{\\ell } } ^ {\\tau _{m} } e^{j 2 \\pi \\left[\\frac{B}{M}(\\ell -m)\\,t-B\\,t \\right]}\\,dt} \\nonumber \\\\&= \\frac{1}{j 2 \\pi (\\ell -m)} \\left[e^{j 2 \\pi (M-\\ell )(\\ell -m)/M}-e^{j 2 \\pi (M-m)(\\ell -m)/M} \\right] \\nonumber \\\\&\\phantom{=}+\\!\\frac{1}{j 2 \\pi (M\\!+m\\!-\\ell )} \\left[\\!e^{j 2 \\pi \\frac{(\\ell -M)(M+m-\\ell )}{M}}\\!- e^{j 2 \\pi \\frac{(m-M)(M+m-\\ell )}{M}}\\!",
"\\right] \\nonumber $ Noting the periodicity of the complex exponential function, we have $C _ { \\ell , m }&=\\frac{1}{j 2 \\pi (\\ell -m)} \\left[e^{j 2 \\pi \\ell (m-\\ell )/M}- e^{j 2 \\pi m(m-\\ell )/M} \\right] \\nonumber \\\\&\\phantom{=}+\\frac{1}{j 2 \\pi (M+m-\\ell )} \\left[e^{j 2 \\pi \\ell (m-\\ell )/M}-e^{j 2 \\pi m(m-\\ell )/M}\\right] \\nonumber \\\\&=\\frac{e^{j 2 \\pi \\ell \\frac{m-\\ell }{M}}- e^{j 2 \\pi m\\frac{m-\\ell }{M}}}{j 2 \\pi } \\left(\\frac{1}{M+m-\\ell }+\\frac{1}{\\ell -m} \\right).",
"$ Similarly, for $m>\\ell $ we have $C _ { \\ell , m }&=\\frac{e^{j 2 \\pi \\ell \\frac{m-\\ell }{M}}- e^{j 2 \\pi m\\frac{m-\\ell }{M}}}{j 2 \\pi } \\left(\\frac{1}{M+\\ell -m}+\\frac{1}{m-\\ell } \\right).$ Putting together (REF ) and (REF ), the complex crosscorrelation, $C _ { \\ell , m }$ , can be derived as in (REF ).",
"The correlation in (REF ) can be zero only if the two exponentials are equal, that requires $\\ell (m-\\ell )/M= m (m-\\ell )/M -k$ , with $k$ an integer.",
"Thus, it must be $|m-\\ell |=\\sqrt{k M}$ .",
"Since this must be an integer, and $M=2^\\text{SF}$ , it follows that $k=2^p$ with $p\\ge 0$ an odd (even) integer for odd (even) $\\text{SF}$ .",
"The real cross-correlation (REF ) follows directly, and the conditions for its zeros are straightforward observing that $\\sin \\alpha = \\sin \\beta $ for $\\alpha =\\beta + k 2 \\pi $ or $\\alpha =\\pi -\\beta + k 2 \\pi $ .",
"In order to find the asymptotic behavior of the complex cross-correlation, we start by upper bounding its absolute value for $\\ell \\ne m$ .",
"From (REF ) we have $C_{\\ell ,m}=M e^{j 2 \\pi \\left(m^2-\\ell ^2\\right)/M} \\, \\frac{e^{-j \\pi (m-\\ell )^2/M}-e^{j \\pi (m-\\ell )^2/M}}{j 2 \\pi (M-|m-\\ell |) |m-\\ell | }$ and therefore $\\left|C_{\\ell ,m}\\right|= M \\frac{\\left|\\sin \\left(\\pi (m-\\ell )^2/M\\right)\\right|}{ \\pi (M-|m-\\ell |) |m-\\ell | }\\,.$ The first maximum for $\\left|C_{\\ell ,m}\\right|$ is in the interval ${1\\le |m-\\ell | \\le \\left\\lfloor {\\sqrt{M/2}}\\right\\rfloor }$ .",
"This is due to the following reasons: $\\left|C_{\\ell ,m}\\right|$ is symmetric around $M/2$ ; the denominator is monotonically increasing for ${1\\le |m-\\ell | \\le M/2}$ ; the numerator is monotonically increasing for $1\\le |m-\\ell | \\le \\left\\lfloor {\\sqrt{M/2}}\\right\\rfloor $ , and starts to decrease after $\\left\\lfloor {\\sqrt{M/2}}\\right\\rfloor $ .",
"Hence, we have $\\max _{\\ell \\ne m} \\left|C_{\\ell ,m}\\right| &=\\max _{1 \\le |m-\\ell |\\le \\left\\lfloor {\\sqrt{M/2}}\\right\\rfloor } M \\frac{\\sin \\left(\\pi (m-\\ell )^2/M\\right)}{ \\pi (M-|m-\\ell |) |m-\\ell | } \\\\&\\le \\max _{1 \\le |m-\\ell |\\le \\left\\lfloor {\\sqrt{M/2}}\\right\\rfloor } \\frac{ |m-\\ell |}{ M-|m-\\ell |}\\\\&\\le \\frac{1}{\\sqrt{2\\,M}-1}$ where for the first inequality $\\sin (x) \\le x$ for $0 \\le x \\le \\pi /2$ is used.",
"For the second inequality, it is noticed that the function is increasing in $ |m-\\ell |$ , so its maximum value is obtained with $|m-\\ell |=\\left\\lfloor {\\sqrt{M/2}}\\right\\rfloor \\le \\sqrt{M/2}$ .",
"Finally, taking the limit when $M\\rightarrow \\infty $ gives (REF ).",
"The correlation among the waveforms of the LoRa modulation has an impact on the error performance for the optimum coherent receiver over AWGN channels [21], [22].",
"In particular, for the pairwise error probability between the $\\ell $ -th and $m$ -th waveforms there is a factor $1-\\Re \\left\\lbrace C_{\\ell ,m}\\right\\rbrace $ in the SNR with respect to orthogonal modulation schemes (see, e.g., equations (4.31) and (4.49) in [21]).",
"In Table REF we report the maximum penalty on the SNR, $\\Delta _{\\max }$ , corresponding to the maximum cross-correlation, to be paid with respect to orthogonal modulation schemes.",
"For example, with $M=2^{7}$ we have $\\max _{\\ell \\ne m} \\left|\\Re \\left\\lbrace C_{\\ell ,m}\\right\\rbrace \\right|=0.045$ and the maximum penalty is $\\Delta _{\\max }=0.2$ dB."
],
[
"Discrete-time description",
"For a simple receiver implementation it has been proposed to sample the received signal at chip rate, i.e., every $T_c= T_{s}/M=1/B$ seconds [16].",
"In this case we have in the interval $[0,T_{s}[$ the samples $x(k T_c;a)&= \\exp \\left\\lbrace j 2 \\pi B \\frac{k T_{s}}{M} \\left[\\frac{a}{M} -\\frac{1}{2}+\\frac{B k T_{s}}{2 M^2} \\right.",
"\\right.",
"\\nonumber \\\\&\\left.\\left.",
"\\hspace{113.81102pt} - u\\left(k \\frac{T_{s}}{M}-\\frac{M-a}{B}\\right) \\right]\\right\\rbrace \\nonumber \\\\&=\\exp \\left\\lbrace j 2 \\pi k \\left[\\frac{a}{M} -\\frac{1}{2} +\\frac{k}{2\\,M} - u\\left(k \\frac{T_{s}}{M}-\\frac{M-a}{B}\\right) \\right]\\right\\rbrace \\nonumber \\\\&=\\exp \\left\\lbrace j 2 \\pi k \\left(\\frac{a}{M} -\\frac{1}{2} +\\frac{k}{2\\,M} \\right)\\right\\rbrace ,\\, k = 0, 1, \\ldots , M-1$ where the last equality is due to the fact that $2 \\pi k \\,u(\\cdot )$ is always an integer multiple of $2 \\pi $ .",
"This observation allows to avoid the modulus operation in the discrete-time description.",
"Then, from (REF ) we have immediately the following property about the orthogonality of the discrete-time waveforms.",
"Property 3 The discrete-time signals $x(k T_c;a)$ are orthogonal in the sense that $\\frac{1}{M} \\sum _{k=0}^{M-1} x(k T_c; \\ell ) x^*(k T_c; m) = \\delta _{\\ell -m}$ From (REF ) we have $\\frac{1}{M} \\sum _{k=0}^{M-1} x(k T_c; \\ell ) x^*(k T_c; m) &= \\frac{1}{M} \\sum _{k=0}^{M-1} e^{j 2 \\pi k \\left(\\frac{\\ell -m}{M} \\right)}\\\\ &=\\delta _{\\ell -m}$ As observed in [16], [18], once we have $x(k T_c;a)$ we can compute the twisted (dechirped) vector $\\tilde{\\bf x}$ with elements $\\tilde{x}_k=\\tilde{x}(k T_c;a)&= x(k T_c;a) e^{-j 2 \\pi \\frac{k^2}{2\\,M}+j \\pi k} \\,.$ Now, substituting (REF ) in (REF ), we see that $\\tilde{x}_k=e^{j 2 \\pi k \\frac{a}{M}}, \\qquad k \\in \\lbrace 0, 1, \\ldots , M-1\\rbrace $ which can be interpreted as a discrete-time complex sinusoid at frequency $a$ .",
"It follows that its Discrete Fourier Transform gives the vector ${\\bf X}=\\text{DFT}(\\tilde{\\bf x})$ with elements $X_q&=\\sum _{k=0}^{M-1} \\tilde{x}(k T_c;a) e^{-j 2 \\pi k q/M} = \\sum _{k=0}^{M-1} e^{-j 2 \\pi k (q-a)/M} \\nonumber \\\\&= M \\delta _{q-a}, \\qquad q \\in \\lbrace 0, 1, \\ldots , M-1\\rbrace \\,.$ Therefore, the DFT of the twisted signal (REF ) has only one non-zero element in the position of the modulating symbol $a$ .",
"This means that a possible way to implement a demodulator is to compute the dechirped vector (REF ), and decide based on its DFT.",
"Remark 1 One could think now that working in the discrete-time domain we can achieve the performance of orthogonal modulations.",
"However, this is not exactly the case, since, as will be shown in the next section, the bandwidth of the signal in (REF ) is larger than $B$ .",
"Therefore, filtering over a bandwidth $B$ will distort the signal, and the resulting samples will not be like in (REF ).",
"As a consequence, they will not obey the orthogonality condition in (REF ).",
"To avoid distortion, in general a bandwidth larger than $B$ should be kept before sampling.",
"In the presence of AWGN, this will produce an increase in the noise power and correlation between noise samples with respect to an orthogonal modulation.",
"However, for large $M$ the bandwidth of the signal stays approximately into a bandwidth $B$ (see next section and Table REF ), and therefore it is possible to implement a receiver based on sampling at rate $B$ , dechirping, and looking for the maximum of the DFT.",
"This is consistent with the observation that for large $M$ the modulation is approximately orthogonal (see Property REF )."
],
[
" Spectral Analysis of the LoRa modulation",
"In this section, the power spectrum of the LoRa modulation is analytically derived in closed form in terms of Fresnel functions, or through the discrete Fourier transform.",
"Then, it is shown that the modulated signal has a discrete spectrum containing a fraction $1/M$ of the overall signal power."
],
[
" Power Spectrum of LoRa Modulated Signals",
"Let us consider a source that emits a sequence of i.i.d.",
"discrete r.v.",
"${{A}}_{n}$ with probability $&\\mathbb {P}\\lbrace {{A}}_{n}=\\ell \\rbrace =\\frac{1}{M}, &\\forall \\ell \\in \\lbrace 0,1,\\cdots ,M-1\\rbrace .$ From (REF ) the modulator output can be represented by the stochastic process $I(t)=\\sum _n x(t- nT_{s};{{A}}_{n}) g_{T_{s}}(t-nT_{s})$ where the random signal $x(t;\\cdot )$ can take values in the set $\\lbrace x(t;\\ell )\\rbrace _{\\ell =0}^{M-1}$ of finite energy deterministic waveforms.",
"The PSD of the random process $I(t)$ can be written as the sum of a continuous and a discrete parts $G_{I}(f)=G_{I}^{\\textrm {c}}(f)+G_{I}^{\\textrm {d}}(f) \\,.$ The expressions of the continuous and discrete spectra in (REF ) can be found by using for the random process (REF ) the frequency domain analysis of randomly modulated signals (see e.g.",
"[21], [22]), obtaining $G_{I}^{\\textrm {c}}(f)&=\\frac{1}{T_{s}\\,M}\\left[\\sum _{\\ell =0}^{M-1} \\left|X(f;\\ell ) \\right|^{2} - \\frac{1}{M} \\left|\\sum _{\\ell =0}^{M-1} X(f;\\ell ) \\right|^{2} \\right] \\\\G_{I}^{\\textrm {d}}(f)&=\\frac{1}{T_{s}^{2}\\,M^{2}} \\sum _{n=-\\infty }^{\\infty } \\left|\\sum _{\\ell =0}^{M-1} X\\left(n\\frac{B}{M};\\ell \\right)\\right|^{2} \\, \\delta \\left(f-n\\frac{B}{M}\\right) $ where $\\lbrace X(f;\\ell )\\rbrace _{\\ell =0}^{M-1}$ are the Fourier transforms of the waveforms $\\lbrace x(t;\\ell )\\rbrace _{\\ell =0}^{M-1}$ given in (REF ).",
"The spectrum can be derived analytically by expressing the Fourier transforms $X(f;\\ell )$ in terms of Fresnel functions.",
"More precisely, we have $\\!\\!\\!X(f;\\ell )&\\!\\!=\\!\\!",
"\\int _{0}^{T_{s}}\\!\\!\\!",
"x(t;\\ell ) e^{-j 2\\pi f t} \\, dt\\!=\\!\\!\\!\\int _{0}^{\\tau _{\\ell }}\\!\\!\\!",
"e^{j 2 \\pi \\left[B t (\\frac{\\ell }{M} \\!-\\!\\frac{1}{2})+\\!",
"\\frac{B^2}{2M} t^2 \\right]} e^{-j 2\\pi f t} dt \\nonumber \\\\&+\\int _{\\tau _{\\ell }}^{T_{s}} e^{j 2 \\pi \\left[B t (\\frac{\\ell }{M} -\\frac{3}{2})+\\frac{B^2}{2\\,M}\\,t^2 \\right]}\\, e^{-j 2\\pi f t} \\, dt.",
"$ Let us define the function $W(a; b; t_1; t_2)&= \\int _{t_1}^{t_2} \\exp \\left({j 2 \\pi \\left[a\\,t+b\\, t^{2}\\right]}\\right)\\, dt $ that can be expressed in terms of the Fresnel functions as $W(a; b; t_1; t_2)&= \\frac{1}{2\\sqrt{b}}\\,e^{-j 2 \\pi \\frac{a^2}{4\\,b}} \\left[K\\left(2\\sqrt{b}\\,\\left(t_{2}+\\frac{a}{2\\,b}\\right) \\right) - \\right.",
"\\nonumber \\\\& \\left.",
"K\\left(2\\sqrt{b}\\,\\,\\left(t_{1}+\\frac{a}{2\\,b}\\right) \\right) \\right] $ where $K(x)\\triangleq C(x)+j\\,S(x)$ .",
"Then, the Fourier transform of the waveforms can be written analytically as $X(f;\\ell )&= W\\left(B \\left(\\frac{\\ell }{M} -\\frac{1}{2}\\right)-f; \\frac{B^2}{2\\,M}; 0; \\frac{M-\\ell }{B}\\right)+ \\nonumber \\\\&W\\left(B \\left(\\frac{\\ell }{M} -\\frac{3}{2}\\right)-f; \\frac{B^2}{2\\,M}; \\frac{M-\\ell }{B};\\frac{M}{B}\\right) \\,$ that used in (REF ) and () gives the signal spectrum.",
"An alternative to the use of the Fresnel functions consists in the standard Discrete Fourier Transform approach, where we take $N$ samples of $x(t;\\ell )$ over the time interval $[0,T_{s}[$ in a vector ${\\bf {x}}(\\ell )=\\lbrace x(0;\\ell ), x(\\Delta _t;\\ell ), \\cdots , x((N-1)\\Delta _t;\\ell )\\rbrace $ , with step $\\Delta _t=T_{s}/N$ .",
"Then, the vector ${\\bf {X}}(\\ell )=\\Delta _t \\, \\text{DFT}({\\bf {x}}(\\ell ))$ gives the samples with frequency step $\\Delta _f=1/T_{s}= B/M$ of the periodic repetition $\\sum _k X(f-k F;\\ell )$ , where $F=N/T_{s}= N B / M$ .",
"For sufficiently large $N$ the effect of aliasing is negligible, so that the elements of ${\\bf {X}}(\\ell )$ are essentially the samples of $X(f;\\ell )$ with step $\\Delta _f$ .",
"For the discrete spectrum this frequency step is exactly what is needed in ().",
"If a finer resolution in frequency is needed (for the continuous spectrum in (REF )) we have to zero-pad the vector ${\\bf {x}}(\\ell )$ before taking the DFT.",
"For example, if we add $(k-1)N$ zeros to ${\\bf {x}}(\\ell )$ the frequency step is $\\Delta _f=1/kT_{s}= B/kM$ ."
],
[
" Total Power of the Discrete spectrum",
"Lines in the spectrum indicates the presence of a non-zero mean value of the signal, which does not carry information.",
"The following property quantifies the power of this mean value with respect to the overall signal power.",
"Property 4 The total power of the discrete spectrum for the LoRa modulation $P_\\textrm {d} &= \\int _{-\\infty }^{\\infty } G_{I}^{\\textrm {d}}(f)\\, df = \\frac{1}{T_{s}^{2}\\,M^{2}} \\sum _{n=-\\infty }^{\\infty } \\left|\\sum _{\\ell =0}^{M-1} X\\left(n\\frac{B}{M};\\ell \\right)\\right|^{2} \\,$ is exactly a fraction $1/M$ of the overall signal power.",
"The discrete spectrum in () is due to the mean value of the signal ${\\mathbb {E}}\\left\\lbrace {I(t)}\\right\\rbrace =\\sum _n {\\mathbb {E}}\\left\\lbrace {x(t- nT_{s};{{A}}_{n})}\\right\\rbrace g_{T_{s}}(t-nT_{s}) \\,.$ This mean value is not zero, implying that there are lines in the spectrum[22], [21].",
"More precisely, since the modulation is memoryless, we have for $0 \\le t < T_{s}$ $&{\\mathbb {E}}\\left\\lbrace {x(t;A_0)}\\right\\rbrace = \\frac{1}{M} \\sum _{\\ell =0}^{M-1} x(t;\\ell )=\\frac{1}{M} \\sum _{\\ell =0}^{M-1} \\sum _{k=0}^{M-1} x(t;\\ell ) \\\\&\\phantom{=}\\times g_{T_{c}}\\left(t-k\\,T_{c}\\right)=\\frac{1}{M} \\sum _{k=0}^{M-1} g_{T_{c}}\\left(t-k\\,T_{c}\\right) \\sum _{\\ell =0}^{M-1} x(t;\\ell )\\\\&=\\frac{1}{M}\\left\\lbrace g_{T_{c}}(t)\\sum _{\\ell =0}^{M-1} x(t;\\ell )+ \\sum _{k=1}^{M-1} g_{T_{c}}\\left(t-k\\,T_{c}\\right) \\sum _{\\ell =0}^{M-1} x(t;\\ell ) \\right\\rbrace $ where $T_{c}=1/B$ is the chip rate.",
"From (REF ) we have $&{\\mathbb {E}}\\left\\lbrace {x(t;A_0)}\\right\\rbrace =\\frac{1}{M} e^{j 2 \\pi \\frac{B}{2 T_{s}}\\, t^2}\\left\\lbrace g_{T_{c}}(t)\\sum _{\\ell =0}^{M-1} e^{j 2 \\pi \\frac{B}{M}\\ell \\, t}\\right.+ \\\\&\\left.",
"\\sum _{k=1}^{M-1} g_{T_{c}}\\left(t-k\\,T_{c}\\right) \\left[\\sum _{\\ell =0}^{M-k-1}e^{j 2 \\pi \\frac{B}{M}\\ell \\, t}+ \\sum _{\\ell =M-k}^{M-1}e^{j 2 \\pi \\frac{B}{M}\\ell \\, t} e^{-j 2 \\pi B t} \\right] \\right\\rbrace \\\\&=\\frac{1}{M} e^{j 2 \\pi \\frac{B}{2 T_{s}}\\, t^2}\\left\\lbrace g_{T_{c}}(t) \\frac{1-e^{j 2 \\pi B t}}{1-e^{j 2 \\pi B t/M}}+\\sum _{k=1}^{M-1} g_{T_{c}}\\left(t-k\\,T_{c}\\right)\\right.\\\\&\\phantom{=}\\left.\\times e^{j 2 \\pi B (M-k)t/M}\\frac{e^{-j 2 \\pi B t}-1}{1-e^{j 2 \\pi B t/M}}\\right\\rbrace .$ After some manipulation we get ${\\mathbb {E}}\\left\\lbrace {x(t;A_0)}\\right\\rbrace &=\\frac{1}{M}e^{j \\frac{\\pi B t}{M} (B t - 1)} \\frac{\\sin \\left(\\pi B t\\right)}{\\sin \\left(\\pi B t /M\\right)} \\\\&\\phantom{=}\\times \\sum _{k=0}^{M-1}g_{T_{c}}\\left(t-k\\,T_{c}\\right)\\, e^{-j 2 \\pi B k t /M}.$ The absolute value of the mean is therefore $|{\\mathbb {E}}\\left\\lbrace {x(t;A_0)}\\right\\rbrace | &= \\frac{1}{M} \\left|\\frac{\\sin \\left(\\pi B t\\right)}{\\sin \\left(\\pi B t /M\\right)} \\right|, && 0 \\le t < T_{s}\\,.$ Now, recalling the following integral for $m$ integer [23] $\\int _{0}^{\\pi /2} \\left(\\frac{\\sin m x}{\\sin x} \\right)^{2}dx=\\frac{\\pi }{2}$ we get the power of the discrete spectrum as $P_\\textrm {d}=\\frac{1}{T_{s}} \\int _0^{T_{s}} |{\\mathbb {E}}\\left\\lbrace {x(t;A_0)}\\right\\rbrace |^2 \\, dt = \\frac{1}{M}\\,.$ Therefore, there are lines in the spectrum of the LoRa modulation, and the power of this discrete spectrum is a fraction $1/M$ of the overall power."
],
[
"Numerical Results",
"We first show in Fig.",
"REF the two-sided power spectrum of the complex envelope for LoRa modulated signals as a function on the normalized frequency $f/B$ , with various spreading factors, i.e., $\\text{SF}\\in \\lbrace 3,7,10,12\\rbrace $ .",
"Since $G_I(-f)=G_I(f)$ we just show $G_I(f)$ for $f\\ge 0$ .",
"Figure: The continuous and discrete spectrum of the complex envelope for LoRa modulation, M=2 SF M=2^\\text{SF}, with SF∈{3,7,10,12}\\text{SF}\\in \\lbrace 3,7,10,12\\rbrace .In the figure we report both the normalized power spectral density, $10\\,\\log _{10} G^{c}_{I}(f)\\, B$ , and the discrete part of the spectrum.",
"For the latter we report the power $\\left|\\sum _{\\ell =0}^{M-1} X\\left(n {B}/{M};\\ell \\right)\\right|^{2} / {T_{s}^{2}\\,M^{2}}$ at frequency $n B/M$ , as given in ().",
"The sum of the power of all lines in the discrete spectrum is equal to $1/M$ , as proved in Property REF .",
"For example, with $\\text{SF}=3$ we have $M=8$ and thus $1/M=12.5\\%$ of the signal power is contained in the discrete spectrum.",
"We can see that the power spectrum becomes more compact for increasing $M$ , so that most of the power for the complex envelope is contained between $-B/2$ and $B/2$ , or, in other words, that the modulated signal bandwidth is close to $B$ for large $M$ .",
"To better quantify this effect, we report in Table REF the bandwidth $B_{99}$ centered on $f_0$ containing $99\\%$ of the power for different spreading factors.",
"It can be seen that, while for $M \\ge 2^7$ almost all of the signal is contained in a bandwidth $B$ , considering just a bandwidth $B$ for smaller spreading factors will leave out a part of the signal, therefore distorting the signal.",
"Moreover, as noted in Section and Section , the spectral efficiency, the maximum real cross-correlation, and the power of the discrete spectrum decrease for increasing $M$ .",
"Figure: The power spectrum of the complex envelope for LoRa modulation using the analytical expressions and the experimental data, for M=2 SF M=2^\\text{SF}, SF∈{7,10}\\text{SF}\\in \\lbrace 7,10\\rbrace , B=125B=125 kHz, Δf=B/256\\Delta f=B/256, and P s =27P_{\\text{s}}=27 dBm.In Fig.",
"REF , we compare the derived analytical power spectrum with that obtained from the IQ samples of a commercially available LoRa transceiver [24].",
"More precisely, IQ samples are provided for LoRa modulated waveforms, which have been created with a randomly generated payload of 16 bytes.",
"The waveforms are obtained for $B=125~$ kHz with sample rate $f_{s}=4\\,B$ [24].",
"The frequency range of interest is divided into several bins with width $\\Delta f=B/256$ , and the power within each bin is computed either analytically via (REF ) and (), or through spectral estimation by implementing the Welch's method on the experimental data[25].",
"It is noticed that the estimated spectrum agrees well with the analytical expression.",
"We can also observe that the tail of the estimated spectrum is slightly higher than the analytical; this is because the experimental samples have been taken at $f_{s}=4\\,B$ , not large enough to completely eliminate frequency aliasing.",
"Figure: The one-sided power spectrum for LoRa modulated passband signals using the analytical expressions, compared with the mask from the ETSI regulation in the G1 sub-band, for M=2 7 M=2^7, Δf=1\\Delta f=1 kHz, and P s =14P_{\\text{s}}=14 dBm.a) One channel with center frequency 868.3868.3 MHz for B=250B=250 kHz.b) Three channels with center frequencies 868.1868.1 MHz, 868.3868.3 MHz, and 868.5868.5 MHz for B=125B=125 kHz.Finally, we investigate the LoRa spectrum along with the ETSI regulations for out-of-band emissions [26].",
"Since LoRa is a chirp spread spectrum technique, it is governed by the regulations for ISM bands that support wideband modulation [26].",
"For example, we consider the G1 sub-band spanning from 868 MHz to $868.6$ MHz[26].",
"There are two possibilities for using LoRa in this sub-band: using a single channel with center frequency $868.3$ MHz for $B=250$ kHz; using three channels with center frequencies $868.1$ , $868.3$ , and $868.5$ MHz for $B=125$ kHz.",
"In Fig.",
"REF we report, for the two cases above, the one-sided power spectrum calculated analytically with bin width (i.e., resolution bandwidth) $\\Delta f=1$ kHz, and $P_{\\text{s}}=14$ dBm, i.e., the maximum allowed transmission power.",
"The spectrum is compared with the spectral mask for the G1 sub-band.",
"It can be noticed that the spectrum meets the regulations of the maximum power limits for adjacent band emissions at the G1 sub-band.",
"The same method can be used to examine the LoRa compliance for various ISM bands, spreading factors, and bandwidths, according to other regional regulations."
],
[
"Conclusions",
"In this paper we investigated the spectral characteristics of the LoRa $M$ -ary modulation, deriving the analytical expression of the spectrum, and comparing it with experimental data and with the spectral limit masks for the ISM bands.",
"We found that there are lines in the spectrum, containing a fraction $1/M$ of the overall power, and that the occupied bandwidth is in general larger than the deviation $B$ .",
"We also derived the waveform cross-correlation function, proving that the LoRa waveforms can be considered orthogonal only for asymptotically large $M$ .",
"-6.8cm"
]
] | 1906.04256 |
[
[
"Fine-grained Event Categorization with Heterogeneous Graph Convolutional\n Networks"
],
[
"Abstract Events are happening in real-world and real-time, which can be planned and organized occasions involving multiple people and objects.",
"Social media platforms publish a lot of text messages containing public events with comprehensive topics.",
"However, mining social events is challenging due to the heterogeneous event elements in texts and explicit and implicit social network structures.",
"In this paper, we design an event meta-schema to characterize the semantic relatedness of social events and build an event-based heterogeneous information network (HIN) integrating information from external knowledge base, and propose a novel Pair-wise Popularity Graph Convolutional Network (PP-GCN) based fine-grained social event categorization model.",
"We propose a Knowledgeable meta-paths Instances based social Event Similarity (KIES) between events and build a weighted adjacent matrix as input to the PP-GCN model.",
"Comprehensive experiments on real data collections are conducted to compare various social event detection and clustering tasks.",
"Experimental results demonstrate that our proposed framework outperforms other alternative social event categorization techniques."
],
[
"Introduction",
"Events are happening in real-world and real-time, which can be planned and organized occasions involving multiple people and objects, such as a social gathering, celebrity activities or a sports competition in some specific location at a particular time.",
"Nowadays, social media platforms have become major sources for publicizing events.",
"Events announced on social media usually attract comments and reposts with opinions and emotions, and such content can reflect public opinion about many social, political, economic issues, etc.",
"Mining of social media posts, such as fine-grained social event categorization, will benefit a lot of real applications, such as information organization, predictive analysis, disaster risk analysis, and others [4], [1], [2].",
"In general, fine-grained social event categorization focus on event detection and event clustering.",
"The tasks of fine-grained social event categorization are more challenging than traditional text mining or social network mining, since social event is a combination of social network and the information flows (in terms of short messages) over it.",
"On the one hand, modeling social events is very complicated and ambiguous.",
"Social events are described in short texts and usually contain different types of entities, such as person, location, organization, number, time, etc [2], [14], [35].",
"Moreover, events are commented or retweeted by social network users.",
"Thus, modeling social event needs to consider heterogeneous elements as well as explicit and implicit social network structures within social posts.",
"On the other hand, models of fine-grained event categorization often have bottlenecks in which the number of the categories is large and the number of samples per class is small.",
"Thus, fine-grained event categorization needs to address the accuracy of the developed algorithms.",
"Currently, fine-grained text classification is more difficult and lacks related research work than the fine-grained object recognition in other fields such as computer vision [37].",
"A handful of studies [26], [9], [6], [27] have investigated leveraging homogeneous graphs or manually defined frames for social event modeling and extracting.",
"The first line of thought is to treat social event as homogeneous words/elements co-occurrence graph [9], [1], [3], [20].",
"Typically, they construct a homogeneous words/elements co-occurrence graph, and then consider different scales of abnormally connected subgraph structures (under different names such as k-clique, motifs or graphlets) as the social events.",
"Despite the compelling results achieved by these studies, their categorization accuracies remain unsatisfactory for building reliable and open domain event detection and clustering systems in practice.",
"The second line of thought is to use manually defined frames-based event definitions applying the well-defined techniques for extracting social event frames from news [15], [14].",
"The frame-based event extraction can extract entities and their relationships, but uses only a limited number of event types, such as earthquake disaster, stock market, venues, politics, etc.",
"Moreover, it uses complicated machine learning models, usually a pipeline of them, to incorporate different levels of annotation and features.",
"Social media events can be regarded as a co-occurrence of event elements including themes, dates, locations, people, organizations, keywords and social behavior participants.",
"The simplest way to monitor social media events is to represent events as bags-of-words, but it will be more semantically meaningful if we can annotate words and multi-word-expressions as entities with types.",
"For example, in the tweet \"China Seismological Network: The earthquake struck at 21:19:46 China Standard Time on 8 August 2017 in Zhangzha Town in Jiuzhaigou County with magnitude 7.0\", there are multiple event elements: Time: 21:19:46; Date: August 8, 2017; Timezone: China Standard Time; Town: Zhangzha; County: Jiuzhaigou; Nation: China; Magnitude: 7.0; Poster: China Seismological Network.",
"Obviously, the above event's elements are of different types.",
"Moreover, in addition to intuitive co-occurrence relationship, after extracting entities, we can make use of external knowledge base [5], [34] to complement more relationships between entities, such as “located-in” relationships with other locations, “attribute-of” relationships with magnitude and earthquake, etc.",
"Thus, a message mentioning an event can be related to its keywords, entities (and their relations), topics, etc.",
"Furthermore, the social network users posting messages are also connected with different relationships, such as following/followed and retweeting.",
"Thus we can model social media events as HIN [29].",
"In this paper, we first present event instance (shown in short text message) as hyper-edge in an HIN, where all the keywords, entities, topics and social users can be connected by this hyper-edge, and define an event meta-schema to characterize the semantic relatedness of social event instances and build event-based HIN.",
"In order to enrich the HIN, we extract some information as a complement of the relationships based on the external knowledge base and algorithms.",
"Based on the event HIN, we define a weighted Knowledgeable meta-paths Instances based Event Similarity measure, namely KIES, from semantically meaningful meta-paths.",
"In order to accurately measure the weights between meta-paths and perform fine-grained event detection, we then design a novel Pairwise Popularity Graph Convolutional Network model, namely PP-GCN, to learn the representation of each event instance.",
"Finally, under the HINs-based event modeling, we present a KIES-measure based fine-grained event clustering.",
"Compared to traditional methods, the proposed models have several advantages: (1) By modeling social events based on a HIN, the proposed framework can integrate event elements, such as keywords, topic, entities, social users and their relations, in a semantically meaningful way, and can also calculate the similarity between any two event instances.",
"(2) By modeling pairwise popularity graph convolutional network, the model achieves state-of-the-art results and avoids overfitting in fine-grained event detection tasks.",
"(3) The proposed KIES with learned weights between meta-paths by the PP-GCN can boost the performance of fine-grained social events clustering compared to existing state-of-the-art baselines methods.",
"The code of this work is publicly available at https://github.com/RingBDStack/PPGCN."
],
[
"Heterogeneous Event Modeling",
"In this section, we define the problem of modeling social events in heterogeneous information network (HIN) and introduce several related concepts and necessary notations."
],
[
"Event Modeling in HIN",
"The definition and characterization of “social event” have received substantial attention across academic fields, from language [22] to cognitive psychology [36].",
"A social event generally refers to influential facts that appear on social networks and occur in the real world, including creators (posters), named entities such as participants, organizations, festival, specific times, places, currency, address, etc., and other elements such as keywords and topics.",
"We name the above elements as event-oriented elements.",
"However, extracting the event-oriented elements from the original social text message with NLP toolshttps://github.com/stanfordnlp/CoreNLP,https://github.com/NLPIR-team/NLPIR is still a prior processing work.",
"Even within most of the events, there are some relationships between event-oriented elements, such as relationships between entities, relationships between keywords, relationships between topics, explicit and implicit relationships between social users, and so on.",
"We name the above relationships as event-elements relationships.",
"Figure: Meta-schema of event-based HINWe use the manually organized synonymshttps://github.com/huyingxi/Synonyms to add synonym relationship among keywords in the event-based HIN.",
"For hierarchical topic structures and the affiliation relationship between keywords and topics in the event-based HIN, we employ the hierarchical latent Dirichlet allocation technologies [11], [8] based on the existing toolboxhttps://github.com/joewandy/hlda (with about 30 most probable words for each topic).",
"In order to build the relationship between entities in the event-based HIN, we consider both accuracy and efficiency, and tackle the problem by following three-steps.",
"First, we retrieve the same entity candidate from knowledge base, such as the Chinses CN-DBpedia [34].",
"Second, we use word embeddings [21] based Word Mover's Distance technology [17] to measure the similarity between context of entity in the social text and description of entity in the candidate, and choose the entity from the candidate with highest similarity.",
"Third, we query the relationship between aligned entities in the knowledge base as the final relationship of entities in event-elements relationships.",
"In order to establish the relationship between entities and keywords in the event-based HIN, we extract the keywords in the relevant description of each entity in the knowledge base and use this affiliation as the relationship between the entity and the keywords.",
"For the relationship between social users, we consider users with a large number of friends, and store the relationship between users in advance.",
"After extracting the above event-oriented elements and event-elements relationships from event instances, we build an event-based HIN, as shown in Figure REF .",
"The social event can be regarded as a co-occurrence of event-oriented elements, and the event-elements relationships are conducive to explaining the relationship between various elements.",
"Thus, an event instance can be treated as a subgraph of the whole HIN.",
"One particular advantage of the HIN is that meta-paths defined over types (e.g., a typical meta-path “event-entity-event” represents the event similarity based on overlapped entities between two event instances) can reflect semantically meaningful information about similarities, and thus can naturally provide explainable results for event modeling."
],
[
"Preliminaries",
"We introduce some basic definitions from previous works [30], [29], and give some event-HIN examples.",
"Definition 2.1 A heterogeneous information network (HIN) is a graph $G = (V,E)$ with an entity type mapping $\\phi : V\\rightarrow A$ and a relation type mapping $\\psi : E\\rightarrow R$ , where $V$ denotes the entity set, $E$ denotes the link set, $R$ denotes the relation type set and $A$ denotes the entity type set.",
"The number of entity types $|A|>1$ or the number of relation types $|R|>1$ .",
"For example, Figure REF shows an example of two event instances connected with different types of entities, keywords, topics, social users and relationships.",
"After giving a complex HIN for event modeling, it is necessary to provide its meta level (i.e., schema-level) description for better understanding.",
"Definition 2.2 Given an HIN $G = (V,E)$ with the entity mapping $\\phi : V\\rightarrow A$ and the relation type mapping $\\psi : E\\rightarrow R$ , the meta-schema (or network schema) for network $G$ , denoted as $T_{G} = (A,R)$ , is a graph with nodes as entity types from $A$ and edges as relation types from $R$ .",
"For example, Figure REF shows an example of the HIN meta-schema characterizing events on social messages.",
"Another important concept is the meta-path which systematically defines relationships between entities at the schema level.",
"Definition 2.3 A meta-path P is a path defined on the graph of network schema $T_{G} = (A,R)$ of the form $A_{I}\\stackrel{R_{1}}{\\longrightarrow }A_{2}\\stackrel{R_{2}}{\\longrightarrow }A_{3}\\cdots A_{L}\\stackrel{R_{L}}{\\longrightarrow }A_{L+1}$ which defines a composite relation $R=R_{1}\\cdot R_{2}\\cdot \\cdots \\cdot R_{L}$ between objects $A_{1},A_{2},A_{3}\\cdots A_{L+1}$ , where $\\cdot $ denotes relation composition operator, and $L+1$ is the length of $P$ .",
"For simplicity, we use object types connected by $\\rightarrow $ to denote the meta-path when there are no multiple relations between a pair of types: $P = (A_{1} - A_{2} - \\cdots - A_{L+1})$ .",
"We say that a meta-path instance $p = (v_{1} - v_{2} - \\cdots - v_{L+1})$ between $v_{1}$ and $v_{L+1}$ in network $G$ follows the meta-path $P$ , if $\\forall l$ , $\\phi (v_{l}) = A_{l}$ and each edge $e_{l} = <v_{l}, v_{l+1}>$ belongs to each relation type $R_{l}\\in P$ .",
"We call these paths as path instances of $P$ , denoted as $p\\in P$ .",
"$R_{l}^{-1}$ represents the reverse order of relation $R_{l}$ .",
"We will introduce more semantically meaningful meta-paths that describe event relations in next section."
],
[
"The Proposed Model",
"In this section, we introduce definitions about knowledgeable meta-paths instances based event similarity measure, and present the technical details about Pairwise Popularity GCN."
],
[
"Event Similarity Measure",
"Before definite the social event similarity, we first present the definition of CouP as following, Definition 3.1 CouP: Given a meta-path $P = (A_{1} - A_{2} \\cdots A_{L+1})$ , CouP is a function of the count of meta-path instances such that $CouP_{P}(v_{i},v_{j}) = M_{P}(v_{i},v_{j})$ where $M_{P}=W_{A_{1}A_{2}}\\cdot W_{A_{2}A_{3}}\\cdots W_{A_{L}A_{L+1}}$ and $W_{A_{k}A_{k+1}}$ is the adjacency matrix between types $A_{k}$ and $A_{k+1}$ in the meta-path $P$ .",
"For example, for event instance similarity based on event-oriented elements and event-element relationships, the composite relation of two event instances containing the same event element and co-occurrence relationship can be described as \"event Instance - Element - event Instance (IEI)\" for simplicity.",
"This meta-path simply gives us $M_{IEI} = W_{IE}W_{EI}^{T}$ , which is the dot product between event instances, where $W_{EI}$ is the event Instance-Element co-occurrence matrix.",
"The similarity based on this meta-path instances is accurate because different elements and lengths are considered.",
"The more meta-paths enumerated by the meta-schema, the higher accuracy of the similarity metric is.",
"We can give more event related meta-paths over different lengths, e.g., $P_1$ : Event-(posted by)-Social user-(post)-Event, $P_2$ : Event-(having)-Washington DC-(capital of)-United States-(contained by)-Event, $P_3$ : Event-(belong to)-Politician-(relevant)-president-(member of)-Ruling Party-(contained by)-Event, etc.",
"$P_1$ means two event instances are similar if they are posted by the same social user.",
"$P_2$ means two event instances are similar if they mention Washington DC and the United States, respectively, where Washington DC is the capital of the United States.",
"$P_3$ means two event instances are similar if they can be associated by a chain of three event elements with meaningful relationships.",
"Note that the meta-path does not need to satisfy symmetry.",
"Here, we enumerate 22 symmetric meta-paths in the meta-schema of event-based HIN.",
"However, if the counts are not normalized for different meta-paths, it is difficult to compare over different meta-path-based similarities.",
"Then, similar to the HIN-based document similarity [33], we also define our knowledgeable meta-paths instances based social event similarity measure, namely KIES.",
"Intuitively, if two event instances are more strongly connected by the important (i.e., highly weighted) meta-paths, they tend to be more similar.",
"Definition 3.2 KIES: a knowledgeable meta-paths instances based social event similarity.",
"Given a collection of meaningful meta-paths, denoted as $\\textbf {P}=\\lbrace P_{m}\\rbrace _{m=1}^{M^{\\prime }}$ , the KIES between two event instances $e_i$ and $e_j$ is defined as: $\\small KIES(e_i,e_j) = \\sum _{m=1}^{M^{\\prime }}\\omega _{m}\\frac{2\\times CouP_{P_m}(e_{i},e_{j})}{CouP_{P_m}(e_{i},e_{i})+CouP_{P_m}(e_{j},e_{j})},$ where $CouP_{P_m}(e_{i},e_{j})$ is a count of meta-path $P_m$ between event instances $e_i$ and $e_j$ , $CouP_{P_m}(e_{i},e_{i})$ is that between event instances $e_i$ and $e_i$ , and $CouP_{P_m}(e_{j},e_{j})$ is that between event instances $e_j$ and $e_j$ .",
"We use a parameter vector $\\vec{\\omega } = [\\omega _{1},\\omega _{2},\\dots ,\\omega _{M^{\\prime }}]$ to denote the meta-path weights, where $\\omega _{m}$ is the weight of meta-path $P_{m}$ .",
"$KIES(e_i,e_j)$ is defined in two parts: (1) the semantic overlap in the numerator, which is defined by the number of meta-paths between event instances $e_i$ and $e_j$ ; and (2) the semantic broadness in the denominator, which is defined by the number of total meta-paths between themselves.",
"Therefore, we can give a KIES distance with weights for any two event instances."
],
[
"Pairwise Popularity GCN Model",
"Next, we show how to implement fine-grained event detection on social message texts through the pairwise popularity GCN model (PP-GCN), and learn the weights $\\vec{\\omega }$ for meta-paths, to overcome the problems of a large number of categories and a small number of samples per class.",
"After computing the distance of any two event instances by the KIES, we can construct a $N\\times N$ weighed adjacent matrix $A$ for manually annotated social event instances, where $N$ is the number of event instances and $A_{ij} = A_{ji} = KIES(e_i,e_j)$ .",
"Then, we train the Doc2vec [18] representation as generalized event instance feature.",
"So, we can construct a $N\\times d$ feature matrix $X$ , where $d$ is the dimension of event instance feature.",
"Obviously, so far, we can use the popular GCN [16] architecture to learn discriminating event representation based on the interactions among event instances and generalized event instance features in node classification task.",
"The input to the GCN model includes the $A$ and $X$ matrices.",
"Here, one class represents one social event class.",
"In order to construct preliminary GCN model, we utilize the popular multi-layer GCN with the following layer-wise propagation rule [16]: $H^{(l+1)} = \\sigma (\\widetilde{D}^{-\\frac{1}{2}}\\widetilde{A}\\widetilde{D}^{-\\frac{1}{2}}H^{(l)}W^{(l)}),$ where $\\widetilde{A} = A + I_{N}$ , $\\widetilde{D}$ is diagonal matrix such that $\\widetilde{D}_{ii}=\\sum _{j}\\widetilde{A}_{ij}$ are the adjacency matrix, $I_{N}$ is the identity matrix, $W$ is the parameter matrix, and $l$ is the number of layers.",
"Let $Z$ be an output $N\\times F$ feature matrix, where $F$ is the dimension of output representation per event instance.",
"The input layer to the GCN is $H^{(0)} = X, X\\in R^{N\\times d}$ , which contains original event instance feature, $H^{(l)} =Z$ , and $Z$ is graph-level output.",
"And $\\sigma $ denotes an activation function such as Sigmoid or ReLU.",
"Figure: An overview of the proposed Pairwise Popularity Graph Convolutional Network (PP-GCN).However, the real-world social events naturally have two problems of sparsity: the small number of event instances for each classification and a large number of categories.",
"So, we sample event instances pair and judge whether the pair belongs to one event to train a pairwise GCN model.",
"As shown in Figure REF , we present the proposed PP-GCN model.",
"Before explaining the PP-GCN model, we show how to implement a pairwise sampling to generate training samples.",
"We assume that if a pair of event instances $e_i$ and $e_j$ belongs to the same event classification, we name the pair $e_i$ and $e_j$ as a positive-pair sample.",
"If a pair of event instances $e_i$ and $e_j$ belongs to two different events classification, we name the pair $e_i$ and $e_j$ as a negative-pair sample.",
"As shown in Figure REF , if the pair is a positive-pair sample, we represent its by two red lines; if the pair is negative-pair sample, we use both gray line and blue line to represent it.",
"After explaining the training samples, we first randomly select $R$ (i.e., 1000) event instances as a preliminary set, then randomly select two event instances for each event instance in the set to form one positive-pair sample and one negative-pair sample, and finally we can construct a $2R$ event instance pairs set from training samples.",
"Here, both the positive-pair and negative-pair samples are equal to $R$ .",
"Second, we randomly sample the $B$ (i.e., 64) samples from the $2R$ (i.e., 2000) event instance pairs set to form a batch to forward propagation of our proposed model.",
"Third, the second step is cycled $E$ (i.e., 32) times to form an epoch.",
"For next epoch, we loop through the above three steps.",
"However, the above pairwise sampling based GCN model can not guarantee that the model avoids over-fitting during training.",
"Suppose that any event classification has an average of $r$ event instances, the probability that any event instance selected into a positive-pair sample is $\\frac{1}{r}$ , and the probability of being selected into a negative-pair sample is about $\\frac{1}{N-r}$ .",
"We note that $\\frac{1}{r} \\gg \\frac{1}{N-r}$ in general.",
"Obviously, the negative-pair samples have more diversity than the positive-pair samples.",
"[24] has observed the phenomenon that the connected probability of a sample determines the popularity of it.",
"Inspired by these observations, we assume that in feature representation learning, the modulus of the learned feature vector is larger if the popularity is greater.",
"So, the two modulus of learned event instances feature vectors of positive-pair will be closer.",
"For discriminate feature learning of our GCN model, we utilize the popularity of the output event instance feature vector in $Z$ to distinguish different classes.",
"As shown in the Figure REF , for any two learned event instance vectors $V_{e_{i}}$ and $V_{e_{j}}$ that satisfy $|V_{e_{i}}|\\ge |V_{e_{j}}|$ , we employ a ratio of modulus $x=\\frac{|V_{e_{i}}|}{|V_{e_{j}}|}$ as the input of a nonlinear mapping function $f(x) = - log(x-1+c)$ , where the coefficient $c$ is 0.01 to avoid no upper bound output.",
"We assume that the ratio of modulus of positive-pair will belong to $[1, 2)$ .",
"So, the nonlinear mapping function $f(x)$ can map the above ratio $x$ from [1, 2) to (0, 2], and [2, +$\\infty $ ) to (-$\\infty $ , 0).",
"Next, we add a Sigmoid function to map the output of the nonlinear mapping layer to 0 or 1 by a threshold 0.5.",
"As shown in the Figure REF , one positive-pair or negative-pair input sample can only be paired with an output of 0 or 1.",
"For one batch (64 pairs) samples, our model can generate one batch size ($1\\times 64$ ) of one-zero output vector.",
"So, we can use a cross entropy function as our model's loss function, and employ the popular stochastic gradient descent (SGD) method to iterate all parameters.",
"The learned weights $\\vec{\\omega }$ will be used to measure similarity for any two social event instances.",
"To verify the avoidance of over-fitting ability of our model, we can perform over 7000 epochs, and observe that the evaluation criteria of the model changes over time in Section .",
"For the testing of any event instance $t$ from the test set of the original $N$ samples, we first assume that there are a total of $C$ event classes in the original $N$ event instances.",
"Secondly, we calculate the ratio of the modulus of the representation vectors for $t$ and the remaining $N-1$ samples, respectively.",
"Then, for each event class, we can get a probability that event instance $t$ most likely belongs to it.",
"If all of the ratios of modulus are 0, the sample itself is a separate event class.",
"Finally, we select the event class with the highest probability as the test output for the event instance $t$ .",
"Note the fact that the event category of the test set may not be included in the event category of the training set.",
"After the previous analysis, we can calculate a similarity for any two event instances under the event-HIN and the weights $\\vec{\\omega }$ .",
"Since our meta-paths have better interpretability, we also can implement a semi-supervised and fine-grained event clustering based on the learned weights of meta-paths and distance-based clustering models."
],
[
"Experiments",
"In this section, we evaluate the proposed PP-GCN model and similarity measure KIES using real surveillance data collected in two enterprise systems.",
"Table: Description of evaluation datasets."
],
[
"Datasets and Settings",
"We select two independent social media platforms, news APP from Tencent (a popular APP for young people) and Sina Weibo (a hybrid of Twitter and Facebook, the Twitter of China and Chinese Social Media), to collect datasets.",
"Each event instance is a non-repeating social message text.",
"One event is a set of event instances that contain semantically identical information revolving around a real world incident.",
"An event always has a specific time of occurrence.",
"It may involve a group of social users, organizations, participating persons, one or several locations, other types of entities, keywords, topics, etc.",
"In our work, social events cover a wide variety of types, including a large number of events that occur in the real world and spread on social networks, such as earthquakes, national policies, economic crises, and so on.",
"Each event class refers to a unique event.",
"For example, social media’s tweet about Tiger Woods winning the 2019 Masters of Golf is an influential event in the real world and unlike Patrick Reid’s 2018 Masters of Golf.",
"These are two different events that happen in the real world and belong to different event categories.",
"The event labels for the Weibo and Tencent datasets are labeled by the outsourcing companies.",
"Both entities and keywords have been manually extracted for the Tencent dataset.",
"Note that the anonymized social users and their friend relationships involved in these two datasets are granted by the two companies for scientific research purposes only.",
"For both of the two datasets, we use 60% of samples as training set, 20% of samples as development set and the remaining 20% of as test set.",
"The statistics of the two datasets is shown in Table REF .",
"We can see that the total number of class is large and the number of samples in per class is small.",
"We conduct the experiments on event detection and event clustering on these two datasets.",
"The operating system and software platforms are Ubuntu 5.4.0, Tensorflow-gpu (1.4.0) and Python 2.7.",
"The metrics used to evaluate the performance of event detection are the accuracy and F1 score.",
"The metric used to evaluate the performance of event clustering is the normalized mutual information (NMI)."
],
[
"Baseline Methods",
"Since the work of fine-grained social event categorization is relatively small, we briefly describe the baseline methods of text matching and text distance.",
"For all the baselines, we use the implementations or open source codes of these models released by authors and other researchers, and report the best performance of the results.",
"Support Vector Machine with TF-IDF feature (SVM): Support Vector Machine with pair document TF-IDF features is the most classical approach for classification task.",
"We extract the TF-IDF features for social messages, and then use the SVM classifier to implement the multi-class event classification.",
"Convolutional Matching Architecture-I (ARC-I) [12]: It encodes text pairs by CNNs, and compares the encoded representations of each text with a MLP.",
"Convolutional Matching Architecture-II (ARC-II) [12]: It builds directly on the interaction space between two texts, and models all the possible combinations of them with 1-D and 2D convolutions.",
"Match by Local and Distributed Representations (DUET) [23]: It matches two texts using both local representation and learned distributed representation.",
"Multiple Positional Semantic Matching (MV-LSTM) [31]: It matches two texts with multiple positional text representations, and aggregates interactions between different positional representations.",
"Convolutional Deep Structured Semantic Models (C-DSSM) [28]: It learns low-dimensional semantic vectors for input text by CNNs.",
"Deep Structured Semantic Model (DSSM) [13]: It utilizes a deep neural network to map high-dimensional sparse features into low-dimensional features, and calculates the semantic similarity of the document pair.",
"Siamese Encoded Graph Convolutional Network (SE-GCN) [19]: It learns vertex representations through a Siamese neural network and aggregates the vertex features though GCNs to generate the document matching.",
"Term Frequency-Inverse Document Frequency (TF-IDF): It uses the bag-of-words representation divided by each word’s document frequency.",
"Latent Dirichlet Allocation (LDA) [7]: is a celebrated generative model for text documents that learns representations for documents as distributions over word topics.",
"Marginalized Stacked Denoising Autoencoder (mSDA) [10]: It is a representation learned from stacked denoting autoencoders.",
"Componential Counting Grid (CCG) [25]: It is a generative model that models documents as a mixture of word distributions and LDA.",
"Word Move Distance (WMD) [17]: It measures the dissimilarity between two documents as the minimum amount of distance that words of one document need to travel to reach words of another document.",
"Knowledge-driven document similarity measure (KnowSim) [32]: It's also a meta-paths instances based document similarity, and hasn't considered the impacts of social users.",
"The weights of meta-paths are estimated by the Laplacian scores of documents."
],
[
"Performance Analysis",
"Table REF shows the accuracy and F1-score of different algorithms on the task of event detection in Tencent and Weibo datasets.",
"Overall, the proposed PP-GCN model consistently and significantly outperforms all baselines in terms of accuracy and F1.",
"In the Tencent dataset, PP-GCN achieves 13%–56% improvements in terms of accuracy and F1 over all baselines.",
"In the Weibo dataset, PP-GCN achieves 10%–33% improvements in terms of accuracy and F1 over all baselines.",
"Table: Accuracy and F1 results of event detection.The improvements can be attributed to the three characteristics of proposed models.",
"First, the knowledgeable HIN is better modeling social events than traditional text modeling methods, such as bag-of-words (SVM), N-gram (ARC-I, ARC-II and C-DSSM) and sequence-of-words (MV-LSTM).",
"Our PP-GCN has improved overall by more than 10% in the event detection over the SE-GCN model incorporating structural and conceptual semantics.",
"Second, the combination of KIES based weighted adjacent matrix and Doc2Vec is better for fine-grained event instance representation learning than for feature extraction on text pairs, such as DUET.",
"Third, the classifier based on the ratio of modulus of generated representations of event instances is better than the traditional pairwise distances.",
"Here, we replace the regression module of SE-GCN model by our proposed popularity based classifier, named by PP-SE-GCN, and the performances can be improved 3%-5% in Tencent and Weibo.",
"The 10%-14% improvements from the SE-GCN to the PP-GCN demonstrate the advantages of knowledgeable HIN modeling and the pairwise popularity based feature learning framework.",
"Figure: Illustration of the Accuracy for PP-GCN and PA-GCN.Table: NMI results of event clustering.Furthermore, our PP-GCN model can avoid over-fitting in training.",
"We replace our classifier in PP-GCN by the angle of generated event instances feature vectors based classifier, namely PA-GCN.",
"In Figure REF , we visualize the test accuracies of the PP-GAN and PA-GCN in Tencent and Weibo in 7000 epochs.",
"From Figure REF , we observe that the overall trend of the accuracies of the PP-GCN model is continuously increasing, but the accuracies of the angle-based PA-GCN model have periodic fluctuations.",
"Essentially, the ratio of the modulus between vectors is more stable than the angle in iterations.",
"Compared to the angle-based classifier, the popularity-based classifier has better ability to learn discriminating and stable event instance feature and prevent overfitting.",
"One advantage of the proposed PP-GCN compared to other methods is that the weights $\\vec{\\omega }$ between the meta-paths can be learned according to the event detection task.",
"Due to the interpretability of the meta-path and similarity measure KIES, the learned weights $\\vec{\\omega }$ can be utilized in other applications.",
"Here, we make use of different similarity measures including KIES and other methods discussed in Section REF , and leverage the popular k-means algorithm to cluster the events.",
"For the KIES distance metric, we use the two weights $\\vec{\\omega }$ learned in the event detection tasks of Tencent and Weibo datasets, and then calculate the KIES distances between event instances by Eq.",
"REF to implement a semi-supervised fine-grained event clustering.",
"Note that the test data did not participate in the training when learned the meta-path weights $\\vec{\\omega }$ in PP-GCN.",
"As shown in Table REF , our proposed similarity measure KIES achieves the best performances on the two clusters tasks in terms of NMI.",
"Moreover, among the baselines, the WMD, mSDA and CGG measures have been verified to achieve state-of-the-art effects in text similarity in [17].",
"Compared to other similarity measures, our KIES based k-means method achieves 6%–24% improvements in terms of NMI.",
"We even implement a meta-path weights transfer experiment between Tencent dataset and Weibo dataset.",
"We see that the performance of the Weibo dataset are improved more than 2% when employing the weights of the Tencent dataset to the Weibo by the KIES(T) based k-means method, but not the other way around.",
"We believe the reason is that Tencent dataset has more applicable meta-path weights by training with manually labeled entities and keywords.",
"Based on the learned weights and the interpretable distance metric KIES, we have achieved the best performance of semi-supervised event clustering."
],
[
"Conclusion",
"In this paper, we propose a knowledgeable HIN based social event modeling framework, and design a novel pairwise popularity GCN model to learn both meta-paths weights and discriminant event instance representation, and achieves fine-grained social event categorization with state-of-the-art performances.",
"By using the proposed PP-GCN model, we are able to overcome the problems of large category size and sparse small number of samples per class and preventing overfitting in our tasks.",
"Experimental results show that our PP-GCN and KIES similarity measure can significantly outperform state-of-the-art baselines methods on two real-world social datasets.",
"In the future, we plan to study the interpretability of the different importance of meta-paths, and extend our framework to other complex parameter leaning and applications."
],
[
"Acknowledgements",
"The corresponding author is Jianxin Li.",
"This work is supported by NSFC program (No.61872022, No.61421003) and SKLSDE-2018ZX-16.",
"Yangqiu Song is supported by the Early Career Scheme (ECS, No.",
"26206717) from Research Grants Council in Hong Kong.",
"Philip S. Yu is supported by NSF through grants IIS-1526499, IIS-1763325, and CNS-1626432, and NSFC No.61672313."
]
] | 1906.04580 |
[
[
"Dynamical System Analysis of Brane Induced Gravity with Tachyon Field"
],
[
"Abstract In this manuscript we use the dynamical system approach to study the linear dynamics of a normal DGP brane-world model with a tachyon field as the dark energy component.",
"Our focus is on a Gaussian tachyonic potential in which the parameter $\\lambda=-V_\\phi/V^{3/2}$, goes to infinity.",
"One of the most important results of this study is that we find critical submanifolds which indicate the effect of extra dimension."
],
[
"Introduction",
"According to the big bang theory which is the prevailing cosmological model, the universe started its evolution and expansion from a state of extremely high temperature and density.",
"In the very early stages, it experienced a very rapidly accelerated expansion phase within a very short period of time, called inflation.",
"Then, it underwent a radiation dominated era, passed through a matter dominated regime and came into another accelerated expansion phase until now, dubbed dark energy (DE) dominated era.",
"Many authors have investigated different stages of the history of the universe, separately.",
"For example, there are many articles about various inflationary models [1]-[12].",
"Also, different DE scenarios have been proposed to explain the late time acceleration [13]-[23].",
"Maybe the most usual theoretical instrument to explain the inflationary era and also the most common candidate for DE, is the concept of scalar field.",
"Among different kinds of scalar fields, tachyon field which originates from string theory is of particular interest.",
"It has frequently been shown that the tachyon field can play the role of the inflaton field, and/or the role of the dark sectors of the universe [24]-[33].",
"It has a positive potential which has a maximum at $\\phi =0$ , and approaches zero when $\\phi \\rightarrow \\infty $ , whilst during the entire of this process, the slope of the potential is negative.",
"In addition to investigating different periods of the evolution of the universe distinctly, it could be very useful to study its whole history, all at once.",
"Dynamical system approach is a well known procedure in this context which provides a mathematical tool for qualitative study of the behavior of complex dynamical systems, usually by employing ordinary differential equations [34]-[46].",
"It has its origins in Newtonian mechanics in which we try to obtain the trajectory of the system.",
"But, the behavior of complex dynamical systems are too complicated to be understood in terms of these individual trajectories, because it may not be possible for instance to know all of the precise values of model parameters.",
"Also, in many situations it is more important to know the type of trajectory than one particular trajectory.",
"Dynamical system theory deals with the long-term qualitative behavior of dynamical systems.",
"This qualitative study is based on stability analysis.",
"The stability of dynamical systems implies the equivalent trajectories related to a class of models or initial conditions and classify all possible trajectories.",
"Several notions of stability have been introduced in dynamical system approach.",
"On the other hand, the idea of higher dimensional theories of gravity has attracted a great deal of attention.",
"Although it was mentioned first in Kaluza-Klein theory but it revived after the advent of string theory in Arkani-Hamed, Dimopoulos and Dvali scenario and Randall and Sundrum (RS) models [47]-[49].",
"Another interesting 5D scenario, proposed by Dvali, Gabadadze and Porrati (DGP), is a brane-induced gravity model which considers a 4D brane, that is embedded into a 5D minkowski bulk [50].",
"This is a phenomenological model that could be considered as an inspiration of string theory.",
"In this model, according to how the 4D brane embeds in the bulk, we obtain two separate branches which are distinguished with a parameter $\\epsilon = \\pm 1$ .",
"The branch with $\\epsilon =+1$ , is called the self-accelerating branch because it results the late time acceleration of the universe naturally, without any need to have a DE component.",
"But the other, $\\epsilon =-1$ , is the normal branch which needs a DE component to explain the late time acceleration.",
"In this manuscript, we will concentrate on the stability analysis of the tachyon DGP model assuming that the tachyon potential has a Gaussian form.",
"The choice of tachyon field as a dark energy term on the brane in DGP cosmology has been studied in several papers [11],[51]-[52].",
"In [52], the authors have investigated the stability of this combination model for two kinds of tachyonic potential: the exponential potential and inverse square potential and they didn't find any attractor critical point analytically, whereas in another article the authors have studied the stability of a DGP model with a quintessence scalar field and obtained the model's attractor critical points [53].",
"They have utilized both a constant potential and an exponential potential and especially revealed that there is an attractor submanifold in the case of a constant potential which indicates the effect of extra dimension.",
"Here, using a Gaussian potential we find attractor critical points as well as such attractor submanifold that shows the effect of extra dimension.",
"Also we describe evolution of the universe according to the critical points.",
"The paper is organized as follows: in Sec., we review the basic equations of the model.",
"In Sec., We rewrite the equations of motion in terms of autonomous differential equations for which we identify the critical points and analyze their stability condition for a inverse square and Gaussian form of the potentials.",
"Finally in Sec., we present a summary and discuss our results."
],
[
"THE MODEL",
"The spatially flat Friedmann equation on the brane in the normal branch of DGP scenario considering tachyon field as the DE component is as follows [54] $H^2+\\frac{H}{r_c}=\\frac{1}{3M_p^2}(\\rho _m+\\rho _{tac})$ Here $r_c$ , which is the relation between the 4D and 5D Planck mass is called the crossover scale, $H$ , is the Hubble parameter, $\\rho _m$ , is the matter energy density, $\\rho _{tac}$ , denotes the energy density of the tachyon field and $M_p$ , is the 4D Planck mass.",
"On the other hand, the energy density and pressure of tachyon field are expressed as $\\rho _{tac} &=& \\frac{V(\\phi )}{\\sqrt{1-\\dot{\\phi }^2}} \\\\P_{tac} &=& -V(\\phi )\\sqrt{1-\\dot{\\phi }^2} $ in which $\\phi $ and $V(\\phi )$ are the tachyon field trapped on the brane and the tachyon potential, respectively and dot means derivative with respect to the cosmic time.",
"In the absence of any interaction between the dark sectors of the universe they satisfy the conservation equations as below $\\dot{\\rho }_m &+& 3H\\rho _m = 0 \\\\\\dot{\\rho }_{tac} &+& 3H(\\rho _{tac}+P_{tac})= 0 $ where we have assumed the matter content of the universe as a perfect fluid with vanishing pressure, called dust.",
"Replacing $\\rho _{tac}$ and $P_{tac}$ , in Eq.",
"(), with Eqs.",
"(REF ) and (), one can obtain the equation of motion of the tachyon field as $\\frac{\\ddot{\\phi }}{1-\\dot{\\phi }^2}+3H\\dot{\\phi }+\\frac{V_\\phi }{V}=0$ in which $V_{\\phi }$ , represents the derivative of the $V(\\phi )$ , with respect to the tachyon scalar field."
],
[
"phase space and stability analysis",
"As we mentioned in introduction we intend to describe the model as an autonomous system of ordinary differential equations.",
"So, we define a new set of dimensionless dynamical variables $x^2=\\frac{\\rho _m}{3M_p^2(H^2+\\frac{H}{r_c})}, \\quad y^2=\\frac{V}{3M_p^2(H^2+\\frac{H}{r_c})}, \\quad d=\\dot{\\phi }, \\quad z^2=1+\\frac{1}{Hr_c}, \\quad \\lambda =\\frac{-M_pV_{\\phi }}{V^{3/2}}$ Since $r_c$ , is always positive, and in an expanding universe we have $H>0$ , then $z\\ge 1$ .",
"On the other hand, in a contracting universe, i.e.",
"$H<0$ , we find $0\\le z\\le 1$ .",
"So, the expanding universe and the contracting universe are independent submanifolds and one can study each of them, separately.",
"Here, we focus on an expanding universe.",
"The common subset $(x, y, d, z = 1)$ , corresponds to the formal limit $r_c\\rightarrow \\infty $ , which represents the standard behavior of 4D Einstein-Hilbert theory coupled to a tachyon field.",
"Using the above new introduced variables, the Friedmann constraint, Eq.",
"(REF ), reads $x^2+\\frac{y^2}{\\sqrt{1-d^2}}=1$ while the Raychaudhury equation can be recast as $\\frac{\\dot{H}}{H^2}=\\frac{-3z^2}{z^2+1}\\left(x^2+y^2\\frac{d^2}{\\sqrt{1-d^2}}\\right)$ Since $d$ , appears under the square root and to have a physical meaning it should satisfy the constraint $-1\\le d\\le 1$ .",
"Also, we can rewrite the first relation in Eq.",
"(REF ), as $x^2z^2=\\Omega _m$, in which $\\Omega _m$ , is the dimensionless density parameter of the matter content of the universe which changes between 0 and 1.",
"Thus, using the constraint of $z$ , we find that $-1\\le x\\le 1$.",
"From the Friedmann constraint and with attention to the range of changes of $x$ , one can conclude that $-1\\le y\\le 1$ .",
"On the other hand, we can obtain the tachyon EoS parameter and the total EoS parameter of the universe in terms of the dimensionless variables as $w_{tac} &=& d^2-1 \\\\w_{tot} &=& -y^2\\sqrt{1-d^2}$ The set of evolution equations of the model under consideration can be obtained using the Eqs.",
"(REF ), (REF ) and (REF ), as below $d^{\\prime } &=& -(3d-\\sqrt{3}zy\\lambda )(1-d^2) \\\\y^{\\prime } &=& -\\frac{\\sqrt{3}}{2}y^2dz\\lambda +\\frac{3}{2}y(1-y^2\\sqrt{1-d^2}) \\\\z^{\\prime } &=& \\frac{3}{2}\\frac{z(z^2-1)}{z^2+1} (1-y^2\\sqrt{1-d^2}) \\\\\\lambda ^{\\prime } &=& -\\sqrt{3}\\lambda ^2dyz(\\Gamma -3/2)$ Here, prime means derivative with respect to $\\ln a$ , and $\\Gamma =VV_{\\phi \\phi }/V_{\\phi }^2$ , where by $V_{\\phi \\phi }$ , we mean the second derivative of the potential with respect to the tachyon field.",
"Also, we have removed the new variable $x$ , in the above equations using the Friedmann constraint.",
"These equations form a four dimensional autonomous system and indicate the evolution of the phase space variables $d$ , $y$ , $z$ and $\\lambda $ , and indirectly the behavior of the DGP model with tachyon field.",
"In stability formalism, we try to solve $d^{\\prime }=y^{\\prime }=z^{\\prime }=\\lambda ^{\\prime }=0$ , simultaneously, and find the critical points of the model and study their existence conditions using respective eigenvalues.",
"The form of the potential $V(\\phi )$ , plays an important role in this scenario.",
"It is more convenient henceforth to divide our discussion into two parts depending on the parameter $\\lambda $ .",
"We consider two general cases: 1.",
"$\\lambda =$ constant.",
"With attention to the definition of $\\lambda $ , this choice yields an inverse square potential, $V(\\phi )\\propto \\phi ^{-2}$ .",
"2.",
"$\\lambda =\\lambda (\\phi )$ .",
"Various potentials lead to a varying $\\lambda $ .",
"But, in the following we are going to consider a Gaussian potential, $V(\\phi )\\propto e^{-\\gamma \\phi ^2}$ which has a maximum value at $\\phi =0$ and decays to zero as $\\phi $ goes to infinity that is suitable for a tachyonic potential."
],
[
"The case $\\lambda =$ constant",
"Obviously, the case $\\lambda =$ constant, relates to $\\lambda ^{\\prime }=0$ .",
"In this situation when we integrate the equation $\\lambda =-V_\\phi /V^{3/2}$ , we conclude that the tachyon potential must have an inverse square form as $V(\\phi )=V_0\\phi ^{-2}$ .",
"This result is just like the one in [55]-[60].",
"Also, in [61], the authors obtained such a potential for a quintessence scalar field in the early stages of a brane-world RSII model.",
"It is easy to show that in this case $\\Gamma =3/2$ , which from Eq.",
"(), results $\\lambda ^{\\prime }=0$ .",
"The fixed points of the system can be obtained by setting $d^{\\prime }=y^{\\prime }=z^{\\prime }=0$ .",
"The results have been given in TABLE REF , in which $y^{\\ast }=\\sqrt{\\alpha /6}$ , $d^{\\ast }=\\lambda y^{\\ast }/\\sqrt{3}$ , $\\alpha =\\sqrt{\\lambda ^4+36}-\\lambda ^2$ and $\\delta =-3+\\alpha \\lambda ^2/72$ .",
"We must note that we have found some other critical points in our model, but we have not mentioned them in TABLE REF , because either they did violate the Friedmann constraint or they did not satisfy the constraints on the dynamical variables.",
"Table: The fixed points of the model for λ=\\lambda = constant.The critical point $P_1$ is a saddle point, because one of its eigenvalues is negative and the others are positive.",
"Since from Friedmann constraint we find $x^2=1$ , using the relation $\\Omega _m=x^2z^2$ , we conclude that this point refers to a matter dominated solution, $\\Omega _m=1$ .",
"One can see in TABLE REF that at this critical point $w_{tot}=0$ , which is in agreement with a matter dominated regime.",
"The critical points, $P_3^\\pm $ , with all positive eigenvalues represent an unstable critical point.",
"Although the universe behaves like a matter dominated era, because $w_{tot}=0$ , but against $P_1$ , the kinetic term of the tachyon field has a remarkable contribution at these points.",
"So, we can call them matter scaling solutions.",
"To specify the characteristics of critical points $P_2^\\pm $ , we need to investigate the parameter $\\lambda $ , more carefully.",
"Using its definition and considering the negative slope of a tachyonic potential, one can find that $\\lambda >0$ .",
"It cannot be zero because $\\lambda =0$ , means $V_\\phi =0$ , which for an inverse square potential, results $V_0=0$ and consequently $V=0$ .",
"Using the constraint $0<\\lambda <\\infty $ , and the definition of $\\alpha $ , one can obtain another constraint as $0<\\alpha <6$ .",
"It is easy to check that for each pair of values of $(\\lambda ,\\alpha )$ , $\\alpha \\lambda ^2\\le 18$ .",
"This also can be deduced from the relation of $w_{tot}$ .",
"Thus, we obtain an interval for the parameter $\\delta $ , as $-3<\\delta <-11/4$ .",
"So, the eigenvalues respective to the critical points $P_2^\\pm $ , are always negative and therefore they are attractor solutions.",
"Also, from Eq.",
"(REF ), we can write $\\Omega _{tac}=\\frac{\\rho _{tac}}{3M_p^2H^2}=\\frac{y^2z^2}{\\sqrt{1-d^2}}$ .",
"One can check that $\\Omega _{tac}=1$ , at points $P_2^\\pm $ .",
"Therefore, we call them tachyon dominated solutions.",
"On the other hand, from $w_{tot}<-1/3$ , we reach the condition of acceleration as $\\alpha \\sqrt{1-\\frac{\\alpha \\lambda ^2}{18}}>2$ .",
"So, one can check that for $0<\\lambda <1.86$ , these critical points show a de Sitter solution, while for $\\lambda \\gtrsim 1.86$ , they are just tachyon dominated solutions, without acceleration.",
"This result is similar to the case reported in [57] and [62].",
"In addition to what we mentioned above about these critical points, their similarity is that all of them belong to a 4D universe, because in all of them $z=1$ .",
"Thus, extra dimension has no significant effect in the case $\\lambda =$ constant.",
"Trajectories may exit the plane $z=1$ , in parts of the history of the universe, but they must return to the 4D universe as these critical points show."
],
[
"The case $\\lambda =\\lambda (\\phi )$",
"When the potential form is something different from the inverse square, $\\lambda $ , will be a dynamically changing quantity.",
"In the following we choose a Gaussian potential.",
"Assuming $\\lambda $ , evolves sufficiently slow such that one can take it to be a constant within a short period of the evolution of the universe, we can regard the critical points in the previous subsection for the constant $\\lambda $ , as the instantaneous critical points of the current dynamical system [56]-[59].",
"This interesting assumption is useful in order to see where the solution tends to at that instant, because according to this assumption, $P_2^\\pm $ , are dynamical critical points.",
"Also, in addition to the fixed points $P_1$ and $P_3^\\pm $ , and the moving critical points $P_2^\\pm $ , which exhibit a standard 4D behavior because of $z=1$ , we find two critical submanifolds in this situation, $P_4^\\pm : (d=0, y=\\pm 1, z)$ , that show the effect of extra dimension and only exist for $\\lambda =0$ , that corresponds to extremum of the Gaussian potential where $V_{\\phi }=0$ .",
"In this case one can check that $\\Omega _{tac}=1$ .",
"On the other hand, using Eq.",
"(), we find $w_{tot}=-1$ .",
"Hence, we conclude that these submanifolds indicate a tachyon dominated de Sitter solution.",
"Further, we obtain the respective eigenvalues for $P_4^\\pm $ submanifolds, as $(0,-3,-3)$ .",
"Since at least there is one zero eigenvalue, we cannot use the linear approximation method to discuss their stability, rather we turn to a method based on center manifold theory [63]-[66].",
"In this scenario, one has to reduce the dimensionality of the dynamical system under consideration and then, the stability of this reduced system is studied instead of the main one.",
"The interesting feature of this method is that the stability of the critical points of the main system can be understood via the stability properties of this reduced system.",
"The mathematical procedure is complicated and out of scope of the current work, but concisely one must calculate the center manifold related to that critical point and investigate the dynamics on the center manifold to comment on the stability of that critical point.",
"But, in the case of a critical line, with only one zero eigenvalue, such as the case here, the respective center manifold is nothing but the critical line itself [34].",
"Therefore, in the case of $P_4^\\pm $ , the $z$ -axis is the center manifold, and we can expect that orbits near the $z$ -axis are perpendicularly attracted or repelled according to the sign of the non vanishing eigenvalues.",
"Since the non vanishing eigenvalues are all negative, then we conclude that $P_4^\\pm $ , are attractors.",
"Since $\\lambda $ , is a varying quantity, its asymptotic behavior has a crucial role in a complete understanding of the nature of the dynamical attractor critical points $P_2^\\pm $ .",
"To this aim, we return to Eq.().",
"As we mentioned above, the case $\\lambda =$ constant, relates to $\\Gamma =3/2$ .",
"Also, it is generally easy to show that for positive values of $d$ , and $y$ , if $\\Gamma >3/2$ , $\\lambda \\rightarrow 0$ , and if $\\Gamma <3/2$ , $\\lambda \\rightarrow \\infty $ , asymptotically.",
"But, for a common Gaussian potential in the form $V=V_0e^{-\\gamma \\phi ^2}$ , we find that $\\Gamma =1-1/(2\\gamma \\phi ^2)$ .",
"So, the maximum value of $\\Gamma $ , that is achieved in the limit $\\phi \\rightarrow \\infty $ , is $\\Gamma =1$ .",
"Therefore, for a Gaussian potential, $\\lambda \\rightarrow \\infty $ , asymptotically.",
"Also, according to discussions in the prior subsection, when $\\lambda $ , increases from zero to infinity, $\\alpha $ , decreases from 6 to zero, so that the relation $\\alpha \\lambda ^2\\le 18$ , is always satisfied.",
"In fact, when $\\lambda =0$ , $\\alpha =6$ , and $P_2^{\\pm }$ start from $(0,\\pm 1,1)$ on the $P_4^{\\pm }$ , with $w_{tot}=-1$ , and when $\\lambda \\rightarrow \\infty $ , $\\alpha \\lambda ^2\\rightarrow 18$ , and consequently $P_2^{\\pm }$ approach $P_3^{\\pm }$ , i.e., $y^{\\ast }\\approx \\sqrt{3}/\\lambda \\rightarrow 0$ , and $d^{\\ast }\\rightarrow 1$ , while they are still attractor solutions with $w_{tot}=0$ .",
"Thus, with attention to descriptions above, it seems that the universe will experience a phase transition from acceleration to deceleration during the evolution of $\\lambda $ .",
"But in fact, it depends crucially on how fast the system reaches a neighborhood of $P_2^{\\pm }$ .",
"If the universe reaches $P_2^{\\pm }$ , while $0<\\lambda <1.86$ , one can conclude that it will experience a temporal acceleration before entering a deceleration phase.",
"Otherwise, it will never come across an accelerating phase.",
"If we correspond the current acceleration of our universe to the one we mentioned above, it will eventually enter a decelerating phase.",
"[56]-[57].",
"Fig.REF , shows the 2D phase plane $(y,d)$ of the model under consideration for $\\lambda =0$ .",
"The universe starts from the unstable matter dominated critical point and tends to reach the stable dark energy dominated critical point.",
"The important property for the case $\\lambda =0$ , is the appearance of $P_4^\\pm $ submanifolds.",
"Apart from the discussion above in utilizing the center manifold theory in investigating the stability of the critical line $P_4^{\\pm }$ , we use numerical techniques to show the behavior of trajectories in the vicinity of this critical line.",
"Fig.REF , indicates the 2D phase plane $(d,z)$ of the model for $y=\\pm 1$ .",
"The line $d=0$ , shows these stable critical submanifolds in this 2D portrait.",
"Figure: The 2D phase space (y,d)(y,d), for λ=0\\lambda =0.",
"The universe starts from matter scaling solution and tends to reach tachyon dominated critical point.Figure: The 2D phase space (d,z)(d,z), for λ=0\\lambda =0 and y=±1y=\\pm 1.",
"The line d=0d=0, refers to P 4 ± P_4^\\pm submanifolds.Fig.REF , consists of three 2D phase portrait $(y,d)$ of our dynamical system for different values of parameter $\\lambda $ .",
"It is obvious from these figures that along with increasing the value of $\\lambda $ , the stable critical points $P_2^\\pm $ , move around and approach $P_3^\\pm $ , while they are still attractors and as well as while universe has experienced a transition from acceleration to deceleration during this process.",
"Figure: The 2D phase space (y,d)(y,d), for different values of λ\\lambda .",
"From left to right: λ=0.1\\lambda =0.1, λ=1.4\\lambda =1.4 and λ=2.4\\lambda =2.4.",
"Attractor solutions P 2 ± P_2^\\pm , are instantaneous critical points.",
"They approach P 3 ± P_3^\\pm and during this process the universe experiences a transition from acceleration to deceleration."
],
[
"Conclusion",
"In the present work we studied a DGP brane-world model with a tachyon scalar field on the brane for two different kinds of tachyon potential.",
"Our main focus were on the Gaussian potential for which $\\lambda $ , is not constant.",
"Then, we followed the dynamical system approach to understand the evolution of the universe in this model.",
"One of our most important results was that we found stable critical submanifolds that depend on the new variable $z$ , which relate to the extra dimension.",
"We should note that in a tachyon DGP model, these attractor submanifolds only exist for those tachyonic potentials that have an extremum, such as the Gaussian potential in the present work.",
"On the other hand, we found two dynamical critical points $P_2^\\pm $ that assuming slowly moving, one can consider them as instantaneous fixed critical points.",
"The most interesting feature of this assumption is that we can study the behavior of the universe for different values of $\\lambda $ .",
"Also, we found that with the expansion of the universe, as $\\phi $ goes to infinity, the tachyonic potential will decrease to zero and critical points $P_2^\\pm $ , approach $P_3^\\pm $ while they remain attractors.",
"During this evolution, the behavior of the total equation of state parameter suggested that the universe may experience a transition from acceleration to deceleration.",
"Future observations may show if such a transition is possible or not.",
"The authors would like to thank Nelson J. Nunes for his valuable and helpful comments."
]
] | 1906.04465 |
[
[
"Multiplicity of subharmonics in a class of periodic predator-prey\n Volterra models"
],
[
"Abstract This paper ascertains the global topological structure of the set of subharmonics of arbitrary order of the periodic predator-prey model introduced in L\\'opez-G\\'omez, Ortega and Tineo in 1996.",
"By constructing the iterates of the monodromy operator of the system, it is shown that the system admits subharmonics of all orders for the appropriate ranges of values of the parameters.",
"Then, some sharp results of topological nature in the context of global bifurcation theory provide us with the fine topological structure of the components of subharmonics emanating from the T-periodic coexistence state."
],
[
"Introduction",
"In this paper we analyze the global structure of the set of subharmonics of the periodic predator-prey model $\\left\\lbrace \\begin{array}{l} u^{\\prime }=\\alpha (t)u(1-v) \\\\ v^{\\prime }=\\beta (t)v(-1+u) \\end{array} \\right.$ where $\\alpha (t)$ and $\\beta (t)$ are real continuous $T$ -periodic functions such that $\\alpha = 0\\quad \\hbox{on}\\;\\; [\\tfrac{T}{2},T], \\quad \\beta = 0 \\quad \\hbox{on}\\;\\; [0,\\tfrac{T}{2}],$ $\\alpha (t)>0$ if $t \\in (0,\\tfrac{T}{2})$ , and $\\beta (t)>0$ if $t\\in (\\tfrac{T}{2},T)$ , which entail $\\alpha \\beta =0$ .",
"This model was introduced by J. López-Gómez, R. Ortega and A. Tineo [14] as a simple example of a predator-prey model with an unstable coexistence state.",
"Later, it was shown in [9] that it actually admits three $T$ -periodic (non-degenerate) coexistence states: one $T$ -periodic and two additional $2T$ -periodic solutions.",
"The non-degeneration of these solutions facilitated the construction of some examples of $T$ -periodic Lotka–Volterra models $\\left\\lbrace \\begin{array}{l} u^{\\prime }=\\lambda (t) u - a(t) u^2 - b(t) uv \\\\ v^{\\prime }=-\\mu (t)v +c(t)uv-d(t)v^2 \\end{array} \\right.$ with at least three coexistence states (see [9]).",
"In (REF ), $\\lambda , \\mu , a, b, c, d$ are smooth positive $T$ -periodic functions.",
"Such multiplicity results contrast very strongly with the main theorem of J. López-Gómez and R. Pardo [15], where it was established the uniqueness of the coexistence state for the boundary value problem $\\left\\lbrace \\begin{array}{ll} \\begin{array}{l} -u^{\\prime \\prime }= \\lambda (x) u-a(x)u^2-b(x)uv\\\\ -v^{\\prime \\prime }=-\\mu (x) v+c(x)uv-d(x)v^2 \\end{array} & \\quad \\hbox{in}\\;\\; (0,L), \\\\u(0)=u(L)=v(0)=v(L)=0, & \\end{array} \\right.$ where $\\lambda , \\mu , a, b, c, d$ are positive (arbitrary) continuous functions in $[0,L]$ .",
"Inheriting the same non-cooperative structure, at first glance causes some perplexity that (REF ) and (REF ) behave so differently.",
"The original theorem of [15] was later refined in a series of papers by A. Casal et al.",
"[2], E. N. Dancer et al.",
"[5] and J. López-Gómez and R. Pardo [16].",
"An important feature of model (REF ) is that it does not fit within the general setting of T. Ding and F. Zanolin [6], where the existence of higher order subharmonics for a general class of predator-prey models was established.",
"Precisely, [6] gives some general conditions on the nonlinearities $f(t,v)$ and $g(t,u)$ so that the Lotka–Volterra predator-prey system $\\left\\lbrace \\begin{array}{ll}u^{\\prime }=uf(t,v)\\\\v^{\\prime }=vg(t,u)\\end{array}\\right.$ can admit higher order subharmonics.",
"In (REF ), $f(t,v)$ and $g(t,u)$ are continuous functions $T$ -periodic in time, $t$ , satisfying certain bounds for the existence of T-periodic solutions and such that, for every $t\\in [0,T]$ , either $v\\mapsto f(t,v)$ is (strictly) decreasing, or $u\\mapsto g(t,u)$ is (strictly) increasing.",
"Under these assumptions, [6] establishes the existence of an integer $m^*\\ge 2$ such that (REF ) admits, at least, one $mT$ -periodic solution for all $m\\ge m^*$ .",
"Although setting $f(t,v):=\\alpha (t)(1-v),\\qquad g(t,u):=\\beta (t)(-1+u),$ (REF ) can be also written down in the form of (REF ), by (REF ), neither $\\alpha (t)(1-v)$ can be decreasing for all $t\\in [0,T]$ , nor $\\beta (t)(-1+u)$ can be increasing for all $t\\in [0,T]$ .",
"Thus, (REF ) remains outside the class of models considered in [6].",
"In particular, [6] cannot be applied to establish the existence of higher order subharmonics for (REF ).",
"The main goal of this paper is to construct the set of all subharmonics of (REF ) in the special, but extremely interesting case, when $A := \\int _0^T \\alpha (t)\\,dt = \\int _0^T \\beta (t)\\,dt>0.$ Precisely, it will be shown that, under assumption (REF ), the model (REF ) admits subharmonics of any order for the appropriate range of values of $A>0$ , which will be regarded as a bifurcation parameter throughout this paper.",
"Actually, our analysis establishes the existence of an integer $m^*(A)\\ge 1$ such that (REF ) possesses, at least, two subharmonic solutions of order $m$ for all $m\\ge m^*(A)$ .",
"In particular, [6] seems to be true in much more general situations than those originally dealt with in [6].",
"Moreover, as a direct consequence of our analysis, $\\lim _{A\\downarrow 0}m^*(A)=+\\infty ,\\quad \\hbox{whereas}\\;\\; m^*(A)=2 \\;\\; \\hbox{if}\\;\\; A>2.$ Figure REF summarizes, at a glance, the main findings of this paper.",
"It is an sketch of the global bifurcation diagram of subharmonics, where we are plotting the value of $A$ in abscisas versus the value of $x=u_0=v_0$ in ordinates.",
"Naturally, $(u_0,v_0)=(u(0),v(0))$ stands for the initial condition of (REF ).",
"Figure: Subharmonics of () under condition () with x=u 0 =v 0 x=u_0=v_0.Each of the curves plotted in Figure REF represents a component of $nT$ -periodic coexistence states of (REF ) for each integer $n\\ge 1$ .",
"By a component it is meant a closed and connected subset of the solution set of (REF ) which is maximal for the inclusion.",
"Each point on the corresponding line, $(A,x)$ , provides us with a value of $A$ for which (REF ) admits a $nT$ -periodic solution with $u_0=v_0=x$ .",
"By the intrinsic nature of (REF ), it turns out that all these components are separated from each other.",
"By some existing results of topological nature in global bifurcation theory, all of them have an unbounded $A$ -projection.",
"However, except for the first three, whose local bifurcation diagrams are described by Theorem REF , the nature of their local bifurcations from $(A,1)$ is not known yet, being possibly random.",
"The problem of ascertaining weather, or not, this occurs, seems extremely challenging.",
"Note that $(A,u,v)=(A,1,1)$ solves (REF ) for all $A>0$ .",
"More precisely, Figure (REF ) shows all the components of subharmonics of order $n$ of (REF ) emanating from the straight line $(A,1)$ for $1\\le n\\le 13$ .",
"It contains the plots of: 1 component of subharmonics of order 1; 1 component of subharmonics of order 2; 1 component of subharmonics of order 3; 2 components of subharmonics of order 4; one of them is actually the component of subharmonics of order 2; 2 components of subharmonics of order 5; 3 components of subharmonics of order 6; one of them is actually the component of subharmonics of order 2 and another must be the component of order 3; 3 components of subharmonics of order 7; 4 components of subharmonics of order 8; one of them must be the component of order 2 and another one is a component of subharmonics with minimal order 4; and so on...",
"The fact that the number of components of subharmonics of order $n\\ge 1$ grows to $+\\infty $ as $n\\uparrow +\\infty $ is rather intriguing and it seems inherent to the non-cooperative character of (REF ) and attributable to the $T$ -periodicity of $\\alpha (t)$ and $\\beta (t)$ .",
"The emergence of secondary bifurcations in any of these components cannot be a priori excluded, however no higher order bifurcations have been represented in Figure REF .",
"Thanks to Theorems REF and Theorem REF , for every $n\\ge 1$ , the bifurcation points from $(A,1)$ to the $nT$ -periodic coexistence states of (REF ) are given by the positive roots of the polynomial $p_n(A):=[2-(-1)^n A]p_{n-1}(A)-p_{n-2}(A),\\qquad n\\ge 3,$ where $p_1(A):= 1,\\qquad p_2(A):=2-A.$ Although, according to Theorem REF , for every $n\\ge 2$ , the positive roots of $p_{2n}(A)$ are separated by the positive roots of $p_{2n-1}(A)$ , the positive roots of $p_{2n+1}(A)$ are separated by those of $p_{2n}(A)$ less than 2, and, for every $n\\ge 1$ , the (even) polynomials $\\tfrac{p_{2n}(A)}{2-A}$ and $p_{2n-1}(A)$ have (exactly) $n-1$ positive roots, which are real and algebraically simple, the problem of ascertaining the sharp ordering structure, if any, of the set of all these positive roots, which is a numerable subset of $(0,2]$ , remains an open problem in this paper.",
"Although there are some serious evidences that this set should be dense in the interval $[0,2]$ , a rigorous proof of this feature is not available yet.",
"The fact that the positive roots of $p_n(A)$ are algebraically simple allows us to apply the main theorem of M. G. Crandall and P. H. Rabinowitz [3] to prove that each of the components of subharmonics in Figure REF must be a real analytic curve about their bifurcation points from $(A,1)$ .",
"The mathematical analysis carried out in this paper has been tremendously facilitated by the fact that $\\alpha \\beta =0$ , which provides us with a rather explicit formula for the iterates, $\\mathcal {P}_n$ , $n\\ge 2$ , of the monodromy operator, $\\mathcal {P}_1$ .",
"Thanks to Proposition REF , for every $ n\\ge 2 $ , the Poincaré map $\\mathcal {P}_n$ can be expressed through $(u_n,v_n)=\\mathcal {P}_n(x,x)=\\left(xE_{2n-1}(x),xE_{2n}(x)\\right),$ where $\\left\\lbrace \\begin{array}{lll}E_0(x):=1,\\quad E_1(x):=e^{(1-x)A}, \\\\[4pt]E_n(x):=\\left\\lbrace \\begin{array}{ll} e^{[x(E_1(x)+E_3(x)+\\cdots +E_{n-1}(x))-\\frac{n}{2}]A} & \\quad \\mathrm {if\\ } \\;\\; n\\in 2\\mathbb {N}, \\\\[4pt]e^{[\\frac{n+1}{2}-x(E_0(x)+E_2(x)+\\cdots +E_{n-1}(x))]A} & \\quad \\mathrm {if\\ } \\;\\; n\\in 2\\mathbb {N}+1.",
"\\end{array} \\right.\\end{array}\\right.$ Thanks to Theorem REF , for every $n\\ge 2$ , the positive fixed points of $\\mathcal {P}_n$ , which provide us with the $nT$ -periodic coexistence states of (REF ), are given by the zeros of the map $\\varphi _n(x)= \\varphi _{n-1}(x) - 1 + xE_{n-1}(x), \\qquad x\\in [0,n].$ Thus, $\\varphi _1(x) & =x-1, \\\\ \\varphi _2(x) & = \\varphi _1(x)-1+x e^{(1-x)A}, \\\\\\varphi _3(x) &= \\varphi _2(x)-1 +xe^{(xe^{(1-x)A}-1)A},\\\\\\varphi _4(x) & =\\varphi _3(x)-1+ x e^{(2-x-xe^{(xe^{(1-x)A}-1)A})A},\\\\\\varphi _5(x) & = \\varphi _4(x)-1+ x e^{\\left(x e^{(1-x)A}+xe^{(2-x-xe^{(xe^{(1-x)A}-1)A})A}-2\\right)A}.$ In particular, the positive fixed points of $\\mathcal {P}_5$ are given by the positive zeros of $\\varphi _5(x)$ , which consists, essentially, in the composition of 4 exponentials functions.",
"This circumstance might provoke dramatic oscillations of $\\varphi _5(x)$ between some consecutive positive zeros.",
"For instance, choosing $A=5$ and $x=0.1$ , it turns out that $\\varphi _5(0.1)\\sim 10^{30}$ , which lies outside the precision range of most of personal computers.",
"Therefore, without no further work, numerics cannot be of any help in constructing the global bifurcation diagram sketched in Figure REF .",
"Lastly, we will consider the associated perturbed $T$ -periodic functions $\\alpha _\\varepsilon :=\\alpha +\\varepsilon ,\\qquad \\beta _\\varepsilon = \\beta +\\varepsilon ,$ where $\\varepsilon >0$ , as well as the associated predator-prey model $\\left\\lbrace \\begin{array}{l} u^{\\prime }=\\alpha _\\varepsilon (t)u(1-v), \\\\ v^{\\prime }=\\beta _\\varepsilon (t)v(-1+u).",
"\\end{array} \\right.$ Taking $\\varepsilon =0$ in (REF ) gives (REF ).",
"Although the $nT$ -periodic coexistence states of (REF ) might degenerate, thanks to a celebrated result of A. Sard [21], most of the subharmonics of order $n$ of (REF ) should provide us with subarmonics of order $n$ of (REF ) for sufficiently small $\\varepsilon >0$ .",
"Therefore, the global topological structure sketched by Figure REF should be essentially preserved, at least for sufficiently small $\\varepsilon >0$ .",
"Note that [6] applies to (REF ) for all $\\varepsilon >0$ , because $\\alpha _\\varepsilon (t)>0$ and $\\beta _\\varepsilon (t)>0$ for all $t\\in [0,T]$ .",
"Thus, it is rather natural to conjecture that, actually, Figure REF provides us with the minimal admissible complexity of the set of subharmonics of (REF ) for sufficiently small $\\varepsilon >0$ .",
"Like in [9], these multiplicity results should provide us with a series of (very intriguing) multiplicity results for (REF ).",
"The distribution of this paper is the following.",
"Section 2 studies the structure and multiplicity of the low order subharmonics of (REF ) in the general case when $0 < A := \\int _0^T\\alpha (t)\\,dt \\ne B := \\int _0^T \\beta (t)\\,dt>0.$ It substantially sharpens some previous findings of [9] by establishing the exact multiplicity of the $2T$ -periodic solutions of (REF ) when $AB>4$ .",
"The rest of the paper focuses attention into the special, but extremely important case, when $A=B$ .",
"In Section 3 we construct the Poincaré maps $\\mathcal {P}_n$ for all $n\\ge 1$ .",
"In Section 4 we introduce the associated polynomials $p_n(A):= \\frac{d\\varphi _n (A,1)}{dx}=\\mathfrak {L}(n;A),\\qquad A>0,$ whose positive roots provide us with the bifurcation points to subharmonics from $(A,1)$ , and analyze some of their most fundamental properties.",
"In Section 5 we establish some fundamental separation properties between the zeros of these polynomials and show that all their positive roots are algebraically simple.",
"This property has important consequences from the point of view of local and global bifurcation theory.",
"Finally, in Section 6 we derive and discuss the global bifurcation diagram sketched in Figure REF ."
],
[
"Multiplicity and structure of $T$ -periodic and {{formula:2d4148fd-28bd-4e1f-8ada-012d8af5bd5e}} -periodic solutions in the model of {{cite:12f64c9a848cce9fd9790fe3f8f2081e0dd77229}}",
"According to [9], $(u,v)=(1,1)$ provides us with the unique $T$ -periodic solution of (REF ), and (REF ) admits, at least, two $2T$ -periodic coexistence states if, and only if, $AB>4$ , where $A := \\int _0^T \\alpha (s)\\,ds>0,\\qquad B:=\\int _0^T \\beta (s)\\,ds>0.$ The next result sharpens these findings.",
"Theorem 2.1 Suppose $AB>4$ .",
"Then, the problem (REF ) possesses exactly two $2T$ -periodic coexistence states (with minimal period $2T$ , of course).",
"We proceed as in the proof [9].",
"Since $\\alpha \\beta =0$ in $\\mathbb {R}$ , the system (REF ) can be solved.",
"Actually, for every $(u_0,v_0)\\in \\mathbb {R}^2$ , the unique solution of (REF ), $(u,v)$ , such that $(u(0),v(0))=(u_0,v_0)$ is given by $u(t)= u_0 e^{(1-v_0)\\int _0^t\\alpha },\\qquad v(t)=v_0 e^{(u(T)-1)\\int _0^t\\beta }, \\qquad t\\in \\mathbb {R}.$ Thus, the associated $T$ -time and $2T$ -time Poincaré maps, $\\mathcal {P}_1$ and $\\mathcal {P}_2$ , are given by $(u_1,v_1) :=\\mathcal {P}_1(u_0,v_0),\\qquad u_1:=u_0e^{(1-v_0)A},\\qquad v_1:= v_0e^{(u_1-1)B},$ and $(u_2,v_2) := \\mathcal {P}_2(u_0,v_0)=\\mathcal {P}^2_1(u_0,v_0)= \\mathcal {P}_1(u_1,v_1)=\\left( u_1e^{(1-v_1)A}, v_1e^{(u_2-1)B}\\right).$ Thus, substituting (REF ) into (REF ) yields $u_2= u_0 e^{(2-v_0-v_1)A},\\qquad v_2= v_0 e^{(u_1+u_2-2)B}.$ A solution with initial data $(u_0,v_0)$ provides us with a componentwise positive fixed point of $\\mathcal {P}_2$ if, and only if, $u_0>0$ , $v_0>0$ , $v_0+v_1=2$ and $u_1+u_2=2$ .",
"Hence, since $u_2=u_0$ , this is equivalent to $u_0>0,\\qquad v_0>0,\\qquad v_0+v_1=2,\\qquad u_0+u_1=2.$ Note that, owing to (REF ), $0<u_0, u_1 <2,\\qquad 0<v_0,v_1<2.$ Since $u_1=2-u_0$ and $v_1=2-v_0$ , from (REF ) it becomes apparent that $2-u_0=u_0e^{(1-v_0)A},\\qquad 2-v_0= v_0e^{(1-u_0)B}.$ Consequently, $u_0=\\frac{2}{e^{(1-v_0)A}+1},\\qquad 2-v_0= v_0e^{\\left(1-\\frac{2}{e^{(1-v_0)A}+1}\\right)B}=v_0e^{\\frac{e^{(1-v_0)A}-1}{e^{(1-v_0)A}+1}B}$ and therefore, the $2T$ -periodic coexistence states are given by the interior zeros of the map $\\varphi (x):= x\\left( e^{\\frac{e^{(1-x)A}-1}{e^{(1-x)A}+1}B}+1\\right)-2,\\qquad x\\in [0,2].$ As this function satisfies $\\varphi (0)=-2<0$ , $\\varphi (1)=0$ , $\\varphi (2)>0$ and $\\varphi ^{\\prime }(1)= 2-\\frac{AB}{2}<0,$ because we are assuming that $AB>4$ , it is easily seen that $\\varphi (x)$ possesses, at least, besides 1, two zeros, $z_1\\in (0,1)$ and $z_2\\in (1,2)$ .",
"Note that 1 provides us with the (unique) $T$ -periodic solution of (REF ).",
"That these zeros are unique is based on the fact that any critical point of $\\varphi $ on $(0,1)$ , $x$ , must satisfy $\\varphi ^{\\prime \\prime }(x)<0$ , and hence, it is a quadratic local maximum, while $\\varphi ^{\\prime \\prime }(y)>0$ for all critical point, $y$ , of $\\varphi $ in $(1,2)$ .",
"In particular, since $\\varphi (0)<0$ and $\\varphi ^{\\prime }(1)<0$ , this entails that $z_1$ is simple and, actually, $\\varphi ^{\\prime }(z_1)>0$ , for as, otherwise, $\\varphi (x)$ should have a local minimum in $(0,1)$ , which is impossible.",
"Similarly, $\\varphi ^{\\prime }(z_2)>0$ .",
"In order to show the previous claim, suppose $\\varphi ^{\\prime }(x)=0 \\quad \\hbox{for some\\;\\;} x\\in (0,2).$ Then, $\\varphi ^{\\prime }(x)= e^{\\tfrac{e^{(1-x)A}-1}{e^{(1-x)A}+1}B}\\left( 1- \\frac{2ABxe^{(1-x)A}}{[e^{(1-x)A}+1]^2}\\right)+1=0.$ Moreover, differentiating $\\varphi ^{\\prime }$ and rearranging terms yields $\\varphi ^{\\prime \\prime }(x)=e^{\\tfrac{e^{(1-x)A}-1}{e^{(1-x)A}+1}B}\\left[x\\left(\\tfrac{2ABe^{(1-x)A}}{[e^{(1-x)A}+1]^2}\\right)^2 - \\tfrac{4ABe^{(1-x)A}}{[e^{(1-x)A}+1]^2} + 2A^2Bxe^{(1-x)A}\\tfrac{1-e^{(1-x)2A}}{[e^{(1-x)A}+1]^4}\\right].$ Now, after some straightforward manipulations, it is easily seen that (REF ) implies $\\tfrac{2 \\Big (e^{-\\tfrac{e^{(1-x)A}-1}{e^{(1-x)A}+1}B}+1\\Big )}{x}=\\tfrac{4ABe^{(1-x)A}}{[e^{(1-x)A}+1]^2}, \\qquad \\Big (\\tfrac{e^{-\\tfrac{e^{(1-x)A}-1}{e^{(1-x)A}+1}B}+1}{x}\\Big )^2=\\left(\\tfrac{2ABe^{(1-x)A}}{[e^{(1-x)A}+1]^2}\\right)^2,$ and substituting (REF ) into (REF ) we find that $\\varphi ^{\\prime \\prime }(x)&=e^{\\tfrac{e^{(1-x)A}-1}{e^{(1-x)A}+1}B}\\Big [ x\\Big (\\tfrac{e^{-\\tfrac{e^{(1-x)A}-1}{e^{(1-x)A}+1}B}+1}{x}\\Big )^2 -\\tfrac{2\\Big (e^{-\\tfrac{e^{(1-x)A}-1}{e^{(1-x)A}+1}B}+1\\Big )}{x}\\Big ] \\\\& \\hspace{170.71652pt} + e^{\\tfrac{e^{(1-x)A}-1}{e^{(1-x)A}+1}B}\\left[ 2A^2Bxe^{(1-x)A} \\tfrac{1-e^{(1-x)2A}}{[e^{(1-x)A}+1]^4}\\right] \\\\&=e^{\\tfrac{e^{(1-x)A}-1}{e^{(1-x)A}+1}B}\\Big [\\tfrac{e^{-2\\tfrac{e^{(1-x)A}-1}{e^{(1-x)A}+1}B} - 1}{x} + 2A^2Bxe^{(1-x)A}\\tfrac{1-e^{(1-x)2A}}{[e^{(1-x)A}+1]^4}\\Big ].$ Suppose $x\\in (0,1) $ .",
"Then, the following holds $e^{-2\\tfrac{e^{(1-x)A}-1}{e^{(1-x)A}+1}B}-1<0,\\qquad 1-e^{(1-x)2A}<0.$ Therefore, $ \\varphi ^{\\prime \\prime }(x)<0 $ , as claimed above.",
"Suppose $x\\in (1,2]$ .",
"Then, $e^{-2\\tfrac{e^{(1-x)A}-1}{e^{(1-x)A}+1}B}-1>0,\\qquad 1-e^{(1-x)2A}>0,$ and hence, $\\varphi ^{\\prime \\prime }(x)>0$ , as requested.",
"The proof is completed.",
"According to the proof of Theorem REF , if $AB>4$ then $\\varphi (x)$ has exactly three (simple) zeros in $(0,2)$ , $z_1, z_2, z_3$ , such that $z_1\\in (0,1)$ , $z_2\\in (1,2)$ and $z_3=1$ , whereas $\\varphi ^{\\prime }(1) = 2-\\frac{AB}{2} \\ge 0 \\quad \\hbox{if}\\;\\; AB\\le 4,$ and hence, 1 is the unique zero of $\\varphi $ in this case.",
"Note that if $AB=4$ , then $\\varphi ^{\\prime }(1)=0$ and $\\varphi ^{\\prime \\prime }(1)=\\left(\\frac{AB}{2}\\right)^2-AB=4-4=0.$ Moreover, differentiating twice yields $\\varphi ^{\\prime \\prime \\prime }(x)=e^{q}\\left(3(q^{\\prime })^2+3q^{\\prime \\prime }+x\\left[(q^{\\prime })^3+3q^{\\prime }q^{\\prime \\prime }+q^{\\prime \\prime \\prime }\\right]\\right),\\quad q(x):=e^{\\tfrac{e^{(1-x)A}-1}{e^{(1-x)A}+1}B}, \\quad x\\in [0,2].$ Thus, $\\varphi ^{\\prime \\prime \\prime }(1)=\\tfrac{AB(5AB+2A^2)}{8}>0$ and therefore, 1 is a treble zero of $\\varphi (x)$ if $AB=4$ .",
"On the other hand, the function $\\varphi (x)$ can be also regarded as an analytic function of $x$ that varies continuously with $B>0$ and does not vanish at the ends of $[0,2]$ .",
"By Rouché's theorem, $\\varphi $ must have three zeros, counting orders, for every $B>0$ .",
"As 1 is the unique real zero of $\\varphi (x)$ if $AB< 4$ and $\\varphi ^{\\prime }(1)>0$ in this range, it becomes apparent that $\\varphi (x)$ possesses two complex zeros if $AB<4$ .",
"Those complex solutions are not going to be taken into account throughout this paper.",
"Subsequently, we are going to regard $B$ as the main continuation parameter in problem (REF ).",
"According to our previous analysis, we already know that $(1,1)$ is the unique $2T$ -periodic solution of (REF ) if $B<4/A$ (note that the minimal period of this solution is $T$ ), whereas (REF ) possesses (exactly) three $2T$ -periodic solutions for every $B>4/A$ .",
"Moreover, two of them, those with minimal period $2T$ , bifurcate from $(1,1)$ as the parameter $B$ crosses the critical value $4/A$ , as it will become apparent later.",
"Precisely, we regard the solutions of (REF ) as solutions of $0=\\varphi (B,x):= x\\left( e^{\\frac{e^{(1-x)A}-1}{e^{(1-x)A}+1}B}+1\\right)-2,\\qquad x\\in [0,2],$ for some $B>0$ .",
"Note that $(B,x)=(B,1)$ is a solution curve of (REF ) defined for all $B>0$ .",
"Moreover, the linearization of (REF ) at $(B,1)$ is $\\mathfrak {L}(B)= \\frac{d\\varphi (B,1)}{dx} =2-\\frac{AB}{2}$ which establishes an isomorphism of $\\mathbb {R}$ , unless $B=4/A$ .",
"Thus, this is the unique value of the parameter where bifurcation to $2T$ -periodic solutions of (REF ) can occur from $(1,1)$ .",
"Since $N[\\mathfrak {L}(4/A)]= \\mathbb {R}= \\mathrm {span\\,}[1]$ and $\\mathfrak {L}_1 := \\frac{d \\mathfrak {L}(4/A)}{d B} = -\\frac{A}{2}\\ne 0,$ it becomes apparent that $\\mathfrak {L}_1 1 \\notin R[\\mathfrak {L}(4/A)]=[0].$ Hence, by the main theorem of M. G. Crandall and P. H. Rabinowitz [3], there exist $s_0>0$ and two analytic maps, $x, B : (-s_0,s_0)\\rightarrow \\mathbb {R}$ such that $x(0)=1$ , $B(0)=4/A$ , $x(s)=1+s+\\mathcal {O}(s^2)$ as $s\\rightarrow 0$ , and $\\varphi (B(s),x(s))=0$ for every $s\\in (-s_0,s_0)$ .",
"Moreover, except for $x=1$ , these are the unique solutions of $\\varphi (B,x)=0$ in a neighborhood of $(B,x)=(4/A,1)$ .",
"As due to Theorem REF , $\\varphi (B,x)=0$ cannot admit a solution $x\\ne 1$ if $B\\le 4/A$ , it becomes apparent that $B(s)>4/A$ for all $s\\in (-s_0,s_0)$ .",
"Note that $x(s)>1$ if $s\\in (0,s_0)$ , while $x(s)<1$ if $s\\in (-s_0,0)$ .",
"On the other hand, it readily follows from (REF ) that $\\varphi (B,x)<0$ if $x\\le 0$ and $\\varphi (B,x)>0$ if $x\\ge 2$ .",
"Thus, any solution of $\\varphi (B,x)=0$ satisfies $x \\in (0,2)$ .",
"In particular, $x(s) \\in (0,2)$ for all $s\\in (-s_0,s_0)$ .",
"Thus, as owing to [9] any solution, $(B,x)$ , of $\\varphi =0$ with $B>4/A$ is non-degenerated, by a rather standard continuation argument involving the Implicit Function Theorem the next result holds true.",
"Theorem 2.2 The set of zeros $(B,x)$ of $\\varphi =0$ with $x\\ne 1$ , consists of a (global) analytic curve, $(B(s),x(s))$ , $s\\in \\mathbb {R}$ , such that $x(s)\\in (0,2)$ for all $s\\in \\mathbb {R}$ and $B(\\mathbb {R})=(4/B,+\\infty )$ , much like illustrated by Figure REF .",
"Actually, each of the two half-branches, the upper and the lower ones, can be globally parameterized by $B\\in (4/A,+\\infty )$ .",
"Figure: The set of 2T2T-periodic solutions of ().Since $\\varphi (B,x)=0$ can be equivalently written down as $\\dfrac{2}{x} -1=e^{\\tfrac{e^{(1-x)A}-1}{e^{(1-x)A}+1}B},$ letting $B\\rightarrow +\\infty $ in this identity, it becomes apparent that $\\lim _{B\\uparrow +\\infty } x = \\left\\lbrace \\begin{array}{ll} 0 & \\quad \\hbox{if}\\;\\; x\\in (0,1), \\\\2 & \\quad \\hbox{if}\\;\\; x\\in (1,2),\\end{array}\\right.$ which is reflected in the global bifurcation diagram of Figure REF ."
],
[
"Constructing the $nT$ -Poincaré maps",
"Throughout the rest of this paper, for every integer $n\\ge 1$ , we denote by $\\mathcal {P}_n$ the $nT$ -Poincaré map of (REF ), and, for every initial data $(u_0,v_0)$ , with $u_0>0$ and $v_0>0$ , we set $(u_n,v_n):=\\mathcal {P}_n (u_0,v_0) = \\mathcal {P}_1^n (u_0,v_0)=(u_0,v_0).$ Then, iterating (REF ) $n$ times, it becomes apparent that $(u_n,v_n)=\\left( u_0 e^{(n-v_0-v_1-\\cdots -v_{n-1})A},v_0e^{(u_1+u_2+\\cdots + u_n -n)B}\\right), \\qquad n\\ge 1.$ Consequently, the solution of (REF ) with initial data $(u_0,v_0)$ , with $u_0>0$ and $v_0>0$ , provides us with a $nT$ -periodic coexistence state of (REF ) if, and only if, $\\left\\lbrace \\begin{array}{ll}n=u_0 + u_1 + \\cdots + u_{n-1}, \\\\[2pt]n=v_0 + v_1 + \\cdots + v_{n-1},\\end{array}\\right.$ where we are using that $u_n=u_0$ .",
"According to (REF ), (REF ) can be equivalently expressed as $\\left\\lbrace \\begin{array}{ll}n=u_0\\left[ 1 + e^{(1-v_0)A} + e^{(2-v_0-v_1)A} + \\cdots + e^{(n-1 - v_0 - v_1 - \\cdots - v_{n-2})A}\\right], \\\\[4pt]n=v_0\\left[ 1 + e^{(u_1-1)B} + e^{(u_1+u_2-2)B} + \\cdots + e^{[u_1+u_2+\\cdots +u_{n-1} - (n-1)]B}\\right].\\end{array}\\right.$ As already shown in the proof of Theorem REF , in the special case when $n=2$ , $u_0$ can be easily obtained as a (explicit) function of $v_0$ , which allowed as to express the system as a single equation of the unknown $x=v_0$ .",
"As this strategy does not work when $n\\ge 3$ , in order to construct the set of $nT$ -periodic solutions of (REF ) for all $n\\ge 3$ , throughout the rest of this paper we will make the additional assumption that $x:=u_0=v_0 \\quad \\hbox{and} \\quad A=B.$ Later, we will analyze their global topological structure through the distribution of their bifurcation points from the trivial curve $x=1$ .",
"Under these assumptions the next result holds.",
"It is a pivotal result to express the Poincaré maps in a manageable way.",
"Lemma 3.1 Suppose (REF ) and (REF ).",
"Then, for every $n\\ge 2$ , $u_h=v_{n-h} \\qquad \\hbox{for all}\\;\\; h\\in \\lbrace 1,\\ldots ,n-1\\rbrace .$ Thus, the two equations of the system (REF ) coincide.",
"Fix $n\\ge 2$ .",
"Then, owing to (REF ), (REF ) and (REF ), we find that $u_1 = u_0e^{(1-v_0)A} = v_0e^{(1-u_0)B} = v_0e^{[u_1+u_2+\\cdots +u_{n-1}-(n-1)]B} =v_{n-1}.$ This relation provides us with the first identity of (REF ) ($h=1$ ).",
"In particular, it shows (REF ) when $n=2$ .",
"More generally, suppose that $n\\ge 3$ and that there exists $k\\ge 1$ such that $u_h=v_{n-h} \\qquad \\hbox{for all}\\;\\; h \\in \\lbrace 1,\\ldots ,n-k-1\\rbrace .$ Then, thanks to (REF ), we have that $u_{n-k}=u_0e^{[(n-k)-v_0-v_1-\\cdots -v_{n-k-1}]A}.$ Thus, by (REF ) and (REF ), $u_{n-k} & =v_0e^{[(n-k)-u_0-u_{n-1}-\\cdots -u_{k+1}]B} \\\\ & =v_0e^{[(n-k)-u_0-u_{n-1}-\\cdots -u_{k+1}-u_k-u_{k-1}-\\cdots -u_1+k+u_1+u_2+\\cdots +u_k-k]B}.$ Thus, due to (REF ), it becomes apparent that $u_{n-k}=v_0e^{(u_1+\\cdots +u_k-k)B}=v_k,$ which concludes the proof of (REF ).",
"Therefore since (REF ) is equivalent to (REF ), the two equtions of (REF ) coincide.",
"The proof is complete.",
"According to Lemma REF , under condition (REF ), to construct the fixed points of the Poincaré map $\\mathcal {P}_n$ , it suffices to consider any of the identities of (REF ) (or (REF )), for instance, the first one.",
"Thus, setting $\\varphi _n(u_0):=u_0+u_1(u_0)+u_2(u_0)+\\cdots +u_{n-1}(u_0)-n, \\qquad u_0>0,$ it becomes apparent that the zeros of $\\varphi _n$ provide us with the positive fixed points of the Poincaré map $\\mathcal {P}_n$ .",
"By (REF ) $u_1(u_0):= u_0e^{(1-u_0)A},$ $u_2(u_0):=u_0e^{[2-u_0-v_1(u_0)]A}=u_0 e^{[2-u_0-v_0e^{(u_1-1)A}]A}=u_0 e^{(2-u_0-u_0e^{[u_0e^{(1-u_0)A}-1]A})A}$ and so on... though, in order to get a manageable expression for $\\varphi _n(u_0)$ , all these terms should be reorganized in a slightly tricky way by using the relationships (REF ), or (REF ), which will be described in the proof of Theorem REF .",
"The next result provides us with the Poincaré maps.",
"Proposition 3.2 Setting $\\left\\lbrace \\begin{array}{lll}E_0(x):=1,\\quad E_1(x):=e^{(1-x)A}, \\\\[4pt]E_n(x):=\\left\\lbrace \\begin{array}{ll} e^{[x(E_1(x)+E_3(x)+\\cdots +E_{n-1}(x))-\\frac{n}{2}]A} & \\quad \\mathrm {if\\ } \\;\\; n\\in 2\\mathbb {N}, \\\\[4pt]e^{[\\frac{n+1}{2}-x(E_0(x)+E_2(x)+\\cdots +E_{n-1}(x))]A} & \\quad \\mathrm {if\\ } \\;\\; n\\in 2\\mathbb {N}+1, \\end{array} \\right.\\end{array}\\right.$ for every $ n\\ge 1 $ the Poincaré map is given through $(u_n,v_n)=\\mathcal {P}_n(x,x)=\\left(xE_{2n-1}(x),xE_{2n}(x)\\right).$ By (REF ) and the definition of $E_1$ and $E_2$ , it is easily seen that $u_1=xe^{(1-x)A}=xE_1(x)\\quad \\hbox{and} \\quad v_1=xe^{[xe^{(1-x)A}-1]A}=xE_2(x).$ Assume, as an induction hypothesis, that, for some integer $n\\ge 1$ , $u_{n-1}=xE_{2(n-1)-1}(x)\\quad \\hbox{and} \\quad v_{n-1}=xE_{2(n-1)}(x).$ To prove (REF )we argue as follows.",
"According to (REF ), $u_n =xe^{(n-v_0-v_1-\\cdots -v_{n-1})A}=u_{n-1}e^{(1-v_{n-1})A}.$ Thus, by the induction hypothesis and (REF ), $u_n& = xE_{2n-3}(x)e^{(1-xE_{2n-2}(x))A}\\\\&=xe^{[\\frac{2n-2}{2}-x(E_0+E_2+\\cdots +E_{2n-4})]A}e^{(1-xE_{2n-2})A}\\\\&=xe^{[n-x(E_0+E_2+\\cdots +E_{2n-4}+E_{2n-2})]A}=xE_{2n-1}(x).$ This provides us with the value of $u_n$ in (REF ).",
"Similarly, $v_n=xe^{(u_1+\\cdots +u_n-n)A}=v_{n-1}e^{(u_n-1)A}.$ Thus, by (REF ), since we already know that $u_n=xE_{2n-1}(x)$ , we can infer that $v_n & =xE_{2n-2}(x)e^{(xE_{2n-1}(x)-1)A}\\\\&=xe^{[x(E_1+E_3+\\cdots +E_{2n-3})-(n-1)]A}e^{(xE_{2n-1}-1)A}\\\\&=xe^{[x(E_1+E_3+\\cdots +E_{2n-3}+E_{2n-1})-n]A}=xE_{2n}(x).$ This ends the proof.",
"As a direct consequence, from Proposition (REF ) one can get the auxiliary maps $\\varphi _n$ , $n\\ge 1$ , introduced in (REF ).",
"Theorem 3.3 For every integer $n\\ge 1$ , $\\varphi _n(x)= \\varphi _{n-1}(x) - 1 + xE_{n-1}(x).$ First note that when $n$ is an odd integer, according to Lemma REF , we have that $\\varphi _n(x)& =u_0+u_1+\\cdots +u_{\\frac{n-1}{2}}+u_{\\frac{n-1}{2}+1}+u_{\\frac{n-1}{2}+2}+\\cdots + u_{n-2}+u_{n-1}-n \\nonumber \\\\ & =u_0+u_1+\\cdots +u_{\\frac{n-1}{2}}+v_{\\frac{n-1}{2}}+v_{\\frac{n-1}{2}-1}+\\cdots + v_{2}+v_{1}-n\\nonumber \\\\ & =u_0+u_1+v_1+u_2+v_2+\\cdots + u_{\\frac{n-1}{2}-1}+v_{\\frac{n-1}{2}-1}+u_{\\frac{n-1}{2}}+v_{\\frac{n-1}{2}}-n.$ Similarly, when $n$ is even, $\\varphi _n(x)& =u_0+u_1+\\cdots +u_{\\frac{n}{2}}+u_{\\frac{n}{2}+1}+u_{\\frac{n}{2}+2}+\\cdots + u_{n-2}+u_{n-1}-n \\nonumber \\\\ & =u_0+u_1+\\cdots +u_{\\frac{n}{2}}+v_{\\frac{n}{2}-1}+v_{\\frac{n}{2}-2}+\\cdots + v_{2}+v_{1}-n\\nonumber \\\\ & =u_0+u_1+v_1+u_2+v_2+\\cdots + u_{\\frac{n}{2}-1}+v_{\\frac{n}{2}-1}+u_{\\frac{n}{2}}-n.$ To prove (REF ) a complete induction argument will be used.",
"When $n=1$ , $\\varphi _1(x)=x-1.$ When $n=2$ , by (REF ), $\\varphi _2(x):=x+u_1-2=x+x e^{(1-x)A}-2=\\varphi _1(x)-1+x E_1(x).$ As the complete induction hypothesis, suppose that, for any given $\\nu \\ge 2$ , (REF ) holds for every $n \\in \\lbrace 1,2,\\ldots ,2\\nu -3,2\\nu -2\\rbrace $ .",
"Then, thanks to (REF ) and (REF ), $\\varphi _{2\\nu -1}(x) & = u_0+u_1+v_1+u_2+v_2+\\cdots + u_{\\nu -2}+v_{\\nu -2}+u_{\\nu -1}+v_{\\nu -1}-2\\nu +1, \\\\\\varphi _{2\\nu }(x)& = u_0+u_1+v_1+u_2+v_2+\\cdots + u_{\\nu -1}+v_{\\nu -1}+u_{\\nu }-2\\nu .$ Thus, thanks to (REF ), $\\begin{split}\\varphi _{2\\nu -1}(x)=x & +xE_1(x) +x E_2(x) +x E_3(x) + x E_4(x)+\\cdots \\\\ & + xE_{2\\nu -5}(x)+xE_{2\\nu -4}(x) +x E_{2\\nu -3}(x)+ x E_{2\\nu -2}(x) -2\\nu +1.\\end{split}$ Similarly, $\\begin{split}\\varphi _{2\\nu }(x)=x & +xE_1(x) +x E_2(x) +x E_3(x) + x E_4(x)+\\cdots \\\\ & + xE_{2\\nu -3}(x)+xE_{2\\nu -2}(x) +x E_{2\\nu -1}(x)-2\\nu .\\end{split}$ Therefore, by the induction hypothesis, $\\varphi _{2\\nu -1}(x)=\\varphi _{2\\nu -2}(x) -1 + x E_{2\\nu -2}(x).$ Similarly, $\\varphi _{2\\nu }(x)= \\varphi _{2\\nu -1}(x)-1+x E_{2\\nu -1}(x).$ The proof is complete.",
"Remark 3.4 By (REF ) and (REF ) it becomes apparent that $\\varphi _n(0)=-n <0 \\quad \\hbox{and}\\quad \\varphi _{n}(n) > 0 \\quad \\hbox{for all integer}\\;\\; n\\ge 2.$ According to Theorem REF , it is easily seen that $\\varphi _1(x) & =x-1, \\\\ \\varphi _2(x) & = \\varphi _1(x)-1+x e^{(1-x)A},\\nonumber \\\\\\varphi _3(x) &= \\varphi _2(x)-1 +xe^{(xe^{(1-x)A}-1)A},\\nonumber \\\\\\varphi _4(x) & =\\varphi _3(x)-1+ xe^{(2-x-xe^{(xe^{(1-x)A}-1)A})A}.\\nonumber $ Crucially, in the formula for $\\varphi _2(x)$ given by Theorem REF it is only required to compose two exponentials, while in (REF ) we had to nest three.",
"Such reduction in the complexity of $\\varphi _2$ is explained by the symmetries revealed by Lemma REF which facilitated the reorganization of the terms of $\\varphi _n$ as to get a function with a minimal number of nested exponentials, much like in the algorithm of the proof of Theorem REF .",
"Using this algorithm, the number of nested exponentials decreases by one when $n$ is odd, and each of the the $E_n$ 's defined by (REF ) consists of a composition of exactly $n$ exponentials.",
"The relevance of this reduction will not be completely understood until the next sections, where the structure of the zeros of the $\\varphi _n$ 's introduced in (REF ) will be analyzed.",
"Those zeros are the positive fixed points of the $nT$ -time Poincaré maps.",
"The main technical difficulty to determine the zeros of the $\\varphi _n$ 's, even from the point of view of numerical analysis, relies on the high sensitivity of these functions to very small variations in the value of the parameter $A=B$ .",
"The higher the number of exponentials nested, the higher the sensitivity in $A$ .",
"As a result, when one tries to determine numerically the zeros of the map $\\varphi _4$ for values of $A$ near 4, the function $\\varphi _4(x)$ takes values of order $10^{31}$ in a neighborhood of zero.",
"So, there is no chance to compute the zeros of these maps assisted by the computer.",
"When dealing with $\\varphi _5$ the value of the parameter $A=B$ should not exceed the value $2.5$ , which is extremely unsatisfactory for our purposes here.",
"These technical troubles inherent to the internal structure of the associated maps $\\varphi _n$ push us to make a direct analysis of the global structure of their zeros.",
"In order to perform this global analysis we first need to ascertain the set of bifurcation points of $\\varphi _n=0$ from the curve $(A,1)$ .",
"This analysis will be carried out in the next section.",
"A canonical chain of associated polynomials Searching for the potential bifurcation points from the curve $(A,1)$ to $nT$ -periodic coexistence states, this section analyzes the spectrum of the linearized family $\\mathfrak {L}(n;A):= \\frac{d\\varphi _n (A,1)}{dx},\\quad n\\in \\mathbb {N},$ i.e., its zero set as a function of the parameter $A$ , as well as the global structure of $\\mathfrak {L}(n;A)$ .",
"Note that, since $(A,1)$ is the $T$ -periodic coexistence state, it also provides us with a $nT$ -periodic solution for all $n\\ge 1$ and, hence, by construction, $\\varphi _n(A,1)=0$ for all $A>0$ and $n\\ge 1$ .",
"The curve $(A,1)$ , $A>0$ , is the trivial curve, as it is known.",
"It is the curve from which are going to bifurcate the $nT$ -periodic coexistence states of (REF ) under assumption (REF ).",
"Note also that, since every $nT$ -periodic solution is $knT$ -periodic for all integer $k\\ge 1$ , $\\varphi _{kn}(x)=0 \\quad \\hbox{for all}\\;\\; x \\in \\varphi _n^{-1}(0)\\quad \\hbox{and}\\quad k\\ge 1.$ Throughout the rest of this paper we will denote $p_n(A):= \\frac{d\\varphi _n (A,1)}{dx}=\\mathfrak {L}(n;A),\\qquad A>0.$ Differentiating with respect to $x$ the identity (REF ) yields $p_n(A)=p_{n-1}(A)+ E_{n-1}(1)+E^{\\prime }_{n-1}(1)\\qquad \\hbox{for all}\\;\\; A>0.$ The next result shows that $p_n \\in \\mathbb {Z}\\left[ A\\right]$ .",
"Lemma 4.1 For every $ n\\in \\mathbb {N}$ , $p_n(A) $ is a polynomial in the variable $A$ with integer coefficients, i.e., $ p_n \\in \\mathbb {Z}[A] $ .",
"By (REF ), it becomes apparent that, since $(1,1)$ is a fixed point of $ \\mathcal {P}_n $ , $(1,1)=\\mathcal {P}_n(1,1)=\\left(E_{2n-1}(1),E_{2n}(1)\\right)$ for all integer $n\\ge 1$ .",
"Thus, $E_n(1)=1 \\quad \\hbox{for all}\\;\\; n\\ge 0.$ Thus, (REF ) becomes $p_n(A)=p_{n-1}(A)+ 1 +E^{\\prime }_{n-1}(1)$ for all $A>0$ and $n\\ge 1$ .",
"Therefore, due to (REF ) , $p_1(A) = \\frac{d\\varphi _1(A,1)}{dx}= 1$ and iterating (REF ) $n-2$ times show that, for every integer $n\\ge 2$ , $p_n(A)= n+E_1^{\\prime }(1)+E_2^{\\prime }(1)+\\cdots +E_{n-1}^{\\prime }(1).$ Consequently, to complete the proof it suffices to show that $ E_n^{\\prime }(1)\\in \\mathbb {Z}[A] $ for all $n\\ge 1$ .",
"Indeed, by (REF ), $E_0^{\\prime }(1)=0$ , $ E_1^{\\prime }(1)=-A $ and $E_n^{\\prime }(1)=\\left\\lbrace \\begin{array}{ll}\\displaystyle { A \\sum _{\\begin{array}{c}j=1\\\\j\\in 2\\mathbb {N}+1\\end{array}}^{n-1} \\left[E_{j}(1) +E_{j}^{\\prime }(1)\\right] } &\\qquad \\hbox{if\\ } n\\in 2\\mathbb {N},\\\\[14pt]\\displaystyle { -A \\sum _{\\begin{array}{c}j=0\\\\ j\\in 2\\mathbb {N}\\end{array}}^{n-1}\\left[ E_{j}(1)+ E_{j}^{\\prime }(1) \\right]} & \\qquad \\hbox{if\\ } n\\in 2\\mathbb {N}+1.\\end{array}\\right.$ Thus, by a complete induction argument it becomes apparent that $ E_n^{\\prime }(1)\\in \\mathbb {Z}[A] $ for all $ n\\in \\mathbb {N}$ .",
"This concludes the proof.",
"Remark 4.2 In Section 5 we will prove that all the roots of the polynomial $p_n(A)$ are simple.",
"In other words, $p^{\\prime }_n(r) = \\frac{d \\mathfrak {L}}{d A}(n;r)\\ne 0$ for all $r\\in p_n^{-1}(0)$ .",
"Thus, the transversality condition of M. G. Crandall and P. H. Rabinowitz [3] holds true.",
"Therefore, by the main theorem of [3], at every positive root of $p_n(A)$ , $r$ , an analytic curve of $nT$ -periodic coexistence states of (REF ) bifurcates from $(A,1)$ at $r$ .",
"This feature explains our interest here in analyzing the nature and the distribution of the positive roots of the polynomials $p_n(A)$ , $n\\in \\mathbb {N}$ .",
"Remark 4.3 Occasionally, we will make explicit the dependence of the function $\\varphi _n(x)$ on the parameter $A$ by setting $\\varphi _n(A,x)$ , instead of $\\varphi _n(x)$ .",
"Similarly, we will set $E_n(A,x):=E_n(x)$ for all $n\\in \\mathbb {N}$ .",
"According to (REF ), $E_n(0,x)=1$ for all $n\\in \\mathbb {N}$ and $x \\in [0,n]$ .",
"Thus, (REF ) yields $\\varphi _n(0,x)= \\varphi _{n-1}(0,x) - 1 + x$ for all $n\\in \\mathbb {N}$ and $x\\in [0,n]$ .",
"Therefore, iterating $n-1$ times, it becomes apparent that $\\varphi _n(0,x)=n(x-1)\\quad \\hbox{for all}\\;\\; n\\in \\mathbb {N}.",
"$ As the zeros of $\\varphi _n(A,x)$ provide us with the $nT$ -periodic positive solutions of (REF ), it follows from (REF ) that $x=1$ is the unique $nT$ -periodic solution, for all $n\\in \\mathbb {N}$ , at the particular value of the parameter $A=0$ .",
"The next list collects the polynomials $p_n(A)$ for $1\\le n\\le 13$ .",
"$p_1(A)&=1\\\\p_2(A)&=-A+2\\\\p_3(A)&=-A^2+3\\\\p_4(A)&=A^3-2A^2-2A+4\\\\p_5(A)&=A^4-5A^2+5\\\\p_6(A)&=-A^5+2A^4+4A^3-8A^2-3A+6\\\\p_7(A)&=-A^6+7A^4-14A^2+7\\\\p_8(A)&=A^7-2A^6-6A^5+12A^4+10A^3-20A^2-4A+8\\\\p_9(A)&=A^8-9A^6+27A^4-30A^2+9\\\\p_{10}(A)&=-A^9+2A^8+8A^7-16A^6-21A^5+42A^4+20A^3-40A^2-5A+10\\\\p_{11}(A)&=-A^{10}+11A^8-44A^6+77A^4-55A^2+11\\\\p_{12}(A)&=A^{11}-2A^{10}-10A^9+20A^8+36A^7-72A^6-56A^5+112A^4+35A^3-70A^2-6A+12\\\\p_{13}(A)&=A^{12}-13A^{10}+65A^8-156A^6+182A^4-91A^2+13.$ The next table collects the coefficients of all the polynomials listed above.",
"Table: First thirteen polynomials coefficients.By simply having a glance to these polynomials, it becomes apparent that the following properties hold: The constant terms of $p_n(A)$ equals $n$ .",
"The degree of $p_n(A)$ equals $n-1$ .",
"The leading coefficients of $p_{4n}(A)$ and $p_{4n+1}(A)$ equal 1, while the leading coefficients of $p_{4n+2}(A)$ and $p_{4n+3}(A)$ equal $-1$ .",
"$p_{2n}(2)=0$ for all integer $n\\ge 1$ .",
"Thus, $p_2|p_{2n}$ for all $n\\ge 1$ .",
"$p_{2n+1}(A)$ is an even function.",
"Besides these properties, it seems all the coefficients of $p_n(A)$ , except the leading one, must be multiples of $n$ if $n$ is a prime integer, though this property will not be used in this paper.",
"The next result shows the property (a).",
"Lemma 4.4 $ p_n(0)=n $ for all $ n\\ge 1$ .",
"By (REF ), $ \\frac{dE_n(0,1)}{dx}=0 $ .",
"Hence, due to (REF ), $p_n(0)=n$ for all $ n\\ge 1$ .",
"The next result establishes the properties (b) and (c).",
"Lemma 4.5 For every integer $n\\ge 1$ , $ {\\rm {deg}}(p_n)=n-1 $ .",
"Moreover, the leading coefficients of $ p_n $ equal 1 if $ n\\in 4\\mathbb {N}\\cup (4\\mathbb {N}+1) $ and $ -1 $ if $ n\\in (4\\mathbb {N}+2)\\cup (4\\mathbb {N}+3) $ .",
"By the proof of Lemma REF , we already know that $E^{\\prime }_n(A,1):=\\frac{d E_n}{dx} (A,1)$ is a polynomial in $A$ for all integer $n\\ge 1$ .",
"Next, we will show that it has degree $ n $ .",
"To prove it, a complete induction argument will be used.",
"According to (REF ), we already know that $ {\\rm {deg}}(E_0^{\\prime }(A,1))={\\rm {deg}}(0)=0 \\quad \\hbox{and}\\quad {\\rm {deg}}(E_1^{\\prime }(A,1))={\\rm {deg}}(-A)=1.$ As the induction assumption, assume that ${\\rm {deg}}(E_j^{\\prime }(A,1))=j \\quad \\hbox{for all}\\; \\; j<n.$ Then, owing to (REF ), it follows that ${\\rm {deg}}(E_n^{\\prime }(A,1))=n,\\qquad n\\ge 0.$ Therefore, by (REF ), ${\\rm {deg}}(p_n)=n-1.$ Subsequently, for any given polynomial, $q \\in \\mathbb {Z}[A]$ , we will denote by $\\ell (q)$ the leading coefficient of $q(A)$ .",
"According to Table REF , we already know that $\\ell (p_5)=1.$ As an induction hypothesis, assume that $\\ell (p_{4(n-1)+1})=1.$ By (REF ), (REF ) and (REF ) $\\ell (p_{4n-2}) & = {\\color {blue} \\ell (E_{4n-3}^{\\prime }(A,1))}=-\\ell (E^{\\prime }_{4(n-1)}(A,1))=-\\ell (p_{4(n-1)+1}),\\\\\\ell ({p_{4n-1}}) & ={\\color {brown} \\ell ({E_{4n-2}^{\\prime }(A,1)})} ={\\color {blue} \\ell ({E_{4n-3}^{\\prime }(A,1)})}=-\\ell (p_{4(n-1)+1}),\\\\\\ell ({p_{4n}}) & ={\\color {cyan} \\ell ({E_{4n-1}^{\\prime }(A,1)})}=-{\\color {brown} \\ell ({E_{4n-2}^{\\prime }(A,1)})}=\\ell (p_{4(n-1)+1}),\\\\\\ell ({p_{4n+1}}) & =\\ell ({E_{4n}^{\\prime }(A,1)})={\\color {cyan} \\ell ({E_{4n-1}^{\\prime }(A,1)})}=\\ell (p_{4(n-1)+1}).\\\\$ By (REF ), the proof is complete.",
"As a consequence of these lemmas, the next result holds.",
"Proposition 4.6 Suppose (REF ).",
"Then, the problem (REF ) possesses infinitely many subharmonics.",
"In other words, there exists a sequence of integers $\\lbrace n_m\\rbrace _{m\\ge 1}$ with $\\lim _{m\\rightarrow +\\infty } n_m =+\\infty ,$ such that (REF ) has at least a $n_mT$ -periodic coexistence state for every $m\\ge 1$ .",
"Since $p_n(0)=n$ for all $n\\in \\mathbb {N}$ and, thanks to Lemma REF , for every integer $n\\ge 1$ , $\\ell (p_{4n+2})=\\ell (p_{4n+3})=-1,$ it becomes apparent that $p_{4n+2}(A)$ (resp.",
"$p_{4n+3}(A)$ ) possesses a root, $A_{4n+2}$ (resp.",
"$A_{4n+3}$ ), where it changes of sign.",
"Thus, for every integer $n\\ge 1$ , there exist two odd integers, $i_n, j_n\\ge 1$ , for which $p^{k)}_{4n+2}(A_{4n+2}) & =0, \\quad 0\\le k \\le i_n-1, \\quad p^{i_n)}_{4n+2}(A_{4n+2}) \\ne 0, \\\\p^{k)}_{4n+3}(A_{4n+3}) & =0, \\quad 0\\le k \\le j_n-1, \\quad p^{j_n)}_{4n+3}(A_{4n+3}) \\ne 0.$ Thus, the algebraic multiplicity of [7] for these polynomials at those roots is given by $\\chi [p_{4n+2}(A);A_{4n+2}]= i_n, \\qquad \\chi [p_{4n+3}(A);A_{4n+3}]= j_n.$ As these integers are odd, by Theorem 5.6.2 of [10], the local topological indexes of $p_{4n+2}(A)$ and $p_{4n+3}(A)$ change as $A$ crosses $A_{4n+2}$ and $A_{4n+3}$ , respectively.",
"Therefore, by Theorem 6.2.1 of [10], there exist two components of $(4n+2)T$ -periodic solutions and $(4n+3)T$ -periodic solutions bifurcating from the trivial solution $(A,1)$ at the roots $A_{4n+2}$ and $A_{4n+3}$ , respectively.",
"This ends the proof.",
"The next result establishes Property (d).",
"Lemma 4.7 $ p_2|p_{2n} $ for all $ n\\ge 1$ .",
"Thus, since $p_2(A)=-A+2$ , $r=2$ is a root of $p_{2n}(A)$ for all integer $n\\ge 1$ .",
"By (REF ), any $2T$ -periodic solution is a $2nT$ -periodic solution for all $n\\ge 1$ .",
"Thus, any bifurcation point from $(A,1)$ to $2T$ -periodic solutions must be a bifurcation point to $2nT$ -periodic solutions.",
"Since the unique bifurcation value to $2T$ -periodic solutions is the root of $p_2(A)=-A+2$ , given by $r=2$ , it becomes apparent that $p_{2n}(2)=0$ for all integer $n\\ge 1$ .",
"Therefore, $p_2|p_{2n}$ for all $n\\ge 1$ .",
"This ends the proof.",
"The next list of polynomials, collecting $p_{2n+1}(A)$ and $\\frac{p_{2n}(A)}{2-A}$ , for $1 \\le n \\le 6$ , might be helpful to understand the (very sharp) identity established by the next result.",
"$\\dfrac{p_2(A)}{2-A}&=1\\\\p_3(A)&=-A^2+3\\\\\\dfrac{p_4(A)}{2-A}&=-A^2+2\\\\p_5(A)&=A^4-5A^2+5\\\\\\dfrac{p_6(A)}{2-A}&=A^4-4A^2+3\\\\p_7(A)&=-A^6+7A^4-14A^2+7\\\\\\dfrac{p_8(A)}{2-A}&=-A^6+6A^4-10A^2+4\\\\p_9(A)&=A^8-9A^6+27A^4-30A^2+9\\\\\\dfrac{p_{10}(A)}{2-A}&=A^8-8A^6+21A^4-20A^2+5\\\\p_{11}(A)&=-A^{10}+11A^8-44A^6+77A^4-55A^2+11\\\\\\dfrac{p_{12}(A)}{2-A}&=-A^{10}+10A^8-36A^6+56A^4-35A^2+6\\\\p_{13}(A)&=A^{12}-13A^{10}+65A^8-156A^6+182A^4-91A^2+13.$ Theorem 4.8 The following identity holds $\\dfrac{p_n(A)}{2-A}=p_{n-1}(A) - \\dfrac{p_{n-2}(A)}{2-A}$ for all $n\\in 2\\mathbb {N}$ , whereas $\\dfrac{p_n(A)}{2+A}=p_{n-1}(A) - \\dfrac{p_{n-2}(A)}{2+A}$ for all $n\\in 2\\mathbb {N}+1$ .",
"First, we will prove the next relationships $\\left\\lbrace \\begin{array}{ll}\\displaystyle {- \\frac{n}{2}-1-\\sum _{\\begin{array}{c}j=1\\\\ j \\in 2\\mathbb {N}\\end{array}}^{n} E_{j}^{\\prime }(A,1) =1-A+\\sum _{j=3}^{n+1}(-1)^jp_j,}&\\quad n\\in 2\\mathbb {N},\\\\[11pt]\\displaystyle { \\left[\\frac{n}{2}\\right]+1+\\sum _{\\begin{array}{c}j=1\\\\ j \\in 2\\mathbb {N}+1\\end{array}}^{n} E_{j}^{\\prime }(A,1) =1-A+\\sum _{j=3}^{n+1}(-1)^jp_j,}&\\quad n\\in 2\\mathbb {N}+1.\\end{array}\\right.$ Since $p_2(A)=2-A$ , particularizing (REF ) at $n=3$ yields $-2-E_2^{\\prime }(A,1)=1-A-p_3(A),$ which is (REF ) for $n=2$ .",
"As the induction assumption, assume that (REF ) holds for some $n=2m$ with $m\\ge 1$ , i.e., $- m -1-\\sum _{\\begin{array}{c}j=1\\\\ j \\in 2\\mathbb {N}\\end{array}}^{2m} E_{j}^{\\prime }(A,1) =1-A+\\sum _{j=3}^{2m+1}(-1)^jp_j(A).$ According to (REF ), $2m+2+E_1^{\\prime }(A,1)+E_2^{\\prime }(A,1)+\\cdots +E_{2m}^{\\prime }(A,1)+E_{2m+1}^{\\prime }(A,1)=p_{2m+2}(A).$ Thus, adding (REF ) and (REF ), we obtain that $m+1 + \\sum _{\\begin{array}{c}j=1\\\\ j \\in 2\\mathbb {N}+1\\end{array}}^{2m+1} E_{j}^{\\prime }(A,1)=1-A+\\sum _{j=3}^{2m+2}(-1)^jp_j(A).$ Equivalently, $\\left[\\frac{2m+1}{2}\\right]+1 + \\sum _{\\begin{array}{c}j=1\\\\ j \\in 2\\mathbb {N}+1\\end{array}}^{2m+1} E_{j}^{\\prime }(A,1)=1-A+\\sum _{j=3}^{2m+2}(-1)^jp_j(A),$ which shows the validity of (REF ) for $n=2m+1$ .",
"To prove the validity of (REF ) for $n=2(m+1)=2m+2$ , we can argue similarly.",
"Again by (REF ), $2m+3+E_1^{\\prime }(A,1)+E_2^{\\prime }(A,1)+\\cdots +E_{2m+1}^{\\prime }(A,1)+E_{2m+2}^{\\prime }(A,1)=p_{2m+3}(A).$ Hence, subtracting (REF ) from (REF ) yields $-m-2 - \\sum _{\\begin{array}{c}j=1\\\\ j \\in 2\\mathbb {N}\\end{array}}^{2m+2} E_{j}^{\\prime }(A,1)=1-A+\\sum _{j=3}^{2m+3}(-1)^jp_j(A).$ Since $-\\frac{2m+2}{2}-1 =-m-2,$ (REF ) provides us with (REF ) for $n=2m+2$ , which ends the proof of (REF ).",
"By (REF ), it follows from (REF ) and (REF ) that $p_n(A)=\\left\\lbrace \\begin{array}{ll}\\displaystyle {p_{n-1}(A)+1+A\\Big [-\\dfrac{n}{2}-\\sum _{\\begin{array}{c}j=1\\\\j\\in 2\\mathbb {N}\\end{array}}^{n-2}E_{j}^{\\prime }(A,1)\\Big ]} & \\quad \\hbox{if\\ } n\\in 2\\mathbb {N},\\\\[15pt]\\displaystyle {p_{n-1}(A) + 1 + A\\Big [\\dfrac{n-1}{2}+\\sum _{\\begin{array}{c}j=1\\\\j\\in 2\\mathbb {N}+1\\end{array}}^{n-2}E_{j}^{\\prime }(A,1)\\Big ]} &\\quad \\hbox{if\\ } n \\in 2\\mathbb {N}+1.",
"\\end{array}\\right.$ On the other hand, when $n\\in 2\\mathbb {N}$ , it follows from (REF ) and (REF ) that $\\begin{split}p_n(A)-p_{n-1}(A)&=1+A\\Big [-\\dfrac{n}{2}-\\sum _{\\begin{array}{c}j=1\\\\j\\in 2\\mathbb {N}\\end{array}}^{n-2}E_{j}^{\\prime }(A,1) \\Big ]\\nonumber \\\\ &=1+A\\Big [-\\dfrac{n-2}{2}-1-\\sum _{\\begin{array}{c}j=1\\\\j\\in 2\\mathbb {N}\\end{array}}^{n-2}E_{j}^{\\prime }(A,1) \\Big ]\\nonumber \\\\ & =1+A\\Big [ 1-A+\\sum _{j=3}^{n-1}(-1)^jp_j\\Big ]\\nonumber \\\\&=1-Ap_{n-1}(A)+A\\Big [1-A+\\sum _{j=3}^{n-2}(-1)^jp_j\\Big ]\\nonumber \\\\&=-Ap_{n-1}(A)+1+A\\Big [\\displaystyle {\\dfrac{n-2}{2}+\\sum _{\\begin{array}{c}j=1\\\\ j\\in 2\\mathbb {N}+1\\end{array}}^{n-3} E_{j}^{\\prime }(A,1)}\\Big ]\\\\&=-Ap_{n-1}(A)+p_{n-1}(A)-p_{n-2}(A).\\end{split}$ Therefore, for every $n\\in 2\\mathbb {N}$ , $p_n(A)= (2-A)p_{n-1}(A)-p_{n-2}(A).$ The proof is complete for $n$ even.",
"Subsequently, we assume that $n$ is odd.",
"Arguing as in the previous case, from (REF ) and (REF ) the following chain of identities holds $p_n(A)-p_{n-1}(A)& = 1+A\\Big [\\dfrac{n-1}{2}+\\sum _{\\begin{array}{c}j=1\\\\j\\in 2\\mathbb {N}+1\\end{array}}^{n-2}E_{j}^{\\prime }(A)\\Big ]\\\\ &=1+A\\Big [1-A+\\sum _{j=3}^{n-1}(-1)^jp_j\\Big ]\\\\&=1+ Ap_{n-1}(A)+A\\Big [1-A+\\sum _{j=3}^{n-2}(-1)^jp_j\\Big ]\\\\&=Ap_{n-1}(A)+1+ A\\Big [-\\dfrac{n-1}{2}-\\sum _{\\begin{array}{c}j=1\\\\j\\in 2\\mathbb {N}\\end{array}}^{n-3}E_{j}^{\\prime }(1)\\Big ]\\\\&=Ap_{n-1}(A)+p_{n-1}(A)-p_{n-2}(A).$ Therefore, for every $n\\in 2\\mathbb {N}+1$ , $p_n(A) =(2+A)p_{n-1}(A)-p_{n-2}(A).$ This ends the proof.",
"Theorem REF can be summarized into the next generalized identity $p_n(A)=[2-(-1)^n A]p_{n-1}(A)-p_{n-2}(A),\\qquad n\\in \\mathbb {N}.$ As a by-product of these identities, the next result, establishing Property (e) at the beginning of the section, holds.",
"Corollary 4.9 For every $n\\ge 1$ , the polynomials $\\dfrac{p_{2n}(A)}{2-A}$ and $p_{2n+1}(A)$ are even.",
"We already know that $\\frac{p_2(A)}{2-A}=1\\qquad \\hbox{and}\\qquad p_3(A)=-A^2+3.$ Arguing by induction, assume that $\\dfrac{p_{2m-2}(A)}{2-A}$ and $p_{2m-1}(A)$ are even polynomials for some $m\\ge 1$ .",
"Then, by (REF ), $\\frac{p_{2m}(A)}{2-A}=p_{2m-1}(A)-\\frac{p_{2m-2}(A)}{2-A}$ must be also even, because it is sum of two even functions.",
"Similarly, since $p_{2m+1}$ can be expressed in the form $p_{2m+1}(A)=(2+A)p_{2m}(A)-p_{2m-1}(A)=(4-A^2)\\dfrac{p_{2m}(A)}{2-A}-p_{2m-1}(A),$ it becomes apparent that $p_{2m+1}(A)$ is also an even polynomial.",
"The proof is completed.",
"Characterizing the bifurcation points from $(A,1)$ The following definition will be used in the statement of the main theorem of this section.",
"Definition 5.1 Given two arbitrary polynomials $q_1, q_2\\in \\mathbb {Z}[A]$ , it is said that the roots of $q_1$ are separated by the roots of $q_2$ if all the roots of $q_2$ lye in between the maximal and minimal roots of $q_1$ and any pair of consecutive roots of $q_2$ contains exactly one root of $q_1$ .",
"The main theorem of this section can be stated as follows.",
"It counts the number of roots of each of the polynomials $p_n(A)$ , $n\\ge 1$ , establishing that there are as many roots as indicated by the degree, that all of them are real and algebraically simple and that the positive roots of $p_{n+1}(A)$ are always separated by the positive roots (less than 2 if $n\\in 2\\mathbb {N}$ ) of $p_n(A)$ .",
"So, it counts all roots establishing their relative positions.",
"Theorem 5.2 For every $n\\ge 2$ , the positive roots of $p_{2n}(A)$ are separated by the positive roots of $p_{2n-1}(A)$ , and the positive roots of $p_{2n+1}(A)$ are separated by those of $p_{2n}(A)$ less than 2.",
"Moreover, for every $n\\ge 1$ , the even polynomials $\\tfrac{p_{2n}(A)}{2-A}$ and $p_{2n-1}(A)$ have (exactly) $n-1$ positive roots.",
"Thus, since they are even with degree $2n-2$ , they must have another $n-1$ negative roots and, therefore, all roots are real and simple.",
"As we have already constructed the associated polynomials above, it is easily seen that all the thesis of Theorem REF hold to be true for $2\\le n\\le 6$ .",
"This task can be easily accomplished by simply looking at Figure REF , where we have plotted all the positive roots of $p_n(A)$ for $2\\le n\\le 13$ .",
"These roots are located in the interval $(0,2]$ and have been represented in abscisas at different levels according to $n$ .",
"As inserting in the same interval $(0,2]$ all the zeros of the first 13 polynomials would not be of any real help for understanding their fine distribution, we have superimposed them at 13 different levels, each of them containing the positive roots of each of the polynomials $p_n$ , $2\\le n\\le 13$ .",
"In total we are representing 42 roots, though some of them are common roots of different polynomials as a result of the fact that any $kT$ -periodic solution must be a $nkT$ -periodic solution for all $n\\ge 1$ .",
"These common roots have been represented in vertical dashed lines to emphasize that all roots on them share the same abscisa value.",
"In such case, the ordinates provide us with the corresponding value of $n$ .",
"By simply having a glance at Figure REF , it is easily realized how the two roots of the polynomial $p_4$ are separated by the root of $p_3$ , the 3 roots of $p_6$ are separated by the 2 roots of $p_5$ , the 4 roots of $p_8$ are separated by the 3 of $p_7$ , and so on...",
"Similarly, the two roots of $p_5$ are separated by the unique root of $p_4$ different from 2, the 3 roots of $p_7$ are separated by the 2 roots of $p_6$ different from 2, and so on...",
"The proof of the theorem will be delivered in two steps by induction in both cases.",
"Since $\\tfrac{p_2(A)}{2-A}=1$ does not admit any root, this is a very special case that will not play any rol in these induction arguments.",
"Figure: Positive roots of p n p_n , 2≤n≤132\\le n \\le 13.",
"Step 1: Passing from $p_{2n}(A)$ to $p_{2n+1}(A)$ , $n\\ge 2$ .",
"According to Figure REF , it becomes apparent that the two positive roots of $p_4(A)$ are separated by the unique root of $p_3(A)$ .",
"Moreover, all these zeros are real and simple and each of the polynomials $p_{3}(A)=-A^2+3,\\qquad \\frac{p_4(A)}{2-A}= -A^2+2,$ has a unique positive root.",
"Arguing by induction, assume that $p_{2n-1}(A)$ and $p_{2n}(A)$ satisfy all the assertions of the statement of the theorem for some $n\\ge 2$ .",
"In other words, all the positive roots of these polynomials are real and algebraically simple, the positive roots of $p_{2n}(A)$ are separated by the positive roots of $p_{2n-1}(A)$ , and the polynomials $\\tfrac{p_{2n}(A)}{2-A}$ and $p_{2n-1}(A)$ have (exactly) $n-1$ positive roots.",
"We claim that the positive roots of the polynomial $p_{2n+1}(A)$ are real and simple, that they are separated by the positive roots of $p_{2n}(A)$ , except for 2, and that it has (exactly) $n$ positive roots.",
"Indeed, by Theorem REF , we already know that $p_{2n+1}(A)=(2+A)p_{2n}(A) - p_{2n-1}(A).$ First, we will show the previous claim in the case when $ 2n\\in 4\\mathbb {N}+2 $ .",
"So, suppose $2n\\in 4\\mathbb {N}+2$ .",
"Figure REF shows the plots of the polynomials $p_{2n-1}(A)$ and $(2+A)p_{2n}(A)$ in one of such cases: Figure: Sketch of the construction of p 2n+1 (A)p_{2n+1}(A).$p_{2n-1}(A)$ has been plotted in brown and $(2+A)p_{2n}(A)$ in blue.",
"According to Lemmas REF and REF , we already know that $2n-1 = p_{2n-1}(0)< 4n = 2p_{2n}(0), \\quad \\mathrm {deg\\,}(p_{2n-1})=2n-2,\\quad \\mathrm {deg\\,}((2+A)p_{2n})=2n,$ and, since $2n\\in 4\\mathbb {N}+2$ , the leading coefficient of $p_{2n-1}(A)$ equals 1, while the leading coefficient of $p_{2n}(A)$ equals $-1$ .",
"Thus, $p_{2n-1}(A)>0$ and $(2+A)p_{2n}(A)<0$ for $A>2$ .",
"By the induction assumption, the polynomials $\\tfrac{p_{2n}(A)}{2-A}$ and $p_{2n-1}(A)$ have (exactly) $n-1$ positive roots.",
"Hence, each of the polynomials $p_{2n-1}(A)$ and $(2+A)p_{2n}(A)$ possesses (exactly) $n-1$ simple roots in the interval $(0,2)$ and, in addition, $p_{2n}(2)=0$ .",
"In Figure REF , we have named by $\\rho _i$ , $1 \\le i \\le n-1$ , the $n-1$ positive roots of $p_{2n-1}(A)$ , $0 < \\rho _1<\\rho _2< \\cdots < \\rho _{n-2}<\\rho _{n-1}<2,$ while those of $p_{2n}(A)$ less than 2 have been named by $r_i$ , $1\\le i \\le n-1$ .",
"So, $0 < r_1< r_2<\\cdots r_{n-1}<r_{n-1}< r_{n}:=2.$ As, again by the induction hypothesis, the positive roots of $(2+A)p_{2n}(A)$ are separated by the positive roots of $p_{2n-1}(A)$ , necessarily $0<r_1<\\rho _1<r_2<\\rho _2<\\cdots < r_{n-2}<\\rho _{n-2}<r_{n-1}<\\rho _{n-1}<r_n=2.$ Consequently, by (REF ), the polynomial $p_{2n+1}(A)$ must have, at least, $n$ different roots in the interval $(0,2)$ .",
"These roots have been named by $z_i$ , $1\\le i \\le n$ , in Figure REF and they satisfy $0 < z_1 < r_1 < \\rho _1 < z_2 < r_2 < \\rho _2 < \\cdots < z_{n-1}<r_{n-1}<\\rho _{n-1}<z_n< 2.$ On the other hand, by Corollary REF , $p_{2n+1}(A)$ is an even polynomial.",
"Thus, since, due to Lemma REF , it has degree $2n$ and, by the previous construction, $\\pm z_i$ , $1\\le i \\le n$ , provides us with a set of $2n$ different roots of $p_{2n+1}(A)$ , necessarily $p_{2n+1}(A)= -\\prod _{j=1}^n(A^2-z_j^2), \\qquad A >0.$ Therefore, all the roots of $p_{2n+1}(A)$ are real and algebraically simple.",
"As a direct consequence of (REF ) it is apparent that the positive roots of $p_{2n+1}(A)$ are separated by the positive roots of $p_{2n}(A)$ , except for 2.",
"Subsequently, we should prove the result in the special case when $2n\\in 4\\mathbb {N}$ .",
"In this situation, owing to Lemmas REF and REF , the plots of the polynomials $p_{2n-1}(A)$ and $(2+A)p_{2n}(A)$ look like illustrated by Figure REF .",
"Apart from the fact that now $p_{2n-1}(A)>0$ and $(2+A)p_{2n}(A)<0$ for all $A>2$ , because the leading coefficients change sign, the previous analysis can be easily adapted to cover the present situation in order to infer that $p_{2n+1}(A)$ satisfies all the requirements also in this case.",
"By repetitive the technical details of the proof are omitted here in.",
"Figure: Sketch of the construction of p 2n+1 (A) p_{2n+1}(A) .",
"Step 2: Passing from $p_{2n+1}(A)$ to $p_{2n+2}(A)$ , $n\\ge 2$ .",
"According to Figure REF , it becomes apparent that the two positive roots of $p_5(A)$ are separated by the unique root of $p_4(A)$ less than 2.",
"Moreover, all their roots are real and simple.",
"Note that the polynomials $\\frac{p_4(A)}{2-A}= -A^2+2, \\qquad p_5(A)=A^4-5A^2+5,$ have one and two positive roots respectively.",
"Arguing by induction, assume that $p_{2n}(A)$ and $p_{2n+1}(A)$ satisfy all the requirements in the statement of the theorem for some $n\\ge 2$ , i.e., all the positive roots of these polynomials are real and algebraically simple, the positive roots of $p_{2n+1}(A)$ are separated by the positive roots less than 2 of $p_{2n}(A)$ , and the polynomials $\\tfrac{p_{2n}(A)}{2-A}$ and $p_{2n+1}(A)$ have, respectively, $n-1$ and $ n $ positive roots.",
"We claim that the roots of the polynomial $p_{2n+2}(A)$ are real and simple, that they are separated by the roots of $p_{2n+1}(A)$ , and that $p_{2n+2}(A)$ possesses $n+1$ positive roots.",
"Indeed, by Theorem REF , $p_{2n+2}(A)=(2-A)p_{2n+1}(A) - p_{2n}(A).$ As in the previous step, we first deal with the case when $2n\\in 4\\mathbb {N}+2$ .",
"By Lemmas REF and REF , we already know that $2n = p_{2n}(0)< 4n+2 = 2p_{2n+1}(0), \\;\\; \\mathrm {deg\\,}(p_{2n})=2n-1,\\;\\;\\mathrm {deg\\,}\\left((2\\!-\\!A)p_{2n+1}\\right)=2n+1,$ and, since $2n\\in 4\\mathbb {N}+2$ , the leading coefficient of $p_{2n}(A)$ equals $-1$ , and the leading coefficient of $p_{2n+1}(A)$ equals also $-1$ .",
"Thus, $p_{2n}(A)<0,\\qquad (2-A)p_{2n+1}(A)>0\\qquad \\hbox{for all}\\;\\; A>2.$ By the induction assumption, the polynomials $\\tfrac{p_{2n}(A)}{2-A}$ and $p_{2n+1}(A)$ have (exactly) $n-1$ and $n$ positive roots, respectively.",
"Thus, each of the polynomials $p_{2n}(A)$ and $(2-A)p_{2n+1}(A)$ possesses (exactly) $n$ simple roots in $(0,2)$ and, obviously, $(2-A)p_{2n+1}(A)$ also vanishes at $A=2$ .",
"Figure REF shows the plots of $p_{2n}(A)$ , in blue, and $(2-A)p_{2n+1}(A)$ , in brown.",
"Figure: Sketch of the construction of p 2n+2 (A) p_{2n+2}(A) in case 2n∈4ℕ+22n\\in 4\\mathbb {N}+2.In Figure REF , we have named by $\\rho _i$ , $1 \\le i \\le n$ , the $n$ positive roots less than 2 of $(2-A)p_{2n+1}(A)$ , $0 < \\rho _1<\\rho _2< \\cdots < \\rho _{n-1}<\\rho _{n}<2:=\\rho _{n+1},$ whereas $r_i$ , $1\\le i \\le n$ , stand for the positive roots of $p_{2n}(A)$ .",
"Since $p_{2n}(2)=0$ , $r_n=2$ .",
"Since the positive roots of $p_{2n+1}(A)$ are separated by the positive roots less than 2 of $p_{2n}(A)$ , the following holds $0 < \\rho _1<r_1<\\rho _2<r_2< \\cdots < \\rho _{n-1}<r_{n-1}<\\rho _{n}<2:=\\rho _{n+1}=r_{n},$ as illustrated by Figure REF .",
"Thanks to (REF ), it becomes apparent that the polynomial $p_{2n+2}(A)$ admits, at least, an interior root in each of the intervals $(\\rho _i,\\rho _{i+1})$ , $i=0,...,n$ , denoted by $z_i$ in Figure REF , plus $z_{n+1}=2$ .",
"Here we are setting $\\rho _0:=0$ .",
"Consequently, $p_{2n+2}(A)$ has, at least, $n+1$ positive roots.",
"On the other hand, thanks to Corollary REF , $\\tfrac{p_{2n+2}(A)}{2-A} $ is an even function and hence, $p_{2n+2}(A)$ has, at least, $2n+1$ different roots.",
"Since, by Lemma REF , $\\mathrm {deg\\,}(p_{2n+2})=2n+1,$ all these roots are real and algebraically simple.",
"By construction, it is apparent that the positive roots of $p_{2n+2}(A)$ are separated by the positive roots of $p_{2n+1}(A)$ (see Figure REF if necessary).",
"If, instead of $2n\\in 4\\mathbb {N}+2$ , we impose $ 2n\\in 4\\mathbb {N}$ , then the previous arguments can be easily adapted to complete the proof of the theorem from Figure REF , where the graphs of $(2+A)p_{2n+1}(A)$ and $p_{2n}(A)$ have been superimposed in order to show their crossing points, which, owing to Theorem REF , are the roots of $p_{2n+2}(A)$ .",
"By repetitive, the technical details of this case are not included here.",
"Figure: Sketch of the construction of p 2n+2 (A) p_{2n+2}(A) in case 2n∈4ℕ2n\\in 4\\mathbb {N}.A careful reading of the proof of Theorem REF reveals that, actually, not only the roots of $p_{n}(A)$ are separated by those of $p_{n-1}(A)$ , but that they are also separated by those of $p_{n-2}(A)$ , taking always into account the exceptional role played by the root 2.",
"Global bifurcation diagram This section analyzes the global structure of the set of zeros of the maps $\\varphi _n$ , $n\\ge 1$ , introduced in (REF ).",
"These zeros are the positive fixed points of the Poincaré maps $\\mathcal {P}_n$ , $n\\ge 1$ , constructed in Section 3.",
"They provide us with the $nT$ -periodic coexistence states of (REF ) under the additional assumption (REF ).",
"It should be remembered that, according to (REF ), for every integer $n\\ge 1$ $p_n(A)= \\frac{d\\varphi _n (A,1)}{dx}=\\mathfrak {L}(n;A),\\qquad A>0,$ provides us with the linearization at the trivial curve, $(A,1)$ , of $\\varphi _n(A,x)$ .",
"In our analysis, $A$ is always regarded as a bifurcation parameter to $nT$ -periodic coexistence states from the $T$ -periodic ones (i.e., from $x=1$ ).",
"As a consequence of the simplicity of all the roots of $p_n(A)$ , $n\\ge 1$ , guaranteed by Theorem REF , the following result holds.",
"Theorem 6.1 For every $n\\ge 1$ and $r \\in p_n^{-1}(0)$ the following algebraic transversality condition holds $\\mathfrak {L}_1 (N\\left[\\mathfrak {L}(n;r) \\right] )\\oplus R\\left[ \\mathfrak {L}(n;r) \\right]=\\mathbb {R},$ where $\\mathfrak {L}_1:= \\frac{d \\mathfrak {L}(n;r)}{d A} , \\qquad n\\ge 1,\\;\\; r\\in p_n^{-1}(0).$ Therefore, by Theorem 1.7 of M. G. Crandall and P. H. Rabinowitz [3], there exists an analytic curve of $nT$ -periodic coexistence states of (REF ) bifurcating from $(A,1)$ at the root $A=r$ .",
"Actually, there exists $\\varepsilon >0$ and a real analytic map $A: (-\\varepsilon ,\\varepsilon )\\rightarrow \\mathbb {R}$ such that $A(0)=r$ and $\\varphi _n(A(s),1+s)=0 \\quad \\hbox{for all}\\;\\; s\\in (-\\varepsilon ,\\varepsilon ).$ Moreover, any non-trivial zero of $\\varphi _n$ , $(A,x)$ with $x\\ne 1$ , in a neighborhood of $(r,1)$ must be of the form $(A(s),1+s)$ for some $s \\in (-\\varepsilon ,\\varepsilon )$ .",
"In other words, there exists $\\varrho >0$ such that $\\left.",
"\\begin{array}{rr} \\varphi _n(A,x)=0 \\\\ |A-r|+|x-1|<\\varrho \\\\ x\\ne 1\\end{array}\\right\\rbrace \\Longrightarrow (A,x)=(A(s),1+s) \\quad \\hbox{for some}\\;\\; s \\in (-\\varepsilon ,\\varepsilon ).$ Furthermore, setting $A(s)= r + A_1 s + A_2 s^2 + \\mathcal {O} (s^3) \\quad \\hbox{as}\\;\\; s\\rightarrow 0,$ one has that $A_1=0$ and $A_2>0$ if $n=2$ and $r=2$ , in complete agreement with Figure REF ; $A_1<0$ if $n=3$ and $r=\\sqrt{3}$ ; $A_1=0$ and $A_2 <0$ (resp.",
"$A_2>0$ ) if $n=4$ and $r = r_{4,1}=\\sqrt{2}$ (resp.",
"$r = r_{4,2}=2$ ).",
"According to (REF ), $\\mathfrak {L}(n;r)=p_n(r)=0$ .",
"Thus, $N[\\mathfrak {L}(n;r)]=\\mathbb {R}$ and (REF ) can be equivalently expressed as $\\mathfrak {L}_1(\\mathbb {R}) =\\mathbb {R}$ , which holds true because, thanks to Theorem REF , we already know that $r$ is an algebraically simple root of $p_n(A)$ , i.e., $\\mathfrak {L}_1 = p_n^{\\prime }(r) \\ne 0.$ So, (REF ) indeed holds and [3] applies to $\\varphi _n(A,x)=0$ at $(A,x)=(r,1)$ .",
"Since we can take $\\psi =1$ as a generator of $N[\\mathfrak {L}(n;r)]=\\mathbb {R}$ and $Y=[0]$ as a supplement of $N[\\mathfrak {L}(n;r)]=\\mathbb {R}$ in $\\mathbb {R}$ , owing to [3], there exist $\\varepsilon >0$ and a real analytic map $(A,y):(-\\varepsilon ,\\varepsilon )\\rightarrow \\mathbb {R}\\times Y$ such that $(A(0),y(0))=(r,0)$ and $\\varphi _n(A(s),1+s(\\psi +y(s)))=0 \\quad \\hbox{for all}\\;\\; s\\in (-\\varepsilon ,\\varepsilon ),$ it becomes apparent, by construction, that $\\varphi _n(A(s),1+s)=0\\quad \\hbox{for all} \\;\\; s \\in (-\\varepsilon ,\\varepsilon ),$ because $y\\equiv 0$ and $\\psi =1$ .",
"This ends the proof of the first two claims of the theorem: the existence of the analytic curve of nontrivial solutions and the uniqueness.",
"As far as concerns to the problem of ascertaining the nature of these local bifurcations at $(r,1)$ , we can proceed as follows.",
"In order to prove Part (a), note that, thanks to (REF ), setting $x(s):=1+s$ and expanding in Taylor series, we have that $0 = \\varphi _2(A(s),x(s))=\\varphi _2(r,1)+\\frac{d\\varphi _2}{ds}(r,1)s+\\frac{1}{2}\\frac{d^2\\varphi _2}{ds^2}(r,1)s^2+\\cdots $ for all $s\\in (-\\varepsilon ,\\varepsilon )$ , where $r=2$ .",
"Moreover, by construction, we already know that $\\varphi _2(r,1)=0,\\qquad \\frac{\\partial \\varphi _2}{\\partial x}(r,1)=p_2(r)=p_2(2)=0$ (see (REF ), if necessary).",
"Thus, since by (REF ) $\\varphi _2(A,x)=x\\left( E_1(A,x)+1\\right) -2,$ it follows from (REF ) and $\\frac{\\partial E_1}{\\partial A}(r,1)=0$ that $\\frac{d\\varphi _2}{ds}(r,1)=\\frac{\\partial \\varphi _2}{\\partial A}(r,1)A^{\\prime }(0)=\\frac{\\partial E_1}{\\partial A}(r,1)A_1=0,$ where $^{\\prime }:=\\frac{d}{ds}$ .",
"Hence, these terms do not provide us with any neat information concerning $A_1$ .",
"So, we must consider higher order terms to find out $A_1$ .",
"As $\\frac{\\partial \\varphi _2}{\\partial A}(r,1)=0=\\frac{\\partial \\varphi _2}{\\partial x}(r,1),$ applying the chain rule it readily follows that $0= \\frac{d^2\\varphi _2}{ds^2}(r,1)=\\frac{\\partial ^2\\varphi _2}{\\partial x^2}(r,1)+2\\frac{\\partial ^2\\varphi _2}{\\partial A\\partial x}(r,1)A_1+\\frac{\\partial ^2\\varphi _2}{\\partial A^2}(r,1)A_1^2.$ On the other hand, differentiating with respect to $x$ the identity (REF ) yields $\\frac{\\partial \\varphi _2}{\\partial x}(A,x) = E_1(A,x)+1+ x \\frac{\\partial E_1}{\\partial x}(A,x).$ So, $\\frac{\\partial ^2 \\varphi _2}{\\partial x^2}(A,x) = 2 \\frac{\\partial E_1}{\\partial x} (A,x)+ x \\frac{\\partial ^2 E_1}{\\partial x^2}(A,x).$ Consequently, particularizing at $(A,x)=(r,1)$ , it follows from (REF ) that $\\frac{\\partial ^2 \\varphi _2}{\\partial x^2}(r,1) =r^2-2r=4-4=0.$ Similarly, differentiating (REF ) with respect to $A$ shows that $\\frac{\\partial ^2 \\varphi _2}{\\partial x\\partial A}(A,x)=\\frac{\\partial E_1}{\\partial A}(A,x)+x\\frac{\\partial ^2 E_1}{\\partial x\\partial A}(A,x)$ and hence, owing to (REF ), $\\frac{\\partial ^2 \\varphi _2}{\\partial x\\partial A}(r,1)=\\frac{\\partial E_1}{\\partial A}(r,1)+\\frac{\\partial ^2 E_1}{\\partial x\\partial A}(r,1) =\\frac{\\partial ^2 E_1}{\\partial x\\partial A}(r,1) = -1.$ Lastly, $\\frac{\\partial ^2\\varphi _2}{\\partial A^2}(A,x)=x\\frac{\\partial ^2 E_1}{\\partial A^2}(A,x)= (1-x)^2E_1(A,x)$ and hence, $\\frac{\\partial ^2\\varphi _2}{\\partial A^2}(A,1)=0.$ Therefore, substituting (REF ), (REF ) and (REF ) into (REF ) it becomes apparent that $A_1=0$ .",
"Thanks to this fact, the third derivative admits the next (simple) expression: $0=\\frac{d^3\\varphi _2}{ds^3}(r,1)=6\\frac{\\partial ^2\\varphi _2}{\\partial x\\partial A}(r,1)A_2+\\frac{\\partial ^3\\varphi _2}{\\partial x^3}(r,1)=-6A_2+3r^2-r^3,$ which implies that $A_2=\\frac{4}{3}>0$ and ends the proof of Part (a).",
"To prove Part (b), note that, much like in Part (a), one has that $0= \\varphi _3(A(s),x(s))=\\varphi _3(r,1)+\\frac{d\\varphi _3}{ds}(r,1)s+\\frac{1}{2} \\frac{d^2\\varphi _3}{ds^2}(r,1)s^2+\\cdots $ for all $s \\in (-\\varepsilon ,\\varepsilon )$ .",
"Similarly, $\\frac{\\partial \\varphi _3}{\\partial A}(r,1)=0=\\frac{\\partial \\varphi _3}{\\partial x}(r,1).$ So, $\\frac{d\\varphi _3}{ds}(r,1)=0.$ Moreover, differentiating twice with respect to $s$ yields $0 & = \\frac{d^2\\varphi _3}{ds^2}(r,1)\\\\[5pt] & =\\frac{\\partial ^2\\varphi _3}{\\partial x^2}(r,1)+2\\frac{\\partial ^2\\varphi _3}{\\partial x\\partial A}(r,1)A_1+\\frac{\\partial ^2\\varphi _3}{\\partial A^2}(r,1)A_1^2\\\\[5pt] & =r^4-r^3-2r^2-4rA_1.$ Consequently, since $r=\\sqrt{3}$ , it follows from this identity that $A_1=\\frac{\\sqrt{3}-3}{4}<0,$ which ends the proof of Part (b).",
"Finally, much like before, we have that $0= \\varphi _4(A(s),x(s))=\\varphi _4(r,1)+\\frac{d\\varphi _4}{ds}(r,1)s+\\frac{1}{2}\\frac{d^2\\varphi _4}{ds^2}(r,1)s^2+\\cdots $ for all $s \\in (-\\varepsilon ,\\varepsilon )$ , and, in addition, $\\frac{\\partial \\varphi _4}{\\partial A}(r,1)=0=\\frac{\\partial \\varphi _4}{\\partial x}(r,1).$ Thus, $\\frac{d\\varphi _4}{ds}(r,1)=0$ .",
"Moreover, differentiating twice yields $0 & =\\frac{d^2\\varphi _4}{ds^2}(r,1)\\\\[5pt] &=\\frac{\\partial ^2\\varphi _4}{\\partial x^2}(r,1)+2\\frac{\\partial ^2\\varphi _4}{\\partial x\\partial A} (r,1)A_1+\\frac{\\partial ^2\\varphi _4}{\\partial A^2}(r,1)A_1^2 \\\\[5pt] &=r^6-3r^5-r^4+8r^3-2r^2-4r+2(r^3-2r^2-2r)A_1.$ Therefore, since $r=\\sqrt{2}$ it follows from this identity that $A_1=0$ .",
"Furthermore, $0 & = \\frac{d^3\\varphi _2}{ds^3}(r,1)\\\\[5pt] &=6\\frac{\\partial ^2\\varphi _2}{\\partial x\\partial A} (r,1)A_2+\\frac{\\partial ^3\\varphi _2}{\\partial x^3}(r,1)\\\\[5pt] &=6(3r^2-4r-2)A_2-r^8+r^7+9r^6-11r^5-10r^4+20r^3-2r^2.$ Consequently, we find from $r=\\sqrt{2}$ that $A_2=-\\frac{2(5+4\\sqrt{2})}{3}<0,$ which ends the proof.",
"Figure REF shows the local bifurcation diagrams of the $2T$ , $3T$ and $4T$ -periodic coexistence states of (REF ) under condition (REF ).",
"We are plotting $x$ , in ordinates, versus $A$ , in abscisas.",
"By the analysis already done at the beginning of Section 2, and, in particular, by Theorem REF , which was sketched in Figure REF , we already know that, under condition (REF ), the problem (REF ) admits a $2T$ -periodic coexistence state if, and only if, $A>2$ .",
"Moreover, the local bifurcation of these solutions must be supercritical.",
"Thus $A_2\\ge 0$ .",
"As a byproduct of Theorem REF , it turns out that $A_2>0$ .",
"So, it is a genuine supercritical pitchfork bifurcation of quadratic type.",
"However, since $A_1<0$ , the bifurcation to $3T$ -periodic coexistence states from $(A,x)=(\\sqrt{3},1)$ is transcritical, whereas the $4T$ -periodic solutions emanate from $(A,x)=(\\sqrt{2},1)$ through a subcritical quadratic pitchfork bifurcation, because $A_1=0$ and $A_2<0$ in this case.",
"The fact that the local nature of the first three bifurcation phenomena possess a completely different character shows that, in general, ascertaining the precise type of these local bifurcations for large $n$ might not be possible, much like happened with the problem of determining the fine structure of the set of bifurcation points from the trivial solution $(A,1)$ .",
"The higher is the order of the bifurcating subharmonics, measured by $n$ , the higher is the complexity of the associated function $\\varphi _n$ and hence, the more involved is finding out the values of $A_1$ and $A_2$ in (REF ) by the intrinsic nature of the functions $E_n$ defined in (REF ).",
"Figure: Local bifurcation diagrams from (A,x)=(A,1)(A,x)=(A,1) of the nTnT-periodic coexistence states for n∈{2,3,4}n\\in \\lbrace 2,3,4\\rbrace .Remark 6.2 Thanks to Theorem REF , it becomes apparent that the set of bifurcation points from $ (A,1) $ to $ nT $ -periodic coexistence states of (REF ) is the set of roots of $ p_n(A) $ .",
"Since the number of roots of a polynomial is finite, the set of bifurcation points is numerable, as it is a numerable union of finite sets.",
"Since every $nT$ -periodic coexistence state of (REF ) provides us with a $knT$ -periodic coexistence state for all $k\\ge 1$ , owing Theorem REF , the roots of $p_n(A)$ must be roots of $p_{kn}(A)$ for all $ n,k\\ge 1 $ , i.e., $p_n|p_{kn}$ for all $n, k \\ge 1$ .",
"Remark 6.3 Thanks to Theorem REF and Remark REF , the set of bifurcation points to a $nT$ -periodic solution is a subset of the interval $(0,2]$ .",
"Complementing [9], where the non-degeneration of the positive $T$ -periodic coexistence states of (REF ) with respect to the $T$ -periodic solutions was established, Theorem REF shows that the $T$ -periodic solutions are degenerated with respect to the $nT$ -periodic solutions of (REF ) for all $n\\ge 2$ at every positive root, $r$ , of $p_n(A)$ .",
"Nevertheless, the $T$ -periodic solutions are non-degenerated with respect to the $nT$ -periodic solutions, $n\\ge 2$ , if $A>2$ , because in this range there is not any bifurcation point from $(A,1)$ .",
"Subsequently, we will discuss the global character of all the local bifurcations documented by Theorem REF in the context of global bifurcation theory.",
"In this discussion, by a (connected) component it is understood any closed and connected subset that is maximal for the inclusion.",
"For any given integer $n\\ge 1$ , the set of non-trivial $nT$ -periodic solutions of (REF ), $\\mathcal {S}_n$ , consists of all $nT$ -periodic coexistence states different from $(A,1)$ plus the set of points $(r,1)$ with $p_n(r)=0$ .",
"In other words, setting $\\mathbb {R}_+:=(0,+\\infty )$ , $\\mathcal {S}_n= \\lbrace (A,x) \\in \\mathbb {R}_+\\times (\\mathbb {R}_+\\setminus \\lbrace 1\\rbrace ) \\;:\\; \\varphi _n(A,x)=0\\rbrace \\cup \\lbrace (r,1)\\;:\\; p_n(r)=0\\rbrace .$ Note that, owing to Theorem REF , $\\lbrace (r,1)\\;:\\; p_n(r)=0\\rbrace $ is the set of bifurcation points to $nT$ -periodic solutions from the trivial curve $(A,1)$ .",
"Thanks to (REF ), the algebraic multiplicity of J. Esquinas and J. López-Gómez [7] equals one, $\\chi [\\mathfrak {L}(n;A);r]=1 \\in 2\\mathbb {N}+1,$ for all $r \\in p_n^{-1}(0)$ , $r>0$ .",
"Thus, by [10], the local degree at $(A,1)$ of the one-dimensional $\\varphi _n(A,\\cdot )$ changes as $A$ crosses $r$ (see also J. López-Gómez and C. Mora-Corral [13] if necessary).",
"Therefore, according to [10], for every integer $n\\ge 2$ and each root $r>0$ of $p_n(A)$ , there is a component of $\\mathcal {S}_n$ , $\\mathfrak {C}_{n,r}$ , such that $(r,1) \\in \\mathfrak {C}_{n,r}\\subset \\mathbb {R}_+\\times \\mathbb {R}.$ Moreover, by the local uniqueness about $(r,1)$ guaranteed by Theorem REF , in a neighborhood of $(r,1)$ the component $\\mathfrak {C}_{n,r}$ consists of an analytic curve, $(A(s),1+s)$ , $|s|<\\varepsilon $ .",
"Note that any real continuous map must be compact.",
"So, the Leray–Schauder degree (see, e.g., N. G. Lloyd [8], or [13], if necessary) can be applied to get these global results.",
"Alternatively, one might use the degree of P. Benevieri and M. Furi [1], as in Theorem 5.4 and Corollary 5.5 of [12] (see [11] for a recent survey on global bifurcation theory).",
"By Remark REF , for every $r \\in p_n^{-1}(0)\\cap \\mathbb {R}_+$ , the component $\\mathfrak {C}_{n,r}$ must be separated away from $x=0$ and hence, all their solutions must be positive, because $\\varphi _n(0)=-n$ .",
"Thus, they indeed provide us with coexistence states of (REF ).",
"Similarly, for every $n\\ge 2$ , since $\\varphi _n(n)>0$ , $\\mathfrak {C}_{n,r}$ is bounded above by $n$ , in the sense that $x<n$ if $(A,x)\\in \\mathfrak {C}_{n,r}$ with $A>0$ .",
"Therefore, $\\mathcal {P}_x(\\mathfrak {C}_{n,r})\\subset (0,n),$ where $\\mathcal {P}_x$ stands for the $x$ -projection operator, $\\mathcal {P}_x(A,x):=x$ .",
"Moreover, due to (REF ), $x=1$ is the unique zero of $\\varphi _n(A,x)$ at $A=0$ .",
"Note that, due to Remark REF , $(A,1)=(0,1)\\notin \\mathfrak {C}_{n,r}$ because $p_n(0)=n>0$ .",
"Throughout the rest of this section, we will also consider the (unilateral) subcomponents $\\mathfrak {C}_{n,r}^+:=\\mathfrak {C}_{n,r}\\cap [x>1], \\qquad \\mathfrak {C}_{n,r}^-:=\\mathfrak {C}_{n,r}\\cap [x<1].$ Thanks to Theorem REF , these subcomponents are non-empty.",
"Moreover, arguing as in [10], it is easily seen that they equal the components $\\mathfrak {C}^+$ and $\\mathfrak {C}^-$ introduced on page [10].",
"This feature heavily relies on the fact that $x$ is a one-dimensional variable.",
"Therefore, the unilateral theorem [10] can be applied to infer that each of the components $\\mathfrak {C}_{n,r}^+$ and $\\mathfrak {C}_{n,r}^-$ satisfies the global alternative of P. H. Rabinowitz [20], because the supplement of $N[\\mathfrak {L}(n;r)]=\\mathbb {R}$ in $\\mathbb {R}$ is $Y=[0]$ and, due to (REF ), $\\mathfrak {C}_{n,r}$ cannot admit an element, $(A,x)$ with $x=0$ .",
"Therefore, $\\mathfrak {C}_{n,r}^+$ (resp.",
"$\\mathfrak {C}_{n,r}^-$ ) satisfies some of the following two conditions, which are far from being excluding: There exists $s \\in p_n^{-1}(0)\\setminus \\lbrace r\\rbrace $ (resp.",
"$t \\in p_n^{-1}(0)\\setminus \\lbrace r\\rbrace $ ) such that $(s,1)\\in \\mathfrak {C}_{n,r}^+$ (resp.",
"$(t,1)\\in \\mathfrak {C}_{n,r}^-$ ).",
"The component $\\mathfrak {C}_{n,r}^+$ (resp.",
"$\\mathfrak {C}_{n,r}^-$ ) is unbounded in $A$ , because of (REF ).",
"Note that the counterexample of E. N. Dancer [4] shows that Theorems 1.27 and 1.40 of P. H. Rabinowitz [20] are not true as originally stated.",
"To show that the second option occurs in both cases we need the next result.",
"Lemma 6.4 Each of the unilateral subcomponents satisfies $\\mathfrak {C}_{n,r}^\\pm \\cap \\lbrace (A,1)\\;:\\; A\\ge 0\\rbrace = \\lbrace (r,1)\\rbrace .$ Thus, also $\\mathfrak {C}_{n,r} \\cap \\lbrace (A,1)\\;:\\; A\\ge 0\\rbrace = \\lbrace (r,1)\\rbrace ,$ i.e., $(r,1)$ is the unique bifurcation point of $\\mathfrak {C}_{n,r}$ from $(A,1)$ .",
"Subsequently, we will denote by $\\nu (n)$ the total number of positive roots of the polynomial $p_n(A)$ .",
"By Theorem REF , we already know that $\\nu (n)=\\frac{n}{2}$ if $n$ is even and $\\nu (n)=\\frac{n-1}{2}$ if $n$ is odd.",
"We will prove the result only for $\\mathfrak {C}_{n,r}^+$ , as the same argument also works out to prove the corresponding assertion for the component $\\mathfrak {C}_{n,r}^-$ .",
"The proof will proceed by contradiction.",
"We already know that $\\mathfrak {C}_{n,r}^+$ can only meet the trivial solution $(A,1)$ at the roots of $p_n(A)$ .",
"Suppose that $r=r_{n,i}$ for some $i \\in \\lbrace 1,...,\\nu (n)\\rbrace $ , and that there exists $j>i$ , $j \\in \\lbrace 1,...,\\nu (n)\\rbrace $ , such that $\\lbrace (r_{n,i},1),(r_{n,j},1)\\rbrace \\subset \\mathfrak {C}_{n,r}^+ \\cap \\lbrace (A,1)\\;:\\; A\\ge 0\\rbrace .$ Then, by the definition of component, it becomes apparent that $\\mathfrak {C}_{n,r_{n,i}}^+=\\mathfrak {C}_{n,r_{n,j}}^+$ as sketched by Figure REF .",
"Thanks to Theorem REF , there exists $r_{n-1,k}\\in (r_{n,i},r_{n,j})\\cap p_{n-1}^{-1}(0).$ By the incommensurability of $nT$ with $(n-1)T$ , $\\mathfrak {C}_{n-1,r_{n-1,k}}^+$ cannot reach the component (REF ).",
"Thus, must be bounded.",
"Consequently, as $\\mathfrak {C}_{n-1,r_{n-1,k}}^+$ also satisfies the global alternative of P. H. Rabinowitz, there exists $r_{n-1,\\ell }\\in p_{n-1}^{-1}(0)$ , with $k\\ne \\ell $ , such that $\\lbrace (r_{n-1,k},1),(r_{n-1,\\ell },1)\\rbrace \\subset \\mathfrak {C}_{n-1,r_{n-1,k}}^+ \\cap \\lbrace (A,1)\\;:\\; A\\ge 0\\rbrace ,$ as sketched in Figure REF .",
"Since the set of roots $\\bigcup _{2\\le \\kappa \\le n} p_{\\kappa }^{-1}(0)$ is finite, it becomes apparent that, after finite many steps, there exists a component, $\\mathfrak {C}_{n-h,r_{n-h,m}}^+$ , for some $2 \\le h \\le n-3$ and $1\\le m\\le \\nu (n-h)$ , that should meet the last component linking two different roots sketched in Figure REF , $\\mathfrak {C}_{n-h+1,r_{n-h+1,v}}^+ = \\mathfrak {C}_{n-h+1,r_{n-h+1,w}}^+,$ because there is no any additional root of $p_{n-h}(A)$ in between $r_{n-h+1,v}$ and $r_{n-h+1,w}$ .",
"But this is impossible, by the incommensurability of $(n-h)T$ with $(n-h+1)T$ .",
"This contradiction ends the proof.",
"Figure: Sketch of the proof of Lemma .As an immediate consequence of the previous analysis, the next result holds.",
"As for the $x$ -projection operator, $\\mathcal {P}_x$ , we will denote by $\\mathcal {P}_A$ the $A$ -projection operator, $\\mathcal {P}_A(A,x):=A.$ Theorem 6.5 For every integer $n\\ge 2$ and each root $r>0$ of $p_n(A)$ , the component $\\mathfrak {C}_{n,r}^+$ satisfies $\\mathcal {P}_x (\\mathfrak {C}_{n,r}^+)\\subset [1,n)$ ; $\\mathcal {P}_A (\\mathfrak {C}_{n,r}^+)=[A^+_{n,r},+\\infty )$ for some $A_{n,r}^+ \\in (0,r]$ .",
"In particular, $\\mathfrak {C}_{n,r}^+$ is unbounded.",
"$\\mathfrak {C}_{n,r}^+ \\cap \\lbrace (A,1)\\;:\\; A\\ge 0\\rbrace = \\lbrace (r,1)\\rbrace $ ; For every $n, m \\ge 2$ , $\\mathfrak {C}_{n,r}^+\\cap \\mathfrak {C}_{m,s}^+=\\emptyset $ if $r \\ne s$ .",
"Moreover, by Theorem REF , in a neighborhood of $(r,1)$ the component $\\mathfrak {C}_{n,r}^+$ consists of an analytic curve, $(A(s),1+s)$ , $0\\le s <\\varepsilon $ .",
"Similarly, the component $\\mathfrak {C}_{n,r}^-$ satisfies (c), (d) and $\\mathcal {P}_x (\\mathfrak {C}_{n,r}^-)\\subset (0,1]$ ; $\\mathcal {P}_A (\\mathfrak {C}_{n,r}^-)=[A^-_{n,r},+\\infty )$ for some $A_{n,r}^- \\in (0,r]$ .",
"In particular, $\\mathfrak {C}_{n,r}^-$ is unbounded.",
"Analogously, in a neighborhood of $(r,1)$ the component $\\mathfrak {C}_{n,r}^-$ consists of an analytic curve, $(A(s),1+s)$ , $-\\varepsilon <s\\le 0$ .",
"At this stage, the only delicate point is Part (d).",
"Suppose that $\\mathfrak {C}_{n,r}^+\\cap \\mathfrak {C}_{m,s}^+\\ne \\emptyset $ for some $r \\ne s$ .",
"Then, by the definition of component, necessarily $\\mathfrak {C}_{n,r}^+ = \\mathfrak {C}_{m,s}^+.$ Thus, $(r,1), (s,1) \\in \\mathfrak {C}_{n,r}^+$ , which contradicts Lemma REF .",
"The proof is complete.",
"Except for the local bifurcations from the trivial line $(A,1)$ , the global diagramas of the components $\\mathfrak {C}_{n,r}^\\pm $ plotted in Figure REF respect the general properties established by Theorem REF .",
"Although the components have been plotted with no secondary bifurcations along them, there are some numerical evidences that $\\mathfrak {C}_{2,2}^-$ possesses a secondary bifurcation to $4T$ -periodic solutions.",
"Nevertheless, thanks to Theorem REF , even in the case that they might occur higher order bifurcations along these components, they must be disjoint.",
"According to Theorems REF and REF , it becomes apparent that some $3T$ and $4T$ -periodic solutions must be degenerated.",
"Namely, those on the turning points of $\\mathfrak {C}_{3,\\sqrt{3}}^+$ , $\\mathfrak {C}_{4,\\sqrt{2}}^+$ and $\\mathfrak {C}_{4,\\sqrt{2}}^-$ in Figure REF .",
"Similarly, the bifurcation points accumulating from the left to $\\sqrt{2}$ and $\\sqrt{3}$ must provide us with additional degenerate solutions: those on the turning points of their corresponding components."
],
[
"A canonical chain of associated polynomials",
"Searching for the potential bifurcation points from the curve $(A,1)$ to $nT$ -periodic coexistence states, this section analyzes the spectrum of the linearized family $\\mathfrak {L}(n;A):= \\frac{d\\varphi _n (A,1)}{dx},\\quad n\\in \\mathbb {N},$ i.e., its zero set as a function of the parameter $A$ , as well as the global structure of $\\mathfrak {L}(n;A)$ .",
"Note that, since $(A,1)$ is the $T$ -periodic coexistence state, it also provides us with a $nT$ -periodic solution for all $n\\ge 1$ and, hence, by construction, $\\varphi _n(A,1)=0$ for all $A>0$ and $n\\ge 1$ .",
"The curve $(A,1)$ , $A>0$ , is the trivial curve, as it is known.",
"It is the curve from which are going to bifurcate the $nT$ -periodic coexistence states of (REF ) under assumption (REF ).",
"Note also that, since every $nT$ -periodic solution is $knT$ -periodic for all integer $k\\ge 1$ , $\\varphi _{kn}(x)=0 \\quad \\hbox{for all}\\;\\; x \\in \\varphi _n^{-1}(0)\\quad \\hbox{and}\\quad k\\ge 1.$ Throughout the rest of this paper we will denote $p_n(A):= \\frac{d\\varphi _n (A,1)}{dx}=\\mathfrak {L}(n;A),\\qquad A>0.$ Differentiating with respect to $x$ the identity (REF ) yields $p_n(A)=p_{n-1}(A)+ E_{n-1}(1)+E^{\\prime }_{n-1}(1)\\qquad \\hbox{for all}\\;\\; A>0.$ The next result shows that $p_n \\in \\mathbb {Z}\\left[ A\\right]$ .",
"Lemma 4.1 For every $ n\\in \\mathbb {N}$ , $p_n(A) $ is a polynomial in the variable $A$ with integer coefficients, i.e., $ p_n \\in \\mathbb {Z}[A] $ .",
"By (REF ), it becomes apparent that, since $(1,1)$ is a fixed point of $ \\mathcal {P}_n $ , $(1,1)=\\mathcal {P}_n(1,1)=\\left(E_{2n-1}(1),E_{2n}(1)\\right)$ for all integer $n\\ge 1$ .",
"Thus, $E_n(1)=1 \\quad \\hbox{for all}\\;\\; n\\ge 0.$ Thus, (REF ) becomes $p_n(A)=p_{n-1}(A)+ 1 +E^{\\prime }_{n-1}(1)$ for all $A>0$ and $n\\ge 1$ .",
"Therefore, due to (REF ) , $p_1(A) = \\frac{d\\varphi _1(A,1)}{dx}= 1$ and iterating (REF ) $n-2$ times show that, for every integer $n\\ge 2$ , $p_n(A)= n+E_1^{\\prime }(1)+E_2^{\\prime }(1)+\\cdots +E_{n-1}^{\\prime }(1).$ Consequently, to complete the proof it suffices to show that $ E_n^{\\prime }(1)\\in \\mathbb {Z}[A] $ for all $n\\ge 1$ .",
"Indeed, by (REF ), $E_0^{\\prime }(1)=0$ , $ E_1^{\\prime }(1)=-A $ and $E_n^{\\prime }(1)=\\left\\lbrace \\begin{array}{ll}\\displaystyle { A \\sum _{\\begin{array}{c}j=1\\\\j\\in 2\\mathbb {N}+1\\end{array}}^{n-1} \\left[E_{j}(1) +E_{j}^{\\prime }(1)\\right] } &\\qquad \\hbox{if\\ } n\\in 2\\mathbb {N},\\\\[14pt]\\displaystyle { -A \\sum _{\\begin{array}{c}j=0\\\\ j\\in 2\\mathbb {N}\\end{array}}^{n-1}\\left[ E_{j}(1)+ E_{j}^{\\prime }(1) \\right]} & \\qquad \\hbox{if\\ } n\\in 2\\mathbb {N}+1.\\end{array}\\right.$ Thus, by a complete induction argument it becomes apparent that $ E_n^{\\prime }(1)\\in \\mathbb {Z}[A] $ for all $ n\\in \\mathbb {N}$ .",
"This concludes the proof.",
"Remark 4.2 In Section 5 we will prove that all the roots of the polynomial $p_n(A)$ are simple.",
"In other words, $p^{\\prime }_n(r) = \\frac{d \\mathfrak {L}}{d A}(n;r)\\ne 0$ for all $r\\in p_n^{-1}(0)$ .",
"Thus, the transversality condition of M. G. Crandall and P. H. Rabinowitz [3] holds true.",
"Therefore, by the main theorem of [3], at every positive root of $p_n(A)$ , $r$ , an analytic curve of $nT$ -periodic coexistence states of (REF ) bifurcates from $(A,1)$ at $r$ .",
"This feature explains our interest here in analyzing the nature and the distribution of the positive roots of the polynomials $p_n(A)$ , $n\\in \\mathbb {N}$ .",
"Remark 4.3 Occasionally, we will make explicit the dependence of the function $\\varphi _n(x)$ on the parameter $A$ by setting $\\varphi _n(A,x)$ , instead of $\\varphi _n(x)$ .",
"Similarly, we will set $E_n(A,x):=E_n(x)$ for all $n\\in \\mathbb {N}$ .",
"According to (REF ), $E_n(0,x)=1$ for all $n\\in \\mathbb {N}$ and $x \\in [0,n]$ .",
"Thus, (REF ) yields $\\varphi _n(0,x)= \\varphi _{n-1}(0,x) - 1 + x$ for all $n\\in \\mathbb {N}$ and $x\\in [0,n]$ .",
"Therefore, iterating $n-1$ times, it becomes apparent that $\\varphi _n(0,x)=n(x-1)\\quad \\hbox{for all}\\;\\; n\\in \\mathbb {N}.",
"$ As the zeros of $\\varphi _n(A,x)$ provide us with the $nT$ -periodic positive solutions of (REF ), it follows from (REF ) that $x=1$ is the unique $nT$ -periodic solution, for all $n\\in \\mathbb {N}$ , at the particular value of the parameter $A=0$ .",
"The next list collects the polynomials $p_n(A)$ for $1\\le n\\le 13$ .",
"$p_1(A)&=1\\\\p_2(A)&=-A+2\\\\p_3(A)&=-A^2+3\\\\p_4(A)&=A^3-2A^2-2A+4\\\\p_5(A)&=A^4-5A^2+5\\\\p_6(A)&=-A^5+2A^4+4A^3-8A^2-3A+6\\\\p_7(A)&=-A^6+7A^4-14A^2+7\\\\p_8(A)&=A^7-2A^6-6A^5+12A^4+10A^3-20A^2-4A+8\\\\p_9(A)&=A^8-9A^6+27A^4-30A^2+9\\\\p_{10}(A)&=-A^9+2A^8+8A^7-16A^6-21A^5+42A^4+20A^3-40A^2-5A+10\\\\p_{11}(A)&=-A^{10}+11A^8-44A^6+77A^4-55A^2+11\\\\p_{12}(A)&=A^{11}-2A^{10}-10A^9+20A^8+36A^7-72A^6-56A^5+112A^4+35A^3-70A^2-6A+12\\\\p_{13}(A)&=A^{12}-13A^{10}+65A^8-156A^6+182A^4-91A^2+13.$ The next table collects the coefficients of all the polynomials listed above.",
"Table: First thirteen polynomials coefficients.By simply having a glance to these polynomials, it becomes apparent that the following properties hold: The constant terms of $p_n(A)$ equals $n$ .",
"The degree of $p_n(A)$ equals $n-1$ .",
"The leading coefficients of $p_{4n}(A)$ and $p_{4n+1}(A)$ equal 1, while the leading coefficients of $p_{4n+2}(A)$ and $p_{4n+3}(A)$ equal $-1$ .",
"$p_{2n}(2)=0$ for all integer $n\\ge 1$ .",
"Thus, $p_2|p_{2n}$ for all $n\\ge 1$ .",
"$p_{2n+1}(A)$ is an even function.",
"Besides these properties, it seems all the coefficients of $p_n(A)$ , except the leading one, must be multiples of $n$ if $n$ is a prime integer, though this property will not be used in this paper.",
"The next result shows the property (a).",
"Lemma 4.4 $ p_n(0)=n $ for all $ n\\ge 1$ .",
"By (REF ), $ \\frac{dE_n(0,1)}{dx}=0 $ .",
"Hence, due to (REF ), $p_n(0)=n$ for all $ n\\ge 1$ .",
"The next result establishes the properties (b) and (c).",
"Lemma 4.5 For every integer $n\\ge 1$ , $ {\\rm {deg}}(p_n)=n-1 $ .",
"Moreover, the leading coefficients of $ p_n $ equal 1 if $ n\\in 4\\mathbb {N}\\cup (4\\mathbb {N}+1) $ and $ -1 $ if $ n\\in (4\\mathbb {N}+2)\\cup (4\\mathbb {N}+3) $ .",
"By the proof of Lemma REF , we already know that $E^{\\prime }_n(A,1):=\\frac{d E_n}{dx} (A,1)$ is a polynomial in $A$ for all integer $n\\ge 1$ .",
"Next, we will show that it has degree $ n $ .",
"To prove it, a complete induction argument will be used.",
"According to (REF ), we already know that $ {\\rm {deg}}(E_0^{\\prime }(A,1))={\\rm {deg}}(0)=0 \\quad \\hbox{and}\\quad {\\rm {deg}}(E_1^{\\prime }(A,1))={\\rm {deg}}(-A)=1.$ As the induction assumption, assume that ${\\rm {deg}}(E_j^{\\prime }(A,1))=j \\quad \\hbox{for all}\\; \\; j<n.$ Then, owing to (REF ), it follows that ${\\rm {deg}}(E_n^{\\prime }(A,1))=n,\\qquad n\\ge 0.$ Therefore, by (REF ), ${\\rm {deg}}(p_n)=n-1.$ Subsequently, for any given polynomial, $q \\in \\mathbb {Z}[A]$ , we will denote by $\\ell (q)$ the leading coefficient of $q(A)$ .",
"According to Table REF , we already know that $\\ell (p_5)=1.$ As an induction hypothesis, assume that $\\ell (p_{4(n-1)+1})=1.$ By (REF ), (REF ) and (REF ) $\\ell (p_{4n-2}) & = {\\color {blue} \\ell (E_{4n-3}^{\\prime }(A,1))}=-\\ell (E^{\\prime }_{4(n-1)}(A,1))=-\\ell (p_{4(n-1)+1}),\\\\\\ell ({p_{4n-1}}) & ={\\color {brown} \\ell ({E_{4n-2}^{\\prime }(A,1)})} ={\\color {blue} \\ell ({E_{4n-3}^{\\prime }(A,1)})}=-\\ell (p_{4(n-1)+1}),\\\\\\ell ({p_{4n}}) & ={\\color {cyan} \\ell ({E_{4n-1}^{\\prime }(A,1)})}=-{\\color {brown} \\ell ({E_{4n-2}^{\\prime }(A,1)})}=\\ell (p_{4(n-1)+1}),\\\\\\ell ({p_{4n+1}}) & =\\ell ({E_{4n}^{\\prime }(A,1)})={\\color {cyan} \\ell ({E_{4n-1}^{\\prime }(A,1)})}=\\ell (p_{4(n-1)+1}).\\\\$ By (REF ), the proof is complete.",
"As a consequence of these lemmas, the next result holds.",
"Proposition 4.6 Suppose (REF ).",
"Then, the problem (REF ) possesses infinitely many subharmonics.",
"In other words, there exists a sequence of integers $\\lbrace n_m\\rbrace _{m\\ge 1}$ with $\\lim _{m\\rightarrow +\\infty } n_m =+\\infty ,$ such that (REF ) has at least a $n_mT$ -periodic coexistence state for every $m\\ge 1$ .",
"Since $p_n(0)=n$ for all $n\\in \\mathbb {N}$ and, thanks to Lemma REF , for every integer $n\\ge 1$ , $\\ell (p_{4n+2})=\\ell (p_{4n+3})=-1,$ it becomes apparent that $p_{4n+2}(A)$ (resp.",
"$p_{4n+3}(A)$ ) possesses a root, $A_{4n+2}$ (resp.",
"$A_{4n+3}$ ), where it changes of sign.",
"Thus, for every integer $n\\ge 1$ , there exist two odd integers, $i_n, j_n\\ge 1$ , for which $p^{k)}_{4n+2}(A_{4n+2}) & =0, \\quad 0\\le k \\le i_n-1, \\quad p^{i_n)}_{4n+2}(A_{4n+2}) \\ne 0, \\\\p^{k)}_{4n+3}(A_{4n+3}) & =0, \\quad 0\\le k \\le j_n-1, \\quad p^{j_n)}_{4n+3}(A_{4n+3}) \\ne 0.$ Thus, the algebraic multiplicity of [7] for these polynomials at those roots is given by $\\chi [p_{4n+2}(A);A_{4n+2}]= i_n, \\qquad \\chi [p_{4n+3}(A);A_{4n+3}]= j_n.$ As these integers are odd, by Theorem 5.6.2 of [10], the local topological indexes of $p_{4n+2}(A)$ and $p_{4n+3}(A)$ change as $A$ crosses $A_{4n+2}$ and $A_{4n+3}$ , respectively.",
"Therefore, by Theorem 6.2.1 of [10], there exist two components of $(4n+2)T$ -periodic solutions and $(4n+3)T$ -periodic solutions bifurcating from the trivial solution $(A,1)$ at the roots $A_{4n+2}$ and $A_{4n+3}$ , respectively.",
"This ends the proof.",
"The next result establishes Property (d).",
"Lemma 4.7 $ p_2|p_{2n} $ for all $ n\\ge 1$ .",
"Thus, since $p_2(A)=-A+2$ , $r=2$ is a root of $p_{2n}(A)$ for all integer $n\\ge 1$ .",
"By (REF ), any $2T$ -periodic solution is a $2nT$ -periodic solution for all $n\\ge 1$ .",
"Thus, any bifurcation point from $(A,1)$ to $2T$ -periodic solutions must be a bifurcation point to $2nT$ -periodic solutions.",
"Since the unique bifurcation value to $2T$ -periodic solutions is the root of $p_2(A)=-A+2$ , given by $r=2$ , it becomes apparent that $p_{2n}(2)=0$ for all integer $n\\ge 1$ .",
"Therefore, $p_2|p_{2n}$ for all $n\\ge 1$ .",
"This ends the proof.",
"The next list of polynomials, collecting $p_{2n+1}(A)$ and $\\frac{p_{2n}(A)}{2-A}$ , for $1 \\le n \\le 6$ , might be helpful to understand the (very sharp) identity established by the next result.",
"$\\dfrac{p_2(A)}{2-A}&=1\\\\p_3(A)&=-A^2+3\\\\\\dfrac{p_4(A)}{2-A}&=-A^2+2\\\\p_5(A)&=A^4-5A^2+5\\\\\\dfrac{p_6(A)}{2-A}&=A^4-4A^2+3\\\\p_7(A)&=-A^6+7A^4-14A^2+7\\\\\\dfrac{p_8(A)}{2-A}&=-A^6+6A^4-10A^2+4\\\\p_9(A)&=A^8-9A^6+27A^4-30A^2+9\\\\\\dfrac{p_{10}(A)}{2-A}&=A^8-8A^6+21A^4-20A^2+5\\\\p_{11}(A)&=-A^{10}+11A^8-44A^6+77A^4-55A^2+11\\\\\\dfrac{p_{12}(A)}{2-A}&=-A^{10}+10A^8-36A^6+56A^4-35A^2+6\\\\p_{13}(A)&=A^{12}-13A^{10}+65A^8-156A^6+182A^4-91A^2+13.$ Theorem 4.8 The following identity holds $\\dfrac{p_n(A)}{2-A}=p_{n-1}(A) - \\dfrac{p_{n-2}(A)}{2-A}$ for all $n\\in 2\\mathbb {N}$ , whereas $\\dfrac{p_n(A)}{2+A}=p_{n-1}(A) - \\dfrac{p_{n-2}(A)}{2+A}$ for all $n\\in 2\\mathbb {N}+1$ .",
"First, we will prove the next relationships $\\left\\lbrace \\begin{array}{ll}\\displaystyle {- \\frac{n}{2}-1-\\sum _{\\begin{array}{c}j=1\\\\ j \\in 2\\mathbb {N}\\end{array}}^{n} E_{j}^{\\prime }(A,1) =1-A+\\sum _{j=3}^{n+1}(-1)^jp_j,}&\\quad n\\in 2\\mathbb {N},\\\\[11pt]\\displaystyle { \\left[\\frac{n}{2}\\right]+1+\\sum _{\\begin{array}{c}j=1\\\\ j \\in 2\\mathbb {N}+1\\end{array}}^{n} E_{j}^{\\prime }(A,1) =1-A+\\sum _{j=3}^{n+1}(-1)^jp_j,}&\\quad n\\in 2\\mathbb {N}+1.\\end{array}\\right.$ Since $p_2(A)=2-A$ , particularizing (REF ) at $n=3$ yields $-2-E_2^{\\prime }(A,1)=1-A-p_3(A),$ which is (REF ) for $n=2$ .",
"As the induction assumption, assume that (REF ) holds for some $n=2m$ with $m\\ge 1$ , i.e., $- m -1-\\sum _{\\begin{array}{c}j=1\\\\ j \\in 2\\mathbb {N}\\end{array}}^{2m} E_{j}^{\\prime }(A,1) =1-A+\\sum _{j=3}^{2m+1}(-1)^jp_j(A).$ According to (REF ), $2m+2+E_1^{\\prime }(A,1)+E_2^{\\prime }(A,1)+\\cdots +E_{2m}^{\\prime }(A,1)+E_{2m+1}^{\\prime }(A,1)=p_{2m+2}(A).$ Thus, adding (REF ) and (REF ), we obtain that $m+1 + \\sum _{\\begin{array}{c}j=1\\\\ j \\in 2\\mathbb {N}+1\\end{array}}^{2m+1} E_{j}^{\\prime }(A,1)=1-A+\\sum _{j=3}^{2m+2}(-1)^jp_j(A).$ Equivalently, $\\left[\\frac{2m+1}{2}\\right]+1 + \\sum _{\\begin{array}{c}j=1\\\\ j \\in 2\\mathbb {N}+1\\end{array}}^{2m+1} E_{j}^{\\prime }(A,1)=1-A+\\sum _{j=3}^{2m+2}(-1)^jp_j(A),$ which shows the validity of (REF ) for $n=2m+1$ .",
"To prove the validity of (REF ) for $n=2(m+1)=2m+2$ , we can argue similarly.",
"Again by (REF ), $2m+3+E_1^{\\prime }(A,1)+E_2^{\\prime }(A,1)+\\cdots +E_{2m+1}^{\\prime }(A,1)+E_{2m+2}^{\\prime }(A,1)=p_{2m+3}(A).$ Hence, subtracting (REF ) from (REF ) yields $-m-2 - \\sum _{\\begin{array}{c}j=1\\\\ j \\in 2\\mathbb {N}\\end{array}}^{2m+2} E_{j}^{\\prime }(A,1)=1-A+\\sum _{j=3}^{2m+3}(-1)^jp_j(A).$ Since $-\\frac{2m+2}{2}-1 =-m-2,$ (REF ) provides us with (REF ) for $n=2m+2$ , which ends the proof of (REF ).",
"By (REF ), it follows from (REF ) and (REF ) that $p_n(A)=\\left\\lbrace \\begin{array}{ll}\\displaystyle {p_{n-1}(A)+1+A\\Big [-\\dfrac{n}{2}-\\sum _{\\begin{array}{c}j=1\\\\j\\in 2\\mathbb {N}\\end{array}}^{n-2}E_{j}^{\\prime }(A,1)\\Big ]} & \\quad \\hbox{if\\ } n\\in 2\\mathbb {N},\\\\[15pt]\\displaystyle {p_{n-1}(A) + 1 + A\\Big [\\dfrac{n-1}{2}+\\sum _{\\begin{array}{c}j=1\\\\j\\in 2\\mathbb {N}+1\\end{array}}^{n-2}E_{j}^{\\prime }(A,1)\\Big ]} &\\quad \\hbox{if\\ } n \\in 2\\mathbb {N}+1.",
"\\end{array}\\right.$ On the other hand, when $n\\in 2\\mathbb {N}$ , it follows from (REF ) and (REF ) that $\\begin{split}p_n(A)-p_{n-1}(A)&=1+A\\Big [-\\dfrac{n}{2}-\\sum _{\\begin{array}{c}j=1\\\\j\\in 2\\mathbb {N}\\end{array}}^{n-2}E_{j}^{\\prime }(A,1) \\Big ]\\nonumber \\\\ &=1+A\\Big [-\\dfrac{n-2}{2}-1-\\sum _{\\begin{array}{c}j=1\\\\j\\in 2\\mathbb {N}\\end{array}}^{n-2}E_{j}^{\\prime }(A,1) \\Big ]\\nonumber \\\\ & =1+A\\Big [ 1-A+\\sum _{j=3}^{n-1}(-1)^jp_j\\Big ]\\nonumber \\\\&=1-Ap_{n-1}(A)+A\\Big [1-A+\\sum _{j=3}^{n-2}(-1)^jp_j\\Big ]\\nonumber \\\\&=-Ap_{n-1}(A)+1+A\\Big [\\displaystyle {\\dfrac{n-2}{2}+\\sum _{\\begin{array}{c}j=1\\\\ j\\in 2\\mathbb {N}+1\\end{array}}^{n-3} E_{j}^{\\prime }(A,1)}\\Big ]\\\\&=-Ap_{n-1}(A)+p_{n-1}(A)-p_{n-2}(A).\\end{split}$ Therefore, for every $n\\in 2\\mathbb {N}$ , $p_n(A)= (2-A)p_{n-1}(A)-p_{n-2}(A).$ The proof is complete for $n$ even.",
"Subsequently, we assume that $n$ is odd.",
"Arguing as in the previous case, from (REF ) and (REF ) the following chain of identities holds $p_n(A)-p_{n-1}(A)& = 1+A\\Big [\\dfrac{n-1}{2}+\\sum _{\\begin{array}{c}j=1\\\\j\\in 2\\mathbb {N}+1\\end{array}}^{n-2}E_{j}^{\\prime }(A)\\Big ]\\\\ &=1+A\\Big [1-A+\\sum _{j=3}^{n-1}(-1)^jp_j\\Big ]\\\\&=1+ Ap_{n-1}(A)+A\\Big [1-A+\\sum _{j=3}^{n-2}(-1)^jp_j\\Big ]\\\\&=Ap_{n-1}(A)+1+ A\\Big [-\\dfrac{n-1}{2}-\\sum _{\\begin{array}{c}j=1\\\\j\\in 2\\mathbb {N}\\end{array}}^{n-3}E_{j}^{\\prime }(1)\\Big ]\\\\&=Ap_{n-1}(A)+p_{n-1}(A)-p_{n-2}(A).$ Therefore, for every $n\\in 2\\mathbb {N}+1$ , $p_n(A) =(2+A)p_{n-1}(A)-p_{n-2}(A).$ This ends the proof.",
"Theorem REF can be summarized into the next generalized identity $p_n(A)=[2-(-1)^n A]p_{n-1}(A)-p_{n-2}(A),\\qquad n\\in \\mathbb {N}.$ As a by-product of these identities, the next result, establishing Property (e) at the beginning of the section, holds.",
"Corollary 4.9 For every $n\\ge 1$ , the polynomials $\\dfrac{p_{2n}(A)}{2-A}$ and $p_{2n+1}(A)$ are even.",
"We already know that $\\frac{p_2(A)}{2-A}=1\\qquad \\hbox{and}\\qquad p_3(A)=-A^2+3.$ Arguing by induction, assume that $\\dfrac{p_{2m-2}(A)}{2-A}$ and $p_{2m-1}(A)$ are even polynomials for some $m\\ge 1$ .",
"Then, by (REF ), $\\frac{p_{2m}(A)}{2-A}=p_{2m-1}(A)-\\frac{p_{2m-2}(A)}{2-A}$ must be also even, because it is sum of two even functions.",
"Similarly, since $p_{2m+1}$ can be expressed in the form $p_{2m+1}(A)=(2+A)p_{2m}(A)-p_{2m-1}(A)=(4-A^2)\\dfrac{p_{2m}(A)}{2-A}-p_{2m-1}(A),$ it becomes apparent that $p_{2m+1}(A)$ is also an even polynomial.",
"The proof is completed.",
"Characterizing the bifurcation points from $(A,1)$ The following definition will be used in the statement of the main theorem of this section.",
"Definition 5.1 Given two arbitrary polynomials $q_1, q_2\\in \\mathbb {Z}[A]$ , it is said that the roots of $q_1$ are separated by the roots of $q_2$ if all the roots of $q_2$ lye in between the maximal and minimal roots of $q_1$ and any pair of consecutive roots of $q_2$ contains exactly one root of $q_1$ .",
"The main theorem of this section can be stated as follows.",
"It counts the number of roots of each of the polynomials $p_n(A)$ , $n\\ge 1$ , establishing that there are as many roots as indicated by the degree, that all of them are real and algebraically simple and that the positive roots of $p_{n+1}(A)$ are always separated by the positive roots (less than 2 if $n\\in 2\\mathbb {N}$ ) of $p_n(A)$ .",
"So, it counts all roots establishing their relative positions.",
"Theorem 5.2 For every $n\\ge 2$ , the positive roots of $p_{2n}(A)$ are separated by the positive roots of $p_{2n-1}(A)$ , and the positive roots of $p_{2n+1}(A)$ are separated by those of $p_{2n}(A)$ less than 2.",
"Moreover, for every $n\\ge 1$ , the even polynomials $\\tfrac{p_{2n}(A)}{2-A}$ and $p_{2n-1}(A)$ have (exactly) $n-1$ positive roots.",
"Thus, since they are even with degree $2n-2$ , they must have another $n-1$ negative roots and, therefore, all roots are real and simple.",
"As we have already constructed the associated polynomials above, it is easily seen that all the thesis of Theorem REF hold to be true for $2\\le n\\le 6$ .",
"This task can be easily accomplished by simply looking at Figure REF , where we have plotted all the positive roots of $p_n(A)$ for $2\\le n\\le 13$ .",
"These roots are located in the interval $(0,2]$ and have been represented in abscisas at different levels according to $n$ .",
"As inserting in the same interval $(0,2]$ all the zeros of the first 13 polynomials would not be of any real help for understanding their fine distribution, we have superimposed them at 13 different levels, each of them containing the positive roots of each of the polynomials $p_n$ , $2\\le n\\le 13$ .",
"In total we are representing 42 roots, though some of them are common roots of different polynomials as a result of the fact that any $kT$ -periodic solution must be a $nkT$ -periodic solution for all $n\\ge 1$ .",
"These common roots have been represented in vertical dashed lines to emphasize that all roots on them share the same abscisa value.",
"In such case, the ordinates provide us with the corresponding value of $n$ .",
"By simply having a glance at Figure REF , it is easily realized how the two roots of the polynomial $p_4$ are separated by the root of $p_3$ , the 3 roots of $p_6$ are separated by the 2 roots of $p_5$ , the 4 roots of $p_8$ are separated by the 3 of $p_7$ , and so on...",
"Similarly, the two roots of $p_5$ are separated by the unique root of $p_4$ different from 2, the 3 roots of $p_7$ are separated by the 2 roots of $p_6$ different from 2, and so on...",
"The proof of the theorem will be delivered in two steps by induction in both cases.",
"Since $\\tfrac{p_2(A)}{2-A}=1$ does not admit any root, this is a very special case that will not play any rol in these induction arguments.",
"Figure: Positive roots of p n p_n , 2≤n≤132\\le n \\le 13.",
"Step 1: Passing from $p_{2n}(A)$ to $p_{2n+1}(A)$ , $n\\ge 2$ .",
"According to Figure REF , it becomes apparent that the two positive roots of $p_4(A)$ are separated by the unique root of $p_3(A)$ .",
"Moreover, all these zeros are real and simple and each of the polynomials $p_{3}(A)=-A^2+3,\\qquad \\frac{p_4(A)}{2-A}= -A^2+2,$ has a unique positive root.",
"Arguing by induction, assume that $p_{2n-1}(A)$ and $p_{2n}(A)$ satisfy all the assertions of the statement of the theorem for some $n\\ge 2$ .",
"In other words, all the positive roots of these polynomials are real and algebraically simple, the positive roots of $p_{2n}(A)$ are separated by the positive roots of $p_{2n-1}(A)$ , and the polynomials $\\tfrac{p_{2n}(A)}{2-A}$ and $p_{2n-1}(A)$ have (exactly) $n-1$ positive roots.",
"We claim that the positive roots of the polynomial $p_{2n+1}(A)$ are real and simple, that they are separated by the positive roots of $p_{2n}(A)$ , except for 2, and that it has (exactly) $n$ positive roots.",
"Indeed, by Theorem REF , we already know that $p_{2n+1}(A)=(2+A)p_{2n}(A) - p_{2n-1}(A).$ First, we will show the previous claim in the case when $ 2n\\in 4\\mathbb {N}+2 $ .",
"So, suppose $2n\\in 4\\mathbb {N}+2$ .",
"Figure REF shows the plots of the polynomials $p_{2n-1}(A)$ and $(2+A)p_{2n}(A)$ in one of such cases: Figure: Sketch of the construction of p 2n+1 (A)p_{2n+1}(A).$p_{2n-1}(A)$ has been plotted in brown and $(2+A)p_{2n}(A)$ in blue.",
"According to Lemmas REF and REF , we already know that $2n-1 = p_{2n-1}(0)< 4n = 2p_{2n}(0), \\quad \\mathrm {deg\\,}(p_{2n-1})=2n-2,\\quad \\mathrm {deg\\,}((2+A)p_{2n})=2n,$ and, since $2n\\in 4\\mathbb {N}+2$ , the leading coefficient of $p_{2n-1}(A)$ equals 1, while the leading coefficient of $p_{2n}(A)$ equals $-1$ .",
"Thus, $p_{2n-1}(A)>0$ and $(2+A)p_{2n}(A)<0$ for $A>2$ .",
"By the induction assumption, the polynomials $\\tfrac{p_{2n}(A)}{2-A}$ and $p_{2n-1}(A)$ have (exactly) $n-1$ positive roots.",
"Hence, each of the polynomials $p_{2n-1}(A)$ and $(2+A)p_{2n}(A)$ possesses (exactly) $n-1$ simple roots in the interval $(0,2)$ and, in addition, $p_{2n}(2)=0$ .",
"In Figure REF , we have named by $\\rho _i$ , $1 \\le i \\le n-1$ , the $n-1$ positive roots of $p_{2n-1}(A)$ , $0 < \\rho _1<\\rho _2< \\cdots < \\rho _{n-2}<\\rho _{n-1}<2,$ while those of $p_{2n}(A)$ less than 2 have been named by $r_i$ , $1\\le i \\le n-1$ .",
"So, $0 < r_1< r_2<\\cdots r_{n-1}<r_{n-1}< r_{n}:=2.$ As, again by the induction hypothesis, the positive roots of $(2+A)p_{2n}(A)$ are separated by the positive roots of $p_{2n-1}(A)$ , necessarily $0<r_1<\\rho _1<r_2<\\rho _2<\\cdots < r_{n-2}<\\rho _{n-2}<r_{n-1}<\\rho _{n-1}<r_n=2.$ Consequently, by (REF ), the polynomial $p_{2n+1}(A)$ must have, at least, $n$ different roots in the interval $(0,2)$ .",
"These roots have been named by $z_i$ , $1\\le i \\le n$ , in Figure REF and they satisfy $0 < z_1 < r_1 < \\rho _1 < z_2 < r_2 < \\rho _2 < \\cdots < z_{n-1}<r_{n-1}<\\rho _{n-1}<z_n< 2.$ On the other hand, by Corollary REF , $p_{2n+1}(A)$ is an even polynomial.",
"Thus, since, due to Lemma REF , it has degree $2n$ and, by the previous construction, $\\pm z_i$ , $1\\le i \\le n$ , provides us with a set of $2n$ different roots of $p_{2n+1}(A)$ , necessarily $p_{2n+1}(A)= -\\prod _{j=1}^n(A^2-z_j^2), \\qquad A >0.$ Therefore, all the roots of $p_{2n+1}(A)$ are real and algebraically simple.",
"As a direct consequence of (REF ) it is apparent that the positive roots of $p_{2n+1}(A)$ are separated by the positive roots of $p_{2n}(A)$ , except for 2.",
"Subsequently, we should prove the result in the special case when $2n\\in 4\\mathbb {N}$ .",
"In this situation, owing to Lemmas REF and REF , the plots of the polynomials $p_{2n-1}(A)$ and $(2+A)p_{2n}(A)$ look like illustrated by Figure REF .",
"Apart from the fact that now $p_{2n-1}(A)>0$ and $(2+A)p_{2n}(A)<0$ for all $A>2$ , because the leading coefficients change sign, the previous analysis can be easily adapted to cover the present situation in order to infer that $p_{2n+1}(A)$ satisfies all the requirements also in this case.",
"By repetitive the technical details of the proof are omitted here in.",
"Figure: Sketch of the construction of p 2n+1 (A) p_{2n+1}(A) .",
"Step 2: Passing from $p_{2n+1}(A)$ to $p_{2n+2}(A)$ , $n\\ge 2$ .",
"According to Figure REF , it becomes apparent that the two positive roots of $p_5(A)$ are separated by the unique root of $p_4(A)$ less than 2.",
"Moreover, all their roots are real and simple.",
"Note that the polynomials $\\frac{p_4(A)}{2-A}= -A^2+2, \\qquad p_5(A)=A^4-5A^2+5,$ have one and two positive roots respectively.",
"Arguing by induction, assume that $p_{2n}(A)$ and $p_{2n+1}(A)$ satisfy all the requirements in the statement of the theorem for some $n\\ge 2$ , i.e., all the positive roots of these polynomials are real and algebraically simple, the positive roots of $p_{2n+1}(A)$ are separated by the positive roots less than 2 of $p_{2n}(A)$ , and the polynomials $\\tfrac{p_{2n}(A)}{2-A}$ and $p_{2n+1}(A)$ have, respectively, $n-1$ and $ n $ positive roots.",
"We claim that the roots of the polynomial $p_{2n+2}(A)$ are real and simple, that they are separated by the roots of $p_{2n+1}(A)$ , and that $p_{2n+2}(A)$ possesses $n+1$ positive roots.",
"Indeed, by Theorem REF , $p_{2n+2}(A)=(2-A)p_{2n+1}(A) - p_{2n}(A).$ As in the previous step, we first deal with the case when $2n\\in 4\\mathbb {N}+2$ .",
"By Lemmas REF and REF , we already know that $2n = p_{2n}(0)< 4n+2 = 2p_{2n+1}(0), \\;\\; \\mathrm {deg\\,}(p_{2n})=2n-1,\\;\\;\\mathrm {deg\\,}\\left((2\\!-\\!A)p_{2n+1}\\right)=2n+1,$ and, since $2n\\in 4\\mathbb {N}+2$ , the leading coefficient of $p_{2n}(A)$ equals $-1$ , and the leading coefficient of $p_{2n+1}(A)$ equals also $-1$ .",
"Thus, $p_{2n}(A)<0,\\qquad (2-A)p_{2n+1}(A)>0\\qquad \\hbox{for all}\\;\\; A>2.$ By the induction assumption, the polynomials $\\tfrac{p_{2n}(A)}{2-A}$ and $p_{2n+1}(A)$ have (exactly) $n-1$ and $n$ positive roots, respectively.",
"Thus, each of the polynomials $p_{2n}(A)$ and $(2-A)p_{2n+1}(A)$ possesses (exactly) $n$ simple roots in $(0,2)$ and, obviously, $(2-A)p_{2n+1}(A)$ also vanishes at $A=2$ .",
"Figure REF shows the plots of $p_{2n}(A)$ , in blue, and $(2-A)p_{2n+1}(A)$ , in brown.",
"Figure: Sketch of the construction of p 2n+2 (A) p_{2n+2}(A) in case 2n∈4ℕ+22n\\in 4\\mathbb {N}+2.In Figure REF , we have named by $\\rho _i$ , $1 \\le i \\le n$ , the $n$ positive roots less than 2 of $(2-A)p_{2n+1}(A)$ , $0 < \\rho _1<\\rho _2< \\cdots < \\rho _{n-1}<\\rho _{n}<2:=\\rho _{n+1},$ whereas $r_i$ , $1\\le i \\le n$ , stand for the positive roots of $p_{2n}(A)$ .",
"Since $p_{2n}(2)=0$ , $r_n=2$ .",
"Since the positive roots of $p_{2n+1}(A)$ are separated by the positive roots less than 2 of $p_{2n}(A)$ , the following holds $0 < \\rho _1<r_1<\\rho _2<r_2< \\cdots < \\rho _{n-1}<r_{n-1}<\\rho _{n}<2:=\\rho _{n+1}=r_{n},$ as illustrated by Figure REF .",
"Thanks to (REF ), it becomes apparent that the polynomial $p_{2n+2}(A)$ admits, at least, an interior root in each of the intervals $(\\rho _i,\\rho _{i+1})$ , $i=0,...,n$ , denoted by $z_i$ in Figure REF , plus $z_{n+1}=2$ .",
"Here we are setting $\\rho _0:=0$ .",
"Consequently, $p_{2n+2}(A)$ has, at least, $n+1$ positive roots.",
"On the other hand, thanks to Corollary REF , $\\tfrac{p_{2n+2}(A)}{2-A} $ is an even function and hence, $p_{2n+2}(A)$ has, at least, $2n+1$ different roots.",
"Since, by Lemma REF , $\\mathrm {deg\\,}(p_{2n+2})=2n+1,$ all these roots are real and algebraically simple.",
"By construction, it is apparent that the positive roots of $p_{2n+2}(A)$ are separated by the positive roots of $p_{2n+1}(A)$ (see Figure REF if necessary).",
"If, instead of $2n\\in 4\\mathbb {N}+2$ , we impose $ 2n\\in 4\\mathbb {N}$ , then the previous arguments can be easily adapted to complete the proof of the theorem from Figure REF , where the graphs of $(2+A)p_{2n+1}(A)$ and $p_{2n}(A)$ have been superimposed in order to show their crossing points, which, owing to Theorem REF , are the roots of $p_{2n+2}(A)$ .",
"By repetitive, the technical details of this case are not included here.",
"Figure: Sketch of the construction of p 2n+2 (A) p_{2n+2}(A) in case 2n∈4ℕ2n\\in 4\\mathbb {N}.A careful reading of the proof of Theorem REF reveals that, actually, not only the roots of $p_{n}(A)$ are separated by those of $p_{n-1}(A)$ , but that they are also separated by those of $p_{n-2}(A)$ , taking always into account the exceptional role played by the root 2.",
"Global bifurcation diagram This section analyzes the global structure of the set of zeros of the maps $\\varphi _n$ , $n\\ge 1$ , introduced in (REF ).",
"These zeros are the positive fixed points of the Poincaré maps $\\mathcal {P}_n$ , $n\\ge 1$ , constructed in Section 3.",
"They provide us with the $nT$ -periodic coexistence states of (REF ) under the additional assumption (REF ).",
"It should be remembered that, according to (REF ), for every integer $n\\ge 1$ $p_n(A)= \\frac{d\\varphi _n (A,1)}{dx}=\\mathfrak {L}(n;A),\\qquad A>0,$ provides us with the linearization at the trivial curve, $(A,1)$ , of $\\varphi _n(A,x)$ .",
"In our analysis, $A$ is always regarded as a bifurcation parameter to $nT$ -periodic coexistence states from the $T$ -periodic ones (i.e., from $x=1$ ).",
"As a consequence of the simplicity of all the roots of $p_n(A)$ , $n\\ge 1$ , guaranteed by Theorem REF , the following result holds.",
"Theorem 6.1 For every $n\\ge 1$ and $r \\in p_n^{-1}(0)$ the following algebraic transversality condition holds $\\mathfrak {L}_1 (N\\left[\\mathfrak {L}(n;r) \\right] )\\oplus R\\left[ \\mathfrak {L}(n;r) \\right]=\\mathbb {R},$ where $\\mathfrak {L}_1:= \\frac{d \\mathfrak {L}(n;r)}{d A} , \\qquad n\\ge 1,\\;\\; r\\in p_n^{-1}(0).$ Therefore, by Theorem 1.7 of M. G. Crandall and P. H. Rabinowitz [3], there exists an analytic curve of $nT$ -periodic coexistence states of (REF ) bifurcating from $(A,1)$ at the root $A=r$ .",
"Actually, there exists $\\varepsilon >0$ and a real analytic map $A: (-\\varepsilon ,\\varepsilon )\\rightarrow \\mathbb {R}$ such that $A(0)=r$ and $\\varphi _n(A(s),1+s)=0 \\quad \\hbox{for all}\\;\\; s\\in (-\\varepsilon ,\\varepsilon ).$ Moreover, any non-trivial zero of $\\varphi _n$ , $(A,x)$ with $x\\ne 1$ , in a neighborhood of $(r,1)$ must be of the form $(A(s),1+s)$ for some $s \\in (-\\varepsilon ,\\varepsilon )$ .",
"In other words, there exists $\\varrho >0$ such that $\\left.",
"\\begin{array}{rr} \\varphi _n(A,x)=0 \\\\ |A-r|+|x-1|<\\varrho \\\\ x\\ne 1\\end{array}\\right\\rbrace \\Longrightarrow (A,x)=(A(s),1+s) \\quad \\hbox{for some}\\;\\; s \\in (-\\varepsilon ,\\varepsilon ).$ Furthermore, setting $A(s)= r + A_1 s + A_2 s^2 + \\mathcal {O} (s^3) \\quad \\hbox{as}\\;\\; s\\rightarrow 0,$ one has that $A_1=0$ and $A_2>0$ if $n=2$ and $r=2$ , in complete agreement with Figure REF ; $A_1<0$ if $n=3$ and $r=\\sqrt{3}$ ; $A_1=0$ and $A_2 <0$ (resp.",
"$A_2>0$ ) if $n=4$ and $r = r_{4,1}=\\sqrt{2}$ (resp.",
"$r = r_{4,2}=2$ ).",
"According to (REF ), $\\mathfrak {L}(n;r)=p_n(r)=0$ .",
"Thus, $N[\\mathfrak {L}(n;r)]=\\mathbb {R}$ and (REF ) can be equivalently expressed as $\\mathfrak {L}_1(\\mathbb {R}) =\\mathbb {R}$ , which holds true because, thanks to Theorem REF , we already know that $r$ is an algebraically simple root of $p_n(A)$ , i.e., $\\mathfrak {L}_1 = p_n^{\\prime }(r) \\ne 0.$ So, (REF ) indeed holds and [3] applies to $\\varphi _n(A,x)=0$ at $(A,x)=(r,1)$ .",
"Since we can take $\\psi =1$ as a generator of $N[\\mathfrak {L}(n;r)]=\\mathbb {R}$ and $Y=[0]$ as a supplement of $N[\\mathfrak {L}(n;r)]=\\mathbb {R}$ in $\\mathbb {R}$ , owing to [3], there exist $\\varepsilon >0$ and a real analytic map $(A,y):(-\\varepsilon ,\\varepsilon )\\rightarrow \\mathbb {R}\\times Y$ such that $(A(0),y(0))=(r,0)$ and $\\varphi _n(A(s),1+s(\\psi +y(s)))=0 \\quad \\hbox{for all}\\;\\; s\\in (-\\varepsilon ,\\varepsilon ),$ it becomes apparent, by construction, that $\\varphi _n(A(s),1+s)=0\\quad \\hbox{for all} \\;\\; s \\in (-\\varepsilon ,\\varepsilon ),$ because $y\\equiv 0$ and $\\psi =1$ .",
"This ends the proof of the first two claims of the theorem: the existence of the analytic curve of nontrivial solutions and the uniqueness.",
"As far as concerns to the problem of ascertaining the nature of these local bifurcations at $(r,1)$ , we can proceed as follows.",
"In order to prove Part (a), note that, thanks to (REF ), setting $x(s):=1+s$ and expanding in Taylor series, we have that $0 = \\varphi _2(A(s),x(s))=\\varphi _2(r,1)+\\frac{d\\varphi _2}{ds}(r,1)s+\\frac{1}{2}\\frac{d^2\\varphi _2}{ds^2}(r,1)s^2+\\cdots $ for all $s\\in (-\\varepsilon ,\\varepsilon )$ , where $r=2$ .",
"Moreover, by construction, we already know that $\\varphi _2(r,1)=0,\\qquad \\frac{\\partial \\varphi _2}{\\partial x}(r,1)=p_2(r)=p_2(2)=0$ (see (REF ), if necessary).",
"Thus, since by (REF ) $\\varphi _2(A,x)=x\\left( E_1(A,x)+1\\right) -2,$ it follows from (REF ) and $\\frac{\\partial E_1}{\\partial A}(r,1)=0$ that $\\frac{d\\varphi _2}{ds}(r,1)=\\frac{\\partial \\varphi _2}{\\partial A}(r,1)A^{\\prime }(0)=\\frac{\\partial E_1}{\\partial A}(r,1)A_1=0,$ where $^{\\prime }:=\\frac{d}{ds}$ .",
"Hence, these terms do not provide us with any neat information concerning $A_1$ .",
"So, we must consider higher order terms to find out $A_1$ .",
"As $\\frac{\\partial \\varphi _2}{\\partial A}(r,1)=0=\\frac{\\partial \\varphi _2}{\\partial x}(r,1),$ applying the chain rule it readily follows that $0= \\frac{d^2\\varphi _2}{ds^2}(r,1)=\\frac{\\partial ^2\\varphi _2}{\\partial x^2}(r,1)+2\\frac{\\partial ^2\\varphi _2}{\\partial A\\partial x}(r,1)A_1+\\frac{\\partial ^2\\varphi _2}{\\partial A^2}(r,1)A_1^2.$ On the other hand, differentiating with respect to $x$ the identity (REF ) yields $\\frac{\\partial \\varphi _2}{\\partial x}(A,x) = E_1(A,x)+1+ x \\frac{\\partial E_1}{\\partial x}(A,x).$ So, $\\frac{\\partial ^2 \\varphi _2}{\\partial x^2}(A,x) = 2 \\frac{\\partial E_1}{\\partial x} (A,x)+ x \\frac{\\partial ^2 E_1}{\\partial x^2}(A,x).$ Consequently, particularizing at $(A,x)=(r,1)$ , it follows from (REF ) that $\\frac{\\partial ^2 \\varphi _2}{\\partial x^2}(r,1) =r^2-2r=4-4=0.$ Similarly, differentiating (REF ) with respect to $A$ shows that $\\frac{\\partial ^2 \\varphi _2}{\\partial x\\partial A}(A,x)=\\frac{\\partial E_1}{\\partial A}(A,x)+x\\frac{\\partial ^2 E_1}{\\partial x\\partial A}(A,x)$ and hence, owing to (REF ), $\\frac{\\partial ^2 \\varphi _2}{\\partial x\\partial A}(r,1)=\\frac{\\partial E_1}{\\partial A}(r,1)+\\frac{\\partial ^2 E_1}{\\partial x\\partial A}(r,1) =\\frac{\\partial ^2 E_1}{\\partial x\\partial A}(r,1) = -1.$ Lastly, $\\frac{\\partial ^2\\varphi _2}{\\partial A^2}(A,x)=x\\frac{\\partial ^2 E_1}{\\partial A^2}(A,x)= (1-x)^2E_1(A,x)$ and hence, $\\frac{\\partial ^2\\varphi _2}{\\partial A^2}(A,1)=0.$ Therefore, substituting (REF ), (REF ) and (REF ) into (REF ) it becomes apparent that $A_1=0$ .",
"Thanks to this fact, the third derivative admits the next (simple) expression: $0=\\frac{d^3\\varphi _2}{ds^3}(r,1)=6\\frac{\\partial ^2\\varphi _2}{\\partial x\\partial A}(r,1)A_2+\\frac{\\partial ^3\\varphi _2}{\\partial x^3}(r,1)=-6A_2+3r^2-r^3,$ which implies that $A_2=\\frac{4}{3}>0$ and ends the proof of Part (a).",
"To prove Part (b), note that, much like in Part (a), one has that $0= \\varphi _3(A(s),x(s))=\\varphi _3(r,1)+\\frac{d\\varphi _3}{ds}(r,1)s+\\frac{1}{2} \\frac{d^2\\varphi _3}{ds^2}(r,1)s^2+\\cdots $ for all $s \\in (-\\varepsilon ,\\varepsilon )$ .",
"Similarly, $\\frac{\\partial \\varphi _3}{\\partial A}(r,1)=0=\\frac{\\partial \\varphi _3}{\\partial x}(r,1).$ So, $\\frac{d\\varphi _3}{ds}(r,1)=0.$ Moreover, differentiating twice with respect to $s$ yields $0 & = \\frac{d^2\\varphi _3}{ds^2}(r,1)\\\\[5pt] & =\\frac{\\partial ^2\\varphi _3}{\\partial x^2}(r,1)+2\\frac{\\partial ^2\\varphi _3}{\\partial x\\partial A}(r,1)A_1+\\frac{\\partial ^2\\varphi _3}{\\partial A^2}(r,1)A_1^2\\\\[5pt] & =r^4-r^3-2r^2-4rA_1.$ Consequently, since $r=\\sqrt{3}$ , it follows from this identity that $A_1=\\frac{\\sqrt{3}-3}{4}<0,$ which ends the proof of Part (b).",
"Finally, much like before, we have that $0= \\varphi _4(A(s),x(s))=\\varphi _4(r,1)+\\frac{d\\varphi _4}{ds}(r,1)s+\\frac{1}{2}\\frac{d^2\\varphi _4}{ds^2}(r,1)s^2+\\cdots $ for all $s \\in (-\\varepsilon ,\\varepsilon )$ , and, in addition, $\\frac{\\partial \\varphi _4}{\\partial A}(r,1)=0=\\frac{\\partial \\varphi _4}{\\partial x}(r,1).$ Thus, $\\frac{d\\varphi _4}{ds}(r,1)=0$ .",
"Moreover, differentiating twice yields $0 & =\\frac{d^2\\varphi _4}{ds^2}(r,1)\\\\[5pt] &=\\frac{\\partial ^2\\varphi _4}{\\partial x^2}(r,1)+2\\frac{\\partial ^2\\varphi _4}{\\partial x\\partial A} (r,1)A_1+\\frac{\\partial ^2\\varphi _4}{\\partial A^2}(r,1)A_1^2 \\\\[5pt] &=r^6-3r^5-r^4+8r^3-2r^2-4r+2(r^3-2r^2-2r)A_1.$ Therefore, since $r=\\sqrt{2}$ it follows from this identity that $A_1=0$ .",
"Furthermore, $0 & = \\frac{d^3\\varphi _2}{ds^3}(r,1)\\\\[5pt] &=6\\frac{\\partial ^2\\varphi _2}{\\partial x\\partial A} (r,1)A_2+\\frac{\\partial ^3\\varphi _2}{\\partial x^3}(r,1)\\\\[5pt] &=6(3r^2-4r-2)A_2-r^8+r^7+9r^6-11r^5-10r^4+20r^3-2r^2.$ Consequently, we find from $r=\\sqrt{2}$ that $A_2=-\\frac{2(5+4\\sqrt{2})}{3}<0,$ which ends the proof.",
"Figure REF shows the local bifurcation diagrams of the $2T$ , $3T$ and $4T$ -periodic coexistence states of (REF ) under condition (REF ).",
"We are plotting $x$ , in ordinates, versus $A$ , in abscisas.",
"By the analysis already done at the beginning of Section 2, and, in particular, by Theorem REF , which was sketched in Figure REF , we already know that, under condition (REF ), the problem (REF ) admits a $2T$ -periodic coexistence state if, and only if, $A>2$ .",
"Moreover, the local bifurcation of these solutions must be supercritical.",
"Thus $A_2\\ge 0$ .",
"As a byproduct of Theorem REF , it turns out that $A_2>0$ .",
"So, it is a genuine supercritical pitchfork bifurcation of quadratic type.",
"However, since $A_1<0$ , the bifurcation to $3T$ -periodic coexistence states from $(A,x)=(\\sqrt{3},1)$ is transcritical, whereas the $4T$ -periodic solutions emanate from $(A,x)=(\\sqrt{2},1)$ through a subcritical quadratic pitchfork bifurcation, because $A_1=0$ and $A_2<0$ in this case.",
"The fact that the local nature of the first three bifurcation phenomena possess a completely different character shows that, in general, ascertaining the precise type of these local bifurcations for large $n$ might not be possible, much like happened with the problem of determining the fine structure of the set of bifurcation points from the trivial solution $(A,1)$ .",
"The higher is the order of the bifurcating subharmonics, measured by $n$ , the higher is the complexity of the associated function $\\varphi _n$ and hence, the more involved is finding out the values of $A_1$ and $A_2$ in (REF ) by the intrinsic nature of the functions $E_n$ defined in (REF ).",
"Figure: Local bifurcation diagrams from (A,x)=(A,1)(A,x)=(A,1) of the nTnT-periodic coexistence states for n∈{2,3,4}n\\in \\lbrace 2,3,4\\rbrace .Remark 6.2 Thanks to Theorem REF , it becomes apparent that the set of bifurcation points from $ (A,1) $ to $ nT $ -periodic coexistence states of (REF ) is the set of roots of $ p_n(A) $ .",
"Since the number of roots of a polynomial is finite, the set of bifurcation points is numerable, as it is a numerable union of finite sets.",
"Since every $nT$ -periodic coexistence state of (REF ) provides us with a $knT$ -periodic coexistence state for all $k\\ge 1$ , owing Theorem REF , the roots of $p_n(A)$ must be roots of $p_{kn}(A)$ for all $ n,k\\ge 1 $ , i.e., $p_n|p_{kn}$ for all $n, k \\ge 1$ .",
"Remark 6.3 Thanks to Theorem REF and Remark REF , the set of bifurcation points to a $nT$ -periodic solution is a subset of the interval $(0,2]$ .",
"Complementing [9], where the non-degeneration of the positive $T$ -periodic coexistence states of (REF ) with respect to the $T$ -periodic solutions was established, Theorem REF shows that the $T$ -periodic solutions are degenerated with respect to the $nT$ -periodic solutions of (REF ) for all $n\\ge 2$ at every positive root, $r$ , of $p_n(A)$ .",
"Nevertheless, the $T$ -periodic solutions are non-degenerated with respect to the $nT$ -periodic solutions, $n\\ge 2$ , if $A>2$ , because in this range there is not any bifurcation point from $(A,1)$ .",
"Subsequently, we will discuss the global character of all the local bifurcations documented by Theorem REF in the context of global bifurcation theory.",
"In this discussion, by a (connected) component it is understood any closed and connected subset that is maximal for the inclusion.",
"For any given integer $n\\ge 1$ , the set of non-trivial $nT$ -periodic solutions of (REF ), $\\mathcal {S}_n$ , consists of all $nT$ -periodic coexistence states different from $(A,1)$ plus the set of points $(r,1)$ with $p_n(r)=0$ .",
"In other words, setting $\\mathbb {R}_+:=(0,+\\infty )$ , $\\mathcal {S}_n= \\lbrace (A,x) \\in \\mathbb {R}_+\\times (\\mathbb {R}_+\\setminus \\lbrace 1\\rbrace ) \\;:\\; \\varphi _n(A,x)=0\\rbrace \\cup \\lbrace (r,1)\\;:\\; p_n(r)=0\\rbrace .$ Note that, owing to Theorem REF , $\\lbrace (r,1)\\;:\\; p_n(r)=0\\rbrace $ is the set of bifurcation points to $nT$ -periodic solutions from the trivial curve $(A,1)$ .",
"Thanks to (REF ), the algebraic multiplicity of J. Esquinas and J. López-Gómez [7] equals one, $\\chi [\\mathfrak {L}(n;A);r]=1 \\in 2\\mathbb {N}+1,$ for all $r \\in p_n^{-1}(0)$ , $r>0$ .",
"Thus, by [10], the local degree at $(A,1)$ of the one-dimensional $\\varphi _n(A,\\cdot )$ changes as $A$ crosses $r$ (see also J. López-Gómez and C. Mora-Corral [13] if necessary).",
"Therefore, according to [10], for every integer $n\\ge 2$ and each root $r>0$ of $p_n(A)$ , there is a component of $\\mathcal {S}_n$ , $\\mathfrak {C}_{n,r}$ , such that $(r,1) \\in \\mathfrak {C}_{n,r}\\subset \\mathbb {R}_+\\times \\mathbb {R}.$ Moreover, by the local uniqueness about $(r,1)$ guaranteed by Theorem REF , in a neighborhood of $(r,1)$ the component $\\mathfrak {C}_{n,r}$ consists of an analytic curve, $(A(s),1+s)$ , $|s|<\\varepsilon $ .",
"Note that any real continuous map must be compact.",
"So, the Leray–Schauder degree (see, e.g., N. G. Lloyd [8], or [13], if necessary) can be applied to get these global results.",
"Alternatively, one might use the degree of P. Benevieri and M. Furi [1], as in Theorem 5.4 and Corollary 5.5 of [12] (see [11] for a recent survey on global bifurcation theory).",
"By Remark REF , for every $r \\in p_n^{-1}(0)\\cap \\mathbb {R}_+$ , the component $\\mathfrak {C}_{n,r}$ must be separated away from $x=0$ and hence, all their solutions must be positive, because $\\varphi _n(0)=-n$ .",
"Thus, they indeed provide us with coexistence states of (REF ).",
"Similarly, for every $n\\ge 2$ , since $\\varphi _n(n)>0$ , $\\mathfrak {C}_{n,r}$ is bounded above by $n$ , in the sense that $x<n$ if $(A,x)\\in \\mathfrak {C}_{n,r}$ with $A>0$ .",
"Therefore, $\\mathcal {P}_x(\\mathfrak {C}_{n,r})\\subset (0,n),$ where $\\mathcal {P}_x$ stands for the $x$ -projection operator, $\\mathcal {P}_x(A,x):=x$ .",
"Moreover, due to (REF ), $x=1$ is the unique zero of $\\varphi _n(A,x)$ at $A=0$ .",
"Note that, due to Remark REF , $(A,1)=(0,1)\\notin \\mathfrak {C}_{n,r}$ because $p_n(0)=n>0$ .",
"Throughout the rest of this section, we will also consider the (unilateral) subcomponents $\\mathfrak {C}_{n,r}^+:=\\mathfrak {C}_{n,r}\\cap [x>1], \\qquad \\mathfrak {C}_{n,r}^-:=\\mathfrak {C}_{n,r}\\cap [x<1].$ Thanks to Theorem REF , these subcomponents are non-empty.",
"Moreover, arguing as in [10], it is easily seen that they equal the components $\\mathfrak {C}^+$ and $\\mathfrak {C}^-$ introduced on page [10].",
"This feature heavily relies on the fact that $x$ is a one-dimensional variable.",
"Therefore, the unilateral theorem [10] can be applied to infer that each of the components $\\mathfrak {C}_{n,r}^+$ and $\\mathfrak {C}_{n,r}^-$ satisfies the global alternative of P. H. Rabinowitz [20], because the supplement of $N[\\mathfrak {L}(n;r)]=\\mathbb {R}$ in $\\mathbb {R}$ is $Y=[0]$ and, due to (REF ), $\\mathfrak {C}_{n,r}$ cannot admit an element, $(A,x)$ with $x=0$ .",
"Therefore, $\\mathfrak {C}_{n,r}^+$ (resp.",
"$\\mathfrak {C}_{n,r}^-$ ) satisfies some of the following two conditions, which are far from being excluding: There exists $s \\in p_n^{-1}(0)\\setminus \\lbrace r\\rbrace $ (resp.",
"$t \\in p_n^{-1}(0)\\setminus \\lbrace r\\rbrace $ ) such that $(s,1)\\in \\mathfrak {C}_{n,r}^+$ (resp.",
"$(t,1)\\in \\mathfrak {C}_{n,r}^-$ ).",
"The component $\\mathfrak {C}_{n,r}^+$ (resp.",
"$\\mathfrak {C}_{n,r}^-$ ) is unbounded in $A$ , because of (REF ).",
"Note that the counterexample of E. N. Dancer [4] shows that Theorems 1.27 and 1.40 of P. H. Rabinowitz [20] are not true as originally stated.",
"To show that the second option occurs in both cases we need the next result.",
"Lemma 6.4 Each of the unilateral subcomponents satisfies $\\mathfrak {C}_{n,r}^\\pm \\cap \\lbrace (A,1)\\;:\\; A\\ge 0\\rbrace = \\lbrace (r,1)\\rbrace .$ Thus, also $\\mathfrak {C}_{n,r} \\cap \\lbrace (A,1)\\;:\\; A\\ge 0\\rbrace = \\lbrace (r,1)\\rbrace ,$ i.e., $(r,1)$ is the unique bifurcation point of $\\mathfrak {C}_{n,r}$ from $(A,1)$ .",
"Subsequently, we will denote by $\\nu (n)$ the total number of positive roots of the polynomial $p_n(A)$ .",
"By Theorem REF , we already know that $\\nu (n)=\\frac{n}{2}$ if $n$ is even and $\\nu (n)=\\frac{n-1}{2}$ if $n$ is odd.",
"We will prove the result only for $\\mathfrak {C}_{n,r}^+$ , as the same argument also works out to prove the corresponding assertion for the component $\\mathfrak {C}_{n,r}^-$ .",
"The proof will proceed by contradiction.",
"We already know that $\\mathfrak {C}_{n,r}^+$ can only meet the trivial solution $(A,1)$ at the roots of $p_n(A)$ .",
"Suppose that $r=r_{n,i}$ for some $i \\in \\lbrace 1,...,\\nu (n)\\rbrace $ , and that there exists $j>i$ , $j \\in \\lbrace 1,...,\\nu (n)\\rbrace $ , such that $\\lbrace (r_{n,i},1),(r_{n,j},1)\\rbrace \\subset \\mathfrak {C}_{n,r}^+ \\cap \\lbrace (A,1)\\;:\\; A\\ge 0\\rbrace .$ Then, by the definition of component, it becomes apparent that $\\mathfrak {C}_{n,r_{n,i}}^+=\\mathfrak {C}_{n,r_{n,j}}^+$ as sketched by Figure REF .",
"Thanks to Theorem REF , there exists $r_{n-1,k}\\in (r_{n,i},r_{n,j})\\cap p_{n-1}^{-1}(0).$ By the incommensurability of $nT$ with $(n-1)T$ , $\\mathfrak {C}_{n-1,r_{n-1,k}}^+$ cannot reach the component (REF ).",
"Thus, must be bounded.",
"Consequently, as $\\mathfrak {C}_{n-1,r_{n-1,k}}^+$ also satisfies the global alternative of P. H. Rabinowitz, there exists $r_{n-1,\\ell }\\in p_{n-1}^{-1}(0)$ , with $k\\ne \\ell $ , such that $\\lbrace (r_{n-1,k},1),(r_{n-1,\\ell },1)\\rbrace \\subset \\mathfrak {C}_{n-1,r_{n-1,k}}^+ \\cap \\lbrace (A,1)\\;:\\; A\\ge 0\\rbrace ,$ as sketched in Figure REF .",
"Since the set of roots $\\bigcup _{2\\le \\kappa \\le n} p_{\\kappa }^{-1}(0)$ is finite, it becomes apparent that, after finite many steps, there exists a component, $\\mathfrak {C}_{n-h,r_{n-h,m}}^+$ , for some $2 \\le h \\le n-3$ and $1\\le m\\le \\nu (n-h)$ , that should meet the last component linking two different roots sketched in Figure REF , $\\mathfrak {C}_{n-h+1,r_{n-h+1,v}}^+ = \\mathfrak {C}_{n-h+1,r_{n-h+1,w}}^+,$ because there is no any additional root of $p_{n-h}(A)$ in between $r_{n-h+1,v}$ and $r_{n-h+1,w}$ .",
"But this is impossible, by the incommensurability of $(n-h)T$ with $(n-h+1)T$ .",
"This contradiction ends the proof.",
"Figure: Sketch of the proof of Lemma .As an immediate consequence of the previous analysis, the next result holds.",
"As for the $x$ -projection operator, $\\mathcal {P}_x$ , we will denote by $\\mathcal {P}_A$ the $A$ -projection operator, $\\mathcal {P}_A(A,x):=A.$ Theorem 6.5 For every integer $n\\ge 2$ and each root $r>0$ of $p_n(A)$ , the component $\\mathfrak {C}_{n,r}^+$ satisfies $\\mathcal {P}_x (\\mathfrak {C}_{n,r}^+)\\subset [1,n)$ ; $\\mathcal {P}_A (\\mathfrak {C}_{n,r}^+)=[A^+_{n,r},+\\infty )$ for some $A_{n,r}^+ \\in (0,r]$ .",
"In particular, $\\mathfrak {C}_{n,r}^+$ is unbounded.",
"$\\mathfrak {C}_{n,r}^+ \\cap \\lbrace (A,1)\\;:\\; A\\ge 0\\rbrace = \\lbrace (r,1)\\rbrace $ ; For every $n, m \\ge 2$ , $\\mathfrak {C}_{n,r}^+\\cap \\mathfrak {C}_{m,s}^+=\\emptyset $ if $r \\ne s$ .",
"Moreover, by Theorem REF , in a neighborhood of $(r,1)$ the component $\\mathfrak {C}_{n,r}^+$ consists of an analytic curve, $(A(s),1+s)$ , $0\\le s <\\varepsilon $ .",
"Similarly, the component $\\mathfrak {C}_{n,r}^-$ satisfies (c), (d) and $\\mathcal {P}_x (\\mathfrak {C}_{n,r}^-)\\subset (0,1]$ ; $\\mathcal {P}_A (\\mathfrak {C}_{n,r}^-)=[A^-_{n,r},+\\infty )$ for some $A_{n,r}^- \\in (0,r]$ .",
"In particular, $\\mathfrak {C}_{n,r}^-$ is unbounded.",
"Analogously, in a neighborhood of $(r,1)$ the component $\\mathfrak {C}_{n,r}^-$ consists of an analytic curve, $(A(s),1+s)$ , $-\\varepsilon <s\\le 0$ .",
"At this stage, the only delicate point is Part (d).",
"Suppose that $\\mathfrak {C}_{n,r}^+\\cap \\mathfrak {C}_{m,s}^+\\ne \\emptyset $ for some $r \\ne s$ .",
"Then, by the definition of component, necessarily $\\mathfrak {C}_{n,r}^+ = \\mathfrak {C}_{m,s}^+.$ Thus, $(r,1), (s,1) \\in \\mathfrak {C}_{n,r}^+$ , which contradicts Lemma REF .",
"The proof is complete.",
"Except for the local bifurcations from the trivial line $(A,1)$ , the global diagramas of the components $\\mathfrak {C}_{n,r}^\\pm $ plotted in Figure REF respect the general properties established by Theorem REF .",
"Although the components have been plotted with no secondary bifurcations along them, there are some numerical evidences that $\\mathfrak {C}_{2,2}^-$ possesses a secondary bifurcation to $4T$ -periodic solutions.",
"Nevertheless, thanks to Theorem REF , even in the case that they might occur higher order bifurcations along these components, they must be disjoint.",
"According to Theorems REF and REF , it becomes apparent that some $3T$ and $4T$ -periodic solutions must be degenerated.",
"Namely, those on the turning points of $\\mathfrak {C}_{3,\\sqrt{3}}^+$ , $\\mathfrak {C}_{4,\\sqrt{2}}^+$ and $\\mathfrak {C}_{4,\\sqrt{2}}^-$ in Figure REF .",
"Similarly, the bifurcation points accumulating from the left to $\\sqrt{2}$ and $\\sqrt{3}$ must provide us with additional degenerate solutions: those on the turning points of their corresponding components."
],
[
"Characterizing the bifurcation points from $(A,1)$",
"The following definition will be used in the statement of the main theorem of this section.",
"Definition 5.1 Given two arbitrary polynomials $q_1, q_2\\in \\mathbb {Z}[A]$ , it is said that the roots of $q_1$ are separated by the roots of $q_2$ if all the roots of $q_2$ lye in between the maximal and minimal roots of $q_1$ and any pair of consecutive roots of $q_2$ contains exactly one root of $q_1$ .",
"The main theorem of this section can be stated as follows.",
"It counts the number of roots of each of the polynomials $p_n(A)$ , $n\\ge 1$ , establishing that there are as many roots as indicated by the degree, that all of them are real and algebraically simple and that the positive roots of $p_{n+1}(A)$ are always separated by the positive roots (less than 2 if $n\\in 2\\mathbb {N}$ ) of $p_n(A)$ .",
"So, it counts all roots establishing their relative positions.",
"Theorem 5.2 For every $n\\ge 2$ , the positive roots of $p_{2n}(A)$ are separated by the positive roots of $p_{2n-1}(A)$ , and the positive roots of $p_{2n+1}(A)$ are separated by those of $p_{2n}(A)$ less than 2.",
"Moreover, for every $n\\ge 1$ , the even polynomials $\\tfrac{p_{2n}(A)}{2-A}$ and $p_{2n-1}(A)$ have (exactly) $n-1$ positive roots.",
"Thus, since they are even with degree $2n-2$ , they must have another $n-1$ negative roots and, therefore, all roots are real and simple.",
"As we have already constructed the associated polynomials above, it is easily seen that all the thesis of Theorem REF hold to be true for $2\\le n\\le 6$ .",
"This task can be easily accomplished by simply looking at Figure REF , where we have plotted all the positive roots of $p_n(A)$ for $2\\le n\\le 13$ .",
"These roots are located in the interval $(0,2]$ and have been represented in abscisas at different levels according to $n$ .",
"As inserting in the same interval $(0,2]$ all the zeros of the first 13 polynomials would not be of any real help for understanding their fine distribution, we have superimposed them at 13 different levels, each of them containing the positive roots of each of the polynomials $p_n$ , $2\\le n\\le 13$ .",
"In total we are representing 42 roots, though some of them are common roots of different polynomials as a result of the fact that any $kT$ -periodic solution must be a $nkT$ -periodic solution for all $n\\ge 1$ .",
"These common roots have been represented in vertical dashed lines to emphasize that all roots on them share the same abscisa value.",
"In such case, the ordinates provide us with the corresponding value of $n$ .",
"By simply having a glance at Figure REF , it is easily realized how the two roots of the polynomial $p_4$ are separated by the root of $p_3$ , the 3 roots of $p_6$ are separated by the 2 roots of $p_5$ , the 4 roots of $p_8$ are separated by the 3 of $p_7$ , and so on...",
"Similarly, the two roots of $p_5$ are separated by the unique root of $p_4$ different from 2, the 3 roots of $p_7$ are separated by the 2 roots of $p_6$ different from 2, and so on...",
"The proof of the theorem will be delivered in two steps by induction in both cases.",
"Since $\\tfrac{p_2(A)}{2-A}=1$ does not admit any root, this is a very special case that will not play any rol in these induction arguments.",
"Figure: Positive roots of p n p_n , 2≤n≤132\\le n \\le 13.Step 1: Passing from $p_{2n}(A)$ to $p_{2n+1}(A)$ , $n\\ge 2$ .",
"According to Figure REF , it becomes apparent that the two positive roots of $p_4(A)$ are separated by the unique root of $p_3(A)$ .",
"Moreover, all these zeros are real and simple and each of the polynomials $p_{3}(A)=-A^2+3,\\qquad \\frac{p_4(A)}{2-A}= -A^2+2,$ has a unique positive root.",
"Arguing by induction, assume that $p_{2n-1}(A)$ and $p_{2n}(A)$ satisfy all the assertions of the statement of the theorem for some $n\\ge 2$ .",
"In other words, all the positive roots of these polynomials are real and algebraically simple, the positive roots of $p_{2n}(A)$ are separated by the positive roots of $p_{2n-1}(A)$ , and the polynomials $\\tfrac{p_{2n}(A)}{2-A}$ and $p_{2n-1}(A)$ have (exactly) $n-1$ positive roots.",
"We claim that the positive roots of the polynomial $p_{2n+1}(A)$ are real and simple, that they are separated by the positive roots of $p_{2n}(A)$ , except for 2, and that it has (exactly) $n$ positive roots.",
"Indeed, by Theorem REF , we already know that $p_{2n+1}(A)=(2+A)p_{2n}(A) - p_{2n-1}(A).$ First, we will show the previous claim in the case when $ 2n\\in 4\\mathbb {N}+2 $ .",
"So, suppose $2n\\in 4\\mathbb {N}+2$ .",
"Figure REF shows the plots of the polynomials $p_{2n-1}(A)$ and $(2+A)p_{2n}(A)$ in one of such cases: Figure: Sketch of the construction of p 2n+1 (A)p_{2n+1}(A).$p_{2n-1}(A)$ has been plotted in brown and $(2+A)p_{2n}(A)$ in blue.",
"According to Lemmas REF and REF , we already know that $2n-1 = p_{2n-1}(0)< 4n = 2p_{2n}(0), \\quad \\mathrm {deg\\,}(p_{2n-1})=2n-2,\\quad \\mathrm {deg\\,}((2+A)p_{2n})=2n,$ and, since $2n\\in 4\\mathbb {N}+2$ , the leading coefficient of $p_{2n-1}(A)$ equals 1, while the leading coefficient of $p_{2n}(A)$ equals $-1$ .",
"Thus, $p_{2n-1}(A)>0$ and $(2+A)p_{2n}(A)<0$ for $A>2$ .",
"By the induction assumption, the polynomials $\\tfrac{p_{2n}(A)}{2-A}$ and $p_{2n-1}(A)$ have (exactly) $n-1$ positive roots.",
"Hence, each of the polynomials $p_{2n-1}(A)$ and $(2+A)p_{2n}(A)$ possesses (exactly) $n-1$ simple roots in the interval $(0,2)$ and, in addition, $p_{2n}(2)=0$ .",
"In Figure REF , we have named by $\\rho _i$ , $1 \\le i \\le n-1$ , the $n-1$ positive roots of $p_{2n-1}(A)$ , $0 < \\rho _1<\\rho _2< \\cdots < \\rho _{n-2}<\\rho _{n-1}<2,$ while those of $p_{2n}(A)$ less than 2 have been named by $r_i$ , $1\\le i \\le n-1$ .",
"So, $0 < r_1< r_2<\\cdots r_{n-1}<r_{n-1}< r_{n}:=2.$ As, again by the induction hypothesis, the positive roots of $(2+A)p_{2n}(A)$ are separated by the positive roots of $p_{2n-1}(A)$ , necessarily $0<r_1<\\rho _1<r_2<\\rho _2<\\cdots < r_{n-2}<\\rho _{n-2}<r_{n-1}<\\rho _{n-1}<r_n=2.$ Consequently, by (REF ), the polynomial $p_{2n+1}(A)$ must have, at least, $n$ different roots in the interval $(0,2)$ .",
"These roots have been named by $z_i$ , $1\\le i \\le n$ , in Figure REF and they satisfy $0 < z_1 < r_1 < \\rho _1 < z_2 < r_2 < \\rho _2 < \\cdots < z_{n-1}<r_{n-1}<\\rho _{n-1}<z_n< 2.$ On the other hand, by Corollary REF , $p_{2n+1}(A)$ is an even polynomial.",
"Thus, since, due to Lemma REF , it has degree $2n$ and, by the previous construction, $\\pm z_i$ , $1\\le i \\le n$ , provides us with a set of $2n$ different roots of $p_{2n+1}(A)$ , necessarily $p_{2n+1}(A)= -\\prod _{j=1}^n(A^2-z_j^2), \\qquad A >0.$ Therefore, all the roots of $p_{2n+1}(A)$ are real and algebraically simple.",
"As a direct consequence of (REF ) it is apparent that the positive roots of $p_{2n+1}(A)$ are separated by the positive roots of $p_{2n}(A)$ , except for 2.",
"Subsequently, we should prove the result in the special case when $2n\\in 4\\mathbb {N}$ .",
"In this situation, owing to Lemmas REF and REF , the plots of the polynomials $p_{2n-1}(A)$ and $(2+A)p_{2n}(A)$ look like illustrated by Figure REF .",
"Apart from the fact that now $p_{2n-1}(A)>0$ and $(2+A)p_{2n}(A)<0$ for all $A>2$ , because the leading coefficients change sign, the previous analysis can be easily adapted to cover the present situation in order to infer that $p_{2n+1}(A)$ satisfies all the requirements also in this case.",
"By repetitive the technical details of the proof are omitted here in.",
"Figure: Sketch of the construction of p 2n+1 (A) p_{2n+1}(A) .Step 2: Passing from $p_{2n+1}(A)$ to $p_{2n+2}(A)$ , $n\\ge 2$ .",
"According to Figure REF , it becomes apparent that the two positive roots of $p_5(A)$ are separated by the unique root of $p_4(A)$ less than 2.",
"Moreover, all their roots are real and simple.",
"Note that the polynomials $\\frac{p_4(A)}{2-A}= -A^2+2, \\qquad p_5(A)=A^4-5A^2+5,$ have one and two positive roots respectively.",
"Arguing by induction, assume that $p_{2n}(A)$ and $p_{2n+1}(A)$ satisfy all the requirements in the statement of the theorem for some $n\\ge 2$ , i.e., all the positive roots of these polynomials are real and algebraically simple, the positive roots of $p_{2n+1}(A)$ are separated by the positive roots less than 2 of $p_{2n}(A)$ , and the polynomials $\\tfrac{p_{2n}(A)}{2-A}$ and $p_{2n+1}(A)$ have, respectively, $n-1$ and $ n $ positive roots.",
"We claim that the roots of the polynomial $p_{2n+2}(A)$ are real and simple, that they are separated by the roots of $p_{2n+1}(A)$ , and that $p_{2n+2}(A)$ possesses $n+1$ positive roots.",
"Indeed, by Theorem REF , $p_{2n+2}(A)=(2-A)p_{2n+1}(A) - p_{2n}(A).$ As in the previous step, we first deal with the case when $2n\\in 4\\mathbb {N}+2$ .",
"By Lemmas REF and REF , we already know that $2n = p_{2n}(0)< 4n+2 = 2p_{2n+1}(0), \\;\\; \\mathrm {deg\\,}(p_{2n})=2n-1,\\;\\;\\mathrm {deg\\,}\\left((2\\!-\\!A)p_{2n+1}\\right)=2n+1,$ and, since $2n\\in 4\\mathbb {N}+2$ , the leading coefficient of $p_{2n}(A)$ equals $-1$ , and the leading coefficient of $p_{2n+1}(A)$ equals also $-1$ .",
"Thus, $p_{2n}(A)<0,\\qquad (2-A)p_{2n+1}(A)>0\\qquad \\hbox{for all}\\;\\; A>2.$ By the induction assumption, the polynomials $\\tfrac{p_{2n}(A)}{2-A}$ and $p_{2n+1}(A)$ have (exactly) $n-1$ and $n$ positive roots, respectively.",
"Thus, each of the polynomials $p_{2n}(A)$ and $(2-A)p_{2n+1}(A)$ possesses (exactly) $n$ simple roots in $(0,2)$ and, obviously, $(2-A)p_{2n+1}(A)$ also vanishes at $A=2$ .",
"Figure REF shows the plots of $p_{2n}(A)$ , in blue, and $(2-A)p_{2n+1}(A)$ , in brown.",
"Figure: Sketch of the construction of p 2n+2 (A) p_{2n+2}(A) in case 2n∈4ℕ+22n\\in 4\\mathbb {N}+2.In Figure REF , we have named by $\\rho _i$ , $1 \\le i \\le n$ , the $n$ positive roots less than 2 of $(2-A)p_{2n+1}(A)$ , $0 < \\rho _1<\\rho _2< \\cdots < \\rho _{n-1}<\\rho _{n}<2:=\\rho _{n+1},$ whereas $r_i$ , $1\\le i \\le n$ , stand for the positive roots of $p_{2n}(A)$ .",
"Since $p_{2n}(2)=0$ , $r_n=2$ .",
"Since the positive roots of $p_{2n+1}(A)$ are separated by the positive roots less than 2 of $p_{2n}(A)$ , the following holds $0 < \\rho _1<r_1<\\rho _2<r_2< \\cdots < \\rho _{n-1}<r_{n-1}<\\rho _{n}<2:=\\rho _{n+1}=r_{n},$ as illustrated by Figure REF .",
"Thanks to (REF ), it becomes apparent that the polynomial $p_{2n+2}(A)$ admits, at least, an interior root in each of the intervals $(\\rho _i,\\rho _{i+1})$ , $i=0,...,n$ , denoted by $z_i$ in Figure REF , plus $z_{n+1}=2$ .",
"Here we are setting $\\rho _0:=0$ .",
"Consequently, $p_{2n+2}(A)$ has, at least, $n+1$ positive roots.",
"On the other hand, thanks to Corollary REF , $\\tfrac{p_{2n+2}(A)}{2-A} $ is an even function and hence, $p_{2n+2}(A)$ has, at least, $2n+1$ different roots.",
"Since, by Lemma REF , $\\mathrm {deg\\,}(p_{2n+2})=2n+1,$ all these roots are real and algebraically simple.",
"By construction, it is apparent that the positive roots of $p_{2n+2}(A)$ are separated by the positive roots of $p_{2n+1}(A)$ (see Figure REF if necessary).",
"If, instead of $2n\\in 4\\mathbb {N}+2$ , we impose $ 2n\\in 4\\mathbb {N}$ , then the previous arguments can be easily adapted to complete the proof of the theorem from Figure REF , where the graphs of $(2+A)p_{2n+1}(A)$ and $p_{2n}(A)$ have been superimposed in order to show their crossing points, which, owing to Theorem REF , are the roots of $p_{2n+2}(A)$ .",
"By repetitive, the technical details of this case are not included here.",
"Figure: Sketch of the construction of p 2n+2 (A) p_{2n+2}(A) in case 2n∈4ℕ2n\\in 4\\mathbb {N}.A careful reading of the proof of Theorem REF reveals that, actually, not only the roots of $p_{n}(A)$ are separated by those of $p_{n-1}(A)$ , but that they are also separated by those of $p_{n-2}(A)$ , taking always into account the exceptional role played by the root 2."
],
[
"Global bifurcation diagram",
"This section analyzes the global structure of the set of zeros of the maps $\\varphi _n$ , $n\\ge 1$ , introduced in (REF ).",
"These zeros are the positive fixed points of the Poincaré maps $\\mathcal {P}_n$ , $n\\ge 1$ , constructed in Section 3.",
"They provide us with the $nT$ -periodic coexistence states of (REF ) under the additional assumption (REF ).",
"It should be remembered that, according to (REF ), for every integer $n\\ge 1$ $p_n(A)= \\frac{d\\varphi _n (A,1)}{dx}=\\mathfrak {L}(n;A),\\qquad A>0,$ provides us with the linearization at the trivial curve, $(A,1)$ , of $\\varphi _n(A,x)$ .",
"In our analysis, $A$ is always regarded as a bifurcation parameter to $nT$ -periodic coexistence states from the $T$ -periodic ones (i.e., from $x=1$ ).",
"As a consequence of the simplicity of all the roots of $p_n(A)$ , $n\\ge 1$ , guaranteed by Theorem REF , the following result holds.",
"Theorem 6.1 For every $n\\ge 1$ and $r \\in p_n^{-1}(0)$ the following algebraic transversality condition holds $\\mathfrak {L}_1 (N\\left[\\mathfrak {L}(n;r) \\right] )\\oplus R\\left[ \\mathfrak {L}(n;r) \\right]=\\mathbb {R},$ where $\\mathfrak {L}_1:= \\frac{d \\mathfrak {L}(n;r)}{d A} , \\qquad n\\ge 1,\\;\\; r\\in p_n^{-1}(0).$ Therefore, by Theorem 1.7 of M. G. Crandall and P. H. Rabinowitz [3], there exists an analytic curve of $nT$ -periodic coexistence states of (REF ) bifurcating from $(A,1)$ at the root $A=r$ .",
"Actually, there exists $\\varepsilon >0$ and a real analytic map $A: (-\\varepsilon ,\\varepsilon )\\rightarrow \\mathbb {R}$ such that $A(0)=r$ and $\\varphi _n(A(s),1+s)=0 \\quad \\hbox{for all}\\;\\; s\\in (-\\varepsilon ,\\varepsilon ).$ Moreover, any non-trivial zero of $\\varphi _n$ , $(A,x)$ with $x\\ne 1$ , in a neighborhood of $(r,1)$ must be of the form $(A(s),1+s)$ for some $s \\in (-\\varepsilon ,\\varepsilon )$ .",
"In other words, there exists $\\varrho >0$ such that $\\left.",
"\\begin{array}{rr} \\varphi _n(A,x)=0 \\\\ |A-r|+|x-1|<\\varrho \\\\ x\\ne 1\\end{array}\\right\\rbrace \\Longrightarrow (A,x)=(A(s),1+s) \\quad \\hbox{for some}\\;\\; s \\in (-\\varepsilon ,\\varepsilon ).$ Furthermore, setting $A(s)= r + A_1 s + A_2 s^2 + \\mathcal {O} (s^3) \\quad \\hbox{as}\\;\\; s\\rightarrow 0,$ one has that $A_1=0$ and $A_2>0$ if $n=2$ and $r=2$ , in complete agreement with Figure REF ; $A_1<0$ if $n=3$ and $r=\\sqrt{3}$ ; $A_1=0$ and $A_2 <0$ (resp.",
"$A_2>0$ ) if $n=4$ and $r = r_{4,1}=\\sqrt{2}$ (resp.",
"$r = r_{4,2}=2$ ).",
"According to (REF ), $\\mathfrak {L}(n;r)=p_n(r)=0$ .",
"Thus, $N[\\mathfrak {L}(n;r)]=\\mathbb {R}$ and (REF ) can be equivalently expressed as $\\mathfrak {L}_1(\\mathbb {R}) =\\mathbb {R}$ , which holds true because, thanks to Theorem REF , we already know that $r$ is an algebraically simple root of $p_n(A)$ , i.e., $\\mathfrak {L}_1 = p_n^{\\prime }(r) \\ne 0.$ So, (REF ) indeed holds and [3] applies to $\\varphi _n(A,x)=0$ at $(A,x)=(r,1)$ .",
"Since we can take $\\psi =1$ as a generator of $N[\\mathfrak {L}(n;r)]=\\mathbb {R}$ and $Y=[0]$ as a supplement of $N[\\mathfrak {L}(n;r)]=\\mathbb {R}$ in $\\mathbb {R}$ , owing to [3], there exist $\\varepsilon >0$ and a real analytic map $(A,y):(-\\varepsilon ,\\varepsilon )\\rightarrow \\mathbb {R}\\times Y$ such that $(A(0),y(0))=(r,0)$ and $\\varphi _n(A(s),1+s(\\psi +y(s)))=0 \\quad \\hbox{for all}\\;\\; s\\in (-\\varepsilon ,\\varepsilon ),$ it becomes apparent, by construction, that $\\varphi _n(A(s),1+s)=0\\quad \\hbox{for all} \\;\\; s \\in (-\\varepsilon ,\\varepsilon ),$ because $y\\equiv 0$ and $\\psi =1$ .",
"This ends the proof of the first two claims of the theorem: the existence of the analytic curve of nontrivial solutions and the uniqueness.",
"As far as concerns to the problem of ascertaining the nature of these local bifurcations at $(r,1)$ , we can proceed as follows.",
"In order to prove Part (a), note that, thanks to (REF ), setting $x(s):=1+s$ and expanding in Taylor series, we have that $0 = \\varphi _2(A(s),x(s))=\\varphi _2(r,1)+\\frac{d\\varphi _2}{ds}(r,1)s+\\frac{1}{2}\\frac{d^2\\varphi _2}{ds^2}(r,1)s^2+\\cdots $ for all $s\\in (-\\varepsilon ,\\varepsilon )$ , where $r=2$ .",
"Moreover, by construction, we already know that $\\varphi _2(r,1)=0,\\qquad \\frac{\\partial \\varphi _2}{\\partial x}(r,1)=p_2(r)=p_2(2)=0$ (see (REF ), if necessary).",
"Thus, since by (REF ) $\\varphi _2(A,x)=x\\left( E_1(A,x)+1\\right) -2,$ it follows from (REF ) and $\\frac{\\partial E_1}{\\partial A}(r,1)=0$ that $\\frac{d\\varphi _2}{ds}(r,1)=\\frac{\\partial \\varphi _2}{\\partial A}(r,1)A^{\\prime }(0)=\\frac{\\partial E_1}{\\partial A}(r,1)A_1=0,$ where $^{\\prime }:=\\frac{d}{ds}$ .",
"Hence, these terms do not provide us with any neat information concerning $A_1$ .",
"So, we must consider higher order terms to find out $A_1$ .",
"As $\\frac{\\partial \\varphi _2}{\\partial A}(r,1)=0=\\frac{\\partial \\varphi _2}{\\partial x}(r,1),$ applying the chain rule it readily follows that $0= \\frac{d^2\\varphi _2}{ds^2}(r,1)=\\frac{\\partial ^2\\varphi _2}{\\partial x^2}(r,1)+2\\frac{\\partial ^2\\varphi _2}{\\partial A\\partial x}(r,1)A_1+\\frac{\\partial ^2\\varphi _2}{\\partial A^2}(r,1)A_1^2.$ On the other hand, differentiating with respect to $x$ the identity (REF ) yields $\\frac{\\partial \\varphi _2}{\\partial x}(A,x) = E_1(A,x)+1+ x \\frac{\\partial E_1}{\\partial x}(A,x).$ So, $\\frac{\\partial ^2 \\varphi _2}{\\partial x^2}(A,x) = 2 \\frac{\\partial E_1}{\\partial x} (A,x)+ x \\frac{\\partial ^2 E_1}{\\partial x^2}(A,x).$ Consequently, particularizing at $(A,x)=(r,1)$ , it follows from (REF ) that $\\frac{\\partial ^2 \\varphi _2}{\\partial x^2}(r,1) =r^2-2r=4-4=0.$ Similarly, differentiating (REF ) with respect to $A$ shows that $\\frac{\\partial ^2 \\varphi _2}{\\partial x\\partial A}(A,x)=\\frac{\\partial E_1}{\\partial A}(A,x)+x\\frac{\\partial ^2 E_1}{\\partial x\\partial A}(A,x)$ and hence, owing to (REF ), $\\frac{\\partial ^2 \\varphi _2}{\\partial x\\partial A}(r,1)=\\frac{\\partial E_1}{\\partial A}(r,1)+\\frac{\\partial ^2 E_1}{\\partial x\\partial A}(r,1) =\\frac{\\partial ^2 E_1}{\\partial x\\partial A}(r,1) = -1.$ Lastly, $\\frac{\\partial ^2\\varphi _2}{\\partial A^2}(A,x)=x\\frac{\\partial ^2 E_1}{\\partial A^2}(A,x)= (1-x)^2E_1(A,x)$ and hence, $\\frac{\\partial ^2\\varphi _2}{\\partial A^2}(A,1)=0.$ Therefore, substituting (REF ), (REF ) and (REF ) into (REF ) it becomes apparent that $A_1=0$ .",
"Thanks to this fact, the third derivative admits the next (simple) expression: $0=\\frac{d^3\\varphi _2}{ds^3}(r,1)=6\\frac{\\partial ^2\\varphi _2}{\\partial x\\partial A}(r,1)A_2+\\frac{\\partial ^3\\varphi _2}{\\partial x^3}(r,1)=-6A_2+3r^2-r^3,$ which implies that $A_2=\\frac{4}{3}>0$ and ends the proof of Part (a).",
"To prove Part (b), note that, much like in Part (a), one has that $0= \\varphi _3(A(s),x(s))=\\varphi _3(r,1)+\\frac{d\\varphi _3}{ds}(r,1)s+\\frac{1}{2} \\frac{d^2\\varphi _3}{ds^2}(r,1)s^2+\\cdots $ for all $s \\in (-\\varepsilon ,\\varepsilon )$ .",
"Similarly, $\\frac{\\partial \\varphi _3}{\\partial A}(r,1)=0=\\frac{\\partial \\varphi _3}{\\partial x}(r,1).$ So, $\\frac{d\\varphi _3}{ds}(r,1)=0.$ Moreover, differentiating twice with respect to $s$ yields $0 & = \\frac{d^2\\varphi _3}{ds^2}(r,1)\\\\[5pt] & =\\frac{\\partial ^2\\varphi _3}{\\partial x^2}(r,1)+2\\frac{\\partial ^2\\varphi _3}{\\partial x\\partial A}(r,1)A_1+\\frac{\\partial ^2\\varphi _3}{\\partial A^2}(r,1)A_1^2\\\\[5pt] & =r^4-r^3-2r^2-4rA_1.$ Consequently, since $r=\\sqrt{3}$ , it follows from this identity that $A_1=\\frac{\\sqrt{3}-3}{4}<0,$ which ends the proof of Part (b).",
"Finally, much like before, we have that $0= \\varphi _4(A(s),x(s))=\\varphi _4(r,1)+\\frac{d\\varphi _4}{ds}(r,1)s+\\frac{1}{2}\\frac{d^2\\varphi _4}{ds^2}(r,1)s^2+\\cdots $ for all $s \\in (-\\varepsilon ,\\varepsilon )$ , and, in addition, $\\frac{\\partial \\varphi _4}{\\partial A}(r,1)=0=\\frac{\\partial \\varphi _4}{\\partial x}(r,1).$ Thus, $\\frac{d\\varphi _4}{ds}(r,1)=0$ .",
"Moreover, differentiating twice yields $0 & =\\frac{d^2\\varphi _4}{ds^2}(r,1)\\\\[5pt] &=\\frac{\\partial ^2\\varphi _4}{\\partial x^2}(r,1)+2\\frac{\\partial ^2\\varphi _4}{\\partial x\\partial A} (r,1)A_1+\\frac{\\partial ^2\\varphi _4}{\\partial A^2}(r,1)A_1^2 \\\\[5pt] &=r^6-3r^5-r^4+8r^3-2r^2-4r+2(r^3-2r^2-2r)A_1.$ Therefore, since $r=\\sqrt{2}$ it follows from this identity that $A_1=0$ .",
"Furthermore, $0 & = \\frac{d^3\\varphi _2}{ds^3}(r,1)\\\\[5pt] &=6\\frac{\\partial ^2\\varphi _2}{\\partial x\\partial A} (r,1)A_2+\\frac{\\partial ^3\\varphi _2}{\\partial x^3}(r,1)\\\\[5pt] &=6(3r^2-4r-2)A_2-r^8+r^7+9r^6-11r^5-10r^4+20r^3-2r^2.$ Consequently, we find from $r=\\sqrt{2}$ that $A_2=-\\frac{2(5+4\\sqrt{2})}{3}<0,$ which ends the proof.",
"Figure REF shows the local bifurcation diagrams of the $2T$ , $3T$ and $4T$ -periodic coexistence states of (REF ) under condition (REF ).",
"We are plotting $x$ , in ordinates, versus $A$ , in abscisas.",
"By the analysis already done at the beginning of Section 2, and, in particular, by Theorem REF , which was sketched in Figure REF , we already know that, under condition (REF ), the problem (REF ) admits a $2T$ -periodic coexistence state if, and only if, $A>2$ .",
"Moreover, the local bifurcation of these solutions must be supercritical.",
"Thus $A_2\\ge 0$ .",
"As a byproduct of Theorem REF , it turns out that $A_2>0$ .",
"So, it is a genuine supercritical pitchfork bifurcation of quadratic type.",
"However, since $A_1<0$ , the bifurcation to $3T$ -periodic coexistence states from $(A,x)=(\\sqrt{3},1)$ is transcritical, whereas the $4T$ -periodic solutions emanate from $(A,x)=(\\sqrt{2},1)$ through a subcritical quadratic pitchfork bifurcation, because $A_1=0$ and $A_2<0$ in this case.",
"The fact that the local nature of the first three bifurcation phenomena possess a completely different character shows that, in general, ascertaining the precise type of these local bifurcations for large $n$ might not be possible, much like happened with the problem of determining the fine structure of the set of bifurcation points from the trivial solution $(A,1)$ .",
"The higher is the order of the bifurcating subharmonics, measured by $n$ , the higher is the complexity of the associated function $\\varphi _n$ and hence, the more involved is finding out the values of $A_1$ and $A_2$ in (REF ) by the intrinsic nature of the functions $E_n$ defined in (REF ).",
"Figure: Local bifurcation diagrams from (A,x)=(A,1)(A,x)=(A,1) of the nTnT-periodic coexistence states for n∈{2,3,4}n\\in \\lbrace 2,3,4\\rbrace .Remark 6.2 Thanks to Theorem REF , it becomes apparent that the set of bifurcation points from $ (A,1) $ to $ nT $ -periodic coexistence states of (REF ) is the set of roots of $ p_n(A) $ .",
"Since the number of roots of a polynomial is finite, the set of bifurcation points is numerable, as it is a numerable union of finite sets.",
"Since every $nT$ -periodic coexistence state of (REF ) provides us with a $knT$ -periodic coexistence state for all $k\\ge 1$ , owing Theorem REF , the roots of $p_n(A)$ must be roots of $p_{kn}(A)$ for all $ n,k\\ge 1 $ , i.e., $p_n|p_{kn}$ for all $n, k \\ge 1$ .",
"Remark 6.3 Thanks to Theorem REF and Remark REF , the set of bifurcation points to a $nT$ -periodic solution is a subset of the interval $(0,2]$ .",
"Complementing [9], where the non-degeneration of the positive $T$ -periodic coexistence states of (REF ) with respect to the $T$ -periodic solutions was established, Theorem REF shows that the $T$ -periodic solutions are degenerated with respect to the $nT$ -periodic solutions of (REF ) for all $n\\ge 2$ at every positive root, $r$ , of $p_n(A)$ .",
"Nevertheless, the $T$ -periodic solutions are non-degenerated with respect to the $nT$ -periodic solutions, $n\\ge 2$ , if $A>2$ , because in this range there is not any bifurcation point from $(A,1)$ .",
"Subsequently, we will discuss the global character of all the local bifurcations documented by Theorem REF in the context of global bifurcation theory.",
"In this discussion, by a (connected) component it is understood any closed and connected subset that is maximal for the inclusion.",
"For any given integer $n\\ge 1$ , the set of non-trivial $nT$ -periodic solutions of (REF ), $\\mathcal {S}_n$ , consists of all $nT$ -periodic coexistence states different from $(A,1)$ plus the set of points $(r,1)$ with $p_n(r)=0$ .",
"In other words, setting $\\mathbb {R}_+:=(0,+\\infty )$ , $\\mathcal {S}_n= \\lbrace (A,x) \\in \\mathbb {R}_+\\times (\\mathbb {R}_+\\setminus \\lbrace 1\\rbrace ) \\;:\\; \\varphi _n(A,x)=0\\rbrace \\cup \\lbrace (r,1)\\;:\\; p_n(r)=0\\rbrace .$ Note that, owing to Theorem REF , $\\lbrace (r,1)\\;:\\; p_n(r)=0\\rbrace $ is the set of bifurcation points to $nT$ -periodic solutions from the trivial curve $(A,1)$ .",
"Thanks to (REF ), the algebraic multiplicity of J. Esquinas and J. López-Gómez [7] equals one, $\\chi [\\mathfrak {L}(n;A);r]=1 \\in 2\\mathbb {N}+1,$ for all $r \\in p_n^{-1}(0)$ , $r>0$ .",
"Thus, by [10], the local degree at $(A,1)$ of the one-dimensional $\\varphi _n(A,\\cdot )$ changes as $A$ crosses $r$ (see also J. López-Gómez and C. Mora-Corral [13] if necessary).",
"Therefore, according to [10], for every integer $n\\ge 2$ and each root $r>0$ of $p_n(A)$ , there is a component of $\\mathcal {S}_n$ , $\\mathfrak {C}_{n,r}$ , such that $(r,1) \\in \\mathfrak {C}_{n,r}\\subset \\mathbb {R}_+\\times \\mathbb {R}.$ Moreover, by the local uniqueness about $(r,1)$ guaranteed by Theorem REF , in a neighborhood of $(r,1)$ the component $\\mathfrak {C}_{n,r}$ consists of an analytic curve, $(A(s),1+s)$ , $|s|<\\varepsilon $ .",
"Note that any real continuous map must be compact.",
"So, the Leray–Schauder degree (see, e.g., N. G. Lloyd [8], or [13], if necessary) can be applied to get these global results.",
"Alternatively, one might use the degree of P. Benevieri and M. Furi [1], as in Theorem 5.4 and Corollary 5.5 of [12] (see [11] for a recent survey on global bifurcation theory).",
"By Remark REF , for every $r \\in p_n^{-1}(0)\\cap \\mathbb {R}_+$ , the component $\\mathfrak {C}_{n,r}$ must be separated away from $x=0$ and hence, all their solutions must be positive, because $\\varphi _n(0)=-n$ .",
"Thus, they indeed provide us with coexistence states of (REF ).",
"Similarly, for every $n\\ge 2$ , since $\\varphi _n(n)>0$ , $\\mathfrak {C}_{n,r}$ is bounded above by $n$ , in the sense that $x<n$ if $(A,x)\\in \\mathfrak {C}_{n,r}$ with $A>0$ .",
"Therefore, $\\mathcal {P}_x(\\mathfrak {C}_{n,r})\\subset (0,n),$ where $\\mathcal {P}_x$ stands for the $x$ -projection operator, $\\mathcal {P}_x(A,x):=x$ .",
"Moreover, due to (REF ), $x=1$ is the unique zero of $\\varphi _n(A,x)$ at $A=0$ .",
"Note that, due to Remark REF , $(A,1)=(0,1)\\notin \\mathfrak {C}_{n,r}$ because $p_n(0)=n>0$ .",
"Throughout the rest of this section, we will also consider the (unilateral) subcomponents $\\mathfrak {C}_{n,r}^+:=\\mathfrak {C}_{n,r}\\cap [x>1], \\qquad \\mathfrak {C}_{n,r}^-:=\\mathfrak {C}_{n,r}\\cap [x<1].$ Thanks to Theorem REF , these subcomponents are non-empty.",
"Moreover, arguing as in [10], it is easily seen that they equal the components $\\mathfrak {C}^+$ and $\\mathfrak {C}^-$ introduced on page [10].",
"This feature heavily relies on the fact that $x$ is a one-dimensional variable.",
"Therefore, the unilateral theorem [10] can be applied to infer that each of the components $\\mathfrak {C}_{n,r}^+$ and $\\mathfrak {C}_{n,r}^-$ satisfies the global alternative of P. H. Rabinowitz [20], because the supplement of $N[\\mathfrak {L}(n;r)]=\\mathbb {R}$ in $\\mathbb {R}$ is $Y=[0]$ and, due to (REF ), $\\mathfrak {C}_{n,r}$ cannot admit an element, $(A,x)$ with $x=0$ .",
"Therefore, $\\mathfrak {C}_{n,r}^+$ (resp.",
"$\\mathfrak {C}_{n,r}^-$ ) satisfies some of the following two conditions, which are far from being excluding: There exists $s \\in p_n^{-1}(0)\\setminus \\lbrace r\\rbrace $ (resp.",
"$t \\in p_n^{-1}(0)\\setminus \\lbrace r\\rbrace $ ) such that $(s,1)\\in \\mathfrak {C}_{n,r}^+$ (resp.",
"$(t,1)\\in \\mathfrak {C}_{n,r}^-$ ).",
"The component $\\mathfrak {C}_{n,r}^+$ (resp.",
"$\\mathfrak {C}_{n,r}^-$ ) is unbounded in $A$ , because of (REF ).",
"Note that the counterexample of E. N. Dancer [4] shows that Theorems 1.27 and 1.40 of P. H. Rabinowitz [20] are not true as originally stated.",
"To show that the second option occurs in both cases we need the next result.",
"Lemma 6.4 Each of the unilateral subcomponents satisfies $\\mathfrak {C}_{n,r}^\\pm \\cap \\lbrace (A,1)\\;:\\; A\\ge 0\\rbrace = \\lbrace (r,1)\\rbrace .$ Thus, also $\\mathfrak {C}_{n,r} \\cap \\lbrace (A,1)\\;:\\; A\\ge 0\\rbrace = \\lbrace (r,1)\\rbrace ,$ i.e., $(r,1)$ is the unique bifurcation point of $\\mathfrak {C}_{n,r}$ from $(A,1)$ .",
"Subsequently, we will denote by $\\nu (n)$ the total number of positive roots of the polynomial $p_n(A)$ .",
"By Theorem REF , we already know that $\\nu (n)=\\frac{n}{2}$ if $n$ is even and $\\nu (n)=\\frac{n-1}{2}$ if $n$ is odd.",
"We will prove the result only for $\\mathfrak {C}_{n,r}^+$ , as the same argument also works out to prove the corresponding assertion for the component $\\mathfrak {C}_{n,r}^-$ .",
"The proof will proceed by contradiction.",
"We already know that $\\mathfrak {C}_{n,r}^+$ can only meet the trivial solution $(A,1)$ at the roots of $p_n(A)$ .",
"Suppose that $r=r_{n,i}$ for some $i \\in \\lbrace 1,...,\\nu (n)\\rbrace $ , and that there exists $j>i$ , $j \\in \\lbrace 1,...,\\nu (n)\\rbrace $ , such that $\\lbrace (r_{n,i},1),(r_{n,j},1)\\rbrace \\subset \\mathfrak {C}_{n,r}^+ \\cap \\lbrace (A,1)\\;:\\; A\\ge 0\\rbrace .$ Then, by the definition of component, it becomes apparent that $\\mathfrak {C}_{n,r_{n,i}}^+=\\mathfrak {C}_{n,r_{n,j}}^+$ as sketched by Figure REF .",
"Thanks to Theorem REF , there exists $r_{n-1,k}\\in (r_{n,i},r_{n,j})\\cap p_{n-1}^{-1}(0).$ By the incommensurability of $nT$ with $(n-1)T$ , $\\mathfrak {C}_{n-1,r_{n-1,k}}^+$ cannot reach the component (REF ).",
"Thus, must be bounded.",
"Consequently, as $\\mathfrak {C}_{n-1,r_{n-1,k}}^+$ also satisfies the global alternative of P. H. Rabinowitz, there exists $r_{n-1,\\ell }\\in p_{n-1}^{-1}(0)$ , with $k\\ne \\ell $ , such that $\\lbrace (r_{n-1,k},1),(r_{n-1,\\ell },1)\\rbrace \\subset \\mathfrak {C}_{n-1,r_{n-1,k}}^+ \\cap \\lbrace (A,1)\\;:\\; A\\ge 0\\rbrace ,$ as sketched in Figure REF .",
"Since the set of roots $\\bigcup _{2\\le \\kappa \\le n} p_{\\kappa }^{-1}(0)$ is finite, it becomes apparent that, after finite many steps, there exists a component, $\\mathfrak {C}_{n-h,r_{n-h,m}}^+$ , for some $2 \\le h \\le n-3$ and $1\\le m\\le \\nu (n-h)$ , that should meet the last component linking two different roots sketched in Figure REF , $\\mathfrak {C}_{n-h+1,r_{n-h+1,v}}^+ = \\mathfrak {C}_{n-h+1,r_{n-h+1,w}}^+,$ because there is no any additional root of $p_{n-h}(A)$ in between $r_{n-h+1,v}$ and $r_{n-h+1,w}$ .",
"But this is impossible, by the incommensurability of $(n-h)T$ with $(n-h+1)T$ .",
"This contradiction ends the proof.",
"Figure: Sketch of the proof of Lemma .As an immediate consequence of the previous analysis, the next result holds.",
"As for the $x$ -projection operator, $\\mathcal {P}_x$ , we will denote by $\\mathcal {P}_A$ the $A$ -projection operator, $\\mathcal {P}_A(A,x):=A.$ Theorem 6.5 For every integer $n\\ge 2$ and each root $r>0$ of $p_n(A)$ , the component $\\mathfrak {C}_{n,r}^+$ satisfies $\\mathcal {P}_x (\\mathfrak {C}_{n,r}^+)\\subset [1,n)$ ; $\\mathcal {P}_A (\\mathfrak {C}_{n,r}^+)=[A^+_{n,r},+\\infty )$ for some $A_{n,r}^+ \\in (0,r]$ .",
"In particular, $\\mathfrak {C}_{n,r}^+$ is unbounded.",
"$\\mathfrak {C}_{n,r}^+ \\cap \\lbrace (A,1)\\;:\\; A\\ge 0\\rbrace = \\lbrace (r,1)\\rbrace $ ; For every $n, m \\ge 2$ , $\\mathfrak {C}_{n,r}^+\\cap \\mathfrak {C}_{m,s}^+=\\emptyset $ if $r \\ne s$ .",
"Moreover, by Theorem REF , in a neighborhood of $(r,1)$ the component $\\mathfrak {C}_{n,r}^+$ consists of an analytic curve, $(A(s),1+s)$ , $0\\le s <\\varepsilon $ .",
"Similarly, the component $\\mathfrak {C}_{n,r}^-$ satisfies (c), (d) and $\\mathcal {P}_x (\\mathfrak {C}_{n,r}^-)\\subset (0,1]$ ; $\\mathcal {P}_A (\\mathfrak {C}_{n,r}^-)=[A^-_{n,r},+\\infty )$ for some $A_{n,r}^- \\in (0,r]$ .",
"In particular, $\\mathfrak {C}_{n,r}^-$ is unbounded.",
"Analogously, in a neighborhood of $(r,1)$ the component $\\mathfrak {C}_{n,r}^-$ consists of an analytic curve, $(A(s),1+s)$ , $-\\varepsilon <s\\le 0$ .",
"At this stage, the only delicate point is Part (d).",
"Suppose that $\\mathfrak {C}_{n,r}^+\\cap \\mathfrak {C}_{m,s}^+\\ne \\emptyset $ for some $r \\ne s$ .",
"Then, by the definition of component, necessarily $\\mathfrak {C}_{n,r}^+ = \\mathfrak {C}_{m,s}^+.$ Thus, $(r,1), (s,1) \\in \\mathfrak {C}_{n,r}^+$ , which contradicts Lemma REF .",
"The proof is complete.",
"Except for the local bifurcations from the trivial line $(A,1)$ , the global diagramas of the components $\\mathfrak {C}_{n,r}^\\pm $ plotted in Figure REF respect the general properties established by Theorem REF .",
"Although the components have been plotted with no secondary bifurcations along them, there are some numerical evidences that $\\mathfrak {C}_{2,2}^-$ possesses a secondary bifurcation to $4T$ -periodic solutions.",
"Nevertheless, thanks to Theorem REF , even in the case that they might occur higher order bifurcations along these components, they must be disjoint.",
"According to Theorems REF and REF , it becomes apparent that some $3T$ and $4T$ -periodic solutions must be degenerated.",
"Namely, those on the turning points of $\\mathfrak {C}_{3,\\sqrt{3}}^+$ , $\\mathfrak {C}_{4,\\sqrt{2}}^+$ and $\\mathfrak {C}_{4,\\sqrt{2}}^-$ in Figure REF .",
"Similarly, the bifurcation points accumulating from the left to $\\sqrt{2}$ and $\\sqrt{3}$ must provide us with additional degenerate solutions: those on the turning points of their corresponding components."
]
] | 1906.04568 |
[
[
"Bilevel Optimization for Planning through Contact: A Semidirect Method"
],
[
"Abstract Many robotics applications, from object manipulation to locomotion, require planning methods that are capable of handling the dynamics of contact.",
"Trajectory optimization has been shown to be a viable approach that can be made to support contact dynamics.",
"However, the current state-of-the art methods remain slow and are often difficult to get to converge.",
"In this work, we leverage recent advances in bilevel optimization to design an algorithm capable of efficiently generating trajectories that involve making and breaking contact.",
"We demonstrate our method's efficiency by outperforming an alternative state-of-the-art method on two benchmark problems.",
"We moreover demonstrate the method's ability to design a simple periodic gait for a quadruped with 15 degrees of freedom and four contact points."
],
[
"Introduction",
"Trajectory optimization is a well-established method to generate motion plans for dynamical systems.",
"One important application for such methods is motion planning and control for systems with contact dynamics, which are ubiquitous in real-world robotic systems.",
"However, trajectory optimization for systems undergoing contact has proven to be inherently difficult due to the drastically changing dynamical behavior arising from making and breaking contact.",
"For this reason, methods have been proposed which generally fall under several categories.",
"Those using continuous models, such as spring-damper models, have been proposed to eliminate the discontinuities that make the problem so challenging.",
"However in these methods approximation error is induced and numerical problems can arise due to stiffness in the resulting differential equations.",
"Additionally, approaches using hybrid models [1] have been introduced which can richly capture the true discontinuous dynamics.",
"However it is well known that trajectory optimization formulations using such models, primarily mixed-integer programs, suffer from computational limitations resulting from a combinatorially large number of mode switching possibilities.",
"An alternative class of approaches, which is the focus of this work, includes the contact forces as decision variables in a trajectory optimization framework.",
"These methods, referred to as planning “through contact”, utilize techniques made popular for simulation of contact-driven dynamical systems.",
"The primary advantage of this category of methods is that rich contact behavior can be captured without the combinatorial explosions in complexity seen in hybrid model based methods.",
"However, even though the formulation of such optimization problems is generally straightforward and the resulting problem non-combinatorial, they remain numerically challenging for optimization algorithms to handle.",
"Therefore, the exploration of different formulations of the embedded contact dynamics problem and exploration of different solution techniques are both tasks of high importance.",
"In this work, we propose a variation of the “through contact” method first put forward in [2].",
"Our approach leverages recent advances in bilevel and differentiable optimization in which a lower level optimization problem is embedded within an upper level optimization.",
"A key aspect of our algorithm is that it gets rid of the complementarity constraints usually needed to model friction forces (which tend to be the cause of many numerical difficulties) and only requires complementarity constraints relating to non-penetration constraints (i.e.",
"for the normal forces).",
"Instead, we compute friction forces as an embedded optimization problem.",
"Note that variations on the idea of modeling contact through bilevel optimization have been explored in the past [3].",
"However, our formulation posses several appealing characteristics such as being able to leverage state-of-the-art off-the-shelf solvers like SNOPT and OSQP in order to outperform alternative approaches.",
"Moreover our method avoids linearizing the non-penetration constraint which can easily lead to infeasible trajectories.",
"Finally, unlike approaches relying on iLQR, our direct transcription based method can easily and effectively be parallelized, which is of particular relevance when modeling contact forces as solutions to embedded optimization problems.",
"An important note the reader: in this work when discussing approaches to trajectory optimization through contact we use the term “direct”, “indirect”, and “semidirect” to refer to the way the contact force is resolved within the trajectory optimization problem, and not the way the problem itself is solved overall (for which we do in fact use a direct method).",
"Specifically, by a “direct” method we mean a method which defines the contact forces in the trajectory optimization problem completely as the result of an embedded optimization problem (i.e.",
"via the maximum dissipation principle), and by an “indirect” method we mean a method which completely resolves the contact forces through the use of complementarity constraints (which can be interpreted as KKT conditions) [2].",
"We refer to our proposed method as “semidirect” because it handles only the friction forces using an embedded optimization problem, and the normal forces through complementarity constraints.",
"The contributions of this paper are as follows.",
"First, by combining different modeling approaches to contact forces, we formulate a “semidirect” trajectory optimization method for planning through contact.",
"Second, we demonstrate how our formulation, a specific bilevel optimization problem, is well suited to leverage recent advances in differentiable optimization as well as state-of-the-art quadratic programming and nonlinear programming solvers.",
"Next, we demonstrate how our method, based on direct transcription, can easily and effectively leverage parallelization to offset the cost incurred by its bilevel structure.",
"Finally we provide evidence that the method offers a promising direction of research by demonstrating it is capable of outperforming a well accepted alternative method on a benchmark problem, and also generate contact rich interactions between a complex robot and its environment."
],
[
"Previous Work",
"Here we survey methods related to “through contact” trajectory optimization and how they differ from our proposed algorithm.",
"One of the first instance of such methods [2] leverages work on rigid body simulation with contact [4] in order to formulate a direct transcription problem capable of generating trajectories that make and break contact.",
"We differ from this approach mainly in that instead of using complemetarity constraint to enforce contact dynamics, we use the maximum dissipation principle to model the friction forces as an embedded quadratic program, and leverage advances in bilevel optimization to handle the newly formed problem.",
"Note that modeling contact forces as optimization problems using the maximum dissipation principle is not an entirely new idea and has been explored in the past.",
"Notably [5] uses this approach to simulate forward contact dynamics and provides a survey of difference instances of this modeling approach.",
"Perhaps closest to our method, [3] demonstrated a method for planning through contact that also leverages a contact dynamics formulation derived from the principle of maximum dissipation.",
"However, our method differentiates itself from this work on a few points.",
"First, we use a direct collocation method instead of iLQR (a shooting method), which allows us to parallelize the computationally more demanding aspects of the method.",
"Second, instead of solving the lower problem with a projected gradient descent algorithm, we use an off-the-shelf quadratic program solver OSQP which provides faster convergence through the Alternating Direction Method of Multipliers (ADMM).",
"Third, we recover the gradients of the embedded problem analytically, instead of autodifferentiating the solver itself, which again is more computationaly efficient.",
"Lastly, we preserve the normal force as decision variables in the upper-level problem.",
"Finally, there has also been some work on using the analytical derivatives of quadratic programs in the context of contact.",
"Notably [6] embeds a non-convex quadratic program corresponding to the Linear Complementarity Problem (LCP) described in [4] to learn dynamics involving contact.",
"Our method differs from this work in that we perform nonlinear optimization with an off-the-shelf constrained nonlinear programming solver to solve the upper problem instead of stochastic gradient descent.",
"Moreover, the formulations of our lower problem differ as we do not use the non-convex LCP contact formulation, but rather the convex maximum dissipation optimization problem (from which the LCP formulation can in fact be derived from as KKT conditions)."
],
[
"Contact Dynamics as Optimization Problems",
"A powerful way of understanding the nature of contact forces is to reason about them using the principle of maximum dissipation.",
"As we show in this section, by performing a first-order discretization of the system dynamics using a backward Euler integration scheme, it is possible to then model the contact forces as the solution of a quadratic program."
],
[
"Maximum Dissipation",
"We start with the rigid body dynamics written in manipulator form $\\begin{aligned}M(q) \\dot{v} + c(q,v) + \\lambda = \\tau ,\\end{aligned}$ where $q$ , $v$ are the configuration and joint velocities of the system.",
"The matrix $M(q)$ is the positive-definite inertia matrix, the function $c(q,v)$ is the dynamics bias (i.e.",
"the Coriolis and potential terms of the manipulator equation combined), $\\lambda (q,v)$ denotes the contact forces and $\\tau $ represents the control input (usually joint torques for robotic systems).",
"This dynamics equation is then discretized in time using a backward Euler integration scheme to yield $\\begin{aligned}M(q_{i+1})(v_{i+1} - v_i) + h c(q_{i+1},v_{i+1}) - h \\tau _{i+1} + h \\lambda _{i+1} = 0, \\\\q_{i+1} = q_i + hv_{i+1}, \\\\\\end{aligned}$ where $h$ is the discretization time step.",
"Next, we define the contact force acting at each contact point as a vector in a frame that is centered at the contact location.",
"The normal force component of the contact force vector at a given point is denoted by $c_n$ , and the (tangential) friction forces by $f_x$ and $f_y$ .",
"Thus the vector $x = [c_n, f_x, f_y]^T$ defines the contact force in the associated contact point frame.",
"With this definition, the dynamics can be written as $\\begin{aligned}M(q_{i+1})(v_{i+1} - v_i) + h c(q_{i+1},v_{i+1}) - h \\tau _{i+1} + h J(q_{i+1})^T x_{i+1} = 0.",
"\\\\q_{i+1} = q_i + hv_{i+1}, \\\\\\end{aligned}$ where $J(q)$ is the Jacobian that maps the contact forces from the contact frame to the joint space.",
"In order for us to be able to use the principle of maximum dissipation when determining the contact forces, we now derive an expression for the change in kinetic energy from one time step to the next $\\begin{aligned}dT &= T_{i+1} - T_i,\\\\&= \\frac{1}{2}v_{i+1}^T M_{i+1} v_{i+1} - \\frac{1}{2}v_i^T M_i v_i,\\\\&= \\frac{1}{2} (v_i - h M_{i+1}^{-1}(J_{i+1}^Tx_{i+1} + c_{i+1} - \\tau _{i+1}))^T M_{i+1} \\\\&\\hspace{56.9055pt}(v_i - h M_{i+1}^{-1}(J_{i+1}^Tx_{i+1} + c_{i+1} - \\tau _{i+1})) - \\frac{1}{2}v_i^T M_i v_i, \\\\&= \\frac{1}{2}h^2(J_{i+1}^Tx_{i+1} + c_{i+1} - \\tau _{i+1})^T M_{i+1}^{-1} (J_{i+1}^Tx_{i+1} + c_{i+1} - \\tau _{i+1}) \\\\&\\hspace{56.9055pt} - h v_i^T(J_{i+1}^T x_{i+1} + c_{i+1} - \\tau _{i+1}) + \\frac{1}{2}v_i^T(M_{i+1} - M_i)v_i.\\end{aligned}$ where $dT$ is the change in kinetic energy, and for notational simplicity we use $M_{i} = M(q_{i})$ , $c_{i}=c(q_{i},v_{i})$ , and $J_{i}=J(q_{i})$ .",
"Note that $dT$ can be expressed quadratically in terms of the contact force $x$ .",
"This now allows us to formulate a quadratic program to find the contact force $x$ that maximizes the energy dissipation."
],
[
"Friction Forces as Quadratic Program",
"We can treat $q_i$ , $v_i$ , $\\tau _{i+1}$ , $M(q_i)$ , $M(q_{i+1})$ , $c(q_{i+1},v_{i+1})$ and $J(q_{i+1})$ as known quantities, because they are for the lower solver in the context of bilevel optimization.",
"Therefore, to find the contact force that provides maximal energy dissipation we seek to minimize the quantity $dT$ in (REF ).",
"This is equivalent to minimizing the following quadratic function.",
"$\\frac{1}{2} x_{i+1}^T Q^d_{i+1} x_{i+1} + (r^{d}_{i+1})^T x_{i+1},$ where $\\begin{aligned}Q^d_{i+1}&= h J_{i+1} M_{i+1}^{-1} J_{i+1}^T, \\\\r^d_{i+1}&=\\big (h(c_{i+1} - \\tau _{i+1})^T M_{i+1}^{-1} - v_i^T\\big )J_{i+1}^T.\\end{aligned}$ Additionally we want to ensure that the contact forces satisfy constraints imposed by a Coulomb friction model.",
"To accomplish this, we first linearize the friction cone, as described in [4], such that $D \\beta = [0,f_x,f_y]^T$ where $D$ is a basis that spans the friction cone and $\\beta $ is a vector of non-negative coefficients.",
"We also define $z = [c_n, \\beta ^T]^T$ , and therefore can express the contact force vector $x$ as $\\begin{aligned}x = Fz, \\quad F:=\\begin{bmatrix} \\hat{n} | D \\end{bmatrix},\\end{aligned}$ where $\\hat{n}$ is the unit length normal vector at the contact point.",
"Then, we impose upon the energy dissipation maximization problem the Coulomb friction model constraint $\\mu c_n - e^T \\beta \\ge 0$ , where $\\mu $ is the friction coefficient and $e = [1,1,\\dots ,1]^T$ .",
"The resulting computation of the friction force is therefore given by the following quadratic program $\\begin{aligned}\\beta = \\;& \\underset{\\beta }{\\text{argmin}}& & \\frac{1}{2} z^T F^T Q^d F z + (r^d)^T F z \\\\&& & \\mu c_n - e^T \\beta \\ge 0,\\\\&& & \\beta \\ge 0,\\end{aligned}$ where the time index notation has been dropped for clarity.",
"Note that the formulation of our “semidirect” method allows us to make the assumption here that the normal force $c_n$ is known and is therefore not a decision variable in (REF ).",
"This will be discussed in further detail next in Section ."
],
[
"Trajectory Optimization Through Contact",
"We now show how to formulate the trajectory optimization problem as a bilevel optimization problem using the results from Section .",
"As mentioned before, this formulation is “semidirect” in the sense that part of the overall contact force (normal forces) are handled via complementarity constraints, and the other part (friction forces) are handled via the maximum dissipation optimization problem given by (REF ).",
"Thus in this formulation the decision variables of the trajectory optimization problem are the control inputs $\\tau $ , configuration variables $q$ , joint velocities $v$ , and the normal force component $c_n$ , but the friction force vector $\\beta $ will not be a decision variable, as it will be implicitly encoded by the embedded optimization problem.",
"Note that handling the non-penetration constraints, and hence the normal force, in a lower problem is also possible.",
"However since it is a constraint that is a function of position that would be imposed on decision variables relating to forces, keeping the embedded constraint linear would introduce compounding approximations (double integration).",
"This is not the case for the problem of friction since this later set of constraints are functions of velocity (not position) imposed on decision variables relating to forces."
],
[
"Dynamics Constraints",
"For a planning horizon $i = 1,\\dots , m$ , from the backward Euler discretized manipulator equation (REF ) with the contact forces in contact space represented by $x=Fz$ where $z=[c_n,\\beta ]^T$ , we have the dynamics constraints $ \\begin{split}M_{i+1}(v_{i+1} - v_i) + h c_{i+1} + h J_{i+1}^T Fz_{i+1} = h \\tau _{i+1}, \\\\q_{i+1} = q_i + hv_{i+1}.",
"\\\\\\end{split}$ The normal force $c_n$ is then constrained by the set of complementarity conditions $\\begin{split}\\phi (q_{i+1}) \\ge 0, \\\\c_{n,i} \\ge 0, \\\\c_{n,i} \\phi (q_{i+1}) = 0,\\end{split}$ where $\\phi (q)$ is a distance function such that $\\phi (q)\\ge 0$ implies non-penetration of the rigid body with its environment.",
"Finally, the friction force vector $\\beta $ is given by (REF )."
],
[
"Trajectory Optimization",
"Given an integrating stage cost function $J(q,v,\\tau )$ , and problem-specific constraints on the configuration and control defined by $g(q,v,\\tau ) \\le 0$ and $h(q,v,\\tau ) = 0$ (i.e.",
"desired initial or final configurations, or actuation limits), the trajectory optimization problem is defined as $\\begin{aligned}& \\underset{q_i,v_i,\\tau _i,c_{n,i};i=1\\;\\ldots m}{\\text{minimize}}& & \\sum _{i=1}^{m}{J(q_i,v_i,\\tau _i)} \\\\& \\text{subject to}& & M_{k+1}(v_{k+1} - v_k) + h c_{k+1} + h J_{k+1}^T Fz_{k+1} = h \\tau _{k+1}, \\\\&& & q_{k+1} = q_k + hv_{k+1}, \\\\&& & z_{k+1} = [c_{n,k+1}, \\beta _{k+1}]^T, \\\\&& & \\phi (q_i) \\ge 0, \\\\&& & c_{n_i} \\ge 0, \\\\&& & c_{n_i} \\phi (q_{i+1}) = 0, \\\\&& & \\beta _{k+1} = \\underset{\\beta _{k+1}}{\\text{argmin}} \\frac{1}{2} z_{k+1}^T F^T Q^d_{k+1} F z_{k+1} + (r^d_{k+1})^T F z_{k+1} \\\\&& & \\hspace{56.9055pt} \\mu c_{n,k+1} - e^T \\beta _{k+1} \\ge 0,\\\\&& & \\hspace{56.9055pt} \\beta _{k+1} \\ge 0.",
"\\\\&& & g(q_i,v_i,\\tau _i) \\le 0, \\\\&& & h(q_i,v_i,\\tau _i) = 0, \\\\\\end{aligned}$ where $k = 1,\\dots ,m-1$ .",
"The above problem forms a nonlinear bilevel trajectory optimization problem that can be solved by off-the-shelf nonlinear solvers as long as special care is taken to handle the friction force embedded problem, as described next in Section ."
],
[
"Solving the Friction Force Lower Problem",
"Now that we have defined the lower problem of our bilevel optimization, namely the quadratic program (REF ) whose solution corresponds to the friction forces at each contact point, we discuss our solution method for it.",
"There exists many solutions methods capable of handling convex quadratic programs similar to the one described in (REF ).",
"However, in the context of bilevel optimization problems, additional requirements are placed on the solvers for the embedded mathematical program.",
"First, solutions are needed very quickly and with little overhead since the embedded solver is called frequently by the primary solver working on the upper problem (i.e.",
"for every evaluation of the constraints of the upper problem by the primary solver).",
"In order to address this requirement, we leverage a state-of-the-art quadratic program solver OSQP [7], that implements the popular Alternating Direction Method of Multipliers (ADMM) algorithm [8].",
"Second and perhaps most importantly, gradients of the solution with respect to the parameters of the problem need to be available."
],
[
"Gradients",
"In order to provide gradients of the solution, we leverage work in sensitivity analysis [9], [10], [11], which has regained traction more recently [12], [13], [14].",
"The approach relies on using first order optimality conditions and the implicit function theorem in order to define a system that can be solved to recover the gradient of the solution with respect to the problem's parameters.",
"The details of this derivation are outside the scope of this paper and we refer the reader to [13] for a thorough treatment of the problem.",
"But for the reader's benefit we include the system that must be solved along with the quadratic program in order to recover the gradients (taken from [14]).",
"Given the following quadratic program: $\\begin{aligned}& \\underset{x}{\\text{minimize}}& & \\frac{1}{2}x^T Q(\\theta ) x + q(\\theta )^T x \\\\& \\text{subject to}& & G(\\theta ) x \\preceq h(\\theta ), \\\\&& & A(\\theta ) x = b(\\theta ),\\end{aligned}$ where $\\theta $ is the parameter vector for which we are interested in getting the gradients with respect to.",
"We can then compute the desired gradient $D_{\\theta }x^*$ by solving $\\Pi \\begin{bmatrix}D_{\\theta }x^* \\\\D_{\\theta }\\lambda ^*\\\\D_{\\theta }\\nu ^*\\end{bmatrix}= z,$ where $\\Pi =\\begin{bmatrix}Q & G^T & A^T \\\\\\textbf {diag}(\\lambda ^*)G & \\textbf {diag}(Gx^* - h) & 0 \\\\A & 0 & 0\\end{bmatrix}, \\quad z =\\begin{bmatrix}dQx^* + D_{\\theta }q + dG^T \\lambda ^* + dA^T \\nu ^* \\\\\\textbf {diag}(\\lambda )(dGx^* - D_{\\theta }h) \\\\dAx^* - D_{\\theta }b\\end{bmatrix}, \\\\$ and where $x^*$ , $\\lambda ^*$ , $\\nu ^*$ are the optimal solution for the primal, and dual variables ($\\lambda $ is the dual for the inequality constraint and $\\nu $ is the dual for the equality constraints).",
"Note that the system must often be solved using a least-squares method."
],
[
"Parallelization",
"One of the main drawback of our approach is that even though the size of the problem (in number of variables and constraints) for the upper problem is reduced, constraint evaluation is now significantly more expensive.",
"Indeed, instead of evaluating a series of inequalities like we would normally be required to, constraint evaluation with our approach requires us to solve a mathematical program.",
"However, because we perform our trajectory optimization using a direct transcription method (and not say iLQR as in [3]), we can trivially run our constraint evaluation in parallel.",
"In our implementation, each dynamic constraint (one per successive knot point) can be evaluated in parallel as its own thread.",
"The performance gain is therefore dependent on the number of knot points used.",
"When using a computer with sufficiently many cores, this parallelization can help offset the cost incurred by the more computationally expensive constraint evaluation as shown in section REF ."
],
[
"Results",
"We now present some results to empirically demonstrate and validate our proposed approach.",
"First, we demonstrate our algorithm on a smaller problem and compare its efficiency with an indirect method which is considered state-of-the-art [2].",
"For thoroughness, we also benchmark a slightly larger problem consisting of a hopping robot.",
"Next, we use our algorithm to design a `stepping forward' motion for a quadruped that can then be used in repetition as a gait."
],
[
"Implementation Details",
"We implement our approach using the Julia programming language [15].",
"Specifically, the rigid body dynamics (excluding contact dynamics) are computed using the package RigidBodyDynamics.jl [16] and we implement the analytical gradient of the embedded quadratic program inside the ForwardDiff.jl framework [17] (allowing for easy integration with other constraints).",
"Additionally, for the upper problem we use the popular sequential quadratic programming (SQP) solver SNOPT [18], which is often regarded as the most performant solver for trajectory optimization.",
"For the embedded problem, we use the solver OSQP (as described in Section ).",
"Our experiments were run on a 16-core 3.0 GHz CPU with 32Gb of memory.",
"All of of our code is made available at https://github.com/blandry/Bilevel.jl."
],
[
"Performance Benchmarks",
"First we solve a simple planning problem involving sliding a single rigid body on a surface to a target location, where the rigid body is modeled as having a single point of contact with the surface.",
"The initial position of the object is chosen such that it is not in contact with the surface, so that the solver must also determine the contact time.",
"Additionally, a final position constraint is defined, the final velocity is constrained to be zero, and there is no control input along the trajectory.",
"The problem therefore corresponds to finding the right initial velocity to throw the object such that it slides across a surface to the target position.",
"Note that this task is more challenging than it might appear at first glance, since it requires the trajectory to include transitions from non-contact to sliding friction, and then to sticking friction.",
"Here we compare the result of solving this trajectory optimization problem using our proposed semidirect method against the indirect method presented in [2].",
"The solver SNOPT was also used for the approach in [2], and in both methods was allowed to run until optimality (with a tolerance of $10^{-5}$ ).",
"Note that even for this simple example, the indirect method required us to introduce slack variables on the complementarity constraints related to friction (a common trick to handle these constraints).",
"Notably however, since our semidirect method handles the friction contact constraints via the embedded problem given by (REF ) no slack variables were necessary.",
"From the trajectory shown in Figure REF and the results presented in Table REF , we can see that our method not only recovers a solution of comparable quality, but it does so in less time.",
"For example, for the largest problem we report here, our semidirect method benchmarked (over several averaged samples) at 1.258 seconds, while the indirect method required more computation time, at 2.247 seconds.",
"Note also that for this example, the semidirect method only required SNOPT to solve a problem with 380 variables while the indirect one contained 652.",
"Table: Comparison of optimization problem size (for the upper problem) and solve time (in seconds) between our proposed semidirect method and the indirect method presented in , when applied to the problem described in Section .Figure: Comparison of the horizontal (left) and vertical (right) position trajectories of the rigid body's center of mass resulting from our proposed semidirect method and the indirect method presented in , when applied to the problem described in Section .",
"Units are in seconds and meters.For thoroughness, we also benchmark a second different toy problem.",
"This time, the problem consists of a hopping robot with a single actuator at the knee and a contact point at its foot.",
"The system has 4 degrees of freedom and one input.",
"The task is to jump to a target height from a given initial configuration, given actuator limits.",
"One of the resulting trajectories is shown in figure REF .",
"Like in the first benchmark, both methods were allowed to run until optimality (with a tolerance of $10^{-5}$ ).",
"The resulting solve times in seconds are reported in table REF .",
"Once again, our semidirect method outperformed the indirect method.",
"Table: Additional benchmark task for computational time comparison with state-of-the-art alternative method.",
"The task consists of getting a hopping robot to reach a target height as shown in .Figure: One of the resulting trajectories of the additional benchmark task for computational time comparison.",
"The goal is for a five degrees-of-freedom robot to jump to a specified height.",
"Only the knee is actuated."
],
[
"Parallelization Benchmark",
"Here, we also report the additional reduction in mean run time that is possible to achieve with our semidirect method by evaluating the constraints (and therefore solving the lower problems) in parallel.",
"The problems solved here are the same ones as described in section REF (with their serial counterparts reported in REF ).",
"A comparison is displayed in table REF .",
"As expected, the computational gains of parallelization increase with problem size.",
"Table: Additional reduction in mean run-time (reported in seconds) of the sliding box benchmark from section when constraints are evaluated (and therefore the lower problems solved) in parallel.",
"As expected, computational gains increase with problem size."
],
[
"Quadruped Gait Design",
"Next, we applied our approach to perform trajectory optimization for Boston Dynamics's “little dog”.",
"The system and the contact points are modeled as 3-dimensional, but the system is constrained such that its center of mass moves in a vertical plane.",
"The system has 15 degrees of freedom (from the position of the center of mass, orientation in the plane, and leg joint angles) and each leg is modeled with a contact point on its tip.",
"The problem consists of moving little dog forward by 20 centimeters, starting and ending with zero velocity, while also respecting actuator upper and lower limits.",
"In order to increase the trajectory's practical implications, we constraint it to describe a periodic gait by also enforcing that the final configuration (minus the forward displacement) matches the initial one.",
"In Figure REF we show snapshots of the trajectory resulting from our proposed semidirect method, and a video of the resulting gait is also available at https://www.youtube.com/playlist?list=PL8-2mtIlFIJpgmWImauC9rXxgN2wWpcIR.",
"Figure: The gait found by our semidirect method for a quadruped, little dog.",
"The robot first lowers itself towards the ground and then quickly moves upwards while pushing backwards (leveraging implicit friction forces computed by the embedded optimization problem).",
"This generates a forward “flight“ phase followed by a landing phase where little dog brings itself to rest (once again leveraging friction with the ground) and returns to its initial configuration.",
"This makes up one cycle of a gait that exploits complex interactions between the robot and its environment through friction and normal forces.Figure REF shows the total normal force on the front and back legs, clearly demonstrating the non-trivial strategy that the algorithm found to design a gait.",
"For this example, the trajectory optimization solve time (over several samples) is benchmarked at 1.8 seconds.",
"After our best effort, we were unable to get the indirect method from [2] to converge to a good solution for this task and therefore cannot report its performance on it.",
"However an alternative method such as [19], which neglects several dynamic constraints in order to make the optimization easier, reports computation times similar to ours for the same quadruped.",
"Figure: Normal contact forces at the legs (top) and position of the center of mass (bottom) of little dog during one cycle of the gait resulting from the trajectory optimization problem described in Section ."
],
[
"Method Shortcomings",
"We note that even though our approach is capable of efficiently generating a vast array of complex trajectories that involve making and breaking contact, it still leaves many open questions.",
"Notably our strategy (least squares) for choosing a subgradient when the gradient of the lower quadratic problem is not uniquely defined seems to work in practice, but does not rest on a strong theoretical foundation yet, and is perhaps far from optimal for this application.",
"Moreover, in practice, we found that trajectories involving sticking contact seemed to pose a bigger challenge to the upper solver (SNOPT) that would sometimes struggle to improve the trajectory past a certain point, most likely due to the gradients of the lower problem.",
"We believe these are great directions for future research in both theoretical and numerical aspects of our method."
],
[
"Conclusion",
"In this work we introduced a bilevel optimization approach to robotic trajectory optimization for systems that can make and break contact with their environment.",
"The approach falls under the category of planning “through contact”, where the contact constraints are directly resolved within the formulated optimization problem.",
"While similar approaches have been proposed in the past, the novelty of our proposed method is that it formulates the contact constraints in the trajectory optimization in a “semidirect” way.",
"Specifically, the normal force contact is handled indirectly via complementarity constraints in the optimization problem and the friction force is handled directly as the solution to an embedded optimization problem.",
"This allows us to avoid to use of additional complementarity constraints for the friction forces and to avoid linearizing the non-penetration constraints.",
"To demonstrate empirically the advantages of our proposed approach we presented results from three problems: two benchmark problems involving sticking and sliding friction and a gait optimization for a 15-degrees-of-freedom quadruped."
],
[
"Acknowledgments",
"This work was supported in part by the Office of Naval Research (Grant N00014-17-1-2749).",
"The authors of this work would also like to thank Hongkai Dai for his guidance at multiple stages of this work, as well as Shane Barratt for his useful pointers."
]
] | 1906.04292 |
[
[
"Chiral twodimensional p-wave superfluid from s-wave pairing in the BEC\n regime"
],
[
"Abstract Twodimensional spin-orbit-coupled Fermi gases subject to s-wave pairing can be driven into a topological phase by increasing the Zeeman spin splitting beyond a critical value.",
"In the topological phase, the system exhibits the hallmarks of chiral p-wave superfluidity, including exotic Majorana excitations.",
"Previous theoretical studies of this realization of a twodimensional topological Fermi superfluid have focused on the BCS regime where the s-wave Cooper pairs are only weakly bound and, hence, the induced chiral p-wave order parameter has a small magnitude.",
"Motivated by the goal to identify potential new ways for the experimental realization of robust topological superfluids in ultra-cold atom gases, we study the BCS-to-BEC crossover driven by increasing the Cooper-pair binding energy for this system.",
"In particular, we obtain phase diagrams in the parameter space of two-particle bound-state energy and Zeeman spin-splitting energy.",
"Ordinary characteristics of the BCS-to-BEC crossover, in particular the shrinking and eventual disappearance of the Fermi surface, are observed in the nontopological phase.",
"In contrast, the topological phase retains all features of chiral p-wave superfluidity, including a well-defined underlying Fermi surface, even for large s-wave pair-binding energies.",
"Compared to the BCS limit, the topological superfluid in the BEC regime turns out to be better realizable even for only moderate magnitude of spin-orbit coupling because the chiral p-wave order parameter is generally larger and remnants of s-wave pairing are suppressed.",
"We identify optimal parameter ranges that can aid further experimental investigations and elucidate the underlying physical reason for the persistence of the chiral p-wave superfluid."
],
[
"Introduction and overview of main results",
"One of the earliest proposed pathways towards realization of a twodimensional (2D) topological superfluid (TSF) [1] is based on s-wave pairing of spin-$\\frac{1}{2}$ fermions subject to spin-orbit coupling and Zeeman spin splitting [2], [3], [4], [5], [6].",
"In the absence of spin-orbit coupling, a population imbalance in the spin components (equivalent to nonzero Zeeman splitting) tends to destroy s-wave superfluidity due to the mismatch of the spin-$\\uparrow $ and spin-$\\downarrow $ Fermi surfaces for weak-coupling superfluids [7], [8].",
"With strong s-wave attraction, phase separation between superfluid and normal phases ensues in this case [9].",
"Adding 2D spin-orbit coupling (e.g., of Rashba form [10], [11], [12]) permits a pairing instability even for unmatched Fermi surfaces and re-establishes a homogeneous superfluid ground state with gapped fermionic quasiparticle excitations.",
"The pairing field for each spin component separately [13] now obtains the characteristics of a chiral 2D p-wave superfluid [14].",
"Increasing the Zeeman coupling energy $h$ beyond the critical value $h_\\mathrm {c} = \\sqrt{\\mu ^2 + |\\Delta |^2}$ quenches one of the Fermi surfaces, and the system enters a TSF phase.",
"In Eq.",
"(REF ), $\\Delta $ and $\\mu $ denote the selfconsistent s-wave pair potential and chemical potential, respectively.",
"Bearing all the characteristics of a 2D spinless p-wave superfluid, a nontrivial topological invariant can be defined [1], and Majorana quasiparticle excitations are present at boundaries [15], [2] and in vortex cores [16], [17], [18], [19], [20] by virtue of an index theorem [21].",
"Majorana zero modes are considered promising candidates for enabling fault-tolerant quantum-information processing [22].",
"Intense efforts towards experimental implementation of 2D TSFs using the above-described route have so far been thwarted by the deleterious effect of Zeeman-splitting-inducing magnetic fields on superconductivity in typical materials [23], as well as basic physical constraints on the magnitude of spin-orbit coupling reachable in solids [11] and ultra-cold atom gases [24], [25].",
"Our present study shows that a possible way around the latter limitation would be to access the strong-coupling regime of the s-wave pairing, which is commonly referred to as the BEC regime [26], [27], [28], [29], [30].",
"Figure: Zero-temperature mean-field phase diagrams, in the parameter space oftwo-particle s-wave bound-state energy E b E_\\mathrm {b} andZeeman energy hh, for a spin-orbit-coupled 2D Fermi gas with fixeddensity n=mE F /(πℏ 2 )≡k F 2 /(2π)n = m E_\\mathrm {F}/(\\pi \\hbar ^2)\\equiv k_\\mathrm {F}^2/(2\\pi ).Panel (a) [(b)] depicts the case where the dimensionless parameterλk F /E F \\lambda k_\\mathrm {F}/E_\\mathrm {F} measuring the spin-orbit-couplingstrength equals 0.500.50 [0.750.75], which illustrates asmall-(large-)spin-orbit-coupling situation.",
"The shaded region h < <h<h > h_< <h < h_>, with h < h_< (h > h_>) indicated by the blue (green) line, is inthe phase-separated first-order-transition regime that emerges forE b >E b (c) E_\\mathrm {b}>E_\\mathrm {b}^{(\\mathrm {c})} and h>h (c) h > h^{(\\mathrm {c})}.The critical Zeeman energy h c h_\\mathrm {c}, defined viaEq.",
"() and indicated by the red curve, delineates thesecond-order transition between an ordinary nontopological superfluid(NSF) and a topological superfluid (TSF).",
"From the point when theh c E b h_\\mathrm {c}\\big (E_\\mathrm {b}\\big )-curve reaches the region wherephase separation occurs, the topological transition is switched fromsecond to first order.",
"The BCS-to-BEC-crossover boundary (dashed line)has been determined via the condition μ(E b ,h)=0\\mu (E_\\mathrm {b},h)=0.The main insights reached in our work are underpinned by zero-temperature phase diagrams in the parameter space of two-particle s-wave bound-state energy $E_\\mathrm {b}$ [31] and Zeeman energy $h$ as illustrated in Fig.",
"REF .",
"These show a second-order transition line (red) between the nontopologial and topological superfluids at small $E_\\mathrm {b}$ being replaced by a first-order phase transition at larger $E_\\mathrm {b}$ .",
"In the phase diagram at constant particle density $n$ , enforced by measuring energies in terms of the Fermi energy $E_\\mathrm {F} = \\pi \\hbar ^2 n/m$ , the first-order phase transition manifests itself as a region without a uniform-density ground state (grey) where phase separation into spatially separated domains ensues.",
"Critical Zeeman-energy values $h_<$ (blue) and $h_>$ (green) delimit the phase-separation region at fixed $E_\\mathrm {b}$ , defining two curves in the phase diagram.",
"The properties of the phase-separation region itself have been the subject of previous work [32], [33], and we also provide a few more details later on.",
"However, the main focus of our present study is the careful determination of the location of the boundaries $h_<$ and $h_>$ and the exploration of the adjacent homogeneous phases, especially in the regime where $E_\\mathrm {b}/E_\\mathrm {F} \\gtrsim 1$ .",
"Comparison of our results with those obtained previously for spin-imbalanced 2D Fermi superfluids See, e.g., Fig.",
"2(a) in Ref. [9].",
"helps to elucidate the physical consequences of finite spin-orbit coupling.",
"The magnitude of the latter is most conveniently measured in terms of the dimensionless parameter $\\lambda k_\\mathrm {F}/E_\\mathrm {F}$ that also involves the density-dependent Fermi wave number $k_\\mathrm {F}\\equiv \\sqrt{2\\pi n}$ .",
"One important effect of finite $\\lambda $ is to shift the low-$E_\\mathrm {b}$ boundary of the phase-separation region from zero to finite values of $E_\\mathrm {b}$ [9], and a second effect is to establish the TSF phase [33] for sufficiently high Zeeman energy $h>\\mathrm {max}\\lbrace h_c, h_>\\rbrace $ in place of the fully polarized normal phase found for $\\lambda =0$ [9].",
"Our present study shows that the character of the TSF phase emerging in the BEC regime of the underpinning s-wave pairing ($E_\\mathrm {b}/E_\\mathrm {F}\\gtrsim 1$ ) is fundamentally similar to the TSF occurring in the BCS limit ($E_\\mathrm {b}/E_\\mathrm {F}\\ll 1$ ).",
"In particular, for the entire TSF region in the phase diagram, the system exhibits canonical signatures of an underlying Fermi surface.",
"As discussed by Sensarma et al.",
"[35], an underlying Fermi surface can be robustly defined even in strong-coupling fermionic superfluids by a number of alternative definitions, such as a zero crossing of the single-particle Greens function $G(\\mathbf {k},0)$ , a drop in the single-particle momentum distribution $n(\\mathbf {k})$ , or, if available, by a minimum of the quasiparticle dispersion relation.",
"Our results for the TSF are in stark contrast to the nontopological superfluid (NSF) phase where the Fermi surface shrinks and eventually disappears as $E_\\mathrm {b}/E_\\mathrm {F}$ increases and the BEC regime is entered.",
"This is expected from the known phenomenology of the BCS-to-BEC crossover for s-wave pairing [36], [37], [38] and illustrated by recent Quantum-Monte-Carlo results [39], [40].",
"Figure: The magnitude of the s-wave pair potential Δ\\Delta in thehomogeneous topological-superfluid phase of a spin-orbit-coupled 2DFermi gas with fixed density n=mE F /(πℏ 2 )≡k F 2 /(2π)n = m E_\\mathrm {F}/(\\pi \\hbar ^2)\\equiv k_\\mathrm {F}^2/(2\\pi ) is maximized, for fixed two-particlebound-state energy E b E_\\mathrm {b} and spin-orbit-coupling strengthλ\\lambda , right after the transition at Zeeman energy h max = max {h c ,h > }h_{\\text{max}}=\\mathrm {max}\\lbrace h_\\mathrm {c}, h_>\\rbrace (see Fig.",
").Panel (a) shows the dependence of this maximum value, |Δ| h=h max |\\Delta |_{h=h_{\\text{max}}}, on E b E_\\mathrm {b} for two fixed values of thedimensionless spin-orbit-coupling strength λk F /E F \\lambda k_\\mathrm {F}/E_\\mathrm {F}.",
"Panel (b) is a plot of the emergentchiral-p-wave gap Δ pw \\Delta _\\mathrm {pw}, extracted from thelow-energy quasiparticle dispersion, for the same parameters.The defining element of the 2D TSF is an emergent chiral p-wave order parameter $\\Delta _\\mathrm {pw}$ whose magnitude provides the energy scale of the quasiparticle-excitation gap.",
"Its value is proportional to the spin-orbit-coupling strength $\\lambda $ and the modulus $|\\Delta |$ of the s-wave pair potential, but inversely related to the spin-splitting (Zeeman) energy scale $h$ [3], [5], [6], [41], [13].",
"Given that increasing $\\lambda $ has adverse side effects such as heating of the atom gas [24] in currently available experimental schemes, maximizing $\\Delta _\\mathrm {pw}$ needs to be pursued by other means.",
"As $|\\Delta |/E_\\mathrm {F}$ is a monotonously increasing function of $E_\\mathrm {b}/E_\\mathrm {F}$ but is suppressed with increasing $h/E_\\mathrm {F}$ (see, e.g., Refs.",
"[42], [13] and below), its practically largest magnitude occurs just after the transition to the homogeneous TSF phase at $h_{\\text{max}}=\\mathrm {max}\\lbrace h_\\mathrm {c}, h_>\\rbrace $ .",
"Figure REF (a) illustrates the dependence of this value, $|\\Delta |_{h=h_{\\text{max}}}$ , on both $E_\\mathrm {b}$ and $\\lambda $ .",
"It reveals a maximum that gets broader and larger as the parameter $\\lambda k_\\mathrm {F}/E_\\mathrm {F}$ increases.",
"As the maximum value of $|\\Delta |$ reaches values up to $\\sim \\!",
"E_\\mathrm {F}$ typically, even for only moderately high values of the spin-orbit-coupling strength, the TSF realized in the BEC regime of s-wave pairing presents a much more favorable platform for useful study and application than would be available in the BCS regime at the same value of $\\lambda $ .",
"This is established even more directly by measuring $\\Delta _\\mathrm {pw}$ as the gap in the low-energy quasiparticle dispersion for the TSF.",
"The values for $\\Delta _\\mathrm {pw}$ found for the same parameter combinations that maximize $|\\Delta |$ are shown in Fig.",
"REF (b).",
"For both fixed values of $\\lambda k_\\mathrm {F}/E_\\mathrm {F}$ , a maximum of $\\Delta _\\mathrm {pw}$ occurs for $E_\\mathrm {b}/E_\\mathrm {F} \\sim 1$ , followed by a broad range for which $\\Delta _\\mathrm {pw}$ is slowly decreasing.",
"All quantitative results in this work were obtained within mean-field theory, even though its validity for a 2D gas, in particular outside of the weakly interacting regime, may not be taken for granted.",
"We nevertheless expect the qualitative physics, and in particular the presence of a Fermi surface in the TSF phase, to be robust because the topological property puts strong constraints on the many-body system.",
"We comment further on the physical reasons below.",
"Mean-field approximations have previously been found to provide useful insight into zero-temperature phases, even when interactions are strong [43], [44], [9], [45].",
"Quantitatively more accurate predictions, in particular for finite temperature, require more sophisticated approaches [46], [43], [44], [47], [48], [49], [50], [51].",
"The expected effects of beyond-mean-field corrections (quantum fluctuations) is to suppress pairing gaps compared to mean-field theory in the strongly interacting regime [49], [51].",
"This fact reinforces the optimal value $E_\\mathrm {b}/E_\\mathrm {F} \\sim 1$ for realizing a robust TSF, as for $E_\\mathrm {b}/E_\\mathrm {F}\\gg 1$ , the true value for the s-wave pairing gap, and therefore also $\\Delta _\\mathrm {pw}$ , are likely to be much smaller than mean-field theory predicts.",
"The physical reasons for the remarkable BCS-like behavior of the TSF even when interactions are strong enough to place s-wave pairs into the BEC regime may be seen from a careful analysis of the relevant low-energy part of the quasiparticle spectrum.",
"A projection of the mean-field equations to the majority-spin component [13] yields a useful approximate expression for the excitation gap and TSF order parameter $\\Delta _\\mathrm {pw} \\approx |\\Delta |\\, \\frac{\\lambda \\,k_\\mathrm {FS}}{h_{k_\\mathrm {FS}}}\\quad ,$ where $h_k = (h + \\sqrt{h^2 + \\lambda ^2 k^2})/2$ , and $k_\\mathrm {FS}\\le \\sqrt{2} k_\\mathrm {F}$ is the radius of the Fermi surface in the TSF phase (see end of Sec. ).",
"The projective approximation is valid when $\\Delta _\\mathrm {pw}$ is small compared to the Fermi energy $E_\\mathrm {F}$ , but this condition will be fulfilled when spin-orbit coupling is not too strong, $\\lambda k_\\mathrm {F} <E_\\mathrm {F}$ , in the TSF regime where $h>|\\Delta |$ due to Eq.",
"(REF ).",
"Note that this means that $\\Delta _\\mathrm {pw}$ is bounded while the binding energy $E_\\mathrm {b}$ may be much larger.",
"Within the same projective approximation [13] and for $\\lambda k_\\mathrm {F} < h$ , one also obtains the estimate $k_\\mathrm {FS} \\approx k_\\mathrm {F} \\left[ \\frac{\\mu + h -\\frac{|\\Delta |^2}{2 h}}{E_\\mathrm {F} \\left( 1 - \\frac{1}{2}\\frac{\\lambda ^2 k_\\mathrm {F}^2}{E_\\mathrm {F} h} \\right)}\\right]^{\\frac{1}{2}} \\quad ,$ which shows that the Fermi surface radius is finite, $k_\\mathrm {FS} >0$ , as long as the Zeeman energy $h$ is sufficiently large.",
"Thus the large magnitude of the Zeeman energy required to reach the uniform TSF phase ultimately ensures the persistence of BCS-like character of chiral p-wave pairing, even as the s-wave interaction is deep in the BEC regime.",
"While the situation becomes slightly more complex for very large spin-orbit-coupling strengths $\\lambda k_\\mathrm {F}/E_\\mathrm {F} > 1$ , we still find signatures of a Fermi surface persisting throughout the TSF phase, and canonical BCS-like behavior being exhibited for $h\\gtrsim h_\\mathrm {max}$ .",
"The remainder of this article is organized as follows.",
"Section introduces the theoretical approach used by us to describe the BCS-to-BEC crossover for the s-wave-paired 2D Fermi gas subject to both spin-orbit coupling and Zeeman spin splitting.",
"Detailed results obtained within this formalism for the system with fixed uniform particle density are presented in the subsequent Sec.",
", together with a discussion of physical implications and limitations inherent in the mean-field approach.",
"Our conclusions are formulated in the final Sec.",
"."
],
[
"Microscopic model of the 2D TSF",
"We utilize a standard Bogoliubov-de Gennes (BdG) mean-field formalism [52] to calculate the quasiparticle spectrum for our system of interest.",
"All relevant thermodynamic quantities can be expressed in terms of the obtained eigenenergies and eigenstates.",
"Throughout this work, we consider the zero-temperature limit.",
"The BdG Hamiltonian of the 2D spin-orbit coupled Fermi gas with $s$ -wave interactions and Zeeman spin splitting $2h$ acting in the four-dimensional Nambu space of spin-$1/2$ fermions is Our notation adheres to that used in Ref. [13].",
"$\\mathcal {H} = \\left( \\begin{array}{cccc} \\epsilon _{{\\mathbf {\\mathrm {k}}}\\uparrow }-\\mu &\\lambda _{\\mathbf {\\mathrm {k}}}& 0& -\\Delta \\\\ {\\lambda _{\\mathbf {\\mathrm {k}}}}^* & \\epsilon _{{\\mathbf {\\mathrm {k}}}\\downarrow } - \\mu &\\Delta & 0 \\\\ 0 & \\Delta ^* & -\\epsilon _{{\\mathbf {\\mathrm {k}}}\\uparrow } + \\mu & {\\lambda _{\\mathbf {\\mathrm {k}}}^*} \\\\ -\\Delta ^* & 0 &\\lambda _{\\mathbf {\\mathrm {k}}}&-\\epsilon _{{\\mathbf {\\mathrm {k}}}\\downarrow } +\\mu \\end{array} \\right) ,$ where ${\\mathbf {\\mathrm {k}}}= (k_x, k_y)$ denotes the 2D wave vector, $\\epsilon _{{\\mathbf {\\mathrm {k}}}\\uparrow (\\downarrow )}=\\epsilon _{\\mathbf {\\mathrm {k}}}\\, \\begin{array}{c}-\\\\(+)\\end{array} \\, h$ with $\\epsilon _{\\mathbf {\\mathrm {k}}}= \\hbar ^2 (k_x^2+k_y^2)/2 m$ , and $\\lambda _{\\mathbf {\\mathrm {k}}}\\equiv \\lambda \\, i(k_x - i k_y)$ is the spin-orbit coupling While we adopt the 2D-Rashba form [10] for $\\lambda _{\\mathbf {\\mathrm {k}}}$ , our results apply also to other types of spin-orbit coupling that depend linearly on the components of ${\\mathbf {\\mathrm {k}}}$ , such as the 2D-Dirac and 2D-Dresselhaus functional forms [11], [12] corresponding to $\\lambda _{\\mathbf {\\mathrm {k}}}=\\lambda (k_x-i k_y)$ and $\\lambda (k_x+ i k_y)$ , respectively..",
"The BdG equation reads $\\mathcal {H} \\left(\\begin{array}{c} u^\\uparrow \\\\ u^\\downarrow \\\\v^\\uparrow \\\\ v^\\downarrow \\end{array}\\right) = E \\left(\\begin{array}{c} u^\\uparrow \\\\ u^\\downarrow \\\\ v^\\uparrow \\\\v^\\downarrow \\end{array}\\right) \\quad .$ Its spectrum consists of four eigenvalue branches [32], [55], $E_{{\\mathbf {\\mathrm {k}}}\\alpha ,<(>)} = \\alpha \\sqrt{(\\epsilon _{\\mathbf {\\mathrm {k}}}-\\mu )^2 +|\\Delta |^2 +h^2 + |\\lambda _{\\mathbf {\\mathrm {k}}}|^2 \\begin{array}{c}-\\\\(+)\\end{array} 2 \\sqrt{(\\epsilon _{\\mathbf {\\mathrm {k}}}-\\mu )^2( h^2 + |\\lambda _{\\mathbf {\\mathrm {k}}}|^2) + |\\Delta |^2 h^2}} \\quad ,$ with associated eigenspinors $(u^\\uparrow _{{\\mathbf {\\mathrm {k}}}\\alpha ,\\eta },u^\\downarrow _{{\\mathbf {\\mathrm {k}}}\\alpha ,\\eta }, v^\\uparrow _{{\\mathbf {\\mathrm {k}}}\\alpha ,\\eta },v^\\downarrow _{{\\mathbf {\\mathrm {k}}}\\alpha ,\\eta })^T$ , where $\\alpha \\in \\lbrace +,-\\rbrace $ and $\\eta \\in \\lbrace <,>\\rbrace $ label the four different energy-dispersion branches.",
"The chemical potential $\\mu $ and magnitude $|\\Delta |$ of the pair potential need to be determined selfconsistently from solutions of the BdG equations in conjunction with the gap equation and the constraint that the uniform particle density is fixed at $n\\equiv k_\\mathrm {F}^2/(2\\pi )$ .",
"Corresponding conditions can be formulated mathematically in terms of the energy spectrum and BdG-Hamiltonian eigenspinor amplitudes.",
"See, e.g., Refs.",
"[52], [13].",
"However, educated by the insights gained from previous work on spin-imbalanced Fermi superfluids [56], we base selfconsistency considerations on the properties of the system's grand-canonical ground-state energy density [57], [44], [56], [32], [55], for which a standard calculation yields $&& \\mathcal {E}_\\mathrm {gs}^\\mathrm {(MF)}(|\\Delta |, \\mu ) = \\nonumber \\\\ && \\hspace{28.45274pt} \\frac{1}{A}\\sum _{\\mathbf {\\mathrm {k}}}\\Big ( \\frac{|\\Delta |^2}{2\\epsilon _{\\mathbf {\\mathrm {k}}}+ E_\\mathrm {b}} +\\epsilon _{\\mathbf {\\mathrm {k}}}-\\mu -\\frac{1}{2} \\sum _\\eta E_{{\\mathbf {\\mathrm {k}}}+,\\eta } \\Big )\\, .",
"\\quad $ Here $A$ denotes the system's volume (area), and $E_\\mathrm {b}>0$ is the magnitude of the two-particle bound-state (i.e., binding) energy [31].",
"The gap and number-density equations can be expressed in terms of derivatives of the ground-state energy density; $\\frac{\\partial \\mathcal {E}_\\mathrm {gs}^\\mathrm {(MF)}}{\\partial |\\Delta |} &=& 0 \\quad , \\\\[0.1cm]\\frac{\\partial \\mathcal {E}_\\mathrm {gs}^\\mathrm {(MF)}}{\\partial \\mu }&=& - n \\quad .",
"$ The lengthy explicit expressions are omitted here.",
"As emphasized previously during the study of spin-imbalanced Fermi superfluids [58], proper application of the condition (REF ) for identifying physical ground states requires ensuring that $\\mathcal {E}_\\mathrm {gs}^\\mathrm {(MF)}(|\\Delta |, \\mu )$ , taken as a function of $|\\Delta |$ at fixed $\\mu $ , has a global minimum at the selfconsistently determined value for $|\\Delta |$ .",
"However, identifying local minima as well as maxima of the ground-state energy at fixed $\\mu $ can also be of interest [59], [60], [61], e.g., to discuss nonequilibrium-dynamic phenomena; hence, we will track these in the following also.",
"The relative magnitude of $E_\\mathrm {b}$ with respect to the Fermi energy $E_\\mathrm {F}$ drives the BCS-to-BEC crossover for s-wave pairing in our system of interest [28].",
"More specifically, we have $\\frac{E_\\mathrm {b}}{E_\\mathrm {F}} \\left\\lbrace \\begin{array}{cl} \\ll 1 &\\mbox{in the BCS limit,} \\\\[0.2cm] \\gtrsim 1 & \\mbox{in the BECregime.}",
"\\end{array} \\right.$ In the following, we absorb any dependence on total particle density $n$ by measuring all energies and wave vectors in units of $E_\\mathrm {F}$ and $k_\\mathrm {F}$ , respectively.",
"Thus the set of externally tuneable parameters comprises $E_\\mathrm {b}/E_\\mathrm {F}$ , $h/E_\\mathrm {F}$ , and $\\lambda k_\\mathrm {F}/E_\\mathrm {F}$ .",
"The system's state is characterized by $|\\Delta |/E_\\mathrm {F}$ and $\\mu /E_\\mathrm {F}$ .",
"The chiral $p$ -wave nature of the superfluid is revealed by the following considerations.",
"Inspection of Eq.",
"(REF ) shows that $E_{{\\mathbf {\\mathrm {0}}} +,<} = | h_\\mathrm {c} - h |$ .",
"In the BCS regime, for $0< h < h_\\mathrm {c}$ , two minima exist in $E_{{\\mathbf {\\mathrm {k}}}+,<}$ at $|{\\mathbf {\\mathrm {k}}}| >0$ , corresponding to effective p-wave pairing around the two spin-split Fermi surfaces for spin-$\\uparrow $ and spin-$\\downarrow $ degrees of freedom.",
"As $h$ is increased, the location of the spin-$\\downarrow $ minimum moves towards $|{\\mathbf {\\mathrm {k}}}|=0$ , with its value shrinking and finally vanishing as it reaches $|{\\mathbf {\\mathrm {k}}}|=0$ at $h =h_\\mathrm {c}$ .",
"For $h > h_\\mathrm {c}$ , the system has only one Fermi surface corresponding to a fully polarized electron system, and the remaining minimum of $E_{{\\mathbf {\\mathrm {k}}}+,<}$ at $|{\\mathbf {\\mathrm {k}}}| \\sim \\sqrt{2}k_\\mathrm {F}$ is associated with an effective pair potential [6], [13] $(\\lambda _{\\mathbf {\\mathrm {k}}}/|\\lambda _{\\mathbf {\\mathrm {k}}}|) \\,\\Delta _\\mathrm {pw}\\equiv i \\mathrm {e}^{- i\\varphi _{\\mathbf {\\mathrm {k}}}}\\, \\Delta _\\mathrm {pw}$ , where $\\varphi _{\\mathbf {\\mathrm {k}}}=\\arctan (k_y/k_x)$ is the polar-angle coordinate for the 2D wave vector ${\\mathbf {\\mathrm {k}}}$ .",
"Proportionality of the superconducting order parameter to the phase factor $\\mathrm {e}^{-i \\varphi _{\\mathbf {\\mathrm {k}}}}$ is the defining property of chiral p-wave pairing [14], and also the origin of its accompanying topological features [14], [1].",
"In contrast, the system has two Fermi surfaces where p-wave pairing with opposite chirality occurs when $h < h_\\mathrm {c}$ , rendering it to be a nontopological superfluid.",
"We now apply the formalism introduced above to investigate the fate of chiral p-wave superfluidity in the BEC regime for the underlying s-wave pairing."
],
[
"Results and discussion",
"To ground ourselves in well-known results [42], [13], we start by fixing a value for $\\lambda k_\\mathrm {F}/E_\\mathrm {F}$ and consider the variation of the chemical potential $\\mu $ and the pair-potential magnitude $|\\Delta |$ as a function of the Zeeman energy $h$ in the BCS limit for s-wave pairing, i.e., for small $E_\\mathrm {b}/E_\\mathrm {F}$ .",
"As illustrated in Fig.",
"REF , both $\\mu (h)$ and $|\\Delta (h)|$ evolve continuously from the nontopological phase where $h < h_\\mathrm {c}$ [defined in Eq.",
"(REF )] via their critical values $\\mu _\\mathrm {c}\\equiv \\mu (h_\\mathrm {c})$ and $|\\Delta (h_\\mathrm {c})| \\equiv \\Delta _\\mathrm {c}$ that satisfy $\\sqrt{\\mu _\\mathrm {c}^2 +\\Delta _\\mathrm {c}^2} = h_\\mathrm {c}$ into the topological phase where $h > h_\\mathrm {c}$ .",
"This reflects the fact that, for any value of $h$ , ${\\mathcal {E}}_\\mathrm {gs}^\\mathrm {(MF)}$ has only a single minimum when plotted as a function of $|\\Delta |$ for fixed $\\mu $ , which occurs at a nonzero $|\\Delta |$ and thus corresponds to a homogeneous superfluid ground state.",
"The search for solutions of the selfconsistency conditions (REF ) and () for larger $E_\\mathrm {b}/E_\\mathrm {F}\\lesssim 1$ continues to yield unique values of $|\\Delta |$ and $\\mu $ .",
"See the examples shown in Fig.",
"REF .",
"However, an intricate complexity associated with selfconsistent solutions starts to develop.",
"As illustrated in Fig.",
"REF , within an intermediate range of Zeeman energies, two additional extrema (specifically, a local minimum and a local maximum) start to appear in the $|\\Delta |$ -dependence of the ground-state energy where $\\mu $ has been fixed to its selfconsistent value.",
"Below the value $E_\\mathrm {b}^{(\\mathrm {c})}$ associated with the critical end-point of the phase-separation region shown in Fig.",
"REF , the unique solution of the selfconsistency conditions still continues to be the global minimum of ${\\mathcal {E}}_\\mathrm {gs}^\\mathrm {(MF)}$ , taken at the selfconsistent $\\mu $ , for any value of $h$ .",
"This is the case, e.g., for the system parameters used to calculate the results shown in Fig.",
"REF (a,c).",
"However, for $E_\\mathrm {b}\\ge E_\\mathrm {b}^{(\\mathrm {c})}$ , which applies to Fig.",
"REF (b,d), the selfconsistently determined value for $|\\Delta |$ ceases to be associated with the global minimum of ${\\mathcal {E}}_\\mathrm {gs}^\\mathrm {(MF)}$ at fixed selfconsistent $\\mu $ for Zeeman energies within a range $h_< < h <h_>$ , corresponding instead to only a local minimum or even a maximum.",
"This implies that no single-phase equilibrium ground state exists in the region $h_< < h < h_>$ .",
"Instead, phase separation into domains of different densities will occur if the system is driven into this region.",
"Even further in the BEC regime when $E_\\mathrm {b}>E_\\mathrm {b}^{(\\mathrm {m})}$ , multiple selfconsistent pairs of values for $|\\Delta |$ and $\\mu $ emerge as illustrated in Fig.",
"REF .",
"Around each of these, additional zeros of the gap equation exist, as seen in Fig.",
"REF .",
"Now the range $h_< < h < h_>$ is defined to be the region where none of the selfconsistent $|\\Delta |$ values is associated with the global minimum of the ground-state energy ${\\mathcal {E}}_\\mathrm {gs}^\\mathrm {(MF)}$ when $\\mu $ is fixed to its corresponding selfconsistent value.",
"Figure: Structure of multiple selfconsistent and associatednonselfconsistent solutions of the gap equation deep in the BECregime.",
"Results shown are obtained for λk F /E F =0.75\\lambda k_\\mathrm {F}/E_\\mathrm {F} = 0.75 and E b /E F =3.0E_\\mathrm {b}/E_\\mathrm {F} = 3.0.",
"Filledsymbols indicate solutions of the selfconsistency conditions thatglobally minimize the ground-state energy and thus correspond toproper equilibrium states of the system.",
"Empty symbols (crosses) areassociated with (non)selfconsistent values corresponding to a localminimum or maximum of the ground-state energy.",
"Circles (a triangle,squares) indicate states where the system is nontopological (critical,topological).",
"Multiple selfconsistent solutions at a given hh aredistinguished by color.",
"The same color is used to indicate theirassociated additional zeros in the gap equation.Figure: Panel (a): Special values E b (c) E_\\mathrm {b}^{(\\mathrm {c})} andE b (m) E_\\mathrm {b}^{(\\mathrm {m})} for the two-particle bound-state energyE b E_\\mathrm {b} plotted as a function of the spin-orbit-couplingstrength λ\\lambda .",
"Here E b (c) E_\\mathrm {b}^{(\\mathrm {c})} is the valueof E b E_\\mathrm {b} associated with the critical end point h ( c), E b (c) \\big (h^\\mathrm {(c)}, E_\\mathrm {b}^{(\\mathrm {c})} \\big ) of thephase-separation region in the E b E_\\mathrm {b}-hh phase diagram wherethe critical-field curves h < h_< and h > h_> merge.",
"The valueE b (m) E_\\mathrm {b}^{(\\mathrm {m})} is the lower limit of bound-stateenergies for which multiple pairs of selfconsistent solutions forμ\\mu and |Δ||\\Delta | exist.",
"Panel (b): Dependence of h ( c)h^\\mathrm {(c)},the hh coordinate of the critical end point of the phase-separationregion in the E b E_\\mathrm {b}-hh phase diagram, on thespin-orbit-coupling strength.",
"For comparison, the critical fieldh c h_\\mathrm {c} [defined in Eq.",
"()] atE b (c) E_\\mathrm {b}^{(\\mathrm {c})} is also shown.The appearance of multiple extrema in the $|\\Delta |$ dependence of ${\\mathcal {E}}_\\mathrm {gs}^\\mathrm {(MF)}$ at fixed $\\mu $ , leading to the selfconsistent minimum ceasing to be the global minimum, indicates the presence of a first-order (noncontinuous) phase transition [9], [55].",
"A proper theoretical description of this situation requires the construction of various phase-coexistence scenarios [32], [33], [41], in analogy with treatments developed for the population-imbalanced Fermi gas without spin-orbit coupling [62], [63], [64], [65], [57], [44], [9], [66].",
"Here we defer the careful determination of the equilibrium ground state in the phase-separation region to future work This task becomes particularly challenging for the part of the phase diagram where multiple selfconsistent solutions of the gap equation exist at fixed $h$ .",
"Generally, two of these correspond to minima of the ground-state energy taken at fixed $\\mu $ , and their combined evolution between global- or local-minimum status needs to be tracked.. Rather, we intend to discuss the properties of the adjacent uniform, single-phase regions for large $E_\\mathrm {b}/E_\\mathrm {F}$ .",
"To this end, we only need to map carefully the boundaries of the phase-separation region, i.e., the critical-Zeeman-energy curves $h_<$ and $h_>$ .",
"Results for representative values of the spin-orbit-coupling strength are given in Fig.",
"REF .",
"We find that the phase-separation region narrows as the spin-orbit-coupling parameter $\\lambda k_\\mathrm {F}/E_\\mathrm {F}$ is increased, while simultaneously the critical end point $\\big (h^{(\\mathrm {c})}, E_\\mathrm {b}^{(\\mathrm {c})} \\big )$ where the $h_<$ and $h_>$ curves merge shifts to larger coordinate values in the phase diagram.",
"The full dependence of $E_\\mathrm {b}^{(\\mathrm {c})}$ (and also of $E_\\mathrm {b}^{(\\mathrm {m})}$ ) as a function of the dimensionless spin-orbit-coupling strength is plotted in Fig.",
"REF (a), with the associated results for $h^{(\\mathrm {c})}$ being provided in Fig.",
"REF (b).",
"Two different regimes, corresponding to small and large values of $\\lambda k_\\mathrm {F}/E_\\mathrm {F}$ , can be identified, where the former (latter) is characterized by the $h^{(\\mathrm {c})}$ values diverging from (coinciding with) the critical field $h_\\mathrm {c}$ for $E_\\mathrm {b}=E_\\mathrm {b}^{(\\mathrm {c})}$ .",
"The curves for $h_<(E_\\mathrm {b})$ and $h_>(E_\\mathrm {b})$ in the phase diagram delimit the phase-separation region associated with a first-order transition between different superfluid states.",
"In those parts of the phase diagram outside this region where only a single pair of selfconsistent values for $\\mu $ and $|\\Delta |$ exists, a curve $h_\\mathrm {c}(E_\\mathrm {b})$ can be defined via Eq.",
"(REF ) that separates the part of the phase diagram where the system is an ordinary nontopological superfluid (NSF, for $h<h_\\mathrm {c}$ ) from the part where the ground state corresponds to a topological superfluid (TSF, for $h>h_\\mathrm {c}$ ).",
"In particular, for $E_\\mathrm {b}<E_\\mathrm {b}^{(\\mathrm {c})}$ , only this second-order topological transition occurs.",
"However, beyond the point where the curve for $h_\\mathrm {c}(E_\\mathrm {b})$ crosses that of $h_>(E_\\mathrm {b})$ , solutions of the selfconsistency conditions that are critical, i.e., satisfy $h=\\sqrt{\\mu ^2 + |\\Delta |^2}$ , continue to exist but are no longer a global minimum of the ground-state energy at fixed $\\mu $ .",
"At the same time, the homogeneous-superfluid states existing for $h>h_>$ satisfy $h>\\sqrt{\\mu ^2 + |\\Delta |^2}$ and are thus in the topological phase.",
"Hence, beyond the crossing point of $h_\\mathrm {c}(E_\\mathrm {b})$ and $h_>(E_\\mathrm {b})$ , the topological transition is of first order.",
"The phase boundary of the homogeneous 2D TSF is therefore delineated by $h_{\\text{max}} (E_\\mathrm {b})=\\mathrm {max}\\lbrace h_\\mathrm {c}(E_\\mathrm {b}), h_>(E_\\mathrm {b})\\rbrace $ .",
"Due to the tendency of $|\\Delta |$ to monotonically decrease with $h$ in regions where a selfconsistent solution is associated with the system's equilibrium ground state (see Figs.",
"REF , REF , and REF ), $h_\\mathrm {max}$ is also the Zeeman energy for which $|\\Delta |$ is maximized in the TSF phase at fixed $E_\\mathrm {b}$ .",
"We now focus on the properties of the single-phase ground states adjacent to the phase-separation region at large $E_\\mathrm {b} > E_\\mathrm {b}^{(\\mathrm {c})}$ .",
"Figure: Expected BCS and BEC characteristics are exhibited in thenontopological-superfluid (NSF) phase of a spin-orbit-coupled 2D Fermigas.",
"All results plotted here have been obtained for fixed λk F /E F =0.75\\lambda k_\\mathrm {F}/E_\\mathrm {F} = 0.75, and k≡|k|k\\equiv |{\\mathbf {\\mathrm {k}}}| is themagnitude of the 2D wave vector of Bogoliubov quasiparticles.",
"Panels(a), (c) and (e) depict the BCS regime, showing results forE b /E F =0.010E_\\mathrm {b}/E_\\mathrm {F}=0.010 and h=h c -0.04E F h = h_\\mathrm {c} - 0.04\\,E_\\mathrm {F}.",
"Juxtaposed are panels (b), (d) and (f) associated withthe BEC regime, specifically for E b /E F =3.0E_\\mathrm {b}/E_\\mathrm {F}=3.0 andh=h < h = h_< (corresponding to the filled circle with maximum hh inFig. ).",
"The presence [absence] of Fermi-surfacefeatures in the momentum-space density distribution n k n_k displayed inpanel (a) [(b)] is typical for the BCS [BEC] regime of s-wavepairing.",
"Also the kk dependence of spin-resolved Bogliubovamplitudes shown in panels (c) and (d) exhibits the familiar pattern,with the spin-↑\\uparrow Nambu-particle component becoming dominant inthe BEC regime.",
"The quasiparticle dispersion E k+,< E_{{\\mathbf {\\mathrm {k}}}+, <} in the BCSregime [shown in panel (e)] has two minima corresponding to excitationgaps at finite kk but, as expected, the excitation gap is at k=0k=0 inthe BEC regime [see panel (f)].",
"Abrupt changes in the spin-resolvedBogoliubov amplitudes in the BCS regime [panel (c)] are the result ofsmall-gap anticrossings between highly spin-polarized dispersionbranches.",
"We illustrate this by showing also the purely Zeeman-splitdispersions |ϵ kσ -μ||\\epsilon _{{\\mathbf {\\mathrm {k}}}\\sigma }-\\mu |, which quite closely resemblethe full dispersions E k+,≶ E_{{\\mathbf {\\mathrm {k}}}+, \\lessgtr } in the BCS regime [panel(e)].",
"In contrast, E k+,< E_{{\\mathbf {\\mathrm {k}}}+, <} is qualitatively different from|ϵ k↑ -μ||\\epsilon _{{\\mathbf {\\mathrm {k}}}\\uparrow }-\\mu | in the BEC regime [panel (f)].Figure: Canonical features of chiral p-wave pairing persist in thetopological-superfluid (TSF) phase of a spin-orbit-coupled 2D Fermigas throughout the BCS-to-BEC crossover of the underlyings-wave pairing.",
"Results plotted here have been obtained forfixed λk F /E F =0.75\\lambda k_\\mathrm {F}/E_\\mathrm {F} = 0.75, and k≡|k|k\\equiv |{\\mathbf {\\mathrm {k}}}|is the magnitude of the 2D wave vector of Bogoliubov quasiparticles.Panels (a), (c) and (e) depict the BCS regime (E b /E F =0.010E_\\mathrm {b}/E_\\mathrm {F}=0.010 and h=h c +0.04E F h = h_\\mathrm {c} + 0.04 \\, E_\\mathrm {F}),whereas panels (b), (d) and (f) are associated with the BEC regime(E b /E F =3.0E_\\mathrm {b}/E_\\mathrm {F}=3.0 and h=h > h = h_>, corresponding tothe filled square with minimum hh in Fig. ).",
"Themomentum-space density distribution n k n_k has the same distinctiveFermi-surface feature in both the BCS and BEC regimes of the TSF[see panels (a) and (b)].",
"Panel (c) [(d)] shows the spin-resolvedparticle and hole probability densities for the lowest positive-energybranch E k+,< E_{{\\mathbf {\\mathrm {k}}}+,<} of Bogoliubov-quasiparticle excitations whoseenergy dispersion is the black solid curve in panel (e) [(f)].",
"Unlikein the BEC regime for the NSF [refer to Fig.",
"(d)],both spin-↑\\uparrow -particle and spin-↑\\uparrow -hole amplitudesdominate in the BEC regime of the TSF [depicted here in panel (d)].The purity of this realization of chiral-p-wave pairingcontrasts with the complicated pattern of the spin-resolved Bogoliubovamplitudes in the BCS regime [panel (c)], which is the result ofsmall-gap anticrossings between several highly spin-polarizeddispersion branches.",
"For illustration, panels (e) and (f) also showthe purely Zeeman-split dispersions |ϵ kσ -μ||\\epsilon _{{\\mathbf {\\mathrm {k}}}\\sigma }-\\mu |.Figure: Radius k FS k_\\mathrm {FS} of the Fermi surface emerging in thetopological-superfluid (TSF) phase of a spin-orbit-coupled 2D Fermigas with density n=mE F /(πℏ 2 )≡k F 2 /(2π)n = m E_\\mathrm {F}/(\\pi \\hbar ^2)\\equiv k_\\mathrm {F}^2/(2\\pi ) subject to s-wave pairing in the BEC regime, definedformally via the condition ().",
"The parameter λk F /E F \\lambda k_\\mathrm {F}/E_\\mathrm {F} measures the spin-orbit-coupling strength,E b E_\\mathrm {b} is the s-wave two-particle bound-state energy,and h max h_\\mathrm {max} corresponds to the critical Zeeman energy abovewhich the system's ground state is a uniform TSF.",
"The two circles(diamonds) indicate values predicted for k FS (h max )k_\\mathrm {FS}(h_\\mathrm {max}) by Eq.",
"(), which is expected to bevalid for λk F <h max \\lambda k_\\mathrm {F} < h_\\mathrm {max}, using λk F /E F =0.50\\lambda k_\\mathrm {F}/E_\\mathrm {F} = 0.50 (0.750.75) and E b /E F E_\\mathrm {b}/E_\\mathrm {F} associated with the correspondingly colored solid(dashed) line.",
"The orange star indicates the value based on theestimate k FS (h max )≈λk F /(2E F )k_\\mathrm {FS}(h_\\mathrm {max}) \\approx \\lambda k_\\mathrm {F}/(2 E_\\mathrm {F}) that is expected to apply when h max <λk F h_\\mathrm {max} <\\lambda k_\\mathrm {F} concomitantly with λk F /E F ≳1\\lambda k_\\mathrm {F}/E_\\mathrm {F} \\gtrsim 1, which is the case for the parameters of theorange dot-dashed curve.The typical phenomenology of the BCS-to-BEC crossover for s-wave pairing entails a shift of the dispersion minimum to $k=0$ , Bogoliubov quasi-particles becoming mostly particle-like, and the momentum-space density distribution loosing its typical Fermi-surface-like shape [36], [37], [38], [35], [39], [40].",
"This exact scenario is played out for our more complicated system of interest in the NSF phase.",
"See Fig.",
"REF and the extensive discussion in its caption.",
"In contrast, as illustrated by Fig.",
"REF , all features associated with effective chiral p-wave pairing in the spin-$\\uparrow $ channel remain present throughout the BCS-to-BEC crossover in the TSF phase.",
"In particular, the momentum-space density distribution shows a distinctive Fermi-surface edge feature even deep in the BEC regime for s-wave pairing, which is unexpected for situations where the chemical potential is negative See, e.g., Ref. [37].",
"Similar behavior to the one found by us here for the 2D TSF seems to also be implicit in results that were presented for the 3D spin-orbit-coupled Fermi superfluid (see, e.g., Fig.",
"6 in Ref.",
"[41]) but whose physical significance was not discussed.. That the effective p-wave pairing retains BCS-like behavior even as the underlying s-wave pairing is in the BEC regime is illustrated most strikingly by the close resemblance between the true Bogoliubov-quasiparticle energies and the dispersions associated with unpaired fermions [see Fig.",
"REF (f)].",
"Although much larger generically than in the BCS regime, $\\Delta _\\mathrm {pw}$ in the TSF phase for large $E_\\mathrm {b}/E_\\mathrm {F}$ is still small enough because of its dependence on the inverse of the Zeeman energy [see Eq.",
"(REF )] that the resulting quasiparticle dispersions are not radically different from those obtained in the absence of pairing.",
"This contrasts with the NSF phase occurring at lower $h$ for the same large value of $E_\\mathrm {b}/E_\\mathrm {F}$ where the unpaired-fermion dispersions are not at all representative of the lowest-energy branch of quasiparticle excitations [see Fig.",
"REF (f)].",
"The stabilization of the Fermi surface in the TSF phase due to the larger Zeeman energy is demonstrated in Fig.",
"REF .",
"Here we plot the $h$ dependence of the Fermi-surface radius $k_\\mathrm {FS}$ , where the latter is defined as the location of the crossing point of the spin-$\\uparrow $ -particle and spin-$\\uparrow $ -hole Bogoliubov-spinor magnitudes for the lowest-energy quasiparticle dispersion, $\\big | u^\\uparrow _{{\\mathbf {\\mathrm {k}}}+, <} \\big |^2_{|{\\mathbf {\\mathrm {k}}}| = k_\\mathrm {FS}} = \\big |v^\\uparrow _{{\\mathbf {\\mathrm {k}}}+, <} \\big |^2_{|{\\mathbf {\\mathrm {k}}}| = k_\\mathrm {FS}} \\quad .$ The condition $|{\\mathbf {\\mathrm {k}}}| = k_\\mathrm {FS}$ clearly defines a surface in wave-vector space that separates states having high and low occupation probabilities, which is the defining property of a Fermi surface [35].",
"We find that a crossing point yielding a definite value of $k_\\mathrm {FS}$ always exists at $h\\ge h_\\mathrm {max}$ for any values of $E_\\mathrm {b}/E_\\mathrm {F}$ and $\\lambda k_\\mathrm {F}/E_\\mathrm {F}$ .",
"For $\\lambda k_\\mathrm {F}/E_\\mathrm {F} < 1$ , the minimum in the dispersion curve $E_{{\\mathbf {\\mathrm {k}}}+,<}$ also occurs at $|{\\mathbf {\\mathrm {k}}}| = k_\\mathrm {FS}$ , and the latter's value turns out to be well-approximated by Eq.",
"(REF ) for $\\lambda k_\\mathrm {F} < h_\\mathrm {max}$ .",
"In situations with very large spin-orbit coupling $\\lambda k_\\mathrm {F}/E_\\mathrm {F} \\gtrsim 1$ , the dispersion minimum is observed to sometimes be absent or appear at $|{\\mathbf {\\mathrm {k}}}| \\ne k_\\mathrm {FS}$ right after the transition to the TSF phase.",
"Nevertheless, the coincidence of the quasiparticle-dispersion minimum and $k_\\mathrm {FS}$ is established for $h\\gtrsim h_\\mathrm {max}$ even in such cases.",
"Application of the approximate two-band-model results from Ref.",
"Brand2018 to the case $\\lambda k_\\mathrm {F} >h_\\mathrm {max}$ yields a conservative estimate for the Fermi-surface radius in this regime, which is given by $\\frac{k_\\mathrm {FS}}{k_\\mathrm {F}} = \\frac{\\lambda k_\\mathrm {F}}{2E_\\mathrm {F}} + \\sqrt{\\frac{\\mu }{E_\\mathrm {F}} + \\frac{\\lambda ^2k_\\mathrm {F}^2}{4 E_\\mathrm {F}^2} + \\frac{|\\Delta |^2}{E_\\mathrm {F}^2}\\left( \\frac{E_\\mathrm {F}}{\\sqrt{2} \\lambda k_\\mathrm {F}} -\\frac{E_\\mathrm {F}}{h} \\right)}$ and only holds when the expression under the square-root is positive.",
"According to results presented in Fig.",
"REF , $k_\\mathrm {FS}$ increases monotonically as a function of $h-h_\\mathrm {max}$ until reaching its asymptotic value $\\sqrt{2}\\,k_\\mathrm {F}$ , which corresponds to the Fermi-surface radius of a spin-polarized 2D Fermi gas with density $n\\equiv k_\\mathrm {F}^2/(2\\pi )$ .",
"As can be seen in Fig.",
"REF , the most visible attributes that distinguish the TSF in the BEC regime from that arising in the BCS regime are the increased magnitude of the low-energy excitation gap $\\Delta _\\mathrm {pw}$ and the strong suppression of the minority-spin degrees of freedom.",
"The clear dominance of the spin-$\\uparrow $ Bogoliubov amplitudes representing chiral p-wave pairing is one of the favorable qualities exhibited by the TSF realized in the BEC regime.",
"In addition, a larger magnitude of $\\Delta _\\mathrm {pw}$ should help to reduce the influence of many experimental nonidealities, including thermal fluctuations, as long as $E_\\mathrm {b}/E_\\mathrm {F}$ is not too large so that beyond-mean-field fluctuations have not yet significantly suppressed the value of the pairing gap.",
"Thus, the TSF realized in the onset of the BEC regime of the underlying s-wave pairing constitutes both a purer and more-robust version of the highly sought-after chiral p-wave order.",
"The results obtained and conclusions drawn in our work are based on the application of mean-field theory.",
"It is well-known that this method can only provide limited insight into the strongly interacting (i.e., the BEC) regime of 2D systems [46], [47], [48], [49], [39], [40], [50].",
"Here we employed the mean-field approach to determine (i) the phase diagram, (ii) the magnitude of the pairing gap, and (iii) momentum-space density distributions.",
"Before concluding, we discuss the reliability of our predictions for these three purposes.",
"(i) Phase diagrams: It is generally accepted that zero-temperature phase diagrams obtained within mean-field theory are qualitatively correct, even in the BEC regime [43], [44], [9], [45], [30].",
"We therefore expect the features presented in our work to be similarly accurate.",
"(ii) Pairing-gap magnitude: Suppression of the pairing gap by beyond-mean-field fluctuations becomes increasingly important for larger $E_\\mathrm {b}$ [49], [51].",
"Therefore, results for gap magnitudes presented, e.g., in Fig.",
"REF are only reliable for $E_\\mathrm {b}/E_\\mathrm {F}\\lesssim 1$ .",
"Nevertheless, the conclusion that $E_\\mathrm {b}/E_\\mathrm {F} \\sim 1$ is optimal for realizing a robust TSF continues to hold.",
"(iii) Momentum-space density distributions: Recent numerical results obtained for our system of interest in the $h=0$ limit (see Supplemental Material for Ref.",
"[40]) indicate that momentum-space density distributions obtained within mean-field theory are accurate to within $\\lesssim 10$ %.",
"Thus our general conclusions about the re-emergence of a Fermi surface and the robustness of chiral p-wave superfluidity in the BEC regime of s-wave pairing are expected to be valid."
],
[
"Conclusions and outlook",
"We have investigated the strongly interacting regime of the 2D Fermi gas with s-wave pairing, with fixed particle density and subject to both spin-orbit coupling and Zeeman spin splitting.",
"Characteristic features of the phase diagram as a function of two-particle binding energy $E_\\mathrm {b}$ and Zeeman energy $h$ are elucidated and the properties of the homogeneous superfluid phases studied in greater detail.",
"In particular, we tracked the boundaries of the homogeneous nontopological and topological superfluids.",
"The second-order topological-transition line $h_\\mathrm {c}\\big (E_\\mathrm {b}\\big )$ , with $h_\\mathrm {c}$ defined via Eq.",
"(REF ), is truncated by a phase-separation region that emerges for $E_\\mathrm {b}$ larger than a critical value $E_\\mathrm {b}^{(\\mathrm {c})}$ that depends on the spin-orbit-coupling strength (see Figs.",
"REF and REF ).",
"As a result, the topological transition is of first order in the limit of large $E_\\mathrm {b}/E_\\mathrm {F}$ .",
"The homogeneous nontopological phase exhibits all of the expected features commonly associated with the BCS-to-BEC crossover for s-wave pairing, especially the shrinking, and eventual disappearance, of an underlying Fermi surface as the Cooper-pair binding energy is increased.",
"See Fig.",
"REF (a,b).",
"In contrast, as illustrated in Fig.",
"REF , the topological superfluid phase always retains the basic properties of the BCS regime, including the Fermi-surface characteristics, even for large $E_\\mathrm {b}/E_\\mathrm {F}$ .",
"This effect demonstrates the continuity of topological protection through the BCS-to-BEC crossover.",
"The larger the value of $\\lambda k_\\mathrm {F}/E_\\mathrm {F}$ , the smaller is the Fermi-surface radius $k_\\mathrm {FS}$ at the transition point $h=h_\\mathrm {max}$ into the uniform topological phase.",
"With increasing $h>h_\\mathrm {max}$ , the Fermi surface is enlarged until its radius reaches the asymptotic value $\\sqrt{2} k_\\mathrm {F}$ expected for a spin-polarized 2D Fermi sea with density $n\\equiv k_\\mathrm {F}^2/(2\\pi )$ .",
"See Fig.",
"REF for an illustration.",
"Promising first steps have recently been made towards physical realization of our system of interest by demonstrating essential ingredients, e.g., in ultra-cold-atom gases [69] and solid-state heterostructures [70], [71].",
"State-of-the-art experimental techniques [72] could be utilized, or related theoretical proposals [73] may be pursued, to confirm the re-appearance of a Fermi surface as the Zeeman energy is tuned across the topological transition when the system is in the BEC regime of the underlying s-wave pairing.",
"Compared to the BCS regime, chiral p-wave superfluidity realized in the BEC regime has a larger excitation gap and is less obscured by minority-spin degrees of freedom, making it the ideal platform for exploring exotic Majorana excitations in vortices [18], [19], [20] and their potential use for topological quantum-information-processing paradigms [22].",
"Future work could focus on elucidating also the evolution and properties of topological superfluids within the phase-separation region.",
"U.Z.",
"thanks W. Belzig, C. Bruder, M. M. Parish, D. M. Stamper-Kurn, and O. P. Sushkov for useful discussions.",
"This work was supported by the Marsden Fund Council (contract no.",
"MAU1604), from NZ government funding managed by the Royal Society Te Apārangi."
]
] | 1906.04461 |
[
[
"Band Attention Convolutional Networks For Hyperspectral Image\n Classification"
],
[
"Abstract Redundancy and noise exist in the bands of hyperspectral images (HSIs).",
"Thus, it is a good property to be able to select suitable parts from hundreds of input bands for HSIs classification methods.",
"In this letter, a band attention module (BAM) is proposed to implement the deep learning based HSIs classification with the capacity of band selection or weighting.",
"The proposed BAM can be seen as a plug-and-play complementary component of the existing classification networks which fully considers the adverse effects caused by the redundancy of the bands when using convolutional neural networks (CNNs) for HSIs classification.",
"Unlike most of deep learning methods used in HSIs, the band attention module which is customized according to the characteristics of hyperspectral images is embedded in the ordinary CNNs for better performance.",
"At the same time, unlike classical band selection or weighting methods, the proposed method achieves the end-to-end training instead of the separated stages.",
"Experiments are carried out on two HSI benchmark datasets.",
"Compared to some classical and advanced deep learning methods, numerical simulations under different evaluation criteria show that the proposed method have good performance.",
"Last but not least, some advanced CNNs are combined with the proposed BAM for better performance."
],
[
"Introduction",
"h yperspectral images (HSIs) contain hundreds of near-continuous spectral bands and this benefits not only attracts the attention in the field of remote sensing, but also arouses great interest in some other fields.",
"With the development of hyperspectral sensors, the intelligent interpretation of HSIs has become an important research issue in the field of remote sensing.",
"However, this problem has not been well solved due to many factors.",
"One important reason for this is that the high-dimensional data of HSIs contains a considerable degree of noise and redundancy, although some traditional band selection methods based on prior knowledge or data characteristics have been extensively studied [1], [2].",
"The deep learning technique, represented by convolutional neural networks (CNNs) [3], has attracted extensive attention due to its good performance in recent years.",
"As a continuation of statistical machine learning [4], deep learning has powerful ability of data fitting under sufficient supervisory information.",
"As this technique drives to maturity, some CNNs based methods [5], [6], [7] have been able to match the accuracy of human recognition in some specific visual tasks.",
"Naturally, we think about using the capability of CNNs to solve the problem of HSI classification.",
"In [8], deep learning method was firstly applied to HSIs for classification.",
"The two-stream and 3D networks [9], [10] used for video processing in computer vision has been widely used in HSIs classification [11].",
"Because HSIs contain spectral dimension and spatial dimension [12], which is similar to the relationship between spatial dimension and temporal dimension in videos.",
"Although some fairly advanced algorithms have been applied to HSI classification, there are still two major problems of deep learning in HSIs.",
"One is the lack of weakly supervised algorithms, which is caused by the difficulty in obtaining data and labels of HSIs.",
"The other, also studied in this letter, is that the network models are mostly borrowed from the mainstream networks in RGB image processing rather than customized for HSIs.",
"The connotation of deep learning lies in that: models are constructed to adapt to the input data in different types, and the deep representations can be obtained through the model so as to get accurate generalization performance.",
"Therefore, it is necessary to design a network model which is highly compatible with the characteristics of HSIs.",
"Based on the above analysis, the purpose of this letter is to construct a classification network for HSIs, which can adapt to the problem of band redundancy.",
"Inspired by the attention mechanism [13], [14], [15], we propose a band attention based HSIs classification framework in this letter.",
"In detail, a band attention module (BAM) embedded in the classification network is proposed.",
"The proposed BAM obtains the global information through a series of convolutions and generates the required weight vector for the processing of input bands.",
"It aims to be a plug-and-play complementary component of the existing HSIs classification networks and also orthogonal and complementary to methods that focus on spatial attention [16].",
"For the input data, the proposed model firstly carries out band selection (or weighting, which is determined by the last activation before the weight vector is obtained) through BAM, then obtains the recognition results through the classification module.",
"The whole network is training end-to-end and the processing of bands shares supervisory information with image classification.",
"The validity of the proposed method is demonstrated by the numerical experiments on HSIs benchmark datasets.",
"The rest of this letter is organized as follows: The proposed strategies are listed in Sections II.",
"Experimental results are exhibited in Section III.",
"Conclusion and future directions are given in Section IV."
],
[
"Proposed Method",
"Fig.",
"1 shows the flowchart of the proposed band attention convolutional neural network (BACNN).",
"Figure: General flow chart of BACNN.",
"The part of red dotted line is unique to the proposed method compared with the ordinary CNNs.As shown in Fig.",
"1, compared with the ordinary CNN, the proposed BACNN has a BAM which can selectively select the input bands.",
"In this way, the adverse effects of redundancy and noise of the bands on CNN classification are reduced and the accuracy is improved.",
"Specifically, a $c\\times 1$ weight is obtained by the BAM for a $h\\times w\\times c$ (c is the number of input bands) HSI and the input is channel-wise multiplied with the weight to obtain the band-processed HSI.",
"Then we use a CNNs based network to classify the band-processed HSI.",
"We think that this end-to-end model has better adaptive ability than the phased classification methods."
],
[
"Band attention module",
"Attention mechanism has been widely used in image processing [14] since it can adaptively stimulate or suppress the input information.",
"Its core lies in infusing global information into the algorithm through the learning of an image mask, so as to accelerate the areas which are beneficial to improving accuracy.",
"We use the attention mechanism as a tool for choosing wanted bands.",
"The structure of BAM is depicted in Fig.",
"2.",
"Figure: The structure of the proposed BAM for attention based band processing.As shown in Fig.",
"2, we use a series of convolutions and sub-sampling.",
"The aim of these operations is to obtain the global information by reducing the resolution while expanding the receptive field and finally obtain the required weight vector.",
"The used BAM consists of five $3\\times 3$ 2D convolution layers which can be divided into three stages by two pooling layers, and each stage with the depth of 16, 32 and 32.",
"Then two 1D convolution layers are used for further nonlinear learning between channels.",
"Here has a hyperparameter $r$ to control the degree of information aggregation in 1D convolution layers.",
"The first 1D kernel should be of size $1\\times 1\\times 32\\times c/r$ and the second with $1\\times 1\\times c/r\\times c$ .",
"Thus, a $c$ dimensional weight vector with global information of bands is learned, which can be seen as the mask of bands.",
"Then the band mask is applied to the input HSI and the band-processed image can be obtained.",
"The forward propagation of BAM can be expressed as: $H_{out}=\\sigma _2(W_{22}\\sigma _1(W_{21}f_{globalpool}(W_1H_{in})))$ where $H_{in},H_{out}$ denote the input HSI and the output of BAM, $W_1,W_2$ represent 2D kernel matrix and 1D kernel matrix, $f_{globalpool}(\\cdot )$ is used to fully fuse the spatial information contained in the feature maps and provides the basis for forming the band masks, which can be defined as: $f_{globalpool}(x_z)=\\frac{1}{h^{\\prime } \\times w^{\\prime }}\\sum _{i=1}^{h^{\\prime }} \\sum _{j=1}^{w^{\\prime }} x_z(i,j)$ where $x_z$ means the $z$ th feature map and $h^{\\prime },w^{\\prime }$ denote its current hight and width.",
"$\\sigma _1$ is rectified linear unit (ReLu) [17].",
"Maybe not the optimal, $\\sigma _2$ is set to be sigmoid activation because this option performs better than other alternatives in the experiments.",
"This design is similar to the “squeeze-and-excitation block” in SENet [15].",
"The difference is that we use convolutions instead of the SE block's global pooling to reduce the spatial resolution of the input.",
"The reason for this change is that the BAM and SE block act on different objects.",
"For an image, the value of each pixel represents only physical meaning, but not the feature.",
"So using SE block with global pooling for band selection can not inject spatial global information.",
"Besides, the goal of SE block is to adaptively select useful features while suppressing less useful ones in channel dimension of the feature maps so SE block is embedded in every convolution layer in SENet.",
"Our aim is to process the input bands in order to reduce the side-effects of noise and redundancy in bands on classification.",
"Therefore, the BAM only appears once in the proposed BACNN.",
"In fact, the design of BAM implies the idea of re-weighting [18] in statistical robust learning.",
"Visual attention mechanism is essentially similar to the idea of re-weighting, the former is currently used in some deep learning models [13], while the latter is mostly used in shallow learning models.",
"This also proves that the BAM embedded classification model has better robustness to the noise of the input bands.",
"Depth fixed vanilla convolution is considered to observe the effectiveness of the BAM in this letter, while various depth settings and advanced convolution operations such as dilated [19], 3D [10] or depthwise separable [20] convolution also can be considered for better classification performance."
],
[
"BAM based HSI Classification",
"With the former module for band selection, a classification module (CM) is also essential to classify the processed HSIs.",
"It is worth to note that although the description is separate, BAM and CM are in the same network and training end-to-end.",
"This integration of separated components can share supervisory information and has better generalization performance.",
"In this letter, a basic CM based on VGGNet [5] is mainly used, which can be seen from Fig.",
"3.",
"Figure: The structure of the CM for classifying the band-processed HSIs.It can be seen from Fig.",
"3 that a eight layers VGGNet is chosen as the CM.",
"In most of the experiments in this letter, we use such a fairly simple classification network to classify the band-processed HSIs.",
"The reason for this is that our aim is to infuse the capability of band selection into existing classification networks, rather than to study a more advanced one.",
"A simple designed CM is enough to observe the improvement of accuracy after adding the proposed BAM.",
"Nevertheless, some advanced CNNs also can be involved to achieve better classification performance, including two-stream CNN [9], [12], [11], ResNet [6], [21], DenseNet [7] and so on."
],
[
"Implementation details",
"For the classical methods which require lots of engineering by hand, band selection or band weighting are two different stories.",
"In the proposed method, we can selectively do band selection or band weighting by changing the last activation function in the BAM.",
"There are several options: ReLu activation to achieve the weights with the value of zero or one for band selection; Sigmoid or softmax activation to let the weights in $[0,1]$ for band weighting.",
"In order to accelerate the convergence of training, we followed some mainstream designs: Before each convolution layer, we add a batch normalization layer [22] and a ReLU layer.",
"Adam optimization method [23] with the learning rate of 0.0001 is chosen to achieve good training.",
"Besides, $20\\%$ neurons of fully connected layers in CM are randomly discarded to prevent overfitting in the training stage."
],
[
"Datasets",
"Numerical experiments are carried out on two benchmark HSI datasets including Indian Pines dataset and Kennedy Space Center (KSC) dataset to evaluate the effectiveness of the proposed method.",
"Several state-of-the-art alternatives are chosen for comparison.",
"The experiment environment: PC with Intel i7-7700 CPU, Nvidia GTX-1060 GPU (6 GB memory), and 16 GB RAM.",
"Overall accuracy (OA), average accuracy (AA), and kappa coefficient are chosen as criteria to evaluate the performance in our experiments.",
"The Indian Pines dataset was acquired by the Airborne Visible/Infrared Imaging Spectrometer (AVIRIS) sensor over the Indian Pines test site in North-western Indiana and contains 200 bands after removing water absorption bands.",
"The image consists of $145\\times 145$ pixels and there are 16 classes of land covers.",
"10249 pixels are selected for manual labeling according to the ground truth map.",
"For the Indian Pines dataset, the number of labeled samples varies greatly among different classes.",
"The smallest class “Oats” with only 20 labeled pixels, and the largest class “Soybean-mintill” with 2455 labeled pixels.",
"The imbalance between categories undoubtedly brings difficulties for subsequent processing.",
"Thus, $30\\%$ samples of the classes with fewer samples and 80 samples of the richer classes are randomly chosen as training set, the remaining as testing set.",
"In addition, for the classes with fewer samples, replication operations have been carried out to mitigate the negative impact of imbalanced classification.",
"The false-color composite image and the corresponding ground reference map are demonstrated in Fig.",
"4.",
"The KSC dataset was acquired by the NASA AVIRIS sensor over the Kennedy Space Center, Florida.",
"The HSI contains 176 bands after removing water absorption and low SNR bands.",
"$512\\times 614$ pixels and 13 classes of land covers exist in the image.",
"The number of labeled samples in this dataset is roughly same among different classes and 5211 pixels are selected for manual labeling according to the ground truth map.",
"$10\\%$ of the total are randomly chosen as training set and the remaining as testing set.",
"The false-color composite image and the corresponding ground reference map are demonstrated in Fig.",
"5.",
"Figure: Indian Pines dataset.",
"(a) Three-channel false-color composition (bands 17, 27, and 50 for RGB).",
"(b) Ground truth map.Figure: KSC dataset.",
"(a) Three-channel false-color composition (bands 10, 19, and 28 for RGB).",
"(b) Ground truth map."
],
[
"Analysis of experimental results",
"Table I shows the classification results of the proposed BACNN and other classification methods on the Indian Pines dataset.",
"The size of the image slice is set to $15\\times 15$ during the experiments.",
"In order to test the effect of BAM on classification, we used an eight layers VGGNet as the CM in the experiments.",
"The CM in the table means directly use VGGNet for classification and not to do any processing for input bands.",
"The other three methods use different ways to select or weight the input bands, including SE block [15] (SE+CM), band weighting module [24] (BW+CM) and the proposed BAM (BAM+CM).",
"Although the band weighting module (BW) also uses the attention mechanism to adapt to HSIs, it only acts on the spectral dimension and ignores the abundant information in the spatial dimension.",
"In order to maintain the persuasiveness of the experiments, we keep a same depth and conduct ten times experiments for all involved models to eliminate randomness.",
"From Table I we know that the proposed BAM achieves the optimal classification performance on Indian Pines dataset.",
"The 1D-BW is not enough to improve the classification results, but its accuracy is inferior to that of 2D no band processing network.",
"Further, since the SE block is designed for feature but not for bands, it does not improve the classification accuracy of CM.",
"What's worse is that it degrades the CM by $1\\%$ of OA and $1.2\\%$ of Kappa.",
"The performance of these two shows that an improper band selection module not only fails to improve the performance of CM, but also has the opposite effect.",
"In contrast, the accuracy of CM has been improved to a certain extent after adding BAM since the BAM is not only tailored for HSI, but also considers spatial and spectral information.",
"The $2\\%$ -$3\\%$ improvement of each criterion confirms the validity of the proposed BAM.",
"Table: Classification Results on Indian Pines Dataset.",
"Numbers in the Parenthesis is the Standard Variances of the Accuracies Obtained in Repeated Experiments.Experiments on KSC datasets are carried out to further validate the proposed method.",
"The same experimental settings are retained and the detailed classification results can be seen in Table II.",
"The optimal results under each class and criterion are shown in bold among the table.",
"Table: Classification Results on KSC Dataset.",
"Numbers in the Parenthesis is the Standard Variances of the Accuracies Obtained in Repeated Experiments.Through the analysis of Table II, we can see that the trend of comparison between the accuracy of each method is similar to the former.",
"The proposed method still achieves the best classification results on KSC dataset.",
"Compared with the CM, the proposed method achieves $4\\%$ -$5\\%$ improvement under involved criteria.",
"Unlike before, although SE block does not consider the application background of band selection, its addition has also achieved a slight performance improvement.",
"However, the effect of SE block is still at least $2\\%$ less than it of BAM.",
"Based on the experimental results of the above two parts, we can see that the proposed BAM can be directly added to a CM without deliberate design to improve its performance.",
"Compared with the existing modules with similar roles, BAM can improve the accuracy more greatly.",
"Thus, the proposed BAM can be regarded as a plug-and-play supplementary component to most of the mainstream CNNs in HSIs classification."
],
[
"Effect of hyperparameters",
"In this section, we present a detailed analysis and evaluation of the influence of hyperparameters on the performance of the proposed method.",
"In the analysis, Indian Pines dataset and the VGGNet based CM are used to explore the changing trend of the classification accuracy.",
"The experimental results are shown in Fig.",
"4.",
"Figure: Effect of two hyperparameters epoch and rr (ratio of information aggregation in 1D convolution layers) of the proposed method on Indian Pines dataset.Fig.",
"4(a)-(b) shows the OA of the proposed method under different value of epoch and $r$ .",
"From Fig.",
"4(a), it can be seen that the loss of the proposed model changes slightly when the epoch is greater than 500, which means that the training process is close to convergence.",
"Besides, too many training times lead to the increase of the loss on both training and testing set, and reduce the testing accuracy to a certain extent.",
"Further, we can see from Fig.",
"4(b) that hyperparameter $r=0.5$ and $r=2$ obtain better classification results.",
"Based on the above empirical knowledge, the training set is reused 1000 times and the information compression ratio $r$ is set to 2 to achieve better performance in the experiments."
],
[
"Combination with advanced CNNs",
"In this section, we test the performance of some advanced CNNs combined with the BAM in order to further verify the general applicability of the proposed BAM, and also to obtain the better results of HSIs classification.",
"Four advanced backbones including two-stream CNN (TSCNN) [9], [11], the network with depthwise separable convolutions (Xception) [20], the network with residual connection (ResNet) [6] and the network with densely connection (DenseNet) [7] are chosen for testing.",
"Figure: Experimental results of different CNN backbones combined with the proposed BAM on Indian Pines and KSC datasets.As shown in Fig.",
"7, the backbone with the BAM achieves higher classification accuracy in both datasets, which proves that the BAM has wider applicability and can be used as a plug-and-play module to improve the performance of most models for HSI classification.",
"In this letter, we propose a novel deep learning based HSIs classification framework which fully considers the redundancy and noise in the band of HSIs.",
"A well-designed BAM is embedded in the ordinary CNN to implement an end-to-end network with the capability of band selection.",
"This module absorbs the experience of visual attention mechanism so it can adaptively stimulate the bands which are beneficial to the improvement of classification accuracy, while suppressing the invalid bands.",
"The proposed band attention based deep classification framework has better adaptability to the task of HSIs classification than the mainstream CNNs since it is customized according to the characteristics of HSIs.",
"Abundant numerical experiments not only reveal the influence of hyperparameters on classification accuracy, but also show that the proposed BAM can be a plug-and-play module to improve the accuracy of CNNs in HSIs classification."
]
] | 1906.04379 |
[
[
"Normal forms and invariant manifolds for nonlinear, non-autonomous PDEs,\n viewed as ODEs in infinite dimensions"
],
[
"Abstract We prove that a general class of nonlinear, non-autonomous ODEs in Fr\\'echet spaces are close to ODEs in a specific normal form, where closeness means that solutions of the normal form ODE satisfy the original ODE up to a residual that vanishes up to any desired order.",
"In this normal form, the centre, stable and unstable coordinates of the ODE are clearly separated, which allows us to define invariant manifolds of such equations in a robust way.",
"In particular, our method empowers us to study approximate centre manifolds, given by solutions of ODEs that are central up to a desired, possibly nonzero precision.",
"The main motivation is the case where the Fr\\'echet space in question is a suitable function space, and the maps involved in an ODE in this space are defined in terms of derivatives of the functions, so that the infinite-dimensional ODE is a finite-dimensional PDE.",
"We show that our methods apply to a relevant class of nonlinear, non-autonomous PDEs in this way."
],
[
"Background and motivation",
"Various invariant manifolds are central to many areas of dynamical systems, including using centre manifolds to construct and justify reduced low-dimensional models of high-dimensional dynamics [29].",
"Many dynamical systems involve pdes in infinite dimensional state spaces of functions, and some applications require infinite dimensional centre manifolds [7], [28].",
"In general we also want to cater for non-autonomous systems, with an aim to subsequently generalise to stochastic/rough dynamics [14].",
"Further, encompassing unstable dynamics with both centre and stable is necessary for application to Saint Venant-like, cylindrical, problems [18], [19], [20], and to deriving boundary conditions for approximate pdes [27].",
"Consequently, here we address the general challenge of constructing and justifying various infinite dimensional invariant manifolds for non-autonomous dynamical systems which have stable, unstable and centre modes.",
"A crucial novel feature of the approach is that we further develop a backward theory recently initiated for finite dimensional systems [30]: analogous backward theory has been very useful in other domains [16].",
"Applications of the extant forward theory in such a general setting is often confounded by impractical preconditions.",
"Non-autonomous invariant manifold theories typically require bounded operators, and Lipschitz and/or uniformly bounded nonlinearities, [3], [4], [5], [10], [17].",
"The extant boundedness requirement [17] arises from the general necessity of both forward and backward time convolutions with the semigroup (e.g., $e^{At}$ for systems that linearise to $\\dot{x}=Ax$ ), convolutions that must be continuous in extant forward theory, but cannot be continuous with unbounded operators.",
"Despite many interesting specific scenarios having rigorous invariant manifolds established via strongly continuous semigroup operators and by mollifying nonlinearity [9], [32], extant non-autonomous forward theory fails to rigorously apply in many practical cases.",
"Our main motivation for studying odes in infinite-dimensional vector spaces is their possible application to analysing invariant manifolds of pdes in finite space-time dimensions.",
"In that setting, the infinite-dimensional vector space in question is a space of functions, and the maps occurring in the ode are differential operators.",
"The linear part of such an ode is a linear partial differential operator, which typically is unbounded in applications.",
"Such an operator can be viewed as a bounded operator between different Banach spaces, with norms adapted to make the operator bounded.",
"(For example, the operator $\\frac{d}{dx}$ is unbounded on $L^2(\\mathbb {R})$ , but becomes bounded if we take its domain to be a first-order Sobolev space.)",
"Centre manifold theory in this setting was developed by several authors [17], [21], [33], and applied to pdes.",
"However, to achieve our goal of developing the desired backward theory, and robustly constructing invariant manifolds via coordinate transformations to approximate normal forms, we need to go beyond this setting.",
"This essentially boils down to the fact that a bounded operator on a single Banach space can be iterated to yield new bounded operators, whereas this is not possible for a bounded operator between different Banach spaces.",
"The necessity of iterating operators in our constructions leads us outside the setting of Banach spaces, to graded Fréchet spaces: intersections of infinite sequences of Banach spaces connected by bounded inclusion maps.",
"These include spaces relevant to the study of pdes, such as spaces of smooth functions.",
"Figure: schematic diagram: blue, new theory and practice established by this article; magenta, for future research; black, mostly established extant theory and practice; red, practically unattainable (in general)."
],
[
"Results",
"The first step in the proposed backward theory is to establish an approximate conjugacy between a given system and a `nearby' system for which we know its invariant manifolds, by its construction.",
"fig:scheme illustrates what this article achieves.",
"Planned future research will then provide novel finite domain and error bounds as illustrated in fig:scheme.",
"That is, instead of proving that there exists a reduced dimensional manifold for a specified system, which is then approximately constructed, our main results, thm normal form,cor special case, establish that there is a system which is both `close' to the specified system, and also has a reduced dimensional manifold which we know exactly.",
"In essence, we invoke an (extended) normal form coordinate transform—related to Hartman–Grobman theory [3], [4], [5]—and use it from a new point of view.",
"An intuitive formulation of our main result on the existence of such a normal form is the following thm normal form intro.",
"thm normal form is a precise formulation, and cor special case is a special case that applies to nonlinear, non-autonomous pdes.",
"(In that special case, the space $V$ is a space of functions, and the maps that occur are partial differential operators.)",
"2 Theorem 1.1 (Normal form theorem, intuitive formulation) Let $I$ be an interval in $t$ , and $V$ a possibly infinite-dimensional vector space.",
"Consider a nonlinear, non-autonomous ode $ \\dot{x}(t) = Ax(t) + f(t, x(t))$ for $x\\colon I \\rightarrow V$ , where $A$ is a linear operator on $V$ (independent of $t$ ), and $f \\colon I \\times V \\rightarrow V$ and its derivative vanish on $I \\times \\lbrace 0\\rbrace $ .",
"For each $p\\ge 2$ , there are both an ode $ \\dot{X}(t) = AX(t) + F_p(t, x(t))$ for $X\\colon I \\rightarrow V$ , and a time-dependent coordinate transformation $\\xi _p\\colon I \\times V \\rightarrow V$ such that if $X(t)$ satisfies eq normal form intro, then setting $x(t) = \\xi _p(t, X(t))$ defines a solution of eq ODE intro up to a residual term that vanishes to order $p$ ; the map $F_p\\colon I \\times V \\rightarrow V$ is of order 2 in its second entry $X$ ; the component of a solution to eq normal form intro in the stable subspace for $A$ decays exponentially quickly to zero as $t$ increases in $I$ .",
"Its component in the unstable subspace for $A$ decays exponentially quickly to zero as $t$ decreases in $I$ .",
"If the solution starts out in either the centre-stable or the centre-unstable subspace for $A$ , then its component in the central subspace for $A$ is bounded by a constant for all $t \\in I$ , or at worst by a specified, small exponential growth rate.",
"The three most important things to make more precise in this intuitive formulation is what `order $p$ ' means, the related question what topology on $V$ is used, and what kinds of maps $A$ , $f$ , $\\xi _p$ and $F_p$ are.",
"The last point in thm normal form intro means that the centre, stable and unstable manifolds (in this case, linear subspaces) of eq normal form intro are exactly the centre, stable and unstable spaces of $A$ , respectively.",
"(And similarly for the centre-stable and centre-unstable subspaces.)",
"The centre, stable, unstable, centre-stable and centre-unstable subspaces for the dynamics in $x$ described by eq normal form intro and $x(t) = \\xi _p(t, X(t))$ (which becomes an ode in $x$ if $\\xi _p(t, )$ is invertible for all $t$ ) are then obtained from these spaces via an application of the coordinate transform $\\xi _p$ .",
"In this way, we show that any (non-autonomous) system of the form eq ODE intro is arbitrarily close to a system with robustly defined invariant manifolds.",
"This definition of these key invariant manifolds is a crucial reformation of the backward theory proposed.",
"Classic definitions of un/stable and centre manifolds require the existence of limits as time goes to $\\pm \\infty $ [2], [6], [17], [26].",
"This consequently requires solutions of the dynamical system to be well-behaved for all time, which requires constraints that in applications are often not found, or are hard to establish.",
"For example, in stochastic systems very rare events will eventually happen over the infinite time requiring global Lipschitz and boundedness that are oppressive in applications.",
"By modifying definitions we establish results for finite times, which are useful in many applications, and for a wider range of non-autonomous systems."
],
[
"Ingredients of the proof",
"The key ingredients of the proofs of our main results are nested sequences of Banach spaces, whose intersections are graded Fréchet spaces; compact polynomial maps between Banach spaces and graded Fréchet spaces; and compactly differentiable maps between such spaces."
],
[
"Sequences of Banach spaces",
"It is important to specify what is meant by a nearby infinite-dimensional mathematical system in fig:scheme.",
"Intuitively, we mean by this that solutions of the nearby system eqs:nfIntro are solutions of the original system eq ODE intro up to any desired order in the magnitude of such a solution, as in thm normal form intro.",
"To be more precise about what that this magnitude is, we need to specify norms or seminorms on spaces containing these solutions.",
"Working with a single Banach space (i.e., a single norm) is too restrictive for applications.",
"This is because in applications to pdes, the maps $A$ and $f$ in thm normal form intro are generally not continuous maps from a single Banach space to itself.",
"This could be remedied by allowing maps between two different Banach spaces, but that would not let us iteratively apply maps involving $A$ and $f$ , which we do in the proof of thm normal form intro.",
"A type of space that is both general enough to apply to various nonlinear pdes, while being close enough to Banach spaces to allow us to define a meaningful notion of a solution of an equation up to a given order, is what are often called graded Fréchet spaces.",
"These are intersections of sequences of Banach spaces, each with a bounded inclusion map into the next.",
"The notion of an operator of a given order on a graded Fréchet space is then defined in terms of the norms on these Banach spaces, see def On.",
"For several convergence questions, it would be useful if the Banach spaces that occur in the definition of a graded Fréchet space are Hilbert spaces.",
"Then we can use orthogonality, for example.",
"However, for applications to nonlinear pdes, it is not enough to use Hilbert spaces.",
"For example, a nonlinear term $u \\mapsto u^2$ is a well-defined (and differentiable) map from the Banach space $L^4(\\mathbb {R})$ to the Hilbert space $L^2(\\mathbb {R})$ .",
"To be able to use Hilbert space techniques in such settings, we use the notion of nested sequences of Banach space that are comparable to nested sequences of Hilbert spaces (see def comparable).",
"This effectively means that a graded Fréchet space that naturally occurs as an intersection of Banach spaces can equivalently be presented as an intersection of Hilbert spaces.",
"Proving such a property in situations relevant to pdes involves the relevant Sobolev embedding theorems."
],
[
"Compact polynomial maps",
"We construct coordinate transformations on graded Fréchet spaces to bring odes in such spaces into normal forms that allow us to define invariant manifolds directly and robustly.",
"These transformations are polynomial maps, which we construct by adding (infinitely many) monomial terms with the right properties together.",
"At the level of Banach spaces, a polynomial map can be naturally defined as a finite sum of restrictions to the diagonal of bounded multilinear maps.",
"For example, Taylor polynomials of differentiable maps between normed vector spaces are polynomials of this type.",
"But not all such polynomial maps (for example, the identity map) can be approximated by sums of monomials.",
"This leads us to define compact polynomial maps, which can be approximated in this way in settings relevant to us.",
"The notion of a compact polynomial map that we use seems natural, but we have not been able to find it elsewhere in the literature.",
"Different notions of compact polynomial maps were developed and studied by Gonzalo, Jaramillo and Pełczyńsky [15], [25]."
],
[
"Compactly differentiable maps",
"Our construction of the required coordinate transforms involves Taylor polynomials of differentiable maps between Banach spaces, and between graded Fréchet spaces.",
"This construction is possible if those polynomials are compact in the sense just mentioned.",
"That is the case for compactly differentiable maps, which we define for this purpose.",
"We will see in sec ex f that, in applications to pdes, the relevant differentiable maps are indeed compactly differentiable.",
"This follows from various Sobolev embedding theorems."
],
[
"Outline of this paper",
"The main results of this paper, on normal forms and invariant manifolds of nonlinear, non-autonomous odes in Fréchet spaces, and of nonlinear, non-autonomous pdes in finite-dimensional spaces, are stated in secPrelim.",
"We illustrate our results by applying them to an example pde in sec ex.",
"In the rest of the paper, we prove our main results.",
"We start by reviewing standard material on differentiable maps and polynomials on normed spaces in sec der.",
"In sec cpt der,sec Frechet, we develop technical tools we need for our proofs.",
"Then in sec co xform,sec transf choice, we use these tools to prove the main thm normal form,thm special case.",
"We prove some properties of the normal form equation, which allow us to identify its invariant manifolds, in sec dynamics.",
"A key ingredient in the proof of a version Taylor's theorem for compactly differentiable maps between Fréchet spaces, mentioned above, is the fact that a compact operator from a Banach space with the approximation property into another Banach space can be approximated by finite-rank operators in a suitable way.",
"This is reviewed in secCompFinite."
],
[
"Notation and conventions",
"We write $\\mathbb {N}$ for the set of positive integers, and $\\mathbb {N}_0$ for the set of nonnegative integers.",
"We write $\\mathbb {N}_0^{\\infty }$ for the set of sequences in $\\mathbb {N}_0$ with finitely many nonzero entries, interpreted as multi-indices.",
"For $q \\in \\mathbb {N}_0^{\\infty }$ , or in $\\mathbb {N}_0^n$ , we denote the (finite) sum of its elements by $|q|$ .",
"We denote spaces of bounded linear operators by the letter $\\mathcal {B}$ , and spaces of compact linear operators by the letter $\\mathcal {K}$ .",
"When we mention a normed vector space $V$ , the implicitly given norm is denoted by $\\Vert \\cdot \\Vert _V$ .",
"Similarly, if $V$ is an inner product space, then the inner product is denoted by $(, )_V$ .",
"Inner products on complex vector spaces are assumed to be linear in their second entries, and antilinear in their first entries.",
"For maps $f,g\\colon V \\rightarrow \\mathbb {R}$ , when we write $f(v) = O(g(v))$ , we implicitly mean that $f(v) = O(g(v))$ as $v \\rightarrow 0$ in $V$ .",
"If $V$ is a normed vector space, and $I$ is an open interval, and $f\\colon I \\rightarrow V$ and $f_j\\colon I \\rightarrow V$ , for $j \\in \\mathbb {N}$ , are maps, then we say that $f_j$ converges to $f$ if $f_j(t)$ converges to $f(t)$ in $V$ uniformly in $t$ in compact subsets of $I$ .",
"If $f$ and $f_j$ are smooth, then we say that $f_j$ converges to $f$ differentiably in $t$ if $f_j^{(n)}$ converges to $f^{(n)}$ for every $n \\in \\mathbb {N}_0$ , in this sense.",
"For maps $f,g\\colon I \\times V \\rightarrow V$ and $h\\colon V \\rightarrow V$ , the maps $f\\circ g\\colon I \\times V \\rightarrow V$ and $f\\circ h\\colon I\\times V \\rightarrow V$ are defined by $(f\\circ g)(t,v):= f(t, g(t,v)), \\quad (f\\circ h)(t,v):= f(t, h(v)),$ for $t \\in I$ and $v \\in V$ .",
"If $\\Omega $ is an open subset of $\\mathbb {R}^d$ and $m \\in \\mathbb {N}$ , then the Sobolev space of functions from $\\Omega $ to $\\mathbb {R}^m$ with weak derivatives up to order $k$ in $L^p$ is denoted by $W^{k,p}(\\Omega ; \\mathbb {R}^m)$ .",
"The norm on this space is $\\Vert u\\Vert _{W^{k,p}} := \\sum _{\\alpha \\in \\mathbb {N}_0^n; |\\alpha |\\le k} \\left\\Vert \\frac{\\partial ^{\\alpha } u}{\\partial x^{\\alpha }}\\right\\Vert _{L^p}.$ If $m = 1$ , we write $W^{k,p}(\\Omega ) = W^{k,p}(\\Omega ; \\mathbb {R})$ ."
],
[
"Preliminaries and results",
"Our main result, thm normal form, asserts that a broad class of nonlinear pdes and odes in infinite-dimensional vector spaces may be effectively approximated by normal form systems via well-chosen, time-dependent, coordinate transformations.",
"In this normal form, the centre, stable and unstable components of the pde and ode are clearly separated, which allows us to define centre manifolds for this class of equations in a robust way (def centre mfd).",
"We first state our result on normal forms, and the definition of centre manifolds, for odes in a class of abstract vector spaces (sec result).",
"Our main reason for developing this theory is to apply it to the study of pdes, for which the vector spaces used are spaces of functions, and the relevant maps between them are defined in terms of derivatives of functions.",
"We discuss a relevant class of examples of such function spaces and maps in sec special."
],
[
"Nested sequences of Banach spaces",
"The normal form we obtain in thm normal form is approximate in the sense that functions satisfying an equation transformed into that form satisfy the original equation up to a residual term.",
"An important point in thm normal form is that this residual vanishes up to a specified order.",
"To make it precise what this vanishing up to a certain order means, we introduce the type of topological vector spaces we consider in this subsection.",
"More details about these spaces and their properties are given in sec Frechet.",
"A concrete class of examples of these spaces relevant to the study of pdes is given in sec special.",
"2 Definition 2.1 By a nested sequence of Banach spaces, we mean a sequence $\\lbrace V_k\\rbrace _{k=1}^{\\infty }$ of Banach spaces such that for every $k$ , $V_{k+1} \\subset V_k$ , where the inclusion map is bounded, and the intersection $V_{\\infty }:=\\bigcap _{l=1}^{\\infty }V_l$ is dense in $V_k$ for every $k\\in \\mathbb {N}$ .",
"We then consider $V_{\\infty }$ as a Fréchet spaceMuch of what we write about Fréchet spaces of this form holds for more general projective limits of Banach spaces connected by bounded operators.",
"But we do not need that degree of generality.",
"with the seminorms (now actual norms) that are the restrictions of the norms on the spaces $V_k$ .",
"A compactly nested sequence of Banach spaces is such a sequence such that for every $l \\in \\mathbb {N}$ , there is a $k \\ge l$ such that the inclusion $V_k \\subset V_l$ is compact.",
"A Fréchet space $V_{\\infty }$ as in this def nested seq is often called a graded Fréchet space.",
"Let $\\lbrace V_k\\rbrace _{k=1}^{\\infty }$ be a nested sequence of Banach spaces.",
"2 Definition 2.2 The space $\\mathcal {B}(V_{\\infty })$ of bounded operators on $V_{\\infty }$ consists of the linear maps $A \\colon V_{\\infty } \\rightarrow V_{\\infty }$ such that for every $l \\in \\mathbb {N}$ , there is a $k \\in \\mathbb {N}$ such that the linear map $A$ extends continuously to a map in $\\mathcal {B}(V_k, V_l)$ .",
"2 Remark 2.3 In def BVinfty, if $k \\le l$ , then the composition $V_l \\hookrightarrow V_k \\xrightarrow{} V_l$ is a bounded operator on $V_l$ .",
"So we may always take $k \\ge l$ in this context, but this does not need to be assumed a priori.",
"Similar remarks apply in analogous situations, such as def On,def Vinfty diffble below.",
"2 Definition 2.4 A map $f\\colon V_{\\infty } \\rightarrow V_{\\infty }$ is of order $n$, written as $f = \\mathcal {O}(n)$ , if for every $l \\in \\mathbb {N}$ , there is a $k \\in \\mathbb {N}$ such that $\\Vert f(v)\\Vert _{V_l} = O(\\Vert v\\Vert _{V_k}^n)$ as $v\\rightarrow 0$ in $V_k$ .",
"If $I$ is an open interval, a map $f\\colon I \\times V_{\\infty } \\rightarrow V_{\\infty }$ is of order $n$, written as $f = \\mathcal {O}(n)$ , if for every $l \\in \\mathbb {N}$ , there is a $k \\in \\mathbb {N}$ such that $\\Vert f(t, v)\\Vert _{V_l} = O(\\Vert v\\Vert _{V_k}^n)$ as $v\\rightarrow 0$ in $V_k$ , uniformly in $t$ in compact subsets of $I$ .",
"2 Definition 2.5 An $n$ times differentiable map from $V_{\\infty }$ to itself is a map $f\\colon V_{\\infty } \\rightarrow V_{\\infty }$ such that for every $l \\in \\mathbb {N}$ , there is a $k \\in \\mathbb {N}$ such that $f$ extends to an $n$ times differentiable map from $V_k$ to $V_l$ .",
"If a map is $n$ times differentiable for every $n \\in \\mathbb {N}$ , then it is infinitely differentiable.",
"Basic material on differentiable maps between normed vector spaces is reviewed in sec der.",
"2 Definition 2.6 Two nested sequences $\\lbrace V_k\\rbrace _{k=1}^{\\infty }$ and $\\lbrace W_k\\rbrace _{k=1}^{\\infty }$ of Banach spaces are comparable if for every $k \\in \\mathbb {N}$ , there are $l_1, l_2, l_3, l_4 \\in \\mathbb {N}$ such that we have bounded inclusions $V_{l_1} \\subset W_k \\subset V_{l_2}$ and $W_{l_3} \\subset V_k \\subset W_{l_4}$ .",
"In the setting of this definition, $V_{\\infty } = W_{\\infty }$ ."
],
[
"Setup and goal",
"Let $\\lbrace V_k\\rbrace _{k=1}^{\\infty }$ be a compactly nested sequence of Banach spaces, such that $V_1$ is a Hilbert space.",
"Let $A \\in \\mathcal {B}(V_{\\infty })$ .",
"Let $I \\subset \\mathbb {R}$ be an open interval in $t$ containing $t=0$ , and let $f\\colon I \\times V_{\\infty } \\rightarrow V_{\\infty }$ be infinitely differentiable with respect to $V_{\\infty }$ and $I$ .",
"Suppose that $f = \\mathcal {O}(2)$ , and that for every $l \\in \\mathbb {N}$ , there is a $k \\in \\mathbb {N}$ such that $f\\colon V_k \\rightarrow V_l$ is differentiable, and $ \\Vert f^{\\prime }_{V_{\\infty }}(t,v)\\Vert _{\\mathcal {B}(V_k, V_l)} = O(\\Vert v\\Vert _{V_k}),$ uniformly in $t$ in compact subsets of $I$ .",
"Suppose that $\\lbrace e_j\\rbrace _{j=1}^{\\infty } \\subset V_{\\infty }$ is a set of eigenvectors of $A$ which is a Hilbert basis of $V_1$ (an orthonormal set that spans a dense subspace of $V_1$ ).",
"We assume that the sequence $\\lbrace V_k\\rbrace _{k=1}^{\\infty }$ is comparable to a nested sequence of separable Hilbert spaces in which the vectors $e_j$ are orthogonal.",
"However, we will see in rem comparable unnecessary that we may equivalently make the seemingly stronger but more concrete assumption that the spaces $V_k$ themselves are separable Hilbert spaces.",
"The assumption that $\\lbrace V_k\\rbrace _{k=1}^{\\infty }$ is comparable to a nested sequence of separable Hilbert spaces is easier to check in practice than the condition that every space $V_k$ can be chosen to be a separable Hilbert space itself.",
"For example, sec special discusses a class of relevant cases where the spaces $V_k$ are not Hilbert spaces for $k \\ge 2$ .",
"In this sense, the notion of comparable sequences of Banach spaces is a tool that makes it easier to check the conditions of thm normal form.",
"We study smooth maps $x\\colon I \\rightarrow V_{\\infty }$ satisfying the non-autonomous dynamical system differential equation $ \\dot{x}(t) = Ax(t) + f(t, x(t))\\quad \\text{for all }t \\in I\\,.$ Since the nonlinearity $f$ satisfies eq est der f, $x=0$ is an equilibrium of the system eq ODE.",
"We provide a novel backward approach to establish invariant manifolds in a finite domain about the equilibrium $x=0$ .",
"For these invariant manifolds to be useful in applications, the time interval $I$ will be long enough for transient dynamics to decay to insignificance in the context of the application.",
"The proofs of our main results simplify considerably if the time interval $I$ is short, or bounded.",
"But we emphasise that we only aim this theory to support the many applications where the time interval $I$ is long enough, or unbounded, so that the theorems are non-trivially useful in the application."
],
[
"Dynamics in a normal form",
"We define invariant manifolds, or sets, for dynamical systems in a particular normal form, and show that this definition captures the essence of such manifolds.",
"In sec result, we show that a very general class of odes of the form eq ODE can be brought into this normal form, modulo residuals that vanish to a desired order."
],
[
"Spectral gap in an exponential trichotomy",
"Let $\\alpha , \\beta , \\gamma ,\\tilde{\\mu }$ be such that $0\\le \\alpha <\\tilde{\\mu }< \\min (\\beta ,\\gamma )$ , and no eigenvalues of $A$ have real parts in the intervals $(-\\beta , -\\alpha )$ and $(\\alpha , \\gamma )$ (depending upon the circumstances, $\\beta $ or $\\gamma $ could be $\\infty $ , and/or $\\alpha $ may be zero).",
"For every $j \\in \\mathbb {N}$ , let $\\alpha _j$ be the eigenvalue of $A$ corresponding to $e_j$ .",
"With respect to the parameters $\\alpha $ , $\\beta $ and $\\gamma $ , we define the sets of indices of central, stable and unstable eigenvalues and eigenvectors, respectively, as $J_c &:= \\lbrace j \\in \\mathbb {N}: |\\Re (\\alpha _j)| \\le \\alpha \\rbrace ;\\\\J_s &:= \\lbrace j \\in \\mathbb {N}: \\Re (\\alpha _j) \\le -\\beta \\rbrace ;\\\\J_u &:= \\lbrace j \\in \\mathbb {N}: \\Re (\\alpha _j) \\ge \\gamma \\rbrace .$ For $a = c,s,u$ , let $V_a$ be the closure in $V_{1}$ of the span of the eigenvectors $e_j$ , for $j \\in J_a$ .",
"For any map $g$ into $V_{1}$ and $a \\in \\lbrace c,s,u\\rbrace $ , we write $g_a$ for its components in $V_a$ .",
"The sets $V_c,V_s,V_u$ are respectively called the centre/stable/unstable subspaces.",
"Further, we define the centre-stable subspace $V_{cs}:=V_c\\oplus V_s$ , and the centre-unstable subspace $V_{cu}:=V_c\\oplus V_u$ .",
"For $v \\in V_{\\infty }$ and a multi-index $q \\in \\mathbb {N}_0^{\\infty }$ , we set $v^q := \\prod _{j=1}^{\\infty } ({e_j},v)_{V_1}^{q_j}.$ For $q \\in \\mathbb {N}_0^{\\infty }$ , write $q = q^c + q^s + q^u$ , for $q^c, q^s, q^u \\in \\mathbb {N}_0^{\\infty }$ such that $q^c_j = 0$ if $j \\notin J_c$ , $q^s_j = 0$ if $j \\notin J_s$ and $q^u_j = 0$ if $j \\notin J_u$ ."
],
[
"Normal form dynamics",
"2 Definition 2.7 A smooth map $F\\colon I \\times V_{\\infty } \\rightarrow V_{\\infty }$ separates invariant subspaces if the components of $F$ in $V_c$ , $V_s$ and $V_u$ are of the forms $F_c (t,v)&= \\sum _{\\parbox {5em}{\\scriptsize \\raggedright q \\in \\mathbb {N}_0^{\\infty }: |q| \\le pand q^s = q^u=0or q^s \\ne 0\\ne q^u}}F^q(t) v^q, \\\\F_s (t,v)&= \\sum _{\\parbox {5em}{\\scriptsize \\raggedright q \\in \\mathbb {N}_0^{\\infty }: |q| \\le pand q^s \\ne 0}}F^q(t) v^q, \\\\F_u (t,v)&= \\sum _{\\parbox {5em}{\\scriptsize \\raggedright q \\in \\mathbb {N}_0^{\\infty }: |q| \\le pand q^u \\ne 0}}F^q(t) v^q,$ for all $t \\in I$ and $v \\in V_{\\infty }$ , for smooth maps $F^q \\colon I \\rightarrow V_{\\infty }$ , where the series converge in $\\operatorname{Pol}(V_{\\infty })$ , differentiably in $t$ .",
"Consider a polynomial map $F\\colon I \\times V_{\\infty } \\rightarrow V_{\\infty }$ that separates invariant subspaces, and the ode $ \\dot{X}(t) = AX(t) + F(t, X(t)),$ in smooth maps $X\\colon I \\rightarrow V_{\\infty }$ .",
"Because $F$ separates invariant subspaces, this ode has very explicit invariant manifolds, by lem Vj invar and prop dynamics below.",
"2 Lemma 2.8 Suppose that $X\\colon I \\rightarrow V_{\\infty }$ satisfies eq ODE XYZ, where $F$ separates invariant subspaces.",
"Let $a \\in \\lbrace c,s,u\\rbrace $ .",
"If there exists a $t \\in I$ such that $X(t) \\in V_a$ , then $X(t) \\in V_a$ for all $t \\in I$ .",
"2 Proposition 2.9 There is a neighbourhood $D_{\\tilde{\\mu }}$ of $I \\times \\lbrace 0\\rbrace $ in $I \\times V_{\\infty }$ , with the following property.",
"Let $X\\colon I \\rightarrow V_{\\infty }$ be a solution of eq ODE XYZ, for some open interval $I$ containing 0, and where $F$ sepearates invariant subspaces.",
"Write $X = X_c + X_s + X_u$ , with $X_a \\in V_a$ for $a = c,s,u$ .",
"If $(t, X(t)) \\in D_{\\tilde{\\mu }}$ for all $ t\\in I$ with $t\\ge 0$ , then for all such $t$ , $\\Vert X_s(t)\\Vert _{V_1} \\le \\Vert X_s(0)\\Vert _{V_1} e^{-(\\beta - \\tilde{\\mu })t}$ .",
"If $(t, X(t)) \\in D_{\\tilde{\\mu }}$ for all $t \\in I$ with $t\\le 0$ , then for all such $t$ , $\\Vert X_u(t)\\Vert _{V_1} \\le \\Vert X_u(0)\\Vert _{V_1} e^{(\\gamma -\\tilde{\\mu })t}$ .",
"Suppose that $X_s(0) = 0$ or $X_u(0)=0$ .",
"If $(t, X(t)) \\in D_{\\tilde{\\mu }}$ for all $t \\in I$ , then for all $t \\in I$ , $\\Vert X_c(t)\\Vert _{V_1} \\le \\Vert X_c(0)\\Vert _{V_1} e^{(\\alpha + \\tilde{\\mu }) |t|}$ .",
"Since $\\beta - \\tilde{\\mu }$ and $\\gamma - \\tilde{\\mu }$ are positive, this proposition in particular states that stable solutions decrease to zero exponentially quickly as $t$ increases in $I$ , while unstable solutions decrease to zero exponentially quickly as $t$ decreases in $I$ .",
"The numbers $\\alpha $ and $\\tilde{\\mu }$ represent bounds on what one takes to be relatively small real parts of eigenvalues of $A$ (classically, these numbers are zero), so that the third point in prop dynamics intuitively states that central solutions, at worst, only grow relatively slowly as $|t|$ increases.",
"lem Vj invar,prop dynamics are proved in sec dynamics.",
"The specific form of the set $D_{\\tilde{\\mu }}$ is also specified there, see eq Dmu."
],
[
"Invariant manifolds",
"lem Vj invar,prop dynamics show that, for every $a=c,s,u$ , the set $D_{\\tilde{\\mu }} \\cap (I\\times V_a)$ is a centre, stable or unstable submanifold of $I \\times V_{\\infty }$ for eq ODE XYZ, respectively.",
"Furthermore, for $a = cs$ and $a = cu$ , we obtain centre-stable and centre-unstable manifolds, respectively.",
"(Here we use the cases of the third point in prop dynamics where $X_u(0)=0$ and $X_s(0)=0$ , respectively.)",
"This motivates def centre mfd of invariant subspaces of dynamical systems of a certain form.",
"To state it precisely, we incorporate existence of solutions of eq ODE XYZ.",
"For $v \\in V_{\\infty }$ , we write $a_v$ for the infimum of the set of all $a > 0$ such that there is a solution $X: (-a, 0] \\rightarrow V_{\\infty }$ of eq ODE XYZ, with $X(0)=v$ .",
"Similarly, $b_v$ is the supremum of the set of all $b > 0$ such that there is a solution $X: [0,b) \\rightarrow V_{\\infty }$ of eq ODE XYZ, with $X(0)=v$ .",
"If such $a$ and $b$ exist, we set $I_v := (-a_v, b_v)$ .",
"(In particular, $I_v = \\mathbb {R}$ if such a solution exists for all $a,b > 0$ .)",
"If such an $a$ exists but no $b$ , we set $I_v := (-a_v, 0)$ , and if such a $b$ exists but no $a$ , we set $I_v := (0, b_v)$ .",
"If there are no such $a,b > 0$ , we set $I_v := \\emptyset $ .",
"2 Definition 2.10 Let $\\xi \\colon I \\times V_{\\infty } \\rightarrow V_{\\infty }$ be a smooth map, and let $F\\colon I \\times V_{\\infty } \\rightarrow V_{\\infty }$ be a polynomial map that separates invariant subspaces.",
"Consider the dynamical system for smooth maps $x\\colon I \\rightarrow V_{\\infty }$ determined by $ x(t) = \\xi (t, X(t)),$ for $t \\in I$ , for a smooth map $X\\colon I \\rightarrow V_{\\infty }$ satisfying eq ODE XYZ.",
"Let $D_{\\tilde{\\mu }}$ be as in prop dynamics.",
"For every $a = c,s,u, cs, cu$ , set $E_{a} := \\bigl \\lbrace (t, \\xi (t,v)): t \\in I_v, v \\in V_a, (t,v) \\in D_{\\tilde{\\mu }} \\bigr \\rbrace \\subset \\mathbb {R}\\times V_{\\infty }.$ The set $E_{c}$ is a centre subset of the dynamical system in $x$ ; the set $E_{s}$ is a stable subset of the system; and the set $E_{u}$ is an unstable subset of the system.",
"The set $E_{cs}$ is a centre-stable subset, and $E_{cu,p}$ is a centre-unstable subset of the system.",
"Such spaces are invariant or integral subsets of the dynamical system in $x$ .",
"2 Remark 2.11 If the map $\\xi _t := \\xi (t, )$ in def centre mfd is invertible for all $t \\in I$ (on a suitable domain), then the dynamical system in $x$ in that definition is equivalent to the ode $\\frac{d}{dt} (\\xi _t^{-1} \\circ x)(t) = A (\\xi _t^{-1} \\circ x)(t) + F(t, (\\xi _t^{-1} \\circ x) (t)).$ 2 Remark 2.12 In general, existence and uniqueness of solutions of eq ODE XYZ is not guaranteed, hence the careful definition of $I_v$ .",
"Existence and uniqueness of solutions is an assumption in previous definitions [17]; see Hypothesis 2.7 in that reference.",
"There are existence and uniqueness results if $f$ satisfies a local Lipschitz condition, but that is not the case in many applications to pdes.",
"Under additional assumptions, Vanderbauwhede & Iooss [33] showed such a local Lipschitz condition holds.",
"2 Remark 2.13 Invariant subsets or submanifolds are not unique in general; here this non-uniqueness is due to various possibilities for $I_v$ , $D_{\\tilde{\\mu }}$ and $\\xi _p$ , and is reflected in the use of the indefinite article in def centre mfd.",
"2 Example 2.14 For one example of the non-uniqueness engendered via $\\xi $ , consider the classic example system of $\\dot{x}=-x^2$ and $\\dot{y}=-y$ in the role of eq ODE (and let the step function $H(x):=1$ when $x>0$ , and $H(x):=0$ when $x\\le 0$ ).",
"This ode system may be given, for every $C$ , as the coordinate transformation, eq def xyz, $x=X$ and $y=Y+CH(X)e^{-1/X}$ together with the system, eq ODE XYZ, $\\dot{X}=-X^2$ and $\\dot{Y}=-Y$ (by design, here symbolically identical to the original $xy$ -system).",
"lem Vj invar identifies $Y=0$ as the centre subspace of this $XY$ -system.",
"def centre mfd then gives the classic non-uniqueness that, for every $C$ , $y=CH(x)e^{-1/x}$ are centre manifolds for the $xy$ -system.",
"2 Remark 2.15 In the setting of def centre mfd, if $\\xi $ is a local diffeomorphism in the Fréchet manifold sense, then the subsets $E_{j}$ in def centre mfd are Fréchet manifolds.",
"This would justify the more specific terminology invariant submanifolds rather than just invariant subsets."
],
[
"Main result: an approximate normal form",
"Our main result, thm normal form, states that for an ode of the form eq ODE, there is a dynamical system in the normal form used to define invariant manifolds in def centre mfd, such that solutions of the normal form system satisfy eq ODE up to a residual term that vanishes to any desired order.",
"In this sense, eq ODE is arbitrarily close to a dynamical system with clearly and robustly defined invariant manifolds.",
"2 Definition 2.16 A function $f\\colon I \\rightarrow \\mathbb {R}$ grows at most polynomially if there are $C,r>0$ such that for all $t \\in I$ , $|f(t)| \\le C(1+|t|^r)$ .",
"(This condition holds for all bounded functions if $I$ is bounded.)",
"An infinitely differentiable map $\\varphi \\colon I \\times V_{\\infty } \\rightarrow V_{\\infty }$ has polynomial growth if for every $v \\in V_{\\infty }$ , every $k \\in \\mathbb {N}$ , and every $l \\in \\mathbb {N}_0$ , the function $\\Vert \\varphi (, v)^{(l)}\\Vert _{V_k}\\colon I \\rightarrow [0, \\infty )$ grows at most polynomially.",
"We use the term $\\tilde{\\mu }$ -regular integral for an integral of the form $\\int _a^{\\infty } e^{-\\mu t} f(t) \\, dt,$ where $\\Re (\\mu ) > \\tilde{\\mu }$ and $f$ grows at most polynomially.",
"The larger $\\tilde{\\mu }$ , the better the convergence properties of $\\tilde{\\mu }$ -regular integrals.",
"2 Theorem 2.17 Let $p \\in \\mathbb {N}$ be such that $p \\ge 2$ , $\\beta - (p+1)\\alpha > \\tilde{\\mu }$ and $\\gamma - (p+1)\\alpha > \\tilde{\\mu }$ .",
"Suppose that $f$ has polynomial growth.",
"Then there are three infinitely differentiable maps $F_p, \\xi _p, R_p\\colon I \\times V_{\\infty } \\rightarrow V_{\\infty }$ , such that $F_p = \\mathcal {O}(2)$ and $F_p$ separates invariant manifolds; $R_p = \\mathcal {O}(p)$ , and if a smooth map $x\\colon I \\rightarrow V_{\\infty }$ is given by $ x(t) = \\xi _p(t,X(t))$ for all $t \\in I$ , for a smooth map $X \\colon I \\rightarrow V_{\\infty }$ satisfying $ \\dot{X}(t) = AX(t) + F_p(t, X(t))$ for all $t \\in I$ , then for all $t \\in I$ , $\\dot{x}(t) = Ax(t) + f(t, x(t)) + R_p(t, X(t)).$ Finally, there is a construction of the map $\\xi _p$ in which all integrals over $I$ that occur are $\\tilde{\\mu }$ -regular.",
"We prove this theorem in sec der,sec cpt der,sec Frechet,sec co xform,sec transf choice; see in particular sec proof main.",
"thm normal form 2 shows that the maps $F_p$ and $\\xi _p$ can be chosen to be polynomials of a certain type.",
"The conclusion that the integrals occurring are $\\tilde{\\mu }$ -regular is more than just convenient: this is clear in the classical case where $\\alpha = 0$ (see rem alpha zero).",
"2 Remark 2.18 In cases where the centre eigenvalue bound $\\alpha $ equals zero, we can always choose $\\tilde{\\mu }$ so small that the conditions on $\\alpha , \\beta , \\gamma , \\tilde{\\mu }$ and $p$ in thm normal form are satisfied.",
"In these cases, the residual $R_p$ can be made to vanish to arbitrarily large order $p$ .",
"Furthermore, the integrals that occur in the construction of $\\xi _p$ (see def conv emu) are $\\tilde{\\mu }$ -regular for some $\\tilde{\\mu }>0$ precisely if they converge.",
"Hence $\\tilde{\\mu }$ -regularity for some $\\tilde{\\mu }>0$ is a necessary condition for the construction to make sense.",
"Choosing the centre eigenvalue bound $\\alpha $ positive, which imposes a positive lower bound on $\\tilde{\\mu }$ , restricts the vanishing order $p$ of $R_p$ , but also makes the construction of the coordinate transform $\\xi _p$ more robust, in the sense that the integrals over $I$ in its construction are $\\tilde{\\mu }$ -regular.",
"Many researchers choose to phrase problems as singular perturbations [8], [24], [34].",
"In such cases the bounds on the hyperbolic rates $\\beta ,\\gamma \\propto \\frac{1}{\\varepsilon }\\rightarrow \\infty $ as the perturbation parameter $\\varepsilon \\rightarrow 0$ .",
"Consequently, choosing $\\tilde{\\mu },p\\propto 1/\\sqrt{\\varepsilon }$ (say) then the residual $R_p$ again can be made to vanish to arbitrarily large order for small enough $\\varepsilon $ .",
"However, in applications we generally require an invariant manifold in some chosen domain of interest that resolve phenomena on chosen time scales of interest.",
"Such subjective choices, informed by the governing equations, generally dictate the chosen bound $\\alpha $ separated by a big enough gap from the bounds $\\beta ,\\gamma $ so that the centre manifold evolution, constructed to a valid order $p$ , provides a useful model over the chosen domain for the desired phenomena.",
"2 Remark 2.19 The derivative at $0 \\in V_{\\infty }$ of the coordinate transformation $\\xi _p$ is the identity map, and hence invertible.",
"If a suitable generalisation of the inverse function theorem applies to $\\xi _p$ , such as a version of the Nash–Moser theorem, then it follows that $\\xi _p$ is a local diffeomorphism at zero.",
"Then it would be justified to call the invariant subsets of def centre mfd invariant submanifolds in this setting (at least in a neighbourhood of zero), see rem space mfd.",
"2 Remark 2.20 In the proof of thm normal form, explicit constructions of the maps $F_p$ and $\\xi _p$ are given.",
"In practice, however, it can be easier to determine these maps in more direct ways.",
"This is illustrated in an example in sec ex.",
"thm normal form implies that one can always find these maps.",
"We prove this by giving a construction that always leads to an answer, even though more direct constructions may exist in specific situations.",
"Similarly, the domain $D_{\\mu }$ in prop dynamics, defined in eq Dmu, is guaranteed to have the properties in prop dynamics.",
"In practice, these properties often hold on much larger domains."
],
[
"A general class of PDEs in bounded domains",
"Because thm normal form applies to abstract Banach spaces $V_k$ , it gives one the flexibility to choose these spaces such that, for specific pde applications, the residual $R_p$ is of order $p$ with respect to norms relevant to the problem, and the spaces $V_k$ incorporate the relevant boundary conditions.",
"This subsection explores a class of nonlinear pdes to which thm normal form, and hence def centre mfd, apply.",
"Let $d \\in \\mathbb {N}$ be the dimension of the domain of the pdes to be considered.",
"Let $\\Omega $ be a bounded, open subset of $\\mathbb {R}^d$ , or of a $d$ -dimensional manifold, with $C^1$ boundary.",
"Let $m \\in \\mathbb {N}$ , and let $1\\le p < \\infty $ .",
"For $k \\in \\mathbb {N}$ , let $V_k$ be the Sobolev space $W^{k-1,k+1}(\\Omega , \\mathbb {R}^m)$ .",
"Let $A\\colon C_c^{\\infty }(\\Omega ; \\mathbb {R}^m)\\rightarrow C_c^{\\infty }(\\Omega ; \\mathbb {R}^m)$ be a linear partial differential operator.",
"(Here the subscript $c$ denotes compactly supported functions.)",
"Let $s \\in \\mathbb {N}$ , with $s \\ge 2$ , be the `polynomial' order of the nonlinearities in the pdes.",
"Let $D_1, \\ldots , D_s\\colon C_c^{\\infty }(\\Omega ; \\mathbb {R}^{m}) \\rightarrow C_c^{\\infty }(\\Omega ; \\mathbb {R})$ be linear partial differential operators.",
"For index-vector $q \\in \\mathbb {N}_0^s$ and $u \\in C^{\\infty }_c(\\Omega , \\mathbb {R}^m)$ , we set $ (Du)^q := (D_1u)^{q_1} \\cdots (D_s u)^{q_s}.$ Let $\\alpha $ , $\\beta $ , $\\gamma $ and $\\tilde{\\mu }$ be as in sec result.",
"Fix smooth functionsThe real line may be replaced by a smaller open interval.",
"$a_q^j\\colon \\mathbb {R}\\rightarrow , for $ q N0s$, with $ |q| s$, such that these functions and all their derivatives grow at most polynomially.Define $ fRCc(; Rm) Cc(; Rm)$ by $ f(t, u) := (f1(t, u), ..., fm(t, u))$, where for each~$ j$,\\begin{equation*}f_j(t, u) = \\sum _{q \\in \\mathbb {N}_0^s, |q| \\le s} a_q^j(t)(Du)^q\\end{equation*}for $ t R$ and $ u Cc(; Rm)$.Suppose that $ f = $\\mathcal {O}$ (2)$.$ We write $W^{\\infty }(\\Omega ; \\mathbb {R}^m) := \\bigcap _{k=1}^{\\infty } W^{k-1, k+1}(\\Omega ; \\mathbb {R}^m).$ Then $C^{\\infty }_c(\\Omega ; \\mathbb {R}^m) \\subset W^{\\infty }(\\Omega ; \\mathbb {R}^m) \\subset C^{\\infty }(\\Omega ; \\mathbb {R}^m)$ .",
"The maps $A$ and $f$ extend continuously to $W^{\\infty }(\\Omega ; \\mathbb {R}^m)$ .",
"Suppose that that the eigenfunctions $\\lbrace e_j\\rbrace _{j=1}^{\\infty }$ of this extension of $A$ form a Hilbert basis of $L^2(\\Omega , \\mathbb {R}^m)$ .",
"2 Theorem 2.21 The spaces $V_k$ and the maps $A$ and $f$ satisfy the hypotheses of thm normal form.",
"We prove thm special case in sec ex f. Together with thm normal form, it has the following immediate consequence.",
"2 Corollary 2.22 Let $p \\in \\mathbb {N}$ be such that $p \\ge 2$ .",
"Suppose that $\\alpha $ , $\\beta $ , $\\gamma $ and $\\tilde{\\mu }$ satisfy $\\beta - (p+1)\\alpha > \\tilde{\\mu }$ and $\\gamma - (p+1)\\alpha > \\tilde{\\mu }$ (as in thm normal form).",
"Then there are infinitely differentiable maps $F_p, \\xi _p, R_p\\colon \\mathbb {R}\\times W^{\\infty }(\\Omega ; \\mathbb {R}^m) \\rightarrow W^{\\infty }(\\Omega ; \\mathbb {R}^m),$ where $F_p$ is a polynomial map that separates invariant subspaces, such that if $X$ and $x$ are as in eq ODE XYZ p and eq def xyz p, then $\\dot{x}(t) = Ax(t) + f(t, x(t)) + R_p(t, X(t))$ for all $t \\in I$ .",
"Further, for every $l \\in \\mathbb {N}$ , there is a $k \\in \\mathbb {N}$ such that for all $u \\in W^{\\infty }(\\Omega ; \\mathbb {R}^m)$ , $\\Vert R_p(t,v) \\Vert _{W^{l-1, l+1}} = O( \\Vert v\\Vert ^p_{W^{k-1, k+1}} )$ as $v \\rightarrow 0$ in $W^{k-1,k+1}(\\Omega , \\mathbb {R}^m)$ .",
"There is a construction of the map $\\xi _p$ in which all integrals over $I$ that occur are $\\tilde{\\mu }$ -regular.",
"This corollary shows that any pde of the form eq ODE, with $A$ and $f$ as in this subsection, is equivalent up to a residual of order $p$ to a pde with clear invariant manifolds, as in Definition REF .",
"2 Example 2.23 Suppose that $\\Omega = S^1$ , the circle.",
"This amounts to imposing periodic boundary conditions.",
"Take $m = 1$ , and let $A\\colon C^{\\infty }(S^1) \\rightarrow C^{\\infty }(S^1)$ by any linear partial differential operator with constant coefficients.",
"Its eigenfunctions, $e_j(\\theta ) = e^{ij\\theta }$ for $j \\in \\mathbb {Z}$ and $\\theta \\in \\mathbb {R}/2\\pi \\mathbb {Z}$ , are orthogonal in the Sobolev spaces $W^{k,2}(S^1)$ .",
"For a map $f$ as in thm special case, that is, a polynomial expression in derivatives of functions, whose polynomial coefficients increase at most polynomially, thm special case implies that the conditions of thm normal form are satisfied in this case, so cor special case applies.",
"This generalises directly to cases where $\\Omega $ is a higher-dimensional torus; that is, to problems in $\\mathbb {R}^d$ with periodic boundary conditions.",
"Here we used the case where the domain $\\Omega $ is a manifold, rather than an open subset of $\\mathbb {R}^d$ .",
"Most of the rest of this paper is devoted to proofs of thm normal form,thm special case, and developing the tools used in these proofs.",
"In sec dynamics, we prove lem Vj invar,prop dynamics.",
"In sec ex we illustrate cor special case by working out an example."
],
[
"Derivatives and polynomials",
"In this section we review standard material on derivatives of maps between normed vector spaces.",
"We also briefly discuss polynomial maps between normed vector spaces.",
"Throughout this section, $(V, \\Vert \\cdot \\Vert _V)$ and $(W, \\Vert \\cdot \\Vert _W)$ are normed vector spaces, possibly infinite-dimensional.",
"Let $U\\subset V$ be an open subset, and let $f\\colon U \\rightarrow W$ be a map.",
"We fix an element $u \\in U$ ."
],
[
"First order derivatives",
"This subsection and the next contain some standard definitions and facts about derivatives of maps between normed vector spaces.",
"Details and proofs can be found in various textbooks [35].",
"For a map $\\varphi \\colon V \\supset \\operatorname{dom}(\\varphi ) \\rightarrow W$ , we use the notation $\\varphi (h) = o(h)$ for the statement $\\lim _{h \\rightarrow 0} \\frac{\\Vert \\varphi (h)\\Vert _W}{\\Vert h\\Vert _V}=0,$ were $h$ runs over $ \\operatorname{dom}(\\varphi ) \\setminus \\lbrace 0\\rbrace $ .",
"2 Definition 3.1 The map $f:U\\rightarrow W$ is differentiable at $u$ , if there is an operator $f^{\\prime }(u) \\in \\mathcal {B}(V,W)$ such that $f(u+h) = f(u) + f^{\\prime }(u)h + o(h).$ Then $f^{\\prime }(u)$ is the derivative of $f$ at $u$ .",
"If $f$ is differentiable at every point in $U$ , then we say that $f$ is differentiable.",
"In that case, the derivative of $f$ is the map $ f^{\\prime }\\colon U \\rightarrow \\mathcal {B}(V,W)$ mapping $u \\in U$ to $f^{\\prime }(u)$ .",
"The derivative of a map at a point is unique, if it exists.",
"2 Lemma 3.2 (Chain rule) Let $(X, \\Vert \\cdot \\Vert _X)$ be a third normed vector space.",
"Let $A \\subset W$ be an open subset containing $f(U)$ .",
"If $g\\colon A \\rightarrow X$ is differentiable at $f(u)$ and $f$ is differentiable at $u$ , then $g\\circ f$ is differentiable at $u$ , and $(g\\circ f)^{\\prime }(u) = g^{\\prime }(f(u)) \\circ f^{\\prime }(u).$ For all $h \\in V$ such that $u+h \\in U$ , differentiability of $f$ at $u$ and of $g$ at $f(u)$ imply that $\\begin{split}(g\\circ f)(u+h) &= g(f(u)+f^{\\prime }(u)h+o(h)) \\\\&= g(f(u)) + g^{\\prime }(f(u))(f^{\\prime }(u)h+o(h)) + o(f^{\\prime }(u)h+o(h)).\\end{split}$ Since $g^{\\prime }(f(u))$ and $f^{\\prime }(u)$ are bounded operators, the second term on the right-hand side equals $g^{\\prime }(f(u))f^{\\prime }(u)h+o(h)$ , while the last term is $o(h)$ .",
"2 Definition 3.3 The map $f$ is a near-identity at $u$ if the map eq def der is continuous in a neighbourhood of $u$ , and $f^{\\prime }(u)h = h +o(h).$"
],
[
"Higher order derivatives",
"Fix a positive integer $n \\in \\mathbb {N}$ .",
"We write $\\mathcal {B}^n(V,W)$ for the space of multilinear maps $\\lambda \\colon V^n \\rightarrow W$ for which the norm $ \\Vert \\lambda \\Vert := \\sup _{\\begin{array}{c}v_1, \\ldots , v_n \\in V\\\\ \\Vert v_1\\Vert _V = \\cdots = \\Vert v_n\\Vert _V=1\\end{array}} \\Vert \\lambda (v_1, \\ldots , v_n)\\Vert _W$ is finite.",
"There is a natural isometric isomorphism $ \\mathcal {B}(V, \\mathcal {B}(V, \\ldots , \\mathcal {B}(V,W) \\cdots )) \\xrightarrow{} \\mathcal {B}^n(V,W)$ mapping an operator $T$ in the left-hand side to the operator $\\lambda \\in \\mathcal {B}^n(V,W)$ given by $\\lambda (v_1, \\ldots , v_n) = T(v_1)(v_2)\\cdots (v_n),$ for $v_1, \\ldots , v_n \\in V$ .",
"Suppose $f:U\\rightarrow W$ is differentiable.",
"The map $f$ is twice differentiable at $u$ if the map eq def der is differentiable at $u$ .",
"Then we write $f^{(2)}(u) :=(f^{\\prime })^{\\prime }(u) \\in \\mathcal {B}(V, \\mathcal {B}(V,W)) \\cong \\mathcal {B}^2(V,W).$ Inductively, for $n \\ge 2$ , $f$ is defined to be $n$ times differentiable at $u$ if it is $n-1$ times differentiable, and the map $f^{(n-1)}\\colon U \\rightarrow \\mathcal {B}^{n-1}(V,W)$ is differentiable at $u$ .",
"We then set $f^{(n)}(u) := (f^{(n-1)})^{\\prime }(u) \\quad \\in \\mathcal {B}^n(V,W).$ In this case, we write $ f^{(n)}(u)h^n := f^{(n)}(u)(h, h, \\ldots , h).$ As before, we say that $f$ is $n$ times differentiable if it is $n$ times differentiable at every point in $U$ .",
"And infinitely differentiable means $n$ times differentiable for every $n\\in \\mathbb {N}$ .",
"2 Theorem 3.4 (Taylor's theorem) Suppose $f$ is $n+1$ times differentiable.",
"Suppose that $\\Vert f^{(n+1)}(\\xi )\\Vert \\le M$ for all $\\xi $ in a closed ball around $u$ contained in $U$ .",
"Then for every $h$ in this ball, $\\Bigl \\Vert f(u+h) - \\sum _{j=0}^n \\frac{1}{j!}",
"f^{(j)}(u)h^j\\Bigr \\Vert _W \\le \\frac{M}{(n+1)!",
"}\\Vert h\\Vert _V^{n+1}.$"
],
[
"Example: Burgers' equation",
"An example of a map to which we would like to apply the material in this section and the next is the nonlinear term $u_x u$ in Burgers' equation $ u_t = u_{xx} - u_x u.$ Let $\\Omega \\subset \\mathbb {R}$ be a bounded, open interval in $x$ .",
"For every $k \\in \\mathbb {N}_0$ , consider the $k$ th $L^2$ -Sobolev space $W^{k,2}(\\Omega )$ , with the inner product $(u_1, u_2)_{W^{k,2}} := \\sum _{j=0}^k (u_1^{(j)}, u_2^{(j)})_{L^2}.$ Consider the map $f\\colon W^{1,2}(\\Omega ) \\rightarrow L^1(\\Omega )$ given by $f(u) = u^{\\prime }u.$ First of all, for $u \\in C^{\\infty }_c(\\Omega )$ , the Cauchy–Schwartz inequality for $L^2(\\Omega )$ (or Hölder's inequality) implies that $ \\Vert u^{\\prime }u\\Vert _{L^1} \\le \\Vert u^{\\prime } \\Vert _{L^2} \\Vert u \\Vert _{L^2} \\le \\Vert u\\Vert _{W^{1,2}}^2.$ So $f$ indeed maps $W^{1,2}(\\Omega )$ into $L^1(\\Omega )$ .",
"We claim that $f$ is infinitely differentiable.",
"Indeed, for $u,h \\in W^{1,2}(\\Omega )$ , $f(u+h) = f(u) + h^{\\prime }u+u^{\\prime }h + h^{\\prime }h.$ And by eq burgers CS, $\\Vert h^{\\prime }h\\Vert _{L^1} \\le \\Vert h\\Vert _{W^{1,2}}^2,$ so $f^{\\prime }(u)h = h^{\\prime }u+u^{\\prime }h.$ The map $f^{\\prime }(u)\\colon W^{1,2}(\\Omega ) \\rightarrow L^1(\\Omega )$ is bounded, because, analogously to eq burgers CS, $ \\Vert h^{\\prime }u+u^{\\prime }h\\Vert _{L^1} \\le 2 \\Vert u\\Vert _{W^{1,2}} \\Vert h\\Vert _{W^{1,2}}.$ If $u,h_1, h_2 \\in W^{1,2}(\\Omega )$ , then $f^{\\prime }(u+h_2)(h_1) = f^{\\prime }(u)h_1 + h_2^{\\prime }h_1 + h_1^{\\prime }h_2.$ So $f^{(2)}(u)$ is the operator in $\\mathcal {B}^2(W^{1,2}(\\Omega ), L^1(\\Omega ))$ given by $f^{(2)}(u)(h_1, h_2) = h_2^{\\prime }h_1 + h_1^{\\prime }h_2.$ The term $o(h_2)$ in the definition of the derivative is zero in this case, and that $f^{(2)}(u)$ does not depend on $u$ .",
"This implies that for every $n \\ge 3$ , $f^{(n)}(u) = 0$ .",
"So $f$ is indeed infinitely differentiable."
],
[
"Bounded polynomial maps",
"An operator in $\\mathcal {B}^n(V,W)$ is said to be symmetric if it is invariant under permutations of its arguments.",
"Let $S\\mathcal {B}^n(V,W)$ be the subspace of symmetric operators in $\\mathcal {B}^n(V,W)$ .",
"An example of such a symmetric operator is the $n$ th derivative of a map.",
"2 Lemma 3.5 If $f$ is $n$ times differentiable at $u$ , then $f^{(n)}(u)$ is symmetric.",
"We denote the permutation group of $\\lbrace 1,\\ldots , n\\rbrace $ by $\\Sigma _n$ .",
"2 Lemma 3.6 The subspace $S\\mathcal {B}^n(V,W) \\subset \\mathcal {B}^n(V,W)$ is closed.",
"If $T \\in \\overline{S\\mathcal {B}^n(V,W)} \\setminus S\\mathcal {B}^n(V,W)$ , and $v_1, \\ldots , v_n \\in V$ and $\\sigma \\in \\Sigma _n$ are such that $T(v_1, \\ldots , v_n) \\ne T(v_{\\sigma (1)}, \\ldots , v_{\\sigma (n)}),$ set $\\varepsilon := \\bigl \\Vert T\\bigl (v_1/\\Vert v_1\\Vert _V, \\ldots , v_n/\\Vert v_n\\Vert _V\\bigr ) - T\\bigl (v_{\\sigma (1)}/\\Vert v_{\\sigma (1)}\\Vert _V, \\ldots , v_{\\sigma (n)}/\\Vert v_{\\sigma (n)}\\Vert _V\\bigr ) \\bigr \\Vert _W\\\\ > 0.$ Let $\\tilde{T} \\in S\\mathcal {B}^n(V,W)$ be such that $\\Vert \\tilde{T} - T\\Vert < \\varepsilon /2$ , for the norm eq norm Bn.",
"Then symmetry of $S$ and the triangle inequality imply that $\\bigl \\Vert T\\bigl (v_1/\\Vert v_1\\Vert _V, \\ldots , v_n/\\Vert v_n\\Vert _V\\bigr ) - T\\bigl (v_{\\sigma (1)}/\\Vert v_{\\sigma (1)}\\Vert _V, \\ldots , v_{\\sigma (n)}/\\Vert v_{\\sigma (n)}\\Vert _V\\bigr ) \\bigr \\Vert _W\\\\< \\bigl \\Vert T\\bigl (v_1/\\Vert v_1\\Vert _V, \\ldots , v_n/\\Vert v_n\\Vert _V\\bigr ) - \\tilde{T}\\bigl (v_1/\\Vert v_1\\Vert _V, \\ldots , v_n/\\Vert v_n\\Vert _V\\bigr ) \\bigr \\Vert _W\\\\+\\bigl \\Vert \\tilde{T}\\bigl (v_{\\sigma (1)}/\\Vert v_{\\sigma (1)}\\Vert _V, \\ldots , v_{\\sigma (n)}/\\Vert v_{\\sigma (n)}\\Vert _V\\bigr ) - T\\bigl (v_{\\sigma (1)}/\\Vert v_{\\sigma (1)}\\Vert _V, \\ldots , v_{\\sigma (n)}/\\Vert v_{\\sigma (n)}\\Vert _V\\bigr )\\bigr \\Vert _W\\\\< \\varepsilon ,$ a contradiction.",
"By this lemma, $S\\mathcal {B}^n(V,W)$ is a Banach space if $V$ and $W$ are.",
"Let $S\\colon \\mathcal {B}^n(V,W) \\rightarrow S\\mathcal {B}^n(V,W)$ be the symmetrisation operator: for every $\\lambda \\in \\mathcal {B}^n(V,W)$ and $v_1, \\ldots , v_n \\in V$ , $(S\\lambda )(v_1, \\ldots , v_n) = \\frac{1}{n!",
"}\\sum _{\\sigma \\in \\Sigma _n} \\lambda (v_{\\sigma (1)}, \\ldots , v_{\\sigma (n)}).$ (An alternative proof of lem SBn closed is to show that $S$ is continuous, and to note that $S\\mathcal {B}^n(V,W)$ is the zero level set of $S$ minus the identity.)",
"The operator $S$ is continuous, and that $S\\mathcal {B}^n(V,W)$ is the zero level set of $S$ minus the identity, and hence closed in $\\mathcal {B}^n(V,W)$ .",
"So $S\\mathcal {B}^n(V,W)$ is a Banach space if $V$ and $W$ are.",
"An element $\\lambda \\in \\mathcal {B}^n(V,W)$ defines a map $p_{\\lambda }\\colon V \\rightarrow W$ by $ p_{\\lambda }(v) = \\lambda (v, \\ldots , v).$ We have $p_{S\\lambda } = p_{\\lambda }$ , and the map $\\lambda \\mapsto p_{\\lambda }$ is injective on $S\\mathcal {B}^n(V,W)$ .",
"2 Definition 3.7 A bounded homogeneous polynomial map of degree $n$ from $V$ to $W$ is a map of the form $p_{\\lambda }$ as in eq def monomial.",
"We write $\\operatorname{Pol}^n(V,W)$ for the space of such maps.",
"It inherits a norm from the space $S\\mathcal {B}^n(V,W)$ via the linear isomorphism $\\lambda \\mapsto p_{\\lambda }$ .",
"If $\\lambda \\ne 0$ , then the degree of $p_{\\lambda }$ is $n$ .",
"A bounded polynomial map from $V$ to $W$ is a finite sum of bounded homogeneous polynomial maps.",
"The degree of a bounded polynomial map is the degree of its highest-degree homogeneous term.",
"We write $\\operatorname{Pol}(V,W)$ for the space of all bounded polynomial maps from $V$ to $W$ .",
"This is the algebraic direct sum of the spaces $\\operatorname{Pol}^n(V,W)$ .",
"By lem SBn closed, $\\operatorname{Pol}^n(V,W)$ is a Banach space if $V$ and $W$ are.",
"Because $S\\mathcal {B}^n(V,W)$ is a Banach space if $V$ and $W$ are, so is $\\operatorname{Pol}^n(V,W)$ .",
"We could define $\\operatorname{Pol}^0(V,W)$ as the space of constant maps into $W$ , but we only consider homogeneous polynomials of order at least one.",
"If $f$ is $n$ times differentiable at $u$ , then we have the map $h \\mapsto f^{(n)}(u)h^n \\in \\operatorname{Pol}^n(V,W).$ lem pol diffble–REF below are basic facts showing that bounded polynomials and their orders and compositions behave as one would expect.",
"Their proofs are short and straightforward.",
"2 Lemma 3.8 Every bounded polynomial map is infinitely differentiable.",
"Let $\\lambda \\in S\\mathcal {B}^n(V,W)$ , for some $n \\ge 2$ .",
"Then for all $u,h \\in V$ , $p_{\\lambda }(u+h) = p_{\\lambda }(u) + n\\lambda (h, u, \\ldots , u) + O(\\Vert h\\Vert ^2).$ Hence $p_{\\lambda }$ is differentiable, and $p_{\\lambda }^{\\prime }(u) = n\\lambda (u, \\ldots , u),$ where on the right-hand side, the operator $\\lambda $ is applied to $n-1$ copies of $u$ , to give an element of $\\mathcal {B}(V,W)$ .",
"Hence $p_{\\lambda }^{\\prime }$ is a bounded polynomial map in $\\operatorname{Pol}^{n-1}(V, \\mathcal {B}(V,W))$ .",
"This proves the claim by induction.",
"2 Lemma 3.9 If $p \\in \\operatorname{Pol}^n(V,W)$ , then there is a constant $C>0$ such that for all $v \\in V$ , $\\Vert p(v)\\Vert _W \\le C \\Vert v\\Vert _V^n.$ Let $\\lambda \\in \\mathcal {B}^n(V,W)$ .",
"By boundedness and multilinearity of $\\lambda $ , we have for all nonzero $v \\in V$ , $\\Vert p_{\\lambda }(v)\\Vert _W = \\Vert v\\Vert _V^n \\cdot \\Vert \\lambda (v/\\Vert v\\Vert _V, \\ldots , v/\\Vert v\\Vert _V)\\Vert _W \\le \\Vert \\lambda \\Vert \\cdot \\Vert v\\Vert _V^n$ 2 Lemma 3.10 If $p$ is a polynomial map from $V$ to $W$ of order lower than $n$ , and $\\Vert p(v)\\Vert _W = O(\\Vert v\\Vert _V^{n}),$ as $v\\rightarrow 0$ in $V$ , then $p=0$ .",
"Let $m,n \\in \\mathbb {N}$ .",
"Let $\\lambda \\in S\\mathcal {B}^m(V,W)$ , and suppose that $\\Vert p_{\\lambda }(v)\\Vert _W = O(\\Vert v\\Vert _V^{n})$ , as $v\\rightarrow 0$ in $V$ .",
"Then there is a $C>0$ such that for all $v \\in V$ with unit norm and $s>0$ small enough, $s^m \\Vert p_{\\lambda }(v)\\Vert _W = \\Vert p_{\\lambda }(sv)\\Vert _W \\le Cs^n.$ If $m<n$ , this implies that $\\Vert p_{\\lambda }(v)\\Vert _W =0$ .",
"2 Lemma 3.11 If $p_1 \\in \\operatorname{Pol}^m(U,V)$ and $p_2 \\in \\operatorname{Pol}^n(V,W)$ , then $p_2 \\circ p_1 \\in \\operatorname{Pol}^{mn}(U,W)$ .",
"For $\\lambda _1 \\in \\mathcal {B}^m(U,V)$ and $\\lambda _2 \\in \\mathcal {B}^n(V,W)$ , define $\\lambda _2 \\circ \\lambda _1\\colon U^{mn} \\rightarrow W$ by $\\lambda _2 \\circ \\lambda _1(u_{11}, \\ldots , u_{1m}; \\ldots ; u_{n1}, \\ldots , u_{nm}):=\\\\\\lambda _2\\bigl (\\lambda _1(u_{11}, \\ldots , u_{1m}), \\ldots , \\lambda _2(\\lambda _1(u_{n1}, \\ldots , u_{nm})\\bigr ),$ for $u_{jk} \\in U$ .",
"Then one checks directly that $\\lambda _2 \\circ \\lambda _1 \\in \\mathcal {B}^{mn}(U,V)$ .",
"This implies the claim about polynomials."
],
[
"Standard monomials",
"Let $V^* := \\mathcal {B}(V,$ be the continuous dual of $V$ .",
"We denote the pairing between $V^*$ and $V$ by $\\langle , \\rangle $ .",
"For every $j \\in \\mathbb {N}$ , let $e^j \\in V^*$ be given.",
"What follows is most natural if $V$ is a Hilbert space and $e^j$ is given by taking inner products with an element $e_j$ of a Hilbert basis, but it applies more generally.",
"Consider a multi-index $q \\in \\mathbb {N}_0^{\\infty }$ .",
"If $|q|=n$ , and $m$ is the largest number for which $q_m \\ne 0$ , then we define the element $ e^q := \\underbrace{e^1 \\otimes \\cdots \\otimes e^1}_{\\text{$q_1$ factors}} \\otimes \\cdots \\otimes \\underbrace{e^m \\otimes \\cdots \\otimes e^m}_{\\text{$q_m$ factors}}\\in \\mathcal {B}^n(V, .$ In other words, for all $v_1, \\ldots , v_n \\in V$ , $&e^q(v_1, \\ldots , v_n) =\\\\&\\langle e^1, v_1 \\rangle \\cdots \\langle e^1, v_{q_1}\\rangle \\langle e^2, v_{q_1 + 1}\\rangle \\cdots \\langle e^2, v_{q_1 + q_2}\\rangle \\cdots \\langle e^m, v_{q_1 + \\cdots + q_{m-1} +1}\\rangle \\cdots \\langle e^m, v_n\\rangle .$ We write $p^q := p_{e^q}$ for the corresponding homogeneous polynomial.",
"One could call this the standard $q$ -monomial with respect to the set $\\lbrace e^j\\rbrace _{j=1}^{\\infty }$ .",
"(If $V = k$ and the elements $e^j$ are the standard coordinates, then the monomial functions in the usual sense are precisely the scalar multiples of the maps $p^q$ .)",
"For $v \\in V$ , we write $ v^q := p^q(v) = \\prod _{j=1}^{\\infty } \\langle e^j, v\\rangle ^{q_j}.$ This product is finite (since $q$ has finitely many nonzero terms) and depends on the set $\\lbrace e^j\\rbrace $ .",
"The following lemma follows from the definition of the derivative.",
"2 Lemma 3.12 The derivative of $p^q$ in eq def vq is given by $(p^q)^{\\prime }(u)(h) = \\sum _{j=1}^{\\infty } q_j\\langle e^j, u\\rangle ^{q_j-1}\\langle e_j, h\\rangle \\Bigl (\\prod _{k\\ne j} \\langle e^{k}, u\\rangle ^{q_{k}}\\Bigr ),$ for all $u,h \\in V$ ."
],
[
"Compact derivatives and polynomials",
"It is a nontrivial question in what sense differentiable maps between normed vector spaces can be approximated by polynomial maps [1], [11], [12], [13], [22], [23].",
"In this section we discuss an approach to this problem that is suitable for our purposes.",
"This discussion includes the further problem of approximating a polynomial by sums of the standard monomials of sec std mon.",
"The polynomials for which this is possible are the compact polynomials introduced in sec cpt pol.",
"sec cpt der def introduces compactly differentiable maps.",
"We combine these with Taylor's theorem to express the lowest order parts of such maps in terms of standard monomials.",
"We discuss a class of examples of compactly differentiable maps relevant to the study of pdes."
],
[
"Compact multilinear maps",
"Let $V$ and $W$ be Banach spaces.",
"Let $\\mathcal {K}^n(V,W) \\subset \\mathcal {B}^n(V,W)$ be the image of the space $\\mathcal {K}(V, \\mathcal {K}(V, \\ldots , \\mathcal {K}(V,W) \\cdots ))$ under the isomorphism eq iso Bn.",
"Using induction on $n$ , one can show that $\\mathcal {K}^n(V,W)$ is closed in $\\mathcal {B}^n(V,W)$ , and hence a Banach space.",
"2 Lemma 4.1 For every $n\\in \\mathbb {N}$ the space $\\mathcal {K}^n(V,W)$ is closed in $\\mathcal {B}^n(V,W)$ .",
"We use induction on $n$ .",
"For $n=1$ the claim is standard.",
"Suppose the claim holds for $n$ .",
"Then $\\mathcal {K}^{n+1}(V,W) = \\mathcal {K}(V, \\mathcal {K}^n(V,W)),$ which is a closed subspace of $\\mathcal {B}(V, \\mathcal {K}^n(V,W))$ .",
"And that space is closed in $\\mathcal {B}^{n+1}(V,W)$ since $\\mathcal {K}^n(V,W)$ is closed in $ \\mathcal {B}^n(V,W)$ by the induction hypothesis.",
"By lem Kn closed, $\\mathcal {K}^n(V,W)$ is a Banach space.",
"Let $\\lbrace e^j\\rbrace _{j=1}^{\\infty } \\subset V^*$ and $\\lbrace f_k\\rbrace _{k=1}^{\\infty } \\subset W$ be countable subsets whose spans are dense.",
"(So $V^*$ and $W$ are separable.)",
"For any $\\alpha \\in \\mathbb {N}^{n}$ , consider the multilinear map $ e^{\\alpha } := e^{\\alpha _1} \\otimes \\cdots \\otimes e^{\\alpha _n}\\colon V \\times \\cdots \\times V \\rightarrow $ A Banach space has the approximation property if every compact operator on the space is a norm-limit of finite-rank operators.",
"This is always true for Hilbert spaces, but we need to consider more general Banach spaces for applications.",
"The following result is standard in the case where $V$ and $W$ are Hilbert spaces.",
"2 Proposition 4.2 If $V^*$ has the approximation property, then the space $\\operatorname{span}\\lbrace e^j \\otimes f_k : j,k \\in \\mathbb {N}\\rbrace $ is dense in $\\mathcal {K}(V,W)$ .",
"Since $V^*$ has the approximation property, the space of finite-rank operators (linear operators with finite-dimensional images) is dense in $\\mathcal {K}(V,W)$ .",
"See for example Proposition 4.12(b) in the book by Ryan [31].",
"The space $\\operatorname{span}\\lbrace e^j \\otimes f_k : j,k \\in \\mathbb {N}\\rbrace $ is dense in the space of finite-rank operators, so the claim follows.",
"2 Lemma 4.3 If $V^*$ has the approximation property, then for every $n \\in \\mathbb {N}$ , the span of $\\lbrace e^{\\alpha } \\otimes f_k: \\alpha \\in \\mathbb {N}^n, k \\in \\mathbb {N}\\rbrace $ is dense in $\\mathcal {K}^n(V,W)$ .",
"We prove this by induction on $n$ .",
"If $n=1$ , then the claim is precisely prop approx cpt in the appendix.",
"Now suppose that the claim holds for a given $n$ .",
"By definition, $\\mathcal {K}^{n+1}(V,W) = \\mathcal {K}(V, \\mathcal {K}^{n}(V,W)).$ By the induction hypothesis, the set $\\lbrace e^{\\alpha } \\otimes f_k : \\alpha \\in \\mathbb {N}^n, k \\in \\mathbb {N}\\rbrace $ has dense span in $\\mathcal {K}^n(V,W)$ .",
"Therefore, prop approx cpt, with $W$ replaced by $\\mathcal {K}^{n}(V,W)$ , implies that the set $\\lbrace e^j \\otimes e^{\\alpha } \\otimes f_k : j,k \\in \\mathbb {N}, \\alpha \\in \\mathbb {N}^n\\rbrace $ has dense span in $\\mathcal {K}^{n+1}(V,W)$ .",
"This is precisely the claim for $n+1$ .",
"A Schauder basis of a Banach space $V$ is a subset $\\lbrace e_j\\rbrace _{j=1}^{\\infty } \\subset V$ such that for each $v \\in V$ , there are unique complex numbers $\\lbrace v^j\\rbrace _{j \\in \\mathbb {N}}$ such that $\\Bigl \\Vert v - \\sum _{j=1}^n v^j e_j\\Bigr \\Vert _V \\rightarrow 0 \\quad \\text{as }n \\rightarrow \\infty .$ A space with a Schauder basis has the approximation property.",
"2 Lemma 4.4 If $\\lbrace e^j\\rbrace _{j=1}^{\\infty }$ is a Schauder basis of $V^*$ and $\\lbrace f_k\\rbrace _{k=1}^{\\infty }$ is a Schauder basis of $W$ , then for every $n \\in N$ , the set $\\lbrace e^{\\alpha } \\otimes f_k : \\alpha \\in \\mathbb {N}^n, k \\in \\mathbb {N}\\rbrace $ is a Schauder basis of $\\mathcal {K}^n(V,W)$ .",
"lem Kn fin dense implies that $\\lbrace e^{\\alpha } \\otimes f_k : \\alpha \\in \\mathbb {N}^n, k \\in \\mathbb {N}\\rbrace $ has dense span.",
"So it remains to show that if $a_{\\alpha }^k \\in are such that$$\\sum _{\\alpha \\in \\mathbb {N}^n} \\sum _{k=1}^{\\infty } a_{\\alpha }^k e^{\\alpha } \\otimes f_k = 0,$$then $ ak = 0$ for all $$ and $ k$.",
"Since $ {fk}k=1$ is a Schauder basis of $ W$, this reduces to the case where $ W = .",
"In that case, one can prove the claim by induction on $n$ , using the fact that $\\lbrace e^j\\rbrace _{j=1}^{\\infty }$ is a Schauder basis of $V^*$ .",
"We prove the claim in that case, by induction on $n$ .",
"If $n = 1$ , then the claim follows since $\\lbrace e^j\\rbrace _{j=1}^{\\infty }$ is a Schauder basis of $V^*$ .",
"So suppose that the claim holds for a given $n$ , and let $a_{\\alpha }^k \\in be such that$$\\sum _{\\alpha \\in \\mathbb {N}^{n+1}}a_{\\alpha }^k e^{\\alpha } = 0.$$Then for all $ v1, ..., vn V$,$$\\sum _{j=1}^{\\infty } \\Bigl [\\sum _{\\alpha \\in \\mathbb {N}^{n}} a_{(j, \\alpha )}^k e^{\\alpha } (v_1, \\ldots , v_n)\\Bigr ] e^j= 0.$$Since, $ {ej}j=1$ is a Schauder basis of $ V*$, this implies that for every $ j N$,$$\\sum _{\\alpha \\in \\mathbb {N}^{n}} a_{(j, \\alpha )}^k e^{\\alpha } (v_1, \\ldots , v_n)= 0.$$Because the sum$$\\sum _{\\alpha \\in \\mathbb {N}^{n}} a_{(j, \\alpha )}^k e^{\\alpha }$$converges in $$\\mathcal {B}$ n(V, $, we find that the sum converges to zero in this space.",
"By the induction hypothesis, this implies that $ aj, = 0$ for every $ j N$ and $ Nn$.$ 2 Remark 4.5 In the induction step in the proof of the special case of lem Kn fin dense where $W$ is a Hillbert space, we still need the general version of prop approx cpt, where $W$ is a Banach space.",
"This is because $\\mathcal {K}^n(V,W)$ is only a Banach space, even if $W$ is a Hilbert space.",
"The subspace $S\\mathcal {K}^n(V,W)$ of symmetric operators in $\\mathcal {K}^n(V,W)$ is closed in $\\mathcal {B}^n(V,W)$ , since it is the intersection of the closed subspaces $S\\mathcal {B}^n(V,W)$ and $\\mathcal {K}^n(V,W)$ (lem SBn closed,lem Kn closed).",
"Hence $S\\mathcal {K}^n(V,W)$ is a Banach space with respect to the norm eq norm Bn.",
"A Schauder basis $\\lbrace e_j\\rbrace _{j=1}^{\\infty }$ of a Banach space $V$ is unconditional if there is a constant $C>0$ such that for all $a^j, \\varepsilon _j \\in with $ |j|=1$, and all $ n N$,$$\\Bigl \\Vert \\sum _{j=1}^{n}\\varepsilon _j a^j e_j \\Bigr \\Vert _V \\le C \\Bigl \\Vert \\sum _{j=1}^{n}a^j e_j \\Bigr \\Vert _V.$$In that case, convergence of $ j=1naj ej$ implies convergence of $ j Aaj ej$, for every $ A N$.$ 2 Lemma 4.6 Suppose that $V$ and $W$ are Hilbert spaces, and that $\\lbrace e_j\\rbrace _{j=1}^{\\infty }$ and $\\lbrace f_k\\rbrace _{k=1}^{\\infty }$ are orthogonal sets in $V$ and $W$ respectively, with dense spans.",
"Let $e^j \\in V^*$ be defined by taking inner products with $e_j$ .",
"Then $\\lbrace e^{\\alpha } \\otimes f_k : \\alpha \\in \\mathbb {N}^n, k \\in \\mathbb {N}\\rbrace $ is an unconditional Schauder basis of $\\mathcal {K}^n(V,W)$ .",
"The set $\\lbrace e^{\\alpha } \\otimes f_k : \\alpha \\in \\mathbb {N}^n, k \\in \\mathbb {N}\\rbrace $ is a Schauder basis of $\\mathcal {K}^n(V,W)$ by lem Kn Schauder.",
"It remains to show that it is unconditional.",
"By rescaling the vectors $e_j$ and $f_k$ , we reduce the proof to the case where $\\lbrace e_j\\rbrace _{j=1}^{\\infty }$ and $\\lbrace f_k\\rbrace _{k=1}^{\\infty }$ are Hilbert bases.",
"In that case, for all finite subsets $A \\subset \\mathbb {N}_0^{n} \\times \\mathbb {N}$ and all $a_{\\alpha }^k \\in ,$$\\Bigl \\Vert \\sum _{(\\alpha , k) \\in A} a_{\\alpha }^k e^{\\alpha }\\otimes f_k\\Bigr \\Vert _{\\mathcal {B}^n(V,W)}^2 =\\sup _{\\alpha \\in \\mathbb {N}_0^m} \\sum _{k \\in \\mathbb {N};\\, (\\alpha , k) \\in A} |a_{\\alpha }^k|^2.$$$ 2 Lemma 4.7 Let $U$ , $V$ and $W$ be normed vector spaces, and $n\\in \\mathbb {N}$ .",
"Let $\\lambda \\in \\mathcal {B}^n(V,W)$ , and $a_1, \\ldots , a_n \\in \\mathcal {K}(U,V)$ .",
"Define $\\nu \\colon U \\times \\cdots \\times U \\rightarrow W$ by $\\nu (u_1, \\ldots , u_n) = \\lambda (a_1u_1, \\ldots , a_n u_n),$ for all $u_1, \\ldots , u_n \\in U$ .",
"Then $\\nu \\in \\mathcal {K}^n(U, W)$ .",
"We use induction on $n$ .",
"For $n = 1$ , $v \\in \\mathcal {K}^1(U,W)$ because the composition of a compact operator and a bounded operator is compact.",
"Suppose that the claim holds for a given $n$ .",
"Let $\\lambda \\in \\mathcal {B}^{n+1}(V,W)$ , and $a_1, \\ldots , a_{n+1} \\in \\mathcal {K}(U,V)$ .",
"For a fixed $u \\in U$ , define $\\nu _u \\in \\mathcal {B}^n(U,W)$ by $\\nu _u(u_1, \\ldots , u_n) = \\lambda (a_1 u_1, \\ldots , a_n u_n, a_{n+1} u),$ for $u_1, \\ldots , u_n \\in U$ .",
"For a fixed $v \\in V$ , define $\\lambda _{v} \\in \\mathcal {B}^n(V,W)$ by $\\lambda _v(v_1, \\ldots , v_n) = \\lambda (v_1, \\ldots , v_n, v),$ for $v_1, \\ldots , v_n \\in V$ .",
"Then for all $u_1, \\ldots , u_n \\in U$ , $\\nu _u(u_1, \\ldots , u_n) = \\lambda _{a_{n+1}u}(a_1 u_1, \\ldots , a_n u_n).$ So by the induction hypothesis, $\\nu _u \\in \\mathcal {K}^n(U,W)$ .",
"In this way, we obtain the map $\\tilde{\\nu }\\colon U \\rightarrow \\mathcal {K}^n(U,W),$ mapping $u \\in U$ to $\\nu _u$ .",
"It remains to show that $\\tilde{\\nu }$ is a compact operator.",
"Define $\\tilde{\\lambda }\\in \\mathcal {B}(V, \\mathcal {K}^n(U,W))$ by $\\tilde{\\lambda }(v)\\colon (u_1, \\ldots , u_n) \\mapsto \\lambda (a_1 u_1, \\ldots , a_n u_n, v),$ for $v \\in V$ and $u_1, \\ldots , u_n \\in U$ .",
"(This map takes values in $\\mathcal {K}^n(U,W)$ by the induction hypothesis.)",
"Since $a_{n+1}$ is compact and $\\tilde{\\lambda }$ is bounded, we find that $\\tilde{\\nu }= \\tilde{\\lambda }\\circ a_{n+1}$ is a compact operator."
],
[
"Compact polynomial maps",
"2 Definition 4.8 A compact homogeneous polynomial map of degree $n$ from $V$ to $W$ is a map of the form $p_{\\lambda }$ as in eq def monomial, for $\\lambda \\in \\mathcal {K}^n(V,W)$ .",
"We write $\\mathcal {K}\\operatorname{Pol}^n(V,W)$ for the space of such maps.",
"This space inherits a norm from the space $\\operatorname{Pol}^n(V,W)$ it is contained in.",
"If $\\lambda \\ne 0$ , then the degree of $p_{\\lambda }$ is $n$ .",
"A compact polynomial map from $V$ to $W$ is a finite sum of compact homogeneous polynomial maps.",
"The degree of a compact polynomial map is the degree of its highest-degree homogeneous term.",
"We write $\\mathcal {K}\\operatorname{Pol}(V,W)$ for the space of all compact polynomial maps between these spaces.",
"The isometric isomorphism $S\\mathcal {B}^n(V,W) \\cong \\operatorname{Pol}^n(V,W)$ restricts to an isometric isomorphism $S\\mathcal {K}^n(V,W) \\cong \\mathcal {K}\\operatorname{Pol}^n(V,W)$ .",
"So $\\mathcal {K}\\operatorname{Pol}^n(V,W)$ is a closed subspace of the Banach space $\\operatorname{Pol}^n(V,W)$ , and hence is a Banach space itself.",
"For every $w \\in W$ , an operator of the form $e^q \\otimes w$ , with $e^q$ as in eq def eq, is an element of $\\mathcal {K}^n(V,W)$ .",
"Indeed, $e^q \\otimes w$ is an iteration of rank-one operators, So $p^q \\otimes w \\in \\mathcal {K}\\operatorname{Pol}^n(V,W)$ .",
"The following proposition is the reason why we are interested in compact polynomial maps.",
"2 Proposition 4.9 Suppose that $V$ and $W$ are Banach spaces, that $V^*$ has a Schauder basis $\\lbrace e^j\\rbrace _{j=1}^{\\infty }$ , and that $W$ has a Schauder basis $\\lbrace f_k\\rbrace _{k=1}^{\\infty }$ .",
"Then the elements $ p^q \\otimes f_k \\quad \\in \\mathcal {K}\\operatorname{Pol}^n(V, W),$ where the multi-index $q$ ranges over the elements of $\\mathbb {N}_0^{\\infty }$ with $|q|=n$ , and $k$ ranges over the positive integers, form a Schauder basis of $\\mathcal {K}\\operatorname{Pol}^n(V,W)$ .",
"Consider the space $X := \\overline{\\operatorname{span}\\lbrace e^\\alpha \\otimes f_k :\\alpha \\in \\mathbb {N}^n, \\alpha _1 \\le \\cdots \\le \\alpha _n, k \\in \\mathbb {N}\\rbrace }.$ lem Kn Schauder implies that the set of $e^{\\alpha } \\otimes f_k$ where $\\alpha _1 \\le \\cdots \\le \\alpha _n$ is a Schauder basis of $X$ .",
"And $S\\colon X \\rightarrow S\\mathcal {K}^n(V,W)$ is a bounded linear isomorphism with bounded inverse.",
"Since such isomorphisms map Schauder bases to Schauder bases, we find that the set $Se^{\\alpha } \\otimes f_k$ , for non-decreasing $\\alpha $ as above, is a Schauder basis of $S\\mathcal {K}^n(V,W)$ .",
"For $\\alpha \\in \\mathbb {N}$ with non-decreasing entries, define $q(\\alpha ) \\in \\mathbb {N}_0^{\\infty }$ by $q(\\alpha )_j= \\# \\lbrace m \\in \\mathbb {N}: \\alpha _m = j\\rbrace .$ Then $e^{\\alpha } = e^{q(\\alpha )}$ .",
"(Note that $e^{\\alpha }$ , for $\\alpha \\in \\mathbb {N}^n$ , and $e^{q}$ , for $q\\in \\mathbb {N}_0^{\\infty }$ , are defined differently; compare eq def eq and eq def ealpha.)",
"Every sequence in $\\mathbb {N}_0^{\\infty }$ occurs in exactly one way as $q(\\alpha )$ , for alpha as above, so $Se^{q} \\otimes f_k$ , where $q \\in \\mathbb {N}^{\\infty }_0$ and $k \\in \\mathbb {N}$ , is a Schauder basis of $S\\mathcal {K}^n(V,W)$ .",
"Since $p^q = p_{Se^q}$ , the claim follows.",
"A reformulation of prop basis is that for every compact polynomial map $p \\in \\mathcal {K}\\operatorname{Pol}^n(V, W)$ , there are unique complex numbers $a^k_q$ such that $p = \\sum _{q, k} a^k_q p^q \\otimes f_k,$ where the sum converges in the norm on $\\operatorname{Pol}^n(V,W)$ .",
"Conversely, all polynomial maps $p$ of this form are compact.",
"2 Lemma 4.10 In the setting of lem uncond Kn, the set $\\lbrace p^q \\otimes f_k : |q| = n, k \\in \\mathbb {N}\\rbrace $ is an unconditional Schauder basis of $\\mathcal {K}\\operatorname{Pol}^n(V,W)$ .",
"The proof is analogous to the proof of prop basis, where we now use lem uncond Kn instead of lem Kn Schauder, and we use the fact that bounded linear isomorphisms with bounded inverses map unconditional Schauder bases to unconditional Schauder bases.",
"2 Lemma 4.11 If $p_1 \\in \\mathcal {K}\\operatorname{Pol}^m(U,V)$ and $p_2 \\in \\mathcal {K}\\operatorname{Pol}^n(V,W)$ , then $p_2 \\circ p_1 \\in \\mathcal {K}\\operatorname{Pol}^{mn}(U,W)$ .",
"The proof is similar to the proof of lem compos bdd pol, with bounded multilinear maps replaced by compact ones.",
"2 Remark 4.12 Other notions of compact polynomial maps were studied by Gonzalo, Jaramillo and Pełczyńsky in [15], [25]."
],
[
"Compactly differentiable maps",
"Let $V$ and $W$ be normed vector spaces, let $U \\subset V$ be an open subset containing a vector $u$ , and let $f\\colon U \\rightarrow W$ be $n$ times differentiable at $u$ .",
"2 Definition 4.13 The map $f$ is $n$ times compactly differentiable at $u$ if $f^{(n)}(u) \\in \\mathcal {K}^n(V,W).$ If $f$ is $n$ times compactly differentiable at $u$ , then by lem symm, $f^{(n)}(u) \\in S\\mathcal {K}^n(V,W).$ Then the map $h \\mapsto f^{(n)}(u)h^n$ of eq fn is the compact polynomial map associated to $f^{(n)}(u)$ .",
"Together with thm Taylor and prop basis, this leads to the following conclusion.",
"2 Corollary 4.14 Suppose that $V$ and $W$ are Banach spaces, that $V^*$ has a Schauder basis $\\lbrace e^j\\rbrace _{j=1}^{\\infty }$ , and that $W$ has a Schauder basis $\\lbrace f_k\\rbrace _{k=1}^{\\infty }$ .",
"Suppose $f$ is $n+1$ times differentiable, and $k$ times compactly differentiable for every $k \\le n$ .",
"Then there are unique complex numbers $a^k_q$ such that $ f(u+h) = \\sum _{q \\in \\mathbb {N}_0^{\\infty };\\, |q|\\le n}\\sum _{k=1}^{\\infty }a^k_q h^q f_k + O(\\Vert h\\Vert _W^{n+1}),$ where the part of the sum where $|q|=m$ converges as a function of $h$ in the norm on $\\operatorname{Pol}^m(V,W)$ , for $m = 0, \\ldots , n$ .",
"(Note that, in eq Taylor cpt 1, on the left-hand side $f$ is a map from $U$ to $W$ , whereas on the right-hand side, $f_k$ is an element of $W$ .)",
"2 Lemma 4.15 A compact polynomial map is infinitely compactly differentiable.",
"We show that the derivative of every homogeneous compact polynomial $p_{\\lambda } \\in \\mathcal {K}\\operatorname{Pol}^n(V,W)$ , for $\\lambda \\in S\\mathcal {K}^n(V,W)$ , is a compact polynomial in $\\mathcal {K}\\operatorname{Pol}^{n-1}(V,\\mathcal {K}(V,W))$ .",
"This implies the claim by induction on $n$ .",
"As in the proof of lem pol diffble, $p^{\\prime }_{\\lambda }(u)h = n\\lambda (h,u,\\ldots , u)$ for all $u,h \\in V$ .",
"In other words, $p^{\\prime }_{\\lambda }(u) = p_{\\nu }$ , with $\\nu = n\\lambda $ , where we view $\\lambda $ as an element of $\\mathcal {K}^{n-1}(V, \\mathcal {K}(V,W))$ .",
"This shows that $p^{\\prime }_{\\lambda }(u) \\in \\mathcal {K}\\operatorname{Pol}^{n-1}(V, \\mathcal {K}(V,W))$ .",
"2 Lemma 4.16 Let $U$ , $V$ and $W$ be normed vector spaces and let $f \\colon U \\rightarrow V$ and $g\\colon V \\rightarrow W$ be differentiable maps.",
"If either $f$ or $g$ is compactly differentiable, then so is $g \\circ f$ .",
"lem chain rule implies that for all $u \\in U$ , $(g\\circ f)^{\\prime }(u) = g^{\\prime }(f(u)) \\circ f^{\\prime }(u).$ If $f$ is compactly differentiable, then $f^{\\prime }(u) \\in \\mathcal {K}(U,V)$ .",
"If $g$ is compactly differentiable, then $g^{\\prime }(f(u)) \\in \\mathcal {K}(V,W)$ .",
"In either case, we find that $(g\\circ f)^{\\prime }(u) \\in \\mathcal {K}(U,V)$ .",
"2 Lemma 4.17 Let $U$ , $V$ and $W$ be normed vector spaces, let $f\\colon V \\rightarrow W$ be $n$ times differentiable, and suppose that $U$ is a subspace of $V$ , with compact inclusion map $j \\colon U \\hookrightarrow V$ .",
"Then $f \\circ j$ is $n$ times compactly differentiable as a map from $U$ to $W$ .",
"Let $u, h_1, \\ldots , h_n \\in U$ .",
"Then $(f \\circ j)^{(n)}(u)(h_1, \\ldots , h_n) = f^{(n)}(j(u))(j(h_1), \\ldots , j(h_n)).$ So the claim follows from lem ex Kn.",
"2 Remark 4.18 It is possible for a map to be $n$ times compactly differentiable, but not $n-1$ times.",
"For example, the first derivative of the identity operator on an infinite-dimensional Banach space $V$ is the identity map itself, and not a compact operator.",
"But its higher-order derivatives are zero, and hence compact."
],
[
"A class of compactly differentiable maps",
"We end this section by discussing a class of compactly differentiable maps (specifically, compact polynomials) that are relevant to the study of nonlinear pdes.",
"These maps are polynomial expressions in derivatives of functions; see prop ex cpt poly Sobolev below.",
"Let $\\Omega \\subset \\mathbb {R}^m$ be a bounded open subset with $C^1$ boundary.",
"For $k \\in \\mathbb {N}_0$ and $p>0$ , consider the Sobolev space $W^{k,p}(\\Omega )$ .",
"2 Lemma 4.19 Let $n \\in \\mathbb {N}$ and $p>1$ .",
"Pointwise multiplication of $n$ functions defines a map $\\mu \\in \\mathcal {B}^n(W^{k, np}(\\Omega ), W^{k,p}(\\Omega ))$ .",
"Let $n \\in \\mathbb {N}$ .",
"By Hölder's inequality, for all $u_1, \\ldots , u_n \\in L^{np}(\\Omega )$ , $ \\Vert u_1 \\cdots u_n\\Vert _{L^{p}(\\Omega )} \\le \\Vert u_1\\Vert _{L^{np}(\\Omega )} \\cdots \\Vert u_n\\Vert _{L^{np}(\\Omega )}.$ For $\\alpha \\in \\mathbb {N}_0^m$ , there are combinatorial constants $c^{\\alpha }_{\\beta }$ , for $\\beta = (\\beta ^{(1)}, \\ldots , \\beta ^{(n)})$ , with $\\beta ^{(1)}, \\ldots , \\beta ^{(n)} \\in \\mathbb {N}_0^m$ such that $|\\beta ^{(1)}|+ \\cdots + |\\beta ^{(n)}| \\le |\\alpha |$ , such that for all $u_1, \\ldots , u_n \\in C^{\\infty }_c(\\Omega )$ , we have the generalised Leibniz rule $\\frac{\\partial ^{\\alpha } (u_1 \\cdots u_n)}{\\partial x^{\\alpha }} = \\sum _{|\\beta ^{(1)}|+ \\cdots + |\\beta ^{(n)}| \\le |\\alpha |} c^{\\alpha }_{\\beta } \\frac{\\partial ^{\\beta ^{(1)}} u_1}{\\partial x^{\\beta ^{(1)}}} \\cdots \\frac{\\partial ^{\\beta ^{(n)}} u_n}{\\partial x^{\\beta ^{(n)}}}.$ Together with eq Holder 1, this implies that $\\Vert u_1 \\cdots u_n\\Vert _{W^{k, p}} \\le \\Bigl ( \\sum _{|\\alpha | \\le k} \\,\\, \\sum _{|\\beta ^{(1)}|+ \\cdots + |\\beta ^{(n)}| \\le |\\alpha |}c^{\\alpha }_{\\beta }\\Bigr ) \\Vert u_1\\Vert _{W^{k,np}} \\cdots \\Vert u_n\\Vert _{W^{k,np}}.$ 2 Proposition 4.20 Let $k, l,m_1, m_2, n \\in \\mathbb {N}$ and $p>1$ , with $k \\le l$ .",
"Let $D_1, \\ldots , D_n\\colon C^{\\infty }_c(\\Omega ; \\mathbb {R}^{m_1}) \\rightarrow C^{\\infty }_c(\\Omega ; \\mathbb {R}^{m_2})$ be linear partial differential operators of orders smaller than $k$ .",
"Fix complex numbers $a_q^j$ , for $q \\in \\mathbb {N}_0^{n}$ with $|q| \\le n$ , and $j \\in \\lbrace 1, \\ldots , m_2\\rbrace $ .",
"Define $f\\colon C_c^{\\infty }(\\Omega ; \\mathbb {R}^{m_1}) \\rightarrow C_c^{\\infty }(\\Omega ; \\mathbb {R}^{m_2})$ by $f(u) = (f_1(u), \\ldots , f_{m_2}(u))$ , where for each $j$ , $ f_j(u) = \\sum _{q \\in \\mathbb {N}_0^n, |q| \\le n} a_q^j(Du)^q$ for $u \\in W^{l,np}(\\Omega ; \\mathbb {R}^{m_1})$ , and with $(Du)^q$ as in eq Duq, defines a compact polynomial map $f \\in \\mathcal {K}\\operatorname{Pol}^n(W^{l,np}(\\Omega ; \\mathbb {R}^{m_1}), W^{l-k, p}(\\Omega ; \\mathbb {R}^{m_2})).$ We first consider the case where $m_2 = 1$ , and $a_q = 1$ if $q = (1,\\ldots , 1)$ , and zero otherwise.",
"By lem mult Sobolev, pointwise multiplication defines a map $\\mu \\in \\mathcal {B}^n(W^{l-k, np}(\\Omega ), W^{l-k, p}(\\Omega ))$ .",
"By Rellich's lemma, boundedness of $\\Omega $ implies that the maps $D_1, \\ldots , D_n$ extend to compact operators $D_1, \\ldots , D_n\\colon W^{l,p}(\\Omega )\\rightarrow W^{l-k,p}(\\Omega ).$ The map $\\nu \\colon W^{l, np}(\\Omega ) \\times \\cdots \\times W^{l, np}(\\Omega ) \\rightarrow W^{l-k, p}(\\Omega )$ defined by $\\nu (u_1, \\ldots , u_n) = \\mu (D_1 u_1, \\ldots , D_n u_n)$ for $u_1, \\ldots , u_n \\in W^{l,np}(\\Omega )$ , is in $\\mathcal {K}^n(W^{l, np}(\\Omega ), W^{l-k, p}(\\Omega ))$ by lem ex Kn.",
"Hence $p_{\\nu }$ is an element of $\\mathcal {K}\\operatorname{Pol}^n(W^{l, np}(\\Omega ), W^{l-k, p}(\\Omega ))$ .",
"Every component of a general map of the form eq f cpt poly Sob is a finite sum of maps of the form $p_{\\nu }$ as above, applied to the components of $f$ .",
"Hence it is in $\\mathcal {K}\\operatorname{Pol}^n(W^{l, np}(\\Omega ; \\mathbb {R}^{m_1}), W^{l-k, p}(\\Omega ; \\mathbb {R}^{m_2}))$ .",
"2 Example 4.21 Consider the map $f$ from sec ex burgers, mapping $u$ to $u^{\\prime } u$ .",
"We now view $f$ as a map from $W^{l,2p}(\\Omega )$ to $W^{l-k, p}(\\Omega )$ , for $k \\ge 1$ , $l \\ge k$ and $p>1$ .",
"Let $k \\ge 1$ , $l \\ge k$ and $p>1$ be integers.",
"Let $\\Omega \\subset \\mathbb {R}$ be a bounded open interval.",
"Consider the map $f$ from $W^{l,2p}(\\Omega )$ to $W^{l-k, p}(\\Omega )$ , mapping $u \\in W^{l,2p}(\\Omega )$ to $u^{\\prime } u$ .",
"Taking $m_1 = m_2 = 1$ , $n = 2$ , $D_1 u = u^{\\prime }$ and $D_2 u = u$ , for $u \\in W^{l,2p}(\\Omega )$ , in prop ex cpt poly Sobolev, we find that $f$ is a compact polynomial in $\\mathcal {K}\\operatorname{Pol}^2(W^{l,2p}(\\Omega ), W^{l-k,p}(\\Omega ))$ for every $k>1$ .",
"Hence, by lem cpt poly diffble, $f$ is in particular infinitely compactly differentiable.",
"For $k=1$ , the map $f$ is only a bounded polynomial in $\\operatorname{Pol}^2(W^{l,2p}(\\Omega ), W^{l-1, p}(\\Omega ))$ .",
"2 Remark 4.22 prop ex cpt poly Sobolev extends directly to relatively compact open subsets $\\Omega $ of manifolds.",
"The latter extension is relevant, for example, if one uses periodic boundary conditions, so that one works with with functions on a torus."
],
[
"Polynomial and differentiable maps on graded Fréchet spaces",
"Apart from polynomial and differentiable maps between normed vector spaces, we also use such maps between graded Fréchet spaces, defined in terms of nested sequences of Banach spaces, as in def nested seq.",
"In this section, we discuss some further properties of such spaces, and in particular what it means for two sequences of Banach spaces defining such a space to be comparable."
],
[
"Properties of nested sequences of Banach spaces",
"Let $\\lbrace V_k\\rbrace _{k=1}^{\\infty }$ be a nested sequence of Banach spaces, as in def nested seq.",
"Their intersection $V_{\\infty }$ is a graded Fréchet space.",
"2 Definition 5.1 The space $\\mathcal {B}^n(V_{\\infty })$ consists of the multilinear maps $\\lambda \\colon V_{\\infty } \\times \\cdots \\times V_{\\infty } \\rightarrow V_{\\infty }$ (with $n$ factors $V_{\\infty }$ ) such that for every $l \\in \\mathbb {N}$ , there is a $k \\in \\mathbb {N}$ such that $\\lambda $ extends continuously to a map in $\\mathcal {B}^n(V_k, V_l)$ .",
"Note that $\\mathcal {B}(V_{\\infty }) = \\mathcal {B}^1(V_{\\infty })$ .",
"2 Definition 5.2 We write $\\operatorname{Pol}^n(V_{\\infty })$ for the space of all maps $p\\colon V_{\\infty } \\rightarrow V_{\\infty }$ such that for every $l \\in \\mathbb {N}$ , there is a $k \\in \\mathbb {N}$ such that $p$ extends continuously to a polynomial in $\\operatorname{Pol}^n(V_k, V_l)$ .",
"The space $\\mathcal {K}\\operatorname{Pol}^n(V_{\\infty })$ is defined analogously.",
"A feature of the spaces $\\operatorname{Pol}^n(V_{\\infty })$ and $\\mathcal {K}\\operatorname{Pol}^n(V_{\\infty })$ that is useful to us, is that they admit natural compositions.",
"If $p_1 \\in \\operatorname{Pol}^m(V_{\\infty })$ and $p_2 \\in \\operatorname{Pol}^n(V_{\\infty })$ , then lem compos bdd pol implies that $p_2 \\circ p_1$ lies in $\\operatorname{Pol}^{mn}(V_{\\infty })$ .",
"Similarly, lem compos cpt pol implies that $p_2 \\circ p_1 \\in \\mathcal {K}\\operatorname{Pol}^{mn}(V_{\\infty })$ if $p_1 \\in \\mathcal {K}\\operatorname{Pol}^m(V_{\\infty })$ and $p_2 \\in \\mathcal {K}\\operatorname{Pol}^n(V_{\\infty })$ .",
"2 Definition 5.3 An $n$ times (compactly) differentiable map from $V_{\\infty }$ to itself is a map $f\\colon V_{\\infty } \\rightarrow V_{\\infty }$ such that for every $l \\in \\mathbb {N}$ , there is a $k \\in \\mathbb {N}$ such that $f$ extends to an $n$ times (compactly) differentiable map from $V_k$ to $V_l$ .",
"lem chain rule,lem cpt chain rule imply that the spaces of differentiable and compactly differentiable maps from $V_{\\infty }$ to itself are closed under composition.",
"lem cpt diff incl implies that if $\\lbrace V_k\\rbrace _{k=1}^{\\infty }$ is a compactly nested sequence of Banach spaces, then all $n$ times differentiable maps from $V_{\\infty }$ to itself are $n$ times compactly differentiable.",
"It follows directly from def On that if $f, g\\colon V_{\\infty } \\rightarrow V_{\\infty }$ are of orders $m$ and $n$ , respectively, then $g\\circ f$ is of order $m+n$ .",
"We also have the following lemma.",
"2 Lemma 5.4 Consider two maps $f, g\\colon V_{\\infty } \\rightarrow V_{\\infty }$ , where $f$ is differentiable and $g$ is of order $m$ .",
"Suppose that for every $l \\in \\mathbb {N}$ , there is a $k \\in \\mathbb {N}$ such that $f\\colon V_k\\rightarrow V_l$ is differentiable, and $\\Vert f^{\\prime }(v)\\Vert _{\\mathcal {B}(V_k, V_l)} = O(\\Vert v\\Vert ^n_{V_k})$ as $v \\rightarrow 0$ in $V_k$ .",
"Then the map $f^{\\prime } \\circ g\\colon V_{\\infty } \\rightarrow V_{\\infty }$ , mapping $v \\in V_{\\infty }$ to $f^{\\prime }(v)(g(v))$ is of order $n+m$ .",
"Let $l \\in \\mathbb {N}$ .",
"Let $k \\in \\mathbb {N}$ be such that $f\\colon V_k \\rightarrow V_l$ is differentiable, and its derivative satisfies the estimate in the lemma.",
"Let $k^{\\prime } \\in \\mathbb {N}$ be such that $\\Vert g(v)\\Vert _{V_k} = O(\\Vert v\\Vert ^m_{V{k^{\\prime }}})$ as $v \\rightarrow 0$ in $V_{k^{\\prime }}$ .",
"Then $\\Vert (f^{\\prime } \\circ g)(v)\\Vert _{V_l} = O(\\Vert v\\Vert _{V_{k^{\\prime }}}^{m+n})$ as $v \\rightarrow 0$ in $V_{k^{\\prime }}$ .",
"An example of a situation where the condition on $f$ in lem der Om+n is satisfied is the following.",
"2 Lemma 5.5 Let $p \\in \\operatorname{Pol}^n(V_{\\infty })$ , for $n \\ge 2$ .",
"Then for every $l \\in \\mathbb {N}$ , there is a $k \\in \\mathbb {N}$ such that $\\Vert p^{\\prime }(v)\\Vert _{\\mathcal {B}(V_k, V_l)} = O(\\Vert v\\Vert ^{n-1}_{V_k})$ as $v \\rightarrow 0$ in $V_k$ .",
"Let $p \\in \\operatorname{Pol}^n(V_{\\infty })$ , and let $k,l \\in \\mathbb {N}$ be such that $p \\in \\operatorname{Pol}^n(V_k, V_l)$ .",
"As in the proof of lem pol diffble, $p^{\\prime } \\in \\operatorname{Pol}^{n-1}(V_k, \\mathcal {B}(V_k, V_l))$ .",
"So the claim follows from lem order pol.",
"2 Remark 5.6 Everything in this subsection generalises directly to polynomial and (compactly) differentiable maps between two different Fréchet spaces that are given as intersections of nested sequences of Banach spaces.",
"We will not need this generalisation, however."
],
[
"Comparable sequences of Banach spaces",
"In the rest of this section, we discuss some relevant properties and examples of comparable sequences of Banach spaces (def comparable).",
"Particularly relevant to thm normal form are sequences of Banach spaces comparable to sequences of separable Hilbert spaces, which we discuss in sec comp Ban Hilb.",
"We will see relevant examples in sec ex Sob Ck.",
"Suppose that $\\lbrace V_k\\rbrace _{k=1}^{\\infty }$ and $\\lbrace W_k\\rbrace _{k=1}^{\\infty }$ are comparable nested sequences of Banach spaces.",
"Then $V_{\\infty } = W_{\\infty }$ as sets.",
"2 Lemma 5.7 The two spaces $V_{\\infty }$ and $W_{\\infty }$ are equal as Frećhet spaces.",
"Let $(v_j)_{j=1}^{\\infty }$ be a sequence in $V_{\\infty }$ such that for every $k \\in \\mathbb {N}$ , $\\lim _{j\\rightarrow \\infty } \\Vert v_j\\Vert _{V_k} = 0$ .",
"Let $k \\in \\mathbb {N}$ , and choose $l \\in \\mathbb {N}$ such that we have a bounded inclusion $V_l \\subset W_k$ .",
"Then there is a constant $C>0$ such that for every $j$ , $\\Vert v_j\\Vert _{W_k} \\le C \\Vert v_j\\Vert _{V_l}$ , which goes to zero as $j \\rightarrow \\infty $ .",
"The following fact follows directly from the definitions, and the fact that the classes of maps in question are closed under composition with bounded linear maps.",
"2 Lemma 5.8 If $f\\colon V_{\\infty } \\rightarrow V_{\\infty }$ is a (compact) polynomial map or a (compactly) differentiable map, then it also defines a map of the same type on $W_{\\infty }$ .",
"This lemma in particular states that $\\operatorname{Pol}^n(V_{\\infty }) = \\operatorname{Pol}^n(W_{\\infty })$ as vector spaces.",
"We will use the fact that this equality includes natural topologies on these spaces (cor PolV PolW below) to prove cor comp conv.",
"2 Lemma 5.9 For all $l \\in N$ , there is an $l^{\\prime } \\in \\mathbb {N}$ such that for every $k^{\\prime } \\in \\mathbb {N}$ with $k^{\\prime } \\ge l^{\\prime }$ , there is a $k \\in \\mathbb {N}$ such that we have a bounded inclusion map $\\mathcal {B}^n(V_{k^{\\prime }}, V_{l^{\\prime }}) \\subset \\mathcal {B}^n(W_{k}, W_{l})$ Let $l \\in \\mathbb {N}$ .",
"Choose $l^{\\prime } \\in \\mathbb {N}$ and $C_1>0$ such that for every $v \\in V_{\\infty }$ , $\\Vert v\\Vert _{W_l} \\le C_1 \\Vert v\\Vert _{V_{l^{\\prime }}}$ .",
"Let $k^{\\prime } \\ge l^{\\prime }$ .",
"Choose $k \\in \\mathbb {N}$ and $C_2>0$ such that for every $v \\in V_{\\infty }$ , $\\Vert v\\Vert _{V_{k^{\\prime }}} \\le C_2\\Vert v\\Vert _{{W_k}}$ .",
"Then for all $\\lambda \\in \\mathcal {B}^n(V_{k^{\\prime }}, V_{l^{\\prime }})$ , $\\sup _{\\Vert w_1\\Vert _{W_k}, \\ldots , \\Vert w_n\\Vert _{W_k} \\le 1} \\Vert \\lambda (w_1, \\ldots , w_n)\\Vert _{W_l}\\le C_1 C_2^n \\sup _{\\Vert v_1\\Vert _{V_{k^{\\prime }}}, \\ldots , \\Vert v_n\\Vert _{V_{k^{\\prime }}} \\le 1} \\Vert \\lambda (v_1, \\ldots , v_n)\\Vert _{V_{l^{\\prime }}}\\\\\\le C_2 C_2^n \\Vert \\lambda \\Vert _{\\mathcal {B}^n(V_{k^{\\prime }}, V_{l^{\\prime }})}\\,.$ For a sequence $(p_j)_{j=1}^{\\infty }$ in $\\operatorname{Pol}^n(V_{\\infty })$ , we define $p_j \\rightarrow 0$ in $\\operatorname{Pol}^n(V_{\\infty })$ to mean that for every $l \\in \\mathbb {N}$ , there is a $k\\in \\mathbb {N}$ such that $p_j \\rightarrow 0$ in $\\operatorname{Pol}^n(V_k, V_l)$ .",
"(This includes the requirement that $p_j \\in \\operatorname{Pol}^n(V_k, V_l)$ for every $j$ .)",
"2 Corollary 5.10 We have $\\operatorname{Pol}^n(V_{\\infty }) = \\operatorname{Pol}^n(W_{\\infty })$ , including topologies.",
"Let $(p_j)_{j=1}^{\\infty }$ be a sequence in $\\operatorname{Pol}^n(V_{\\infty })$ converging to zero.",
"Let $l \\in \\mathbb {N}$ .",
"Choose $l^{\\prime }$ as in lem Bn Vk Wk.",
"Choose $k^{\\prime } \\in \\mathbb {N}$ such that $p_j \\rightarrow 0$ in $\\operatorname{Pol}^n(V_{k^{\\prime }}, V_{l^{\\prime }})$ .",
"Choose $k \\in \\mathbb {N}$ as in lem Bn Vk Wk.",
"For each $j$ , write $p_j = p_{\\lambda _j}$ , for $\\lambda _j \\in S\\mathcal {B}^n(V_{k^{\\prime }}, V_{l^{\\prime }})$ .",
"By lem Bn Vk Wk, there is a $C>0$ such that for every $j$ , $\\Vert \\lambda _j\\Vert _{\\mathcal {B}(W_k, W_l)} \\le C \\Vert \\lambda _j\\Vert _{\\mathcal {B}(V_{k^{\\prime }}, V_{l^{\\prime }})},$ which goes to zero as $j\\rightarrow \\infty $ .",
"Hence $p_j \\rightarrow 0$ in $\\operatorname{Pol}^n(W_{\\infty })$ ."
],
[
"Sequences of Banach spaces comparable to sequences of Hilbert spaces",
"As before, we suppose that $\\lbrace V_k\\rbrace _{k=1}^{\\infty }$ and $\\lbrace W_k\\rbrace _{k=1}^{\\infty }$ are comparable nested sequences of Banach spaces.",
"Now we make the additional assumption that the spaces $W_k$ are separable Hilbert spaces.",
"Suppose that $\\lbrace e_j\\rbrace _{j=1}^{\\infty }$ is a subset of $W_{\\infty }$ that is orthogonal in all spaces $W_k$ , with dense span.",
"Then taking inner products with $e_j$ defines bounded functionals, all denoted by $e^j$ , on all spaces $W_k$ , and hence in $V_k$ for $k$ large enough.",
"2 Corollary 5.11 Suppose $f$ is an $n+1$ times differentiable map from $V_{\\infty }$ to itself, and that $f$ is $m$ times compactly differentiable for every $m \\le n$ .",
"Then there are unique complex numbers $a^k_q$ such that $ f(u+h) = \\sum _{q \\in \\mathbb {N}_0^{\\infty };\\, |q|\\le n}\\sum _{j=1}^{\\infty }a^j_q h^q e_j + \\rho (h),$ where $\\rho \\colon V_{\\infty } \\rightarrow V_{\\infty }$ is of order $n+1$ , and the part of the sum where $|q|=m$ converges as a function of $h$ in $\\operatorname{Pol}^m(V_{\\infty })$ , for $m = 0, \\ldots , n$ .",
"Let $l \\in \\mathbb {N}$ .",
"Choose $k, k^{\\prime }, l^{\\prime } \\in \\mathbb {N}$ be such that we have bounded inclusions $V_{l^{\\prime }} \\subset W_l$ and $W_k \\subset V_{k^{\\prime }}$ , and $f\\colon V_{k^{\\prime }} \\rightarrow V_{l^{\\prime }}$ is $n+1$ times differentiable, and $m$ times compactly differentiable for every $m \\le n$ .",
"Then the same is true for $f\\colon W_{k} \\rightarrow W_{l}$ .",
"So cor Taylor cpt implies that eq Taylor cpt holds, for unique $a_q^j$ , where the sum converges in $\\operatorname{Pol}(W_{\\infty })$ , and hence in $\\operatorname{Pol}(V_{\\infty })$ by cor PolV PolW, and $\\rho $ is of order $n+1$ as a map from $W_{\\infty }$ to itself, and hence as a map from $V_{\\infty }$ to itself.",
"2 Remark 5.12 A useful feature of cor Taylor cpt Vinfty is that it is not assumed that the spaces $V_k$ have the approximation property.",
"The point is that the separable Hilbert spaces $W_k$ do have this property.",
"The following corollary is an important way in which we use comparable sequences of Banach spaces.",
"It is used in the proof of lem conv hat psi hat F. 2 Corollary 5.13 Let $a_q^j \\in be given such that\\begin{equation} \\sum _{q \\in \\mathbb {N}_0^{\\infty };\\, |q|=n} \\sum _{j=1}^{\\infty } a_q^j p^q \\otimes e_j\\end{equation}converges in $ Poln(V)$.",
"Then for all subsets $ A {q N0 : |q|=n} N$, the series\\begin{equation} \\sum _{(q,j) \\in A} a_q^j p^q \\otimes e_j\\end{equation}converges in $ Poln(V)$.$ By cor PolV PolW, the series eq sum all converges in $\\operatorname{Pol}^n(W_{\\infty })$ , and it is enough to show that eq sum A converges in $\\operatorname{Pol}^n(W_{\\infty })$ .",
"And that follows from lem uncond KPn.",
"2 Remark 5.14 In cor comp conv, the two series converge to elements of $\\mathcal {K}\\operatorname{Pol}^n(V_{\\infty })$ ."
],
[
"Example: Sobolev spaces and $C^k$ -spaces",
"Let $\\Omega $ be a bounded open subset of $\\mathbb {R}^d$ or of an $d$ -dimensional Riemannian manifold, and suppose that the boundary of $\\Omega $ is $C^1$ .",
"Set $V_k := W^{k-1,2}(\\Omega ) \\quad \\text{and}\\quad W_k := W^{k-1, k+1}(\\Omega ).$ 2 Lemma 5.15 The above sequences $\\lbrace V_k\\rbrace _{k=1}^{\\infty }$ and $\\lbrace W_k\\rbrace _{k=1}^{\\infty }$ of Banach spaces are comparable.",
"Set $r:= \\lceil d/2\\rceil $ .",
"By a Sobolev embedding theorem, we have bounded inclusions $W^{l+ \\frac{d}{p}(1-p/2) , p}(\\Omega ) \\subset W^{l,2}(\\Omega )\\quad \\text{and}\\quad W^{l+r,2}(\\Omega ) \\subset W^{l+r-\\frac{d}{2}(1-2/q), q}(\\Omega ),$ for all $1\\le p<2<q<\\infty $ and every $l \\in \\mathbb {N}_0$ .",
"Now for all such $p,q$ and $l$ , $l +\\frac{d}{p}(1-p/2) < l+r\\quad \\text{and}\\quad l+r-\\frac{d}{2}(1-2/q) > l \\ge 1.$ So we have bounded inclusions $W^{ l+r, p}(\\Omega ) \\subset W^{l,2}(\\Omega )\\quad \\text{and}\\quad W^{l+r,2}(\\Omega ) \\subset W^{l, q}(\\Omega ).$ Furthermore, since $\\Omega $ has finite volume, we have bounded inclusions $W^{l, p^{\\prime }}(\\Omega ) \\subset W^{ l, p^{\\prime \\prime }}(\\Omega )$ for every $l$ and all $p^{\\prime }\\ge p^{\\prime \\prime }$ .",
"The above arguments imply that we have bounded inclusions $W_{k+r} \\subset V_k \\subset W_1\\quad \\text{and}\\quad V_{k+r}\\subset W_k \\subset V_k.$ 2 Remark 5.16 In this example, the spaces $V_k$ are separable Hilbert spaces.",
"For another example of comparable sequences of Banach spaces, fix $p>n$ .",
"For $k \\in \\mathbb {N}$ , set $V_k := W^{k, p}(\\Omega )\\quad \\text{and}\\quad W_k := C^{k}(\\overline{\\Omega }).$ The space $C^k(\\overline{\\Omega })$ is complete in the norm given by the maximum of the sup-norms of the partial derivatives of functions up to order $k$ .",
"2 Lemma 5.17 The sequences $\\lbrace V_k\\rbrace _{k=1}^{\\infty }$ and $\\lbrace W_k\\rbrace _{k=1}^{\\infty }$ of Banach spaces are comparable.",
"We have a bounded inclusion $W_k \\subset V_{k}$ for every $k$ .",
"So it remains to show that for every $k \\in \\mathbb {N}$ , there are $l_1, l_2 \\in \\mathbb {N}$ such that we have bounded inclusions $ \\begin{split}V_{l_1} &\\subset W_k;\\\\V_k &\\subset W_{l_2}.\\end{split}$ By a Sobolev embedding theorem, we have a bounded inclusion $W^{k,p}(\\Omega ) \\subset C^l(\\overline{\\Omega })$ for all $k,l \\in \\mathbb {N}$ such that $l+\\frac{n}{p}<k \\le l+1+\\frac{n}{p}.$ For $k \\in \\mathbb {N}$ , set $l_1 := k+1+\\lceil n/p\\rceil $ and $l_2 := \\max \\lbrace k-1-\\lfloor n/p \\rfloor , 1\\rbrace $ .",
"Then this Sobolev embedding theorem yields the desired inclusions eq Sob Ck incl.",
"2 Remark 5.18 If $n=1$ , then we may take $p=2$ , so that the spaces $V_k$ are Hilbert spaces.",
"prop ex cpt poly Sobolev and lem Sobolev comparable together imply thm special case.",
"The extension of prop ex cpt poly Sobolev to coeficients $a_q$ depending on a real (time) parameter $t$ in a smooth way, and the extension of lem Sobolev comparable to vector-valued functions, are straightforward."
],
[
"A residual",
"Recall the setting of sec setup.",
"In this section and the next, based upon the details of some given dynamical system eq ODE we construct both a coordinate transformation eq def xyz p and a corresponding `normal form' system eq ODE XYZ p, such that solutions $X$ to eq ODE XYZ p, transformed by eq def xyz p, satisfy the given dynamical system eq ODE up to residuals of a specified order $p$ .",
"See thm normal form.",
"We do this inductively, by showing how to construct such a transformed system to satisfy eq ODE with residual of order $p+1$ from a version with residual of order $p$ .",
"In sec transf choice, we construct a more specific choice of the general coordinate transform constructed in this section, in order to establish exact invariant manifolds, and study their properties, for constructed systems arbitrarily close to the given system eq ODE.",
"2 Remark 6.1 In sec setup, we assumed that the sequence $\\lbrace V_k\\rbrace _{k=1}^{\\infty }$ is comparable to a nested sequence $\\lbrace W_k\\rbrace _{k=1}^{\\infty }$ of separable Hilbert spaces in which the vectors $e_j$ are orthogonal.",
"lem comparable same maps implies that we may equivalently assume that $\\lbrace V_k\\rbrace _{k=1}^{\\infty }$ itself is a nested sequence of separable Hilbert spaces, because all maps from $V_{\\infty }$ to itself we use transfer to maps from $W_{\\infty }$ to itself of the same type (e.g.",
"compact polynomial and compactly differentiable maps).",
"However, the formulation where $\\lbrace V_k\\rbrace _{k=1}^{\\infty }$ is only comparable to a nested sequence of separable Hilbert spaces makes it clearer that we have the flexibility to consider maps between Banach spaces.",
"This is natural for example in the context of prop ex cpt poly Sobolev.",
"Let $p \\in \\mathbb {N}$ , with $p \\ge 2$ .",
"Let $\\xi _p, F_p\\colon I \\times V_{\\infty }\\rightarrow V_{\\infty }$ be such that $\\xi _p - \\operatorname{id}$ and $F_p$ are compact polynomial maps of order at most $p-1$ in the $V_{\\infty }$ component, and infinitely differentiable in $I$ .",
"Suppose, furthermore, that $\\xi _p$ is a near-identity at zero, and that $F_p = \\mathcal {O}(2)$ .",
"Recall that our goal is to relate the dynamics of maps $x$ satisfying eq ODE to the dynamics of maps $X\\colon I \\rightarrow V_{\\infty }$ satisfying eq ODE XYZ p when $x$ and $X$ are related by the coordinate transform $\\xi _p$ as in eq def xyz p. For maps $f,g\\colon I \\times V_{\\infty } \\rightarrow V_{\\infty }$ , with $f$ differentiable, we write $f^{\\prime }_{V_{\\infty }} \\circ g$ for the map from $I \\times V_{\\infty }$ to $V_{\\infty }$ given by $(f^{\\prime }_{V_{\\infty }} \\circ g)(t,v) = f^{\\prime }_{V_{\\infty }}(t,v)(g(t,v)),$ for all $t \\in I$ and $v \\in V_{\\infty }$ .",
"(Note that this is different from $(f\\circ g)^{\\prime }_{V_{\\infty }}(t,v) = f^{\\prime }_{V_{\\infty }}(t, g(t,v))(g(t,v))$ .)",
"If $g$ is a map from $V_{\\infty }$ to $V_{\\infty }$ to itself, then the composition $f^{\\prime }_{V_{\\infty }} \\circ g$ is defined analogously.",
"Also recall the notation for compositions of maps to and from $I \\times V_{\\infty }$ and $V_{\\infty }$ under Notation and conventions in secNota.",
"Define the maps $\\Phi _p, R_p\\colon I \\times V_{\\infty } \\rightarrow V_{\\infty }$ by $\\Phi _p:= (\\xi _p)^{\\prime }_I + (\\xi _p)^{\\prime }_{V_{\\infty }} \\circ (A + F_p)\\quad \\text{and}\\quad R_p := -A\\circ \\xi _p-f \\circ \\xi _p+\\Phi _p.$ The map $R_p$ is the residual of the transformed ode, in the following sense.",
"2 Lemma 6.2 For all smooth maps $X\\colon I \\rightarrow V_{\\infty }$ satisfying eq ODE XYZ p, and with $x\\colon I \\rightarrow V_{\\infty }$ determined from $X$ by eq def xyz p, $ \\dot{x}(t) = Ax(t) + f(t, x(t)) + R_p(t, X(t)).$ For $X$ and $x$ as in the lemma, the chain rule (lem chain rule) and eq ODE XYZ p imply that for all $t \\in I$ , $\\dot{x}(t) = \\Phi _p(t, X(t)) = Ax(t) + f(t, x(t)) + R_p(t, X(t)).$ 2 Lemma 6.3 The maps $\\Phi _p$ and $R_p$ are infinitely compactly differentiable.",
"Because the Banach spaces $V_k$ are compactly nested, it is enough to show that $\\Phi _p$ and $R_p$ are infinitely differentiable.",
"And that is true by the chain rule, because $f$ is infinitely differentiable, and so are $F_p$ and $\\xi _p$ , by lem pol diffble.",
"To recursively construct eq ODE XYZ p and eq def xyz p, suppose that $R_p = \\mathcal {O}(p)$ .",
"We proceed to show that eq ODE XYZ p and eq def xyz p may be refined to make the new residual of $\\mathcal {O}(p+1)$ .",
"By lem Rp diffble, cor Taylor cpt Vinfty (where $u=0$ and $h=v$ ), and lem order pol,lem order pol lower, there are unique, infinitely differentiable maps $a^{q}\\colon I \\rightarrow V_{\\infty }$ for all multi-indices $q \\in \\mathbb {N}_0^{\\infty }$ with $|q| = p$ , and a map $\\rho _p\\colon I \\times V_{\\infty } \\rightarrow V_{\\infty }$ such that for all $t \\in I$ , and $v \\in V_{\\infty }$ , $ R_p(t, v) = - \\sum _{q \\in \\mathbb {N}_0^{\\infty };\\, |q|= p} a^{q}(t)v^q + \\rho _p(t,v),$ where the sum converges in $\\operatorname{Pol}^p(V_{\\infty })$ , differentiably in $t$ , and $\\rho _p = \\mathcal {O}(p+1)$.",
"The monomial $v^q$ is defined as in eq def vq, with respect to the set of functionals $e^j = (e_j, )_{V_1}$ , for $j \\in \\mathbb {N}$ .",
"In sec coord transf, we construct maps $\\xi _{p+1}, F_{p+1}$ such that the order $p$ term $R_p(t, X(t))$ in eq ODE xyz may be replaced by an order $p+1$ term $R_{p+1}(t, X(t))$ , if eq ODE XYZ p and eq def xyz p hold with $p$ replaced by $p+1$ ."
],
[
"Construction of the coordinate transform",
"For each $j \\in \\mathbb {N}$ , let $\\alpha _j$ be the eigenvalue of $A$ corresponding to $e_j$ .",
"For all $q \\in \\mathbb {N}_0^{\\infty }$ with $|q| = p$ , define $\\mu ^q \\in by \\begin{equation} \\mu ^{q} = \\sum _{j=1}^{\\infty } q_j \\alpha _j \\quad \\in \\end{equation}(This sum has at most $ p$ nonzero terms.",
")For such $ q N0$, let $ aq$ be as in {eq aqrs}.",
"Let$ q, Fq I V $be smooth maps such that\\begin{equation} \\hat{F}^{q} + (\\hat{\\xi }^{q})^{\\prime } + \\mu ^{q}\\hat{\\xi }^{q} - A\\hat{\\xi }^{q} = a^{q}.\\end{equation}Suppose thatthe sums$$\\hat{F}(t,v) =\\sum _{q \\in \\mathbb {N}_0^{\\infty };\\, |q| = p} \\hat{F}^{q}(t) v^q\\quad \\text{and}\\quad \\hat{\\xi }(t,v) =\\sum _{q \\in \\mathbb {N}_0^{\\infty };\\, |q| = p} \\hat{\\xi }_{q}(t)v^q$$converge in $ Polp(V)$, differentiably in $ t$.$ Define a new coordinate transform map $\\xi _{p+1}\\colon I \\times V_{\\infty } \\rightarrow V_{\\infty }$ and corresponding map $F_{p+1}\\colon I \\times V_{\\infty } \\rightarrow V_{\\infty }$ that replaces $F_p$ in eq ODE XYZ p, by $ \\xi _{p+1}= \\xi _{p}+ \\hat{\\xi }\\quad \\text{and}\\quad F_{p+1} = F_{p}+ \\hat{F}.$ The following result is the main step in the construction of the coordinate transform we are looking for.",
"2 Proposition 6.4 If $X$ and $x$ are as in eq ODE XYZ p and eq def xyz p, with $p$ replaced by $p+1$ , then eq ODE xyz holds, with the residual $R_p$ replaced by a residual $R_{p+1}$ satisfying $R_{p+1}= \\mathcal {O}(p+1).$ The maps $\\xi _{p+1} - \\operatorname{id}$ and $F_{p+1}$ are compact polynomials in $\\mathcal {K}\\operatorname{Pol}(V_{\\infty })$ of order at most $p$ , and $\\xi _{p+1}$ is a near-identity.",
"2 Remark 6.5 The maps $\\hat{\\xi }^q$ and $\\hat{F}^q$ can be found explicitly if we decompose eq DE iteration with respect to the basis $\\lbrace e_j\\rbrace _{j=1}^{\\infty }$ .",
"This will be done in sec transf choice.",
"One solution to eq DE iteration is $\\hat{F}^q = a^q$ and $\\hat{\\xi }^q = 0$ .",
"However, for our purposes, we need the function $F^q$ to be of a specific form.",
"The main purpose of this work is to find $\\hat{F}^q$ such that the $e_j$ -component of $F^{q}$ is zero for certain combinations of $q$ and $j$ , in such a way that an exact separation of stable, centre and unstable modes is maintained.",
"See prop csu."
],
[
"Proof of prop coord transf",
"Define the map $\\hat{\\Phi }\\colon I \\times V_{\\infty } \\rightarrow V_{\\infty }$ by $ \\hat{\\Phi }= \\hat{\\xi }^{\\prime }_I+ \\hat{\\xi }^{\\prime }_{V_{\\infty }}\\circ (A + F_p)+ (\\xi _p)^{\\prime }_{V_{\\infty }}\\circ \\hat{F} +\\hat{\\xi }^{\\prime }_{V_{\\infty }}\\circ \\hat{F}.$ 2 Lemma 6.6 We have $ \\hat{\\Phi }- (\\hat{\\xi }^{\\prime }_I + \\hat{\\xi }^{\\prime }_{V_{\\infty }} \\circ A + \\hat{F}) = \\mathcal {O}(p+1).$ The left-hand side of eq diff hat Phi equals $\\hat{\\xi }^{\\prime }_{V_{\\infty }} \\circ F_p + \\hat{\\xi }^{\\prime }_{V_{\\infty }} \\circ \\hat{F} + ((\\xi _p)^{\\prime }_{V_{\\infty }} -\\operatorname{id})\\circ \\hat{F}.$ By lem der poly On-1, the derivative $\\hat{\\xi }^{\\prime }_{V_{\\infty }}$ satisfies the condition of lem der Om+n, with $n=p-1$ .",
"Since $F_p = \\mathcal {O}(2)$ , lem der Om+n implies that $\\hat{\\xi }^{\\prime }_{V_{\\infty }} \\circ F_p = \\mathcal {O}(p+1)$ .",
"Similarly, $\\hat{\\xi }^{\\prime }_{V_{\\infty }} \\circ \\hat{F} = \\mathcal {O}(2p-1)$ .",
"Now $\\xi _p$ is a polynomial map, and a near-identity.",
"By lem der poly On-1, this implies that $(\\xi _p)^{\\prime }_{V_{\\infty }} -\\operatorname{id}$ satisfies the condition of lem der Om+n, with $n=1$ .",
"So lem der Om+n implies that $((\\xi _p)^{\\prime }_{V_{\\infty }} -\\operatorname{id})\\circ \\hat{F} = \\mathcal {O}(p+1)$ .",
"2 Lemma 6.7 For $q \\in \\mathbb {N}_0^{\\infty }$ such that $|q| = p$ , let $a^{q}$ be as in eq aqrs.",
"Then for all $t \\in I$ and $v \\in V_{\\infty }$ , $ \\hat{\\xi }_I^{\\prime }(t, v) + \\hat{\\xi }^{\\prime }_{V_{\\infty }}(t, v) Au \\\\+ \\hat{F}(t, v) = A\\hat{\\xi }(t, v) +\\sum _{q \\in \\mathbb {N}_0^{\\infty };\\, |q| = p} a^{q}(t)v^q.$ First of all, $\\hat{\\xi }^{\\prime }_I(t, v) = \\sum _{q \\in \\mathbb {N}_0^{\\infty };\\, |q|= p} (\\hat{\\xi }^{q})^{\\prime }(t)v^q.$ By lem der poly, $\\hat{\\xi }^{\\prime }_{V_{\\infty }}(t, v) Av =\\\\\\sum _{q \\in \\mathbb {N}_0^{\\infty };\\, |q|= p} \\hat{\\xi }^{q}(t) \\sum _{j=1}^{\\infty } q_j(e_j, v)_{V_1}^{q_j-1} (e_j, Av)_{V_1}\\prod _{j^{\\prime }\\ne j} (e_{j^{\\prime }}, v)_{V_1}^{q_{j^{\\prime }}}.$ Now, because the vectors $\\lbrace e_j\\rbrace _{j=1}^{\\infty }$ are orthogonal with respect to $(, )_{V_1}$ , we have $(e_j, Av)_{V} = \\alpha _j (e_j, v)_{V_1}$ for every $j$ .",
"So $\\hat{\\xi }^{\\prime }_{V_{\\infty }}(t, v) Av =\\sum _{q\\in \\mathbb {N}_0^{\\infty };\\, |q| = p} \\hat{\\xi }^{q}(t) \\Bigl (\\sum _{j=1}^{\\infty } q_j \\alpha _j \\Bigr )v^q.$ We find that the left-hand side of eq hat phi aqrs equals $\\sum _{q \\in \\mathbb {N}_0^{\\infty };\\, |q|= p} \\Bigl ((\\hat{\\xi }^{q})^{\\prime }(t) + \\hat{F}^{q}(t) + \\mu ^{q} \\hat{\\xi }^{q}(t)\\Bigr )v^q,$ with $\\mu ^{q}$ as in eq def nuqrs.",
"So the claim follows from eq DE iteration.",
"Define the maps $\\Phi _{p+1}, R_{p+1}\\colon I \\times V_{\\infty } \\rightarrow V_{\\infty }$ by $\\Phi _{p+1} := \\Phi _p + \\hat{\\Phi }\\quad \\text{and}\\quad R_{p+1} := -A\\circ \\xi _{p+1}-f \\circ \\xi _{p+1}+\\Phi _{p+1}.$ 2 Lemma 6.8 The residual $R_{p+1}$ satisfies $R_{p+1} = \\mathcal {O}(p+1).$ By lem hat Phi p,lem hat phi aqrs, $ \\begin{split}R_{p+1} &= -A \\circ \\xi _p- f\\circ \\xi _{p+1} + \\Phi _p - \\tilde{R}_p+ \\mathcal {O}(p+1)\\\\&= R_p - f \\circ \\xi _{p+1} + f\\circ \\xi _p - \\tilde{R}_p+ \\mathcal {O}(p+1),\\end{split}$ where, for $t \\in I$ and $v \\in V_{\\infty }$ , $\\tilde{R}_p(t,v) := -\\sum _{q \\in \\mathbb {N}_0^{\\infty };\\, |q| = p} a^{q}(t)v^q.$ By eq aqrs, the last expression in eq Rp+1 Op+1 equals $f \\circ \\xi _{p+1} - f \\circ \\xi _{p} + \\mathcal {O}(p+1).$ Let $l \\in \\mathbb {N}$ be given, and choose $k, k^{\\prime } \\in \\mathbb {N}$ such that $f\\colon I \\times V_{k^{\\prime }} \\rightarrow V_l$ is differentiable, and $\\xi _p, \\xi _{p+1} \\in \\operatorname{Pol}(V_{k}, V_{k^{\\prime }})$ .",
"Using thm Taylor, we write $\\bigl \\Vert f(t,\\xi _{p+1}(t,v)) - f(t,\\xi _{p}(t,v)) - f^{\\prime }(t, \\xi _p(t,v)) \\hat{\\xi }(t,v)\\bigr \\Vert _{V_l} = O(\\Vert \\hat{\\xi }(t,v)\\Vert ^2),$ for $t \\in I$ and $v \\in V_{k^{\\prime }}$ .",
"lem order pol implies that $\\Vert \\hat{\\xi }(t,v)\\Vert _{V_{k^{\\prime }}} = O(\\Vert v\\Vert _{V_k})$ uniformly in $t$ in compact sets.",
"So $\\bigl \\Vert f(t,\\xi _{p+1}(t,v)) - f(t,\\xi _{p}(t,v))\\bigr \\Vert _{V_l}= \\bigl \\Vert f^{\\prime }(t, \\xi _p(t,v)) \\hat{\\xi }(t,v)\\bigr \\Vert _{V_l} + O(\\Vert v\\Vert _{V_k}^{2p}),$ uniformly in $t$ in compact sets.",
"And then, as in the proof of lem der Om+n, the assumption eq est der f and lem der Om+n imply that $\\Vert f^{\\prime }(t, \\xi _p(t,v)) \\hat{\\xi }(t,v)\\Vert _{V_l} = O(\\Vert v\\Vert _{k^{\\prime }}^{p+1})$ , uniformly in $t$ in compact sets.",
"[Proof of prop coord transf] The correction terms $\\hat{\\xi }$ and $\\hat{F}$ lie in $\\mathcal {K}\\operatorname{Pol}^p(V_{\\infty })$ .",
"Hence $\\xi _{p+1}-\\operatorname{id}$ and $F_{p+1}$ are compact polynomials, because $\\xi _{p}-\\operatorname{id}$ and $F_{p}$ are.",
"By lem order pol, this also implies that $\\xi _{p+1}$ is a near-identity because $\\xi _p$ is.",
"The desired property of $R_{p+1}$ is lem Rp+1."
],
[
"Centre, stable and unstable coordinates",
"There is considerable flexibility in choosing the maps $\\hat{\\xi }^{q}$ and $\\hat{F}^{q}$ in sec coord transf.",
"In this section, we discuss how to make specific choices, in terms of the eigenvalues of $A$ , so that the normal form eq ODE XYZ p is useful for detecting invariant manifolds."
],
[
"Centre, stable and unstable components of $\\hat{F}^q$",
"We use the notation from sec result.",
"In particular, let $\\alpha $ , $\\beta $ , $\\gamma $ and $\\tilde{\\mu }$ be spectral gap parameters defined there.",
"Recall the definition of polynomial growth in def pol growth.",
"2 Proposition 7.1 Suppose that $\\beta - (p+1)\\alpha > \\tilde{\\mu }$ and $\\gamma - (p+1)\\alpha > \\tilde{\\mu }$ .",
"Suppose that $R_p$ has polynomial growth.",
"The maps $\\hat{\\xi }^q$ and $\\hat{F}^q$ in sec coord transf can be chosen such that if either $q^s = 0$ and $q^u \\ne 0$ , or $q^u = 0$ and $q^s \\ne 0$ , then $\\hat{F}^q_c = 0$ ; if $q^s = 0$ , then $\\hat{F}^q_s = 0$ ; and if $q^u = 0$ , then $\\hat{F}^q_u = 0$ .",
"prop csu is proved in sec update, after some preparation done in this subsection.",
"2 Definition 7.2 Let $\\mu \\in , such that $ |()| > $.",
"Set $ a:= I$ and $ b:= I$.Let $ u$ be a continuous function on $ R$ such that $ u(t) = O(e|t|)$ as $ t $ if $ b = $ and as $ t -$ if $ a = -$.Then we define the function $ e() u$ on~$ I$ by$$(e^{\\mu (\\cdot )} \\star u)(t) := {\\left\\lbrace \\begin{array}{ll}\\int _{a}^t e^{\\mu (t-\\tau )}u(\\tau )\\, d\\tau & \\text{if $\\Re (\\mu ) < -\\tilde{\\mu }$};\\\\\\int _{t}^{b} e^{\\mu (t-\\tau )}u(\\tau )\\, d\\tau & \\text{if $\\Re (\\mu ) > \\tilde{\\mu }$}.\\end{array}\\right.",
"}$$$ The integrals occurring in this definition are $\\tilde{\\mu }$ -regular, in the sense defined in sec result.",
"2 Lemma 7.3 In the setting of def conv emu, $(e^{\\mu (\\cdot )} \\star u)^{\\prime } = \\mu (e^{\\mu (\\cdot )} \\star u) - \\operatorname{sgn}(\\Re (\\mu ))u.$ This is a straightforward computation.",
"2 Lemma 7.4 Let $u\\colon I \\rightarrow be a smooth function, and suppose that $ u$ and all its derivatives grow at most polynomially.",
"Then, for every $$ as in {def conv emu}, $ e() u$ and all its derivatives grow at most polynomially.$ First of all, because $(e^{\\mu (\\cdot )} \\star u)^{(k)} = e^{\\mu (\\cdot )}\\star (u^{(k)})$ , it is enough to consider the function $u$ itself rather than all its derivatives.",
"Let $C > 0$ and $n \\in \\mathbb {N}_0$ be such that for all $t \\in I$ , $|u(t)| \\le C(1+ |t|^{n})$ .",
"We prove by induction on $n$ that there is a constant $C^{\\prime } > 0$ such that for all $t \\in I$ , $|(e^{\\mu (\\cdot )} \\star u)(t)| \\le C^{\\prime }(1+| t|^{n})$ .",
"We consider the case where $\\Re (\\mu ) < -\\tilde{\\mu }$ ; the case where $\\Re (\\mu ) > \\tilde{\\mu }$ is similar.",
"If $n = 0$ , then for all $t \\in I$ , $\\bigl | (e^{\\mu (\\cdot )} \\star u)(t) \\bigr | \\le 2C \\int _{-\\infty }^{t} e^{\\mu (t-\\tau )}\\, d\\tau = \\frac{-2C}{\\mu }.$ Suppose that the claim holds for $n$ , and suppose that $|u(t)| \\le C(1+ |t|^{n+1})$ for a constant $C$ .",
"Using integration by parts, we find that $\\bigl | (e^{\\mu (\\cdot )} \\star u)(t) \\bigr |\\le C \\int _{-\\infty }^{t} e^{\\mu (t-\\tau )}(1+|\\tau |^{n+1})\\, d\\tau \\\\= \\frac{-C}{\\mu }\\Bigl (1+|t|^{n+1} - (n+1) \\int _{-\\infty }^{t} e^{\\mu (t-\\tau )}\\operatorname{sgn}(\\tau )|\\tau |^{n}\\, d\\tau \\Bigr ),$ which implies the claim by the induction hypothesis.",
"Let $q \\in \\mathbb {N}_0^{\\infty }$ , with $|q|=p$ .",
"Consider the differentiable maps $a^q_j\\colon I \\rightarrow \\text{such that}\\quad a^q(t) = \\sum _{j=1}^{\\infty }a^q_j(t) e_j,$ where the sum converges in $V_1$ , uniformly and differentiably in $t$ in compact sets in $I$ .",
"For $q \\in \\mathbb {N}_0^{\\infty }$ with $|q|=p$ , let $J^q \\subset \\mathbb {N}$ be the set of $j \\in \\mathbb {N}$ such that either $j \\in J_c$ and either $q^s = 0$ and $q^u \\ne 0$ , or $q^u = 0$ and $q^s \\ne 0$ ; $j \\in J_s$ and $q^s = 0$ ; or $j \\in J_u$ and $q^u = 0$ .",
"For $q \\in \\mathbb {N}_0^{\\infty }$ with $|q|=p$ , and $j \\in \\mathbb {N}$ , write $\\mu ^q_j := \\mu ^{q}- \\alpha _j$ , with $\\mu ^q$ as in eq def nuqrs.",
"2 Lemma 7.5 If $\\beta - (p+1)\\alpha > \\tilde{\\mu }$ and $\\gamma - (p+1)\\alpha > \\tilde{\\mu }$ , then for every $j \\in J^q$ , $|\\Re (\\mu ^q_j)| > \\tilde{\\mu }$ .",
"If $q^s = 0$ and $q^u \\ne 0$ , and $j \\in J_c$ , then $\\Re (\\mu _j^q) = \\sum _{k \\in J_c} q_k \\Re (\\alpha _k) + \\sum _{k \\in J_u} q_k \\Re (\\alpha _k) - \\Re (\\alpha _j) \\ge \\gamma -(p+1) \\alpha >\\tilde{\\mu }.$ If $q^u = 0$ and $q^s \\ne 0$ , and $j \\in J_c$ , then $\\Re (\\mu _j^q) =\\sum _{k \\in J_c} q_k \\Re (\\alpha _k) + \\sum _{k \\in J_s} q_k \\Re (\\alpha _k) - \\Re (\\alpha _j)\\le -\\beta + (p+1) \\alpha < -\\tilde{\\mu }.$ If $q^s = 0$ , and $j \\in J_s$ , then $\\Re (\\mu _j^q) = \\sum _{k \\in J_c} q_k \\Re (\\alpha _k) + \\sum _{k \\in J_u} q_k \\Re (\\alpha _k) - \\Re (\\alpha _j) \\ge \\beta - p\\alpha > \\tilde{\\mu }.$ And if $q^u = 0$ , and $j \\in J_u$ , then $\\Re (\\mu _j^q) = \\sum _{k \\in J_c} q_k \\Re (\\alpha _k) + \\sum _{k \\in J_s} q_k \\Re (\\alpha _k) - \\Re (\\alpha _j) \\le -\\gamma + p\\alpha < \\tilde{\\mu }.$"
],
[
"Update terms for $\\xi _p$ and {{formula:40947836-bcf1-4f48-9b3e-2af2053a0a89}}",
"Suppose that $R_p$ has polynomial growth.",
"Then the functions $a^q_j$ and their derivatives grow at most polynomially, uniformly in $q$ and $j$ .",
"For every $q \\in \\mathbb {N}_0^{\\infty }$ with $|q|=p$ and $j \\in \\mathbb {N}$ , consider the ode for $\\hat{\\xi }^{q}_j$ and $\\hat{F}^{q}_j$ $ \\hat{F}^{q}_j + (\\hat{\\xi }^{q}_j)^{\\prime } + \\mu ^q_j \\hat{\\xi }^{q}_j = a^{q}_j.$ Define the maps $\\hat{\\xi }^{q}_j, \\hat{F}^{q}_j \\colon I \\rightarrow as follows.",
"If $ j Jq$, then\\begin{equation} \\hat{\\xi }^{q}_j =\\operatorname{sgn}(\\Re (\\mu _j^q)) e^{-\\mu _j^q (\\cdot )}\\star a^q_j\\quad \\text{and}\\quad \\hat{F}^{q}_j = 0.\\end{equation}This definition makes sense because of {lem Re muqj} and the growth behaviour of the functions $ aqj$.If $ j Jq$, then we set\\begin{equation} \\hat{\\xi }^{q}_j = 0\\quad \\text{and}\\quad \\hat{F}^{q}_j = a^q_j.\\end{equation}\\begin{lemma} With the above definitions, the \\textsc {ode}\\ {eq ODE j} is satisfied for all q and j.\\end{lemma}\\begin{proof}If j \\notin J^q, the statement is immediate from the definitions.",
"If j \\in J^q, itfollows from {lem conv}.\\end{proof}$ 2 Lemma 7.6 Suppose that $\\beta - (p+1)\\alpha > \\tilde{\\mu }$ and $\\gamma - (p+1)\\alpha > \\tilde{\\mu }$ .",
"Then the sums $ \\hat{F}(t,v) =\\sum _{q \\in \\mathbb {N}_0^{\\infty };\\, |q| = p} \\sum _{j \\in \\mathbb {N}} \\hat{F}^{q}_j(t) e_j v^q\\quad \\text{and}\\quad \\hat{\\xi }(t,v) =\\sum _{q \\in \\mathbb {N}_0^{\\infty };\\, |q| = p} \\sum _{j \\in \\mathbb {N}} \\hat{\\xi }^{q}_j(t) e_j v^q$ converge in $\\operatorname{Pol}^p(V_{\\infty })$ , differentiably in $t$ .",
"The first sum in eq sums Fq psiq 2 equals $ \\sum _{q \\in \\mathbb {N}_0^{\\infty };\\, |q| = p} \\sum _{j \\in \\mathbb {N}\\setminus J^q} a^{q}_j(t) e_j v^q.$ Since $\\lbrace V_{k}\\rbrace _{k=1}^{\\infty }$ is comparable to a nested sequence of separable Hilbert spaces in which the set $\\lbrace e_j\\rbrace _{j=1}^{\\infty }$ is orthogonal, cor comp conv implies that this series converges in $\\operatorname{Pol}^p(V_{\\infty })$ .",
"Write $J^q_{\\pm } := \\lbrace j \\in J^q : \\pm \\Re (\\mu ^q_j) > \\tilde{\\mu }\\rbrace .$ lem Re muqj states that $J^q = J^q_+ \\cup J^q_-$ .",
"So the second sum in eq sums Fq psiq 2 equals $\\sum _{q \\in \\mathbb {N}_0^{\\infty };\\, |q| = p} \\sum _{j \\in J^q_+}\\int _{-\\infty }^t e^{-\\mu ^q_j (t-\\tau )} a^q_j(\\tau ) \\, d\\tau \\, e_j v^q\\\\+ \\sum _{q \\in \\mathbb {N}_0^{\\infty };\\, |q| = p} \\sum _{j \\in J^q_-}\\int _{t}^{\\infty } e^{-\\mu ^q_j (t-\\tau )} a^q_j(\\tau ) \\, d\\tau \\, e_j v^q.$ Tonelli's theorem implies that convergence of the first of these sums is equivalent to convergence of $ \\int _{-\\infty }^te^{-\\mu ^q_j (t-\\tau )}\\Bigl (\\sum _{q \\in \\mathbb {N}_0^{\\infty };\\, |q| = p} \\sum _{j \\in J^q_+}a^q_j(\\tau ) e_j v^q \\Bigr ) \\, d\\tau $ The sum inside the brackets converges in $\\operatorname{Pol}^p(V_{\\infty })$ , uniformly in $\\tau $ , by convergence of eq aqrs and cor comp conv.",
"Since the functions $a^q_j$ grow at most polynomially, uniformly in $q$ and $j$ , the value of that sum grows at most polynomially as well, when viewed as a convergent series in $\\operatorname{Pol}^p(V_k, V_l)$ .",
"So the integral over $\\tau $ converges in $\\operatorname{Pol}^p(V_{\\infty })$ , by completeness of the spaces $\\operatorname{Pol}^p(V_k, V_l)$ .",
"By continuity of eq int J+ in $t$ , the convergence is uniform in $t$ on compact subsets of $I$ .",
"The derivatives of eq int J+ with respect to $t$ are linear combinations of eq int J+ and $\\sum _{q \\in \\mathbb {N}_0^{\\infty };\\, |q| = p} \\sum _{j \\in J^q_+}a^q_j(t) e_j v^q$ and therefore converge as well.",
"By an analogous argument, the second sum in eq sums Jpm converges as well, differentiably in $t$ .",
"prop csu follows from lem ODE j,lem conv hat psi hat F."
],
[
"Proof of thm normal form",
"2 Lemma 7.7 If $f, \\xi _p, F_p, \\Phi _p$ and $R_{p}$ have polynomial growth, then so do $\\xi _{p+1}$ , $F_{p+1}$ , $\\Phi _{p+1}$ and $R_{p+1}$ .",
"If $R_p$ has polynomial growth, then the functions $a^q_j$ and all their derivatives grow at most polynomially, uniformly in $q$ and $j$ .",
"Hence, by eq hat psi hat F Jq and eq hat psi hat F Jqc, the map $\\hat{F}$ has polynomial growth.",
"By lem conv pol growth, the same is true for $\\hat{\\xi }$ .",
"So $F_{p+1}$ and $\\xi _{p+1}$ have polynomial growth.",
"Polynomial growth is preserved under composition and derivatives in the $I$ and $V_{\\infty }$ directions.",
"Hence the map $\\hat{\\Phi }$ as in eq def hat Phi has polynomial growth, and therefore so do $\\Phi _{p+1}$ and $R_{p+1}$ .",
"Combining lem pol growth with prop coord transf,prop csu, we prove the following slightly more explicit version of thm normal form.",
"2 Theorem 7.8 Let $p \\in \\mathbb {N}$ be such that $p \\ge 2$ , $\\beta - (p+1)\\alpha > \\tilde{\\mu }$ and $\\gamma - (p+1)\\alpha > \\tilde{\\mu }$ .",
"Suppose that $f$ has polynomial growth.",
"Then there are infinitely differentiable maps $F_p, \\xi _p, R_p\\colon I \\times V_{\\infty } \\rightarrow V_{\\infty },$ where $R_p = \\mathcal {O}(p)$ , $F_p$ is a polynomial map that separates invariant subspaces, $\\xi _p$ is a near-identity and $\\xi _p - \\operatorname{id}$ and $F_p$ are compact polynomials of orders at most $p-1$ , such that if $X$ and $x$ are as in eq ODE XYZ p and eq def xyz p, then eq ODE xyz holds.",
"Finally, there is a construction of the map $\\xi _p$ in which all integrals over $I$ that occur are $\\tilde{\\mu }$ -regular.",
"We use induction on $p$ to prove that the claim holds for every $p$ , including the auxiliary statement that $\\xi _p, F_p, \\Phi _p$ and $R_{p}$ have polynomial growth.",
"If $p=2$ , then we may take $F_{p}(t,v) = 0$ and $\\xi _{p}(t,v) = v$ for all $t \\in I$ and $v \\in V_0$ .",
"Then $R_2 = f$ , so $\\xi _p, F_p, \\Phi _p$ and $R_{p}$ have polynomial growth because $f$ does.",
"The induction step follows from lem pol growth,prop coord transf,prop csu."
],
[
"Dynamics of the normal form equation",
"It remains to prove lem Vj invar and prop dynamics, which we use to justify def centre mfd based on thm normal form.",
"Throughout this section, we suppose that $F \\colon I \\times V_{\\infty } \\rightarrow V_{\\infty }$ is a smooth map that separates invariant subspaces.",
"[Proof of lem Vj invar.]",
"First, suppose that $a=c$ .",
"For all $v \\in V_c$ and all $q \\in \\mathbb {N}_0^{\\infty }$ with $|q| \\le p$ and $q^s \\ne 0$ or $q^u \\ne 0$ , we have $v^q = 0$ .",
"So the properties eq Fcsu of the map $F$ imply that $F(I \\times V_c) \\subset V_c$ .",
"This, in turn, implies that for all maps $X\\colon I \\rightarrow V_{\\infty }$ satisfying eq ODE XYZ p, if $X(t) \\in V_c$ for a given $t$ then $\\dot{X}(t) \\in V_c$ .",
"So $X(t) \\in V_c$ for all $t \\in I$ .",
"Next, suppose that $a=s$ .",
"If $v \\in V_s$ and $q \\in \\mathbb {N}_0^{\\infty }$ , then $v^q=0$ if $q^u \\ne 0$ .",
"So $F_u(t,v)=0$ for all $t \\in I$ .",
"And the components of $F_c(t,v)$ with $q^u \\ne 0$ are zero for the same reason, while its components with $q^s = 0$ are zero since $v \\in V_s$ .",
"Hence $F_c(t,v)=0$ .",
"We conclude that $F(I \\times V_s) \\subset V_s$ .",
"As in the case $a=c$ , this implies the claim for $a=s$ .",
"The argument for $a=u$ is entirely analogous to the case $a=s$ .",
"To prove prop dynamics, we start with a general comparison estimate for solutions of odes in Hilbert spaces.",
"2 Lemma 8.1 Let $V$ be a Hilbert space, $W \\subset V$ a subspace, $I$ an open interval containing 0, and $g$ a map from $I \\times V$ into the space of linear operators from $W$ to $V$ .",
"Let $X \\colon I \\rightarrow W$ be a differentiable map (as a map into $V$ ), such that for all $t \\in I$ , $\\dot{X}(t) = g(t, X(t))X(t).$ If $\\zeta \\in \\mathbb {R}$ is such that $g(t,w)+\\zeta $ is negative semidefinite for all $t \\in I$ and $w \\in W$ , then for all $t \\in I$ with $t\\ge 0$ , $\\Vert X(t)\\Vert _V \\le \\Vert X(0)\\Vert _V e^{-\\zeta t}$ First, suppose that $\\zeta = 0$ .",
"Then for all $t\\in I$ , $\\frac{d}{dt} \\Vert X(t)\\Vert _V^2 = 2\\Re (\\dot{X}(t), X(t))_V = 2\\Re (g(t, X(t))X(t), X(t))_V \\le 0.$ So $\\Vert X\\Vert _V^2$ is a nonnegative, non-increasing function on $I$ , and the claim for $\\zeta = 0$ follows.",
"Next, let $\\zeta \\in \\mathbb {R}$ be arbitrary.",
"Then $\\frac{d}{dt}(X(t)e^{\\zeta t}) = \\bigl (g(t, X(t))+\\zeta \\bigr )X(t)e^{\\zeta t}.$ Applying the claim for $\\zeta = 0$ , with $X(t)$ replaced by $X(t)e^{\\zeta t}$ and $g(t,w)$ by $g(t,w)+\\zeta $ , now yields the claim for $\\zeta $ .",
"For any homogeneous polynomial map $p = p_{\\lambda }$ between normed vector spaces $V$ and $W$ , where $\\lambda \\in S\\mathcal {B}^n(V,W)$ , define the map $\\tilde{p}\\colon V \\rightarrow \\mathcal {B}(V,W)$ by $ \\tilde{p}(v_1)v_2 := \\lambda (v_1, \\ldots , v_1, v_2).$ Here $n-1$ copies of $v_1$ are inserted into $\\lambda $ on the right hand side.",
"For all $t \\in I$ , the map $F(t, )$ lies in $\\operatorname{Pol}(V_{k}, V_1)$ for some $k$ .",
"The operator $A$ lies in $\\mathcal {B}(V_l, V_1)$ for some $l$ .",
"By replacing the smaller of $k$ or $l$ by the larger of these two numbers, we henceforth assume $k=l$ .",
"Applying the construction eq def tilde p to each homogeneous term of $F(t, )$ and adding the resulting maps, we obtain a map $\\tilde{F}\\colon V_k \\rightarrow \\mathcal {B}(V_k, V_1)$ , such that for all $v \\in V_k$ , $F(t,v) = \\tilde{F}(t,v)v.$ For $a \\in \\lbrace c,s,u\\rbrace $ , we write $\\tilde{F}_a$ for $\\tilde{F}$ composed with orthogonal projection onto $V_a$ .",
"For $j \\in \\mathbb {N}$ , let $q^{(j)} \\in \\mathbb {N}_0^{\\infty }$ be defined by $q^{(j)}_m = 1$ if $m=j$ , and $q^{(j)}_m = 0$ otherwise.",
"2 Lemma 8.2 Let $v \\in V_k$ .",
"Write $v = v_c + v_s+v_u$ , where $v_a \\in V_a$ for $a \\in \\lbrace c,s,u\\rbrace $ .",
"Then for all $t \\in I$ , the components of $F(t,v)$ in $V_s$ , $V_u$ and $V_c$ satisfy $(F(t,v))_s &= \\tilde{F}(t,v)v_s; \\\\(F(t,v))_u &= \\tilde{F}(t,v)v_u; \\\\(F(t,v))_c &= \\tilde{F}(t,v_c)v_c \\quad \\text{if $v_s = 0$ or $v_u = 0$.",
"}$ Let $t \\in I$ and $v \\in V_k$ .",
"To prove eq tilFp s, we use the fact that by eq sep invar s, $F_s (t,v)=\\sum _{j \\in J_s} \\,\\, \\sum _{q \\in \\mathbb {N}_0^{\\infty }: |q| \\le p-1} F^{q+ q^{(j)}}(t) v^q v_j.$ So $\\tilde{F}_s(t,v) = \\sum _{j \\in J_s} \\,\\, \\sum _{q \\in \\mathbb {N}_0^{\\infty }: |q| \\le p-1} F^{q+ q^{(j)}}(t) v^q e^j,$ where $\\lbrace e^j\\rbrace _{j \\in \\mathbb {N}}$ is the basis of $V_1^*$ dual to $\\lbrace e_j\\rbrace _{j \\in \\mathbb {N}}$ .",
"Hence $\\tilde{F}_s(t,v)v = \\tilde{F}(t,v) v_s,$ which implies eq tilFp s. The equality eq tilFp u can be proved anaogously.",
"To prove eq tilFp c, we note that by eq sep invar c, $F_c (t,v)= F_{c,1}(t,v) + F_{c,2}(t,v),$ where $F_{c,1}(t,v)&= \\sum _{q \\in \\mathbb {N}_0^{\\infty }: |q| \\le p,q^s = q^u=0} F^q(t) v^q; \\\\F_{c,2}(t,v)&= \\sum _{q \\in \\mathbb {N}_0^{\\infty }: |q| \\le p,q^s \\ne 0 \\ne q^u} F^q(t) v^q.",
"$ The right hand side of eq Fpc1 only depends on $v_c$ , and the right hand side of eq Fpc2 is zero if $v_s = 0$ or $v_u = 0$ .",
"So, under that condition, $F_c (t,v) = F_{c}(t,v_c) = \\tilde{F}(t,v_c)v_c.$ [Proof of prop dynamics.]",
"Set $ D_{\\tilde{\\mu }} := \\bigl \\lbrace (t,v) \\in I \\times V_{\\infty } : \\Vert \\tilde{F}(t,v)\\Vert _{\\mathcal {B}(V_k, V_1)} < \\tilde{\\mu }\\bigr \\rbrace .$ Because $F$ is a sum of polynomials of degrees at least two, we have $\\tilde{F}(t,0) = 0$ for all $t$ .",
"So $D_{\\tilde{\\mu }}$ contains $I \\times \\lbrace 0\\rbrace $ .",
"It is open by continuity of $\\tilde{F}$ .",
"Let $X\\colon I \\rightarrow V_{\\infty }$ be a solution of the constructed system eq ODE XYZ p. As in the proof of lem Vj invar, $ \\dot{X}_s(t) = AX(t) + F_s(t, X(t)) = \\bigl (A + \\tilde{F}(t, X(t) \\bigr ) X_s(t),$ where we used the first equality in lem tilFp and the fact that $A$ preserves $V_s$ .",
"For all $(t,v)\\in D_{\\tilde{\\mu }} $ , the operator $A + \\tilde{F}_s(t, v) + \\beta - \\tilde{\\mu }\\colon V_k \\cap V_s \\rightarrow V_1 \\cap V_s$ is negative semidefinite.",
"Hence the claim about $X_s$ follows from the second part of lem decay to Vc.",
"The claim about $X_u$ can be proved similarly, via a version of lem decay to Vc for positive-definite operators.",
"Next, suppose that $X_s(0) = 0$ or $X_u(0) = 0$ .",
"By lem Vj invar, either $X_s(t) = 0$ for all $t \\in I$ or $X_u(t) = 0$ for all $t \\in I$ .",
"Similarly to eq X dot dynamics, the third equality in lem tilFp implies that $\\dot{X}_c(t) = \\bigl (A + \\tilde{F}(t, X_c(t) \\bigr ) X_c(t),$ for all $t \\in I$ .",
"And for all $(t,v)\\in D_{\\tilde{\\mu }}$ , the operator $A + \\tilde{F}_c(t, v) - \\alpha - \\tilde{\\mu }\\colon V_k \\cap V_c \\rightarrow V_1 \\cap V_c$ is negative semidefinite.",
"So by lem decay to Vc, $\\Vert X_c(t)\\Vert _{V_1} \\le e^{(\\alpha + \\tilde{\\mu })t}\\Vert X_c(0)\\Vert _{V_1}$ for all $t\\ge 0$ in $I$ .",
"It similarly follows that for all $t\\le 0$ in $I$ , $\\Vert X_c(t)\\Vert _{V_1} \\le e^{-(\\alpha + \\tilde{\\mu })t} \\Vert X_c(0)\\Vert _{V_1}.$"
],
[
"Example: a non-autonomous version of Burgers' equation",
"Let $r \\in \\mathbb {R}$ , and consider the non-autononomous, nonlinear pde $ \\partial _t u (t, \\theta )= \\partial _\\theta ^2 u(t,\\theta ) + ru(t,\\theta ) - \\frac{t}{2}(\\partial _\\theta u (t,\\theta ))^2,$ with $2\\pi $ -periodic boundary conditions in $\\theta $ .",
"Then thm special case applies, where $\\Omega $ is the circle.",
"Using thm normal form, we compute the centre manifold of the normal form system approximating eq modified Burgers up to residuals of order three, in sec ex thm.",
"Via a direct approach, we compute all invariant manifolds for residuals of orders three and four, in sec ex direct.",
"We find that the order three centre manifolds computed in the two ways agree.",
"These computations illustrate rem direct constr, that the construction from thm normal form is guaranteed to give a result, while a direct computation may be more efficient in concrete situations."
],
[
"Centre manifold via thm normal form",
"In this setting, $Au = u^{\\prime \\prime } + ru\\quad \\text{and}\\quad f(t,u) = - \\frac{t}{2}(u^{\\prime })^2,$ where a prime denotes the derivative in the $\\theta $ -direction.",
"The eigenfunctions of $A$ are $e_j$ , for $j \\in \\mathbb {Z}$ , given by $e_j(\\theta ) := e^{ij\\theta }$ .",
"The eigenvalue corresponding to $e_j$ is $\\alpha _j = r-j^2$ (which has multiplicity two when $j\\ne 0$ ).",
"Choose $\\alpha , \\beta , \\gamma $ and $\\tilde{\\mu }$ such that $0 \\le \\alpha < \\tilde{\\mu }< \\beta = \\gamma < 1$ , and $\\alpha < \\frac{1}{2}$ .",
"Suppose that $r$ lies within $\\alpha $ of an integer of the form $n^2$ , for a nonzero $n \\in \\mathbb {Z}$ .",
"Then the eigenvalue $\\alpha _n$ is central up to precision $\\alpha $ .",
"We determine a corresponding centre manifold for a system that approximates eq modified Burgers up to a third-order residual.",
"This involves the coordinate transform $\\xi _3$ .",
"To compute this centre manifold, we only need to apply $\\xi _3$ to elements of $V_c = \\operatorname{span}\\lbrace e_n,e_{-n}\\rbrace $ .",
"In other words, we only need to compute $\\xi _3 (t, X_ne_n+X_{-n} e_{-n})$ , for $t \\in \\mathbb {R}$ and $X_n,X_{-n} \\in .",
"(We do not determine the domain $ D$ here.",
")$ For $p=2$ , the map $\\xi _2$ is the identity map.",
"So $\\xi _3 (t, X_ne_n+X_{-n}e_{-n}) = X_ne_n+X_{-n}e_{-n} + \\hat{\\xi }(X_ne_n + X_{-n}e_{-n}),$ where $\\hat{\\xi }(X_ne_n+ X_{-n}e_{-n}) = \\sum _{q \\in \\mathbb {Z}^{\\infty };\\, |q|=2} \\sum _{j \\in \\mathbb {Z}} \\hat{\\xi }^q_j(t) e_j (X_ne_n + X_{-n}e_{-n})^q.$ For $j \\in \\mathbb {Z}$ , let $q^{(j)} \\in \\mathbb {Z}^{\\infty }$ be defined by $q^{(j)}_m = 1$ if $m=j$ , and $q^{(j)}_m = 0$ otherwise.",
"Then, for $q \\in \\mathbb {Z}^{\\infty }$ with $|q|=2$ , $(X_ne_n + X_{-n}e_{-n})^q ={\\left\\lbrace \\begin{array}{ll}X_n^2 & \\text{if }q = 2q^{(n)}; \\\\X_{-n}^2 & \\text{if }q = 2q^{(-n)}; \\\\X_nX_{-n} & \\text{if }q = q^{(n)} + q^{(-n)}\\\\0 & \\text{otherwise}.\\end{array}\\right.",
"}$ So $\\hat{\\xi }(X_ne_n+ X_{-n}e_{-n}) = \\sum _{j \\in \\mathbb {Z}}\\Bigl (X_n^2\\hat{\\xi }^{2q^{(n)}}_j(t) + X_{-n}^2 \\hat{\\xi }^{2q^{(-n)}}_j(t) + X_nX_{-n} \\hat{\\xi }^{q^{(n)} + q^{(-n)}}_j(t)\\Bigr )e_j.$ The map $\\hat{\\xi }^{2q^{(n)}}_j$ is expressed in terms of the map $a^{2q^{(n)}}_j$ in $R_2(t,u) = -\\sum _{j \\in \\mathbb {Z}} \\sum _{q \\in \\mathbb {Z}^{\\infty };\\, |q|= 2} a^{q}_j(t) e_ju^q.$ (The order three term in eq aqrs now equals zero.)",
"See eq hat psi hat F Jq and eq hat psi hat F Jqc.",
"If $u = \\sum _{l \\in \\mathbb {Z}} x_l e_l$ , then $R_2(t,u) = -\\frac{t}{2}(u^{\\prime })^2 =- \\frac{t}{2}\\sum _{j \\in \\mathbb {Z}} \\Bigl ( \\sum _{k \\in \\mathbb {Z}} k(j-k)x_k x_{j-k} \\Bigr ) e_j.$ The equality $x_k x_{j-k} = u^{2q^{(n)}} = x_n^2$ holds precisely if $k = n$ and $j=2n$ .",
"Hence $a^{2q^{(n)}}_{2n}(t) = \\frac{t}{2} n^2$ and $a^{2q^{(n)}}_{j} = 0$ if $j \\ne 2n$ .",
"An analogous argument shows that $a^{2q^{(-n)}}_{-2n}(t) = \\frac{t}{2} n^2$ and $a^{2q^{(-n)}}_{j} = 0$ if $j \\ne -2n$ .",
"The equality $x_k x_{j-k} = u^{q^{(n)} + q^{(-n)}} = x_n x_{-n}$ holds precisely if $j=0$ and either $k = n$ or $k=-n$ .",
"Hence $a^{q^{(n)} + q^{(-n)}}_{0}(t) = -t n^2$ and $a^{q^{(n)} + q^{(-n)}}_{j} = 0$ if $j \\ne 0$ .",
"The relevant numbers $\\mu ^q_j$ as in sec comp Fq equal $\\begin{split}\\mu ^{2q^{(n)}}_{2n} &= 2 \\alpha _n - \\alpha _{2n} = r+2n^2;\\\\\\mu ^{2q^{(-n)}}_{-2n} &= 2 \\alpha _{-n} - \\alpha _{-2n} = r+2n^2; \\\\\\mu ^{q^{(n)} + q^{(-n)}}_{0} &= \\alpha _n + \\alpha _{-n} - \\alpha _0 = r-2n^2\\end{split}$ (note that $\\alpha _j =\\alpha _{-j}$ for every $j$ ).",
"Because $n^2 \\ge 1$ and $\\alpha < \\frac{1}{2}$ , the real parts of $\\mu ^{2q^{(n)}}_{2n}$ and $\\mu ^{2q^{(-n)}}_{2n} $ are greater than $\\alpha $ , whereas the real part of $\\mu ^{q^{(n)} + q^{(-n)}}_{0}$ is smaller than $-\\alpha $ .",
"And with $J^q$ as in sec comp Fq, we have $2n \\in J^{2q^{(n)}}$ .",
"Indeed, $J_s = \\lbrace j \\in \\mathbb {Z}: |j| \\ge n+1 \\rbrace ,$ so $2n \\in J_s$ and $(2q^{(n)})^s = 0$ .",
"Similarly, $2n \\in J^{2q^{(-n)}}$ .",
"And $0 \\in J_u$ and $(q^{(n)} + q^{(-n)})^u = 0$ , so $0 \\in J^{q^{(n)} + q^{(-n)}}$ .",
"Hence, by eq hat psi hat F Jq, $\\begin{split}\\hat{\\xi }^{2q^{(n)}}_{2n}(t) &= \\int _{-\\infty }^t e^{-(r+2n^2)(t-\\tau )} \\frac{\\tau }{2} n^2 \\, d\\tau \\\\&= \\frac{n^2}{2(r+2n^2)} \\Bigl (t- \\frac{1}{r+2n^2}\\Bigr ).\\end{split}$ The integral converges since $\\Re (r+2n^2)>0$ , and is $\\tilde{\\mu }$ -regular.",
"Similarly, $\\hat{\\xi }^{2q^{(-n)}}_{-2n}(t) = \\frac{n^2}{2(r+2n^2)} \\Bigl (t- \\frac{1}{r+2n^2}\\Bigr ).$ And because $\\Re (\\mu ^{q^{(n)} + q^{(-n)}}_{0}) < -\\alpha $ , $\\begin{split}\\hat{\\xi }^{q^{(n)}+ q^{(-n)}}_{0}(t) &= -\\int _t^{\\infty } e^{-(r-2n^2)(t-\\tau )} ({-\\tau } n^2) \\, d\\tau \\\\&= \\frac{-n^2}{r-2n^2} \\Bigl (t- \\frac{1}{r-2n^2}\\Bigr ).\\end{split}$ We conclude that for all $t \\in \\mathbb {R}$ and $X_n, X_{-n} \\in ,\\begin{multline} \\xi _3(t,X_ne_n + X_{-n}e_{-n}) = \\\\X_ne_n +X_{-n}e_{-n} + \\frac{n^2}{2(r+2n^2)} \\Bigl (t- \\frac{1}{r+2n^2}\\Bigr )(X_n^2e_{2n}+ X_{-n}^2e_{-2n}) \\\\-\\frac{n^2}{r-2n^2} \\Bigl (t- \\frac{1}{r-2n^2}\\Bigr ) X_nX_{-n}.\\end{multline}(The last term is a scalar multiple of the constant function $ e0$.",
")If $ r = n2$, this simplifies to\\begin{multline*}\\xi _3(t,X_ne_n + X_{-n}e_{-n}) = \\\\X_ne_n+X_{-n}e_{-n} + \\frac{1}{6} \\Bigl (t- \\frac{1}{3n^2}\\Bigr )(X_n^2 e_{2n}+X_{-n}^2e_{-2n}) + \\Bigl (t+ \\frac{1}{n^2}\\Bigr ) X_nX_{-n}.\\end{multline*}$"
],
[
"Invariant manifolds via direct computations",
"For order of residual $p=2$ , the map $\\xi _2$ is the identity map, $x_j=X_j$ .",
"Proceeding to order of residual $p=3$ we construct quadratic corrections to the identity $\\xi _2$ to form $\\xi _3$ .",
"In the eigenvector basis the field $u(t,\\theta )=\\sum _{j} x_j(t)e^{i j\\theta }$ (all sums in this section are over $\\mathbb {Z}$ ), and the pde eq modified Burgers becomes $\\dot{x}_j=\\alpha _jx_j +\\frac{t}{2}\\sum _{k} b_{jk}x_{j-k}x_k\\quad \\text{where }b_{jk}:=k(j-k).$ Writing $x_j(t) = \\xi _3(t, X(t))_j = X_j + \\sum _{k,l \\in \\mathbb {Z}}g^{kl}_j(t) X_k X_l,$ and solvingThe computer algebra code used for the computations in this section is available on http://www.maths.adelaide.edu.au/anthony.roberts/pBurgers.txt.",
"for $g^{kl}_j$ such that $x_j$ satisfies eqBurgPODEs up to terms of order three if $\\dot{X}_k = \\alpha _k X_k$ , we find that $x(t)=\\xi _3(t,X(t))$ is given by $&x_j=X_j+\\frac{1}{2}\\sum _{k:|d_{jk}^{-1}|>\\tilde{\\mu }} b_{jk}[d_{jk}t-d_{jk}^2]X_{j-k}X_k\\,,\\\\&\\text{where }d_{jk}:=1/[-\\alpha _j+\\alpha _k+\\alpha _{j-k}]=1/[r+2jk-k^2].\\nonumber $ For $r\\approx n^2$ and odd $n$ , the denominators in $d_{jk}$ are not small.",
"Then this map, combined with the linear $\\dot{X}_j=\\alpha _j X_j$ , matches the pde eq modified Burgers to third-order errors.",
"However, for $r\\approx n^2$ and even $n>0$ , some denominators are small, becoming zero when $r=n^2$ .",
"Then the divisor being zero becomes $k(k-j)=n^2/2$ and hence has zeros for every pair of integer factors of $n^2/2$ (including negative pairs).",
"Consequently these terms are excluded from the sum eqBurgP3, and instead lead to nonlinearly modifying the evolution for some $j$ via $\\dot{X}_j=\\alpha _j X_j+\\frac{t}{2}\\sum _{k:|d_{jk}^{-1}|<\\tilde{\\mu }} b_{jk}X_{j-k}X_k\\,.$ Often the centre manifold is of most interest, so in $\\xi _3$ setting all $X_j=0$ except $X_{\\pm n}$ , gives the quadratic approximate centre manifold to be $x_j=X_j$ for all $j$ except $&x_0=X_0-n^2\\left[\\frac{1}{r-2n^2}t-\\frac{1}{(r-2n^2)^2}\\right]X_nX_{-n}\\,,\\\\&x_{\\pm 2n}=X_{\\pm 2n}+\\frac{1}{2}n^2\\left[\\frac{1}{r+2n^2}t-\\frac{1}{(r+2n^2)^2}\\right]X_{\\pm n}^2\\,.$ This is the same result as eq xi3 Burgers centre.",
"Proceeding to order of residual $p=4$ we may construct cubic corrections to $\\xi _2$ to form $\\xi _4$ .",
"For simplicity, restrict attention to the cases of $n$ odd.",
"It is straightforward but tedious to construct that for $\\xi _4$ $&x_j=\\xi _{3,j}+\\sum _{k,l:|d_{jkl}^{-1}|>\\tilde{\\mu }} b_{jl}b_{lk}c_{jkl}(t)X_{j-l}X_{l-k}X_k\\,,\\\\\\text{where }&c_{jkl}:=\\tfrac{1}{2}d_{lk}d_{jkl}t^2-(d_{lk}d_{jkl}^2+\\tfrac{1}{2}d_{lk}^2d_{jkl})t+(d_{lk}d_{jkl}^3+\\tfrac{1}{2}d_{lk}^2d_{jkl}^2),\\nonumber \\\\&d_{jkl}:=1/[-\\alpha _j+\\alpha _k+\\alpha _{l-k}+\\alpha _{j-l}]=1/[2r +2jl+2kl -2k^2-2l^2].\\nonumber $ The terms excluded from $(k,l)$ in the sum eqBurgP4 must cause cubic terms in the evolution.",
"For example, when $r=n^2=1$ thenThe apparent pattern in these odes becomes more complicated—at $\\dot{X}_{\\pm 6}$ for example.",
"$&\\dot{X}_0=X_0,\\\\&\\dot{X}_{\\pm 1}=(\\tfrac{1}{9}t-\\tfrac{1}{3}t^2)X_{-1}X_{\\pm 1}^2\\,,\\\\&\\dot{X}_{\\pm 2}=-3X_{\\pm 2}+(\\tfrac{104}{225}t-\\tfrac{8}{15}t^2)X_{\\mp 1}X_{\\pm 1}X_{\\pm 2}\\,,\\\\&\\dot{X}_{\\pm 3}=-8X_{\\pm 2}+(\\tfrac{594}{1225}t-\\tfrac{18}{35}t^2)X_{\\mp 1}X_{\\pm 1}X_{\\pm 3}\\,,\\\\&\\dot{X}_{\\pm 4}=-15X_{\\pm 3}+(\\tfrac{1952}{3969}t-\\tfrac{32}{63}t^2)X_{\\mp 1}X_{\\pm 1}X_{\\pm 3}\\,,\\\\&\\qquad \\vdots \\nonumber $ By construction, in this case of $r=1$ , the coordinate transform eqBurgP4 together with the odes eqsBurgP4n1 creates a dynamical system in $u(t,\\theta )=\\sum _j x_je^{ij\\theta }$ which is the same as the pde eq modified Burgers to a residual of order four.",
"In the combined system eqBurgP4,eqsBurgP4n1, by definition def centre mfd three invariant manifolds are: the 1D unstable manifold parametrised by $X_0$ with all other $X_j=0$ ; the 2D centre manifold parametrised by $X_{\\pm 1}$ with all other $X_j=0$ ; and the stable manifold with $X_0=X_{\\pm 1}=0$ .",
"See sec code Burgers for the computer algebra code used the for the computations in this subsection.",
"It is also available on http://www.maths.adelaide.edu.au/anthony.roberts/pBurgers.txt."
],
[
"Acknowledgement",
"Part of this research was supported by the Australian Research Council grant DP150102385."
],
[
"Compact and finite-rank operators into Banach spaces",
"Let $V$ and $W$ be Banach spaces, and suppose that $V^*$ has the approximation property (this is true for example if $V$ is a Hilbert space).",
"Let $\\lbrace e^j\\rbrace _{j\\in \\mathbb {N}} \\subset V^*$ and $\\lbrace f_k\\rbrace _{k \\in \\mathbb {N}} \\subset W$ be countable subsets with dense spans.",
"(So $V^*$ and $W$ are separable.)",
"In the main text, we use the following, which is standard in the case where $V$ and $W$ are Hilbert spaces.",
"2 Proposition A.1 The space $\\operatorname{span}\\lbrace e^j \\otimes f_k : j,k \\in \\mathbb {N}\\rbrace $ is dense in $\\mathcal {K}(V,W)$ .",
"Let $\\mathcal {F}(V,W)$ be the space of finite-rank linear operators from $V$ to $W$ ; that is, operators whose images are finite-dimensional.",
"2 Lemma A.2 The space $\\operatorname{span}\\lbrace e^j \\otimes f_k : j,k \\in \\mathbb {N}\\rbrace $ is dense in $\\mathcal {F}(V,W)$ .",
"Let $T \\in \\mathcal {F}(V,W)$ .",
"Since the image of $T$ is finite-dimensional, there are $v^1, \\ldots , v^n \\in V^*$ and $w_1, \\ldots , w_n \\in W$ such that $T = \\sum _{l=1}^n v^l \\otimes w_l\\,.$ Let $\\varepsilon > 0$ .",
"For every $l$ , choose $r \\in \\mathbb {N}$ and $a_l^1, \\ldots , a_l^r \\in and $ bl1, ..., blr such that $\\Bigl \\Vert v^l- \\sum _{j=1}^r a_l^j e^j \\Bigr \\Vert _{V^*} \\le \\sqrt{\\varepsilon /n}\\quad \\text{and}\\quad \\Bigl \\Vert w_l-\\sum _{k=1}^r b_l^k f_k \\Bigr \\Vert _W \\le \\sqrt{\\varepsilon /n}.$ Using the triangle and Cauchy–Schwartz inequalities, one finds that for all $v \\in V$ , $\\bigl \\Vert Tv - \\Bigl (\\sum _{j,k = 1}^r a_l^j b_l^k e^j \\otimes f_k\\Bigr ) (v) \\bigr \\Vert _W=\\Bigl \\Vert \\sum _{l=1}^n \\bigl \\langle v^l - \\sum _{j=1}^r a^j_l e^j, v \\bigr \\rangle \\Bigl (w_l - \\sum _{k=1}^r b^j_l f_k\\Bigr )\\Bigr \\Vert _W\\\\\\le \\Vert v\\Vert _{V} \\sum _{l=1}^n \\Bigl (\\bigl \\Vert v^l - \\sum _{j=1}^r a^j_l e^j\\bigr \\Vert _{V^*} \\cdot \\bigl \\Vert w_l - \\sum _{k=1}^r b^j_l f_k\\bigr \\Vert _W\\Bigr )\\le \\varepsilon \\Vert v\\Vert _{V}.$ [Proof of prop approx cpt.]",
"Since $V^*$ has the approximation property, $\\mathcal {F}(V,W)$ is dense in $\\mathcal {K}(V,W)$ .",
"See for example Proposition 4.12(b) in the book by Ryan [31].",
"So the claim follows from lem span F."
],
[
"Computer algebra code for Burgers example computation",
"[numbers=left]pBurgers.txt"
],
[
"A residual",
"Recall the setting of sec setup.",
"In this section and the next, based upon the details of some given dynamical system eq ODE we construct both a coordinate transformation eq def xyz p and a corresponding `normal form' system eq ODE XYZ p, such that solutions $X$ to eq ODE XYZ p, transformed by eq def xyz p, satisfy the given dynamical system eq ODE up to residuals of a specified order $p$ .",
"See thm normal form.",
"We do this inductively, by showing how to construct such a transformed system to satisfy eq ODE with residual of order $p+1$ from a version with residual of order $p$ .",
"In sec transf choice, we construct a more specific choice of the general coordinate transform constructed in this section, in order to establish exact invariant manifolds, and study their properties, for constructed systems arbitrarily close to the given system eq ODE.",
"2 Remark 6.1 In sec setup, we assumed that the sequence $\\lbrace V_k\\rbrace _{k=1}^{\\infty }$ is comparable to a nested sequence $\\lbrace W_k\\rbrace _{k=1}^{\\infty }$ of separable Hilbert spaces in which the vectors $e_j$ are orthogonal.",
"lem comparable same maps implies that we may equivalently assume that $\\lbrace V_k\\rbrace _{k=1}^{\\infty }$ itself is a nested sequence of separable Hilbert spaces, because all maps from $V_{\\infty }$ to itself we use transfer to maps from $W_{\\infty }$ to itself of the same type (e.g.",
"compact polynomial and compactly differentiable maps).",
"However, the formulation where $\\lbrace V_k\\rbrace _{k=1}^{\\infty }$ is only comparable to a nested sequence of separable Hilbert spaces makes it clearer that we have the flexibility to consider maps between Banach spaces.",
"This is natural for example in the context of prop ex cpt poly Sobolev.",
"Let $p \\in \\mathbb {N}$ , with $p \\ge 2$ .",
"Let $\\xi _p, F_p\\colon I \\times V_{\\infty }\\rightarrow V_{\\infty }$ be such that $\\xi _p - \\operatorname{id}$ and $F_p$ are compact polynomial maps of order at most $p-1$ in the $V_{\\infty }$ component, and infinitely differentiable in $I$ .",
"Suppose, furthermore, that $\\xi _p$ is a near-identity at zero, and that $F_p = \\mathcal {O}(2)$ .",
"Recall that our goal is to relate the dynamics of maps $x$ satisfying eq ODE to the dynamics of maps $X\\colon I \\rightarrow V_{\\infty }$ satisfying eq ODE XYZ p when $x$ and $X$ are related by the coordinate transform $\\xi _p$ as in eq def xyz p. For maps $f,g\\colon I \\times V_{\\infty } \\rightarrow V_{\\infty }$ , with $f$ differentiable, we write $f^{\\prime }_{V_{\\infty }} \\circ g$ for the map from $I \\times V_{\\infty }$ to $V_{\\infty }$ given by $(f^{\\prime }_{V_{\\infty }} \\circ g)(t,v) = f^{\\prime }_{V_{\\infty }}(t,v)(g(t,v)),$ for all $t \\in I$ and $v \\in V_{\\infty }$ .",
"(Note that this is different from $(f\\circ g)^{\\prime }_{V_{\\infty }}(t,v) = f^{\\prime }_{V_{\\infty }}(t, g(t,v))(g(t,v))$ .)",
"If $g$ is a map from $V_{\\infty }$ to $V_{\\infty }$ to itself, then the composition $f^{\\prime }_{V_{\\infty }} \\circ g$ is defined analogously.",
"Also recall the notation for compositions of maps to and from $I \\times V_{\\infty }$ and $V_{\\infty }$ under Notation and conventions in secNota.",
"Define the maps $\\Phi _p, R_p\\colon I \\times V_{\\infty } \\rightarrow V_{\\infty }$ by $\\Phi _p:= (\\xi _p)^{\\prime }_I + (\\xi _p)^{\\prime }_{V_{\\infty }} \\circ (A + F_p)\\quad \\text{and}\\quad R_p := -A\\circ \\xi _p-f \\circ \\xi _p+\\Phi _p.$ The map $R_p$ is the residual of the transformed ode, in the following sense.",
"2 Lemma 6.2 For all smooth maps $X\\colon I \\rightarrow V_{\\infty }$ satisfying eq ODE XYZ p, and with $x\\colon I \\rightarrow V_{\\infty }$ determined from $X$ by eq def xyz p, $ \\dot{x}(t) = Ax(t) + f(t, x(t)) + R_p(t, X(t)).$ For $X$ and $x$ as in the lemma, the chain rule (lem chain rule) and eq ODE XYZ p imply that for all $t \\in I$ , $\\dot{x}(t) = \\Phi _p(t, X(t)) = Ax(t) + f(t, x(t)) + R_p(t, X(t)).$ 2 Lemma 6.3 The maps $\\Phi _p$ and $R_p$ are infinitely compactly differentiable.",
"Because the Banach spaces $V_k$ are compactly nested, it is enough to show that $\\Phi _p$ and $R_p$ are infinitely differentiable.",
"And that is true by the chain rule, because $f$ is infinitely differentiable, and so are $F_p$ and $\\xi _p$ , by lem pol diffble.",
"To recursively construct eq ODE XYZ p and eq def xyz p, suppose that $R_p = \\mathcal {O}(p)$ .",
"We proceed to show that eq ODE XYZ p and eq def xyz p may be refined to make the new residual of $\\mathcal {O}(p+1)$ .",
"By lem Rp diffble, cor Taylor cpt Vinfty (where $u=0$ and $h=v$ ), and lem order pol,lem order pol lower, there are unique, infinitely differentiable maps $a^{q}\\colon I \\rightarrow V_{\\infty }$ for all multi-indices $q \\in \\mathbb {N}_0^{\\infty }$ with $|q| = p$ , and a map $\\rho _p\\colon I \\times V_{\\infty } \\rightarrow V_{\\infty }$ such that for all $t \\in I$ , and $v \\in V_{\\infty }$ , $ R_p(t, v) = - \\sum _{q \\in \\mathbb {N}_0^{\\infty };\\, |q|= p} a^{q}(t)v^q + \\rho _p(t,v),$ where the sum converges in $\\operatorname{Pol}^p(V_{\\infty })$ , differentiably in $t$ , and $\\rho _p = \\mathcal {O}(p+1)$.",
"The monomial $v^q$ is defined as in eq def vq, with respect to the set of functionals $e^j = (e_j, )_{V_1}$ , for $j \\in \\mathbb {N}$ .",
"In sec coord transf, we construct maps $\\xi _{p+1}, F_{p+1}$ such that the order $p$ term $R_p(t, X(t))$ in eq ODE xyz may be replaced by an order $p+1$ term $R_{p+1}(t, X(t))$ , if eq ODE XYZ p and eq def xyz p hold with $p$ replaced by $p+1$ ."
],
[
"Construction of the coordinate transform",
"For each $j \\in \\mathbb {N}$ , let $\\alpha _j$ be the eigenvalue of $A$ corresponding to $e_j$ .",
"For all $q \\in \\mathbb {N}_0^{\\infty }$ with $|q| = p$ , define $\\mu ^q \\in by \\begin{equation} \\mu ^{q} = \\sum _{j=1}^{\\infty } q_j \\alpha _j \\quad \\in \\end{equation}(This sum has at most $ p$ nonzero terms.",
")For such $ q N0$, let $ aq$ be as in {eq aqrs}.",
"Let$ q, Fq I V $be smooth maps such that\\begin{equation} \\hat{F}^{q} + (\\hat{\\xi }^{q})^{\\prime } + \\mu ^{q}\\hat{\\xi }^{q} - A\\hat{\\xi }^{q} = a^{q}.\\end{equation}Suppose thatthe sums$$\\hat{F}(t,v) =\\sum _{q \\in \\mathbb {N}_0^{\\infty };\\, |q| = p} \\hat{F}^{q}(t) v^q\\quad \\text{and}\\quad \\hat{\\xi }(t,v) =\\sum _{q \\in \\mathbb {N}_0^{\\infty };\\, |q| = p} \\hat{\\xi }_{q}(t)v^q$$converge in $ Polp(V)$, differentiably in $ t$.$ Define a new coordinate transform map $\\xi _{p+1}\\colon I \\times V_{\\infty } \\rightarrow V_{\\infty }$ and corresponding map $F_{p+1}\\colon I \\times V_{\\infty } \\rightarrow V_{\\infty }$ that replaces $F_p$ in eq ODE XYZ p, by $ \\xi _{p+1}= \\xi _{p}+ \\hat{\\xi }\\quad \\text{and}\\quad F_{p+1} = F_{p}+ \\hat{F}.$ The following result is the main step in the construction of the coordinate transform we are looking for.",
"2 Proposition 6.4 If $X$ and $x$ are as in eq ODE XYZ p and eq def xyz p, with $p$ replaced by $p+1$ , then eq ODE xyz holds, with the residual $R_p$ replaced by a residual $R_{p+1}$ satisfying $R_{p+1}= \\mathcal {O}(p+1).$ The maps $\\xi _{p+1} - \\operatorname{id}$ and $F_{p+1}$ are compact polynomials in $\\mathcal {K}\\operatorname{Pol}(V_{\\infty })$ of order at most $p$ , and $\\xi _{p+1}$ is a near-identity.",
"2 Remark 6.5 The maps $\\hat{\\xi }^q$ and $\\hat{F}^q$ can be found explicitly if we decompose eq DE iteration with respect to the basis $\\lbrace e_j\\rbrace _{j=1}^{\\infty }$ .",
"This will be done in sec transf choice.",
"One solution to eq DE iteration is $\\hat{F}^q = a^q$ and $\\hat{\\xi }^q = 0$ .",
"However, for our purposes, we need the function $F^q$ to be of a specific form.",
"The main purpose of this work is to find $\\hat{F}^q$ such that the $e_j$ -component of $F^{q}$ is zero for certain combinations of $q$ and $j$ , in such a way that an exact separation of stable, centre and unstable modes is maintained.",
"See prop csu."
],
[
"Proof of prop coord transf",
"Define the map $\\hat{\\Phi }\\colon I \\times V_{\\infty } \\rightarrow V_{\\infty }$ by $ \\hat{\\Phi }= \\hat{\\xi }^{\\prime }_I+ \\hat{\\xi }^{\\prime }_{V_{\\infty }}\\circ (A + F_p)+ (\\xi _p)^{\\prime }_{V_{\\infty }}\\circ \\hat{F} +\\hat{\\xi }^{\\prime }_{V_{\\infty }}\\circ \\hat{F}.$ 2 Lemma 6.6 We have $ \\hat{\\Phi }- (\\hat{\\xi }^{\\prime }_I + \\hat{\\xi }^{\\prime }_{V_{\\infty }} \\circ A + \\hat{F}) = \\mathcal {O}(p+1).$ The left-hand side of eq diff hat Phi equals $\\hat{\\xi }^{\\prime }_{V_{\\infty }} \\circ F_p + \\hat{\\xi }^{\\prime }_{V_{\\infty }} \\circ \\hat{F} + ((\\xi _p)^{\\prime }_{V_{\\infty }} -\\operatorname{id})\\circ \\hat{F}.$ By lem der poly On-1, the derivative $\\hat{\\xi }^{\\prime }_{V_{\\infty }}$ satisfies the condition of lem der Om+n, with $n=p-1$ .",
"Since $F_p = \\mathcal {O}(2)$ , lem der Om+n implies that $\\hat{\\xi }^{\\prime }_{V_{\\infty }} \\circ F_p = \\mathcal {O}(p+1)$ .",
"Similarly, $\\hat{\\xi }^{\\prime }_{V_{\\infty }} \\circ \\hat{F} = \\mathcal {O}(2p-1)$ .",
"Now $\\xi _p$ is a polynomial map, and a near-identity.",
"By lem der poly On-1, this implies that $(\\xi _p)^{\\prime }_{V_{\\infty }} -\\operatorname{id}$ satisfies the condition of lem der Om+n, with $n=1$ .",
"So lem der Om+n implies that $((\\xi _p)^{\\prime }_{V_{\\infty }} -\\operatorname{id})\\circ \\hat{F} = \\mathcal {O}(p+1)$ .",
"2 Lemma 6.7 For $q \\in \\mathbb {N}_0^{\\infty }$ such that $|q| = p$ , let $a^{q}$ be as in eq aqrs.",
"Then for all $t \\in I$ and $v \\in V_{\\infty }$ , $ \\hat{\\xi }_I^{\\prime }(t, v) + \\hat{\\xi }^{\\prime }_{V_{\\infty }}(t, v) Au \\\\+ \\hat{F}(t, v) = A\\hat{\\xi }(t, v) +\\sum _{q \\in \\mathbb {N}_0^{\\infty };\\, |q| = p} a^{q}(t)v^q.$ First of all, $\\hat{\\xi }^{\\prime }_I(t, v) = \\sum _{q \\in \\mathbb {N}_0^{\\infty };\\, |q|= p} (\\hat{\\xi }^{q})^{\\prime }(t)v^q.$ By lem der poly, $\\hat{\\xi }^{\\prime }_{V_{\\infty }}(t, v) Av =\\\\\\sum _{q \\in \\mathbb {N}_0^{\\infty };\\, |q|= p} \\hat{\\xi }^{q}(t) \\sum _{j=1}^{\\infty } q_j(e_j, v)_{V_1}^{q_j-1} (e_j, Av)_{V_1}\\prod _{j^{\\prime }\\ne j} (e_{j^{\\prime }}, v)_{V_1}^{q_{j^{\\prime }}}.$ Now, because the vectors $\\lbrace e_j\\rbrace _{j=1}^{\\infty }$ are orthogonal with respect to $(, )_{V_1}$ , we have $(e_j, Av)_{V} = \\alpha _j (e_j, v)_{V_1}$ for every $j$ .",
"So $\\hat{\\xi }^{\\prime }_{V_{\\infty }}(t, v) Av =\\sum _{q\\in \\mathbb {N}_0^{\\infty };\\, |q| = p} \\hat{\\xi }^{q}(t) \\Bigl (\\sum _{j=1}^{\\infty } q_j \\alpha _j \\Bigr )v^q.$ We find that the left-hand side of eq hat phi aqrs equals $\\sum _{q \\in \\mathbb {N}_0^{\\infty };\\, |q|= p} \\Bigl ((\\hat{\\xi }^{q})^{\\prime }(t) + \\hat{F}^{q}(t) + \\mu ^{q} \\hat{\\xi }^{q}(t)\\Bigr )v^q,$ with $\\mu ^{q}$ as in eq def nuqrs.",
"So the claim follows from eq DE iteration.",
"Define the maps $\\Phi _{p+1}, R_{p+1}\\colon I \\times V_{\\infty } \\rightarrow V_{\\infty }$ by $\\Phi _{p+1} := \\Phi _p + \\hat{\\Phi }\\quad \\text{and}\\quad R_{p+1} := -A\\circ \\xi _{p+1}-f \\circ \\xi _{p+1}+\\Phi _{p+1}.$ 2 Lemma 6.8 The residual $R_{p+1}$ satisfies $R_{p+1} = \\mathcal {O}(p+1).$ By lem hat Phi p,lem hat phi aqrs, $ \\begin{split}R_{p+1} &= -A \\circ \\xi _p- f\\circ \\xi _{p+1} + \\Phi _p - \\tilde{R}_p+ \\mathcal {O}(p+1)\\\\&= R_p - f \\circ \\xi _{p+1} + f\\circ \\xi _p - \\tilde{R}_p+ \\mathcal {O}(p+1),\\end{split}$ where, for $t \\in I$ and $v \\in V_{\\infty }$ , $\\tilde{R}_p(t,v) := -\\sum _{q \\in \\mathbb {N}_0^{\\infty };\\, |q| = p} a^{q}(t)v^q.$ By eq aqrs, the last expression in eq Rp+1 Op+1 equals $f \\circ \\xi _{p+1} - f \\circ \\xi _{p} + \\mathcal {O}(p+1).$ Let $l \\in \\mathbb {N}$ be given, and choose $k, k^{\\prime } \\in \\mathbb {N}$ such that $f\\colon I \\times V_{k^{\\prime }} \\rightarrow V_l$ is differentiable, and $\\xi _p, \\xi _{p+1} \\in \\operatorname{Pol}(V_{k}, V_{k^{\\prime }})$ .",
"Using thm Taylor, we write $\\bigl \\Vert f(t,\\xi _{p+1}(t,v)) - f(t,\\xi _{p}(t,v)) - f^{\\prime }(t, \\xi _p(t,v)) \\hat{\\xi }(t,v)\\bigr \\Vert _{V_l} = O(\\Vert \\hat{\\xi }(t,v)\\Vert ^2),$ for $t \\in I$ and $v \\in V_{k^{\\prime }}$ .",
"lem order pol implies that $\\Vert \\hat{\\xi }(t,v)\\Vert _{V_{k^{\\prime }}} = O(\\Vert v\\Vert _{V_k})$ uniformly in $t$ in compact sets.",
"So $\\bigl \\Vert f(t,\\xi _{p+1}(t,v)) - f(t,\\xi _{p}(t,v))\\bigr \\Vert _{V_l}= \\bigl \\Vert f^{\\prime }(t, \\xi _p(t,v)) \\hat{\\xi }(t,v)\\bigr \\Vert _{V_l} + O(\\Vert v\\Vert _{V_k}^{2p}),$ uniformly in $t$ in compact sets.",
"And then, as in the proof of lem der Om+n, the assumption eq est der f and lem der Om+n imply that $\\Vert f^{\\prime }(t, \\xi _p(t,v)) \\hat{\\xi }(t,v)\\Vert _{V_l} = O(\\Vert v\\Vert _{k^{\\prime }}^{p+1})$ , uniformly in $t$ in compact sets.",
"[Proof of prop coord transf] The correction terms $\\hat{\\xi }$ and $\\hat{F}$ lie in $\\mathcal {K}\\operatorname{Pol}^p(V_{\\infty })$ .",
"Hence $\\xi _{p+1}-\\operatorname{id}$ and $F_{p+1}$ are compact polynomials, because $\\xi _{p}-\\operatorname{id}$ and $F_{p}$ are.",
"By lem order pol, this also implies that $\\xi _{p+1}$ is a near-identity because $\\xi _p$ is.",
"The desired property of $R_{p+1}$ is lem Rp+1."
],
[
"Centre, stable and unstable coordinates",
"There is considerable flexibility in choosing the maps $\\hat{\\xi }^{q}$ and $\\hat{F}^{q}$ in sec coord transf.",
"In this section, we discuss how to make specific choices, in terms of the eigenvalues of $A$ , so that the normal form eq ODE XYZ p is useful for detecting invariant manifolds."
],
[
"Centre, stable and unstable components of $\\hat{F}^q$",
"We use the notation from sec result.",
"In particular, let $\\alpha $ , $\\beta $ , $\\gamma $ and $\\tilde{\\mu }$ be spectral gap parameters defined there.",
"Recall the definition of polynomial growth in def pol growth.",
"2 Proposition 7.1 Suppose that $\\beta - (p+1)\\alpha > \\tilde{\\mu }$ and $\\gamma - (p+1)\\alpha > \\tilde{\\mu }$ .",
"Suppose that $R_p$ has polynomial growth.",
"The maps $\\hat{\\xi }^q$ and $\\hat{F}^q$ in sec coord transf can be chosen such that if either $q^s = 0$ and $q^u \\ne 0$ , or $q^u = 0$ and $q^s \\ne 0$ , then $\\hat{F}^q_c = 0$ ; if $q^s = 0$ , then $\\hat{F}^q_s = 0$ ; and if $q^u = 0$ , then $\\hat{F}^q_u = 0$ .",
"prop csu is proved in sec update, after some preparation done in this subsection.",
"2 Definition 7.2 Let $\\mu \\in , such that $ |()| > $.",
"Set $ a:= I$ and $ b:= I$.Let $ u$ be a continuous function on $ R$ such that $ u(t) = O(e|t|)$ as $ t $ if $ b = $ and as $ t -$ if $ a = -$.Then we define the function $ e() u$ on~$ I$ by$$(e^{\\mu (\\cdot )} \\star u)(t) := {\\left\\lbrace \\begin{array}{ll}\\int _{a}^t e^{\\mu (t-\\tau )}u(\\tau )\\, d\\tau & \\text{if $\\Re (\\mu ) < -\\tilde{\\mu }$};\\\\\\int _{t}^{b} e^{\\mu (t-\\tau )}u(\\tau )\\, d\\tau & \\text{if $\\Re (\\mu ) > \\tilde{\\mu }$}.\\end{array}\\right.",
"}$$$ The integrals occurring in this definition are $\\tilde{\\mu }$ -regular, in the sense defined in sec result.",
"2 Lemma 7.3 In the setting of def conv emu, $(e^{\\mu (\\cdot )} \\star u)^{\\prime } = \\mu (e^{\\mu (\\cdot )} \\star u) - \\operatorname{sgn}(\\Re (\\mu ))u.$ This is a straightforward computation.",
"2 Lemma 7.4 Let $u\\colon I \\rightarrow be a smooth function, and suppose that $ u$ and all its derivatives grow at most polynomially.",
"Then, for every $$ as in {def conv emu}, $ e() u$ and all its derivatives grow at most polynomially.$ First of all, because $(e^{\\mu (\\cdot )} \\star u)^{(k)} = e^{\\mu (\\cdot )}\\star (u^{(k)})$ , it is enough to consider the function $u$ itself rather than all its derivatives.",
"Let $C > 0$ and $n \\in \\mathbb {N}_0$ be such that for all $t \\in I$ , $|u(t)| \\le C(1+ |t|^{n})$ .",
"We prove by induction on $n$ that there is a constant $C^{\\prime } > 0$ such that for all $t \\in I$ , $|(e^{\\mu (\\cdot )} \\star u)(t)| \\le C^{\\prime }(1+| t|^{n})$ .",
"We consider the case where $\\Re (\\mu ) < -\\tilde{\\mu }$ ; the case where $\\Re (\\mu ) > \\tilde{\\mu }$ is similar.",
"If $n = 0$ , then for all $t \\in I$ , $\\bigl | (e^{\\mu (\\cdot )} \\star u)(t) \\bigr | \\le 2C \\int _{-\\infty }^{t} e^{\\mu (t-\\tau )}\\, d\\tau = \\frac{-2C}{\\mu }.$ Suppose that the claim holds for $n$ , and suppose that $|u(t)| \\le C(1+ |t|^{n+1})$ for a constant $C$ .",
"Using integration by parts, we find that $\\bigl | (e^{\\mu (\\cdot )} \\star u)(t) \\bigr |\\le C \\int _{-\\infty }^{t} e^{\\mu (t-\\tau )}(1+|\\tau |^{n+1})\\, d\\tau \\\\= \\frac{-C}{\\mu }\\Bigl (1+|t|^{n+1} - (n+1) \\int _{-\\infty }^{t} e^{\\mu (t-\\tau )}\\operatorname{sgn}(\\tau )|\\tau |^{n}\\, d\\tau \\Bigr ),$ which implies the claim by the induction hypothesis.",
"Let $q \\in \\mathbb {N}_0^{\\infty }$ , with $|q|=p$ .",
"Consider the differentiable maps $a^q_j\\colon I \\rightarrow \\text{such that}\\quad a^q(t) = \\sum _{j=1}^{\\infty }a^q_j(t) e_j,$ where the sum converges in $V_1$ , uniformly and differentiably in $t$ in compact sets in $I$ .",
"For $q \\in \\mathbb {N}_0^{\\infty }$ with $|q|=p$ , let $J^q \\subset \\mathbb {N}$ be the set of $j \\in \\mathbb {N}$ such that either $j \\in J_c$ and either $q^s = 0$ and $q^u \\ne 0$ , or $q^u = 0$ and $q^s \\ne 0$ ; $j \\in J_s$ and $q^s = 0$ ; or $j \\in J_u$ and $q^u = 0$ .",
"For $q \\in \\mathbb {N}_0^{\\infty }$ with $|q|=p$ , and $j \\in \\mathbb {N}$ , write $\\mu ^q_j := \\mu ^{q}- \\alpha _j$ , with $\\mu ^q$ as in eq def nuqrs.",
"2 Lemma 7.5 If $\\beta - (p+1)\\alpha > \\tilde{\\mu }$ and $\\gamma - (p+1)\\alpha > \\tilde{\\mu }$ , then for every $j \\in J^q$ , $|\\Re (\\mu ^q_j)| > \\tilde{\\mu }$ .",
"If $q^s = 0$ and $q^u \\ne 0$ , and $j \\in J_c$ , then $\\Re (\\mu _j^q) = \\sum _{k \\in J_c} q_k \\Re (\\alpha _k) + \\sum _{k \\in J_u} q_k \\Re (\\alpha _k) - \\Re (\\alpha _j) \\ge \\gamma -(p+1) \\alpha >\\tilde{\\mu }.$ If $q^u = 0$ and $q^s \\ne 0$ , and $j \\in J_c$ , then $\\Re (\\mu _j^q) =\\sum _{k \\in J_c} q_k \\Re (\\alpha _k) + \\sum _{k \\in J_s} q_k \\Re (\\alpha _k) - \\Re (\\alpha _j)\\le -\\beta + (p+1) \\alpha < -\\tilde{\\mu }.$ If $q^s = 0$ , and $j \\in J_s$ , then $\\Re (\\mu _j^q) = \\sum _{k \\in J_c} q_k \\Re (\\alpha _k) + \\sum _{k \\in J_u} q_k \\Re (\\alpha _k) - \\Re (\\alpha _j) \\ge \\beta - p\\alpha > \\tilde{\\mu }.$ And if $q^u = 0$ , and $j \\in J_u$ , then $\\Re (\\mu _j^q) = \\sum _{k \\in J_c} q_k \\Re (\\alpha _k) + \\sum _{k \\in J_s} q_k \\Re (\\alpha _k) - \\Re (\\alpha _j) \\le -\\gamma + p\\alpha < \\tilde{\\mu }.$"
],
[
"Update terms for $\\xi _p$ and {{formula:40947836-bcf1-4f48-9b3e-2af2053a0a89}}",
"Suppose that $R_p$ has polynomial growth.",
"Then the functions $a^q_j$ and their derivatives grow at most polynomially, uniformly in $q$ and $j$ .",
"For every $q \\in \\mathbb {N}_0^{\\infty }$ with $|q|=p$ and $j \\in \\mathbb {N}$ , consider the ode for $\\hat{\\xi }^{q}_j$ and $\\hat{F}^{q}_j$ $ \\hat{F}^{q}_j + (\\hat{\\xi }^{q}_j)^{\\prime } + \\mu ^q_j \\hat{\\xi }^{q}_j = a^{q}_j.$ Define the maps $\\hat{\\xi }^{q}_j, \\hat{F}^{q}_j \\colon I \\rightarrow as follows.",
"If $ j Jq$, then\\begin{equation} \\hat{\\xi }^{q}_j =\\operatorname{sgn}(\\Re (\\mu _j^q)) e^{-\\mu _j^q (\\cdot )}\\star a^q_j\\quad \\text{and}\\quad \\hat{F}^{q}_j = 0.\\end{equation}This definition makes sense because of {lem Re muqj} and the growth behaviour of the functions $ aqj$.If $ j Jq$, then we set\\begin{equation} \\hat{\\xi }^{q}_j = 0\\quad \\text{and}\\quad \\hat{F}^{q}_j = a^q_j.\\end{equation}\\begin{lemma} With the above definitions, the \\textsc {ode}\\ {eq ODE j} is satisfied for all q and j.\\end{lemma}\\begin{proof}If j \\notin J^q, the statement is immediate from the definitions.",
"If j \\in J^q, itfollows from {lem conv}.\\end{proof}$ 2 Lemma 7.6 Suppose that $\\beta - (p+1)\\alpha > \\tilde{\\mu }$ and $\\gamma - (p+1)\\alpha > \\tilde{\\mu }$ .",
"Then the sums $ \\hat{F}(t,v) =\\sum _{q \\in \\mathbb {N}_0^{\\infty };\\, |q| = p} \\sum _{j \\in \\mathbb {N}} \\hat{F}^{q}_j(t) e_j v^q\\quad \\text{and}\\quad \\hat{\\xi }(t,v) =\\sum _{q \\in \\mathbb {N}_0^{\\infty };\\, |q| = p} \\sum _{j \\in \\mathbb {N}} \\hat{\\xi }^{q}_j(t) e_j v^q$ converge in $\\operatorname{Pol}^p(V_{\\infty })$ , differentiably in $t$ .",
"The first sum in eq sums Fq psiq 2 equals $ \\sum _{q \\in \\mathbb {N}_0^{\\infty };\\, |q| = p} \\sum _{j \\in \\mathbb {N}\\setminus J^q} a^{q}_j(t) e_j v^q.$ Since $\\lbrace V_{k}\\rbrace _{k=1}^{\\infty }$ is comparable to a nested sequence of separable Hilbert spaces in which the set $\\lbrace e_j\\rbrace _{j=1}^{\\infty }$ is orthogonal, cor comp conv implies that this series converges in $\\operatorname{Pol}^p(V_{\\infty })$ .",
"Write $J^q_{\\pm } := \\lbrace j \\in J^q : \\pm \\Re (\\mu ^q_j) > \\tilde{\\mu }\\rbrace .$ lem Re muqj states that $J^q = J^q_+ \\cup J^q_-$ .",
"So the second sum in eq sums Fq psiq 2 equals $\\sum _{q \\in \\mathbb {N}_0^{\\infty };\\, |q| = p} \\sum _{j \\in J^q_+}\\int _{-\\infty }^t e^{-\\mu ^q_j (t-\\tau )} a^q_j(\\tau ) \\, d\\tau \\, e_j v^q\\\\+ \\sum _{q \\in \\mathbb {N}_0^{\\infty };\\, |q| = p} \\sum _{j \\in J^q_-}\\int _{t}^{\\infty } e^{-\\mu ^q_j (t-\\tau )} a^q_j(\\tau ) \\, d\\tau \\, e_j v^q.$ Tonelli's theorem implies that convergence of the first of these sums is equivalent to convergence of $ \\int _{-\\infty }^te^{-\\mu ^q_j (t-\\tau )}\\Bigl (\\sum _{q \\in \\mathbb {N}_0^{\\infty };\\, |q| = p} \\sum _{j \\in J^q_+}a^q_j(\\tau ) e_j v^q \\Bigr ) \\, d\\tau $ The sum inside the brackets converges in $\\operatorname{Pol}^p(V_{\\infty })$ , uniformly in $\\tau $ , by convergence of eq aqrs and cor comp conv.",
"Since the functions $a^q_j$ grow at most polynomially, uniformly in $q$ and $j$ , the value of that sum grows at most polynomially as well, when viewed as a convergent series in $\\operatorname{Pol}^p(V_k, V_l)$ .",
"So the integral over $\\tau $ converges in $\\operatorname{Pol}^p(V_{\\infty })$ , by completeness of the spaces $\\operatorname{Pol}^p(V_k, V_l)$ .",
"By continuity of eq int J+ in $t$ , the convergence is uniform in $t$ on compact subsets of $I$ .",
"The derivatives of eq int J+ with respect to $t$ are linear combinations of eq int J+ and $\\sum _{q \\in \\mathbb {N}_0^{\\infty };\\, |q| = p} \\sum _{j \\in J^q_+}a^q_j(t) e_j v^q$ and therefore converge as well.",
"By an analogous argument, the second sum in eq sums Jpm converges as well, differentiably in $t$ .",
"prop csu follows from lem ODE j,lem conv hat psi hat F."
],
[
"Proof of thm normal form",
"2 Lemma 7.7 If $f, \\xi _p, F_p, \\Phi _p$ and $R_{p}$ have polynomial growth, then so do $\\xi _{p+1}$ , $F_{p+1}$ , $\\Phi _{p+1}$ and $R_{p+1}$ .",
"If $R_p$ has polynomial growth, then the functions $a^q_j$ and all their derivatives grow at most polynomially, uniformly in $q$ and $j$ .",
"Hence, by eq hat psi hat F Jq and eq hat psi hat F Jqc, the map $\\hat{F}$ has polynomial growth.",
"By lem conv pol growth, the same is true for $\\hat{\\xi }$ .",
"So $F_{p+1}$ and $\\xi _{p+1}$ have polynomial growth.",
"Polynomial growth is preserved under composition and derivatives in the $I$ and $V_{\\infty }$ directions.",
"Hence the map $\\hat{\\Phi }$ as in eq def hat Phi has polynomial growth, and therefore so do $\\Phi _{p+1}$ and $R_{p+1}$ .",
"Combining lem pol growth with prop coord transf,prop csu, we prove the following slightly more explicit version of thm normal form.",
"2 Theorem 7.8 Let $p \\in \\mathbb {N}$ be such that $p \\ge 2$ , $\\beta - (p+1)\\alpha > \\tilde{\\mu }$ and $\\gamma - (p+1)\\alpha > \\tilde{\\mu }$ .",
"Suppose that $f$ has polynomial growth.",
"Then there are infinitely differentiable maps $F_p, \\xi _p, R_p\\colon I \\times V_{\\infty } \\rightarrow V_{\\infty },$ where $R_p = \\mathcal {O}(p)$ , $F_p$ is a polynomial map that separates invariant subspaces, $\\xi _p$ is a near-identity and $\\xi _p - \\operatorname{id}$ and $F_p$ are compact polynomials of orders at most $p-1$ , such that if $X$ and $x$ are as in eq ODE XYZ p and eq def xyz p, then eq ODE xyz holds.",
"Finally, there is a construction of the map $\\xi _p$ in which all integrals over $I$ that occur are $\\tilde{\\mu }$ -regular.",
"We use induction on $p$ to prove that the claim holds for every $p$ , including the auxiliary statement that $\\xi _p, F_p, \\Phi _p$ and $R_{p}$ have polynomial growth.",
"If $p=2$ , then we may take $F_{p}(t,v) = 0$ and $\\xi _{p}(t,v) = v$ for all $t \\in I$ and $v \\in V_0$ .",
"Then $R_2 = f$ , so $\\xi _p, F_p, \\Phi _p$ and $R_{p}$ have polynomial growth because $f$ does.",
"The induction step follows from lem pol growth,prop coord transf,prop csu."
],
[
"Dynamics of the normal form equation",
"It remains to prove lem Vj invar and prop dynamics, which we use to justify def centre mfd based on thm normal form.",
"Throughout this section, we suppose that $F \\colon I \\times V_{\\infty } \\rightarrow V_{\\infty }$ is a smooth map that separates invariant subspaces.",
"[Proof of lem Vj invar.]",
"First, suppose that $a=c$ .",
"For all $v \\in V_c$ and all $q \\in \\mathbb {N}_0^{\\infty }$ with $|q| \\le p$ and $q^s \\ne 0$ or $q^u \\ne 0$ , we have $v^q = 0$ .",
"So the properties eq Fcsu of the map $F$ imply that $F(I \\times V_c) \\subset V_c$ .",
"This, in turn, implies that for all maps $X\\colon I \\rightarrow V_{\\infty }$ satisfying eq ODE XYZ p, if $X(t) \\in V_c$ for a given $t$ then $\\dot{X}(t) \\in V_c$ .",
"So $X(t) \\in V_c$ for all $t \\in I$ .",
"Next, suppose that $a=s$ .",
"If $v \\in V_s$ and $q \\in \\mathbb {N}_0^{\\infty }$ , then $v^q=0$ if $q^u \\ne 0$ .",
"So $F_u(t,v)=0$ for all $t \\in I$ .",
"And the components of $F_c(t,v)$ with $q^u \\ne 0$ are zero for the same reason, while its components with $q^s = 0$ are zero since $v \\in V_s$ .",
"Hence $F_c(t,v)=0$ .",
"We conclude that $F(I \\times V_s) \\subset V_s$ .",
"As in the case $a=c$ , this implies the claim for $a=s$ .",
"The argument for $a=u$ is entirely analogous to the case $a=s$ .",
"To prove prop dynamics, we start with a general comparison estimate for solutions of odes in Hilbert spaces.",
"2 Lemma 8.1 Let $V$ be a Hilbert space, $W \\subset V$ a subspace, $I$ an open interval containing 0, and $g$ a map from $I \\times V$ into the space of linear operators from $W$ to $V$ .",
"Let $X \\colon I \\rightarrow W$ be a differentiable map (as a map into $V$ ), such that for all $t \\in I$ , $\\dot{X}(t) = g(t, X(t))X(t).$ If $\\zeta \\in \\mathbb {R}$ is such that $g(t,w)+\\zeta $ is negative semidefinite for all $t \\in I$ and $w \\in W$ , then for all $t \\in I$ with $t\\ge 0$ , $\\Vert X(t)\\Vert _V \\le \\Vert X(0)\\Vert _V e^{-\\zeta t}$ First, suppose that $\\zeta = 0$ .",
"Then for all $t\\in I$ , $\\frac{d}{dt} \\Vert X(t)\\Vert _V^2 = 2\\Re (\\dot{X}(t), X(t))_V = 2\\Re (g(t, X(t))X(t), X(t))_V \\le 0.$ So $\\Vert X\\Vert _V^2$ is a nonnegative, non-increasing function on $I$ , and the claim for $\\zeta = 0$ follows.",
"Next, let $\\zeta \\in \\mathbb {R}$ be arbitrary.",
"Then $\\frac{d}{dt}(X(t)e^{\\zeta t}) = \\bigl (g(t, X(t))+\\zeta \\bigr )X(t)e^{\\zeta t}.$ Applying the claim for $\\zeta = 0$ , with $X(t)$ replaced by $X(t)e^{\\zeta t}$ and $g(t,w)$ by $g(t,w)+\\zeta $ , now yields the claim for $\\zeta $ .",
"For any homogeneous polynomial map $p = p_{\\lambda }$ between normed vector spaces $V$ and $W$ , where $\\lambda \\in S\\mathcal {B}^n(V,W)$ , define the map $\\tilde{p}\\colon V \\rightarrow \\mathcal {B}(V,W)$ by $ \\tilde{p}(v_1)v_2 := \\lambda (v_1, \\ldots , v_1, v_2).$ Here $n-1$ copies of $v_1$ are inserted into $\\lambda $ on the right hand side.",
"For all $t \\in I$ , the map $F(t, )$ lies in $\\operatorname{Pol}(V_{k}, V_1)$ for some $k$ .",
"The operator $A$ lies in $\\mathcal {B}(V_l, V_1)$ for some $l$ .",
"By replacing the smaller of $k$ or $l$ by the larger of these two numbers, we henceforth assume $k=l$ .",
"Applying the construction eq def tilde p to each homogeneous term of $F(t, )$ and adding the resulting maps, we obtain a map $\\tilde{F}\\colon V_k \\rightarrow \\mathcal {B}(V_k, V_1)$ , such that for all $v \\in V_k$ , $F(t,v) = \\tilde{F}(t,v)v.$ For $a \\in \\lbrace c,s,u\\rbrace $ , we write $\\tilde{F}_a$ for $\\tilde{F}$ composed with orthogonal projection onto $V_a$ .",
"For $j \\in \\mathbb {N}$ , let $q^{(j)} \\in \\mathbb {N}_0^{\\infty }$ be defined by $q^{(j)}_m = 1$ if $m=j$ , and $q^{(j)}_m = 0$ otherwise.",
"2 Lemma 8.2 Let $v \\in V_k$ .",
"Write $v = v_c + v_s+v_u$ , where $v_a \\in V_a$ for $a \\in \\lbrace c,s,u\\rbrace $ .",
"Then for all $t \\in I$ , the components of $F(t,v)$ in $V_s$ , $V_u$ and $V_c$ satisfy $(F(t,v))_s &= \\tilde{F}(t,v)v_s; \\\\(F(t,v))_u &= \\tilde{F}(t,v)v_u; \\\\(F(t,v))_c &= \\tilde{F}(t,v_c)v_c \\quad \\text{if $v_s = 0$ or $v_u = 0$.",
"}$ Let $t \\in I$ and $v \\in V_k$ .",
"To prove eq tilFp s, we use the fact that by eq sep invar s, $F_s (t,v)=\\sum _{j \\in J_s} \\,\\, \\sum _{q \\in \\mathbb {N}_0^{\\infty }: |q| \\le p-1} F^{q+ q^{(j)}}(t) v^q v_j.$ So $\\tilde{F}_s(t,v) = \\sum _{j \\in J_s} \\,\\, \\sum _{q \\in \\mathbb {N}_0^{\\infty }: |q| \\le p-1} F^{q+ q^{(j)}}(t) v^q e^j,$ where $\\lbrace e^j\\rbrace _{j \\in \\mathbb {N}}$ is the basis of $V_1^*$ dual to $\\lbrace e_j\\rbrace _{j \\in \\mathbb {N}}$ .",
"Hence $\\tilde{F}_s(t,v)v = \\tilde{F}(t,v) v_s,$ which implies eq tilFp s. The equality eq tilFp u can be proved anaogously.",
"To prove eq tilFp c, we note that by eq sep invar c, $F_c (t,v)= F_{c,1}(t,v) + F_{c,2}(t,v),$ where $F_{c,1}(t,v)&= \\sum _{q \\in \\mathbb {N}_0^{\\infty }: |q| \\le p,q^s = q^u=0} F^q(t) v^q; \\\\F_{c,2}(t,v)&= \\sum _{q \\in \\mathbb {N}_0^{\\infty }: |q| \\le p,q^s \\ne 0 \\ne q^u} F^q(t) v^q.",
"$ The right hand side of eq Fpc1 only depends on $v_c$ , and the right hand side of eq Fpc2 is zero if $v_s = 0$ or $v_u = 0$ .",
"So, under that condition, $F_c (t,v) = F_{c}(t,v_c) = \\tilde{F}(t,v_c)v_c.$ [Proof of prop dynamics.]",
"Set $ D_{\\tilde{\\mu }} := \\bigl \\lbrace (t,v) \\in I \\times V_{\\infty } : \\Vert \\tilde{F}(t,v)\\Vert _{\\mathcal {B}(V_k, V_1)} < \\tilde{\\mu }\\bigr \\rbrace .$ Because $F$ is a sum of polynomials of degrees at least two, we have $\\tilde{F}(t,0) = 0$ for all $t$ .",
"So $D_{\\tilde{\\mu }}$ contains $I \\times \\lbrace 0\\rbrace $ .",
"It is open by continuity of $\\tilde{F}$ .",
"Let $X\\colon I \\rightarrow V_{\\infty }$ be a solution of the constructed system eq ODE XYZ p. As in the proof of lem Vj invar, $ \\dot{X}_s(t) = AX(t) + F_s(t, X(t)) = \\bigl (A + \\tilde{F}(t, X(t) \\bigr ) X_s(t),$ where we used the first equality in lem tilFp and the fact that $A$ preserves $V_s$ .",
"For all $(t,v)\\in D_{\\tilde{\\mu }} $ , the operator $A + \\tilde{F}_s(t, v) + \\beta - \\tilde{\\mu }\\colon V_k \\cap V_s \\rightarrow V_1 \\cap V_s$ is negative semidefinite.",
"Hence the claim about $X_s$ follows from the second part of lem decay to Vc.",
"The claim about $X_u$ can be proved similarly, via a version of lem decay to Vc for positive-definite operators.",
"Next, suppose that $X_s(0) = 0$ or $X_u(0) = 0$ .",
"By lem Vj invar, either $X_s(t) = 0$ for all $t \\in I$ or $X_u(t) = 0$ for all $t \\in I$ .",
"Similarly to eq X dot dynamics, the third equality in lem tilFp implies that $\\dot{X}_c(t) = \\bigl (A + \\tilde{F}(t, X_c(t) \\bigr ) X_c(t),$ for all $t \\in I$ .",
"And for all $(t,v)\\in D_{\\tilde{\\mu }}$ , the operator $A + \\tilde{F}_c(t, v) - \\alpha - \\tilde{\\mu }\\colon V_k \\cap V_c \\rightarrow V_1 \\cap V_c$ is negative semidefinite.",
"So by lem decay to Vc, $\\Vert X_c(t)\\Vert _{V_1} \\le e^{(\\alpha + \\tilde{\\mu })t}\\Vert X_c(0)\\Vert _{V_1}$ for all $t\\ge 0$ in $I$ .",
"It similarly follows that for all $t\\le 0$ in $I$ , $\\Vert X_c(t)\\Vert _{V_1} \\le e^{-(\\alpha + \\tilde{\\mu })t} \\Vert X_c(0)\\Vert _{V_1}.$"
],
[
"Example: a non-autonomous version of Burgers' equation",
"Let $r \\in \\mathbb {R}$ , and consider the non-autononomous, nonlinear pde $ \\partial _t u (t, \\theta )= \\partial _\\theta ^2 u(t,\\theta ) + ru(t,\\theta ) - \\frac{t}{2}(\\partial _\\theta u (t,\\theta ))^2,$ with $2\\pi $ -periodic boundary conditions in $\\theta $ .",
"Then thm special case applies, where $\\Omega $ is the circle.",
"Using thm normal form, we compute the centre manifold of the normal form system approximating eq modified Burgers up to residuals of order three, in sec ex thm.",
"Via a direct approach, we compute all invariant manifolds for residuals of orders three and four, in sec ex direct.",
"We find that the order three centre manifolds computed in the two ways agree.",
"These computations illustrate rem direct constr, that the construction from thm normal form is guaranteed to give a result, while a direct computation may be more efficient in concrete situations."
],
[
"Centre manifold via thm normal form",
"In this setting, $Au = u^{\\prime \\prime } + ru\\quad \\text{and}\\quad f(t,u) = - \\frac{t}{2}(u^{\\prime })^2,$ where a prime denotes the derivative in the $\\theta $ -direction.",
"The eigenfunctions of $A$ are $e_j$ , for $j \\in \\mathbb {Z}$ , given by $e_j(\\theta ) := e^{ij\\theta }$ .",
"The eigenvalue corresponding to $e_j$ is $\\alpha _j = r-j^2$ (which has multiplicity two when $j\\ne 0$ ).",
"Choose $\\alpha , \\beta , \\gamma $ and $\\tilde{\\mu }$ such that $0 \\le \\alpha < \\tilde{\\mu }< \\beta = \\gamma < 1$ , and $\\alpha < \\frac{1}{2}$ .",
"Suppose that $r$ lies within $\\alpha $ of an integer of the form $n^2$ , for a nonzero $n \\in \\mathbb {Z}$ .",
"Then the eigenvalue $\\alpha _n$ is central up to precision $\\alpha $ .",
"We determine a corresponding centre manifold for a system that approximates eq modified Burgers up to a third-order residual.",
"This involves the coordinate transform $\\xi _3$ .",
"To compute this centre manifold, we only need to apply $\\xi _3$ to elements of $V_c = \\operatorname{span}\\lbrace e_n,e_{-n}\\rbrace $ .",
"In other words, we only need to compute $\\xi _3 (t, X_ne_n+X_{-n} e_{-n})$ , for $t \\in \\mathbb {R}$ and $X_n,X_{-n} \\in .",
"(We do not determine the domain $ D$ here.",
")$ For $p=2$ , the map $\\xi _2$ is the identity map.",
"So $\\xi _3 (t, X_ne_n+X_{-n}e_{-n}) = X_ne_n+X_{-n}e_{-n} + \\hat{\\xi }(X_ne_n + X_{-n}e_{-n}),$ where $\\hat{\\xi }(X_ne_n+ X_{-n}e_{-n}) = \\sum _{q \\in \\mathbb {Z}^{\\infty };\\, |q|=2} \\sum _{j \\in \\mathbb {Z}} \\hat{\\xi }^q_j(t) e_j (X_ne_n + X_{-n}e_{-n})^q.$ For $j \\in \\mathbb {Z}$ , let $q^{(j)} \\in \\mathbb {Z}^{\\infty }$ be defined by $q^{(j)}_m = 1$ if $m=j$ , and $q^{(j)}_m = 0$ otherwise.",
"Then, for $q \\in \\mathbb {Z}^{\\infty }$ with $|q|=2$ , $(X_ne_n + X_{-n}e_{-n})^q ={\\left\\lbrace \\begin{array}{ll}X_n^2 & \\text{if }q = 2q^{(n)}; \\\\X_{-n}^2 & \\text{if }q = 2q^{(-n)}; \\\\X_nX_{-n} & \\text{if }q = q^{(n)} + q^{(-n)}\\\\0 & \\text{otherwise}.\\end{array}\\right.",
"}$ So $\\hat{\\xi }(X_ne_n+ X_{-n}e_{-n}) = \\sum _{j \\in \\mathbb {Z}}\\Bigl (X_n^2\\hat{\\xi }^{2q^{(n)}}_j(t) + X_{-n}^2 \\hat{\\xi }^{2q^{(-n)}}_j(t) + X_nX_{-n} \\hat{\\xi }^{q^{(n)} + q^{(-n)}}_j(t)\\Bigr )e_j.$ The map $\\hat{\\xi }^{2q^{(n)}}_j$ is expressed in terms of the map $a^{2q^{(n)}}_j$ in $R_2(t,u) = -\\sum _{j \\in \\mathbb {Z}} \\sum _{q \\in \\mathbb {Z}^{\\infty };\\, |q|= 2} a^{q}_j(t) e_ju^q.$ (The order three term in eq aqrs now equals zero.)",
"See eq hat psi hat F Jq and eq hat psi hat F Jqc.",
"If $u = \\sum _{l \\in \\mathbb {Z}} x_l e_l$ , then $R_2(t,u) = -\\frac{t}{2}(u^{\\prime })^2 =- \\frac{t}{2}\\sum _{j \\in \\mathbb {Z}} \\Bigl ( \\sum _{k \\in \\mathbb {Z}} k(j-k)x_k x_{j-k} \\Bigr ) e_j.$ The equality $x_k x_{j-k} = u^{2q^{(n)}} = x_n^2$ holds precisely if $k = n$ and $j=2n$ .",
"Hence $a^{2q^{(n)}}_{2n}(t) = \\frac{t}{2} n^2$ and $a^{2q^{(n)}}_{j} = 0$ if $j \\ne 2n$ .",
"An analogous argument shows that $a^{2q^{(-n)}}_{-2n}(t) = \\frac{t}{2} n^2$ and $a^{2q^{(-n)}}_{j} = 0$ if $j \\ne -2n$ .",
"The equality $x_k x_{j-k} = u^{q^{(n)} + q^{(-n)}} = x_n x_{-n}$ holds precisely if $j=0$ and either $k = n$ or $k=-n$ .",
"Hence $a^{q^{(n)} + q^{(-n)}}_{0}(t) = -t n^2$ and $a^{q^{(n)} + q^{(-n)}}_{j} = 0$ if $j \\ne 0$ .",
"The relevant numbers $\\mu ^q_j$ as in sec comp Fq equal $\\begin{split}\\mu ^{2q^{(n)}}_{2n} &= 2 \\alpha _n - \\alpha _{2n} = r+2n^2;\\\\\\mu ^{2q^{(-n)}}_{-2n} &= 2 \\alpha _{-n} - \\alpha _{-2n} = r+2n^2; \\\\\\mu ^{q^{(n)} + q^{(-n)}}_{0} &= \\alpha _n + \\alpha _{-n} - \\alpha _0 = r-2n^2\\end{split}$ (note that $\\alpha _j =\\alpha _{-j}$ for every $j$ ).",
"Because $n^2 \\ge 1$ and $\\alpha < \\frac{1}{2}$ , the real parts of $\\mu ^{2q^{(n)}}_{2n}$ and $\\mu ^{2q^{(-n)}}_{2n} $ are greater than $\\alpha $ , whereas the real part of $\\mu ^{q^{(n)} + q^{(-n)}}_{0}$ is smaller than $-\\alpha $ .",
"And with $J^q$ as in sec comp Fq, we have $2n \\in J^{2q^{(n)}}$ .",
"Indeed, $J_s = \\lbrace j \\in \\mathbb {Z}: |j| \\ge n+1 \\rbrace ,$ so $2n \\in J_s$ and $(2q^{(n)})^s = 0$ .",
"Similarly, $2n \\in J^{2q^{(-n)}}$ .",
"And $0 \\in J_u$ and $(q^{(n)} + q^{(-n)})^u = 0$ , so $0 \\in J^{q^{(n)} + q^{(-n)}}$ .",
"Hence, by eq hat psi hat F Jq, $\\begin{split}\\hat{\\xi }^{2q^{(n)}}_{2n}(t) &= \\int _{-\\infty }^t e^{-(r+2n^2)(t-\\tau )} \\frac{\\tau }{2} n^2 \\, d\\tau \\\\&= \\frac{n^2}{2(r+2n^2)} \\Bigl (t- \\frac{1}{r+2n^2}\\Bigr ).\\end{split}$ The integral converges since $\\Re (r+2n^2)>0$ , and is $\\tilde{\\mu }$ -regular.",
"Similarly, $\\hat{\\xi }^{2q^{(-n)}}_{-2n}(t) = \\frac{n^2}{2(r+2n^2)} \\Bigl (t- \\frac{1}{r+2n^2}\\Bigr ).$ And because $\\Re (\\mu ^{q^{(n)} + q^{(-n)}}_{0}) < -\\alpha $ , $\\begin{split}\\hat{\\xi }^{q^{(n)}+ q^{(-n)}}_{0}(t) &= -\\int _t^{\\infty } e^{-(r-2n^2)(t-\\tau )} ({-\\tau } n^2) \\, d\\tau \\\\&= \\frac{-n^2}{r-2n^2} \\Bigl (t- \\frac{1}{r-2n^2}\\Bigr ).\\end{split}$ We conclude that for all $t \\in \\mathbb {R}$ and $X_n, X_{-n} \\in ,\\begin{multline} \\xi _3(t,X_ne_n + X_{-n}e_{-n}) = \\\\X_ne_n +X_{-n}e_{-n} + \\frac{n^2}{2(r+2n^2)} \\Bigl (t- \\frac{1}{r+2n^2}\\Bigr )(X_n^2e_{2n}+ X_{-n}^2e_{-2n}) \\\\-\\frac{n^2}{r-2n^2} \\Bigl (t- \\frac{1}{r-2n^2}\\Bigr ) X_nX_{-n}.\\end{multline}(The last term is a scalar multiple of the constant function $ e0$.",
")If $ r = n2$, this simplifies to\\begin{multline*}\\xi _3(t,X_ne_n + X_{-n}e_{-n}) = \\\\X_ne_n+X_{-n}e_{-n} + \\frac{1}{6} \\Bigl (t- \\frac{1}{3n^2}\\Bigr )(X_n^2 e_{2n}+X_{-n}^2e_{-2n}) + \\Bigl (t+ \\frac{1}{n^2}\\Bigr ) X_nX_{-n}.\\end{multline*}$"
],
[
"Invariant manifolds via direct computations",
"For order of residual $p=2$ , the map $\\xi _2$ is the identity map, $x_j=X_j$ .",
"Proceeding to order of residual $p=3$ we construct quadratic corrections to the identity $\\xi _2$ to form $\\xi _3$ .",
"In the eigenvector basis the field $u(t,\\theta )=\\sum _{j} x_j(t)e^{i j\\theta }$ (all sums in this section are over $\\mathbb {Z}$ ), and the pde eq modified Burgers becomes $\\dot{x}_j=\\alpha _jx_j +\\frac{t}{2}\\sum _{k} b_{jk}x_{j-k}x_k\\quad \\text{where }b_{jk}:=k(j-k).$ Writing $x_j(t) = \\xi _3(t, X(t))_j = X_j + \\sum _{k,l \\in \\mathbb {Z}}g^{kl}_j(t) X_k X_l,$ and solvingThe computer algebra code used for the computations in this section is available on http://www.maths.adelaide.edu.au/anthony.roberts/pBurgers.txt.",
"for $g^{kl}_j$ such that $x_j$ satisfies eqBurgPODEs up to terms of order three if $\\dot{X}_k = \\alpha _k X_k$ , we find that $x(t)=\\xi _3(t,X(t))$ is given by $&x_j=X_j+\\frac{1}{2}\\sum _{k:|d_{jk}^{-1}|>\\tilde{\\mu }} b_{jk}[d_{jk}t-d_{jk}^2]X_{j-k}X_k\\,,\\\\&\\text{where }d_{jk}:=1/[-\\alpha _j+\\alpha _k+\\alpha _{j-k}]=1/[r+2jk-k^2].\\nonumber $ For $r\\approx n^2$ and odd $n$ , the denominators in $d_{jk}$ are not small.",
"Then this map, combined with the linear $\\dot{X}_j=\\alpha _j X_j$ , matches the pde eq modified Burgers to third-order errors.",
"However, for $r\\approx n^2$ and even $n>0$ , some denominators are small, becoming zero when $r=n^2$ .",
"Then the divisor being zero becomes $k(k-j)=n^2/2$ and hence has zeros for every pair of integer factors of $n^2/2$ (including negative pairs).",
"Consequently these terms are excluded from the sum eqBurgP3, and instead lead to nonlinearly modifying the evolution for some $j$ via $\\dot{X}_j=\\alpha _j X_j+\\frac{t}{2}\\sum _{k:|d_{jk}^{-1}|<\\tilde{\\mu }} b_{jk}X_{j-k}X_k\\,.$ Often the centre manifold is of most interest, so in $\\xi _3$ setting all $X_j=0$ except $X_{\\pm n}$ , gives the quadratic approximate centre manifold to be $x_j=X_j$ for all $j$ except $&x_0=X_0-n^2\\left[\\frac{1}{r-2n^2}t-\\frac{1}{(r-2n^2)^2}\\right]X_nX_{-n}\\,,\\\\&x_{\\pm 2n}=X_{\\pm 2n}+\\frac{1}{2}n^2\\left[\\frac{1}{r+2n^2}t-\\frac{1}{(r+2n^2)^2}\\right]X_{\\pm n}^2\\,.$ This is the same result as eq xi3 Burgers centre.",
"Proceeding to order of residual $p=4$ we may construct cubic corrections to $\\xi _2$ to form $\\xi _4$ .",
"For simplicity, restrict attention to the cases of $n$ odd.",
"It is straightforward but tedious to construct that for $\\xi _4$ $&x_j=\\xi _{3,j}+\\sum _{k,l:|d_{jkl}^{-1}|>\\tilde{\\mu }} b_{jl}b_{lk}c_{jkl}(t)X_{j-l}X_{l-k}X_k\\,,\\\\\\text{where }&c_{jkl}:=\\tfrac{1}{2}d_{lk}d_{jkl}t^2-(d_{lk}d_{jkl}^2+\\tfrac{1}{2}d_{lk}^2d_{jkl})t+(d_{lk}d_{jkl}^3+\\tfrac{1}{2}d_{lk}^2d_{jkl}^2),\\nonumber \\\\&d_{jkl}:=1/[-\\alpha _j+\\alpha _k+\\alpha _{l-k}+\\alpha _{j-l}]=1/[2r +2jl+2kl -2k^2-2l^2].\\nonumber $ The terms excluded from $(k,l)$ in the sum eqBurgP4 must cause cubic terms in the evolution.",
"For example, when $r=n^2=1$ thenThe apparent pattern in these odes becomes more complicated—at $\\dot{X}_{\\pm 6}$ for example.",
"$&\\dot{X}_0=X_0,\\\\&\\dot{X}_{\\pm 1}=(\\tfrac{1}{9}t-\\tfrac{1}{3}t^2)X_{-1}X_{\\pm 1}^2\\,,\\\\&\\dot{X}_{\\pm 2}=-3X_{\\pm 2}+(\\tfrac{104}{225}t-\\tfrac{8}{15}t^2)X_{\\mp 1}X_{\\pm 1}X_{\\pm 2}\\,,\\\\&\\dot{X}_{\\pm 3}=-8X_{\\pm 2}+(\\tfrac{594}{1225}t-\\tfrac{18}{35}t^2)X_{\\mp 1}X_{\\pm 1}X_{\\pm 3}\\,,\\\\&\\dot{X}_{\\pm 4}=-15X_{\\pm 3}+(\\tfrac{1952}{3969}t-\\tfrac{32}{63}t^2)X_{\\mp 1}X_{\\pm 1}X_{\\pm 3}\\,,\\\\&\\qquad \\vdots \\nonumber $ By construction, in this case of $r=1$ , the coordinate transform eqBurgP4 together with the odes eqsBurgP4n1 creates a dynamical system in $u(t,\\theta )=\\sum _j x_je^{ij\\theta }$ which is the same as the pde eq modified Burgers to a residual of order four.",
"In the combined system eqBurgP4,eqsBurgP4n1, by definition def centre mfd three invariant manifolds are: the 1D unstable manifold parametrised by $X_0$ with all other $X_j=0$ ; the 2D centre manifold parametrised by $X_{\\pm 1}$ with all other $X_j=0$ ; and the stable manifold with $X_0=X_{\\pm 1}=0$ .",
"See sec code Burgers for the computer algebra code used the for the computations in this subsection.",
"It is also available on http://www.maths.adelaide.edu.au/anthony.roberts/pBurgers.txt."
],
[
"Acknowledgement",
"Part of this research was supported by the Australian Research Council grant DP150102385."
],
[
"Compact and finite-rank operators into Banach spaces",
"Let $V$ and $W$ be Banach spaces, and suppose that $V^*$ has the approximation property (this is true for example if $V$ is a Hilbert space).",
"Let $\\lbrace e^j\\rbrace _{j\\in \\mathbb {N}} \\subset V^*$ and $\\lbrace f_k\\rbrace _{k \\in \\mathbb {N}} \\subset W$ be countable subsets with dense spans.",
"(So $V^*$ and $W$ are separable.)",
"In the main text, we use the following, which is standard in the case where $V$ and $W$ are Hilbert spaces.",
"2 Proposition A.1 The space $\\operatorname{span}\\lbrace e^j \\otimes f_k : j,k \\in \\mathbb {N}\\rbrace $ is dense in $\\mathcal {K}(V,W)$ .",
"Let $\\mathcal {F}(V,W)$ be the space of finite-rank linear operators from $V$ to $W$ ; that is, operators whose images are finite-dimensional.",
"2 Lemma A.2 The space $\\operatorname{span}\\lbrace e^j \\otimes f_k : j,k \\in \\mathbb {N}\\rbrace $ is dense in $\\mathcal {F}(V,W)$ .",
"Let $T \\in \\mathcal {F}(V,W)$ .",
"Since the image of $T$ is finite-dimensional, there are $v^1, \\ldots , v^n \\in V^*$ and $w_1, \\ldots , w_n \\in W$ such that $T = \\sum _{l=1}^n v^l \\otimes w_l\\,.$ Let $\\varepsilon > 0$ .",
"For every $l$ , choose $r \\in \\mathbb {N}$ and $a_l^1, \\ldots , a_l^r \\in and $ bl1, ..., blr such that $\\Bigl \\Vert v^l- \\sum _{j=1}^r a_l^j e^j \\Bigr \\Vert _{V^*} \\le \\sqrt{\\varepsilon /n}\\quad \\text{and}\\quad \\Bigl \\Vert w_l-\\sum _{k=1}^r b_l^k f_k \\Bigr \\Vert _W \\le \\sqrt{\\varepsilon /n}.$ Using the triangle and Cauchy–Schwartz inequalities, one finds that for all $v \\in V$ , $\\bigl \\Vert Tv - \\Bigl (\\sum _{j,k = 1}^r a_l^j b_l^k e^j \\otimes f_k\\Bigr ) (v) \\bigr \\Vert _W=\\Bigl \\Vert \\sum _{l=1}^n \\bigl \\langle v^l - \\sum _{j=1}^r a^j_l e^j, v \\bigr \\rangle \\Bigl (w_l - \\sum _{k=1}^r b^j_l f_k\\Bigr )\\Bigr \\Vert _W\\\\\\le \\Vert v\\Vert _{V} \\sum _{l=1}^n \\Bigl (\\bigl \\Vert v^l - \\sum _{j=1}^r a^j_l e^j\\bigr \\Vert _{V^*} \\cdot \\bigl \\Vert w_l - \\sum _{k=1}^r b^j_l f_k\\bigr \\Vert _W\\Bigr )\\le \\varepsilon \\Vert v\\Vert _{V}.$ [Proof of prop approx cpt.]",
"Since $V^*$ has the approximation property, $\\mathcal {F}(V,W)$ is dense in $\\mathcal {K}(V,W)$ .",
"See for example Proposition 4.12(b) in the book by Ryan [31].",
"So the claim follows from lem span F."
],
[
"Computer algebra code for Burgers example computation",
"[numbers=left]pBurgers.txt"
],
[
"Dynamics of the normal form equation",
"It remains to prove lem Vj invar and prop dynamics, which we use to justify def centre mfd based on thm normal form.",
"Throughout this section, we suppose that $F \\colon I \\times V_{\\infty } \\rightarrow V_{\\infty }$ is a smooth map that separates invariant subspaces.",
"[Proof of lem Vj invar.]",
"First, suppose that $a=c$ .",
"For all $v \\in V_c$ and all $q \\in \\mathbb {N}_0^{\\infty }$ with $|q| \\le p$ and $q^s \\ne 0$ or $q^u \\ne 0$ , we have $v^q = 0$ .",
"So the properties eq Fcsu of the map $F$ imply that $F(I \\times V_c) \\subset V_c$ .",
"This, in turn, implies that for all maps $X\\colon I \\rightarrow V_{\\infty }$ satisfying eq ODE XYZ p, if $X(t) \\in V_c$ for a given $t$ then $\\dot{X}(t) \\in V_c$ .",
"So $X(t) \\in V_c$ for all $t \\in I$ .",
"Next, suppose that $a=s$ .",
"If $v \\in V_s$ and $q \\in \\mathbb {N}_0^{\\infty }$ , then $v^q=0$ if $q^u \\ne 0$ .",
"So $F_u(t,v)=0$ for all $t \\in I$ .",
"And the components of $F_c(t,v)$ with $q^u \\ne 0$ are zero for the same reason, while its components with $q^s = 0$ are zero since $v \\in V_s$ .",
"Hence $F_c(t,v)=0$ .",
"We conclude that $F(I \\times V_s) \\subset V_s$ .",
"As in the case $a=c$ , this implies the claim for $a=s$ .",
"The argument for $a=u$ is entirely analogous to the case $a=s$ .",
"To prove prop dynamics, we start with a general comparison estimate for solutions of odes in Hilbert spaces.",
"2 Lemma 8.1 Let $V$ be a Hilbert space, $W \\subset V$ a subspace, $I$ an open interval containing 0, and $g$ a map from $I \\times V$ into the space of linear operators from $W$ to $V$ .",
"Let $X \\colon I \\rightarrow W$ be a differentiable map (as a map into $V$ ), such that for all $t \\in I$ , $\\dot{X}(t) = g(t, X(t))X(t).$ If $\\zeta \\in \\mathbb {R}$ is such that $g(t,w)+\\zeta $ is negative semidefinite for all $t \\in I$ and $w \\in W$ , then for all $t \\in I$ with $t\\ge 0$ , $\\Vert X(t)\\Vert _V \\le \\Vert X(0)\\Vert _V e^{-\\zeta t}$ First, suppose that $\\zeta = 0$ .",
"Then for all $t\\in I$ , $\\frac{d}{dt} \\Vert X(t)\\Vert _V^2 = 2\\Re (\\dot{X}(t), X(t))_V = 2\\Re (g(t, X(t))X(t), X(t))_V \\le 0.$ So $\\Vert X\\Vert _V^2$ is a nonnegative, non-increasing function on $I$ , and the claim for $\\zeta = 0$ follows.",
"Next, let $\\zeta \\in \\mathbb {R}$ be arbitrary.",
"Then $\\frac{d}{dt}(X(t)e^{\\zeta t}) = \\bigl (g(t, X(t))+\\zeta \\bigr )X(t)e^{\\zeta t}.$ Applying the claim for $\\zeta = 0$ , with $X(t)$ replaced by $X(t)e^{\\zeta t}$ and $g(t,w)$ by $g(t,w)+\\zeta $ , now yields the claim for $\\zeta $ .",
"For any homogeneous polynomial map $p = p_{\\lambda }$ between normed vector spaces $V$ and $W$ , where $\\lambda \\in S\\mathcal {B}^n(V,W)$ , define the map $\\tilde{p}\\colon V \\rightarrow \\mathcal {B}(V,W)$ by $ \\tilde{p}(v_1)v_2 := \\lambda (v_1, \\ldots , v_1, v_2).$ Here $n-1$ copies of $v_1$ are inserted into $\\lambda $ on the right hand side.",
"For all $t \\in I$ , the map $F(t, )$ lies in $\\operatorname{Pol}(V_{k}, V_1)$ for some $k$ .",
"The operator $A$ lies in $\\mathcal {B}(V_l, V_1)$ for some $l$ .",
"By replacing the smaller of $k$ or $l$ by the larger of these two numbers, we henceforth assume $k=l$ .",
"Applying the construction eq def tilde p to each homogeneous term of $F(t, )$ and adding the resulting maps, we obtain a map $\\tilde{F}\\colon V_k \\rightarrow \\mathcal {B}(V_k, V_1)$ , such that for all $v \\in V_k$ , $F(t,v) = \\tilde{F}(t,v)v.$ For $a \\in \\lbrace c,s,u\\rbrace $ , we write $\\tilde{F}_a$ for $\\tilde{F}$ composed with orthogonal projection onto $V_a$ .",
"For $j \\in \\mathbb {N}$ , let $q^{(j)} \\in \\mathbb {N}_0^{\\infty }$ be defined by $q^{(j)}_m = 1$ if $m=j$ , and $q^{(j)}_m = 0$ otherwise.",
"2 Lemma 8.2 Let $v \\in V_k$ .",
"Write $v = v_c + v_s+v_u$ , where $v_a \\in V_a$ for $a \\in \\lbrace c,s,u\\rbrace $ .",
"Then for all $t \\in I$ , the components of $F(t,v)$ in $V_s$ , $V_u$ and $V_c$ satisfy $(F(t,v))_s &= \\tilde{F}(t,v)v_s; \\\\(F(t,v))_u &= \\tilde{F}(t,v)v_u; \\\\(F(t,v))_c &= \\tilde{F}(t,v_c)v_c \\quad \\text{if $v_s = 0$ or $v_u = 0$.",
"}$ Let $t \\in I$ and $v \\in V_k$ .",
"To prove eq tilFp s, we use the fact that by eq sep invar s, $F_s (t,v)=\\sum _{j \\in J_s} \\,\\, \\sum _{q \\in \\mathbb {N}_0^{\\infty }: |q| \\le p-1} F^{q+ q^{(j)}}(t) v^q v_j.$ So $\\tilde{F}_s(t,v) = \\sum _{j \\in J_s} \\,\\, \\sum _{q \\in \\mathbb {N}_0^{\\infty }: |q| \\le p-1} F^{q+ q^{(j)}}(t) v^q e^j,$ where $\\lbrace e^j\\rbrace _{j \\in \\mathbb {N}}$ is the basis of $V_1^*$ dual to $\\lbrace e_j\\rbrace _{j \\in \\mathbb {N}}$ .",
"Hence $\\tilde{F}_s(t,v)v = \\tilde{F}(t,v) v_s,$ which implies eq tilFp s. The equality eq tilFp u can be proved anaogously.",
"To prove eq tilFp c, we note that by eq sep invar c, $F_c (t,v)= F_{c,1}(t,v) + F_{c,2}(t,v),$ where $F_{c,1}(t,v)&= \\sum _{q \\in \\mathbb {N}_0^{\\infty }: |q| \\le p,q^s = q^u=0} F^q(t) v^q; \\\\F_{c,2}(t,v)&= \\sum _{q \\in \\mathbb {N}_0^{\\infty }: |q| \\le p,q^s \\ne 0 \\ne q^u} F^q(t) v^q.",
"$ The right hand side of eq Fpc1 only depends on $v_c$ , and the right hand side of eq Fpc2 is zero if $v_s = 0$ or $v_u = 0$ .",
"So, under that condition, $F_c (t,v) = F_{c}(t,v_c) = \\tilde{F}(t,v_c)v_c.$ [Proof of prop dynamics.]",
"Set $ D_{\\tilde{\\mu }} := \\bigl \\lbrace (t,v) \\in I \\times V_{\\infty } : \\Vert \\tilde{F}(t,v)\\Vert _{\\mathcal {B}(V_k, V_1)} < \\tilde{\\mu }\\bigr \\rbrace .$ Because $F$ is a sum of polynomials of degrees at least two, we have $\\tilde{F}(t,0) = 0$ for all $t$ .",
"So $D_{\\tilde{\\mu }}$ contains $I \\times \\lbrace 0\\rbrace $ .",
"It is open by continuity of $\\tilde{F}$ .",
"Let $X\\colon I \\rightarrow V_{\\infty }$ be a solution of the constructed system eq ODE XYZ p. As in the proof of lem Vj invar, $ \\dot{X}_s(t) = AX(t) + F_s(t, X(t)) = \\bigl (A + \\tilde{F}(t, X(t) \\bigr ) X_s(t),$ where we used the first equality in lem tilFp and the fact that $A$ preserves $V_s$ .",
"For all $(t,v)\\in D_{\\tilde{\\mu }} $ , the operator $A + \\tilde{F}_s(t, v) + \\beta - \\tilde{\\mu }\\colon V_k \\cap V_s \\rightarrow V_1 \\cap V_s$ is negative semidefinite.",
"Hence the claim about $X_s$ follows from the second part of lem decay to Vc.",
"The claim about $X_u$ can be proved similarly, via a version of lem decay to Vc for positive-definite operators.",
"Next, suppose that $X_s(0) = 0$ or $X_u(0) = 0$ .",
"By lem Vj invar, either $X_s(t) = 0$ for all $t \\in I$ or $X_u(t) = 0$ for all $t \\in I$ .",
"Similarly to eq X dot dynamics, the third equality in lem tilFp implies that $\\dot{X}_c(t) = \\bigl (A + \\tilde{F}(t, X_c(t) \\bigr ) X_c(t),$ for all $t \\in I$ .",
"And for all $(t,v)\\in D_{\\tilde{\\mu }}$ , the operator $A + \\tilde{F}_c(t, v) - \\alpha - \\tilde{\\mu }\\colon V_k \\cap V_c \\rightarrow V_1 \\cap V_c$ is negative semidefinite.",
"So by lem decay to Vc, $\\Vert X_c(t)\\Vert _{V_1} \\le e^{(\\alpha + \\tilde{\\mu })t}\\Vert X_c(0)\\Vert _{V_1}$ for all $t\\ge 0$ in $I$ .",
"It similarly follows that for all $t\\le 0$ in $I$ , $\\Vert X_c(t)\\Vert _{V_1} \\le e^{-(\\alpha + \\tilde{\\mu })t} \\Vert X_c(0)\\Vert _{V_1}.$"
],
[
"Example: a non-autonomous version of Burgers' equation",
"Let $r \\in \\mathbb {R}$ , and consider the non-autononomous, nonlinear pde $ \\partial _t u (t, \\theta )= \\partial _\\theta ^2 u(t,\\theta ) + ru(t,\\theta ) - \\frac{t}{2}(\\partial _\\theta u (t,\\theta ))^2,$ with $2\\pi $ -periodic boundary conditions in $\\theta $ .",
"Then thm special case applies, where $\\Omega $ is the circle.",
"Using thm normal form, we compute the centre manifold of the normal form system approximating eq modified Burgers up to residuals of order three, in sec ex thm.",
"Via a direct approach, we compute all invariant manifolds for residuals of orders three and four, in sec ex direct.",
"We find that the order three centre manifolds computed in the two ways agree.",
"These computations illustrate rem direct constr, that the construction from thm normal form is guaranteed to give a result, while a direct computation may be more efficient in concrete situations."
],
[
"Centre manifold via thm normal form",
"In this setting, $Au = u^{\\prime \\prime } + ru\\quad \\text{and}\\quad f(t,u) = - \\frac{t}{2}(u^{\\prime })^2,$ where a prime denotes the derivative in the $\\theta $ -direction.",
"The eigenfunctions of $A$ are $e_j$ , for $j \\in \\mathbb {Z}$ , given by $e_j(\\theta ) := e^{ij\\theta }$ .",
"The eigenvalue corresponding to $e_j$ is $\\alpha _j = r-j^2$ (which has multiplicity two when $j\\ne 0$ ).",
"Choose $\\alpha , \\beta , \\gamma $ and $\\tilde{\\mu }$ such that $0 \\le \\alpha < \\tilde{\\mu }< \\beta = \\gamma < 1$ , and $\\alpha < \\frac{1}{2}$ .",
"Suppose that $r$ lies within $\\alpha $ of an integer of the form $n^2$ , for a nonzero $n \\in \\mathbb {Z}$ .",
"Then the eigenvalue $\\alpha _n$ is central up to precision $\\alpha $ .",
"We determine a corresponding centre manifold for a system that approximates eq modified Burgers up to a third-order residual.",
"This involves the coordinate transform $\\xi _3$ .",
"To compute this centre manifold, we only need to apply $\\xi _3$ to elements of $V_c = \\operatorname{span}\\lbrace e_n,e_{-n}\\rbrace $ .",
"In other words, we only need to compute $\\xi _3 (t, X_ne_n+X_{-n} e_{-n})$ , for $t \\in \\mathbb {R}$ and $X_n,X_{-n} \\in .",
"(We do not determine the domain $ D$ here.",
")$ For $p=2$ , the map $\\xi _2$ is the identity map.",
"So $\\xi _3 (t, X_ne_n+X_{-n}e_{-n}) = X_ne_n+X_{-n}e_{-n} + \\hat{\\xi }(X_ne_n + X_{-n}e_{-n}),$ where $\\hat{\\xi }(X_ne_n+ X_{-n}e_{-n}) = \\sum _{q \\in \\mathbb {Z}^{\\infty };\\, |q|=2} \\sum _{j \\in \\mathbb {Z}} \\hat{\\xi }^q_j(t) e_j (X_ne_n + X_{-n}e_{-n})^q.$ For $j \\in \\mathbb {Z}$ , let $q^{(j)} \\in \\mathbb {Z}^{\\infty }$ be defined by $q^{(j)}_m = 1$ if $m=j$ , and $q^{(j)}_m = 0$ otherwise.",
"Then, for $q \\in \\mathbb {Z}^{\\infty }$ with $|q|=2$ , $(X_ne_n + X_{-n}e_{-n})^q ={\\left\\lbrace \\begin{array}{ll}X_n^2 & \\text{if }q = 2q^{(n)}; \\\\X_{-n}^2 & \\text{if }q = 2q^{(-n)}; \\\\X_nX_{-n} & \\text{if }q = q^{(n)} + q^{(-n)}\\\\0 & \\text{otherwise}.\\end{array}\\right.",
"}$ So $\\hat{\\xi }(X_ne_n+ X_{-n}e_{-n}) = \\sum _{j \\in \\mathbb {Z}}\\Bigl (X_n^2\\hat{\\xi }^{2q^{(n)}}_j(t) + X_{-n}^2 \\hat{\\xi }^{2q^{(-n)}}_j(t) + X_nX_{-n} \\hat{\\xi }^{q^{(n)} + q^{(-n)}}_j(t)\\Bigr )e_j.$ The map $\\hat{\\xi }^{2q^{(n)}}_j$ is expressed in terms of the map $a^{2q^{(n)}}_j$ in $R_2(t,u) = -\\sum _{j \\in \\mathbb {Z}} \\sum _{q \\in \\mathbb {Z}^{\\infty };\\, |q|= 2} a^{q}_j(t) e_ju^q.$ (The order three term in eq aqrs now equals zero.)",
"See eq hat psi hat F Jq and eq hat psi hat F Jqc.",
"If $u = \\sum _{l \\in \\mathbb {Z}} x_l e_l$ , then $R_2(t,u) = -\\frac{t}{2}(u^{\\prime })^2 =- \\frac{t}{2}\\sum _{j \\in \\mathbb {Z}} \\Bigl ( \\sum _{k \\in \\mathbb {Z}} k(j-k)x_k x_{j-k} \\Bigr ) e_j.$ The equality $x_k x_{j-k} = u^{2q^{(n)}} = x_n^2$ holds precisely if $k = n$ and $j=2n$ .",
"Hence $a^{2q^{(n)}}_{2n}(t) = \\frac{t}{2} n^2$ and $a^{2q^{(n)}}_{j} = 0$ if $j \\ne 2n$ .",
"An analogous argument shows that $a^{2q^{(-n)}}_{-2n}(t) = \\frac{t}{2} n^2$ and $a^{2q^{(-n)}}_{j} = 0$ if $j \\ne -2n$ .",
"The equality $x_k x_{j-k} = u^{q^{(n)} + q^{(-n)}} = x_n x_{-n}$ holds precisely if $j=0$ and either $k = n$ or $k=-n$ .",
"Hence $a^{q^{(n)} + q^{(-n)}}_{0}(t) = -t n^2$ and $a^{q^{(n)} + q^{(-n)}}_{j} = 0$ if $j \\ne 0$ .",
"The relevant numbers $\\mu ^q_j$ as in sec comp Fq equal $\\begin{split}\\mu ^{2q^{(n)}}_{2n} &= 2 \\alpha _n - \\alpha _{2n} = r+2n^2;\\\\\\mu ^{2q^{(-n)}}_{-2n} &= 2 \\alpha _{-n} - \\alpha _{-2n} = r+2n^2; \\\\\\mu ^{q^{(n)} + q^{(-n)}}_{0} &= \\alpha _n + \\alpha _{-n} - \\alpha _0 = r-2n^2\\end{split}$ (note that $\\alpha _j =\\alpha _{-j}$ for every $j$ ).",
"Because $n^2 \\ge 1$ and $\\alpha < \\frac{1}{2}$ , the real parts of $\\mu ^{2q^{(n)}}_{2n}$ and $\\mu ^{2q^{(-n)}}_{2n} $ are greater than $\\alpha $ , whereas the real part of $\\mu ^{q^{(n)} + q^{(-n)}}_{0}$ is smaller than $-\\alpha $ .",
"And with $J^q$ as in sec comp Fq, we have $2n \\in J^{2q^{(n)}}$ .",
"Indeed, $J_s = \\lbrace j \\in \\mathbb {Z}: |j| \\ge n+1 \\rbrace ,$ so $2n \\in J_s$ and $(2q^{(n)})^s = 0$ .",
"Similarly, $2n \\in J^{2q^{(-n)}}$ .",
"And $0 \\in J_u$ and $(q^{(n)} + q^{(-n)})^u = 0$ , so $0 \\in J^{q^{(n)} + q^{(-n)}}$ .",
"Hence, by eq hat psi hat F Jq, $\\begin{split}\\hat{\\xi }^{2q^{(n)}}_{2n}(t) &= \\int _{-\\infty }^t e^{-(r+2n^2)(t-\\tau )} \\frac{\\tau }{2} n^2 \\, d\\tau \\\\&= \\frac{n^2}{2(r+2n^2)} \\Bigl (t- \\frac{1}{r+2n^2}\\Bigr ).\\end{split}$ The integral converges since $\\Re (r+2n^2)>0$ , and is $\\tilde{\\mu }$ -regular.",
"Similarly, $\\hat{\\xi }^{2q^{(-n)}}_{-2n}(t) = \\frac{n^2}{2(r+2n^2)} \\Bigl (t- \\frac{1}{r+2n^2}\\Bigr ).$ And because $\\Re (\\mu ^{q^{(n)} + q^{(-n)}}_{0}) < -\\alpha $ , $\\begin{split}\\hat{\\xi }^{q^{(n)}+ q^{(-n)}}_{0}(t) &= -\\int _t^{\\infty } e^{-(r-2n^2)(t-\\tau )} ({-\\tau } n^2) \\, d\\tau \\\\&= \\frac{-n^2}{r-2n^2} \\Bigl (t- \\frac{1}{r-2n^2}\\Bigr ).\\end{split}$ We conclude that for all $t \\in \\mathbb {R}$ and $X_n, X_{-n} \\in ,\\begin{multline} \\xi _3(t,X_ne_n + X_{-n}e_{-n}) = \\\\X_ne_n +X_{-n}e_{-n} + \\frac{n^2}{2(r+2n^2)} \\Bigl (t- \\frac{1}{r+2n^2}\\Bigr )(X_n^2e_{2n}+ X_{-n}^2e_{-2n}) \\\\-\\frac{n^2}{r-2n^2} \\Bigl (t- \\frac{1}{r-2n^2}\\Bigr ) X_nX_{-n}.\\end{multline}(The last term is a scalar multiple of the constant function $ e0$.",
")If $ r = n2$, this simplifies to\\begin{multline*}\\xi _3(t,X_ne_n + X_{-n}e_{-n}) = \\\\X_ne_n+X_{-n}e_{-n} + \\frac{1}{6} \\Bigl (t- \\frac{1}{3n^2}\\Bigr )(X_n^2 e_{2n}+X_{-n}^2e_{-2n}) + \\Bigl (t+ \\frac{1}{n^2}\\Bigr ) X_nX_{-n}.\\end{multline*}$"
],
[
"Invariant manifolds via direct computations",
"For order of residual $p=2$ , the map $\\xi _2$ is the identity map, $x_j=X_j$ .",
"Proceeding to order of residual $p=3$ we construct quadratic corrections to the identity $\\xi _2$ to form $\\xi _3$ .",
"In the eigenvector basis the field $u(t,\\theta )=\\sum _{j} x_j(t)e^{i j\\theta }$ (all sums in this section are over $\\mathbb {Z}$ ), and the pde eq modified Burgers becomes $\\dot{x}_j=\\alpha _jx_j +\\frac{t}{2}\\sum _{k} b_{jk}x_{j-k}x_k\\quad \\text{where }b_{jk}:=k(j-k).$ Writing $x_j(t) = \\xi _3(t, X(t))_j = X_j + \\sum _{k,l \\in \\mathbb {Z}}g^{kl}_j(t) X_k X_l,$ and solvingThe computer algebra code used for the computations in this section is available on http://www.maths.adelaide.edu.au/anthony.roberts/pBurgers.txt.",
"for $g^{kl}_j$ such that $x_j$ satisfies eqBurgPODEs up to terms of order three if $\\dot{X}_k = \\alpha _k X_k$ , we find that $x(t)=\\xi _3(t,X(t))$ is given by $&x_j=X_j+\\frac{1}{2}\\sum _{k:|d_{jk}^{-1}|>\\tilde{\\mu }} b_{jk}[d_{jk}t-d_{jk}^2]X_{j-k}X_k\\,,\\\\&\\text{where }d_{jk}:=1/[-\\alpha _j+\\alpha _k+\\alpha _{j-k}]=1/[r+2jk-k^2].\\nonumber $ For $r\\approx n^2$ and odd $n$ , the denominators in $d_{jk}$ are not small.",
"Then this map, combined with the linear $\\dot{X}_j=\\alpha _j X_j$ , matches the pde eq modified Burgers to third-order errors.",
"However, for $r\\approx n^2$ and even $n>0$ , some denominators are small, becoming zero when $r=n^2$ .",
"Then the divisor being zero becomes $k(k-j)=n^2/2$ and hence has zeros for every pair of integer factors of $n^2/2$ (including negative pairs).",
"Consequently these terms are excluded from the sum eqBurgP3, and instead lead to nonlinearly modifying the evolution for some $j$ via $\\dot{X}_j=\\alpha _j X_j+\\frac{t}{2}\\sum _{k:|d_{jk}^{-1}|<\\tilde{\\mu }} b_{jk}X_{j-k}X_k\\,.$ Often the centre manifold is of most interest, so in $\\xi _3$ setting all $X_j=0$ except $X_{\\pm n}$ , gives the quadratic approximate centre manifold to be $x_j=X_j$ for all $j$ except $&x_0=X_0-n^2\\left[\\frac{1}{r-2n^2}t-\\frac{1}{(r-2n^2)^2}\\right]X_nX_{-n}\\,,\\\\&x_{\\pm 2n}=X_{\\pm 2n}+\\frac{1}{2}n^2\\left[\\frac{1}{r+2n^2}t-\\frac{1}{(r+2n^2)^2}\\right]X_{\\pm n}^2\\,.$ This is the same result as eq xi3 Burgers centre.",
"Proceeding to order of residual $p=4$ we may construct cubic corrections to $\\xi _2$ to form $\\xi _4$ .",
"For simplicity, restrict attention to the cases of $n$ odd.",
"It is straightforward but tedious to construct that for $\\xi _4$ $&x_j=\\xi _{3,j}+\\sum _{k,l:|d_{jkl}^{-1}|>\\tilde{\\mu }} b_{jl}b_{lk}c_{jkl}(t)X_{j-l}X_{l-k}X_k\\,,\\\\\\text{where }&c_{jkl}:=\\tfrac{1}{2}d_{lk}d_{jkl}t^2-(d_{lk}d_{jkl}^2+\\tfrac{1}{2}d_{lk}^2d_{jkl})t+(d_{lk}d_{jkl}^3+\\tfrac{1}{2}d_{lk}^2d_{jkl}^2),\\nonumber \\\\&d_{jkl}:=1/[-\\alpha _j+\\alpha _k+\\alpha _{l-k}+\\alpha _{j-l}]=1/[2r +2jl+2kl -2k^2-2l^2].\\nonumber $ The terms excluded from $(k,l)$ in the sum eqBurgP4 must cause cubic terms in the evolution.",
"For example, when $r=n^2=1$ thenThe apparent pattern in these odes becomes more complicated—at $\\dot{X}_{\\pm 6}$ for example.",
"$&\\dot{X}_0=X_0,\\\\&\\dot{X}_{\\pm 1}=(\\tfrac{1}{9}t-\\tfrac{1}{3}t^2)X_{-1}X_{\\pm 1}^2\\,,\\\\&\\dot{X}_{\\pm 2}=-3X_{\\pm 2}+(\\tfrac{104}{225}t-\\tfrac{8}{15}t^2)X_{\\mp 1}X_{\\pm 1}X_{\\pm 2}\\,,\\\\&\\dot{X}_{\\pm 3}=-8X_{\\pm 2}+(\\tfrac{594}{1225}t-\\tfrac{18}{35}t^2)X_{\\mp 1}X_{\\pm 1}X_{\\pm 3}\\,,\\\\&\\dot{X}_{\\pm 4}=-15X_{\\pm 3}+(\\tfrac{1952}{3969}t-\\tfrac{32}{63}t^2)X_{\\mp 1}X_{\\pm 1}X_{\\pm 3}\\,,\\\\&\\qquad \\vdots \\nonumber $ By construction, in this case of $r=1$ , the coordinate transform eqBurgP4 together with the odes eqsBurgP4n1 creates a dynamical system in $u(t,\\theta )=\\sum _j x_je^{ij\\theta }$ which is the same as the pde eq modified Burgers to a residual of order four.",
"In the combined system eqBurgP4,eqsBurgP4n1, by definition def centre mfd three invariant manifolds are: the 1D unstable manifold parametrised by $X_0$ with all other $X_j=0$ ; the 2D centre manifold parametrised by $X_{\\pm 1}$ with all other $X_j=0$ ; and the stable manifold with $X_0=X_{\\pm 1}=0$ .",
"See sec code Burgers for the computer algebra code used the for the computations in this subsection.",
"It is also available on http://www.maths.adelaide.edu.au/anthony.roberts/pBurgers.txt."
],
[
"Acknowledgement",
"Part of this research was supported by the Australian Research Council grant DP150102385."
],
[
"Compact and finite-rank operators into Banach spaces",
"Let $V$ and $W$ be Banach spaces, and suppose that $V^*$ has the approximation property (this is true for example if $V$ is a Hilbert space).",
"Let $\\lbrace e^j\\rbrace _{j\\in \\mathbb {N}} \\subset V^*$ and $\\lbrace f_k\\rbrace _{k \\in \\mathbb {N}} \\subset W$ be countable subsets with dense spans.",
"(So $V^*$ and $W$ are separable.)",
"In the main text, we use the following, which is standard in the case where $V$ and $W$ are Hilbert spaces.",
"2 Proposition A.1 The space $\\operatorname{span}\\lbrace e^j \\otimes f_k : j,k \\in \\mathbb {N}\\rbrace $ is dense in $\\mathcal {K}(V,W)$ .",
"Let $\\mathcal {F}(V,W)$ be the space of finite-rank linear operators from $V$ to $W$ ; that is, operators whose images are finite-dimensional.",
"2 Lemma A.2 The space $\\operatorname{span}\\lbrace e^j \\otimes f_k : j,k \\in \\mathbb {N}\\rbrace $ is dense in $\\mathcal {F}(V,W)$ .",
"Let $T \\in \\mathcal {F}(V,W)$ .",
"Since the image of $T$ is finite-dimensional, there are $v^1, \\ldots , v^n \\in V^*$ and $w_1, \\ldots , w_n \\in W$ such that $T = \\sum _{l=1}^n v^l \\otimes w_l\\,.$ Let $\\varepsilon > 0$ .",
"For every $l$ , choose $r \\in \\mathbb {N}$ and $a_l^1, \\ldots , a_l^r \\in and $ bl1, ..., blr such that $\\Bigl \\Vert v^l- \\sum _{j=1}^r a_l^j e^j \\Bigr \\Vert _{V^*} \\le \\sqrt{\\varepsilon /n}\\quad \\text{and}\\quad \\Bigl \\Vert w_l-\\sum _{k=1}^r b_l^k f_k \\Bigr \\Vert _W \\le \\sqrt{\\varepsilon /n}.$ Using the triangle and Cauchy–Schwartz inequalities, one finds that for all $v \\in V$ , $\\bigl \\Vert Tv - \\Bigl (\\sum _{j,k = 1}^r a_l^j b_l^k e^j \\otimes f_k\\Bigr ) (v) \\bigr \\Vert _W=\\Bigl \\Vert \\sum _{l=1}^n \\bigl \\langle v^l - \\sum _{j=1}^r a^j_l e^j, v \\bigr \\rangle \\Bigl (w_l - \\sum _{k=1}^r b^j_l f_k\\Bigr )\\Bigr \\Vert _W\\\\\\le \\Vert v\\Vert _{V} \\sum _{l=1}^n \\Bigl (\\bigl \\Vert v^l - \\sum _{j=1}^r a^j_l e^j\\bigr \\Vert _{V^*} \\cdot \\bigl \\Vert w_l - \\sum _{k=1}^r b^j_l f_k\\bigr \\Vert _W\\Bigr )\\le \\varepsilon \\Vert v\\Vert _{V}.$ [Proof of prop approx cpt.]",
"Since $V^*$ has the approximation property, $\\mathcal {F}(V,W)$ is dense in $\\mathcal {K}(V,W)$ .",
"See for example Proposition 4.12(b) in the book by Ryan [31].",
"So the claim follows from lem span F."
],
[
"Computer algebra code for Burgers example computation",
"[numbers=left]pBurgers.txt"
]
] | 1906.04420 |
[
[
"Halpha Nuclear Geyser (Bipolar Outflow) from the Barred Galaxy NGC 1415\n (ESO 482-G033)"
],
[
"Abstract A long slit spectrum from the barred galaxy NGC 1415 has been obtained with the 2.1m Guillermo Haro telescope in Cananea, Mexico at position angle 155d (EofN) and shows the kinematics of Na I D lines (in absorption) and Halpha, 6562.8 A, [NII] 6548 A, [NII] 6584 A, [SII] 6716 A, and [SII] 6731 A lines in emission from the central regions and the disk.",
"Our previous Halpha continuum-free imaging of the central region showed mainly two central bright Halpha knots straddling the nucleus, and Halpha emission regions along the south-east and north-west inner spiral arms.",
"Velocities of the Na I D absorption lines are taken as representative of the rotation curve of NGC 1415.",
"Our kinematical data indicates that the central bright Halpha straddling the nucleus have velocities in excess of the Na I D velocities.",
"We interpret these velocity excesses of the central bright Halpha knots as due to a geyser (nuclear bipolar outflow) with V_outflow 140 km/s at a P.A.+165d (EofN).",
"The axis of this outflow, is not along the rotation axis of the disk of NGC 1415 (if it were, it would be at P.A.+238d (EofN)).",
"Additionally we have determined Omega_{gas}, (radial resonances) kappa(R) and estimated the value of the pattern angular speed of an inner boxy stellar bar in NGC 1415, Omega_{bar}, from the Na I D rotation curve assuming Ratio of CR/a_{bar} = 1, Omega_{bar} = 134 km/s."
],
[
"Introduction",
"To understand the kinematics of the H$\\alpha $ bright regions straddling the nucleus of the barred galaxy NGC 1415 and the gas in the inner 300 region, we have performed long slit optical spectral line observation at a PA$\\sim 155$ (very close to the PA$\\sim 148$ of the disk) mainly including the red portion of the optical spectrum detecting Na I D (unresolved $\\lambda $ 5895.92, 5889.95 Å) lines, in absorption, H$\\alpha $ ($\\lambda $ 6562.8 Å), [N II] ($\\lambda $ 6548, 6584 Å), and [S II] ($\\lambda $ 6717, 6731 Å) in emission.",
"In our original survey of H$\\alpha $ emission from strong barred galaxies, within the Revised Shapley Ames Catalog, with IRAS f(IRAS)$_{60} \\ge 5$ Jy, and colors characteristic of star-forming galaxies, we included the disk barred galaxy NGC 1415 [32].",
"Our optical red continuum, filter I $\\lambda 8040$ Å, observation of the inner 1150 of the barred galaxy NGC 1415 shows elongated and boxy-shaped isophotes in the region around the nucleus [32], [34].",
"Figure 1 is a reproduction of our filter I $\\lambda 8040$ Å, image of the inner 115$^{\\prime \\prime }$ regions of NGC 1415 (in contours).",
"The continuum optical red (filter I) had not been flux calibrated, so the isophotes are in arbitrary units proportional to the equivalent of noise [32], [34].",
"Figure 2 shows the H$\\alpha $ continuum-free images of the inner region of NGC 1415.",
"Left plot shows the emission in contours in arbitrary units proportional to the equivalent of noise [32], [34] Right plot shows the H$\\alpha $ continuum-free emission (in grey scale) superposed on the optical red continuum (filter I, in contours).",
"The letter $\\mathcal {A}$ indicates the SE bright H$\\alpha $ knot, and letter $\\mathcal {B}$ indicates the NW bright H$\\alpha $ knot.",
"NGC 1415 (ESO 482-G033) is classified as an SBa in the Revised Shapley Ames catalog [69], as RSXS0 in RC3 [18], as (R)SAB0/a(s) in homogenized NED, and as (RL)SAB$_a$ (r'l,nr)0$^+$ in the paper Near-IR atlas of S0-Sa galaxies [48], see Table 1.",
"NGC 1415 has galactic coordinates $b \\sim 215.7$ and $l \\sim -51.4$ , namely, it is in the third quadrant of our galaxy and far below from the plane of our galaxy.",
"The values of galactic extinction are $A(R)_V \\sim 0.052$ in R band, and $A(I)_V \\sim 0.036$ in I band [70].",
"NGC 1415 is a member of a nearby poor cluster, the Eridanus Group [80], [62], which consists of 54 galaxies within the approximate limits of 3$^h~17^m$ and 4$^h~02^m$ in right ascension (J2000) and -25$~49^{\\prime }$ and -14$~52^{\\prime }$ in declination (J2000) [80], [62].",
"Twenty five out of 32 S galaxies in the Eridanus group are barred, Hubble type SB, that is 78%.",
"This is a high percentage of barred galaxies and it might indicate the physical conditions for galaxy formation and evolution on that part of the nearby universe.",
"For the Eridanus group with $\\sigma \\simeq 265$ km s$^{-1}$ [80], the predicted X ray luminosity would be L$_X(Eridanus)\\sim 4\\times 10^{43}$ erg s$^{-1}$ and the predicted X ray temperature would be T$_X(Eridanus)\\sim 1.12$ keV or T$_X(Eridanus)\\sim 1.3\\times 10^7$ K [21].",
"Radio continuum radiation has been detected from VLA maps from NGC 1415 at 20cm [12].",
"NGC 1415 has 2 galaxy companions within 10 diameters (less than 200 kpc away) and 5 galaxy companions within 10 and 20 diameters (between 200 kpc and 350 kpc away, [29].",
"Figure: NO_CAPTIONLong exposure photographs of NGC 1415, like those in the Hubble Atlas [68], indicate a rectangular bar at a PA $\\sim 130$ with two main inner spiral arms originating from the SE and NW.",
"Here we adopt a distance of to NGC 1415 as D$_{N1415} = 17.7$ Mpc (H$_{\\circ }=75$ km s$^{-1}$ Mpc$^{-1}$ , [76]), the linear scale is $10 \\sim 85.81$ pc.",
"Numerical N-body simulations of disk galaxies have shown that they may develop a high eccentricity bar with an elliptical shape (Martin 1995; Martin & Friedli 1997).",
"Sparke & Sellwood (1987), Combes et al.",
"(1990) and Athanassoula et al.",
"(1990) have shown, however, that bars in some disk galaxies are more rectangular than elliptical.",
"Figure: NO_CAPTIONImportant for the present study of the barred galaxy NGC 1415 are: a) the position of the optical nucleus $\\alpha (J2000.0)$ and $\\delta (J2000.0)$ of the galaxy.",
"b) the relative spatial positions and distances of the structures observed in the H$\\alpha $ gas continuum-free optical brightness distribution (from our optical observations [32], [34]).",
"c) The systemic velocity of NGC 1415. d) The inclination of the disk of NGC 1415 with respect to the plane of the sky.",
"See Table 1 for adopted parameters.",
"In this paper we anchored the position of the nucleus in our red optical continuum (filter I), and in our H$\\alpha $ , continuum-free image, with the optical position of the nucleus (from palomar plates [28], see Table 1).",
"The barred galaxy NGC 1415 appears to be one of several normal disk galaxies with a geyser (or bipolar outflow) from the nucleus, see $§$ 6 below.",
"In this paper, we present the observed kinematics of the gas distribution in the central region of the galaxy using primarily the Na I D absorption lines, and H$\\alpha $ emission lines from a long slit spectrum along P.A.$\\sim 155$ .",
"In $§$ 2 we describe the long slit optical spectral observation.",
"In $§$ 3 we describe the surface brightness distribution of the optical red continuum , filter I $\\lambda 8040$ Å and the H$\\alpha $ continuum-free.",
"In $§$ 4 we describe the kinematics of the Na I D absorption lines, and the H$\\alpha $ emission lines, and we describe our estimate for the rotation curve from the unresolved Na I D absorption lines at a P.A.$\\sim 155$ (E of N) using the redshifted and blueshifted velocities from the systemic heliocentric velocity of NGC 1415.",
"In $§$ 5 we describe the high values of the H$\\alpha $ velocities from $\\mathcal {A}$ , and $\\mathcal {B}$ knots.",
"In $§$ 6 we compute, from the rotation curve, $\\Omega _{gas}$ , the radial epicycle frequencies $\\kappa (r)$ , estimate the values of $\\Omega _{gas} - \\kappa /2$ , $\\Omega _{gas} + \\kappa /2$ , and estimate the angular velocity $\\Omega _{bar}$ for the boxy stellar bar in NGC 1415.",
"In $§$ 7 we summarize our findings.",
"lcc General Properties of the barred disk galaxy NGC 1415 Characteristic Value Reference Hubble Type (RSA) SBa 1 Distance 17.7 Mpc 2 Spatial scale $10 \\sim 85.81$ pc 2 IRAS $12\\mu $ 0.26 (0.53)$^a$ Jy 3, 4, 5 IRAS $25\\mu $ 0.55 (0.53)$^a$ Jy 3, 4, 5 IRAS $60\\mu $ 5.28 (6.72)$^a$ Jy 3, 4, 5 IRAS $100\\mu $ 12.32 (12.64)$^a$ Jy 3, 4, 5 T$(IRAS)_{dust}$ 33$$ K 4 log L$_B/L_{\\odot }$ 9.73 2 Heliocentric Systemic Velocity V$(hel)_{sys} \\sim 1564 \\pm 9$ km s${-1}$ 6,7 Photometric major axis P.A.",
"148$E of N $ 8 Stellar boxy bar P.A.",
"130$\\pm 5$ 4, 9 Stellar boxy bar semi major diameter 17$5 = 1.5$ kpc 9 M(HI) $5.75 - 7.1\\times 10^8$ M$_\\odot $ 10 Total radio continuum (20 cm) $\\theta _{fwhm}\\sim 8^{\\prime \\prime }$ , S$_T\\sim 18$ mJy 11 Peak SE knot radio continuum (20 cm) $\\theta _{fwhm}\\sim 8^{\\prime \\prime }$ , S$_p$ , 6.1 mJy 11 Right Ascension (J2000) $3^h ~~ 40^m ~~ 56^s.921$ 12 Declination (J2000) $-22~~ 33^{\\prime } ~~ 49.507$ 12 Inclination of the disk $i$ 65$$ 13 aTotal IRAS fluxes using ADDSCAN (Rush et al.",
"[1993] 1) RSA Sandage & Tammann, (1987), 2) Tully (1988), 3) IRAS Point Source Catalog, 4) Garcia-Barreto et al.",
"(1996), 5) Rush et al.",
"(1993), 6) Lauberts (1982), 7) Da Costa et al.",
"(1998), 8) de Vaucouleurs et al., RC3 (1993) 9) Garcia-Barreto & Moreno (2000), 10) Huchtmeier (1982), 11) Condon et al.",
"(1990), 12) Gallouët et al.",
"(1975) 13) Tully (1988)"
],
[
"Optical Spectral Observations",
"An optical line emission spectrum was obtained at the 2.1m optical telescope of the Observatorio Astrofisico Guillermo Haro (OAGH) in Cananea, Sonora, México, operated by the Instituto Nacional de Astrofisica Optica y Electrónica (INAOE), during the nights of December 29 and 30, 2000.",
"The observatory location is latitude +31$$ 03$^{\\prime }$ 10$^{\\prime \\prime }$ and longitude 110$$ 23$^{\\prime }$ 05$^{\\prime \\prime }$ west at an altitude of 2480m above the mean sea level.",
"A grating with 300 groves mm$^{-1}$ was used which resulted in a spectral sampling of about 7.6 Å and spectral coverage of $\\approx 5000 \\rightarrow 6800$ Å.",
"The spectral and spatial samplings were 1.66 Å pixel$^{-1}$ and 0463 pixel$^{-1}$ , respectively.",
"Slit width was $\\sim 18$ with lenght of about 80$^{\\prime \\prime }$ .",
"The slit P.A.",
"was $\\sim 155$ close to the P.A.$\\sim 148$ of the major axis of the optical disk.",
"The air mass was 1.75.",
"Three spectra of 30 minutes each were taken.",
"The instrumental response was calibrated by the observation of the standard star Feige 25.",
"Wavelength calibration was established via observations of He/Ar lamp [54].",
"Each frame was bias-corrected and divided by a normalized flat field, using various tasks in the IRAF optical image analysis package.",
"Three wavelength-calibrated frames at each P.A.",
"were averaged (task imcombine in IRAF), in the process of removing cosmic-ray events.",
"Sky spectrum was extracted from the slit from the object-free regions and subtracted from the NGC 1415's spectrum.",
"The uncertainity of the peak of the emission line determination was about 10%, or about $\\pm 0.38$ Å (this would correspond to about $\\pm 17$ km s$^{-1}$ at H$\\alpha $ , and to about $\\pm 19$ km s$^{-1}$ at Na I D$_1$ line $\\lambda 5895.92$ Å).",
"We were able to detect the neutral gas through sodium (Na I D lines, see appendix) in absorption throughout the disk of NGC 1415, in the slit.",
"Our long slit spectrum observation was unable to resolve each Na I D$_1$ and D$_2$ lines ($\\lambda _{D2} \\sim 5889.951$ , and $\\lambda _{D1} \\sim 5895.924$ ) in velocity, since our instrumental spectral sampling was about 7.6 Å.",
"Thus we fitted gaussian profiles for the combined symmetrical absorption Na I D lines and determine their peak velocities in space and velocity from the spectrum.",
"In the case of our kinematical study of the gas in NGC 1415, the Na I D lines are also redshifted (similar to the redshifted of emission lines [40]) and thus one confidently may assume that the absorbing gas shares the orbital rotational motion of the gas around the center of the disk."
],
[
"Red Optical Continuum, filter ",
"Figure 1 shows a reproduction of our optical red continuum image (innermost $\\sim 115^{\\prime \\prime }$ region) of NGC 1415, filter I $\\lambda _c\\sim 8040$ Å, $\\Delta \\lambda \\sim 1660$ Å from our survey work of bright and nearby RSA barred disk galaxies (with the criteria f$(IRAS)_{60\\mu } \\ge 5$ Jy) where details about the observations and the data reduction can be found [32].",
"The image was not flux calibrated (the contours are in arbitrary units of relative intensity (r.i.) of the equivalent noise, starting with first contour at 3$\\sigma $ ) [32], [34].",
"From our previous analysis, four basic mass structures were semi analytically modeled to fit the relative intensities of the isophotal observations of the optical red continumm [34], namely, (1) a compact bulge of radius $\\sim 300$ pc, (2) an elongated elliptical inner inner stellar bar of radius ${\\sim }11.6^{\\prime \\prime }$ ($\\sim 1$ kpc), at a P.A.",
"$\\sim 150$ , (3) a inner boxy contours outside the inner inner elliptical bar delineating an inner boxy bar, with radius $\\sim 17.5$ (1.5 kpc), and (4) a disk with semimajor axis $\\sim 9$ kpc [34].",
"Notice that the boxy bar is not seen edge-on, but at an inclination of $i \\sim 65$ .",
"The outer spiral arms in our optical red image start from the boxy-shaped isophotal contours radius $\\sim 18.6$ (1.6 kpc) in the NW and in the SE directions suggesting (if they trail) that the disk galaxy is rotating clockwise."
],
[
"Structure decomposition of optical (BVRI), $K_s$ , and UV continuum images from the literature",
"Two dimensional structural surface brightness BVRI decomposition of 605 galaxies with $B_T \\le 12.9$ mag and $\\delta \\le 0$ has been done by the Carnegie-Irvine group [43], [49].",
"NGC 1415 is among the galaxies that they analyzed.",
"They report among other results, B, V, R, and I magnitudes, radial profiles, color index maps, isophotal and photometric parameters, radius enclosing 20%, 50%, and 80% of the light in the B band, ellipticity, $e$ , and P.A.",
"of the photometric major axis in the I band, the inclination angle [43].",
"Additionallythey performed Fourier decomposition of the isophotes to quantify non-axisymmetric deviations in the light distribution (namely, bars) [49].",
"They used the task ellipse in IRAF to determine the values of the $e$ and P.A.",
"of a stellar bar with the following arbitrary criteria to decide if a galaxy has or has not a stellar bar: if none of the points in the $e$ profiles exceeds 0.2 or if $\\Delta e \\le 0.1$ throughout the entire $e$ profile they classified a galaxy as non barred [49].",
"On the other hand if $e \\ge 0.2$ and P.A.",
"is constant then they decide that a galaxy is barred with a length of the bar at a radius when $e$ and P.A.",
"begin to show large deviations ($\\Delta P.A.",
"\\ge 10$ ) [49].",
"Unfortunately for the case of the galaxy NGC 1415 they report it as non barred [49].",
"However, we belive that their result is in error, since the inner stellar bar in NGC 1415 has boxy isophotal shapeA rectangular, or boxy isophotal bar cannot ever be described by an ellipse, that is, how can a simple ellipse account for the corners of a boxy (rectangular) shape bar?",
"Is there any meaning of $e$ for rectangular isophotes?.",
"Two dimensional structural surface brightness of $K_s$ -band images of 206 galaxies has been done for S0-Sa galaxies (NGC 1415 is included) in order to report a detailed morphological classification [48].",
"The decomposition were made by fitting Sérsic functions for the bulge, an exponential function for the disk and a Ferrers function for the stellar bar [48].",
"For measuring bar lengths, they used two main methods: (1) visual estimation by marking the outskirts of the bar and drawing an ellipse to that distance, (2) radial profiles of the $e$ were used where the bar length was taken to be the radial distance where maximum $e$ in the bar region appeared [48].",
"For NGC 1415 they report a stellar bar of type AB$_a$ , P.A.$_{bar}\\sim 133$ , a$_{bar}^{ell} \\sim 274$ and $a_{bar}^{visual} \\sim 350$ , $b_{bar}/a_{bar} \\sim 0.43$ [48].",
"This NIR $K_s$ bar would include the inner optical stellar bar and the inner optical spiral arms (see Fig.",
"1) Two dimensional structural surface brightness of NGC 1415 Spitzer 3.6$\\mu $ image has been done [67], modeling a bulge with a radius of $\\sim 427$ pc, an exponential disk (with an exponential scale length of $\\sim 9.4$ kpc and a P.A.$_{disk}\\sim 152$ E of N), and a modified ferrer2 profile for a only one component bar, $a_{bar} \\sim 1.35$ kpc ($\\sim 157$ at a P.A.$_{bar}\\sim +134$ E of N) [67].",
"Structure decomposition from UV ($\\lambda _{W2} \\sim 2030~Å$ $\\lambda _{M2} \\sim 2231~Å$ , $\\lambda _{W1} \\sim 2030~Å$ ), and optical U, $\\lambda _{U} \\sim 3501~Å$ , B, $\\lambda _{B} \\sim 4329~Å$ , and V, $\\lambda _{V} \\sim 5402~Å$ , continuum images from Swift-UVOT has been done for 11 galaxies including NGC 1415 [64].",
"In particular for NGC 1415 they report a nuclear ring ($nr$ ) with a$_{nr}\\sim 108$ , b$_{nr} \\sim 52$ , and P.A.$_{nr}\\sim 166\\pm 2$ .",
"In their reporting of a nuclear ring from the continuum images, they included the $\\mathcal {A}$ and $\\mathcal {B}$ knots.",
"They did that without kinematical information.",
"With kinematical information, as discussed in $§$ 5, $\\mathcal {A}$ is moving away from the center (blueshifted), and $\\mathcal {B}$ is moving away from the center also (redshifted), and thus we do not think that $\\mathcal {A}$ nor $\\mathcal {B}$ knots are members of the nuclear ring."
],
[
"Spatial Distribution of H${\\alpha }$ Emission Regions",
"In NGC 1415 the inner H$\\alpha $ emission originates from regions within the inner $\\pm 20^{\\prime \\prime }$ .",
"The H$\\alpha $ +N[II] continuum-free image was obtained with a set of two narrowband filters with $\\lambda \\sim 6459$ Å with $\\Delta \\lambda \\sim 101$ Å and $\\lambda \\sim 6607$ Å with $\\Delta \\lambda \\sim 89$ Å [32], [34].",
"$\\mathcal {A}$ indicates (see right plot in Fig.",
"2) the SE H$\\alpha $ (bright by $\\sim 3.6$ times, in relative units, than the weaker NW component) component, and $\\mathcal {B}$ indicates the NW H$\\alpha $ (weak) component.",
"There are four characteristics, of the H$\\alpha $ emission from the central region of NGC 1415, worth emphasizing : 1) there are two bright regions straddling the nucleus $\\mathcal {A}$ and $\\mathcal {B}$ with weak emission connecting to what might be an apparent inner nuclear ring, the P.A.",
"of an imaginary line joining $\\mathcal {A}$ and the nucleus and the P.A.",
"of an imaginary line joining the nucleus with $\\mathcal {B}$ are the same $\\sim 161\\pm 4$ E of N, 2) there is H$\\alpha $ emission west of the nucleus, which might apparently be part of a circumnuclear structure (or nuclear ring), 3) there is no H${\\alpha }$ emission from the compact nucleus (assuming the subtraction of the red continuum was done correctly [32]), 4) there is emission from several regions in the disk.",
"The P.A.",
"of $\\mathcal {A}$ and $\\mathcal {B}$ regions is off by $\\sim 13\\rightarrow 17$ from the P.A.",
"of the major axis of the galaxy P.A.$_{A-B} \\sim 161\\pm 4$ , versus P.A.$_{optical-disk} \\sim 148 \\pm 4$ .",
"With kinematic information (see $§4$ and $§5$ ) and the spatial orientation of the inner spiral arms, the disk is rotating clockwise, assuming that the inner spiral arms are trailing.",
"The peak of the weak H$\\alpha $ emission just to the W of the nucleus at PA$\\sim 90$ lies at a radius of $4^{\\prime \\prime }$ or about 345 pc.",
"The peak of $\\mathcal {A}$ H$\\alpha $ knot is relatively brighter than the $\\mathcal {B}$ knot.",
"The peak of $\\mathcal {B}$ lies at a distance of ${\\sim }103$ or $\\sim 884$ pc.",
"$\\mathcal {A}$ lies at a distance of $\\sim 88$ or $\\sim 755$ pc.",
"It is now well accepted that gas in a non-axisymmetric bar potential loses angular momentum when gas inside the corotation radius (CR) is transfered inwards, and could accumulate near an Inner Lindblad Resonance (ILR), while gas outside CR is transfered outwards and could accumulate near an Outer Lindblad Resonance (OLR)x$_1$ orbits in a non-axisymmetrical gravitational potential (stellar bar) is one family of orbits that are elongated and sustain the stellar bar along the major axis[13], [14].",
"x$_2$ orbits is another family of orbits closer to the nucleus but are elongated perpendicular to the x$_1$ orbits along the minor axis of the stellar bar [50], [13], [14].",
"[50], [72], [10], [5].",
"Density enhancements (with star formation) near an IILR might form a circumnuclear structure (CNS) [72], [10].",
"Other RSA bright and near barred galaxies with clear CNS at r$\\sim 6 \\rightarrow 9 0$ are, for example, NGC 1326 [30], and NGC 4314 [31].",
"Figure: NO_CAPTIONWithout kinematic information two plausibles interpretations might have been considered for the existence of $\\mathcal {A}$ and $\\mathcal {B}$ regions.",
"First, they might have been regions of density enhancement and recent star formation due to shocks when gas meets the stellar bars in NGC 1415 [3] and try to follow x$_1$ orbits along the major axis of the bars.",
"However, as seen in the plane of the sky, the spatial locations of $\\mathcal {A}$ and $\\mathcal {B}$ are in the expected spatial positions if the stellar bars were rotating counter-clockwise [3].",
"However, as mentioned earlier, from the spatial location, curvature of spiral arms (trailing), and kinematics, the disk rotates clockwise and thus we assume that the bars do also rotate likewise, with the NE side being closer to the observer.",
"Thus this first interpretation is not valid with kinematic information.",
"Second, $\\mathcal {A}$ and $\\mathcal {B}$ might be considered knots of a nuclear ring with a$_{nuclear-ring} \\sim 108$ , b$_{nuclear-ring} \\sim 52$ at P.A.$\\sim 166 \\pm 2 $ [64].",
"This is the interpretation of the continuum UV and optical U, B and V imaging observations but without any kinematical analysis[64].",
"However with information of the kinematics of the gas (see $§4$ , and $§5$ ) $\\mathcal {A}$ and $\\mathcal {B}$ represent regions that are not in the rotating plane of the galaxy, but instead they form a different system, namely, gas moving away from the nucleus.",
"The imaginary line joining $\\mathcal {A}$ and $\\mathcal {B}$ is at P.A.$\\sim 161\\pm 4$ , making an angle $\\Delta \\theta \\sim 13\\rightarrow 17$ from the P.A.$\\sim 148$ of the major axis of the galaxy, and making an angle $\\Delta \\theta \\sim 73$ from the P.A.",
"of the rotation axis of the galaxy (P.A.$_{rotation-axis} \\sim 238$ ).",
"The nuclear ring would have r$_{nuclear-ring} \\sim 40 \\rightarrow 50$ [64] (see previous paragraph)."
],
[
"Velocities of gas in the gravitational field of a disk galaxy",
"The velocity of gas under the gravitational potential of a disk galaxy could be expressed in cylindrical coordinates as $ \\vec{V}(R, \\theta , z) = V_R \\hat{r} + V_{\\theta } \\hat{\\theta } + V_z \\hat{z}.$ For an outside observer, the heliocentric velocity from different regions in the disk of a galaxy with a heliocentric velocity, V$_{sys}$ , inclined an angle $i$ with respect to the plane of the sky, and from an azimuthal angle $\\theta $ from the major axis is given by $ V(obs)_{hel} = V(hel)_{sys} + V_R(R, \\theta ) sin {\\it i} sin \\theta + V_{\\theta }(R, \\theta ) sin {\\it i} cos \\theta + V_{z}(R, \\theta ) cos {\\it i}.$ In the first approximation that gas is on circular orbits, with V$_R(R, \\theta )$ = 0, and V$_z(R, \\theta )$ = 0, and the above expression becomes $ V(obs)_{hel} = V(hel)_{sys} + V_{\\theta }(R, \\theta ) sin ({\\it i}) cos (\\theta ).",
"$ For a long slit spectrum observation along the major axis of a disk galaxy $\\theta = 0$ , the velocities are: $ V_{\\theta }(R, \\theta =0) = \\frac{V(obs)_{hel} - V(hel)_{sys}}{sin ({\\it i})}, $ and for a long slit spectral observation at different position angle, the velocities are: $ V_{\\theta }(R, \\theta ) = \\frac{V(obs)_{hel} - V(hel)_{sys}}{sin ({\\it i}) cos (\\Delta \\theta )}, $ where $\\Delta \\theta $ is the angle difference between the P.A.",
"of the slit and the P.A.",
"of the major axis of the galaxy.",
"In our long slit spectroscopy study of NGC 1415, we consider V(hel)$_{sys} = 1564$ km s$^{-1}$ , i = 65$$ , P.A.$_{phot-axis} = 148$ , and $\\Delta \\theta _{slit155} = 7$ (the difference in P.A.",
"of the slit and the P.A.",
"of the photometric major axis)."
],
[
"Velocity Curve from a slit spectrum at P.A.$\\sim 155$",
"The NW distances from the center show redshifted velocities, while the SE distances from the center show blueshifted velocities.",
"The north SE-NE edge of NGC 1415 is closer to the observer.",
"Assuming the inner spiral arms are trailing, then the direction of galaxy inner rotation is clockwise.",
"The inner stellar boxy bar in NGC 1415 is at P.A.$_{bar}\\sim 130$ .",
"A line joining $\\mathcal {A}$ and $\\mathcal {B}$ knots, on the plane of the sky, straddling the nuclear region has P.A.$_{AB}\\sim 165$ .",
"Neutral sodium, Na I, D lines correspond to its doublet-resonance transition (see Appendix) are seen in absorption in the interstellar medium [74], [20].",
"Observations of Na I D lines, in our galaxy, indicate that the diffuse neutral gas is confined to a disk thickness approximately 250 pc [74], [20].",
"Since, in NGC 1415, Na I D lines (optically thin, cold, and most probably on the plane of the galaxy) are detected in absorption, it is a reasonable assumption that they represent well the kinematics of gas in rotation orbiting around the center of NGC 1415, as was first detected in Cen A [40].",
"Figure 3 left plot shows the neutral sodium, Na I D line heliocentric velocities respect to V$_{SYS}$ versus distance at a PA155 EofN.",
"Negative relative distances, left on figure, correspond to SE of NGC 1415, while positive relative distance correspond to NW of NGC 1415.",
"The velocities shown take into account $\\Delta \\theta = 7$ , they are smoothly rising with abrupt changes.",
"Right plot shows the line at P.A.$\\sim +155$ over the red optical continuum (in contours) and H$\\alpha $ image (in grey scale) indicating approximately the spatial position of the long slit.",
"Letter B on north axis, and A on bottom axis denote approximately the space locations of $\\mathcal {B}$ and $\\mathcal {A}$ knots.",
"Notice that the long slit mainly shows velocities associated with $\\mathcal {A}$ knot, the north of a plausible nuclear ring and $\\mathcal {B}$ knot.",
"This spectral slit P.A.",
"is very close to the photometric PA$\\sim 148$ of the major axis of the disk.",
"Figure: NO_CAPTIONFigure 4 shows H$\\alpha $ heliocentric velocities versus distance at a P.A.",
"$\\sim +155$ (EofN).",
"Negative relative distances, left on plot, correspond to SE of NGC 1415, while positive relative distances correspond to NW of NGC 1415.",
"The SE (negative distances) velocities shown are smoothly rising and reach V$\\sim 100$ km s$^{-1}$ at d$\\sim 0.22$ kpc, then from d$\\sim 0.43$ kpc the velocities increase with distance and reach V$\\sim 211$ km s$^{-1}$ at d$\\sim 0.8$ kpc), then slowly decrease to a value (V$\\sim 180$ km s$^{-1}$ at d$\\sim 1.65$ kpc, continue decreasing to a value (V$\\sim 138$ km s$^{-1}$ at d$\\sim 2.2$ kpc, V$\\sim 164$ km s$^{-1}$ at d$\\sim 2.78$ kpc, and finally V$\\sim 194$ km s$^{-1}$ at d$\\sim 3.4$ kpc.",
"Letter B on north axis, and A on bottom axis denote approximately the space locations of $\\mathcal {B}$ and $\\mathcal {A}$ knots.",
"A similar behavior is shown by the redshifted (NW) velocities, namely, The NW (positive distances) velocities shown smoothly rise and reach V$\\sim 130$ km s$^{-1}$ at d$\\sim 0.17$ kpc, then they rise upto V$\\sim 259$ km s$^{-1}$ at d$\\sim 1.18$ kpc, then decrese to a value V$\\sim 216$ km s$^{-1}$ at d$\\sim 1.8$ kpc, V$\\sim 238$ kkm s$^{-1}$ at d$\\sim 2.4$ kpc, and finally reach V$\\sim 229$ km s$^{-1}$ at d$\\sim 2.95$ kpc.",
"Notice that the center of the slit the velocity is redshifted (by about V$\\sim 34$ km s$^{-1}$ ).",
"Figure: NO_CAPTIONFig.",
"5 (left plot shows velocity versus distance in arcsec, while right plot shows velocities versus distance in kpc) shows the Na I D velocities (crosses) from long slit at P.A.155$$ (which is only 7$$ from the photometric P.A.148$$ of the disk of NGC 1415), simultaneously with H$\\alpha $ heliocentric velocities (open circles).",
"The velocity values at R$ = 0$ show the uncertainities in both H$\\alpha $ and Na I D lines.",
"Letter B on north axis, and A on bottom axis denote approximately the space locations of $\\mathcal {B}$ and $\\mathcal {A}$ knots.",
"Notice the higher H$\\alpha $ velocities (in emission) compared with the Na I D velocities (in absortion) at the locations shown by letter B and A.",
"There is a large difference in H$\\alpha $ (open circles) compared with the Na I D (crosses) in both the SE (left on plots) blueshifted velocities and the NW (right on plots) redshifted velocities in the sense that the absolute values show higher velocities for H$\\alpha $ than Na I D. This difference is large in the redshifted velocities in the inner $0 \\le d \\le 1.2$ kpc (right side of plots), and in the blueshifted velocities in the inner $0.43 \\le d \\le 1.0$ kpc (left side of plots).",
"$\\mathcal {A}$ knot is located at a SE distance $\\sim 755$ pc from the center of the galaxy, while $\\mathcal {B}$ knot is located at a NW distance $\\sim 884$ pc from the center of the galaxy."
],
[
"Rotation Curve in NGC 1415",
"As briefly described in previous sections, our long slit spectroscopic observation detected lines of N[II] $\\lambda 6548$ Å, H$\\alpha \\lambda 6562.8$ Å, [NII] $\\lambda 6584$ Å, [SII] $\\lambda 6716$ Å and [SII] $\\lambda 6731$ Å in emission, while the convolved Na I D lines (spectrally unresolved due to our instrumental spectral sampling) $\\lambda 5895,92$ Å, and $\\lambda 5889.95$ Å were detected in absorption.",
"Since the absorption Na I D line is optically thin, cold, residing most likely in the plane of the NGC 1415 galaxy, it most certainly shares the orbital motion of gas in the disk.",
"We assume that the Na I D velocities versus distance from the slit at P.A.$\\sim 155$ shown in Fig.",
"3 and in Fig.",
"5 is representative of the innermost rotation curve in the barred galaxy NGC 1415."
],
[
"Kinematics of $\\mathcal {A}$ and {{formula:26df1742-2900-42a3-9bf2-3cadda248a1a}}",
"As mentioned earlier, the Na I D line velocities at the center of the spectrum at P.A.$\\sim 155$ EofN in Fig.",
"3 and Fig 5 show negative velocity (blueshifted with respect to systemic) at distance 0.",
"Carefull observation of Fig.",
"3, Fig.4 and specially in Fig.",
"5 indicates that the H$\\alpha $ velocity at SE distance from $0.4 \\le d \\le 1.5$ kpc are higher (blueshifted) than the Na I D line.",
"The value of the Na I D line at about $\\mathcal {A}$ , d$\\sim 800$ pc (left side of plots in Fig.5), the velocity is V$(Na I)\\sim 140$ km s$^{-1}$ , while the value of the H$\\alpha $ line is V$(H\\alpha )\\sim 212$ km s$^{-1}$ .",
"However, as we mentioned in the previous paragraph, the Na I D line (from Fig.",
"3, and Fig.",
"5) shows V$(Na I D line)_{d=0} \\sim -46$ km s$^{-1}$ (blueshifted).",
"If we were to force V$(Na I D line)_{d=0} \\sim 0$ km s$^{-1}$ , then we would be adding 46 km s$^{-1}$ , that is, the new value of Na I D line at $\\mathcal {A}$ , d$\\sim 800$ pc would be V$(Na I D line) \\sim 94$ km s$^{-1}$ , and the difference is $\\Delta V(H\\alpha - Na I D line)_{SE} \\sim 118$ km s$^{-1}$ .",
"Similarly, the value of the Na I D line velocity at about $\\mathcal {B}$ , d$\\sim 800$ pc (crosses at right side in Fig.",
"5), is V$(Na I D line)\\sim 11$ km s$^{-1}$ , while the value of the H$\\alpha $ line is V$(H\\alpha )\\sim 217$ km s$^{-1}$ .",
"If we were to force V$(Na I D line)_{d=0} \\sim 0$ km s$^{-1}$ , then we would be adding 46 km s$^{-1}$ , that is, the new value of Na I D line at $\\mathcal {B}$ , d$\\sim 800$ pc would be V$(Na I D line) \\sim 57$ km s$^{-1}$ .",
"Then the difference is $\\Delta V(H\\alpha - Na I D line)_{NW} \\sim 160$ km s$^{-1}$ .",
"A value similar to the one obtained from $\\mathcal {A}$ .",
"We believe that the larger velocities seen in H$\\alpha $ (compared with the rotation curve shown by the Na I D velocities in the disk of NGC 1415) suggest non circular kinematics in the sense that gas in $\\mathcal {A}$ is moving away from the center (approaching us) and gas in $\\mathcal {B}$ is moving away from the center (receding from us)."
],
[
"Observational evidence of a H$\\alpha $ Geyser (Bipolar Outflow) from the nucleus in NGC 1415",
"Our estimated spatial location and kinematical analysis of $\\mathcal {A}$ and $\\mathcal {B}$ (left plot of Fig.",
"3, Fig.",
"4 and Fig.",
"5) indicates that H$\\alpha $ gas shows larger (absolute) velocities compared with Na I D velocities.",
"Thus $\\mathcal {A}$ shows bluer velocities and thus gas is moving away from the nucleus (approaching us) at a velocity of about 118 km s$^{-1}$ , while $\\mathcal {B}$ shows redder velocities and thus gas is moving away from the nucleus (receding from us) a velocity of about 160 km s$^{-1}$ .",
"The P.A.",
"of a line joining $\\mathcal {A}$ with the nucleus and the P.A.",
"of a line joining $\\mathcal {B}$ with the nucleus are the same.",
"Additionally, radio continuum radiation (20 cm, mostly synchrotron emission) has been detected with the VLA with FWHM$_{beam}80$ angular resolution from the central region of NGC 1415 [12].",
"The central radio continuum map covers approximately the same area as our innermost H$\\alpha $ emission and is elongated SE - NW at P.A.$\\sim 162$ very similar to the P.A.",
"joining $\\mathcal {A}$ and $\\mathcal {B}$ .",
"Surprisingly the peak of the radio continuum emission does not come from the nucleus of NGC 1415, but instead it coincides with $\\mathcal {A}$ .",
"This detection of apparently symmetric synchrotron 20 cm radio continuum emission at a similar P.A.",
"as the line joining $\\mathcal {A}$ and $\\mathcal {B}$ , together with our long slit kinematical data, suggests that this is a bipolar outflow from the nucleus, with the brighter knot $\\mathcal {A}$ being in front of the galaxy and the fainter knot $\\mathcal {B}$ being behind the disk.",
"Polarization studies of radio continuum emission would be very useful to confirm this fact.",
"The kinematical age might be estimated by $\\tau \\sim d/v$ , which for $d\\sim 800$ pc and $v \\sim 140$ km s$^{-1}$ , $\\tau \\sim 5.6 $ Myrs, a very recent event in the life of a galaxy."
],
[
"Orientation of the P.A. of the outflow in comparison with P.A. of rotation axis of NGC 1415",
"The P.A.$\\sim 161\\pm 4$ of the line joining $\\mathcal {A}$ and $\\mathcal {B}$ knots is considerably different from the expected nuclear outflow perpendicular to the plane of a galaxy.",
"If it were along the rotation axis of NGC 1415 (perpendicular to the plane), the expected outflow direction would be at P.A.$_{rotation-axis}\\sim +238$ .",
"Thus, the source of the bipolar outflow must be inclined to the plane of the disk of NGC 1415.",
"The unified model suggests that there is a black hole with an accretion disk and dusty torus associated with different types of AGN (Liners, being the low luminosity end of AGNs [2]).",
"The spatial location of $\\mathcal {A}$ and $\\mathcal {B}$ knots indicates that the accretion disk (which might be responsible of collimated bipolar outflow) is highly inclined $\\sim 73$ with respect to the plane of the disk galaxy.",
"It is known that radio continuum emission P.A.",
"off the nucleus does not necessarily agrees with photometric minor axis P.A.",
"of host Seyfert galaxies [71], [45].",
"Different P.A.",
"of the radio continuum emission from blobs straddling the nucleus have been detected with values different than the P.A.",
"of the photometric minor axis of Sy 2 galaxies [71], [45].",
"One plausible interpretation of the orientation difference between nuclear bipolar outflow and rotation axis of the galaxy (at least in Seyfert galaxies) might be that it is a result of a minor galaxy merger [61]."
],
[
"Origin of the Nuclear Geyser (Bipolar Outflow) in NGC 1415",
"Neither our angular resolution of the H$\\alpha $ continuum-free image nor the angular resolutiion of the radio continuum emission is sufficient to resolve any long and narrow structure that links $\\mathcal {A}$ or $\\mathcal {B}$ knots with the center that could be identified as a jet [12].",
"So following the notation for the outflow in the center of the disk galaxy M101 [60], we may call the cause of the outflow a geyser.",
"In the barred galaxy NGC 1415, could the geyser be caused by a recent activity of a low mass extragalactic black hole?",
"or by a compact burst of thousands of O, B star formation ?",
"Nuclear compact bursts of thousands of O, B star formation give rise to bubbles that are usually perpendicular to the plane, that is, parallel to the rotation axis of the disk [7], [78]In the case that the outflow from NGC 3079 parallel to the rotation axis were due to an AGN, the black hole mass would be M(NGC 3079$_{BH} \\sim 8.4 \\times 10^7$ M$_{\\odot }$ , see section below., and thus is not a plausible source in NGC 1415 because the outflow P.A.",
"is quite different than the P.A.",
"of the rotation axis of the disk.",
"It does seem that the barred galaxy NGC 1415 is one of several normal disk galaxies with weak optical and radio continumm emission from its nucleus.",
"Other normal galaxies are M51 (NGC 5194 Sbc(s)I-II) [25], [15], [8], M81 (NGC 3031, SA(s)ab) [39], M101 (NGC 5457 SAB(rs)cd I) [60], NGC 3367 (SBc(s)II) [37], [33], see below.",
"Studies of the innermost central region in M51 of radio continuum VLA high resolution mapping [25], [15] indicate the presence of a radio continuum nuclear source (with spectral index $\\alpha \\sim -0.67$ indicating non-thermal emission), a jet and two radio continuum bubbles straddling the nucleus [25].",
"M51 has an inner bar with a$_{bar} \\sim 20^{\\prime \\prime }$ at a P.A.$\\sim 135$ [63] (832 pc, at a distance of 8.58 Mpc [56]).",
"Additionally, reported optical spectroscopy and Fabry Perot H$\\alpha $ +[N II 6584] [8] show the spatial distribution and kinematics of the innermost central ionized gas in M51 [8].",
"The optical emission spatial distribution is very similar to the radio continuum emission, where the southern bubble seems to be a working surface (of a nuclear outflow) moving at $\\sim 200 -- 500$ km s$^{-1}$ [8].",
"16 Spectograms were centered near H$\\alpha $ and [N II 6583.4] and the slits always intersected the nucleus of M81 [39].",
"The nucleus contains a point source radio continumm source (non-thermal) from a diameter less than 20 ($\\sim 31$ pc) [17].",
"The kinematics of the ionized gas in the innermost central region of M81 reveals gas (with non-circular velocities) with nuclear outflow velociy of about $v_{outflow} \\sim 38$ km s$^{-1}$ .",
"Another example of two H$\\alpha $ , continuum-free, knots have been reported straddling the nucleus, in the north-south direction, from the disk galaxy M101 [60].",
"Several long spectroscopic slits observations of the innermost central region of M101 reveal that kinematics of the two H$\\alpha $ knots straddling the nucleus is most likely a geyser or bipolar outflow with velocity less than 100 km s$^{-1}$ [60].",
"The barred galaxy NGC 3367 (SBc) shows bipolar radio continuum 20cm and 6cm emissions from interferometric observations with synchrotron emission lobes extending upto 6 kpc straddling the compact nucleus [37], [33].",
"All of these examples of weak optical and radio continuum emissions from the nuclei of normal disk galaxies may be explained by having an active nucleus with a low mass black hole.",
"With the relation M$_{BH}$ versus $\\sigma _*$ , by now well accepted for extragalactic massive black holes [38], [58], [59] one may estimate the masses of the each black hole of the normal disk galaxies briefly described earlier.",
"The values of their central velocity dispersions $\\sigma _*$ are taken from Ho et al.",
"(2009).",
"The mass is obtained from the expresion M$_{BH} \\sim 1.2 \\times 10^8$ M$_{\\odot }$ $(\\sigma _*/200 km/s)^{3.75}$ [38].",
"Thus, M(M51)$_{BH} \\sim 7.6 \\times 10^6$ M$_{\\odot }$ , M(M81)$_{BH} \\sim 5.4 \\times 10^7$ M$_{\\odot }$ , M(M101)$_{BH} \\sim 3.97 \\times 10^4$ M$_{\\odot }$ , and M(NGC3367)$_{BH} \\sim 1.4 \\times 10^6$ M$_{\\odot }$ .",
"As a comparison, NGC 5548 a Seyfert 1.5 disk galaxy (Hubble type R$^{\\prime }$ SA(0)/as) shows a unresolved radio continuum source from the nucleus (at 025 angular resolution) and two unresolved radio continuum sources straddling the nucleus at d$\\le 20$ (d $\\le 1.4$ kpc [46]), shows X-ray variable emission and it has been detected as a X-ray warm absorber.",
"M(NGC5548)$_{BH} \\sim 5 \\times 10^8$ M$_{\\odot }$ is a more massive black hole.",
"It is the result of a galaxy merger event.",
"Figure: NO_CAPTION"
],
[
"Angular Velocity of the stellar bar, $\\Omega _{bar}$ , in NGC 1415",
"The angular velocity of a stellar bar ($\\Omega _{bar}$ ) in a SB galaxy is among the most important parameters that govern the galaxy's internal dynamics, kinematics and the morphology of its internal structure [5].",
"Hubble early type barred galaxies tend to have flat light profiles, while Hubble late type SB galaxies tend to have exponential light profiles along the ellipsoidal bars [11], [22].",
"For slightly non circular star and gas orbits perturbed by a non-axisymmetrical gravitational potential, orbits have natural resonant frequencies.",
"If the gravitational field generated by the stellar bar perturbs a gas orbit at or near one of its resonant frequencies, then the response of the orbit will be large."
],
[
"Methods for estimating $\\Omega _{bar}$ from gas kinematics",
"One method to estimate the angular pattern speed ($\\Omega _{bar}$ ) for stellar bars is through the resonances method [5], [22], where it is based on gas kinematics, the locations of resonance rings, and ILR and OLR structures [5], [22].",
"Another method for estimating a bar pattern speed is that proposed by Tremaine & Weinberg (1984).",
"In this paper we will utilize the first method.",
"Resonances occur when the circular angular velocity $\\Omega _{gas}$ and the radial epicycle frequency, $\\kappa (R)$ , in the unperturbed orbit satisfies one of the three conditions: $\\Omega _{bar} = \\Omega _{gas}$ at CR, ($\\mathcal {R} = R_{CR} / R_{bar}$ ), $\\Omega _{bar} = \\Omega _{gas} - \\kappa /2$ (near an ILR), and $\\Omega _{bar} = \\Omega _{gas} + \\kappa /2$ (near an OLR) [50], [13], [14], [5].",
"Gas outside CR drifts towards near an OLR, while gas inside CR drifts towards near an ILR [50], [13], [14], [5].",
"The radial epicycle frequency in terms of the effective gravitational potential is given by the expression $\\kappa ^2(R) = \\left(\\partial ^2 (\\Phi _{eff})/ \\partial (R^2) \\right)$ , where $\\Phi _{eff}$ is the sum of the gravitational potential energy of the orbiting gas and the kinetic energy associated with its motion in the azimuthal direction [5].",
"In terms of $\\Omega _{gas}$ , $d(\\Omega _{gas})/dR$ , the radial epicycle frequency can be estimated by the expression $\\kappa ^2(R) = \\left( 4\\Omega ^2(R)_{gas} + 2R\\Omega (R) (d\\Omega (R)_{gas})/dR \\right)$ [5].",
"The derived observed values of $\\Omega (Na I D lines)_{gas}$ (crosses), $\\Omega _{gas} - \\kappa /2$ (open circles), and $\\Omega _{gas} + \\kappa /2$ (open triangles) are shown in Fig.",
"6.",
"In a disk galaxy with an additional non-axisymmetric gravitational potential as is a stellar bar, CR, $\\Omega _{bar} = \\Omega _{gas}$ , occurs at the end of the ellipsoidal stellar bar ($\\mathcal {R} = 1$ ) [13], [14].",
"In the disk barred galaxy NGC 1415, R$_{boxy bar} \\sim 1.5$ kpc [34], then from Fig.",
"6, $\\Omega _{gas}$ at R$ \\sim 1.5$ kpc has a value of $\\Omega _{gas} \\sim 134 \\pm 7$ km s$^{-1}$ kpc$^{-1}$ and thus $\\Omega _{bar} \\sim 134$ km s$^{-1}$ kpc$^{-1}$ .",
"$\\Omega _{bar}$ does not cross the derived $\\Omega _{gas} - \\kappa (R)/2$ , nor $\\Omega _{gas} + \\kappa (R)/2$ .",
"We notice, from optical continuum image (see contours in Fig.1) that at a distance just at the end of the boxy bar ($\\sim 2$ kpc) there are elongated isophotes indicating the start of the inner spiral arms (see Fig 1).",
"It is difficult to tell at which distance OLR were to occur, since the values of $\\Omega _{gas}$ changed rapidly at large distances and $\\kappa (R)$ was thus not easy to estimate.",
"This constant value of $\\Omega _{bar}$ for this boxy bar in NGC 1415 is very large at least a factor from 3 to 5 with respect to values of other ellipsoidal bars [41], [24], [23].",
"From numerical simulations, large values of $\\Omega _{bar}$ are expected for an inner bar (see computer simulated model 2 values of doubly barred galaxies [51]).",
"Is the large NIR bar in NGC 1415 [48] the primary bar?",
"This value of $\\Omega _{bar} \\sim 134 \\pm 7$ km s$^{-1}$ kpc$^{-1}$ in NGC 1415 is large, however it is similar to the estimated large values in NGC 4303 for $\\Omega _{bar}$ when $\\mathcal {R} = 1$ [36]In the case of NGC 4303, if one were to take $a_{bar} = 20$ ($\\sim 1.5$ kpc) [52] and $\\mathcal {R} =1$ , then $\\Omega _{bar} \\sim 160$ km s$^{-1}$ kpc$^{-1}$ (CR $\\sim 1.5$ kpc), it would not cross $\\Omega _{gas} - \\kappa (R)/2$ , but it would cross $\\Omega _{gas} + \\kappa (R)/2$ at about 2.4 kpc (OLR) which coincides with the location of the inner southern spiral arm in NGC 4303 [36].",
"While if one were to take $a_{bar} \\sim 286$ ($\\sim 2.1$ kpc) [27] and $\\mathcal {R} =1$ , then $\\Omega _{bar} \\sim 120$ km s$^{-1}$ kpc$^{-1}$ (CR $\\sim 2.1$ kpc), it also does not cross $\\Omega _{gas} - \\kappa /2$ , but it crosses $\\Omega _{gas} + \\kappa /2$ at R$_{OLR} \\sim 2.7$ kpc which would approximately also coincide with the spatial location of the inner southern spiral arm in NGC 4303 [36].. Notice that the bright inner spiral arms [at about d$\\sim 40^{\\prime \\prime }$ , or d$\\sim 3.43$ kpc] are in the inner bright part of the inner disk of NGC 1415, and the disk is much larger, as reported in NED, with ESO-LV quick blue IIa-O semi major axis extending to 1694 or 14.53 kpc."
],
[
"Comparison of $\\mathcal {R}$ in NGC 1415 with values from other galaxies",
"How does the ratio $\\mathcal {R} = 1$ in NGC 1415 compare with the values in other barred galaxies?",
"Seven out of nine Hubble early type SB galaxies, $\\mathcal {R} = 1.2 \\rightarrow 2.2$ [22].",
"In a model-based study of 38 SB galaxies (using NIR and optical images from the OSUBGS), $\\mathcal {R}$ is near 1.15 in types SB0/a-SBab [65].",
"Values have been obtained for $\\Omega _{bar}$ from 10 barred galaxies with large ellipsoidal stellar bars and $0.8 \\le \\mathcal {R} \\le 1.1 $ .",
"In a study of 15 CALIFA SB galaxies, using the TW method, $\\langle \\mathcal {R} = 1.3 \\rangle $ for SB0-SB0/a galaxies [1].",
"Stellar bars observed almost edge on may show inner boxy optical isophotes (for example NGC 1415 [32], [34], and NGC 4569 [77], [36]).",
"Yet there is no statistical observation, to our knowledge, of values of $\\Omega _{bar}$ for these boxy small inner bars.",
"In NGC 1415 the stellar bar is boxy with R$_{bar} \\sim 1.5$ kpc small as compared with the radius of the large disk blue IIa-O semi major axis of 14.5 kpc, $a_{bar}/R_{25} \\sim 0.1$ which is smaller than any value reported from other barred galaxies from the S$^4G$ 3.6$\\mu m$ [19]."
],
[
"Summary and Conclusions",
"Our previous imaging H$\\alpha $ continuum-free data from the central $\\pm 200$ of the barred galaxy NGC 1415 showed a) two central bright H$\\alpha $ , $\\mathcal {A}$ , and $\\mathcal {B}$ knots neither of them coinciding with the optical continuum nucleus, b) emission from regions around the nucleus, and c) emission from knots in the SE and NW inner spiral arms.",
"From our previous optical red continumm (filter I $\\lambda 8040$ Å$~$ a mass distribution model was made which included a two stellar bars: one to reproduce the observed inner ellipsoidal isophotes (R$_{bar I} \\sim 1$ kpc) and a second to reproduce the boxy-shaped isophotes (R$_{bar II} \\sim 1.5$ kpc) [34].",
"In this study we have obtained kinematical data from the disk of NGC 1415 with a long slit spectrum at P.A.$\\sim 155$ .",
"We were able to detect the convolved Na I D lines in absorption, [NII] $\\lambda 6548, 6584$ , H$\\alpha \\lambda 6562.8$ , and [SII] $\\lambda 6716, 6731$ lines in emission.",
"We estimated that the heliocentric velocities of Na I D absorption lines from a long slit spectrum at P.A.+155$$ (which is closest to the photometric P.A.",
"+148$$ of the disk of NGC 1415) versus distance [taken as V$_{sys} \\sim 1564$ km s$^{-1}$ ] may be representative of the rotation curve in NGC 1415.",
"From a comparison of the heliocentric velocities of Na I D absorption lines and H$\\alpha $ emission lines we observed an excess in velocity of the H$\\alpha $ emission lines associated with $\\mathcal {A}$ (bluer velocities) and $\\mathcal {B}$ (redder velocities) knots as compared with the velocities shown by the Na I D lines at the same distances, $|\\Delta V| \\sim 140$ km s$^{-1}$ .",
"From reported VLA radio continuum (20 cm) mapping of the inner $30^{\\prime \\prime }$ , with an angular resolution of $8^{\\prime \\prime }$ [12], the peak of the 20 cm radio continuum emission surprisingly does not coincide with the nucleus but instead it coincides with $\\mathcal {A}$ .",
"Our interpretation of the spatial location (very symmetric the positions of $\\mathcal {A}$ and $\\mathcal {B}$ ) and blueshifted velocities of $\\mathcal {A}$ , and redshifted velocities of $\\mathcal {B}$ and the detected radio continuum 20 cm emission at an angular resolution of 80(synchrotron), is that they are the result of gas moving away from the nucleus.",
"We interpret that there may be a geyser (bipolar outflow from the nucleus) with V$_{geyser} \\sim 140$ km s$^{-1}$ .",
"The geyser lies at a P.A.+165$$ which is approximately 73$$ from the rotation axis of the disk of NGC 1415, namely it is not perpendicular to the disk of NGC 1415.",
"The cause of the geyser could be a low mass black hole with an accretion disk highly inclined to the plane of the disk of NGC 1415 (similar nuclear activity from other normal disk galaxies has been reported in M51, M81, M101, and NGC 3367 that may be explained by the existence of low mass black hole in each of them).",
"From our estimated rotation curve we were able to estimate the angular velocity of the gas, $\\Omega _{gas}$ , the radial epicycle frequencies $\\kappa (r)$ , $\\Omega _{gas} - \\kappa /2$ , and $\\Omega _{gas} + \\kappa /2$ .",
"A value of $\\Omega _{bar}$ is considered in the barred galaxy NGC 1415 with $\\mathcal {R} = 1$ , that is, CR is at 1.5 kpc (radius of the fitted boxy stellar bar), with an estimated value $\\Omega _{bar} \\sim 134$ km s$^{-1}$ kpc$^{-1}$ .",
"This value of $\\Omega _{bar}$ is high and does not cross $\\Omega _{gas} - \\kappa /2$ , nor $\\Omega _{gas} + \\kappa /2$ .",
"We would like to thank constructive comments to improve this version of the text to an anonymous referee.",
"We also would like to thank for their useful comments, suggestions and references to previous published articles related to lengths and angular velocities of stellar bars to F. Combes (France), E. Athanassuola (France), and B. Elmegreen (USA).",
"This work has made extensive use of the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration.",
"We also would like to thank the operator control of the 2.1m GH telescope in Cananea, México for his help with the observations in December of 2000.",
"This research has made use of the VizieR catalogue access tool, CDS, Strasbourg, France (DOI:10.26093/cds/vizier).",
"The original description of the VizieR service was published in A&AS, 143, 23, and the astronomy package IRAF (developed by NOAO.",
"NOAO is managed by the Association of Universities for Research in Astronomy under Cooperative Agreement with the National Science Foundation, USA)."
],
[
"Neutral Sodium Optical interstellar Absoprtion/Emission Lines",
"Sodium atom (Na) has 11 protons, 11 neutrons and 11 electrons.",
"Na I first ionization potential is 5.14 eV.",
"The electron distribution in the sodium atom is Na I [$1S^2 2S^2 2P^6$ 3$S^1$ ].",
"The neutral sodium D lines in absorption arise from the hyperfine structure from the levels $3 ^2S \\rightarrow 3 ^2P$ [57].",
"In 1937 the transition $3 ^2S_{1/2} \\rightarrow 3 ^2P_{1/2}$ reported a wavelength $\\lambda _{D1} \\sim 5895.932$ Åand was denoted as D$_1$ line, while the transition $3 ^2S_{1/2} \\rightarrow 3 ^2P_{3/2}$ has the wavelength $\\lambda _{D2} \\sim 5889.965$ Åand was denoted as D$_2$ line.",
"More recent work show that the wavelength $\\lambda _{D1} \\sim 5895.924$ Å, while the transition $3 ^2S_{1/2} \\rightarrow 3 ^2P_{3/2}$ has the wavelength $\\lambda _{D2} \\sim 5889.951$ Å[55].",
"The energy difference between the upper and lower states in the Na D$_1$ line is $\\Delta E_{D1} \\sim 2.103$ eV, while it is $\\Delta E_{D2} \\sim 2.105$ eV.",
"They are the so called unresolved D$_1$ , and D$_2$ lines of neutral sodium, Na I D [55].",
"The wavelength separation between the Na I D$_1$ line and the Na I D$_2$ line is $\\Delta \\lambda \\sim 5.973$ Å.",
"Galactic and extragalactic Na I D lines are detected in absorption, where the underlying continuum is strong, in the interstellar medium [74], [40], [20].",
"The cosmic composition of Na, in our galaxy, is $12 + [log N_{Na}/N_H] \\sim 6.25 \\rightarrow 6.3$ [74], [20].",
"Studies in our galaxy indicate that Na I abundance is proportional to the square of the local hydrogen density, and thus it will be concentrated in the colder, denser portions of the clouds [79].",
"Neutral hydrogen, HI, and sodium, Na I, are constituents of diffuse neutral clouds at a typical temperature of about $70\\rightarrow 80$ K. These clouds also contains ions such as C$^+$ , Ca$^+$ , etc.",
"produced by photoionization by starlight [74], [20].",
"The mean electron density is about 0.1 cm$^{-3}$ , corresponding to $n_H \\sim 10^3$ cm$^{-3}$ (Ca atom, in general, is depleted by a factor of 100 to 1000 relative to Na atom [74]).",
"The median Na I column density, from observation of absorption towards 38 bright stars in our galaxy is $log N[(Na I)] \\sim 11.09$ [79], while it is $log N[(Na I)] \\sim 12.8$ towards $\\alpha $ Cygni line of sight [6].",
"If neutral sodium, Na I, and neutral hydrogen, HI, are characterized by the same temperature and turbulent velocity in diffuse clouds, the optical depth $\\tau $ is less than 1, namely, the Na I D absorption lines are optically thin [79].",
"Na I D lines have been detected in absorption towards the galaxy NGC 5128 (Cen A) [40], where it is reported that within the errors of measurements, the wavelenghts are redshifted, in Na I D lines, the same as those measured for NGC 5128 emission lines at each point, and it is clear that the absorbing material shares the motion of the gas which is responsible for the emission lines.",
"Haro Observatory, OAGH(2.1m) Cananea, México"
]
] | 1906.04564 |
[
[
"Top-Hat Spherical Collapse with Clustering Dark Energy. I. Radius\n Evolution and Critical Contrast Density"
],
[
"Abstract Understanding the influence of dark energy on the formation of structures is currently a major challenge in Cosmology, since it can distinguish otherwise degenerated viable models.",
"In this work we consider the Top-Hat Spherical-Collapse (SC) model with dark energy, which can partially (or totally) cluster, according to a free parameter $\\gamma$.",
"The {\\it lack of} energy conservation has to be taken into account accordingly, as we will show.",
"We determine characteristic quantities for the SC model, such as the critical contrast density and radius evolution, with particular emphasis on their dependence on the clustering parameter $\\gamma$."
],
[
"Introduction",
"Recent results [1], [2] from independent cosmological observations — such as anisotropies in the Cosmic Microwave Background (CMB), Baryon Acoustic Oscillations (BAO), type-Ia Supernovae (SNe Ia) and the Large-Scale Structure of the Universe (LSS) — imply that the Universe is speeding up.",
"[3], [4].",
"The responsible for this effect is dubbed “dark energy” (DE), whose physical nature is still unknown.",
"If we model dark energy as a fluid, according to General Relativity, it needs to have negative pressure.",
"In particular, the cosmological model that better fits observations is the cold-dark-matter with Cosmological-Constant model ($\\Lambda $ CDM).",
"However, this model presents difficulties at theoretical level [5], [6], motivating the search for alternatives such as quintessence [7], [8], [9], [10], phantom dark energy [11], k-essence [12], decaying vacuum models [13], [14] or even modifications of General Relativity, such as $f(R)$ theories [15], among others.",
"A great difficulty is that many of these models behave very similarly to $\\Lambda $ CDM at the background level, making it difficult to distinguish them through cosmological kinematical tests (those that depend essentially only on distance).",
"Therefore, it is crucially important to study the evolution of perturbations and the structure formation in those models, where they are expected to have different (and measurable) consequences from those obtained by $\\Lambda $ CDM.",
"The simplest way to study the structure formation with dark energy is through the Top-Hat Spherical-Collapse (SC) approach, which was initially used in Einstein-de Sitter (EdS) background (as an useful benchmark since it yields an exact analytical result for the critical density), in the standard cold-dark-matter scenario [16], and later in $\\Lambda $ CDM [17].",
"The SC model has also been extended to quintessence fields [18], [19], [20], decaying vacuum models [21], $f(R)$ theories [22], [23], [24], [25], [26], DE with constant equation-of-state (EoS) models [27], [28], [29], [30], coupled DE models [31], [32], and agegraphic DE cosmologies [33].",
"In particular, Ref.",
"[27] investigated constant phantom, constant non-phantom and varying DE EoS parameter, always assuming that the latter is the same both inside and outside the collapsed regionAs we will show further down, this assumption is equivalent to requiring that the DE EoS parameter is equal to its speed of sound squared: $w=c_s^2$ ..",
"These authors have focused only in the limiting cases, namely, fully clustered and completely homogeneous DE.",
"In Ref.",
"[30] the SC model with fully clustered DE is considered assuming a linear relation between the matter contrast density and the DE one, according to a free parameter $r$ .",
"In Ref.",
"[30], as well as in [28], it is also assumed that the DE EoS is the same inside and outside the collapsed region.",
"In this work we relax the aforementioned hypotheses and generalize some of those results.",
"Following the Ansatz suggested in Ref.",
"[34] (see also [18]), we investigate the SC model with DE, assuming that it can cluster partially or totally, according to a normalized parameter: if $\\gamma =0$ , DE is fully clustered; if $\\gamma =1$ , DE is completely homogeneous.",
"This paper is organized as follows.",
"In Section we show the basic equations that describe the SC model with dark-energy perturbations.",
"We apply the so-called differential-radius method, which has been shown [25] (see also [35]) to be more robust than the constant-infinity method — which uses a fixed large value for the local overdensity as a threshold for indicating a collapsed structure.",
"The former method, on the other hand, follows the difference between the background scale factor and the collapsing bubble radius (also known as local scale factor).",
"In Subsection REF we analyze the radius evolution of the collapsing spherical region.",
"In Section REF we determine the critical contrast density.",
"We conclude in Section ."
],
[
"Spherical collapse with dark energy perturbations",
"For a flat, homogeneous and isotropic universe with dark matter and dark energy, the Einstein equations are given by: $\\left(\\frac{\\dot{a}}{a}\\right)^{2} &\\equiv & H^{2} = \\frac{8\\pi G}{3}\\left(\\bar{\\rho }_{m}+ \\bar{ \\rho }_{de}\\right), \\\\\\frac{\\ddot{a}}{a} &=& -\\frac{4\\pi G}{3}\\left[\\bar{\\rho }_{m}+ (1+ 3 w) \\bar{\\rho }_{de}\\right].$ In the equations above, $a$ is the scale factor, $H$ is the Hubble parameter, $w\\equiv \\bar{p}_{de}/\\bar{\\rho }_{de}$ is the EoS parameter of DE (assumed to be constant), and $\\bar{\\rho }_{m}$ , $\\bar{\\rho }_{de}$ and $\\bar{p}_{de}$ are the (background) energy densities of matter and DE and the DE pressure, respectively.",
"A dot over a given quantity denotes its time derivative.",
"Assuming that both dark matter and DE interact only gravitationally and are separately conserved, we get $&& \\dot{\\bar{\\rho }}_{m} +3H\\bar{\\rho }_{m}=0, \\\\&& \\dot{\\bar{\\rho }}_{de} +3H(1+w)\\bar{\\rho }_{de}=0.$ Here we investigate the nonlinear evolution of the gravitational collapse and, to this aim, we consider the Top-Hat Spherical-Collapse (SC) model.",
"The SC model considers a spherical region with a top-hat profile and uniform density $\\rho (t) = \\bar{\\rho }(t) + \\delta \\rho (t)$ , immersed in a homogeneous universe with energy density $ \\bar{\\rho }(t)$ .",
"Here $\\delta \\rho $ initially is a small perturbation of the background fluid energy density.",
"We suppose that this region also contains nonrelativistic matter ($p_m=\\bar{p}_m=0$ ) and DE.",
"Such a spherical region can be described as a separated universe with (local) scale factor $r$ .",
"The acceleration equation for this region is given by: $\\frac{\\ddot{r}}{r}=-\\frac{4\\pi G}{3}\\left(\\rho _{m}+ \\rho _{de}+3 p_{de}\\right),$ where $p_{de} (t)= \\bar{p}_{de}(t)+ \\delta p(t)$ is the DE pressure inside the spherical region and $ \\delta p(t)$ a small pressure perturbation.",
"The DE EoS parameter in the spherical region is given by [36] $w^{c} \\equiv \\frac{p_{de}}{\\rho _{de}} = w+\\frac{(c_s^2 -w)\\delta _{de}}{1+\\delta _{de}},$ where the superscript “$^c$ ” stands for “clustered”, $c_s^2\\equiv \\delta p_{de}/\\delta \\rho _{de}$ is the DE sound speed squared (assumed to be constant) and $\\delta _{de}$ is the DE density contrast (see its definition below).",
"Note that only if $c_s^2=w$ (or homogeneous DE, i.e., $\\delta _{de}=0$ ) the DE EoS parameter in the collapsing region is equal to that of the background ($w^{c}= w$ ).",
"Due to its standard attractive character, dark matter always tends to cluster, so the local continuity equation takes a similar form as the continuity equation for the background fluid, that is: $\\dot{\\rho }_{m}+ 3\\frac{\\dot{r}}{r}\\rho _{m}=0,$ where $r$ is the local scale factor.",
"Of course, it is clear that dark matter will actually cluster only if the initial $\\delta \\rho _{m}$ is large enough to overcome the effects from both the background expansion and DE.",
"In the present work we assume that DE can also collapse — although not necessarily together with the matter content, since it can flow away from the collapsing sphere.",
"This is precisely the reason for the lack of energy conservation in the perturbed region.",
"Therefore, we parameterize such physical phenomenon writing the local continuity equation for DE as [34] (see also [18]) : $\\dot{\\rho }_{de}+ 3(1+w^{c})\\frac{\\dot{r}}{r}\\rho _{de}=\\gamma \\Gamma \\, ,\\quad \\textup {with} \\quad 0 \\le \\gamma \\le 1 \\, , \\\\$ where $\\Gamma \\equiv 3(1+w^{c}) \\left( \\frac{\\dot{r}}{r} -\\frac{\\dot{a}}{a} \\right) \\rho _{de}\\, .$ Here, $\\Gamma $ describes the leaking of DE away from the spherical collapsing region and $0\\le \\gamma \\le 1$ is the aforementioned clustering parameter.",
"The non-clustering, i.e, homogeneous DE corresponds to $ \\gamma = 1 $ .",
"Notice that in this case, we have $\\rho _{de} \\propto \\exp [-3\\int (1+w^c) da/a]$ while $ \\bar{\\rho }_{de}$ scales as $ \\bar{\\rho }_{de} \\propto a^{-3(1+w)}$ .",
"So, in principle, even if the DE energy densities were initially equal, they would evolve differently.",
"However, as we will show further down, when $\\gamma =1$ , in the linear regime, there is no growing mode and $\\delta \\rho _{de}$ rapidly tends to zero.",
"Therefore, it is not possible to distinguish in this case the behavior of the DE inside and outside the spherical region: $\\rho _{de} = \\bar{\\rho }_{de}$ and consequently $w^{c}= w$ .",
"In this case (and also for $\\gamma >0$ ) the total energy of the system is not conserved [34].",
"In contrast, the case of full clustering, i.e, when $\\gamma =0$ , ensures that $\\rho _{de}\\ne \\bar{\\rho }_{de}$ , such that the spherical region is completely segregated from the background and it is considered an isolated system, which conserves energy.",
"We shall also consider intermediate values of $\\gamma $ in our analysis.",
"Notice that, differently from Ref.",
"[34], we are not assuming that the DE EoS is the same inside and outside the collapsing spherical region.",
"As remarked above, this is only the case when dark energy is homogeneous ($\\gamma = 1$ ) or $c_s^2=w$ .",
"Differentiating twice the density contrast $ \\delta _{j} \\equiv \\rho _{j} / \\bar{\\rho }_{j} - 1 $ for both dark matter ($\\delta _m$ ) and dark energy ($\\delta _{de}$ ) and using the equations above we obtain the following nonlinear evolution equations : $\\ddot{\\delta }_{m} +2 H\\dot{\\delta }_{m} - \\frac{4 \\dot{\\delta }^{2}_{m}}{3(1+\\delta _{m})} =\\frac{3 H^{2}}{2}(1+\\delta _{m})\\left(\\Omega _{m}\\delta _{m} + (1-\\Omega _{m})\\delta _{de}(1+3 c_s^2)\\right),$ $\\ddot{\\delta }_{de} &=&-3 (h-H) (1+w) (1-\\gamma ) \\dot{\\delta } _{de}-3 (h (1-\\gamma )+\\gamma H) \\delta w \\dot{\\delta } _{de}-\\nonumber \\\\&-& 3 (1+w) (1-\\gamma ) \\left(\\dot{h}-\\dot{H} \\right)\\left(1+\\delta _{de} \\right)-\\nonumber \\\\&-& 3 \\delta w(1-\\gamma )\\left( \\dot{h}(1-\\gamma )+\\gamma \\dot{H} \\right) \\left(1+\\delta _{de} \\right) -\\nonumber \\\\&-& 3 \\left(h (1-\\gamma )+\\gamma H\\right) \\dot{\\delta } w \\left(1+\\delta _{de}\\right).$ In the expression above, $\\delta w \\equiv w^{c} - w$ (see eq.",
"(REF )), $h \\equiv \\frac{\\dot{r}}{r} =\\frac{\\dot{\\delta } _{de}+3 H \\left((1+w) (-1+\\gamma \\ )+\\left(-1-w+\\gamma +\\gamma c_s^2\\right) \\delta _{de}\\right)}{3 \\ (-1+\\gamma ) \\left(1+w+\\left(1+c_s^2\\right)\\delta _{de}\\right)},$ $\\dot{h}=\\frac{\\ddot{r}}{r}-h^2,$ $\\frac{\\ddot{r}}{r}=-\\frac{H^2}{2}\\bigg [\\Omega _m (1+\\delta _m)+(1-\\Omega _m)\\big ((1+3c_s^2)\\delta _{de}+1+3w\\big )\\bigg ]$ and $\\dot{H}=-\\frac{3}{2} H^2 \\left(1+w \\left(1-\\Omega _m\\right)\\right).$ Here $\\Omega _{m}=\\Omega _{m}(t)$ is the background nonrelativistic matter energy-density parameter at the instant $t$ .",
"In the expressions above we assume, obviously, that $\\gamma \\ne 1$ , since, as mentioned before, if $\\gamma = 1$ DE does not cluster.",
"We note that in the particular case in which $c_s^2=w$ [27], [28], [30], such that $\\delta w=0$ , eq.",
"(REF ) reduces to $\\ddot{\\delta }_{de} &+& 2 H\\dot{\\delta }_{de} -\\frac{4+3w-3\\gamma (1+w)}{3(1+w)(1-\\gamma )} \\frac{\\dot{\\delta }^{2}_{de}}{(1+\\delta _{de})} = \\nonumber \\\\&=& \\frac{3 H^{2}}{2}(1+\\delta _{de})(1-\\gamma )(1+w)[\\Omega _{m}\\delta _{m} + (1-\\Omega _{m})\\delta _{de}(1+3 w)].$ If we further impose $\\gamma =0$ , we then recover Eq.",
"(7) of Ref.",
"[27] for the case in which $w$ is constant.",
"To determine the initial conditions for $\\delta _ {m} $ and $\\delta _{de}$ , we consider the linear approximation of Eqs.",
"(REF ) and (REF ) in a matter-dominated universe ($\\Omega _m \\sim 1$ and $\\Omega _{de}\\sim 0$ ): $\\delta ^{\\prime \\prime }_{m} &+& \\frac{3}{2}\\frac{\\delta ^{\\prime }_{m}}{a}-\\frac{3}{2 a^2}\\delta _{m}=0 \\\\\\delta ^{\\prime \\prime }_{de} &+& \\left(\\frac{3}{2}- 3(w-c_s^2)\\right)\\frac{\\delta ^{\\prime }_{de}}{a}- \\nonumber \\\\&-& \\frac{3}{2 a^{2}}\\left((1+w)(1-\\gamma )\\delta _{m}+(w-c_s^2)\\delta _{de})\\right)=0,$ where $^{\\prime }\\equiv d/da$ .",
"Since we are interested in the formation of structures, the decreasing mode of the above equations will not be considered.",
"The growing mode solutions are: $\\delta _{m}(a) &=& C \\, a \\qquad \\qquad \\qquad {\\rm and}\\\\\\delta _{de}(a) &=& \\frac{(1+w)(1-\\gamma )}{1-3(w-c_s^2)}\\delta _{m}(a).$ As remarked above, if $\\gamma =1$ we obtain $\\delta _{de}=0$ .",
"We assume in our analysis that $1-3(w-c_s^2)>0$ which implies that for phantom models ($w<-1$ ) $\\delta _{de}<0$ (i.e, there is less dark energy inside the bubble than in the background).",
"Note that if $\\delta _{de}<-1$ , then $\\rho _{de}<0$ .",
"Although such case is exotic, in principle, it is allowed in some modified gravity models [37].",
"Whenever $\\delta _{de}$ crosses $-1$ , which happens only if $w<-1$ , then $w^c$ goes from $-\\infty $ to $+\\infty $ (see Eq.",
"(REF ) and Fig.",
"REF ).",
"Note, however, that such divergence does not affect the evolution of the bubble, since $w^c$ does not appear explicitly in the equations of motion for the radius $r$ (or, actually, for the variable $y$ ), as we will show next.",
"Figure: Behavior of the equation-of-state parameter inside the bubble (w c w^c) with respect to time, for the labeled parameters.Given the contrast density for each fluid, the evolution of the local scale factor is given by Eq.",
"(REF ), which in terms of $y \\equiv \\frac{r}{r_{i}}-\\frac{a}{a_{i}}$ can be written as: $y^{\\prime \\prime } &+& \\left(y^{\\prime }+\\frac{1}{a_{i}}\\right) \\left(\\frac{H^{\\prime }}{H}+\\frac{1}{a}\\right) = \\\\ \\nonumber & & = -\\frac{1}{2}\\left(\\frac{H_{0}}{ H a}\\right)^{2}\\left(y+ \\frac{a}{a_{i}}\\right) \\times \\\\ \\nonumber & & \\quad \\times \\bigg [\\Omega _{m0}a^{-3}\\Big (1+\\delta _{m}\\Big ) + \\nonumber \\\\&& \\quad + \\Big (1+3w+(1+3c_s^2)\\delta _{de}\\Big )\\Big (1-\\Omega _{m0}\\Big )a^{-3(1+w)}\\Big (1+\\delta _{de}\\Big )\\bigg ] , \\nonumber $ where $\\Omega _{m0}$ is the present value of the matter density parameter.",
"An initial condition for Eq.",
"(REF ) is naturally $y (a_{i}) = 0 $ .",
"To obtain $y ^{\\prime }(a_{i})$ , we consider that, initially, the mass of the spherical region is given only by the contribution from dark matter: $M_i=\\frac{4}{3} \\pi R_i^3 (1+ \\delta _{mi})\\bar{\\rho }_{mi}.$ The (possible) contribution from DE is negligible since $\\rho _{de}\\ll \\rho _m$ when the initial conditions are set, in a matter-dominated universe.",
"In the above equations, $R_i \\equiv r(t_i) X$ is the physical radius of the collapsing sphere at instant $t_i$ , X is its coordinate radius and $\\delta _{mi}$ is the initial matter density contrast.",
"Since dark matter always collapses (depending, of course, on the initial conditions of the matter perturbations), the mass $M$ inside the spherical region will always be a constant.",
"Thus, we have $y^{\\prime }(a_{i})= -\\delta ^{\\prime }_{mi}/ [3(1+\\delta _{mi})]$ .",
"We adopt in our numerical calculations $a_i=10^{-5}$ ."
],
[
"Bubble Evolution",
"In this section we investigate the bubble evolution, namely its radius as a function of time, and one of the main results from the SC model: the critical density contrast — a crucial quantity to determine the number of collapsed objects.",
"Throughout the paper we assume that $\\Omega _{m0}=0.3$ .",
"We also keep the same initial conditions for dark-matter perturbations, such that the collapse in $\\Lambda $ CDM always occurs at the present time.",
"We pay special attention to the dependence of the outcomes in the free parameters of our model: $\\gamma $ , $c_s^2$ and $w$ .",
"Some situations are particularly interesting and express the richness of the present parametrization: $c^{2}_{s}=0$ , in which there is no DE pressure perturbation.",
"It is interesting to point out that, in this case, in the final stages of the collapse ($\\delta _{de} \\rightarrow \\infty $ ), the local dark energy does behave as dark matter, since $w^c \\rightarrow 0$ — see eq.",
"(REF ).",
"Note also, from eq.",
"(), that for phantom dark energy one will always get $\\delta _{de}<0$ : there is less dark energy inside the bubble than in the background.",
"$c^{2}_{s}=w$ , which indicates that the clustered DE EoS parameter ($w^c$ ) and the background one ($w$ ) are equal.",
"$c_s^2=\\gamma $ .",
"We intend to model a continuous “turning on” of the clustering in scalar field models [34].",
"In quintessence and k-essence models, usually, two choices are made: a) $c_s^2=1$ , in which case the standard quintessence scalar field (i.e, a minimally coupled scalar field with a canonical kinetic term) does not cluster, remaining homogeneous on subhorizon scales [10], and b) $c_s^2=0$ (or more generally sub-luminal behaviour) are considered in k-essence scalar fields [19].",
"The new parameter $\\gamma $ models the lack of energy conservation, which happens whenever a fraction of DE does not cluster.",
"Note that when $\\gamma =1$ , results from different $c_s^2$ should coincide, since the latter does not play a role if DE is homogeneous."
],
[
"Radius",
"We now investigate the evolution of the spherical-region radius, as given by eq.",
"(REF ).",
"As mentioned before, the initial conditions for dark-matter perturbations in all models are fixed such that the collapse in $\\Lambda $ CDM model always occurs at the present time.",
"The initial conditions for dark-energy perturbations are given by Eq.",
"().",
"We point out some noteworthy features in a few particular cases: $c_s^2 = w$ ($w^c= w$ ) The collapsing time $t_{col}$ is earlier than $\\Lambda $ CDM $t_{col,\\Lambda CDM}$ only for phantom DE.",
"This is a reasonable outcome, since $\\delta _{de}<0$ if $w<-1$ (as mentioned above): the lack of DE in the clustered region accelerates the collapse.",
"For non-phantom, DE starts to dominate earlier when compared to $\\Lambda $ CDM for any $\\gamma $ .",
"On the other hand, a smaller $\\gamma $ corresponds to a larger $\\delta _{de}$ , which will delay the collapse, since in this case $\\delta _{de}>0$ .",
"There is no strong dependence on $\\gamma $ , except for a small drift towards $\\Lambda $ CDM when $\\gamma \\rightarrow 1$ (homogeneous DE), as expected.",
"Besides, the term that inhibits the collapse in Eq.",
"(REF ), namely $\\delta \\rho _{de}+ 3\\delta p_{de}$ , although always present, will be less important in this limit.",
"See Fig.",
"REF .",
"$c_s^2 =0$ We also get $t_{col}<t_{col,\\Lambda CDM}$ only for phantom DE, as anticipated.",
"The dependence on $\\gamma $ is very weak.",
"There is a slight drift away from $\\Lambda $ CDM as $\\gamma \\rightarrow 1$ .",
"Such opposite behavior (as compared to the previous case) happens because here $\\delta p_{de}\\equiv c_s^2 \\delta \\rho _{de} = 0$ .",
"Without any pressure support, the collapse is expedited if $\\gamma \\rightarrow 0$ and $w>-1$ .",
"Nevertheless, with phantom DE ($w<-1$ ), one has $\\delta \\rho _{de}<0$ and the clustering of DE (slightly) delays the collapse — one can (barely) see the tiny shift to larger $t_{col}$ when $\\gamma $ decreases from $0.8$ to 0 in Fig.",
"REF .",
"$c_s^2=1$ (standard quintessence-like DE) As before, $t_{col}<t_{col,\\Lambda CDM}$ for phantom DE but one can also expedite the collapse if $w>-1$ .",
"See Fig.",
"REF .",
"The most striking feature is the possibility to entirely prevent the collapse.",
"This is not completely unexpected if there is enough stiff DE in the initial perturbation.The other ingredients for the bounce are phantom dark energy and no energy leaking.",
"The full consequences of such behavior will be the subject of a future work.",
"Here, the collapse time $t_{col}$ is defined as: ${}t_{col}(w)= \\int _{0}^{a_{c}}\\frac{da}{H(w,a)a},$ where $a_{c}$ is the scale factor at collapse and $H(w,a)$ is the Hubble parameter of the $w$ CDM model.",
"Of course, $t_{col}(w=-1)$ represents the collapse time of $\\Lambda $ CMD model, $t_{col,\\Lambda CDM}$ .",
"The curves $\\gamma =1$ (homogeneous DE) from all the panels coincide, regardless of $c_s^2$ , as expected.",
"Figure: Evolution of the scale radius of the collapsing sphere for c s 2 =w={-0.9,-1.1}c_s^2=w=\\lbrace -0.9, -1.1\\rbrace and different values of γ\\gamma .",
"The solid blue line corresponds to the Λ\\Lambda CDM model.Figure: Evolution of the scale radius of the collapsing sphere for c s 2 =0c_s^2=0, w={-0.9,-1.1}w=\\lbrace -0.9, -1.1\\rbrace and different values of γ\\gamma .",
"The solid blue line corresponds to the Λ\\Lambda CDM model.Figure: Evolution of the scale radius of the collapsing sphere for c s 2 =1c_s^2=1, w={-0.9,-1.1}w=\\lbrace -0.9, -1.1\\rbrace and different values of γ\\gamma .",
"The solid blue line corresponds to the Λ\\Lambda CDM model.",
"Note the non-collapsing curve (γ=0,w=-1.1\\gamma =0, w=-1.1).",
"We also point out that the curve given by γ=0\\gamma =0, c s 2 =1c_s^2=1 and w=-0.9w=-0.9, that crosses Λ\\Lambda CDM close to the collapse, is also dissonant in Fig.",
".Figure: Evolution of the scale radius of the collapsing sphere for c s 2 =γc_s^2=\\gamma , w={-0.9,-1.1}w=\\lbrace -0.9, -1.1\\rbrace and different values of γ\\gamma ={0,0.8}=\\lbrace 0, 0.8\\rbrace .",
"The solid blue line corresponds to the Λ\\Lambda CDM model."
],
[
"The critical contrast density",
"As can be seen in Eq.",
"(REF ) and from the discussions in the previous section, the DE perturbations do contribute to the collapse.",
"Therefore, the definition of the critical density contrast must be modified in order to take this contribution into account.",
"So, let us consider the expression [29], [38] $\\delta _{tot}= \\delta _{m}+ \\frac{\\Omega _{de}}{\\Omega _{m}}\\delta _{de},$ as the total perturbation.",
"Note that, when $\\Omega _{de} \\rightarrow 0$ , the conventional definition for the critical contrast is recovered.",
"As usual, the critical contrast $\\delta _c$ is determined by its linear evolution — given by Eqs.",
"(REF ) and () — at the collapse redshift $z_c$ (obtained from requiring that $r(z=z_c) \\rightarrow 0 $ ): $\\delta _{c}=\\delta _{tot}^{lin}(z_{c}).$ Using the differential-radius method [25], the dependence of $\\delta _{c}$ with $z_c$ is shown in Fig.",
"REF , REF , REF , and REF for different values of the free parameters $c_s^2$ , $w$ and $\\gamma $ , and fixed $\\Omega _{m0}=0.3$ .",
"Dark-energy overdensities ($\\delta _{de}>0$ ) inhibits the growth of dark-matter perturbations ($\\delta _m$ ) due to its repulsive nature.",
"On the other hand, dark-energy underdensities ($\\delta _{de}<0$ ) enhance the growth of $\\delta _m$ .",
"The former case occurs in non-phantom models ($w>-1$ ), while the latter generally happens when $w<-1$ .",
"Indeed, as one can see in Figs.",
"REF to REF , the critical overdensity for a collapsing structure ($\\delta _c$ ) is smaller in phantom cases.",
"Therefore, one should expect an enhancement on the number of collapsed objects in this case.",
"The choice $\\gamma =0$ yields extreme variations of $\\delta _c$ with respect to $\\Lambda $ CDM, because, in this case, there is no leakage of DE away from the collapsing regions, which maximizes its effects.",
"In all the presented cases, $\\delta _c$ tends to the expected EdS value at high $z_c$ .",
"Note also that $\\delta _{c}$ is always larger (smaller) than the standard $\\Lambda $ CDM value for $w<-1$ ($w>-1$ ) and $\\gamma \\ne 1$ (i.e, in the presence of DE perturbations).",
"When $\\gamma =1$ (homogeneous DE), this behavior is inverted.",
"The most striking feature in Fig.",
"REF ($c_s^2=0$ ) is the strong dependence of $\\delta _c(z_c=0)$ on $w$ alone.",
"That piece of information by itself reassures the importance of studying the critical density for breaking the degeneracy among different DE models.",
"The dependence on $\\gamma $ alone is not so strong ($\\sim 2\\%$ ).",
"Changing both parameters at a time yields larger modifications on the curves, of course.",
"The possibility of constraints on this parameters from observational data is beyond the scope of this paper.",
"In Fig.",
"REF , where we keep $c_s^2=1$ , we note once again the dependence on $w$ , although about half as strong as in the previous case.",
"One can notice a dissonant curve ($\\gamma =0$ , $c_s^2=1$ , $w=-0.9$ ), which corresponds to the one that crosses over $\\Lambda $ CDM in Fig.",
"REF .",
"It might be a sign of incompatibility of such parameters, since $\\gamma =0$ means that there is no DE leaking away from the collapsing matter bubble, but at the same time $c_s^2=1$ corresponds to a stiff behavior of the former, which should (at least) delay the DE collapsing process.",
"The strongest dependence of $\\delta _c(z_c=0)$ on the parameters is observed in Fig.",
"REF , where we keep $c_s^2=w$ .",
"Observe also that, for larger $w$ , $\\delta _c(z_c=0)$ rapidly increases.",
"For (non)phantom DE, a (larger) smaller $\\gamma $ decreases $\\delta _c(z_c=0)$ .",
"On the other hand, a larger failure on energy conservation (i.e, larger $\\gamma $ ) in the collapsing region does move any of the curves towards $\\Lambda $ CDM.",
"The cases $c_s^2=\\gamma $ are depicted in Fig.",
"REF .",
"As expected, the curves tend to $\\Lambda $ CDM whenever $\\gamma \\rightarrow 1$ , regardless of the values of $c_s^2$ .",
"We also notice that if $\\gamma = 1$ (without DE perturbation), the phantom-DE curve is slightly above $\\Lambda $ CDM, as opposed to all the other cases presented here.",
"The non-phantom is also inverted (below $\\Lambda $ CDM in this case alone)."
],
[
"Conclusions",
"In summary, we have shown the non-linear equations that describe the evolution of the perturbations for both the dark matter and dark energy in the SC model when the clustering fraction of the latter is defined by a parameter $\\gamma $ , which consequently also models the lack of energy conservation in the collapsing region.",
"We have determined the critical contrast density $\\delta _c$ for different values of $\\gamma $ , obtaining larger values for stronger DE clustering.",
"The largest discrepancies from $\\Lambda $ CDM happen when $c_s^2=w$ (both clustered and smooth DE have the same EoS) and $\\gamma =0$ (fully clustered DE).",
"In a next paper, we will explore the consequences of the results presented here, namely deviations on the number density of collapsed objects, and the possibility of constraining the free parameters with current and future observational data."
],
[
"ACKNOWLEDGMENTS",
"D.H. acknowledges financial support from CAPES.",
"Figure: Evolution of the critical contrast density for c s 2 =0c^{2}_{s}=0 and different values of ww and γ\\gamma .",
"The solid black line corresponds to Λ\\Lambda CDM model.Figure: Evolution of the critical contrast density for c s 2 =1c^{2}_{s}=1 and different values of ww and γ\\gamma .",
"The solid black line corresponds to Λ\\Lambda CDM model.Figure: Evolution of the critical contrast density for c s 2 =wc^{2}_{s}=w and different values of ww and γ\\gamma .",
"The solid black line corresponds to Λ\\Lambda CDM model in all panels.",
"As before, here we find the largest deviations from Λ\\Lambda CDM.Figure: Evolution of the critical contrast density for : c s 2 =γc^{2}_{s}=\\gamma and different values of ww and γ\\gamma .",
"The solid black line corresponds to Λ\\Lambda CDM model in all panels."
]
] | 1906.04326 |
[
[
"Odd Order Group Actions on Alternating Knots"
],
[
"Abstract Let K be a an alternating prime knot in the 3-sphere.",
"We investigate the category of flypes between reduced alternating diagrams for K. As a consequence, we show that any odd prime order action on K is isotopic through maps of pairs to a single flype.",
"This implies that for any odd prime order action on K there is either a reduced alternating periodic diagram or a reduced alternating free periodic diagram.",
"Finally, we deduce that the quotient of an odd periodic alternating knot is also alternating."
],
[
"Introduction",
"A diagram $D$ for a knot $K \\subset S^3$ is alternating if the crossings in $D$ alternate between under and over crossings as you follow $K$ around the diagram, and a knot $K$ is alternating if it has an alternating diagram.",
"See Figure REF for an example.",
"Recently, Greene [3] and Howie [4] each showed that an alternating knot can instead be characterized by the existence of certain spanning surfaces.",
"In light of this more geometric interpretation, it is interesting to consider how the property of being alternating interacts with finite order group actions on $K$ .",
"Specifically, we have the following conjecture.",
"Conjecture 1.1 The quotient of an alternating periodic knot $K \\subset S^3$ is alternating.",
"To approach this conjecture, we use a theorem of Menasco and Thistlethwaite [6] that any homeomorphism $f:(S^3,K) \\rightarrow (S^3,K)$ , can be realized up to isotopy through maps of pairs by a sequence of certain diagrammatic moves called flypes.",
"In particular, we study the category of flypes on an alternating knot, and as a consequence classify odd prime order group actions on alternating knots.",
"Specifically, we prove the following main result.",
"Theorem 1.2 Let $\\tau $ be an odd prime order $p$ action on a prime alternating knot $K$ .",
"Then there exists a reduced alternating diagram $D$ for $K$ , and a flype $f$ from $D$ to $D$ such that $\\tau $ is isotopic through maps of pairs $(S^3,K)$ to $f$ .",
"We then obtain the following consequences for periodic and free periodic actions, proving Conjecture REF when the order of the period is odd.",
"Corollary 1.3 If $K$ is a $p$ periodic prime alternating knot for an odd prime $p$ , then $K$ has a reduced alternating periodic diagram.",
"Corollary 1.4 If $K$ is an odd prime $p$ free periodic alternating hyperbolic knot, then $K$ has a reduced alternating free periodic diagram.",
"See Figure REF .",
"While preparing this paper, the author discovered that Costa and Hongler released [1], which contains overlapping results.",
"In particular, Corollary REF , one of the main goals of this paper, also appears there.",
"However, Corollary REF does not appear in [1], while their paper also discusses the case of certain 2 periodic actions.",
"Both this paper and [1] use flypes as a main tool, but differ in their techniques.",
"Throughout this paper, all knots are prime, alternating, and contained in $S^3$ ."
],
[
"Organization",
"Section defines the relevant notions of flypes and their equivalences, Section proves the main results, and Section gives a few example applications."
],
[
"Acknowledgments",
"The author would like to thank Liam Watson and Joshua Greene for helpful conversations, and Robert Lipshitz for his support and belief in this project."
],
[
"The Category of Flypes ",
"In this section we define the category of flypes for a given prime alternating knot $K \\subset S^3$ , which has objects roughly corresponding to diagrams for $K$ , and morphisms generated by flypes.",
"Definition 2.1 The standard crossing ball $B_{\\mbox{\\tiny {std}}} = (B^3, D^2, a_1 \\cup a_2)$ is the triple of the 3-ball $\\lbrace (x,y,z) \\in \\mathbb {R}^3 \\mid |(x,y,z)| \\le 1\\rbrace $ , the horizontal unit disk inside this ball $\\lbrace (x,y,z) \\in \\mathbb {R}^3 \\mid z=0 \\mbox{ and } |(x,y,z)| \\le 1\\rbrace $ , and the union of the two arcs $a_1 = \\lbrace (x,y,z) \\in \\mathbb {R}^3 \\mid x=0, z \\ge 0 \\mbox{ and } y^2 + z^2 = 1\\rbrace $ and $a_2 = \\lbrace (x,y,z) \\in \\mathbb {R}^3 \\mid y=0, z\\le 0 \\mbox{ and } x^2 + z^2 = 1\\rbrace $ .",
"Definition 2.2 A realized diagram $\\lambda (D)$ for a knot $K \\subset S^3$ is smooth embeddings $S^2 \\hookrightarrow S^3$ , the projection sphere, $K: S^1 \\hookrightarrow S^3$ , the knot, and $\\lbrace B_i\\rbrace \\hookrightarrow S^3$ , the crossing balls, such that the $\\lbrace B_i\\rbrace $ are disjoint and $K \\subset S^2 \\cup \\lbrace B_i\\rbrace $ , along with homeomorphisms of triples $c_i: (B_i, B_i \\cap S^2, B_i \\cap K)\\rightarrow B_{\\mbox{\\tiny {std}}}$ , the crossing ball identification maps.",
"The diagram $D$ is the labeled graph in $S^2$ which is the projection of $K$ with vertices labeled to reflect under and over crossings.",
"Definition 2.3 An isomorphism of realized diagrams $f: \\lambda (D) \\rightarrow \\lambda (\\overline{D})$ is a homeomorphism of pairs $f: (S^3, K) \\rightarrow (\\overline{S}^3, \\overline{K})$ such that $f(S^2)$ is isotopic to $\\overline{S}^2$ relative to $\\overline{K}$ , $f(B_i) = \\overline{B}_i$ and $\\overline{c}_i \\circ f= c_i$ .",
"It is immediate that if $\\lambda (D)$ and $\\lambda ^{\\prime }(D)$ are realized diagrams for isomorphic labeled graphs $D$ , then there is an isomorphism of realized diagrams $f: \\lambda (D) \\rightarrow \\lambda ^{\\prime }(D)$ , and vice versa.",
"Definition 2.4 A standard flype is a transformation between realized diagrams $\\lambda (D) \\rightarrow \\lambda (E)$ of the form shown in Figure REF , where both tangles $T_1$ and $T_2$ are required to be non-trivial.",
"That is, a homeomorphism $f:S^3 \\rightarrow S^3$ which restricts to the identity on a round ball containing $T_2$ (shown as the exterior of $\\alpha _2$ ), $\\pi $ rotation around the horizontal axis on a ball containing $T_1$ (shown as the interior of $\\alpha _1$ ) and a linear homotopy between them to get a homeomorphism.",
"We further fix once and for all a homeomorphism $c_{\\mbox{\\tiny {std}}}: c_1 \\rightarrow B_{\\mbox{\\tiny {std}}}$ , and require that this be the crossing ball identification map used in $\\lambda (D)$ and its 180 degree rotation be the crossing ball identification map used in $\\lambda (E)$ .",
"Definition 2.5 A flype $f: \\lambda (D) \\rightarrow \\lambda (D^{\\prime })$ is any composition $f = g_1 \\circ s \\circ g_2$ where $g_1$ and $g_2$ are isomorphisms of realized diagrams, and $s$ is a standard flype.",
"We will refer to the crossing ball in $\\lambda (D^{\\prime })$ created by $f$ as $c_f$ , and the crossing ball in $\\lambda (D)$ removed by $f$ as $c^f$ .",
"The ball $\\alpha _1$ containing the tangle $T_1$ will be referred to as the domain of $f$ .",
"We also consider an isomorphism of realized diagrams to be a flype, and refer to it as the trivial flype.",
"Now consider a flype $f: \\lambda (D) \\rightarrow \\lambda (D^{\\prime })$ .",
"Then in the planar graph projection of $D$ we get a distinguished crossing $c^f$ and a distinguished pair of edges $(e^1_f, e^2_f)$ which will cross to form $c_f$ .",
"The following lemma states that this is enough to reconstruct the flype.",
"Lemma 2.6 Let $f: \\lambda (D) \\rightarrow \\lambda (D^{\\prime })$ and $g: \\lambda ^{\\prime }(D) \\rightarrow \\lambda ^{\\prime }(D^{\\prime })$ be flypes such that $(e^1_f, e^2_f) = (e_1^g,e_2^g)$ and $c^f = c^g$ .",
"Then there exists a pair of isomorphisms of realized diagrams $g_1,g_2$ such that $f = g_1 \\circ g \\circ g_2$ .",
"To begin, note that there is an isomorphism between $\\lambda (D)$ and $\\lambda ^{\\prime }(D)$ , so that we may consider $f$ and $g$ to start at the same realized diagram.",
"Similarly, there is an isomorphism from $\\lambda (D^{\\prime })$ to $\\lambda ^{\\prime }(D^{\\prime })$ , so we may assume $f$ and $g$ end at the same realized diagram.",
"Now note that $f$ and $g$ induce maps on underlying graphs in $S^2$ which are homotopic relative to the vertices of the graph.",
"In particular, $f$ and $g$ restrict to the same map on crossing balls since both are determined by the crossing ball identification maps for $\\lambda (D)$ and $\\lambda (D^{\\prime })$ .",
"From there we have a unique extension to the rest of $S^3$ up to an isomorphism of realized diagrams, as desired.",
"Figure: A standard flype fixes the exterior of α 2 \\alpha _2 and reflects the interior of α 1 \\alpha _1 across the horizontal axis, with a linear homotopy in between.",
"It removes the crossing ball at c 1 c_1 and creates a crossing ball at c 2 c_2.Lemma 2.7 Let $\\lambda (D)$ and $\\lambda (D^{\\prime })$ be realized reduced alternating diagrams for $K$ , and let $N(K)$ be a neighborhood of $K$ .",
"Then if $f: \\lambda (D) \\rightarrow \\lambda (D^{\\prime })$ is a homeomorphism $S^3 \\rightarrow S^3$ such that $f$ agrees with a flype when restricted to $(N(K)\\cap S^2)\\cup \\lbrace c_i\\rbrace $ , then $f$ is a flype.",
"That is, if $f$ restricts to a flype on the underlying diagram, then $f$ is a flype.",
"Let $\\varphi $ be a flype from $\\lambda (D) \\rightarrow \\lambda (D^{\\prime })$ , so that $f$ and $\\varphi $ agree when restricted to both the crossing balls and a neighborhood of $K$ in $S^2$ .",
"Now let $g = f \\circ \\varphi ^{-1}:\\lambda (D^{\\prime }) \\rightarrow \\lambda (D^{\\prime })$ , and observe that $g$ is the identity map on each crossing ball, and on $N(K) \\cap S^2$ .",
"In particular, this determines the relative isotopy class of $g(S^2)^{\\prime }$ so that $g$ is an isomorphism of realized diagram.",
"But then $f = g \\circ \\varphi $ , so $f$ is a flype, as desired.",
"By combining Lemmas REF and REF , we see that a diagrammatic description of a flype is sufficient since it corresponds to a unique flype up to isomorphism of realized diagrams.",
"Now given a composition of two flypes, the following lemmas will allow us to either combine them into a single flype, or else (roughly) commute them past each other.",
"Lemma 2.8 If $f_1: \\lambda (D_1) \\rightarrow \\lambda (D_2)$ and $f_2: \\lambda (D_2) \\rightarrow \\lambda (D_3)$ are flypes such that the crossing created by $f_1$ is the crossing removed by $f_2$ , and $K$ is prime, then there exists a flype $f_{1,2}: \\lambda (D_1) \\rightarrow \\lambda (D_3)$ with $f_2 \\circ f_1 = f_{1,2}$ .",
"By Lemmas REF and REF , it is enough to consider diagrammatic flypes.",
"We first note that there are three possible configurations for $f_2$ relative to $f_1$ .",
"See Figure REF .",
"Observe, however, that configuration (A) is impossible.",
"Indeed, if $T_3$ or $T_4$ is a non-trivial tangle, then $K$ cannot be prime.",
"On the other hand, in configuration (C), the composition will flip the tangle $T_1$ over twice so that the composition is simply a flype with domain $T_2$ .",
"Similarly, in configuration (B), the composition will flip both tangles $T_1$ and $T_2$ over once, which can be realized as a single flype on their sum.",
"Figure: Three potential configurations for the composition of two flypes f 1 :λ(D)→λ(D ' )f_1:\\lambda (D) \\rightarrow \\lambda (D^{\\prime }) and f 2 :λ(D ' )→λ(D '' )f_2:\\lambda (D^{\\prime }) \\rightarrow \\lambda (D^{\\prime \\prime }) with c f 1 =c f 2 c_{f_1} = c^{f_2}.",
"The diagrams shown are DD (as opposed to D ' D^{\\prime } or D '' D^{\\prime \\prime }), and the domain α f 2 \\alpha _{f_2} shown for f 2 f_2 is the preimage of the domain under f 1 f_1.Lemma 2.9 If $f_1: \\lambda (D_1) \\rightarrow \\lambda (D_2)$ and $f_2: \\lambda (D_2) \\rightarrow \\lambda (D_3)$ are flypes such that the crossing created by $f_1$ is not the crossing removed by $f_2$ , and $K$ is prime, then there exists a pair of flypes $f_2^{\\prime }: \\lambda (D_1) \\rightarrow \\lambda (D_2^{\\prime })$ and $f_1^{\\prime }: \\lambda (D_2^{\\prime }) \\rightarrow \\lambda (D_3)$ such that $f_2 \\circ f_1 = f_1^{\\prime } \\circ f_2^{\\prime }$ , $f_2(c_{f_1}) = c_{f_1^{\\prime }}$ , and $f_1^{\\prime }(c_{f_2^{\\prime }}) = c_{f_2}$ .",
"Furthermore, the domains of $f_1^{\\prime }$ and $f_2^{\\prime }$ are either disjoint or nested (See Figure REF ).",
"Informally, we will use this lemma to say that $f_1$ and $f_2$ commute, and by abuse of notation we will refer to $f_1^{\\prime }$ as $f_1$ and to $f_2^{\\prime }$ as $f_2$ .",
"As a further abuse of notation, given a crossing ball $c$ in $\\lambda (D)$ and a flype $f:\\lambda (D) \\rightarrow \\lambda (E)$ which does not remove $c$ , we will refer to $f(c)$ as just $c$ .",
"Remark 2.10 Note that while $f^{\\prime }_1(c_{f^{\\prime }_2}) = c_{f_2}$ , the crossings created by $f_1$ and $f^{\\prime }_1$ may be different.",
"However since this replacement process only reduces the domains of $f_1,f_2$ , a single replacement can be done for an arbitrary composition of flypes after which commuting them does not affect which crossings they create and remove.",
"Again, by Lemmas REF and REF it is enough to prove this lemma diagrammatically on the underlying graphs.",
"In the case where the domain for $f_1$ is contained in the domain for $f_2$ , and the case where the domains for $f_1$ and $f_2$ are disjoint, this lemma is clear by defining $f_1^{\\prime }$ and $f_2^{\\prime }$ to have the same domains as $f_1$ and $f_2$ respectively.",
"On the other hand, suppose that the domains intersect but are not nested.",
"Then we have the configuration shown in Figure REF .",
"In this case, define $f_1^{\\prime }$ as the flype with domain $T_1$ and define $f_2^{\\prime }$ as the flype with domain $T_3$ , and the result is again clear.",
"Figure: A possible configuration for two flypes f 1 :λ(D)→λ(D ' )f_1: \\lambda (D) \\rightarrow \\lambda (D^{\\prime }) and f 2 :λ(D ' )→λ(D)f_2:\\lambda (D^{\\prime }) \\rightarrow \\lambda (D) which overlap.",
"The shown diagram is DD, α f 1 \\alpha _{f_1} is the domain for f 1 f_1, and the domain α f 2 \\alpha _{f_2} shown is the preimage under f 1 f_1 of the domain for f 2 f_2."
],
[
"Odd Prime Actions ",
"In this section we will apply the structure of the category of flypes developed in Section to periodic actions using the following theorem of Menasco and Thistlethwaite.",
"Theorem 3.1 [6] For any reduced alternating diagram $D$ for $K$ and realization $\\lambda (D)$ , any homeomorphism of pairs $(S^3, K) \\cong (S^3, K)$ is isotopic through maps of pairs to an isomorphism of realized diagrams which is equal to a composition of flypes from $\\lambda (D)$ to $\\lambda (D)$ .",
"With this theorem in hand, let $K \\subset S^3$ be an alternating prime knot with a group action ${\\mathbb {Z}}/p$ for $p$ an odd prime and with a reduced alternating diagram $D$ .",
"Let $\\tau : S^3 \\rightarrow S^3$ be a generator for the action so that $\\tau ^p$ is the identity map.",
"Then by Theorem REF , $\\tau $ is isotopic to a composition of flypes $\\tau \\cong f:= f_n \\circ f_{n-1} \\circ \\dots \\circ f_1: \\lambda (D) \\rightarrow \\lambda (D)$ , so that $\\tau ^p = \\mbox{identity} \\cong f^p$ .",
"In order to keep track of these flypes, we will refer to the $i$ th iteration of $f_k$ as $f_{k,i}$ so that $f^p = f_{n,p} \\circ f_{n-1,p} \\circ \\dots \\circ f_{2,1} \\circ f_{1,1}$ .",
"Note that since $f^p$ is an isomorphism $\\lambda (D) \\rightarrow \\lambda (D)$ , it takes crossing balls to crossing balls and hence induces a permutation $\\sigma _{f^p}$ on the set of crossing balls $\\lbrace c_i\\rbrace $ for $\\lambda (D)$ .",
"To proceed we need the following lemma.",
"Lemma 3.2 If $K$ is an alternating knot which is not a torus knot, then as defined above, $\\sigma _{f^p}$ is the identity permutation.",
"To begin, we claim that $f^p$ is isotopic through maps of pairs $(S^3,K)$ preserving the crossing balls to a finite order map $g$ .",
"Clearly the permutation of crossing balls has finite order.",
"We can then perform an homotopy on the edges of the diagram $D$ to get a map isotopic to $f$ but which has finite order when restricted to $K$ .",
"This homotopy can then be extended to the projection sphere $S^2$ since its isotopy class relative to $K$ is fixed by $f$ , and from there to a map $g$ on all of $S^3$ .",
"But now, since $f^p$ is isotopic to the identity so is $g$ , and so we have a finite order map which is isotopic to the identity.",
"Now if $K$ is an alternating knot, it is not a satellite knot [5], and so it is either a torus knot or a hyperbolic knot.",
"By assumption $K$ is not a torus knot, so $K$ must be hyperbolic.",
"But by Mostow rigidity, any finite order map on a hyperbolic knot which is isotopic through maps of pairs to the identity is the identity map.",
"Hence $\\sigma _{f^p}$ is isotopic through maps preserving the crossing balls to the identity map, and hence $\\sigma _{f^p}$ is the identity permutation.",
"Now, $(f_{n} \\circ \\dots \\circ f_1)^p$ induces the identity map on each crossing ball of $\\lambda (D)$ , so each crossing ball created by a flype must later be destroyed by another flype, or else remain in the final diagram.",
"We may then separate the set of flypes into orbits $\\lbrace f_{r_1,s_1} \\dots f_{r_j,s_j}\\rbrace $ such that the crossing ball created by $f_{r_i,s_i}$ is destroyed by $f_{r_{i+1},s_{i+1}}$ .",
"That is, $c_{f_{r_i,s_i}} = c^{f_{r_{i+1},s_{i+1}}}$ .",
"Lemma 3.3 There is a reduced alternating diagram $D^{\\prime }$ and a choice of flypes $f_1, \\dots f_{m}$ such that $\\tau \\cong f_{m}^{\\prime } \\circ \\dots \\circ f_1^{\\prime } : \\lambda (D^{\\prime }) \\rightarrow \\lambda (D^{\\prime })$ and the orbit of the flype $f_{i,1}$ is $\\lbrace f_{i,1},f_{i,2}, \\dots , f_{i,p}\\rbrace $ .",
"In order to keep track of our diagram, we will refer to $f_n\\circ \\dots f_1: \\lambda (D) \\rightarrow \\lambda (D)$ as $\\tau _D$ .",
"Additionally, by Remark REF , we may once and for all make a replacement of flypes so that commuting them will not change which crossings they create and remove.",
"Now consider a crossing ball $c_{f_{i,j}}$ created by a flype $f_{i,j}$ which does not survive to the final diagram, and let the flype that destroys $c_{f_{i,j}}$ be $f_{k,l}$ with $k \\ne i$ .",
"First, suppose $f_{k,l}$ and $f_{i,j}$ occur in the same iteration of $\\tau $ so that $l = j$ .",
"Then we can simply commute the flypes in $\\tau _D$ by Lemma REF until $f_{i,j}$ and $f_{k,j} = f_{k,l}$ are adjacent, and then combine them into a single flype by Lemma REF .",
"Second, suppose that $l = j+1$ so that $f_{k,l}$ occurs in the iteration of $\\tau _D$ directly succeeding that of $f_{i,j}$ .",
"Then by Lemma REF we can commute $f_k$ in $\\tau _D$ until $f_k$ comes just before $f_j$ .",
"That is, $\\tau _D \\cong f_n \\circ \\dots \\circ f_j \\circ f_k \\circ f_{j-1} \\circ \\dots \\circ f_1$ .",
"In this case, consider the realized diagram $\\lambda (D^{\\prime })$ obtained by applying $f_k \\circ f_{j-1} \\circ \\dots \\circ f_1$ to $\\lambda (D)$ .",
"We then have $\\tau _{D^{\\prime }} = f_k \\circ f_{j-1} \\circ \\dots \\circ f_1 \\circ f_n \\circ \\dots \\circ f_j$ , so that $f_{i,j}$ and $f_{k,l}$ appear in the same iteration of $\\tau _{D^{\\prime }}$ and we can apply the argument above to combine them.",
"Finally, if $l > j+1$ , then we can iterate the above change of diagram until $f_{i,j}$ and $f_{k,l}$ appear in the same iteration of $\\tau _{D^{\\prime \\prime }}$ on some diagram $D^{\\prime \\prime }$ and then combine them.",
"The only flypes remaining will now have orbits only containing their own later iterations, and since $p$ is prime, the orbits are exactly as stated.",
"Using this lemma, we now reduce to the case that $\\tau $ is given by a single flype and prove Theorem REF .",
"To begin, reduce to the case that $\\tau _D$ has flypes with orbits as described in Lemma REF .",
"That is, the crossing ball created by each flype is only removed by a later iteration of the same flype.",
"Now focus on a particular orbit, that of the flype $f_i$ .",
"We know that $c_{f_{i,1}} = c^{f_{i,j}}$ for some $j$ , and from the proof of Lemma REF there are two choices for the orientation of the composition: either $f_{i,j}$ continues in the direction of $f_{i,1}$ as in configuration (B) in Figure REF , or else it flips back in the opposite direction of $f_{i,1}$ as in configuration (C) in Figure REF .",
"However, if it is the opposite direction, then since both of these flypes are $f_1$ they have the same tangle as their domains and so $f_{i,j}$ will exactly undo $f_{i,1}$ .",
"In other words, the orbit will consist of exactly the pair $\\lbrace f_{i,1}, f_{i,j}\\rbrace $ .",
"However, since $p \\ne 2$ this is impossible, so $f_{i,j}$ must continue in the same direction as $f_{i,1}$ .",
"Hence each orbit of flypes will form a loop, and have the effect of rotating $K$ around an axis contained in the projection sphere.",
"Now consider the orbits for $f_1$ and $f_2$ .",
"Note that the replacement via Lemma REF of $f_1$ and $f_2$ with $f_1^{\\prime }$ and $f_2^{\\prime }$ only reduces the domains of $f_1$ and $f_2$ , and hence we may make a replacement which ensures that $f_{1,i}$ and $f_{2,j}$ commute for all $i,j$ .",
"In particular $c_{f_{1,i}}$ is not contained in the domain of any $f_{2,j}$ .",
"However, all crossings except for the $\\lbrace c_{f_{2,j}}\\rbrace $ must be contained in the domain of some $f_{1,i}$ since the collection of these flypes forms a loop which rotates the entire knot.",
"This is a contradiction, so that in fact $\\tau _D$ can be written as a single flype.",
"We now return to the case that $K$ is $p$ -periodic (as opposed to free periodic).",
"Then iterating a single flype cannot rotate the knot around an axis in the diagram as well as an axis perpendicular to the diagram since that would be a free period and so we obtain a proof of Corollary REF .",
"If $K$ is a torus knot, then it is $T(2,p)$ which has a unique reduced alternating diagram, and it is $p$ periodic.",
"If $K$ is a hyperbolic knot, then by Theorem REF the periodic action is generated by a single flype $f$ on some reduced alternating diagram, and by Lemma REF the $p$ th power of this flype is the identity map on the crossings.",
"Then we can cut $K$ into the $p$ domains of the $f_i$ , and describe $\\tau $ as rotation around an axis permuting these domains, plus a rotation around an axis in the plane of the diagram.",
"However, this describes a free periodic action unless one of these rotations is trivial.",
"Note that the rotation around an axis in the plane of the diagram needs to be either 2-periodic or trivial, and since $p$ is odd, it is trivial.",
"Hence $f$ is a trivial flype, and just an isomorphism of realized diagrams.",
"In particular, $\\tau $ is just a rotational symmetry of the diagram, as desired.",
"On the other hand, when $K$ has a ${\\mathbb {Z}}/p$ -action which is a free period, we get Corollary REF .",
"Figure: A (p=3,n)(p=3,n) free periodic alternating diagram.",
"TT is any alternating tangle, and F n F^n is nn full twists.Again by Theorem REF the free periodic action is generated by a single flype $f$ on some reduced alternating diagram.",
"If $f$ is a trivial flype, then the the action is a period of the knot $K$ , so $f$ must be a non-trivial flype.",
"As above, we can then cut $K$ into the domains of the $f_i$ , and describe $\\tau $ as a permutation of these domains with a twist around an axis in the plane of the diagram in between.",
"In particular, the diagram must already be a free periodic diagram, see Figure REF .",
"Finally, we use Corollary REF to consider the quotient of an alternating periodic knot.",
"Corollary 3.4 Let $K$ be an odd-periodic alternating prime knot.",
"Then the quotient knot of $K$ is alternating.",
"By Corollary REF , we can find a reduced alternating periodic diagram $D$ for $K$ .",
"Then the quotient knot is obtained by cutting a fundamental domain of the rotation from $D$ and connecting the free ends without crossings.",
"In particular, a strand leaving an under crossing and going to an over crossing still does so."
],
[
"Examples ",
"Corollaries REF and REF give an elementary method to determine if an alternating prime knot $K$ has a $p$ period or a free $p$ period.",
"Since all reduced alternating diagrams are related by flypes, you can simply list all reduced alternating diagrams for $K$ and check what symmetries they have.",
"Example 4.1 Consider $4_1$ , the figure eight knot.",
"See Figure REF , which shows a reduced alternating diagram.",
"It has no possible non-trivial flypes, and it is not a periodic or free periodic diagram, so $4_1$ is not $p$ periodic or free periodic for any odd prime $p$ , and hence for any odd integer $p$ .",
"Figure: The unique reduced alternating diagram for 4 1 4_1.More generally, the number of crossings in a reduced alternating diagram for a knot obstructs the existence of periods, although the existence of the full twists in a free periodic diagram means there is no such obstruction for free periods.",
"Example 4.2 Suppose $K$ is $p$ periodic for an odd prime $p$ , and has a reduced alternating diagram with $n$ crossings.",
"Then since any other reduced alternating diagram is related by a sequence of flypes which does not change the crossing number, $K$ must have a $p$ periodic diagram with $n$ crossings by Corollary REF .",
"In particular, $n$ must be a multiple of $p$ ."
]
] | 1906.04308 |
[
[
"Feedback Linearization for Quadrotors UAV"
],
[
"Abstract In the paper \"Control Design for UAV Quadrotors via Embedded Model Control\" [1], the authors designed a complete control unit for a UAV Quadrotor, based on the Embedded Model Control (EMC) methodology, in combination with the Feedback Linearization (FL); when applied to non-linear systems.",
"Specifically, [1] proposes to use the FL as a novel way to design the internal model for the EMC state and disturbance predictor.",
"To support the treatise in [1], in this report the feedback-linearized model of the UAV quadrotor leveraged in [1] is step-by-step derived."
],
[
"Introduction",
"Feedback Linearization (FL) technique allows transforming a command-affine non-linear model of the UAV quadrotor into an equivalent (fully or partly) linear one.",
"Specifically, FL: (i) pursues the collection of all the model non-linearities in specific points, e.g.",
"at the command level, and (ii) achieves an input-output linearization by means of a non-linear feedback, performing a perfect cancellation of non-linearities [2].",
"Nevertheless, model uncertainties make non-linear terms uncertain, and their use into the FL feedback may imply performance degradation and/or instability.",
"Hence, this study proposes to use FL in combination with the Embedded Model Control (EMC) framework.",
"In short, the designed FL-EMC approach let us to treat non-linearities as known and unknown disturbances to be estimated and then rejected, thus enhancing control robustness and performance.",
"To this purpose, the study in [1] focused on the so-called normal form representation of the non-linear model, where the non-linearities are collected at the command level, which perfectly fits the EMC internal model design rationale [1]."
],
[
"Feedback Linearization",
"The feedback linearization (FL) technique is an effective resource to linearize a non-linear model, by introducing a proper state transformation and a non-linear feedback [3].",
"Then, starting from the new linear model (i.e.",
"the feedback-linearized one), a linear controller can be designed.",
"In practice, within the FL, the model input-output linearization is obtained by differentiating each model output many times, until a control input component appears in the resulting equation.",
"Generally speaking, the feedback-linearized model is obtained by means of a system state transformation (diffeomorphism) and a non-linear feedback [3].",
"The state variables of the transformed model are the Lie derivatives of the system output $\\mathbf {y}$ .",
"This implies that the choice of the output vector is extremely important to accomplish the input-output linearization.",
"Let us consider a command-affine square non-linear system, with state vector $\\mathbf {x}\\,{\\in }\\,R^{n}$ , input $\\mathbf {u}\\,{\\in }\\,R^{m}$ , and output $\\mathbf {y}\\,{\\in }\\,R^{m}$ : $\\begin{split}\\dot{\\mathbf {x}}(t) &= \\mathbf {f}(\\mathbf {x}(t)) + G(\\mathbf {x}(t))\\mathbf {u}(t) , \\: \\mathbf {x}(0)=\\mathbf {x}_0, \\\\\\mathbf {y}(t) &= \\mathbf {h}(\\mathbf {x}(t)), \\end{split}$ where $\\mathbf {f}$ and $\\mathbf {h}$ represent smooth vector fields [2], while $G\\,{\\in }\\,R^{{n{\\times }m}}$ is a matrix with smooth vector fields as columns.",
"Denoting with $r_i$ the relative degree of the $i{-}th$ output, we aimed to obtain an equivalent non-linear model, where all non-linearities have been collected at the command level, i.e.",
": $\\begin{bmatrix}y_1^{(r_1)} \\\\ y_2^{(r_2)} \\\\ \\dots \\\\ y_m^{(r_m)} \\\\\\end{bmatrix}(t) = E(\\mathbf {x}(t))\\mathbf {u}(t) + \\mathbf {b}(\\mathbf {x}(t)),$ where $y^{(n)}(t)$ denotes the time derivative of order $n$ .",
"Specifically, (REF ) represents a cascade of integrators in parallel, where the output and its derivatives are the new state variables $\\mathbf {z}$ defined as: $\\begin{split}\\mathbf {z}\\,{=}\\,T(\\mathbf {x})\\,{=}\\,[ & y_1 \\:\\: y_1^{(1)} \\:\\: \\dots \\:\\: y_1^{(r_1-1)} \\:\\: \\dots \\\\& y_2 \\:\\: y_2^{(1)} \\:\\: \\dots \\:\\: y_2^{(r_2-1)} \\:\\: \\dots \\\\& \\dots \\\\& y_m \\:\\: y_m^{(1)} \\:\\: \\dots \\:\\: y_m^{(r_m-1)} ]^T,\\end{split}$ where $T(\\mathbf {x})$ represents a diffeomorfism.",
"Whenever this state transformation is applicable and the decoupling matrix $E(\\cdot )$ is invertible, it is possible to linearize the equivalent model (REF ) via the non-linear feedback: $\\begin{split}\\mathbf {u}(t) = E(\\mathbf {x}(t))^{-1} \\left( \\mathbf {v}(t) - \\mathbf {b}(\\mathbf {x}(t)) \\right).\\end{split}$ where $\\mathbf {v}$ is a new equivalent command.",
"Hence, by applying the feedback (REF ) to the model (REF ), a parallel of $m$ decoupled input-output channels, represented by cascaded integrators, is obtained, viz.",
": $\\begin{split}\\begin{bmatrix}y_1^{(r_1)} \\\\ y_2^{(r_2)} \\\\ \\dots \\\\ y_m^{(r_m)} \\\\\\end{bmatrix}(t) &= \\mathbf {v}(t)\\end{split},$ where the new command $\\mathbf {v}$ is used for the design of a linear controller.",
"A “full\" input-output linearization is achieved if the sum of the output relative degrees is equal to the order of the model (REF ) [3].",
"When this condition is not verified, some dynamics are hidden by the state transformation.",
"This so-called “internal\" dynamics could be unstable [3].",
"In this case the feedback linearization process fails.",
"Finally, when $E(\\cdot )$ is not invertible in $R^{n}$ , it is possible to adopt a new command vector by considering the derivative of some of the command components.",
"This solution, called dynamic extension [3], enables the applicability of the feedback linearization by making the $E(\\cdot )$ invertible, yet at a cost: the introduction of a dynamics in the command.",
"Furthermore, let us remark that the non-singularity condition of $E(\\cdot )$ is not necessary verified in the complete state space.",
"As a matter of fact, in some applications, these singularities can be avoided by applying proper constrains to the state trajectories."
],
[
"The Quadrotor UAV Case-study",
"As above depicted, the input-output linearization is achieved by differentiating each output many times until a control input appears.",
"Hence, the successful application of the feedback linearization is strongly dependent on the output vector.",
"As a matter of fact, the adopted Quadrotor UAV dynamics, which encompasses four commands and three outputs [1], is characterized by a non-square decoupling matrix $E(\\cdot )\\,{\\in }\\,\\mathcal {R}^{3,4}$ [4].",
"Therefore, in [1], the quadrotor heading angle $\\psi $ was selected as an additional output to be controlled, so to make the FL applicable to the Quadrotor UAV model [1].",
"However, in the quadrotor UAV case-study treated in [1], the problem is not solvable by means of a static feedback [4].",
"Conversely, a full linearization may be only achieved by introducing additional states to the standard quadrotor model, thus obtaining the so-called extended model [4].",
"In fact, in order to obtain a non-singular decoupling matrix $E(\\cdot )$ , the second derivative of the first command component, $u_1$ in REF , with respect to time was also needed.",
"As a result, the introduction of two more states, i.e.",
"$\\zeta \\,{=}\\,u_1$ and its derivative $\\chi \\,{=}\\,\\dot{u}_1$ , lead to define the Quadrotor UAV extended model, whose state vector is $\\mathbf {x}\\,{=}\\,\\left[ \\mathbf {r}^T \\:\\: \\mathbf {v}^T \\:\\: \\theta ^T \\:\\: \\omega _b^T \\:\\: \\zeta \\:\\: \\chi \\right]^T$ , where $\\mathbf {r}$ and $\\mathbf {v}$ are respectively the inertial position and velocity of the CoM, $\\theta =[\\phi \\:\\: \\theta \\:\\: \\psi ]^T$ are the attitude angles, and $\\omega _b$ is the body angular rate vector.",
"Hence, the model, sketched in REF , holds [1]: $\\begin{split}\\dot{\\theta }(t) &= A(\\theta (t)) \\omega _b(t), \\quad \\theta (0)= \\theta _0, \\\\A(\\theta ) &=\\frac{1}{c_{\\theta }}\\begin{bmatrix}c_{\\psi } & -s_{\\psi } & 0 \\\\c_{\\theta }s_{\\psi } & c_{\\theta }c_{\\psi } & 0 \\\\-s_{\\theta }c_{\\psi } & s_{\\theta }s_{\\psi } & c_{\\theta }\\end{bmatrix}, \\\\ \\dot{\\omega }_b(t) &= \\mathbf {u}(t) - J^{-1}(\\omega _b(t) \\times J\\omega _b(t)) + \\mathbf {d}(t), \\\\\\omega _b(0) &= \\omega _{b0}, \\\\\\dot{\\mathbf {r}}(t) &= \\mathbf {v}(t), \\quad \\mathbf {r}(0) = \\mathbf {r}_0, \\\\\\dot{\\mathbf {v}}(t) &= R_{b}^i(\\theta )\\begin{bmatrix} 0 & 0 & \\zeta (t) \\end{bmatrix}^T- \\mathbf {g} + \\mathbf {a}_d(t), \\\\\\mathbf {v}(0) &= \\mathbf {v}_0, \\\\\\dot{\\zeta }(t) &= \\chi (t), \\\\\\dot{\\chi }(t) &= \\ddot{u}_1(t), \\\\\\mathbf {y}(t) &= \\begin{bmatrix} \\mathbf {r} \\\\ \\psi \\end{bmatrix}(t), \\quad \\overline{\\mathbf {u}} = \\begin{bmatrix} \\ddot{u}_1 \\\\ \\mathbf {u} \\end{bmatrix}(t).\\end{split}$ In (REF ), $\\mathbf {u}$ is the command torque along the three body axes, while $J$ is the quadrotor inertia matrix.",
"In addition, $R^i_{b}(\\theta )$ describes the body-to-inertial attitude, $\\mathbf {g}\\,{=}\\,\\left[ 0 \\:\\: 0 \\:\\: 9.81 \\right]^T$ is the gravity vector, while $\\mathbf {d}$ and $\\mathbf {a}_d$ represent all the external disturbances (e.g.",
"wind, rotor aerodynamics, mechanical vibration, actuator noise) affecting the model dynamics.",
"Finally, $\\overline{\\mathbf {u}}$ and $\\mathbf {y}$ are the new command vector and the extended model output, respectively.",
"Figure: Quadrotor UAV extended Model.The extended model (REF ) is the foremost building block of the feedback linearization process, performed to obtain the feedback-linearized model leveraged in [1] to design the internal model for the EMC state and disturbance predictor.",
"To this aim, starting from the overall model in (REF ), we will derive the feedback-linearized heading and CoM dynamics, through the procedure outlined in Sec.",
"II."
],
[
"The Quadrotor UAV Dynamics",
"Considering the quadrotor attitude kinematics in (REF ), let us start from the heading dynamics.",
"The first order time-derivative of the heading angle holds: $\\begin{split}\\psi ^{(1)}(t) &= \\eta (t) = \\mathbf {b}_{\\psi }(\\theta (t))\\omega _b(t), \\\\\\mathbf {b}_{\\psi }(\\theta (t)) &= [ -t_\\theta c_\\psi \\quad t_\\theta s_\\psi \\quad 1].\\end{split}$ A further derivative is needed in order to relate the output with the command, viz.",
": $\\begin{split}\\psi ^{(2)}(t) &= \\mathbf {b}_{\\psi }(\\theta (t))(\\mathbf {u}(t) - \\mathbf {h}_g(t) + \\mathbf {d}(t)) + \\dot{\\mathbf {b}}_{\\psi }(\\theta (t))\\omega _b(t)= \\\\&= \\mathbf {b}_{\\psi }(\\theta (t))\\mathbf {u}(t) + h_\\psi (\\mathbf {x}(t)) + d_\\psi (t),\\end{split}$ where we defined $h_\\psi $ and $d_\\psi $ as: $\\begin{split}h_\\psi (\\mathbf {x}(t)) &= \\dot{\\mathbf {b}}_{\\psi }(\\theta (t))\\omega _b(t) - \\mathbf {b}_{\\psi } (\\theta (t))\\mathbf {h}_g(t), \\\\d_\\psi (t) &= \\mathbf {b}_{\\psi }(\\theta (t))\\mathbf {d}(t).\\end{split}$ Concerning the CoM dynamics, to relate the output with the command (cf.",
"Sec.",
"), two derivatives will be needed; starting from the CoM acceleration.",
"Specifically, the CoM position third derivative, i.e.",
"the jerk $\\mathbf {s}$ , holds: $\\mathbf {r}^{(3)}(t) = \\mathbf {s}(t) = \\\\= R_b^i(\\theta (t))\\left( \\begin{bmatrix}0 \\\\ 0 \\\\ \\chi (t)\\end{bmatrix} +S(\\omega _b(t))\\begin{bmatrix} 0 \\\\ 0 \\\\ \\zeta (t) \\end{bmatrix} \\right) + \\dot{\\mathbf {a}}_d(t),$ being $S(\\omega _b)$ : $\\begin{split}S(\\omega _b) = \\begin{bmatrix}0 & -\\omega _z & \\omega _y \\\\ \\omega _z & 0 & -\\omega _x \\\\ -\\omega _y & \\omega _x & 0\\end{bmatrix}\\end{split}.$ In (REF ), the first term on the right-hand side represents the contribution of the vertical jerk command, whereas the second term is the jerk component due to the quadrotor angular rates.",
"Moreover, a further derivative of (REF ) is necessary in order to have a full-rank decoupling matrix $E(\\cdot )$ .",
"Hence, it holds: $\\mathbf {r}^{(4)}(t) = R_b^i(\\theta (t))\\begin{bmatrix}\\dot{\\omega }_y \\zeta \\\\ -\\dot{\\omega }_x \\zeta \\\\ \\ddot{u}_1 \\end{bmatrix}(t) + \\\\+ 2\\chi (t)R_b^i(\\theta (t))\\begin{bmatrix}\\omega _y \\\\ -\\omega _x \\\\ 0 \\end{bmatrix}(t) + \\\\+ \\zeta (t)R_b^i(\\theta (t))\\begin{bmatrix}\\omega _x\\omega _z \\\\ \\omega _y\\omega _z \\\\ -(\\omega _x^2 + \\omega _y^2)\\end{bmatrix}(t) + \\ddot{\\mathbf {a}}_d(t).$ As a result, by introducing the quadrotor attitude dynamics from (REF ) in (REF ), we obtain: $\\begin{split}\\mathbf {r}^{(4)}(t) &= R_b^i(\\theta (t))\\begin{bmatrix}\\zeta u_3 \\\\ - \\zeta u_2 \\\\ \\ddot{u}_1\\end{bmatrix}(t) + \\mathbf {h}_r(\\mathbf {x}(t)) + \\mathbf {d}_r(\\mathbf {x}(t),t),\\end{split}$ where $\\mathbf {h}_r$ and $\\mathbf {d}_r$ are the known and the unknown disturbance terms, respectively.",
"Specifically, $\\mathbf {h}_r(\\mathbf {x}(t))$ was defined as: $\\mathbf {h}_r(t) = \\zeta (t)R_b^i(\\theta (t))\\begin{bmatrix}\\omega _x\\omega _z -h_{gy} \\\\ \\omega _y\\omega _z + h_{gx} \\\\ - (\\omega _x^2 + \\omega _y^2)\\end{bmatrix}(t) + \\\\+ 2\\chi (t)R_b^i(\\theta (t))\\begin{bmatrix}\\omega _y \\\\ -\\omega _x \\\\ 0 \\end{bmatrix}(t),$ whereas $\\mathbf {d}_r(\\mathbf {x}(t))$ holds: $\\begin{split}\\mathbf {d}_r(\\mathbf {x}(t)) = R_b^i(\\theta (t))\\begin{bmatrix}\\zeta d_{ry} \\\\ -\\zeta d_{rx} \\\\ 0\\end{bmatrix}(t) + \\ddot{\\mathbf {a}}_d(t).\\end{split}$ From (REF ), it is interesting to notice how only tilt disturbances ($d_{rx}$ , $d_{ry}$ ) act on the CoM dynamics, and how they are amplified by the body vertical acceleration $\\zeta \\,{=}\\,u_1$ , which is close to the gravity value for non aggressive manoeuvres [5]."
],
[
"The Quadrotor UAV Case-study: Complete Model",
"As a further step, putting together the heading dynamics (REF ) and the CoM dynamics (REF ), the whole input-output relation is found: $\\begin{bmatrix} \\mathbf {r}^{(4)} \\\\ \\psi ^{(2)} \\end{bmatrix}(t) =E(\\theta (t),\\zeta (t))\\overline{\\mathbf {u}}(t) + \\begin{bmatrix} \\mathbf {h}_r \\\\ h_\\psi \\end{bmatrix}(t) +\\begin{bmatrix} \\mathbf {d}_r \\\\ d_\\psi \\end{bmatrix}(t),$ where $\\begin{split}E(\\theta (t),\\zeta (t)) &= \\left[ \\begin{array}{c;{2pt/2pt}c}B_r(\\theta (t),\\zeta (t)) & \\begin{array}{c}0 \\\\ 0 \\\\ 0\\end{array} \\\\ [2pt/2pt]\\begin{array}{ccc}0 & -t_\\theta c_\\psi & \\quad t_\\theta s_\\psi \\end{array} & 1\\end{array} \\right] \\\\ B_r(\\theta (t),\\zeta (t)) &= R_b^i(\\theta (t))\\begin{bmatrix}0 & 0 & \\zeta (t) \\\\ 0 & -\\zeta (t) & 0 \\\\ 1 & 0 & 0 \\end{bmatrix}.\\end{split}$ As shown in (REF ), the state vector of the new equivalent model (REF ) is defined by $\\mathbf {z}\\,{=}\\,T(\\mathbf {x})\\,{=}\\,[\\mathbf {r} \\:\\: \\mathbf {v} \\:\\: \\mathbf {a} \\:\\: \\mathbf {s} \\:\\: \\psi \\:\\: \\eta ]^T$ .",
"More interestingly, the total relative degree of the model in (REF ) is equal to the order of the extended model in (REF ), namely $r_1\\,{+}\\,r_2\\,{+}\\,r_3\\,{+}\\,r_4\\,{=}\\,n\\,{=}\\,14$ : this implies that no internal dynamics exists, and a full input-output linearization has been achieved.",
"Furthermore, the decoupling matrix $E(\\mathbf {x}(t))$ is non-singular in $D\\,{=}\\,\\lbrace \\mathbf {x}(t) \\subset R^n : \\zeta (t) \\ne 0, |\\phi (t)|<\\pi /2, |\\theta (t)|<\\pi /2 \\rbrace $ .",
"This implies that aggressive manoeuvres may be performed, although with tilt angles lower than $\\pi /2$ (acrobatic manoeuvres, such as 360-loops, were not in the scope of [1]).",
"On the other side, since the actuators effect is lower-bounded by a minimum saturation thrust, the total vertical acceleration is always positive.",
"For these practical reasons, the state trajectories was constrained to the domain $D$ , where the invertibility of the decoupling matrix is guaranteed.",
"To conclude, REF sketches the final model (REF ) (cf.",
"also [1]), where the non-linear couplings with the commands $u_2$ and $u_3$ in $\\mathbf {b}_\\psi $ were collected in $h_\\psi ^*(\\mathbf {x}(t),u_2(t),u_3(t))\\,{=}\\,-t_\\theta c_\\psi u_2(t)\\,{+}\\,t_\\theta s_\\psi u_3(t)$ .",
"On the other hand, $\\mathbf {h}_r$ and $h_\\psi $ collect all the non-linearities, collocated at command level.",
"Finally, consistently with the EMC design framework, the terms $\\mathbf {d}_r$ and $d_\\psi $ represent the non-explicitly modelled effects and the external disturbances.",
"Figure: Global scheme of the quadrotor UAV input-output linearized model, as result of the FL technique (Courtesy: ).As per REF , for the purpose of the linear control design, the model (REF ) can be rewritten as: $\\dot{\\mathbf {r}}(t) &= \\mathbf {v}(t), \\quad \\mathbf {r}(0) = \\mathbf {r}_0, \\\\\\dot{\\mathbf {v}}(t) &= \\mathbf {a}(t), \\quad \\mathbf {v}(0) = \\mathbf {v}_0, \\\\\\dot{\\mathbf {a}}(t) &= \\mathbf {s}(t), \\quad \\mathbf {a}(0) = \\mathbf {a}_0, \\\\\\dot{\\mathbf {s}}(t) &= \\mathbf {u}_r(t) + \\mathbf {h}_r(\\mathbf {x}(t)) + \\mathbf {d}_r(t), \\quad \\mathbf {s}(0) = \\mathbf {s}_0, \\\\\\dot{\\psi }(t) &= \\eta (t), \\quad \\psi (0) = \\psi _0, \\\\\\dot{\\eta }(t) &= u_4(t) + h_\\psi (\\mathbf {x}(t)) + h_\\psi ^*(\\cdot ) + d_\\psi (t), \\quad \\eta (0) = \\eta _0, $ where $\\mathbf {u}_r$ is a transformed command, defined as: $\\begin{split}\\mathbf {u}_r(t) &= \\mathbf {B}_r(\\mathbf {x}(t)) \\begin{bmatrix} \\ddot{u}_1 & u_2 & u_3 \\end{bmatrix}^T(t).\\end{split}$ As a result, eq:flA1,eq:flA2,eq:flA3,eq:flA4,eq:flB1,eq:flB2 represent the UAV quadrotor model where all the non-linearities have been collocated at the command level and therefore can be cancelled by a non-linear feedback in the form expressed in (REF ).",
"Nevertheless, this approach relies on the perfect knowledge of the model non-linearities ($\\mathbf {h}_r$ , $h_{psi}$ ) which may considerably limit the controller performance as well as its practical applicability.",
"To this aim, the novel approach proposed in [1] (namely, the FL-EMC design) considers the model (REF ) from a different point of view.",
"More precisely, the non-linear components are treated as generic unknown disturbances which are real-time estimated by a proper extended state observer.",
"Thus, implementing a direct disturbance rejection, jointly with a linear control law, it is possible to completely neglect model non-linearities and, at same time, to enhance the controller robustness against model uncertainties."
]
] | 1906.04263 |
[
[
"Applying the tempered Lefschetz thimble method to the Hubbard model away\n from half-filling"
],
[
"Abstract The tempered Lefschetz thimble method is a parallel-tempering algorithm towards solving the numerical sign problem.",
"It uses the flow time of the gradient flow as a tempering parameter and is expected to tame both the sign and multimodal problems simultaneously.",
"In this paper, we further develop the algorithm so that the expectation values can be estimated precisely with a criterion ensuring global equilibrium and the sufficiency of the sample size.",
"To demonstrate that this algorithm works well, we apply it to the quantum Monte Carlo simulation of the Hubbard model away from half-filling on a two-dimensional lattice of small size, and show that the numerical results agree nicely with exact values."
],
[
"Introduction\n",
"The sign problem is one of the major obstacles when performing numerical calculations in various fields of physics.",
"Typical examples include finite density QCD [1], quantum Monte Carlo (QMC) calculations of quantum statistical systems [2], [3], [4], and the numerical simulations of real-time quantum field theories.",
"Among a variety of approaches, two algorithms have taken attention as potential candidates to generically solve the sign problem for systems with complex action; one is the complex Langevin method [5], and the other is a class of algorithms utilizing the Lefschetz thimbles [6], [7], [8], [9], [10], [11], [12], [13], [14], [15].",
"Although both the algorithms make use of complexification of variables and analytic continuation of integrands, their methodologies are fairly different; the former algorithm attempts to replace the complex Boltzmann weight by a real positive weight defined in the whole complex space, while the latter deforms the integration region in the complex space so as to reduce the phase oscillation.",
"At this stage, each algorithm has its own advantage and disadvantage.",
"The former is advantageous in that it is relatively fast with computational cost $O(N)$ ($N$ : the degrees of freedom), but it suffers from the so-called wrong convergence problem [16], [17], [18], [19].",
"The latter is generally free from the wrong convergence problem if only a single thimble is relevant in evaluating the expectation values of physical observables of interest.",
"The disadvantage is its expensive numerical cost, which is $O(N^3)$ because of the need to calculate the Jacobian determinant.",
"When multiple thimbles are relevant, one needs to take care of the multimodality of the distribution.",
"The tempered Lefschetz thimble method (TLTM) was thus proposed in [14] to tame both the sign and multimodal problems simultaneously, where the system is tempered by the flow time of the antiholomorphic gradient flow (see also [15] for a similar idea).",
"In this paper, we further develop the TLTM, proposing an algorithm which allows the precise estimation of expectation values with a criterion ensuring global equilibrium and the sufficiency of the sample size.",
"The key is the use of the fact that the expectation values should be the same for all flow times.",
"To demonstrate that this algorithm works well, we apply it to the QMC simulation of the Hubbard model away from half-filling.",
"The application of Lefschetz thimble methods to the Hubbard model has already been considered by several groups [20], [21], [22] (see also [23], [24] for recent study), and the relevance of the contributions from multiple thimbles has been reported.",
"In this paper, we consider a two-dimensional periodic square lattice of size $N_s=2\\times 2$ with the inverse temperature decomposed to $N_\\tau =5$ pieces, and numerically evaluate the expectation values of observables as functions of the chemical potential with other parameters fixed to some values.",
"We show that the TLTM (the implementation of tempering combined with the above algorithm for precise estimation) give results that agree nicely with exact values, simultaneously resolving the sign and multimodal problems.",
"We comment that the extent of seriousness of the sign problem in the QMC simulation of the Hubbard model depends heavily on the choice of the Hubbard-Stratonovich variables.",
"In this paper, in order to apply the Lefschetz thimble method, we exclusively consider a Gaussian Hubbard-Stratonovich variable that leads to a complex action.",
"There the sign problem is actually severe as we will see below, and one needs to seriously consider a dilemma between the sign and multimodal problems, which can be solved by the TLTM as stated above.",
"However, the temporal size considered here is still small ($N_\\tau =5$ ), and for such a high temperature regime one can resort to other methods than the Lefschetz thimble methods with a different type of Hubbard-Stratonovich variables (see discussions in section ).",
"This paper is organized as follows.",
"In section after briefly reviewing the TLTM [14], we give a new algorithm which allows the precise estimation of expectation values with a criterion ensuring global equilibrium and the sufficiency of the sample size.",
"This algorithm is applied to the Hubbard model in section , and we discuss about the obtained numerical results.",
"We there also make a comment on the sign averages obtained by other methods.",
"Section is devoted to conclusion and outlook.",
"Five appendices are given for more detailed discussions on various topics."
],
[
"Tempered Lefschetz thimble method\n",
"Let $x=(x^i)\\in {\\mathbb {R}}^N$ be a real $N$ -dimensional dynamical variable with action $S(x)$ which may take complex values.",
"Our main concern is to estimate the expectation values $\\langle \\mathcal {O}(x) \\rangle _S\\equiv \\frac{\\int _{{\\mathbb {R}}^N} dx\\, e^{-S(x)}\\,\\mathcal {O}(x)}{\\int _{{\\mathbb {R}}^N} dx\\, e^{-S(x)}}.$ We assume that $e^{-S(z)}$ and $e^{-S(z)}\\,\\mathcal {O}(z)$ are entire functions over ${\\mathbb {C}}^N$ when $x$ is complexified to $z=(z^i)\\in {\\mathbb {C}}^N$ .",
"Then, due to Cauchy's theorem for higher dimensions, the right-hand side does not change under continuous deformations of the integration region as long as the boundary at infinity is kept fixed so that the integrals converge.",
"The sign problem will get reduced if ${\\rm Im}\\, S(z)$ is almost constant on the new integration region.",
"In [11], [11], [12], [13], [14], [15] such a deformation $x \\rightarrow z_t(x)$ ($t\\ge 0$ ) is made according to the antiholomorphic gradient flow: $\\dot{z}_t^i &= [\\partial _i S(z_t)]^\\ast ,\\quad z^i_{t=0} = x^i.$ Equation (REF ) can then be rewritten as $\\langle \\mathcal {O}(x) \\rangle _S= \\frac{\\int _{\\Sigma _t} d z\\, e^{-S(z)}\\,\\mathcal {O}(z)}{\\int _{\\Sigma _t} d z\\, e^{-S(z)}}\\quad (\\Sigma _t \\equiv z_t({\\mathbb {R}}^N)),$ which can be further rewritten as a ratio of reweighted integrals over ${\\mathbb {R}}^N$ by using the Jacobian matrix $J_t(x)\\equiv \\bigl (\\partial z_t^i(x)/\\partial x^j\\bigr )$ [11]: $\\langle \\mathcal {O}(x) \\rangle _S&= \\frac{\\int _{{\\mathbb {R}}^N} d x\\,\\det \\!J_t(x)\\,e^{-S(z_t(x))}\\,\\mathcal {O} (z_t(x))}{\\int _{{\\mathbb {R}}^N} d x\\, \\det \\!J_t(x)\\,e^{-S(z_t(x))}}\\nonumber \\\\&=\\frac{\\bigl \\langle e^{i \\theta _t(x)}\\mathcal {O}(z_t(x))\\bigr \\rangle _{S^{\\rm eff}_t}}{\\bigl \\langle e^{i \\theta _t(x)}\\bigr \\rangle _{S^{\\rm eff}_t}}.$ Here, $S^{\\rm eff}_t(x)$ and $\\theta _t(x)$ are defined by $e^{-S^{\\rm eff}_t(x)} &\\equiv e^{-{\\rm Re}\\,S(z_t(x))}\\, |\\det J_t(x)|,\\\\e^{i \\theta _t(x)} &\\equiv e^{-i\\,{\\rm Im}\\,S(z_t(x))}\\,e^{i\\, {\\rm arg}\\, \\det J_t(x)},$ and $J_t(x)$ obeys the following differential equation [11] (see also footnote 2 of [14]): $\\dot{J}_t &= [ H(z_t(x))\\cdot J_t]^\\ast ,\\quad J_{t=0} = 1$ with $H(z)\\equiv (\\partial _i \\partial _j S(z))$ .",
"Under the flow (REF ), the action changes as $(d/dt) S(z_t(x))=\\bigl |\\partial _i S(z_t(x))\\bigr |^2\\ge 0$ , and thus ${\\rm Re}\\,S(z_t(x))$ increases except at the critical points $z_\\ast $ ($\\partial _i S(z_\\ast )=0$ ), while ${\\rm Im}\\,S(z_t(x))$ is kept constant.",
"In particular, in the limit $t\\rightarrow \\infty $ , the deformed region will approach a union of $N$ -dimensional submanifolds (Lefschetz thimbles) on each of which ${\\rm Im}\\,S(z)$ is constant, and thus the sign problem is expected to disappear there (except for a possible residual sign problem arising from the phase of the complex measure $dz$ and a possible global sign problem caused by phase cancellations among different thimbles).",
"However, in the Monte Carlo calculation one cannot take the $t\\rightarrow \\infty $ limit naïvely, because the potential barriers between different thimbles become infinitely high so that the whole configuration space cannot be explored sufficiently.",
"This multimodality of distribution makes the Monte Carlo calculation impractical, especially when contributions from more than one thimble are relevant to estimating expectation values.",
"A key proposal in [12] is to use a finite value of flow time that is large enough to avoid the sign problem but simultaneously is not too large so that the exploration in the configuration space is still possible.",
"However, it is a difficult task to find such value of flow time in a systematic way, as we will discuss at the end of section and in Appendix .",
"The TLTM [14] is a tempering algorithm that uses the flow time as a tempering parameter.",
"There, the global relaxation of the multimodal distribution is prompted by enabling configurations around different modes to easily communicate through transitions in ensembles at smaller flow times.",
"Among other possible tempering algorithms, the parallel tempering algorithm [25], [26] (also known as the replica exchange MCMC method; see [27] for a review) is adopted in the TLTM [14] because it does not need to introduce the probability weight factors of ensembles at various flow times and because most of relevant steps can be done in parallel processes.",
"In the TLTM (see Appendix for the summary of the algorithm), we first fix the maximum flow time $T$ which should be sufficiently large such that the sign problem is reduced there.",
"A possible criterion is that the sign average $|\\langle e^{i \\theta _{T}(x)} \\rangle _{S^{\\rm eff}_T}|$ is $O(1)$ in the absence of tempering.",
"This process can be carried out by a test run with small statistics.",
"We then enlarge the configuration space from ${\\mathbb {R}}^N=\\lbrace x\\rbrace $ to the set of $A+1$ replicas, $({\\mathbb {R}}^N)^{A+1}=\\lbrace (x_0,x_1,\\ldots ,x_A)\\rbrace $ .",
"We assign to replicas $a$ ($a=0,1,\\ldots ,A$ ) the flow times $t_a$ with $t_0= 0 < t_1 < \\cdots < t_A=T$ .",
"The action at replica $a$ , $S^{\\rm eff}_{t_a}(x_a)$ , is obtained by solving (REF ) and (REF ) with its own initial conditions $z_{t=0}^i=x_a^i$ , $J_{t=0}=1$ .",
"We set up an irreducible, aperiodic Markov chain for the enlarged configuration space such that the probability distribution for $\\lbrace (x_0,x_1,\\ldots ,x_A)\\rbrace $ eventually approaches the equilibrium distribution proportional to $\\prod _a \\exp [-S^{\\rm eff}_{t_a}(x_a)].$ This can be realized by combining (a) the Metropolis algorithm (or the Hybrid Monte Carlo algorithm) in the $x$ direction at each fixed flow time and (b) the swap of configurations at two adjacent replicas.",
"Each of the steps (a) and (b) can be done in parallel processes.",
"After the system is well relaxed to global equilibrium, we estimate the expectation value at flow time $t_a$ [see (REF )] by using the subsample at replica $a$ , $\\lbrace x_a^{(k)}\\rbrace _{k=1,2,\\ldots ,N_{\\rm conf}}$ , that is retrieved from the total sample $\\lbrace (x_0^{(k)},x_1^{(k)},\\ldots ,x_A^{(k)})\\rbrace _{k=1,2,\\ldots ,N_{\\rm conf}}$ : $&\\frac{\\bigl \\langle e^{i \\theta _{t_a}(x)}\\mathcal {O}(z_{t_a}(x))\\bigr \\rangle _{S^{\\rm eff}_{t_a}}}{\\bigl \\langle e^{i \\theta _{t_a}(x)}\\bigr \\rangle _{S^{\\rm eff}_{t_a}}}\\nonumber \\\\&\\approx \\frac{\\sum _{k=1}^{N_{\\rm conf}}\\exp [i \\theta _{t_a}(x_a^{(k)})]\\,\\mathcal {O} (z_{t_a}(x_a^{(k)}))}{\\sum _{k=1}^{N_{\\rm conf}}\\exp [i \\theta _{t_a}(x_a^{(k)})]}\\equiv \\bar{\\mathcal {O}}_a.$ The original proposal in [14] is to use (REF ) at the maximum flow time, $\\bar{\\mathcal {O}}_{a=A}$ , as an estimate of $\\langle \\mathcal {O} \\rangle _S$ .",
"Recall here that the left-hand side of (REF ) is independent of $a$ due to Cauchy's theorem, and thus the ratio $\\bar{\\mathcal {O}}_a$ should yield the same value within the statistical error margin if the system is well in global equilibrium.",
"In practice, this is not true for small $a$ 's due to the sign problem, where the estimate of the sign average, $\\bigl |\\overline{e^{i\\theta _{t_a}}}\\bigr |\\equiv \\bigl |(1/N_{\\rm conf})\\sum _k e^{i \\theta _{t_a}(x^{(k)}_a)}\\bigr |$ , can be smaller than its statistical error ($\\simeq 1/\\sqrt{2N_{\\rm conf}}$ ; the value for the uniform distribution of phases).",
"In this case, the statistical error of the ratio $\\bar{\\mathcal {O}}_a$ cannot be trusted, which means that such $\\bar{\\mathcal {O}}_a$ should not be used as an estimate of $\\langle \\mathcal {O} \\rangle _S$ .",
"Based on the observation above, we now propose an algorithm which allows a precise estimation of $\\langle \\mathcal {O} \\rangle _S$ with a criterion ensuring global equilibrium and the sufficiency of the sample size.",
"First, we continue the sampling until we find some range of $a$ (to be denoted by $a=a_{\\rm min},\\ldots ,a_{\\rm max}(=A)$ ) in which $\\bigl |\\overline{e^{i\\theta _{t_a}}}\\bigr |$ are well above $1/\\sqrt{2N_{\\rm conf}}$ and $\\bar{\\mathcal {O}}_a$ take the same value within the statistical error margin.",
"We will require that the $1 \\sigma $ intervals around $\\bigl |\\overline{e^{i\\theta _{t_a}}}\\bigr |$ be above $3/\\sqrt{2N_{\\rm conf}}$ .",
"Then, we estimate $\\langle \\mathcal {O} \\rangle _S$ by using the $\\chi ^2$ fit for $\\lbrace \\bar{\\mathcal {O}}_a\\rbrace _{a=a_{\\rm min},\\ldots ,a_{\\rm max}}$ with a constant function of $a$ .",
"Global equilibrium and the sufficiency of the sample size are checked by looking at the optimized value of $\\chi ^2/{\\rm DOF}=\\chi ^2/(a_{\\max }-a_{\\min })$ .",
"Note that the parameters determined by this procedure (such as $N_{\\rm conf}$ , $a_{\\rm min}$ , $a_{\\rm max}=A$ ) can vary depending on the choice of observable $\\mathcal {O}$ .",
"We close this section with a few comments.",
"First, in the TLTM a sufficient overlap of the distributions at adjacent replicas is expected even for large flow times as long as the spacings are not too large.",
"This is because the distributions at large $a$ 's ($\\propto \\exp [-S^{\\rm eff}_{t_a}(x)]$ ) have peaks at the same points in ${\\mathbb {R}}^N$ that flow to critical points in ${\\mathbb {C}}^N$ .",
"This is in sharp contrast with the situation in other tempered systems, where the distribution often changes rapidly as a function of the tempering parameter so that an enough overlap cannot be achieved for realistically meaningful small spacings.",
"Second, the optimal form of $t_a$ is a linear function of $a$ when flowed configurations are close to a critical point.",
"This is because the optimal choice for the overall coefficients in tempering algorithms is exponential (see, e.g., [28], [29]) and because the real part of the action grows exponentially in flow time near critical points.",
"Finally, the computational cost in the TLTM is expected to be $O(N^{3-4})$ due to the increase caused by the tempering algorithm (which will be $O(N^{0-1})$ ).",
"Note that this growth of computational cost can be compensated by increasing the number of parallel processes."
],
[
"Application to the Hubbard model away from half-filling\n",
"Let $\\Lambda =\\lbrace {\\bf x} \\rbrace $ be a $d$ -dimensional lattice with $N_s$ lattice points.",
"The Hubbard model describes nonrelativistic lattice fermions of spin one-half, and is defined by the Hamiltonian (including the chemical potential) $H &= {} -\\kappa \\,\\sum _{{\\textbf {x}},{\\textbf {y}}}\\sum _\\sigma \\,K_{{\\textbf {x}}{\\textbf {y}}}\\,c_{{\\textbf {x}},\\sigma }^\\dagger c_{{\\textbf {y}},\\sigma }- \\mu \\,\\sum _{\\textbf {x}}\\,(n_{{\\textbf {x}},\\uparrow } + n_{{\\textbf {x}},\\downarrow } - 1)\\nonumber \\\\& ~~~~ + U \\sum _{\\textbf {x}}\\,(n_{{\\textbf {x}},\\uparrow }-1/2)\\,(n_{{\\textbf {x}},\\downarrow }-1/2).$ Here, $c_{{\\textbf {x}},\\sigma }$ and $c_{{\\textbf {x}},\\sigma }^\\dagger $ are the annihilation and creation operators on site ${\\textbf {x}}\\in \\Lambda $ with spin $\\sigma $ $(=\\uparrow ,\\downarrow )$ obeying the anticommutation relations $\\lbrace c_{{\\textbf {x}},\\sigma },c_{{\\textbf {y}},\\tau }^\\dagger \\rbrace = \\delta _{{\\textbf {x}}{\\textbf {y}}}\\,\\delta _{\\sigma \\tau }$ and $\\lbrace c_{{\\textbf {x}},\\sigma },c_{{\\textbf {y}},\\tau }\\rbrace = \\lbrace c_{{\\textbf {x}},\\sigma }^\\dagger ,c_{{\\textbf {y}},\\tau }^\\dagger \\rbrace = 0$ , and $n_{{\\textbf {x}},\\sigma }\\equiv c_{{\\textbf {x}},\\sigma }^\\dagger c_{{\\textbf {x}},\\sigma }$ .",
"$K_{{\\textbf {x}}{\\textbf {y}}}$ is the adjacency matrix that takes a nonvanishing value ($\\equiv 1$ ) only for nearest neighbors, and we assume the lattice to be bipartite.",
"$\\kappa \\,(>0)$ is the hopping parameter, $\\mu $ is the chemical potential, and $U\\,(>0)$ represents the strength of the on-site repulsive potential.",
"We have shifted $n_{{\\textbf {x}},\\sigma }$ as $n_{{\\textbf {x}},\\sigma }\\rightarrow n_{{\\textbf {x}},\\sigma }-1/2$ so that $\\mu =0$ corresponds to the half-filling state, $\\sum _\\sigma \\langle n_{{\\textbf {x}},\\sigma }-1/2 \\rangle = 0$ .",
"We approximate the grand partition function ${\\rm tr}\\,e^{-\\beta H}$ by using the Trotter decomposition with equal spacing $\\epsilon $ ($\\beta = N_\\tau \\epsilon $ ), and rewrite it as a path integral over a Gaussian Hubbard-Stratonovich variable $\\phi =(\\phi _{\\ell ,{\\textbf {x}}})$ .",
"Then the expectation value of the number density $n\\equiv (1/N_s)\\sum _{{\\textbf {x}}}(n_{{\\textbf {x}},\\uparrow }+n_{{\\textbf {x}},\\downarrow }-1)$ is expressed as (see Appendix for the derivation) $\\langle n \\rangle _S&\\equiv \\frac{\\int [d\\phi ]\\,e^{-S[\\phi ]}\\,n[\\phi ]}{\\int [d\\phi ]\\,e^{-S[\\phi ]}}\\quad \\Bigl ( [d\\phi ] \\equiv \\prod _{\\ell ,{\\textbf {x}}} d\\phi _{\\ell ,{\\textbf {x}}} \\Bigr ),\\\\e^{-S[\\phi ]} &\\equiv e^{-(1/2)\\,\\sum _{\\ell ,{\\textbf {x}}} \\phi _{\\ell ,{\\textbf {x}}}^2}\\,\\det M^a[\\phi ]\\,\\det M^b[\\phi ],\\\\M^{a/b}[\\phi ]&\\equiv 1 + e^{\\pm \\beta \\mu }\\,\\prod _\\ell e^{\\epsilon \\kappa K}\\,e^{\\pm \\,i\\sqrt{\\epsilon U} \\phi _\\ell },\\\\n[\\phi ] &\\equiv (i\\sqrt{\\epsilon U} N_s)^{-1}\\,\\sum _{\\textbf {x}}\\phi _{\\ell =0,{\\textbf {x}}},$ where $\\phi _\\ell \\equiv (\\phi _{\\ell ,{\\textbf {x}}}\\,\\delta _{{\\textbf {x}}{\\textbf {y}}})$ and $\\prod _\\ell $ is a product in descending order.",
"Note that $n[\\phi ]$ in () can be replaced by $(i\\sqrt{\\epsilon U} N_\\tau N_s)^{-1}\\,\\sum _{\\ell ,{\\textbf {x}}} \\phi _{\\ell ,{\\textbf {x}}}$ , which is more preferable in Monte Carlo calculations because statistical errors will be reduced due to the averaging over $\\ell $ .",
"The charge-charge correlation, $\\langle n_{\\textbf {x}}\\,n_{\\textbf {y}}\\rangle _S$ ($n_{\\textbf {x}}\\equiv n_{{\\textbf {x}},\\uparrow }+n_{{\\textbf {x}},\\downarrow }-1$ ), can also be evaluated as a path integral by simply replacing $n_{\\textbf {x}}$ by $(i\\sqrt{\\epsilon U})^{-1}\\,\\phi _{\\ell =0,{\\textbf {x}}}$ when ${\\textbf {x}}\\ne {\\textbf {y}}$ .",
"As for the observables that are not directly constructed from $n_{\\textbf {x}}$ , the expectation values can be evaluated by using the formula (REF ).",
"We now apply the TLTM to the Hubbard model on a two-dimensional periodic square lattice of size $2\\times 2$ (thus $N_s=4$ ) with $N_\\tau =5$ .",
"We first estimate $\\langle n \\rangle _S$ numerically by using the expressions (REF )–() for various values of $\\beta \\mu $ with other parameters fixed to be $\\beta \\kappa =3$ , $\\beta U=13$ .",
"Note that the physical quantities depend only on the dimensionless parameters $\\beta \\mu $ , $\\beta \\kappa $ , $\\beta U$ for fixed $N_\\tau $ .",
"The complex action () gives rise to a serious sign problem, as can be seen in the left panel of Fig.",
"REF .",
"Figure: (Left) the sign averages obtained by the reweighting method (flow time T=0T=0)for the complex action ()with the Gaussian Hubbard-Stratonovich variable.",
"(Right) the sign averages obtained by using ALFwith the M z M_z parametrization.However, we should note that the extent of the seriousness of the sign problem heavily depends on the choice of the Hubbard-Stratonovich variables, and actually, the sign problem can be avoided for the above parameters within the BSS (Blankenbecler, Scalapino and Sugar)-QMC method [30].",
"In fact, the right panel of Fig.",
"REF shows the sign averages calculated by using a public code called ALF (Algorithms for Lattice Fermions) [31] that is based on the discrete variables introduced in [32], [33].",
"We see that the sign averages are above 0.98 for all the range of $\\beta \\mu $ studied here.We thank a referee for suggesting us to investigate this point.",
"Following the general prescription and writing $x=(x^i)=(\\phi _{\\ell ,{\\textbf {x}}})$ $(i=1,\\ldots ,N)$ with $N=N_\\tau N_s$ , we introduce the enlarged configuration space $(\\mathbb {R}^N)^{A+1}=\\lbrace (x_0,x_1,\\ldots ,x_A)\\rbrace $ .",
"We here brief the setup of the parameters relevant to the TLTM (see Appendix for more details).",
"We set $t_a$ to be piecewise linear in $a$ with a single breakpoint whose position will be tuned such that the acceptance rates of the swapping process at adjacent replicas are almost the same for all pairs (being roughly above 40%).This functional form of $t_a$ is best suited to the case where the deformed region reaches the vicinity of all the relevant Lefschetz thimbles at almost the same flow time and such a linear form is effective also for the transient period.",
"For each value of $\\beta \\mu $ , we make a test run with small statistics to adjust parameters.",
"This gives the values $T/(\\beta \\mu )=1/12$ –$1/10$ , $A=8$ –12, $N_{\\rm conf}=5,000$ –$25,000$ , varying on the value of $\\beta \\mu $ .",
"We make a sampling after discarding 5,000 configurations, and from the obtained data $\\lbrace \\bar{n}_a\\rbrace _{a=a_{\\rm min},\\ldots ,a_{\\rm max}}$ we estimate $\\langle n \\rangle _S$ by using the $\\chi ^2$ fit.",
"As an example, let us see Fig.",
"REF , which shows $\\bigl |\\overline{e^{i\\theta _{t_a}}}\\bigr |$ and $\\bar{n}_a$ at various replicas for $\\beta \\mu =5$ .",
"Figure: With tempering (βμ=5\\beta \\mu =5).",
"(Left) the sign averagesat various replicas.The horizontal dashed line represents 3/2N conf =0.0173/\\sqrt{2N_{\\rm conf}}= 0.017.",
"(Right) the data n ¯ a \\bar{n}_a.The solid red line with a shaded band representsthe estimate of 〈n〉 S \\langle n \\rangle _S with 1σ1 \\sigma interval.The gray dashed line represents the exact value.The left panel shows that the $1\\sigma $ intervals around $\\bigl |\\overline{e^{i\\theta _{t_a}}}\\bigr |$ are above $3/\\sqrt{2 N_{\\rm conf}}$ for $a=5,\\ldots ,11$ (and thus we set $a_{\\rm min}=5$ and $a_{\\rm max}=11$ ).",
"The right panel shows that the data $\\lbrace \\bar{n}_a\\rbrace $ in this range give the same value within the statistical error margin.",
"The $\\chi ^2$ fit gives the estimate $\\langle n\\rangle _S \\approx 0.221 \\pm 0.012$ (exact value: 0.212) with $\\chi ^2/{\\rm DOF}=0.45$ .",
"Figure REF shows the thus-obtained numerical estimates of $\\langle n \\rangle _S$ as a function of $\\beta \\mu $ .",
"Figure: The expectation values of the number density operator,〈n〉 S \\langle n \\rangle _S (N τ =5)(N_\\tau =5).The results obtained with temperingcorrectly reproduce the exact values.The exact values for N τ =∞N_\\tau =\\infty are also displayed for comparison.We also display the estimates obtained without tempering (at the same maximum flow times $T$ ) and those from the original reweighting method (i.e.",
"$T=0$ ), together with the values obtained by the explicit evaluation of the trace under the Trotter decomposition with $N_\\tau =5$ and for the continuum imaginary time (i.e.",
"$N_\\tau =\\infty $ ) (see Appendix ).",
"We see that the exact values are correctly reproduced when the tempering is implemented, while there are significant deviations when not implemented.",
"As in the $(0+1)$ -dimensional massive Thirring model [14], the deviation reflects the fact that the relevant thimbles are not sampled sufficiently.",
"In fact, from Fig.",
"REF , which shows the distribution of averaged flowed configurations $\\hat{z}\\equiv (1/N)\\sum _i z_T^i$ at $T=0.5$ for $\\beta \\mu =5$ , Figure: The distribution of z ^\\hat{z}.",
"(Left) with tempering.",
"(Right) without tempering.we see that, although the flowed configurations are widely distributed over many thimbles when the tempering is implemented, they are restricted to only a small number of thimbles when not implemented.",
"Three comments are in order.",
"First, a larger value of the sign average does not necessarily mean a better resolution of the sign problem, as can be seen from Fig.",
"REF .",
"In fact, when only a very few thimbles are sampled, the sign average can become larger than the value in the correct sampling due to the absence of phase mixtures among different thimbles.",
"Figure: The sign averages at TT,|〈e iθ T (x) 〉 S T eff ||\\langle e^{i\\theta _T(x)}\\rangle _{S^{\\rm eff}_T}|Second, whether the multimodality can affect the estimates of expectation values depends on the choice of observables.",
"In fact, from the discrepancies of the sign averages in Fig.",
"REF , we see that the multimodality must be severe in the region $\\beta \\mu \\le 9$ .",
"However, the estimates of $\\langle n \\rangle _S$ almost agree between the two methods with and without tempering in the range $7\\le \\beta \\mu \\le 9$ .",
"This means that the operator $n$ is not sensitive to the multimodality in this range.",
"To find an observable that is sensitive to the multimodality, we estimated the nearest-neighbor charge-charge correlation $\\langle n_{\\textbf {x}}\\, n_{\\textbf {y}}\\rangle _S$ with the same sample.We thank the referee for suggesting us to investigate the expectation values of observables other than the number density operator.",
"The results are shown in Fig.",
"REF , where we see a significant discrepancy at $\\beta \\mu =9$ between the two methods.",
"Figure: The nearest-neighbor charge-charge correlations〈n 𝐱 n 𝐲 〉 S \\langle n_{\\textbf {x}}n_{\\textbf {y}}\\rangle _S (N τ =5N_\\tau =5)Such discrepancies become more manifest if we look at the observables that are not directly constructed from the number density operator $n_{\\textbf {x}}$ .",
"As an example, we show in Fig.",
"REF the expectation values of the kinetic energy operator (without the factor “$-\\kappa $ ”) $K\\equiv \\sum _{{\\textbf {x}},{\\textbf {y}}}\\sum _\\sigma K_{{\\textbf {x}}{\\textbf {y}}}\\, c^\\dag _{{\\textbf {x}},\\sigma }c_{{\\textbf {y}},\\sigma }$ , which are estimated for the same sample as above by using the formula (REF ).",
"Figure: The kinetic energies 〈K〉 S \\langle K \\rangle _S (N τ =5N_\\tau =5)We there notice two things.",
"One is that the discrepancies between the two methods now become significant for all the range $7\\le \\beta \\mu \\le 9$ .",
"The other is that the precision of the TLTM becomes worse compared to the case for the observables that are constructed solely from $n_{\\textbf {x}}$ .",
"In fact, those observables that are not directly constructed from $n_{\\textbf {x}}$ (such as $K$ ) contain matrix elements of $M^{a/b}[\\phi ]^{-1}$ , and may have divergently large values in the vicinity of zeros of the fermion determinants $\\det M^{a/b}[\\phi ]$ .",
"In this case, precise estimation will require a larger sample size and a more accuracy in integrating flow equations compared with operators constructed solely from $n_{\\textbf {x}}$ .",
"We expect that a similar attention must be paid when one applies the TLTM to finite density QCD.",
"We leave a further investigation of this point as a future investigation.",
"Finally, from Fig.",
"REF , we see that it should be a difficult task to find an intermediate flow time (without tempering) that avoids both the sign problem (severe at smaller flow times) and the multimodal problem (severe at larger flow times) (see Appendix for more detailed discussions).",
"Figure: Without tempering (βμ=5\\beta \\mu =5).",
"(Left) the sign averages.",
"(Right) the estimates n ¯ a \\bar{n}_a.There is no such flow time that clearly avoidsboth the sign and multimodal problems simultaneously(at least for the present spacings).Generically, flowed configurations repeatedly experience the event at which they get trapped to fewer number of Lefschetz thimbles, so that there is a large ambiguity in distinguishing the larger and smaller flow times in the first place."
],
[
"Conclusion and outlook\n",
"In this paper, we proposed an algorithm for the TLTM which allows a precise estimation of expectation values.",
"We confirm the effectiveness by applying it to the two-dimensional Hubbard model away from half-filling.",
"We should stress that our study in this paper is still at an exploratory level.",
"In fact, the lattice must be enlarged much more both in the spatial and imaginary time directions to claim the validity of our method for the sign problem in the Hubbard model, revealing the phase structure of the model.",
"In doing this, it should be important to check whether the computational scaling is actually $O(N^{3-4})$ as expected.",
"More generally, we should keep developing the algorithm further so that it can be more easily applied to the three major problems listed in Introduction.",
"There should also be other interesting branches of fields where the TLTM may shed new light on the theoretical understanding through a numerical analysis, such as the Chern-Simons theory [34] and matrix models that generate random volumes [35].",
"When we were preparing the first version of the manuscript, there appeared an interesting paper [23] (see also its detailed version [24]), where the sign and ergodicity problems are also studied for the Lefschetz thimble method applied to the Hubbard model away from half-filling.",
"In our method (TLTM), the two problems are solved simultaneously by tempering the system with the flow time, where one does not need to know a detailed structure of thimbles.",
"In contrast, in [23], [24] they redundantly introduce two continuous Gaussian Hubbard-Stratonovich variables with a parameter representing the mixture of the two variables (see also [22]).",
"With knowledge of thimble structures, they tune the parameter in such a way that only a few number of thimbles become relevant to the evaluation, and obtain results for a $2\\times 2$ hexagonal lattice $(N_s=8)$ with $N_\\tau = 384$ and $\\beta =30$ .",
"It would be interesting to introduce such redundant integration variables also in the TLTM so as to reduce the global sign problem (possible cancellation of phases among different thimbles), which we observe also depends heavily on the choice of integration variables.",
"The authors thank Yoshimasa Hidaka, Issaku Kanamori, Norio Kawakami, Yoshio Kikukawa, Jun Nishimura, Akira Ohnishi, Masaki Tezuka, Asato Tsuchiya and Urs Wenger for useful discussions.",
"They also thank an anonymous referee of Physical Review D for giving us valuable comments, which were very helpful in improving the first version of the manuscript.",
"This work was partially supported by JSPS KAKENHI (Grant Numbers 16K05321, 18J22698 and 17J08709) and by SPIRITS 2019 of Kyoto University (PI: M.F.",
")."
],
[
"Summary of the algorithm\n",
"We summarize the algorithm of the TLTM (we partially repeat the presentation of [14]): $\\bullet $ Step 0.",
"We fix the maximum flow time $T$ which should be sufficiently large such that the sign problem is reduced there.",
"A possible criterion is that the sign average $|\\langle e^{i \\theta _{T}(x)} \\rangle _{S^{\\rm eff}_T}|$ is $O(1)$ in the absence of tempering.",
"This can be carried out by a test run with small statistics.",
"We then pick up flow times $\\lbrace t_a\\rbrace $ from the interval $[0,T]$ with $t_0=0 < t_1 < \\cdots < t_A = T$ .",
"The values of $A$ and $t_a$ are determined manually or adaptively to optimize the acceptance rate in Step 3 below.",
"Practically, once $A$ is determined, $t_a$ can be chosen to be a piecewise linear function of $a$ [see the argument for (REF )].",
"$\\bullet $ Step 1.",
"For each replica $a$ , we choose an initial value $x_a \\in {\\mathbb {R}}^N$ and numerically solve the differential equations (REF ) and (REF ) to obtain the triplet $(x_a,z_a\\equiv z_{t_a}(x_a),J_a\\equiv J_{t_a}(x_a))$ .",
"$\\bullet $ Step 2.",
"For each replica $a$ , we use the Metropolis algorithm to update the value of $x_a$ .",
"To be explicit, we take a value $x^{\\prime }_a$ from $x_a$ using a symmetric proposal distribution, and recalculate the triplet $(x^{\\prime }_a,z^{\\prime }_a,J^{\\prime }_a)$ using the $x^{\\prime }_a$ as the initial value.",
"We then update $x_a$ to $x^{\\prime }_a$ with the probability ${\\rm min}(1,e^{-\\Delta S_a})$ , where $\\Delta S_a&\\equiv S^{\\rm eff}_{t_a}(x^{\\prime }_a)-S^{\\rm eff}_{t_a}(x_a)\\nonumber \\\\&=({\\rm Re}\\,S(z^{\\prime }_a)-\\ln \\,\\bigl |\\det \\!J^{\\prime }_a\\bigr |)-({\\rm Re}\\,S(z_a)-\\ln \\,\\bigl |\\det \\!J_a\\bigr |).$ We repeat the process sufficiently many times such that local equilibrium is realized for each $a$ .",
"Step 1 and Step 2 can be performed in parallel processes.",
"$\\bullet $ Step 3.",
"We swap the configurations at two adjacent replicas $a$ and $a+1$ by updating $(x_a,x_{a+1})=(x,y)$ to $(x^{\\prime }_a,x^{\\prime }_{a+1})=(y,x)$ with the probability $w_a(x,y)={\\rm min}\\Bigl (1,\\,e^{-S^{\\rm eff}_{t_a}(y)-S^{\\rm eff}_{t_{a+1}}(x)+S^{\\rm eff}_{t_a}(x)+S^{\\rm eff}_{t_{a+1}}(y)}\\Bigr ).$ One can easily see that this satisfies the detailed balance condition with respect to the global equilibrium distribution (REF ) because $w_a(x,y)\\,e^{-S^{\\rm eff}_{t_a}(x)-S^{\\rm eff}_{t_{a+1}}(y)}= w_a(y,x)\\,e^{-S^{\\rm eff}_{t_a}(y)-S^{\\rm eff}_{t_{a+1}}(x)}.$ We repeat the process several times so as to reduce autocorrelations.",
"This procedure can also be performed in parallel processes by choosing a set of independent pairs.",
"$\\bullet $ Step 4.",
"By repeating Step 2 and Step 3, we obtain a sequence of triplets, $\\lbrace (x_a^{(k)},z_a^{(k)},J_a^{(k)})\\rbrace _{k=1,2,\\ldots ,N_{\\rm conf}},$ for each $a$ , with which we estimate the expectation value at flow time $t_a$ : $\\frac{\\bigl \\langle e^{i \\theta _{t_a}(x)}\\mathcal {O}(z_{t_a}(x))\\bigr \\rangle _{S^{\\rm eff}_{t_a}}}{\\bigl \\langle e^{i \\theta _{t_a}(x)}\\bigr \\rangle _{S^{\\rm eff}_{t_a}}}&\\approx \\frac{\\sum _{k=1}^{N_{\\rm conf}}e^{i \\theta ^{(k)}_a}\\mathcal {O}\\bigl (z_a^{(k)}\\bigr )}{\\sum _{k=1}^{N_{\\rm conf}}e^{i \\theta ^{(k)}_a}}\\equiv \\bar{\\mathcal {O}}_a\\nonumber \\\\&\\qquad [ \\theta _a^{(k)} \\equiv \\theta _{t_a}(x_a^{(k)}) ].$ Here, $N_{\\rm conf}$ is chosen to be large enough so that we find some range of $a$ (to be denoted by $a=a_{\\rm min},\\ldots ,a_{\\rm max}$ with $a_{\\rm max}=A$ ) in which the $1 \\sigma $ intervals around $\\bigl |\\overline{e^{i\\theta _{t_a}}}\\bigr |=\\bigl |(1/N_{\\rm conf})\\sum _k e^{i \\theta _{t_a}(x^{(k)}_a)}\\bigr |$ are above $3/\\sqrt{2N_{\\rm conf}}$ and $\\bar{\\mathcal {O}}_a$ take the same value within the statistical error margin.",
"$\\bullet $ Step 5.",
"The expectation value of $\\langle \\mathcal {O} \\rangle _S$ is estimated by the $\\chi ^2$ fit from the data $\\lbrace \\bar{\\mathcal {O}}_a\\rbrace _{a=a_{\\rm min},\\ldots ,a_{\\rm max}}$ with a constant function of $a$ .",
"Global equilibrium and the sufficiency of the sample size $N_{\\rm conf}$ is checked by looking at the optimized value of $\\chi ^2/{\\rm DOF}=\\chi ^2/(a_{\\max }-a_{\\min })$ .",
"In the above algorithm, we have implicitly assumed that the action at $t_0=0$ does not exhibit multimodality.",
"If this is not the case, we further introduce other parameters (such as the overall coefficient of the action) as extra tempering parameters or prepare flow times $\\lbrace t_a\\rbrace $ with $t_0 < 0$ [14]."
],
[
"Derivation of eqs. (",
"For a bipartite lattice, we specify which sublattice ${\\textbf {x}}$ belongs to by the sign $(-1)^{\\textbf {x}}=\\pm 1$ .",
"We first make the so-called particle-hole transformation, $c_{{\\textbf {x}},\\uparrow }=a_{\\textbf {x}}$ and $c_{{\\textbf {x}},\\downarrow }=(-1)^{\\textbf {x}}\\,b_{\\textbf {x}}^\\dagger $ .",
"Then the one-body part $H_1$ and the two-body part $H_2$ of the Hamiltonian (REF ) are rewritten, respectively, as $H_1&={} -\\sum _{{\\textbf {x}},{\\textbf {y}}}\\, (\\kappa K + \\mu \\, 1)_{{\\textbf {x}}{\\textbf {y}}}\\,a_{\\textbf {x}}^\\dagger a_{\\textbf {y}}-\\sum _{{\\textbf {x}},{\\textbf {y}}}\\, (\\kappa K - \\mu \\, 1)_{{\\textbf {x}}{\\textbf {y}}}\\,b_{\\textbf {x}}^\\dagger b_{\\textbf {y}},\\\\H_2 &={} -U\\,\\sum _{\\textbf {x}}(n^a_{\\textbf {x}}-1/2)(n^b_{\\textbf {x}}-1/2)\\nonumber \\\\&= (U/2)\\,\\sum _{\\textbf {x}}\\,(n^a_{\\textbf {x}}-n^b_{\\textbf {x}})^2 - N_s U/4.$ In the last equation, we have used the identity $n^a_{\\textbf {x}}\\,(\\equiv a_{\\textbf {x}}^\\dagger a_{\\textbf {x}}) = (n^a_{\\textbf {x}})^2$ and $n^b_{\\textbf {x}}\\,(\\equiv b_{\\textbf {x}}^\\dagger b_{\\textbf {x}}) = (n^b_{\\textbf {x}})^2$ .",
"Note that the number density operator is written as $n \\equiv (1/N_s)\\,\\sum _{\\textbf {x}}\\,(n_{{\\textbf {x}},\\uparrow }+n_{{\\textbf {x}},\\downarrow }-1)= (1/N_s)\\,\\sum _{\\textbf {x}}\\,(n^a_{\\textbf {x}}- n^b_{\\textbf {x}}).$ In order to perform a Monte Carlo simulation, we approximate $e^{-\\beta H}$ in the grand partition function by using the Trotter decomposition with equal spacing $\\epsilon $ ($\\beta =N_\\tau \\epsilon $ ): $e^{-\\beta H} = (e^{-\\epsilon (H_1+H_2)})^{N_\\tau }\\simeq (e^{-\\epsilon H_1}\\,e^{-\\epsilon H_2})^{N_\\tau },$ and rewrite $e^{-\\epsilon H_2}$ at the $\\ell $ -th position from the right to the exponential of a fermion bilinear by using a Gaussian Hubbard-Stratonovich variable $\\phi _{\\ell ,{\\textbf {x}}}$ : $e^{-\\epsilon H_2}&= e^{N_s \\epsilon U/4}\\,e^{-(\\epsilon U/2)\\,\\sum _{\\textbf {x}}(n^a_{\\textbf {x}}-n^b_{\\textbf {x}})^2}\\nonumber \\\\&= e^{N_s \\epsilon U/4}\\,\\prod _{\\textbf {x}}\\int \\frac{d\\phi _{\\ell ,{\\textbf {x}}}}{\\sqrt{2\\pi }}\\,e^{-(1/2)\\,\\phi _{\\ell ,{\\textbf {x}}}^2+ \\,i\\sqrt{\\epsilon U}\\,\\phi _{\\ell ,{\\textbf {x}}} (n^a_{\\textbf {x}}-n^b_{\\textbf {x}}) }.$ Then, the approximated grand partition function takes the following path integral form: $Z_{\\rm QMC}&\\equiv {\\rm tr}\\,\\bigl [(e^{-\\epsilon H_1}\\,e^{-\\epsilon H_2})^{N_\\tau }\\bigr ]\\nonumber \\\\&= (e^{\\epsilon U/4}/\\sqrt{2\\pi })^{N_\\tau N_s}\\int [d\\phi ]\\,e^{-S[\\phi ]}.$ Here, $[d\\phi ] \\equiv \\prod _{\\ell ,{\\textbf {x}}} d\\phi _{\\ell ,{\\textbf {x}}}$ , and the action $S[\\phi ]$ is given by $e^{-S[\\phi ]}&= e^{-\\sum _{\\ell ,{\\textbf {x}}} (1/2)\\,\\phi _{\\ell ,{\\textbf {x}}}^2}\\,{\\rm tr}_a\\,\\prod _\\ell \\,e^{\\epsilon \\sum _{{\\textbf {x}},{\\textbf {y}}}(\\kappa K + \\mu \\, 1)_{{\\textbf {x}}{\\textbf {y}}}\\,a_{\\textbf {x}}^\\dagger a_{\\textbf {y}}}\\,e^{\\sum _{\\textbf {x}}\\,(i\\sqrt{\\epsilon U}\\,\\phi _{\\ell ,{\\textbf {x}}})\\,a_{\\textbf {x}}^\\dagger a_{\\textbf {x}}}\\nonumber \\\\&~~~~~\\times {\\rm tr}_b\\,\\prod _\\ell \\,e^{\\epsilon \\sum _{{\\textbf {x}},{\\textbf {y}}}(\\kappa K - \\mu \\, 1)_{{\\textbf {x}}{\\textbf {y}}}\\,b_{\\textbf {x}}^\\dagger b_{\\textbf {y}}}\\,e^{\\sum _{\\textbf {x}}\\,(-i\\sqrt{\\epsilon U}\\,\\phi _{\\ell ,{\\textbf {x}}})\\,b_{\\textbf {x}}^\\dagger b_{\\textbf {x}}},$ where $\\prod _\\ell $ is an ordered product ($\\prod _\\ell f_\\ell \\equiv f_{N_\\tau -1}\\cdots f_1 f_0$ ), and ${\\rm tr}_a$ (or ${\\rm tr}_b$ ) represents the trace over the Fock space created by $a_{\\textbf {x}}^\\dagger $ (or by $b_{\\textbf {x}}^\\dagger $ ).",
"The fermion trace in (REF ) can be evaluated explicitly by using the following formulas that hold for the operator $\\hat{A}\\equiv \\sum _{{\\textbf {x}},{\\textbf {y}}}\\,A_{{\\textbf {x}}{\\textbf {y}}}\\,a_{\\textbf {x}}^\\dagger a_{\\textbf {y}}$ constructed from an $N_s\\times N_s$ matrix $A=(A_{{\\textbf {x}}{\\textbf {y}}})$ : $&e^A\\,e^B = e^C ~~\\Rightarrow ~~ e^{\\hat{A}}\\,e^{\\hat{B}} = e^{\\hat{C}},\\\\&{\\rm tr}\\,e^{\\hat{A}} = \\det \\, (1+e^A).$ (The first equation can be readily proved by the fact that $A\\mapsto \\hat{A}$ is a Lie algebra homomorphism.",
"The second equation can be easily understood by moving to a diagonalizing basis for $A$ .)",
"We thus find that the action becomes $e^{-S[\\phi ]} &= e^{-(1/2)\\,\\sum _{\\ell ,{\\textbf {x}}} \\phi _{\\ell ,{\\textbf {x}}}^2}\\,\\det M^a[\\phi ]\\,\\det M^b[\\phi ],\\\\M^{a/b}[\\phi ]&= 1 + e^{\\pm \\beta \\mu }\\,\\prod _\\ell e^{\\epsilon \\kappa K}e^{\\pm \\,i\\sqrt{\\epsilon U} \\phi _\\ell },$ where $\\phi _\\ell $ is a diagonal matrix of the form $\\phi _\\ell =(\\phi _{\\ell ,{\\textbf {x}}}\\,\\delta _{{\\textbf {x}}{\\textbf {y}}})$ .",
"Note that, while the action is real-valued for the half-filling case $(\\mu =0)$ due to the identity $M^b[\\phi ]|_{\\mu =0}=(M^a[\\phi ]|_{\\mu =0})^\\ast $ , it is generically complex-valued when $\\mu \\ne 0$ .",
"The expectation values of such observables that are made solely from the number density operators $n_{\\textbf {x}}\\equiv n_{{\\textbf {x}},\\uparrow }+n_{{\\textbf {x}},\\downarrow }-1=n^a_{\\textbf {x}}-n^b_{\\textbf {x}}$ can be evaluated as a path integral over $\\phi $ by simply replacing $n_{\\textbf {x}}$ by $(i\\sqrt{\\epsilon U})^{-1}\\,\\phi _{\\ell =0,{\\textbf {x}}}$ , as easily proved by using the operator identity $&\\int d\\phi \\, e^{-(1/2)\\,\\phi ^2 + \\,i\\sqrt{\\epsilon U}\\,\\phi \\,(n^a_{\\textbf {x}}-n^b_{\\textbf {x}})}\\,(n^a_{\\textbf {x}}-n^b_{\\textbf {x}})\\nonumber \\\\&= \\int d\\phi \\, e^{-(1/2)\\,\\phi ^2+ \\,i\\sqrt{\\epsilon U}\\,\\phi \\,(n^a_{\\textbf {x}}-n^b_{\\textbf {x}})}\\,\\phi /(i\\sqrt{\\epsilon U}).$ For example, the expectation value of the number density operator, $\\langle n\\rangle _S$ , can be rewritten to a path integral form as in (REF ).",
"As for observables of general form, one can resort to the Wick-Bloch-de Dominicis theorem, $&{\\rm tr}\\,\\bigl [ e^{\\hat{A}}\\,a_{{\\textbf {x}}_m}\\cdots a_{{\\textbf {x}}_1}\\,a^\\dag _{{\\textbf {x}}^{\\prime }_1} \\cdots a^\\dag _{{\\textbf {x}}^{\\prime }_{m^{\\prime }}} \\bigr ]\\nonumber \\\\&= \\delta _{m m^{\\prime }}\\,\\det (1+e^A)\\,\\left|\\begin{array}{ccc}(1+e^A)^{-1}_{{\\textbf {x}}_1 {\\textbf {x}}^{\\prime }_1} & \\cdots & (1+e^A)^{-1}_{{\\textbf {x}}_1 {\\textbf {x}}^{\\prime }_m} \\\\\\vdots & \\ddots & \\vdots \\\\(1+e^A)^{-1}_{{\\textbf {x}}_m {\\textbf {x}}^{\\prime }_1} & \\cdots & (1+e^A)^{-1}_{{\\textbf {x}}_m {\\textbf {x}}^{\\prime }_m}\\end{array} \\right|,$ to obtain the following expression: $&\\frac{{\\rm tr}\\,\\bigl [(e^{-\\epsilon H_1}\\,e^{-\\epsilon H_2})^{N_\\tau }\\,a_{{\\textbf {x}}_m}\\cdots a_{{\\textbf {x}}_1}\\,a^\\dag _{{\\textbf {x}}^{\\prime }_1} \\cdots a^\\dag _{{\\textbf {x}}^{\\prime }_{m^{\\prime }}}\\,b_{{\\textbf {y}}_n}\\cdots b_{{\\textbf {y}}_1}\\,b^\\dag _{{\\textbf {y}}^{\\prime }_1} \\cdots b^\\dag _{{\\textbf {y}}^{\\prime }_{n^{\\prime }}}\\bigr ]}{{\\rm tr}\\,\\bigl [ (e^{-\\epsilon H_1}\\,e^{-\\epsilon H_2})^{N_\\tau } \\bigr ]}\\nonumber \\\\&= \\frac{\\delta _{m m^{\\prime }}\\,\\delta _{n n^{\\prime }}}{Z}\\,\\int [d\\phi ]\\,e^{-S[\\phi ]}\\,\\left|\\begin{array}{ccc}\\Delta ^a_{{\\textbf {x}}_1 {\\textbf {x}}^{\\prime }_1} & \\cdots & \\Delta ^a_{{\\textbf {x}}_1 {\\textbf {x}}^{\\prime }_m} \\\\\\vdots & \\ddots & \\vdots \\\\\\Delta ^a_{{\\textbf {x}}_m {\\textbf {x}}^{\\prime }_1} & \\cdots & \\Delta ^a_{{\\textbf {x}}_m {\\textbf {x}}^{\\prime }_m}\\end{array} \\right| \\cdot \\left|\\begin{array}{ccc}\\Delta ^b_{{\\textbf {y}}_1 {\\textbf {y}}^{\\prime }_1} & \\cdots & \\Delta ^b_{{\\textbf {y}}_1 {\\textbf {y}}^{\\prime }_n} \\\\\\vdots & \\ddots & \\vdots \\\\\\Delta ^b_{{\\textbf {y}}_n {\\textbf {y}}^{\\prime }_1} & \\cdots & \\Delta ^b_{{\\textbf {y}}_n {\\textbf {y}}^{\\prime }_n}\\end{array} \\right|\\quad \\Bigl ( Z = \\int [d\\phi ]\\,e^{-S[\\phi ]} \\Bigr ),$ where $\\Delta ^{a/b}[\\phi ]=M^{a/b}[\\phi ]^{-1}$ ."
],
[
"Evaluation of the trace under the Trotter decomposition\n",
"The Hilbert space $\\mathbb {V}$ of the Hubbard model after the particle-hole transformation is the tensor product of two Fock spaces, $\\mathbb {V} = \\mathbb {V}^a \\otimes \\mathbb {V}^b$ , each constructed by acting $a_{\\textbf {x}}^\\dag $ or $b_{\\textbf {x}}^\\dag $ on the Fock vacuum $|0\\rangle $ .",
"In this appendix, we give the explicit forms of the matrix elements that appear in the trace under the Trotter decomposition: $\\langle n \\rangle _S= \\frac{{\\rm tr}\\,\\bigl [(e^{-\\epsilon H_1}\\,e^{-\\epsilon H_2})^{N_\\tau }\\,n \\bigr ]}{{\\rm tr}\\,\\bigl [(e^{-\\epsilon H_1}\\,e^{-\\epsilon H_2})^{N_\\tau } \\bigr ]}= \\frac{{\\rm tr}\\,\\bigl [(T_1 T_2)^{N_\\tau }\\,n \\bigr ]}{{\\rm tr}\\,\\bigl [(T_1 T_2)^{N_\\tau } \\bigr ]}.$ Here, the one-body part $H_1$ and the two-body part $H_2$ of the Hamiltonian are given by [see (REF ) and ()] $H_1 &= H^a_1 \\otimes 1 + 1 \\otimes H^b_1,\\\\H^{a}_1 &= \\sum _{{\\textbf {x}},{\\textbf {y}}}\\,h^a_{{\\textbf {x}}{\\textbf {y}}}\\,a_{\\textbf {x}}^\\dag a_{\\textbf {y}}\\equiv {}-\\sum _{{\\textbf {x}},{\\textbf {y}}}\\, (\\kappa K + \\mu \\, 1)_{{\\textbf {x}}{\\textbf {y}}}\\,a_{\\textbf {x}}^\\dagger a_{\\textbf {y}},\\\\H^{b}_1 &= \\sum _{{\\textbf {x}},{\\textbf {y}}}\\,h^b_{{\\textbf {x}}{\\textbf {y}}}\\,b_{\\textbf {x}}^\\dag b_{\\textbf {y}}\\equiv {}-\\sum _{{\\textbf {x}},{\\textbf {y}}}\\, (\\kappa K - \\mu \\, 1)_{{\\textbf {x}}{\\textbf {y}}}\\,b_{\\textbf {x}}^\\dagger b_{\\textbf {y}},\\\\H_2 &={} -U\\,\\sum _{\\textbf {x}}(n^a_{\\textbf {x}}-1/2)\\otimes (n^b_{\\textbf {x}}-1/2).$ The number density operator is given by $n &= \\frac{1}{N_s}\\,\\sum _{\\textbf {x}}(n^a_{\\textbf {x}}\\otimes 1 - 1 \\otimes n^b_{\\textbf {x}}),$ and we have introduced the transfer matrices corresponding to $H_1$ and $H_2$ : $T_1 &\\equiv e^{-\\epsilon H_1} = e^{-\\epsilon H^a_1}\\otimes e^{-\\epsilon H^b_1}\\equiv T^a_1\\otimes T^b_1,\\nonumber \\\\T_2 &\\equiv e^{-\\epsilon H_2}.$ We first introduce a one-dimensional ordering to the set of all spatial coordinates, $\\Lambda =\\lbrace {\\textbf {x}}\\rbrace $ , and take a basis of $\\mathbb {V}$ to be $\\lbrace | X \\rangle \\otimes | Y \\rangle \\rbrace ,$ where the states $|X\\rangle &\\equiv a_{{\\textbf {x}}_1}^\\dag a_{{\\textbf {x}}_2}^\\dag \\cdots a_{{\\textbf {x}}_m}^\\dag |0\\rangle \\in \\mathbb {V}^a,\\\\|Y\\rangle &\\equiv b_{{\\textbf {y}}_1}^\\dag b_{{\\textbf {y}}_2}^\\dag \\cdots b_{{\\textbf {y}}_n}^\\dag |0\\rangle \\in \\mathbb {V}^b,$ are labeled by the subsets of ordered coordinates, $X=\\lbrace {\\textbf {x}}_1,{\\textbf {x}}_2,\\ldots ,{\\textbf {x}}_m\\rbrace \\subset \\Lambda $ (with ${\\textbf {x}}_1<{\\textbf {x}}_2<\\cdots <{\\textbf {x}}_m$ ), $Y=\\lbrace {\\textbf {y}}_1,{\\textbf {y}}_2,\\ldots ,{\\textbf {y}}_n\\rbrace \\subset \\Lambda $ (with ${\\textbf {y}}_1<{\\textbf {y}}_2<\\cdots <{\\textbf {y}}_n$ ).",
"We will denote their sizes by $|X|=m$ , $|Y|=n$ .",
"The matrix elements of $T^a_1 = e^{-\\epsilon H^a_1}$ are then given by the following determinants: $&(T^a_1)_{X X^{\\prime }} = \\left|\\begin{array}{ccc}(e^{-\\epsilon h^a})_{{\\textbf {x}}_1 {\\textbf {x}}_1^{\\prime }} & \\cdots & (e^{-\\epsilon h^a})_{{\\textbf {x}}_1 {\\textbf {x}}_m^{\\prime }} \\\\\\vdots & \\ddots & \\vdots \\\\(e^{-\\epsilon h^a})_{{\\textbf {x}}_m {\\textbf {x}}_1^{\\prime }} & \\cdots & (e^{-\\epsilon h^a})_{{\\textbf {x}}_m {\\textbf {x}}_m^{\\prime }}\\end{array} \\right| \\,\\delta _{|X|,|X^{\\prime }|}\\nonumber \\\\&= e^{\\epsilon \\mu |X|}\\,\\left|\\begin{array}{ccc}(e^{\\epsilon \\kappa K})_{{\\textbf {x}}_1 {\\textbf {x}}_1^{\\prime }} & \\cdots & (e^{\\epsilon \\kappa K})_{{\\textbf {x}}_1 {\\textbf {x}}_m^{\\prime }} \\\\\\vdots & \\ddots & \\vdots \\\\(e^{\\epsilon \\kappa K})_{{\\textbf {x}}_m {\\textbf {x}}_1^{\\prime }} & \\cdots & (e^{\\epsilon \\kappa K})_{{\\textbf {x}}_m {\\textbf {x}}_m^{\\prime }}\\end{array} \\right| \\,\\delta _{|X|,|X^{\\prime }|}\\nonumber \\\\& \\qquad (m \\equiv |X|=|X^{\\prime }|),$ as can be easily proven by investigating the action of $T^a_1$ on the state $|X^{\\prime }\\rangle $ : $T^a_1\\,|X^{\\prime }\\rangle &= e^{-\\epsilon \\sum _{{\\textbf {x}}{\\textbf {y}}} h^a_{{\\textbf {x}}{\\textbf {y}}} a_{\\textbf {x}}^\\dag a_{\\textbf {y}}}\\,a_{{\\textbf {x}}_1^{\\prime }}^\\dag \\cdots a_{{\\textbf {x}}_m^{\\prime }}^\\dag |0\\rangle \\nonumber \\\\&\\equiv \\sum _X |X\\rangle \\, (T^a_1)_{X X^{\\prime }},$ where the coefficients do not vanish only when $|X|=|X^{\\prime }|\\,(=m)$ .",
"The matrix elements of $T^b_1 = e^{-\\epsilon H^b_1}$ can also be given in the forms of determinant, $&(T^b_1)_{Y Y^{\\prime }}\\nonumber \\\\&= e^{-\\epsilon \\mu |Y|}\\,\\left|\\begin{array}{ccc}(e^{\\epsilon \\kappa K})_{{\\textbf {y}}_1 {\\textbf {y}}_1^{\\prime }} & \\cdots & (e^{\\epsilon \\kappa K})_{{\\textbf {y}}_1 {\\textbf {y}}_n^{\\prime }} \\\\\\vdots & \\ddots & \\vdots \\\\(e^{\\epsilon \\kappa K})_{{\\textbf {y}}_n {\\textbf {y}}_1^{\\prime }} & \\cdots & (e^{\\epsilon \\kappa K})_{{\\textbf {y}}_n {\\textbf {y}}_n^{\\prime }}\\end{array} \\right| \\,\\delta _{|Y|,|Y^{\\prime }|}\\nonumber \\\\& \\qquad (n \\equiv |Y|=|Y^{\\prime }|).$ We thus obtain the explicit forms of the matrix elements $(T_1)_{XY,X^{\\prime }Y^{\\prime }} = (T^a_1)_{X X^{\\prime }}\\,(T^b_1)_{Y Y^{\\prime }}$ .",
"As for $T_2 = e^{-\\epsilon H_2}$ , we note that $H_2$ acts on $|X\\rangle \\otimes |Y\\rangle $ diagonally: $&H_2\\,|X\\rangle \\otimes |Y\\rangle \\nonumber \\\\&={}-U\\,\\sum _{\\textbf {z}}\\, \\Bigl (a_{\\textbf {z}}^\\dag a_{\\textbf {z}}-\\frac{1}{2}\\Bigr ) |X\\rangle \\otimes \\Bigl (b_{\\textbf {z}}^\\dag b_{\\textbf {z}}-\\frac{1}{2}\\Bigr ) |Y\\rangle \\nonumber \\\\&\\equiv (h_2)_{X Y}\\,|X\\rangle \\otimes |Y\\rangle .$ The coefficients $(h_2)_{X Y}$ can be calculated easily to be $&(h_2)_{X Y}\\nonumber \\\\&= {}-\\frac{U}{4}\\,\\sum _{\\textbf {z}}\\,\\bigl [ \\theta ({\\textbf {z}}\\in X)\\,\\theta ({\\textbf {z}}\\in Y)- \\theta ({\\textbf {z}}\\in X)\\,\\theta ({\\textbf {z}}\\notin Y)\\nonumber \\\\&~~~~{} - \\theta ({\\textbf {z}}\\notin X)\\,\\theta ({\\textbf {z}}\\in Y)+ \\theta ({\\textbf {z}}\\notin X)\\,\\theta ({\\textbf {z}}\\notin Y) \\bigr ]\\nonumber \\\\&={} -\\frac{U}{4}\\,( 2|X\\cap Y| + 2|\\bar{X}\\cap \\bar{Y}| - N_s ),$ where $\\theta ()$ is the logical step function and $\\bar{X}$ stands for the complement of the set $X$ , $\\bar{X}=\\Lambda \\setminus X$ .",
"The matrix elements of $T_2$ is then given by $(T_2)_{XY,X^{\\prime }Y^{\\prime }} = e^{-\\epsilon (h_2)_{X Y}}\\,\\delta _{X X^{\\prime }}\\,\\delta _{Y Y^{\\prime }}$ .",
"Finally, the matrix elements of $n$ are given by $n_{XY,X^{\\prime }Y^{\\prime }} = \\frac{1}{N_s}\\,(|X|-|Y|)\\,\\delta _{X X^{\\prime }}\\,\\delta _{Y Y^{\\prime }}.$ With the matrix elements given above, $\\langle n \\rangle _S$ can be expressed as $\\langle n \\rangle _S = \\frac{1}{N_s}\\,\\frac{\\sum _{X,Y\\subset \\Lambda } \\bigl [(T_1\\,T_2)^{N_\\tau }\\bigr ]_{XY,XY}\\,(|X|-|Y|)}{\\sum _{X,Y\\subset \\Lambda } \\bigl [(T_1\\,T_2)^{N_\\tau }\\bigr ]_{XY,XY}}.$"
],
[
"Summary of the parameters in the computation\n",
"We summarize the parameters relevant to the TLTM in the estimation of $\\langle n \\rangle _S$ .",
"We order the termination times $t_a$ for replicas $a$ as $t_0=0 < t_1 < \\cdots < t_A=T$ ($T$ : the largest flow time), and set $t_a$ to be a piecewise linear function of $a$ with a single breakpoint at $a=a_c$ , by assuming that the deformed region reaches the vicinity of all the relevant Lefschetz thimbles at almost the same flow time and that the linear form is effective also for the transient period: $t_a =\\left\\lbrace \\begin{array}{cl}t_c\\,a/a_c & (0 \\le a \\le a_c)\\\\t_c + (T-t_c)\\,(a-a_c)/(A-a_c) & (a_c < a \\le A)\\end{array}\\right..$ Each Monte Carlo step consists of 50 Metropolis tests in the $x$ direction and $N_{\\rm swap}$ swaps of configurations at adjacent replicas, and the flow equations (REF ) and (REF ) are integrated numerically by using the adaptive Runge-Kutta of 7-8th order.",
"For each value of $\\beta \\mu $ , we make a test run with small statistics and adjust the parameters $A$ , $t_c$ , $a_c$ in such a way that the acceptance rates of the swapping process at adjacent replicas are almost the same for all pairs (being roughly above 40%).",
"After this, we make another test run of 1,000 data points to adjust the width of the Gaussian proposal in the Metropolis test in the $x$ direction so that the acceptance rate is in the range 50%–80%.",
"This width varies depending on replicas $a$ and the values of $\\beta \\mu $ .",
"Using the adjusted parameters, we get a sample of size $N_{\\rm conf}$ after discarding 5,000 configurations, and analyze the data by using the Jackknife method, with bins whose sizes are adjusted by taking account of autocorrelations.",
"Finally, from the obtained data $\\lbrace \\bar{n}_a\\rbrace $ $(a=a_{\\rm min},\\ldots ,a_{\\rm max}(=A))$ [see (REF )], we estimate the expectation value $\\langle n \\rangle _S$ by using the $\\chi ^2$ fit with a constant function of $a$ .",
"We confirm that the system is in global equilibrium and the sample size is sufficient by looking at the optimized value of $\\chi ^2/{\\rm DOF}$ with ${\\rm DOF}=a_{\\rm max}-a_{\\rm min}$ .",
"The obtained results are summarized in Table REF .",
"Table: TLTM parameters and the results"
],
[
"More on the fine-tuning of flow time without tempering\n",
"In order to understand the difficulty to find such an intermediate value of flow time that avoids both the sign and multimodal problems (without tempering), let us see the right panel of Fig.",
"REF , which is the counterpart of Fig.",
"REF (with tempering) for the same $\\beta \\mu =5$ .",
"We see that the estimated values have large statistical errors at smaller flow times (due to the sign problem) while they have small statistical errors around incorrect values at larger flow times (due to the trapping of configurations at a small number of thimbles).",
"The best flow time must be at the boundary between the two regions, but it should be a difficult task to find such value out of the set of flow times with finite spacings.",
"In fact, if one takes a flow time from the smaller region, then, although the obtained estimate may happen to be close to the correct value, it must have a large statistical error.",
"On the other hand, if a flow time is taken from the larger region, it will give an incorrect value (but with a small statistical error because only a small number of thimbles are sampled).",
"In order to understand Fig.",
"REF and Fig.",
"REF as reflecting the extent of the sign and multimodal problems, let us see Fig.",
"REF , which depicts the normalized histograms of phases $\\theta _{t_a}(x)$ for $\\beta \\mu =5$ with tempering (top) and without tempering (bottom).",
"Figure: Normalized histograms of θ t a (x)/π\\theta _{t_a}(x)/\\pi for βμ=5\\beta \\mu =5.",
"(Top) with tempering.",
"(Bottom) without tempering.We see that at smaller flow times the histograms are almost flat for the both cases (giving rise to the sign problem), but at larger flow times those without tempering become almost unimodal (reflecting the trapping at a small number of thimbles) while those with tempering correctly come to have various peaks (which may not be so obvious from the figure because there are many peaks and each peak is broadened by the Jacobian determinant)."
]
] | 1906.04243 |
[
[
"Temporal and spectral X-ray properties of magnetar SGR 1900+14 derived\n from observations with NuSTAR and XMM-Newton"
],
[
"Abstract X-ray observations play a crucial role in understanding the emission mechanism and relevant physical phenomena of magnetars.",
"We report X-ray observations of a young magnetar SGR 1900+14 made in 2016, which is famous for a giant flare in 1998 August.",
"Simultaneous observations were conducted with XMM-Newton and NuSTAR on 2016 October 20 with 23 and 123 ks exposures, respectively.",
"The NuSTAR hard X-ray coverage enabled us to detect the source up to 70 keV.",
"The 1-10 keV and 15-60 keV fluxes were $3.11(3)\\times10^{-12}\\;{\\rm erg\\;s^{-1}\\;cm^{-2}}$ and $6.8(3)\\times10^{-12}\\;{\\rm erg\\;s^{-1}\\;cm^{-2}}$, respectively.",
"The 1-70 keV spectra were well fitted by a blackbody plus power-law model with a surface temperature of $kT=0.52(2)\\;{\\rm keV}$, a photon index of the hard power-law of $\\Gamma=1.21(6)$, and a column density of $N_{\\rm H}=1.96(11)\\times10^{22}\\;{\\rm cm^{-2}}$.",
"Compared with previous observations with Suzaku in 2006 and 2009, the 1-10 keV flux showed a decrease by 25-40%, while the spectral shape did not show any significant change with differences of $kT$ and $N_{\\rm H}$ being within 10% of each other.",
"Through timing analysis, we found that the rotation period of SGR 1900+14 on 2016 October 20 was $5.22669(3)\\;{\\rm s}$.",
"The long-term evolution of the rotation period shows a monotonic decrease in the spin-down rate $\\dot{P}$ lasting for more than 15 years.",
"We also found a characteristic behavior of the hard-tail power-law component of SGR 1900+14.",
"The energy-dependent pulse profiles vary in morphology with a boundary of 10 keV.",
"The phase-resolved spectra show the differences between photon indices ($\\Gamma=1.02$-$1.44$) as a function of the pulse phase.",
"Furthermore, the photon index is positively correlated with the X-ray flux of the hard power-law component, which could not be resolved by the previous hard X-ray observations."
],
[
"Introduction",
"Soft gamma-ray repeaters (SGRs) and anomalous X-ray pulsars (AXPs) have recently been considered to form a class of young neutron stars with extremely strong magnetic fields [35], [28], which we call magnetars (for a recent review, see [27], [47]).",
"These objects exhibit rather slow rotation periods of $P=2$ –$12\\;{\\rm s}$ and large spin-down rates of $\\dot{P}=10^{-15}$ –$10^{-10}\\;{\\rm s\\;s^{-1}}$ ([36]; McGill Online Magnetar Cataloghttp://www.physics.mcgill.ca/~pulsar/magnetar/main.html), which lead to huge dipole magnetic fields of $B_{\\rm d}=10^{14}$ –$10^{15}\\;{\\rm G}$ , exceeding the quantum critical magnetic field $B_{\\rm QED}=m_{\\rm e}^{2}c^{3}/(e\\hbar )=4.4\\times 10^{13}\\;{\\rm G}$ [23], where $m_{\\rm e}$ , $c$ , $e$ , and $\\hbar $ denote the electron mass, speed of light, elementary charge, and Planck's constant, respectively.",
"The radiation of magnetars is mainly emitted in the X-ray frequency and typically gives a luminosity of $10^{34}$ –$10^{35}\\;{\\rm erg\\;s^{-1}}$ [36], which is much higher than the typical spin-down luminosity of magnetars of $10^{32}$ –$10^{34}\\;{\\rm erg\\;s^{-1}}$ .",
"The small rotation power compared to magnetar luminosity and the absence of evidence of accretion suggest that the magnetar is powered by liberating a part of its huge magnetic energy, but its mechanism is still unknown.",
"X-ray observation of magnetars is a crucial step in explaining how they convert magnetic energy into radiation as well as what physical phenomena take place in their extremely strong magnetic fields (especially above $B_{\\rm QED}$ ).",
"It is widely known that magnetars typically show unstable fluctuations of spin-down rates $\\dot{P}$ (e.g., CXOU J171405.7-381031: [22]; Swift J1822.3-1606: [44]), while normal radio pulsars have constant $\\dot{P}$ (e.g., the Crab pulsar: [41]).",
"The common mechanism of spin-down fluctuations of magnetars is still unclear.",
"However, recent studies have suggested a common trend of the $\\dot{P}$ fluctuations after experiencing outburst activities, which are also characteristic of magnetars.",
"They seemed to show such unstable fluctuations of $\\dot{P}$ followed by monotonic decreases (e.g., 1E 1048.1-5937: [4]).",
"A twisted magnetic field in the magnetosphere was proposed to explain the monotonic decreases in $\\dot{P}$ by the decay of the twist [43], [8].",
"This seems to be a valuable common property of magnetars, but more samples are required to confirm it.",
"Magnetars are also known for their hard-tail power-law components, which are dominant above $\\sim 10\\;{\\rm keV}$ with hard photon indices of $\\Gamma \\sim 1$ , coexisting with soft blackbody components with temperatures of $kT\\sim 0.5\\;{\\rm keV}$ .",
"Although the origin of the power-law component is still unknown, systematic studies on magnetar hard-tails have been conducted.",
"They showed a trend that a magnetar with a younger characteristic age and stronger magnetic field displays a softer hard-tail [15], [14].",
"Photon splitting under extremely strong magnetic fields is a possible mechanism of the radiation [6], [7], [15]; however, this has not been confirmed owing to a lack of hard-tail observations.",
"We require more magnetar samples with continuous broad-band X-ray monitoring to reveal these temporal and spectral properties.",
"Compared to other magnetars, SGR 1900+14 has a rather young characteristic age of $\\tau _{\\rm c}\\sim 0.9\\;{\\rm kyr}$ and a strong dipole magnetic field of $B_{\\rm d}\\sim 7\\times 10^{14}\\;{\\rm G}$ [36].",
"It experienced a giant flare in 1998 August [25], emitting a peak luminosity of $\\gtrsim 10^{44}\\;{\\rm erg\\;s^{-1}}$ [32], [17], which is much higher than ordinary outbursts of other magnetars [12].",
"Although it has been continuously observed for more than 20 years, the long-term variability of its rotation period after the giant flare has been poorly investigated.",
"Its hard-tail power-law component was detected by INTEGRAL [21], BeppoSAX [16] and Suzaku HXD [15], [14], but they were not able to precisely determine its spectral and temporal properties because of large uncertainties.",
"SGR 1900+14 could be a valuable resource for studying common properties of magnetars if it is observed for a long period of time and with sufficient exposures.",
"In this paper, we present analysis results of the simultaneous observations of SGR 1900+14 with XMM-Newton and NuSTAR.",
"The observation was conducted after sufficient time had passed for tracking the long-term evolution since the last observation in 2009.",
"Making full use of the wide-band coverage of XMM-Newton and NuSTAR, we performed a detailed analysis, particularly on its hard-tail, for the first time.",
"We also compare our results with previous ones and discuss the long-term evolution of SGR 1900+14.",
"The remainder of this paper is organized as follows.",
"In Section 2, we describe our observations and data reductions.",
"Section 3 is devoted to the results of our observations.",
"Then, we discuss our results in Section 4 and present our conclusions in Section 5."
],
[
"Observation and data reduction",
"We observed SGR 1900+14 simultaneously with XMM-Newton and NuSTAR on 2016 October 20.",
"Table REF shows details of the observations.",
"Analyses were conducted using XSPEC 12.10.0 and Xronos 5.22.",
"Table: NO_CAPTION"
],
[
"XMM-Newton",
"SGR 1900+14 was observed for $23\\;{\\rm ks}$ with XMM-Newton, which is an X-ray telescope sensitive to 0.1–15 keV [26]; it carries three X-ray detectors: MOS1, MOS2, and pn [46], [39].",
"Throughout the observation, the MOSs were set in the large window mode, while pn was set in the full frame mode (time resolutions of $0.9\\;{\\rm s}$ and $73.4\\;{\\rm ms}$ , respectively).",
"All data were processed using XMM-Newton Science Analysis System (SAS) version 17.0.0, following the “Users Guide to the XMM-Newton Science Analysis System”https://xmm-tools.cosmos.esa.int/external/xmm_user_support/documentation/sas_usg/USG/.",
"We omitted high background intervals by setting thresholds of 0.35 counts ${\\rm s^{-1}}$ ($>$ 10 keV, single pixel events only) for the MOSs and 0.40 counts ${\\rm s^{-1}}$ (10–12 keV, single pixel events only) for pn.",
"As a result, the net exposure times became $21.4\\;{\\rm ks}$ for the MOSs and $11.3\\;{\\rm ks}$ for pn, as shown in Table REF .",
"We selected a circular source extraction region with a radius of 40 centered on SGR 1900+14.",
"For the MOSs, the background region was an annulus centered on the object with an inner radius of 40 and an outer radius of 144, while for pn, it was a circle with a radius of 108 at the source-free region on the same segment.",
"We used rmfgen and arfgen in SAS to obtain the redistribution matrix files and ancillary response files, respectively.",
"The spectra were re-binned by grppha to have at least 50 counts in each bin.",
"Barycentric corrected light curves were also generated using barycen in SAS."
],
[
"NuSTAR",
"SGR 1900+14 was observed for an elapsed time of $242\\;{\\rm ks}$ with NuSTAR, which is the first focusing hard X-ray telescope covering 3–78 keV [24].",
"All data were processed using nupipeline and nuproducts in HEASoft 6.23, following the “NuSTAR Data Analysis Software Guide”https://heasarc.gsfc.nasa.gov/docs/nustar/analysis/nustar_swguide.pdf.",
"The net exposure time became 122.6 ks after the standard pipeline processes, as shown in Table REF .",
"Figure REF shows 3–78 keV images obtained from both detectors of NuSTAR, namely, FPMA and FPMB.",
"The source region is the same as that in the XMM-Newton analysis.",
"As shown in the upper left regions of both images, bright signals were caused by stray light from the nearby object GRS 1915+105.",
"Due to heavy contamination by the stray light in the source region, we decided not to use the FPMB data in this work.",
"Other than GRS 1915+105, there are two fainter sources of stray light called IGR J19140+0951 and 4U 1908+075.",
"Referring to the stray light simulation by the NuSTAR help deskhttps://heasarc.gsfc.nasa.gov/cgi-bin/Feedback, which showed background fluctuations, we selected the background region as shown in the left panel of Figure REF to avoid stray light contamination and to have the source and background region positioned in the same stray light area.",
"We also checked the spectra generated with other background sets and found that there were no significant changes in the analysis.",
"Setting the source and background region, we extracted spectrum and light curves.",
"The spectrum was binned at minimum counts of 50 bin$^{-1}$ by grppha, as described in Section REF .",
"The barycentric corrected light curve was also generated using barycorr by adopting the following coordinates for the source: RA=286.7891, DEC=9.3079."
],
[
"Spectral analysis",
"Figures REF and REF show the XMM-Newton and NuSTAR spectra of SGR 1900+14.",
"These spectra are featureless, and the NuSTAR spectrum extends up to 70 keV.",
"We conducted spectral fittings assuming a typical magnetar spectrum, a BB plus power-law (PL) [34], [15].",
"Photoelectric absorption was also taken into account.",
"In XSPEC, we employed a model phabs*(bbody+pegpwrlw) to perform chi-squared fittings.",
"We employed the phabs model with solar metallicity abundance angr [2] and photoelectric absorption cross-section vern [48].",
"We also tried another cross-section model, bcmc [5], and confirmed no significant changes in our results.",
"The free parameters of the spectral fitting consist of hydrogen column density $N_{\\rm H}$ , BB surface temperature $kT$ , BB normalization factor expressed in terms of luminosity, PL photon index $\\Gamma $ , and PL normalization factor in terms of the 2–10 keV unabsorbed flux.",
"When conducting the spectral fitting, we omitted data below $1\\;{\\rm keV}$ due to poor statistics.",
"We also discarded data above $8\\;{\\rm keV}$ for the MOSs due to poor statistics.",
"As a result, the energy ranges for the fitting were 1–8 keV and 1–10 keV for the MOSs and pn, respectively.",
"The fitting returned a good reduced chi-squared $\\chi _{\\nu }^{2}$ (d.o.f.)",
"of 1.08 (256) without large residuals.",
"The best-fit model is presented in the left panel of Figure REF and the first row of Table REF .",
"When all the five parameters are set free, the obtained $N_{\\rm H}=(2.6\\pm 0.3)\\times 10^{22}\\;{\\rm cm^{-2}}$ is significantly different from those obtained in studies by Suzaku, which were $(1.8\\pm 0.3)\\times 10^{22}\\;{\\rm cm^{-2}}$ and $(1.9\\pm 0.1)\\times 10^{22}\\;{\\rm cm^{-2}}$ [14].",
"This could be due to the absence of data above 10 keV, which leads to a failure in determining the photon index and thus $N_{\\rm H}$ .",
"We checked the correlation contour of $N_{\\rm H}$ and $kT$ , and confirmed that the two parameters are significantly coupled.",
"We thus conducted another spectral fitting with a fixed $N_{\\rm H}$ of $1.9\\times 10^{22}\\;{\\rm cm^{-2}}$ in accordance with the previous studies by Suzaku [14].",
"This fitting yielded a similar acceptable $\\chi _{\\nu }^{2}$ (d.o.f.)",
"of 1.15 (257) without large residuals.",
"This best-fit model is shown in the right panel of Figure REF and the second row of Table REF .",
"Figure: XMM-Newton spectra fitted with blackbody (BB) + power-law (PL) model.",
"N H N_{\\rm H} is set free in the left panel, while it is fixed to 1.9×10 22 cm -2 1.9\\times 10^{22}\\;{\\rm cm^{-2}} in the right panel.",
"Crosses are background-subtracted data and error bars represent 1σ1\\sigma confidence level.",
"Each cross is binned with a minimum of 50 counts bin -1 ^{-1}.",
"Magenta, black, and red crosses and lines represent results for MOS1, MOS2, and pn, respectively.",
"Dotted, dashed, and solid lines are BB component, PL component, and the aggregation of the two, respectively."
],
[
"NuSTAR",
"We detected X-ray emissions from SGR 1900+14 in the energy range of 3–70 keV.",
"The signal significance was $6.5\\sigma $ in the range of 60–70 keV, while the Suzaku observation in 2006 detected the source only up to 50 keV [15].",
"Our observation realizes the first detection of SGR 1900+14 above 50 keV after the detections by INTEGRAL in 2003 and 2004 [21].",
"In the fitting to NuSTAR data, $N_{\\rm H}$ was fixed to $1.9\\times 10^{22}\\;{\\rm cm^{-2}}$ , which was reported in the previous Suzaku studies [14], because the photoelectric absorption does not have significant influence on the spectrum above 3 keV.",
"The fitting applied to 3–70 keV yielded a good $\\chi _{\\nu }^{2}$ (d.o.f.)",
"of 1.17 (118) without any distinctive structure in the residuals.",
"Figure REF and the third row of Table REF describe the best-fit model.",
"The obtained parameters are roughly consistent with those yielded with XMM-Newton (section REF ), but $\\Gamma $ was determined more precisely due to the hard X-ray coverage of NuSTAR.",
"Figure: NuSTAR FPMA spectrum fitted with blackbody (BB) + power-law (PL) model.",
"Crosses are background-subtracted data and error bars represent 1σ1\\sigma confidence level.",
"Each cross is binned with a minimum of 50 counts bin -1 ^{-1}.",
"Dotted, dashed, and solid lines are BB component, PL component, and the aggregation of the two, respectively."
],
[
"Joint fitting",
"We fitted the spectra of both XMM-Newton and NuSTAR simultaneously.",
"We used the same energy ranges employed for each detector in the separate analyses (see Sections REF and REF ).",
"Although inter-calibration uncertainties between different instruments exist (e.g., see [45]), the inter-calibration uncertainties between XMM-Newton and NuSTAR is up to 10% [30], and we fixed the cross normalization to 1 because it does not affect our results.",
"The parameter $N_{\\rm H}$ was set free in this fitting.",
"The fitting yielded an acceptable $\\chi _{\\nu }^{2}$ (d.o.f.)",
"of 1.18 (378).",
"Although the residuals may show a distinctive structure, this does not affect the results significantly.",
"The best-fit model is presented in Figure REF and the fourth row of Table REF .",
"The absorbed 1–70 keV flux was $(1.21\\pm 0.04)\\times 10^{-11}\\;{\\rm erg\\;s^{-1}\\;cm^{-2}}$ , where the error denotes 1$\\sigma $ confidence level.",
"Owing to the wide-band spectral fitting, parameters $kT$ and $\\Gamma $ were both successfully determined precisely.",
"Figure: XMM-Newton + NuSTAR spectra fitted with blackbody (BB) + power-law (PL) model.",
"Crosses are background-subtracted data and error bars represent 1σ1\\sigma confidence level.",
"Each cross is binned with a minimum of 50 counts bin -1 ^{-1}.",
"Magenta, black, red, and blue crosses and lines represent results for MOS1, MOS2, pn, and FPMA, respectively.",
"Dotted, dashed, and solid lines are BB component, PL component, and the aggregation of the two, respectively.Table: NO_CAPTION"
],
[
"Time variability",
"Figure REF shows the light curves obtained from the observations.",
"Chi-squared tests against being constant were conducted for each light curve, where the bin time was set to $1000\\;{\\rm s}$ for XMM-Newton and $5000\\;{\\rm s}$ for NuSTAR.",
"For MOS1, MOS2, pn, and FPMA, the $\\chi ^{2}_{\\nu }$ values are $1.22\\;({\\rm d.o.f.",
"}=21)$ , $0.99\\;({\\rm d.o.f.",
"}=21)$ , $0.50\\;(\\rm {d.o.f.",
"}=15)$ , and $1.02\\;({\\rm d.o.f.",
"}=48)$ , respectively, indicating no significant time variability in the XMM-Newton and NuSTAR data at the timescales of the binning times.",
"We also tried a smaller bin of $1\\;{\\rm s}$ and $0.1\\;{\\rm s}$ to the NuSTAR data and $1\\;{\\rm s}$ to the aggregation of the XMM-Newton data, and confirmed that there were no short bursts during the observation.",
"Figure: Background-subtracted light curves of MOS1, MOS2, pn, and FPMA.",
"The energy ranges employed are 1–10 keV for the MOSs and pn and 3–78 keV for FPMA.",
"Each light curve is binned at 5000s bin -1 5000\\;{\\rm s\\;bin^{-1}}.",
"Errors indicate 1σ1\\sigma confidence level."
],
[
"Coherent pulsation",
"Figure REF presents the power spectra of SGR 1900+14 generated by applying Fourier transforms to the light curves with Xronos.",
"All the four power spectra show peaks at similar frequencies of $0.1913\\;{\\rm Hz}$ to $0.1914\\;{\\rm Hz}$ .",
"In addition, we performed an epoch-folding search on the NuSTAR data.",
"As a result, we found a coherent pulsation of SGR 1900+14 at a period of $5.22669\\pm 0.00003\\;{\\rm s}$ on 2016 October 20 to 23.",
"Note that we were not able to detect any change of the pulsation period within this observation and conducted the epoch-folding search under the assumption that $\\dot{P}=0$ .",
"We also confirmed that the results do not change significantly when adopting an appropriate value of $\\dot{P}=10^{-10}\\;{\\rm s\\;s^{-1}}$ , which is roughly equal to one, as derived by a previous XMM-Newton study with the same source [34].",
"Figure: Power spectra of MOS1, MOS2, pn, and FPMA.",
"The energy ranges employed are 1–10 keV for the MOSs and pn, and 3–78 keV for FPMA.",
"Binning time is 1.0s1.0\\;{\\rm s} for MOS1 and MOS2 and 0.5s0.5\\;{\\rm s} for pn and FPMA.",
"Errors indicate 1σ1\\sigma confidence level."
],
[
"Pulse profiles",
"Here, we define the strength of pulsations by two different methods with reference to [40].",
"First, we define the rms pulse fraction as ${\\rm PF_{rms}}=\\frac{2\\sqrt{\\sum _{k=1}^{k_{\\rm max}}\\left(\\left(a_{k}^{2}+b_{k}^{2}\\right)-\\left(\\sigma _{a_{k}}^{2}+\\sigma _{b_{k}}^{2}\\right)\\right)}}{a_{0}},$ where $a_{k}$ and $b_{k}$ are Fourier coefficients expressed by $a_{k}&=&\\frac{1}{N}\\sum _{j=1}^{N}p_{j}\\cos \\left(\\frac{2\\pi kj}{N}\\right) \\\\b_{k}&=&\\frac{1}{N}\\sum _{j=1}^{N}p_{j}\\sin \\left(\\frac{2\\pi kj}{N}\\right),$ and $\\sigma _{a_{k}}^{2}$ and $\\sigma _{b_{k}}^{2}$ are the uncertainties in $a_{k}$ and $b_{k}$ , respectively, expressed by $\\sigma _{a_{k}}^{2}&=&\\frac{1}{N^{2}}\\sum _{j=1}^{N}\\sigma _{p_{j}}^{2}\\cos ^{2}\\left(\\frac{2\\pi kj}{N}\\right) \\\\\\sigma _{b_{k}}^{2}&=&\\frac{1}{N^{2}}\\sum _{j=1}^{N}\\sigma _{p_{j}}^{2}\\sin ^{2}\\left(\\frac{2\\pi kj}{N}\\right).",
"$ $N$ , $p_{j}$ , and $\\sigma _{p_{j}}$ denote the number of phase bins, number of photons in each phase bin, and the Poisson variance in each phase bin, respectively.",
"Note that the definition of ${\\rm PF_{rms}}$ is modified from [40] so that ${\\rm PF_{rms}}=B/A$ when the input signal is $A+B\\sin \\phi $ (see Appendix 1 of [1] for more details).",
"In this work, we set $k_{\\rm max}=5$ .",
"We also employ the area pulse fraction ${\\rm PF_{area}}$ described by ${\\rm PF_{area}}=\\frac{\\sum _{j=1}^{N}p_{j}-N*{\\rm min}(p_{j})}{\\sum _{j=1}^{N}p_{j}}, $ which is defined in [19].",
"Figure REF shows the pulse profiles of all four detectors, which are folded by the best-fit pulsation period $5.22669\\;{\\rm s}$ .",
"All four profiles show their maxima at a phase of 0.6–0.8 and minima at a phase of 0.1–0.3.",
"Although we did not consider the change of pulsation period $\\dot{P}$ , we have checked that the results do not change significantly when it is considered.",
"We calculated the pulse fractions of all four profiles by employing the above equations ((REF )–() and (REF )).",
"The ${\\rm PF_{rms}}$ (${\\rm PF_{area}}$ ) values measured by MOS1, MOS2, pn, and FPMA are $18.6\\pm 2.6\\%\\;(20.9\\pm 4.9\\%)$ , $19.0\\pm 2.7\\%\\;(22.3\\pm 4.7\\%)$ , $15.6\\pm 2.0\\%\\;(18.7\\pm 4.0\\%)$ , and $17.0\\pm 2.5\\%\\;(19.0\\pm 5.0\\%)$ , respectively, with the error denoting $1\\sigma $ confidence.",
"We made energy-selected pulse profiles and conducted chi-squared tests against being constant.",
"We found signals of periodic fluctuations below 20 keV with a confidence level of $99\\%$ , while we were not able to detect any pulse above 20 keV, presumably owing to poor statistics.",
"The left panel of Figure REF shows NuSTAR pulse profiles of 3–5, 5–10, and 10–20 keV.",
"This suggests that the pulse shape differs below and above 10 keV.",
"To quantify this feature, we applied Fourier transforms to the pulse profiles, as shown in the right panel of Figure REF .",
"The $i$ -th harmonic power relative to the total power is calculated as $\\frac{P_{i}}{P_{\\rm total}}=\\frac{a_{i}^{2}+b_{i}^{2}}{\\sum _{k=1}^{k_{\\rm max}}(a_{k}^{2}+b_{k}^{2})}.$ While the 3–5 keV and 5–10 keV profiles display almost sinusoidal profiles with the fundamental frequency dominantly contributing to the whole powers, the 10–20 keV profile shows contributions from higher harmonics.",
"This result clearly indicates a change of the pulse profile at $\\sim 10\\;{\\rm keV}$ .",
"Because the power-law (PL) component is prominent above 10 keV (see Figure REF ), the appearance of the higher harmonics can be interpreted as a characteristic behavior of the PL component, while the blackbody component seems to have the sinusoidal profile.",
"Figure: Pulse profiles of MOS1, MOS2, pn, and FPMA folded by the rotation period 5.22669s5.22669\\;{\\rm s}.",
"Background is subtracted.",
"The energy ranges employed are 1–10 keV for the MOSs and pn and 3–78 keV for FPMA.",
"The horizontal axis represents two cycles of pulsation.",
"Errors indicate 1σ1\\sigma confidence level.Figure: Pulse profiles of FPMA 3–5, 5–10, and 10–20 keV (left).",
"Fourier transforms of each pulse profile (right).",
"Background is subtracted.",
"Green, black, and red represent 3–5, 5–10, and 10–20 keV, respectively.",
"Errors indicate 1σ1\\sigma confidence level.To explore the counterpart of the pulse profile difference among energy bands (confirmed in Section REF ), we extracted phase-resolved spectra and conducted spectral fittings to investigate the differences in the spectrum among different pulse phases.",
"Only pn and FPMA data were used, but we have confirmed that there is no significant change in results when we include MOS data in the analysis.",
"The definition of the phase is the same as that employed in the analysis of the pulse profiles (Section REF ).",
"We first divided the whole spectrum into two pieces, namely, phases 0.0–0.5 and 0.5–1.0.",
"In addition, we divided the whole spectrum into five pieces.",
"Throughout the entire analysis, the parameter $N_{\\rm H}$ was fixed, and either $kT$ or $\\Gamma $ was also fixed, adopting the value obtained in Section REF .",
"The best-fit parameters of each fitting are presented in Table REF .",
"All 14 fittings show acceptable values of $\\chi ^{2}/({\\rm d.o.f})$ .",
"When we fix $kT$ , we clearly see that the photon index $\\Gamma $ varies among phases beyond their error ranges.",
"Similarly, when we fix $\\Gamma $ , we see that the surface temperature $kT$ varies among phases beyond their error ranges.",
"These properties are presented in Figure REF and REF .",
"These results show that the spectrum changes with the pulse phase, which corresponds to the differences in pulse profiles among energies.",
"Table: NO_CAPTIONFigure: (left) BB temperature versus pulse phase.",
"The photon index of the PL component is fixed in the fitting.",
"The vertical error bars denote single-parameter 90% confidence level.",
"(right) BB temperature versus BB normalization, which is proportional to BB flux.",
"Both vertical and horizontal error bars denote single-parameter 90% confidence level.Figure: (left) Photon index versus pulse phase.",
"The temperature of the BB component is fixed in the fitting.",
"The vertical error bars denote single-parameter 90% confidence level.",
"(right) Photon index versus 2–10 keV unabsorbed PL flux.",
"Both vertical and horizontal error bars denote single-parameter 90% confidence level."
],
[
"Broad-band spectra of SGR 1900+14",
"Our simultaneous broad-band observation with XMM-Newton and NuSTAR successfully measured the spectrum of SGR 1900+14 much more precisely than previous observations.",
"The obtained spectral properties can be compared with those obtained in previous observations performed in the quiescent stages of the object, where bursting activities were not seen.",
"Table REF presents a comparison of our spectral analysis results with the previous XMM-Newton and Suzaku results in 2005, 2006, and 2009 [34], [14].",
"We found a clear decrease in the flux of SGR 1900+14, which can be attributed to a continuous decline since the giant flare.",
"We confirm that the 1-10 keV flux $F_{1-10}$ decreases by $\\sim 25$ –$40\\%$ .",
"The 15–60 keV flux $F_{15-60}$ also shows a possible decrease, but this cannot be confirmed because of background model uncertainties of Suzaku HXD [18]: when we employed the results of another analysis of the same data set [15], the decrease of $F_{15-60}$ was not found.",
"Although the flux shows a clear decrease, the spectral shape does not show significant changes from those observed in 2005, 2006, and 2009.",
"The obtained $N_{\\rm H}$ and $kT$ values are perfectly matched with those of Suzaku results; they are also comparable with those obtained from the XMM-Newton observations.",
"As for the photon index $\\Gamma $ , our results do not agree with previous studies in terms of XMM-Newton or Suzaku, which is presumably due to the absence of the hard X-ray observation for XMM-Newton and uncertainties of the background model of Suzaku HXD [18].",
"Another analysis performed on the same Suzaku observation gives $\\Gamma =1.2(5)$ and $1.4(3)$ for the 2006 and 2009 data [15], respectively, both of which agreeing with our result within the errors.",
"Therefore, we suggest that SGR 1900+14 has been quiescent for more than 10 years.",
"In addition, we found a clear decrease in the 1–60 keV unabsorbed soft component (BB) flux $F_{\\rm s}$ over 10 years.",
"We also found a trend where the PL component may decrease even faster than the BB component because the hardness ratios, which are defined in two ways as $\\eta =F_{15-60}/F_{1-10}$ or $\\xi =F_{\\rm h}/F_{\\rm s}$ , show decreases, where $F_{\\rm h}$ denotes the 1–60 keV unabsorbed PL flux.",
"However, this result cannot be confirmed because the evaluation of the hardness ratios largely depends on the uncertainties of the HXD background model[18].",
"Table: NO_CAPTIONThe detection of coherent pulsation (Section REF ) indicates that the rotation period of SGR 1900+14 was $5.22669(3)\\;{\\rm s}$ on 2016 October 20 to 23.",
"Figure REF and Table REF present the long-term evolution of the rotation period of SGR 1900+14, where we see a monotonic spin-down over the past 20 years.",
"As shown in Figure REF , the spin-down rate $\\dot{P}$ has a large fluctuation.",
"This behavior is typical for magnetars (e.g., CXOU J171405.7-381031: [22]; Swift J1822.3-1606: [44]) and is in clear contrast with the behavior of normal radio pulsars (e.g., the Crab pulsar: [41]).",
"Here, we define the “post-outburst phase” of SGR 1900+14 as the hatched area in Figure REF .",
"In this phase, the unstable fluctuations of $\\dot{P}$ soon after the giant flare [49] cease and $\\dot{P}$ decreases monotonically.",
"Assuming a constant decay of $\\dot{P}$ , we determined a quadratic function (black curve in Figure REF ) that describes the trend of $\\dot{P}$ in the post-outburst phase.",
"We obtained $\\ddot{P}=-3.1\\times 10^{-19}\\;{\\rm s^{-1}}$ , which suggests a monotonic decrease in the spin-down rate.",
"$\\dot{P}$ shows a drastic decrease from $2.0\\times 10^{-10}\\;{\\rm s\\;s^{-1}}$ in 2000 April (MJD=51660) to $3.3\\times 10^{-11}\\;{\\rm s\\;s^{-1}}$ in 2016 October (MJD=57681).",
"The trend of $\\dot{P}$ in the post-outburst phase could be explained by the decrease in the toroidal component (or decrease in the twist) of the magnetic fields in the magnetosphere (twisted magnetosphere model: [43], [8]), which is considered to cause a decrease in the spin-down torque.",
"Since the spin-down torque is proportional to the spin-down rate $\\dot{P}$ , the observed monotonic decrease in $\\dot{P}$ during the post-outburst phase can imply that the twist of the magnetic fields has declined monotonically for more than 15 years since it reached its maximum, which was soon after the giant flare.",
"Similar behaviors of other magnetars have been reported.",
"For example, PSR J1622-4950, which entered its outburst in or before 2007 June, showed an unstable fluctuation of the spin-down rate soon after its outburst and then a monotonic decrease with a constant rate [38].",
"XTE J1810-197, which is one of the few magnetars that display radio emissions, experienced an outburst in 2003 August along with a subsequent large fluctuation of $\\dot{P}$ , which was followed by a slow and gradual decrease of $\\dot{P}$ [37].",
"Another example is SGR J1745-2900, which also showed an unstable behavior of $\\dot{P}$ soon after its outburst in 2013 April and subsequent gradual monotonic decrease of $\\dot{P}$ [10], [11].",
"The values of $\\ddot{P}$ in the post-outburst phases are $-2.0\\times 10^{-20}\\;{\\rm s^{-1}}$ , $-5.5\\times 10^{-21}\\;{\\rm s^{-1}}$ , and $-1.8\\times 10^{-19}\\;{\\rm s^{-1}}$ for PSR J1622-4950, XTE J1810-197, and SGR J1745-2900, respectively.",
"All three values of $\\ddot{P}$ are negative, as is that for SGR 1900+14, whose $|\\ddot{P}|=3.1\\times 10^{-19}\\;{\\rm s^{-1}}$ is larger than those of the three above-mentioned examples.",
"We point out the possibility that the decay of the twist of magnetic fields in the magnetosphere of SGR 1900+14 has been lasting for more than 15 years, which is much longer than the duration reported for other magnetars.",
"This may be due to the scale of the outburst; we call the outburst of SGR 1900+14 a giant flare with a peak luminosity of $\\gtrsim 10^{44}\\;{\\rm erg\\;s^{-1}}$ , while other magnetars that have such an outburst have peak luminosities of $\\gtrsim 10^{35}\\;{\\rm erg\\;s^{-1}}$ [3], [20], [10].",
"Still, another magnetar SGR 1806-20, which experienced a giant flare and is often regarded as a set with SGR 1900+14, was reported to have a similar trend of the timing evolution [52], [51].",
"It should be noted that since observations have not been conducted so frequently for SGR 1900+14, this is not necessarily the case.",
"For example, the monotonic decrease of $\\dot{P}$ may have ceased at some point between 2006 and 2016, where we have no data of the rotation period, and thereafter began linear spin-down behavior like normal radio pulsars.",
"Employing the newly obtained $P$ and $\\dot{P}$ , we can evaluate two important parameters: the dipole magnetic field $B_{\\rm d}$ and the characteristic age $\\tau _{\\rm c}$ of SGR 1900+14.",
"These are defined as $B_{\\rm d}&=&3.2\\times 10^{19}\\left(\\frac{P}{\\rm s}\\cdot \\frac{\\dot{P}}{\\rm s\\;s^{-1}}\\right)^{1/2}\\;{\\rm G},\\\\\\tau _{\\rm c}&=&\\frac{(P/{\\rm s})}{2(\\dot{P}/{\\rm s\\;s^{-1}})}.$ The $P$ and $\\dot{P}$ values on 2016 October 20 give $B_{\\rm d}=4.3\\times 10^{14}\\;{\\rm G}$ and $\\tau _{\\rm c}=2.4\\;{\\rm kyr}$ , respectively.",
"This result means that SGR 1900+14 is older and has a smaller dipole magnetic field than previously reported ($B_{\\rm d}=7\\times 10^{14}\\;{\\rm G}$ and $\\tau _{\\rm c}=0.9\\;{\\rm kyr}$ reported by [36] and based on [34]).",
"This is because the previous studies determined these parameters over a short duration without considering the long-term evolution of the rotation period.",
"Figure: Long-term evolution of the rotation period of SGR 1900+14.",
"Red crosses denote the rotation period of SGR 1900+14 at each time, among which the right one is obtained from our work.",
"Other data were obtained from previous studies , , , , .",
"Errors are much smaller than the crosses.",
"The red arrow denotes the epoch of the giant flare (MJD=51052).",
"The blue hatched area denotes the post-outburst phase, which we defined in this work (see text for details), covering the data since MJD=51660.",
"The black curve denotes the quadratic model fitted to the data in the post-outburst phase.Table: NO_CAPTION"
],
[
"Pulse profiles",
"We found no significant variation in the pulse fraction over 10 years.",
"The pulse profiles obtained with XMM-Newton and NuSTAR all give pulse fractions of 15–20%, which agrees with the previous XMM-Newton observations in 2005 and 2006 [34].",
"The pulse profile shape is almost sinusoidal in the 3–10 keV energy band, while it is more structured in the 10–20 keV.",
"Since the power-law component is dominant in the range 10–20 keV (see Figure REF ), we may see contributions of higher multipolar components of the magnetic field on the stellar surface."
],
[
"Phase-dependent fluctuation of power-law component",
"Our phase-resolved spectral analysis (Section REF ) suggests that the spectral shape varies with the pulse phase.",
"Figure REF shows the BB temperature as a function of the pulse phase and the relation between the BB temperature and the BB flux, which was obtained with a fixed PL photon index.",
"Although the BB temperature shows variations with the pulse phase, we did not find its correlation with the BB flux, as shown in Figure REF .",
"As a next step, we explored how the PL component varies with the pulse phase.",
"Figure REF shows the photon index as a function of the pulse phase and the relation between the photon index and the unabsorbed PL flux.",
"We found a positive correlation between the photon index and the unabsorbed PL flux.",
"Note that although these two parameters can be covariant and positively coupled, we confirmed that the positive correlation is significant by checking their error contours.",
"The relation between the photon index and the PL flux is consistent with the trend of the systematic analysis of various magnetars [15], [14].",
"They reported a signature that the hard PL component above 10 keV shows softer spectral photon index as dipole magnetic fields of a magnetar become stronger.",
"This trend was interpreted as a process that the higher magnetic field leads to more photon splittings into lower energy photon, i.e., the softer power-law spectrum.",
"The positive correlation between the photon index and the PL flux means that the spectrum gets softer when the emitting area, which is usually a hot spot, is better oriented to the observer.",
"We can interpret this trend in terms of photon splitting, which is a nonlinear effect of QED under extremely strong magnetic fields [6], [7], [9].",
"When the PL flux is at the pulse maximum, we can see the region with the strongest magnetic fields, which cause more photon splittings and thus a softer spectrum.",
"Another possibility is that we may see the difference of the path length of splitting photons: the photons from the direction of the magnetic pole travel longer in magnetosphere and experience more splittings.",
"In both cases, our results are consistent with the trend of the photon splitting model."
],
[
"Conclusions",
"We performed the first simultaneous broad-band observations of the magnetar SGR 1900+14 covering 0.1–78 keV, making full use of the high sensitivity of XMM-Newton and NuSTAR.",
"The NuSTAR hard X-ray coverage enabled us to detect the source up to 70 keV, with a 60–70 keV source significance of $6.5\\sigma $ .",
"The spectrum of SGR 1900+14 was well fitted by a typical magnetar spectral model: BB plus PL.",
"We have successfully determined the properties of the spectrum with a higher accuracy than any previous studies, especially for the hard-tail power-law component.",
"The obtained parameters are $N_{\\rm H}=(1.96\\pm 0.11)\\times 10^{22}\\;{\\rm cm^{-2}}$ , $kT=0.52\\pm 0.02\\;{\\rm keV}$ , and $\\Gamma =1.21\\pm 0.06$ .",
"We have found that the flux has decreased for more than 10 years, presumably because of the decline from the giant flare, while the spectral shape exhibited no significant variations.",
"Timing analysis allowed us to determine that the rotation period of SGR 1900+14 on 2016 October 20 to 23 was $5.22669(3)\\;{\\rm s}$ .",
"The long-term evolution of the rotation period shows a monotonic decrease in $\\dot{P}$ in the post-outburst phase, which suggests that the twist of the magnetic fields in the magnetosphere has been decaying for more than 15 years.",
"Its pulse fraction was in the range 15–20%, showing no variation in the past 10 years.",
"The energy-dependent pulse profiles present an interesting trend that the 3–10 keV band is almost perfectly sinusoidal, while the 10–20 keV band contains higher harmonics.",
"Combining the spectral and temporal analyses, we succeeded in obtaining the phase-resolved spectra.",
"It shows an interesting feature that the photon index and unabsorbed PL flux have a positive correlation, which suggests that we may see differences in the process of photon splitting within one phase cycle.",
"We thank the anonymous referee for his/her valuable comments.",
"We thank Shinpei Shibata, Kuniaki Masai, Hiromasa Suzuki, Takahiro Matsumoto, and Kazuo Makishima for their helpful advice and discussions.",
"This work was also supported by the Grant-in-Aid for Scientific Research on Innovative Areas “Toward new frontiers: Encounter and synergy of state-of-the-art astronomical detectors and exotic quantum beams” (18H05459; AB).",
"We acknowledge support from JSPS/MEXT KAKENHI grant numbers 18H05459 (AB), 19K03908 (AB), 18H05861 (HO), 16H03954 (HO), 15H00845 (TE), 17K18776 (TE), and 18H04584 (TE)."
]
] | 1906.04406 |
[
[
"Spontaneous symmetry breaking and the flat phase of crystalline\n membranes"
],
[
"Abstract Crystalline membranes are one of the rare examples of bidimensional systems in which long-range order can stabilise an ordered phase in the thermodynamic limit.",
"By a careful analysis of the Goldstone modes counting, we propose a symmetry breaking mechanism associated with the generation of the flat phase and show how it highlights the crucial role played by the crystalline lattice in the establishment of long-range order in these objects.",
"Comparison with other symmetry breaking mechanisms in membrane physics is also used to unveil the links between symmetry breaking patterns and the physical properties of the flat phase."
],
[
"Goldstone modes counting without Lorentz invariance",
"Different low-energy states, that can be discriminated from each other, must have the same energy if they are related by a symmetry of the free energy.",
"In particular, if a ground state breaks a symmetry of the free energy, all states that can be obtained by recursively applying this symmetry to the ground state are also possible ground states.",
"This is how the comparative study of the symmetries of the states that the system can reach, and those of the free energy, gives insight into the degeneracies of the spectrum of the theory.",
"In the following, we shall restrict ourselves to systems simple enough that more refined arguments about Higgs mechanisms, or other subtle ways to generate gaps [26], [27] are not necessary.",
"Then the low-energy spectrum can be directly read from the spontaneous symmetry-breaking pattern.",
"Obviously, not all the symmetries need to be broken by the ground state, i.e., there can be residual symmetries.",
"In the case when the broken symmetries are associated with continuous transformations, it is possible to relate any pair of ground states by a continous path of other ground states.",
"That is, the spectrum includes massless excitations, which are the Goldstone modes.",
"Because of their massless nature, Goldstone modes are not secreened at large distances, and therefore they play a crucial role in the infrared physics of the system.",
"In relativistic systems, Lorentz symmetry imposes on these modes a dispersion relation of the form $\\omega =q c$ ; however this no longer holds for non-relativistic systems.",
"For example, the ferromagnetic magnon has a low-energy dispersion relation of the form $\\omega \\sim q^2$ [18].",
"Hence for systems without Lorentz invariance, the counting rule should give access not only to the number of Goldstone modes, but also to their type of dispersion relation.",
"The counting of Goldstone modes can be related to the algebra of the symmetry groups of the free energy and the ground states by Goldstone's theorem [17] : let $G$ be the symmetry group of the free energy, and let $H$ be the group of symmetries of the ground states, namely the group of residual symmetries, with $H\\subset G$ in light of the above.",
"Goldstone's theorem (in its original version) states that the number of Goldstone modes is simply given by dim$(G/H)$ .",
"However, we must note the following: this rule cannot be trivially extended to non-relativistic systems.",
"Indeed, in ferromagnets for example, $G=$ O$(3)$ is broken into $H=$ O$(2)$ (the ground state is still invariant with respect to rotations of axis in the direction of the spontaneous magnetization).",
"The ferromagnetic state thus breaks two generators of $G$ , but there is only one magnon [18].",
"As Goldstone's theorem is intended only for relativistic systems, it does not give any information about the dispersion relations, which are crucial characteristics of the massless modes in non-relativistic theories.",
"It was first pointed out by Nielsen and Chadha that the infrared behavior of the Goldstone bosons has a direct influence on the number of generated massless modes : to continue with the example of magnets, ferromagnets have one magnon with a quadratic dispersion relation $\\omega \\sim q^2$ , whereas antiferromagnets have two magnons with a linear dispersion relation $\\omega \\sim q$ , while in both cases two rotation generators are broken by the ground state [18].",
"the knowledge of the algebra of broken generators is not sufficient in general.",
"Indeed, two different broken generators can generate the same excitation on a given ground state (see Fig.",
"REF for a simple example), and therefore they are not associated with different Goldstone modes [28].",
"This is all the more important for systems that possess spacetime symmetries which frequently generate non-independent transformations of the ground states [29].",
"Figure: Left: effect of a rotation on a line.",
"Right: the action, on the same line, of a combinationof local translations induces the exact same transformation of the line.",
"It is one of the simplestexamples in which mathematically independent transformations can induce similar excitationsof a given state.This example is discussed in .It can be understood in light of the above that the central quantity involved in the counting of Goldstone bosons should include more information than the mere algebra of the broken generators.",
"Let us define the matrix $\\rho $ of the commutators of independent broken generators $\\left\\lbrace Q_i\\right\\rbrace $ evaluated in the ground state $\\left|0\\right>$ [21] : $\\rho _{ab}=\\left<[Q_a,Q_b]\\right>\\ .$ It is also important to discriminate the generated massless modes according to their dispersion relation: we will call type-A Goldstone bosons those that have a linear dispersion relation, and type-B those whose dispersion relation is quadratic (the rigorous definition is a little bit more subtle, but the difference is irrelevant to our purpose [22]).",
"The numbers $n_A$ and $n_B$ of each type of Goldstone modes is then given by the following formulas [21]: $\\left\\lbrace \\begin{split}&n_A = \\text{dim}(G/H)-\\text{rank}(\\rho )\\\\&n_B = \\frac{\\text{rank}(\\rho )}{2}\\ .\\end{split}\\right.$ Note that in the relativistic case, the existence of a non zero expectation value of the commutator in the ground state would break Lorentz invariance [22], thus Lorentz symmetry requires $\\rho =0$ , and we recover the original Goldstone's theorem with dim$(G/H)$ type-A massless modes.",
"As a less obvious example, consider an ideal crystalline solid in $D$ dimensions.",
"The ground state of the system is given by the periodic lattice that breaks both the translation and rotation symmetries of the free-energy (given in that case by the theory of elasticity [30]).",
"The residual symmetry group $H$ is the discrete subgroup of symmetry of the crystalline lattice denoted $\\mathcal {C}$ .",
"The spontaneous symmetry breaking mechanism characterizing the crystalline ground state can be written as follows: $\\text{\\underline{\\textbf {Mechanism 1:}}} \\quad \\text{ISO}(D)\\rightarrow \\mathcal {C}\\ ,$ where ISO$(D)$ is the group of isometries.",
"There are $D$ broken translation generators, as well as $D(D-1)/2$ broken rotation generators.",
"However, the action of translations and rotations on the ground state do not generate independent excitations [31], [32], and the group of independent broken generators reduces to the broken translations.",
"Because translations commute with each other, $\\rho =0$ and the counting rule Eq.",
"(REF ) gives $D$ type-A Goldstone bosons, corresponding to the well-known $D$ acoustic phonons of crystals, and not $D+D(D-1)/2=D(D+1)/2$ Goldstone bosons as a naive application of Goldstone's theorem would have given.",
"This simple example sheds light onto two properties of the counting rule Eq.",
"(REF ).",
"First, for non-Lorentz-invariant systems, the total number of Goldstone modes does not need to be equal to the total number of broken generators, even if all of them have a linear dispersion relation (the equality occurs only if we suppose further that the actions of each of the generators on the ground state are independent).",
"Second a non-trivial algebra of independent broken generators is required for the existence of type-B Goldstone bosons."
],
[
"Spontaneous symmetry breaking in the flat phase",
"In this section, we study the application of the Goldstone counting rule Eq.",
"(REF ) to the flat phase.",
"Note that we are concerned here with the symmetries of the zero-temperature ground state of the crystalline membrane, which is still a periodic crystal.",
"The influence of thermal fluctuations on the stability of such a state is discussed in the next section.",
"In addition to usual crystals, a fundamental property of crystalline membranes is their ability to fluctuate in a bigger embedding space, whose dimension will be called $d$ .",
"This paves the way to more complex spacetime symmetry breaking patterns, which, as we can already anticipate will be of paramount importance to explain the quadratic dispersion relation of flexurons (see below).",
"The massless fluctuation spectrum of the flat phase of crystalline membranes is well-known.",
"It includes the following[11]: $D$ acoustic phonons, which are type-A Goldstone bosons, as in any crystal.",
"$d-D$ flexurons which are type-B Goldstone bosons, and therefore much stronger than the phonons in the infrared limit.",
"There is no clear consensus in the literature on the spontaneous symmetry-breaking pattern associated with the flat phase to the best of our knowledge [9], [24], [12].",
"Note that most of these patterns were proposed at a time when the Goldstone counting rule was not known.",
"In the following we will only discuss the more widely used mechanism, which is also the one that best respects the symmetries of the flat phase (see Fig.",
"REF ) and argue why we think it is not correct.",
"Consider the following symmetry breaking pattern [12]: $\\text{\\underline{\\textbf {Mechanism 2:}}} \\quad \\text{ISO}(d)\\rightarrow \\text{ISO}(D)\\times \\text{SO}(d-D)\\ .$ The free-energy is invariant under all isometries of the embedding space, forming the group ISO$(d)$ , and the flat phase configuration is an infinite plane, which is still invariant under the plane isometries ISO$(D)$ as well as the rotations of SO$(d-D)$ which act only on directions of the embedding space that are all orthogonal to the flat phase plane (see Fig.",
"REF for a picture for the physical dimensions $D=2$ , $d=3$ ).",
"Figure: Symmetries of the infinite plane in three dimensions : the plane is invariant undertwo translations and one rotation (in blue in the figure), but it breaks the other two rotations and theremaining translation (in red).The SO(d-D)(d-D) group reduces to the discrete ℤ 2 \\mathbb {Z}_2 group corresponding to reversingthe up and down sides of the plane.The set of broken symmetry generators thus contains the following: the $d-D$ translations in the directions orthogonal to the flat phase plane, denoted $\\left\\lbrace P_\\alpha \\right\\rbrace _{\\alpha \\in [\\![D+1;d]\\!",
"]}$ .",
"In the following, greek letters denote indices in $[\\![1;d]\\!",
"]$ , whereas latin ones only run into $[\\![1;D]\\!",
"]$ .",
"the $D\\times (d-D)$ mixed rotations that bring the vector giving the $i^{th}$ direction inside the flat phase plane to the direction given by the $\\alpha ^{th}$ vector of the embedding space outside the membrane's plane $\\left\\lbrace J_{i\\alpha }\\right\\rbrace _{i\\in [\\![1;D]\\!",
"],\\alpha \\in [\\![D+1;d]\\!",
"]}$ .",
"The commutation relations between these generators are given by the $\\text{iso}(d)$ algebra of the isometry group [33]: $\\begin{split}[J_{\\mu \\nu },J_{\\alpha \\beta }]&=\\delta _{\\nu \\alpha }J_{\\mu \\beta }-\\delta _{\\mu \\alpha }J_{\\nu \\beta }-\\delta _{\\nu \\beta }J_{\\mu \\alpha }+\\delta _{\\mu \\beta }J_{\\nu \\alpha }\\\\[0.2cm][J_{\\mu \\nu },P_{\\theta }]\\ \\,&=\\delta _{\\nu \\theta }P_{\\mu }-\\delta _{\\mu \\theta }P_{\\nu }\\\\[0.2cm][P_\\mu ,P_\\nu ]\\ \\ \\,&=0\\ .\\end{split}$ The next question one should answer is which of these generators act independently on the ground state $\\left|0\\right>$ .",
"Consider the action of a mixed rotation on the ground state : $J_{i\\alpha }\\left|0\\right>=\\big (x_iP_\\alpha -x_\\alpha P_i\\big )\\left|0\\right>=x_i P_\\alpha \\left|0\\right>\\ .$ The second equality follows from the fact that the translations inside the flat phase plane are not broken.",
"Hence the action of a mixed rotation can always be canceled by a combination of broken translations.",
"The set of independent broken generators thus reduces to $\\left\\lbrace P_\\alpha \\right\\rbrace _{\\alpha \\in [\\![D+1;d]\\!",
"]}$ , and the counting rule Eq.",
"(REF ) yields: $\\left\\lbrace \\begin{split}&n_A=d-D\\\\&n_B=0\\ ,\\end{split}\\right.$ which is not consistent with the expected spectrum for the flat phase of crystalline membranes.",
"This computation is quite enlightening though, since it underlines the necessity of having a non-trivial algebra of broken generators to describe type-B Goldstone bosons such as the flexurons in crystalline membranes.",
"Looking more thoroughly at Eq.",
"(REF ), we notice that the number of generated Goldstone bosons is equal to the number of directions of the embedding space orthogonal to the flat phase plane, which indicates that they must be flexurons (but not with the expected dispersion relation): the mechanism in Eq.",
"(REF ) misses the phonons.",
"Thus, this mechanism seems more appropriate to describe the flat phase of fluid membranes Such an ordered phase only exists at $T=0$ however, because of Hohenberg-Mermin-Wagner's theorem.",
"in which only flexurons are at play.",
"As a matter of fact, the constituents of an incompressible fluid membrane are free to diffuse on its surface, thereby forbidding the definition of any particular reference state by the position of the molecules, and therefore any kind of positional order.",
"Such materials thus do not have acoustic phonons.",
"This assertion can indeed be checked for consistency with usual models of fluid membranes: from the Canham-Helfrich free-energy [34], [35], the flexuron's propagator $G(q)$ has the following asymptotic behavior in the infrared limit: $G(q)=\\left<h(q)h(-q)\\right>\\underset{q\\rightarrow 0}{\\sim }\\frac{1}{\\sigma q^2}\\ ,$ where $\\sigma $ is the tension of the membrane.",
"Such behavior is typical of type-A Goldstone bosons.",
"One could argue that, according to Eq.",
"(REF ), another type of behavior could be expected in tensionless fluid membranes, but this does not hold since even if not present at the microscopic scale, $\\sigma $ is generated by the renormalization group flow when going towards the infrared regime [11].",
"Our Goldstone modes analysis leads to the following complementary argument: the tension term is not protected by the symmetries of the system, and therefore cannot be consistently enforced to be zero.",
"As we have seen in Eq.",
"(REF ), the origin of acoustic phonons in crystals is the breaking of the isometry group by the discrete group of the crystalline lattice $\\mathcal {C}$ .",
"This must also hold for crystalline membranes in which, although not well preserved by the thermal fluctuations, a crystalline lattice is still present in the flat phase.",
"In light of these arguments, it seems reasonable to take a closer look at the following symmetry-breaking pattern: $\\text{\\underline{\\textbf {Mechanism 3:}}} \\quad \\text{ISO}(d)\\rightarrow \\mathcal {C}\\times \\text{SO}(d-D)\\ .$ Note that the track of the discrete symmetry group $\\mathcal {C}$ is still present even in the continuum theory of crystalline membranes: in [36] for example, the transition to the flat phase is presented as the generation of a non-zero expectation value for the metric of the flat phase $g_{ij}^0$ which then characterizes the ground state.",
"Because the membrane is assumed to be homogeneous and isotropic, its metric in the ground state has to be proportional to $\\delta _{ij}$ .",
"The proportionality coefficient $\\zeta ^2$ , called the extension parameter, characterizes the unit of length inside the membrane's plane.",
"The presence of this fixed reference metric is one of the key differences between crystalline membranes and fluid ones.",
"The broken symmetry generators in the mechanism given by Eq.",
"(REF ) are thus as follows: the $d-D$ external translations $\\left\\lbrace P_\\alpha \\right\\rbrace _{\\alpha \\in [\\![D+1;d]\\!",
"]}$ .",
"the $D$ internal translations $\\left\\lbrace P_i\\right\\rbrace _{i\\in [\\![1;D]\\!",
"]}$ .",
"the $D\\times (d-D)$ mixed rotations, mixing the internal and external spaces $\\left\\lbrace J_{i\\alpha }\\right\\rbrace _{i\\in [\\![1;D]\\!",
"],\\alpha \\in [\\![D+1;d]\\!",
"]}$ .",
"the $\\frac{D(D+1)}{2}$ internal rotations $\\left\\lbrace J_{ij}\\right\\rbrace _{(i,j)\\in [\\![1;D]\\!",
"]^2}$ .",
"The action of internal translations and rotations are not independent, as in usual crystals.",
"This time however, the action of mixed rotations on the ground state can no longer be canceled by a carefully chosen combination of external translations (see Fig.",
"REF for a picture in the one dimensional case).",
"Indeed, whereas rotations always preserve the ground state metric, a combination of translations bringing the system into a similar plane would induce a dilation of the system (as an obvious application of Pythagoras' theorem shows) and therefore a flat phase state with a different extension parameter $\\zeta $ .",
"This is a direct consequence of the presence of a microscopic lattice, or equivalently of the presence of phonons in the system.",
"Figure: Left: action of a mixed rotation on a one-dimensional lattice.Right: a linear combination of translations can align the lattice on the same line,but the state of the system is different because of the dilation that the translations induceon the lattice due to Pythagoras' theorem.The internal space isometries can be used to relate the mixed rotations with the same external index $\\alpha $ , but different internal indices $i$ .",
"All in all, the total number of independently acting broken symmetry generators is as follows: $D$ internal translations $+$ $(d-D)$ external translations $+$ $(d-D)$ mixed rotations, so that finally: $\\text{dim}(G/H)=2d-D\\ .$ The commutations relations between these generators evaluated in a ground state are given as usual by the algebra of isometries $\\text{iso}(d)$ : $\\begin{split}&\\left<[J_{\\alpha i},J_{\\beta i}]\\right> \\propto \\left<J_{\\alpha \\beta }\\right>=0\\\\&\\left<[P_{\\alpha },P_{\\beta }]\\right>=0\\\\&\\left<[J_{\\alpha i},P_{\\gamma }]\\right>\\underset{\\gamma \\ne i,\\gamma \\ne \\alpha }{=}0\\\\&\\left<[J_{\\alpha i},P_{\\alpha }]\\right>=\\left<P_i\\right>\\ne 0\\\\&\\left<[J_{\\alpha i},P_{i}]\\right>=\\left<P_\\alpha \\right>\\ne 0\\ .\\end{split}$ It is then possible to write $\\rho $ is a basis where the determination of its rank is simple, even without knowing the precise value of the non-zero matrix coefficients, denoted \"$*$ \" below.",
"We choose to write in a basis in which the $d$ translations are displayed first, and then the $d-D$ rotations : $\\rho =\\left[\\begin{array}{ccccc|ccccc}0 & 0 & 0 & 0 & ... & * & * & 0 & 0 & ... \\\\0 & 0 & 0 & 0 & ... & * & 0 & * & 0 & ... \\\\0 & 0 & 0 & 0 & ... & * & 0 & 0 & * & ... \\\\0 & 0 & 0 & 0 & ... & * & 0 & 0 & 0 & ... \\\\... & ... & ... & ... & ... & ... & ... & ... & ... & ... \\\\\\hline * & * & * & * & ... & 0 & 0 & 0 & 0 & ... \\\\* & 0 & 0 & 0 & ... & 0 & 0 & 0 & 0 & ... \\\\0 & * & 0 & 0 & ... & 0 & 0 & 0 & 0 & ... \\\\0 & 0 & * & 0 & ... & 0 & 0 & 0 & 0 & ... \\\\... & ... & ... & ... & ... & ... & ... & ... & ... & ...\\end{array}\\right]\\ ,$ which leads to: $\\text{rank}(\\rho )=2(d-D)\\ ,$ and finally, thanks to the counting rule Eq.",
"(REF ), $\\left\\lbrace \\begin{split}&n_A=D\\\\&n_B=d-D\\ .\\end{split}\\right.$ Finally, we get $D$ type-A Goldstone modes with linear dispersion relation, corresponding to the acoustic phonons, and $d-D$ type-B Goldstone bosons with quadratic dispersion relation corresponding to the flexurons; which is the exact spectrum of crystalline membranes that we recalled at the beginning of the section.",
"The crucial difference between the second and third proposed mechanisms – Eq.",
"(REF ) and Eq.",
"(REF ) respectively – is the presence of the crystalline lattice discrete group in the latter, that allows to keep some independent rotation generators required to get a non trivial algebra Eq.",
"(REF ) and thus a non zero rank for $\\rho $ , which allows for the existence of the type-B flexurons.",
"The consequences are far-reaching as among the proposed mechanisms, only the third one is associated with a stable ordered phase in two dimensions: phonons Eq.",
"(REF ) or flexurons Eq.",
"(REF ) alone cannot stabilize a long range order of positional or orientational nature.",
"Already at this stage, we can notice two main differences between mechanisms 1 and 2 on the one hand and mechanism 3 Eq.",
"(REF ) on the other hand (although they are all three built from the same groups), namely that mechanism 3 has two different types of fluctuation modes, and it has type-B Goldstone modes.",
"The rest of this paper is dedicated to the analysis of the consequences of these two differences."
],
[
"Hohenberg-Mermin-Wagner's theorem",
"The spontaneous symmetry breaking pattern combined with the counting rule Eq.",
"(REF ) gives access to the number of massless modes as well as their dispersion relation in the large distance limit in the ordered phase predicted by mean-field theory, but it is not sufficient to know if such an ordered phase is robust to thermal fluctuations.",
"For that last purpose, the most useful tool is the Hohenberg-Mermin-Wagner's theorem [15], [14].",
"In their original paper, Mermin and Wagner stress an important hypothesis for their theorem to apply: the interaction needs to have a short enough range.",
"Namely, if $J$ is the coupling constant describing the strength of the interaction between the order parameter fields, $J(x)x^2$ must be an integrable function in $D$ dimensions [14].",
"Let us test this hypothesis in the case of crystalline membranes.",
"First, we need the action describing the small fluctuations around the flat phase.",
"As stated before, crystalline membranes can be described as an elastic medium fluctuating in an embedding space, and their action thus contains both a curvature term, proportional to the bending energy $\\kappa $ of the membrane, and an elastic term quadratic in the strain tensor $\\varepsilon _{ij}$ : $S=\\int _x\\left[\\frac{\\kappa }{2}\\big (\\partial ^2\\vec{r}\\big )^2+\\frac{c_{ijab}}{2}\\varepsilon _{ij}\\varepsilon _{ab}\\right]\\ ,$ where $\\int _x=\\int d^Dx$ is an integral over the internal space of the membrane, $\\vec{r}$ is the position vector describing the membrane, and $c_{ijab}$ is the elastic tensor, which can be expressed for example in terms of the Lamé coefficients as $\\lambda \\delta _{ij}\\delta _{ab}+\\mu (\\delta _{ia}\\delta _{jb}+\\delta _{ib}\\delta _{ja})$ .",
"The strain tensor $\\varepsilon _{ij}$ can be expressed as half the difference between the metric in the current state of the membrane $g_{ij}=\\partial _i\\vec{r}.\\partial _j\\vec{r}$ and that of the flat phase reference state $g_{ij}^0=\\zeta ^2\\delta _{ij}$ .",
"To build a theory of small fluctuations around the flat phase, it is necessary to expand $\\vec{r}$ around the equilibrium configuration with extension parameter $\\zeta $ : $\\vec{r}=\\zeta x_i\\vec{e}_i+\\vec{u}+\\vec{h}\\ ,$ where a basis of the flat phase plane $\\lbrace \\vec{e}_i\\rbrace $ has been introduced.",
"This expansion causes the phonons $\\vec{u}$ and the flexurons $\\vec{h}$ to appear explicitely.",
"Using the fact that the phonons and flexurons vibrate in orthogonal spaces, the action writes in terms of the most relevant terms reads [8] : $\\begin{split}S=\\int _x\\bigg [&\\frac{\\kappa }{2}\\big (\\partial ^2\\vec{h}\\big )^2+\\frac{c_{abcd}}{2}u_{ab}u_{cd}+\\frac{c_{abcd}}{2}u_{ab}(\\partial _c\\vec{h}.\\partial _d\\vec{h})\\\\&+\\frac{c_{abcd}}{8}(\\partial _a\\vec{h}.\\partial _b\\vec{h})(\\partial _c\\vec{h}.\\partial _d\\vec{h})\\bigg ]\\ ,\\end{split}$ where the symmetric tensor $u_{ab}=(\\partial _au_b+\\partial _bu_a)/2$ has been introduced.",
"At first glance, it seems that the action Eq.",
"(REF ) contains only local interactions between phonons and flexurons, what could lead one to conclude that the long-range orientational order is broken by thermal fluctuations.",
"But in the original argument of Mermin and Wagner, there is only one fluctuation mode.",
"We have seen that flexurons are the dominant modes in the infrared limit.",
"Moreover, the action Eq.",
"(REF ) is quadratic in the phonons, it is thus possible to perform an exact integration over the phonons to define an effective action with the flexurons as only fields [37].",
"For sake of simplicity, we give it here in the Fourier space, with implicit momentum conservation, and the shorthand notation $\\vec{h}(k_i)=\\vec{h}_i$ : $\\begin{split}S_{\\rm eff}&=\\int _{k_1,k_2}\\frac{\\kappa }{2}\\,k_1^4\\big (\\vec{h}_1.\\vec{h}_2\\big )\\\\+&\\int _{k_1,k_2,k_3,k_4}\\bigg [\\frac{\\mathcal {R}_{abcd}(q)}{4}k_1^ak_2^bk_3^ck_4^d\\big (\\vec{h}_1.\\vec{h}_2\\big )\\big (\\vec{h}_3.\\vec{h}_4\\big )\\bigg ]\\ ,\\end{split}$ with $\\int _k=\\int \\frac{d^Dk}{(2\\pi )^D}$ according to our convention for Fourier transforms, and $q=k_1+k_2=-k_3-k_4$ .",
"The price for working with one type of field only is that now the interaction vertex $\\mathcal {R}$ is non local.",
"It depends only on two coupling constants, exactly like the elasticity tensor $c$ , which can be made explicit by decomposing it onto the following set of orthogonal projectors: $\\begin{split}& N_{abcd}(q) = \\frac{1}{D-1}P^T_{ab}(q)P^T_{cd}(q)\\\\& M_{abcd}(q) = \\frac{1}{2}\\big (P^T_{ac}(q)P^T_{bd}(q)+P^T_{ad}(q)P^T_{bc}(q)\\big )-N_{abcd}(q)\\ ,\\end{split}$ where $P^T_{ij}(q)=\\delta _{ij}-q_iq_j/q^2$ is the projector in the direction orthogonal to $q$ .",
"The effective interaction vertex is then [37] : $\\mathcal {R}_{abcd}(q)=\\frac{\\mu (D \\lambda +2\\mu )}{\\lambda +2\\mu }N_{abcd}(q)+\\mu M_{abcd}(q)\\ .$ Note that in the particular case $D=2$ , corresponding to physical membranes, the two projectors Eq.",
"(REF ) are equal, and $\\mathcal {R}$ depends on only one elastic constant, which turns out to be Young's modulus $\\mathcal {Y}$ [16].",
"The first proof of the presence of long-range content of the interaction in Eq.",
"(REF ) has been given by Nelson and Peliti [16], they showed that in two dimensions, the interaction term in Eq.",
"(REF ) $S_{int}$ can be rewritten as an interaction between the local Gaussian curvature $s(x)=\\text{det}(\\partial _i\\partial _j h)$ : $S_{int}=\\frac{\\mathcal {Y}}{16\\pi }\\int _{x,y}\\mathcal {G}(x-y)s(x)s(y)\\ ,$ where the (non-local) interaction vertex between the Gaussian curvature $\\mathcal {G}$ behaves as $\\mathcal {G}(x)\\simeq x^2 \\ln (x/a)$ at large distance ($a$ being the lattice spacing), which is clearly a long-range type of interaction.",
"To make the connection with the original work of Mermin and Wagner [14], we must first find the equivalent to the $J(x)$ interaction term.",
"An order parameter associated to the flat phase is given by the extension parameter $\\zeta $ which is always non zero in the flat phase, and equal to zero in a completely disordered crumpled configuration.",
"It can also be expressed as a function of the corraltion between the tangent vectors to the surface generated by the membrane [12]: $\\zeta ^2=\\frac{1}{D}\\left<\\partial _i\\vec{r}\\right>.\\left<\\partial _i\\vec{r}\\right>\\ .$ Finally, taking into account the fact that the action in Eq.",
"(REF ) is generated after an integration over the phonon fields, the analog of $J(x)$ in our model is the interaction between the $\\partial h$ terms, which turn out to be $\\mathcal {R}$ .",
"To test if the range of $\\mathcal {R}$ is short enough for Hohenberg-Mermin-Wagner's theorem to apply, it must be reexpressed in direct space.",
"We do not give here the full expression of $\\mathcal {R}(x-y)$ , but we rather analyze the following elementary block: $P^T_{ab}(q)P_{cd}^T(q)=\\delta _{ab}\\delta _{cd}-\\delta _{ab}\\frac{q_cq_d}{q^2}-\\delta _{cd}\\frac{q_aq_b}{q^2}+\\frac{q_aq_bq_cq_d}{q^4}\\ .$ Each term can be expressed in direct space thanks to the following formula of the Fourier transform of power laws (see, for example, Ref.",
"[38]): $\\frac{1}{\\big (p^2\\big )^a}=\\frac{1}{4^a\\pi ^\\frac{D}{2}}\\frac{\\Gamma \\Big ({\\small \\frac{D}{2}-a}\\Big )}{\\Gamma (a)}\\int _x\\frac{e^{ip.x}}{\\big (x^2\\big )^{\\frac{D}{2}-a}}\\ ,$ which finally gives: $\\begin{split}& \\int _q\\delta _{ab}\\delta _{cd}e^{i\\,q.",
"(x-y)}=\\delta _{ab}\\delta _{cd}\\,\\delta ^{(D)}(x-y)\\\\& \\int _q\\frac{q_aq_b}{q^2}e^{i\\,q.",
"(x-y)}=\\frac{\\delta _{ab}}{2\\pi |x-y|^2}-\\frac{(x_a-y_a)(x_b-y_b)}{\\pi |x-y|^4}\\\\& \\int _q\\frac{q_aq_bq_cq_d}{q^4}e^{i\\,q.",
"(x-y)}=\\frac{X_{abcd}}{4\\pi |x-y|^2}-\\frac{Y_{abcd}(x-y)}{2\\pi |x-y|^4}\\\\&\\quad +\\frac{2(x_a-y_a)(x_b-y_b)(x_c-y_c)(x_d-y_d)}{\\pi |x-y|^6}\\ ,\\end{split}$ with the following tensors being defined: $\\begin{split}& X_{\\mu \\nu \\rho \\sigma }=\\delta _{\\mu \\nu }\\delta _{\\rho \\sigma }+\\delta _{\\mu \\rho }\\delta _{\\nu \\sigma }+\\delta _{\\mu \\sigma }\\delta _{\\nu \\rho }\\ ,\\\\& Y_{\\mu \\nu \\rho \\sigma }(\\vec{x})=x_\\mu x_\\nu \\delta _{\\rho \\sigma }+x_\\mu x_\\rho \\delta _{\\nu \\sigma }+x_\\mu x_\\sigma \\delta _{\\nu \\rho }\\\\&\\qquad \\qquad +x_\\nu x_\\rho \\delta _{\\mu \\sigma }+x_\\nu x_\\sigma \\delta _{\\mu \\rho }+x_\\rho x_\\sigma \\delta _{\\mu \\nu }\\ .\\end{split}$ Among the different terms of Eq.",
"(REF ), only the first one is local.",
"The other ones are not integrable over the membrane's internal space once multiplied by $(x-y)^2$ .",
"Hence, Hohenberg-Mermin-Wagner's theorem does not apply, and the orientational order in crystalline membranes can be long-range.",
"In the previous argument, it is the non-local structure of the effective interaction vertex between flexurons that is at the origin of the stabilization of long range order in two dimensions.",
"In light of our analysis of the spontaneous symmetry-breaking pattern Eq.",
"(REF ), we can add the following: in the flat phase, even if flexurons are the modes that dominate in the infrared limit, they are not the only important fluctuation modes.",
"In particular, acoustic phonons carry a non-local effective interaction between flexurons at various locations in the flat phase's plane.",
"The mechanism in Eq.",
"(REF ) moreover guarantees that the phonons are Goldstone modes, i.e., thanks to the massless nature they posses by symmetry, they are not efficiently screened out at large distances.",
"Thus, the effective interaction they carry is a true long range one, which explains why Hohenberg-Mermin-Wagner's theorem does not apply, and the flat phase is robust against thermal fluctuations."
],
[
"The origin of long-range order in the flat phase",
"In the previous sections, we identified two main differences between the third mechanism Eq.",
"(REF ) and the two other ones Eq.",
"(REF ) and Eq.",
"(REF ).",
"In order to refine our comprehension of the necessary ingredients to generate a stable order phase in the system, we propose to analyze a fourth mechanism.",
"Whenever a crystalline membrane undergoes a strong enough stress, it buckles.",
"This buckling phenomenon can be understood as a (second-order) phase transition [9], [12], [39], [40], [13] : depending on the type of applied forces, the membrane will either become buckled if compressed – in this state, the membrane appears as a mosaic of locally flat domains the orientation of which is random [12] – or overstretched if sufficiently dilated.",
"Because our main concern is the origin of long-range order in two-dimensional systems, we will focus here on the overstretched phase in which a strong long-range orientational order is present.",
"In the overstretched phase, the external stress screens the effects of flexurons [39], [41] which become energetically disfavored.",
"As a result, the massless infrared spectrum of overstretched membranes contains only type-A Goldstone bosons.",
"Macroscopically, the shape of the overstretched membrane is still flat, with weaker height fluctuations compared to the flat phase.",
"Microscopically, the stress dominates the local rearrangement of atoms, and the ancient lattice positional order is broken.",
"Contrary to the flat phase examined in Sec.",
", the ground state in the overstretched phase cannot be uniquely characterized by its metric [36].",
"Indeed, the local arrangement of atoms depends both on the intrinsic properties of the material (captured by $\\zeta $ ) and the external stress.",
"Consequently, the previous argument that allowed us to disentangle mixed rotations and external translations does not hold anymore: the situation is analogous to that of Fig.",
"REF rather than Fig.",
"REF .",
"To avoid confusion, and to ensure the disentanglement between the action of translations and mixed rotations on the ground state, the symmetry-breaking mechanism will be written as : $\\text{\\underline{\\textbf {Mechanism 4:}}} \\quad \\text{ISO}(d)\\rightarrow \\text{SO}(d-D)\\ .$ The set broken generators is the same as for the third mechanism, but now the unit of length at the surface of the membrane is determined by the applied stress rather than the sole extension parameter $\\zeta $ .",
"The set of independent broken generators hence reduces to the $d$ translations, so that $\\text{dim}(G/H)=d\\,,\\quad \\rho =0\\ ,$ and the counting rule Eq.",
"(REF ) leads to: $\\left\\lbrace \\begin{split}&n_A=d\\\\&n_B=0\\ ,\\end{split}\\right.$ namely $D$ type-A acoustic phonons and $d-D$ type-A flexurons, as expected.",
"Finally, mechanism 4 Eq.",
"(REF ) provides an example in which long-range orientational order can be maintained in two dimensions without requiring the help of type-B Goldstone bosons.",
"Indeed in the overstretched phase, the previous argument with regard to the Hohenberg-Mermin-Wagner's theorem still holds: despite the fact that the flexurons are screened by the stress, they remain Goldstone bosons, and an effective theory of interacting flexurons can still be built, in which the phonons, which are also massless, carry an effective long-range interaction between flexurons.",
"Note, however, that in the ground state of overstretched membranes the local pseudo-ordering persists, which causes the breaking of the group isometries inside the membrane's plane, which is in strong contrast with the second mechanism Eq.",
"(REF ), in which the microcopic constituents are free to move and no phonon is generated (and therefore no long range interaction can occur and the ordered phase is destroyed by the thermal fluctuations).",
"A remaining question involves the role of the type-B Goldstone bosons.",
"To address it, we must compare the mechanisms 3 Eq.",
"(REF ) and 4 Eq.",
"(REF ), which differ only by the dispersion relation of the flexurons.",
"The most striking difference between the flat phase and the overstretched phase is the presence in the former of a strong anomalous exponent $\\eta \\simeq 0.85$ [11], [42], [43], [44] , at the origin of the highly anharmonic behavior of the thermal fluctuations, which leads to many unusual effects, as well as a modified elasticity theory, which is in total contrast with the overstretched phase in which conventional elasticity is restored and $\\eta =0$ (see [39], [40], [41] and [45] for a comparative study).",
"The role of the type-B Goldstone bosons is thus probably related to the generation of such an anharmonic behavior and unusual scaling relations.",
"To sum up on the Goldstone physics, we have analyzed various non-trivial symmetry breaking patterns related to the physics of crystalline membranes, which have highlighted a number of general features of the physics of Goldstone modes in theories without Lorentz invariance.",
"First, mechanism 1 Eq.",
"(REF ) illustrates the fact that whenever different broken symmetry generators generate linearly dependent transformations of the ground state, the total number of associated Goldstone modes is smaller than the total number of broken generators, which is the main lesson of reference [28].",
"This feature is quite common in theories presenting spacetime symmetry breakings, since the action of rotations and translations are rarely independent (it is much more difficult to see this if an internal symmetry group is broken).",
"Mechanism 2 Eq.",
"(REF ) teaches us the importance of the underlying microscopics, even in continuum theories.",
"Indeed, taking into account the mere overall shape of the membrane leads to a spontaneous symmetry breaking mechanism without phonons, which is much more sensitive to thermal fluctuations (as such a system cannot sustain long range order in two dimensions).",
"The comparison between mechanisms 3 Eq.",
"(REF ) and 4 Eq.",
"(REF ) shows that the presence of two different types of interacting Goldstone modes is a sufficient condition to generate a stable ordered phase in two dimensions, as whenever one can reexpress the theory as an effective theory of a single mode, the effective interaction carried by the second Goldstone mode must be long range, because of its massless character.",
"Having two different types of Goldstone modes, however, requires particular patterns of symmetry breaking.",
"Finally, we also showed the necessity of having a non-trivial algebra of independently acting broken generators to generate type-B Goldstone bosons, which can be seen as an obvious consequence of the Goldstone counting rule Eq.",
"(REF ), but which we have shown is not so easy to achieve in practice.",
"We also give hints at the possible link between the presence of such type-B Goldstone bosons and unusual scaling behaviors in the ordered phase, related to the generation of a non-trivial field anomalous dimension $\\eta $ ."
],
[
"Conclusion",
"To conclude, the corrections of the symmetry breaking mechanism at the origin of the flat phase teach us two main lessons on the physics of crystalline membranes.",
"First, acoustic phonons cannot be overlooked, even though crytalline order is broken by thermal fluctuations, and by the fact that they are subdominant at large distances.",
"It is all the more important that the presence of a massless effective interaction carrier is required to ensure that long-range orientational order is not destroyed by fluctuations.",
"This is why, genuine bidimensional crystals or fluid membranes, in which only phonons alone, or flexurons alone survive at large distances, do not present any stable ordered phase in two dimensions, but crystalline membranes, which possess both modes, also present long-range order.",
"Second, the origin of the flat phase anomalous scaling laws can be traced back to a delicate geometrical interplay between the intrinsic properties of the material, and its embedding in the three dimensional space.",
"In the presence of a strong enough external stress field – which drives the system into the overstretched phase – this subtle balance is broken, and conventional elasticity is restored, probably due to the absence of type-B Goldstone bosons.",
"The key to understanding all these results is the Goldstone counting rule Eq.",
"(REF ).",
"We hope that our work will motivate further studies in the context of condensed matter physics, in which Lorentz invariance is frequently absent, spacetime symmetries are often at play, and therefore such tools are certainly of interest."
],
[
"Acknowledgements",
"The author thanks D. Mouhanna for useful discussions and a careful reading of this manuscript."
]
] | 1906.04455 |
[
[
"On a series of Darboux integrable discrete equations on the square\n lattice"
],
[
"Abstract We present a series of Darboux integrable discrete equations on the square lattice.",
"Equations of the series are numbered with natural numbers $M$.",
"All the equations have a first integral of the first order in one of directions of the two-dimensional lattice.",
"The minimal order of a first integral in the other direction is equal to $3M$ for an equation with the number $M$.",
"In the cases $M=1,\\ 2,\\ 3$ we show that those equations are integrable in quadratures.",
"More precisely, we construct their general solutions in terms of the discrete integrals.",
"We also construct a modified series of Darboux integrable discrete equations which have in different directions the first integrals of the orders $2$ and $3M-1$, where $M$ is the equation number in series.",
"Both first integrals are unobvious in this case."
],
[
" Introduction",
"We consider here discrete equations of the form: $(u_{n+1,m+1}+1)(u_{n,m+1}-1)=\\theta (u_{n+1,m}+1)(u_{n,m}-1),$ where $n,m\\in \\mathbb {Z}$ and $\\theta $ is a constant coefficient.",
"Two integrable equations of this form are known.",
"The case $\\theta =1$ is presented in [12] and the case $\\theta =-1$ has been found in [2].",
"Equations in both the cases are Darboux integrable.",
"Both equations have the first order first integral in the $n$ -direction, while in the $m$ -direction the minimal orders of first integrals are 3 and 6.",
"We present a series of equations of the form (REF ) with special coefficients $\\theta =\\theta _M,\\ M\\in \\mathbb {N}$ , including two above examples.",
"All the equations are Darboux integrable and have a first integral of the first order in the $n$ -direction.",
"The minimal order of a first integral $W_{2,M}$ in the $m$ -direction is equal to $3M$ for an equation with the number $M$ .",
"So, those equations may have in the $m$ -direction first integrals of an arbitrarily high minimal order.",
"A few similar series of integrable equations are known in the literature.",
"In [5] a series of Darboux integrable discrete equations is discussed which, however, are of Burgers type.",
"The minimal orders of first integrals in both directions may be arbitrarily high there.",
"An analogous series of the continuous hyperbolic type equations is discussed in [14].",
"In [7] a series of the sine-Gordon type autonomous discrete equations is presented.",
"Autonomous generalized symmetries and conservation laws in both directions may have arbitrarily high minimal orders in this case.",
"Some series of non-autonomous discrete equations of sine-Gordon type are studied in [3].",
"It is interesting to construct the general solutions for equations of the form (REF ) under investigation with such a high minimal order of first integrals.",
"Following [6], [8], [9], we succeed to do it in the cases $M=1,2,3$ , where first integrals $W_{2,M}$ have the minimal orders 3,6 and 9.",
"In the case $M=1$ the general solution is constructed in an explicit way and coincides with the known [12].",
"In the cases $M=2,3$ we show that those equations are integrable in quadratures.",
"This means that one can construct general solutions in terms of the discrete integrals.",
"Using non-point transformations invertible on the solutions of discrete equations [13], [12], we construct one more series of Darboux integrable discrete equations.",
"Equations of that series have in different directions first integrals of the minimal orders 2 and $3M-1$ , where $M$ is the equation number in the series.",
"Both first integrals are not obvious in this case.",
"In Section we introduce a series of discrete equations, prove the Darboux integrability of those equations, and discuss the minimal orders of first integrals.",
"The general solutions in the cases $M=1,2,3$ are constructed in Section .",
"A modified series of integrable discrete equations is discussed in Section .",
"english"
],
[
"Darboux integrability",
"We are going to study the following series of discrete equations: $(u_{n+1,m+1}+1)(u_{n,m+1}-1)=\\theta _M(u_{n+1,m}+1)(u_{n,m}-1),$ where $\\theta _M$ is the primitive root of unit of the degree $M \\in \\mathbb {N}$ .",
"More precisely, $\\theta _1=1$ and for $M>1$ one has: $\\theta _M^M=1,\\quad \\theta _M^j\\ne 1,\\ 1\\le j\\le M-1.$ For example, all the primitive roots of the degree $M\\le 4$ read: $\\begin{split}\\theta _1=1,\\quad \\theta _2= -1,\\quad \\theta _3=-\\frac{1}{2}\\pm i\\frac{\\sqrt{3}}{2},\\quad \\theta _4=\\pm i.\\end{split}$ For every $M>2$ we have at least two values of $\\theta _M$ : $\\theta _M=\\exp (\\pm 2\\pi i/M)=\\cos \\frac{2\\pi }{M}\\pm i\\sin \\frac{2\\pi }{M}.$ An equation of the form $F(u_{n+1,m+1},u_{n+1,m},u_{n,m+1},u_{n,m})=0$ is called Darboux integrable if it has two first integrals $W_1,W_2$ , such that $(T_n-1)W_2=0,\\quad W_2=w_{n,m}^{(2)}(u_{n,m+l},u_{n,m+l-1},\\ldots ,u_{n,m}),$ $(T_m-1)W_1=0,\\quad W_1=w_{n,m}^{(1)}(u_{n+k,m},u_{n+k-1,m},\\ldots ,u_{n,m}).$ Here $l,k$ are some positive integers, and $T_n,T_m$ are operators of the shift in the $n$ - and $m$ -directions, respectively: $T_n h_{n,m}=h_{n+1,m}$ , $T_m h_{n,m}=h_{n,m+1}$ .",
"We suppose that the relations (REF ,REF ) are satisfied identically on the solutions of the corresponding equation (REF ).",
"The functions $W_1$ and $W_2$ will be called the first integrals in the $n$ - and $m$ -directions, respectively.",
"We assume here that each of the conditions $\\frac{\\partial W_1}{\\partial u_{n,m}}\\ne 0,\\qquad \\frac{\\partial W_1}{\\partial u_{n+k,m}}\\ne 0,\\qquad \\frac{\\partial W_2}{\\partial u_{n,m}}\\ne 0,\\qquad \\frac{\\partial W_2}{\\partial u_{n,m+l}}\\ne 0$ is satisfied for at least some $n,m$ .",
"The numbers $k,l$ are called the orders of these first integrals $W_1, W_2$ , respectively.",
"The case $M=1$ is known, see equation (4.6) in [12].",
"That equation (4.6) is obtained from (REF ) with $\\theta _M=1$ by the point transformation $v_{n,m}=\\frac{1-u_{n,m}}{1+u_{n,m}}.$ It is shown in [12] that that equation is Darboux integrable by constructing the first integrals in both directions, and its general solution has been found.",
"The case $M=2$ is known, see equation (51a) in [2].",
"The first integrals in both directions have been found there for this equation, see relations (53) in [2].",
"For any $M$ equation (REF ) has the following first integral in the $n$ -direction: $ W_{1,M} =(\\theta _M)^{-m}(u_{n+1,m}+1)(u_{n,m}-1).$ This is true, as equation (REF ) is equivalent to the relation $(T_m-\\theta _M)[(u_{n+1,m}+1)(u_{n,m}-1)]=0.$ Moreover, formula (REF ) with $\\theta $ instead of $\\theta _M$ provides the first integral for equation (REF ) for any $\\theta $ .",
"As for the $m$ -direction, we succeed to find the following formula: $ W_{2,M} =\\frac{(u_{n,m+3M}-u_{n,m+M})(u_{n,m+2M}-u_{n,m})}{(u_{n,m+3M}-u_{n,m+2M})(u_{n,m+M}-u_{n,m})},$ which provides first integrals for all the equations (REF ).",
"It is easy to see that these integrals (REF ) and (REF ) have the orders 1 and $3M$ , respectively.",
"Conditions (REF ) are satisfied for all $n,m$ in this case.",
"The fact that formula (REF ) defines a first integral in the case $M=1$ is checked by direct calculation.",
"Theorem 1 The function $W_{2,M}$ defined by (REF ) is the first integral of equation (REF ,REF ) in the $m$ -direction for any $M>1$ .",
"Proof.",
"At first we denote $\\Psi _{n,m}=(u_{n+1,m}+1)(u_{n,m}-1).$ From (REF ) we have $T_m \\Psi _{n,m}=\\theta _M\\Psi _{n,m},$ and therefore $T^M_m \\Psi _{n,m}=\\Psi _{n,m}$ for equations (REF ) satisfying condition (REF ).",
"By using the notation $u^{(j)}_{n,k}=u_{n,Mk+j},\\quad 1\\le j\\le M,$ we rewrite (REF ) as a system: $(u^{(j)}_{n+1,k+1}+1)(u^{(j)}_{n,k+1}-1)=(u^{(j)}_{n+1,k}+1)(u^{(j)}_{n,k}-1),\\quad 1\\le j\\le M.$ We see that all the equations in (REF ) are independent, and each of them coincides with (REF ), in which $M=1$ and $m$ is replaced by $k$ .",
"For this reason, for any equation of system (REF ), we can use the first integral $W_{2,1}$ of equation (REF ).",
"As a result we get for those equations: $W_{2}^{(j)} =\\frac{(u^{(j)}_{n,k+3}-u^{(j)}_{n,k+1})(u^{(j)}_{n,k+2}-u^{(j)}_{n,k})}{(u^{(j)}_{n,k+3}-u^{(j)}_{n,k+2})(u^{(j)}_{n,k+1}-u^{(j)}_{n,k})}.$ Taking into account (REF ) we are led to: $W_{2}^{(j)}=\\frac{(u_{n,M(k+3)+j}-u_{n,M(k+1)+j})(u_{n,M(k+2)+j}-u_{n,Mk+j})}{(u_{n,M(k+3)+j}-u_{n,M(k+2)+j})(u_{n,M(k+1)+j}-u_{n,Mk+j})}.$ Denoting $m=Mk+j$ , we see that the relation $T_n W_{2}^{(j)}= W_{2}^{(j)}$ implies $T_n W_{2,M}=W_{2,M}$ for any $n,m$ .",
"As (REF ) is equivalent to (REF ), the last relation is satisfied on any solution of equation (REF ) and hence of ( REF).",
"The first integral $W_{1,M}$ obviously has the lowest possible order for any $M$ .",
"As it is known from [12], [2], in the cases $M=1$ and $M=2$ the integral $W_{2,M}$ also has the lowest possible order in its direction.",
"The same is true for the case $M=3$ as it follows from: Theorem 2 Equation (REF ,REF ) with $M=3$ does not have in the $m$ -direction any first integral of the order $l<9$ .",
"In order to prove this theorem, we apply a method described in detail in [4].",
"That method uses so-called annihilation operators introduced in [10] and allows one to find the first integrals.",
"In the framework of that method, the proof comes to direct but cumbersome calculation.",
"The following hypothesis seems to be true: first integral (REF ) of equation (REF ,REF ) has the lowest possible order for any $M>1$ ."
],
[
"General solutions",
"Here we use and improve a method developed in [6],[9],[8].",
"We construct the general solutions for equations (REF ) with $M=1,2,3.$ In the case $M=1$ a solution will be explicit and will coincide with a solution of [12] up to the Möbius transformation $v_{n,m}=\\frac{1-u_{n,m}}{1+u_{n,m}}$ .",
"Corresponding calculation will be needed, however, for the cases $M=2,3$ .",
"In the cases $M=2,3$ , the general solutions will be given in terms of discrete integrals in terminology of [8], i.e.",
"it will be shown that equations (REF ) with $M=2,3$ are solved by quadrature.",
"Let us consider an ordinary discrete equation $a_{n+1}-a_{n}=A_n,$ where $a_n$ is an unknown function and $A_n$ is given.",
"We will say that $a_n$ is found by the discrete integration, in analogy with the ordinary differential equation $a^{\\prime }(x)=A(x),$ and the solution $a_n$ of equation (REF ) will be called the discrete integral of $A_n$ .",
"The explicit general solution of equation (REF ) will be called a function of the form $u_{n,m}=\\Phi _{n,m}[a_n,b_m], $ where $a_n,b_m$ are arbitrary functions of one variable.",
"Here the square brackets mean that the function $\\Phi _{n,m}$ depends on a finite number of the shifts $a_{n+j},b_{m+j}.$ Such a solution must identically satisfy equation (REF ) for all values of the functions $a_n,b_m$ .",
"For example, the discrete wave equation $u_{n+1,m+1}-u_{n+1,m}-u_{n,m+1}+u_{n,m}=0$ has the following general solution: $u_{n,m}=a_n+b_m.$ Equation (REF ) is solved by quadrature if it has a solution of the form: $u_{n,m}=\\Phi _{n,m}\\left[a_n,b_m,a_n^{(1)},a_n^{(2)},\\ldots ,a_n^{(j_1)},b_m^{(1)},b_m^{(2)},\\ldots ,b_m^{(j_2)}\\right],$ where $a_n,b_m$ are arbitrary functions and the square brackets mean, as above, that the function $\\Phi _{n,m}$ depends on a finite number of the shifts of its arguments.",
"The functions $a_n^{(j)}$ are obtained from $a_n$ by a finite number of applications of the shift operator $T_n$ , of the functions of many variables, and of the discrete integrations.",
"The functions $b_m^{(j)}$ are obtained from $b_m$ analogously.",
"So, the functions $a_n^{(j)}, b_m^{(j)}$ and therefore solution (REF ) are implicit in a sense."
],
[
"Case $M=1$",
"Equation (REF ) is equivalent to $(u_{n+1,m}+1)(u_{n,m}-1)=\\lambda _n,$ where $\\lambda _n$ is an arbitrary function.",
"This is the discrete Riccati equation, and we need to know a particular solution of it in order to linearize and then to solve it.",
"We cannot solve this equation for a given $\\lambda _n$ in general case.",
"We use the fact that the function $\\lambda _n$ is arbitrary and replace $\\lambda _n$ by another arbitrary function $\\alpha _n$ which plays the role of particular solution: $\\lambda _n=(\\alpha _{n+1}+1)(\\alpha _n-1).$ In accordance with the known method of solving the Riccati equation, we use the transformation $u_{n,m}=\\alpha _n+\\frac{\\alpha _n-1}{v_{n,m}}$ to get a linear equation for $v_{n,m}$ : $\\frac{\\alpha _{n+1}+1}{\\alpha _{n+1}-1}v_{n+1,m}+v_{n,m}+1=0.$ To solve this equation, it is convenient to introduce a new arbitrary function $\\beta _n$ instead of $\\alpha _n$ as: $\\frac{\\alpha _{n+1}+1}{\\alpha _{n+1}-1}=-\\frac{\\beta _{n+2}-\\beta _{n+1}}{\\beta _{n+1}-\\beta _{n}}.$ Here we follow [7], see (39), (40).",
"Representing equation (REF ) in the form $(T_n-1)[(\\beta _{n}-\\beta _{n+1})v_{n,m}+\\beta _n]=0,$ we find $v_{n,m}=\\frac{\\beta _n+\\omega _m}{\\beta _{n+1}-\\beta _n},$ where $\\omega _m$ is another arbitrary function.",
"Finally, using (REF ,REF ,REF ), we find $u_{n,m}$ : $u_{n,m}=\\frac{\\beta _{n+1}-2\\beta _n+\\beta _{n-1}}{\\beta _{n+1}-\\beta _{n-1}}-2\\frac{(\\beta _{n+1}-\\beta _n)(\\beta _n-\\beta _{n-1})}{(\\beta _{n+1}-\\beta _{n-1})(\\beta _n+\\omega _m)}.$ It is easy to check that the function (REF ) satisfies equation (REF ) with $M=1$ for any values of the arbitrary functions $\\beta _n,\\omega _m.$ So we have got the explicit general solution of (REF ) with $M=1$ ."
],
[
"Cases $M=2$ and {{formula:820a4f1f-a440-4741-95e2-15a505045cb7}}",
"Equation (REF ) is equivalent to $(u_{n+1,m}+1)(u_{n,m}-1)=\\theta _M^m\\lambda _n,$ where $\\lambda _n$ is an arbitrary function.",
"It is convenient to go from (REF ) to an equivalent system by using the transformation (REF ): $(u^{(j)}_{n+1,k}+1)(u^{(j)}_{n,k}-1)=\\theta _M^j\\lambda _n,\\quad 1\\le j\\le M.$ Let us note that $j$ is a number of the function $u^{(j)}_{n,k}$ , while $n$ and $k$ are the discrete variables.",
"Unlike (REF ), the right hand side of equations (REF ) depends on one discrete variable $n$ only, as in the case of (REF ).",
"By analogy with the previous case $M=1$ , we can introduce functions $\\alpha _n^{(j)}$ , so that: $\\theta _M^j\\lambda _n=(\\alpha ^{(j)}_{n+1}+1)(\\alpha ^{(j)}_{n}-1),\\quad 1\\le j\\le M.$ Now we can apply the transformations $u^{(j)}_{n,k}=\\alpha ^{(j)}_n+\\frac{\\alpha ^{(j)}_n-1}{v^{(j)}_{n,k}}$ to get the linear equations for $v^{(j)}_{n,k}$ : $\\frac{\\alpha ^{(j)}_{n+1}+1}{\\alpha ^{(j)}_{n+1}-1}v^{(j)}_{n+1,k}+v^{(j)}_{n,k}+1=0.$ As above, we introduce functions $\\beta ^{(j)}_{n}$ , such that $\\frac{\\alpha ^{(j)}_{n+1}+1}{\\alpha ^{(j)}_{n+1}-1}=-\\frac{\\beta ^{(j)}_{n+2}-\\beta ^{(j)}_{n+1}}{\\beta ^{(j)}_{n+1}-\\beta ^{(j)}_{n}},$ and we obtain $v^{(j)}_{n,k}=\\frac{\\beta ^{(j)}_n+\\omega ^{(j)}_k}{\\beta ^{(j)}_{n+1}-\\beta ^{(j)}_n},$ where $\\omega ^{(j)}_k$ are arbitrary functions on $k$ .",
"We can write down for $u_{n,k}^{(j)}$ formulae analogues to (REF ).",
"The problem is that, instead of one $n$ -dependent arbitrary function $\\beta _n$ in the case $M=1$ , we have now $M$ functions $\\beta _n^{(j)}$ with a complex relationship between them defined by (REF ) and (REF ).",
"We can solve this problem for $M=2$ and $M=3$ in terms of the discrete integrals."
],
[
"A relation between the functions $\\alpha _{n}^{(1)}$ and $\\alpha _{n}^{(2)}$ is obtained from system (REF ) by excluding $\\lambda _n$ : $(\\alpha ^{(1)}_{n+1}+1)(\\alpha ^{(1)}_{n}-1)=-(\\alpha ^{(2)}_{n+1}+1)(\\alpha ^{(2)}_{n}-1).$ If one of these functions is known, then the second one is found from the Riccati equation.",
"If we replace $\\alpha _n^{(j)}$ by $\\beta _{n}^{(j)}$ by using $\\alpha _n^{(j)}=\\frac{\\beta _{n+1}^{(j)}-2\\beta _{n}^{(j)}+\\beta _{n-1}^{(j)}}{\\beta _{n+1}^{(j)}-\\beta _{n-1}^{(j)}},$ then a relation between the functions $\\beta _{n}^{(1)}$ and $\\beta _{n}^{(2)}$ becomes even more complex.",
"In order to solve this problem, we rewrite (REF ) in the form: $\\frac{\\alpha ^{(1)}_{n}-1}{\\alpha ^{(2)}_{n}-1}=-\\frac{\\alpha ^{(2)}_{n+1}+1}{\\alpha ^{(1)}_{n+1}+1}.$ Denoting the left hand side by $\\gamma _{n+1}$ , we derive a system for $\\alpha ^{(1)}_{n}$ and $\\alpha ^{(2)}_{n}$ : $\\frac{\\alpha ^{(1)}_{n}-1}{\\alpha ^{(2)}_{n}-1}=\\gamma _{n+1}, \\quad \\frac{\\alpha ^{(2)}_{n}+1}{\\alpha ^{(1)}_{n}+1}=-\\gamma _n,$ which is solved as follows: $\\alpha ^{(1)}_n=-\\frac{\\gamma _{n+1}\\gamma _n+2\\gamma _{n+1}-1}{\\gamma _{n+1}\\gamma _n+1},\\quad \\alpha ^{(2)}_n=\\frac{\\gamma _{n+1}\\gamma _n-2\\gamma _{n}-1}{\\gamma _{n+1}\\gamma _n+1}.$ Now we consider $\\gamma _n$ as a new arbitrary function, then the functions $\\alpha _n^{(1)}$ and $\\alpha _n^{(2)}$ are found explicitly by (REF ).",
"The functions $\\beta _n^{(1)}$ and $\\beta _n^{(2)}$ are found from (REF ) by two discrete integrations, as relations (REF ) can be rewritten in the form: $(T_n-1)\\log (\\beta ^{(j)}_{n+1}-\\beta ^{(j)}_{n})=\\log \\frac{1+\\alpha ^{(j)}_{n+1}}{1-\\alpha ^{(j)}_{n+1}} .$ Let us use (REF ) and (REF ) to derive a formula for the solution $u_{n,m}$ : $u_{n,m}=\\chi _{m+1}\\left(\\alpha ^{(1)}_n+\\frac{(\\alpha ^{(1)}_n-1)(\\beta _{n+1}^{(1)}-\\beta _{n}^{(1)})}{\\beta _{n}^{(1)}+\\omega _m}\\right)+\\chi _{m}\\left(\\alpha ^{(2)}_n+\\frac{(\\alpha ^{(2)}_n-1)(\\beta _{n+1}^{(2)}-\\beta _{n}^{(2)})}{\\beta _{n}^{(2)}+\\omega _m}\\right),$ where $\\chi _m=\\frac{1+(-1)^m}{2},\\quad \\omega _{2k+1}=\\omega ^{(1)}_k,\\quad \\omega _{2k+2}=\\omega ^{(2)}_k.$ In formula (REF ) we have two arbitrary functions $\\gamma _n$ and $\\omega _m$ , and the functions $\\alpha _{n}^{(j)}$ and $\\beta _{n}^{(j)}$ are defined as explained above.",
"The functions $\\alpha _{n}^{(j)}$ are found explicitly, while the functions $\\beta _{n}^{(j)}$ are in quadratures."
],
[
"Relations between the functions $\\alpha _{n}^{(j)},\\ \\ j=1,2,3,$ are in this case: $\\begin{split}(\\alpha ^{(2)}_{n+1}+1)(\\alpha ^{(2)}_{n}-1)=\\theta _3 (\\alpha ^{(1)}_{n+1}+1)(\\alpha ^{(1)}_{n}-1),\\\\ (\\alpha ^{(3)}_{n+1}+1)(\\alpha ^{(3)}_{n}-1)=\\theta _3 (\\alpha ^{(2)}_{n+1}+1)(\\alpha ^{(2)}_{n}-1).\\end{split}$ We rewrite these relations to introduce new functions $\\gamma ^{(1)}_n$ and $\\gamma ^{(2)}_n$ : $\\begin{split}\\frac{\\alpha ^{(2)}_{n+1}+1}{\\alpha ^{(1)}_{n+1}+1}=\\theta _3 \\frac{\\alpha ^{(1)}_{n}-1}{\\alpha ^{(2)}_{n}-1}=\\gamma ^{(1)}_{n+1},\\\\ \\frac{\\alpha ^{(3)}_{n+1}+1}{\\alpha ^{(2)}_{n+1}+1}=\\theta _3 \\frac{\\alpha ^{(2)}_{n}-1}{\\alpha ^{(3)}_{n}-1}=\\gamma ^{(2)}_{n+1}.\\end{split}$ Using the shift operator $T_n$ , we get two systems for three functions $\\alpha _{n}^{(j)}$ .",
"Solutions of these systems read: $\\alpha ^{(1)}_n=\\frac{2(\\gamma ^{(1)}_{n+1}-\\theta _3)}{\\gamma ^{(1)}_{n+1}\\gamma ^{(1)}_{n}-\\theta _3}-1,\\quad \\alpha ^{(2)}_n=\\frac{2\\theta _3(1-\\gamma ^{(1)}_{n})}{\\gamma ^{(1)}_{n+1}\\gamma ^{(1)}_{n}-\\theta _3}+1,\\\\\\alpha ^{(2)}_n=\\frac{2(\\gamma ^{(2)}_{n+1}-\\theta _3)}{\\gamma ^{(2)}_{n+1}\\gamma ^{(2)}_{n}-\\theta _3}-1,\\quad \\alpha ^{(3)}_n=\\frac{2\\theta _3(1-\\gamma ^{(2)}_{n})}{\\gamma ^{(2)}_{n+1}\\gamma ^{(2)}_{n}-\\theta _3}+1.$ We have to agree two different formulae for the function $\\alpha ^{(2)}_n$ .",
"It is convenient to do it for the following function of $\\alpha _{n}^{(2)}$ : $\\frac{\\alpha ^{(2)}_n+1}{\\alpha ^{(2)}_n-1}=\\frac{\\gamma ^{(1)}_{n}(\\theta _3-\\gamma ^{(1)}_{n+1})}{\\theta _3(\\gamma ^{(1)}_{n}-1)}=\\frac{\\theta _3-\\gamma ^{(2)}_{n+1}}{\\gamma ^{(2)}_{n+1}(\\gamma ^{(2)}_{n}-1)}.$ We rewrite the last equality in the form $\\frac{\\gamma ^{(1)}_{n}(\\gamma ^{(2)}_{n}-1)}{\\gamma ^{(1)}_{n}-1}=\\theta _3\\frac{\\gamma ^{(2)}_{n+1}-\\theta _3}{\\gamma ^{(2)}_{n+1}(\\gamma ^{(1)}_{n+1}-\\theta _3)}$ and denote the left hand side by $\\delta _{n+1}-1.$ Using the shift $T_n$ , we get a system for $\\gamma ^{(1)}_{n}$ and $\\gamma ^{(2)}_{n}$ , which can be expressed as: $\\gamma ^{(1)}_{n}=\\frac{\\delta _{n+1}-1}{\\delta _{n+1}-\\gamma ^{(2)}_{n}},$ $\\theta _3\\delta _n(\\gamma ^{(2)}_{n})^2-[(\\theta _3-1)\\delta _{n+1}\\delta _n+\\delta _{n+1}+\\delta _n+\\theta _3^2-1]\\gamma ^{(2)}_{n}+\\theta _3^2\\delta _{n+1}=0.$ Let us now consider $\\delta _n$ as a new arbitrary function.",
"All the other $n$ -dependent functions are expressed via it.",
"The functions $\\gamma ^{(1)}_{n},\\gamma ^{(2)}_{n}$ are given by (REF ,REF ), the functions $\\alpha _n^{(1)},\\alpha _n^{(2)}$ and $\\alpha _n^{(3)}$ are found from (REF ,), and the functions $\\beta _n^{(1)},\\beta _n^{(2)}$ and $\\beta _n^{(3)}$ are found from (REF ).",
"Let us note that the functions $\\alpha _n^{(j)}$ and $\\gamma _n^{(1)}$ are found explicitly, while the functions $\\beta _n^{(j)}$ are found by two discrete integrations, and $\\gamma _n^{(2)}$ is defined implicitly by the quadratic equation.",
"Solutions $u_{n,k}^{(1)},u_{n,k}^{(2)}$ and $u_{n,k}^{(3)}$ of the system (REF ) are given by (REF ,REF ).",
"These solutions depend on the arbitrary functions $\\delta _n$ and $\\omega ^{(1)}_k,\\omega ^{(2)}_k,\\omega ^{(3)}_k$ , see (REF ).",
"Let us come back to a solution $u_{n,m}$ of equation (REF ) with $M=3$ , equivalent to equation (REF ), which is given by transformation (REF ).",
"From the viewpoint of this solution we have two arbitrary functions $\\delta _n$ and $\\omega _m$ , where $\\omega _{3k+1}=\\omega ^{(1)}_k,\\quad \\omega _{3k+2}=\\omega ^{(2)}_k,\\quad \\omega _{3k+3}=\\omega ^{(3)}_k.$"
],
[
"Modified series",
"Here we use a transformation theory developed in [13], [12].",
"Let us rewrite equation (REF ) in the form: $\\frac{u_{n,m+1}-1}{u_{n,m}-1}=\\theta _M\\frac{u_{n+1,m}+1}{u_{n+1,m+1}+1}.$ This allows us to introduce a new function $v_{n,m}$ , so that: $v_{n,m}=\\theta _M\\frac{u_{n,m}+1}{u_{n,m+1}+1},\\quad v_{n+1,m}=\\frac{u_{n,m+1}-1}{u_{n,m}-1}.$ Resulting relations can be rewritten as: $u_{n,m}=\\frac{v_{n+1,m}v_{n,m}-2v_{n,m}+\\theta _M}{v_{n+1,m}v_{n,m}-\\theta _M}\\quad u_{n,m+1}=-\\frac{v_{n+1,m}v_{n,m}-2\\theta _Mv_{n+1,m}+\\theta _M}{v_{n+1,m}v_{n,m}-\\theta _M}.$ Rewriting these formulae at the same point $u_{n,m+1}$ , we get an equation for $v_{n,m}$ : $(v_{n+1,m+1}-1)(v_{n,m}-\\theta _M)=\\theta _M(1-v_{n+1,m}^{-1})(1-\\theta _Mv_{n,m+1}^{-1}),$ where $\\theta _M$ is the primitive root of unit.",
"For all the equations of the form (REF ) we have got transformation (REF ) which is invertible on the solutions of (REF ).",
"Two series of equations (REF ) and (REF ) are equivalent up to this transformation.",
"The particular case $M=2$ of equation (REF ) is presented in [2] up to $v_{n,m}\\rightarrow -v_{n,m}$ together with the first integrals.",
"In the case $M=1$ we can apply the point transformation $v_{n,m}=1+w_{n,m}^{-1}$ and get the following equation: $w_{n+1,m+1}w_{n,m}=(w_{n+1,m}+1)(w_{n,m+1}+1).$ This is nothing but the discrete Liouville equation found in [11].",
"Its first integrals and general solution have been constructed in [1].",
"In the general case, by using transformation (REF ), we can rewrite the first integrals.",
"Theorem 3 For any $M\\ge 1$ equation (REF ) has the following first integrals in the $n$ - and $m$ -directions, respectively: $W_{1,M}&=&\\theta _M^{-m}\\frac{(v_{n+2,m}-1)v_{n+1,m}(v_{n,m}-\\theta _M)}{(v_{n+2,m}v_{n+1,m}-\\theta _M)(v_{n+1,m}v_{n,m}-\\theta _M)},\\\\W_{2,M}&=&\\frac{\\left(V_{n,m+2M}^{(M)}V^{(M)}_{n,m+M}-1\\right)\\left(V^{(M)}_{n,m+M}V^{(M)}_{n,m}-1\\right)}{\\left(V^{(M)}_{n,m+2M}-1\\right)V^{(M)}_{n,m+M}\\left(V^{(M)}_{n,m}-1\\right)},\\\\\\nonumber &V^{(M)}_{n,m}&=v_{n,m}v_{n,m+1}\\ldots v_{n,m+M-1}.$ Proof.",
"First integral (REF ) is obtained from (REF ) by direct calculation, using transformation (REF ).",
"In order to derive (), we need auxiliary relation.",
"It follows from the first relation of (REF ) that for any $k\\ge 1$ one has: $\\frac{u_{n,m}+1}{u_{n,m+k}+1}=\\frac{u_{n,m}+1}{u_{n,m+1}+1}\\frac{u_{n,m+1}+1}{u_{n,m+2}+1}\\ldots \\frac{u_{n,m+k-1}+1}{u_{n,m+k}+1}=\\theta _M^{-k}v_{n,m}v_{n,m+1}\\ldots v_{n,m+k-1}.$ Now first integral (REF ) can be rewritten in the form: $\\begin{split}W_{2,M}=&\\frac{[(u_{n,m+3M}+1)-(u_{n,m+M}+1)][(u_{n,m+2M}+1)-(u_{n,m}+1)]}{[(u_{n,m+3M}+1)-(u_{n,m+2M}+1)][(u_{n,m+M}+1)-(u_{n,m}+1)]}\\\\=&\\frac{\\left(\\frac{u_{n,m+3M}+1}{u_{n,m+M}+1}-1\\right)\\left(1-\\frac{u_{n,m}+1}{u_{n,m+2M}+1}\\right)}{\\left(\\frac{u_{n,m+3M}+1}{u_{n,m+2M}+1}-1\\right)\\left(1-\\frac{u_{n,m}+1}{u_{n,m+M}+1}\\right)}\\\\=&\\frac{\\left(\\frac{\\theta _M^{2M}}{V^{(M)}_{n,m+M}V^{(M)}_{n,m+2M}}-1\\right)\\left(1-\\theta _M^{-2M}V^{(M)}_{n,m}V^{(M)}_{n,m+M}\\right)}{\\left(\\frac{\\theta _M^M}{V^{(M)}_{n,m+2M}}-1\\right)\\left(1-\\theta _M^{-M}V^{(M)}_{n,m}\\right)}.\\end{split}$ As $\\theta _M^M=1$ , we are led to first integral ().",
"The order of first integral (REF ) is equal to two.",
"Let us show that this order is minimally possible.",
"If in the $n$ -direction there exists a first integral $\\widetilde{W}_{1,n,m}(v_{n+1,m},v_{n,m})$ for equation (REF ), then we use transformation (REF ) and get a first integral for equation (REF ) of the nonstandard form $\\widehat{W}_{1,n,m}(u_{n,m+1},u_{n,m})$ satisfying the relation $(T_m-1)\\widehat{W}_{1,n,m}=0$ .",
"It is easy to check that this is impossible.",
"The order of first integral () is equal to $3M-1$ .",
"In the cases $M=1$ and $M=2$ , the fact that this order $3M-1$ is minimally possible follows from [1] and [2], respectively.",
"In the case $M=3$ we can prove the same, using the fact that the order 9 of corresponding first integral (REF ) of equation (REF ) is minimal, see Theorem REF .",
"In fact, in the case $M=3$ let us suppose that equation (REF ) has a first integral in the $m$ -direction $\\widetilde{W}_{2,n,m}(v_{n,m+k},v_{n,m+k-1},\\ldots ,v_{n,m})$ of an order $1\\le k\\le 7$ .",
"This means that for some $n,m$ $\\frac{\\partial \\widetilde{W}_{2,n,m}}{\\partial v_{n,m}}\\ne 0,\\quad \\frac{\\partial \\widetilde{W}_{2,n,m}}{\\partial v_{n,m+k}}\\ne 0.$ By using the first relation of (REF ), we rewrite $\\widetilde{W}_{2,n,m}$ in terms $u_{n,m+j}$ and get a first integral for equation (REF ) of the following form: $\\widehat{W}_{2,n,m}(u_{n,m+k+1},u_{n,m+k},\\ldots ,u_{n,m}).$ It easy to prove that its order equals $k+1$ , where $2\\le k+1\\le 8<9$ , but this is impossible.",
"Finally we remark that, using the results of Section and the first of transformations (REF ), we can construct the general solutions for equations (REF ) with $M=1,2,3$ ."
]
] | 1906.04503 |
[
[
"Aerogel-based metasurfaces for perfect acoustic energy absorption"
],
[
"Abstract The unusual viscoelastic properties of silica aerogel plates are efficiently used to design subwavelength perfect sound absorbers.",
"We theoretically, numerically and experimentally report a perfect absorbing metamaterial panel made of periodically arranged resonant building blocks consisting of a slit loaded by a clamped aerogel plate backed by a closed cavity.",
"The impedance matching condition is analyzed using the Argand diagram of the reflection coefficient, i.e., the trajectory of the reflection coefficient as a function of frequency in the complex plane.",
"The lack or excess of losses in the system can be identified via this Argand diagram in order to achieve the impedance matching condition.",
"The universality of this tool can be further exploited to design more complex metasurfaces for perfect sound absorption, thus allowing the rapid design of novel and efficient absorbing metamaterials."
],
[
"[]Aerogel-based metasurfaces for perfect acoustic energy absorption Antonio A. Fernández-Marín Wave Phenomena Group, Departamento de Ingeniería Electrónica, Universitat Politècnica de València, Camino de Vera s/n, 46022 València, Spain Laboratoire d'Acoustique de l'Université du Mans, LAUM - UMR 6613 CNRS, Le Mans Université, Avenue Olivier Messiaen, 72085 LE MANS CEDEX 9, France Noé Jiménez Laboratoire d'Acoustique de l'Université du Mans, LAUM - UMR 6613 CNRS, Le Mans Université, Avenue Olivier Messiaen, 72085 LE MANS CEDEX 9, France Instituto de Instrumentación para Imagen Molecular (i3M), CSIC-UPV, Camino de Vera s/n, 46022 València, Spain Jean-Philippe Groby Laboratoire d'Acoustique de l'Université du Mans, LAUM - UMR 6613 CNRS, Le Mans Université, Avenue Olivier Messiaen, 72085 LE MANS CEDEX 9, France José Sánchez-Dehesa Wave Phenomena Group, Departamento de Ingeniería Electrónica, Universitat Politècnica de València, Camino de Vera s/n, 46022 València, Spain Vicente Romero-García Laboratoire d'Acoustique de l'Université du Mans, LAUM - UMR 6613 CNRS, Le Mans Université, Avenue Olivier Messiaen, 72085 LE MANS CEDEX 9, France The unusual viscoelastic properties of silica aerogel plates are efficiently used to design subwavelength perfect sound absorbers.",
"We theoretically, numerically and experimentally report a perfect absorbing metamaterial panel made of periodically arranged resonant building blocks consisting of a slit loaded by a clamped aerogel plate backed by a closed cavity.",
"The impedance matching condition is analyzed using the Argand diagram of the reflection coefficient, i.e., the trajectory of the reflection coefficient as a function of frequency in the complex plane.",
"The lack or excess of losses in the system can be identified via this Argand diagram in order to achieve the impedance matching condition.",
"The universality of this tool can be further exploited to design more complex metasurfaces for perfect sound absorption, thus allowing the rapid design of novel and efficient absorbing metamaterials.",
"Silica aerogels are extremely-lightweight nanoporous materials[1].",
"The frame of these materials consists of an assembly of connected small cross-sections beam-like elements resulting from fused nanoparticles.",
"This particular assembly provides silica aerogel a very low elastic stiffness when compared to rigid silica structure of identical porosity.",
"Aerogels possess a wide variety of exceptional properties such as low thermal conductivity, low dielectric constant, low index of refraction or a very large porosity (80-99.8$\\%$ ) thus providing these materials an extremely low density[2].",
"Because of this large porosity and therefore of the very large available contact area, they have been used as filters, absorbent media or waste containment (see Refs.",
"[Cooper89,Gesser89,Komarneni93] and references therein).",
"They have also been applied as catalysts or even to capture cosmic dust [5], [6].",
"Similarly, their low thermal conductivity, which so far seems to be their most interesting property compared to any other elastic or poroelastic material, has been exploited in various commercial applications including thermal insulation [7], heat and cold storage devices [5], [8].",
"In acoustics, aerogels are used as impedance matching materials to develop efficient ultrasonic devices [9], [10] or sound absorbing materials for anechoic chambers[5], [11].",
"Beyond these properties, silica aerogel plates are excellent candidates to design new types of membrane metamaterials, since they exhibit subwavelength resonances and present absorption efficiency[12], [13].",
"Effectively, the use of membrane metamaterials to control acoustic waves has shown an increasing interest in recent years[14].",
"Membrane and plate metamaterials have been employed in the past to design efficient absorbers[15], [16], [17], e.g., using a single membrane backed by a cavity [16], [18], which can present deeper subwavelength resonances as compared with absorbing metamaterials based in air cavities[19], [20], [21], [22].",
"Moreover, double negative acoustic metamaterials[23] can be achieved by combining a lattice of membranes, which provides a negative effective mass density [24], with subwavelength side-branch resonators, which provides a negative effective bulk modulus[25].",
"In addition, periodic arrangements of clamped plates have been efficiently used to control harmonic generation[26] or solitary waves in the nonlinear acoustic regime[27].",
"In this work, we make use of the efficient and unusual attenuating properties of silica aerogel plates to design subwavelength perfect sound absorbers.",
"The analyzed system is depicted in Fig.",
"REF (a) and consists of a periodic repetition of the resonant building units illustrated in Fig.",
"REF (b).",
"These units are made of a slit loaded by a clamped aerogel plate backed by a closed cavity.",
"The perfect absorption of this system is comprehensively analyzed both theoretically and experimentally.",
"In a first stage, we model the system using the Transfer Matrix Method (TMM) accounting for the contribution of the losses to the problem, i.e., the viscothermal losses from the slit and cavity and the viscoelastic losses from the aerogel plate.",
"In a second stage, we analyze the impedance matching condition, also known as critical coupling condition, which is obtained once the inherent losses exactly compensates the leakage of the system[28].",
"The Argand diagram of the reflection coefficient [29] is further employed to evaluate either the lack or excess of inherent losses in the system, thus providing important information on the impedance matching condition.",
"The Argand diagram is revealed as a universal and powerful tool to design perfect absorbers.",
"We consider a slotted panel of thickness $L$ , whose slits, of height $h_s$ , are loaded by a circular clamped aerogel plate of radius $r_m$ and thickness $h_m$ backed by a cylindrical air cavity of the same radius and depth $l_c$ .",
"The clamped aerogel plate plays the central role, as it represents the main source of intrinsic losses and mainly governs the resonance of the system.",
"Although the theoretical calculations are only carried out for the building block of size $L\\times a\\times a$ shown in Fig.",
"REF (b), the generalization to the case of $N$ resonator unit cell is straightforward.",
"Figure: (color online) (a) Scheme of the panel under consideration constructed by a slit, the aerogel plate, and the cavity.",
"(b) Schematic description of the unit cell.For wavelengths $\\lambda $ large enough compared to the thickness of the aerogel plate $h_m$ and neglecting the effects of rotary inertia and additional deflections caused by shear forces, the transverse plate displacement $w_m$ satisfies the Kirchhoff-Love wave equation [30].",
"The plate can be described by $\\rho $ , the density and $D={Eh_m^3}/{12(1-\\nu ^2)}$ , the bending stiffness.",
"$E$ is the Young modulus and $\\nu $ is the Poisson's ratio of the plate.",
"Assuming an implicit time dependence $e^{i\\omega t}$ , with $\\omega $ the angular frequency, the viscoelastic behavior of silica aerogel can be modeled via a complex Young modulus, $E=E_0(1+i\\eta \\omega )$ , where $E_0$ and $\\eta $ are the unrelaxed Young modulus and the loss factor respectively.",
"In the subwavelength regime, the silica aerogel disk can be considered as a punctual resonant element located at $(x,y)=(L/2,a/2)$ (note that $h_m\\ll \\lambda _0$ , where $\\lambda _0$ is the wavelength in air).",
"The acoustic impedance of the clamped circular cross-sectional plate thus takes the form [31], [32], $ Z_p = \\frac{-i\\omega \\rho h_m}{\\pi r_m^2}\\frac{ I_1(k r_m) J_0(k r_m) + J_1(k r_m)I_0(k r_m)}{I_1(k r_m)J_2(k r_m)-J_1(k r_m)I_2(k r_m)},$ where $J_n$ and $I_n$ are the of $n$ -th order regular and modified Bessel's functions of the first kind respectively and the wavenumber in the plate satisfies $k^2=\\omega \\sqrt{\\rho h_m/D}$ .",
"Viscothermal losses also occurs in the narrow slits[33] and in the cavity, also offering a useful degree of freedom to tune the losses of the system.",
"Assuming only plane wave propagates in these channels, the viscothermal losses are modeled by effective parameters: complex and frequency-dependent wavenumbers $k_s=\\omega \\sqrt{\\rho _s/\\kappa _s}$ and $k_c=\\omega \\sqrt{\\rho _c/\\kappa _c}$ , and impedances $Z_s=\\sqrt{\\kappa _s\\rho _s}/h_s a$ and $Z_c=\\sqrt{\\kappa _c\\rho _c}/\\pi r_m^2$ in the slit and in the cavity respectively.",
"Note that we make use of the effective density $\\rho _s$ and bulk modulus $\\kappa _s$ of a slit [34] for the slotted channel, while we make use of the effective density $\\rho _c$ and bulk modulus $\\kappa _c$ of a cylindrical duct[34] for the cavity.",
"Figure: (color online) Reflection coefficient, in logarithmic scale, as a function of the frequency and (a) slit length, LL, and (b) cavity depth, l c l_c.",
"The rest of parameters are fixed to the optimal geometry (see main text).",
"(c) Representation of the reflection coefficient in the complex frequency plane for the sample with optimized parameters.The lines show the trajectory of the zero when the corresponding geometrical parameter is modified.",
"(d) and (e) show the absorption coefficient and the phase of the reflection coefficient respectively for three values of the slit thickness, h s ' h^\\prime _s, h s h_s (black line) being the optimum value.",
"(f) Argand diagram of the complex reflection coefficient from 0 to 1200 Hz, for the lossless structure (dashed circle), the case of perfect absorption (PA) geometry (black circle), the case h s ' =h s /2h_s^{\\prime }=h_s/2, (red circle) and the case h s ' =2h s h_s^{\\prime }=2h_s (blue circle).",
"The small arrows indicate the trajectory from low to high frequencies.The scattering properties of the system are studied through the reflection coefficient $R$ obtained by TMM.",
"We relate the sound pressure and normal particle velocity at the surface of the system, $[P_0,V_0] = [P(x), V_x(x)]_{x=0}$ to the ones at the end of the system, $[P_L,V_L] = [P(x), V_x(x)]_{x=L}$ , by a transfer matrix as $\\left[\\begin{array}{c}P_0 \\\\V_0 \\\\\\end{array}\\right] = {\\bf T} \\left[\\begin{array}{c}P_L \\\\V_L \\\\\\end{array}\\right],\\,\\text{where}\\quad {\\bf T}={\\bf M}_{\\Delta l}{\\bf M}_S{\\bf M}_{R}{\\bf M}_S.$ In Eq.",
"(REF ), transfer matrix over half the slit length ${\\bf M}_S$ reads as ${\\bf M}_S=\\left[\\begin{array}{cc}\\cos \\left(k_s{L}/{2}\\right) & i Z_s\\sin \\left(k_s{L}/{2}\\right) \\\\{i}\\sin \\left(k_s{L}/{2}\\right)/{Z_s} & \\cos \\left(k_s{L}/{2}\\right) \\\\\\end{array}\\right],$ and ${\\bf M}_{R}$ accounts for the local effect of the aerogel plate together with the back cavity as ${\\bf M}_{R}=\\left[\\begin{array}{cc}1 & 0 \\\\{1}/{Z_{R}} & 1 \\\\\\end{array}\\right],$ where $Z_{R}=Z_p - i Z_c\\cot (k_c l_c)$ .",
"The matrix ${\\bf M}_{\\Delta l}$ provides the radiation correction of the slit to the free space as ${\\bf M}_{\\Delta l}=\\left[\\begin{array}{cc}1 & Z_{\\Delta l} \\\\0 & 1\\end{array}\\right],$ where $Z_{\\Delta l}=-i\\omega \\Delta l\\rho _0/\\phi _sa^2$ , with $\\phi _s=h_s/a$ the surface porosity of the metasurface, $\\rho _0$ the air density and $\\Delta l$ the end correction length that can be approximated as[35] $\\Delta l=h_s\\phi _s\\sum _{n=1}^{\\infty }\\frac{\\sin ^2(n\\pi \\phi _s)}{\\left(n\\pi \\phi _s\\right)^3}.$ The surface impedance at $x=0$ can thus be directly obtained using Eq.",
"(REF ) and considering the rigid backing condition ($V_L = 0$ ) as $Z_T = \\frac{P_{0}}{V_{0}}= \\dfrac{Z_s (Z_{\\Delta l}+Z_{R})+i \\tan \\left({k_s L}/{2}\\right) \\left[i Z_s Z_{R} \\tan \\left({k_s L}/{2}\\right)+2 Z_{\\Delta l} Z_{R}+Z_s^2\\right]}{Z_s+ 2 iZ_{R} \\tan \\left({k_s L}/{2}\\right)}.$ Finally, we calculate the reflection and absorption coefficient using Eq.",
"(REF ) as $R=\\frac{Z_T-Z_0}{Z_T+Z_0},\\quad \\text{and}\\quad \\alpha =1-|R|^2,$ where $Z_0$ is the impedance of the surrounding medium, i.e., the airFor the air medium at room temperature and ambient pressure $P_0 = 101325$ Pa, we used an adiabatic coefficient $\\gamma = 1.4$ , a density $\\rho _0 = 1.213$ kg/m$^3$ , a bulk modulus $\\kappa _0 = \\gamma P_0$ , a Prandtl number $\\mathrm {Pr} = 0.71$ , a viscosity $\\eta _0 = 1.839\\times 10^{-5}$ Pa$\\cdot $ s, a sound speed $c_0 = \\sqrt{\\gamma P_0/\\rho _0}$ m/s, and an acoustic impedance $Z_0 = \\rho _0 c_0 / a^2$ .. For the aerogel plate, we used an unrelaxed Young modulus $E_0=197.92$ kPa and a loss factor $\\eta =4.47\\times 10^{-6}$ Pa$\\cdot $ s, a density $\\rho = 80$ kg/m$^3$ and a Poison's ratio $\\nu = 0.12$ , as characterized in Ref.",
"[Geslaina2018].",
"Aerogel plates of $r_m=19.5$ mm and $h_m=10.5$ mm were selected and $a=42$ mm was fixed by the width of the square cross-sectional impedance tube that was used for the experimental validation.",
"The procedure begins with looking for the geometric parameters giving the most efficient absorption at the lowest resonance frequency.",
"The geometric parameters of the system are thus optimized numerically using a sequential quadratic programming (SQP) method [37].",
"The following parameters were obtained: $L=44$ mm, $h_s=1.285$ mm, and $l_c=29.8$ mm.",
"The highly efficient absorption peak also appears at 591.2 Hz and is associated with a reflection coefficient amplitude of $10\\log _{10}|R| = -62$ dB.",
"The corresponding perfectly absorbed wavelength is $\\lambda /L=13.1$ times larger than he depth of the structure.",
"This subwavelength feature is due to the slow sound properties induced by the presence of the slit loading resonators[38], [39], [20].",
"Effectively, the resonance frequency of the slit in the absence of these loading resonators is around 1900 Hz.",
"Note that the slow sound properties can be improved by using thinner aerogel plates, thereby allowing a strongly reduced ratio, e.g., $\\lambda /L=30$ at $f=259$ Hz using $h_m = 1.32$ and $h_s= 0.95$ mm.",
"Nevertheless, we are constrained by the available aerogel plates ($h_m=10.5$ mm) for the experimental validation.",
"Figure: (color online) (a) Photograph of the experimental configuration.",
"(b) Absorption as a function of the frequency.",
"(c) Complex plane representation of the reflection coefficient.",
"The small arrows indicate the trajectory from low to high frequencies.Figures REF (a) and REF (b) show a parametric study of the system reflection coefficient around the optimal configuration.",
"The reflection coefficient is significantly reduced when the parameters correspond to the optimal parameters (marked by white crosses).",
"However, the balance between the inherent losses and the leakage of the system is difficult to identify by using this parametric analysis.",
"A first approach to ensure that these parameters led to perfect absorption of the sound energy consists in representing the reflection coefficient in the complex frequency plane, as shown in Fig.",
"REF (c).",
"Using this representation, the locations of the zero/pole pairs of the reflection coefficient can be studied.",
"In the lossless case, the zeros are complex conjugates of their corresponding poles, both appearing in the opposite half spaces of complex frequency plane (zeros in the lower half space and poles in the upper one with our time Fourier convention).",
"However, the zeros follows a given trajectory towards the pole half space when losses are introduced in the system.",
"Note that the losses are modified when modifying the system geometry.",
"In this way, the trajectory of the lowest frequency zero is depicted Fig.",
"REF (c), when $h_s$ , $l_c$ and $L$ are modified.",
"For a given set of geometric parameters, the trajectory of the zero crosses the real frequency axis, ensuring the balance of the leakage by the inherent losses, therefore providing the perfect absorption[28].",
"The complex frequency plane also gives useful insights to design and tune open lossy resonant systems[18], [20], [41].",
"Such system is characterized by its leakage and the inherent losses; the impedance matching condition corresponds to the critical coupling of the system, i.e., the perfect balance of the leakage by the inherent losses.",
"On the one hand, the intrinsic losses of the system are too large (too small) compared to the leakage of the system when the zero has (has not) already crossed the real axis, meaning that the absorption is not optimal as the impedance condition is not satisfied.",
"These situations are illustrated Fig.",
"REF (d), where the absorption coefficient is depicted for different values of $h_s$ .",
"The red curve corresponds to a narrow slit (height $h_s^{\\prime }=h_s/2$ ) providing an excess of losses, while the blue curve corresponds to a wide slit (height $h_s^{\\prime }=2 h_s$ ) providing a lack of losses.",
"The location of the corresponding zero in the complex frequency plane is marked with red and blue crosses in Fig.",
"REF (c).",
"The perfect balance between the leakage by the losses is the situation depicted on the color map Fig.",
"REF (c); the zero of the reflection coefficient is exactly located on the real frequency axis.",
"Therefore, we can conclude that the complex frequency plane is very useful to immediately identify if one particular configuration has a lack or excess of losses.",
"This approach has been recently used to design absorbing materials ranging from porous media[42], [43] to different kind of metamaterials[44], [41], [45], [39], [18].",
"However, the acoustic behavior of all systems cannot necessarily be assessed in the complex frequency plane.",
"Effectively, numerical methods do not usually allow to calculate solutions for complex frequencies, and more importantly, experimental results are usually only provided for real frequencies.",
"A useful approach to overcome this problem consists in analyzing the reflection coefficient $R=|R|e^{i\\varphi }$ , with $\\varphi = \\arctan [\\textrm {Im}(R)/\\textrm {Re}(R)]$ , in the complex plane.",
"Figure REF (e) and (f) depict respectively the phase of the reflection coefficient and the corresponding Argand diagram from 400 to 800 Hz.",
"The reflection coefficient is necessarily inscribed within the unitary circle, i.e., $|R|\\le 1$ .",
"In the lossless case, the reflection coefficient follows the unitary circle counter-clockwise with increasing frequency starting from $\\varphi =0$ at 0 Hz, as $R=e^{i\\varphi }$ .",
"When losses are accounted for, the trajectory of $R$ is modified and follows an elliptical trajectory around the resonance, contained inside of the unitary circle and displaced along the real axis in the diagram.",
"On the one hand, the reflection coefficient describes a loop that does not encompass the origin if the losses exceed the optimal ones, e.g.",
"the red ellipse Figure REF (f) calculated for $h_s^{\\prime }=h_s/2$ .",
"On the other hand, the ellipse encompasses the origin if the losses lacks, e.g.",
"the blue ellipse Figure REF (f) calculated for $h_s^{\\prime }=2h_s$ .",
"Finally, the ellipse must pass through the origin, i.e., $R=0$ , when perfect absorption is reached, e.g.",
"the black ellipse Figure REF (f) calculated for $h_s^{\\prime }=h_s$ .",
"In this situation, the impedance matching condition is satisfied.",
"The designed optimal structure was validated experimentally in a square cross-sectional impedance tube.",
"The circular aerogel plate was cut by laser cutting and then inserted in a 3D-printed support manufactured by stereolithography (Form 2, Formlabs, UK).",
"In addition, full wave numerical simulations by finite element method (FEM) were performed.",
"For the FEM simulations, the plate was modeled as an elastic bulk plate of thickness $h_m$ considering a Kelvin-Voigt viscoelastic model and viscothermal losses were accounted for in the ducts using effective parameters as previously introduced for TMM calculations.",
"Figure REF (a) shows the 3D printed system together with the aerogel plate before assembling.",
"The measured absorption is shown in Fig.",
"REF (b).",
"A good agreement is observed between the measurements, FEM simulations and TMM predictions.",
"The ripples observed in the experimental data are probably due to non-symmetrical errors during the manufacturing of the circular plate, as well as to the fact that the plate is not perfectly clamped.",
"Finally, Fig.",
"REF (c) presents the Argand diagram of the reflection coefficient measured and calculated with the TMM.",
"Both curves passe through the origin at a specific frequency that corresponds to the one at which the system is impedance matched.",
"In summary, we have designed and manufactured a resonant building block made of cavity backed aerogel clamped plates that is suitable and efficient for perfect sound absorbing panel.",
"The experimental data agree with those predicted by both the one-dimensional TMM model and the FEM simulations.",
"We have presented a universal methodology based on the complex representation of the reflection coefficient, i.e., its Argand diagram, to identify the lack or the excess of losses in the system.",
"This tool can be used further to design complex metasurfaces for perfect sound absorption when the system cannot be evaluated at complex frequencies, thus helping in the rapid design of novel and efficient absorbing metamaterials.",
"This work has been funded by the RFI Le Mans Acoustique, Région Pays de la Loire.",
"N.J. acknowledges financial support from Generalitat Valenciana through grant APOSTD/2017/042.",
"J.-P.G and V.R.G.",
"gratefully acknowledge the ANR-RGC METARoom (ANR-18-CE08-0021) project and the HYPERMETA project funded under the program Étoiles Montantes of the Région Pays de la Loire.",
"J.S-D. acknowledges the support of the Ministerio de Economía y Competitividad of the Spanish government, and the European Union FEDER through project TEC2014-53088-C3-1-R."
]
] | 1906.04496 |
[
[
"Study of semi-linear $\\sigma$-evolution equations with frictional and\n visco-elastic damping"
],
[
"Abstract In this article, we study semi-linear $\\sigma$-evolution equations with double damping including frictional and visco-elastic damping for any $\\sigma\\ge 1$.",
"We are interested in investigating not only higher order asymptotic expansions of solutions but also diffusion phenomenon in the $L^p-L^q$ framework, with $1\\le p\\le q\\le \\infty$, to the corresponding linear equations.",
"By assuming additional $L^{m}$ regularity on the initial data, with $m\\in [1,2)$, we prove the global (in time) existence of small data energy solutions and indicate the large time behavior of the global obtained solutions as well to semi-linear equations.",
"Moreover, we also determine the so-called critical exponent when $\\sigma$ is integers."
],
[
"Introduction and main results",
"In this paper, let us consider the following Cauchy problem for semi-linear $\\sigma $ -evolution equations with frictional and visco-elastic damping terms: ${\\left\\lbrace \\begin{array}{ll}u_{tt}+ (-\\Delta )^\\sigma u+ u_t+ (-\\Delta )^{\\sigma } u_t= |u|^p, \\\\u(0,x)= u_0(x),\\quad u_t(0,x)=u_1(x),\\end{array}\\right.",
"}$ where $\\sigma \\ge 1$ and a given real number $p>1$ .",
"The corresponding linear equation with vanishing right-hand side is ${\\left\\lbrace \\begin{array}{ll}u_{tt}+ (-\\Delta )^\\sigma u+ u_t+ (-\\Delta )^{\\sigma } u_t= 0, \\\\u(0,x)= u_0(x),\\quad u_t(0,x)=u_1(x).\\end{array}\\right.",
"}$ At first, let us recall some recent results concerning the study of typical important problems of (REF ) and (REF ) with $\\sigma =1$ , the so-called wave equations with frictional damping and visco-elastic damping.",
"Of special interest are the following Cauchy problems: ${\\left\\lbrace \\begin{array}{ll}u_{tt}- \\Delta u+ u_t- \\Delta u_t= |u|^p, \\\\u(0,x)= u_0(x),\\quad u_t(0,x)=u_1(x).\\end{array}\\right.",
"}$ Namely, in [14] the authors derived the asymptotic profile of the solutions in the $L^2$ setting to the corresponding linear equations of (REF ) by assuming weighted $L^{1,1}$ initial data from the energy space.",
"In comparison with the two types of damping terms, they analyzed the interesting properties which tell us that the effect of the frictional damping is really more dominant than that of the visco-elastic one, the so-called strong damping (see, for example, [12], [16]), by the study of asymptotic profile as $t\\rightarrow \\infty $ .",
"In addition, the higher order (up to the first order) asymptotic profiles of the solutions to the linear corresponding equations of (REF ) were discussed in the space dimension $n=1$ only.",
"Quite recently, the authors in [13] have succeeded in obtaining some higher order (greater than the second order) asymptotic expansions of the solutions to this linear equation under more heavy moment conditions on the initial data for any space dimensions by applying Taylor expansion theorem effectively (see more [17], [18], [20]).",
"For the treatment of the semi-linear equations (REF ), in [1] some obtained energy estimates combined with $L^1-L^1$ estimates come into play to prove the global (in time) solutions for any space dimensions.",
"Moreover, taking into consideration the effect of the two damping types as mentioned in [14] to the corresponding linear problem the authors in [15] pointed out again this effect which is still true for the semi-linear problems (REF ).",
"In particular, they indicated that the critical exponent $p_{crit}=1+\\frac{2}{n}$ coincides with the so-called Fujita exponent which is well-known to be the critical exponent for the semi-linear heat equations and the semi-linear classical damped wave equations as well with nonlinearity term $|u|^p$ .",
"Besides, not only the existence of the global solutions to (REF ) has been investigated but also the large time behavior of the obtained global solutions has been established in low space dimensions in [15].",
"Hence, related to the more general cases of (REF ) and (REF ) with $\\sigma \\ge 1$ , a natural question is whether or not the frictional damping is still more dominant than the visco-elastic one for any $\\sigma \\ge 1$ as it happened for the case $\\sigma =1$ .",
"One of the main goals of this paper is to give a positive answer to this question.",
"More recently, the authors in [6], [7] have studied the following Cauchy problem for structurally damped $\\sigma $ -evolution equations (see also [3], [4], [8]): ${\\left\\lbrace \\begin{array}{ll}u_{tt}+ (-\\Delta )^\\sigma u+ (-\\Delta )^{\\delta } u_t= 0, \\\\u(0,x)= u_0(x),\\quad u_t(0,x)=u_1(x).\\end{array}\\right.",
"}$ From the point of view of decay estimates, they emphasized that the properties of the solutions change completely from the case $\\delta =0$ , corresponding to the frictional damping, to the case $\\delta =\\sigma $ , corresponding to the visco-elastic damping.",
"More in detail, they proposed to distinguish between “parabolic like models\" in the former case (see [5], [9] for the classical damped wave equations in the case $\\sigma =1$ ) and “$\\sigma $ -evolution like models\" in the latter case, the so-called “hyperbolic like models\" or “wave like models\" in the case $\\sigma =1$ (see [5], [10]).",
"Roughly speaking, the asymptotic profile of the solutions to (REF ) with $\\delta =0$ , as $t \\rightarrow \\infty $ , is same as that of the following anomalous diffusion equations: $v_t+ (-\\Delta )^\\sigma v= 0, \\qquad v(0,x)= v_0(x),$ for a suitable choice of data $v_0$ (see, for instance, [3], [8]).",
"Meanwhile, for the case $\\delta =\\sigma $ this phenomenon is no longer true, that is, some kind of wave structure appears and oscillations come into play from the asymptotic profile of the solutions to (REF ).",
"Furthermore, compared with the regularity of the initial data we can see that a smoothing effect appears for some derivatives of the solutions to (REF ) with respect to the time variable (see [7]) in the latter case.",
"This brings some benefits in treament of the corresponding semi-linear equations.",
"Otherwise, in the former case this effect does not happen (see [6]).",
"In the connection between the two types of damping terms appearing in (REF ), the asymptotic profile of the solutions inherits both these above mentioned properties of the two kind of models to give new results.",
"For this reason, the second main goal of the present paper is to conclude a diffusion phenomenon not only in the $L^2-L^2$ theory (see more [8], [9]) but also in the $L^p-L^q$ framework (see also [2], [19]), where $1\\le p\\le q\\le \\infty $ .",
"Moreover, we also establish some higher order asymptotic expansions of the difference between the solutions to (REF ) and those to (REF ) by developing several techniques in [13].",
"In order to explain these results more precisely, one knows that these results come from estimates for small-frequency part of the solutions to (REF ) whose profile is modified by the presence of the fractional damping, whereas their large-frequency profile is modified by the presence of the visco-elastic damping.",
"Our third main goal of this paper is to prove the global (in time) existence of small data energy solutions to (REF ) and analyze the large time behavior of these global solutions as well by mixing additional $L^{m}$ regularity for the data with $m\\in [1,2)$ .",
"Finally, when $\\sigma $ is integers, a blow-up result is shown to find the critical exponent $p_{crit}=1+\\frac{2\\sigma }{n}$ ."
],
[
"Notations",
"Throughout the present paper, we use the following notations.",
"[leftmargin=*] We write $f\\lesssim g$ when there exists a constant $C>0$ such that $f\\le Cg$ , and $f \\approx g$ when $g\\lesssim f\\lesssim g$ .",
"We denote $\\hat{f}(t,\\xi ):= \\mathcal {F}_{x\\rightarrow \\xi }\\big (f(t,x)\\big )$ as the Fourier transform with respect to the space variable of a function $f(t,x)$ .",
"As usual, $H^{a}$ and $\\dot{H}^{a}$ , with $a \\ge 0$ , denote Bessel and Riesz potential spaces based on $L^2$ spaces.",
"Here $\\big <D\\big >^{a}$ and $|D|^{a}$ stand for the pseudo-differential operators with symbols $\\big <\\xi \\big >^{a}$ and $|\\xi |^{a}$ , respectively.",
"For any $\\gamma >0$ , the weighted spaces $L^{1,\\gamma }(\\mathbb {R}^n)$ are defined by $ L^{1,\\gamma }(\\mathbb {R}^n):= \\Big \\lbrace f\\in L^1(\\mathbb {R}^n) \\text{ such that } \\Vert f\\Vert _{L^{1,\\gamma }}:= \\int _{\\mathbb {R}^n} (1+|x|)^\\gamma f(x)\\,dx<+\\infty \\Big \\rbrace .",
"$ For any $s \\in \\mathbb {R}$ , we denote $[s]^+:= \\max \\lbrace s,0\\rbrace $ as its positive part, and $[s]:= \\max \\big \\lbrace k \\in \\mathbb {Z}\\,\\, : \\,\\, k\\le s \\big \\rbrace $ as its integer part.",
"We denote $\\mathbb {N}_0:= \\mathbb {N}\\cup \\lbrace 0\\rbrace $ .",
"For later convenience, we put $ G_\\sigma (t,x):=\\mathcal {F}^{-1}\\big (e^{-t|\\xi |^{2\\sigma }}\\big )(x), $ and denote the following two quantities: $P_0:= \\int _{\\mathbb {R}^n}u_0(x)dx \\quad \\text{ and }\\quad P_1:= \\int _{\\mathbb {R}^n}u_1(x)dx.",
"$ Let $\\chi (|\\xi |)$ be $\\mathcal {C}_0^\\infty (\\mathbb {R}^n)$ a smooth cut-off function equal to 1 for small $|\\xi |$ and vanishing for large $|\\xi |$ .",
"We decompose a function $f(t,x)$ into two parts localized separately at low and high frequencies as follows: $ f(t,x)= f_{\\text{\\fontshape {n}\\selectfont low}}(t,x)+ f_{\\text{\\fontshape {n}\\selectfont high}}(t,x), $ where we denote $f_{\\text{\\fontshape {n}\\selectfont low}}(t,x)= \\mathcal {F}^{-1}\\big (\\chi (|\\xi |)\\hat{f}(t,\\xi )\\big )\\quad \\text{ and }\\quad f_{\\text{\\fontshape {n}\\selectfont high}}(t,x)= \\mathcal {F}^{-1}\\Big (\\big (1-\\chi (|\\xi |)\\big )\\hat{f}(t,\\xi )\\Big ).",
"$ Applying the Fourier transform to (REF ) we have ${\\left\\lbrace \\begin{array}{ll}\\hat{u}_{tt}+ |\\xi |^{2\\sigma } \\hat{u}+ \\hat{u}_t+|\\xi |^{2\\sigma } \\hat{u}_t= 0, \\\\\\hat{u}(0,\\xi )= \\widehat{u_0}(\\xi ),\\quad \\hat{u}_t(0,\\xi )=\\widehat{u_1}(\\xi ).\\end{array}\\right.",
"}$ The characteristic equation of (REF ) is $\\lambda ^2+ (1+|\\xi |^{2\\sigma }) \\lambda + |\\xi |^{2\\sigma }= 0.$ The solution to (REF ) can be given by $\\lambda _\\pm & =\\frac{-(1+|\\xi |^{2\\sigma })\\pm \\sqrt{(1+|\\xi |^{2\\sigma })^2-4|\\xi |^{2\\sigma }}}{2}=\\frac{-(1+|\\xi |^{2\\sigma })\\pm \\big |1-|\\xi |^{2\\sigma }\\big |}{2},$ i.e., ${\\left\\lbrace \\begin{array}{ll}\\lambda _+= -|\\xi |^{2\\sigma }, \\quad \\lambda _-= -1 &\\text{ if } |\\xi |\\le 1, \\\\\\lambda _+= -1, \\quad \\lambda _-= -|\\xi |^{2\\sigma } &\\text{ if } |\\xi |\\ge 1.\\end{array}\\right.",
"}$ So we explicitly write down the solution formula in the Fourier space as follows: $\\hat{u}(t,\\xi )& =\\frac{1}{1-|\\xi |^{2\\sigma }}e^{-t|\\xi |^{2\\sigma }}\\big (\\widehat{u_0}(\\xi )+\\widehat{u_1}(\\xi )\\big )-\\frac{1}{1-|\\xi |^{2\\sigma }}e^{-t}\\big (|\\xi |^{2\\sigma } \\widehat{u_0}(\\xi )+\\widehat{u_1}(\\xi )\\big ) \\\\& =\\frac{e^{-t|\\xi |^{2\\sigma }}-|\\xi |^{2\\sigma } e^{-t}}{1-|\\xi |^{2\\sigma }}\\widehat{u_0}(\\xi )+\\frac{e^{-t|\\xi |^{2\\sigma }}-e^{-t}}{1-|\\xi |^{2\\sigma }}\\widehat{u_1}(\\xi ) \\\\& =\\frac{1}{|\\xi |^{2\\sigma }-1}e^{-t}\\big (|\\xi |^{2\\sigma } \\widehat{u_0}(\\xi )+\\widehat{u_1}(\\xi )\\big )-\\frac{1}{|\\xi |^{2\\sigma }-1}e^{-t|\\xi |^{2\\sigma }}\\big (\\widehat{u_0}(\\xi )+\\widehat{u_1}(\\xi )\\big ).",
"$ Note that $\\lbrace \\xi \\in \\mathbb {R}^n : |\\xi |=1\\rbrace $ is not a singular set.",
"Indeed, we can give an equivalent formula: $\\hat{u}(t,\\xi )=e^{-t}\\widehat{u_0}(\\xi )+\\left(e^{-t}\\int _0^te^{-s(|\\xi |^{2\\sigma }-1)} \\,ds\\right)(\\widehat{u_0}(\\xi )+\\widehat{u_1}(\\xi )).$ For purpose of this paper, we write $v_0:= u_0+ u_1$ .",
"So we fix the initial data $v_0= u_0+ u_1$ to (REF )."
],
[
"Main results",
"Let us state the main results that will be proved in this paper.",
"At first, we indicate higher order asymptotic expansions of the solutions to (REF ) with weighted initial data.",
"Theorem 1.1 Let $n\\ge 1$ and $u_0,\\,u_1\\in L^{1,\\gamma }(\\mathbb {R}^n)\\cap L^2(\\mathbb {R}^n)$ with $\\gamma \\ge 0$ .",
"Take $k\\in \\mathbb {N}_0$ satisfying $\\lambda (k)\\le \\gamma <\\lambda (k+1)$ .",
"Then, the function $u$ defined by (REF )-(REF ) satisfies $\\big \\Vert \\hat{u}(t,\\xi )-A_k^{\\sigma ,v_0}(\\xi )e^{-t|\\xi |^{2\\sigma }}\\big \\Vert _{L^2}\\lesssim t^{-\\frac{n}{4\\sigma }-\\frac{\\gamma }{2\\sigma }}\\big (\\Vert v_0\\Vert _{L^{1,\\gamma }}+\\Vert u_0\\Vert _{L^1\\cap L^2}+\\Vert u_1\\Vert _{L^1\\cap L^2}\\big ),\\qquad t\\ge 1.$ Here, we introduce $A_k^{\\sigma ,v_0}(\\xi )$ as in the expression (REF ) (see Definition REF ).",
"Furthermore, it holds $\\lim _{t\\rightarrow \\infty }t^{\\frac{n}{4\\sigma }+\\frac{\\gamma }{2\\sigma }}\\big \\Vert \\hat{u}(t,\\xi )-A_k^{\\sigma ,v_0}(\\xi )e^{-t|\\xi |^{2\\sigma }}\\big \\Vert _{L^2}=0.$ The second result is concerned with the diffusion phenomenon in the $L^p-L^q$ framework with $1\\le p\\le q\\le \\infty $ to (REF ).",
"Theorem 1.2 Let $1\\le p\\le q\\le \\infty $ .",
"Let $u$ be the solution to (REF ) and let $v$ be the solution to (REF ).",
"Then, for large $t \\ge 1$ we have the following $L^p-L^q$ estimate: $ \\big \\Vert \\partial _t^j |D|^a \\big (u_{\\text{\\fontshape {n}\\selectfont low}}(t,\\cdot )- v_{\\text{\\fontshape {n}\\selectfont low}}(t,\\cdot )\\big )\\big \\Vert _{L^q} \\lesssim (1+t)^{-\\frac{n}{2\\sigma }(\\frac{1}{p}- \\frac{1}{q})- \\frac{a}{2\\sigma }-j-1} \\big (\\Vert u_0\\Vert _{L^p}+ \\Vert u_1\\Vert _{L^p}\\big ), $ for all $a\\ge 0$ , $j=0,\\,1$ and for all space dimensions $n\\ge 1$ .",
"The third result contains the global (in time) existence of small data energy solutions to (REF ).",
"Theorem 1.3 Let $m \\in [1,2)$ .",
"We assume the conditions $&\\frac{2}{m} \\le p< \\infty &\\quad \\text{ if }\\,\\,\\, &n \\le 2\\sigma , \\\\&\\frac{2}{m} \\le p\\le \\frac{n}{n- 2\\sigma } &\\quad \\text{ if }\\,\\,\\, &n \\in \\Big (2\\sigma , \\frac{4\\sigma }{2-m}\\Big ].",
"$ Moreover, we suppose the following condition: $ p> 1+\\frac{2m\\sigma }{n}.$ Then, there exists a constant $\\varepsilon >0$ such that for any small data $(u_0,u_1) \\in \\mathcal {A}^{\\sigma }_{m}:= (L^m \\cap H^{\\sigma }) \\times (L^m \\cap L^{2}) \\text{ satisfying the assumption } \\Vert (u_0,u_1)\\Vert _{\\mathcal {A}^{\\sigma }_{m}} \\le \\varepsilon ,$ we have a uniquely determined global (in time) small data energy solution $ u \\in C([0,\\infty ),H^{\\sigma })\\cap C^1([0,\\infty ),L^2) $ to (REF ).",
"The following estimates hold: $\\Vert u(t,\\cdot )\\Vert _{L^2}& \\lesssim (1+t)^{-\\frac{n}{2\\sigma }(\\frac{1}{m}-\\frac{1}{2})} \\Vert (u_0,u_1)\\Vert _{\\mathcal {A}^{\\sigma }_{m}}, \\\\\\big \\Vert |D|^{\\sigma } u(t,\\cdot )\\big \\Vert _{L^2}& \\lesssim (1+t)^{-\\frac{n}{2\\sigma }(\\frac{1}{m}-\\frac{1}{2})- \\frac{1}{2}} \\Vert (u_0,u_1)\\Vert _{\\mathcal {A}^{\\sigma }_{m}}, \\\\\\Vert u_t(t,\\cdot )\\Vert _{L^2}& \\lesssim (1+t)^{-\\frac{n}{2\\sigma }(\\frac{1}{m}-\\frac{1}{2})- 1} \\Vert (u_0,u_1)\\Vert _{\\mathcal {A}^{\\sigma }_{m}}.$ Next, we obtain the large time behavior of the global solutions to (REF ).",
"Theorem 1.4 Under the assumptions of Theorem REF with $m=1$ , the global (in time) small data energy solutions to (REF ) satisfy the following estimate: $\\big \\Vert \\partial _t^j |D|^{k\\sigma }\\big (u(t,\\cdot )- M\\,G_\\sigma (t,\\cdot )\\big )\\big \\Vert _{L^2}= o\\big (t^{-\\frac{n}{4\\sigma }- \\frac{k}{2}-j}\\big ),$ for $j,\\,k=0,\\,1$ and $(j,k)\\ne (1,1)$ .",
"Here, we denote the quantity $M:= \\int _{\\mathbb {R}^n} \\big (u_0(y)+u_1(y)\\big )dy+ \\int _0^\\infty \\int _{\\mathbb {R}^n} |u(\\tau ,y)|^p dyd\\tau .$ Finally, we obtain the blow-up result to (REF ).",
"Theorem 1.5 Let $\\sigma \\ge 1$ be an integer.",
"We assume the initial data $u_0=0$ and $u_1 \\in L^1 \\cap L^{2}$ satisfying the following relation: $ \\liminf _{R\\longrightarrow \\infty } \\int _{|x| < R} u_1(x) dx >0.$ Moreover, we suppose the condition $ 1< p\\le 1+\\frac{2\\sigma }{n}.$ Then, there is no global (in time) energy solution to (REF ).",
"In other words, we have only local (in time) energy solutions to (REF ), that is, there exists $T_\\varepsilon < \\infty $ such that $ \\lim _{t \\rightarrow T_\\varepsilon -0} \\Vert (u, u_t)\\Vert _{H^{\\sigma } \\times L^2}= +\\infty .",
"$ Remark 1.1 If we choose $m=1$ into Theorem REF , then from Theorem REF it is clear to see that the exponent $p$ given by $p=p(n,\\sigma )=1+\\frac{2\\sigma }{n}$ is really critical.",
"The structure of this paper is organized as follows: Section is devoted to estimates for the solutions to (REF ).",
"In particular, we present some $(L^m \\cap L^2)- L^2$ and $L^2- L^2$ estimates for the solutions with $m\\in [1,2)$ , give the proof of higher order asymptotic expansions of the solutions with weighted initial data, and prove the diffusion phenomenon in the $L^p-L^q$ framework with $1\\le p\\le q\\le \\infty $ to (REF ) in Sections REF , REF and REF , respectively.",
"We prove the global (in time) existence of small data energy solutions to (REF ) in Section REF , and derive their large time behavior in Section REF .",
"Finally, in Section , we show the blow-up result and find the critical exponent as well."
],
[
"$L^m \\cap L^2-L^2$ and {{formula:e625d8fb-f96a-491f-a0fa-f70a0a107848}} estimates",
"Proposition 2.1 Let $m \\in [1,2)$ .",
"Then, the Sobolev solutions to (REF ) satisfy the $(L^m \\cap L^2)-L^2$ estimates $\\big \\Vert \\partial _t^j |D|^a u(t,\\cdot )\\big \\Vert _{L^2} &\\lesssim (1+t)^{-\\frac{n}{2\\sigma }(\\frac{1}{m}-\\frac{1}{2})- \\frac{a}{2\\sigma }-j} \\big (\\Vert u_0\\Vert _{L^m \\cap H^a}+ \\Vert u_1\\Vert _{L^m \\cap H^{[a+2(j-1)\\sigma ]^+}}\\big ),$ and the $L^2-L^2$ estimates $ \\big \\Vert \\partial _t^j |D|^a u(t,\\cdot )\\big \\Vert _{L^2} \\lesssim (1+t)^{-\\frac{a}{2\\sigma }-j}\\big (\\Vert u_0\\Vert _{H^a}+ \\Vert u_1\\Vert _{H^{[a+2(j-1)\\sigma ]^+}}\\big ), $ for any $a\\ge 0$ , $j=0,1$ and for all space dimensions $n\\ge 1$ .",
"To derive $(L^m \\cap L^2)- L^2$ estimates, our strategy is to control $L^2$ norm of the low-frequency part of the solution by $L^m$ norm of the data, whereas its high-frequency part is estimated by using $L^2-L^2$ estimates with a suitable regularity of the data $u_0$ and $u_1$ .",
"We shall divide our considerations into two steps.",
"In the first step, let us devote to estimates for low frequencies.",
"We denote $m^{\\prime }$ as a conjugate number of $m$ , this is, $\\frac{1}{m}+\\frac{1}{m^{\\prime }}=1$ and $m_0$ satisfying $\\frac{1}{m_0}= \\frac{1}{m}- \\frac{1}{2}$ .",
"By (REF ), using the formula of Parseval-Plancherel and Hölder's inequality leads to $\\big \\Vert \\partial _t^j |D|^a u_{\\text{\\fontshape {n}\\selectfont low}}(t,\\cdot )\\big \\Vert _{L^2}&= \\big \\Vert |\\xi |^a \\partial _t^j \\widehat{u}(t,\\xi ) \\chi (|\\xi |) \\big \\Vert _{L^2} \\nonumber \\\\&\\le \\Big \\Vert \\frac{(-1)^j |\\xi |^{a+2j\\sigma }e^{-t|\\xi |^{2\\sigma }}}{1-|\\xi |^{2\\sigma }}\\big (\\widehat{u_0}(\\xi )+ \\widehat{u_1}(\\xi )\\big ) \\chi (|\\xi |)\\Big \\Vert _{L^2} \\nonumber \\\\&\\qquad + \\Big \\Vert \\frac{(-1)^{j+1}|\\xi |^a e^{-t}}{1-|\\xi |^{2\\sigma }}\\big (|\\xi |^{2\\sigma } \\widehat{u_0}(\\xi )+ \\widehat{u_1}(\\xi )\\big )\\chi (|\\xi |)\\Big \\Vert _{L^2} \\nonumber \\\\&\\lesssim \\Big \\Vert \\frac{|\\xi |^{a+2j\\sigma }e^{-t|\\xi |^{2\\sigma }}\\chi (|\\xi |)}{1-|\\xi |^{2\\sigma }}\\Big \\Vert _{L^{m_0}} \\Vert \\widehat{u_0}+ \\widehat{u_1}\\Vert _{L^{m^{\\prime }}} \\\\&\\qquad + e^{-t}\\,\\Big \\Vert \\frac{|\\xi |^{a+2\\sigma } \\chi (|\\xi |)}{1-|\\xi |^{2\\sigma }}\\Big \\Vert _{L^{m_0}} \\Vert \\widehat{u_0}\\Vert _{L^{m^{\\prime }}}+ e^{-t}\\,\\Big \\Vert \\frac{|\\xi |^a \\chi (|\\xi |)}{1-|\\xi |^{2\\sigma }}\\Big \\Vert _{L^{m_0}} \\Vert \\widehat{u_1}\\Vert _{L^{m^{\\prime }}}.",
"$ For the sake of Young-Hausdorff inequality, we can control $ \\Vert \\widehat{u_0}+ \\widehat{u_1}\\Vert _{L^{m^{\\prime }}}$ , $\\Vert \\widehat{u_0}\\Vert _{L^{m^{\\prime }}}$ and $\\Vert \\widehat{u_1}\\Vert _{L^{m^{\\prime }}}$ by $\\Vert u_0+ u_1\\Vert _{L^m}$ , $\\Vert u_0\\Vert _{L^m}$ and $\\Vert u_1\\Vert _{L^m}$ , respectively.",
"Hence, we only have to estimate $L^{m_0}$ norm of the multipliers.",
"It is clear to see that the last two $L^{m_0}$ norms are bounded.",
"Taking account of the first $L^{m_0}$ norm, we apply Lemma REF to obtain immediately the following estimate: $\\Big \\Vert \\frac{|\\xi |^{a+2j\\sigma }e^{-t|\\xi |^{2\\sigma }}\\chi (|\\xi |)}{1-|\\xi |^{2\\sigma }}\\Big \\Vert _{L^{m_0}} \\lesssim \\big \\Vert |\\xi |^{a+2j\\sigma }e^{-t|\\xi |^{2\\sigma }}\\chi (|\\xi |)\\big \\Vert _{L^{m_0}} \\lesssim (1+t)^{-\\frac{n}{2\\sigma }(\\frac{1}{m}-\\frac{1}{2})- \\frac{a}{2\\sigma }-j}.$ Therefore, from (REF ) to (REF ) we arrive at $\\big \\Vert \\partial _t^j |D|^a u_{\\text{\\fontshape {n}\\selectfont low}}(t,\\cdot )\\big \\Vert _{L^2}\\lesssim (1+t)^{-\\frac{n}{2\\sigma }(\\frac{1}{m}-\\frac{1}{2})- \\frac{a}{2\\sigma }-j}\\Vert u_0+ u_1\\Vert _{L^m}+ e^{-t}\\,\\big (\\Vert u_0\\Vert _{L^m}+ \\Vert u_1\\Vert _{L^m}\\big ).$ Next, let us turn to estimate the solution and some of its derivatives to (REF ) for large frequencies.",
"Thanks to (), we apply again the formula of Parseval-Plancherel and use a suitable regularity of the data $u_0$ and $u_1$ to find the following estimate: $\\big \\Vert \\partial _t^j |D|^a u_{\\text{\\fontshape {n}\\selectfont high}}(t,\\cdot )\\big \\Vert _{L^2} \\lesssim e^{-t}\\big (\\Vert u_0\\Vert _{H^a}+ \\Vert u_1\\Vert _{H^{[a- 2\\sigma ]^+}}\\big )+ e^{-t}\\Vert u_0+ u_1\\Vert _{H^{[a+2(j-1)\\sigma ]^+}}.$ From (REF ) and (REF ) we may conclude all the desired estimates.",
"Summarizing, the proof of Proposition REF is completed.",
"Remark 2.1 Here we want to underline that the exponential decay $e^{-t}$ appearing in the proof of Proposition REF is better than the potential decay.",
"Since we have in mind that the characteristic roots $\\lambda _{\\pm }$ are negative in the middle zone $|\\xi | \\in \\big \\lbrace \\varepsilon ,\\, \\frac{1}{\\varepsilon }\\big \\rbrace $ with a sufficiently small positive $\\varepsilon $ , the corresponding estimates yield an exponential decay in this zone, too."
],
[
"Asymptotic profile and higher order asymptotic expansions",
"In this subsection we obtain higher order asymptotic expansions of the solution to (REF ) in the Fourier space (see Theorem REF ).",
"It is necessary to analyze the solution formulas (REF )-(REF ) in the low-frequency region.",
"In order to state Lemma REF , which is a key to derive Theorem REF , we prepare the following notation and definition.",
"For $f\\in L^{1,\\gamma }(\\mathbb {R}^n)$ , we put $M_\\alpha (f):=\\frac{(-1)^{|\\alpha |}}{\\alpha !",
"}\\int _{\\mathbb {R}^n} x^\\alpha f(x)\\,dx,\\qquad |\\alpha |\\le [\\gamma ].$ Definition 2.1 For $0<\\sigma \\in \\mathbb {R}$ , set $\\mathfrak {S}:=\\lbrace 2\\sigma \\ell +j : \\ell , j\\in \\mathbb {N}_0\\rbrace $ .",
"We define the following function inductively: $\\lambda (k):={\\left\\lbrace \\begin{array}{ll}\\min \\mathfrak {S}=0 &\\text{ for }\\,\\,\\, k=0, \\\\\\min \\mathfrak {S}\\setminus \\lbrace \\lambda (j):0\\le j\\le k-1\\rbrace &\\text{ for }\\,\\,\\, k\\in \\mathbb {N}.\\end{array}\\right.",
"}$ Remark 2.2 Now we deal with the case of $\\sigma \\ge 1$ and thus it holds that $\\lambda (k)=k,\\qquad k=0,1,2.$ Since $\\mathbb {N}_0\\subset \\mathfrak {S}$ , it holds that $0<\\lambda (k+1)-\\lambda (k)\\le 1$ for all $k\\in \\mathbb {N}_0$ .",
"Definition 2.2 Let $\\sigma \\ge 1$ and $k\\in \\mathbb {N}_0$ .",
"We define $A^{\\sigma ,v_0}_k(\\xi ):=\\sum _{0\\le 2\\sigma \\ell +j\\le \\lambda (k)}\\left(|\\xi |^{2\\sigma \\ell }\\sum _{|\\alpha |=j}M_\\alpha (v_0)(i\\xi )^\\alpha \\right).$ The sum $\\displaystyle {\\sum _{0\\le 2\\sigma \\ell +j\\le \\lambda (k)}}$ is taken over all $\\ell , j\\in \\mathbb {N}_0$ satisfying $0\\le 2\\sigma \\ell +j\\le \\lambda (k)$ .",
"Remark 2.3 The function (REF ) itself can be defined for $v_0 \\in L^{1,\\gamma }(\\mathbb {R}^n)$ with $\\gamma \\ge [\\lambda (k)]$ .",
"For later necessarity, it suffices to consider (REF ) for $v_0 \\in L^{1,\\gamma }(\\mathbb {R}^n)$ with $\\gamma \\ge \\lambda (k)$ .",
"Recall Remark REF to confirm $A_0^{\\sigma , v_0}(\\xi )=M_0(v_0),\\qquad A_1^{\\sigma , v_0}(\\xi )=\\sum _{|\\alpha |\\le 1}M_\\alpha (v_0)(i\\xi )^\\alpha ,$ $A_2^{\\sigma , v_0}(\\xi )={\\left\\lbrace \\begin{array}{ll}\\displaystyle {\\sum _{|\\alpha |\\le 2}M_\\alpha (v_0)(i\\xi )^\\alpha }&\\text{ if }\\,\\,\\, \\sigma >1, \\\\[22pt]|\\xi |^2 M_0(v_0)+\\displaystyle {\\sum _{|\\alpha |\\le 2}M_\\alpha (v_0)(i\\xi )^\\alpha }&\\text{ if }\\,\\,\\, \\sigma =1.\\end{array}\\right.",
"}$ We can easily see that the difference between the heat flow and equation (REF ) will come out first in the $k$ -th order expansion $A_k^{\\sigma ,v_0}$ with $k\\le \\sigma <k+1$ .",
"However, it seems difficult to write down $A_k^{\\sigma , v_0}$ for large $k\\in \\mathbb {N}_0$ .",
"In [13], the case of $\\sigma =1$ was completely investigated.",
"Lemma 2.1 Let $n\\ge 1$ and $v_0 \\in L^{1,\\gamma }(\\mathbb {R}^n)$ with $\\gamma \\ge 0$ .",
"For this $\\gamma $ , there exists a unique number $k\\in \\mathbb {N}_0$ satisfying $\\lambda (k)\\le \\gamma <\\lambda (k+1)$ .",
"Then, it holds that $\\big |F^{\\sigma ,v_0}(\\xi )-A^{\\sigma ,v_0}_k(\\xi )\\big |&\\lesssim |\\xi |^\\gamma \\Vert v_0\\Vert _{L^{1,\\gamma }}$ for $\\xi \\in \\mathbb {R}^n$ with $|\\xi |\\le 1/2$ .",
"For given $\\gamma \\ge 0$ , we can find $k\\in \\mathbb {N}_0$ satisfying $\\lambda (k)\\le \\gamma <\\lambda (k+1)$ .",
"In this setting we have $[\\gamma ]=[\\lambda (k)].$ If not, then there exists an integer $b\\in \\mathbb {N}$ such that $\\lambda (k)<b\\le \\gamma <\\lambda (k+1)$ .",
"All natural numbers are included in $\\lbrace \\lambda (j) : j\\in \\mathbb {N}_0\\rbrace $ and so this is a contradiction.",
"It follows that $F^{\\sigma ,v_0}(\\xi )& =\\left(\\sum _{\\ell =0}^{[\\lambda (k)]} |\\xi |^{2\\sigma \\ell }+\\frac{|\\xi |^{2\\sigma ([\\lambda (k)]+1)}}{1-|\\xi |^{2\\sigma }}\\right)\\left\\lbrace \\sum _{j=0}^{[\\lambda (k)]}\\sum _{|\\alpha |=j}M_\\alpha (v_0)(i\\xi )^\\alpha +\\left(\\widehat{v_0}-\\sum _{|\\alpha |\\le [\\gamma ]}M_\\alpha (v_0)(i\\xi )^\\alpha \\right)\\right\\rbrace \\\\& =\\left(\\sum _{\\ell =0}^{[\\lambda (k)]} |\\xi |^{2\\sigma \\ell }\\right)\\left(\\sum _{j=0}^{[\\lambda (k)]}\\sum _{|\\alpha |=j}M_\\alpha (v_0)(i\\xi )^\\alpha \\right) \\\\& \\qquad +\\left(\\sum _{\\ell =0}^{[\\lambda (k)]} |\\xi |^{2\\sigma \\ell }\\right)\\left(\\widehat{v_0}-\\sum _{|\\alpha |\\le [\\gamma ]}M_\\alpha (v_0)(i\\xi )^\\alpha \\right)+\\frac{|\\xi |^{2\\sigma ([\\gamma ]+1)}}{1-|\\xi |^{2\\sigma }}\\widehat{v_0}.$ From (REF ), we see that $\\left|\\left(\\sum _{\\ell =0}^{[\\lambda (k)]} |\\xi |^{2\\sigma \\ell }\\right)\\left(\\widehat{v_0}-\\sum _{|\\alpha |\\le [\\gamma ]}M_\\alpha (v_0)(i\\xi )^\\alpha \\right)\\right|\\lesssim |\\xi |^\\gamma \\Vert v_0\\Vert _{L^{1,\\gamma }}$ for $\\xi \\in \\mathbb {R}^n$ with $|\\xi |\\le 1/2$ .",
"Since $2\\sigma ([\\gamma ]+1)>\\gamma ,$ one easily sees that $\\left|\\frac{|\\xi |^{2\\sigma ([\\gamma ]+1)}}{1-|\\xi |^{2\\sigma }}\\widehat{v_0}\\right|\\lesssim |\\xi |^\\gamma \\Vert v_0\\Vert _{L^1}$ for $\\xi \\in \\mathbb {R}^n$ with $|\\xi |\\le 1/2$ .",
"Thus, we arrive at $\\left|F^{\\sigma ,v_0}(\\xi )-\\left(\\sum _{\\ell =0}^{[\\lambda (k)]} |\\xi |^{2\\sigma \\ell }\\right)\\left(\\sum _{j=0}^{[\\lambda (k)]}\\sum _{|\\alpha |=j}M_\\alpha (v_0)(i\\xi )^\\alpha \\right)\\right|& \\lesssim |\\xi |^\\gamma \\Vert v_0\\Vert _{L^{1,\\gamma }}+|\\xi |^{2\\sigma ([\\gamma ]+1)} \\Vert v_0\\Vert _{L^1} \\\\& \\lesssim |\\xi |^\\gamma \\Vert v_0\\Vert _{L^{1,\\gamma }}$ for $\\xi \\in \\mathbb {R}^n$ with $|\\xi |\\le 1/2$ .",
"If $[\\lambda (k)]=0$ , i.e., $k=0$ , then $\\left(\\sum _{\\ell =0}^{[\\lambda (k)]} |\\xi |^{2\\sigma \\ell }\\right)\\left(\\sum _{j=0}^{[\\lambda (k)]}\\sum _{|\\alpha |=j}M_\\alpha (v_0)(i\\xi )^\\alpha \\right)=M_0(v_0)=A_0^{\\sigma , v_0}(\\xi ).$ So in this case we obtain the lemma.",
"On the other hand, if $[\\lambda (k)]\\ge 1$ , we have $\\lambda (k)<(2\\sigma +1)[\\lambda (k)]\\in \\mathfrak {S}.$ Hence, it follows that $& \\left(\\sum _{\\ell =0}^{[\\lambda (k)]} |\\xi |^{2\\sigma \\ell }\\right)\\left(\\sum _{j=0}^{[\\lambda (k)]}\\sum _{|\\alpha |=j}M_\\alpha (v_0)(i\\xi )^\\alpha \\right) \\\\&\\qquad =\\sum _{\\begin{array}{c}0\\le 2\\sigma \\ell +j\\le \\lambda (k), \\\\0\\le j\\le [\\lambda (k)]\\end{array}}\\left(|\\xi |^{2\\sigma \\ell }\\sum _{|\\alpha |=j}M_\\alpha (v_0)(i\\xi )^\\alpha \\right)+\\sum _{\\begin{array}{c}\\lambda (k)<2\\sigma \\ell +j\\le (2\\sigma +1)[\\lambda (k)], \\\\0\\le j\\le [\\lambda (k)]\\end{array}}\\left(|\\xi |^{2\\sigma \\ell }\\sum _{|\\alpha |=j}M_\\alpha (v_0)(i\\xi )^\\alpha \\right) \\\\&\\qquad =A_k^{\\sigma , v_0}(\\xi )+\\sum _{\\begin{array}{c}\\lambda (k+1)\\le 2\\sigma \\ell +j\\le (2\\sigma +1)[\\lambda (k)], \\\\0\\le j\\le [\\gamma ]\\end{array}}\\left(|\\xi |^{2\\sigma \\ell }\\sum _{|\\alpha |=j}M_\\alpha (v_0)(i\\xi )^\\alpha \\right).$ Thus, it holds $\\left|\\sum _{\\begin{array}{c}\\lambda (k+1)\\le 2\\sigma \\ell +j\\le (2\\sigma +1)[\\lambda (k)], \\\\0\\le j\\le [\\gamma ]\\end{array}}\\left(|\\xi |^{2\\sigma \\ell }\\sum _{|\\alpha |=j}M_\\alpha (v_0)(i\\xi )^\\alpha \\right)\\right|\\lesssim |\\xi |^{\\lambda (k+1)}\\Vert v_0\\Vert _{L^{1,[\\gamma ]}}\\lesssim |\\xi |^\\gamma \\Vert v_0\\Vert _{L^{1,[\\gamma ]}}$ for $\\xi \\in \\mathbb {R}^n$ with $|\\xi |\\le 1/2$ .",
"Therefore, we obtain (REF ).",
"It follows from Lemma REF with Lemma REF that $\\big \\Vert F^{\\sigma ,v_0}(\\xi )e^{-t|\\xi |^{2\\sigma }}-A_k^{\\sigma ,v_0}(\\xi )e^{-t|\\xi |^{2\\sigma }}\\big \\Vert _{L^2(|\\xi |\\le 1/2)}\\lesssim (1+t)^{-\\frac{n}{4\\sigma }-\\frac{\\gamma }{2\\sigma }}\\Vert v_0\\Vert _{L^{1,\\gamma }},\\qquad t\\ge 0,$ which implies $\\big \\Vert \\hat{u}(t,\\xi )-A_k^{\\sigma ,v_0}(\\xi )e^{-t|\\xi |^{2\\sigma }}\\big \\Vert _{L^2(|\\xi |\\le 1/2)}\\lesssim (1+t)^{-\\frac{n}{4\\sigma }-\\frac{\\gamma }{2\\sigma }}(\\Vert v_0\\Vert _{L^{1,\\gamma }}+\\Vert u_0\\Vert _{L^1}+\\Vert u_1\\Vert _{L^1}),\\qquad t\\ge 0,$ We employ (REF ) to derive $|\\hat{u}(t,\\xi )|\\lesssim e^{-t} |\\widehat{u_0}(\\xi )|+te^{-t} (|\\widehat{u_0}(\\xi )|+|\\widehat{u_1}(\\xi )|)$ for $t\\ge 0$ and $\\xi \\in \\mathbb {R}^n$ with $|\\xi |\\ge 1/2$ .",
"We can easily see that $\\big |A_k^{\\sigma ,v_0}(\\xi )e^{-t|\\xi |^{2\\sigma }}\\big |\\lesssim |\\xi |^{\\lambda (k)}\\exp \\left(-\\frac{|\\xi |^{2\\sigma }}{2}\\right)\\exp \\left(-\\frac{t}{2^{2\\sigma +1}}\\right)\\Vert v_0\\Vert _{L^{1,[\\gamma ]}}$ for $t\\ge 1$ and $\\xi \\in \\mathbb {R}^n$ with $|\\xi |\\ge 1/2$ .",
"Thus, there exists a constant $c\\in (0,2^{-(2\\sigma +1)}]$ such that $\\big \\Vert \\hat{u}(t,\\xi )-A_k^{\\sigma ,v_0}(\\xi )e^{-t|\\xi |^{2\\sigma }}\\big \\Vert _{L^2(|\\xi |\\ge 1/2)}\\lesssim e^{-ct}\\big (\\Vert u_0\\Vert _{L^2}+\\Vert u_1\\Vert _{L^2}+\\Vert v_0\\Vert _{L^{1,[\\gamma ]}}\\big ),\\qquad t\\ge 1.$ Inequalities (REF ) and (REF ) give (REF ).",
"By (REF ), we can also obtain (REF ).",
"See the corresponding proof in [13] and details are left to the reader.",
"Recalling (REF ) in Theorem REF with $\\gamma =1$ we can obtain the following corollary since $\\left\\Vert \\sum _{|\\alpha |=1}M_\\alpha (v_0)(i\\xi )^\\alpha e^{-t|\\xi |^{2\\sigma }}\\right\\Vert _{L^2}\\lesssim t^{-\\frac{n}{4\\sigma }- \\frac{1}{2\\sigma }}\\Vert v_0\\Vert _{L^{1,1}},\\qquad t>0.$ Corollary 2.1 Let $n\\ge 1$ and $u$ be the function defined by (REF )-(REF ).",
"If $u_0,\\, u_1\\in L^{1,1}(\\mathbb {R}^n)\\cap L^2(\\mathbb {R}^n)$ , it holds $ \\big \\Vert u(t,\\cdot )- (P_0+P_1)G_\\sigma (t,\\cdot )\\big \\Vert _{L^2}\\lesssim t^{-\\frac{n}{4\\sigma }- \\frac{1}{2\\sigma }} \\big (\\Vert u_0\\Vert _{L^{1,1} \\cap L^2}+ \\Vert u_1\\Vert _{L^{1,1} \\cap L^2}\\big ),\\qquad t\\ge 1.",
"$ We may conclude the following optimal result at the end of this subsection.",
"Corollary 2.2 Let $n\\ge 1$ and $u$ be the function defined by (REF )-(REF ).",
"If $u_0,\\, u_1\\in L^{1}(\\mathbb {R}^n)\\cap L^2(\\mathbb {R}^n)$ , it holds $ C_1(|P_0+ P_1|)t^{-\\frac{n}{4\\sigma }} \\le \\Vert u(t,\\cdot )\\Vert _{L^2}\\le C_2 t^{-\\frac{n}{4\\sigma }} \\big (\\Vert u_0\\Vert _{L^{1} \\cap L^2}+ \\Vert u_1\\Vert _{L^{1} \\cap L^2}\\big ),\\qquad t\\ge 1.",
"$ Here, $C_1>0$ and $C_2 >0$ are constants independent of $t$ and the initial data.",
"The second inequality can be easily given with the aid of (REF )-(REF ) or (REF ) with $\\gamma =0$ .",
"So we confirm the first inequality.",
"To do so, it suffices to consider the case that $P_0+P_1\\ne 0$ .",
"In this situation one has $\\Vert u(t,\\cdot )\\Vert _2& \\ge \\Vert u(t,\\xi )\\Vert _{L^2(|\\xi |\\le 1/2)} \\\\& \\ge |P_0+P_1| \\Vert e^{-t|\\xi |^{2\\sigma }}\\Vert _{L^2(|\\xi |\\le 1/2)}-o(t^{-\\frac{n}{4\\sigma }})$ as $t\\rightarrow \\infty $ .",
"Here, we used (REF ) with $\\gamma =0$ , that is, $\\lim _{t\\rightarrow \\infty }t^{\\frac{n}{4\\sigma }}\\big \\Vert \\hat{u}(t,\\xi )- (P_0+P_1)e^{-t|\\xi |^{2\\sigma }}\\big \\Vert _{L^2(|\\xi |\\le 1/2)}=0.$ For $t\\ge 1$ , we have $\\Vert e^{-t|\\xi |^{2\\sigma }}\\Vert _{L^2(|\\xi |\\le 1/2)}=t^{-\\frac{n}{4\\sigma }}\\left(\\int _{|\\eta |\\le t^\\frac{1}{2\\sigma }/2}e^{-2|\\eta |^{2\\sigma }}\\,d\\eta \\right)^{\\frac{1}{2}}\\ge \\left(\\int _{|\\eta |\\le 1/2}e^{-2|\\eta |^{2\\sigma }}\\,d\\eta \\right)^{\\frac{1}{2}}t^{-\\frac{n}{4\\sigma }}$ and thus the corollary is obtained."
],
[
"Diffusion phenomenon in the $L^p-L^q$ framework",
"In this section, we shall discuss a relation between the solutions to (REF ) and to the anomalous diffusion equation (REF ).",
"Clearly, if we consider the power $\\sigma = 1$ , then it corresponds to the classical heat equation.",
"Let $(u_0,\\,u_1)$ be given data to (REF ).",
"If we can find an appropriate data $v_0$ to (REF ) such that the difference of the corresponding solutions $u(t,\\cdot )- v(t,\\cdot )$ possesses a decay rate as $t \\rightarrow \\infty $ in a suitable norm, then one says that the asymptotic behavior of both the solutions is the same for large time.",
"This effect is the so-called diffusion phenomenon.",
"The application of partial Fourier transform to (REF ) leads to ${\\left\\lbrace \\begin{array}{ll}\\hat{v}_t+ |\\xi |^{2\\sigma } \\hat{v}= 0, \\\\\\hat{v}(0,\\xi )= \\widehat{v_0}(\\xi ).\\end{array}\\right.",
"}$ Hence, the solution to (REF ) is written by the formula $\\hat{v}(t,\\xi )= e^{-t|\\xi |^{2\\sigma }}\\widehat{v_0}(\\xi ).$ Recalling the abbreviation $G_\\sigma (t,x)$ gives $v(t,x)= G_\\sigma (t,x) \\ast _x v_0(x).$ Because of the presence of the frictional damping term in (REF ), considering large frequencies $|\\xi |$ and large time $t$ we can conclude some exponential decay estimates for the solutions to (REF ) as we derived (REF ).",
"Moreover, it holds that we may arrive at an exponential decay for the solutions to (REF ) for large frequencies $|\\xi |$ and for large times $t$ .",
"This means that the difference between the solutions to (REF ) and (REF ) decays.",
"Hence, the obtained decay rate of the difference is optimal.",
"For this reason, it is suitable to focus our attentions on estimates for the difference localized to small frequencies.",
"Our approach to prove Theorem REF is based on applying the following two auxiliary results.",
"The first one is a result for radial convolution kernels.",
"Lemma 2.2 (Lemma 3.1 in [2]) Let $K(t,x)$ be a radial convolution kernel of the form $ K(t,x) \\ast _{(x)} h(x):= F^{-1}\\big (f(|\\xi |)\\,e^{-g(|\\xi |)t}\\,\\hat{h}(\\xi )\\big ), $ with compactly supported $h$ , where $f$ and $g$ satisfy the following conditions: $\\big |f^{(k)}(\\rho )\\big |&\\lesssim \\rho ^{\\alpha -k}, \\\\\\big |g^{(k)}(\\rho )\\big |&\\lesssim \\rho ^{-k}g(\\rho ), \\\\g(\\rho )&\\approx \\rho ^{\\beta },$ for some $\\alpha > -1$ , $\\beta >0$ and $k\\le [(n+3)/2]$ .",
"Then, it holds $ \\Vert K_{\\text{\\fontshape {n}\\selectfont low}}(t,\\cdot ) \\ast _{(x)} h\\Vert _{L^q} \\lesssim (1+t)^{-\\frac{n}{\\beta }(\\frac{1}{p}- \\frac{1}{q})- \\frac{\\alpha }{\\beta }}\\Vert h\\Vert _{L^p}, $ provided that for any ${\\left\\lbrace \\begin{array}{ll}1\\le p\\le q\\le \\infty &\\text{ if }\\,\\,\\, \\alpha >0 \\text{ or } f \\text{ is a nonzero constant}, \\\\1\\le p< q\\le \\infty &\\text{ if }\\,\\,\\, \\alpha =0 \\text{ and } f \\text{ is not constant}, \\\\1\\le p\\le q\\le \\infty &\\text{ if }\\,\\,\\, -1<\\alpha <0 \\text{ such that } \\frac{1}{p}- \\frac{1}{q}\\ge \\frac{-\\alpha }{n}.\\end{array}\\right.",
"}$ The second result is related to $L^r$ estimates for multipliers.",
"Lemma 2.3 Let $n\\ge 1$ and $r\\in [1,\\infty ]$ .",
"Then, for all $a>0$ it holds $K:= \\mathcal {F}^{-1}\\Big (\\frac{|\\xi |^a}{1-|\\xi |^{2\\sigma }}\\chi (|\\xi |)\\Big ) \\in L^r.$ First it is clear to see that $K \\in L^{\\infty }$ .",
"For this reason, in order to prove $K \\in L^r$ for all $r\\in [1,\\infty ]$ , we only indicate $K \\in L^1$ and apply an interpolation argument.",
"Indeed, we shall split our considerations into two cases.",
"In the first case of $|x|\\le 1$ , it is obvious to conclude the desired statement.",
"Let us devote to the second case of $|x|\\ge 1$ .",
"Because the function in the parenthesis in (REF ) is radially symmetric with respect to $\\xi $ , the inverse Fourier transform is radially symmetric with respect to $x$ , too.",
"Applying the modified Bessel functions from Proposition REF we obtain $K(x)= c\\int _0^\\infty \\frac{r^a}{1-r^{2\\sigma }}\\chi (r) r^{n-1} \\tilde{J}_{\\frac{n}{2}-1}(r|x|) dr. $ Let us consider odd spatial dimensions $n=2m+1,\\, m \\ge 1$ .",
"Introducing the vector field $Xf(r):= \\frac{d}{dr}\\big (\\frac{1}{r}f(r)\\big )$ and carrying out $m+1$ steps of partial integration we derive $K(x)= -\\frac{c}{|x|^{n}}\\int _0^\\infty \\partial _r \\Big (X^m \\Big ( \\frac{1}{1-r^{2\\sigma }} \\chi (r) r^{a+2m}\\Big )\\Big ) \\sin (r|x|)dr. $ A straightforward computation gives $K(x)&= \\sum _{j=0}^m \\sum _{k=0}^{j+1}\\frac{c_{jk}}{|x|^{n}}\\int _0^\\infty \\partial _r^{j+1-k} \\Big (\\frac{1}{1-r^{2\\sigma }}\\Big )\\, \\chi ^{(k)}(r) r^{a+j} \\sin (r|x|)dr\\\\&\\quad + \\sum _{j=0}^m \\sum _{k=0}^j \\frac{c_{jk}}{|x|^{n}}\\int _0^\\infty \\partial _r^{j-k} \\Big (\\frac{1}{1-r^{2\\sigma }}\\Big )\\, \\chi ^{(k+1)}(r) r^{a+j} \\sin (r|x|)dr\\\\&\\quad + \\sum _{j=1}^m \\sum _{k=0}^j \\frac{c_{jk}}{|x|^{n}}\\int _0^\\infty \\partial _r^{j-k} \\Big (\\frac{1}{1-r^{2\\sigma }}\\Big )\\, \\chi ^{(k)}(r) r^{a+j-1} \\sin (r|x|)dr$ with some constants $c_{jk}$ .",
"Hence, it is reasonable to estimate the integrals $K_{j,k}(x):= \\int _0^\\infty \\partial _r^{j+1-k} \\Big (\\frac{1}{1-r^{2\\sigma }}\\Big )\\, \\chi ^{(k)}(r) r^{a+j} \\sin (r|x|)dr. $ For $k\\ge 1$ , due to the fact that for all $l\\ge 0$ $ \\Big |\\partial _r^{l} \\Big (\\frac{1}{1-r^{2\\sigma }}\\Big )\\Big | \\le C_l $ on the support of the derivatives of $\\chi $ , we perform one more step of partial integration to get $K_{j,k}(x)|\\lesssim |x|^{-(n+1)}.$ For $k=0$ , because of the small values of $r$ , we arrive at $ \\Big |\\partial _r^{l} \\Big (\\frac{1}{1-r^{2\\sigma }}\\Big )\\Big | \\lesssim {\\left\\lbrace \\begin{array}{ll}1 &\\text{ if }\\,\\,\\, l=0, \\\\r^{2\\sigma -l} &\\text{ if }\\,\\,\\, l\\ge 1,\\end{array}\\right.}",
"$ on the support of $\\chi (r)$ .",
"Consequently, it deduces for small $r$ and $j= 0,\\cdots ,m$ the estimates $\\Big |\\partial _r^{j+1} \\Big (\\frac{1}{1-r^{2\\sigma }}\\Big )\\, \\chi (r) r^{a+j}\\Big | \\lesssim r^{a+2\\sigma -1} $ on the support of $\\chi (r)$ .",
"By dividing the integral (REF ) into two parts, on the one hand, we have $\\Big |\\int _0^{\\frac{\\pi }{2|x|}} \\partial _r^{j+1} \\Big (\\frac{1}{1-r^{2\\sigma }}\\Big )\\, \\chi (r) r^{a+j} \\sin (r|x|)dr \\Big | \\lesssim \\frac{1}{|x|^{a+2\\sigma }}.",
"$ On the other hand, after carrying out one more step of partial integration in the remaining integral we can proceed as follows: $&\\Big |\\int _{\\frac{\\pi }{2|x|}}^\\infty \\partial _r^{j+1} \\Big (\\frac{1}{1-r^{2\\sigma }}\\Big )\\, \\chi (r) r^{a+j} \\sin (r|x|)dr \\Big | \\nonumber \\\\&\\qquad \\lesssim \\frac{1}{|x|}\\Big |\\partial _r^{j+1} \\Big (\\frac{1}{1-r^{2\\sigma }}\\Big )\\, \\chi (r) r^{a+j} \\cos (r|x|) \\Big |_{r=\\frac{\\pi }{2|x|}}^\\infty \\nonumber \\\\&\\qquad \\quad + \\frac{1}{|x|}\\int _{\\frac{\\pi }{2|x|}}^\\infty \\Big |\\partial _r \\Big ( \\partial _r^{j+1} \\Big (\\frac{1}{1-r^{2\\sigma }}\\Big )\\, \\chi (r) r^{a+j}\\Big ) \\cos (r|x|)\\Big | \\,dr \\lesssim \\frac{1}{|x|}\\int _{\\frac{\\pi }{2|x|}}^1 r^{a+2\\sigma -2}\\,dr \\lesssim \\frac{1}{|x|}.",
"$ Here we notice that for all $j= 0,\\cdots ,m$ and for small $|\\xi |$ it holds $\\Big |\\partial _r \\Big ( \\partial _r^{j+1} \\Big (\\frac{1}{1-r^{2\\sigma }}\\Big )\\, \\chi (r) r^{a+j}\\Big )\\Big | \\lesssim r^{a+2\\sigma -2}.",
"$ From (REF ) to (REF ) we have produced term $|x|^{-(n+1)}$ which guarantees the $L^1$ property in $x$ .",
"Therefore, we may conclude $K \\in L^1$ for all $n=2m+1$ .",
"Let us consider even spatial dimensions $n=2m,\\, m \\ge 1$ .",
"After carrying out $m-1$ steps of partial integration we derive $K(x)&= \\frac{c}{|x|^{2m-2}}\\int _0^\\infty X^{m-1}\\Big ( \\frac{1}{1-r^{2\\sigma }} \\chi (r) r^{a+2m-1} \\Big ) \\tilde{J}_0(r|x|) dr \\nonumber \\\\&= \\sum _{j=0}^{m-1}\\frac{c_j}{|x|^{2m-2}}\\int _0^\\infty \\partial _r^j \\Big ( \\frac{1}{1-r^{2\\sigma }} \\chi (r) r^a \\Big ) r^{j+1} \\tilde{J}_0(r|x|) dr =:\\sum _{j=0}^{m-1} c_j K_j(x).",
"$ Using the first rule of the modified Bessel functions for $\\mu =1$ and the fifth rule for $\\mu =0$ from Proposition REF , after two more steps of partial integration we arrive at $K_0(x)= -\\frac{1}{|x|^{n}}\\int _0^1 \\partial _r \\Big ( \\partial _r \\Big ( \\frac{1}{1-r^{2\\sigma }} \\chi (r) r^a \\Big ) r \\Big ) \\tilde{J}_0(r|x|) dr. $ Due to small $r$ , it implies the following inequality: $ \\Big |\\partial _r \\Big ( \\partial _r \\Big ( \\frac{1}{1-r^{2\\sigma }} \\chi (r) r^a \\Big ) r \\Big )\\Big | \\lesssim r^{a-1} $ on the support of $\\chi (r)$ .",
"By the aid of the estimates $|\\tilde{J}_0(s)| \\le C$ for $s \\in [0,1]$ , and $|\\tilde{J}_0(s)| \\le Cs^{-\\frac{1}{2}}$ for $s>1$ , we get $\\Big | \\int _0^{\\frac{1}{|x|}} \\partial _r \\Big ( \\partial _r \\Big ( \\frac{1}{1-r^{2\\sigma }} \\chi (r) r^a \\Big ) r \\Big ) \\tilde{J}_0(r|x|) dr \\Big | \\lesssim \\int _0^{\\frac{1}{|x|}} r^{a-1} dr \\lesssim \\frac{1}{|x|^a},$ and $&\\Big | \\int _{\\frac{1}{|x|}}^1 \\partial _r \\Big ( \\partial _r \\Big ( \\frac{1}{1-r^{2\\sigma }} \\chi (r) r^a \\Big ) r \\Big ) \\tilde{J}_0(r|x|) dr \\Big | \\nonumber \\\\&\\qquad \\lesssim \\frac{1}{|x|^{\\frac{1}{2}}}\\int _{\\frac{1}{|x|}}^1 r^{a-\\frac{3}{2}} dr \\lesssim {\\left\\lbrace \\begin{array}{ll}\\frac{1}{|x|^{\\frac{1}{2}}}\\Big (1+ \\frac{1}{|x|^{a- \\frac{1}{2}}}\\Big ) &\\text{ if }\\,\\,\\, a \\ne \\frac{1}{2} \\\\\\frac{1}{|x|^{\\frac{1}{2}}}\\log (|x|) &\\text{ if }\\,\\,\\, a= \\frac{1}{2}\\end{array}\\right.",
"}\\,\\,\\lesssim \\frac{1}{|x|^{\\varepsilon }}, $ with a sufficiently small positive constant $\\varepsilon $ , respectively.",
"As a result, from (REF ) to (REF ) we obtain $K_0 \\in L^1$ .",
"Let $j \\in [1,m-1]$ be an integer.",
"By applying again the first rule of the modified Bessel functions for $\\mu =1$ and the fifth rule for $\\mu =0$ from Proposition REF and carrying out partial integration we can re-write $K_j(x)$ in (REF ) as follows: $K_j(x)= &-\\frac{1}{|x|^{2m}}\\int _0^\\infty \\partial _r \\Big ( \\partial _r^{j+1}\\Big ( \\frac{1}{1-r^{2\\sigma }} \\chi (r) r^a \\Big ) r^{j+1}\\Big ) \\tilde{J}_0(r|x|) dr\\\\&-\\frac{j}{|x|^{2m}}\\int _0^\\infty \\partial _r \\Big ( \\partial _r^j \\Big ( \\frac{1}{1-r^{2\\sigma }} \\chi (r) r^a \\Big ) r^j \\Big ) \\tilde{J}_0(r|x|) dr.$ Repeating an analogous treatment as we did for $K_0=K_0(x)$ we derive $K_j \\in L^1$ for $j=1,\\cdots ,m-1$ .",
"Therefore, we may conclude the desired estimate $K \\in L^1$ for all $n=2m$ .",
"Summarizing, this completes the proof of Lemma REF .",
"Thanks to the solution formulas (REF ) and (REF ), we obtain $\\big \\Vert \\partial _t^j |D|^a \\big (u_{\\text{\\fontshape {n}\\selectfont low}}(t,\\cdot )- v_{\\text{\\fontshape {n}\\selectfont low}}(t,\\cdot )\\big )\\big \\Vert _{L^q}&= \\big \\Vert \\mathcal {F}^{-1}\\big (|\\xi |^a \\partial _t^j \\big (\\hat{u}(t,\\xi )- \\hat{v}(t,\\xi )\\big )\\chi (|\\xi |)\\big )\\big \\Vert _{L^q} \\nonumber \\\\&\\lesssim e^{-t}\\,\\Big \\Vert \\mathcal {F}^{-1}\\Big (\\frac{|\\xi |^{a+2\\sigma }}{1- |\\xi |^{2\\sigma }}\\chi (|\\xi |)\\widehat{u_0}(|\\xi |)+ \\frac{|\\xi |^a}{1- |\\xi |^{2\\sigma }}\\chi (|\\xi |)\\widehat{u_1}(|\\xi |)\\Big )\\Big \\Vert _{L^q} \\\\&\\quad + \\Big \\Vert \\mathcal {F}^{-1}\\Big (e^{-t|\\xi |^{2\\sigma }}\\frac{|\\xi |^{a+2(j+1)\\sigma }}{1-|\\xi |^{2\\sigma }}\\chi (|\\xi |)\\big (\\widehat{u_0}(|\\xi |)+\\widehat{u_1}(|\\xi |)\\big )\\Big )\\Big \\Vert _{L^q}.",
"$ Applying Young's convolution inequality and Lemma REF we can proceed (REF ) as follows: $e^{-t}\\,\\Big \\Vert \\mathcal {F}^{-1}\\Big (\\frac{|\\xi |^{a+2\\sigma }}{1- |\\xi |^{2\\sigma }}\\chi (|\\xi |)\\widehat{u_0}(|\\xi |)\\Big )\\Big \\Vert _{L^q}\\lesssim e^{-t}\\,\\Big \\Vert \\mathcal {F}^{-1}\\Big (\\frac{|\\xi |^{a+2\\sigma }}{1- |\\xi |^{2\\sigma }}\\chi (|\\xi |)\\Big )\\Big \\Vert _{L^{r_1}}\\, \\Vert u_0\\Vert _{L^p}\\lesssim e^{-t} \\Vert u_0\\Vert _{L^p},$ where $r_1 \\in [1,\\infty ]$ fulfills $\\frac{1}{r_1}+\\frac{1}{p}= 1+\\frac{1}{q}$ , and $e^{-t}\\,\\Big \\Vert \\mathcal {F}^{-1}\\Big (\\frac{|\\xi |^a}{1- |\\xi |^{2\\sigma }}\\chi (|\\xi |)\\widehat{u_1}(|\\xi |)\\Big )\\Big \\Vert _{L^q}&\\lesssim {\\left\\lbrace \\begin{array}{ll}e^{-t}\\,\\Big \\Vert \\mathcal {F}^{-1}\\Big (\\frac{|\\xi |^a}{1- |\\xi |^{2\\sigma }}\\chi (|\\xi |)\\Big )\\Big \\Vert _{L^{r_1}}\\, \\Vert u_1\\Vert _{L^{p}} &\\text{ if }\\,\\,\\, a>0 \\\\e^{-t}\\,\\Big \\Vert \\mathcal {F}^{-1}\\Big (\\frac{|\\xi |^{\\varepsilon }}{1- |\\xi |^{2\\sigma }}\\chi (|\\xi |)\\Big )\\Big \\Vert _{L^{r_2}}\\, \\big \\Vert |\\xi |^{-\\varepsilon } u_1\\big \\Vert _{L^{p^*}} &\\text{ if }\\,\\,\\, a=0\\end{array}\\right.}",
"\\nonumber \\\\&\\lesssim e^{-t} \\Vert u_1\\Vert _{L^p}, $ where $\\varepsilon $ is a sufficiently small positive constant and $r_2 \\in [1,\\infty ]$ satisfies $1+\\frac{1}{q}= \\frac{1}{r_2}+ \\frac{1}{p^*}$ .",
"Here we used the property of the normalized Riez potential in Remark REF below.",
"In order to control (), we re-write $ \\frac{1}{1- |\\xi |^{2\\sigma }}= \\sum _{k=0}^\\infty |\\xi |^{2k\\sigma } $ due to the small value of $|\\xi |$ .",
"Hence, using Lemma REF we arrive at the following estimate: $&\\Big \\Vert \\mathcal {F}^{-1}\\Big (e^{-t|\\xi |^{2\\sigma }}\\frac{|\\xi |^{a+2(j+1)\\sigma }}{1-|\\xi |^{2\\sigma }}\\chi (|\\xi |)\\big (\\widehat{u_0}(|\\xi |)+\\widehat{u_1}(|\\xi |)\\big )\\Big )\\Big \\Vert _{L^q} \\nonumber \\\\&\\qquad \\lesssim \\sum _{k=0}^\\infty \\Big \\Vert \\mathcal {F}^{-1}\\Big (|\\xi |^{a+2(k+j+1)\\sigma } e^{-t|\\xi |^{2\\sigma }} \\chi (|\\xi |)\\big (\\widehat{u_0}(|\\xi |)+\\widehat{u_1}(|\\xi |)\\big )\\Big )\\Big \\Vert _{L^q} \\nonumber \\\\&\\qquad \\lesssim \\sum _{k=0}^\\infty (1+t)^{-\\frac{n}{2\\sigma }(\\frac{1}{p}- \\frac{1}{q})- \\frac{a}{2\\sigma }-(k+j+1)}\\Vert u_0+ u_1\\Vert _{L^p}\\lesssim (1+t)^{-\\frac{n}{2\\sigma }(\\frac{1}{p}- \\frac{1}{q})- \\frac{a}{2\\sigma }-j-1}\\Vert u_0+ u_1\\Vert _{L^p} $ for large $t$ .",
"Therefore, from (REF ) to (REF ) we may conclude the desired estimates.",
"This completes the proof of Theorem REF .",
"Remark 2.4 Here we want to underline that in the proof of (REF ) we used the property of the normalized Riez potential (see more Remark 2.1 in [2]) $ I_\\varepsilon f(x):= \\mathcal {F}^{-1}\\big (|\\xi |^{-\\varepsilon }\\hat{f}(\\xi )\\big )= C_{n,\\varepsilon }\\int _{\\mathbb {R}^n}\\frac{f(y)}{|x-y|^{n-\\varepsilon }}dy, $ where $\\varepsilon \\in (0,n)$ .",
"In particular, if $f \\in L^p$ for some $p \\in \\big (1,\\frac{n}{\\varepsilon }\\big )$ , then the following properties hold: $ I_\\varepsilon f \\in L^{p^*} \\quad \\text{ and } \\quad \\Vert I_\\varepsilon f\\Vert _{L^{p^*}} \\lesssim \\Vert f\\Vert _{L^p}, \\quad \\text{ where }\\quad \\frac{1}{p}- \\frac{1}{p^*}= \\frac{\\varepsilon }{n}.",
"$"
],
[
"Global (in time) existence of the solution",
"We choose introduce the solution space $X(t):= C([0,t],H^{\\sigma }) \\cap C^1([0,t],L^2), $ with the norm $\\Vert u\\Vert _{X(t)}:= \\sup _{0\\le \\tau \\le t} \\Big ( (1+\\tau )^{\\frac{n}{2\\sigma }(\\frac{1}{m}-\\frac{1}{2})}\\Vert u(\\tau ,\\cdot )\\Vert _{L^2} &+ (1+\\tau )^{\\frac{n}{2\\sigma }(\\frac{1}{m}-\\frac{1}{2})+ \\frac{1}{2}}\\big \\Vert |D|^{\\sigma } u(\\tau ,\\cdot )\\big \\Vert _{L^2} \\\\&+ (1+\\tau )^{\\frac{n}{2\\sigma }(\\frac{1}{m}-\\frac{1}{2})+ 1}\\Vert u_t(\\tau ,\\cdot )\\Vert _{L^2}\\Big ).$ By recalling the fundamental solutions from (), we can write the solutions of the corresponding linear Cauchy problems with vanishing right-hand sides to (REF ) as follows: $ u^{ln}(t,x)= K_0(t,x) \\ast _{x} u_0(x)+ K_1(t,x) \\ast _{x} u_1(x), $ where $ K_0(t,x):= \\mathcal {F}^{-1}\\Big (\\frac{e^{-t|\\xi |^{2\\sigma }}-|\\xi |^{2\\sigma } e^{-t}}{1-|\\xi |^{2\\sigma }}\\Big ) \\quad \\text{ and }\\quad K_1(t,x):= \\mathcal {F}^{-1}\\Big (\\frac{e^{-t|\\xi |^{2\\sigma }}-e^{-t}}{1-|\\xi |^{2\\sigma }}\\Big ).",
"$ Using Duhamel's principle we get the formal implicit representation of the solutions to (REF ) in the following form: $ u(t,x)= u^{ln}(t,x) + \\int _0^t K_1(t-\\tau ,x) \\ast _x |u(\\tau ,x)|^p d\\tau =: u^{ln}(t,x)+ u^{nl}(t,x).",
"$ We define for all $t>0$ the operator $N: \\, u \\in X(t) \\longrightarrow Nu \\in X(t)$ by $Nu(t,x)= u^{ln}(t,x)+ u^{nl}(t,x).",
"$ We will show that the operator $N$ fulfills the following two inequalities: $\\Vert Nu\\Vert _{X(t)} &\\lesssim \\Vert (u_0,u_1)\\Vert _{\\mathcal {A}^{\\sigma }_m}+ \\Vert u\\Vert ^p_{X(t)}, \\\\\\Vert Nu- Nv\\Vert _{X(t)} &\\lesssim \\Vert u- v\\Vert _{X(t)} \\big (\\Vert u\\Vert ^{p-1}_{X(t)}+ \\Vert v\\Vert ^{p-1}_{X(t)}\\big ).",
"$ After that, applying Banach's fixed point theorem we gain local (in time) existence results of large data solutions and global (in time) existence results of small data solutions as well.",
"From the definition of the norm in $X(t)$ , by plugging $a=0$ , $a=\\sigma $ and $j=0,\\,1$ into the statements from Proposition REF we arrive at $\\big \\Vert u^{ln}\\big \\Vert _{X(t)} \\lesssim \\Vert (u_0,u_1)\\Vert _{\\mathcal {A}^{\\sigma }_{m}}.",
"$ Thus, it is reasonable to indicate the following inequality instead of (REF ): $\\Vert u^{nl}\\Vert _{X(t)} \\lesssim \\Vert u\\Vert ^p_{X(t)}.",
"$ At the first stage, let us prove the inequality (REF ).",
"In order to deal with some estimates for $u^{nl}$ , we use the $(L^m \\cap L^2)- L^2$ estimates if $\\tau \\in [0,t/2]$ and the $L^2-L^2$ estimates if $\\tau \\in [t/2,t]$ from Proposition REF .",
"As a result, we derive the following estimates for $k=0,\\,1$ : $\\big \\Vert |D|^{k\\sigma } u^{nl}(t,\\cdot )\\big \\Vert _{L^2} &\\lesssim \\int _0^{t/2}(1+t-\\tau )^{-\\frac{n}{2\\sigma }(\\frac{1}{m}-\\frac{1}{2})- \\frac{k}{2}}\\big \\Vert |u(\\tau ,\\cdot )|^p\\big \\Vert _{L^m \\cap L^2}d\\tau \\\\&\\qquad + \\int _{t/2}^t (1+t-\\tau )^{-\\frac{k}{2}}\\big \\Vert |u(\\tau ,\\cdot )|^p\\big \\Vert _{L^2}d\\tau .$ Hence, it is clear to estimate $|u(\\tau ,x)|^p$ in $L^m \\cap L^2$ and $L^2$ .",
"Namely, we can proceed as follows: $\\big \\Vert |u(\\tau ,\\cdot )|^p\\big \\Vert _{L^m \\cap L^2} \\lesssim \\Vert u(\\tau ,\\cdot )\\Vert ^p_{L^{mp}}+ \\Vert u(\\tau ,\\cdot )\\Vert ^p_{L^{2p}} \\quad \\text{ and } \\quad \\big \\Vert |u(\\tau ,\\cdot )|^p\\big \\Vert _{L^2}= \\Vert u(\\tau ,\\cdot )\\Vert ^p_{L^{2p}}.$ By applying the fractional Gagliardo-Nirenberg inequality from Proposition REF we obtain $\\big \\Vert |u(\\tau ,\\cdot )|^p\\big \\Vert _{L^m \\cap L^2} &\\lesssim (1+\\tau )^{-\\frac{n}{2m\\sigma }(p-1)}\\Vert u\\Vert ^p_{X(\\tau )}, \\\\\\big \\Vert |u(\\tau ,\\cdot )|^p\\big \\Vert _{L^2} &\\lesssim (1+\\tau )^{-\\frac{n}{2m\\sigma }(p-\\frac{m}{2})}\\Vert u\\Vert ^p_{X(\\tau )}, $ provided that the conditions (REF ) and () are satisfied.",
"Consequently, we arrive at $\\big \\Vert |D|^{k\\sigma } u^{nl}(t,\\cdot )\\big \\Vert _{L^2} &\\lesssim (1+t)^{-\\frac{n}{2\\sigma }(\\frac{1}{m}-\\frac{1}{2})- \\frac{k}{2}}\\Vert u\\Vert ^p_{X(t)} \\int _0^{t/2}(1+\\tau )^{-\\frac{n}{2m\\sigma }(p-1)} d\\tau \\\\&\\qquad + (1+t)^{-\\frac{n}{2m\\sigma }(p-\\frac{m}{2})}\\Vert u\\Vert ^p_{X(t)} \\int _{t/2}^t (1+t-\\tau )^{-\\frac{k}{2}}d\\tau .$ Here we notice that $(1+t-\\tau ) \\approx (1+t)$ if $\\tau \\in [0,t/2]$ and $(1+\\tau ) \\approx (1+t)$ if $\\tau \\in [t/2,t]$ .",
"Since the condition (REF ) holds, it is equivalent to $-\\frac{n}{2m\\sigma }(p-1)< -1$ .",
"Moreover, it is clear to see that $-\\frac{k}{2}>-1$ for $k=0,\\,1$ .",
"Therefore, we get $ (1+t)^{-\\frac{n}{2\\sigma }(\\frac{1}{m}-\\frac{1}{2})- \\frac{k}{2}}\\int _0^{t/2}(1+\\tau )^{-\\frac{n}{2m\\sigma }(p-1)} d\\tau \\lesssim (1+t)^{-\\frac{n}{2\\sigma }(\\frac{1}{m}-\\frac{1}{2})- \\frac{k}{2}}, $ and $(1+t)^{-\\frac{n}{2m\\sigma }(p-\\frac{m}{2})} \\int _{t/2}^t (1+t-\\tau )^{-\\frac{k}{2}}d\\tau \\lesssim (1+t)^{-\\frac{n}{2m\\sigma }(p-\\frac{m}{2})+1- \\frac{k}{2}}\\lesssim (1+t)^{-\\frac{n}{2\\sigma }(\\frac{1}{m}-\\frac{1}{2})- \\frac{k}{2}}.",
"$ From both the above estimates we may conclude $\\big \\Vert |D|^{k\\sigma } u^{nl}(t,\\cdot )\\big \\Vert _{L^2} \\lesssim (1+t)^{-\\frac{n}{2\\sigma }(\\frac{1}{m}-\\frac{1}{2})- \\frac{k}{2}}\\Vert u\\Vert ^p_{X(t)}, $ for $k=0,\\,1$ .",
"In the similar way we also derive the following estimate: $\\big \\Vert \\partial _t u^{nl}(t,\\cdot )\\big \\Vert _{L^2} &\\lesssim (1+t)^{-\\frac{n}{2\\sigma }(\\frac{1}{m}-\\frac{1}{2})- 1}\\Vert u\\Vert ^p_{X(t)} \\int _0^{t/2}(1+\\tau )^{-\\frac{n}{2m\\sigma }(p-1)} d\\tau \\\\&\\qquad + (1+t)^{-\\frac{n}{2m\\sigma }(p-\\frac{m}{2})}\\Vert u\\Vert ^p_{X(t)} \\int _{t/2}^t (1+t-\\tau )^{-1}d\\tau .$ It is obvious that the first integral will be handled as before.",
"For this reason, we only need to estimate the second one.",
"In particular, we have $(1+t)^{-\\frac{n}{2m\\sigma }(p-\\frac{m}{2})} \\int _{t/2}^t (1+t-\\tau )^{-1}d\\tau &\\lesssim (1+t)^{-\\frac{n}{2m\\sigma }(p-\\frac{m}{2})}\\log (1+t) \\nonumber \\\\&\\lesssim (1+t)^{-\\frac{n}{2m\\sigma }(p-\\frac{m}{2})+\\varepsilon } \\lesssim (1+t)^{-\\frac{n}{2\\sigma }(\\frac{1}{m}-\\frac{1}{2})- 1}.",
"$ Here because of $-\\frac{n}{2m\\sigma }(p-1)< -1$ , we choose a sufficiently small positive number $\\varepsilon $ satisfying $\\varepsilon < -1+\\frac{n}{2m\\sigma }(p-1)$ .",
"Therefore, we arrive at $ \\big \\Vert \\partial _t u^{nl}(t,\\cdot )\\big \\Vert _{L^2} \\lesssim (1+t)^{-\\frac{n}{2\\sigma }(\\frac{1}{m}-\\frac{1}{2})- 1}\\Vert u\\Vert ^p_{X(t)}.",
"$ From the definition of the norm in $X(t)$ , we may conclude immediately the inequality (REF ).",
"Next, let us prove the inequality ().",
"Taking account of two elements $u$ and $v$ from $X(t)$ , we get $Nu(t,x)- Nv(t,x)= u^{nl}(t,x)- v^{nl}(t,x).",
"$ Using again the $(L^m \\cap L^2)- L^2$ estimates if $\\tau \\in [0,t/2]$ and the $L^2-L^2$ estimates if $\\tau \\in [t/2,t]$ from Proposition REF , we obtain the following estimates: $\\big \\Vert |D|^{k\\sigma } (u^{nl}- v^{nl})(t,\\cdot )\\big \\Vert _{L^2} &\\lesssim \\int _0^{t/2}(1+t-\\tau )^{-\\frac{n}{2\\sigma }(\\frac{1}{m}-\\frac{1}{2})- \\frac{k}{2}}\\big \\Vert |u(\\tau ,\\cdot )|^p- v(\\tau ,\\cdot )|^p\\big \\Vert _{L^m \\cap L^2}d\\tau \\\\&\\qquad + \\int _{t/2}^t (1+t-\\tau )^{-\\frac{k}{2}}\\big \\Vert |u(\\tau ,\\cdot )|^p- v(\\tau ,\\cdot )|^p\\big \\Vert _{L^2}d\\tau ,$ and $\\big \\Vert \\partial _t (u^{nl}- v^{nl})(t,\\cdot )\\big \\Vert _{L^2} &\\lesssim \\int _0^{t/2}(1+t-\\tau )^{-\\frac{n}{2\\sigma }(\\frac{1}{m}-\\frac{1}{2})- 1}\\big \\Vert |u(\\tau ,\\cdot )|^p- v(\\tau ,\\cdot )|^p\\big \\Vert _{L^m \\cap L^2}d\\tau \\\\&\\qquad + \\int _{t/2}^t (1+t-\\tau )^{-1}\\big \\Vert |u(\\tau ,\\cdot )|^p- v(\\tau ,\\cdot )|^p\\big \\Vert _{L^2}d\\tau .$ Applying Hölder's inequality leads to $\\big \\Vert |u(\\tau ,\\cdot )|^p- |v(\\tau ,\\cdot )|^p\\big \\Vert _{L^2}& \\lesssim \\Vert u(\\tau ,\\cdot )- v(\\tau ,\\cdot )\\Vert _{L^{2p}} \\big (\\Vert u(\\tau ,\\cdot )\\Vert ^{p-1}_{L^{2p}}+ \\Vert v(\\tau ,\\cdot )\\Vert ^{p-1}_{L^{2p}}\\big ),\\\\\\big \\Vert |u(\\tau ,\\cdot )|^p- |v(\\tau ,\\cdot )|^p\\big \\Vert _{L^m}& \\lesssim \\Vert u(\\tau ,\\cdot )- v(\\tau ,\\cdot )\\Vert _{L^{mp}} \\big (\\Vert u(\\tau ,\\cdot )\\Vert ^{p-1}_{L^{mp}}+ \\Vert v(\\tau ,\\cdot )\\Vert ^{p-1}_{L^{mp}}\\big ).$ Analogously to the proof of (REF ), we employ the fractional Gagliardo-Nirenberg inequality from Proposition REF to the terms $ \\Vert u(\\tau ,\\cdot )- v(\\tau ,\\cdot )\\Vert _{L^{\\eta }},\\quad \\Vert u(\\tau ,\\cdot )\\Vert _{L^{\\eta }},\\quad \\Vert v(\\tau ,\\cdot )\\Vert _{L^{\\eta }} $ with $\\eta =2p$ or $\\eta =mp$ to conclude the inequality ().",
"Summarizing, the proof of Theorem REF is completed."
],
[
"Large time behavior of the global solution",
"In order to prove Theorem REF , we need some auxiliary estimates as follows: Proposition 3.1 The Sobolev solutions to (REF ) satisfy the following estimate for $j,\\,k=0,\\,1$ and $(j,k)\\ne (1,1)$ : $ \\big \\Vert \\partial _t^j |D|^{k\\sigma } \\big (u(t,\\cdot )- (P_0+P_1)G_\\sigma (t,\\cdot )\\big )\\big \\Vert _{L^2} \\lesssim (1+t)^{-\\frac{n}{4\\sigma }-\\frac{k}{2}- j- \\frac{1}{2\\sigma }} \\big (\\Vert u_0\\Vert _{L^{1,1} \\cap H^\\sigma }+ \\Vert u_1\\Vert _{L^{1,1} \\cap L^2}\\big ), $ for large $t \\ge 1$ and for all space dimensions $n\\ge 1$ .",
"Following the proof of Corollary REF with a minor modification we can conclude the proof of Proposition REF .",
"Proposition 3.2 The Sobolev solutions to (REF ) satisfy the following estimate for large $t \\ge 1$ : $\\big \\Vert \\partial _t^j |D|^a \\big (u(t,\\cdot )- v(t,\\cdot )\\big )\\big \\Vert _{L^2} \\lesssim t^{-\\frac{n}{4\\sigma }-\\frac{a}{2\\sigma }-j-1} \\Vert (u_0,u_1)\\Vert _{L^1}+ e^{-t} \\big (\\Vert u_0\\Vert _{H^a}+ \\Vert u_1\\Vert _{H^{[a-2\\sigma ]^+}}\\big ),$ for any $a\\ge 0$ , $j=0,1$ and for all space dimensions $n\\ge 1$ .",
"For small frequencies, we can repeat exactly the same way as we did in the proof of Theorem REF to derive $ \\big \\Vert \\partial _t^j |D|^a \\big (u_{\\text{\\fontshape {n}\\selectfont low}}(t,\\cdot )- v_{\\text{\\fontshape {n}\\selectfont low}}(t,\\cdot )\\big )\\big \\Vert _{L^2} \\lesssim (1+t)^{-\\frac{n}{4\\sigma }- \\frac{a}{2\\sigma }-j-1}\\Vert (u_0,u_1)\\Vert _{L^1}+ e^{-t}\\Vert (u_0,u_1)\\Vert _{L^2}.",
"$ Taking account of large frequencies we can proceed as follows: $&\\big \\Vert \\partial _t^j |D|^a \\big (u_{\\text{\\fontshape {n}\\selectfont high}}(t,\\cdot )- v_{\\text{\\fontshape {n}\\selectfont high}}(t,\\cdot )\\big )\\big \\Vert _{L^2} \\nonumber \\\\&\\qquad \\lesssim e^{-t}\\,\\Big \\Vert \\frac{|\\xi |^{a+2\\sigma }}{|\\xi |^{2\\sigma }-1}\\big (1-\\chi (|\\xi |)\\big )\\widehat{u_0}(|\\xi |)+ \\frac{|\\xi |^a}{|\\xi |^{2\\sigma }-1}\\big (1-\\chi (|\\xi |)\\big )\\widehat{u_1}(|\\xi |)\\Big \\Vert _{L^2} \\nonumber \\\\&\\qquad \\quad + \\Big \\Vert e^{-t|\\xi |^{2\\sigma }}\\frac{|\\xi |^{a+2(j+1)\\sigma }}{|\\xi |^{2\\sigma }-1}\\big (1-\\chi (|\\xi |)\\big )\\big (\\widehat{u_0}(|\\xi |)+\\widehat{u_1}(|\\xi |)\\big )\\Big \\Vert _{L^2} \\\\&\\qquad \\lesssim e^{-t}\\,\\big \\Vert |\\xi |^a\\widehat{u_0}(|\\xi |)+ |\\xi |^{[a- 2\\sigma ]^+}\\widehat{u_1}(|\\xi |)\\big \\Vert _{L^2} \\nonumber \\\\&\\qquad \\quad + \\big \\Vert e^{-t|\\xi |^{2\\sigma }}|\\xi |^{a+2(j+1)\\sigma }\\big (1-\\chi (|\\xi |)\\big )\\big \\Vert _{L^2}\\Vert \\widehat{u_0}+\\widehat{u_1}\\Vert _{L^\\infty } \\\\&\\qquad \\lesssim e^{-t}\\,\\big (\\Vert u_0\\Vert _{H^a}+ \\Vert u_1\\Vert _{H^{[a- 2\\sigma ]^+}}\\big ) + t^{-\\frac{n}{4\\sigma }-\\frac{a}{2\\sigma }-j-1} \\Vert (u_0,u_1)\\Vert _{L^1} $ Here we notice that we used Parseval-Plancherel formula in (REF ).",
"Moreover, for the second term in () and () we applied Hölder's inequality and Lemma REF combined with Young-Hausdorff inequality, respectively.",
"Therefore, from the above estimates we may conclude the desired statement what we wanted to prove.",
"Proposition 3.3 The Sobolev solutions to (REF ) satisfy the following estimate for large $t \\ge 1$ : $\\big \\Vert \\partial _t^j |D|^a v(t,\\cdot )\\big \\Vert _{L^2} \\lesssim t^{-\\frac{n}{4\\sigma }- \\frac{a}{2\\sigma }-j} \\Vert (u_0,u_1)\\Vert _{L^1},$ for any $a\\ge 0$ , $j=0,1$ and for all space dimensions $n\\ge 1$ .",
"Moreover, the following estimate holds for any $t>0$ : $\\big \\Vert \\partial _t^j |D|^a G_\\sigma (t,\\cdot )\\big \\Vert _{L^2} \\lesssim t^{-\\frac{n}{4\\sigma }- \\frac{a}{2\\sigma }-j},$ for any $a\\ge 0$ , $j=0,1$ and for all space dimensions $n\\ge 1$ .",
"To derive (REF ), we repeat some arguments as we did in the proof of the second term in ().",
"By the aid of Parseval-Plancherel formula and a change of variables when needed, we may conclude the proof of (REF ).",
"Hence, this completes the proof of Proposition REF .",
"Thanks to the statement in Proposition REF , we only indicate the following estimate in place of (REF ): $\\Big \\Vert \\partial _t^j |D|^{k\\sigma }\\Big (\\int _0^t K_1(t-\\tau ,x) \\ast _x |u(\\tau ,x)|^p d\\tau - &\\Big (\\int _0^\\infty \\int _{\\mathbb {R}^n} |u(,\\tau ,y)|^p dyd\\tau \\Big )G_\\sigma (t,x)\\Big )\\Big \\Vert _{L^2} \\\\&= o\\big (t^{-\\frac{n}{4\\sigma }- \\frac{k}{2}-j}\\big ),$ by recalling the presentation of the solutions $u(t,x)= u^{ln}(t,x)+u^{nl}(t,x)$ to (REF ) as in Theorem REF .",
"Due to the fact that $K_1(0,x) \\ast _x |u(t,x)|^p= 0$ , we can re-write the above estimate in the equivalent form $\\Big \\Vert \\int _0^t \\partial _t^j |D|^{k\\sigma }\\big (K_1(t-\\tau ,x) \\ast _x |u(\\tau ,x)|^p\\big ) d\\tau - &\\Big (\\int _0^\\infty \\int _{\\mathbb {R}^n} |u(\\tau ,y)|^p dyd\\tau \\Big )\\partial _t^j |D|^{k\\sigma }G_\\sigma (t,x)\\Big \\Vert _{L^2} \\nonumber \\\\&= o\\big (t^{-\\frac{n}{4\\sigma }- \\frac{k}{2}-j}\\big ).",
"$ Now we shall separate the left-hand side term of (REF ) in $L^2$ norm into five sub-terms as follows: $&\\int _0^t \\partial _t^j |D|^{k\\sigma }\\big (K_1(t-\\tau ,x) \\ast _x |u(\\tau ,x)|^p\\big ) d\\tau - \\Big (\\int _0^\\infty \\int _{\\mathbb {R}^n} |u(\\tau ,y)|^p dyd\\tau \\Big )\\partial _t^j |D|^{k\\sigma }G_\\sigma (t,x) \\nonumber \\\\&\\qquad = \\int _0^{t/2} \\partial _t^j |D|^{k\\sigma }\\Big (\\big (K_1(t-\\tau ,x)- G_\\sigma (t-\\tau ,\\cdot )\\big ) \\ast _x |u(\\tau ,x)|^p\\Big ) d\\tau \\nonumber \\\\&\\qquad \\quad + \\int _{t/2}^t \\partial _t^j |D|^{k\\sigma }\\big (K_1(t-\\tau ,x) \\ast _x |u(\\tau ,x)|^p\\big ) d\\tau \\nonumber \\\\&\\qquad \\quad + \\int _0^{t/2} \\partial _t^j |D|^{k\\sigma }\\Big (\\big (G_\\sigma (t-\\tau ,x)- G_\\sigma (t,x) \\big )\\ast _x |u(\\tau ,x)|^p\\Big ) d\\tau \\nonumber \\\\&\\qquad \\quad + \\int _0^{t/2} \\partial _t^j |D|^{k\\sigma }\\Big (G_\\sigma (t,x)\\ast _x |u(\\tau ,x)|^p - \\Big (\\int _{\\mathbb {R}^n} |u(\\tau ,y)|^p dy\\Big )G_\\sigma (t,x) \\Big )d\\tau \\nonumber \\\\&\\qquad \\quad - \\Big (\\int _{t/2}^\\infty \\int _{\\mathbb {R}^n} |u(\\tau ,y)|^p dyd\\tau \\Big )\\partial _t^j |D|^{k\\sigma }G_\\sigma (t,x) \\nonumber \\\\&\\qquad =: I_1(t,x)+ I_2(t,x)+ I_3(t,x)+ I_4(t,x)- I_5(t,x).",
"$ At first, let us estimate $I_1(t,x)$ .",
"Namely, applying the statement (REF ) leads to $\\Vert I_1(t,\\cdot )\\Vert _{L^2}& \\lesssim \\int _0^{t/2} \\Big \\Vert \\partial _t^j |D|^{k\\sigma }\\Big (\\big (K_1(t-\\tau ,x)- G_\\sigma (t-\\tau ,\\cdot )\\big ) \\ast _x |u(\\tau ,x)|^p\\Big )\\Big \\Vert _{L^2} d\\tau \\nonumber \\\\&\\lesssim \\int _0^{t/2} (t-\\tau )^{-\\frac{n}{4\\sigma }-\\frac{k}{2}-j-1} \\big \\Vert |u(\\tau ,\\cdot )|^p\\big \\Vert _{L^1}d\\tau + \\int _0^{t/2} e^{-(t-\\tau )} \\big \\Vert |u(\\tau ,\\cdot )|^p\\big \\Vert _{L^2}d\\tau \\nonumber \\\\&\\lesssim t^{-\\frac{n}{4\\sigma }-\\frac{k}{2}-j-1} \\int _0^{t/2} (1+\\tau )^{-\\frac{n}{2\\sigma }(p-1)}d\\tau + e^{-t/2} \\int _0^{t/2} (1+\\tau )^{-\\frac{n}{2\\sigma }(p-\\frac{1}{2})}d\\tau \\\\&\\lesssim t^{-\\frac{n}{4\\sigma }-\\frac{k}{2}-j-1}+ e^{-t/2}\\lesssim t^{-\\frac{n}{4\\sigma }-\\frac{k}{2}-j-1}.",
"$ Here we used (REF ), () and the relation $t-\\tau \\approx t$ if $\\tau \\in [0,t/2]$ in (REF ).",
"Moreover, in order to derive () we notice that the condition (REF ) implies the integrability of both the above integrals.",
"In the second step, taking account of $I_2(t,x)$ we repeat exactly the arguments as we did in the proofs of (REF ) and (REF ) to obtain $\\Vert I_2(t,\\cdot )\\Vert _{L^2}\\lesssim (1+t)^{-\\frac{n}{4\\sigma }- \\frac{k}{2}- j-\\varepsilon }$ where $\\varepsilon $ is a sufficiently small positive satisfying $2\\varepsilon < -1+\\frac{n}{2\\sigma }(p-1)$ .",
"To control $I_3(t,x)$ , by using the mean value theorem on $t$ we get the following representation: $ G_\\sigma (t-\\tau ,x)- G_\\sigma (t,x)= -\\tau \\,\\partial _t G_\\sigma (t- \\omega _1 \\tau ,x) $ with a constant $\\omega _1 \\in [0,1]$ .",
"Hence, we can proceed as follows: $\\Vert I_3(t,\\cdot )\\Vert _{L^2} &\\lesssim \\int _0^{t/2} \\Big \\Vert \\partial _t^j |D|^{k\\sigma }\\Big (\\big (G_\\sigma (t-\\tau ,x)- G_\\sigma (t,x) \\big )\\ast _x |u(\\tau ,x)|^p\\Big )\\Big \\Vert _{L^2} d\\tau \\\\&\\lesssim \\int _0^{t/2} \\tau \\, \\big \\Vert \\partial _t^{j+1} |D|^{k\\sigma }\\big (G_\\sigma (t- \\omega _1 \\tau ,x)\\ast _x |u(\\tau ,x)|^p\\big )\\big \\Vert _{L^2} d\\tau $ Employing (REF ) gives $\\Vert I_3(t,\\cdot )\\Vert _{L^2}&\\lesssim \\int _0^{t/2} \\tau \\,(t- \\omega _1 \\tau )^{-\\frac{n}{4\\sigma }- \\frac{a}{2\\sigma }-j-1} \\big \\Vert |u(\\tau ,\\cdot )|^p\\big \\Vert _{L^1} d\\tau \\\\&\\lesssim t^{-\\frac{n}{4\\sigma }- \\frac{a}{2\\sigma }-j-1}\\int _0^{t/2} \\tau \\, (1+\\tau )^{-\\frac{n}{2\\sigma }(p-1)} d\\tau ,$ where we used (REF ) and the relation $t- \\omega _1 \\tau \\approx t$ if $\\tau \\in [0,t/2]$ .",
"Thus, we may arrive at $\\Vert I_3(t,\\cdot )\\Vert _{L^2}&\\lesssim t^{-\\frac{n}{4\\sigma }- \\frac{a}{2\\sigma }-j-1}\\int _0^{t/2} (1+\\tau )^{-\\frac{n}{2\\sigma }(p-1)+1} d\\tau \\nonumber \\\\&\\lesssim t^{-\\frac{n}{4\\sigma }- \\frac{a}{2\\sigma }-j-1}\\Big (1+ (1+t)^{-\\frac{n}{2\\sigma }(p-1)+2}+ \\log (1+t)\\Big ) \\nonumber \\\\&\\lesssim t^{-\\frac{n}{4\\sigma }- \\frac{a}{2\\sigma }-j-1}(1+t)^{-\\varepsilon +1} \\lesssim t^{-\\frac{n}{4\\sigma }- \\frac{a}{2\\sigma }-j-\\varepsilon } $ as $t \\rightarrow \\infty $ , with a sufficiently small positive number $\\varepsilon $ such that $-\\frac{n}{2\\sigma }(p-1)+1< -\\varepsilon $ .",
"Let us now devote to the estimate for $I_4(t,x)$ .",
"To do this, we shall divide our attention into two parts.",
"In particular, we write $I_4(t,x)&= \\int _0^{t/2} \\partial _t^j |D|^{k\\sigma }\\Big (\\int _{\\mathbb {R}^n} G_\\sigma (t,x-y) |u(\\tau ,y)|^p dy - \\int _{\\mathbb {R}^n} G_\\sigma (t,x) |u(\\tau ,y)|^p dy\\Big )d\\tau \\\\&= \\int _0^{t/2} \\int _{|y|\\le t^{\\frac{1}{4\\sigma }}} \\partial _t^j |D|^{k\\sigma }\\big (G_\\sigma (t,x-y)- G_\\sigma (t,x)\\big ) |u(\\tau ,y)|^p dyd\\tau \\\\&\\qquad + \\int _0^{t/2} \\int _{|y|\\ge t^{\\frac{1}{4\\sigma }}} \\partial _t^j |D|^{k\\sigma }\\big (G_\\sigma (t,x-y)- G_\\sigma (t,x)\\big ) |u(\\tau ,y)|^p dy d\\tau =: I_{41}(t,x)+ I_{42}(t,x).$ For the first integral $I_{41}(t,x)$ , using the mean value theorem on $x$ we derive $ G_\\sigma (t,x-y)- G_\\sigma (t,x)= -y\\,\\partial _x G_\\sigma (t,x- \\omega _2 y) $ with a constant $\\omega _2 \\in [0,1]$ .",
"For this reason, we may conclude the following estimate: $\\Vert I_{41}(t,\\cdot )\\Vert _{L^2}&\\lesssim \\Big \\Vert \\int _0^{t/2} \\int _{|y|\\le t^{\\frac{1}{4\\sigma }}} (-y)\\,\\partial _t^j |D|^{k\\sigma + 1} G_\\sigma (t,x- \\omega _2 y) |u(\\tau ,y)|^p dyd\\tau \\Big \\Vert _{L^2} \\nonumber \\\\&\\lesssim \\int _0^{t/2} \\int _{|y|\\le t^{\\frac{1}{4\\sigma }}} |y|\\,\\big \\Vert \\partial _t^j |D|^{k\\sigma + 1} G_\\sigma (t,\\cdot ) \\big \\Vert _{L^2} |u(\\tau ,y)|^p dyd\\tau \\nonumber \\\\&\\lesssim t^{-\\frac{n}{4\\sigma }- \\frac{k}{2}-j- \\frac{1}{2\\sigma }} \\int _0^{t/2} \\int _{|y|\\le t^{\\frac{1}{4\\sigma }}} |y|\\,|u(\\tau ,y)|^p dyd\\tau \\qquad \\big (\\text{by } (\\ref {pro2.1.2.3})\\big ) \\nonumber \\\\&\\lesssim t^{-\\frac{n}{4\\sigma }- \\frac{k}{2}-j- \\frac{1}{2\\sigma }}\\int _0^{t/2} t^{\\frac{1}{4\\sigma }} \\big \\Vert |u(\\tau ,\\cdot )|^p\\big \\Vert _{L^1}d\\tau \\nonumber \\\\&\\lesssim t^{-\\frac{n}{4\\sigma }- \\frac{k}{2}-j- \\frac{1}{4\\sigma }}\\int _0^{t/2} (1+\\tau )^{-\\frac{n}{2\\sigma }(p-1)}d\\tau \\qquad \\big (\\text{by } (\\ref {the3.1.5})\\big ) \\nonumber \\\\&\\lesssim t^{-\\frac{n}{4\\sigma }- \\frac{k}{2}-j- \\frac{1}{4\\sigma }}.",
"$ In order to deal with the other interesting integral $I_{42}(t,x)$ , we notice that by (REF ) again it holds $ \\int _0^\\infty \\int _{\\mathbb {R}^n}|u(\\tau ,y)|^p dyd\\tau = \\int _0^\\infty \\big \\Vert |u(\\tau ,\\cdot )|^p\\big \\Vert _{L^1}d\\tau \\lesssim \\int _0^\\infty (1+\\tau )^{-\\frac{n}{2\\sigma }(p-1)}d\\tau \\lesssim 1.",
"$ As a result, this deduces immediately the relation $ \\lim _{t \\rightarrow \\infty } \\int _0^\\infty \\int _{|y|\\ge t^{\\frac{1}{4\\sigma }}}|u(\\tau ,y)|^p dyd\\tau = 0, $ that is, there exist a sufficiently small positive $\\varepsilon $ such that $ \\int _0^\\infty \\int _{|y|\\ge t^{\\frac{1}{4\\sigma }}}|u(\\tau ,y)|^p dyd\\tau \\lesssim t^{-\\varepsilon }, $ as $t \\rightarrow \\infty $ .",
"Therefore, by (REF ) we can estimate $I_{42}(t,x)$ in the following way: $\\Vert I_{42}(t,\\cdot )\\Vert _{L^2}&\\le 2\\int _0^{t/2} \\int _{|y|\\ge t^{\\frac{1}{4\\sigma }}} \\big \\Vert \\partial _t^j |D|^{k\\sigma }G_\\sigma (t,\\cdot )\\big \\Vert _{L^2} |u(\\tau ,y)|^p dy d\\tau \\nonumber \\\\&\\lesssim t^{-\\frac{n}{4\\sigma }- \\frac{k}{2}-j}\\int _0^{t/2} \\int _{|y|\\ge t^{\\frac{1}{4\\sigma }}}|u(\\tau ,y)|^p dy d\\tau \\lesssim t^{-\\frac{n}{4\\sigma }- \\frac{k}{2}-j-\\varepsilon }.",
"$ From (REF ) and (REF ), we arrive at $\\Vert I_{42}(t,\\cdot )\\Vert _{L^2}\\lesssim t^{-\\frac{n}{4\\sigma }- \\frac{k}{2}-j-\\varepsilon }.$ Finally, we need to control $I_5(t,x)$ to complete our proof.",
"For this purpose, by (REF ) again we have $\\Vert I_5(t,\\cdot )\\Vert _{L^2}&\\lesssim \\big \\Vert \\partial _t^j |D|^{k\\sigma }G_\\sigma (t,\\cdot )\\big \\Vert _{L^2} \\int _{t/2}^\\infty \\int _{\\mathbb {R}^n} |u(\\tau ,y)|^p dyd\\tau \\nonumber \\\\&\\lesssim t^{-\\frac{n}{4\\sigma }- \\frac{a}{2\\sigma }-j} \\int _{t/2}^\\infty \\big \\Vert |u(\\tau ,\\cdot )|^p\\big \\Vert _{L^1} d\\tau \\lesssim t^{-\\frac{n}{4\\sigma }- \\frac{a}{2\\sigma }-j} \\int _{t/2}^\\infty (1+\\tau )^{-\\frac{n}{2\\sigma }(p-1)} d\\tau \\nonumber \\\\&\\lesssim t^{-\\frac{n}{4\\sigma }- \\frac{a}{2\\sigma }-j} (1+t)^{-\\frac{n}{2\\sigma }(p-1)+1} \\lesssim t^{-\\frac{n}{4\\sigma }- \\frac{a}{2\\sigma }-j-\\varepsilon } $ as $t \\rightarrow \\infty $ and $\\varepsilon $ is again chosen as a sufficiently small positive to guarantee $-\\frac{n}{2\\sigma }(p-1)+1< -\\varepsilon $ .",
"Consequently, combining (REF ) to (REF ) we may conclude (REF ).",
"Summarizing, Theorem REF is proved."
],
[
"Blow-up result",
"In this section, our aim is to verify the critical exponent to (REF ).",
"To state our result, we recall the definition of the weak solution to (REF ) (see, for instance, [15]).",
"Definition 4.1 Let $p>1$ and $T>0$ .",
"We say that $u \\in L^p_{\\text{\\fontshape {n}\\selectfont loc}}([0,T)\\times \\mathbb {R}^n)$ is a local weak solution to (REF ) if for any test function $\\phi (t,x) \\in \\mathcal {C}_0^\\infty ([0,T)\\times \\mathbb {R}^n)$ it holds $&\\int _0^T \\int _{\\mathbb {R}^n}|u(t,x)|^p \\phi (t,x)dxdt+ \\int _{\\mathbb {R}^n}u_0(x)\\big (\\phi (0,x)+ (-\\Delta )^{\\sigma }\\phi (0,x)- \\phi _t(0,x)\\big )dx+ \\int _{\\mathbb {R}^n} u_1(x) \\phi (0,x)dx \\nonumber \\\\&\\qquad = \\int _0^T \\int _{\\mathbb {R}^n}u(t,x)\\big (\\phi _{tt}(t,x)- (-\\Delta )^{\\sigma }\\phi _t(t,x)+ (-\\Delta )^{\\sigma }\\phi (t,x)- \\phi _t(t,x) \\big )dxdt.",
"$ If $T= \\infty $ , we say that $u$ is a global weak solution to (REF ).",
"The main ideas of our proof of blow-up result are based on a contradiction argument by using the test function method (see, for example, [4], [21]).",
"Due to the fact that this method, in general, cannot be directly applied to the fractional Laplacian operators $(-\\Delta )^\\sigma $ as well-known non-local operators, the assumption for integer $\\sigma $ comes into play in our proof.",
"At first, let us introduce the test functions $\\eta = \\eta (t)$ and $\\varphi =\\varphi (x)$ satisfying the following properties: $&1.\\quad \\eta \\in \\mathcal {C}_0^\\infty ([0,\\infty )) \\text{ and }\\eta (t)={\\left\\lbrace \\begin{array}{ll}1 \\quad \\text{ for }0 \\le t \\le 1/2, \\\\0 \\quad \\text{ for }t \\ge 1,\\end{array}\\right.}",
"& \\nonumber \\\\&2.\\quad \\varphi \\in \\mathcal {C}_0^\\infty (\\mathbb {R}^n) \\text{ and }\\varphi (x)= {\\left\\lbrace \\begin{array}{ll}1 \\quad \\text{ for } |x|\\le 1/2, \\\\0 \\quad \\text{ for }|x|\\ge 1,\\end{array}\\right.}",
"& \\nonumber \\\\&3.\\quad \\eta ^{-\\frac{p^{\\prime }}{p}}\\big (|\\eta ^{\\prime }|^{p^{\\prime }}+|\\eta ^{\\prime \\prime }|^{p^{\\prime }}\\big ) \\text{ and } \\varphi ^{-\\frac{p^{\\prime }}{p}} |\\Delta ^{\\sigma }\\varphi |^{p^{\\prime }} \\text{ are bounded, } & $ where $p^{\\prime }$ is the conjugate of $p$ .",
"In addtion, we suppose that $\\eta (t)$ is a decreasing function and that $\\varphi =\\varphi (|x|)$ is a radial function fulfilling $\\varphi (|x|) \\le \\varphi (|y|)$ for any $|x|\\ge |y|$ .",
"Let $R$ be a large parameter in $[0,\\infty )$ .",
"We define the following test function: $ \\phi _R(t,x):= \\eta _R(t) \\varphi _R(x), $ where $\\eta _R(t):= \\eta (R^{-2\\sigma }t)$ and $\\varphi _R(x):= \\varphi (R^{-1}x)$ .",
"Moreover, we define the funtional $ I_R:= \\int _0^{\\infty }\\int _{\\mathbb {R}^n}|u(t,x)|^p \\phi _R(t,x) dxdt= \\int _{Q_R}|u(t,x)|^p \\phi _R(t,x) d(x,t), $ where $Q_R:= [0,R^{2\\sigma }] \\times B_R,\\qquad B_R:= \\big \\lbrace x\\in \\mathbb {R}^n: |x|\\le R \\big \\rbrace .",
"$ Let us now assume that $u= u(t,x)$ is a global (in time) energy solution to (REF ).",
"Replacing $\\phi (t,x)= \\phi _R(t,x)$ in (REF ) we arrive at $&I_R+ \\int _{B_R} u_1(x) \\varphi _R(x)dx \\nonumber \\\\&\\,\\,\\,= \\int _{Q_R}u(t,x) \\Big (\\eta ^{\\prime \\prime }_R(t) \\varphi _R(x)- \\eta ^{\\prime }_R(t) (-\\Delta )^{\\sigma }\\varphi _R(x)+ \\eta _R(t) (-\\Delta )^{\\sigma }\\varphi _R(x)- \\eta ^{\\prime }_R(t) \\varphi _R(x)\\Big )d(x,t).",
"$ By applying Hölder's inequality with $\\frac{1}{p}+\\frac{1}{p^{\\prime }}=1$ we obtain $&\\int _{Q_R} |u(t,x)|\\, \\big |\\eta ^{\\prime \\prime }_R(t) \\varphi _R(x)\\big | d(x,t) \\\\&\\qquad \\le \\Big (\\int _{Q_R} \\Big |u(t,x)\\phi ^{\\frac{1}{p}}_R(t,x)\\Big |^p d(x,t)\\Big )^{\\frac{1}{p}} \\Big (\\int _{Q_R} \\Big |\\phi ^{-\\frac{1}{p}}_R(t,x) \\eta ^{\\prime \\prime }_R(t) \\varphi _R(x)\\Big |^{p^{\\prime }} d(x,t)\\Big )^{\\frac{1}{p^{\\prime }}} \\\\&\\qquad \\le I_R^{\\frac{1}{p}} \\Big ( \\int _{Q_R}\\eta _R^{-\\frac{p^{\\prime }}{p}}(t) \\big |\\eta ^{\\prime \\prime }_R(t)\\big |^{p^{\\prime }} \\varphi _R(x) d(x,t)\\Big )^{\\frac{1}{p^{\\prime }}}.$ Using the change of variables $\\tilde{t}:= R^{-2\\sigma }t$ and $\\tilde{x}:= R^{-1}x$ gives $ \\int _{Q_R} |u(t,x)|\\, \\big |\\eta ^{\\prime \\prime }_R(t) \\varphi _R(x)\\big | d(x,t) \\lesssim I_R^{\\frac{1}{p}}\\, R^{-4\\sigma + \\frac{n+2\\sigma }{p^{\\prime }}}.$ Here we notice that $ \\eta ^{\\prime \\prime }_R(t)= R^{-4\\sigma }\\eta ^{\\prime \\prime }(\\tilde{t})$ and the assumption (REF ) holds.",
"In the analogous treament, we also derive the following estimates: $\\int _{Q_R}|u(t,x)|\\, \\big |\\eta ^{\\prime }_R(t) (-\\Delta )^{\\sigma }\\varphi _R(x)\\big | d(x,t) &\\lesssim I_R^{\\frac{1}{p}}\\, R^{-4\\sigma + \\frac{n+2\\sigma }{p^{\\prime }}}, \\\\\\int _{Q_R}|u(t,x)|\\, \\big |\\eta _R(t) (-\\Delta )^{\\sigma }\\varphi _R(x)\\big | d(x,t) &\\lesssim I_R^{\\frac{1}{p}}\\, R^{-2\\sigma + \\frac{n+2\\sigma }{p^{\\prime }}}, \\\\\\int _{Q_R}|u(t,x)|\\, \\big |\\eta ^{\\prime }_R(t) \\varphi _R(x)\\big | d(x,t) &\\lesssim I_R^{\\frac{1}{p}}\\, R^{-2\\sigma + \\frac{n+2\\sigma }{p^{\\prime }}}, $ where we used $ \\eta ^{\\prime }_R(t)= R^{-2\\sigma }\\eta ^{\\prime }(\\tilde{t}) \\quad \\text{ and }\\quad (-\\Delta )^{\\sigma }\\varphi _R(x)= R^{-2\\sigma }(-\\Delta )^{\\sigma }\\varphi (\\tilde{x}), $ since $\\sigma $ is a integer.",
"Due to the assumption (REF ), there exists a constant $R_0>0$ such that it holds $ \\int _{B_R} u_1(x) \\varphi _R(x)dx >0,$ for any $R>R_0$ .",
"As a result, from (REF ) to (REF ) we may conclude the following estimate: $ I_R \\lesssim I_R^{\\frac{1}{p}}\\, R^{-2\\sigma + \\frac{n+2\\sigma }{p^{\\prime }}}, $ this is, $ I_R^{\\frac{1}{p^{\\prime }}} \\lesssim R^{-2\\sigma + \\frac{n+\\alpha }{p^{\\prime }}}.$ It is obvious to see that the assumption (REF ) is re-written in the equivalent form as follows: $-2\\sigma + \\frac{n+\\alpha }{p^{\\prime }} \\le 0.",
"$ Hence, it is reasonable to separate our consideration into two subcases.",
"Taking account of the first subcase $-2\\sigma + \\frac{n+\\alpha }{p^{\\prime }}< 0$ , we let $R \\rightarrow \\infty $ in (REF ) to get $ \\int _0^{\\infty }\\int _{\\mathbb {R}^n}|u(t,x)|^p dxdt= 0, $ which implies immediately $u \\equiv 0$ .",
"This is a contradiction to the assumption (REF ).",
"Let us now devote our attention to the second subcase $-2\\sigma + \\frac{n+\\alpha }{p^{\\prime }}= 0$ .",
"By (REF ) it follows $ I_R= \\int _{Q_R}|u(t,x)|^p \\phi _R(t,x) d(x,t) \\le C, $ for a sufficiently large $R$ and a suitable positive constant $C$ .",
"For this reason, we derive $ \\int _{\\tilde{Q}_R}|u(t,x)|^p \\phi _R(t,x) d(x,t) \\rightarrow 0 \\text{ as } R \\rightarrow \\infty ,$ where we introduce notations $\\tilde{Q}_R:= Q_R \\setminus \\big ([0,R^{2\\sigma }/2] \\times B_{R/2}\\big ),\\qquad B_{R/2}:= \\big \\lbrace x\\in \\mathbb {R}^n: 0\\le |x|\\le R/2 \\big \\rbrace .",
"$ Because of $\\partial ^2_t \\phi _R(t,x)= \\partial _t\\phi _R(t,x)= (-\\Delta )^{\\sigma }\\partial _t\\phi _R(t,x)= (-\\Delta )^{\\sigma }\\phi _R(t,x)=0$ in $(\\mathbb {R}^1_+ \\times \\mathbb {R}^n) \\setminus \\tilde{Q}_R$ , we repeat the steps of the proofs from (REF ) to () to arrive at the following estimate: $ I_R+ \\int _{B_R} u_1(x) \\varphi _R(x)dx \\lesssim \\Big (\\int _{\\tilde{Q}_R}|u(t,x)|^p \\phi _R(t,x) d(x,t)\\Big )^{\\frac{1}{p}}\\, R^{-2\\sigma + \\frac{n+2\\sigma }{p^{\\prime }}}.",
"$ Due to $-2\\sigma + \\frac{n+\\alpha }{p^{\\prime }}= 0$ , from the above estimate and (REF ) we obtain $ I_R< I_R+ \\int _{B_R} u_1(x) \\varphi _R(x)dx \\lesssim \\Big (\\int _{\\tilde{Q}_R}|u(t,x)|^p \\phi _R(t,x) d(x,t)\\Big )^{\\frac{1}{p}},$ for any $R>R_0$ .",
"By combining (REF ) and (REF ), we let $R \\rightarrow \\infty $ to conclude $ \\int _0^{\\infty }\\int _{\\mathbb {R}^n}|u(t,x)|^p dxdt= 0, $ which is again a contradiction to the assumption (REF ).",
"Summarizing, Theorem REF is proved.",
"Remark 4.1 Here we want to emphasize that for all small positive constants $\\varepsilon $ the lifespan $T_\\varepsilon $ of the solution to the given data $(0,\\varepsilon u_1)$ in Theorem REF can be estimated as follows: $ T_\\varepsilon \\le C\\,\\varepsilon ^{-\\frac{2\\sigma (p-1)}{2\\sigma - n(p-1)}} \\quad \\text{ with }C>0.$ Indeed, let us now consider the case of subcritical exponents.",
"We suppose that $u= u(t,x)$ is a local (in time) energy solution to (REF ) in $([0,T)\\times \\mathbb {R}^n)$ .",
"To varify the lifespan estimate, we take the initial data $(0,\\varepsilon u_1)$ in place of $(0,u_1)$ with a small positive constant $\\varepsilon $ where $u_1\\in L_1 \\cap L_2$ satisfies the assumption (REF ).",
"In the same way as we did in the steps of the proof of Theorem REF , we obtain the following estimte: $I_R+ c\\varepsilon \\le C\\, I_R^{\\frac{1}{p}}\\, R^{-2\\sigma + \\frac{n+2\\sigma }{p^{\\prime }}}.$ Here we notice that due to the assumption (REF ), we choose a suitable constant $c$ such that it holds $\\int _{B_R} u_1(x) \\varphi _R(x)dx> c> 0 $ for any $R> R_0$ .",
"From (REF ) we arrive at $c\\varepsilon \\le C\\, I_R^{\\frac{1}{p}}\\, R^{-2\\sigma + \\frac{n+2\\sigma }{p^{\\prime }}}- I_R.",
"$ By the aid of the elementary inequality $ A\\,y^\\gamma - y \\le A^{\\frac{1}{1-\\gamma }} \\text{ for any } A>0,\\, y \\ge 0 \\text{ and } 0< \\gamma < 1, $ a straightforward computation gives from (REF ) $ \\varepsilon \\le C\\,R^{-2\\sigma p^{\\prime }+ n+ 2\\sigma }= C\\,T^{-\\frac{2\\sigma p^{\\prime }- n- 2\\sigma }{2\\sigma }}= C\\,T^{-\\frac{2\\sigma - n(p-1)}{2\\sigma (p-1)}}, $ with $R= T^{\\frac{1}{2\\sigma }}$ .",
"Summarizing, letting $T \\rightarrow T_\\varepsilon - 0$ we may conclude (REF ).",
"Acknowledgments The PhD study of the second author is supported by Vietnamese Government's Scholarship (Grant number: 2015/911).",
"Appendix A A.1.",
"Fractional Gagliardo-Nirenberg inequality Proposition 4.1 Let $1<p,\\, p_0,\\, p_1<\\infty $ , $\\sigma >0$ and $s\\in [0,\\sigma )$ .",
"Then, it holds the following fractional Gagliardo-Nirenberg inequality for all $u\\in L^{p_0} \\cap \\dot{H}^\\sigma _{p_1}$ : $ \\Vert u\\Vert _{\\dot{H}^{s}_p}\\lesssim \\Vert u\\Vert _{L^{p_0}}^{1-\\theta }\\,\\, \\Vert u\\Vert _{\\dot{H}^{\\sigma }_{p_1}}^\\theta , $ where $\\theta =\\theta _{s,\\sigma }(p,p_0,p_1)=\\frac{\\frac{1}{p_0}-\\frac{1}{p}+\\frac{s}{n}}{\\frac{1}{p_0}-\\frac{1}{p_1}+\\frac{\\sigma }{n}}$ and $\\frac{s}{\\sigma }\\le \\theta \\le 1$ .",
"For the proof one can see [11].",
"A.2.",
"Modified Bessel functions Proposition 4.2 Let $f \\in L^p(\\mathbb {R}^n)$ , $p\\in [1,2]$ , be a radial function.",
"Then, the Fourier transform $F(f)$ is also a radial function and it satisfies $ F(f)(\\xi )= c \\int _0^\\infty g(r) r^{n-1} \\tilde{J}_{\\frac{n}{2}-1}(r|\\xi |)dr,\\quad g(|x|):= f(x), $ where $\\tilde{J}_\\mu (s):=\\frac{J_\\mu (s)}{s^\\mu }$ is called the modified Bessel function with the Bessel function $J_\\mu (s)$ and a non-negative integer $\\mu $ .",
"Proposition 4.3 The following properties of the modified Bessel function hold: $s\\,d_s\\tilde{J}_\\mu (s)= \\tilde{J}_{\\mu -1}(s)-2\\mu \\tilde{J}_\\mu (s)$ , $d_s\\tilde{J}_\\mu (s)= -s\\tilde{J}_{\\mu +1}(s)$ , $\\tilde{J}_{-\\frac{1}{2}}(s)= \\sqrt{\\frac{2}{\\pi }}\\cos s$ and $\\tilde{J}_{\\frac{1}{2}}(s)= \\sqrt{\\frac{2}{\\pi }} \\frac{\\sin s}{s}$ , $|\\tilde{J}_\\mu (s)| \\le Ce^{\\pi |\\fontshape {n}\\selectfont \\text{Im}\\mu |} \\text{ if } s \\le 1, $ and $\\tilde{J}_\\mu (s)= Cs^{-\\frac{1}{2}}\\cos \\big ( s-\\frac{\\mu }{2}\\pi - \\frac{\\pi }{4} \\big ) +\\mathcal {O}(|s|^{-\\frac{3}{2}}) \\text{ if } |s|\\ge 1$ , $\\tilde{J}_{\\mu +1}(r|x|)= -\\frac{1}{r|x|^2}\\partial _r \\tilde{J}_\\mu (r|x|)$ , $r \\ne 0$ , $x \\ne 0$ .",
"A.3.",
"Useful lemmas Lemma 4.1 Let $n\\ge 1$ , $c>0$ , $\\alpha >0$ and $\\beta \\in \\mathbb {R}$ satisfy $n+\\beta >0$ .",
"The following estimates hold for $t>0$ : $ \\int _{|\\xi |\\le 1} |\\xi |^\\beta e^{-c|\\xi |^\\alpha t}d\\xi \\lesssim (1+t)^{-\\frac{n+\\beta }{\\alpha }} \\quad \\text{ and } \\quad \\int _{|\\xi |\\ge 1} |\\xi |^\\beta e^{-c|\\xi |^\\alpha t}d\\xi \\lesssim t^{-\\frac{n+\\beta }{\\alpha }}.",
"$ In order to prove the first desired estimate, we shall split our consideration into two cases.",
"In the first case $t \\in (0,1]$ , we get immediately the following estimate: $ \\int _{|\\xi | \\le 1} |\\xi |^\\beta e^{-c|\\xi |^\\alpha t}d\\xi \\lesssim \\int _0^1 |\\xi |^{n+\\beta -1} e^{-c|\\xi |^\\alpha t}d|\\xi | \\lesssim 1.",
"$ For the second case $t \\in [1,\\infty )$ , we carry out the change of variables $\\xi ^\\alpha t= \\eta ^\\alpha $ , that is, $\\xi = t^{-\\frac{1}{\\alpha }} \\eta $ to dervie $ \\int _{|\\xi | \\le 1} |\\xi |^\\beta e^{-c|\\xi |^\\alpha t}d\\xi \\lesssim t^{-\\frac{n+\\beta }{\\alpha }} \\int _0^\\infty |\\eta |^{n+\\beta -1} e^{-c|\\eta |^\\alpha }d|\\eta | \\lesssim t^{-\\frac{n+\\beta }{\\alpha }}.",
"$ Hence, from the above two estimates we can conclude the desired statement.",
"In the same treatment we can prove the second estimate.",
"This completes our proof.",
"Lemma 4.2 ([13], [17]) Let $n\\ge 1$ and $f\\in L^{1,\\gamma }(\\mathbb {R}^n)$ with $\\gamma \\ge 0$ .",
"It holds that $\\biggr |\\hat{f}(\\xi )-\\sum _{|\\alpha |\\le [\\gamma ]}M_\\alpha (f)(i\\xi )^\\alpha \\biggr |\\lesssim |\\xi |^\\gamma \\Vert f\\Vert _{L^{1,\\gamma }}$ for $\\xi \\in \\mathbb {R}^n$ .",
"It is true that $\\lim _{t\\rightarrow \\infty }t^{\\frac{n}{4\\sigma }+\\frac{\\gamma }{2\\sigma }}\\biggr \\Vert e^{-t|\\xi |^{2\\sigma }}\\hat{f}(\\xi )-\\sum _{|\\alpha |\\le [\\gamma ]}M_\\alpha (f)(i\\xi )^\\alpha e^{-t|\\xi |^{2\\sigma }}\\biggr \\Vert _{L^2}=0.$ The proof of inequality (REF ) was given in [13] and [17].",
"With a slight modification, we can also obtain (REF ) and thus the proof is omitted."
]
] | 1906.04471 |
[
[
"The Non-m-Positive Dimension of a Positive Linear Map"
],
[
"Abstract We introduce a property of a matrix-valued linear map $\\Phi$ that we call its \"non-m-positive dimension\" (or \"non-mP dimension\" for short), which measures how large a subspace can be if every quantum state supported on the subspace is non-positive under the action of $I_m \\otimes \\Phi$.",
"Equivalently, the non-mP dimension of $\\Phi$ tells us the maximal number of negative eigenvalues that the adjoint map $I_m \\otimes \\Phi^*$ can produce from a positive semidefinite input.",
"We explore the basic properties of this quantity and show that it can be thought of as a measure of how good $\\Phi$ is at detecting entanglement in quantum states.",
"We derive non-trivial bounds for this quantity for some well-known positive maps of interest, including the transpose map, reduction map, Choi map, and Breuer--Hall map.",
"We also extend some of our results to the case of higher Schmidt number as well as the multipartite case.",
"In particular, we construct the largest possible multipartite subspace with the property that every state supported on that subspace has non-positive partial transpose across at least one bipartite cut, and we use our results to construct multipartite decomposable entanglement witnesses with the maximum number of negative eigenvalues."
],
[
"=1 in:"
]
] | 1906.04517 |
[
[
"Maximal and Typical Topology of Real Polynomial Singularities"
],
[
"Abstract Given a polynomial map $\\psi:S^m\\to \\mathbb{R}^k$ with components of degree $d$, we investigate the structure of the semialgebraic set $Z\\subseteq S^m$ consisting of those points where $\\psi$ and its derivatives satisfy a given list of polynomial equalities and inequalities (we call such a set a \"singularity\").",
"Concerning the upper estimate on the topological complexity of a polynomial singularity, we sharpen the classical bound $b(Z)\\leq O(d^{m+1})$, proved by Milnor, with \\begin{equation}\\label{eq:abstract} b(Z)\\leq O(d^{m}),\\end{equation} which holds for the generic polynomial map.",
"For what concerns the \"lower bound\" on the topology of $Z$, we prove a general semicontinuity result for the Betti numbers of the zero set of $\\mathcal{C}^0$ perturbations of smooth maps -- the case of $\\mathcal{C}^1$ perturbations is the content of Thom's Isotopy Lemma (essentially the Implicit Function Theorem).",
"This result is of independent interest and it is stated for general maps (not just polynomial); this result implies that small continuous perturbations of $\\mathcal{C}^1$ manifolds have a richer topology than the one of the original manifold.",
"We then compare the extremal case with a random one and prove that on average the topology of $Z$ behaves as the \"square root\" of its upper bound: for a random Kostlan map $\\psi:S^m\\to \\mathbb{R}^k$ with components of degree $d$, we have: \\begin{equation} \\mathbb{E}b(Z)=\\Theta(d^{\\frac{m}{2}}).\\end{equation} This generalizes classical results of Edelman-Kostlan-Shub-Smale from the zero set of a random map, to the structure of its singularities."
],
[
"Introduction",
"In this paper we deal with the problem of understanding the structure of the singularities of polynomial maps $\\psi :S^m\\rightarrow \\mathbb {R}^k,$ where each component of $\\psi =(\\psi _1, \\ldots , \\psi _k)$ is the restriction to the sphere of a homogeneous polynomial of degree $d$ .",
"For us “singularity” means the set of points in the sphere where the $r$ -jet extension $j^r\\psi :S^n\\rightarrow J^{r}(S^n, \\mathbb {R}^k)$ meets a given semialgebraic set $W\\subseteq J^{r}(S^n, \\mathbb {R}^k).$ Example of these type of singularities are: zero sets of polynomial functions, critical points of a given Morse index of a real valued function or the set of Whitney cusps of a planar map.",
"Because we are looking at polynomial maps, this problem has two different quantitative faces, which we both investigate in this paper.",
"(1) From one hand we are interested in understanding the extremal cases, meaning that, for fixed $m, d$ and $k$ we would like to know how complicated can the singularity be, at least in the generic case.",
"(2) On the other hand, we can ask what is the typical complexity of such a singularity.",
"Here we adopt a measure-theoretic point of view and endow the space of polynomial maps with a natural Gaussian probability measure, for which it makes sense to ask about expected properties of these singularities, such as their Betti numbers."
],
[
"Quantitative bounds, the h-principle and the topology semicontinuity",
"Measuring the complexity of $Z=j^r\\psi ^{-1}(W)$ with the sum $b(Z)$ of its Betti numbers, problem (1) above means producing a-priori upper bounds for $b(Z)$ (as a function of $m, d, k$ ) as well as trying to realize given subsets of the sphere as $j^r\\psi ^{-1}(W)$ for some $W$ and some map $\\psi $ .",
"For the case of the zero set $Z=\\psi ^{-1}(0)$ of a polynomial function $\\psi :S^m\\rightarrow \\mathbb {R}$ of degree $d$ , the first problem is answered by a Milnor's type boundMilnor's bound [21] would give $b(Z)\\le O(d^{m+1})$ , whereas [19] gives the improvement $b(Z)\\le O(d^m)$ .",
"In the context of this paper the difference between these two bounds is relevant, especially because when switching to the probabilistic setting it will give the so called “generalized square root law”.",
"$b(Z)\\le O(d^m)$ and the second problem by Seifert's theorem: every smooth hypersurface in the sphere can be realized (up to ambient diffeomorphisms) as the zero set of a polynomial function.",
"In the case of more general singularities, both problems are more subtle.",
"The problem of giving a good upper bound on the complexity of $Z=j^r\\psi ^{-1}(W)$ will require us to develop a quantitative version of stratified Morse Theory for semialgebraic maps (Theorem ).",
"We use the word “good” because there is a vast literature on the subject of quantitative semialgebraic geometry, and it is not difficult to produce a bound of the form $b(Z)\\le O(d^{m+1})$ ; instead here (Theorem REF and Theorem REF ) we prove the following result.",
"Theorem 1 For the generic polynomial map $\\psi :S^m\\rightarrow \\mathbb {R}^k$ with components of degree $d$ , and for $W\\subseteq J^{r}(S^m, \\mathbb {R}^k)$ semialgebraic, we have: $b(j^{r}\\psi ^{-1}(W))\\le O(d^m).$ (The implied constant depends on $W$ .)",
"In the case $W$ is algebraic we do not need the genericity assumption on $\\psi $ for proving (REF ), but in the general semialgebraic case some additional complications arise and this assumption allows to avoid them through the use of Theorem .",
"We believe, however, that (REF ) is still true even in the general case.",
"Moreover, for our scopes the genericity assumption is not restrictive, as it fits in the probabilistic point of view of the second part of the paper, where a generic property is a property holding with probability one.",
"For what concerns the realizability problem, as simple as it might seem at first glance, given $W\\subseteq J^{r}(S^m, \\mathbb {R}^k)$ it is not even trivial to find a map $f:S^m\\rightarrow \\mathbb {R}^k$ whose jet is transversal to $W$ and such that $b(j^rf^{-1}(W))>0$ (we prove this in Corollary REF ).",
"Let us try to explain carefully what is the subtlety here.",
"In order to produce such a map, one can certainly produce a section of the jet bundle $\\sigma :S^m\\rightarrow J^{r}(S^m, \\mathbb {R}^k)$ which is transversal to $W$ and such that $b(\\sigma ^{-1}(W))>0$ (this is easy).",
"However, unless $r=0$ , this section needs not to be holonomic, i.e.",
"there might not exist a function $f:S^m\\rightarrow \\mathbb {R}^k$ such that $\\sigma =j^{r}f$ .",
"We fix this first issue using an h-principle argument: the Holonomic Approximation Theorem [8] guarantees that, after a small $\\mathcal {C}^0$ perturbation of the whole picture, we can assume that there is a map $f:S^m\\rightarrow \\mathbb {R}^k$ whose jet $j^{r}f$ is $\\mathcal {C}^0$ close to $\\sigma $ .",
"There is however a second issue that one needs to address.",
"In fact, if the jet perturbation was $\\mathcal {C}^1$ small (i.e.",
"if $\\sigma $ and $j^rf$ were $\\mathcal {C}^1$ close), Thom's Isotopy Lemma would guarantee that $\\sigma ^{-1}(W)\\sim j^rf^{-1}(W)$ (i.e.",
"the two sets are ambient diffeomorphic), but the perturbation that we get from the Holonomic Approximation Theorem is guaranteed to be only $\\mathcal {C}^0$ small!",
"To avoid this problem we prove the following general result on the semicontinuity of the topology of small $\\mathcal {C}^0$ perturbations (see Theorem REF below for a more precise statement).",
"Theorem 2 Let $S, J$ be smooth manifolds, $W\\subseteq J$ be a submanifoldIn fact here we can take $W$ to be stratified in the sense of [15].",
"and $\\sigma \\in \\mathcal {C}^{1}(S, J)$ such that $\\sigma \\pitchfork W$ .",
"Then for every $\\gamma \\in \\mathcal {C}^{1}(S, J) $ which is sufficiently close to $\\sigma $ in the $\\mathcal {C}^0$ -topology and such that $\\gamma \\pitchfork W$ , we have: $b(\\gamma ^{-1}(W))\\ge b(\\sigma ^{-1}(W)).$ In particular we see that if small $\\mathcal {C}^1$ perturbations of a regular equation preserve the topology of the zero set, still if we take just small $\\mathcal {C}^0$ perturbations the topology of such zero set can only increase.",
"To apply Theorem REF to our original question we consider $S=S^m$ and $J=J^{r}(S^m, \\mathbb {R}^k)$ , $W\\subseteq J^{r}(S^m, \\mathbb {R}^m)$ is the semialgebraic set defining the singularity and $\\sigma :S^m\\rightarrow J^r(S^m, \\mathbb {R}^k)$ is the (possibly non-holonomic) section such that $\\sigma \\pitchfork W$ and $b(\\sigma ^{-1}(W))>0$ .",
"Then for every $f\\in \\mathcal {C}^{r+1}(S^m, \\mathbb {R}^k) $ with $\\tau =j^rf$ sufficiently close to $\\sigma $ and such that $j^rf\\pitchfork W$ , we have: $b(j^rf^{-1}(W))\\ge b(\\sigma ^{-1}(W))>0.$ (We will use the content of Corollary REF and the existence of a function $f$ such that (REF ) holds in the second part of the paper for proving the convergence of the expected Betti numbers of a random singularity.)"
],
[
"The random point of view and the generalized square-root law",
"Switching to the random point of view offers a new perspective on these problems: from Theorem REF we have an extremal bound (REF ) for the complexity of polynomial singularities, but it is natural to ask how far is this bound from the typical situation.",
"Of course, in order to start talking about randomness, we need to choose a probability distribution on the space of (homogeneous) polynomials.",
"It is natural to require that this distribution is gaussian, centered, and that it is invariant under orthogonal changes of variables (in this way there are no preferred points or directions in the sphere).",
"If we further assume that the monomials are independent, this distribution is unique (up to multiples), and called the Kostlan distribution.",
"To be more precise, this probability distribution is the measure on $\\mathbb {R}[x_0,\\dots ,x_{m}]_{(d)}$ (the space of homogeneous polynomials of degree $d$ ) induced by the gaussian random polynomial: $P(x)=\\sum _{ |\\alpha |=d}\\xi _\\alpha \\cdot \\left(\\frac{d!",
"}{\\alpha _0!\\cdots \\alpha _m!",
"}\\right)^{1/2} x_0^{\\alpha _0}\\cdots x_m^{\\alpha _m},$ where $\\lbrace \\xi _\\alpha \\rbrace $ is a family of standard independent gaussian variables.",
"A list $P=(P_1, \\ldots , P_k)$ of $k$ independent Kostlan polynomials defines a random polynomial map: $\\psi =P|_{S^m}\\rightarrow \\mathbb {R}^k.$ In particular, it is now natural to view such a $\\psi $ as a random variable in the space $\\mathcal {C}^{\\infty }(S^m, \\mathbb {R}^k)$ and to study the differential topology of this map, such as the behavior of its singularities, described a preimages of jet submanifolds $W\\subseteq J^{r}(S^m, \\mathbb {R}^k)$ in the previous section.",
"In this direction, it has already been observed by several authors, in different contexts, that random real algebraic geometry seems to behave as the “square root” of generic complex geometry.",
"Edelman and Kostlan [17], [7] were the first to observe this phenomenon: a random Kostlan polynomial of degree $d$ in one variable has $\\sqrt{d}$ many real zeroes, on averageIn the notation of the current paper this correspond to the case of $\\psi :S^1\\rightarrow \\mathbb {R}$ of degree $d$ , whose expected number of zeroes is $2\\sqrt{d}$ .",
"The multiplicative constant “2” appears when passing from the projective to the spherical picture.",
"Shub and Smale [24] generalized this result and proved that the expected number of zeroes of a system of $m$ Kostlan equations of degrees $(d_1, \\ldots , d_m)$ in $m$ variables is $\\sqrt{d_1\\cdots d_m}$ (the bound coming from complex algebraic geometry would be $d_1\\cdots d_m$ ).",
"Moving a bit closer to topology, Bürgisser [3] and Podkorytov [22] proved that the expectation of the Euler characteristic of a random Kostlan algebraic set has the same order of the square-root of the Euler characteristic of its complex part (when the dimension is even, otherwise it is zero).",
"A similar result for the Betti numbers has also been proved by Gayet and Welschinger [11], [13], [12], and by Fyodorov, Lerario and Lundberg [10] for invariant distributions.",
"Using the language of the current paper, these results correspond to the case of a polynomial map $\\psi :S^m\\rightarrow \\mathbb {R}^k$ and to the “singularity” $Z=j^0\\psi ^{-1}(W)$ , where $W=S^m\\times \\lbrace 0\\rbrace \\subset J^{0}(S^m, \\mathbb {R}^k)=S^m\\times \\mathbb {R}^k$ and $j^0\\psi (x)=(x, \\psi (x))$ is the section given by the map $\\psi $ itself.",
"Here we generalize these results and prove that a similar phenomenon is a very general fact of Kostlan polynomial maps.",
"Theorem 3 Let $W\\subset J^r(S^m,\\mathbb {R}^k)$ be a closed semialgebraic setHere we actually need an additional technical assumption on $W$ , i.e.",
"we want it to be “intrinsic”.",
"Roughly speaking, this means being invariant under diffeomorphisms of $S^m$ .",
"This property it is satisfied in all natural examples, see Definition REF .",
"Moreover, for the statement of this Theorem recall that we write $f(d)=\\Theta (g(d))$ if there exist constants $a_1, a_2>0$ such that $a_1 g(d)\\le f(d)\\le a_2f(d)$ for all $d\\ge d_0$ sufficiently large.",
"of positive codimension.",
"If $\\psi :S^m\\rightarrow \\mathbb {R}^k$ is a random Kostlan polynomial map, then $\\mathbb {E}b(j^r\\psi ^{-1}(W))=\\Theta (d^{\\frac{m}{2}}).$ (The implied constants depend on $W$ .)",
"We call the previous Theorem REF the “generalized square root law” after comparing it with the extremal inequality $b(j^{r}\\psi ^{-1}(W))\\le O(d^m)$ from Theorem REF , whose proof is ultimately based on bounds coming from complex algebraic geometryThe reader can now appreciate the estimate $O(d^m)$ instead of $O(d^{m+1})$ from Theorem REF ..",
"In the case $W$ has codimension $m$ (i.e.",
"when we expect $j^r\\psi ^{-1}(W)$ to consist of points), we actually sharpen (REF ) and get the explicit asymptotic to the leading order, see Theorem REF below.",
"Moreover, a similar result holds for every fixed Betti number $b_i(j^r\\psi ^{-1}(W))$ when $i$ is in the range $0\\le i\\le m-\\mathrm {codim}(W),$ see Theorem REF for a detailed statement.",
"Remark 4 The ingredients for the proof of Theorem REF are: Theorem for the upper bound and Corollary REF for the lower bound.",
"The main property that we use in this context is the fact that a Kostlan map $\\psi :S^m\\rightarrow \\mathbb {R}^k$ has a rescaling limit when restricted to a small disk $D_d=D(x, d^{-1/2})$ around any point $x\\in S^m$ .",
"In other words, one can fix a diffeomorphism $a_d:\\mathbb {D}^m\\rightarrow D_d$ of the standard disk $\\mathbb {D}^m$ with the small spherical disk $D(x, d^{-1/2})\\subset S^m$ and see that the sequence of random functions: $X_d=\\psi \\circ a_d:\\mathbb {D}^m\\rightarrow \\mathbb {R}^k$ converges to the Bargmann-Fock field, see Theorem REF .",
"In a recent paper [20] we introduced a general framework for dealing with random variables in the space of smooth functions and their differential topology – again we can think of $X_d\\in C^{\\infty }(\\mathbb {D}^m, \\mathbb {R}^k)$ as a sequence of random variables of this type.",
"The results from [20], applied to the setting of random Kostlan polynomial maps are collected in Theorem REF below, which lists the main properties of the rescaled Kostlan polynomial $X_d$ .",
"Some of these properties are well-known to experts working on random fields, but some of them seem to have been missed.",
"Moreover, we believe that our language is more flexible and well-suited to the setting of differential topology, whereas classical references look at these random variables from the point of view of functional analysis and stochastic calculus.",
"Of special interest from Theorem REF are properties (2), (5) and (7), which are closely related.",
"In fact (2) and (5) combined together tells that open sets $U\\subset \\mathcal {C}^{\\infty }(\\mathbb {D}^m, \\mathbb {R}^k)$ which are defined by open conditions on the $r$ -jet of $X_d$ , have a positive limit probability when $d\\rightarrow \\infty $ .",
"Property (7), tells that the law for Betti numbers of a random singularity $Z_d=j^rX_d^{-1}(W)$ has a limit.",
"(Even in the case of zero sets this property was not noticed before, see Example REF .)",
"We consider Theorem REF as a practical tool that people interested in random algebraic geometry can directly use, and we will show how to concretely use this tool in a list of examples that we give in Appendix REF .",
"Remark 5 The current paper, and in particular the generalized square-root law Theorem REF , complement recent work of Diatta and Lerario [6] and Breiding, Keneshlou and Lerario [2], where tail estimates on the probabilities of the maximal configurations are proved."
],
[
"Structure of the paper",
"In Section REF we prove a quantitative semialgebraic version of stratified Morse Theory, which is a technical tool needed in the sequel, and in Section REF we prove Theorem REF and Theorem REF (whose combination give Theorem REF ).",
"In section REF we discuss the semicontinuity of topology under holonomic approximation and prove Theorem REF (which is Theorem REF from the Introduction).",
"In Section we introduce the random point of view and prove the generalized square-root law.",
"Appendix 1 contains three short examples of use the random techniques."
],
[
"Stratified Morse Theory",
"Let us fix a Whitney stratification $W=\\sqcup _{S\\in {S}}S$ (see [14] for the definition) of the semialgebraic subset $W\\subset J^r(S^m,\\mathbb {R}^k)=:J$ , with each stratum $S\\in {S}$ being semialgebraic and smooth (such decomposition exists [14]), so that, by definition a smooth map $f\\colon M\\rightarrow J$ , is transverse to $W$ if $f\\mathrel {\\text{\\m@th {#\\crcr \\smash{-}\\crcr \\pitchfork \\crcr }}}$ S$ for all strata $ SS$.When this is the case, we write $ $-$$\\pitchfork $ W$ and implicitly consider the subset $ -1(W)M$ to be equipped with the Whitney stratification given by $ -1S={-1(S)}SS$.\\begin{defi}Given a Whitney stratified subset Z=\\cup _{i\\in I}S_i of a smoooth manifold M (without boundary), we say that a function g\\colon Z\\rightarrow \\mathbb {R} is a Morse function if g is the restriction of a smooth function \\tilde{g}\\colon M\\rightarrow \\mathbb {R} such that\\begin{enumerate}[(a)]\\item g|_{S_i} is a Morse function on S_i.\\item For every critical point p\\in S_i and every generalized tangent space Q\\subset T_pM (defined as in \\cite [p. 44]{GoreskyMacPherson})we have d_p\\tilde{g}(Q)\\ne 0, except for the case Q=T_pS_i.\\end{enumerate}\\end{defi}Note that the condition of being a Morse function on a stratified space $ ZM$ depends on the given stratification of $ Z$.\\begin{remark}The definition above is slightly different than the one given in the book \\cite [p. 52]{GoreskyMacPherson} by Goresky and MacPherson, where a Morse function, in addition, must be proper and have distinct critical values.\\end{remark}\\begin{thm}Let W\\subset J be a semialgebraic subset of a real algebraic smooth manifold J, with a given semialgebraic Whitney stratification W=\\sqcup _{S\\in {S}}S. Let M be a real algebraic smooth manifold and let \\psi \\colon M\\rightarrow J, g\\colon M\\rightarrow \\mathbb {R} be smooth maps.\\begin{enumerate}\\item There is a semialgebraic subset \\hat{W}\\subset J^{1}(M,J\\times \\mathbb {R}) with codimension larger or equal than \\dim M, equipped with a semialgebraic Whitney stratification such that if j^1(\\psi ,g)\\mathrel {\\text{\\m@th {#\\crcr \\smash{-}\\crcr \\pitchfork \\crcr }}}\\end{enumerate}\\hat{W} then \\psi \\mathrel {\\text{\\m@th {#\\crcr \\smash{-}\\crcr \\pitchfork \\crcr }}}\\end{thm}W$ and $g|_{\\psi ^{-1}(W)}$ is a Morse function with respect to the stratification $\\psi ^{-1}{S}$ .",
"In this case $\\text{Crit}(g|_{\\psi ^{-1}(W)})=\\left(j^{1}(\\psi ,g)\\right)^{-1}(\\hat{W}).$ There is a constant $N_W>0$ depending only on $W$ and ${S}$ , such that if $\\psi ^{-1}(W)$ is compact and $j^1(\\psi ,g)\\mathrel {\\text{\\m@th {#\\crcr \\smash{-}\\crcr \\pitchfork \\crcr }}}$ W$, then\\begin{equation}b_i(\\psi ^{-1}(W))\\le N_W\\#\\text{Crit}(g|_{\\psi ^{-1}(W)}),\\end{equation}for all $ i=0,1,2...$$ Let $S\\in {S}$ .",
"Let us consider the set $D_pS$ of degenerate covectors at a point $p\\in S$ that are conormal to $S$ (defined as in [14]), in other words: $D_pS=\\lbrace \\xi \\in T^*_p{J}\\colon \\xi \\in T_pS^\\perp ,\\ \\xi \\in Q^\\perp \\text{ for some $Q$ generalized tangent space at $p$}\\rbrace .$ It is proved in [14] that $DS=\\cup _{p\\in S}D_pS$ is a semialgebraic subset of codimension greater than 1 of the conormal bundle $TS^\\perp $$TS^\\perp =T_S^*J$ , in the notation of [14] to the stratum S. Define the $\\hat{S}=\\lbrace j^1_p(\\psi ,g)\\in J^1(M,J\\times \\mathbb {R})\\colon \\psi (p)\\in S \\text{ and }d_pg\\in d_p\\psi ^*(TS^\\perp )\\rbrace .$ It is easy to see that $j^1(\\psi ,g)\\mathrel {\\text{\\m@th {#\\crcr \\smash{-}\\crcr \\pitchfork \\crcr }}}$ S$ if and only if $ $-$$\\pitchfork $ S$ and $ g|-1(S)$ is a Morse function, indeed $ dpgdp*(TpS)$ if and only if $ dpg|Tp(-1(S))=0$.",
"In particular the codimension of $ S$ is equal to $ m$, the dimension of $ M$.Now define\\begin{equation}D_p\\hat{S}=\\lbrace j^1_p(\\psi ,g)\\in J^1(M,J\\times \\mathbb {R})\\colon \\psi (p)\\in S \\text{ and }d_pg\\in d_p\\psi ^*(DS)\\rbrace .\\end{equation}By Definition \\ref {def:mors}, we have that $ j1(,g) $-$$\\pitchfork $ S$ and $ j1(,g)DS$ if and only if $ $-$$\\pitchfork $ S$ and $ g|-1(W)$ is a Morse function along $ -1(S)$.Note that $ DS$ is a subset of $ S$ of codimension $ 1$, therefore the codimension of $ DS$ in $ J1(M,JR)$ is $ m+1$.",
"It follows that $ j1(,g)DS$ if and only if $ j1(,g) $-$$\\pitchfork $ DS$.$ Define $\\hat{W}=\\cup _{S\\in {S}}\\hat{S}\\backslash D\\hat{S}$ .",
"Since $\\hat{S}$ and $D\\hat{S}$ are clearly semialgebraic, $\\hat{W}$ is semialgebraic and admits a semialgebraic Whitney stratification $\\hat{{S}}$ such that all $\\hat{S}$ and $D\\hat{S}$ are union of strata.",
"With this stratification, $\\hat{W}$ satisfies condition $(1)$ of the Theorem.",
"Let us prove condition (2).",
"Let $Z=\\psi ^{-1}(W)\\subset M$ be compact.",
"Without loss of generality we can assume that each critical values $c_1,\\dots , c_n$ of $g|_{Z}$ corresponds to only one critical point.",
"Consider a sequence of real numbers $a_1,\\dots a_{n+1}$ such that $a_1<c_1 <a_2<c_2<\\dots <a_{n}<c_n<a_{n+1}.$ by the main Theorem of stratified Morse theory [14], there is an homeomorphism $Z\\cap \\lbrace g\\le a_{l+1}\\rbrace \\cong (Z\\cap \\lbrace g\\le a_l\\rbrace )\\sqcup _B A,$ with $(A,B)= TMD_p(g)\\times NMD_p(g),$ where $TMD_p(g)$ is the tangential Morse data and $NMD_p(g)$ is the normal Morse data.",
"A fundamental result of classical Morse theory is that the tangential Morse data is homeomorphic to the pair $TMD_p(g)\\cong (\\mathbb {D}^\\lambda \\times \\mathbb {D}^{m-\\lambda },(\\partial \\mathbb {D}^\\lambda )\\times \\mathbb {D}^{m-\\lambda }),$ while the normal Morse data is defined as the local Morse data of $g|_{N_p}$ for a normal slice (see [14]) at $p$ .",
"A consequence of the transversality hypothesis $\\psi \\mathrel {\\text{\\m@th {#\\crcr \\smash{-}\\crcr \\pitchfork \\crcr }}}$ W$ is that there is a small enough normal slice $ Np$ such that $ | NpNpJ$ is the embedding of a normal slice at $ (p)$ for $ W$.",
"Therefore the normal data $ NMDp(g)$ belongs (up to homeomeorphisms) to the set $ (W)$ of all possible normal Morse data that can be realized by a critical point of a Morse function on $ W$.",
"By Corollary $ 7.5.3$ of \\cite [p. 95]{GoreskyMacPherson} it follows that the cardinality of the set $ (W)$ is smaller or equal to the number of connected components of the semialgebraic set $ SS(TSDS)$, hence finite\\footnote {In the book this is proved only for any fixed point p, as a corollary of Theorem 7.5.1 \\cite [p.93]{GoreskyMacPherson}.",
"However it is easy to understand that the latter theorem is still true under the additional assumption that the point is moving.",
"}.Let\\begin{equation}N_W:=\\max _{\\nu \\in \\nu (W),\\ \\lambda \\in \\lbrace 0,\\dots , m\\rbrace } b_i\\left(\\left(\\mathbb {D}^\\lambda \\times \\mathbb {D}^{m-\\lambda },(\\partial \\mathbb {D}^\\lambda )\\times \\mathbb {D}^{m-\\lambda }\\right)\\times \\nu \\right)\\in \\mathbb {N}.\\end{equation}From the long exact sequence of the pair $ (Z{gal+1},(Z{gal})$ we deduce that\\begin{equation}\\begin{aligned}b_i(Z\\cap \\lbrace g\\le a_{l+1}\\rbrace )-b_i(Z\\cap \\lbrace g\\le a_{l}\\rbrace )&\\le b_i\\left(Z\\cap \\lbrace g\\le a_{l+1}\\rbrace ,Z\\cap \\lbrace g\\le a_{l}\\rbrace \\right)\\\\&= b_i\\left(A,B\\right) \\\\&= b_i\\left(TMD_p(g)\\times NMD_p(g)\\right) \\\\&\\le N_W.\\end{aligned}\\end{equation}Since $ Z$ is compact, the set $ Z{ga1}$ is empty, hence by repeating the inequality (\\ref {eq:lesbetti}) for each critical value, we finally get\\begin{equation}b_i(Z)=b_i(Z\\cap g\\le a_{n+1})\\le N_Wn=N_W\\#\\text{Crit}\\left(g|_{\\psi ^{-1}(W)}\\right).\\end{equation}$ Below we will restrict to those semialgebraic sets $W\\subset J^{r}(S^m, \\mathbb {R}^k)$ that have a differential geometric meaning, as specified in the next definition.",
"Definition 6 A submanifold $W\\subset J^r(M,\\mathbb {R}^k)$ is said to be intrinsic if there is a submaniolfd $W_0\\subset J^r(\\mathbb {D}^m,\\mathbb {R}^k)$ , called the model, such that for any embedding $\\mathbb {D}^m\\hookrightarrow M$ , one has that $j^r*(W)=W_0$ , where $j^r*\\colon J^r\\left(\\mathbb {D}^m),\\mathbb {R}^k\\right)\\xrightarrow{}J^r\\left(\\mathbb {D}^m,\\mathbb {R}^k\\right), \\qquad j^r_{p)}f\\mapsto j^r_p(f\\circ .$ Intrinsic submanifolds are, in other words, those that have the same shape in every coordinate charts, as in the following examples.",
"$W=\\lbrace j^r_pf\\colon f(p)=0\\rbrace $ ; $W=\\lbrace j^r_pf\\colon j^sf(p)=0\\rbrace $ for some $s\\le r$ ; $W=\\lbrace j^r_pf\\colon \\text{rank}(df(p))=s\\rbrace $ for some $s\\in \\mathbb {N}$ .",
"Remark 7 In the case when $J=J^{r}(M,\\mathbb {R}^k)$ we can consider $\\hat{W}$ to be a subset of $ J^{r+1}(M,\\mathbb {R}^{k+1})$ taking the preimage via the natural submersion $J^{r+1}(M,\\mathbb {R}^{k+1})\\rightarrow J^1\\left(M,J^{r}(M,\\mathbb {R}^k)\\times \\mathbb {R}\\right), \\qquad j^{r+1}(f,g)\\mapsto j^1(j^rf,g),$ then Theorem holds for any $\\psi $ of the form $\\psi =j^rf$ .",
"Moreover, in this case, observe that if $W$ is intrinsic (in the sense of Definition REF below), then $\\hat{W}$ is intrinsic as well.",
"Quantitative bounds In this section we prove Theorem REF , which actually immediately follows by combining Theorem REF and Theorem REF .",
"Next theorem gives a deterministic bound for on the complexity of $Z=j^r\\psi ^{-1}(W)$ when the codimension of $W$ is $m$ .",
"Theorem 8 Let $P\\in \\mathbb {R}[x_0, \\ldots , x_m]_{(d)}^k$ be a polynomial map and consider its restriction $\\psi =P|_{S^m}$ to the unit sphere: $\\psi :S^m\\rightarrow \\mathbb {R}^k.$ Let also $j^r\\psi :S^m\\rightarrow J^r(S^m, \\mathbb {R}^k)$ be the associated jet map and $W\\subset J^{r}(S^m, R^k) $ be a semialgebraic set of codimension $m$ .",
"There exists a constant $c>0$ (which only depends on $W$ , $m$ and $k$ ) such that, if $j^r\\psi \\mathrel {\\text{\\m@th {#\\crcr \\smash{-}\\crcr \\pitchfork \\crcr }}}$ W$, then:\\begin{equation} \\#j^r\\psi ^{-1}(W)\\le c \\cdot d^m.\\end{equation}$ Let us make the identification $J^{r}(\\mathbb {R}^{m+1}, \\mathbb {R}^k)\\simeq \\mathbb {R}^{m+1}\\times \\mathbb {R}^N$ , so that the restricted jet bundle $J^{r}(\\mathbb {R}^{m+1}, \\mathbb {R}^k)|_{S^m}$ corresponds to the semialgebraic subset $S^m\\times \\mathbb {R}^N$ .",
"Observe that the inclusion $S^m\\hookrightarrow \\mathbb {R}^{m+1}$ induces a semialgebraic map: $J^{r}(\\mathbb {R}^{m+1}, \\mathbb {R}^k)|_{S^m}\\stackrel{i^*}{\\longrightarrow } J^r(S^m, \\mathbb {R}^k),$ that, roughly speaking, forgets the normal derivatives.",
"Notice that while the map $j^{r}\\psi =j^r(P|_{S^m})$ is a section of $J^r(S^m, \\mathbb {R}^k)$ , $(j^rP)|_{S^m}$ is a section of $J^{r}(\\mathbb {R}^{m+1}, \\mathbb {R}^k)|_{S^m}$ .",
"These sections are related by the identity $i^*\\circ (j^rP)|_{S^m}=j^{r}\\psi .$ Thus, defining $ \\overline{W}=(i^*)^{-1}(W)$ , we have $j^r\\psi ^{-1}(W)=\\left((j_rP)|_{S^m}\\right)^{-1}(\\overline{W}).$ Since $\\overline{W}$ is a semialgebraic subset of $\\mathbb {R}^{m+1}\\times \\mathbb {R}^N$ , it can be written as: $\\overline{W}=\\bigcup _{j=1}^\\ell \\left\\lbrace f_{j, 1}=0, \\ldots , f_{j, \\alpha _j}=0, g_{j, 1}>0,\\ldots , g_{j, \\beta _j}>0\\right\\rbrace ,$ where the $f_{j,i}$ s and the $g_{j, i}$ s are polynomials of degree bounded by a constant $b>0.$ For every $j=1, \\ldots , \\ell $ we can write: $\\left\\lbrace f_{j, 1}=0, \\ldots , f_{j, \\alpha _j}=0, g_{j, 1}>0,\\ldots , g_{j, \\beta _j}>0\\right\\rbrace =Z_j\\cap A_j,$ where $Z_j$ is algebraic (given by the equations) and $A_j$ is open (given by the inequalities).",
"Observe also that the map $(j^rP)|_{S^m}$ is the restriction to the sphere $S^m$ of a polynomial map $Q:\\mathbb {R}^{m+1}\\rightarrow \\mathbb {R}^{m+1}\\times \\mathbb {R}^N$ whose components have degree smaller than $d$ .",
"Therefore for every $j=1\\ldots , \\ell $ the set $((j^rP)|_{S^m})^{-1}(Z_j)=(Q|_{S^m})^{-1}(Z_j)$ is an algebraic set on the sphere defined by equations of degree less than $b \\cdot d$ and, by [19] we have that: $ b_0(Q|_{S^m})^{-1}(Z_j))\\le B d^m$ for some constant $B>0$ depending on $b$ and $m$ .",
"The set $(Q|_{S^m})^{-1}(Z_j)$ consists of several components, some of which are zero dimensional (points): $(Q|_{S^m})^{-1}(Z_j)=\\underbrace{\\lbrace p_{j, 1}, \\ldots , p_{j, \\nu _j}\\rbrace }_{P_j}\\cup \\underbrace{X_{j,1}\\cup \\cdots \\cup X_{j, \\mu _j}}_{Y_j}.$ The inequality (REF ) says in particular that: $\\#P_j\\le Bd^n.$ Observe now that if $j^r\\psi \\mathrel {\\text{\\m@th {#\\crcr \\smash{-}\\crcr \\pitchfork \\crcr }}}$ W$ then, because the codimension of $ W$ is $ m$, the set $ jr-1(W)=(Q|Sm)-1(W)$ consists of finitley many points and therefore, since $ (Q|Sm)-1(Aj)$ is open, we must have:\\begin{equation}j^{r}\\psi ^{-1}(W)\\subset \\bigcup _{j=1}^\\ell P_j.\\end{equation}(Otherwise $ jr-1(W)$ would contain an open, nonempty set of a component of codimension smaller than $ m$.",
")Inequality (\\ref {ineqbb}) implies now that:\\begin{equation}\\#j^{r}\\psi ^{-1}(W)\\le \\sum _{j=1}^\\ell \\#P_j\\le \\ell b d^m\\le cd^m.\\end{equation}$ Using Theorem it is now possible to improve Theorem REF to the case of any codimension, replacing the cardinality with any Betti number.",
"Theorem 9 Let $P\\in \\mathbb {R}[x_0, \\ldots , x_m]_{(d)}^k$ be a polynomial map and consider its restriction $\\psi =P|_{S^m}$ to the unit sphere: $\\psi :S^m\\rightarrow \\mathbb {R}^k.$ Let also $j^r\\psi :S^m\\rightarrow J^r(S^m, \\mathbb {R}^k)$ be the associated jet map and $W\\subset J^{r}(S^m, R^k) $ be a closed semialgebraic set (of arbitrary codimension).",
"There exists a constant $c>0$ (which only depends on $W$ , $m$ and $k$ ) such that, if $j^r\\psi \\mathrel {\\text{\\m@th {#\\crcr \\smash{-}\\crcr \\pitchfork \\crcr }}}$ W$, then:\\begin{equation}b_i\\left(j^r\\psi ^{-1}(W)\\right)\\le c \\cdot d^m.\\end{equation}$ Let $J=J^r(S^m,\\mathbb {R}^k)$ and let $\\hat{W}$ be the (stratified according to a chosen stratification of $W$ ) subset of $J^{r+1}(S^m,\\mathbb {R}^{k+1})$ coming from Theorem and Remark REF .",
"Let $g$ be a homogeneous polynomial of degree $d$ such that $\\Psi =(\\psi ,g)\\in \\mathbb {R}[x_0, \\ldots , x_m]_{(d)}^{k+1}$ satisfies the condition $j^{r+1}\\Psi \\mathrel {\\text{\\m@th {#\\crcr \\smash{-}\\crcr \\pitchfork \\crcr }}}$ W$ (almost every polynomial $ g$ has this property by standard arguments) and $ (jr)-1(W)$ is closed in $ Sm$, hence compact.",
"Then by Theorem \\ref {thm:strat}, there is a constant $ NW$, such that we have that\\begin{equation}b_i\\left(j^r\\psi ^{-1}(W)\\right)\\le N_W \\#\\lbrace (j^{r+1}\\Psi )^{-1}(\\hat{W})\\rbrace \\end{equation}and by Theorem \\ref {thm:bound}, the right hand side is bounded by $ cdm$.$ Given $P=(P_1, \\ldots , P_k)$ with each $P_i$ a homogeneous polynomial of degree $d$ in $m+1$ variables, we denote by $\\psi _d:S^m\\rightarrow \\mathbb {R}^k$ its restriction to the unit sphere (the subscript keeps track of the dependence on $d$ ).",
"Example 10 (Real algebraic sets) Let us take $W=S^m\\times \\lbrace 0\\rbrace \\subset J^{0}(S^m, \\mathbb {R}^k),$ then $j^0\\psi ^{-1}(W)$ is the zero set of $\\psi _d:S^m\\rightarrow \\mathbb {R}^k$ , i.e.",
"the set of solutions of a system of polynomial equations of degree $d$ .",
"In this case the inequality () follows from [19].",
"Example 11 (Critical points) If we pick $W=\\lbrace j^1f=0\\rbrace \\subset J^1(S^m, \\mathbb {R}),$ then $j^1\\psi _d^{-1}(W)$ is the set of critical points of $\\psi _d:S^m\\rightarrow \\mathbb {R}$ .",
"In 2013 Cartwright and Sturmfels [4] proved that $\\#Z_d\\le 2(d-1)^m+\\dots +(d-1)+1$ (this bounds follows from complex algebraic geometry), and this estimate was recently proved to be sharp by Kozhasov [18].",
"Of course one can also fix the index of a nondegenerate critical point (in the sense of Morse Theory); for example we can take $W=\\lbrace df=0, d^2f>0\\rbrace \\subset J^2(S^m, \\mathbb {R}),$ and $j^2\\psi _d^{-1}(W)$ is the set of nondegenerate minima of $\\psi _d:S^m\\rightarrow \\mathbb {R}$ (similar estimates of the order $d^{m}$ holds for the fixed Morse index, but the problem of of finding a sharp bound is very much open).",
"Example 12 (Whitney cusps) When $W=\\lbrace \\textrm {Whitney cusps}\\rbrace \\subset J^3(S^2, \\mathbb {R}^2),$ then $\\psi _d^{3}f^{-1}(W)$ consists of the set of points where the polynomial map $\\psi _d:S^2\\rightarrow \\mathbb {R}^2$ has a critical point which is a Whitney cusp.",
"In this case () controls the number of possible Whitney cusps (the bound is of the order $O(d^2)$ ).",
"Semicontinuity of topology under holonomic approximation Consider the following setting: $M$ and $J$ are smooth manifolds, $M$ is compact, and $W\\subset J$ is a smooth cooriented submanifold.",
"Given a smooth map $F\\colon M\\rightarrow N$ which is transversal to $W$ , it follows from standard transversality arguments that there exists a small $\\mathcal {C}^1$ neighborhood $U_1$ of $F$ such that for every map $\\tilde{F}\\in U_1$ the pairs $(M, F^{-1}(W))$ and $(M, \\tilde{F}^{-1}(W))$ are isotopic (in particular $F^{-1}(W)$ and $\\tilde{F}^{-1}(W)$ have the same Betti numbers, this is the so-called “Thom's isotopy Lemma”).",
"The question that we address is the behavior of the Betti numbers of $\\tilde{F}^{-1}(W)$ under small $\\mathcal {C}^0$ perturbations, i.e.",
"how the Betti number can change under modifications of the map $F$ without controlling its derivative.",
"Figure: A small 𝒞 0 \\mathcal {C}^0 perturbation of a regular equation can only increase the topology of its zero set.In this direction we prove the following result.",
"Theorem 13 Let $M,J$ be smooth manifolds and let $W\\subset J$ be a smooth cooriented submanifold.",
"Let $F\\colon M\\rightarrow N$ be a smooth map such that $F\\mathrel {\\text{\\m@th {#\\crcr \\smash{-}\\crcr \\pitchfork \\crcr }}}$ W$.",
"If a smooth map $ F$ is strongly\\footnote {Meaning: in Whitney strong topology.",
"In particular if C\\subset M is closed and U\\subset J is open, then the set \\lbrace f\\in \\mathcal {C}^0(M,J)\\colon f(C)\\subset U\\rbrace is open, see \\cite {Hirsch}.}",
"$ C0-$close to $ F$ such that $ F $-$$\\pitchfork $ W$, then there is an algebra isomorphism\\begin{equation}H^*\\left(\\tilde{F}^{-1}(W)\\right)\\cong H^*\\left(F^{-1}(W)\\right)\\oplus K\\end{equation}for some algebra $ K$.$ Call $A=F^{-1}(W)$ and $\\tilde{A}=\\tilde{F}^{-1}(W)$ .",
"$E\\subset M$ be a tubular neighbourhood of $A$ such that $\\tilde{A}\\subset B$ , where $B$ is a strictly smaller (in the sense that $\\overline{B}\\subset E$ ) tubular neighbourhood of $A$ .",
"Denote by $\\pi \\colon E\\rightarrow A$ the retraction map.",
"Since $\\tilde{F}$ is $\\mathcal {C}^0-$ close to $F$ we can assume that there is an homotopy $F_t$ connecting $F=F_0$ and $\\tilde{F}=F_1$ such that $F_t^{-1}(M\\backslash E)\\subset J\\backslash \\overline{W}$ .",
"Define analogously $\\tilde{\\pi }:\\tilde{E}\\rightarrow \\tilde{A}$ and $\\tilde{B}$ in such a way that $\\tilde{E}\\subset B$ .",
"It follows that there is an inclusion of pairs $u:(E,E\\backslash B)\\rightarrow (E,E\\backslash \\tilde{B})$ .",
"The fact that $W$ is cooriented guarantees the existence of a Thom class $\\phi \\in H^r(J,J\\backslash W)$ , where $r$ is the codimension of $W$ .",
"By transversality we have that also $A$ and $\\tilde{A}$ are cooriented with Thom classes $f_0^*\\phi =\\phi _B\\in H^r(E,E\\backslash B)\\cong H^r(E,E\\backslash A)$ and $f_1^*\\phi =\\phi _{\\tilde{B}}\\in H^r(\\tilde{E},\\tilde{E}\\backslash \\tilde{B})\\cong H^r(\\tilde{E},\\tilde{E}\\backslash \\tilde{A})$ .",
"From the commutative diagram it follows that there exists an algebra homomorphism $U\\colon H^*(\\tilde{A})\\rightarrow H^*(A)$ such that $U\\circ \\pi ^*=$ id.",
"$\\begin{tikzcd}& {H^{*+r}(J, J\\backslash W)}[ldd, \"f_1^*\"^{\\prime }] [rdd, \"f_1^*=f_0^*\"] & \\\\& & \\\\{H^{*+r}(\\tilde{E}, \\tilde{E}\\backslash \\tilde{B})} [r, \"\\eta ^{-1}\"] & {H^{*+r}(E, E\\backslash \\tilde{B})} [r, \"u^*\"] & {H^{*+r}(E, E\\backslash B)} \\\\H^{*}(\\tilde{A}) [u, \"\\tilde{\\pi }^*(\\cdot )\\cup \\phi _{\\tilde{B}}\"] & & H^*(A) [ll, \"\\pi ^*\"] [u, \"\\pi ^*(\\cdot )\\cup \\phi _B\"^{\\prime }]\\end{tikzcd}$ (where $\\eta $ is the excision isomorphism) it follows that there exists an algebra homomorphism $U\\colon H^*(\\tilde{A})\\rightarrow H^*(A)$ such that $U\\circ \\pi ^*=$ id.",
"Corollary 14 Let $M$ be a compact manifold of dimension $m$ .",
"Let $W\\subset J^r(M,\\mathbb {R}^k)$ be a smooth stratified submanifold of codimension $1\\le l\\le m$ being transverse to the fibers of the canonical projection $\\pi \\colon J^r(M,\\mathbb {R}^k)\\rightarrow M$ .",
"Then for any number $n \\in \\mathbb {N}$ there exists a smooth function $\\psi \\in \\mathcal {C}^{\\infty }(M,\\mathbb {R}^k)$ such that $j^r\\psi \\mathrel {\\text{\\m@th {#\\crcr \\smash{-}\\crcr \\pitchfork \\crcr }}}$ W$ and\\begin{equation}b_i\\left((j^r\\psi )^{-1}(W)\\right)\\ge n, \\quad \\forall i=0,\\dots , m-l.\\end{equation}$ Let $B\\subset J^r(M,\\mathbb {R}^k)$ be a small neighbourhood of a regular point $j^r_pf$ of $W$ so that $(B,B\\cap W)\\cong (\\mathbb {R}^{k+l},\\mathbb {R}^k\\times \\lbrace 0\\rbrace )$ .",
"Moreover we can assume that there is a neighbourhood $U\\cong \\mathbb {R}^m$ of $p\\in M$ and a commutative diagram of smooth maps $\\begin{tikzcd}& \\mathbb {R}^m\\times \\mathbb {R}^k\\times \\lbrace 0\\rbrace [rr, hook] & & \\mathbb {R}^m\\times \\mathbb {R}^k\\times \\mathbb {R}^l [d] \\\\B\\cap W [rr, hook] [ru, \"\\cong \" description] & & B [d, \"\\pi \"^{\\prime }] [ru, \"\\cong \" description] & \\mathbb {R}^m \\\\& & U [ru, \"\\cong \" description] &\\end{tikzcd}$ This follows from the fact that $\\pi |_{W}$ is a submersion, because of the transversality assumption.",
"For any $0\\le i\\le m-l$ consider the smooth map $i\\colon \\mathbb {R}^m\\rightarrow \\mathbb {R}^l, \\quad u\\mapsto \\left(\\sum _{\\ell =1}^{i+1} (u_\\ell )^2-1,\\sum _{\\ell =i+2}^{m} (u_\\ell )^2-1 ,u_{m-l+2},\\dots ,u_m\\right)$ Clearly 0 is a regular value for $i$ , with preimageExcept for the case $l=1$ .",
"Here one should adjust the defintion of $i$ in order to have $b_i(i^{-1}(0))>0$ .",
"$i^{-1}(0)\\cong S^i\\times S^{m-l-i}$ and it is contained in the unit ball of radius 2.",
"Let $C\\subset \\mathbb {R}^m$ be a set of $n(m-l+1)$ points such that $|c-c^{\\prime }|\\ge 5$ for all pair of distinct elements $c,c^{\\prime }\\in C$ .",
"Now choose a partition $C=C_0\\sqcup C_1\\sqcup \\dots C_{m-l}$ in sets of cardinality $n$ and define a smooth map $\\mathbb {R}^m\\rightarrow \\mathbb {R}^l$ such that $x)=i(x-c)$ whenever $\\text{dist}(x,C_i)\\le 2$ .",
"We may also assume that 0 is a regular value for $.",
"Notice that $ -1(0)$ has a connected component\\begin{equation}S\\cong \\lbrace 1,\\dots , n\\rbrace \\times \\left( S^0\\times S^{m-l}\\sqcup S^1\\times S^{m-l-1}\\sqcup \\dots S^{m-l}\\times S^{0}\\right).\\end{equation}Construct a smooth (non necessarily holonomic) section $ FMJr(M,Rk)$ such that $ F|U(u)=(u,0,$ on a neighbourhood of $ S$, so that $ F-1(W)$ still contains $ S$ as a connected component, hence $ bi(F-1(W))n$ for all $ i=0,..., m-l$.$ To conclude we use the holonomic approximation theorem [8], saying that after a $\\mathcal {C}^0$ small perturbation of both $S$ and $F$ we can find a new section $\\tilde{F}$ that is holonomic in a neighbourhood of a submanifold $\\tilde{S}$ isotopic to $S$ , meaning that $\\tilde{F}=j^r\\psi $ in a neighbourhood of $\\tilde{S}$ for some smooth map $\\psi \\colon M\\rightarrow \\mathbb {R}^k$ .",
"Moreover, we can assume that $j^r\\psi \\mathrel {\\text{\\m@th {#\\crcr \\smash{-}\\crcr \\pitchfork \\crcr }}}$ W$, by Thom transversality Theorem (see \\cite {Hirsch} or \\cite {eliash}).",
"Applying Theorem \\ref {thm:semiconttop} we finally get that $ bi((jr)-1(W))bi(S)n$ for all $ i=0,..., m-l$.$ Random Algebraic Geometry Kostlan maps In this section we give the definition of a random Kostlan polynomial map $P:\\mathbb {R}^{m+1}\\rightarrow \\mathbb {R}^{k}$ , which is a Gaussian Random Field (GRF) that generalizes the notion of Kostlan polynomial.",
"Definition 15 (Kostlan polynomial maps) Let $d,m,k\\in \\mathbb {N}$ .",
"We define the degree $d$ homogeneous Kostlan random map as the measure on $\\mathbb {R}[x]_{(d)}^k=\\mathbb {R}[x_0,\\dots ,x_{m}]_{(d)}^k$ induced by the gaussian random polynomial: $P_d^{m,k}(x)=\\sum _{\\alpha \\in \\mathbb {N}^{m+1},\\ |\\alpha |=d} \\xi _\\alpha x^\\alpha ,$ where $x^\\alpha =x_0^{\\alpha _0}\\dots x_m^{\\alpha _m}$ and $\\lbrace \\xi _\\alpha \\rbrace $ is a family of independent gaussian random vectors in $\\mathbb {R}^k$ with covariance matrix $K_{\\xi _\\alpha }={d\\atopwithdelims ()\\alpha }\\mathbb {1}_k=\\left(\\frac{d!",
"}{\\alpha _0!\\dots \\alpha _m!",
"}\\right)\\mathbb {1}_k.$ We will call $P_d^{m,k}$ the Kostlan polynomial of type $(d,m,k)$ (we will simply write $P_d=P_d^{m,k}$ when the dimensions are understood).",
"(In other words, a Kostlan polynomial map $P_d^{m, k}$ is given by a list of $k$ independent Kostlan polynomials of degree $d$ in $m+1$ homogeneous variables.)",
"There is a non-homogeneous version of the Kostlan polynomial, which we denote as $p_d(u)=P_d(1,u)=\\sum _{\\beta \\in \\mathbb {N}^{m},\\ |\\beta |\\le d} \\xi _\\beta u^\\beta \\in \\mathcal {G}^{\\infty }(\\mathbb {R}^m,\\mathbb {R}^{k}),$ where $u=(u_1,\\dots , u_m)\\in \\mathbb {R}^m$ and $\\xi _\\beta \\sim N\\left(0,{d\\atopwithdelims ()\\beta }\\mathbb {1}_k\\right)$ are independent.",
"Here we use the notation of [20], where $\\mathcal {G}^{\\infty }(\\mathbb {R}^m,\\mathbb {R}^{k})$ denotes the space of gaussian random field on $\\mathbb {R}^m$ with values in $\\mathbb {R}^k$ which are $\\mathcal {C}^{\\infty }$ .",
"Next Proposition collects some well known facts on the Kostlan measure.",
"Proposition 16 Let $P_d$ be the Kostlan polynomial of type $(d,m,k)$ and $p_d$ be its dehomogenized version, as defined in (REF ).",
"For every $x,y\\in \\mathbb {R}^{m+1}$ : $K_{P_d}(x,y)=\\left(x^Ty\\right)^d\\mathbb {1}_k.$ Moreover, given $R\\in O(m+1)$ and $S\\in O(k)$ and defined the polynomial $\\tilde{P}_d(x)=SP_d(Rx)$ , then $P_d$ and $\\tilde{P}_d$ are equivalentTwo random fields are said to be equivalent if they induce the same probability measure on $\\mathcal {C}^{\\infty }(\\mathbb {R}^m,\\mathbb {R}^{k})$.",
"For every $u, v\\in \\mathbb {R}^n$ $K_{p_d}(u,v)=(1+u^Tv)^d\\mathbb {1}_k.$ Moreover, if $R\\in O(m)$ and $S\\in O(k)$ and defined the polynomial $\\tilde{p}_d(x)=Sp_d(Rx)$ , then $p_d$ and $\\tilde{p}_d$ are equivalent.",
"The proof of this proposition simply follows by computing explicitly the covariance functions and observing that they are invariant under orthogonal change of coordinates in the target and the source.",
"For example, in the case of $P_{d}$ we have: $\\begin{aligned}K_{P_d}(x,y)&= \\mathbb {E}\\lbrace P_d(x)P_d(y)^T\\rbrace = \\\\&= \\sum _{|\\alpha |,|\\alpha ^{\\prime }|=d}\\mathbb {E}\\left\\lbrace \\xi _\\alpha \\xi _{\\alpha ^{\\prime }}^T\\right\\rbrace x^\\alpha y^{\\alpha ^{\\prime }}=\\\\&=\\sum _{|\\alpha |=d}{d\\atopwithdelims ()\\alpha } (x_0 y_0)^{\\alpha _0}\\dots (x_m y_m)^{\\alpha _m}\\mathbb {1}_k=\\\\&= (x_0y_0+\\dots +x_my_m)^d\\mathbb {1}_k,\\end{aligned}$ from which the orthogonal invariance is clear.",
"The case of $p_d$ from the identity: $K_{p_d}(u,v)=K_{P_d}\\left((1,u),(1,v)\\right).$ .",
"Properties of the rescaled Kostlan The main feature here is the fact that the local model of a Kostlan polynomial has a rescaling limit.",
"The orthogonal invariance is used to prove that the limit does not depend on the point where we center the local model, hence it is enough to work around the point $(1, 0, \\ldots , 0)\\in S^m$ .",
"These considerations lead to introduce the Gaussian Random Field $X_d:\\mathbb {D}^m\\rightarrow \\mathbb {R}^{k}$ (we call it the rescaled Kostlan) defined by: $X_d(u)=P_d^{m,k}\\left(1, \\frac{u_1}{\\sqrt{d}}, \\ldots , \\frac{u_m}{\\sqrt{d}}\\right).$ Next result gives a description of the properties of the rescaled Kostlan polynomial, in particular its convergence in law as a random element of the space of smooth functions, space which, from now, on we will always assume to be endowed with the weak Whitney's topology as in [20].",
"Theorem 17 (Properties of the rescaled Kostlan) Let $X_d:\\mathbb {R}^m\\rightarrow \\mathbb {R}^{k}$ be the Gaussian random field defined in (REF ).",
"(The limit) Given a family of independent gaussian random vectors $\\xi _\\beta \\sim N\\left(0,\\frac{1}{\\beta !",
"}\\mathbb {1}_k\\right)$ , the series $X_\\infty (u)=\\sum _{\\beta \\in \\mathbb {N}^{m}} \\xi _\\beta u^\\beta ,$ is almost surely convergent in $\\mathcal {C}^{\\infty }(\\mathbb {R}^m,\\mathbb {R}^{k})$ to the Gaussian Random Field$X_\\infty $ is indeed a random analytic function, commonly known as the Bargmann-Fock ensemble.",
"$X_\\infty \\in \\mathcal {G}^{\\infty }(\\mathbb {R}^m,\\mathbb {R}^{k})$ .",
"(Convergence) $X_d\\Rightarrow X_\\infty $ in $\\mathcal {G}^{\\infty }(\\mathbb {R}^m,\\mathbb {R}^{k})$ , that is: $\\lim _{d\\rightarrow +\\infty }\\mathbb {E}\\lbrace F(X_d)\\rbrace =\\mathbb {E}\\lbrace F(X_\\infty )\\rbrace $ for any bounded and continuous function $F\\colon \\mathcal {C}^{\\infty }(\\mathbb {R}^m,\\mathbb {R}^{k})\\rightarrow \\mathbb {R}$ .",
"Equivalently, we have $\\mathbb {P}\\lbrace X_\\infty \\in \\emph {int}(A)\\rbrace \\le \\liminf _{d\\rightarrow +\\infty }\\mathbb {P}\\lbrace X_d\\in A\\rbrace \\le \\limsup _{d\\rightarrow +\\infty }\\mathbb {P}\\lbrace X_d\\in A\\rbrace \\le \\mathbb {P}\\lbrace X_d\\in \\overline{A}\\rbrace $ for any Borel subset $A\\subset \\mathcal {C}^{\\infty }(\\mathbb {R}^m,\\mathbb {R}^{k})$ .",
"(Nondegeneracy of the limit) The support of $X_\\infty $ is the whole $\\mathcal {C}^{\\infty }(\\mathbb {R}^m,\\mathbb {R}^{k})$ .",
"In other words, for any non empty open set $U\\subset \\mathcal {C}^{\\infty }(\\mathbb {R}^m,\\mathbb {R}^{k})$ we have that $\\mathbb {P}\\lbrace X_\\infty \\in U\\rbrace >0$ .",
"(Probabilistic Transversality) For $d\\ge r$ and $d=\\infty $ , we have $\\mathrm {supp}(j^r_pX_d)=J_p^r(\\mathbb {R}^m,\\mathbb {R}^k)$ for every $p\\in \\mathbb {R}^m$ and consequently for every submanifold $W\\subset J^r(\\mathbb {R}^m,\\mathbb {R}^k)$ , we have $\\mathbb {P}\\lbrace j^rX_d\\pitchfork W\\rbrace =1.$ (Existence of limit probability) Let $V\\subset J^{r}(\\mathbb {R}^m, \\mathbb {R}^k)$ be an open set whose boundary is a (possibly stratified) submanifoldFor example $V$ could be a semialgebraic set.",
"Then $\\lim _{d\\rightarrow +\\infty }\\mathbb {P}\\lbrace j^r_p X_d\\in V,\\ \\forall p\\in \\mathbb {R}^m\\rbrace = \\mathbb {P}\\lbrace j^r_pX_\\infty (\\mathbb {R}^m) \\in V,\\ \\forall p\\in \\mathbb {R}^m\\rbrace .$ In other words, we have equality in (REF ) for sets of the form $U=\\lbrace f\\colon j^rf\\in V\\rbrace $ .",
"(Kac-Rice densities) Let $W\\subset J^r(\\mathbb {R}^m,\\mathbb {R}^k)$ be a semialgebraic subset of codimension $m$ , such thatIn this paper the symbol $\\mathrel {\\text{\\m@th {#\\crcr \\smash{-}\\crcr \\pitchfork \\crcr }}}$$ stands for ``it is transverse to^{\\prime \\prime }.$ $W\\mathrel {\\text{\\m@th {#\\crcr \\smash{-}\\crcr \\pitchfork \\crcr }}}$ Jpr(Rm,Rk)$ for all $ pM$ (i.e.",
"$ W$ is transverse to fibers of the projection of the jet space).",
"Then for all $ dr$ and for $ d=+$ there exists a locally bounded function $ dWLloc (Rm)$ such that\\footnote {A formula for \\rho _d^W is presented in \\cite {DTGRF1}, as a generalization of the classical Kac-Rice formula.",
"}\\begin{equation}\\mathbb {E}\\#\\lbrace u\\in A\\colon j^r_uX_d\\in W\\rbrace =\\int _A\\rho ^W_d,\\end{equation}for any Borel subset $ ARm$.",
"Moreover $ Wd W$ in $ Lloc$.\\item \\emph {(Limit of Betti numbers)} Let $ WJr(Rm,Rk)$ be any closed semialgebraic subset transverse to fibers.",
"Then:\\begin{equation}\\lim _{d\\rightarrow +\\infty }\\mathbb {E}\\left\\lbrace b_i\\left((j^{r}X_d)^{-1}(W)\\cap \\mathbb {D}^m\\right)\\right\\rbrace =\\mathbb {E}\\left\\lbrace b_i\\left((j^{r}X_\\infty )^{-1}(W)\\cap \\mathbb {D}^m\\right)\\right\\rbrace ,\\end{equation}where $ bi(Z)=Hi(Z,R)$.",
"Moreover, if the codimension of $ W$ is $ l1$, then the r.h.s.",
"in equation (\\ref {eq:localEbetti}) is strictly positive for all $ i=0,..., m-l$.$ The proof uses a combination of results from [20].",
"Let $S_d=\\sum _{|\\beta |\\le d}\\xi _\\beta u^\\beta \\in \\mathcal {G}^{\\infty }(M,\\mathbb {R}^{k})$ .",
"The covariance function of $S_d$ converges in Whitney's weak topology: $K_{S_d}(u,v)=\\sum _{|\\beta |\\le d}\\frac{u^\\beta v^\\beta }{\\beta !",
"}\\mathbb {1}_k\\xrightarrow{} \\exp (u^Tv)\\mathbb {1}_k.$ It follows by [20] that $S_d$ converges in $\\mathcal {G}^{\\infty }(M,\\mathbb {R}^{k})$ , moreover since all the terms in the series are independent we can conclude with the Ito-Nisio It may not be trivial to apply the standard Ito-Nisio theorem, which actually regards convergence of series in a Banach space.",
"See Theorem 36 of [20] for a statement that is directly applicabile to our situation Theorem [16] that indeed the convergence holds almost surely.",
"By [20] it follows from convergence of the covariance functions: $K_{X_d}(u,v)=\\left(1+\\frac{u^Tv}{d}\\right)^d\\mathbb {1}_k \\quad \\xrightarrow{}\\quad K_{X_\\infty }(u,v)=\\exp (u^Tv)\\mathbb {1}_k$ The support of $X_\\infty $ contains the set of polynomial functions $\\mathbb {R}[u]^k$ , which is dense in $\\mathcal {C}^{\\infty }(\\mathbb {R}^m,\\mathbb {R}^{k})$ , hence the thesis follows from [20].",
"Let $d\\ge r$ or $d=+\\infty $ .",
"We have that $\\begin{aligned}\\text{supp}(j_u^rX_d)&=\\lbrace j^r_uf\\colon f\\in \\mathbb {R}[u]^k \\text{ of degree $\\le d$}\\rbrace =\\\\&=\\textrm {span}\\lbrace j^r_uf\\colon f(v)=(v-u)^\\beta \\text{ with $|\\beta |\\le d$}=\\\\&=\\textrm {span}\\lbrace j^r_uf\\colon f(v)=(v-u)^\\beta \\text{ with $|\\beta |\\le r$}\\rbrace =\\\\&=J^r_u(\\mathbb {R}^m,\\mathbb {R}^k).\\end{aligned}$ The fact that $\\mathbb {P}\\lbrace j^rX_d\\mathrel {\\text{\\m@th {#\\crcr \\smash{-}\\crcr \\pitchfork \\crcr }}}$ W}=1$ follows \\cite [Theorem 8]{DTGRF1}.\\item Let $ A={fC(Rm,Rk)jrfV}$.",
"If $ fA$, then $ jrfV$ and there is a point $ uRm$ such that $ jrufV$.",
"Let $ V$ be stratified as $ V=Zi$ with each $ Zi$ a submanifold.",
"If $ jrf $-$$\\pitchfork $ V$ then it means that $ jr f$ is transversal to all the $ Zi$ and there exists one of them which contains $ jruf$ (i.e.",
"the jet of $ f$ intersect $ V$).",
"Therefore the intersection would be transversal \\emph {and nonempty}, and then there exists a small Whitney-neighborhood of $ f$ such that for every $ g$ in this neighborhood $ jrg$ still intersects $ V.$ This means that there is a neighborhood of $ f$ consisting of maps that are not in $ A$, which means $ f$ has a neighborhood contained in $ Ac$.",
"It follows that $ fA$ and consequently $ fA$, which is a contradiction.",
"Therefore we have that\\begin{equation}\\partial A\\subset \\lbrace f\\in \\mathcal {C}^{\\infty }(R^m,\\mathbb {R}^{k})\\colon f \\text{ is not transverse to }\\partial V\\rbrace .\\end{equation}It follows by point $ (4)$ that $ P{XA}=0$, so that we can conclude by points $ (2)$ and $ (3)$.\\item By previous points, we deduce that we can apply the results described in section $ 7$ of \\cite {DTGRF1}.\\item This proof is postponed to Section \\ref {sec:betti}.$ Given a $\\mathcal {C}^{\\infty }$ Gaussian Random Field $X:\\mathbb {R}^m\\rightarrow \\mathbb {R}^k$ , let us denote by $[X]$ the probability measure induced on $\\mathcal {C}^{\\infty }(\\mathbb {R}^m, \\mathbb {R}^k)$ and defined by: $[X](U)=\\mathbb {P}(X\\in U),$ for every $U$ belonging to the Borel $\\sigma -$ algebra relative to the weak Whitney topology, see [15] for details on this topology.",
"Combining Theorem REF with Skorohod Theorem [1] one gets that it is possible to represent $[X_d]$ with equivalent fields $\\tilde{X}_d$ such that $\\tilde{X}_d\\rightarrow \\tilde{X}_\\infty $ almost surely in $\\mathcal {C}^{\\infty }(\\mathbb {R}^m,\\mathbb {R}^{k})$ .",
"This is in fact equivalent to point $(2)$ of Theorem REF .",
"In other words there is a (not unique) choice of the gaussian coefficients of the random polynomials in $(\\ref {eq:Kd})$ , for which the covariances $\\mathbb {E}\\lbrace \\tilde{X}_d\\tilde{X}_{d^{\\prime }}^T\\rbrace $ are such that the sequence converges almost surely.",
"We leave to the reader to check that a possible choice is the following.",
"Let $\\lbrace \\gamma _\\beta \\rbrace _{\\beta \\in \\mathbb {N}^m}$ be a family of i.i.d.",
"gaussian random vectors $\\sim N(0,\\mathbb {1}_k)$ and define for all $d<\\infty $ $\\tilde{X}_d=\\sum _{|\\beta |\\le d}{d\\atopwithdelims ()\\beta }^\\frac{1}{2} \\gamma _\\beta \\left(\\frac{u}{\\sqrt{d}}\\right)^{\\beta }$ and $\\tilde{X}_\\infty =\\sum _{\\beta }{\\left(\\frac{1}{\\beta !",
"}\\right)}^\\frac{1}{2} \\gamma _\\beta u^{\\beta }$ Proposition 18 $\\tilde{X}_d\\rightarrow \\tilde{X}_{\\infty }$ in $\\mathcal {C}^{\\infty }(\\mathbb {R}^m,\\mathbb {R}^{k})$ almost surely.",
"However, we stress the fact that in most situations: when one is interested in the sequence of probability measures $[X_d]$ , it is sufficient to know that such a sequence exists.",
"Limit laws for Betti numbers and the generalized square-root law Figure: The random set S d ={X d =0}⊂mS_d=\\lbrace X_d=0\\rbrace \\subset m is a rescaled version of Z d ∩D(p,d -1/2 )Z_d\\cap D(p, d^{-1/2}), where Z d ={ψ d =0}Z_d=\\lbrace \\psi _d=0\\rbrace .Let $W_0\\subset J^r(\\mathbb {R}^m,\\mathbb {R}^k)$ be a semialgebraic subset.",
"Consider the random set $S_d= \\lbrace p\\in \\mathbb {D}^m\\colon j_p^rX_d\\in W_0\\rbrace ,$ where $X_d\\colon \\mathbb {R}^m\\rightarrow \\mathbb {R}^k$ is the rescaled Kostlan polynomial from Theorem REF (see Figure REF ).",
"We are now in the position of complete the proof of Theorem REF by showing point (7).",
"Let us start by proving the following Lemma.",
"Lemma 19 Let $r$ be the codimension of $W_0$ and suppose $0\\le i\\le m-r\\le m-1$ .",
"Then $\\mathbb {E}\\lbrace b_i(S_\\infty )\\rbrace >0 .$ From Corollary REF we deduce that there exists a function $f\\in \\mathcal {C}^{\\infty }(\\mathbb {D}^m,\\mathbb {R}^{k})$ such that $j^rf\\mathrel {\\text{\\m@th {#\\crcr \\smash{-}\\crcr \\pitchfork \\crcr }}}$ W0$ and $ bi((jrf)-1(W0))0$.Since the condition on $ f$ is open, there is an open neighbourhood $ O$ of $ f$ where $ bi((jrg)-1(W0))=c>0$ for all $ gO$.",
"Thus $ P{bi(S)=c}>0$ because every open set has positive probability for $ X$, by \\ref {thm:Kostlan}.3. therefore $ E{bi(S)}>0$.$ We complete the proof of Theorem REF with the next Proposition.",
"Proposition 20 $\\lim _{d\\rightarrow \\infty }\\mathbb {E}\\lbrace b_i(S_d)\\rbrace =\\mathbb {E}\\lbrace b_i(S_\\infty )\\rbrace $ Let $b_i(S_d)=b_d$ .",
"Define a random field $Y_d=(X_d,x_d)\\colon \\mathbb {R}^m\\rightarrow \\mathbb {R}^{k}\\times \\mathbb {R}$ to be the rescaled Kostlan polynomial of type $(m,k+1)$ .",
"Consider the semialgebraic subset $W^{\\prime }=W\\cap J^r(\\mathbb {D}^m,\\mathbb {R}^k)$ of the real algebraic smooth manifold $J^r(\\mathbb {R}^m,\\mathbb {R}^k)$ and observe that $S_d=(j^rX_d)^{-1}(W^{\\prime })$ is compact.",
"Now Theorem , along with Remark REF , implies the existence of a semialgebraic submanifold $\\hat{W^{\\prime }}\\subset J^{r+1}(\\mathbb {R}^m,\\mathbb {R}^{k+1})$ of codimension $m$ and a constant $C$ , such that $b_d\\le C\\#\\left\\lbrace \\left(j^{r+1}(Y_d)\\right)^{-1}(\\hat{W^{\\prime }})\\right\\rbrace =:N_d$ whenever $j^{r+1}Y_d\\mathrel {\\text{\\m@th {#\\crcr \\smash{-}\\crcr \\pitchfork \\crcr }}}$ W'$ and hence almost surely, because of Theorem \\ref {thm:Kostlan}.4.Since $ YdY$ by \\ref {thm:Kostlan}.2, we see that $ [bd,Nd][b,N]$ and it is not restrictive to assume that $ bi(Zd),Ndbi(Z),N$ almost surely, by Skorokhod^{\\prime }s theorem (see \\cite [Theorem 6.7]{Billingsley}).",
"Moreover $ E{Nd}E{N}$ by Theorem \\ref {thm:Kostlan}.6.Now we can conclude with Fatou^{\\prime }s Lemma as follows\\begin{equation}\\begin{aligned}2\\mathbb {E}\\lbrace N_\\infty \\rbrace &=\\mathbb {E}\\lbrace \\liminf _d N_d+N_\\infty -|b_d-b_\\infty |\\rbrace \\le \\\\&\\le \\liminf _d\\mathbb {E}\\lbrace N_d+N_\\infty -|b_d-b_\\infty |\\rbrace =\\\\&=2\\mathbb {E}\\lbrace N_\\infty \\rbrace -\\limsup _d\\mathbb {E}\\lbrace |b_d-b_\\infty |\\rbrace ,\\end{aligned}\\end{equation}so that\\begin{equation}\\limsup _d\\mathbb {E}\\lbrace |b_d-b_\\infty |\\rbrace \\le 0.\\end{equation}$ In the sequel, with the scope of keeping a light notation, for a given $W\\subset J^{r}(S^m, \\mathbb {R}^k)$ and $\\psi :S^m\\rightarrow \\mathbb {R}^k$ we will denote by $Z_d\\subseteq S^m$ the set $Z_d=j^r\\psi ^{-1}(W).$ If $W$ is of codimension $m$ , then by Theorem REF , $Z_d$ is almost surely a finite set of points and the expectation of this number is given by next result.",
"Theorem 21 (Generalized square-root law for cardinality) Let $W\\subset J^r(S^m,\\mathbb {R}^k)$ be a semialgebraic intrinsic subset of codimension $m$ .",
"Then there is a constant $C_W>0$ such that: $\\mathbb {E}\\lbrace \\#Z_d\\rbrace =C_W d^{\\frac{m}{2}}+ O(d^{\\frac{m}{2}-1}).$ Moreover, the value of $C_W$ can be computed as follows.",
"Let $Y_\\infty =e^{-\\frac{|u|^2}{2}}X_\\infty \\in \\mathcal {G}^{\\infty }(\\mathbb {D}^m,\\mathbb {R}^{k})$ and let $W_0\\subset J^r(\\mathbb {D}^m,\\mathbb {R}^k)$ be the local model for $W$ .",
"Then $C_W=m\\frac{\\mathrm {vol}(S^m)}{\\mathrm {vol}(S^{m-1})} \\mathbb {E}\\#\\lbrace u\\in \\mathbb {D}^m\\colon j^r_uY_{\\infty }\\in W_0\\rbrace .$ In order to prove Theorem REF , we will need a preliminary Lemma, which ensures that we will be in the position of using the generalized Kac-Rice formula of point (6) from Theorem REF .",
"Lemma 22 If $W\\subset J^r(M,\\mathbb {R}^k)$ is intrinsic, then $W$ is transverse to fibers.",
"Since the result is local it is sufficient to prove it in the case when $M=\\mathbb {R}^m$ .",
"In this case we have a canonical identification $J^r(\\mathbb {R}^m,\\mathbb {R}^k)\\cong \\mathbb {R}^m\\times J^r_0(\\mathbb {R}^m,\\mathbb {R}^k), \\qquad j^r_uf\\rightarrow (u,j^r_{h=0}f(u+h))$ Consider the embedding $ i_u \\colon \\mathbb {D}^m\\rightarrow \\mathbb {R}^m$ obtained as the isometric inclusion in the disk with center $u$ and let $\\tau _h\\colon \\mathbb {R}^m\\rightarrow \\mathbb {R}^m$ be the translation map $u\\mapsto u+h$ .",
"Let $(v,j^r_0f)\\in W$ , then $j^r(\\tau _{v-u}\\circ i_u)^*(v,j^r_0f)\\in W_0$ .",
"Given any other point $u$ , we have $\\begin{aligned}j^r(\\tau _{v-u}\\circ i_u)^*(v,j^r_0g)&= j^ri_u^*\\left(j^r(\\tau _{v-u})^*(v,j^r_0g)\\right)=\\\\&= j^ri_u^*\\left(j^r(\\tau _{v-u})^*(v,j^r_{h=0}f(v+h))\\right)=\\\\&= j^ri_u^*\\left(u,j^r_{h=0} f\\circ \\tau _{v-u}(u+h)\\right)=\\\\&= j^ri_u^*\\left(u,j^r_0g\\right)\\in W_0\\end{aligned}$ so that $\\left(u,j^r_0f\\right)\\in W$ .",
"It follows that, calling $\\lbrace 0\\rbrace \\times \\bar{W}= W\\cap J^r_0(\\mathbb {R}^m,\\mathbb {R}^k)$ , $W=\\lbrace (u,j_0^rf)\\colon j^r_0f \\in W_0\\rbrace =\\mathbb {R}^m\\times \\bar{W},$ which is clearly transverse to each fiber $\\lbrace u\\rbrace \\times J^r_0(\\mathbb {R}^m,\\mathbb {R}^k)$ .",
"Figure: A family of shrinking embedding of the unit disk.The reason why we consider intrinsic submanifold is to be able to easily pass to the rescaled Kostlan polynomial $X_d\\in \\mathcal {G}^{\\infty }(\\mathbb {D}^m,\\mathbb {R}^{k})$ by composing $\\psi _d$ with the embedding of the disk $a_d^R$ defined by: $a_d^R\\colon \\mathbb {D}^m\\hookrightarrow S^m,\\quad u\\mapsto \\frac{R\\begin{pmatrix}1 \\\\ \\frac{u}{\\sqrt{d}}\\end{pmatrix}}{\\sqrt{\\left(1+\\frac{|u|^2}{d}\\right)}}$ for any $R\\in O(m+1)$ (see Figure REF ).",
"Let us consider the set function $\\mu _d\\colon \\mathcal {B}(S^m)\\mapsto \\mathbb {R}$ such that $A\\mapsto \\mathbb {E}\\lbrace \\#(j^rX_d)^{-1}(W)\\cap A\\rbrace $ .",
"It is explained in [20] that $\\mu _d$ is a Radon measure on $S^m$ .",
"Because of the invariance under rotation of $P_d$ , by Haar's theorem $\\mu $ needs to be proportional to the volume measure.",
"Therefore for any Borel subset $A\\subset S^m$ we have $\\mathbb {E}\\lbrace \\#Z_d\\rbrace =\\mu _d(S^m)=\\mu _d(A)\\mathrm {vol}(A)^{-1}\\mathrm {vol}(S^m)$ .",
"Define $Y_d\\in \\mathcal {G}^{\\infty }(\\mathbb {D}^m,\\mathbb {R}^{k})$ as $Y_d= \\left(1+\\frac{|u|^2}{d}\\right)^{-\\frac{d}{2}}X_d.$ Observe that $Y_d\\Rightarrow Y_\\infty =\\exp (-\\frac{|u|^2}{2})X_\\infty $ and that $Y_d$ is equivalent to the GRF $\\psi _d\\circ a^R_d$ for any $R\\in O(m+1)$ .",
"Now let $W_0\\subset J^r(\\mathbb {D}^m,\\mathbb {R}^k)$ be the (semialgebraic) model of $W$ .",
"By the same proof of point (7) from Theorem REF , adapted to $Y_d$ , there is a convergent sequence of functions $\\rho _d\\rightarrow \\rho _{+\\infty }\\in L^1(\\mathbb {D}^m)$ such that $\\mathbb {E}\\lbrace \\#(j^rY_d)^{-1}(W_0)\\rbrace =\\int _{\\mathbb {D}^m}\\rho _d\\rightarrow \\int _{\\mathbb {D}^m}\\rho _\\infty = \\mathbb {E}\\lbrace \\#(j^rY_\\infty )^{-1}(W_0)\\rbrace .$ In conclusion we have for $A=a^R_d(\\mathbb {D}^m)$ , as $d\\rightarrow +\\infty $ $\\begin{aligned}\\mathbb {E}\\lbrace \\#Z_d\\rbrace &=\\mu _d(A)\\mathrm {vol}(A)^{-1}\\mathrm {vol}(S^m)\\\\&=\\mathbb {E}\\lbrace \\#(j^rY_d)^{-1}(j^r*(W))\\rbrace \\mathrm {vol}(A)^{-1}\\mathrm {vol}(S^m)\\\\&=\\mathbb {E}\\lbrace \\#(j^rY_d)^{-1}(W_0)\\rbrace \\left(\\frac{\\int _0^{\\pi }|\\sin \\theta |^{m-1}d\\theta }{\\int _0^{\\arctan \\left(d^{-\\frac{1}{2}}\\right)}|\\sin \\theta |^{m-1}d\\theta } \\right)\\\\&=\\mathbb {E}\\lbrace \\#(j^rY_\\infty )^{-1}(W_0)\\rbrace m\\frac{\\mathrm {vol}(S^m)}{\\mathrm {vol}(S^{m-1})}d^{\\frac{m}{2}}+O(d^{\\frac{m}{2}-1}).\\end{aligned}$ Building on the previous results, we can now prove the general case for Betti numbers of a random singualrity.",
"Theorem 23 (Generalized square-root law for Betti numbers) Let $W\\subset J^r(S^m,\\mathbb {R}^k)$ be a closed semialgebraic intrinsic (as defined in Definition REF ) of codimension $1\\le l\\le m$ .",
"Then there are constants $b_W, B_W > 0$ depending only on $W$ such that $b_W d^{\\frac{m}{2}}\\le \\mathbb {E}\\lbrace b_i(Z_d)\\rbrace \\le B_W d^{\\frac{m}{2}}\\quad \\forall i=0,\\dots , m-l$ and $\\mathbb {E}\\lbrace b_i(Z_d)\\rbrace =0$ for all other $i$ .",
"The proof is divided in two parts, first we prove the upper bound, using the square-root law from Theorem REF , then the we use Theorem to deduce the lower bound.",
"1.",
"Assume $W$ is smooth with codimension $s$ .",
"Let us consider $P^{m, k+1}_d|_{S^m}=\\Psi _d=(\\psi _d,\\psi _d^1)\\in \\mathcal {G}^{\\infty }(S^m,\\mathbb {R}^{k+1})$ and Let $\\hat{W}\\subset J^{r+1}(S^m,\\mathbb {R}^{k+1})$ be the intrinsic semialgebraic submanifold coming from Theorem and Remark REF .",
"Thus, using Theorems and REF , we get $\\mathbb {E}\\lbrace b_i(Z_d)\\rbrace \\le N_W\\mathbb {E}\\#\\lbrace (j^{r+1}\\Psi _d)^{-1}(\\hat{W})\\rbrace \\le N_WC_{\\hat{W}}d^{\\frac{m}{2}}.$ 2.",
"Consider the embeddings of the $m$ dimensional disk $a_d^R\\colon \\mathbb {D}^m\\hookrightarrow S^m$ defined in (REF ).",
"For any fixed $d\\in \\mathbb {N}$ , choose a finite subset $F_d\\subset O(m+1)$ such that the images of the corresponding embeddings $\\lbrace a_d^R(\\mathbb {D}^m)\\rbrace _{R\\in F_d}$ are disjoint.",
"Denoting by $Z_d^R$ the union of all connected components of $Z_d$ that are entirely contained in $a_d^R(\\mathbb {D}^m)$ , we have $b_i(Z_d)\\ge \\sum _{R\\in F_d}b_i(Z_d^R).$ Let $W_0\\subset J^r(\\mathbb {D}^m,\\mathbb {R}^k)$ be the model of $W$ as an intrinsic submanifold, it is closed and semialgebraic.",
"By Definition REF , we have $(a_d^R)^{-1}\\left((j^r\\psi _d)^{-1}(W)\\right)= \\left(j^r(\\psi _d\\circ a_d^R)\\right)^{-1}(W_0)\\subset \\mathbb {D}^m.$ Recall that for any $R\\in O(m+1)$ , the GRF $\\psi _d\\circ a^R_d$ is equivalent to $Y_d\\in \\mathcal {G}^{\\infty }(\\mathbb {D}^m,\\mathbb {R}^{k})$ defined in REF , hence taking expectation in Equation (REF ) we find $\\mathbb {E}\\lbrace b_i(Z_d)\\rbrace \\ge \\#(F_d)\\mathbb {E}\\lbrace b_i(S_d)\\rbrace ,$ where $S_d=\\left(j^r(Y_d)\\right)^{-1}(W_0)$ .",
"is easy to see (repeating the same proof) that Theorem REF .7 holds also for the sequence $Y_d\\Rightarrow Y_\\infty $ , so that $\\mathbb {E}\\lbrace S_d\\rbrace \\rightarrow \\mathbb {E}\\lbrace S_\\infty \\rbrace $ .",
"We can assume that $\\mathbb {E}\\lbrace S_\\infty \\rbrace >0$ , because of Lemma REF , thus for big enough $d$ , the numbers $\\mathbb {E}\\lbrace b_i(S_d)\\rbrace $ are bounded below by a constant $C>0$ .",
"Now it remains to estimate the number $\\#(F_d)$ .",
"Notice that $a_d^R(\\mathbb {D}^m)$ is a ball in $S^m$ of a certain radius $\\varepsilon _d$ , hence it is possible to choose $F_d$ to have at least $N_m \\varepsilon _d^{-1}$ elements, for some dimensional constant $N_m>0$ depending only on $m$ .",
"We conclude by observing that $\\varepsilon _d\\approx d^{-\\frac{m}{2}}.$ Appendix 1: Examples of applications of Theorem REF Example 24 (Zero sets of random polynomials) Consider the zero set $Z_d\\subset \\mathbb {R}\\mathbb {P}^m$ of a random Kostlan polynomial $P_d=P_{d}^{m+1,1}$ .",
"Recently Gayet and Welschinger [11] have proved that given a compact hypersurface $Y\\subset \\mathbb {R}^{m}$ there exists a positive constant $c=c(\\mathbb {R}^m, Y)>0$ and $d_0=d_0(\\mathbb {R}^m,Y)\\in \\mathbb {N}$ such that for every point $x\\in \\mathbb {R}\\mathbb {P}^m$ and every large enough degree $d\\ge d_0$ , denoting by $B_d$ any open ball of radius $d^{-1/2}$ in $\\mathbb {R}\\mathbb {P}^m$ , we have: $\\left(B_d, B_d\\cap Z_d\\right)\\cong (\\mathbb {R}^m, Y) $ (i.e.",
"the two pairs are diffeomorphic) with probability larger than $c$ .",
"This result follows from Theorem REF as follows.",
"Let $\\mathbb {D}^m\\subset \\mathbb {R}^m$ be the unit disk, and let $U\\subset \\mathcal {C}^{\\infty }(\\mathbb {D}^m, \\mathbb {R})$ be the open set consisting of functions $g:\\mathbb {D}^m\\rightarrow \\mathbb {R}$ whose zero set is regular (an open $\\mathcal {C}^1$ condition satisfied almost surely by $X_d$ , because of point (4)), entirely contained in the interior of $\\mathbb {D}^m$ (an open $\\mathcal {C}^0$ condition) and such that, denoting by $\\mathbb {B}\\subset \\mathbb {R}^m$ the standard unit open ball, the first two conditions hold and $(\\mathbb {B}, \\mathbb {B}\\cap \\lbrace g=0\\rbrace )$ is diffeomorphic to $(\\mathbb {R}^m, Y)$ (this is another open $\\mathcal {C}^1$ condition).",
"Observe that, using the notation above: $\\left(B_d, B_d\\cap Z_d\\right)\\sim (\\mathbb {B}, \\mathbb {B}\\cap \\lbrace X_d=0\\rbrace ) $ (this is simply because $X_d(u)=P_d(1, ud^{-1/2})$ ).",
"Consequently point (5) of Theorem REF implies that: $\\lim _{d\\rightarrow +\\infty }\\mathbb {P}\\lbrace (\\ref {eq:isotopic})\\rbrace &=\\lim _{d\\rightarrow \\infty } \\mathbb {P}\\left\\lbrace (\\mathbb {B}, \\mathbb {B}\\cap \\lbrace X_d=0\\rbrace )\\sim (\\mathbb {R}^m, Y)\\right\\rbrace \\\\&=\\lim _{d\\rightarrow \\infty } \\mathbb {P}\\left\\lbrace X_d\\in U\\right\\rbrace \\\\&=\\mathbb {P}\\left\\lbrace X_\\infty \\in U \\right\\rbrace >0.$ We stress that, as an extra consequence of Theorem REF , compared to [11] what we get is the existence of the limit of the probability of seeing a given diffeomorphism type.",
"Example 25 (Discrete properties of random maps) Let $[X_d]\\Rightarrow [X_\\infty ]$ be a converging family of gaussian random fields.",
"In this example we introduce a useful tool for studying the asymptotic probability induced by $X_d$ on discrete sets as $d\\rightarrow \\infty $ .",
"The key example that we have in mind is the case when we consider a codimension-one “discriminant” $\\Sigma \\subset \\mathcal {C}^\\infty (S^m, \\mathbb {R}^k)$ which partitions the set of functions into many connected open sets.",
"For instance $\\Sigma $ could be the set of maps for which zero is not a regular value: the complement of $\\Sigma $ consists of countably many open connected sets, each one of which corresponds to a rigid isotopy class of embedding of a smooth codimension-$k$ submanifold $Z\\subset S^m$ .",
"The following Lemma gives a simple technical tool for dealing with these situations.",
"Lemma 26 Let $E$ be a metric space and let $[X_d], [X_\\infty ]$ be a random fields such that $[X_d]\\Rightarrow [X_\\infty ]$ .",
"Let also $Z$ be a discrete space and $\\nu \\colon U\\subset E\\rightarrow Z$ be a continuous function defined on an open subset $U\\subset E$ such thatOf course, $E\\setminus U=\\Sigma $ is what we called “discriminant” in the previous discussion.",
"Note that we do not require that $\\mathbb {P}\\lbrace X_d\\in U\\rbrace =1$ , however it will follow that $\\lim _d \\mathbb {P}\\lbrace X_d\\in U\\rbrace =1$ .",
"$\\mathbb {P}\\lbrace X_\\infty \\in U\\rbrace =1$ .",
"Then, for any $A\\subset Z$ we have: $\\exists \\lim _{d\\rightarrow \\infty }\\mathbb {P}\\left\\lbrace X_d\\in U,\\ \\nu (X_d)\\in A\\right\\rbrace =\\mathbb {P}\\left\\lbrace \\nu (X_\\infty )\\in A\\right\\rbrace .$ Since $\\nu ^{-1}(A)$ is closed and open by continuity of $\\nu $ , it follows that $\\partial \\nu ^{-1}(A)\\subset E\\backslash U$ .",
"Therefore $\\mathbb {P}\\lbrace X_\\infty \\in \\partial \\nu ^{-1}(A)\\rbrace =0$ and by Portmanteau's Theorem [1], we conclude that $\\mathbb {P}\\lbrace X_d\\in \\nu ^{-1}(A)\\rbrace \\xrightarrow[d\\rightarrow \\infty ]{}\\mathbb {P}\\lbrace X_\\infty \\in \\nu ^{-1}(A)\\rbrace , \\quad \\ \\forall \\ A\\subset Z.$ Equation (REF ), in the case of a discrete topological space such as $Z$ , is equivalent to narrow convergence $\\nu (X_d)\\Rightarrow \\nu (X)$ , by Portmanteau's Theorem, because $\\partial A=\\emptyset $ for all subsets $A\\subset Z$ .",
"Note also that to prove narrow convergence of a sequence of measures on $Z$ , it is sufficient to show (REF ) for all $A=\\lbrace z\\rbrace $ , indeed in that case the inequality $\\liminf _{d\\rightarrow \\infty }\\mathbb {P}\\lbrace \\nu _d\\in A\\rbrace =\\liminf _{d\\rightarrow \\infty }\\sum _{z\\in A}\\mathbb {P}\\lbrace \\nu _d=z\\rbrace \\ge \\sum _{z\\in A}\\mathbb {P}\\lbrace \\nu =z\\rbrace =\\mathbb {P}\\lbrace \\nu \\in A\\rbrace $ follows automatically from Fatou's lemma.",
"Following Sarnak and Wigman [23], let us consider one simple application of this Lemma.",
"Let $H_{m-1}$ be the set of diffeomorphism classes of smooth closed connected hypersurfaces of $\\mathbb {R}^{m}$ .",
"Consider $U=\\lbrace f\\in \\mathcal {C}^{\\infty }(\\mathbb {D}^m,\\mathbb {R}^{})\\,\\colon \\, f\\mathrel {\\text{\\m@th {#\\crcr \\smash{-}\\crcr \\pitchfork \\crcr }}}$ 0}$ and let $ (f)$ be the number of connected components of $ f-1(0)$ entirely contained in the interior of $ Dm$.",
"For $ hHm-1$ let $ h(f)$ be the number of those components which are diffeomorphic to $ hRm$.",
"In the spirit of \\cite {SarnakWigman}, we define the probability measure $ (f)P(Hm-1)$ as\\begin{equation}\\mu (f)=\\frac{1}{\\nu (f)}\\sum _{h\\in H_{m-1}}\\nu _h(f)\\delta _h.\\end{equation}Let us consider now the rescaled Kostlan polynomial $ Xd:DmR$ as in Theorem \\ref {thm:Kostlan}.",
"The diffeomorphism type of each internal component of $ f-1(0)$ remains the same after small perturbations of $ f$ inside $ U$, hence $ UP(Hm-1)$ is a locally constant map, therefore by Lemma \\ref {discretelemma} we obtain that for any subset $ AP(Hm-1)$,\\begin{equation}\\exists \\lim _{d\\rightarrow \\infty }\\mathbb {P}\\lbrace X_d\\in U \\text{ and }\\mu (X_d)\\in A\\rbrace =\\mathbb {P}\\lbrace \\mu (X_\\infty )\\in A\\rbrace .\\end{equation}Moreover since in this case $ XdU$ with $ P=1$, for all $ dN$ and the support of $ X$ is the whole $ C(Dm,R)$, we have\\begin{equation}\\exists \\lim _{d\\rightarrow \\infty }\\mathbb {P}\\lbrace \\mu (X_d)\\in A\\rbrace =\\mathbb {P}\\lbrace \\mu (X_\\infty )\\in A\\rbrace >0.\\end{equation}$ Example 27 (Random rational maps) The Kostlan polynomial $P_d^{m, k+1}$ can be used to define random rational maps.",
"In fact, writing $P_{d}^{m, k+1}=(p_0, \\ldots , p_k)$ , then one can consider the map $\\varphi _{d}^{m,k}:\\mathbb {R}\\mathrm {P}^m\\dashrightarrow \\mathbb {R}\\mathrm {P}^k$ defined by: $ \\varphi _d^{m,k}([x_0, \\ldots , x_m])=[p_0(x), \\ldots , p_m(x)].$ (When $m>k$ , with positive probability, this map might not be defined on the whole $\\mathbb {R}\\mathrm {P}^m$ ; when $m\\le k$ with probability one we have that the list $(p_0, \\ldots , p_k)$ has no common zeroes, and we get a well defined map $\\varphi _{d}^{m,k}:\\mathbb {R}\\mathrm {P}^m\\rightarrow \\mathbb {R}\\mathrm {P}^k.$ ) Given a point $x\\in \\mathbb {R}\\mathrm {P}^m$ and a small disk $D_d=D(x, d^{-1/2})$ centered at this point, the behavior of $\\varphi _{d}^{m,k}|_{D_d}$ is captured by the random field $X_d$ defined in (REF ): one can therefore apply Theorem REF and deduce, asymptotic local properties of this map.",
"For example, when $m\\le k$ for any given embedding of the unit disk $q:\\mathbb {D}^m\\hookrightarrow \\mathbb {R}\\mathrm {P}^k$ and for every neighborhood $U$ of $q(\\partial \\mathbb {D}^m)$ there exists a positive constant $c=c(q)>0$ such that for big enough degree $d$ and with probability larger than $c$ the map $X_d=\\varphi _{d}^{m,k}\\circ a_d:\\mathbb {D}^m\\rightarrow \\mathbb {R}\\mathrm {P}^k$ (defined by composing $\\varphi $ with the rescaling diffeomorphism $a_d:\\mathbb {D}^m\\rightarrow D_d$ ) is isotopic to $q$ thorugh an isotopy $\\lbrace q_t:\\mathbb {D}^m\\rightarrow \\mathbb {R}\\mathrm {P}^k\\rbrace _{t\\in I}$ such that $q_t(\\partial \\mathbb {D}^m)\\subset U$ for all $t\\in I$ .",
"The random map $d^{m,k}$ is strictly related to the random map $\\psi ^{m,k}_d\\colon S^m\\rightarrow \\mathbb {R}^k$ : $\\psi ^{m,k}_d(x)=P^{m,k}_d(x),$ which is an easier object to work with.",
"For example the random algebraic variety $\\lbrace d=0\\rbrace $ is the quotient of $\\lbrace \\psi _d=0\\rbrace $ modulo the antipodal map.",
"If we denote by $D_d$ any sequence of disks of radius $d^{-\\frac{1}{2}}$ in the sphere, then $\\psi _d|_{D_d}\\approx X_d$ , so that we can understand the local asymptotic behaviour of $\\psi _d$ using Theorem REF (see Figure REF ).",
"For instance, from point $(7)$ it follows that $\\mathbb {E}\\left\\lbrace b_i\\left(\\left\\lbrace \\psi _d=0\\right\\rbrace \\cap D_d\\right)\\right\\rbrace \\rightarrow \\mathbb {E}\\left\\lbrace b_i\\left(\\left\\lbrace X_\\infty =0\\right\\rbrace \\cap \\mathbb {D}^m\\right)\\right\\rbrace .$ Example 28 (Random knots) Kostlan polynomials offer different possible ways to define a “random knot”.",
"The first is by considering a random rational map: $\\varphi _{d}^{1,3}:\\mathbb {R}\\mathrm {P}^1\\rightarrow \\mathbb {R}\\mathrm {P}^3,$ to which the discussion from Example REF applies.",
"(Observe that this discussion has to do with the local structure of the knot.)",
"Another interesting example of random knots, with a more global flavour, can be obtained as follows.",
"Take the random Kostlan map $X_d: \\mathbb {R}^2\\rightarrow \\mathbb {R}^3$ (as in (REF ) with $m=2$ and $k=3$ ) and restrict it to $S^1=\\partial \\mathbb {D}^m$ to define a random knot: $k_d=X_{d}|_{\\partial \\mathbb {D}^m}:S^1\\rightarrow \\mathbb {R}^3.$ The difference between this model and the previous one is that this is global, in the sense that as $d\\rightarrow \\infty $ we get a limit global model $k_\\infty =X_\\infty |_{\\partial D}:S^1\\rightarrow \\mathbb {R}^3$ .",
"What is interesting for this model is that the Delbruck–Frisch–Wasserman conjecture [5], [9], that a typical random knot is non-trivial, does not hold: in fact $k_\\infty $ charges every knot (included the unknot) with positive probability.",
"Proposition 29 The random map: $k_d=X_d|_{\\partial 2}:S^1\\rightarrow \\mathbb {R}.$ is almost surely a topological embedding (i.e.",
"a knot).",
"Similarly, the random rational map $\\varphi _{d}^{1,3}:\\mathbb {R}\\mathrm {P}^1\\rightarrow \\mathbb {R}\\mathrm {P}^3$ is almost surely an embedding.",
"We prove the statement for $k_d$ , the case of $\\varphi _d^{1, 3}$ is similar.",
"Since $S^1$ is compact, it is enough to prove that $k_d$ is injective with probability one.",
"Let $F_d=\\mathbb {R}[x_0, x_1, x_2]_{(d)}^3$ be the space of triples of homogeeous polynomials of degree $d$ in 3 variables.",
"Recall that $k_d=X_d|_{\\partial \\mathbb {D}^2}$ , where, if $P\\in F_d$ , we have set: $X_d(u)=P\\left(1, \\frac{u}{\\sqrt{d}}\\right),\\quad u=(u_1, u_2)\\in \\mathbb {R}^2.$ Let now $S^1=\\partial \\mathbb {D}^2\\subset \\mathbb {R}^2$ and $\\phi :\\left((S^1\\times S^1)\\backslash \\Delta \\right)\\times F_d\\rightarrow \\mathbb {R}^3$ be the map defined by $\\phi (x,y, P)=P\\left(1, \\frac{x}{\\sqrt{d}}\\right)-P\\left(1, \\frac{y}{\\sqrt{d}}\\right).$ Observe that $\\phi \\pitchfork \\lbrace 0\\rbrace .$ By the parametric transversality theorem we conclude that $\\phi $ is almost surely transversal to $W=\\lbrace 0\\rbrace $ .",
"This imples that, with probability one, the set $\\lbrace x\\ne y\\in S^1\\times S^1\\,|\\, k_d(x)=k_d(y)\\rbrace $ is a codimension-three submanifold of $S^1\\times S^1$ hence it is empty, so that $k_d$ is injective.",
"Theorem REF implies now that the random variable $k_d\\in C^\\infty (S^1, \\mathbb {R}^3)$ converges narrowly to $k_\\infty \\in C^{\\infty }(S^1, \\mathbb {R}^3)$ , which is the restriction to $S^1=\\partial \\mathbb {D}^2$ of $X_\\infty .$ Note that, since the support of $X_\\infty $ is all $C^\\infty (\\mathbb {D}^2, \\mathbb {R}^3)$ , it follows that the support of $k_\\infty $ is all $C^\\infty (S^1, \\mathbb {R}^3)$ and in particular every knot (i.e.",
"isotopy class of topological embeddings $S^1\\rightarrow \\mathbb {R}^3$ , a set with nonempty interior in the $C^\\infty $ topology) has positive probability by Theorem REF .3.",
"Moreover, denoting by $\\gamma _1\\sim \\gamma _2$ two isotopic knots, we have that $\\mathbb {P}\\left(\\partial \\lbrace k_\\infty \\sim \\gamma \\rbrace \\right)\\le \\mathbb {P}\\lbrace k_\\infty \\text{ is not an immersion}\\rbrace =0$ by Theorem REF .4, because the condition of being an immersion is equivalent to that of being transverse to the zero section of $J^1(S^1,\\mathbb {R}^3)\\rightarrow S^1\\times \\mathbb {R}^3$ .",
"Theorem REF .2, thus implies that for every knot $\\gamma :S^1\\rightarrow \\mathbb {R}^3$ we have: $\\lim _{d\\rightarrow \\infty }\\mathbb {P}\\lbrace k_d\\sim \\gamma \\rbrace =\\mathbb {P}\\lbrace k_\\infty \\sim \\gamma \\rbrace >0.$"
],
[
"Kostlan maps",
"In this section we give the definition of a random Kostlan polynomial map $P:\\mathbb {R}^{m+1}\\rightarrow \\mathbb {R}^{k}$ , which is a Gaussian Random Field (GRF) that generalizes the notion of Kostlan polynomial.",
"Definition 15 (Kostlan polynomial maps) Let $d,m,k\\in \\mathbb {N}$ .",
"We define the degree $d$ homogeneous Kostlan random map as the measure on $\\mathbb {R}[x]_{(d)}^k=\\mathbb {R}[x_0,\\dots ,x_{m}]_{(d)}^k$ induced by the gaussian random polynomial: $P_d^{m,k}(x)=\\sum _{\\alpha \\in \\mathbb {N}^{m+1},\\ |\\alpha |=d} \\xi _\\alpha x^\\alpha ,$ where $x^\\alpha =x_0^{\\alpha _0}\\dots x_m^{\\alpha _m}$ and $\\lbrace \\xi _\\alpha \\rbrace $ is a family of independent gaussian random vectors in $\\mathbb {R}^k$ with covariance matrix $K_{\\xi _\\alpha }={d\\atopwithdelims ()\\alpha }\\mathbb {1}_k=\\left(\\frac{d!",
"}{\\alpha _0!\\dots \\alpha _m!",
"}\\right)\\mathbb {1}_k.$ We will call $P_d^{m,k}$ the Kostlan polynomial of type $(d,m,k)$ (we will simply write $P_d=P_d^{m,k}$ when the dimensions are understood).",
"(In other words, a Kostlan polynomial map $P_d^{m, k}$ is given by a list of $k$ independent Kostlan polynomials of degree $d$ in $m+1$ homogeneous variables.)",
"There is a non-homogeneous version of the Kostlan polynomial, which we denote as $p_d(u)=P_d(1,u)=\\sum _{\\beta \\in \\mathbb {N}^{m},\\ |\\beta |\\le d} \\xi _\\beta u^\\beta \\in \\mathcal {G}^{\\infty }(\\mathbb {R}^m,\\mathbb {R}^{k}),$ where $u=(u_1,\\dots , u_m)\\in \\mathbb {R}^m$ and $\\xi _\\beta \\sim N\\left(0,{d\\atopwithdelims ()\\beta }\\mathbb {1}_k\\right)$ are independent.",
"Here we use the notation of [20], where $\\mathcal {G}^{\\infty }(\\mathbb {R}^m,\\mathbb {R}^{k})$ denotes the space of gaussian random field on $\\mathbb {R}^m$ with values in $\\mathbb {R}^k$ which are $\\mathcal {C}^{\\infty }$ .",
"Next Proposition collects some well known facts on the Kostlan measure.",
"Proposition 16 Let $P_d$ be the Kostlan polynomial of type $(d,m,k)$ and $p_d$ be its dehomogenized version, as defined in (REF ).",
"For every $x,y\\in \\mathbb {R}^{m+1}$ : $K_{P_d}(x,y)=\\left(x^Ty\\right)^d\\mathbb {1}_k.$ Moreover, given $R\\in O(m+1)$ and $S\\in O(k)$ and defined the polynomial $\\tilde{P}_d(x)=SP_d(Rx)$ , then $P_d$ and $\\tilde{P}_d$ are equivalentTwo random fields are said to be equivalent if they induce the same probability measure on $\\mathcal {C}^{\\infty }(\\mathbb {R}^m,\\mathbb {R}^{k})$.",
"For every $u, v\\in \\mathbb {R}^n$ $K_{p_d}(u,v)=(1+u^Tv)^d\\mathbb {1}_k.$ Moreover, if $R\\in O(m)$ and $S\\in O(k)$ and defined the polynomial $\\tilde{p}_d(x)=Sp_d(Rx)$ , then $p_d$ and $\\tilde{p}_d$ are equivalent.",
"The proof of this proposition simply follows by computing explicitly the covariance functions and observing that they are invariant under orthogonal change of coordinates in the target and the source.",
"For example, in the case of $P_{d}$ we have: $\\begin{aligned}K_{P_d}(x,y)&= \\mathbb {E}\\lbrace P_d(x)P_d(y)^T\\rbrace = \\\\&= \\sum _{|\\alpha |,|\\alpha ^{\\prime }|=d}\\mathbb {E}\\left\\lbrace \\xi _\\alpha \\xi _{\\alpha ^{\\prime }}^T\\right\\rbrace x^\\alpha y^{\\alpha ^{\\prime }}=\\\\&=\\sum _{|\\alpha |=d}{d\\atopwithdelims ()\\alpha } (x_0 y_0)^{\\alpha _0}\\dots (x_m y_m)^{\\alpha _m}\\mathbb {1}_k=\\\\&= (x_0y_0+\\dots +x_my_m)^d\\mathbb {1}_k,\\end{aligned}$ from which the orthogonal invariance is clear.",
"The case of $p_d$ from the identity: $K_{p_d}(u,v)=K_{P_d}\\left((1,u),(1,v)\\right).$ ."
],
[
"Properties of the rescaled Kostlan",
"The main feature here is the fact that the local model of a Kostlan polynomial has a rescaling limit.",
"The orthogonal invariance is used to prove that the limit does not depend on the point where we center the local model, hence it is enough to work around the point $(1, 0, \\ldots , 0)\\in S^m$ .",
"These considerations lead to introduce the Gaussian Random Field $X_d:\\mathbb {D}^m\\rightarrow \\mathbb {R}^{k}$ (we call it the rescaled Kostlan) defined by: $X_d(u)=P_d^{m,k}\\left(1, \\frac{u_1}{\\sqrt{d}}, \\ldots , \\frac{u_m}{\\sqrt{d}}\\right).$ Next result gives a description of the properties of the rescaled Kostlan polynomial, in particular its convergence in law as a random element of the space of smooth functions, space which, from now, on we will always assume to be endowed with the weak Whitney's topology as in [20].",
"Theorem 17 (Properties of the rescaled Kostlan) Let $X_d:\\mathbb {R}^m\\rightarrow \\mathbb {R}^{k}$ be the Gaussian random field defined in (REF ).",
"(The limit) Given a family of independent gaussian random vectors $\\xi _\\beta \\sim N\\left(0,\\frac{1}{\\beta !",
"}\\mathbb {1}_k\\right)$ , the series $X_\\infty (u)=\\sum _{\\beta \\in \\mathbb {N}^{m}} \\xi _\\beta u^\\beta ,$ is almost surely convergent in $\\mathcal {C}^{\\infty }(\\mathbb {R}^m,\\mathbb {R}^{k})$ to the Gaussian Random Field$X_\\infty $ is indeed a random analytic function, commonly known as the Bargmann-Fock ensemble.",
"$X_\\infty \\in \\mathcal {G}^{\\infty }(\\mathbb {R}^m,\\mathbb {R}^{k})$ .",
"(Convergence) $X_d\\Rightarrow X_\\infty $ in $\\mathcal {G}^{\\infty }(\\mathbb {R}^m,\\mathbb {R}^{k})$ , that is: $\\lim _{d\\rightarrow +\\infty }\\mathbb {E}\\lbrace F(X_d)\\rbrace =\\mathbb {E}\\lbrace F(X_\\infty )\\rbrace $ for any bounded and continuous function $F\\colon \\mathcal {C}^{\\infty }(\\mathbb {R}^m,\\mathbb {R}^{k})\\rightarrow \\mathbb {R}$ .",
"Equivalently, we have $\\mathbb {P}\\lbrace X_\\infty \\in \\emph {int}(A)\\rbrace \\le \\liminf _{d\\rightarrow +\\infty }\\mathbb {P}\\lbrace X_d\\in A\\rbrace \\le \\limsup _{d\\rightarrow +\\infty }\\mathbb {P}\\lbrace X_d\\in A\\rbrace \\le \\mathbb {P}\\lbrace X_d\\in \\overline{A}\\rbrace $ for any Borel subset $A\\subset \\mathcal {C}^{\\infty }(\\mathbb {R}^m,\\mathbb {R}^{k})$ .",
"(Nondegeneracy of the limit) The support of $X_\\infty $ is the whole $\\mathcal {C}^{\\infty }(\\mathbb {R}^m,\\mathbb {R}^{k})$ .",
"In other words, for any non empty open set $U\\subset \\mathcal {C}^{\\infty }(\\mathbb {R}^m,\\mathbb {R}^{k})$ we have that $\\mathbb {P}\\lbrace X_\\infty \\in U\\rbrace >0$ .",
"(Probabilistic Transversality) For $d\\ge r$ and $d=\\infty $ , we have $\\mathrm {supp}(j^r_pX_d)=J_p^r(\\mathbb {R}^m,\\mathbb {R}^k)$ for every $p\\in \\mathbb {R}^m$ and consequently for every submanifold $W\\subset J^r(\\mathbb {R}^m,\\mathbb {R}^k)$ , we have $\\mathbb {P}\\lbrace j^rX_d\\pitchfork W\\rbrace =1.$ (Existence of limit probability) Let $V\\subset J^{r}(\\mathbb {R}^m, \\mathbb {R}^k)$ be an open set whose boundary is a (possibly stratified) submanifoldFor example $V$ could be a semialgebraic set.",
"Then $\\lim _{d\\rightarrow +\\infty }\\mathbb {P}\\lbrace j^r_p X_d\\in V,\\ \\forall p\\in \\mathbb {R}^m\\rbrace = \\mathbb {P}\\lbrace j^r_pX_\\infty (\\mathbb {R}^m) \\in V,\\ \\forall p\\in \\mathbb {R}^m\\rbrace .$ In other words, we have equality in (REF ) for sets of the form $U=\\lbrace f\\colon j^rf\\in V\\rbrace $ .",
"(Kac-Rice densities) Let $W\\subset J^r(\\mathbb {R}^m,\\mathbb {R}^k)$ be a semialgebraic subset of codimension $m$ , such thatIn this paper the symbol $\\mathrel {\\text{\\m@th {#\\crcr \\smash{-}\\crcr \\pitchfork \\crcr }}}$$ stands for ``it is transverse to^{\\prime \\prime }.$ $W\\mathrel {\\text{\\m@th {#\\crcr \\smash{-}\\crcr \\pitchfork \\crcr }}}$ Jpr(Rm,Rk)$ for all $ pM$ (i.e.",
"$ W$ is transverse to fibers of the projection of the jet space).",
"Then for all $ dr$ and for $ d=+$ there exists a locally bounded function $ dWLloc (Rm)$ such that\\footnote {A formula for \\rho _d^W is presented in \\cite {DTGRF1}, as a generalization of the classical Kac-Rice formula.",
"}\\begin{equation}\\mathbb {E}\\#\\lbrace u\\in A\\colon j^r_uX_d\\in W\\rbrace =\\int _A\\rho ^W_d,\\end{equation}for any Borel subset $ ARm$.",
"Moreover $ Wd W$ in $ Lloc$.\\item \\emph {(Limit of Betti numbers)} Let $ WJr(Rm,Rk)$ be any closed semialgebraic subset transverse to fibers.",
"Then:\\begin{equation}\\lim _{d\\rightarrow +\\infty }\\mathbb {E}\\left\\lbrace b_i\\left((j^{r}X_d)^{-1}(W)\\cap \\mathbb {D}^m\\right)\\right\\rbrace =\\mathbb {E}\\left\\lbrace b_i\\left((j^{r}X_\\infty )^{-1}(W)\\cap \\mathbb {D}^m\\right)\\right\\rbrace ,\\end{equation}where $ bi(Z)=Hi(Z,R)$.",
"Moreover, if the codimension of $ W$ is $ l1$, then the r.h.s.",
"in equation (\\ref {eq:localEbetti}) is strictly positive for all $ i=0,..., m-l$.$ The proof uses a combination of results from [20].",
"Let $S_d=\\sum _{|\\beta |\\le d}\\xi _\\beta u^\\beta \\in \\mathcal {G}^{\\infty }(M,\\mathbb {R}^{k})$ .",
"The covariance function of $S_d$ converges in Whitney's weak topology: $K_{S_d}(u,v)=\\sum _{|\\beta |\\le d}\\frac{u^\\beta v^\\beta }{\\beta !",
"}\\mathbb {1}_k\\xrightarrow{} \\exp (u^Tv)\\mathbb {1}_k.$ It follows by [20] that $S_d$ converges in $\\mathcal {G}^{\\infty }(M,\\mathbb {R}^{k})$ , moreover since all the terms in the series are independent we can conclude with the Ito-Nisio It may not be trivial to apply the standard Ito-Nisio theorem, which actually regards convergence of series in a Banach space.",
"See Theorem 36 of [20] for a statement that is directly applicabile to our situation Theorem [16] that indeed the convergence holds almost surely.",
"By [20] it follows from convergence of the covariance functions: $K_{X_d}(u,v)=\\left(1+\\frac{u^Tv}{d}\\right)^d\\mathbb {1}_k \\quad \\xrightarrow{}\\quad K_{X_\\infty }(u,v)=\\exp (u^Tv)\\mathbb {1}_k$ The support of $X_\\infty $ contains the set of polynomial functions $\\mathbb {R}[u]^k$ , which is dense in $\\mathcal {C}^{\\infty }(\\mathbb {R}^m,\\mathbb {R}^{k})$ , hence the thesis follows from [20].",
"Let $d\\ge r$ or $d=+\\infty $ .",
"We have that $\\begin{aligned}\\text{supp}(j_u^rX_d)&=\\lbrace j^r_uf\\colon f\\in \\mathbb {R}[u]^k \\text{ of degree $\\le d$}\\rbrace =\\\\&=\\textrm {span}\\lbrace j^r_uf\\colon f(v)=(v-u)^\\beta \\text{ with $|\\beta |\\le d$}=\\\\&=\\textrm {span}\\lbrace j^r_uf\\colon f(v)=(v-u)^\\beta \\text{ with $|\\beta |\\le r$}\\rbrace =\\\\&=J^r_u(\\mathbb {R}^m,\\mathbb {R}^k).\\end{aligned}$ The fact that $\\mathbb {P}\\lbrace j^rX_d\\mathrel {\\text{\\m@th {#\\crcr \\smash{-}\\crcr \\pitchfork \\crcr }}}$ W}=1$ follows \\cite [Theorem 8]{DTGRF1}.\\item Let $ A={fC(Rm,Rk)jrfV}$.",
"If $ fA$, then $ jrfV$ and there is a point $ uRm$ such that $ jrufV$.",
"Let $ V$ be stratified as $ V=Zi$ with each $ Zi$ a submanifold.",
"If $ jrf $-$$\\pitchfork $ V$ then it means that $ jr f$ is transversal to all the $ Zi$ and there exists one of them which contains $ jruf$ (i.e.",
"the jet of $ f$ intersect $ V$).",
"Therefore the intersection would be transversal \\emph {and nonempty}, and then there exists a small Whitney-neighborhood of $ f$ such that for every $ g$ in this neighborhood $ jrg$ still intersects $ V.$ This means that there is a neighborhood of $ f$ consisting of maps that are not in $ A$, which means $ f$ has a neighborhood contained in $ Ac$.",
"It follows that $ fA$ and consequently $ fA$, which is a contradiction.",
"Therefore we have that\\begin{equation}\\partial A\\subset \\lbrace f\\in \\mathcal {C}^{\\infty }(R^m,\\mathbb {R}^{k})\\colon f \\text{ is not transverse to }\\partial V\\rbrace .\\end{equation}It follows by point $ (4)$ that $ P{XA}=0$, so that we can conclude by points $ (2)$ and $ (3)$.\\item By previous points, we deduce that we can apply the results described in section $ 7$ of \\cite {DTGRF1}.\\item This proof is postponed to Section \\ref {sec:betti}.$ Given a $\\mathcal {C}^{\\infty }$ Gaussian Random Field $X:\\mathbb {R}^m\\rightarrow \\mathbb {R}^k$ , let us denote by $[X]$ the probability measure induced on $\\mathcal {C}^{\\infty }(\\mathbb {R}^m, \\mathbb {R}^k)$ and defined by: $[X](U)=\\mathbb {P}(X\\in U),$ for every $U$ belonging to the Borel $\\sigma -$ algebra relative to the weak Whitney topology, see [15] for details on this topology.",
"Combining Theorem REF with Skorohod Theorem [1] one gets that it is possible to represent $[X_d]$ with equivalent fields $\\tilde{X}_d$ such that $\\tilde{X}_d\\rightarrow \\tilde{X}_\\infty $ almost surely in $\\mathcal {C}^{\\infty }(\\mathbb {R}^m,\\mathbb {R}^{k})$ .",
"This is in fact equivalent to point $(2)$ of Theorem REF .",
"In other words there is a (not unique) choice of the gaussian coefficients of the random polynomials in $(\\ref {eq:Kd})$ , for which the covariances $\\mathbb {E}\\lbrace \\tilde{X}_d\\tilde{X}_{d^{\\prime }}^T\\rbrace $ are such that the sequence converges almost surely.",
"We leave to the reader to check that a possible choice is the following.",
"Let $\\lbrace \\gamma _\\beta \\rbrace _{\\beta \\in \\mathbb {N}^m}$ be a family of i.i.d.",
"gaussian random vectors $\\sim N(0,\\mathbb {1}_k)$ and define for all $d<\\infty $ $\\tilde{X}_d=\\sum _{|\\beta |\\le d}{d\\atopwithdelims ()\\beta }^\\frac{1}{2} \\gamma _\\beta \\left(\\frac{u}{\\sqrt{d}}\\right)^{\\beta }$ and $\\tilde{X}_\\infty =\\sum _{\\beta }{\\left(\\frac{1}{\\beta !",
"}\\right)}^\\frac{1}{2} \\gamma _\\beta u^{\\beta }$ Proposition 18 $\\tilde{X}_d\\rightarrow \\tilde{X}_{\\infty }$ in $\\mathcal {C}^{\\infty }(\\mathbb {R}^m,\\mathbb {R}^{k})$ almost surely.",
"However, we stress the fact that in most situations: when one is interested in the sequence of probability measures $[X_d]$ , it is sufficient to know that such a sequence exists.",
"Limit laws for Betti numbers and the generalized square-root law Figure: The random set S d ={X d =0}⊂mS_d=\\lbrace X_d=0\\rbrace \\subset m is a rescaled version of Z d ∩D(p,d -1/2 )Z_d\\cap D(p, d^{-1/2}), where Z d ={ψ d =0}Z_d=\\lbrace \\psi _d=0\\rbrace .Let $W_0\\subset J^r(\\mathbb {R}^m,\\mathbb {R}^k)$ be a semialgebraic subset.",
"Consider the random set $S_d= \\lbrace p\\in \\mathbb {D}^m\\colon j_p^rX_d\\in W_0\\rbrace ,$ where $X_d\\colon \\mathbb {R}^m\\rightarrow \\mathbb {R}^k$ is the rescaled Kostlan polynomial from Theorem REF (see Figure REF ).",
"We are now in the position of complete the proof of Theorem REF by showing point (7).",
"Let us start by proving the following Lemma.",
"Lemma 19 Let $r$ be the codimension of $W_0$ and suppose $0\\le i\\le m-r\\le m-1$ .",
"Then $\\mathbb {E}\\lbrace b_i(S_\\infty )\\rbrace >0 .$ From Corollary REF we deduce that there exists a function $f\\in \\mathcal {C}^{\\infty }(\\mathbb {D}^m,\\mathbb {R}^{k})$ such that $j^rf\\mathrel {\\text{\\m@th {#\\crcr \\smash{-}\\crcr \\pitchfork \\crcr }}}$ W0$ and $ bi((jrf)-1(W0))0$.Since the condition on $ f$ is open, there is an open neighbourhood $ O$ of $ f$ where $ bi((jrg)-1(W0))=c>0$ for all $ gO$.",
"Thus $ P{bi(S)=c}>0$ because every open set has positive probability for $ X$, by \\ref {thm:Kostlan}.3. therefore $ E{bi(S)}>0$.$ We complete the proof of Theorem REF with the next Proposition.",
"Proposition 20 $\\lim _{d\\rightarrow \\infty }\\mathbb {E}\\lbrace b_i(S_d)\\rbrace =\\mathbb {E}\\lbrace b_i(S_\\infty )\\rbrace $ Let $b_i(S_d)=b_d$ .",
"Define a random field $Y_d=(X_d,x_d)\\colon \\mathbb {R}^m\\rightarrow \\mathbb {R}^{k}\\times \\mathbb {R}$ to be the rescaled Kostlan polynomial of type $(m,k+1)$ .",
"Consider the semialgebraic subset $W^{\\prime }=W\\cap J^r(\\mathbb {D}^m,\\mathbb {R}^k)$ of the real algebraic smooth manifold $J^r(\\mathbb {R}^m,\\mathbb {R}^k)$ and observe that $S_d=(j^rX_d)^{-1}(W^{\\prime })$ is compact.",
"Now Theorem , along with Remark REF , implies the existence of a semialgebraic submanifold $\\hat{W^{\\prime }}\\subset J^{r+1}(\\mathbb {R}^m,\\mathbb {R}^{k+1})$ of codimension $m$ and a constant $C$ , such that $b_d\\le C\\#\\left\\lbrace \\left(j^{r+1}(Y_d)\\right)^{-1}(\\hat{W^{\\prime }})\\right\\rbrace =:N_d$ whenever $j^{r+1}Y_d\\mathrel {\\text{\\m@th {#\\crcr \\smash{-}\\crcr \\pitchfork \\crcr }}}$ W'$ and hence almost surely, because of Theorem \\ref {thm:Kostlan}.4.Since $ YdY$ by \\ref {thm:Kostlan}.2, we see that $ [bd,Nd][b,N]$ and it is not restrictive to assume that $ bi(Zd),Ndbi(Z),N$ almost surely, by Skorokhod^{\\prime }s theorem (see \\cite [Theorem 6.7]{Billingsley}).",
"Moreover $ E{Nd}E{N}$ by Theorem \\ref {thm:Kostlan}.6.Now we can conclude with Fatou^{\\prime }s Lemma as follows\\begin{equation}\\begin{aligned}2\\mathbb {E}\\lbrace N_\\infty \\rbrace &=\\mathbb {E}\\lbrace \\liminf _d N_d+N_\\infty -|b_d-b_\\infty |\\rbrace \\le \\\\&\\le \\liminf _d\\mathbb {E}\\lbrace N_d+N_\\infty -|b_d-b_\\infty |\\rbrace =\\\\&=2\\mathbb {E}\\lbrace N_\\infty \\rbrace -\\limsup _d\\mathbb {E}\\lbrace |b_d-b_\\infty |\\rbrace ,\\end{aligned}\\end{equation}so that\\begin{equation}\\limsup _d\\mathbb {E}\\lbrace |b_d-b_\\infty |\\rbrace \\le 0.\\end{equation}$ In the sequel, with the scope of keeping a light notation, for a given $W\\subset J^{r}(S^m, \\mathbb {R}^k)$ and $\\psi :S^m\\rightarrow \\mathbb {R}^k$ we will denote by $Z_d\\subseteq S^m$ the set $Z_d=j^r\\psi ^{-1}(W).$ If $W$ is of codimension $m$ , then by Theorem REF , $Z_d$ is almost surely a finite set of points and the expectation of this number is given by next result.",
"Theorem 21 (Generalized square-root law for cardinality) Let $W\\subset J^r(S^m,\\mathbb {R}^k)$ be a semialgebraic intrinsic subset of codimension $m$ .",
"Then there is a constant $C_W>0$ such that: $\\mathbb {E}\\lbrace \\#Z_d\\rbrace =C_W d^{\\frac{m}{2}}+ O(d^{\\frac{m}{2}-1}).$ Moreover, the value of $C_W$ can be computed as follows.",
"Let $Y_\\infty =e^{-\\frac{|u|^2}{2}}X_\\infty \\in \\mathcal {G}^{\\infty }(\\mathbb {D}^m,\\mathbb {R}^{k})$ and let $W_0\\subset J^r(\\mathbb {D}^m,\\mathbb {R}^k)$ be the local model for $W$ .",
"Then $C_W=m\\frac{\\mathrm {vol}(S^m)}{\\mathrm {vol}(S^{m-1})} \\mathbb {E}\\#\\lbrace u\\in \\mathbb {D}^m\\colon j^r_uY_{\\infty }\\in W_0\\rbrace .$ In order to prove Theorem REF , we will need a preliminary Lemma, which ensures that we will be in the position of using the generalized Kac-Rice formula of point (6) from Theorem REF .",
"Lemma 22 If $W\\subset J^r(M,\\mathbb {R}^k)$ is intrinsic, then $W$ is transverse to fibers.",
"Since the result is local it is sufficient to prove it in the case when $M=\\mathbb {R}^m$ .",
"In this case we have a canonical identification $J^r(\\mathbb {R}^m,\\mathbb {R}^k)\\cong \\mathbb {R}^m\\times J^r_0(\\mathbb {R}^m,\\mathbb {R}^k), \\qquad j^r_uf\\rightarrow (u,j^r_{h=0}f(u+h))$ Consider the embedding $ i_u \\colon \\mathbb {D}^m\\rightarrow \\mathbb {R}^m$ obtained as the isometric inclusion in the disk with center $u$ and let $\\tau _h\\colon \\mathbb {R}^m\\rightarrow \\mathbb {R}^m$ be the translation map $u\\mapsto u+h$ .",
"Let $(v,j^r_0f)\\in W$ , then $j^r(\\tau _{v-u}\\circ i_u)^*(v,j^r_0f)\\in W_0$ .",
"Given any other point $u$ , we have $\\begin{aligned}j^r(\\tau _{v-u}\\circ i_u)^*(v,j^r_0g)&= j^ri_u^*\\left(j^r(\\tau _{v-u})^*(v,j^r_0g)\\right)=\\\\&= j^ri_u^*\\left(j^r(\\tau _{v-u})^*(v,j^r_{h=0}f(v+h))\\right)=\\\\&= j^ri_u^*\\left(u,j^r_{h=0} f\\circ \\tau _{v-u}(u+h)\\right)=\\\\&= j^ri_u^*\\left(u,j^r_0g\\right)\\in W_0\\end{aligned}$ so that $\\left(u,j^r_0f\\right)\\in W$ .",
"It follows that, calling $\\lbrace 0\\rbrace \\times \\bar{W}= W\\cap J^r_0(\\mathbb {R}^m,\\mathbb {R}^k)$ , $W=\\lbrace (u,j_0^rf)\\colon j^r_0f \\in W_0\\rbrace =\\mathbb {R}^m\\times \\bar{W},$ which is clearly transverse to each fiber $\\lbrace u\\rbrace \\times J^r_0(\\mathbb {R}^m,\\mathbb {R}^k)$ .",
"Figure: A family of shrinking embedding of the unit disk.The reason why we consider intrinsic submanifold is to be able to easily pass to the rescaled Kostlan polynomial $X_d\\in \\mathcal {G}^{\\infty }(\\mathbb {D}^m,\\mathbb {R}^{k})$ by composing $\\psi _d$ with the embedding of the disk $a_d^R$ defined by: $a_d^R\\colon \\mathbb {D}^m\\hookrightarrow S^m,\\quad u\\mapsto \\frac{R\\begin{pmatrix}1 \\\\ \\frac{u}{\\sqrt{d}}\\end{pmatrix}}{\\sqrt{\\left(1+\\frac{|u|^2}{d}\\right)}}$ for any $R\\in O(m+1)$ (see Figure REF ).",
"Let us consider the set function $\\mu _d\\colon \\mathcal {B}(S^m)\\mapsto \\mathbb {R}$ such that $A\\mapsto \\mathbb {E}\\lbrace \\#(j^rX_d)^{-1}(W)\\cap A\\rbrace $ .",
"It is explained in [20] that $\\mu _d$ is a Radon measure on $S^m$ .",
"Because of the invariance under rotation of $P_d$ , by Haar's theorem $\\mu $ needs to be proportional to the volume measure.",
"Therefore for any Borel subset $A\\subset S^m$ we have $\\mathbb {E}\\lbrace \\#Z_d\\rbrace =\\mu _d(S^m)=\\mu _d(A)\\mathrm {vol}(A)^{-1}\\mathrm {vol}(S^m)$ .",
"Define $Y_d\\in \\mathcal {G}^{\\infty }(\\mathbb {D}^m,\\mathbb {R}^{k})$ as $Y_d= \\left(1+\\frac{|u|^2}{d}\\right)^{-\\frac{d}{2}}X_d.$ Observe that $Y_d\\Rightarrow Y_\\infty =\\exp (-\\frac{|u|^2}{2})X_\\infty $ and that $Y_d$ is equivalent to the GRF $\\psi _d\\circ a^R_d$ for any $R\\in O(m+1)$ .",
"Now let $W_0\\subset J^r(\\mathbb {D}^m,\\mathbb {R}^k)$ be the (semialgebraic) model of $W$ .",
"By the same proof of point (7) from Theorem REF , adapted to $Y_d$ , there is a convergent sequence of functions $\\rho _d\\rightarrow \\rho _{+\\infty }\\in L^1(\\mathbb {D}^m)$ such that $\\mathbb {E}\\lbrace \\#(j^rY_d)^{-1}(W_0)\\rbrace =\\int _{\\mathbb {D}^m}\\rho _d\\rightarrow \\int _{\\mathbb {D}^m}\\rho _\\infty = \\mathbb {E}\\lbrace \\#(j^rY_\\infty )^{-1}(W_0)\\rbrace .$ In conclusion we have for $A=a^R_d(\\mathbb {D}^m)$ , as $d\\rightarrow +\\infty $ $\\begin{aligned}\\mathbb {E}\\lbrace \\#Z_d\\rbrace &=\\mu _d(A)\\mathrm {vol}(A)^{-1}\\mathrm {vol}(S^m)\\\\&=\\mathbb {E}\\lbrace \\#(j^rY_d)^{-1}(j^r*(W))\\rbrace \\mathrm {vol}(A)^{-1}\\mathrm {vol}(S^m)\\\\&=\\mathbb {E}\\lbrace \\#(j^rY_d)^{-1}(W_0)\\rbrace \\left(\\frac{\\int _0^{\\pi }|\\sin \\theta |^{m-1}d\\theta }{\\int _0^{\\arctan \\left(d^{-\\frac{1}{2}}\\right)}|\\sin \\theta |^{m-1}d\\theta } \\right)\\\\&=\\mathbb {E}\\lbrace \\#(j^rY_\\infty )^{-1}(W_0)\\rbrace m\\frac{\\mathrm {vol}(S^m)}{\\mathrm {vol}(S^{m-1})}d^{\\frac{m}{2}}+O(d^{\\frac{m}{2}-1}).\\end{aligned}$ Building on the previous results, we can now prove the general case for Betti numbers of a random singualrity.",
"Theorem 23 (Generalized square-root law for Betti numbers) Let $W\\subset J^r(S^m,\\mathbb {R}^k)$ be a closed semialgebraic intrinsic (as defined in Definition REF ) of codimension $1\\le l\\le m$ .",
"Then there are constants $b_W, B_W > 0$ depending only on $W$ such that $b_W d^{\\frac{m}{2}}\\le \\mathbb {E}\\lbrace b_i(Z_d)\\rbrace \\le B_W d^{\\frac{m}{2}}\\quad \\forall i=0,\\dots , m-l$ and $\\mathbb {E}\\lbrace b_i(Z_d)\\rbrace =0$ for all other $i$ .",
"The proof is divided in two parts, first we prove the upper bound, using the square-root law from Theorem REF , then the we use Theorem to deduce the lower bound.",
"1.",
"Assume $W$ is smooth with codimension $s$ .",
"Let us consider $P^{m, k+1}_d|_{S^m}=\\Psi _d=(\\psi _d,\\psi _d^1)\\in \\mathcal {G}^{\\infty }(S^m,\\mathbb {R}^{k+1})$ and Let $\\hat{W}\\subset J^{r+1}(S^m,\\mathbb {R}^{k+1})$ be the intrinsic semialgebraic submanifold coming from Theorem and Remark REF .",
"Thus, using Theorems and REF , we get $\\mathbb {E}\\lbrace b_i(Z_d)\\rbrace \\le N_W\\mathbb {E}\\#\\lbrace (j^{r+1}\\Psi _d)^{-1}(\\hat{W})\\rbrace \\le N_WC_{\\hat{W}}d^{\\frac{m}{2}}.$ 2.",
"Consider the embeddings of the $m$ dimensional disk $a_d^R\\colon \\mathbb {D}^m\\hookrightarrow S^m$ defined in (REF ).",
"For any fixed $d\\in \\mathbb {N}$ , choose a finite subset $F_d\\subset O(m+1)$ such that the images of the corresponding embeddings $\\lbrace a_d^R(\\mathbb {D}^m)\\rbrace _{R\\in F_d}$ are disjoint.",
"Denoting by $Z_d^R$ the union of all connected components of $Z_d$ that are entirely contained in $a_d^R(\\mathbb {D}^m)$ , we have $b_i(Z_d)\\ge \\sum _{R\\in F_d}b_i(Z_d^R).$ Let $W_0\\subset J^r(\\mathbb {D}^m,\\mathbb {R}^k)$ be the model of $W$ as an intrinsic submanifold, it is closed and semialgebraic.",
"By Definition REF , we have $(a_d^R)^{-1}\\left((j^r\\psi _d)^{-1}(W)\\right)= \\left(j^r(\\psi _d\\circ a_d^R)\\right)^{-1}(W_0)\\subset \\mathbb {D}^m.$ Recall that for any $R\\in O(m+1)$ , the GRF $\\psi _d\\circ a^R_d$ is equivalent to $Y_d\\in \\mathcal {G}^{\\infty }(\\mathbb {D}^m,\\mathbb {R}^{k})$ defined in REF , hence taking expectation in Equation (REF ) we find $\\mathbb {E}\\lbrace b_i(Z_d)\\rbrace \\ge \\#(F_d)\\mathbb {E}\\lbrace b_i(S_d)\\rbrace ,$ where $S_d=\\left(j^r(Y_d)\\right)^{-1}(W_0)$ .",
"is easy to see (repeating the same proof) that Theorem REF .7 holds also for the sequence $Y_d\\Rightarrow Y_\\infty $ , so that $\\mathbb {E}\\lbrace S_d\\rbrace \\rightarrow \\mathbb {E}\\lbrace S_\\infty \\rbrace $ .",
"We can assume that $\\mathbb {E}\\lbrace S_\\infty \\rbrace >0$ , because of Lemma REF , thus for big enough $d$ , the numbers $\\mathbb {E}\\lbrace b_i(S_d)\\rbrace $ are bounded below by a constant $C>0$ .",
"Now it remains to estimate the number $\\#(F_d)$ .",
"Notice that $a_d^R(\\mathbb {D}^m)$ is a ball in $S^m$ of a certain radius $\\varepsilon _d$ , hence it is possible to choose $F_d$ to have at least $N_m \\varepsilon _d^{-1}$ elements, for some dimensional constant $N_m>0$ depending only on $m$ .",
"We conclude by observing that $\\varepsilon _d\\approx d^{-\\frac{m}{2}}.$ Appendix 1: Examples of applications of Theorem REF Example 24 (Zero sets of random polynomials) Consider the zero set $Z_d\\subset \\mathbb {R}\\mathbb {P}^m$ of a random Kostlan polynomial $P_d=P_{d}^{m+1,1}$ .",
"Recently Gayet and Welschinger [11] have proved that given a compact hypersurface $Y\\subset \\mathbb {R}^{m}$ there exists a positive constant $c=c(\\mathbb {R}^m, Y)>0$ and $d_0=d_0(\\mathbb {R}^m,Y)\\in \\mathbb {N}$ such that for every point $x\\in \\mathbb {R}\\mathbb {P}^m$ and every large enough degree $d\\ge d_0$ , denoting by $B_d$ any open ball of radius $d^{-1/2}$ in $\\mathbb {R}\\mathbb {P}^m$ , we have: $\\left(B_d, B_d\\cap Z_d\\right)\\cong (\\mathbb {R}^m, Y) $ (i.e.",
"the two pairs are diffeomorphic) with probability larger than $c$ .",
"This result follows from Theorem REF as follows.",
"Let $\\mathbb {D}^m\\subset \\mathbb {R}^m$ be the unit disk, and let $U\\subset \\mathcal {C}^{\\infty }(\\mathbb {D}^m, \\mathbb {R})$ be the open set consisting of functions $g:\\mathbb {D}^m\\rightarrow \\mathbb {R}$ whose zero set is regular (an open $\\mathcal {C}^1$ condition satisfied almost surely by $X_d$ , because of point (4)), entirely contained in the interior of $\\mathbb {D}^m$ (an open $\\mathcal {C}^0$ condition) and such that, denoting by $\\mathbb {B}\\subset \\mathbb {R}^m$ the standard unit open ball, the first two conditions hold and $(\\mathbb {B}, \\mathbb {B}\\cap \\lbrace g=0\\rbrace )$ is diffeomorphic to $(\\mathbb {R}^m, Y)$ (this is another open $\\mathcal {C}^1$ condition).",
"Observe that, using the notation above: $\\left(B_d, B_d\\cap Z_d\\right)\\sim (\\mathbb {B}, \\mathbb {B}\\cap \\lbrace X_d=0\\rbrace ) $ (this is simply because $X_d(u)=P_d(1, ud^{-1/2})$ ).",
"Consequently point (5) of Theorem REF implies that: $\\lim _{d\\rightarrow +\\infty }\\mathbb {P}\\lbrace (\\ref {eq:isotopic})\\rbrace &=\\lim _{d\\rightarrow \\infty } \\mathbb {P}\\left\\lbrace (\\mathbb {B}, \\mathbb {B}\\cap \\lbrace X_d=0\\rbrace )\\sim (\\mathbb {R}^m, Y)\\right\\rbrace \\\\&=\\lim _{d\\rightarrow \\infty } \\mathbb {P}\\left\\lbrace X_d\\in U\\right\\rbrace \\\\&=\\mathbb {P}\\left\\lbrace X_\\infty \\in U \\right\\rbrace >0.$ We stress that, as an extra consequence of Theorem REF , compared to [11] what we get is the existence of the limit of the probability of seeing a given diffeomorphism type.",
"Example 25 (Discrete properties of random maps) Let $[X_d]\\Rightarrow [X_\\infty ]$ be a converging family of gaussian random fields.",
"In this example we introduce a useful tool for studying the asymptotic probability induced by $X_d$ on discrete sets as $d\\rightarrow \\infty $ .",
"The key example that we have in mind is the case when we consider a codimension-one “discriminant” $\\Sigma \\subset \\mathcal {C}^\\infty (S^m, \\mathbb {R}^k)$ which partitions the set of functions into many connected open sets.",
"For instance $\\Sigma $ could be the set of maps for which zero is not a regular value: the complement of $\\Sigma $ consists of countably many open connected sets, each one of which corresponds to a rigid isotopy class of embedding of a smooth codimension-$k$ submanifold $Z\\subset S^m$ .",
"The following Lemma gives a simple technical tool for dealing with these situations.",
"Lemma 26 Let $E$ be a metric space and let $[X_d], [X_\\infty ]$ be a random fields such that $[X_d]\\Rightarrow [X_\\infty ]$ .",
"Let also $Z$ be a discrete space and $\\nu \\colon U\\subset E\\rightarrow Z$ be a continuous function defined on an open subset $U\\subset E$ such thatOf course, $E\\setminus U=\\Sigma $ is what we called “discriminant” in the previous discussion.",
"Note that we do not require that $\\mathbb {P}\\lbrace X_d\\in U\\rbrace =1$ , however it will follow that $\\lim _d \\mathbb {P}\\lbrace X_d\\in U\\rbrace =1$ .",
"$\\mathbb {P}\\lbrace X_\\infty \\in U\\rbrace =1$ .",
"Then, for any $A\\subset Z$ we have: $\\exists \\lim _{d\\rightarrow \\infty }\\mathbb {P}\\left\\lbrace X_d\\in U,\\ \\nu (X_d)\\in A\\right\\rbrace =\\mathbb {P}\\left\\lbrace \\nu (X_\\infty )\\in A\\right\\rbrace .$ Since $\\nu ^{-1}(A)$ is closed and open by continuity of $\\nu $ , it follows that $\\partial \\nu ^{-1}(A)\\subset E\\backslash U$ .",
"Therefore $\\mathbb {P}\\lbrace X_\\infty \\in \\partial \\nu ^{-1}(A)\\rbrace =0$ and by Portmanteau's Theorem [1], we conclude that $\\mathbb {P}\\lbrace X_d\\in \\nu ^{-1}(A)\\rbrace \\xrightarrow[d\\rightarrow \\infty ]{}\\mathbb {P}\\lbrace X_\\infty \\in \\nu ^{-1}(A)\\rbrace , \\quad \\ \\forall \\ A\\subset Z.$ Equation (REF ), in the case of a discrete topological space such as $Z$ , is equivalent to narrow convergence $\\nu (X_d)\\Rightarrow \\nu (X)$ , by Portmanteau's Theorem, because $\\partial A=\\emptyset $ for all subsets $A\\subset Z$ .",
"Note also that to prove narrow convergence of a sequence of measures on $Z$ , it is sufficient to show (REF ) for all $A=\\lbrace z\\rbrace $ , indeed in that case the inequality $\\liminf _{d\\rightarrow \\infty }\\mathbb {P}\\lbrace \\nu _d\\in A\\rbrace =\\liminf _{d\\rightarrow \\infty }\\sum _{z\\in A}\\mathbb {P}\\lbrace \\nu _d=z\\rbrace \\ge \\sum _{z\\in A}\\mathbb {P}\\lbrace \\nu =z\\rbrace =\\mathbb {P}\\lbrace \\nu \\in A\\rbrace $ follows automatically from Fatou's lemma.",
"Following Sarnak and Wigman [23], let us consider one simple application of this Lemma.",
"Let $H_{m-1}$ be the set of diffeomorphism classes of smooth closed connected hypersurfaces of $\\mathbb {R}^{m}$ .",
"Consider $U=\\lbrace f\\in \\mathcal {C}^{\\infty }(\\mathbb {D}^m,\\mathbb {R}^{})\\,\\colon \\, f\\mathrel {\\text{\\m@th {#\\crcr \\smash{-}\\crcr \\pitchfork \\crcr }}}$ 0}$ and let $ (f)$ be the number of connected components of $ f-1(0)$ entirely contained in the interior of $ Dm$.",
"For $ hHm-1$ let $ h(f)$ be the number of those components which are diffeomorphic to $ hRm$.",
"In the spirit of \\cite {SarnakWigman}, we define the probability measure $ (f)P(Hm-1)$ as\\begin{equation}\\mu (f)=\\frac{1}{\\nu (f)}\\sum _{h\\in H_{m-1}}\\nu _h(f)\\delta _h.\\end{equation}Let us consider now the rescaled Kostlan polynomial $ Xd:DmR$ as in Theorem \\ref {thm:Kostlan}.",
"The diffeomorphism type of each internal component of $ f-1(0)$ remains the same after small perturbations of $ f$ inside $ U$, hence $ UP(Hm-1)$ is a locally constant map, therefore by Lemma \\ref {discretelemma} we obtain that for any subset $ AP(Hm-1)$,\\begin{equation}\\exists \\lim _{d\\rightarrow \\infty }\\mathbb {P}\\lbrace X_d\\in U \\text{ and }\\mu (X_d)\\in A\\rbrace =\\mathbb {P}\\lbrace \\mu (X_\\infty )\\in A\\rbrace .\\end{equation}Moreover since in this case $ XdU$ with $ P=1$, for all $ dN$ and the support of $ X$ is the whole $ C(Dm,R)$, we have\\begin{equation}\\exists \\lim _{d\\rightarrow \\infty }\\mathbb {P}\\lbrace \\mu (X_d)\\in A\\rbrace =\\mathbb {P}\\lbrace \\mu (X_\\infty )\\in A\\rbrace >0.\\end{equation}$ Example 27 (Random rational maps) The Kostlan polynomial $P_d^{m, k+1}$ can be used to define random rational maps.",
"In fact, writing $P_{d}^{m, k+1}=(p_0, \\ldots , p_k)$ , then one can consider the map $\\varphi _{d}^{m,k}:\\mathbb {R}\\mathrm {P}^m\\dashrightarrow \\mathbb {R}\\mathrm {P}^k$ defined by: $ \\varphi _d^{m,k}([x_0, \\ldots , x_m])=[p_0(x), \\ldots , p_m(x)].$ (When $m>k$ , with positive probability, this map might not be defined on the whole $\\mathbb {R}\\mathrm {P}^m$ ; when $m\\le k$ with probability one we have that the list $(p_0, \\ldots , p_k)$ has no common zeroes, and we get a well defined map $\\varphi _{d}^{m,k}:\\mathbb {R}\\mathrm {P}^m\\rightarrow \\mathbb {R}\\mathrm {P}^k.$ ) Given a point $x\\in \\mathbb {R}\\mathrm {P}^m$ and a small disk $D_d=D(x, d^{-1/2})$ centered at this point, the behavior of $\\varphi _{d}^{m,k}|_{D_d}$ is captured by the random field $X_d$ defined in (REF ): one can therefore apply Theorem REF and deduce, asymptotic local properties of this map.",
"For example, when $m\\le k$ for any given embedding of the unit disk $q:\\mathbb {D}^m\\hookrightarrow \\mathbb {R}\\mathrm {P}^k$ and for every neighborhood $U$ of $q(\\partial \\mathbb {D}^m)$ there exists a positive constant $c=c(q)>0$ such that for big enough degree $d$ and with probability larger than $c$ the map $X_d=\\varphi _{d}^{m,k}\\circ a_d:\\mathbb {D}^m\\rightarrow \\mathbb {R}\\mathrm {P}^k$ (defined by composing $\\varphi $ with the rescaling diffeomorphism $a_d:\\mathbb {D}^m\\rightarrow D_d$ ) is isotopic to $q$ thorugh an isotopy $\\lbrace q_t:\\mathbb {D}^m\\rightarrow \\mathbb {R}\\mathrm {P}^k\\rbrace _{t\\in I}$ such that $q_t(\\partial \\mathbb {D}^m)\\subset U$ for all $t\\in I$ .",
"The random map $d^{m,k}$ is strictly related to the random map $\\psi ^{m,k}_d\\colon S^m\\rightarrow \\mathbb {R}^k$ : $\\psi ^{m,k}_d(x)=P^{m,k}_d(x),$ which is an easier object to work with.",
"For example the random algebraic variety $\\lbrace d=0\\rbrace $ is the quotient of $\\lbrace \\psi _d=0\\rbrace $ modulo the antipodal map.",
"If we denote by $D_d$ any sequence of disks of radius $d^{-\\frac{1}{2}}$ in the sphere, then $\\psi _d|_{D_d}\\approx X_d$ , so that we can understand the local asymptotic behaviour of $\\psi _d$ using Theorem REF (see Figure REF ).",
"For instance, from point $(7)$ it follows that $\\mathbb {E}\\left\\lbrace b_i\\left(\\left\\lbrace \\psi _d=0\\right\\rbrace \\cap D_d\\right)\\right\\rbrace \\rightarrow \\mathbb {E}\\left\\lbrace b_i\\left(\\left\\lbrace X_\\infty =0\\right\\rbrace \\cap \\mathbb {D}^m\\right)\\right\\rbrace .$ Example 28 (Random knots) Kostlan polynomials offer different possible ways to define a “random knot”.",
"The first is by considering a random rational map: $\\varphi _{d}^{1,3}:\\mathbb {R}\\mathrm {P}^1\\rightarrow \\mathbb {R}\\mathrm {P}^3,$ to which the discussion from Example REF applies.",
"(Observe that this discussion has to do with the local structure of the knot.)",
"Another interesting example of random knots, with a more global flavour, can be obtained as follows.",
"Take the random Kostlan map $X_d: \\mathbb {R}^2\\rightarrow \\mathbb {R}^3$ (as in (REF ) with $m=2$ and $k=3$ ) and restrict it to $S^1=\\partial \\mathbb {D}^m$ to define a random knot: $k_d=X_{d}|_{\\partial \\mathbb {D}^m}:S^1\\rightarrow \\mathbb {R}^3.$ The difference between this model and the previous one is that this is global, in the sense that as $d\\rightarrow \\infty $ we get a limit global model $k_\\infty =X_\\infty |_{\\partial D}:S^1\\rightarrow \\mathbb {R}^3$ .",
"What is interesting for this model is that the Delbruck–Frisch–Wasserman conjecture [5], [9], that a typical random knot is non-trivial, does not hold: in fact $k_\\infty $ charges every knot (included the unknot) with positive probability.",
"Proposition 29 The random map: $k_d=X_d|_{\\partial 2}:S^1\\rightarrow \\mathbb {R}.$ is almost surely a topological embedding (i.e.",
"a knot).",
"Similarly, the random rational map $\\varphi _{d}^{1,3}:\\mathbb {R}\\mathrm {P}^1\\rightarrow \\mathbb {R}\\mathrm {P}^3$ is almost surely an embedding.",
"We prove the statement for $k_d$ , the case of $\\varphi _d^{1, 3}$ is similar.",
"Since $S^1$ is compact, it is enough to prove that $k_d$ is injective with probability one.",
"Let $F_d=\\mathbb {R}[x_0, x_1, x_2]_{(d)}^3$ be the space of triples of homogeeous polynomials of degree $d$ in 3 variables.",
"Recall that $k_d=X_d|_{\\partial \\mathbb {D}^2}$ , where, if $P\\in F_d$ , we have set: $X_d(u)=P\\left(1, \\frac{u}{\\sqrt{d}}\\right),\\quad u=(u_1, u_2)\\in \\mathbb {R}^2.$ Let now $S^1=\\partial \\mathbb {D}^2\\subset \\mathbb {R}^2$ and $\\phi :\\left((S^1\\times S^1)\\backslash \\Delta \\right)\\times F_d\\rightarrow \\mathbb {R}^3$ be the map defined by $\\phi (x,y, P)=P\\left(1, \\frac{x}{\\sqrt{d}}\\right)-P\\left(1, \\frac{y}{\\sqrt{d}}\\right).$ Observe that $\\phi \\pitchfork \\lbrace 0\\rbrace .$ By the parametric transversality theorem we conclude that $\\phi $ is almost surely transversal to $W=\\lbrace 0\\rbrace $ .",
"This imples that, with probability one, the set $\\lbrace x\\ne y\\in S^1\\times S^1\\,|\\, k_d(x)=k_d(y)\\rbrace $ is a codimension-three submanifold of $S^1\\times S^1$ hence it is empty, so that $k_d$ is injective.",
"Theorem REF implies now that the random variable $k_d\\in C^\\infty (S^1, \\mathbb {R}^3)$ converges narrowly to $k_\\infty \\in C^{\\infty }(S^1, \\mathbb {R}^3)$ , which is the restriction to $S^1=\\partial \\mathbb {D}^2$ of $X_\\infty .$ Note that, since the support of $X_\\infty $ is all $C^\\infty (\\mathbb {D}^2, \\mathbb {R}^3)$ , it follows that the support of $k_\\infty $ is all $C^\\infty (S^1, \\mathbb {R}^3)$ and in particular every knot (i.e.",
"isotopy class of topological embeddings $S^1\\rightarrow \\mathbb {R}^3$ , a set with nonempty interior in the $C^\\infty $ topology) has positive probability by Theorem REF .3.",
"Moreover, denoting by $\\gamma _1\\sim \\gamma _2$ two isotopic knots, we have that $\\mathbb {P}\\left(\\partial \\lbrace k_\\infty \\sim \\gamma \\rbrace \\right)\\le \\mathbb {P}\\lbrace k_\\infty \\text{ is not an immersion}\\rbrace =0$ by Theorem REF .4, because the condition of being an immersion is equivalent to that of being transverse to the zero section of $J^1(S^1,\\mathbb {R}^3)\\rightarrow S^1\\times \\mathbb {R}^3$ .",
"Theorem REF .2, thus implies that for every knot $\\gamma :S^1\\rightarrow \\mathbb {R}^3$ we have: $\\lim _{d\\rightarrow \\infty }\\mathbb {P}\\lbrace k_d\\sim \\gamma \\rbrace =\\mathbb {P}\\lbrace k_\\infty \\sim \\gamma \\rbrace >0.$"
],
[
"Appendix 1: Examples of applications of Theorem ",
"Example 24 (Zero sets of random polynomials) Consider the zero set $Z_d\\subset \\mathbb {R}\\mathbb {P}^m$ of a random Kostlan polynomial $P_d=P_{d}^{m+1,1}$ .",
"Recently Gayet and Welschinger [11] have proved that given a compact hypersurface $Y\\subset \\mathbb {R}^{m}$ there exists a positive constant $c=c(\\mathbb {R}^m, Y)>0$ and $d_0=d_0(\\mathbb {R}^m,Y)\\in \\mathbb {N}$ such that for every point $x\\in \\mathbb {R}\\mathbb {P}^m$ and every large enough degree $d\\ge d_0$ , denoting by $B_d$ any open ball of radius $d^{-1/2}$ in $\\mathbb {R}\\mathbb {P}^m$ , we have: $\\left(B_d, B_d\\cap Z_d\\right)\\cong (\\mathbb {R}^m, Y) $ (i.e.",
"the two pairs are diffeomorphic) with probability larger than $c$ .",
"This result follows from Theorem REF as follows.",
"Let $\\mathbb {D}^m\\subset \\mathbb {R}^m$ be the unit disk, and let $U\\subset \\mathcal {C}^{\\infty }(\\mathbb {D}^m, \\mathbb {R})$ be the open set consisting of functions $g:\\mathbb {D}^m\\rightarrow \\mathbb {R}$ whose zero set is regular (an open $\\mathcal {C}^1$ condition satisfied almost surely by $X_d$ , because of point (4)), entirely contained in the interior of $\\mathbb {D}^m$ (an open $\\mathcal {C}^0$ condition) and such that, denoting by $\\mathbb {B}\\subset \\mathbb {R}^m$ the standard unit open ball, the first two conditions hold and $(\\mathbb {B}, \\mathbb {B}\\cap \\lbrace g=0\\rbrace )$ is diffeomorphic to $(\\mathbb {R}^m, Y)$ (this is another open $\\mathcal {C}^1$ condition).",
"Observe that, using the notation above: $\\left(B_d, B_d\\cap Z_d\\right)\\sim (\\mathbb {B}, \\mathbb {B}\\cap \\lbrace X_d=0\\rbrace ) $ (this is simply because $X_d(u)=P_d(1, ud^{-1/2})$ ).",
"Consequently point (5) of Theorem REF implies that: $\\lim _{d\\rightarrow +\\infty }\\mathbb {P}\\lbrace (\\ref {eq:isotopic})\\rbrace &=\\lim _{d\\rightarrow \\infty } \\mathbb {P}\\left\\lbrace (\\mathbb {B}, \\mathbb {B}\\cap \\lbrace X_d=0\\rbrace )\\sim (\\mathbb {R}^m, Y)\\right\\rbrace \\\\&=\\lim _{d\\rightarrow \\infty } \\mathbb {P}\\left\\lbrace X_d\\in U\\right\\rbrace \\\\&=\\mathbb {P}\\left\\lbrace X_\\infty \\in U \\right\\rbrace >0.$ We stress that, as an extra consequence of Theorem REF , compared to [11] what we get is the existence of the limit of the probability of seeing a given diffeomorphism type.",
"Example 25 (Discrete properties of random maps) Let $[X_d]\\Rightarrow [X_\\infty ]$ be a converging family of gaussian random fields.",
"In this example we introduce a useful tool for studying the asymptotic probability induced by $X_d$ on discrete sets as $d\\rightarrow \\infty $ .",
"The key example that we have in mind is the case when we consider a codimension-one “discriminant” $\\Sigma \\subset \\mathcal {C}^\\infty (S^m, \\mathbb {R}^k)$ which partitions the set of functions into many connected open sets.",
"For instance $\\Sigma $ could be the set of maps for which zero is not a regular value: the complement of $\\Sigma $ consists of countably many open connected sets, each one of which corresponds to a rigid isotopy class of embedding of a smooth codimension-$k$ submanifold $Z\\subset S^m$ .",
"The following Lemma gives a simple technical tool for dealing with these situations.",
"Lemma 26 Let $E$ be a metric space and let $[X_d], [X_\\infty ]$ be a random fields such that $[X_d]\\Rightarrow [X_\\infty ]$ .",
"Let also $Z$ be a discrete space and $\\nu \\colon U\\subset E\\rightarrow Z$ be a continuous function defined on an open subset $U\\subset E$ such thatOf course, $E\\setminus U=\\Sigma $ is what we called “discriminant” in the previous discussion.",
"Note that we do not require that $\\mathbb {P}\\lbrace X_d\\in U\\rbrace =1$ , however it will follow that $\\lim _d \\mathbb {P}\\lbrace X_d\\in U\\rbrace =1$ .",
"$\\mathbb {P}\\lbrace X_\\infty \\in U\\rbrace =1$ .",
"Then, for any $A\\subset Z$ we have: $\\exists \\lim _{d\\rightarrow \\infty }\\mathbb {P}\\left\\lbrace X_d\\in U,\\ \\nu (X_d)\\in A\\right\\rbrace =\\mathbb {P}\\left\\lbrace \\nu (X_\\infty )\\in A\\right\\rbrace .$ Since $\\nu ^{-1}(A)$ is closed and open by continuity of $\\nu $ , it follows that $\\partial \\nu ^{-1}(A)\\subset E\\backslash U$ .",
"Therefore $\\mathbb {P}\\lbrace X_\\infty \\in \\partial \\nu ^{-1}(A)\\rbrace =0$ and by Portmanteau's Theorem [1], we conclude that $\\mathbb {P}\\lbrace X_d\\in \\nu ^{-1}(A)\\rbrace \\xrightarrow[d\\rightarrow \\infty ]{}\\mathbb {P}\\lbrace X_\\infty \\in \\nu ^{-1}(A)\\rbrace , \\quad \\ \\forall \\ A\\subset Z.$ Equation (REF ), in the case of a discrete topological space such as $Z$ , is equivalent to narrow convergence $\\nu (X_d)\\Rightarrow \\nu (X)$ , by Portmanteau's Theorem, because $\\partial A=\\emptyset $ for all subsets $A\\subset Z$ .",
"Note also that to prove narrow convergence of a sequence of measures on $Z$ , it is sufficient to show (REF ) for all $A=\\lbrace z\\rbrace $ , indeed in that case the inequality $\\liminf _{d\\rightarrow \\infty }\\mathbb {P}\\lbrace \\nu _d\\in A\\rbrace =\\liminf _{d\\rightarrow \\infty }\\sum _{z\\in A}\\mathbb {P}\\lbrace \\nu _d=z\\rbrace \\ge \\sum _{z\\in A}\\mathbb {P}\\lbrace \\nu =z\\rbrace =\\mathbb {P}\\lbrace \\nu \\in A\\rbrace $ follows automatically from Fatou's lemma.",
"Following Sarnak and Wigman [23], let us consider one simple application of this Lemma.",
"Let $H_{m-1}$ be the set of diffeomorphism classes of smooth closed connected hypersurfaces of $\\mathbb {R}^{m}$ .",
"Consider $U=\\lbrace f\\in \\mathcal {C}^{\\infty }(\\mathbb {D}^m,\\mathbb {R}^{})\\,\\colon \\, f\\mathrel {\\text{\\m@th {#\\crcr \\smash{-}\\crcr \\pitchfork \\crcr }}}$ 0}$ and let $ (f)$ be the number of connected components of $ f-1(0)$ entirely contained in the interior of $ Dm$.",
"For $ hHm-1$ let $ h(f)$ be the number of those components which are diffeomorphic to $ hRm$.",
"In the spirit of \\cite {SarnakWigman}, we define the probability measure $ (f)P(Hm-1)$ as\\begin{equation}\\mu (f)=\\frac{1}{\\nu (f)}\\sum _{h\\in H_{m-1}}\\nu _h(f)\\delta _h.\\end{equation}Let us consider now the rescaled Kostlan polynomial $ Xd:DmR$ as in Theorem \\ref {thm:Kostlan}.",
"The diffeomorphism type of each internal component of $ f-1(0)$ remains the same after small perturbations of $ f$ inside $ U$, hence $ UP(Hm-1)$ is a locally constant map, therefore by Lemma \\ref {discretelemma} we obtain that for any subset $ AP(Hm-1)$,\\begin{equation}\\exists \\lim _{d\\rightarrow \\infty }\\mathbb {P}\\lbrace X_d\\in U \\text{ and }\\mu (X_d)\\in A\\rbrace =\\mathbb {P}\\lbrace \\mu (X_\\infty )\\in A\\rbrace .\\end{equation}Moreover since in this case $ XdU$ with $ P=1$, for all $ dN$ and the support of $ X$ is the whole $ C(Dm,R)$, we have\\begin{equation}\\exists \\lim _{d\\rightarrow \\infty }\\mathbb {P}\\lbrace \\mu (X_d)\\in A\\rbrace =\\mathbb {P}\\lbrace \\mu (X_\\infty )\\in A\\rbrace >0.\\end{equation}$ Example 27 (Random rational maps) The Kostlan polynomial $P_d^{m, k+1}$ can be used to define random rational maps.",
"In fact, writing $P_{d}^{m, k+1}=(p_0, \\ldots , p_k)$ , then one can consider the map $\\varphi _{d}^{m,k}:\\mathbb {R}\\mathrm {P}^m\\dashrightarrow \\mathbb {R}\\mathrm {P}^k$ defined by: $ \\varphi _d^{m,k}([x_0, \\ldots , x_m])=[p_0(x), \\ldots , p_m(x)].$ (When $m>k$ , with positive probability, this map might not be defined on the whole $\\mathbb {R}\\mathrm {P}^m$ ; when $m\\le k$ with probability one we have that the list $(p_0, \\ldots , p_k)$ has no common zeroes, and we get a well defined map $\\varphi _{d}^{m,k}:\\mathbb {R}\\mathrm {P}^m\\rightarrow \\mathbb {R}\\mathrm {P}^k.$ ) Given a point $x\\in \\mathbb {R}\\mathrm {P}^m$ and a small disk $D_d=D(x, d^{-1/2})$ centered at this point, the behavior of $\\varphi _{d}^{m,k}|_{D_d}$ is captured by the random field $X_d$ defined in (REF ): one can therefore apply Theorem REF and deduce, asymptotic local properties of this map.",
"For example, when $m\\le k$ for any given embedding of the unit disk $q:\\mathbb {D}^m\\hookrightarrow \\mathbb {R}\\mathrm {P}^k$ and for every neighborhood $U$ of $q(\\partial \\mathbb {D}^m)$ there exists a positive constant $c=c(q)>0$ such that for big enough degree $d$ and with probability larger than $c$ the map $X_d=\\varphi _{d}^{m,k}\\circ a_d:\\mathbb {D}^m\\rightarrow \\mathbb {R}\\mathrm {P}^k$ (defined by composing $\\varphi $ with the rescaling diffeomorphism $a_d:\\mathbb {D}^m\\rightarrow D_d$ ) is isotopic to $q$ thorugh an isotopy $\\lbrace q_t:\\mathbb {D}^m\\rightarrow \\mathbb {R}\\mathrm {P}^k\\rbrace _{t\\in I}$ such that $q_t(\\partial \\mathbb {D}^m)\\subset U$ for all $t\\in I$ .",
"The random map $d^{m,k}$ is strictly related to the random map $\\psi ^{m,k}_d\\colon S^m\\rightarrow \\mathbb {R}^k$ : $\\psi ^{m,k}_d(x)=P^{m,k}_d(x),$ which is an easier object to work with.",
"For example the random algebraic variety $\\lbrace d=0\\rbrace $ is the quotient of $\\lbrace \\psi _d=0\\rbrace $ modulo the antipodal map.",
"If we denote by $D_d$ any sequence of disks of radius $d^{-\\frac{1}{2}}$ in the sphere, then $\\psi _d|_{D_d}\\approx X_d$ , so that we can understand the local asymptotic behaviour of $\\psi _d$ using Theorem REF (see Figure REF ).",
"For instance, from point $(7)$ it follows that $\\mathbb {E}\\left\\lbrace b_i\\left(\\left\\lbrace \\psi _d=0\\right\\rbrace \\cap D_d\\right)\\right\\rbrace \\rightarrow \\mathbb {E}\\left\\lbrace b_i\\left(\\left\\lbrace X_\\infty =0\\right\\rbrace \\cap \\mathbb {D}^m\\right)\\right\\rbrace .$ Example 28 (Random knots) Kostlan polynomials offer different possible ways to define a “random knot”.",
"The first is by considering a random rational map: $\\varphi _{d}^{1,3}:\\mathbb {R}\\mathrm {P}^1\\rightarrow \\mathbb {R}\\mathrm {P}^3,$ to which the discussion from Example REF applies.",
"(Observe that this discussion has to do with the local structure of the knot.)",
"Another interesting example of random knots, with a more global flavour, can be obtained as follows.",
"Take the random Kostlan map $X_d: \\mathbb {R}^2\\rightarrow \\mathbb {R}^3$ (as in (REF ) with $m=2$ and $k=3$ ) and restrict it to $S^1=\\partial \\mathbb {D}^m$ to define a random knot: $k_d=X_{d}|_{\\partial \\mathbb {D}^m}:S^1\\rightarrow \\mathbb {R}^3.$ The difference between this model and the previous one is that this is global, in the sense that as $d\\rightarrow \\infty $ we get a limit global model $k_\\infty =X_\\infty |_{\\partial D}:S^1\\rightarrow \\mathbb {R}^3$ .",
"What is interesting for this model is that the Delbruck–Frisch–Wasserman conjecture [5], [9], that a typical random knot is non-trivial, does not hold: in fact $k_\\infty $ charges every knot (included the unknot) with positive probability.",
"Proposition 29 The random map: $k_d=X_d|_{\\partial 2}:S^1\\rightarrow \\mathbb {R}.$ is almost surely a topological embedding (i.e.",
"a knot).",
"Similarly, the random rational map $\\varphi _{d}^{1,3}:\\mathbb {R}\\mathrm {P}^1\\rightarrow \\mathbb {R}\\mathrm {P}^3$ is almost surely an embedding.",
"We prove the statement for $k_d$ , the case of $\\varphi _d^{1, 3}$ is similar.",
"Since $S^1$ is compact, it is enough to prove that $k_d$ is injective with probability one.",
"Let $F_d=\\mathbb {R}[x_0, x_1, x_2]_{(d)}^3$ be the space of triples of homogeeous polynomials of degree $d$ in 3 variables.",
"Recall that $k_d=X_d|_{\\partial \\mathbb {D}^2}$ , where, if $P\\in F_d$ , we have set: $X_d(u)=P\\left(1, \\frac{u}{\\sqrt{d}}\\right),\\quad u=(u_1, u_2)\\in \\mathbb {R}^2.$ Let now $S^1=\\partial \\mathbb {D}^2\\subset \\mathbb {R}^2$ and $\\phi :\\left((S^1\\times S^1)\\backslash \\Delta \\right)\\times F_d\\rightarrow \\mathbb {R}^3$ be the map defined by $\\phi (x,y, P)=P\\left(1, \\frac{x}{\\sqrt{d}}\\right)-P\\left(1, \\frac{y}{\\sqrt{d}}\\right).$ Observe that $\\phi \\pitchfork \\lbrace 0\\rbrace .$ By the parametric transversality theorem we conclude that $\\phi $ is almost surely transversal to $W=\\lbrace 0\\rbrace $ .",
"This imples that, with probability one, the set $\\lbrace x\\ne y\\in S^1\\times S^1\\,|\\, k_d(x)=k_d(y)\\rbrace $ is a codimension-three submanifold of $S^1\\times S^1$ hence it is empty, so that $k_d$ is injective.",
"Theorem REF implies now that the random variable $k_d\\in C^\\infty (S^1, \\mathbb {R}^3)$ converges narrowly to $k_\\infty \\in C^{\\infty }(S^1, \\mathbb {R}^3)$ , which is the restriction to $S^1=\\partial \\mathbb {D}^2$ of $X_\\infty .$ Note that, since the support of $X_\\infty $ is all $C^\\infty (\\mathbb {D}^2, \\mathbb {R}^3)$ , it follows that the support of $k_\\infty $ is all $C^\\infty (S^1, \\mathbb {R}^3)$ and in particular every knot (i.e.",
"isotopy class of topological embeddings $S^1\\rightarrow \\mathbb {R}^3$ , a set with nonempty interior in the $C^\\infty $ topology) has positive probability by Theorem REF .3.",
"Moreover, denoting by $\\gamma _1\\sim \\gamma _2$ two isotopic knots, we have that $\\mathbb {P}\\left(\\partial \\lbrace k_\\infty \\sim \\gamma \\rbrace \\right)\\le \\mathbb {P}\\lbrace k_\\infty \\text{ is not an immersion}\\rbrace =0$ by Theorem REF .4, because the condition of being an immersion is equivalent to that of being transverse to the zero section of $J^1(S^1,\\mathbb {R}^3)\\rightarrow S^1\\times \\mathbb {R}^3$ .",
"Theorem REF .2, thus implies that for every knot $\\gamma :S^1\\rightarrow \\mathbb {R}^3$ we have: $\\lim _{d\\rightarrow \\infty }\\mathbb {P}\\lbrace k_d\\sim \\gamma \\rbrace =\\mathbb {P}\\lbrace k_\\infty \\sim \\gamma \\rbrace >0.$"
]
] | 1906.04444 |
[
[
"Central charge and topological invariant of Calabi-Yau manifolds"
],
[
"Abstract F-theory, as a 12-dimensional theory that is a contender of the Theory of Everything, should be compactified into elliptically fibered threefolds or fourfolds of Calabi-Yau.",
"Such manifolds have an elliptic curve as a fiber, and their bases may have singularities.",
"We considered orbifold as simplest non-flat construction.",
"Blow up modes of orbifold singularities can be considered as coordinates of complexified Kahler moduli space.",
"Quiver diagrams are used for discribing D-branes near orbifold point.",
"In this case it is possible to calculate Euler character defined through $\\mbox{Ext}^i(A,B)$ groups and coherent sheaves $A, B$ over projective space, which are representations of orbifold space after blowing up procedure.",
"These fractional sheaves are characterized by $Q_0$, $Q_2$ and $Q_4$ Ramon-Ramon charges, which have special type, calculated for $C^3/Z_3$ case.",
"BPS central charge for $C^3/Z_3$ orbifold is calculated through Ramon-Ramon charges and Picard-Fuchs periods."
],
[
"Introduction",
"Modern high energy theoretical physics is a unified theory of all particles and all interactions.",
"It is Theory of Everything, because it gives a universal description of the processes occurring on modern accelerators, and processes in the Universe.",
"Theory of everything (abbr.",
"TOE) - hypothetical combined physical and mathematical theory describing all known fundamental interactions.",
"This theory unifies all four fundamental interactions in nature.",
"The main problem of building TOE is that quantum mechanics and general theory of relativity have different applications.",
"Quantum mechanics is mainly used to describe the microworld, and general relativity is applicable to the macro world.",
"But it does not mean that such theory cannot be constructed.",
"Modern physics requires from TOE the unification of four fundamental interactions: $\\bullet $ gravitational interaction; $\\bullet $ electromagnetic interaction; $\\bullet $ strong nuclear interaction; $\\bullet $ weak nuclear interaction.",
"The first step towards this was the unification of the electromagnetic and weak interactions in the theory of electro-weak interaction created by in 1967 by Stephen Weinberg, Sheldon Glashow and Abdus Salam.",
"In 1973, the theory of strong interaction was proposed.",
"The main candidate as TOE is F-theory, which operates with a large number of dimensions.",
"Thanks to the ideas of Kaluza and Klein it became possible to create theories operating with large extra dimensions.",
"The use of extra dimensions prompted the answer to the question about why the effect of gravity appears much weaker than other types of interactions.",
"The generally accepted answer is that gravity exists in extra dimensions, therefore its effect on observable measurements weakened.",
"F-theory is a string twelve-dimensional theory defined on energy scale of about 10$^{19}$ GeV [1].",
"F-theory compactification leads to a new type of vacuum, so to study supersymmetry we must compactify the F-theory on Calabi-Yau manifolds.",
"Since there are many Calabi-Yau manifolds, we are dealing with a large number of new models implemented in low-energy approximation.",
"Studying the singularities of Calabi manifold determines the physical characteristics of topological solitonic states which plays the role of particles in high energy physics.",
"Compactification of F-theory on different Calabi-Yau manifolds allows to calculate topological invariants.",
"Let us consider in more detail the compactification of F-theory on threefolds Calabi Yau."
],
[
"Calabi-Yau threefold compactification",
"Twelve-dimensional space describing space-time and internal degrees of freedom, we compactify as follows: $R^6 \\times X^6 \\ ,$ where $R^6$ - six-dimensional space-time, on which acts conformal group SO(4, 2), and $X^6 $ - threefold, which is three-dimensional Calabi Yau complex manifold [2]."
],
[
"Toric representation of threefolds",
"Let's consider weighted projective space defined as follows: $P^4_{\\omega _1,\\ldots ,\\omega _5 }=P^4/Z_{\\omega _1}\\times \\ldots \\times Z_{\\omega _5}\\ , $ where $P^4$ - four-dimensional projective space, $Z_{\\omega _i}$ - cyclic group of order $\\omega _i$ .",
"On weighted projective space $P^4_{\\omega _1,\\ldots ,\\omega _5 }$ is defined polynomial $W(\\varphi _1, \\ldots , \\varphi _5)$ , called superpotential which satisfies the homogeneity condition $W(x^{\\omega _1}\\varphi _1, \\ldots , x^{\\omega _5}\\varphi _5)=x^d W(\\varphi _1, \\ldots , \\varphi _5)\\ ,$ where $d=\\sum \\limits _{i=1}^5\\omega _i$ , $\\varphi _1, \\ldots , \\varphi _5 \\in P^4_{\\omega _1,\\ldots ,\\omega _5 } $ .",
"The set of points $p\\in P^4_{\\omega _1,\\ldots ,\\omega _5 } $ , satisfying the condition $W(p)=0$ forms Calabi-Yau threefold $X_d(\\omega _1, \\ldots , \\omega _5)$ .",
"The simplest examples of toric varieties [3] are projective spaces.",
"Let's consider $P^{2}$ defined as follows: $P^{2} = \\frac{C^{3}/{0}}{C/{0}}, $ where dividing by $ C/{0}$ means identification of points connected by equivalence relation $(x, y, z)\\sim (\\lambda x, \\lambda y, \\lambda z) $ $\\lambda \\in C/{0}, $ $ x, y, z $ are homogeneous coordinates.",
"Elliptic curve in $P^{2}$ is described by the Weierstrass equation $y^2z = x^3 + axz^2 + bz^3.",
"$ In general Calabi-Yau manifold can be described by Weierstrass form $y^2=x^3+xf+g,$ which describes an elliptic fibration (parametrized by $(y, x)$ ) over the base, where $f, g$ - functions defined on the base.",
"In some divisors $D_i$ the layer are degenerated.",
"Such divisors are zeros of discriminant $\\Delta =4f^3+27g^2.$ The singularities of Calabi-Yau manifold are singularities of its elliptic fibrations.",
"These singularities are coded in polynomials $f, g$ and their type determines the gauge group and matter content of compactified F-theory.",
"The classification of singularities of elliptic fibrations was given by Kodaira and presented table 1.",
"Table 1.",
"Kodaira classification of singularities of elliptic fibrations Table: NO_CAPTION The classification of elliptic fibers is presented in Figure 1.",
"Figure: NO_CAPTION"
],
[
"Ramon-Ramon charges",
"One of the most interesting problems of modern high-energy physics is the calculation of topological invariants - analogs of high-energy observables in physics.",
"In this aspect, symmetries and the use of the apparatus of algebraic geometry play an indispensable role.",
"We considered orbifold as simplest non-flat constructions.",
"For D3-branes on such internal space $C^n/\\Gamma $ the representations are characterized by gauge groups $G=\\oplus _iU(N_i)$ .",
"In this case the superpotential is of N=4 $U(N)$ super Yang-Mills, $W_{N=4}=\\mbox{tr}X^1[X^2,X^3],$ where $X^i$ are chiral matter fields in production of fundamental representation $V^i\\cong C^{N_i}$ of the group $U(N_i)$ .",
"Blow up modes of orbifold singularities can be considered as coordinates of complexified Kahler moduli space.",
"Quiver diagrams are used for discribing D-branes near orbifold point.",
"Figure: NO_CAPTIONIn this case it is possible to calculate Euler character defined as $\\chi (A,B)=\\sum _i(-1)^i\\mbox{dimExt}^i(A,B),$ where $\\mbox{Ext}^0(A,B)\\equiv \\mbox{Hom}(A,B)$ and $A, B$ are coherent sheaves over projective space, $P^N$ (general case), which are representations of orbifold space after blowing up procedure.",
"Since we will deal with orbifolds $C^3/Z_3$ in the future, it is necessary to emphasize the following equivalence relation $(x_1x_2x_3)\\sim (e^{2i\\pi /3}x_1, e^{2i\\pi /3}x_2, e^{2i\\pi /3}x_3), \\ e^{2i\\pi /3}\\in Z_3$ Orbifold is not a manifold, since it has singularities at a point $(0, 0, 0)$ .",
"Blowing up the singularity of the orbifold $C^3/Z_3$ , we obtain a sheave ${\\cal {O}}_{P^2}(-3)$ with which we will work further.",
"In particular, the Euler matrix for sheaves ${\\cal {O}}_{P^2}$ , ${\\cal {O}}_{P^2}(1)$ , ${\\cal {O}}_{P^2}(2)$ over projective space, $P^2$ looks like $ \\chi ({\\cal {O}}_{P^2}(1), {\\cal {O}}_{P^2}(2)) = \\left( \\begin{array}{ccc}1 & 3& 6 \\\\0& 1 &3 \\\\0 & 0 & 1 \\end{array} \\right).$ Transposed matrix has the form $ \\left( \\begin{array}{ccc}1 & 3& 6 \\\\0& 1 &3 \\\\0 & 0 & 1 \\end{array} \\right)\\Rightarrow \\left( \\begin{array}{ccc}1 & 0& 0 \\\\3& 1 &0 \\\\6& 3 & 1 \\end{array} \\right).$ The rows of matrices are RR-charges characterizing the sheaves: ${\\cal {O}}_{P^2}(-3)=(6\\ 3\\ 1), {\\cal {O}}_{P^2}(-2)=(3\\ 1\\ 0),{\\cal {O}}_{P^2}(-1)= (1\\ 0\\ 0),$ ${\\cal {O}}_{P^2}=(0\\ 0\\ 1), {\\cal {O}}_{P^2}(1)=(0\\ 1\\ 3),{\\cal {O}}_{P^2}(2)= (1\\ 3\\ 6),$ which can be written through large volume charges $(Q_4, Q_2, Q_0)$ : $Q_4=n_1-2n_2+n_3, \\ \\ Q_2=-n_1+n_2, \\ \\ Q_0=\\frac{n_1+n_2}{2}$ included in the definition of the Chern character $ch(n_1n_2n_3)$ $ch(n_1n_2n_3)=Q_4+Q_2w+Q_0w^2,$ where $w$ - Wu number.",
"Then sheaves (1), (2) describe fractional branes [4] ${\\cal {O}}_{P^2}(-3)=(1\\ -3\\ \\ \\frac{9}{2}), {\\cal {O}}_{P^2}(-2)=(1\\ -2\\ \\ \\frac{4}{2}), {\\cal {O}}_{P^2}(-1)= (1\\ -1\\ \\ \\frac{1}{2}), $ ${\\cal {O}}_{P^2}=(1\\ 0\\ 0), {\\cal {O}}_{P^2}(1)=(1\\ 1\\ \\ \\frac{1}{2}),{\\cal {O}}_{P^2}(2)= (1\\ 2\\ \\ \\frac{4}{2}),$ General formula for Chern character of bundle $E$ : $ch(E)=k+c_1(E)+\\frac{1}{2}(c_1(E)^2-2c_2(E))+\\ldots ,$ where $c_i(E)$ are the Chern classes of line bundle $E$ .",
"In our case of a line bundle ${\\cal {O}}_{P^2}(k)$ , only the first Chern class is nonzero, and therefore the formula for the Chern character is following $ch(E)=k+c_1(E)+\\frac{1}{2}c_1^2$ As $1+c_1(E)+\\ldots + c_n(E)=\\prod \\limits _{i=1}^{n}(1+w_i),$ then $c_1(E)=w_1=w$ and formula (3) can be rewritten $ch(n_1n_2n_3)=Q_4+Q_2w+Q_0w^2,$ where Ramon-Ramon charges $(n_1n_2n_3)$ characterize the bundle $E$ , the rank of the line bundle $Q_4=1, Q_2=c_1$ by the fundamental cycle, $Q_0=\\frac{c_1^2}{2}$ from a comparison of formulas (3) and (4).",
"Thus fractional sheaves ${\\cal {O}}_{P^2}(k)$ are characterized by $Q_0, Q_2, Q_4$ Ramon-Ramon charges, which have special type, calculated for $C^3/Z_3$ case."
],
[
"BPS central charge",
"As we are interested in the moduli spaces, we give them a visual definition.",
"Suppose we have a cube curve with the parameter $\\lambda $ $y^2-x(x-1)(x-\\lambda )=0$ As $\\lambda $ - the variable value, then the equation (5) describes a continuous family of cubic curves.",
"The parameter spaces describing continuous families of manifolds are called moduli spaces.",
"We form $\\frac{dx}{y}$ the form where $y$ are determined from equation (5).",
"It turns out that periods $\\pi _1(\\lambda ),\\ pi_2(\\lambda )$ : $\\pi _1(\\lambda )=2\\int \\limits _0^1\\frac{dx}{[x(x-1)(x-\\lambda )]^{1/2}}, \\ \\ \\pi _2(\\lambda )=2\\int \\limits _1^{\\lambda }\\frac{dx}{[x(x-1)(x-\\lambda )]^{1/2}}$ satisfy Picard-Fuchs equation $\\frac{1}{4}\\pi _i+(2\\lambda -1)\\frac{d\\pi _i}{d\\lambda }+\\lambda (\\lambda -1)\\frac{d^2\\pi _i}{d\\lambda ^2}=0\\ .$ Periods that satisfy equation (6) describe the moduli space of a cubic curves.",
"For the moduli space of a line bundle ${\\cal {O}}_{P^2}(-3)$ , Picard-Fuchs equation and its solutions are written as $\\Biggl (z\\frac{d}{dz}\\Biggr )^3+27z\\Biggl (z\\frac{d}{dz}\\Biggr )\\Biggl (z\\frac{d}{dz}+\\frac{1}{3}\\Biggr )\\Biggl (z\\frac{d}{dz}+\\frac{2}{3}\\Biggr )\\Pi =0$ $\\Pi _0=1,$ $\\Pi _1=\\frac{1}{2i\\pi }log\\ z=t=w_0,$ $\\Pi _2=t^2-t-\\frac{1}{6}=-\\frac{2}{3}(w_0-w_1) \\ .$ The BPS central charge [5] associated with the D-brane over $C^3/Z_3$ with Ramon-Ramon-charge $n=(n_1n_2n_3)$ and with the Picard-Fuchs period $\\Pi =(\\Pi _0\\Pi _1\\Pi _2)$ is given by the formula $Z(n)=n\\cdot \\Pi $ The central charge associated with the sheave ${\\cal {O}}_{P^2}(k)$ is given by the formula $Z({\\cal {O}}_{P^2}(k))=-(k+\\frac{1}{3}w_0)+\\frac{1}{3}w_1+\\frac{1}{2}k^2 +\\frac{1}{2}k+\\frac{1}{3}\\ .$"
],
[
"Conclusion",
"In the framework of F-theory we prsented the ideology of extra dimensional spaces.",
"It was stressed the exceptional role of topological invariants for Calabi-Yau manifolds.",
"We have considered the special type of the space of extra dimensions - orbifold $C^/Z_3$ .",
"Using blowing up procedure of singularity we calculated special type of topological invariant - Ramon-Ramon central charges of fractional sheaves, in which is encoded the information about the structure of line bundles.",
"Consideration of moduli space of orbifold leads us to the equation of Picard-Fuchs periods, through which we calculated central charge for sheave ${\\cal {O}}_{P^2}(k)$ .",
"This topological invariant is of importance because of information of stability of D-branes as bound states of fractional branes or sheaves presented in this paper."
]
] | 1906.04247 |
[
[
"Continuous Time Analysis of Momentum Methods"
],
[
"Abstract Gradient descent-based optimization methods underpin the parameter training of neural networks, and hence comprise a significant component in the impressive test results found in a number of applications.",
"Introducing stochasticity is key to their success in practical problems, and there is some understanding of the role of stochastic gradient descent in this context.",
"Momentum modifications of gradient descent such as Polyak's Heavy Ball method (HB) and Nesterov's method of accelerated gradients (NAG), are also widely adopted.",
"In this work our focus is on understanding the role of momentum in the training of neural networks, concentrating on the common situation in which the momentum contribution is fixed at each step of the algorithm.",
"To expose the ideas simply we work in the deterministic setting.",
"Our approach is to derive continuous time approximations of the discrete algorithms; these continuous time approximations provide insights into the mechanisms at play within the discrete algorithms.",
"We prove three such approximations.",
"Firstly we show that standard implementations of fixed momentum methods approximate a time-rescaled gradient descent flow, asymptotically as the learning rate shrinks to zero; this result does not distinguish momentum methods from pure gradient descent, in the limit of vanishing learning rate.",
"We then proceed to prove two results aimed at understanding the observed practical advantages of fixed momentum methods over gradient descent.",
"We achieve this by proving approximations to continuous time limits in which the small but fixed learning rate appears as a parameter.",
"Furthermore in a third result we show that the momentum methods admit an exponentially attractive invariant manifold on which the dynamics reduces, approximately, to a gradient flow with respect to a modified loss function."
],
[
"Background and Literature Review",
"At the core of many machine learning tasks is solution of the optimization problem $\\mathop {\\mathrm {arg\\,min\\,\\,}}_{u \\in \\mathbb {R}^d} \\Phi (u)$ where $\\Phi : \\mathbb {R}^d\\rightarrow \\mathbb {R}$ is an objective (or loss) function that is, in general, non-convex and differentiable.",
"Finding global minima of such objective functions is an important and challenging task with a long history, one in which the use of stochasticity has played a prominent role for many decades, with papers in the early development of machine learning [12], [42], together with concomitant theoretical analyses for both discrete [2] and continuous problems [22], [23].",
"Recent successes in the training of deep neural networks have built on this older work, leveraging the enormous computer power now available, together with empirical experience about good design choices for the architecture of the networks; reviews may be found in [14], [25].",
"Gradient descent plays a prominent conceptual role in many algorithms, following from the observation that the equation $\\frac{du}{dt}=-\\nabla \\Phi (u)$ will decrease $\\Phi $ along trajectories.",
"The most widely adopted methods use stochastic gradient decent (SGD), a concept introduced in [37]; the basic idea is to use gradient decent steps based on a noisy approximation to the gradient of $\\Phi $ .",
"Building on deep work in the convex optimization literature, momentum-based modifications to stochastic gradient decent have also become widely used in optimization.",
"Most notable amongst these momentum-based methods are the Heavy Ball Method (HB), due to [34], and Nesterov's method of accelerated gradients (NAG) [30].",
"To the best of our knowledge, the first application of HB to neural network training appears in [38].",
"More recent work, such as [44], has even argued for the indispensability of such momentum based methods for the field of deep learning.",
"From these two basic variants on gradient decent, there have come a plethora of adaptive methods, incorporating momentum-like ideas, such as Adam [20], Adagrad [7], and RMSProp [45].",
"There is no consensus on which method performs best and results vary based on application.",
"The recent work of [49] argues that the rudimentary, non-adaptive schemes SGD, HB, and NAG result in solutions with the greatest generalization performance for supervised learning applications with deep neural network models.",
"There is a natural physical analogy for momentum methods, namely that they relate to a damped second order Hamiltonian dynamic with potential $\\Phi $ : $m\\frac{d^2u}{dt^2}+\\gamma (t)\\frac{du}{dt}+\\nabla \\Phi (u)=0.$ This perspective goes back to Polyak's original work [34], [35] and was further expanded on in [36], although no proof was given.",
"For NAG, the work of [43] proves that the method approximates a damped Hamiltonian system of precisely this form, with a time-dependent damping coefficient.",
"The analysis in [43] holds when the momentum factor is chosen according to the rule $\\lambda = \\lambda _n = \\frac{n}{n + 3},$ where $n$ is the iteration count; this choice was proposed in the original work of [30] and results in a choice of $\\lambda $ which is asymptotic to 1.",
"In the setting where $\\Phi $ is $\\mu $ -strongly convex, it is proposed in [31] that the momentum factor is fixed and chosen close to 1; specifically it is proposed that $\\lambda = \\frac{1 - \\sqrt{\\mu h}}{1+ \\sqrt{\\mu h}}$ where $h > 0$ is the time-step (learning rate).",
"In [48], a limiting equation for both HB and NAG of the form $\\ddot{u} + 2\\sqrt{\\mu }\\dot{u} + \\nabla \\Phi (u) = 0$ is derived under the assumption that $\\lambda $ is fixed with respect to iteration number $n$ , and dependent on the time-step $h$ as specified in (REF ); convergence is obtained to order $\\mathcal {O}(h^{1/2})$ .",
"Using insight from this limiting equation it is possible to choose the optimal value of $\\mu $ to maximize the convergence rate in the neighborhood of a locally strongly convex objective function.",
"Further related work is developed in [40] where separate limiting equations for HB and NAG are derived both in the cases of $\\lambda $ given by (REF ) and (REF ), obtaining convergence to order $\\mathcal {O}(h^{3/2})$ .",
"Much work has also gone into analyzing these methods in the discrete setting, without appeal to the continuous time limits, see [19], [26], as well as in the stochastic setting, establishing how the effect on the generalization error, for example, [11], [28], [50].",
"In this paper, however, our focus is on the use of continuous time limits as a methodology to explain optimization algorithms.",
"In many machine learning applications, especially for deep learning, NAG and HB are often used with a constant momentum factor $\\lambda $ that is chosen independently of the iteration count $n$ (contrary to (REF )) and independently of the learning rate $h$ (contrary to (REF )).",
"In fact, popular books on the subject such as [14] introduce the methods in this way, and popular articles, such as [16] to name one of many, simply state the value of the constant momentum factor used in their experiments.",
"Widely used deep learning libraries such as Tensorflow [1] and PyTorch [32] implement the methods with a fixed choice of momentum factor.",
"Momentum based methods used in this way, with fixed momentum, have not been carefully analyzed.",
"We will undertake such an analysis, using ideas from numerical analysis, and in particular the concept of modified equations [15], [5] and from the theory of attractive invariant manifolds [18], [47]; both ideas are explained in the text [41].",
"It is noteworthy that the high resolution ODE approximation described in [40] may be viewed as a rediscovery of the method of modified equations.",
"We emphasize the fact that our work is not at odds with any previous analyses of these methods, rather, we consider a setting which is widely adopted in deep learning applications and has not been subjected to continuous time analysis to date.",
"Since publication of this article in [21], we became aware of related, and earlier, work by [8].",
"Farazmand starts from the Bregman Lagrangian introduced in [46] and uses ideas from geometric singular perturbation theory to derive an invariant manifold.",
"The work leads to a more general description of the invariant manifold than the one given by our equation (REF ).",
"Farazmand's work was published in [9]."
],
[
"Our Contribution",
"We study momentum-based optimization algorithms for the minimization task (REF ), with learning rate independent momentum, fixed at every iteration step, focusing on deterministic methods for clarity of exposition.",
"Our approach is to derive continuous time approximations of the discrete algorithms; these continuous time approximations provide insights into the mechanisms at play within the discrete algorithms.",
"We prove three such approximations.",
"The first shows that the asymptotic limit of the momentum methods, as learning rate approaches zero, is simply a rescaled gradient flow (REF ).",
"The second two approximations include small perturbations to the rescaled gradient flow, on the order of the learning rate, and give insight into the behavior of momentum methods when implemented with momentum and fixed learning rate.",
"Through these approximation theorems, and accompanying numerical experiments, we make the following contributions to the understanding of momentum methods as often implemented within machine learning: We show that momentum-based methods with a fixed momentum factor, satisfy, in the continuous-time limit obtained by sending the learning rate to zero, a rescaled version of the gradient flow equation (REF ).",
"We show that such methods also approximate a damped Hamiltonian system of the form (REF ), with small mass $m$ (on the order of the learning rate) and constant damping $\\gamma (t)=\\gamma $ ; this approximation has the same order of accuracy as the approximation of the rescaled equation (REF ) but provides a better qualitative understanding of the fixed learning rate momentum algorithm in its transient phase.",
"We also show that, for the approximate Hamiltonian system, the dynamics admit an exponentially attractive invariant manifold, locally representable as a graph mapping co-ordinates to their velocities.",
"The map generating this graph describes a gradient flow in a potential which is a small (on the order of the learning rate) perturbation of $\\Phi $ – see (REF ); the correction to the potential is convexifying, does not change the global minimum, and provides insight into the fixed learning rate momentum algorithm beyond its initial transient phase.",
"We provide numerical experiments which illustrate the foregoing considerations, for simple linear test problems, and for the MNIST digit classification problem; in the latter case we consider SGD and thereby demonstrate that the conclusions of our theory have relevance for understanding the stochastic setting as well.",
"Taken together our results are interesting because they demonstrate that the popular belief that (fixed) momentum methods resemble the dynamics induced by (REF ) is misleading.",
"Whilst it is true, the mass in the approximating equation is small and as a consequence understanding the dynamics as gradient flows (REF ), with modified potential, is more instructive.",
"In fact, in the first application of HB to neural networks described in [38], the authors state that [their] experience has been that [one] get[s] the same solutions by setting [the momentum factor to zero] and reducing the size of [the learning rate].",
"However our theorems should not be understood to imply that there is no practical difference between momentum methods (with fixed learning rate) and SGD.",
"There is indeed a practical difference as has been demonstrated in numerous papers throughout the machine learning literature, and our experiments in Section further confirm this.",
"We show that while these methods have the same transient dynamics, they are approximated differently.",
"Our results demonstrate that, although momentum methods behave like a gradient descent algorithm, asymptotically, this algorithm has a modified potential.",
"Furthermore, although this modified potential (REF ) is on the order of the learning rate, the fact that the learning rate is often chosen as large as possible, constrained by numerical stability, means that the correction to the potential may be significant.",
"Our results may be interpreted as indicating that the practical success of momentum methods stems from the fact that they provide a more stable discretization to (REF ) than the forward Euler method employed in SGD.",
"The damped Hamiltonian dynamic (REF ), as well the modified potential, give insight into how this manifests.",
"Our work gives further theoretical justification for the exploration of the use of different numerical integrators for the purposes of optimization such as those performed in [39], [3], [51].",
"While our analysis is confined to the non-stochastic case to simplify the exposition, the results will, with some care, extend to the stochastic setting using ideas from averaging and homogenization [33] as well as continuum analyses of SGD as in [27], [10]; indeed, in the stochastic setting, sharp uniform in time error estimates are to be expected for empirical averages [29], [6].",
"To demonstrate that our analysis is indeed relevant in the stochastic setting, we train a deep autoencoder with mini-batching (stochastic) and verify that our convergence results still hold.",
"The details of this experiment are given in section .",
"Furthermore we also confine our analysis to fixed learning rate, and impose global bounds on the relevant derivatives of $\\Phi $ ; this further simplifies the exposition of the key ideas, but is not essential to them; with considerably more analysis the ideas exposed in this paper will transfer to adaptive time-stepping methods and much less restrictive classes of $\\Phi $ .",
"The paper is organized as follows.",
"Section introduces the optimization procedures and states the convergence result to a rescaled gradient flow.",
"In section we derive the modified, second-order equation and state convergence of the schemes to this equation.",
"Section asserts the existence of an attractive invariant manifold, demonstrating that it results in a gradient flow with respect to a small perturbation of $\\Phi $ .",
"In section , we train a deep autoencoder, showing that our results hold in a stochastic setting with Assumption REF violated.",
"We conclude in section .",
"All proofs of theorems are given in the appendices so that the ideas of the theorems can be presented clearly within the main body of the text."
],
[
"Notation",
"We use $|\\cdot |$ to denote the Euclidean norm on $\\mathbb {R}^d.$ We define $f : \\mathbb {R}^d\\rightarrow \\mathbb {R}^d$ by $f(u) - \\nabla \\Phi (u)$ for any $u \\in \\mathbb {R}^d$ .",
"Given parameter $\\lambda \\in [0,1)$ we define $\\bar{\\lambda } (1-\\lambda )^{-1}$ .",
"For two Banach spaces $A,B$ , and $A_0$ a subset in $A$ , we denote by $C^k(A_0;B)$ the set of $k$ -times continuously differentiable functions with domain $A_0$ and range $B$ .",
"For a function $u \\in C^k(A_0;B)$ , we let $D^ju$ denote its $j$ -th (total) Fréchet derivative for $j=1,\\dots ,k$ .",
"For a function ${u \\in C^k([0,\\infty ), \\mathbb {R}^d)}$ , we denote its derivatives by $\\frac{du}{dt}, \\frac{d^2 u}{dt^2},$ etc.",
"or equivalently by $\\dot{u}, \\ddot{u},$ etc.",
"To simplify our proofs, we make the following assumption about the objective function.",
"Assumption Suppose $\\Phi \\in C^3(\\mathbb {R}^d;\\mathbb {R})$ with uniformly bounded derivatives.",
"Namely, there exist constants $B_0,B_1,B_2 > 0$ such that $\\Vert D^{j-1}f\\Vert = \\Vert D^j \\Phi \\Vert \\le B_{j-1}$ for $j=1,2,3$ where $\\Vert \\cdot \\Vert $ denotes any appropriate operator norm.",
"We again stress that this assumption is not key to developing the ideas in this work, but is rather a simplification used to make our results global.",
"Without Assumption REF , and no further assumption on $\\Phi $ such as convexity, one could only hope to give local results i.e.",
"in the neighborhood of a critical point of $\\Phi $ .",
"Such analysis could indeed be carried out (see for example [4]), but we choose not to do so here for the sake of clarity of exposition.",
"In section , we give a practical example where this assumption is violated and yet the behavior is as predicted by our theory.",
"Finally we observe that the nomenclature “learning rate” is now prevalent in machine learning, and so we use it in this paper; it refers to the object commonly referred to as “time-step” in the field of numerical analysis."
],
[
"Momentum Methods and Convergence to Gradient Flow",
"In subsection REF we state Theorem REF concerning the convergence of a class of momentum methods to a rescaled gradient flow.",
"Subsection REF demonstrates that the HB and NAG methods are special cases of our general class of momentum methods, and gives intuition for proof of Theorem REF ; the proof itself is given in Appendix A. Subsection REF contains a numerical illustration of Theorem REF ."
],
[
"Main Result",
"The standard Euler discretization of (REF ) gives the discrete time optimization scheme $\\mathsf {u}_{n+1} = \\mathsf {u}_n + h f(\\mathsf {u}_n), \\quad n=0,1,2,\\dots \\,.$ Implementation of this scheme requires an initial guess $\\mathsf {u}_0 \\in \\mathbb {R}^d$ .",
"For simplicity we consider a fixed learning rate $h > 0$ .",
"Equation (REF ) has a unique solution $u \\in C^3([0,\\infty );\\mathbb {R}^d)$ under Assumption REF and for $u_n=u(nh)$ $\\sup _{0 \\le nh \\le T} |\\mathsf {u}_n-u_n| \\le C(T)h;$ see [41], for example.",
"In this section we consider a general class of momentum methods for the minimization task (REF ) which can be written in the form, for some $a \\ge 0$ and $\\lambda \\in (0,1)$ , $\\begin{split}\\mathsf {u}_{n+1} &= \\mathsf {u}_n + \\lambda (\\mathsf {u}_n - \\mathsf {u}_{n-1}) + hf(\\mathsf {u}_n + a(\\mathsf {u}_n - \\mathsf {u}_{n-1})), \\quad n=0,1,2,\\dots \\,, \\\\\\mathsf {u}_1 &= \\mathsf {u}_0 + hf(\\mathsf {u}_0)\\,.\\end{split}$ Again, implementation of this scheme requires an an initial guess $\\mathsf {u}_0 \\in \\mathbb {R}^d$ .",
"The parameter choice $a=0$ gives HB and $a = \\lambda $ gives NAG.",
"In Appendix A we prove the following: Suppose Assumption REF holds and let $u \\in C^3([0,\\infty );\\mathbb {R}^d)$ be the solution to $\\begin{split}&\\frac{d u}{dt} = -(1-\\lambda )^{-1}\\nabla \\Phi (u) \\\\&u(0) = \\mathsf {u}_0\\end{split}$ with $\\lambda \\in (0,1)$ .",
"For $n=0,1,2,\\dots $ let $\\mathsf {u}_n$ be the sequence given by (REF ) and define $u_n u(nh)$ .",
"Then for any $T \\ge 0$ , there is a constant $C = C(T) > 0$ such that $\\sup _{0 \\le nh \\le T} |u_n - \\mathsf {u}_n| \\le Ch.$ Note that (REF ) is simply a sped-up version of (REF ): if $v$ solves (REF ) and $w$ solves (REF ) then $v(t) = w((1-\\lambda )t)$ for any $t \\in [0,\\infty )$ .",
"This demonstrates that introduction of momentum in the form used within both HB and NAG results in numerical methods that do not differ substantially from gradient descent."
],
[
"Link to HB and NAG",
"The HB method is usually written as a two-step scheme taking the form ([44]) $\\mathsf {v}_{n+1} &= \\lambda \\mathsf {v}_n + h f(\\mathsf {u}_n) \\\\\\mathsf {u}_{n+1} &= \\mathsf {u}_n + \\mathsf {v}_{n+1}$ with $\\mathsf {v}_0 = 0$ , $\\lambda \\in (0,1)$ the momentum factor, and $h > 0$ the learning rate.",
"We can re-write this update as $\\mathsf {u}_{n+1} &= \\mathsf {u}_n + \\lambda \\mathsf {v}_n + hf(\\mathsf {u}_n) \\\\&= \\mathsf {u}_n + \\lambda (\\mathsf {u}_n - \\mathsf {u}_{n-1}) + h f(\\mathsf {u}_n)$ hence the method reads $\\begin{split}\\mathsf {u}_{n+1} &= \\mathsf {u}_n + \\lambda (\\mathsf {u}_n - \\mathsf {u}_{n-1}) + h f(\\mathsf {u}_n) \\\\\\mathsf {u}_1 &= \\mathsf {u}_0 + hf(\\mathsf {u}_0).\\end{split}$ Similarly NAG is usually written as ([44]) $\\mathsf {v}_{n+1} &= \\lambda \\mathsf {v}_n + h f(\\mathsf {u}_n + \\lambda \\mathsf {v}_n) \\\\\\mathsf {u}_{n+1} &= \\mathsf {u}_n + \\mathsf {v}_{n+1}$ with $\\mathsf {v}_0 = 0$ .",
"Define $\\mathsf {w}_n \\mathsf {u}_n + \\lambda \\mathsf {v}_n$ then $\\mathsf {w}_{n+1} &= \\mathsf {u}_{n+1} + \\lambda \\mathsf {v}_{n+1} \\\\&= \\mathsf {u}_{n+1} + \\lambda (\\mathsf {u}_{n+1} - \\mathsf {u}_n)$ and $\\mathsf {u}_{n+1} &= \\mathsf {u}_n + \\lambda \\mathsf {v}_n + hf(\\mathsf {u}_n + \\lambda \\mathsf {v}_n) \\\\&= \\mathsf {u}_n + (\\mathsf {w}_n - \\mathsf {u}_n) + h f(\\mathsf {w}_n) \\\\&= \\mathsf {w}_n + h f(\\mathsf {w}_n).$ Hence the method may be written as $\\begin{split}\\mathsf {u}_{n+1} &= \\mathsf {u}_n + \\lambda (\\mathsf {u}_n - \\mathsf {u}_{n-1}) + hf(\\mathsf {u}_n + \\lambda (\\mathsf {u}_n - \\mathsf {u}_{n-1})) \\\\\\mathsf {u}_1 &= \\mathsf {u}_0 + hf(\\mathsf {u}_0).\\end{split}$ It is clear that (REF ) and (REF ) are special cases of (REF ) with $a=0$ giving HB and $a = \\lambda $ giving NAG.",
"To intuitively understand Theorem REF , re-write (REF ) as $\\frac{du}{dt} - \\lambda \\frac{du}{dt} = f(u).$ If we discretize the $du/dt$ term using forward differences and the $-\\lambda du/dt$ term using backward differences, we obtain $\\frac{u(t+h) - u(t)}{h} - \\lambda \\frac{u(t) - u(t-h)}{h} \\approx f(u(t)) \\approx f \\left( u(t) + ha \\frac{u(t) - u(t-h)}{h} \\right)$ with the second approximate equality coming from the Taylor expansion of $f$ .",
"This can be rearranged as $u(t+h) \\approx u(t) + \\lambda (u(t) - u(t-h)) + h f(u(t) + a(u(t)-u(t-h)))$ which has the form of (REF ) with the identification $\\mathsf {u}_n \\approx u(nh)$ .",
"Figure: Comparison of trajectories for HB and NAG with the gradient flow () on the two-dimensional problemΦ(u)=1 2〈u,Qu〉\\Phi (u) = \\frac{1}{2} \\langle u, Q u \\rangle with λ=0.9\\lambda =0.9 fixed.",
"We vary the conditionnumber of QQ as well as the learning rate hh.Figure: The numerical rate of convergence, as a function of the learning rate hh, of HB and NAG to the gradient flow ()for the problem described in Figure ."
],
[
"Numerical Illustration",
"Figure REF compares trajectories of the momentum numerical method (REF ) with the rescaled gradient flow (REF ), for the two-dimensional problem $\\Phi (u) = \\frac{1}{2} \\langle u, Q u \\rangle $ .",
"We pick $Q$ to be positive-definite so that the minimum is achieved at the point $(0,0)^T$ and make it diagonal so that we can easily control its condition number.",
"In particular, the condition number of $Q$ is given as $\\kappa = \\frac{\\max \\lbrace Q_{11}, Q_{22}\\rbrace }{\\min \\lbrace Q_{11}, Q_{22}\\rbrace }.$ We see that, as the condition number is increased, both HB and NAG exhibit more pronounced transient oscillations and are thus further away from the trajectory of (REF ), however, as the learning rate $h$ is decreased, the oscillations dampen and the trajectories match more and more closely.",
"This observation from Figure REF is quantified in Figure REF where we estimate the rate of convergence, as a function of $h$ , which is defined as $\\Delta = \\log _2 \\frac{\\Vert \\mathsf {u}^{(h)} - u\\Vert _\\infty }{\\Vert \\mathsf {u}^{(h/2)} - u\\Vert _\\infty }$ where $\\mathsf {u}^{(\\alpha )}$ is the numerical solution using time-step $\\alpha $ .",
"The figure shows that the rate of convergence is indeed close to 1, as predicted by our theory.",
"In summary the behavior of the momentum methods is precisely that of a rescaled gradient flow, but with initial transient oscillations which capture momentum effects, but disappear as the learning rate is decreased.",
"We model these oscillations in the next section via use of a modified equation."
],
[
"Modified Equations",
"The previous section demonstrates how the momentum methods approximate a time rescaled version of the gradient flow (REF ).",
"In this section we show how the same methods may also be viewed as approximations of the damped Hamiltonian system (REF ), with mass $m$ on the order of the learning rate, using the method of modified equations.",
"In subsection REF we state and discuss the main result of the section, Theorem REF .",
"Subsection REF gives intuition for proof of Theorem REF ; the proof itself is given in Appendix B.",
"And the section also contains comments on generalizing the idea of modified equations.",
"In subsection REF we describe a numerical illustration of Theorem REF ."
],
[
"Main Result",
"The main result of this section quantifies the sense in which momentum methods do, in fact, approximate a damped Hamiltonian system; it is proved in Appendix B.",
"Fix $\\lambda \\in (0,1)$ and assume that $a \\ge 0$ is chosen so that $\\alpha \\frac{1}{2}(1 + \\lambda - 2a(1-\\lambda ))$ is strictly positive.",
"Suppose Assumption REF holds and let $u \\in C^4([0,\\infty );\\mathbb {R}^d)$ be the solution to $\\begin{split}& h \\alpha \\frac{d^2u}{dt^2} + (1-\\lambda )\\frac{du}{dt} = -\\nabla \\Phi (u) \\\\&u(0) = \\mathsf {u}_0, \\quad \\frac{du}{dt}(0) = \\mathsf {u}_0^{\\prime }.\\end{split}$ Suppose further that $h \\le (1-\\lambda )^2/2 \\alpha B_1$ .",
"For $n=0,1,2,\\dots $ let $\\mathsf {u}_n$ be the sequence given by (REF ) and define $u_n u(nh)$ .",
"Then for any $T \\ge 0$ , there is a constant $C = C(T) > 0$ such that $\\sup _{0 \\le nh \\le T} |u_n - \\mathsf {u}_n| \\le Ch.$ Theorem REF demonstrates the same order of convergence, namely ${\\mathcal {O}}(h)$ , to the rescaled gradient flow equation (REF ), obtained from (REF ) simply by setting $h=0.$ In the standard method of modified equations the limit system (here (REF )) is perturbed by small terms (in terms of the assumed small learning rate) and an increased rate of convergence is obtained to the modified equation (here (REF )).",
"In our setting however, because the small modification is to a higher derivative (here second) than appears in the limit equation (here first order), an increased rate of convergence is not obtained.",
"This is due to the nature of the modified equation, whose solution has derivatives that are inversely proportional to powers of $h$ ; this fact is quantified in Lemma REF from Appendix B.",
"It is precisely because the modified equation does not lead to a higher rate of convergence that the initial parameter $\\mathsf {u}_0^{\\prime }$ is arbitrary; the same rate of convergence is obtained no matter what value it takes.",
"It is natural to ask, therefore, what is learned from the convergence result in Theorem REF .",
"The answer is that, although the modified equation (REF ) is approximated at the same order as the limit equation (REF ), it actually contains considerably more qualitative information about the dynamics of the system, particularly in the early transient phase of the algorithm; this will be illustrated in subsection REF .",
"Indeed we will make a specific choice of $\\mathsf {u}_0^{\\prime }$ in our numerical experiments, namely $\\frac{du}{dt}(0) = \\frac{1 - 2\\alpha }{2\\alpha - \\lambda + 1} f(\\mathsf {u}_0),$ to better match the transient dynamics."
],
[
"Idea Behind The Modified Equations",
"In this subsection, we show that the scheme (REF ) exhibits momentum, in the sense of approximating a momentum equation, but the size of the momentum term is on the order of the step size $h$ .",
"To see this intuitively, we add and subtract $\\mathsf {u}_n - \\mathsf {u}_{n-1}$ to the right hand size of (REF ) then we can rearrange it to obtain $h \\frac{\\mathsf {u}_{n+1} - 2\\mathsf {u}_n + \\mathsf {u}_{n-1}}{h^2} + (1-\\lambda ) \\frac{\\mathsf {u}_n - \\mathsf {u}_{n-1}}{h} = f(\\mathsf {u}_n + a(\\mathsf {u}_n - \\mathsf {u}_{n-1})).$ This can be seen as a second order central difference and first order backward difference discretization of the momentum equation $h \\frac{d^2u}{dt^2} + (1-\\lambda )\\frac{du}{dt} = f(u)$ noting that the second derivative term has size of order $h$ ."
],
[
"Higher Order Modified Equations For HB",
"We will now show that, for HB, we may derive higher order modified equations that are consistent with (REF ).",
"Taking the limit of these equations yields an operator that agrees with with our intuition for discretizing (REF ).",
"To this end, suppose $\\Phi \\in C^\\infty _b (\\mathbb {R}^d, \\mathbb {R})$ and consider the ODE(s), $\\sum _{k=1}^p \\frac{h^{k-1}(1 + (-1)^k \\lambda )}{k!}",
"\\frac{d^k u}{dt^k} = f(u)$ noting that $p = 1$ gives (REF ) and $p=2$ gives (REF ).",
"Let $u \\in C^\\infty ([0,\\infty ),\\mathbb {R}^d)$ be the solution to (REF ) and define $u_n u(nh)$ , $u_n^{(k)} \\frac{d^k u}{dt^k} (nh)$ for $n=0,1,2,\\dots $ and $k=1,2,\\dots ,p$ .",
"Taylor expanding yields $u_{n \\pm 1} = u_n + \\sum _{k=1}^p \\frac{(\\pm 1)^k h^k}{k!}",
"u^{(k)}_n + h^{p+1} I^{\\pm }_n$ where $I^{\\pm }_n = \\frac{(\\pm 1)^{p+1}}{p!}",
"\\int _0^1 (1-s)^p \\frac{d^{p+1}u}{dt^{p+1}} ((n \\pm s)h) ds.$ Then $u_{n+1} - u_n - \\lambda (u_n - u_{n-1}) &= \\sum _{k=1}^p \\frac{h^k}{k!}",
"u^{(k)}_n + \\lambda \\sum _{k=1}^p \\frac{ (-1)^k h^k}{k!}",
"u^{(k)}_n + h^{p+1}(I^+_n - \\lambda I^-_n) \\\\&= h \\sum _{k=1}^p \\frac{h^{k-1}(1 + (-1)^k \\lambda )}{k!}",
"u^{(k)}_n + h^{p+1}(I^+_n - \\lambda I^-_n) \\\\&= h f(u_n) + h^{p+1}(I^+_n - \\lambda I^-_n)$ showing consistency to order $p+1$ .",
"As is the case with (REF ) however, the $I^\\pm _n$ terms will be inversely proportional to powers of $h$ hence global accuracy will not improve.",
"We now study the differential operator on the l.h.s.",
"of (REF ) as $p \\rightarrow \\infty $ .",
"Define the sequence of differential operators $T_p : C^\\infty ([0,\\infty ),\\mathbb {R}^d) \\rightarrow C^\\infty ([0,\\infty ),\\mathbb {R}^d)$ by $T_p u = \\sum _{k=1}^p \\frac{h^{k-1}(1 + (-1)^k \\lambda )}{k!}",
"\\frac{d^k u}{dt^k}, \\quad \\forall u \\in C^\\infty ([0,\\infty ),\\mathbb {R}^d).$ Taking the Fourier transform yields $\\mathcal {F}(T_p u)(\\omega ) = \\sum _{k=1}^p \\frac{h^{k-1}(1 + (-1)^k \\lambda )(i \\omega )^k}{k!}",
"\\mathcal {F}(u)(\\omega )$ where $i = \\sqrt{-1}$ denotes the imaginary unit.",
"Suppose there is a limiting operator $T_p \\rightarrow T$ as $p \\rightarrow \\infty $ then taking the limit yields $\\mathcal {F}(Tu)(\\omega ) = \\frac{1}{h} (e^{ih\\omega } + \\lambda e^{-ih\\omega } - \\lambda -1) \\mathcal {F}(u)(\\omega ).$ Taking the inverse transform and using the convolution theorem, we obtain $(Tu)(t) &= \\frac{1}{h} \\mathcal {F}^{-1} ( e^{ i h \\omega } + \\lambda e^{- i h \\omega } - \\lambda - 1 )(t) * u(t) \\\\&= \\frac{1}{h} \\left( -(1+\\lambda ) \\delta (t) + \\lambda \\delta (t + h) + \\delta (t - h) \\right) * u(t) \\\\&= \\frac{1}{h} \\int _{-\\infty }^\\infty \\left( -(1+\\lambda ) \\delta (t - \\tau ) + \\lambda \\delta (t - \\tau + h) + \\delta (t - \\tau - h) \\right)u(\\tau ) \\: d\\tau \\\\&= \\frac{1}{h} \\left( -(1+\\lambda )u(t) + \\lambda u(t - h) + u(t + h) \\right) \\\\&= \\frac{u(t+h) - u(t)}{h} - \\lambda \\left( \\frac{u(t) - u(t-h)}{h} \\right)$ where $\\delta (\\cdot )$ denotes the Dirac-delta distribution and we abuse notation by writing its action as an integral.",
"The above calculation does not prove convergence of $T_p$ to $T$ , but simply confirms our intuition that (REF ) is a forward and backward discretization of (REF ).",
"Figure: Comparison of trajectories for HB and NAG with the Hamiltonian dynamic () on the two-dimensional problemΦ(u)=1 2〈u,Qu〉\\Phi (u) = \\frac{1}{2} \\langle u, Q u \\rangle with λ=0.9\\lambda =0.9 fixed.",
"We vary the conditionnumber of QQ as well as the learning rate hh.Figure: The numerical rate of convergence, as a function of the learning rate hh, of HB and NAG to the momentum equation ()for the problem described in Figure ."
],
[
"Numerical Illustration",
"Figure REF shows trajectories of (REF ) and (REF ) for different values of $a$ and $h$ on the two-dimensional problem $\\Phi (u) = \\frac{1}{2} \\langle u , Q u \\rangle $ , varying the condition number of $Q$ .",
"We make the specific choice of $\\mathsf {u}_0^{\\prime }$ implied by the initial condition (REF ).",
"Figure REF shows the numerical order of convergence as a function of $h$ , as defined in Section REF , which is near 1, matching our theory.",
"We note that the oscillations in HB are captured well by (REF ), except for a slight shift when $h$ and $\\kappa $ are large.",
"This is due to our choice of initial condition which cancels the maximum number of terms in the Taylor expansion initially, but the overall rate of convergence remains $\\mathcal {O}(h)$ due to Lemma REF .",
"Other choices of $\\mathsf {u}_0^{\\prime }$ also result in $\\mathcal {O}(h)$ convergence and can be picked on a case-by-case basis to obtain consistency with different qualitative phenomena of interest in the dynamics.",
"Note also that $\\alpha |_{a = \\lambda } < \\alpha |_{a = 0}$ .",
"As a result the transient oscillations in (REF ) are more quickly damped in the NAG case than in the HB case; this is consistent with the numerical results.",
"However panels (d)-(f) in Figure REF show that (REF ) is not able to adequately capture the oscillations of NAG when $h$ is relatively large.",
"We leave for future work, the task of finding equations that are able to appropriately capture the oscillations of NAG in the large $h$ regime."
],
[
"Invariant Manifold",
"The key lessons of the previous two sections are that the momentum methods approximate a rescaled gradient flow of the form (REF ) and a damped Hamiltonian system of the form (REF ), with small mass $m$ which scales with the learning rate, and constant damping $\\gamma .$ Both approximations hold with the same order of accuracy, in terms of the learning rate, and numerics demonstrate that the Hamiltonian system is particularly useful in providing intuition for the transient regime of the algorithm.",
"In this section we link the two theorems from the two preceding sections by showing that the Hamiltonian dynamics with small mass from section has an exponentially attractive invariant manifold on which the dynamics is, to leading order, a gradient flow.",
"That gradient flow is a small, in terms of the learning rate, perturbation of the time-rescaled gradient flow from section ."
],
[
"Main Result",
"Define $\\mathsf {v}_n (\\mathsf {u}_n - \\mathsf {u}_{n-1})/h$ noting that then (REF ) becomes $\\mathsf {u}_{n+1} = \\mathsf {u}_n + h \\lambda \\mathsf {v}_n + hf(\\mathsf {u}_n + ha\\mathsf {v}_n)$ and $\\mathsf {v}_{n+1} = \\frac{\\mathsf {u}_{n+1} - \\mathsf {u}_n}{h} = \\lambda \\mathsf {v}_n + f(\\mathsf {u}_n + ha\\mathsf {v}_n).$ Hence we can re-write (REF ) as $\\begin{split}\\mathsf {u}_{n+1} &= \\mathsf {u}_n + h \\lambda \\mathsf {v}_n + hf(\\mathsf {u}_n + ha\\mathsf {v}_n) \\\\\\mathsf {v}_{n+1} &= \\lambda \\mathsf {v}_n + f(\\mathsf {u}_n + ha \\mathsf {v}_n).\\end{split}$ Note that if $h=0$ then (REF ) shows that $\\mathsf {u}_n=\\mathsf {u}_0$ is constant in $n$ , and that $\\mathsf {v}_n$ converges to $(1-\\lambda )^{-1}f(\\mathsf {u}_0).$ This suggests that, for $h$ small, there is an invariant manifold which is a small perturbation of the relation $\\mathsf {v}_n=\\bar{\\lambda }f(\\mathsf {u}_n)$ and is representable as a graph.",
"Motivated by this, we look for a function $g: \\mathbb {R}^d\\rightarrow \\mathbb {R}^d$ such that the manifold $\\mathsf {v}=\\bar{\\lambda } f(\\mathsf {u}) + hg(\\mathsf {u})$ is invariant for the dynamics of the numerical method: $\\mathsf {v}_n = \\bar{\\lambda } f(\\mathsf {u}_n) + hg(\\mathsf {u}_n) \\Longleftrightarrow \\mathsf {v}_{n+1} = \\bar{\\lambda } f(\\mathsf {u}_{n+1}) + hg(\\mathsf {u}_{n+1}).$ We will prove the existence of such a function $g$ by use of the contraction mapping theorem to find fixed point of mapping $T$ defined in subsection REF below.",
"We seek this fixed point in set $\\Gamma $ which we now define: Let $\\gamma , \\delta > 0$ be as in Lemmas REF , REF .",
"Define $\\Gamma \\Gamma (\\gamma ,\\delta )$ to be the closed subset of $C(\\mathbb {R}^d;\\mathbb {R}^d)$ consisting of $\\gamma $ -bounded functions: $\\Vert g\\Vert _\\Gamma \\sup _{\\xi \\in \\mathbb {R}^d} |g(\\xi )| \\le \\gamma , \\quad \\forall g \\in \\Gamma $ that are $\\delta $ -Lipshitz: $|g(\\xi ) - g(\\eta )| \\le \\delta |\\xi - \\eta |, \\quad \\forall g \\in \\Gamma , \\xi , \\eta \\in \\mathbb {R}^d.$ Fix $\\lambda \\in (0,1).$ Suppose that $h$ is chosen small enough so that Assumption REF holds.",
"For $n=0,1,2,\\dots $ , let $\\mathsf {u}_n$ , $\\mathsf {v}_n$ be the sequences given by (REF ).",
"Then there is a $\\tau >0$ such that, for all $h \\in (0,\\tau )$ , there is a unique $g \\in \\Gamma $ such that (REF ) holds.",
"Furthermore, $|\\mathsf {v}_n - \\bar{\\lambda }f(\\mathsf {u}_n) - hg(\\mathsf {u}_n)| \\le (\\lambda + h^2 \\lambda \\delta )^n |\\mathsf {v}_0 - \\bar{\\lambda }f(\\mathsf {u}_0) - hg(\\mathsf {u}_0)|$ where $\\lambda + h^2 \\lambda \\delta < 1$ .",
"The statement of Assumption REF , and the proof of the preceding theorem, are given in Appendix C. The assumption appears somewhat involved at first glance but inspection reveals that it simply places an upper bound on the learning rate $h,$ as detailed in Lemmas REF , REF .",
"The proof of the theorem rests on the Lemmas REF , REF and REF which establish that the operator $T$ is well-defined, maps $\\Gamma $ to $\\Gamma $ , and is a contraction on $\\Gamma $ .",
"The operator $T$ is defined, and expressed in a helpful form for the purposes of analysis, in the next subsection.",
"In the next subsection we obtain the leading order approximation for $g$ , given in equation (REF ).",
"Theorem REF implies that the large-time dynamics are governed by the dynamics on the invariant manifold.",
"Substituting the leading order approximation for $g$ into the invariant manifold (REF ) and using this expression in the definition (REF ) shows that $\\mathsf {v}_n&=-(1-{\\lambda })^{-1} \\nabla \\left( \\Phi (\\mathsf {u}_n) + \\frac{1}{2} h \\bar{\\lambda }(\\bar{\\lambda } - a) |\\nabla \\Phi (\\mathsf {u}_n)|^2 \\right),\\\\\\mathsf {u}_{n}&=\\mathsf {u}_{n-1}- h(1-{\\lambda })^{-1} \\nabla \\left( \\Phi (\\mathsf {u}_n) + \\frac{1}{2} h \\bar{\\lambda }(\\bar{\\lambda } - a) |\\nabla \\Phi (\\mathsf {u}_n)|^2 \\right).$ Setting $c=\\bar{\\lambda } \\left( \\bar{\\lambda }-a+\\frac{1}{2} \\right)$ we see that for large time the dynamics of momentum methods, including HB and NAG, are approximately those of the modified gradient flow $\\frac{d u}{dt}= - (1-{\\lambda })^{-1} \\nabla \\Phi _h(u)$ with $\\Phi _h(u)=\\Phi (u)+\\frac{1}{2} hc|\\nabla \\Phi (u)|^2.$ To see this we proceed as follows.",
"Note that from (REF ) $\\frac{d^2u}{dt^2} = - \\frac{1}{2} (1-\\lambda )^{-2} \\nabla |\\nabla \\Phi (u)|^2 + \\mathcal {O}(h)$ then Taylor expansion shows that, for $u_n=u(nh)$ , $u_n &= u_{n-1} + h \\dot{u}_n - \\frac{h^2}{2} \\ddot{u}_n + \\mathcal {O}(h^3) \\\\&=u_{n-1} - h \\bar{\\lambda } \\left( \\nabla \\Phi (u_n) + \\frac{1}{2}hc\\nabla |\\nabla \\Phi (u_n)|^2 \\right) + \\frac{1}{4}h^2 \\bar{\\lambda }^2 \\nabla |\\nabla \\Phi (u_n)|^2 + \\mathcal {O}(h^3)$ where we have used that $Df(u)f(u)=\\frac{1}{2} \\nabla \\left(|\\nabla \\Phi (u)|^2\\right).$ Choosing $c=\\bar{\\lambda }(\\bar{\\lambda }-a+1/2)$ we see that $u_n=u_{n-1}- h(1-{\\lambda })^{-1} \\nabla \\left( \\Phi (u_n) + \\frac{1}{2} h \\bar{\\lambda }(\\bar{\\lambda } - a) |\\nabla \\Phi (u_n)|^2 \\right) + \\mathcal {O}(h^3).$ Notice that comparison of () and (REF ) shows that, on the invariant manifold, the dynamics are to $\\mathcal {O}(h^2)$ the same as the equation (REF ); this is because the truncation error between () and (REF ) is $\\mathcal {O}(h^3)$ .",
"Thus we have proved: Suppose that the conditions of Theorem REF hold.",
"Then for initial data started on the invariant manifold and any $T \\ge 0$ , there is a constant $C = C(T) > 0$ such that $\\sup _{0 \\le nh \\le T} |u_n - \\mathsf {u}_n| \\le Ch^2,$ where $u_n=u(nh)$ solves the modified equation (REF ) with $c=\\bar{\\lambda }(\\bar{\\lambda }-a+1/2)$ ."
],
[
"Intuition",
"We will define mapping $T: C(\\mathbb {R}^d;\\mathbb {R}^d) \\rightarrow C(\\mathbb {R}^d;\\mathbb {R}^d)$ via the equations $\\begin{split}p = \\xi + h \\lambda \\bigl (\\bar{\\lambda } f(\\xi ) + hg(\\xi )\\bigr ) +hf\\Bigl (\\xi + ha\\bigl (\\bar{\\lambda } f(\\xi ) + hg(\\xi )\\bigr )\\Bigr ) \\\\\\bar{\\lambda } f(p)+h(Tg)(p) = \\lambda \\bigl (\\bar{\\lambda } f(\\xi ) + hg(\\xi )\\bigr ) +f\\Bigl (\\xi + ha\\bigl (\\bar{\\lambda } f(\\xi ) + hg(\\xi )\\bigr )\\Bigr ).\\end{split}$ A fixed point of the mapping $g \\mapsto Tg$ will give function $g$ so that, under (REF ), identity (REF ) holds.",
"Later we will show that, for $g$ in $\\Gamma $ and all $h$ sufficiently small, $\\xi $ can be found from (REF a) for every $p$ , and that thus (REF b) defines a mapping from $g \\in \\Gamma $ into $Tg \\in C(\\mathbb {R}^d;\\mathbb {R}^d).$ We will then show that, for $h$ sufficiently small, $T: \\Gamma \\mapsto \\Gamma $ is a contraction.",
"For any $g \\in C(\\mathbb {R}^d;\\mathbb {R}^d)$ and $\\xi \\in \\mathbb {R}^d$ define $w_g(\\xi ) &\\bar{\\lambda } f(\\xi ) + h g(\\xi ) \\\\z_g(\\xi ) &\\lambda w_g(\\xi ) + f\\bigl (\\xi + haw_g(\\xi )\\bigr ).$ With this notation the fixed point mapping (REF ) for $g$ may be written $\\begin{split}p = \\xi + hz_g(\\xi ),\\\\\\bar{\\lambda } f(p)+h(Tg)(p) = z_g(\\xi ).\\end{split}$ Then, by Taylor expansion, $\\begin{split}f\\Bigl (\\xi + ha\\bigl (\\bar{\\lambda } f(\\xi ) + hg(\\xi )\\bigr )\\Bigr ) &= f\\bigl (\\xi + haw_g(\\xi )\\bigr ) \\\\&= f(\\xi ) + ha \\int _0^1 Df\\bigl (\\xi + shaw_g(\\xi )\\bigr )w_g(\\xi ) ds \\\\&= f(\\xi ) + ha I^{(1)}_g (\\xi )\\end{split}$ where the last line defines $I^{(1)}_g$ .",
"Similarly $\\begin{split}f(p) &= f(\\xi + hz_g(\\xi )) \\\\&= f(\\xi ) + h \\int _0^1 Df\\bigl (\\xi + shz_g(\\xi )\\bigr )z_g(\\xi )ds \\\\&= f(\\xi ) + h I^{(2)}_g(\\xi ),\\end{split}$ where the last line now defines $I^{(2)}_g$ .",
"Then (REF b) becomes $\\bar{\\lambda }\\bigl (f(\\xi ) + h I^{(2)}_g(\\xi )\\bigr ) + h (Tg)(p) = \\lambda \\bar{\\lambda } f(\\xi ) + h \\lambda g(\\xi ) + f(\\xi ) + haI^{(1)}_g(\\xi )$ and we see that $(Tg)(p) = \\lambda g(\\xi ) + a I^{(1)}_g(\\xi ) - \\bar{\\lambda }I^{(2)}_g(\\xi ).$ In this light, we can rewrite the defining equations (REF ) for $T$ as $p &= \\xi + hz_g(\\xi ), \\\\(Tg)(p) &= \\lambda g(\\xi ) + a I^{(1)}_g(\\xi ) - \\bar{\\lambda }I^{(2)}_g(\\xi ).$ for any $\\xi \\in \\mathbb {R}^d$ .",
"Perusal of the above definitions reveals that, to leading order in $h$ , $w_g(\\xi )=z_g(\\xi )=\\bar{\\lambda }f(\\xi ), I^{(1)}_g(\\xi )=I^{(2)}_g(\\xi )=\\bar{\\lambda }Df(\\xi )f(\\xi ).$ Thus setting $h=0$ in (REF ), () shows that, to leading order in $h$ , $g(p) = \\bar{\\lambda }^2(a-\\bar{\\lambda })Df(p)f(p).$ Note that since $f(p) = - \\nabla \\Phi (p)$ , $Df$ is the negative Hessian of $\\Phi $ and is thus symmetric.",
"Hence we can write $g$ in gradient form, leading to $g(p) = \\frac{1}{2} \\bar{\\lambda }^2(a-\\bar{\\lambda })\\nabla \\bigl (|\\nabla \\Phi (p)|^2\\bigr ).$ This modified potential (REF ) also arises in the construction of Lyapunov functions for the one-stage theta method – see Corollary 5.6.2 in [41].",
"Figure: Invariant manifold for HB and NAG with h=2 -6 h=2^{-6} and λ=0.9\\lambda = 0.9 on the two-dimensional problem Φ(u)=1 2〈u,Qu〉\\Phi (u) = \\frac{1}{2} \\langle u, Q u \\rangle , varying the condition number of QQ.",
"Panels (c), (f) show the distance from the invariant manifold for the largest condition number κ=20\\kappa = 20."
],
[
"Numerical Illustration",
"In Figure REF panels (a),(b),(d),(e), we plot the components $\\mathsf {u}_n$ and $\\mathsf {v}_n$ found by solving (REF ) with initial conditions $\\mathsf {u}_0=(1,1)^T$ and $\\mathsf {v}_n=(0,0)^T$ in the case where $\\Phi (u) = \\frac{1}{2} \\langle u, Q u \\rangle $ .",
"These initial conditions correspond to initializing the map off the invariant manifold.",
"To leading order in $h$ the invariant manifold is given by (see equation (REF )) $v= - (1-{\\lambda })^{-1} \\nabla \\left( \\Phi (u) + \\frac{1}{2} h \\bar{\\lambda }(\\bar{\\lambda } - a) |\\nabla \\Phi (u)|^2 \\right).$ To measure the distance of the trajectory shown in panels (a),(b),(d),(e) from the invariant manifold we define $\\mathsf {e}_n= \\left| \\mathsf {v}_n+ (1-{\\lambda })^{-1} \\nabla \\left( \\Phi (\\mathsf {u}_n) + \\frac{1}{2} h \\bar{\\lambda }(\\bar{\\lambda } - a) |\\nabla \\Phi (\\mathsf {u}_n)|^2 \\right) \\right|.$ Panels (c),(f) show the evolution of $\\mathsf {e}_n$ as well as the (approximate) bound on it found from substituting the leading order approximation of $g$ into the following upper bound from Theorem REF : $(\\lambda + h^2 \\lambda \\delta )^n |\\mathsf {v}_0 - \\bar{\\lambda }f(\\mathsf {u}_0) - hg(\\mathsf {u}_0)|.$"
],
[
"Deep Learning Example",
"Our theory is developed under quite restrictive assumptions, in order to keep the proofs relatively simple and to allow a clearer conceptual development.",
"The purpose of the numerical experiments in this section is twofold: firstly to demonstrate that our theory sheds light on a stochastic version of gradient descent applied, furthermore, to a setting in which the objective function does not satisfy the global assumptions which facilitate our analysis; and second to show that methods implemented as we use them here (with learning-rate independent momentum, fixed at every step of the iteration) can out-perform other choices on specific problems.",
"Our numerical experiments in this section are undertaken with in the context of the example given in [44].",
"We train a deep autoencoder, using the architecture of [17] on the MNIST dataset [24].",
"Since our work is concerned only with optimization and not generalization, we present our results only on the training set of 60,000 images and ignore the testing set.",
"We fix an initialization of the autoencoder following [13] and use it to test every optimization method.",
"Furthermore, we fix a batch size of 200 and train for 500 epochs, not shuffling the data set during training so that each method sees the same realization of the noise.",
"We use the mean-squared error as our loss function.",
"Figure: Final training errors for the autoencoder on MNIST for sixtraining methods over different learning rates.",
"GF refers to equation () whileHB and NAG to () all with fixed λ=0.9\\lambda = 0.9.Figure: The numerical rate of convergence for the parameters ofthe autoencoder, as a function of the learning rate h, of HBand NAG to () (a), as well as of HB-μ\\mu and NAG-μ\\mu to ()(b).We compare HB and NAG given by (REF ) to the re-scaled gradient flow (REF ) which we discretize in the standard way to yield the numerical method $\\mathsf {u}_{n+1} = \\mathsf {u}_n - \\frac{h}{(1-\\lambda )} \\nabla \\Phi (\\mathsf {u}_n),$ hence the momentum term $\\lambda $ only acts to re-scale the learning rate.",
"We do not test against equation (REF ) because, to discretize it faithfully, we would need to use a time-step much lower than $h$ (because (REF ) contains a term of order $h$ ), but doing so would mean that we need to train for many more epochs compared to HB and NAG so that the same final time is reached.",
"This, in turn, implies that the methods would see different realization of the noise.",
"Thus, to compare them well, we would need to perform a Monte Carlo simulation, however, since we do not state any of our results in a stochastic setting, we leave this for future work.",
"We also compare our results to those of [48] which analyze HB and NAG in the setting where $\\Phi $ is $\\mu $ -strongly convex and $\\lambda $ is given by (REF ) that is $\\lambda = \\frac{1 - \\sqrt{\\mu h}}{1 + \\sqrt{\\mu h}}.$ They obtain the limiting equation $\\ddot{u} + 2\\sqrt{\\mu } \\dot{u} + \\nabla \\Phi (u) = 0$ which we discretize via a split-step method to yield $\\begin{split}\\mathsf {u}_{n+1} &= \\mathsf {u}_n + \\frac{1}{2\\sqrt{\\mu }} \\left( 1- e^{-2\\sqrt{ \\mu h}} \\right) \\mathsf {v}_n \\\\\\mathsf {v}_{n+1} &= e^{-2\\sqrt{ \\mu h}} \\mathsf {v}_n - \\sqrt{h} \\nabla \\Phi (\\mathsf {u}_{n+1})\\end{split}$ where we have mapped the the time-step $h$ in HB and NAG to $\\sqrt{h}$ as in done in [48].",
"We choose this discretization because it allows us to directly solve for the linear parts of the ODE (in the enlarged state-space), yielding a more accurate approximation than the forward-Euler method used to obtain (REF ).",
"A detailed derivation is given in Appendix D. We will refer to the method in equation (REF ) as Wilson.",
"Further we refer to equation (REF ) with $\\lambda $ given by (REF ) and $a=0$ as HB-$\\mu $ and equation (REF ) with $\\lambda $ given by (REF ) and $a=\\lambda $ as NAG-$\\mu $ .",
"Since deep neural networks are not strongly convex, there is no single optimal choice of $\\mu $ ; we simply set $\\mu =1$ in our experiments.",
"Figure REF gives the final training errors for each method for several learning rates.",
"We were unable to train the autoencoder using (REF ) with $h=1$ since $\\lambda =0.9$ implies an effective learning rate of 10 for which the system blows up.",
"In general, NAG is the best performing method for relatively large $h$ which is an observation that is consistently made in the deep learning literature.",
"Further, we note that as the learning rate decreases, the final errors become closer indicating convergence to the appropriate limiting equations.",
"Figure REF showcases the practical effectiveness of momentum methods as they provide a way of discretizing the gradient flow (REF ) with a large effective learning rate that forward Euler cannot accommodate.",
"From this perspective, we can view momentum methods as providing a more stable discretization to gradient flows in a manner illustrated by (REF ).",
"Such a viewpoint informs the works [39], [3], [51].",
"To further illustrate the point of convergence to the limiting equation, we compute the numerical rate of convergence, defined in Section REF , as a function of $h$ for the neural network parameters between (REF ) and HB and NAG as well as between (REF ) and HB-$\\mu $ and NAG-$\\mu $ .",
"Figure REF gives the results.",
"We note that this rate is around 1 as predicted by our theory while the rate for (REF ) is around 0.5 which is also consistent with the theory in [48]."
],
[
"Conclusion",
"Together, equations (REF ), (REF ) and (REF ) describe the dynamical systems which are approximated by momentum methods, when implemented with fixed momentum, in a manner made precise by the four theorems in this paper.",
"The insight obtained from these theorems sheds light on how momentum methods perform optimization tasks.",
"Both authors are supported, in part, by the US National Science Foundation (NSF) grant DMS 1818977, the US Office of Naval Research (ONR) grant N00014-17-1-2079, and the US Army Research Office (ARO) grant W911NF-12-2-0022.",
"Both authors are also grateful to the anonymous reviewers for their invaluable suggestions which have helped to significantly strengthen this work."
],
[
"Appendix A",
"[of Theorem REF ] Taylor expanding yields $u_{n+1} = u_n + h \\bar{\\lambda } f(u_n) + \\mathcal {O}(h^2)$ and $u_n = u_{n-1} + h \\bar{\\lambda } f(u_n) + \\mathcal {O}(h^2).$ Hence $(1+\\lambda )u_n - \\lambda u_{n-1} = u_n + h \\lambda \\bar{\\lambda } f(u_n) + \\mathcal {O}(h^2).$ Subtracting the third identity from the first, we find that $u_{n+1} - \\left( (1+\\lambda ) u_n - \\lambda u_{n-1} \\right) = hf(u_n) + \\mathcal {O}(h^2)$ by noting $\\bar{\\lambda } - \\bar{\\lambda } \\lambda = 1$ .",
"Similarly, $a(u_n - u_{n-1}) = h a \\bar{\\lambda } f(u_n) + \\mathcal {O}(h^2)$ hence Taylor expanding yields $f(u_n + a(u_n-u_{n-1})) &= f(u_n) + a Df(u_n)(u_n - u_{n-1}) \\\\&\\,\\,\\,\\,\\, + a^2 \\int _0^1 (1-s) D^2f(u_n + sa(u_n-u_{n-1}))[u_n-u_{n-1}]^2 ds \\\\&= f(u_n) + h a \\bar{\\lambda } Df(u_n)f(u_n) + \\mathcal {O}(h^2).$ From this, we conclude that $hf(u_n + a(u_n-u_{n-1})) = hf(u_n) + \\mathcal {O}(h^2)$ hence $u_{n+1} = (1+\\lambda )u_n - \\lambda u_{n-1} + hf(u_n + a(u_n-u_{n-1})) + \\mathcal {O}(h^2).$ Define the error $e_n u_n - \\mathsf {u}_n$ then $e_{n+1} &= (1 + \\lambda )e_n - \\lambda e_{n-1} + h \\left( f(u_n + a(u_n - u_{n-1})) - f(\\mathsf {u}_n + a(\\mathsf {u}_n - \\mathsf {u}_{n-1})) \\right) + \\mathcal {O}(h^2) \\\\&= (1 + \\lambda )e_n - \\lambda e_{n-1} + h \\mathsf {M}_n((1+a)e_n - a e_{n-1}) + \\mathcal {O}(h^2)$ where, from the mean value theorem, we have $\\mathsf {M}_n = \\int _0^1 Df \\Bigl ( s \\bigl ( u_n + a(u_n-u_{n-1}) \\bigr ) + \\bigl ( 1-s \\bigr ) \\bigl ( \\mathsf {u}_n + a(\\mathsf {u}_n-\\mathsf {u}_{n-1}) \\bigr ) \\Bigr ) ds.$ Now define the concatenation $E_{n+1} [e_{n+1}, e_n] \\in \\mathbb {R}^{2d}$ then $E_{n+1} = A^{(\\lambda )} E_n + h A^{(a)}_n E_n + \\mathcal {O}(h^2)$ where $A^{(\\lambda )}, A^{(a)}_n \\in \\mathbb {R}^{2d \\times 2d}$ are the block matrices $A^{(\\lambda )} \\begin{bmatrix}(1+\\lambda )I & - \\lambda I \\\\I & 0I\\end{bmatrix}, \\quad A^{(a)}_n \\begin{bmatrix}(1+a) \\mathsf {M}_n & - a\\mathsf {M}_n \\\\0I & 0 I\\end{bmatrix}$ with $I \\in \\mathbb {R}^{d \\times d}$ the identity.",
"We note that $A^{(\\lambda )}$ has minimal polynomial $\\mu _{A^{(\\lambda )}}(z) = (z-1)(z-\\lambda )$ and is hence diagonalizable.",
"Thus there is a norm on $\\Vert \\cdot \\Vert $ on $\\mathbb {R}^{2d}$ such that its induced matrix norm $\\Vert \\cdot \\Vert _m$ satifies $\\Vert A^{(\\lambda )}\\Vert _m = \\rho (A^{(\\lambda )})$ where $\\rho : \\mathbb {R}^{2d \\times 2d} \\rightarrow \\mathbb {R}_+$ maps a matrix to its spectral radius.",
"Hence, since $\\lambda \\in (0,1)$ , we have $\\Vert A^{(\\lambda )}\\Vert _m = 1$ .",
"Thus $\\Vert E_{n+1}\\Vert \\le (1 + h \\Vert A^{(a)}_n\\Vert _m) \\Vert E_n\\Vert + \\mathcal {O}(h^2).$ Then, by finite dimensional norm equivalence, there is a constant $\\alpha > 0$ , independent of $h$ , such that $\\Vert A^{(a)}_n\\Vert _m &\\le \\alpha \\left\\Vert \\begin{bmatrix} 1+a & -a \\\\ 0 & 0 \\end{bmatrix} \\otimes \\mathsf {M}_n \\right\\Vert _2 \\\\&= \\alpha \\sqrt{2a^2 + 2a + 1} \\Vert \\mathsf {M}_n\\Vert _2$ where $\\Vert \\cdot \\Vert _2$ denotes the spectral 2-norm.",
"Using Assumption REF , we have $\\Vert \\mathsf {M}_n\\Vert _2 \\le B_1$ thus, letting $c \\alpha \\sqrt{2a^2 + 2a + 1} B_1$ , we find $\\Vert E_{n+1}\\Vert \\le (1+hc)\\Vert E_n\\Vert + \\mathcal {O}(h^2).$ Then, by Grönwall lemma, $\\Vert E_{n+1}\\Vert &\\le (1+hc)^n \\Vert E_1\\Vert _n + \\frac{(1+hc)^{n+1}-1}{ch} \\mathcal {O}(h^2) \\\\&= (1+hc)^n \\Vert E_1\\Vert _n + \\mathcal {O}(h)$ noting that the constant in the $\\mathcal {O}(h)$ term is bounded above in terms of $T$ , but independently of $h$ .",
"Finally, we check the initial condition $E_1 =\\begin{bmatrix}u_1 - \\mathsf {u}_1 \\\\u_0 - \\mathsf {u}_0\\end{bmatrix} =\\begin{bmatrix}h(\\bar{\\lambda }-1)f(\\mathsf {u}_0) + \\mathcal {O}(h^2) \\\\0\\end{bmatrix} = \\mathcal {O}(h)$ as desired.",
"[of Theorem REF ] Taylor expanding yields $u_{n \\pm 1} = u_n \\pm h \\dot{u}_n + \\frac{h^2}{2} \\ddot{u}_n \\pm \\frac{h^3}{2} I^{\\pm }_n$ where $I^\\pm _n = \\int _0^1 (1-s)^2 \\dddot{u}((n \\pm s)h)ds.$ Then using equation (REF ) $\\begin{split}u_{n+1} - u_n - \\lambda (u_n - u_{n-1}) &= h(1-\\lambda )\\dot{u}_n + \\frac{h^2}{2} (1+\\lambda ) \\ddot{u}_n + \\frac{h^3}{2}(I^+_n - \\lambda I^-_n) \\\\&= hf(u_n) + h^2a(1-\\lambda ) \\ddot{u}_n + \\frac{h^3}{2}(I^+_n - \\lambda I^-_n).\\end{split}$ Similarly $a(u_n - u_{n-1}) = ha\\dot{u}_n - \\frac{h^2}{2}a \\ddot{u}_n + \\frac{h^3}{2}aI^-_n$ hence $f(u_n + a(u_n-u_{n-1})) = f(u_n) + haDf(u_n) \\dot{u}_n - Df(u_n) \\left( \\frac{h^2}{2}a \\ddot{u}_n - \\frac{h^3}{2}a I_n^- \\right) + I^f_n$ where $I^f_n = a^2 \\int _0^1 (1-s) D^2f(u_n + sa(u_n-u_{n-1}))[u_n - u_{n-1}]^2 ds.$ Differentiating (REF ) yields $h \\alpha \\frac{d^3u}{dt^3} + (1-\\lambda )\\frac{d^2u}{dt^2} = Df(u) \\frac{du}{dt}$ hence $hf(u_n + a(u_n-u_{n-1})) &= hf(u_n) + h^2 a \\left( h \\alpha \\dddot{u}_n + (1-\\lambda ) \\ddot{u}_n \\right) - Df(u_n) \\left( \\frac{h^3}{2}a \\ddot{u}_n - \\frac{h^4}{2}a I_n^- \\right) + hI^f_n \\\\&=hf(u_n) + h^2 a(1-\\lambda ) \\ddot{u}_n + h^3 a \\alpha \\dddot{u}_n - Df(u_n) \\left( \\frac{h^3}{2}a \\ddot{u}_n - \\frac{h^4}{2}a I_n^- \\right) + hI^f_n.$ Rearranging this we obtain an expression for $hf(u_n)$ which we plug into equation (REF ) to yield $u_{n+1} - u_n - \\lambda (u_n - u_{n-1}) = hf(u_n + a(u_n-u_{n-1})) + \\text{LT}_n$ where $\\text{LT}_n = \\underbrace{\\frac{h^3}{2} (I^+_n - \\lambda I^-_n)}_{\\mathcal {O} \\left( h \\text{exp} \\left(-\\frac{(1-\\lambda )}{2 \\alpha } n \\right) \\right)} - \\underbrace{h^3 a \\alpha \\dddot{u}_n}_{\\mathcal {O} \\left( h \\text{exp} \\left(-\\frac{(1-\\lambda )}{2 \\alpha } n \\right) \\right)} + \\underbrace{Df(u_n) \\left( \\frac{h^3}{2}a \\ddot{u}_n - \\frac{h^4}{2}a I_n^- \\right)}_{\\mathcal {O}(h^2)} - \\underbrace{hI^f_n}_{\\mathcal {O}(h^3)}.$ The bounds (in braces) on the four terms above follow from employing Assumption REF and Lemma REF .",
"From them we deduce the existence of constants $K_1,K_2 > 0$ independent of $h$ such that $|\\text{LT}_n| \\le h K_1 \\text{exp} \\left( -\\frac{(1-\\lambda )}{2 \\alpha } n \\right) + h^2 K_2.$ We proceed similarly to the proof of Theorem REF , but with a different truncation error structure, and find the error satsifies $\\Vert E_{n+1}\\Vert \\le (1+hc)\\Vert E_n\\Vert + h K_1 \\text{exp} \\left( -\\frac{(1-\\lambda )}{2 \\alpha } n \\right) + h^2 K_2$ where we abuse notation and continue to write $K_1,K_2$ when, in fact, the constants have changed by use of finite-dimensional norm equivalence.",
"Define $K_3 K_2/c$ then summing this error, we find $\\Vert E_{n+1}\\Vert &\\le (1+hc)^n \\Vert E_1\\Vert + h K_3 ((1 + hc)^{n+1} - 1) + h K_1 \\sum _{j=0}^n (1+hc)^j \\text{exp} \\left( - \\frac{(1-\\lambda )}{2 \\alpha } (n-j) \\right) \\\\&= (1+hc)^n \\Vert E_1\\Vert + hK_3 ((1 + hc)^{n+1} - 1) + hK_1 S_n.$ where $S_n = \\text{exp} \\left( - \\frac{(1-\\lambda )}{2\\alpha }n \\right) \\left( \\frac{(1+hc)^{n+1} \\text{exp} \\left( \\frac{(1-\\lambda )}{2\\alpha } (n+1) \\right) - 1}{(1+hc) \\text{exp} \\left( \\frac{1-\\lambda }{2\\alpha } \\right) - 1} \\right).$ Let $T = nh$ then $S_n &\\le \\frac{(1+hc)^{n+1} \\text{exp} \\left( \\frac{1-\\lambda }{2\\alpha } \\right)}{(1+hc) \\text{exp} \\left( \\frac{1-\\lambda }{2\\alpha } \\right) - 1} \\\\&\\le \\frac{ 2 \\text{exp} \\left( cT + \\frac{1-\\lambda }{2\\alpha } \\right)}{\\text{exp} \\left( \\frac{1-\\lambda }{2\\alpha }\\right) - 1}$ From this we deduce that $\\Vert E_{n+1}\\Vert \\le (1+hc)^n \\Vert E_1\\Vert + \\mathcal {O}(h)$ noting that the constant in the $\\mathcal {O}(h)$ term is bounded above in terms of $T$ , but independently of $h$ .",
"For the initial condition, we check $u_1 - \\mathsf {u}_1 = h (\\mathsf {u}_0^{\\prime } - f(\\mathsf {u}_0)) + \\frac{h^2}{2} \\ddot{u}_0 + \\frac{h^3}{2} I^+_0$ which is $\\mathcal {O}(h)$ by Lemma REF .",
"Putting the bounds together we obtain $\\sup _{0 \\le nh \\le T} \\Vert E_n\\Vert \\le C(T)h.$ Suppose Assumption REF holds and let $u \\in C^3([0,\\infty );\\mathbb {R}^d)$ be the solution to $&h \\alpha \\frac{d^2u}{dt^2} + (1-\\lambda )\\frac{du}{dt} = f(u) \\\\&u(0) = \\mathsf {u}_0, \\quad \\frac{du}{dt}(0) = \\mathsf {v}_0$ for some $\\mathsf {u}_0, \\mathsf {v}_0 \\in \\mathbb {R}^d$ and $\\alpha > 0$ independent of $h$ .",
"Suppose $h \\le (1-\\lambda )^2/2 \\alpha B_1$ then there are constants $C^{(1)}, C^{(2)}_1, C^{(2)}_2, C^{(3)}_1, C^{(3)}_2 > 0$ independent of $h$ such that for any $t \\in [0,\\infty )$ , $|\\dot{u}(t)| &\\le C^{(1)}, \\\\|\\ddot{u}(t)| &\\le \\frac{C^{(2)}_1}{h} \\text{exp} \\left( -\\frac{(1-\\lambda )}{2h\\alpha } t \\right) + C^{(2)}_2, \\\\|\\dddot{u}(t)| &\\le \\frac{C^{(3)}_1}{h^2} \\text{exp} \\left( -\\frac{(1-\\lambda )}{2h\\alpha } t \\right) + C^{(3)}_2.$ One readily verifies that the result of Lemma REF is tight by considering the one-dimensional case with $f(u) = - u$ .",
"This implies that the result of Theorem REF cannot be improved without further assumptions.",
"[of Lemma REF ] Define $v \\dot{u}$ then $\\dot{v} = - \\frac{1}{h \\alpha } \\left( (1-\\lambda ) v - f(u) \\right).$ Define $w (1-\\lambda )v - f(u)$ hence $\\dot{v} = -(1/h\\alpha )w$ and $\\dot{u} = v = \\bar{\\lambda }(w + f(u))$ .",
"Thus $\\dot{w} &= (1-\\lambda )\\dot{v} - Df(u)\\dot{u} \\\\&= - \\frac{(1-\\lambda )}{h \\alpha } w - Df(u)(\\bar{\\lambda }(w + f(u))).$ Hence we find $\\frac{1}{2} \\frac{d}{dt} |w|^2 &= - \\frac{(1-\\lambda )}{h\\alpha } |w|^2 - \\bar{\\lambda } \\langle w, Df(u) w \\rangle - \\bar{\\lambda } \\langle w, Df(u)f(u) \\rangle \\\\&\\le - \\frac{(1-\\lambda )}{h\\alpha } |w|^2 + \\bar{\\lambda } |\\langle w, Df(u) w \\rangle | + \\bar{\\lambda } |\\langle w, Df(u)f(u) \\rangle | \\\\&\\le - \\frac{(1-\\lambda )}{h\\alpha } |w|^2 + \\bar{\\lambda } B_1 |w|^2 + \\bar{\\lambda } B_0 B_1 |w| \\\\&\\le - \\frac{(1-\\lambda )}{h\\alpha } |w|^2 + \\frac{(1-\\lambda )}{2h\\alpha } |w|^2 + \\bar{\\lambda } B_0 B_1 |w| \\\\&= -\\frac{(1-\\lambda )}{2h\\alpha } |w|^2 + \\bar{\\lambda }B_0 B_1 |w|$ by noting that our assumption $h \\le (1-\\lambda )^2 / 2 \\alpha B_1$ implies $\\bar{\\lambda }B_1 \\le (1-\\lambda ) / 2 h \\alpha $ .",
"Hence $\\frac{d}{dt} |w| \\le -\\frac{(1-\\lambda )}{2h\\alpha } |w| + \\bar{\\lambda }B_0 B_1$ so, by Grönwall lemma, $|w(t)| &\\le \\text{exp} \\left( -\\frac{(1-\\lambda )}{2h\\alpha } t \\right) |w(0)| + 2h \\bar{\\lambda }^2 \\alpha B_0 B_1 \\left(1 - \\text{exp} \\left( -\\frac{(1-\\lambda )}{2h\\alpha } t \\right) \\right) \\\\&\\le \\text{exp} \\left( -\\frac{(1-\\lambda )}{2h\\alpha } t \\right) |w(0)| + h \\beta _1$ where we define $\\beta _1 2 \\bar{\\lambda }^2 \\alpha B_0 B_1$ .",
"Hence $|\\ddot{u}(t)| &= |\\dot{v}(t)| \\\\&= \\frac{1}{h \\alpha } |w(t)| \\\\&\\le \\frac{1}{h \\alpha } \\text{exp} \\left( -\\frac{(1-\\lambda )}{2h\\alpha } t \\right) |w(0)| + \\frac{\\beta _1}{\\alpha } \\\\&= \\frac{|(1-\\lambda )\\mathsf {v}_0 - f(\\mathsf {u}_0)|}{h \\alpha } \\text{exp} \\left( -\\frac{(1-\\lambda )}{2h\\alpha } t \\right) + \\frac{\\beta _1}{\\alpha }$ thus setting $C^{(2)}_1 = |(1-\\lambda )\\mathsf {v}_0 - f(\\mathsf {u}_0)|/\\alpha $ and $C^{(2)}_1 = \\beta _1 / \\alpha $ gives the desired result.",
"Further, $|\\dot{u}(t)| &= |v(t)| \\\\&\\le \\bar{\\lambda }(|w(t)| + |f(u(t))|) \\\\&\\le \\bar{\\lambda }(|w(0)| + h \\beta _1 + B_0)$ hence we deduce the existence of $C^{(1)}$ .",
"Now define $z \\dot{w}$ then $\\dot{z} = - \\frac{(1-\\lambda )}{h \\alpha } z - \\bar{\\lambda } Df(u)z + G(u,v,w)$ where we define $G(u,v,w) - \\bar{\\lambda }(Df(u)(Df(u)v) + D^2f(u)[v,w] + D^2f(u)[Df(u)v,f(u)])$ .",
"Using Assumption REF and our bounds on $w$ and $v$ , we deduce that there is a constant $C > 0$ independent of $h$ such that $|G(u,v,w)| \\le C$ hence $\\frac{1}{2} \\frac{d}{dt} |z|^2 &= - \\frac{(1-\\lambda )}{h \\alpha } |z|^2 - \\bar{\\lambda } \\langle z, Df(u)z \\rangle + \\langle z, G(u,v,w) \\rangle \\\\&\\le - \\frac{(1-\\lambda )}{h \\alpha } |z|^2 + \\bar{\\lambda }B_1 |z|^2 + C|z| \\\\&\\le - \\frac{(1-\\lambda )}{2 h \\alpha } |z|^2 + C|z|$ as before.",
"Thus we find $\\frac{d}{dt} |z| \\le - \\frac{(1-\\lambda )}{2 h \\alpha } |z| + C$ so, by Grönwall lemma, $|z(t)| \\le \\text{exp} \\left( -\\frac{(1-\\lambda )}{2h\\alpha } t \\right) |z(0)| + h \\beta _2$ where we define $\\beta _2 2 \\bar{\\lambda } \\alpha C $ .",
"Recall that $\\dddot{u} = \\ddot{v} = - \\frac{1}{h \\alpha } \\dot{w} = - \\frac{1}{h \\alpha } z$ and note $|z(0)| \\le \\frac{(1-\\lambda )|(1-\\lambda )\\mathsf {v}_0 - f(\\mathsf {u}_0)|}{h \\alpha } + B_1 |\\mathsf {v}_0|$ hence we find $|\\dddot{u}(t)| \\le \\left( \\frac{(1-\\lambda )|(1-\\lambda )\\mathsf {v}_0 - f(\\mathsf {u}_0)}{h^2 \\alpha ^2} + \\frac{B_1 |\\mathsf {v}_0|}{h \\alpha } \\right) \\text{exp} \\left( -\\frac{(1-\\lambda )}{2h\\alpha } t \\right) + \\frac{\\beta _2}{\\alpha }.$ Thus we deduce that there is a constant $C^{(3)}_1 > 0$ independent of $h$ such that $|\\dddot{u}(t)| \\le \\frac{C^{(3)}_1}{h^2} \\text{exp} \\left( -\\frac{(1-\\lambda )}{2h\\alpha } t \\right) + C^{(3)}_2$ as desired where $C^{(3)}_2 = \\beta _2 / \\alpha $ .",
"For the results of Section we make the following assumption on the size of $h$ .",
"Recall first that by Assumption REF there are constants $B_0,B_1,B_2 > 0$ such that $\\Vert D^{j-1}f\\Vert = \\Vert D^j \\Phi \\Vert \\le B_{j-1}$ for $j=1,2,3$ .",
"Suppose $h>0$ is small enough such that $\\lambda + hB_1(a + \\lambda \\bar{\\lambda }) < 1$ then there is a $\\tau _1 > 0$ such that for any $\\gamma \\in [\\tau _1,\\infty )$ $(\\lambda + hB_1(a + \\lambda \\bar{\\lambda })) \\gamma + \\bar{\\lambda }B_0B_1(a + \\bar{\\lambda }) \\le \\gamma .$ Using Lemma REF fix $\\gamma \\in [\\tau _1, \\infty )$ and define the constants $\\begin{split}K_1 &\\bar{\\lambda }B_0 + h\\gamma \\\\K_3 &B_0 + \\lambda K_1 \\\\\\alpha _2 &h^2 (\\lambda + h a B_1), \\\\\\alpha _1 &\\lambda - 1 + h \\left( B_1(\\bar{\\lambda } + a(1+h \\bar{\\lambda }B_1)) + \\lambda \\bar{\\lambda } (B_1 + hB_2K_3) + ha(aB_2K_1 + B_1 \\bar{\\lambda }(B_1 + hB_2K_3) \\right), \\\\\\alpha _0 &aB_2K_1(1 + ha\\bar{\\lambda }B_1) + \\bar{\\lambda }(aB_1^2 + B_2K_3) + \\bar{\\lambda }^2B_1(1+haB_1)(B_1 + hB_2K_3).\\end{split}$ Suppose $h>0$ is small enough such that $\\alpha _1^2 > 4 \\alpha _2 \\alpha _0, \\quad \\alpha _1 < 0$ then there are $\\tau _2^{\\pm } > 0$ such that for any $\\delta \\in (\\tau _2^-,\\tau _2^+]$ $\\alpha _2 \\delta ^2 + \\alpha _1 \\delta + \\alpha _0 \\le 0.$ Using Lemma REF fix $\\delta \\in (\\tau _2^-,\\tau _2^+]$ .",
"We make the following assumption on the size of the learning rate $h$ which is achievable since $\\lambda \\in (0,1)$ .",
"Assumption Let Assumption REF hold and suppose $h > 0$ is small enough such that the assumptions of Lemmas REF , REF hold.",
"Define $K_2 \\bar{\\lambda }B_1 + h \\delta $ and suppose $h > 0$ is small enough such that $c h(\\lambda K_2 + B_1(1+haK_2)) < 1.$ Define constants $\\begin{split}Q_1 &\\lambda \\delta + a( B_1 K_2 + B_2 K_1 (1 + haK_2 )) + \\bar{\\lambda }( ( B_1 + hB_2K_3 )(\\lambda K_2 + B_1 (1+ haK_2)) + B_2K_3 ), \\\\Q_2 &h(a(B_1 + haB_2K_1) + \\bar{\\lambda } (\\lambda + haB_1)(B_1+hB_2K_3)), \\\\Q_3 &h(\\lambda K_2 + B_1(1+haK_2)), \\\\\\mu &\\lambda + Q_2 + \\frac{h^2(\\lambda + haB_1)Q_1}{1-Q_3}.\\end{split}$ Suppose $h > 0$ is small enough such that $Q_3 < 1, \\quad \\mu < 1.$ Lastly assume $h > 0$ is small enough such that $\\lambda + h^2 \\lambda \\delta < 1.$ [of Lemma REF .]",
"Since $\\lambda + hB_1(a + \\lambda \\bar{\\lambda }) < 1$ and $\\bar{\\lambda }B_0B_1(a + \\bar{\\lambda }) > 0$ the line defined by $(\\lambda + hB_1(a + \\lambda \\bar{\\lambda })) \\gamma + \\bar{\\lambda }B_0B_1(a + \\bar{\\lambda })$ will intersect the identity line at a positive $\\gamma $ and lie below it thereafter.",
"Hence setting $\\tau _1 = \\frac{\\bar{\\lambda }B_0B_1(a + \\bar{\\lambda })}{1 - \\lambda + hB_1(a + \\lambda \\bar{\\lambda })}$ completes the proof.",
"[of Lemma REF .]",
"Note that since $\\alpha _2 > 0$ , the parabola defined by $\\alpha _2 \\delta ^2 + \\alpha _1 \\delta + \\alpha _0$ is upward-pointing and has roots $\\zeta _{\\pm } = \\frac{-\\alpha _1 \\pm \\sqrt{\\alpha _1^2 - 4\\alpha _2\\alpha _0}}{2\\alpha _2}.$ Since $\\alpha _1^2 > 4\\alpha _2\\alpha _0$ , $\\zeta _{\\pm } \\in \\mathbb {R}$ with $\\zeta _+ \\ne \\zeta _-$ .",
"Since $\\alpha _1 < 0$ , $\\zeta _+ > 0$ hence setting $\\tau _2^+ = \\zeta _+$ and $\\tau _2^- = \\max \\lbrace 0, \\zeta _-\\rbrace $ completes the proof.",
"The following proof refers to four lemmas whose statement and proof follow it.",
"[of Theorem REF .]",
"Define $\\tau > 0$ as the maximum $h$ such that Assumption REF holds.",
"The contraction mapping principle together with Lemmas REF , REF , and REF show that the operator $T$ defined by (REF ) and () has a unique fixed point in $\\Gamma $ .",
"Hence, from its definition and equation (REF b), we immediately obtain the existence result.",
"We now show exponential attractivity.",
"Recall the definition of the operator $T$ namely equations (REF ), (): $p &= \\xi + h z_g (\\xi ) \\\\(Tg)(p) &= \\lambda g(\\xi ) + a I^{(1)}_g(\\xi ) - \\bar{\\lambda } I^{(2)}_g(\\xi ).$ Let $g \\in \\Gamma $ be the fixed point of $T$ and set $p &= \\mathsf {u}_n + h z_g(\\mathsf {u}_n) \\\\g(p) &= \\lambda g(\\mathsf {u}_n) + a I^{(1)}_g(\\mathsf {u}_n) - \\bar{\\lambda }I^{(2)}_g(\\mathsf {u}_n).$ Then $|\\mathsf {v}_{n+1} - \\bar{\\lambda }f(\\mathsf {u}_{n+1}) - hg(\\mathsf {u}_{n+1})| &\\le |\\mathsf {v}_{n+1} - \\bar{\\lambda }f(\\mathsf {u}_{n+1}) - hg(p)| + h|g(p) - g(\\mathsf {u}_{n+1})| \\\\&\\le |\\mathsf {v}_{n+1} - \\bar{\\lambda }f(\\mathsf {u}_{n+1}) - hg(p)| + h \\delta |p - \\mathsf {u}_{n+1}|$ since $g \\in \\Gamma $ .",
"Since, by definition, $\\mathsf {v}_{n+1} = \\lambda \\mathsf {v}_n + f(\\mathsf {u}_n + ha\\mathsf {v}_n)$ we have, $|\\mathsf {v}_{n+1} - \\bar{\\lambda }f(\\mathsf {u}_{n+1}) - hg(p)| &= |\\lambda \\mathsf {v}_n + f(\\mathsf {u}_n + ha\\mathsf {v}_n) - \\bar{\\lambda }f(\\mathsf {u}_{n+1}) - h(\\lambda g(\\mathsf {u}_n) + a I^{(1)}_g(\\mathsf {u}_n) - \\bar{\\lambda }I^{(2)}_g(\\mathsf {u}_n))| \\\\&= \\lambda |\\mathsf {v}_n - \\bar{\\lambda }f(\\mathsf {u}_n) - hg(\\mathsf {u}_n)|$ by noting that $&f(\\mathsf {u}_n + ha\\mathsf {v}_n) = f(\\mathsf {u}_n) + ha I^{(1)}_g(\\mathsf {u}_n) \\\\&f(\\mathsf {u}_{n+1}) = f(\\mathsf {u}_n) + h I^{(2)}_g(\\mathsf {u}_n).$ From definition, $\\mathsf {u}_{n+1} = \\mathsf {u}_n + h \\lambda \\mathsf {v}_n + h f(\\mathsf {u}_n + ha \\mathsf {v}_n)$ thus $|p-\\mathsf {u}_{n+1}| &= |\\mathsf {u}_n + hz_g(\\mathsf {u}_n) - \\mathsf {u}_n - h\\lambda \\mathsf {v}_n - hf(\\mathsf {u}_n + ha\\mathsf {v}_n)| \\\\&= h | \\lambda (\\bar{\\lambda } f(\\mathsf {u}_n) + hg(\\mathsf {u}_n)) + f(\\mathsf {u}_n + ha\\mathsf {v}_n) - \\lambda \\mathsf {v}_n - f(\\mathsf {u}_n + ha\\mathsf {v}_n)| \\\\&= h \\lambda |\\mathsf {v}_n - \\bar{\\lambda }f(\\mathsf {u}_n) - hg(\\mathsf {u}_n)|.$ Hence $|\\mathsf {v}_{n+1} - \\bar{\\lambda }f(\\mathsf {u}_{n+1}) - hg(\\mathsf {u}_{n+1})| \\le (\\lambda + h^2 \\lambda \\delta ) |\\mathsf {v}_n - \\bar{\\lambda }f(\\mathsf {u}_n) - hg(\\mathsf {u}_n)|$ as desired.",
"By Assumption REF , $\\lambda + h^2 \\lambda \\delta < 1$ .",
"The following lemma gives basic bounds which are used in the proof of Lemmas REF , REF , REF .",
"Let $g,q \\in \\Gamma $ and $\\xi ,\\eta \\in \\mathbb {R}^d$ then the quantities defined by (REF ), (), (REF ), (REF ) satisfy the following: $|w_g(\\xi )| &\\le K_1, \\\\|w_g(\\xi ) - w_g(\\eta )| &\\le K_2 |\\xi -\\eta |, \\\\|w_g(\\xi ) - w_q(\\xi )| &\\le h|g(\\xi ) - q(\\xi )|, \\\\|z_g(\\xi )| &\\le K_3, \\\\|z_g(\\xi ) - z_g(\\eta )| &\\le \\left( \\lambda K_2 + B_1 \\left(1 + haK_2 \\right) \\right)|\\xi - \\eta |, \\\\|z_g(\\xi ) - z_q(\\xi )| &\\le h \\left( \\lambda + h a B_1 \\right)|g(\\xi ) - q(\\xi )|, \\\\|I_g^{(1)}(\\xi )| &\\le B_1 K_1, \\\\|I_g^{(1)}(\\xi ) - I_g^{(1)}(\\eta )| &\\le ( B_1 K_2 + B_2 K_1 (1 + haK_2 ))|\\xi - \\eta |, \\\\|I_g^{(1)}(\\xi ) - I_q^{(1)}(\\xi )| &\\le h ( B_1 + haB_2 K_1 )|g(\\xi ) - q(\\xi )|, \\\\|I_g^{(2)}(\\xi )| &\\le B_1 K_3 \\\\|I_g^{(2)}(\\xi ) - I_g^{(2)}(\\eta )| &\\le ( ( B_1 + hB_2K_3 )(\\lambda K_2 + B_1 (1+ haK_2)) + B_2K_3 )|\\xi -\\eta |, \\\\|I_g^{(2)}(\\xi ) - I_q^{(2)}(\\xi )| &\\le h(\\lambda + hB_1a)(B_1 + hB_2K_3)|g(\\xi ) - q(\\xi )|.$ These bounds relay on applications of the triangle inequality together with boundedness of $f$ and its derivatives as well as the fact that functions in $\\Gamma $ are bounded and Lipschitz.",
"To illustrate the idea, we will prove the bounds for $w_g, w_q, I^{(1)}_g,$ and $I^{(1)}_q$ .",
"To that end, $|w_g(\\xi )| &= |\\bar{\\lambda } f(\\xi ) + hg(\\xi )| \\\\&\\le \\bar{\\lambda } |f(\\xi )| + h |g(\\xi )| \\\\&\\le \\bar{\\lambda } B_0 + h \\gamma \\\\&= K_1$ establishing the first bound.",
"For the second, $|w_g(\\xi ) - w_g(\\eta )| &\\le \\bar{\\lambda }|f(\\xi ) - f(\\eta )| + h |g(\\xi ) - g(\\eta )| \\\\&\\le \\bar{\\lambda } B_1 |\\xi - \\eta | + h \\delta |\\xi - \\eta | \\\\&= K_2 |\\xi - \\eta |$ as desired.",
"Finally, $|w_g(\\xi ) - w_q(\\xi )| &= |\\bar{\\lambda }f(\\xi ) + h g(\\xi ) - \\bar{\\lambda }f(\\xi ) - h q(\\xi )| \\\\&= h |g(\\xi ) - q(\\xi )|$ as desired.",
"We now turn to the bounds for $I^{(1)}_g, I^{(1)}_q$ , $|I^{(1)}_g(\\xi )| &\\le \\int _0^1 | Df(\\xi + shaw_g(\\xi ))||w_g(\\xi )|ds \\\\&\\le \\int _0^1 B_1 K_1 ds \\\\&= B_1 K_1$ establishing the first bound.",
"For the second bound, $|I^{(1)}_g(\\xi ) - I^{(1)}_g(\\eta )| &\\le \\int _0^1 |Df(\\xi + shaw_g(\\xi ))w_g(\\xi ) - Df(\\eta + shaw_g(\\eta ))w_g(\\xi )|ds \\\\&\\;\\;\\;\\;+ \\int _0^1 |Df(\\eta + shaw_g(\\eta ))w_g(\\xi ) - Df(\\eta + shaw_g(\\eta ))w_g(\\eta )|ds \\\\&\\le K_1 B_2 \\int _0^1 (|\\xi - \\eta | + sha|w_g(\\xi ) - w_g(\\eta )|)ds + B_1 |w_g(\\xi ) - w_g(\\eta )| \\\\&\\le K_1 B_2 (|\\xi - \\eta | + ha K_2 |\\xi - \\eta | ) + B_1 K_2 |\\xi - \\eta | \\\\&= (B_1 K_2 + B_2 K_1(1 + haK_2))|\\xi - \\eta |$ as desired.",
"Finally $|I^{(1)}_g(\\xi ) - I^{(1)}_q(\\xi )| &\\le \\int _0^1 |Df(\\xi + shaw_g(\\xi ))w_g(\\xi ) - Df(\\xi + shaw_g(\\xi ))w_q(\\xi )|ds \\\\&\\;\\;\\;\\;+ \\int _0^1 |Df(\\xi + shaw_g(\\xi ))w_q(\\xi ) - Df(\\xi + shaw_q(\\xi ))w_q(\\xi )|ds \\\\&\\le B_1 \\int _0^1 |w_g(\\xi ) - w_q(\\xi )|ds + K_1 B_2 \\int _0^1 |\\xi + shaw_g(\\xi ) - \\xi - shaw_q(\\xi )|ds \\\\&\\le h B_1 |g(\\xi ) - q(\\xi )| + h^2 a B_2 K_1 |g(\\xi ) - q(\\xi )| \\\\&= h(B_1 + ha B_2 K_1)|g(\\xi ) - q(\\xi )|$ as desired.",
"The bounds for $z_g,z_q,I^{(2)}_g,$ and $I^{(2)}_q$ follow similarly.",
"We also need the following three lemmas: Suppose Assumption REF holds.",
"For any $g \\in \\Gamma $ and $p \\in \\mathbb {R}^d$ there exists a unique $\\xi \\in \\mathbb {R}^d$ satisfying (REF ).",
"Suppose Assumption REF holds.",
"The operator $T$ defined by () satisfies ${T : \\Gamma \\rightarrow \\Gamma }$ .",
"Suppose Assumption REF holds.",
"For any $g_1,g_2 \\in \\Gamma $ , we have $\\Vert Tg_1 - Tg_2\\Vert _\\Gamma \\le \\mu \\Vert g_1 - g_2\\Vert _\\Gamma $ where $\\mu < 1$ .",
"Now we prove these three lemmas.",
"[of Lemma REF .]",
"Consider the iteration of the form $\\xi ^{k+1} = p - h z_g(\\xi ^k).$ For any two sequences $\\lbrace \\xi ^k\\rbrace $ , $\\lbrace \\eta ^k\\rbrace $ generated by this iteration we have, by Lemma REF , $|\\xi ^{k+1} - \\eta ^{k+1}| &\\le h |z_g(\\eta ^k) - z_g(\\xi ^k)| \\\\&\\le h(\\lambda K_2 + B_1(1+haK_2))|\\xi ^k - \\eta ^k| \\\\&= c |\\xi ^k - \\eta ^k|$ which is a contraction by (REF ).",
"[of Lemma REF .]",
"Let $g \\in \\Gamma $ and $p \\in \\mathbb {R}^d$ then by Lemma REF there is a unique $\\xi \\in \\mathbb {R}^d$ such that (REF ) is satisfied.",
"Then $|(Tg)(p)| &\\le \\lambda |g(\\xi )| + a |I^{(1)}_g(\\xi )| + \\tilde{\\lambda } |I^{(2)}_g(\\xi )| \\\\&\\le \\lambda \\gamma + a B_1(\\tilde{\\lambda }B_0 + h \\gamma ) + \\tilde{\\lambda }B_1(\\lambda (\\tilde{\\lambda }B_0 + h\\gamma ) + B_0) \\\\&= (\\lambda + hB_1(a + \\lambda \\tilde{\\lambda })) \\gamma + \\tilde{\\lambda }B_0B_1(a + \\tilde{\\lambda }) \\\\&\\le \\gamma $ with the last inequality following from (REF ).",
"Let $p_1, p_2 \\in \\mathbb {R}^d$ then, by Lemma REF , there exist $\\xi _1,\\xi _2 \\in \\mathbb {R}^d$ such that (REF ) is satisfied with $p=\\lbrace p_1,p_2\\rbrace $ .",
"Hence, by Lemma REF , $|(Tg)(p_1) - (Tg)(p_2)| &\\le \\lambda |g(\\xi _1) - g(\\xi _2)| + a |I^{(1)}_g(\\xi _1) - I^{(1)}_g(\\xi _2)| + \\tilde{\\lambda } |I^{(2)}_g(\\xi _1) - I^{(2)}_g(\\xi _2)| \\\\&\\le K |\\xi _1 - \\xi _2|$ where we define $K \\lambda \\delta + a( B_1 K_2 + B_2 K_1 (1 + haK_2 )) + \\tilde{\\lambda } ( ( B_1 + hB_2K_3 )(\\lambda K_2 + B_1 (1+ haK_2)) + B_2K_3 ).$ Now, using (REF ) and the proof of Lemma REF , $|\\xi _1 - \\xi _2| &\\le |p_1 - p_2| + h |z_g(\\xi _1) - z_g(\\xi _2)| \\\\&\\le |p_1 - p_2| + c |\\xi _1 - \\xi _2|.$ Since $c < 1$ by (REF ), we obtain $|\\xi _1 - \\xi _2| \\le \\frac{1}{1-c}|p_1-p_2|$ thus $|(Tg)(p_1) - (Tg)(p_2)| \\le \\frac{K}{1-c}|p_1-p_2| \\le \\delta |p_1 - p_2|.$ To see the last inequality, we note that $\\frac{K}{1-c} \\le \\delta \\iff K - \\delta (1-c) \\le 0$ and $K - \\delta (1-c) = \\alpha _2 \\delta ^2 + \\alpha _1 \\delta + \\alpha _0$ by (REF ) hence (REF ) gives the desired result.",
"[of Lemma REF .]",
"By Lemma REF , for any $p \\in \\mathbb {R}^d$ and $g_1,g_2 \\in \\Gamma $ , there are $\\xi _1,\\xi _2 \\in \\mathbb {R}^d$ such that $p &= \\xi _j + h z_{g_j}(\\xi _j)\\\\(Tg_j)(p) &= \\lambda g_j (\\xi _j) + a I^{(1)}_{g_j}(\\xi _j) - \\tilde{\\lambda } I^{(2)}_{g_j}(\\xi _j)$ for $j=1,2$ .",
"Then $|(Tg_1)(p) - (Tg_2)(p)| \\le \\lambda |g_1(\\xi _1) - g_2(\\xi _2)| + a |I^{(1)}_{g_1}(\\xi _1) - I^{(1)}_{g_2}(\\xi _2)| + \\tilde{\\lambda }|I^{(2)}_{g_1}(\\xi _1) - I^{(2)}_{g_2}(\\xi _2)|.$ Note that $|g_1(\\xi _1) - g_2(\\xi _2)| &= |g_1(\\xi _1) - g_2(\\xi _2) - g_2(\\xi _1) + g_2(\\xi _1)| \\\\&\\le |g_1(\\xi _1) - g_2(\\xi _1)| + \\delta |\\xi _1 - \\xi _2|.$ Similarly, by Lemma REF , $|I^{(1)}_{g_1}(\\xi _1) - I^{(1)}_{g_2}(\\xi _2)| &= |I^{(1)}_{g_1}(\\xi _1) - I^{(1)}_{g_2}(\\xi _2) - I^{(1)}_{g_2}(\\xi _1) + I^{(1)}_{g_2}(\\xi _1)| \\\\&\\le |I^{(1)}_{g_1}(\\xi _1) - I^{(1)}_{g_2}(\\xi _1)| + |I^{(1)}_{g_2}(\\xi _1) - I^{(1)}_{g_2}(\\xi _2)| \\\\&\\le h ( B_1 + haB_2 K_1)|g_1(\\xi _1) - g_2(\\xi _1)| + ( B_1 K_2 + B_2 K_1 (1 + haK_2 ))|\\xi _1 - \\xi _2|$ Finally, $|I^{(2)}_{g_1}(\\xi _1) - I^{(2)}_{g_2}(\\xi _2)| &= |I^{(2)}_{g_1}(\\xi _1) - I^{(2)}_{g_2}(\\xi _2) - I^{(2)}_{g_2}(\\xi _1) + I^{(2)}_{g_2}(\\xi _1)| \\\\&\\le |I^{(2)}_{g_1}(\\xi _1) - I^{(2)}_{g_2}(\\xi _1)| + |I^{(2)}_{g_2}(\\xi _1) - I^{(2)}_{g_2}(\\xi _2)| \\\\&\\le h(\\lambda + hB_1a)(B_1 + hB_2K_3)|g_1(\\xi _1) - g_2(\\xi _1)| + \\\\&+( ( B_1 + hB_2K_3 )(\\lambda K_2 + B_1 (1+ haK_2)) + B_2K_3 )|\\xi _1 - \\xi _2|$ Putting these together and using (REF ), we obtain $|(Tg_1)(p) - (Tg_2)(p)| \\le ( \\lambda + Q_2 )|g_1(\\xi _1) - g_2(\\xi _1)| + Q_1 |\\xi _1 - \\xi _2|.",
"$ Now, by Lemma REF , $|\\xi _1 - \\xi _2| &\\le h |z_{g_1}(\\xi _1) - z_{g_2}(\\xi _2) - z_{g_2}(\\xi _1) + z_{g_2}(\\xi _1)| \\\\&\\le h (|z_{g_1}(\\xi _1) - z_{g_2}(\\xi _1)| + |z_{g_2}(\\xi _1) - z_{g_2}(\\xi _2)|) \\\\&\\le h^2(\\lambda + haB_1)|g_1(\\xi ) - g_2(\\xi _1)| + h(\\lambda K_2 + B_1(1+haK_2))|\\xi _1 - \\xi _2| \\\\&= h^2(\\lambda + haB_1)|g_1(\\xi ) - g_2(\\xi _1)| + Q_3|\\xi _1 - \\xi _2|$ using (REF ).",
"Since, by (REF ), $Q_3 < 1$ , we obtain $|\\xi _1 - \\xi _2| \\le \\frac{h^2(\\lambda + haB_1)}{1-Q_3} |g_1(\\xi _1) - g_2(\\xi _1)|$ and thus $|(Tg_1)(p) - (Tg_2)(p)| &\\le \\left( \\lambda + Q_2 + \\frac{h^2(\\lambda + haB_1)Q_1}{1-Q_3} \\right) |g_1(\\xi _1) - g_2(\\xi _1)| \\\\&= \\mu |g_1(\\xi _1) - g_2(\\xi _1)|$ by (REF ).",
"Taking the supremum over $\\xi _1$ then over $p$ gives the desired result.",
"Since $\\mu < 1$ by (REF ), we obtain that $T$ is a contraction on $\\Gamma $ .",
"We consider the equation $&\\ddot{u} + 2 \\sqrt{\\mu } \\dot{u} + \\nabla \\Phi (u) = 0 \\\\&u(0) = \\mathsf {u}_0, \\quad \\dot{u}(0) = \\mathsf {v}_0.$ Set $v = \\dot{u}$ then we have $\\begin{bmatrix}\\dot{u} \\\\\\dot{v}\\end{bmatrix}=\\begin{bmatrix}v \\\\-2\\sqrt{\\mu } v - \\nabla \\Phi (u)\\end{bmatrix}.$ Define the maps $f_1(u,v) \\begin{bmatrix}v \\\\-2 \\sqrt{\\mu } v\\end{bmatrix},\\quad f_2(u,v) \\begin{bmatrix}0 \\\\- \\nabla \\Phi (u)\\end{bmatrix}$ then $\\begin{bmatrix}\\dot{u} \\\\\\dot{v}\\end{bmatrix}= f_1(u,v) + f_2(u,v).$ We first solve the system $\\begin{bmatrix}\\dot{u} \\\\\\dot{v}\\end{bmatrix}= f_1(u,v).$ Clearly $v(t) = e^{-2\\sqrt{\\mu }t} \\mathsf {v}_0$ hence $u(t) &= \\mathsf {u}_0 + \\int _0^t e^{-2\\sqrt{\\mu }s} \\mathsf {v}_0 \\: ds \\\\&= \\mathsf {u}_0 + \\frac{1}{2 \\sqrt{\\mu }} \\left( 1 - e^{-2\\sqrt{\\mu }t} \\right) \\mathsf {v}_0.$ This gives us the flow map $\\psi _1(\\mathsf {u}, \\mathsf {v}; t) = \\begin{bmatrix}\\mathsf {u}+ \\frac{1}{2 \\sqrt{\\mu }} \\left( 1 - e^{-2\\sqrt{\\mu }t} \\right) \\mathsf {v}\\\\e^{-2\\sqrt{\\mu }t} \\mathsf {v}\\end{bmatrix}.$ We now solve the system $\\begin{bmatrix}\\dot{u} \\\\\\dot{v}\\end{bmatrix}= f_2(u,v).$ Clearly $u(t) = \\mathsf {u}_0$ hence $v(t) = \\mathsf {v}_0 - t \\nabla \\Phi (\\mathsf {u}_0).$ This gives us the flow map $\\psi _2(\\mathsf {u},\\mathsf {v};t) = \\begin{bmatrix}\\mathsf {u}\\\\\\mathsf {v}- t \\nabla \\Phi (\\mathsf {u})\\end{bmatrix}.$ The composition of the flow maps is then $(\\psi _2 \\circ \\psi _1)(\\mathsf {u}, \\mathsf {v}; t) = \\begin{bmatrix}\\mathsf {u}+ \\frac{1}{2 \\sqrt{\\mu }} \\left( 1 - e^{-2\\sqrt{\\mu }t} \\right) \\mathsf {v}\\\\e^{-2\\sqrt{\\mu }t} \\mathsf {v}- t \\nabla \\Phi \\left( \\mathsf {u}+ \\frac{1}{2 \\sqrt{\\mu }} \\left( 1 - e^{-2\\sqrt{\\mu }t} \\right) \\mathsf {v}\\right)\\end{bmatrix}.$ Mapping $t$ to the time-step $\\sqrt{h}$ gives the numerical method (REF )."
]
] | 1906.04285 |
[
[
"Normalization of Hamiltonian and Nonlinear Stability of the Triangular\n Equilibrium Points in Non-resonance Case with Perturbations"
],
[
"Abstract For the study of nonlinear stability of a dynamical system, normalized Hamiltonian of the system is very important to discuss the dynamics in the vicinity of invariant objects.",
"In general, it represents a nonlinear approximation to the dynamics, which is very helpful to obtain the information about realistic solution of the problem.",
"Present paper reflects about normalization of the Hamiltonian and analysis of nonlinear stability in non-resonance case, in the Chermnykh-like problem under the influence of perturbations in the form of radiation pressure, oblateness, and a disc.",
"To describe nonlinear stability, initially, quadratic part of the Hamiltonian is normalized in the neighborhood of triangular equilibrium point and then higher order normalization is performed.",
"Due to the presence of perturbations and a tedious huge algebraic computation for intermediate terms, we have computed only up to the fourth order normalized Hamiltonian using Lie transforms.",
"In non-resonance case, nonlinear stability of the system is discussed with the help of Arnold-Moser theorem.",
"Again, the effects of radiation pressure, oblateness and presence of the disc are analyzed, separately and it is observed that in the absence as well as presence of perturbation parameters, triangular equilibrium point is unstable in nonlinear sense within the stability range $0<\\mu<\\mu_1=\\bar{\\mu_c}$ due to failure of Arnold-Moser theorem.",
"However, perturbation parameters affect the values of $\\mu$ at which $D_4=0$, significantly.",
"This study may help to analyze more generalized cases of the problem in the presence of some other types of perturbations such as P-R drag and solar wind drag.",
"The results are limited to the regular symmetric disc but in future it can be extended."
],
[
"Introduction",
"Study of stability property of a dynamical system is a necessary step which brings not only the system to tackle many realistic problems of the world but also helps to understand the motion of test particle for a long time of evolution.",
"The stability of the system for a long time of evolution is an important and critical issue and hence, a number of researchers are studying the Hamiltonian system of the problem in the vicinity of elliptic equilibrium point in many fields such as mathematical physics, dynamical astronomy, astronomy, celestial mechanics etc.",
"Many researchers [9], [31], [33], [6], [14], [22] have studied restricted problem of three bodies in the context of stability in classical cases and some of the researchers [3], [32], [17], [44], [26], [1] have discussed the stability for generalized cases.",
"However, a very little attention has been given to the problem with the effect of perturbations such as radiation pressure, oblateness, drag forces, and presence of a disc like structure in the problem.",
"In the present paper, we consider Chermnykh-like problem under the influence of perturbations in the form of radiation pressure, oblateness and presence of a disc, which is rotating about common center of mass of the system.",
"Chermnykh-like problem is a result of some modification in original Chermnykh's problem which consists with the motion of a point mass in a plane under the influence of gravitational effect of a uniformly rotating dumb-bell and it was first time studied by [5].",
"This problem has a number of applications in different areas such as celestial mechanics, dynamical astronomy, extra solar planetary system and chemistry [16], [41], [18], [19], [20].",
"The different aspects of the problem such as existence of equilibrium points, stability analysis in resonance and non-resonance cases, computation of orbits, Lyapunov characteristic exponent of trajectories etc.",
"have been studied by many authors [14], [16], [15], [36], [37], [21], [50], [25], [27], [24].",
"[12] studied KAM stability of Trojan asteroid under the frame of planar restricted three body problem and [2] described condition of the applicability of the Nekhoroshev stability theorem.",
"Moreover, nonlinear stability of Trojan asteroid in the sense of Nekhoroshev stability has described by many researcher [30], [29], [13], [11], [28].",
"In the present paper, we are interested to discuss nonlinear stability of triangular equilibrium point in non-resonance case under the influence of perturbations in the form of radiation pressure, oblateness and the disc with the help of Arnold-Moser theorem [33], [14].",
"In order to discuss, nonlinear stability of triangular equilibrium point in non-resonance case with the help of Arnold- Moser theorem, we obtain normal forms of the Hamiltonian of the system up to a finite order, which are very important to discuss the dynamics in the neighborhood of invariant objects.",
"Several researchers [38], [39], [4], [8], [46], [45], [47], [48], [22], [6] have described the different normalization processes and also, they have utilized the normalized Hamiltonian to analyze the nonlinear stability of the dynamical system.",
"The main idea behind the normal form is to construct a suitable transformation of phase space which yields the simplest form up to a certain order of accuracy of a given system of differential equations.",
"In short, it can be used to approximate the dynamics and hence, study of real world problems.",
"There are several approaches [4], [7], [46], [45], [10], [48] to find the transformation equations to reduce the Hamiltonian into simplest form.",
"We have performed the normalization of Hamiltonian of the system up to fourth order by the method of Lie transforms which are described well in [22] and [6].",
"The paper is organized as follows: Section- described the formulation of problem whereas, diagonalization of the Hamiltonian is discussed in detail under Section-.",
"Section- is devoted to non-linear stability in non-resonance case by the use of Arnold-Moser theorem, whereas Subsection-REF contains computation of coefficients of the normalized Hamiltonian up to order four on the basis of Lie transform.",
"Finally, the results are concluded in Section-.",
"Algebraic as well as numerical computation has been performed with the help of Mathematica$^{®}$ [49] software package."
],
[
"Formulation of the problem",
"Mathematical formulation of the problem is similar to [27] whereas, for self sufficient paper it is as follows.",
"We consider the motion of infinitesimal mass under the influence of gravitational field of massive bodies (also known as primaries, here bigger primary is taken as radiating body and smaller is an oblate spheroid) and perturbations in the form of radiation pressure of bigger primary, oblateness of smaller primary and a disc, which is rotating about the common center of mass of the system having power-law density profile $\\rho (r)=\\dfrac{c}{r^{p}}$ , where $p\\in \\mathbb {N}$ (here, we have taken $p=3$ ) and $c$ is a constant which depends on total mass of the disc.",
"It is assumed that the effect of infinitesimal mass on the motions of both the primaries as well as of the disc, is negligible.",
"The proposed model can be realize by considering, a disc about the common center of mass of Sun-Planet system and an infinitesimal body such as spacecraft or satellite moves under the influence of celestial forces.",
"Units of mass and distance are taken as the sum of masses of the primaries and separation between them, respectively whereas, unit of time is taken as time period of rotating frame.",
"Under these assumption, Hamiltonian function of the Chermnykh-like problem in the presence of radiation pressure, oblateness and the disc, in the phase coordinate $(x, \\, y, \\, p_x, \\, p_y)$ , is written as [24]: $H\\left(x,y,p_x,p_y\\right)&=&\\frac{1}{2}\\left(p^2_x+p^2_y\\right)+\\mathbf {n}\\left(yp_x-xp_y\\right)\\nonumber \\\\&&-\\frac{(1-\\mu )q_1}{r_1}-(1+\\frac{A_2}{2r_{2}^{3}})\\frac{\\mu }{r_2}\\nonumber \\\\&&-\\pi c h\\left[\\frac{2(b-a)}{abr}+\\frac{7\\log \\frac{b}{a}}{8r^2}\\right],$ where $p_x=\\dot{x}-\\mathbf {n}y$ and $p_y=\\dot{y}+\\mathbf {n}x$ are momenta coordinate.",
"Last term on the right side is due to presence of the disc.",
"Mean motion of the system is given as: $\\mathbf {n}&=&\\sqrt{q_1+\\frac{3}{2}A_2-2f_b(r)},$ where mass reduction factor $q_1=(1-\\frac{F_p}{F_g})$ [40] with $F_p$ and $F_g$ as the radiation pressure and gravitational attraction forces respectively (here, $0< q_1< 1$ because in solar planetary system for radiating body as the Sun, $\\frac{F_p}{F_g}< 1$ )[42]; oblateness coefficients $A_2=\\frac{R^{2}_e-R^{2}_p}{5R^{2}}$ [34], with $R_e$ and $R_p$ be the equatorial and polar radii of the oblate body, respectively and $R$ is the separation between both the primaries (here, $0< A_2< 1$ for oblate body but for prolate body $-1< A_2< 0$ ); $f_b(r)$ is the gravitational force due to the disc which is given as [27] $f_b(r)&=&-\\pi c h\\left[\\frac{2(b-a)}{abr}+\\frac{7\\log \\frac{b}{a}}{8r^2}\\right] ,$ where $a, b$ are inner and outer radii respectively, of the radially symmetric disc (here, the dimension of the disc is taken in such a way that disc width $b-a< 0.3$ whereas, thickness of the disc is $h=10^{-4}$ and constant $c=1910.83$ ).",
"$\\mu =\\frac{m_J}{M_S+m_J}$ be the mass parameter in the Sun-Jupiter system ($M_S$ and $m_J$ are masses of the Sun and the Jupiter, respectively).",
"The coordinates of triangular equilibrium points of the problem are given as [27]: $x_e&=&\\frac{q_1^{\\frac{2}{3}}}{2}-\\mu +(q_1^{\\frac{2}{3}} \\delta _1-\\delta _2),\\\\y_e&=&\\pm q_1^{\\frac{1}{3}}[1-\\frac{q_1^{\\frac{2}{3}}}{4}+(2-q_1^{\\frac{2}{3}}) \\delta _1+\\delta _2]^{\\frac{1}{2}}, $ where $\\delta _1&=&\\frac{1}{3}\\left[1-\\mathbf {n}^2 +\\frac{2 \\pi c h (b-a)}{a b \\lbrace \\mu ^2 +q_1^{\\frac{2}{3}}(1-\\mu )\\rbrace ^{\\frac{3}{2}}}\\right.\\nonumber \\\\&&\\left.+\\frac{3 \\pi c h {\\log \\frac{b}{a}}}{8\\lbrace \\mu ^2 +q_1^{\\frac{2}{3}}(1-\\mu )\\rbrace ^2}\\right], \\\\\\delta _2&=&\\frac{1}{3(1+\\frac{5}{2}A_2)}\\left[1-\\mathbf {n}^2+\\frac{3A_2}{2}+\\right.\\nonumber \\\\&&\\left.\\frac{2\\pi c h(b-a)}{a b\\lbrace \\mu ^2 +q_1^{\\frac{2}{3}}(1-\\mu )\\rbrace ^{\\frac{3}{2}}}+\\frac{3 \\pi c h{\\log \\frac{b}{a}}}{8\\lbrace \\mu ^2 +q_1^{\\frac{2}{3}}(1-\\mu )\\rbrace ^2}\\right]$ are small quantities."
],
[
"Diagonalization of Hamiltonian",
"For the simplicity and to over come the expression computation's complexity, we have considered only linear order terms in perturbing parameters through the computations in the paper.",
"Therefore, before diagonalization of the Hamiltonian, we obtain mean motion and hence, triangular equilibrium points $L_{4,5}(x_e, \\, y_e)$ as a linear function of parameters $\\mu $ , $q_1, \\, A_2$ and $b$ .",
"Since, $q_1<1, \\, A_2<1$ and $b>1$ so, we have supposed that $q_1=1-\\epsilon _{1}$ and $b=1+\\epsilon _{2}$ , $0<\\epsilon _{1}, \\epsilon _{2}\\ll 1$ .",
"First, we have expanded mean motion $(\\mathbf {n})$ and then coordinates of triangular equilibrium points $(x_e, \\, y_e)$ about $\\mu =0, \\, \\epsilon _{1}=0, \\, A_2=0$ and $\\epsilon _{2}=0$ , respectively and finally, taking linear order terms of $\\mu , \\, \\epsilon _{1}, \\, A_2$ and $\\epsilon _{2}$ , we get $\\mathbf {n}&=& 1+\\frac{3 A_2}{4}-\\frac{\\epsilon _1}{2}+\\frac{3 \\epsilon _2}{2},\\\\x_e&=&\\frac{1}{2}-\\mu -\\frac{\\epsilon _1}{3},\\\\y_e&=&\\pm \\frac{\\sqrt{3}}{2}\\left( 1-\\frac{2A_2}{3}+\\frac{\\epsilon _1}{4}-\\frac{2\\epsilon _2}{3}\\right),$ where $‘+’$ sign corresponds to $L_4$ point and $‘-’$ for $L_5$ .",
"We have discussed the stability of $L_4$ point whereas, dynamics of $L_5$ is similar to that of $L_4$ .",
"For the convenience, we shift the origin at equilibrium point $L_4$ using simple translation.",
"$&&x^{*}=x-x_e, \\quad y^{*}=y-y_e, \\nonumber \\\\&& p^{*}_{x}=p_{x}+y_e, \\quad p^{*}_{y}=p_{y}-x_e.$ Applying this change of variable to the Hamiltonian (REF ), we obtain $H^{*}&=&\\frac{1}{2}\\left[(p^{*}_{x}-y_e)^2+(p^{*}_{y}+x_e)^2\\right]+\\nonumber \\\\&& n\\left[(y^{*}+y_e)(p^{*}_{x}-y_e)-(x^{*}+x_e)(p^{*}_{y}+x_e)\\right]\\nonumber \\\\&&-\\frac{(1-\\mu )q_1}{[(x^{*}+x_e+\\mu )^2+(y^{*}+y_e)^2]^{\\frac{1}{2}}}\\nonumber \\\\&&-\\frac{\\mu }{(x^{*}+x_e+\\mu -1)^2+(y^{*}+y_e)^2]^{\\frac{1}{2}}}\\nonumber \\\\&&-\\frac{\\mu A_2}{2[(x^{*}+x_e+\\mu )^2+(y^{*}+y_e)^2]^{\\frac{3}{2}}}\\nonumber \\\\&&-\\pi c h\\left[\\frac{2(b-a)}{ab[(x^{*}+x_e)^2+(y^{*}+y_e)^2]^{\\frac{1}{2}}}\\right.\\nonumber \\\\&&\\left.+\\frac{7\\log \\frac{b}{a}}{8[(x^{*}+x_e)^2+(y^{*}+y_e)^2]}\\right].$ Expanding the resulting Hamiltonian in Taylor series about origin (which is actually the triangular equilibrium point), as follows $&&H^*=H_0+H_1+H_2+H_3+H_4+\\dots ,$ where $&&H_n=\\sum {H_{jkls}}{x^{*}}^{j}{y^{*}}^{k}{p^{*}_{x}}^{l}{p^{*}_{y}}^{s}\\quad \\text{with}\\nonumber \\\\&&j+k+l+s=n .$ Since, origin is an equilibrium point therefore, first order term $H_1$ must vanish whereas, constant term $H_0$ drop out because it is irrelevant to the dynamics.",
"The quadratic term $H_2$ , is useful for higher order normal forms, around the triangular point $L_4$ , and given as $H_2&=&\\frac{1}{2}\\left({p^{*}_{x}}^{2}+{p^{*}_{y}}^{2}\\right)+\\mathbf {n}\\left(y^*p^{*}_{x}-x^*p^{*}_{y}\\right)-\\nonumber \\\\&&\\frac{1}{2}\\left(P {x^{*}}^2 +Q {y^{*}}^2+S x^{*}y^{*}\\right),$ where coefficients $P, \\, Q$ and $S$ are given as $&&P=-\\frac{1}{4}\\left( 1+\\frac{49\\epsilon _1}{16}-\\frac{3A_2}{2}-\\frac{\\epsilon _2}{4}\\right),\\\\&&Q=\\frac{5}{4}\\left( 1-\\frac{19\\epsilon _1}{80}+\\frac{9A_2}{10}+\\frac{5\\epsilon _2}{4}\\right),\\\\&&S=\\frac{3\\sqrt{3}(1-2\\mu )}{4}\\left( 1+\\frac{73\\epsilon _1}{48}+\\frac{11A_2}{6}-\\frac{53\\epsilon _2}{12}\\right).$ Thus, $H_2$ becomes $H_2&=&\\frac{1}{2}\\left({p^{*}_{x}}^{2}+{p^{*}_{y}}^{2}\\right)+\\mathbf {n}\\left(y^*p^{*}_{x}-x^*p^{*}_{y}\\right)\\nonumber \\\\&&-\\frac{1}{2}\\left(\\frac{-1}{4} {x^{*}}^2 +\\frac{5}{4} {y^{*}}^2+a_0 x^{*}y^{*}\\right)\\nonumber \\\\&&-\\frac{1}{2}\\left(\\frac{-49}{64} {x^{*}}^2 -\\frac{19}{64}{y^{*}}^2+a_{1} x^{*}y^{*} \\right)\\epsilon _{1}\\nonumber \\\\&&-\\frac{1}{2}\\left(\\frac{3}{8} {x^{*}}^2 +\\frac{9}{8} {y^{*}}^2+a_{2} x^{*}y^{*}\\right)A_2\\nonumber \\\\&&-\\frac{1}{2}\\left(\\frac{1}{16} {x^{*}}^2 +\\frac{25}{16} {y^{*}}^2+a_{3} x^{*}y^{*}\\right)\\epsilon _2,$ where $&&a_0=\\frac{3\\sqrt{3}(1-2\\mu )}{4}, \\, a_1=\\frac{73\\sqrt{3}(1-2\\mu )}{64}, \\nonumber \\\\&&a_2=\\frac{11\\sqrt{3}(1-2\\mu )}{8}, \\, a_3=\\frac{-53\\sqrt{3}(1-2\\mu )}{16}$ Since, we are dealing the problem in the presence of three type of perturbations in the form of radiation pressure, obletness and the disc, therefore, for simplicity, coefficients $H_{jkls}$ in equation (REF ) are splitted into four parts such as $g_{jkls}, \\, g_{jklse1}$ , $g_{jklsA}$ and $g_{jklse2}$ , which correspond to the terms due to classical case (i.e.",
"absence of perturbations), radiation effects $\\epsilon _1$ , oblateness $A_2$ and presence of the disc $\\epsilon _2$ , respectively.",
"In other words, $H_{jkls}=g_{jkls}+g_{jklse1}+g_{jklsA}+g_{jklse2},$ where $g_{jkls}$ represents coefficients for classical part, $g_{jklse1}$ indicates coefficients for radiation pressure terms, $g_{jklsA}$ used for oblateness and $g_{jklse2}$ corresponds to the disc with $j, \\, k, \\, l, \\, s=0, \\, 1, \\, 2,\\,3,\\,4$ such that $j+k+l+s=4$ .",
"However, in the absence of perturbations i.e.",
"when $A_2=\\epsilon _1=\\epsilon _2=0$ then $H_{jkls}=g_{jkls}$ , which is equivalent to the coefficients of the Hamiltonian in classical case.",
"Since, Hamilton's equations of motion of the infinitesimal mass are written as $\\begin{bmatrix}\\dot{x^{*}}\\\\\\dot{y^{*}}\\\\\\dot{p^{*}_x} \\\\\\dot{p^{*}_y}\\end{bmatrix}=J_4.\\nabla H_2= J_4.",
"Hess[H_2]\\begin{bmatrix}x^*\\\\y^*\\\\p^{*}_x \\\\p^{*}_y\\end{bmatrix},$ where $J_4=\\begin{bmatrix}0&0&1&0 \\\\0&0&0&1 \\\\-1&0&0&0\\\\0&-1&0&0\\end{bmatrix},$ and $J_4.Hess[H_2]=\\mathbf {M}=\\begin{bmatrix}0&\\mathbf {n}&1&0 \\\\-\\mathbf {n}&0&0&1 \\\\P&S&0&\\mathbf {n}\\\\S&Q&-\\mathbf {n}&0\\end{bmatrix}.$ The characteristic equation of the monodromy matrix $\\mathbf {M}$ is $&& \\lambda ^4+\\left(-P^2-Q^2+2\\mathbf {n}^2\\right)\\lambda ^2+\\left(\\mathbf {n}^2P+\\mathbf {n}^2Q\\right.\\nonumber \\\\&&\\left.+PQ-S^2+\\mathbf {n}^4\\right)=0.$ From equation (REF ), it can be easily obtained that system (REF ) is stable, if the mass parameter $\\mu $ satisfy the condition $0<\\mu <\\bar{\\mu }_c$ , where $\\bar{\\mu }_c$ is the Routh value of the mass ratio of the problem [23].",
"Since, we are studying the same case $0<\\mu <\\bar{\\mu }_c$ so, it is assumed that four roots of characteristic equation (REF ) are purely imaginary say, $\\lambda _{1,2}=\\pm i \\omega _1$ and $\\lambda _{3,4}=\\pm i \\omega _2$ .",
"As, the real values of $\\omega _{1,2}$ are frequencies of the linear oscillations of the infinitesimal mass at the equilibrium point $L_4$ and it is obvious that they differ for the stability region $0<\\mu <\\bar{\\mu }_c$ .",
"Now, our aim is to obtain a real symplectic change of variable due to which one can find the real diagonalize Hamiltonian from (REF ).",
"For that, first step is to obtain the characteristic vectors of the matrix $\\mathbf {M}$ corresponding to the characteristic roots.",
"If we denote the matrix $\\mathbf {M-\\lambda I_4}$ by $\\mathbf {M_\\lambda }$ [22], then $\\mathbf {M_\\lambda }=\\begin{bmatrix}\\mathbf {M_\\lambda ^{^{\\prime }}}&\\mathbf {I_2} \\\\\\mathbf {M^{^{\\prime }}}&\\mathbf {M_\\lambda ^{^{\\prime }}}\\end{bmatrix},$ where $\\mathbf {M_\\lambda ^{^{\\prime }}}=\\begin{bmatrix}-\\lambda &\\mathbf {n} \\\\-\\mathbf {n}&-\\lambda \\end{bmatrix}, \\quad \\mathbf {M^{^{\\prime }}}=\\begin{bmatrix}P&S \\\\S&Q\\end{bmatrix}$ and $\\mathbf {I_2}$ is identity matrix of $2\\times 2$ .",
"Since, $\\lambda $ is a root of the matrix $\\mathbf {M}$ and hence, the kernel of $\\mathbf {M_\\lambda }$ is obtained by solving the system $\\begin{bmatrix}\\mathbf {M_\\lambda ^{^{\\prime }}}&\\mathbf {I_2} \\\\\\mathbf {M^{^{\\prime }}}&\\mathbf {M_\\lambda ^{^{\\prime }}}\\end{bmatrix}.\\begin{bmatrix}\\mathbf {X_1} \\\\\\mathbf {X_2}\\end{bmatrix}=\\begin{bmatrix}0 \\\\0\\end{bmatrix},$ where $\\,\\mathbf {X_1}=\\begin{bmatrix}x^* \\\\y^*\\end{bmatrix}$ and $\\mathbf {X_2}=\\begin{bmatrix}p^{*}_x \\\\p^{*}_y\\end{bmatrix}$ i.e.",
"$&&\\mathbf {M_\\lambda ^{^{\\prime }} X_1}+\\mathbf {I_2 X_2}={0},\\\\&&\\mathbf {M^{^{\\prime }} X_1}+\\mathbf {M_\\lambda ^{^{\\prime }} X_2}={0}.$ From above equations, we have $\\begin{bmatrix}\\mathbf {n}^2+P-\\lambda &S+2\\mathbf {n}\\lambda \\\\S+2\\mathbf {n}\\lambda &\\mathbf {n}^2+Q-\\lambda ^2\\end{bmatrix}.\\begin{bmatrix}x^* \\\\y^*\\end{bmatrix}=\\begin{bmatrix}0 \\\\0\\end{bmatrix},$ which gives either $&&x^*=S+2\\mathbf {n}\\lambda , \\, y^*=-(\\mathbf {n}^2+P-\\lambda ^2)\\quad \\\\\\text{or} &&x^*=(\\mathbf {n}^2+Q-\\lambda ^2), \\,y^*=-S+2\\mathbf {n}\\lambda .$ If we take first set (REF ) of $\\,x^*\\,$ and $\\,y^*\\,$ , then final symplectic matrix is similar to that of [22], whereas if we use second set () of $\\,x^*\\,$ and $\\,y^*\\,$ , results agree with that of [9] and [33].",
"Here, we use second set () of$x^*$ and $y^*$ to proceed further.",
"Putting, these $x^*$ and $y^*$ into equation (), we get $&&p^{*}_x=-\\lambda ^3-(\\mathbf {n}^2-Q)\\lambda +\\mathbf {n}S,\\\\&&p^{*}_y=\\mathbf {n}\\lambda ^2-S\\lambda +\\mathbf {n}Q+\\mathbf {n}^3.$ Thus, from the equations (-), the characteristic vector of the matrix $\\mathbf {M}$ is given as $\\begin{bmatrix}x^*\\\\y^*\\\\p^{*}_x \\\\p^{*}_y\\end{bmatrix}=\\begin{bmatrix}(\\mathbf {n}^2+Q-\\lambda ^2)\\\\-S+2\\mathbf {n}\\lambda \\\\-\\lambda ^3-(\\mathbf {n}^2-Q)\\lambda +\\mathbf {n}S\\\\\\mathbf {n}\\lambda ^2-S\\lambda +\\mathbf {n}Q+\\mathbf {n}^3\\end{bmatrix}.$ Since, the characteristic roots of the matrix are pure imaginary and given as $\\lambda =i\\omega , \\, \\omega \\in \\mathbb {R}$ , which can be obtained with the help of equation $&&\\omega ^4-\\left(-P^2-Q^2+2\\mathbf {n}^2\\right)\\omega ^2+\\left(\\mathbf {n}^2P+\\mathbf {n}^2Q\\right.\\nonumber \\\\&&\\left.+PQ-S^2+\\mathbf {n}^4\\right)=0,$ which provide $\\lambda _{1,2}=\\pm i\\omega _{1}$ and $\\lambda _{3,4}=\\pm i\\omega _{2}$ .",
"The frequencies $\\omega _{1,2}$ in terms of $\\epsilon _1, \\, A_2, \\, \\epsilon _2$ and $\\mu $ , are obtained with the help of equations (REF ), (REF -) and (REF ) under linear approximation of $\\epsilon _1, \\, A_2, \\, \\epsilon _2$ , and these are given as $&&\\omega _1=\\left[\\frac{B_1+\\sqrt{B_{1}^2-4B_2}}{2}\\right]^{\\frac{1}{2}},\\\\&&\\omega _2=\\left[\\frac{B_1-\\sqrt{B_{1}^2-4B_2}}{2}\\right]^{\\frac{1}{2}}$ with $&&B_1=1-\\frac{15\\epsilon _1}{16}+\\frac{3A_2}{2}+\\frac{35\\epsilon _2}{8},\\\\&&B_2=\\frac{27}{16}-\\frac{27(1-2\\mu )^2}{16}-\\frac{633\\epsilon _1}{128}\\nonumber \\\\&&+\\frac{99A_2}{16}+\\frac{165\\epsilon _2}{16}.$ Again, if we put $\\lambda =i\\omega $ into characteristic vector (REF ) and then separating real and imaginary parts, say $u$ and $v$ , of resulting characteristic vectors, then we obtain $u=\\begin{bmatrix}\\mathbf {n}^2+Q+\\omega ^2\\\\-S\\\\\\mathbf {n}S\\\\-\\mathbf {n}\\omega ^2+\\mathbf {n}Q+\\mathbf {n}^3\\end{bmatrix},$ $v=\\begin{bmatrix}0\\\\2\\mathbf {n}\\omega \\\\-\\mathbf {n}^2\\omega +Q\\omega +\\omega ^3\\\\-S\\omega \\end{bmatrix}.$ Now, consider the required symplectic change of phase variable is given by the matrix $\\mathbf {C}=(v_1, \\, v_2, \\, u_1, \\, u_2)$ , where $u_j, \\, v_j, \\,j=1,2$ represent the values of $u, \\, v$ correspond to frequencies $\\omega _j, \\,j=1,2$ , respectively.",
"Thus, it is obvious the symplectic change satisfy the property $\\mathbf {C}^{T}\\mathbf {J_4}\\mathbf {C}=\\mathbf {J_4}$ .",
"Substituting $&&\\omega ^2_{1}+\\omega ^2_{2}=2\\mathbf {n}^2-P-Q,\\\\&&\\omega ^2_{1}\\omega ^2_{2}=\\mathbf {n}^4+\\mathbf {n}^2(P+Q)+PQ-S^2,$ and simplifying, we obtain $\\mathbf {C}^{T}\\mathbf {J_4}\\mathbf {C}=\\begin{bmatrix}\\mathbf {0}&\\mathbf {D} \\\\-\\mathbf {D}&\\mathbf {0}\\end{bmatrix},\\quad \\mathbf {D}=\\begin{bmatrix}d(\\omega _1)&0 \\\\0&d(\\omega _2)\\end{bmatrix},$ where $d(\\omega )&=&2\\omega \\left[2\\omega ^4+(P+3Q)\\omega ^2+PQ+Q^2\\right.\\nonumber \\\\&&\\left.+\\mathbf {n}^2(P-Q-2\\mathbf {n}^4)\\right].$ Since, in order to satisfy the symplectic property, $d(\\omega )$ should be one, if it is not then we need to scale the columns of $\\mathbf {C}$ by the quantity $\\sqrt{d(\\omega _j)}, \\, j=1,2$ i.e.",
"$&&\\mathbf {C}=\\left(\\frac{v_1}{\\sqrt{d(\\omega _1)}}, \\, \\frac{v_2}{\\sqrt{d(\\omega _2)}}, \\, \\frac{u_1}{\\sqrt{d(\\omega _1)}}, \\, \\frac{u_2}{\\sqrt{d(\\omega _2)}}\\right).$ Now, this matrix is symplectic but, we require real symplectic change, so it is necessary to take $d(\\omega _j)>0, \\, j=1,2$ which is possible when we take $\\omega _1>0$ and $\\omega _2<0$ .",
"Thus, the transformation obtained is real and symplectic and gives diagonalize form of the Hamiltonian (REF ) as $&&H_2=\\frac{\\omega _{1}}{2}({\\mathbf {x}}^2+{\\mathbf {p_{x}}}^2)+\\frac{\\omega _{2}}{2}({\\mathbf {y}}^2+{\\mathbf {p_{y}}}^2).$ In order to solve homological equations [22], which determine the generating function, in an easier way, we have to change the normalized Hamiltonian (REF ) into complex normal form with the help of an other symplectic change of variable, which are given as follows: $&&\\mathbf {x}=\\frac{X+iP_X}{\\sqrt{2}},\\quad \\mathbf {y}=\\frac{-Y+iP_Y}{\\sqrt{2}},\\\\&&\\mathbf {p_x}=\\frac{iX+P_X}{\\sqrt{2}}, \\quad \\mathbf {p_y}=\\frac{iY-P_Y}{\\sqrt{2}}.$ Thus, if we express the Hamiltonian (REF ) in eigen coordinates (REF -), we obtain normal form of $H_2$ i.e.",
"$&&H_2=i\\omega _1 XP_X-i\\omega _2YP_Y.$ The above complexification gives the final change used in this article as follows $&&\\mathbf {C}=[c_{ij}], \\, 1\\le i, j\\le 4,$ where $&&c_{11}=0;c_{12}=0;c_{13}=\\frac{\\mathbf {n}^2+Q+\\omega _1{}^2}{\\surd d(\\omega _1)};\\\\&&c_{14}=\\frac{\\mathbf {n}^2+Q+\\omega _2{}^2}{\\surd d(\\omega _2)};c_{21}=\\frac{2\\mathbf {n} \\omega _1}{\\surd d(\\omega _1)};\\\\&&c_{22}=\\frac{2\\mathbf {n} \\omega _2}{\\surd d(\\omega _2)};c_{23}=\\frac{-S}{\\surd d(\\omega _1)};\\\\&& c_{24}=\\frac{-S}{\\surd d(\\omega _2)};c_{31}=\\frac{(-\\mathbf {n}^2+Q) \\omega _1+\\omega _1{}^3}{\\surd d(\\omega _1)};\\\\&&c_{32}=\\frac{(-\\mathbf {n}^2+Q) \\omega _2+\\omega _2{}^3}{\\surd d(\\omega _2)};c_{33}=\\frac{\\mathbf {n} S}{\\surd d(\\omega _1)};\\\\&&c_{34}=\\frac{\\mathbf {n} S}{\\surd d(\\omega _2)};c_{41}=\\frac{- S \\omega _1}{\\surd d(\\omega _1)};c_{42}=\\frac{- S \\omega _2}{\\surd d(\\omega _2)};\\\\&&c_{43}=\\frac{\\mathbf {n}^3+\\mathbf {n} Q -\\mathbf {n} \\omega _1{}^2}{\\surd d(\\omega _1)};c_{44}=\\frac{\\mathbf {n}^3+\\mathbf {n} Q -\\mathbf {n} \\omega _2{}^2}{\\surd d(\\omega _2)},$ where $d(\\omega _{1,2})$ are obtain from equation (REF ).",
"If we ignore the perturbations $A_2, \\, \\epsilon _1$ and $\\epsilon _2$ then above symplectic change agree with that of [9] and [33]."
],
[
"Nonlinear Stability of Triangular Point in Non-resonance Case",
"To study the nonlinear stability of the equilibrium points, there are two cases: (i) resonance case and (ii) non-resonance case.",
"Nonlinear stability in resonance case would be discussed with the help of theorems of Sokolsky (1974), Markeev (1978) and Grebenikov (1986) as in [14] whereas, in later case it is studied with the help of Arnold-Moser theorem.",
"Here, authors are interested only to examine the nonlinear stability in non-resonance case by the use of Arnold-Moser theorem under the influence of perturbations.",
"The general form of the theorem is presented in Appendix (REF .)",
"as in [33], [35] however, for self sufficient paper, authors have described the theorem in their own notations as follows: Consider the Hamiltonian expressed in action-angle variables $(I_1,\\,I_2,\\,\\phi _1,\\,\\phi _2$ ) as $K=K_2+K_4+K_6+\\dots +K_{2n}+K^{*}_{2n+1},$ where $K_{2r}$ are homogeneous polynomials in action variables $I_1,\\,I_2$ , of degree $r$ , $K_{2n+1}$ are polynomials containing terms of higher order than $n$ and $K_2=\\omega _1I_1-\\omega _2I_2$ with $\\omega _i,\\, i=1,\\,2$ positive constants.",
"$K_4=-\\left(K_{2020}I_1^{2}+K_{1111}I_1I_2+K_{0202}I_2^{2}\\right)$ , where $K_{2020}\\,K_{1111},\\,K_{0202}$ are constants.",
"Since, $K_2,\\,K_4,\\,\\dots ,\\, K_{2n}$ are function of only action variables $I_1,\\,I_2$ , so, the Hamiltonian is assumed to be in Birkhoff's normal form up to terms of degree $2n$ which can be obtained with few non-resonance assumptions on the frequencies $\\omega _1$ and $\\omega _2$ , but in order to state the theorem, assume that $K$ is in the required form.",
"Now, Arnold-Moser theorem stated as: Theorem 4.1 (Arnold-Moser) The origin is stable for the system whose Hamiltonian is (REF ) provided for some $r,\\,1\\le r\\le n,\\, D_{2r}=K_{2r}(\\omega _2,\\,\\omega _1)\\ne 0$ or equivalently provided $K_2$ does not divide $K_{2r}.$ Since, Arnold-Moser theorem requires Birkhoff's normal form of the Hamiltonian and Birkhoff's normal form requires some non-resonance assumptions on the frequencies $\\omega _1$ and $\\omega _2$ , which is described as follows [9].",
"Suppose, $\\omega _1$ and $\\omega _2$ are frequencies in case of linear dynamics of the infinitesimal mass, and $n$ is an integer such that $n\\ge 2$ , then $l_1\\omega _1+l_2\\omega _2\\ne 0,$ for all $l_1, \\, l_2 \\in \\mathbb {Z}$ such that $|l_1|+|l_2|\\le 2n$ known as irrationality condition.",
"This condition ensures that there is an analytic symplectic normalizing transformation such that the Hamiltonian (REF ) takes the form (REF ).",
"Coefficients in the normalized Hamiltonian are independent on the integer $n$ and also independent to the manner of transformation is obtained.",
"In particular, the determinant $\\mbox{det}\\begin{bmatrix}\\frac{\\partial ^{2} K}{\\partial I^2_{1}}&\\frac{\\partial ^{2} K}{\\partial I_{1}\\partial I_{2}}&\\frac{\\partial K}{\\partial I_1} \\\\\\frac{\\partial ^{2} K}{\\partial I_{2}\\partial I_{1}}&\\frac{\\partial ^{2} K}{\\partial I^2_{2}}&\\frac{\\partial K}{\\partial I_2}\\\\\\frac{\\partial K}{\\partial I_1}&\\frac{\\partial K}{\\partial I_2}&0\\end{bmatrix}_{I_1, \\, I_2=0}$ is an invariant of the Hamiltonian (REF ) with respect to the symplectic transformation used.",
"Arnold-Moser theorem decides the stability of equilibrium points under these two conditions.",
"Here, we are interested to implement this procedure to the problem in question for $n=2$ .",
"That is, we have to compute Birkhoff's normal form of Hamiltonian (REF ) up to degree 2 in action variables and then analyze the quantity $D_4$ with respect to the perturbations in question."
],
[
"Birkhoff's Normal Form",
"In order to apply Arnold-Moser theorem, we have to compute Brikhoff's normal form up to 4th order normal form of the Hamiltonian in the vicinity of equilibrium point which will be the function of action-angle variables $(I_1, \\, I_2, \\, \\phi _1, \\,\\phi _2)$ .",
"To obtain the Brikhoff's normal form, we have used Lie transform described in [6] and [22].",
"Since, higher order normalized Hamiltonian is given as [6]: $&&K=K_2+K_3+K_4\\dots , $ where $&&K_n=\\sum {K_{jkls}}X^{j}Y^{k}P^{l}_XP^{s}_Y\\quad \\text{with}\\nonumber \\\\&& j+k+l+s=n$ is known as Kamiltonian.",
"Now, the quadratic part of $n$ -th order normal form $K_n$ is $K_2=H_2$ .",
"At the $n^{th}$ step of the Lie transform method, Kamiltonian $K_n$ is obtained by the expression $K_n=\\frac{1}{n}\\left\\lbrace H_2, \\, W_n\\right\\rbrace +\\text{previously known terms}.$ Now, to determine the generating function $W_n$ , which provides the best simplified form of Kamiltonian $K_n$ , first we determine the Lie bracket $\\left\\lbrace H_2, \\, W_n\\right\\rbrace $ with $H_2$ from equation (REF ) as follows $\\left\\lbrace H_2, \\, W_n\\right\\rbrace &=&\\frac{\\partial H_2}{\\partial X}\\frac{\\partial W_n}{\\partial P_X}+\\frac{\\partial H_2}{\\partial Y}\\frac{\\partial W_n}{\\partial P_Y}\\nonumber \\\\&&-\\frac{\\partial H_2}{\\partial P_X}\\frac{\\partial W_n}{\\partial X}-\\frac{\\partial H_2}{\\partial P_Y}\\frac{\\partial W_n}{\\partial Y}\\nonumber \\\\&=&i\\omega _1\\left[P_X\\frac{\\partial W_n}{\\partial P_X}-X\\frac{\\partial W_n}{\\partial X}\\right]\\nonumber \\\\&&-i\\omega _2\\left[P_Y\\frac{\\partial W_n}{\\partial P_Y}-Y\\frac{\\partial W_n}{\\partial Y}\\right].$ We need $W_n$ such that results of this partial linear differential operator on $W_n$ remove out as many terms as possible in the expression of Kamiltonian $K_n$ .",
"It is clear from the expression of $K_n$ that each term to be canceled will be of the form $P_0X^{j}Y^{k}P_X^{l}P_Y^{s}$ , where $P_0$ is constant.",
"Let us take, $W_n$ as the sum of terms of the form $Q_0X^{j}Y^{k}P_X^{l}P_Y^{s}$ , where $Q_0$ is an undetermined constant and hence, we get $\\frac{\\left\\lbrace H_2, \\, W_n\\right\\rbrace }{n}=\\frac{i\\left[\\omega _1(l-j)-\\omega _2(s-k)\\right]}{n}Q_0X^{j}Y^{k}P_X^{l}P_Y^{s},$ which gives $Q_0&=&\\frac{inP_0}{\\omega _1(l-j)-\\omega _2(s-k)}\\quad \\text{with}\\nonumber \\\\&&j+k+l+s=n.$ Now, from equation (REF ), it is clear that above scheme fails if denominator vanishes.",
"If we assume that frequencies $\\omega _{1,2}$ are non-resonant then denominator will vanish only in case of $l=j$ and $s=k$ .",
"In other words, in the expression of $K_n$ , the terms of the type $(X^{l}Y^{s}P_X^{l}P_Y^{s})$ can not be removed.",
"Hence, in case of non-resonance, the problem always can be reduced to the form of (REF ), where $K_2&=&H_2=i\\omega _1 XP_X-i\\omega _2YP_Y,\\\\K_3&=&0,\\\\K_4&=&K_{2020}X^{2}P_X^{2}+K_{1111}XYP_XP_Y\\nonumber \\\\&&+K_{0202}Y^{2}P_Y^{2}.$ Thus, in case of non-resonance, normalized (transformed) Hamiltonian is a function of only action variables, $I_1=iXP_X$ and $I_2=iYP_Y$ and hence, both the coordinates are ignorable which infer that system is consistent and system is said to be in Brikhoff's normal form.",
"Now, in term of action variable $I_1, \\, I_2$ , 4th order part of normalized Hamiltonian is written as $K_4(I_1, \\, I_2)&=&-\\left(K_{2020}I_1^{2}+K_{1111}I_1I_2\\right.\\nonumber \\\\&&\\left.+K_{0202}I_2^{2}\\right).$ On the other hand, in case of resonant values of frequencies $\\omega _{1,2}$ , some additional non-removable terms occur while solving the generating function $W_n$ .",
"For simplicity, coefficients $K_{jkls}$ in equation (REF ) are split-up into four parts such as $\\mathbf {k}_{jkls}, \\, \\mathbf {k}_{jklse1}$ , $\\mathbf {k}_{jklsA}$ and $\\mathbf {k}_{jklse2}, \\, j, \\, k, \\, l, \\, s=0, \\, 1, \\, 2,\\,3,\\,4$ such that $j+k+l+s=4$ .",
"These coefficients correspond to the terms due to classical case, radiation effects $\\epsilon _1$ , oblateness $A_2$ and presence of the disc $\\epsilon _2$ , respectively.",
"In other words $K_{2020}&=& \\mathbf {k}_{2020}+\\mathbf {k}_{2020e1}+\\mathbf {k}_{2020A}\\nonumber \\\\&&+\\mathbf {k}_{2020e2},$ $K_{1111}&=& \\mathbf {k}_{1111}+\\mathbf {k}_{1111e1}+\\mathbf {k}_{1111A}\\nonumber \\\\&&+\\mathbf {k}_{1111e2},$ $K_{0202}&=& \\mathbf {k}_{0202}+\\mathbf {k}_{0202e1}+\\mathbf {k}_{0202A}\\nonumber \\\\&&+\\mathbf {k}_{0202e2},$ where $\\mathbf {k}_{jkls}$ represents coefficients for classical part, $\\mathbf {k}_{jklse1}$ indicates coefficients for radiation pressure terms, $\\mathbf {k}_{jklsA}$ used for oblateness and $\\mathbf {k}_{jklse2}$ for the disc with $j, \\, k, \\, l, \\, s=0, \\, 1, \\, 2,\\,3,\\,4$ such that $j+k+l+s=4$ .",
"But, in the absence of perturbations i.e.",
"$A_2=\\epsilon _1=\\epsilon _2=0$ , $K_{jkls}=\\mathbf {k}_{jkls}$ , which are equivalent to the coefficients of the normalized Hamiltonian of the classical problem.",
"Further, due to very large expressions of all the coefficients $\\mathbf {k}_{jkls}$ , $\\mathbf {k}_{jklse1}$ , $\\mathbf {k}_{jklsA}$ and $\\mathbf {k}_{jklse2}$ with $j, \\, k, \\, l, \\, s=0, \\, 1, \\, 2,\\,3,\\,4$ such that $j+k+l+s=4$ , in equations (REF -REF ), these are given in Appendix REF .",
"As, we have the fourth order normalized Hamiltonian given in equation (REF ), so, we have computed the determinant $D_4$ to use Arnold-Moser theorem which will decide nonlinear stability of the triangular equilibrium point in non-resonance case.",
"In our case, we found $D_4$ as: $&&D_4=D_{40}+D_{41}\\epsilon _1+D_{{42}} A_2+D_{43}\\epsilon _2, $ where $D_{40}, \\, D_{41}, \\, D_{42}$ and $D_{43}$ correspond to the classical parts, radiation pressure term, oblateness and the terms due to presence of the disc, respectively, which are obtained with the help of equations (REF ,REF -REF ) and have very large expression hence, these are given in Appendix REF .",
"Substituting the values of $D_{40}, \\, D_{41}, \\, D_{42}$ and $D_{43}$ in terms of frequencies $\\omega _{1,2}$ from Appendix REF into equation (REF ) and after a tedious simplification at several stages, we get $D_4&=&\\frac{-36+541 \\omega _1^2 \\omega _2^2-644 \\omega _1^4 \\omega _2^4}{8 \\left(1-4\\omega _1^2 \\omega _2^2\\right) \\left(-4+25 \\omega _1^2 \\omega _2^2\\right)}+\\frac{G_1 \\epsilon _1}{G_2}\\nonumber \\\\&&+\\frac{G_3 A_2}{G_4}+\\frac{G_5 \\epsilon _2}{G_6},$ where $G_1&=&\\left(691394292-5589532701 \\omega _1^2 \\omega _2^2\\right.\\nonumber \\\\&&\\left.+7636835570 \\omega _1^4 \\omega _2^4-4919234984 \\omega _1^6 \\omega _2^6\\right.\\nonumber \\\\&&\\left.+373115136 \\omega _1^8 \\omega _2^8\\right.\\nonumber \\\\&&\\left.+597854208 \\omega _1^{10} \\omega _2^{10}\\right),$ $G_2&=&12288 \\omega _1^2 \\omega _2^2 \\left[\\left(1-4 \\omega _1^2 \\omega _2^2\\right){}^2\\left(4-25 \\omega _1^2 \\omega _2^2\\right)\\right.\\nonumber \\\\&&\\left.\\left(117+16 \\omega _1^2 \\omega _2^2\\right)\\right],$ $G_3&=&\\left(103351788-881740119 \\omega _1^2 \\omega _2^2\\right.\\nonumber \\\\&&\\left.+1442778388 \\omega _1^4 \\omega _2^4-916084992 \\omega _1^6 \\omega _2^6\\right.\\nonumber \\\\&&\\left.+193442816 \\omega _1^8 \\omega _2^8\\right),$ $G_4&=&384 \\omega _1 \\omega _2 \\left[\\left(1-4 \\omega _1^2 \\omega _2^2\\right){}^2\\left(-4+25 \\omega _1^2 \\omega _2^2\\right)\\right.\\nonumber \\\\&&\\left.",
"\\left(117+16 \\omega _1^2 \\omega _2^2\\right)\\right],$ $G_5&=&\\left(341649252-2463280353 \\omega _1^2 \\omega _2^2\\right.\\nonumber \\\\&&\\left.+1827505940 \\omega _1^4 \\omega _2^4-1159472852 \\omega _1^6 \\omega _2^6\\right.\\nonumber \\\\&&\\left.+210281632 \\omega _1^8 \\omega _2^8\\right.\\nonumber \\\\&&\\left.+205320704 \\omega _1^{10} \\omega _2^{10}\\right),$ $G_6&=&1536 \\omega _1^2 \\omega _2^2 \\left(1-4 \\omega _1^2 \\omega _2^2\\right){}^2 \\left(-8910\\right.\\nonumber \\\\&&\\left.+2861 \\omega _1^2 \\omega _2^2+400 \\omega _1^4 \\omega _2^4\\right).$ Figure: Stability condition from the normalized Hamiltonian of order four.Figure: Change in critical value μ ϵ 1 \\mu _{\\epsilon _1} with respect to radiation pressure parameter ϵ 1 =1-q 1 \\epsilon _1=1-q_1.",
"Curves are drawn at A 2 =0.0,ϵ 2 =0.0A_2=0.0,\\, \\epsilon _2=0.0 and I: ϵ 1 =0.0,\\epsilon _1=0.0, II :ϵ 1 =0.01,\\epsilon _1=0.01, III: ϵ 1 =0.02,\\epsilon _1=0.02, IV: ϵ 1 =0.03,\\epsilon _1=0.03, V: ϵ 1 =0.04.\\epsilon _1=0.04.Figure: Change in critical value μ A 2 \\mu _{A_2} with respect to oblateness parameter A 2 A_2.",
"(a) Curves are drawn at ϵ 1 =0.0,ϵ 2 =0.0\\epsilon _1=0.0,\\, \\epsilon _2=0.0 and I: A 2 =0.0,A_2=0.0, II :A 2 =0.001,A_2=0.001, III: A 2 =0.002,A_2=0.002, IV: A 2 =0.003,A_2=0.003, V: A 2 =0.004.A_2=0.004.",
"(b) Zoom of figure (a).Figure: Change in critical value μ ϵ 2 \\mu _{\\epsilon _2} with respect to disc parameter ϵ 2 =b-1\\epsilon _2=b-1.",
"(a) Curves are drawn at ϵ 1 =0.0,A 2 =0.0\\epsilon _1=0.0,\\, A_2=0.0 and I: ϵ 2 =0.0,\\epsilon _2=0.0, II :ϵ 2 =0.001,\\epsilon _2=0.001, III: ϵ 2 =0.002,\\epsilon _2=0.002, IV: ϵ 2 =0.003,\\epsilon _2=0.003, V: ϵ 2 =0.004.\\epsilon _2=0.004.",
"(b) Zoom-I of figure (a).",
"(c) Zoom-II of figure (a).rrrrr Stability range and critical values of $\\,\\mu \\,$ at which $\\,D_4=0\\,$ for different values of perturbation parameters.",
"0pt $\\epsilon _1$ $A_2$ $\\epsilon _2$ $\\text{Value of}\\,\\mu \\, \\text{at which}\\, D_4=0$ $\\text{stability range}\\,0<\\mu <\\mu _1=\\bar{\\mu _c}$ 0.0 0.0 0.00.01095$(0.0,\\, 0.03852)$ 0.01 0.0 0.00.02949$(0.0,\\, 0.03861)$ 0.02 0.0 0.00.03799$(0.0,\\, 0.03870)$ 0.03 0.0 0.00.04626$(0.0,\\, 0.03879)$ 0.04 0.0 0.00.05447$(0.0,\\, 0.03888)$ 0.0 0.001 0.00.02999$(0.0,\\, 0.03838)$ 0.0 0.002 0.00.02553$(0.0,\\, 0.03825)$ 0.0 0.003 0.00.02305$(0.0,\\, 0.03811)$ 0.0 0.004 0.00.02132$(0.0,\\, 0.03798)$ 0.0 0.0 0.0010.00269$(0.0,\\, 0.03853)$ 0.0 0.0 0.0010.00697$(0.0,\\, 0.03853)$ 0.0 0.0 0.0010.03761$(0.0,\\, 0.03853)$ 0.0 0.0 0.0020.03678$(0.0,\\, 0.03854)$ 0.0 0.0 0.0030.03608$(0.0,\\, 0.03855)$ 0.0 0.0 0.0040.03551$(0.0,\\, 0.03855)$ 0.0 0.0 0.0070.03506$(0.0,\\, 0.03858)$ 0.0 0.0 0.0090.03706$(0.0,\\, 0.03859)$ 0.0 0.0 0.0100.03925$(0.0,\\, 0.03860)$ 0.0 0.0 0.0300.05261$(0.0,\\, 0.03877)$ 0.0 0.0 0.0600.07054$(0.0,\\, 0.03901)$ 0.0 0.0 0.0900.05604$(0.0,\\, 0.03926)$ 0.0 0.0 0.1000.05757$(0.0,\\, 0.03934)$ 0.0 0.0 0.200—$(0.0,\\, 0.04015)$ 0.0 0.0 0.3000.08642$(0.0,\\, 0.04097)$ 0.02 0.003 0.00.03486$(0.0,\\, 0.03830)$ 0.02 0.0 0.0020.03474$(0.0,\\, 0.03873)$ 0.0 0.003 0.0020.01978$(0.0,\\, 0.03813)$ 0.02 0.003 0.0020.03155$(0.0,\\, 0.03833)$ From the expression (REF ) of $D_4$ , it can be easily seen that if perturbations are ignored then it coincides with that of [9], [6], [43] and [1].",
"To see the effect of perturbations on the nonlinear stability, we plot the expression of $D_4$ against mass ratio $\\mu $ at different values of perturbations, individually as well as simultaneously (Fig.",
"REF -REF ), in the stability region $0<\\mu <\\mu _1=\\bar{\\mu _c}$ .",
"From figure (REF a), it is seen that there is only one zero of $D_4$ at $\\mu _0=0.01095$ whereas, $\\mu _\\kappa , \\, \\kappa =1,2,3$ indicate the values of mass ratio under three main resonance cases of linear stability [23], which change their values with respect to the values of $\\epsilon _1, A_2, \\epsilon _2$ (Fig.",
"REF a-e).",
"Thus, Arnold-Moser theorem infer that the triangular equilibrium point $L_4$ is stable in the interval $0<\\mu <\\mu _1$ except at $\\mu _0=0.01095$ , in addition to the values $\\mu _\\kappa , \\, \\kappa =1,2,3$ for three main resonance cases of linear stability.",
"This result is similar to that of [9] in the absence of perturbations.",
"The presence of radiation pressure (Fig.",
"REF b) as well as oblateness (Fig.",
"REF c) cause the theorem fail at $\\mu _{\\epsilon _1}=0.03795$ and $\\mu _{A_2}=0.02297$ , respectively.",
"The effect of disc seen only at very small values of $\\epsilon _2$ (Fig.",
"REF d) and the value of $\\mu $ at which $D_4$ vanishes is $0.03678$ (Fig.",
"REF d) whereas, at larger value of $\\epsilon _2$ , it is beyond the stability range $0<\\mu <\\mu _1$ .",
"If, we analyze, by taking two perturbations at a time such as $(\\epsilon _1, \\, A_2), \\, (\\epsilon _1, \\, \\epsilon _2)$ and $(A_2, \\, \\epsilon _2)$ then zeros of $D_4$ i.e.",
"the value of mass ratio $\\mu $ at which $D_4$ vanish, are obtained as $0.03486, \\, 0.03474$ and $0.01978$ , respectively.",
"If all the perturbations taken at a time then the value of the mass ratio is $\\mu _{\\epsilon _1, A_2, \\epsilon _2}=0.03155$ at which theorem fails.",
"Thus, the triangular equilibrium point within the stability range, in the absence as well as presence of perturbation parameters, is unstable due to failure of Arnold-Moser theorem.",
"In order to see the effect of perturbation, we have drawn the figures (REF -REF ) by varying individual parameter taking remaining two of them zero.",
"From, figure (REF ), critical value $\\mu _{\\epsilon _1}$ increases with the increment in the value of $\\epsilon _1$ .",
"The critical values $\\mu _{\\epsilon _1}$ corresponding to the curve I, II and III lie within the stability range whereas for curve IV and V, it is beyond the stability range (see Table-REF ).",
"From figure (REF a,b) and Table-REF , it is clear that the critical value $\\mu _{A_2}$ decreasing with the increment in the value of $A_2$ (also, see Table-REF ).",
"Figure (REF a,b) shows that there are three critical value $\\mu _{\\epsilon _2}$ at $\\epsilon _2=0.001$ but for other values it reduces to one.",
"It is also, clear that value of $\\mu _{\\epsilon _2}$ decreasing slowly with increment in the value of $\\epsilon _2$ , but when, value of $\\epsilon _2$ increases after $0.009$ , the value of critical $\\mu _{\\epsilon _2}$ go beyond the corresponding stability range (Table-REF )."
],
[
"Conclusion",
"The analysis of nonlinear stability of triangular equilibrium point in the Chermnykh-like problem under the influence of perturbations have been preformed for non-resonance case.",
"The diagonalization and higher order normalization of Hamiltonian of the problem are made by the method of Lie transform under the influence of perturbations in the form of radiation pressure force, oblateness and the disc.",
"Due to perturbations, transformation equations take a complicated form but in the absence of perturbations these equations agree with the classical results.",
"Next, we have analyzed nonlinear stability with the help of Arnold-Moser theorem.",
"After a huge and tedious computation at several intermediate steps, we have obtained determinant $D_4$ in the presence of perturbations which is agree with that of [9], [33], [6] and [43] under the vanishing condition of perturbations.",
"Due to Arnold-Moser theorem, it is found that under the influence of perturbations, the motion of infinitesimal mass in the vicinity of triangular equilibrium point get affected.",
"In other words, in classical case, triangular equilibrium point is stable within the stability range $0<\\mu <\\mu _1=0.03852$ except for the value of $\\mu _2, \\, \\mu _3$ and $\\mu _0=0.01095$ at which $D_4$ vanishes and hence, Arnold-Moser theorem fails there.",
"But under the effect of radiation pressure and oblateness, nonlinear stability of the triangular equilibrium point fail at $\\mu _{\\epsilon _1}=0.03795$ and $\\mu _{A_2}=0.02297$ due to the same reason whereas, in the presence of all three perturbations, it fails at $\\mu _{\\epsilon _1,A_2,\\epsilon _2}=0.03155$ .",
"The effect of disc is seen either at very small values of disc outer radius or very large values of disc outer radius.",
"It is observed that for small values of disc outer radius, theorem fails at $\\mu _{\\epsilon _2}=0.03678$ , whereas for large value of disc outer radius, effect is beyond the stability range $0<\\mu <\\mu _1$ .",
"Moreover, in case of taking two perturbations at a time such as $(\\epsilon _1, \\, A_2), \\, (\\epsilon _1, \\, \\epsilon _2)$ and $(A_2, \\, \\epsilon _2)$ , the zeros of $D_4$ are $0.03486, \\, 0.03474$ and $0.01978$ , respectively.",
"The nature of variation in the value of critical mass $\\mu $ at which the value of $D_4=0$ at different values of individual perturbation parameters is observed and significant variation are found.",
"The critical value $\\mu _{\\epsilon _1}$ first, increases with the increment in the value of $\\epsilon _1$ , and then this value go beyond the stability range (see figure REF and Table-REF ).",
"Again, the critical value $\\mu _{A_2}$ decreasing with the increment in the value of $A_2$ (see figure REF a,b and Table-REF ).",
"There are three critical value $\\mu _{\\epsilon _2}$ at $\\epsilon _2=0.001$ Figure (REF a,b) but for other values it reduces to one.",
"It is also, clear that value of $\\mu _{\\epsilon _2}$ decreasing slowly with increment in the value of $\\epsilon _2$ , but for larger values of $\\epsilon _2$ , $\\mu _{\\epsilon _2}$ do not lie in the corresponding stability range (Table-REF ).",
"The results, which are obtained, are very helpful to observe the motion of infinitesimal mass such as spacecraft, asteroid or satellite in the Sun-Jupiter system.",
"The present study and observations are applicable to the analysis of more generalized problems and would be extended up to higher order in addition with some other type of perturbations like P-R drag, solar wind drag.",
"On the other hand results are limited up to radially symmetric disc but in future it would be extended.",
"The authors are very thankful for the referee's comments and suggestions; they have been very useful and have greatly improved the manuscript.",
"The financial support by the Department of Science and Technology, Govt.",
"of India through the SERC-Fast Track Scheme for Young Scientist [SR/FTP /PS-121/2009] is duly acknowledged.",
"Some of the important references in addition to basic facilities are provided by IUCAA Library, Pune, India."
],
[
"Arnold-Moser theorem {{cite:614041d0c54856b379c0b060aa6a1677d1806aa6}}, {{cite:910b28f8815208e0703d8d2388925b514e41e635}}",
"Consider a Hamiltonian, which is the function of canonical coordinates $x_i, \\, y_i,\\, i=1,\\, 2,$ expressed as $H=H_2+H_4+H_6+\\dots +H_{2n}+H^{*}_{2n+1},$ where $H$ is real analytic in the a neighborhood of the origin in $\\mathbb {R}^4$ ; $H_{2k},\\, 1\\le k\\le n,$ is a homogeneous function of degree $k$ in $I_i=\\frac{1}{2}(x^{2}_i+y^{2}_i),\\, i=1,\\,2$ ; $H^*$ has a series expansion which starts with terms at least of order $2n+1$ ; $H_2=\\omega _1I_1-\\omega _2I_2$ with $\\omega _i,\\, i=1,\\,2$ positive constants; $H_4=\\frac{1}{2}\\left(AI^{2}_1-2BI_1I_2+CI^{2}_2\\right),\\, A,\\,B,\\,C$ constants.",
"There are several implicit assumptions in stating that Hamiltonian $H$ in the form of (REF ).",
"As, $H$ is at least quadratic in canonical coordinates $x_i, \\, y_i,\\, i=1,\\, 2,$ the origin is assumed to be the equilibrium point in question.",
"Again, $H_2=\\omega _1I_1-\\omega _2I_2$ is the Hamiltonian of two harmonic oscillators with frequency $\\omega _1$ and $\\omega _2$ , the linearization at the origin of the system of equations whose Hamiltonian is $H,$ is two harmonic oscillators.",
"Since, $H_2$ is not sign definite, a simple appeal to stability theorem of Lyapunov can not be made.",
"Again, $H_2,\\,H_4,\\,\\dots ,\\, H_{2n}$ are function of only $I_i=\\frac{1}{2}(x_i+y_i),\\, i=1,\\,2,$ the Hamiltonian is assumed to be in Birkhoff's normal form up to terms of degree $2n$ .",
"The Birkhoff's normal form usually requires some non-resonance assumptions on the frequencies $\\omega _1$ and $\\omega _2$ , but in order to state the theorem, assume that $H$ is in the required form.",
"Theorem 6.1 (Arnold-Moser) The origin is stable for the system whose Hamiltonian is (REF ) provided for some $k,\\,2\\le k\\le n,\\,\\, D_{2k}=H_{2k}(\\omega _2,\\,\\omega _1)\\ne 0$ or equivalently provided $H_2$ does not divide $H_{2k}.$"
]
] | 1906.04495 |
[
[
"Testing the gravitational field generated by a quantum superposition"
],
[
"Abstract What gravitational field is generated by a massive quantum system in a spatial superposition?",
"Despite decades of intensive theoretical and experimental research, we still do not know the answer.",
"On the experimental side, the difficulty lies in the fact that gravity is weak and requires large masses to be detectable.",
"However, it becomes increasingly difficult to generate spatial quantum superpositions for increasingly large masses, in light of the stronger environmental effects on such systems.",
"Clearly, a delicate balance between the need for strong gravitational effects and weak decoherence should be found.",
"We show that such a trade off could be achieved in an optomechanics scenario that allows to determine whether the gravitational field generated by a quantum system in a spatial superposition is in a coherent superposition or not.",
"We estimate the magnitude of the effect and show that it offers perspectives for observability."
],
[
"Framework",
"We consider a setup formed of two systems interacting gravitationally.",
"All non-gravitational interactions are considered, for all practical purposes, negligible.",
"The first system (S1) has a mass $m_1$ , and it is initially prepared in a spatial superposition along the $x$ direction.",
"Its wave-function is $\\psi ({\\bf r}_1)=\\tfrac{1}{\\sqrt{2}}(\\alpha ({\\bf r}_1)+\\beta ({\\bf r}_1))$ , where $\\alpha ({\\bf r}_1)$ and $\\beta ({\\bf r}_1)$ are sufficiently well localized states in position, far from each other in order to prevent any overlap.",
"Thus, we can consider them as distinguishable (in a macroscopic sense), and we approximate $\\mathinner {\\langle {\\alpha |\\beta }\\rangle }\\simeq 0$ .",
"The second system (S2) will serve as a probe of the gravitational field generated by S1, it has mass $m_2$ and state $\\phi ({\\bf r}_2)$ .",
"The state $\\phi ({\\bf r}_2)$ is initially assumed to be localized in position and centered along the $y$ direction [cf. Fig.",
"REF ].",
"The question we address is: which is the gravitational field, generated by the quantum superposition of S1, that S2 experiences?",
"We probe the following two different scenarios.",
"Figure: Schematic representation of the two-body setup.",
"S1 is prepared in a spatial superposition along the xx direction (red balls).",
"S2 is initially prepared in a localized wavepacket (blue ball), and it probes the gravitational field generated by S1.",
"(a) The gravitational field acting on S2 is a linear combination of gravitational fields produced by S1 being in a superposed state.",
"(b) The semi-classical treatment of gravity, where the gravitational field acting on S2 is that produced by a total mass m 1 m_1 with density 1 2|α(𝐫)| 2 +|β(𝐫)| 2 \\frac{1}{2}\\left(|\\alpha ({\\bf r})|^2+|\\beta ({\\bf r})|^2\\right).Quantum gravity scenario.– Although we do not have a quantum theory of gravity so far, one can safely claim that it would manifest in S1 generating a superposition of gravitational fields.",
"The linearity, which is the characteristic trait of quantum theory, is preserved, as one would expect in any quantum theory of gravity.",
"The reaction of S2 is then to go in a superposition of being attracted towards the region where $\\mathinner {|{\\alpha }\\rangle }$ sits and where $\\mathinner {|{\\beta }\\rangle }$ does.",
"The final two-body state will have the following entangled form $\\Psi ^\\text{\\tiny final}_\\text{\\tiny QG}({\\bf r}_1,{\\bf r}_2)=\\frac{\\alpha ({\\bf r}_1)\\phi _\\alpha ({\\bf r}_2)+\\beta ({\\bf r}_1)\\phi _\\beta ({\\bf r}_2)}{\\sqrt{2}},$ where $\\phi _\\alpha ({\\bf r}_2)$ ($\\phi _\\beta ({\\bf r}_2)$ ) represents the state of S2 attracted towards the region where $\\mathinner {|{\\alpha }\\rangle }$ ($\\mathinner {|{\\beta }\\rangle }$ ) rests.",
"The latter superposition of motions for S2 is produced by the following potential $\\hat{V}_\\gamma (\\hat{{\\bf r}}_2) = -G m_1 m_2\\,\\int _1\\frac{|\\gamma ({ {\\bf r}}_1)|^2}{| {{\\bf r}}_1-\\hat{ {\\bf r}}_2|},~~~~~(\\gamma =\\alpha ,\\beta ).$ Moreover, we assume that the quantum fluctuations around the mean values for S1 are small, so that the gravitational interaction can be approximated by $\\hat{V}_{\\gamma } (\\hat{{\\bf r}}_2) \\approx -\\frac{G m_1 m_2}{|\\mathinner {\\langle {\\hat{\\bf r}_1(t)}\\rangle }_{\\gamma }-\\hat{\\bf r}_2(t)|},~~~~~~(\\gamma =\\alpha ,\\beta ),$ where $\\mathinner {\\langle {\\hat{{\\bf r}}_1}\\rangle }_\\gamma =\\mathinner {\\langle {\\gamma |\\hat{{\\bf r}}_1|\\gamma }\\rangle }$ with $\\gamma =\\alpha , \\beta $ .",
"Semiclassical gravity scenario.– The second scenario sees gravity as fundamentally classical.",
"In this case, it is not clear which characteristics one should expect from the gravitational field generated by a superposition.",
"However, in analogy with classical mechanics, one can assume that is the mass density of the system in superposition that produces the gravitational field.",
"This is also what is predicted by the Schrödinger-Newton equation , , [5], , , .",
"In such a case, what matters is the full wavefunction of S1 and not its single parts.",
"Consequently, the generated gravitational field is not in a quantum superposition, but it manifests as that produced by a classical object with total mass $m_1$ with density $|\\psi ({\\bf r}_1)|^2\\simeq \\frac{1}{2}\\left(|\\alpha ({\\bf r}_1)|^2+|\\beta ({\\bf r}_1)|^2\\right)$ .",
"Clearly, S2 reacts as driven by a classical gravitational field.",
"The final two-body state will be of the form $\\Psi ^\\text{\\tiny final}_\\text{\\tiny CG}({\\bf r}_1,{\\bf r}_2)=\\frac{\\alpha ({\\bf r}_1)+\\beta ({\\bf r}_1)}{\\sqrt{2}}\\phi ({\\bf r}_2),$ where the difference with Eq.",
"(REF ) is clear.",
"The gravitational potential becomes $\\hat{V}_\\text{\\tiny cl} (\\hat{{\\bf r}}_2) \\approx \\tfrac{1}{2}\\sum _{\\gamma =\\alpha ,\\beta } \\hat{V}_{\\gamma } (\\hat{{\\bf r}}_2),$ where $\\hat{V}_{\\gamma } (\\hat{{\\bf r}}_2)$ can be eventually approximated as in Eq.",
"(REF ).",
"In the next Section, we investigate the difference between the two scenarios by exploiting the sophisticated and powerful machinery provided by optomechanics.",
"Figure: The proposed set-up for the optomechanical falsification of quantum/classical gravity.",
"A system S1 is prepared in a superposition of two localised states at ±d x \\pm d_x along the xx axis.",
"An optomechanical cavity acts as transducer and probe of (potentially quantum) gravity effects S2: the effect of the gravitational coupling between S1 and the mechanical oscillator of an optomechanical cavity induces an effect on the variance of the position fluctuations of the oscillator.",
"The mean position of the latter along the xx axis is x ¯ 2 \\bar{x}_2.",
"The cavity is pumped by an external field (frequency ω 0 \\omega _0 and coupling rate ℰ{\\cal E})."
],
[
"Theoretical model",
"To describe the dynamics that follow the first or second scenario, we take advantage of the quantum Langevin equations, which is the typical description for optomechanical systems.",
"Moreover, we assume that the mass of S1 is sufficiently large to consider an adiabatic approach: S1 is stable and S2 evolves in the gravitational potential produced by S1.",
"Clearly, such a situation can be assumed only as long as the S1 superposition lives.",
"We assume S2 as trapped harmonically in ${\\bf r}_\\text{\\tiny osc}=(r_{x,\\text{\\tiny osc}},r_{y,\\text{\\tiny osc}},0)$ along the $x$ and $y$ directons by means of the cavity fields.",
"The corresponding quantum Langevin equations for the position ${\\hat{r}}_i$ and momentum $\\hat{p}_i$ operator of S2 read $\\begin{aligned}{\\frac{_i(t)}{t̥} }&=\\dfrac{{\\hat{p}}_i(t)}{m_2},\\\\{\\frac{_i(t)}{t̥} }&= -m_2\\omega _i^2 \\left({\\hat{r}}_i(t) -r_{i,\\text{\\tiny osc}}\\right)-\\gamma _i {\\hat{p}}_i(t)+{\\hat{\\xi }}_i(t)\\\\&+\\hbar \\chi _i\\hat{a}_i^\\dag (t)\\hat{a}_i(t)+\\dfrac{i}{\\hbar }[\\hat{V}_{{\\nu }},\\hat{p}_i(t)],\\end{aligned}$ where $i=x,y$ (we do not consider the motion along $z$ ) and $\\nu =\\alpha , \\beta , \\text{cl}$ .",
"Here, $\\omega _i$ is the harmonic frequency of the mechanical oscillator, $\\gamma _i$ is the damping rate for the vibrations, which are characterized by the noise operator $\\hat{\\xi }_i$ , having the correlation functions defined as $\\mathinner {\\langle {\\hat{\\xi }_i(t)}\\rangle }=0$ and $\\mathinner {\\langle {\\hat{\\xi }_i(t)\\hat{\\xi }_j(s)}\\rangle }=\\hbar m\\gamma _i\\delta _{ij}\\!\\int \\!\\frac{}{2\\pi }e^{-i\\omega (t-s)}\\omega [1+\\coth (\\tfrac{\\hbar \\omega }{2k_\\text{\\tiny B}T})].$ The position of S2 is measured by means of the cavity field, whose creation and annihilation operator are $\\hat{a}_i^\\dag $ and $\\hat{a}_i$ .",
"The dynamical equation of the latter is given by ${\\frac{(t)}{t̥} }= -i\\left[\\Delta _{0,i}- \\chi _i{\\hat{r}}_i(t)\\right] {\\hat{a}_i}(t)-\\kappa _i\\hat{a}_i(t)+\\sqrt{2\\kappa _i} {\\hat{a}}_{i,\\text{\\tiny in}}(t),\\\\$ where we defined $\\Delta _{0,i}=\\omega _{c,i}-\\omega _{0,i}$ , with $\\omega _{0,i}$ denoting the frequency of the external laser, $\\omega _{c,i}$ the frequency of the cavity mode derived by the laser, $\\chi _i=\\omega _{c,i}/L_i$ the optomechanical coupling constant between the cavity and the mechanical oscillator with $L_i$ the size of the cavity, and ${\\cal E}_i=\\sqrt{2\\kappa _i{\\cal P}_i/\\hbar \\omega _{0,i}}$ .",
"Here, ${\\cal P}_i$ is the laser power and $\\kappa _i$ is the cavity photon decay rate.",
"Moreover, we defined ${\\hat{a}}_{i,\\text{\\tiny in}}$ as the annihilation operator of external laser field, whose only non-zero correlation reads $\\mathinner {\\langle {\\hat{a}_{i,\\text{\\tiny in}}(t)\\hat{a}^\\dag _{j,\\text{\\tiny in}}(s)}\\rangle }=\\delta _{ij}\\delta (t-s)$ .",
"The last term in Eq.",
"(REF ) describes the gravitational interaction with S1, whose action is described below.",
"To be quantitative, we define the mean positions of the two systems in interaction.",
"We consider S1 as holding a steady position that can be approximated to its average value on $\\alpha $ or $\\beta $ respectively: $\\mathinner {\\langle {\\hat{{\\bf r}}_1(t)}\\rangle }_{\\gamma }\\approx (s_\\gamma d_x,0,0)$ , with $s_\\alpha =1$ , $s_\\beta =-1$ .",
"Conversely, we consider the position of S2 as an operator, center in $(\\bar{x}_2,d_y,0)$ [cf. Fig.",
"REF ].",
"Thus, we have $\\hat{{\\bf r}}_2(t)=(\\hat{r}_x(t),\\hat{r}_y(t), 0)=(\\bar{x}_2+\\hat{\\delta }_x(t),d_y+\\hat{\\delta }_y(t),0)$ and $\\hat{{\\bf p}}_2(t)=(\\hat{p}_x(t), \\hat{p}_y(t),0)$ is its momentum operator.",
"Assuming that the quantum fluctuations $\\delta \\hat{\\bf r}_2(t)=(\\hat{\\delta }_x(t),\\hat{\\delta }_y(t),0)$ around the initial mean values for S2 are small, we can expand the commutator in the last term of Eq.",
"(REF ) up to the first order in the fluctuations.",
"Thus, we have $\\frac{i}{\\hbar }[\\hat{V}_{{\\nu }},\\hat{p}_i(t)]=C^{(\\nu )}_{0,i}+C^{(\\nu )}_{1,i}\\hat{\\delta }_i(t)+C^{(\\nu )}_{2,i}\\hat{\\delta }_j(t),~~\\text{with }j\\ne i.$ In the quantum scenario, the coefficients $C^{(\\nu )}_{n,i}$ entering in Eq.",
"(REF ) are defined in Table REF , while those in the classical scenario are given by $C^\\text{(cl)}_{n,i}=\\tfrac{1}{2}(C^{(\\alpha )}_{n,i}+C^{(\\beta )}_{n,i})$ .",
"Table: Explicit form of the coefficients C n,i (γ) C^{(\\gamma )}_{n,i} entering in Eq.",
"() for the quantum scenario, with 𝒢 γ =Gm 1 m 2 /h γ 3 \\mathcal {G}_\\gamma ={G m_1m_2}/{h_\\gamma ^3} and h γ =(x ¯ 2 -s γ d x ) 2 +d y 2 h_\\gamma =\\sqrt{(\\bar{x}_2-s_\\gamma d_x)^2+d_y^2}.",
"For the classical scenario we have C i,x (cl) =1 2(C i,x (α) +C i,x (β) )C^\\text{(cl)}_{i,x}=\\tfrac{1}{2}(C^{(\\alpha )}_{i,x}+C^{(\\beta )}_{i,x}).In the limit of $d_x\\gg \\bar{x}_2$ , they become $C_{1,x}=\\dfrac{G m_1m_2}{d^5}(2d_x^2-d_y^2),\\\\C_{1,y}=\\dfrac{G m_1m_2}{d^5}(2d_y^2-d_x^2),\\\\C_{2}^{(\\gamma )}=-\\frac{3Gm_1m_2}{d^5}d_xd_ys_\\gamma ,~~\\text{and}~~C_{2}^{\\text{(}cl)}=0,$ where $d^2=({d_x^2+d_y^2})$ .",
"Here only $C_{2}^{(\\nu )}$ depends on the specific scenario (quantum or semi-classical) we are considering.",
"Following conventional approach, one finds: $\\bar{r}_i^{(\\nu )}=\\frac{\\hbar \\chi _i|\\bar{a}_i|^2+C_{0,i}^{\\nu }}{m_2\\omega _i^2}+r_{i,\\text{\\tiny osc}},\\quad \\text{and}\\quad \\bar{p}^{(\\nu )}_i=0.$ We can remove the radiation pressure contribution by setting the center of the harmonic trap to $r_{i,\\text{\\tiny osc}}=-\\hbar \\chi _i|\\bar{a}_i|^2/m_2\\omega _i^2$ .",
"Moreover, we assume that $d_x\\gg \\bar{x}_2$ , such that one can approximate $h_\\gamma \\simeq d=(d_x^2+d_y^2)^{1/2}$ [cf.",
"Table REF ], thus finding $\\bar{x}_2^{(\\gamma )}=\\frac{Gm_1d_x}{\\omega _x^2d^3}s_\\gamma ,~~~\\bar{y}_2^{(\\gamma )}=\\frac{Gm_1d_y}{\\omega ^2_yd^3}.$ These expressions show the first difference between the quantum and the classical scenario.",
"In the quantum scenario S2 is pulled towards positive (or negative) $x$ while in the classical scenario it remains at the center $\\bar{x}_2^{\\text{(}cl)}=\\bar{x}_2^{(\\alpha )}+\\bar{x}_2^{(\\beta )}=0$ .",
"However, it also highlights the difficulties one has in discerning the two scenarios.",
"Once the average is taken in the quantum scenario, we have $\\mathinner {\\langle {\\hat{x}_2}\\rangle }_\\text{(qu)}=\\tfrac{1}{2}\\sum _\\gamma \\bar{x}_2^{(\\gamma )}=0$ , which corresponds to the classical result.",
"Equation (REF ) shows that the difference between the quantum and the semi-classical scenario is embedded in the coupling between the motions along $x$ and $y$ of S2.",
"Indeed, in the quantum scenario, the gravitation attraction of S1 pulls S2 towards one of the branches of the superposition of S1, leading to correlations between the $x$ and $y$ motions.",
"Conversely, in the semi-classical scenario, for which $C_2^{\\text{(}cl)}=0$ , the dynamics along the two direction is decoupled, due to the symmetrical attraction of S1 along $y$ .",
"The verification of a coupling of the motion along $x$ with that along $y$ would be sufficient to prefer the quantum scenario over the semi-classical one.",
"Next we discuss possible mechanisms that can be exploited for this task."
],
[
"Revelation strategies",
"There are different measurements that one can exploit for witnessing the correlations between the $x$ and $y$ motions, and thus providing a verification of the quantum scenario over the semi-classical one.",
"1) Direct measurement of the Density Noise Spectrum.",
"To quantify the difference between the two scenarios, we consider the Density Noise Spectrum (DNS) corresponding to the motion of S2 along the $x$ axis.",
"By working under conditions such that $d_x\\gg \\bar{x}_2$ , the Langevin equations for the fluctuations read $\\begin{aligned}{\\frac{(t)}{t̥} }&={\\frac{{\\delta \\hat{p}}_i(t)}{m_2},}\\\\{\\frac{_i(t)}{t̥} }&= -m_2\\omega _i^2\\hat{\\delta }_i(t)-\\gamma _i {\\delta \\hat{p}}_i(t){+{\\hat{\\xi }}_i(t)}+C_{1,i} \\hat{\\delta }_i(t)\\\\&+C_{2,i}^{(\\nu )} \\hat{\\delta }_j(t)+\\hbar \\chi _i[\\bar{a}_i^*\\delta \\hat{a}_i(t)+\\bar{a}_i\\delta a_i^\\dag (t)],\\\\{\\frac{(t)}{t̥} }&{=} -i\\Delta _{i}^{(\\nu )} {\\delta \\hat{a}_i}(t){+}i \\chi _i\\bar{a}_i{\\hat{\\delta }_i}(t){-}\\kappa _i\\delta \\hat{a}_i(t){+}\\sqrt{2\\kappa _i} {\\hat{a}}_{i,{\\text{\\tiny in}}}(t)\\end{aligned}$ for $j\\ne i$ .",
"The coefficients $C_{n,i}^{(\\nu )}$ are approximated as in Eqs.",
"(), $\\Delta _i^{(\\nu )}=\\Delta _{0,i}-\\chi _i\\bar{a}_i\\bar{r}_i^{(\\nu )}$ , which becomes $\\Delta _i\\simeq \\Delta _{0,i}$ in light of the weakness of the optomechanical coupling.",
"Eqs.",
"(REF ) can be solved in the frequency domain by using the standard approach .",
"By defining $\\tilde{r}_i(\\omega )$ as the Fourier transform of $\\hat{\\delta }_i(t)$ , after lengthly yet straightforward calculations, we find $\\tilde{r}_i(\\omega )=\\frac{1}{m_2\\left[\\omega _{i,\\text{\\tiny eff}}^2(\\omega )-\\omega ^2-i\\gamma _{i,\\text{\\tiny eff}}(\\omega )\\omega \\right]}\\left[\\tilde{\\xi }_i(\\omega )+C_2^{(\\nu )}\\tilde{r}_j(\\omega )+\\hbar \\chi _i\\sqrt{2\\kappa _i}\\left(\\frac{\\bar{a}_i^*\\tilde{a}_{i,\\text{\\tiny in}}(\\omega )}{\\kappa _i+i(\\Delta _i-\\omega )} +\\frac{\\bar{a}_i\\tilde{a}^\\dag _{i,\\text{\\tiny in}}(\\omega )}{\\kappa _i-i(\\Delta _i+\\omega )} \\right) \\right],$ where we defined the following effective frequencies and dampings $\\omega _{i,\\text{\\tiny eff}}^2(\\omega )=\\omega _i^2+\\frac{2\\hbar \\chi _i^2|\\bar{a}_i|^2\\Delta _i(\\omega ^2-\\kappa _i^2-\\Delta _i^2)}{m_2\\left[(\\kappa _i^2+\\Delta _i^2+\\omega ^2)^2-4\\Delta _i^2\\omega ^2\\right]}-\\frac{C_{1,i}}{m_2},$ $\\gamma _{i,\\text{\\tiny eff}}(\\omega )=\\gamma _i+\\frac{4\\hbar \\chi _i^2|\\bar{a}_i|^2\\Delta _i\\kappa _i}{m_2\\left[(\\kappa _i^2+\\Delta _i^2+\\omega ^2)^2-4\\Delta _i^2\\omega ^2\\right]}.$ The effect of such correlation can be seen in the DNS, which can be derived from Eq.",
"(REF ) by applying its definition $\\mathcal {S}_{ii}(\\omega )=\\tfrac{1}{4\\pi }\\int \\,\\mathinner {\\langle {\\lbrace \\tilde{r}_i(\\omega ),\\tilde{r}_i(\\Omega )\\rbrace }\\rangle }$ .",
"Then we find $\\mathcal {S}_{xx}(\\omega )=\\frac{m_2g_y(\\omega )\\left[\\left(\\hbar m_2\\gamma _x\\omega \\coth \\left(\\tfrac{\\hbar \\omega }{2k_\\text{\\tiny B}T}\\right)+\\mathcal {S}^{x}_\\text{\\tiny L}(\\omega )\\right)+\\frac{(C_2^{(\\nu )})^2}{m_2^2g_y^2(\\omega )}\\left(\\hbar m_2\\gamma _y\\omega \\coth \\left(\\tfrac{\\hbar \\omega }{2k_\\text{\\tiny B}T}\\right)+\\mathcal {S}^{y}_\\text{\\tiny L}(\\omega )\\right) \\right]}{m_2^4g_x(\\omega )g_y(\\omega )-2m_2^2(C_2^{(\\nu )})^2f(\\omega )+(C_2^{(\\nu )})^4},$ where $g_i(\\omega )=(\\omega _{i,\\text{\\tiny eff}}^2(\\omega )-\\omega ^2)^2+\\gamma _{i,\\text{\\tiny eff}}^2(\\omega )\\omega ^2,\\\\\\mathcal {S}^{i}_\\text{\\tiny L}(\\omega )=\\frac{2\\hbar ^2\\chi _i^2\\kappa _i|\\bar{a}_i|^2(\\kappa _i^2+\\Delta _i^2+\\omega ^2)}{\\left[(\\kappa _i^2+\\Delta _i^2+\\omega ^2)^2-4\\Delta _i^2\\omega ^2\\right]},$ and $\\begin{aligned}f(\\omega )&=(\\omega _{x,\\text{\\tiny eff}}^2(\\omega ){-}\\omega ^2)(\\omega _{y,\\text{\\tiny eff}}^2(\\omega ){-}\\omega ^2){-}\\gamma _{x,\\text{\\tiny eff}}(\\omega )\\gamma _{y,\\text{\\tiny eff}}(\\omega )\\omega ^2.\\end{aligned}$ with $\\omega _\\text{\\tiny eff}$ and $\\gamma _\\text{\\tiny eff}$ denoting the effective frequency and damping respectively.",
"Eq.",
"(REF ) shows that in the quantum scenario the gravitational interaction leads to an extra contribution in the DNS (last term in squared brackets), which is directly connected to the motion along $y$ .",
"Such a term appears as an extra peak centred in the effective oscillation frequency of the $y$ motion.",
"The amplitude of the peak is related to the coupling between S2 and the cavity field along $y$ .",
"Clearly, the larger the coupling the bigger is the amplitude of the peak.",
"An example of the presence of this second peak is shown in Fig.",
"REF .",
"Figure: Comparison between the DNS for the classical (in green) and the quantum (in red) scenario.",
"We have taken m 1 =5×10 -14 m_1=5\\times 10^{-14}\\,kg, m 2 =9.5×10 -19 m_2=9.5\\times 10^{-19} kg, d x =10 -9 d_{x}=10^{-9} m, d y =2.9×10 -4 d_{y}=2.9 \\times 10^{-4} m, ω x =2π×10 4 \\omega _x=2\\pi \\times 10^4 Hz, ω y =2π×9.5×10 3 \\omega _y=2\\pi \\times 9.5\\times 10^3 Hz, γ x =2π×100\\gamma _x=2\\pi \\times 100\\,Hz, γ y =2π×3×10 -3 \\gamma _y=2\\pi \\times 3\\times 10^{-3}\\,Hz, T=4×10 -3 T=4\\times 10^{-3}\\,K, ℰ y =2×10 4 ℰ x =8×10 14 \\mathcal {E}_y=2\\times 10^{4}\\,\\mathcal {E}_x=8\\times 10^{14} Hz, κ x =10 3 κ y =9×10 8 \\kappa _x=10^3\\kappa _y=9\\times 10^8\\,Hz, ω c,y =10 5 ω c,x =2π×3.7×10 15 \\omega _{c,y}=10^5\\,\\omega _{c,x}=2\\pi \\times 3.7\\times 10^{15}\\,Hz.2) Indirect measurement of non-classical correlation between cavity fields.",
"A viable strategy for the inference of the potentially non-classical nature of gravitational interaction goes through the assessment of possible non-classical correlations induced by the latter, according to the following rationale: The potential non-classical nature of gravity would induce a coupling between the $x$ and $y$ degrees of freedom, which might induce non-classical correlations in their joint state.",
"Such a coupling disappears for classical gravity as $C^{(cl)}_2=0$ .",
"The induced all-mechanical correlations could in turn translate into analogous all-optical ones in light of the optomechanical coupling.",
"In an experiment where all other plausible sources of correlations are carefully characterised, the possibility to detect all-optical quantum correlations would pave the way to the inference of the non-classical nature of gravity.",
"It is important to stress that such correlations do not need to be as strong as entanglement: any non-zero value of $C^{(\\gamma )}_2$ results in non-diagonal elements in the covariance matrix of the overall optomechanical system.",
"The entries of such matrix are $\\sigma _{ij}=\\langle \\lbrace \\delta \\hat{O}_i,\\delta \\hat{O}_j\\rbrace \\rangle $ , where the expectation value is taken over the state of the system.",
"Within the validity of the first-order expansion in the fluctuations invoked before, the presence of such non-diagonal elements entails non-classical correlations of the discord form .",
"It is thus sufficient to ascertain the non-nullity of the non-diagonal entries of the covariance matrix of the all-optical system embodied by the cavity fields only to infer, indirectly, the non classical nature of their correlations, and thus the quantum nature of the gravitational interaction.",
"In Fig.",
"REF (a) we report the total norm $\\sigma _{tot}=\\sum _j |\\sigma ^{f}_{jj}|$ of the non-diagonal part of the covariance matrix $\\sigma ^{f}$ of the two cavity fields (i.e.",
"we take only the fluctuation operators $\\delta \\hat{O}_i$ pertaining to the cavity fields) against $C_{1,x}$ for parameters such that $C_{1,x}=C_{1,y}$ .",
"We observe a linear growth of the covariances with the strength of the gravity-induced interaction.",
"This gives rise to non-zero values of the discord between such fields, a illustrated in panel (b).",
"Needless to say, the experimental ascertainment of a non-zero value of all-optical discord would pose significant experimental challenges, in light of its weakness.",
"Nevertheless, the link with the strength of the non-diagonal entries of the corresponding covariance matrix offers a potentially viable route towards the goal of this paper: the reconstruction of the entries of an all-optical covariance matrix can indeed be accurately performed via high-efficiency homodyne measurements, as routinely implemented in many laboratories.",
"Figure: Total norm of the non-diagonal entries of the all-optical covariance matrix [panel (a)] and all-optical discord [panel (b)] plotted against C 1,x C_{1,x}.",
"We have taken d x,y ∼10 -6 d_{x,y}\\sim 10^{-6}\\,m, m 1,2 =5×10 -10 m_{1,2}=5\\times 10^{-10}\\,Kg, mechanical modes of frequency 2π×10 7 2\\pi \\times 10^7\\,Hz, T=4T=4\\,mK, γ x,y =2π×100\\gamma _{x,y}=2\\pi \\times 100\\,Hz.",
"The cavity has length of 1mm and finesse of 1.07×10 4 1.07\\times 10^4.3) Experimental feasibility.",
"To reduce the decoherence rates from gas collisions and blackbody photons to be smaller than the expected gravity effects, experiments should be done a low temperature and ultra-high vacuum.",
"The calculation of the expected non-classical correlations quantified by discord have been done with typical parameters for optomechanical cantilever or membrane systems .",
"The calculation for the direct observation of the DNS assumes parameters typical for levitated mechanical systems , , , , , , , , .",
"The challenge for the direct DNS test will be to realise the strongly asymmetric double-cavity setup, where the two cavity frequencies are different.",
"The biggest challenge for the presented experimental geometry will be the handling of the effect of short-range interactions such as van der Waals and Casimir-Polder (CP) , which can overtone the gravity interaction between the two masses - given their close proximity."
],
[
"Conclusions",
"We have illustrated the dynamics of an optomechanical system probing the gravitational field of a massive quantum system in a spatial superposition.",
"Two different dynamics are found whether gravity is treated quantum mechanically or classically.",
"Here, we propose two distinct methods to infer which of the two dynamics rules the motion of the quantum probe, thus discerning the intrinsic nature of the gravitational field.",
"Such methods will be then eventually able to falsify one of the two treatments of gravity.",
"Recently other interferometric and non-interferometric tests of nature of gravity were proposed.",
"They are based on the detection of entanglement between two probes, respectively coupled to two different massive systems, which interact through gravity (NV center spins for and cavity fields for ).",
"Clearly, to have such entanglement, each of the three couples of interconnected systems (probe 1, system 1, system 2 and probe 2) needs to be entangled on their own.",
"Moreover, the entanglement between the two massive systems is inevitably small due to its gravitational character.",
"Conversely, our proposal profits of having only a single massive system involved in the interconnection, which reduces correlation losses.",
"In addition, we provide a second method for discerning the nature of gravity: the individuation of a second peak in the DNS.",
"The latter does not rely on delicate measurements of quantum correlations but can be assessed through standard optomechanical detection schemes.",
"Other experimental proposals were presented in , , , .",
"Acknowledgements.– The authors acknowledge financial support from the H2020 FET Project TEQ (grant n. 766900) and the COST Action QTSpace (CA15220).",
"AB acknowledges financial support from INFN.",
"MP is supported by the SFI-DfE Investigator Programme through project QuNaNet (grant 15/IA/2864), the Leverhulme Trust through the Research Project Grant UltraQuTe (grant nr.",
"RGP-2018-266) and the Royal Society Wolfson Fellowship scheme through project ExTraQCT (RSWF\\R3\\183013)."
]
] | 1906.04513 |
[
[
"Anisotropic evolution of 4-brane in a 6D generalized Randall-Sundrum\n model"
],
[
"Abstract We investigate a 6d generalized Randall-Sundrum brane world scenario with a bulk cosmological constant.",
"It is shown that each stress-energy tensor $T_{ab}^{i}$ on the brane is similar to a constant vacuum energy.",
"This is consistent with the Randall-Sundrum model in which each 3-brane Lagrangian separated out a constant vacuum energy.",
"By adopting an anisotropic metric ansatz, we obtain the 5d Friedmann-Robertson-Walker field equations.",
"At a little later period, the expansion of the universe is proportional to $t^{\\frac{1}{2}}$ which is as similar as the period of the radiation-dominated.",
"We also investigate the case with two $a(t)$ and two $b(t)$.",
"In a large region of $t$, we obtain the 3d effective cosmological constant $\\Lambda_{eff}=-2\\Omega/3>0$ which is independent of the integral constant.",
"Here the scale factor is exponential expansion which is consistent with our present observation of the universe.",
"Our results demonstrate that it is possible to construct a model which solves the dark energy problem, meanwhile guaranteeing a positive brane tension."
],
[
"Introduction",
"In the early 1920s, Kaluza and Klein attempted to establish a more fundamental theory which unifies the forces of electromagnetism and gravitation by introducing extra dimension(s) into general relativity [1].",
"The Kaluza-Klein (KK) theory attracted a lot of attention to explore extra dimensions in various observable phenomena [2], [3], [6], [7], [4], [5].",
"In the middle of last century, this interest in extra dimensions has been enhanced because of the emergence of string/M theory in which the extra dimensional space appear naturally.",
"Inspired by the concept of brane in string theory [8], the braneworld scenario is proposed.",
"This theory can well explain some difficult problems in physics, such as the hierarchy problem (the problem of why the electroweak scale/Higgs mass $M_{EW}\\sim 1$ TeV is so different from the Planck scale $M_{pl}\\sim 10^{16}$ TeV) and the cosmological constant problem [9], [10], [7], [11].",
"The most successful resolution of hierarchy problem in the above theories is Randall-Sundrum (RS) two-brane model [9].",
"The RS model takes into account the tension of the brane which causes the spacetime outside the brane to be curved.",
"It consists of a 5d AdS bulk with a negative cosmological constant $\\Lambda $ and a single extra dimension satisfying $S1/Z_{2}$ orbifold symmetry.",
"In such a scenario, our universe is described by a 5d metric $ds^{2}=e^{-2\\sigma (\\phi )}\\eta _{\\mu \\nu }dx^{\\mu }dx^{\\nu }+r_{c}^{2}d\\phi ^{2},$ where $\\phi $ is the coordinate for an extra dimension, $r_{c}$ is the compactification radius, $e^{-2\\sigma }$ is the warp factor with $\\sigma =kr_{c}|\\phi |$ , $k=\\sqrt{-\\Lambda /24M^{3}}$ with $M$ being the 5d Planck mass.",
"In this model, the weak scale is generated from the Planck scale through the warp factor which originates from the background metric.",
"But the visible brane in RS model have a negative tension which is intrinsically unstable.",
"Furthermore the visible 3-brane (four-dimensional spacetime) has zero cosmological constant, which is not consistent with presently observed small value [11], [12].",
"Such a braneworld model has been widely studied.",
"It is shown that the induced cosmological constant and the brane tension of the visible brane can be both positive or negative [13], [14], [15].",
"By replacing $\\eta _{\\mu \\nu }$ with $g_{\\mu \\nu }$ , a generalized RS braneworld scenario is achieved [11].",
"In this model, the negative brane cosmological constant is analysed in detail [16], [19], [17], [20], [18].",
"It shows that $N$ has a minimum value $N_{min}=2n(n\\approx 16)$ which leads to an upper bound for the induced negative cosmological constants.",
"Furthermore, there are two different solutions to the hierarchical problem for a tiny value of cosmological constant.",
"One solution corresponds to the visible and hidden brane both with positive tension.",
"This is very interesting because both branes are stable.",
"In another case, the induced positive cosmological constant corresponds to a negative tension visible brane which is instable, so we do not considered this case anymore.",
"In above anti-de Sitter brane region, a large part of the parameter space corresponds to a positive value for the visible brane tension.",
"But our universe is currently undergoing accelerated expansion which is indicated by recent observations of type Ia supernovae [21], [22] and measurements of the anisotropies of the cosmic microwave background [23], [25], [24].",
"To explain this late-time epoch of accelerating expansion of the universe, we assume that there is a cosmological constant component in 4d Einstein's field equation [26].",
"The cosmological constant is a very small value ($\\simeq 10^{-124}$ in Planck unit) which is restricted by the above experiments.",
"So we need to cancel the induced negative cosmological constants in order to be consistent with observations.",
"In this paper, we focus on a 6d braneworld models because there is no special reason to restrict the number of dimensions to five.",
"For solving the above problem of the induced cosmological constant on the visible brane being negative, we consider the 4-brane (a extra dimension on the brane) in 6d generalized RS model instead of the 3-brane in 5d generalized RS model.",
"Then, we obtain the effective induced positive cosmological constant of 4d spacetime with an anisotropic metric ansatz.",
"At a little later period, the expansion of the 3d scale factor is as similar as the period of the radiation-dominated.",
"Our work is organized as follows: In Sec.",
"II, by considering the 4-brane with the matter field Lagrangian in 6d generalized RS model, we obtain a 5d Einstein field equation.",
"In Sec.",
"III, we focus on the evolution of 4-brane solved from the above field equation with an anisotropic metric ansatz.",
"Finally, the summary and conclusion are presented in Sec.",
"IV."
],
[
"6d Generalized Randall-Sundrum Model",
"We start with a 6d generalized Randall-Sundrum model action: $S=S_{bulk}+S_{vis}+S_{hid}.$ The bulk action, the visible brane action and the hidden brane action are respectively: $S_{bulk}=\\int d^{5}xdy\\sqrt{-G}(M^{4}_{6}R-\\Lambda ),\\\\S_{vis}=\\int d^{5}x\\sqrt{-g_{vis}}(\\mathcal {L}_{vis}-V_{vis}),\\\\S_{hid}=\\int d^{5}x\\sqrt{-g_{hid}}(\\mathcal {L}_{hid}-V_{hid}),$ where $\\Lambda $ is a bulk cosmological constant, $M_{6}$ denotes 6d fundamental mass scale, $G_{AB}$ is the 6d metric tensor, $R$ is the 6d Ricci scalar, $\\mathcal {L}_{vis}(\\mathcal {L}_{hid})$ and $V_{vis}(V_{hid})$ are the matter field Lagrangian and the tension of the visible(hidden) brane, respectively.",
"Variation of the above action with respect to the 6d metric tensor $G_{AB}$ led to Einstein¡¯s equations: $R_{AB}-\\frac{1}{2}G_{AB}R=\\frac{1}{2M^{4}_{6}}\\lbrace -G_{AB}\\Lambda +\\sum _{i}[T^{i}_{AB}\\nonumber \\\\\\times \\delta (y-y_{i})-G_{ab}\\delta _{A}^{a}\\delta _{B}^{b}V_{i}\\delta (y-y_{i})]\\rbrace ,$ where Capital Latin $A,B$ indices run over all spacetime dimensions, Lowercase Latin $a,b=0,1,2,3,4$ $R_{AB}$ and $T^{i}_{AB}$ are the 6d Ricci and the energy-momentum tensors respectively, $y_{i}$ represents the position of the $i$ -th brane in the sixth coordinate, $i=hid$ or $vis$ .",
"The 6d stress-energy tensor $T^{iA}_{B}$ will be assumed to be that of an anisotropic perfect fluid and of the form $T^{iA}_{B}=diag[-\\rho _{i}(t),p_{i1}(t),p_{i2}(t),p_{i3}(t),p_{i4}(t),0].$ The metric ansatz in the generalized RS scenario, satisfying the 6d Einstein equations is $ds^{2}=G_{AB}dx^{A}dx^{B}=e^{-2A(y)}g_{ab}dx^{a}dx^{b}+r^{2}dy^{2},$ where $g_{ab}$ is the 5d metric tensor.",
"The corresponding Einstein equations are given by: $\\widetilde{R}=e^{-2A}(20A^{\\prime 2}+\\frac{\\Lambda }{M^{4}_{6}}),$ and $\\widetilde{R}_{ab}-\\frac{1}{2}g_{ab}\\widetilde{R}=g_{ab}e^{-2A}\\lbrace (4A^{\\prime \\prime }-10A^{\\prime 2})-\\frac{1}{2M^{4}_{6}}[\\Lambda \\nonumber \\\\+\\sum _{i}\\delta (y-y_{i})V_{i}]\\rbrace +\\frac{e^{-2A}}{M^{4}_{6}}\\sum _{i}T_{ab}^{i}\\delta (y-y_{i}),$ where $\\widetilde{R}_{ab}$ and $\\widetilde{R}$ are the five-dimensional Ricci tensor and Ricci scalar respectively, defined with respect to $g_{\\mu \\nu }$ .",
"One side of Eq.",
"(REF ) contains the derivatives of $A(y)$ , depending on the extra coordinate $y$ alone, while the other side depends on the brane coordinates $x_{\\mu }$ alone.",
"Thus each side is equal to an arbitrary constant.",
"For convenience, we take this arbitrary constant to be $10\\Omega /3$ .",
"Thus, we get from Eq.",
"(REF ): $e^{-2A}(20A^{\\prime 2}+\\frac{\\Lambda }{M^{4}_{6}})=\\frac{10}{3}\\Omega ,$ and $\\widetilde{R}=\\frac{10}{3}\\Omega .$ Multiply both sides of Eq.",
"(REF ) by $g^{ab}$ , and rearranging terms, we get: $\\widetilde{R}=-\\frac{2}{3}e^{-2A}\\lbrace (4A^{\\prime \\prime }-10A^{\\prime 2})-\\frac{1}{2M^{4}_{6}}[\\Lambda \\nonumber \\\\+\\sum _{i}\\delta (y-y_{i})V_{i}]\\rbrace -\\frac{e^{-2A}}{3M^{4}_{6}}\\sum _{i}T^{i}\\delta (y-y_{i}),$ where $T^{i}=g^{ab}T^{i}_{ab}$ .",
"Using Eqs.",
"(REF ) and (REF ) cancel $A^{\\prime 2}$ and $\\widetilde{R}$ in Eq.",
"(REF ), yielding a simplified expression for $A^{\\prime \\prime }$ , $A^{\\prime \\prime }=\\frac{\\Omega }{6}e^{2A}+\\frac{1}{8M^{4}_{6}}\\sum _{i}\\delta (y-y_{i})(V_{i}-\\frac{T^{i}}{5}).$ The left side and the first term of the right depending on the extra coordinate $y$ alone, while the other term appeare only when the extra coordinate $y=y_{i}$ alone.",
"Thus we get $T^{i}=constant$ .",
"For convenience, we define $T^{i}\\equiv 5C_{i}$ , where the the $C_{i}$ is a constant.",
"Eq.",
"(REF ) can be written: $A^{\\prime \\prime }=\\dfrac{\\Omega }{6}e^{2A}+\\frac{1}{8M^{4}_{6}}\\sum _{i}\\delta (y-y_{i})(V_{i}-C_{i}).$ Rearrange Eq.",
"(REF ), we get an expression for $A^{\\prime 2}$ : $A^{\\prime 2}=\\dfrac{\\Omega }{6}e^{2A}+k^{2},$ where $k^{2}\\equiv -\\Lambda /20M^{4}_{6}>0$ (for Ads bulk i.e.",
"$\\Lambda <0$ ).",
"We cancel the $A^{\\prime 2}$ and $A^{\\prime \\prime }$ in eq.",
"(REF ) by Eqs.",
"(REF ) and (REF ), then get: $\\widetilde{R}_{ab}-\\frac{1}{2}g_{ab}\\widetilde{R}=-\\Omega g_{ab}+\\frac{1}{2M^{4}_{6}}\\nonumber \\\\\\times \\sum _{i}(T_{ab}^{i}-g_{ab}C_{i})\\delta (y-y_{i}).$ From the above equation, we can see that $T_{ab}^{i}-g_{ab}C_{i}=0$ , then we get $\\rho _{i}=-p_{i1}=-p_{i2}=-p_{i3}=-p_{i4}=-C_{i}$ .",
"So each stress-energy tensor $T_{ab}^{i}$ is similar to a constant vacuum energy.",
"This is consistent with the RS model [9] in which each 3-brane Lagrangian separated out a constant vacuum energy.",
"We define the $\\mathcal {V}_{i}\\equiv V_{i}-C_{i}$ .",
"Thus, we get a 5d Einstein field equation: $\\widetilde{R}_{ab}-\\frac{1}{2}g_{ab}\\widetilde{R}=-\\Omega g_{ab},$ and the system of equations of $A(y)^{\\prime \\prime }$ and $A^{\\prime 2}$ : $\\left\\lbrace \\begin{aligned}A^{\\prime \\prime }&=\\dfrac{\\Omega }{6}e^{2A}+\\frac{1}{8M^{4}_{6}}\\sum _{i}\\delta (y-y_{i})\\mathcal {V}_{i}, \\\\A^{\\prime 2}&=\\dfrac{\\Omega }{6}e^{2A}+k^{2}.",
"\\\\\\end{aligned}\\right.$ The above corresponds to the induced cosmological constant $\\Omega $ on the visible brane.",
"For the induced brane cosmological constant $\\Omega >0$ and $\\Omega <0$ , the brane metric $g_{ab}$ may correspond to dS-Schwarzschild and AdS-Schwarzschild spacetimes respectively [27].",
"We first consider the induced negative cosmological constant on the visible brane, the following solution for the warp factor is obtained: $A=-\\ln [\\omega \\cosh (k|y|+C_{-})],$ where $\\omega \\equiv -\\Omega /6k^{2}$ , and the constant $C_{-}=\\ln \\frac{1-\\sqrt{1-\\omega ^{2}}}{\\omega }$ for considering the normalization of this factor at the orbifold fixed point $y = 0$ .",
"Note in the limit $\\omega \\sim 0$ , the RS solution $A = ky$ can be recovered.",
"This is consistent with the results in Ref. [11].",
"The other solution $C_{-}=\\ln \\frac{1+\\sqrt{1-\\omega ^{2}}}{\\omega }$ is excluded because the RS solution can not be recovered in the $\\omega ^{2}\\rightarrow 0$ limit.",
"We can get the 5d effective theory from the original action Eq.",
"(REF ).",
"We focus on the curvature term from which we can derive the scale of gravitational interactions: $S_{eff}\\supset \\int d^{5}x\\int _{-\\pi }^{\\pi }dy\\sqrt{-g}M^{4}_{6}re^{-3A(kry)}\\tilde{R},$ where we only focus on the coefficient proportional to five-dimensional Ricci scalar $\\tilde{R}$ .",
"The Legendre term [28] is not proportional to $\\tilde{R}$ when the metric was substituted inside the action.",
"So we do not consider this term here.",
"We can perform the $y$ integral to obtain a 5d action.",
"From this, we get $M^{3}_{5pl}=M^{4}_{6}[\\frac{\\omega ^{6}}{12kc_{1}^{3}}(e^{3kr\\pi }-1)+\\frac{c_{1}^{3}}{12k}(1-e^{-3kr\\pi })\\nonumber \\\\+\\frac{3\\omega ^{4}}{4kc_{1}}(e^{kr\\pi }-1)+\\frac{3\\omega ^{2}c_{1}}{4k}(1-e^{-kr\\pi })],$ where $c_{1}\\equiv 1+\\sqrt{1-\\omega ^{2}}$ .",
"We find that if $\\omega ^{6}\\ll e^{-3kr\\pi }$ , then $M_{5pl}$ depends only weakly on $r$ in the large $kr$ limit.",
"From this, Eq.",
"(REF ) can be simplified to $M^{3}_{5pl}=\\frac{2M^{4}_{6}}{3k}(1-e^{-3kr\\pi }).$ Then we can get $M^{3}\\simeq 2M^{4}_{6}/3k$ in the large $kr$ limit.",
"In this 4-brane model, note the 5d components of the bulk metric is $g^{vis}_{ab}=G_{\\mu \\nu }(x^{a},y=r\\pi )$ , we obtain: $g^{vis}_{ab}=g_{ab}e^{-2A(kr\\pi )}, \\\\\\sqrt{-g_{vis}}=\\sqrt{-g}e^{-3A(kr\\pi )}.$ From the above equations, we can not determine the physical masses by properly normalizing the fields, namely the hierarchy problem cannot be solved in this 4-brane model.",
"Take the second derivative of Eq.",
"(REF ) with respect to $y$ , we get: $A^{\\prime \\prime }=\\frac{\\Omega }{6}e^{2A}-2k\\tanh (k|y|+\\ln \\frac{1-\\sqrt{1-\\omega ^{2}}}{\\omega })\\nonumber \\\\\\times (\\delta (y)-\\delta (y-y_{vis})).$ Note the orbifold fixed point $y_{hid}=0$ .",
"Comparing the above equation with Eq.",
"(REF ), we get the tension of the visible(hidden) $V_{vis}$ ($V_{hid}$ ): $V_{vis}=16M^{4}_{6}k\\big [\\frac{e^{2kr\\pi }\\frac{\\omega ^{2}}{c_{1}^{2}}-1}{e^{2kr\\pi }\\frac{\\omega ^{2}}{c_{1}^{2}}+1}\\big ],$ and $V_{hid}=16M^{4}_{6}k\\big [\\frac{1-\\frac{\\omega ^{2}}{c_{1}^{2}}}{1+\\frac{\\omega ^{2}}{c_{1}^{2}}}\\big ].$ Setting $e^{-A(r\\pi )}=10^{-n}$ , then we get from Eq.",
"(REF ): $10^{-N}=4(10^{-n}e^{-x}-e^{-2x}),$ $e^{-x}=\\frac{10^{-n}}{2}[1\\pm \\sqrt{1-10^{-(N-2n)}}],$ where $x\\equiv \\pi kr$ , $\\omega ^{2}\\equiv 10^{-N}$ .",
"For $1-10^{2n}\\omega ^{2}\\ge 0$ , we find $\\omega ^{2}\\le 10^{2n}$ which leads to an upper bound for the cosmological constant ($\\sim 10^{-N}$ ) given by $N_{min}=2n$ .",
"Eq.",
"(REF ) have two values of $x$ instead of one, the both values give rise to the required warping.",
"For $(N-2n)\\gg 1$ , the first solution of $x$ corresponds to the RS value plus a minute correction which is given by $x_{1}=n\\ln 10+\\frac{1}{4}10^{-(N-2n)}$ , while the second solution of $x$ is given by $x_{2}=(N-n)\\ln 10+\\ln 4$ [11].",
"Obviously, the $x_{2}$ is greater than the $x_{1}$ .",
"Rewriting Eq.",
"(REF ) with $n$ and $N$ , we get: $\\mathcal {V}_{vis}=16M^{4}_{6}k\\frac{1-10^{N-2n}[1\\pm \\sqrt{1-10^{-(N-2n)}}]}{10^{N-2n}[1\\pm \\sqrt{1-10^{-(N-2n)}}]},$ where the visible brane tension $\\mathcal {V}_{vis}$ is different from Eq.",
"(23) worked out in Ref. [11].",
"The two brane tensions are approximately given as: $\\mathcal {V}_{vis-1}=-16M^{4}_{6}k,$ $\\mathcal {V}_{vis-2}=16M^{4}_{6}k.$ The visible brane tension in Eq.",
"(REF ) is greater than Eq.",
"(23) in Ref.",
"[11] because that the denominator of Eq.",
"(REF ) is different from that of Eq.",
"(23) in Ref. [11].",
"We see that a small negative cosmological constant suffices to render the tension positive, provided the distance between the branes is somewhat larger than the value predicted by RS model.",
"Note the tension $\\mathcal {V}_{vis-2}$ on the visible brane is inconsistent with Eq.",
"(25) in Ref. [11].",
"Because of $\\omega \\equiv 10^{-N}\\ll 0$ , we get that the hidden brane tension $V_{hid}$ is always positive.",
"For $\\Omega >0$ , the warp factor which satisfies Eq.",
"(REF ) is given by: $A=-\\ln [\\omega \\sinh (-k|y|+C_{+})],$ where $\\omega \\equiv \\Omega /6k^{2}$ , $C_{+}=\\ln \\frac{1+\\sqrt{1+\\omega ^{2}}}{\\omega }$ .",
"In this case, the value of $\\omega ^{2}$ is unbounded, so the positive brane cosmological constant $\\Omega $ can be of arbitrary value.",
"The solution of $kr\\pi $ is a single solution instead of two solutions for $\\Omega <0$ .",
"And the above solution is depend on $\\omega ^{2}$ and $n$ .",
"For $\\Omega >0$ , the visible brane tension $\\mathcal {V}_{vis}$ and the hidden brane tension are always negative and positive respectively [11].",
"The negative tension visible brane is instable, so we do not considered this case anymore."
],
[
"Anisotropic evolution of 4-brane",
"For $\\Omega <0$ , interestingly one can obtain the upper bound ($\\sim -10^{-2n}$ in Planck units) of the induced negative cosmological constant on the visible 4-brane and the 4-brane tension can be positive for the second solution.",
"In this paper, we only consider two different spatial scaling factors $a(t)$ and $b(t)$ ."
],
[
"Case I",
"First, we investigate the case with three $a(t)$ and one $b(t)$ which is most in line with the presently observed 3d space universe.",
"We choose an anisotropic metric ansatz of the form $g_{ab}=diag[-1,a^{2}(t),a^{2}(t),a^{2}(t),b^{2}(t)]$ [26].",
"We allow the scale factor of the extra dimension on the visible brane $b(t)$ to evolve at a rate different from that of the 3d scale factor $a(t)$ .",
"This metric describes a flat, homogeneous, and isotropic 3d space and a flat extra dimension on the visible brane.",
"In this case, by adopting the above metric ansatz, we obtain the 5d FRW field equations from the Einstein field equations Eq.",
"(REF ): $H_{a}^{2}+H_{a}H_{b}=\\frac{1}{3}\\Omega ,$ $\\dot{H}_{a}+2H_{a}^{2}=\\frac{1}{3}\\Omega ,$ $2\\dot{H}_{a}+\\dot{H}_{b}+3H_{a}^{2}+H_{b}^{2}+2H_{a}H_{b}=\\Omega ,$ where a dot denotes a time derivative, $H_{a}\\equiv \\dot{a}/a$ and $H_{b}\\equiv \\dot{b}/b$ are the Hubble parameters of the 3d space and extra dimension respectively.",
"Eq.",
"(REF ) can be rewritten as: $\\frac{dH_{a}}{H_{a}^{2}-\\frac{1}{6}\\Omega }=-2dt.$ Eq.",
"(REF ) can be integrated, then we get the following solution for the 3d Hubble parameter: $H_{a}=-\\sqrt{-\\frac{\\Omega }{6}}\\tan (2\\sqrt{-\\frac{\\Omega }{6}}t+c),$ where $c$ is a arbitrary constants of integration.",
"Performing the integration of Eq.",
"(REF ), one find the solution of 3d space scale factor $a(t)$ : $a=c_{a}\\big |\\cos (2\\sqrt{-\\frac{\\Omega }{6}}t+c)\\big |^{\\frac{1}{2}},$ where $c_{a}$ is a arbitrary constants of integration also.",
"We set that at the initial time $t=0$ , $a=a_{0}$ .",
"We can get $c_{a}=a_{0}|\\cos c|^{-\\frac{1}{2}}$ , Eq.",
"(REF ) may then be rewritten as: $a=a_{0}\\frac{|\\cos (2\\sqrt{-\\frac{\\Omega }{6}}t+c)|^{\\frac{1}{2}}}{|\\cos c|^{\\frac{1}{2}}}.$ where the scale factor $a(t)$ increases with the increasing of $t$ when $-\\frac{\\pi }{2}<2\\sqrt{-\\frac{\\Omega }{6}}t+c<0$ .",
"For $\\Omega <0$ , the induced negative cosmological constant is bounded from below by $\\sim -10^{-2n}$ .",
"In order that the 3d space factor changes with time as smooth as possible, we get the second solution $x_{2}\\simeq (N-n)\\ln 10+\\ln 4\\simeq 172$ with $n\\simeq 50$ and $N\\simeq 124$ .",
"Note here the above case $n\\simeq 50$ and $N\\simeq 124$ is satisfied both conditions $N-n\\ll 0$ from $\\omega ^{6}\\ll e^{-3kr\\pi }$ and $(N-2n)\\gg 1$ .",
"In this case of $\\Omega \\simeq -10^{-124}$ , we obtain that $2\\sqrt{-\\frac{\\Omega }{6}}t\\ll 1$ when $t$ is not very large (for today $t\\sim 10^{60}$ in Planck unit).",
"We set the constants $c$ in Eq.",
"(REF ) equal to $-\\pi /2$ plus a small positive constant $\\chi $ to make sure that the scale factor $a(t)$ is increasing from $t=0$ to the present $t\\sim 10^{60}$ .",
"Then the scale factor $a(t)$ can be written: $a&=&a_{0}\\frac{|\\cos [2\\sqrt{-\\frac{\\Omega }{6}}t-\\frac{\\pi }{2}+\\chi ]|^{\\frac{1}{2}}}{|\\cos (-\\frac{\\pi }{2}+\\chi )|^{\\frac{1}{2}}}\\nonumber \\\\&=&a_{0}\\frac{\\sin ^{\\frac{1}{2}}(2\\sqrt{-\\frac{\\Omega }{6}}t+\\chi )}{\\sin ^{\\frac{1}{2}}\\chi }.$ The Hubble parameter $H_{a}$ is rewritten as: $H_{a}&=&-\\sqrt{-\\frac{\\Omega }{6}}\\tan (2\\sqrt{-\\frac{\\Omega }{6}}t-\\frac{\\pi }{2}+\\chi )\\nonumber \\\\&=&\\sqrt{-\\frac{\\Omega }{6}}\\cot (2\\sqrt{-\\frac{\\Omega }{6}}t+\\chi ).$ When $\\sqrt{-\\frac{\\Omega }{6}}t\\ll \\chi \\ll 1$ , the Hubble parameter $H_{a}$ is obtained: $H_{a}\\simeq \\sqrt{-\\frac{\\Omega }{6}}\\frac{1}{\\chi }.$ Here $H_{a}$ is a constant leading to an exponential expansion of the 3d scale factor.",
"Comparing with the FRW equation in which the gaussian curvature $K=0$ , and contains only the cosmological constant, we obtain the 4d effective cosmological constant $\\Lambda _{eff}=-\\Omega /2\\chi ^{2}>0$ .",
"But the above period is so short that the 3d space scale factor only increase from $a_{0}$ to $a_{0}(1+\\sqrt{\\frac{-\\Omega }{6\\chi ^{2}}}t)$ .",
"After that, we obtain the Hubble parameter $H_{a}$ and the 3d scale factor $a(t)$ when $\\chi \\ll \\sqrt{-\\frac{\\Omega }{6}}t\\ll 1$ : $H_{a}&\\simeq &\\frac{1}{2t},\\\\a(t)&\\simeq &a_{0}(-\\frac{2\\Omega }{3\\chi ^2})^{\\frac{1}{4}}t^{\\frac{1}{2}},$ where $a(t)$ is proportional to $t^{\\frac{1}{2}}$ which is as similar as the period of the radiation-dominated.",
"The deceleration parameter $q\\equiv -\\ddot{a}a/\\dot{a}^{2}=1-\\Omega /3H_{a}^{2}>1$ .",
"It is unsatisfactory because the aforementioned deceleration parameter is not consistent with currently undergoing accelerated expansion.",
"Using Eqs.",
"(REF ) and (REF ), the extra dimension Hubble parameter $H_{b}$ is given by: $H_{b}&=&\\frac{\\Omega }{3H_{a}}-H_{a}\\nonumber \\\\&=&\\frac{\\Omega }{3\\sqrt{-\\frac{\\Omega }{6}}\\cot (2\\sqrt{-\\frac{\\Omega }{6}}t+\\chi )}\\nonumber \\\\&&-\\sqrt{-\\frac{\\Omega }{6}}\\cot (2\\sqrt{-\\frac{\\Omega }{6}}t+\\chi ).$ Note when $H_{a}>0$ , we obtain $H_{b}<0$ , and vice versa.",
"Performing the integration of Eq.",
"(REF ), one find the solution of the extra dimension scale factor $b(t)$ : $b=b_{0}\\frac{\\sin ^{\\frac{1}{2}}\\chi \\cos (2\\sqrt{-\\frac{\\Omega }{6}}t+\\chi )}{\\cos \\chi \\sin ^{\\frac{1}{2}}(2\\sqrt{-\\frac{\\Omega }{6}}t+\\chi )},$ where we considered the initial conditions that when time $t=0$ , $b=b_{0}$ .",
"Form Eqs.",
"(REF ) and (REF ), it is obvious that the scale factor $a(t)$ and $b(t)$ are impossible to increase or reduce at the same time.",
"When the scale factor $a(t)$ increases, $b(t)$ decreases, and vice versa.",
"In other words, the decrease of $b(t)$ provides a driving force for the increasing of $a(t)$ .",
"The above investigation is the case of the increasing of the 3d scale factor $a(t)$ .",
"In the following, we investigate the case the scale factor $a(t)$ decrease with the increasing of time $t$ when $0<2\\sqrt{-\\frac{\\Omega }{6}}t+c<\\frac{\\pi }{2}$ in Eq.",
"(REF ).",
"The analysis is similar to the previous one, we substitute a small positive constant $\\psi $ into the constants $c$ in Eq.",
"(REF ).",
"The Hubble parameter $H_{a}$ and $H_{b}$ are rewritten as: $H_{a}=-\\sqrt{-\\frac{\\Omega }{6}}\\tan (2\\sqrt{-\\frac{\\Omega }{6}}t+\\psi ),$ $H_{b}&=&\\frac{\\Omega }{3H_{a}}-H_{a}\\nonumber \\\\&=&-\\frac{\\Omega }{\\sqrt{-\\frac{\\Omega }{6}}\\tan (2\\sqrt{-\\frac{\\Omega }{6}}t+\\psi )}\\nonumber \\\\&&-\\sqrt{-\\frac{\\Omega }{6}}\\tan (2\\sqrt{-\\frac{\\Omega }{6}}t+\\psi ).$ Then we obtain the scale factors $a(t)$ and $b(t)$ : $a=a_{0}\\frac{\\cos ^{\\frac{1}{2}}(2\\sqrt{-\\frac{\\Omega }{6}}t+\\psi )}{\\cos ^{\\frac{1}{2}}\\psi },$ $b=b_{0}\\frac{\\cos ^{\\frac{1}{2}}\\psi \\sin (2\\sqrt{-\\frac{\\Omega }{6}}t+\\psi )}{\\sin \\psi \\cos ^{\\frac{1}{2}}(2\\sqrt{-\\frac{\\Omega }{6}}t+\\psi )}.$ When $\\sqrt{-\\frac{\\Omega }{6}}t\\ll \\psi \\ll 1$ , the Hubble parameter $H_{a}\\simeq -\\sqrt{-\\frac{\\Omega }{6}}\\psi $ is a negative constant.",
"The Hubble parameter $H_{b}$ is obtained: $H_{b}\\simeq 2\\sqrt{-\\frac{\\Omega }{6}}\\frac{1}{\\psi }.$ Then we obtain the Hubble parameter $H_{b}$ and $b(t)$ when $\\psi \\ll \\sqrt{-\\frac{\\Omega }{6}}t\\ll 1$ : $&&H_{b}\\simeq \\frac{1}{t},\\\\&&b\\simeq b_{0}(-\\frac{2\\Omega }{3\\psi ^2})^{\\frac{1}{2}}t,$ where $b(t)$ is proportional to $t$ which is faster than the $a(t)$ in the case of the increasing of $a(t)$ .",
"Because the decreasing of three dimensions instead of one provides dynamic.",
"In Case I, the decreasing of scale factor(s) on the brane does(do) not provide sufficient impetus for the other scale factors(factor) to expand exponentially."
],
[
"Case II",
"Similarly to the Case I, we investigate the case with two $a(t)$ and two $b(t)$ .",
"We choose an anisotropic metric ansatz of the form $g_{ab}=diag[-1,a^{2}(t),a^{2}(t),b^{2}(t),b^{2}(t)]$ , the 5d FRW field equations are of the form: $H_{a}^{2}+4H_{a}H_{b}+H_{b}^{2}=\\Omega ,$ $\\dot{H}_{a}+2\\dot{H}_{b}+H_{a}^{2}+3H_{b}^{2}+2H_{a}H_{b}=\\Omega ,$ $\\dot{H}_{b}+2\\dot{H}_{a}+H_{b}^{2}+3H_{a}^{2}+2H_{a}H_{b}=\\Omega .$ where the $H_{a}$ and $H_{b}$ are symmetric.",
"Setting $H_{a}$ positive and $H_{b}$ negative, we obtain the following solutions for the Hubble parameters $H_{a}$ and $H_{b}$ respectively: $H_{a}&=&-\\sqrt{\\frac{-\\Omega }{6}}[\\tan (2\\sqrt{\\frac{-2\\Omega }{3}}t+c_{2})\\nonumber \\\\&&-\\sqrt{3}|\\sec (2\\sqrt{\\frac{-2\\Omega }{3}}t+c_{2})|],\\\\H_{b}&=&-\\sqrt{\\frac{-\\Omega }{6}}[\\tan (2\\sqrt{\\frac{-2\\Omega }{3}}t+c_{2})\\nonumber \\\\&&+\\sqrt{3}|\\sec (2\\sqrt{\\frac{-2\\Omega }{3}}t+c_{2})|].$ Figure: (Color online).",
"The Hubble parameter H a H_{a} (solid curve) varies as a function of 2-2Ω/3t+c 2 2\\sqrt{-2\\Omega /3}t+c_{2} in Case II.",
"The dashed curve is a constant H≃2H\\simeq 2.As show in Fig.",
"REF , the Hubble parameter $H_{a}$ is close to a constant $H\\simeq 2$ in a large region $-0.8<2\\sqrt{-2\\Omega /3}t+c_{2}<1.2$ .",
"It is very different from the Case I in which the Hubble parameter $H_{a}$ is a constant in a very tiny interval $\\sqrt{-\\frac{\\Omega }{6}}t\\ll \\chi \\ll 1$ .",
"Considering the constraints in the Eqs.",
"(REF ) and () as in Case I, $2\\sqrt{-\\frac{2\\Omega }{3}}t$ changes slowly with $t$ when we also set the second solution $x_{2}\\simeq (N-n)\\ln 10+\\ln 4\\simeq 172$ with $n\\simeq 50$ and $N\\simeq 124$ .",
"We obtain the 3d effective cosmological constant $\\Lambda _{eff}=-2\\Omega /3>0$ which is independent of the integral constant.",
"This is an important result.",
"It tells us that we can obtain an exponential expansion solution which is consistent with our presently observed universe when we start from a induced negative cosmological constants on the brane.",
"It is unsatisfactory because that the numbers of the expansion scale factor is two.",
"But this problem should be solved in a higher dimensional brane.",
"Finally, we consider an isotropic metric ansatz of the form $g_{ab}=diag[-1,a^{2}(t),a^{2}(t),a^{2}(t),a^{2}(t)]$ in the 5d Einstein field equations Eq.",
"(REF ), then we obtain the time-time component of 5d FRW field equations: $H_{a}^{2}=\\frac{1}{6}\\Omega .$ Note there is no solution to the above equation because $\\Omega <0$ ."
],
[
"Summary and Conclusion",
"To summarize, in this paper we investigate a 6d theory with a 4-brane in order to solve the cosmological fine tuning problem.",
"We find that each stress-energy tensor $T_{ab}^{i}$ on the brane is similar to a constant vacuum energy.",
"Note the hierarchy problem cannot be solved well in this model.",
"This is consistent with the RS model [9] in which each 3-brane Lagrangian separates out a constant vacuum energy.",
"The visible brane tension obtained in our paper is greater than the result in Ref. [11].",
"For $\\Omega <0$ , the induced negative cosmological constant on the visible 4-brane has an upper bound ($\\sim -10^{-32}$ in Planck units), and the 4-brane tension is positive for the second solution.",
"In above case, we obtain the 5d FRW field equations from the Einstein field equations by adopting an anisotropic metric ansatz.",
"In Case I, we find that the 3d space scale factor is increasing from $t=0$ to the present $t\\sim 10^{60}$ .",
"The constant Hubble parameter resulted in exponential expansion of the 3d scale factor slightly after the initial time $t=0$ .",
"But the period is so short that 3d space scale factor only increases from $a_{0}$ to $a_{0}(1+\\sqrt{\\frac{-\\Omega }{6\\chi ^{2}}}t)$ .",
"When $\\chi \\ll \\sqrt{-\\frac{\\Omega }{6}}t\\ll 1$ , 3d space scale factor $a(t)$ is proportional to $t^{\\frac{1}{2}}$ which is as similar as the period of the radiation-dominated.",
"In Case II, we investigate the case with two $a(t)$ and two $b(t)$ .",
"In a large region of $t$ , we obtain the 3d effective cosmological constant $\\Lambda _{eff}=-2\\Omega /3>0$ which is independent of the integral constant.",
"Here the scale factor is of exponential expansion which is consistent with our presently observed universe.",
"It is shown that the expansion rate of scale factor is not directly related to the numbers of the scale factor of decrease.",
"It is unsatisfactory because that the numbers of the expansion scale factor is two.",
"But this problem should be solved in a higher dimensional brane.",
"It will now be interesting to study whether the extra dimensions on the brane in this kind of generalised RS model with higher dimension (e.g.",
"10d spacetime as required by superstring theory) would provide enough impetus for 3d spcae exponential expansion.",
"We hope to report these in future works.",
"This paper is to be published in Chinese Physics C. We wish to acknowledge the support of the Key Program of National Natural Science Foundation of China (under Grant No.",
"11535005), the National Natural Science Foundation of China (under Grant No.",
"11647087 and No.",
"11805097), the Natural Science Foundation of Yangzhou Polytechnic Institute (under Grant No.",
"201917), and the Natural Science Foundation of Changzhou Institute of Technology (Grant No.",
"YN1509)."
]
] | 1906.04425 |
[
[
"Superfast Refinement of Low Rank Approximation of a Matrix"
],
[
"Abstract Low rank approximation (LRA) of a matrix is a hot subject of modern computations.",
"In application to Big Data mining and analysis the input matrices are usually so immense that one must apply superfast algorithms, which only access a tiny fraction of the input entries and involve much fewer memory cells and flops than an input matrix has entries.",
"Recently we devised and analyzed some superfast LRA algorithms; in this paper we extend a classical algorithm of iterative refinement of the solution of linear systems of equations to superfast refinement of a crude but reasonably close LRA; we also list some heuristic recipes for superfast a posteriori estimation of the errors of LRA and support our superfast refinement algorithm with some superfast heuristic recipes for a posteriori error estimation of LRA and with superfast back and forth transition between any LRA of a matrix and its SVD.",
"Our algorithm of iterative refinement of LRA is the first attempt of this kind and should motivate further effort in that direction, but already our initial tests are in good accordance with our formal study."
],
[
"Introduction",
"(a) Superfast accurate LRA: the problem and our recent progress.",
"Low rank approximation (LRA) of a matrix is a hot subject of Numerical Linear and Multilinear Algebra and Data Mining and Analysis, with applications ranging from machine learning theory and neural networks to term document data and DNA SNP data (see [12], [5], [15], [13], [2], [16], [17], [33], [34], and pointers to the huge bibliography therein).",
"Matrices representing Big Data are usually so immense that realistically one can only access a tiny fraction of all their entries, but quite typically these matrices admit LRA, that is, are close to low rank matrices,Here and throughout we use such concepts as “low\", “small\", “nearby\", “much fewer\" etc.",
"defined in context.",
"with which one can operate superfast, aka at sublinear cost, that is, by using much fewer memory cells and flops than an input matrix has entries.",
"For an $m\\times n$ input a superfast algorithm involves $f=o(mn)$ flops, and then it accesses at most $2f$ input entries – at most two per flop.",
"Unfortunately, every superfast algorithm fails to compute close LRA of a worst case input matrix and even of the matrices of the small families of our Appendix , but in the paper [26] the authors propose a superfast variation of the randomized algorithm of [32] where its dense random test matrices are replaced by fixed sparse ones and where an input matrix is random.",
"Under this dual randomization model, taken from [29], [28], [30], [24], [25], [23], [27], the authors prove that the output LRAs of the resulting algorithms are reasonably accurate with a high probability (whp), and hence are reasonably accurate for a large input class.Similar results have been obtained for some other superfast LRA algorithms in [27].",
"The output error bounds were not very close to the optimal LRA bound, defined by the Eckart-Young-Mirsky theorem, but small enough in order to motivate iterative refinement of LRA, which we propose in our present paper.",
"Namely we extend to LRA the popular methods for iterative refinement of the solution as well as least squares solution of a linear system of equations (see [31], [11], [6], [10], [3]).",
"Such refinement algorithms are also known for other linear and nonlinear matrix computations (see [31]), but to the best of our knowledge our iterative refinement of LRA is the first attempt of this kind, possibly because superfast LRA has not been studied as much as it deserves.",
"Our work promises an extension to computing LRA of an unknown rank (see our brief comments in Remark REF ).",
"Our initial tests provide some empirical support for our approach, and we recall that historically iterative refinement algorithms have become highly efficient in practice before their adequate formal support have been elaborated upon.",
"Organization of our paper.",
"In the next section we recall some background material.",
"In Section and Appendix we discuss superfast transition between LRA and SVD or rank-revealing QRP factorization.",
"We describe a superfast algorithm for iterative refinement of an LRA in Section and cover superfast a posteriori estimates in Section and Appendix .",
"We devote Section , the contribution of the first author, to numerical tests.",
"In Appendix we describe some small families of hard inputs for superfast computation of their LRA."
],
[
"LRA background",
"Hereafter $|||\\cdot |||$ stands for both spectral norm $||\\cdot ||_2$ and Frobenius norm $||\\cdot ||_F$ (cf.",
"[12]).",
"Definition 2.1 (i) An $m\\times n$ matrix $\\tilde{M}$ has rank at most $\\rho $ if $\\tilde{M}=AB,~A\\in \\mathbb {C}^{m\\times \\rho },~{\\rm and}~B\\in \\mathbb {C}^{\\rho \\times n},$ and in this case we write $\\operatorname{rank}(\\tilde{M})\\le \\rho $ .",
"(ii) An $m\\times n$ matrix $M$ has $\\epsilon $ -rank at most $\\rho $ for a fixed tolerance $\\epsilon $ if $M$ admits its approximation within an error norm $\\epsilon $ by a matrix $\\tilde{M}$ of rank at most $\\rho $ or equivalently if there exist three matrices $A$ , $B$ and $E$ such that $M=\\tilde{M}+E~{\\rm where}~|||E|||\\le \\epsilon ~|||M|||,~\\tilde{M}=AB,~A\\in \\mathbb {C}^{m\\times \\rho },~{\\rm and}~B\\in \\mathbb {C}^{\\rho \\times n}.$ 0-rank is the rank, and it is quite common to fix tolerance $\\epsilon $ in context and then to call $\\epsilon $ -rank numerical rank, $\\operatorname{nrank}(M)$ (cf.",
"[10]); then a matrix admits its close approximation by a matrix of rank $\\rho $ if and only if it has numerical rank $\\rho $ .",
"One can generalize a 2-factor LRA $AB$ of $M$ of (REF ) to a 3-factor LRA, $M=\\tilde{M}+E,~|||E|||\\le \\epsilon ,~\\tilde{M}=XTY,~X\\in \\mathbb {C}^{m\\times k},~T\\in \\mathbb {C}^{k\\times l},~Y\\in \\mathbb {C}^{l\\times n},$ $\\rho =\\operatorname{rank}(\\tilde{M})\\le k\\le m,~\\rho \\le l\\le n.$ An important 3-factor LRA of a matrix $M$ of rank at least $\\rho $ is its $\\rho $ -top SVD $M_{\\rho }=U_{\\rho }\\Sigma _{\\rho } V_{\\rho }^*$ where $\\Sigma _{\\rho }$ is the diagonal matrix of the $\\rho $ top (largest) singular values of $M$ and where $U_{\\rho }$ and $V_{\\rho }$ are the unitary (orthogonal) matrices of the associated top singular spaces of $M$ .",
"$M_{\\rho }$ is said to be the $\\rho $ -truncation of $M$ , obtained from $M$ by setting to zero all its singular values but the $\\rho $ largest ones.",
"$M_{\\rho }=M$ for a matrix $M$ of rank $\\rho $ , and then its $\\rho $ -top SVD is just its compact SVD $M=U\\Sigma V^*,~{\\rm for}~U=U_{\\rho },~\\Sigma =\\Sigma _{\\rho },~{\\rm and}~V=V_{\\rho }.$ By virtue of the Eckart-Young-Mirsky theorem, the $\\rho $ -top SVD of a matrix defines its optimal rank-$\\rho $ approximation under both spectral and Frobenius norms: Theorem 2.1 [10].",
"Write $\\tau _{\\rho +1}(M):=|||M_{\\rho }-M|||=\\min _{N:~\\operatorname{rank}(N)=\\rho } |||M-N|||.$ Then $\\tau _{\\rho +1}(M)=\\sigma _{\\rho +1}(M)$ under the spectral norm and $\\tau _{\\rho +1}(M)=\\sigma _{F,\\rho +1}(M):=\\sqrt{\\sum _{j> \\rho }\\sigma _j^2(M)}$ under the Frobenius norm.",
"Lemma 2.1 [10].",
"For $m\\ge n$ and a pair of ${m\\times n}$ matrices $M$ and $M+E$ it holds that $|\\sigma _j(M+E)-\\sigma _j(M)|\\le ||E||_2~{\\rm for}~j=1,\\dots ,n. $ Hereafter $M^+$ denotes the Moore–Penrose pseudo inverse of $M$ .",
"We use the following well-known property.",
"Lemma 2.2 [The norm of the pseudo inverse of a matrix product.]",
"Suppose that $A\\in \\mathbb {C}^{k\\times r}$ , $B\\in \\mathbb {C}^{r\\times l}$ and the matrices $A$ and $B$ have full rank $r\\le \\min \\lbrace k,l\\rbrace $ .",
"Then $|||(AB)^+|||\\le |||A^+|||~|||B^+|||$ ."
],
[
"Computation of a $\\rho $ -top SVD\nof an LRA",
"The following classical algorithm (cf., e.g., [19]) computes $\\rho $ -top SVD of an LRA.",
"Algorithm 3.1 [Computation of a $\\rho $ -top SVD of an LRA.]",
"Input: Four integers $\\rho $ , $k$ , $m$ , and $n$ such that $0<\\rho \\le k\\le \\min \\lbrace m,n\\rbrace $ and two matrices $A\\in \\mathbb {C}^{m\\times k}$ and $B\\in \\mathbb {C}^{k\\times n}$ .",
"Output: Three matrices $U\\in \\mathbb {C}^{m\\times \\rho }$ (unitary), $\\Sigma \\in \\mathbb {R}^{\\rho \\times \\rho }$ (diagonal), and $V\\in \\mathbb {C}^{n\\times \\rho }$ (unitary) such that $(AB)_{\\rho }=U\\Sigma V^*$ is a $\\rho $ -top SVD of $AB$ .",
"Computations: Compute SVDs $A=U_A\\Sigma _AV^*_A$ and $B=U_B\\Sigma _BV^*_B$ where $U_B\\in \\mathbb {C}^{m\\times k}$ , $V_B^*\\in \\mathbb {C}^{k\\times n}$ , and $\\Sigma _A,V^*_A,U_B,\\Sigma _B\\in \\mathbb {C}^{k\\times k}$ .",
"Compute $k\\times k$ matrices $W=\\Sigma _AV^*_AU_B\\Sigma _B$ , $U_W$ , $\\Sigma _W$ , and $V^*_W$ such that $W=U_W\\Sigma _WV_W^*$ is SVD, $\\Sigma _W=\\operatorname{diag}(\\Sigma ,\\Sigma ^{\\prime })$ , and $\\Sigma =\\operatorname{diag}(\\sigma _j)_{j=1}^{\\rho }$ and $\\Sigma ^{\\prime }=\\operatorname{diag}(\\sigma _j)_{j=\\rho +1}^{k}$ are the matrices of the $\\rho $ top (largest) and the $k-\\rho $ trailing (smallest) singular values of the matrix $W$ , respectively.",
"Output the matrix $\\Sigma $ .",
"Compute and output the matrices $U$ and $V$ made up of the first $\\rho $ columns of the matrices $U_AU_W$ and $V_BV_W$ , respectively.",
"The algorithm involves $O((m+n)k^2)$ flops; it is superfast if $k^2\\ll \\min \\lbrace m,n\\rbrace $ .",
"Its correctness follows from equations $AB=U_AWV^*_B$ , $W=U_W\\Sigma _W V^*_W$ , and $\\Sigma _W=\\operatorname{diag}(\\Sigma ,\\Sigma ^{\\prime })$ .",
"For every matrix $M$ triangle inequality implies that $|||M-(AB)_{\\rho }|||\\le |||M-AB|||+|||AB-(AB)_{\\rho }|||=|||M-AB|||+\\tau _{\\rho }(AB).$ Furthermore $\\tau _{\\rho }(AB)\\le |||AB-M_{\\rho }|||\\le |||AB-M|||+\\tau _{\\rho }(M)$ , and so (cf.",
"[32]) $|||M-(AB)_{\\rho }|||\\le 2~||||M-AB|||+\\tau _{\\rho }(M).$ The reader can readily extend Algorithm REF to the computation of $\\rho $ -top SVD of a 3-factor LRA $M=XTY$ .",
"Furthermore one can quite closely approximates $\\rho $ -top SVD of an LRA $AB$ (and similarly of a 3-factor LRA) based on rank-revealing QRP factorization of the factors $AB$ rather than on their SVD (cf.",
"[1]).",
"Algorithm 3.2 [Approximation of a $\\rho $ -top SVD of an LRA.]",
"Input: as in Algorithm REF .",
"Output: Three matrices $U\\in \\mathbb {C}^{m\\times \\rho }$ (unitary), $\\Sigma \\in \\mathbb {R}^{\\rho \\times \\rho }$ (diagonal), and $V\\in \\mathbb {C}^{n\\times \\rho }$ (unitary) such that the matrix $U\\Sigma V^*$ approximates a $\\rho $ -top SVD of $AB$ .",
"Computations: Compute Rank-revealing QRP and P$^{\\prime }$ LQ$^{\\prime }$ factorization $A=QRP$ and $B=P^{\\prime }LQ;$ where $Q\\in \\mathbb {C}^{m\\times k}$ , $Q^{\\prime }\\in \\mathbb {C}^{k\\times n}$ , $R$ and $L^*$ are $k\\times k$ upper triangular matrices, and $P$ and $P^{\\prime }$ are $k\\times k$ permutation matrices.",
"Define $\\rho \\times \\rho $ leading principal submatrices $R_{\\rho }$ of $R$ and $L^{\\prime }_{\\rho }$ of $L^{\\prime }$ .",
"Define $\\rho \\times \\rho $ matrix $P_{\\rho }$ by first deleting the $k-\\rho $ last rows of $P$ and then deleting the $k-\\rho $ columns of the resulting matrix filled with 0s.",
"Define $\\rho \\times \\rho $ matrix $P_{\\rho }^{\\prime }$ by first deleting the $k-\\rho $ last columns of $P^{\\prime }$ and then deleting the $k-\\rho $ rows of the resulting matrix filled with 0s.",
"Compute the $\\rho \\times \\rho $ nonsingular matrices $\\bar{R}_{\\rho }:=R_{\\rho }P_{\\rho }$ , $\\bar{L}^{\\prime }_{\\rho }:=P^{\\prime }_{\\rho }L^{\\prime }_{\\rho }$ , and $A:=\\bar{R}_{\\rho }\\bar{L}^{\\prime }_{\\rho }$ .",
"Compute SVD $A=U_A\\Sigma V^*_A$ where $U_A$ , $\\Sigma $ , and $V^*_A$ are $\\rho \\times \\rho $ matrices, $U_A$ and $V^*_A$ are unitary, and $\\Sigma _A$ is a diagonal matrix filled with the singular values of $A$ .",
"Output the matrix $\\Sigma $ .",
"Compute and output the unitary matrices $U:=QU_A$ and $V:=V_AQ^{\\prime }$ .",
"The asymptotic complexity of the algorithm is $O((m+n)k^2)$ flops, the same as for Algorithm REF , but the overhead constant in the \"O\" notation is a little smaller because at stage 1 we substitute rank revealing QRP factorization for SVD.",
"One can compute this factorization by applying the algorithm of [9], which keeps the errors of the approximation of SVD by means of QRP factorization within the error bound $\\sqrt{1+h^2(k-\\rho )\\rho }$ for any $h>1$ , e.g., $h=1.01$ .",
"Alternatively one can use rank-revealing $LUP$ factorization of [20] instead of $QRP$ factorization – staying within the same arithmetic cost bound but increasing the error bound to $1+h^2(k-\\rho )\\rho $ ."
],
[
"Superfast iterative refinement of an LRA",
"Next we describe iterative refinement of a sufficiently close LRA of $M$ towards nearly optimal level.",
"We assume that instead of $mn$ entries of an $m\\times n$ input matrix $M$ we are only allowed to access its sketches, $FM$ , $MH$ , and $FMH$ , defined by $\\rho \\times m$ matrix $F$ and $n\\times \\rho $ matrix $H$ (cf.",
"[12] and [32]).",
"We call such a representation of a matrix $M$ by its sketches sublinear if one can compute the sketches superfast; this only possible where the test matrices $F$ and $H$ are sparse.",
"We further assume that together with a large class of $m\\times n$ matrices represented sublinearly, we are given a black box Subalgorithm APPROX(r): by using $O((m+n)r^2)$ flops, it computes a rank-$r$ approximation of a matrix of this class.",
"One may think of the superfast variation of [32] in [26], cited in our Introduction, as an example of such a Subalgorithm.",
"For iterative refinement of an LRA, we first apply the Subalgorithm to an input matrix $E^{(0)}=M$ and then at the $i$ th step improve a current LRA $\\tilde{M}^{(i)}$ by applying the subalgorithm to the current error matrix $E^{(i)}=\\tilde{M}^{(i)}-M$ .",
"At every iteration the rank of the new tentative LRA is at most doubled, and we periodically cut it back to the value $\\operatorname{nrank}(M)$ (see Remark REF ).",
"We also assume that we are given a Stopping Criterion, e.g., based on a posteriori error estimation (see Section ).",
"Algorithm 4.1 (Superfast iterative refinement of a CUR LRA.",
"See Remarks REF and REF .)",
"Input: Three integers $m$ , $n$ , and $\\rho $ , $\\rho \\le \\min \\lbrace m, n\\rbrace $ , sublinear representation of an $m\\times n$ matrix $M$ , Subalgorithm APPROX(r), which for a fixed positive integer $r$ computes a rank-$r$ approximation of its input matrix, and a Stopping Criterion, which signals that the current candidate LRA is accepted as satisfactory.",
"Initialization: $\\tilde{M}^{(0)}=O^{m\\times n}$ .",
"Computations: Recursively for $i =0, 1, 2,\\dots $ do: Apply Subalgorithms APPROX($r_{i}$ ) to the matrix $E^{(i)}=M-\\tilde{M}^{(i)}$ for $r_{i}=\\operatorname{rank}(\\tilde{M}^{(i)})+\\rho $ .",
"Let $\\Delta ^{(i)}$ denote its output matrix of rank at most $r_{i}$ .",
"Compute a new approximation $\\tilde{M}^{(i+1)}=\\tilde{M}^{(i)}+ \\Delta ^{(i)}$ of $M$ and the matrix $E^{(i+1)}=M-\\tilde{M}^{(i+1)}$ of numerical rank at most $2^i \\rho $ .",
"Replace $i$ by $i + 1$ and reapply stages 1 and 2 until either $i$ exceeds the allowed tolerance, and then stop and output FAILURE, or until Stopping Criterion is satisfied for some integer $i$ , and then stop and output the matrix $\\tilde{M}=\\tilde{M}^{(i+1)}$ .",
"Remark 4.1 Progress in refinement.",
"Write $e_{i}=|||E^{(i)}|||$ for all $i$ and observe that $E^{(i)}=E^{(i-1)}-\\Delta ^{(i-1)}$ , and so $e_{i}<e_{i-1}$ if $\\Delta ^{(i-1)}$ approximates $E^{(i-1)}$ closer than the matrix $O$ filled with 0s.",
"Furthermore equation $r_{i}=\\operatorname{rank}(\\tilde{M}^{(i)})+\\rho $ at stage 1 implies that $\\tau _{r_i}(M-\\tilde{M}^{(i)})\\le \\tau _{\\rho }(M)$ .",
"Remark 4.2 Management of the precision of computing.",
"As this is customary in iterative refinement we apply mixed precision technique, that is, perform the subtraction stage 2 with a higher precision than stage 1.",
"Remark 4.3 Management of the rank growth.",
"The bound on the rank of the matrix $M^{(i)}$ is doubled at every iteration of Algorithm REF ; by allowing its increase we obtain more accurate LRA, but increase the complexity of an iteration.",
"In order to bound the complexity we can periodically compress the computed LRA $\\tilde{M}^{(i)}$ into its $\\rho $ -truncation $\\tilde{M}^{(i)}_{\\rho }$ or into a rank-$\\rho $ matrix approximating $\\tilde{M}^{(i)}_{\\rho }$ by applying Algorithm REF or REF , respectively (cf.",
"a similar re-compression technique in another context in [20]).",
"The errors of LRA grow in compression within bounds (REF ) and (REF ) for $AB=\\tilde{M}^{(i)}$ and $\\rho =r_i$ , that is, less significantly if an initial LRA $\\tilde{M}^{(0)}$ is close to an input matrix $M$ .",
"Remark 4.4 One can try to extend our algorithm to determination of numerical rank by means of observing at which iteration of refinement and for re-compression to which rank $\\rho $ we arrive at closer approximation to an input matrix."
],
[
"Superfast heuristic a posteriori error estimation for LRA",
"The customary randomized a posteriori error estimation [12] is practically reliable but not superfast; moreover superfast accurate a posteriori error estimation is impossible for the small input families of Example REF , as one can readily prove by extending the argument in that example.",
"Next we list some superfast heuristic recipes for this task.",
"(i) Clearly, the value $|e_{i,j}|$ for every entry $e_{i,j}$ of the error matrix $E=(e_{i,j})_{i,j}=\\tilde{M}-M$ of an LRA of (REF ) is a lower bound on the norm $|E|$ of this matrix.",
"(ii) The above deterministic lower bound on the LRA error norm $|||E|||$ also implies its a posteriori randomized upper bound if the error matrix $E$ is filled with i.i.d.",
"values of a single random variable and has sufficiently many entries, e.g., 100 entries or more (see Appendix ).",
"(iii) By generalizing the technique of part (i) we obtain deterministic lower bounds on the error norm $|||E|||$ given by the ratios $\\frac{|||FE|||}{|||F|||}$ , $\\frac{|||EH|||}{|||H|||}$ or $\\frac{|||FEH|||}{|||F|||~|||H|||}$ for any pair of matrices $F\\in \\mathbb {C}^{k\\times m}$ and $H\\in \\mathbb {C}^{n\\times l}$ .",
"The computation also defines randomized upper bounds on the error norm $|||E|||$ for random matrices $F$ and/or $H$ and sufficiently large integers $k$ and/or $l$ (see [8]) and runs superfast if $k=o(m)$ or $l=o(n)$ or if $F$ and $H$ are sufficiently sparse matrices, e.g., full rank submatrices of permutation matrices.",
"(iv) We can use the following heuristic recipes in testing convergence of an iterative refinement algorithm.",
"Suppose that the algorithm output a sequence of LRAs $\\tilde{M}^{(0)},\\tilde{M}^{(1)},\\tilde{M}^{(2)},\\dots $ for a matrix $M$ and suppose that we have estimated the norms of the residuals $R^{(j)}=|||\\tilde{M}^{(j)}-\\tilde{M}^{(j-1)}|||$ for $j=0,1,2,\\dots $ .",
"Then we can stop the algorithm where a fixed tolerance TOL exceeds such a residual norm.",
"Various recipes can be suggested for superfast estimation of the values $R^{(j)}$ (e.g., based on recursively updating SVD, cf.",
"[19]), but one can skip all or part of that estimation based on computing the values of $|{\\bf f}^T(\\tilde{M}^{(j)}-\\tilde{M}^{(j-1)}){\\bf h}|$ for a fixed pair of vectors ${\\bf f}$ and ${\\bf h}$ or for the set of a small number of such pairs.",
"(v) The above recipes for superfast a posteriori error estimation can be applied to any LRA or any sequence of LRAs.",
"In [27] and [26] the authors deduce such estimates for LRA output by some specified algorithms.",
"In the next section we specify superfast a posteriori error estimation in the case of iterative refinement with Subalgorithm from [26]."
],
[
"Numerical Experiments",
"In this subsection we present the test results for Algorithm REF on inputs of five types made up of synthetic and real-world data with various spectra.",
"Our table displays the relative error $r = \\frac{||M - \\tilde{M}^{(i)}||_2}{||M - M_\\rho ||_2}$ where $M$ denotes an input matrix, $\\tilde{M}^{(i)}$ denotes the computed approximation output by Algorithm REF in the $i$ -th iteration, $M_\\rho $ denotes the $\\rho $ truncation of $M$ , and $\\rho $ is a target rank for each input matrix.",
"Unless the output of the algorithm is corrupted by rounding errors, the ratio $r$ is not supposed to be noticeably exceeded by 1, and it was not in our experiments.",
"The algorithm was implemented in Python, and we run all experiments on a 64bit MacOS 10.15.7 machine with 2.6GHz CPU and 16GB Memory.",
"We called scipy.linalg version 1.5.4 for numerical linear algebra routines such as QR factorization with pivoting, Moore-Penrose matrix inversion, and SVD.",
"Synthetic Input: We used random synthetic $1024\\times 1024$ input matrices of two kinds – with fast and slowly decaying spectra.",
"In both cases we generated these matrices as the products $U\\Sigma V^T$ , where $U$ and $V$ were the matrices of the left and right singular vectors of a random Gaussian matrix, respectively.",
"By letting $\\Sigma = \\textrm {diag}(v)$ , where $v_i = 1$ for $i = 1, 2, 3,\\dots 20$ , $v_i = \\frac{1}{2}^{i-20}$ for $i = 21, \\dots , 100$ , and $v_i = 0$ for $i > 100$ , we generated input matrices with fast decaying spectra and target rank $\\rho = 20$ .",
"By letting $\\Sigma = \\textrm {diag}(u)$ , where $u_i = 1$ for $i = 1, 2, 3,\\dots 20$ , and $u_i = \\frac{1}{(1 + i - 20)^2}$ for $i > 20$ , we generated input matrices with slowly decaying spectra and target rank $\\rho = 20$ .",
"Real-world Input: We used three different input matrices coming from different applications.",
"The first matrix, which we call Shaw, has target rank $\\rho = 20$ and comes from a one-dimensional image restoration model problem.",
"The second matrix, which we call Gravity, has target rank $\\rho = 45$ and comes from a one-dimensional gravity surveying model problem.",
"Both of these input matrices were $1000\\times 1000$ dense real matrices having low numerical rank, and we padded them with 0s in order to increase their size to $1024\\times 1024$ .",
"They represent discretization of Integral Equations, provided in the built-in problems of the Regularization ToolsSee http://www.math.sjsu.edu/singular/matrices and http://www2.imm.dtu.dk/$\\sim $ pch/Regutools For more details see Ch4 of http://www.imm.dtu.dk/$\\sim $ pcha/Regutools/RTv4manual.pdf .",
"The third input matrix is generated from the discretization of a single layer potential operator.",
"We set its size to $1024\\times 1024$ and the target rank $\\rho $ to 11.",
"For more details about the generation of these matrices, we refer readers to [12].",
"We display the distribution of their top 50 singular values in Figure REF .",
"Figure: Spectra of real world input matricesSubalgorithm: Our subalgorithm Algorithm REF modifies the sketching algorithm of [32] by replacing the Gaussian random left and right multipliers with sparse and orthogonal abridged Hadamard multipliers of [24] and [25] having recursion depth 3 and size $2r\\times 1024$ and $1024\\times r$ , respectively.",
"For comparison with the main results of our tests performed with sparse orthogonal multipliers, we also experimented with Gaussian random multipliers $F\\in \\mathbb {R}^{2r\\times 1024}$ and $H\\in \\mathbb {R}^{1024\\times r}$ , by following [32], where $F$ and $H$ are denoted ${\\bf \\Omega }$ and ${\\bf \\Psi }$ , respectively.",
"We applied the following modification of the algorithm of [32].",
"Algorithm 6.1 Input: Three integers $m$ , $n$ , and $r$ satisfying $2r \\le \\min (m ,n)$ ; an $m\\times n$ matrix $E$ given explicitly or implicitly.",
"Initialization: Generate abridged Hadamard multipliers (see [24] and [25]) $F$ and $H$ of sizes $2r\\times m$ and $n\\times r$ independently; Compute the sketches $FE$ and $EH$ at sublinear cost.",
"Computations: Compute matrix $Q$ as the Q factor of the thin QR factorizationRecall the uniqueness of the thin QR factorization of an $m\\times k$ matrix where an $k\\times k$ upper triangular factor $R$ has positive diagonal entries [10].",
"of $EH$ .",
"Compute matrices $U$ and $T$ of the thin QR factorization $UT$ of $FQ$ .",
"Compute $\\Delta = QT^+U^T(FE)$ .",
"Output: $\\Delta $ as the LRA of $E$ .",
"Namely, we recall that the $(i-1)$ -th step of Algorithm REF outputs the matrix $\\tilde{M}^{(i-1)}$ and at the $i$ -th iterative refinement step of Algorithm REF apply Algorithm REF to the error matrix $E^{(i-1)} = M - \\tilde{M}^{(i-1)}$ .",
"Notice that the algorithm does not explicitly compute the matrices $E^{(i-1)}$ but only involves the sketches $FE^{(i-1)} = FM - F\\tilde{M}^{(i-1)}$ and $E^{(i-1)}H = MH - \\tilde{M}^{(i-1)}H$ .",
"At the end of the $i$ -th iterative refinement step, we compute LRA $\\tilde{M}^{(i)} = \\tilde{M}^{(i-1)} + \\Delta ^{(i-1)}$ or $\\tilde{M}^{(i)} = (\\tilde{M}^{(i-1)} + \\Delta ^{(i-1)})_\\rho $ where $\\Delta ^{(i-1)}$ is output by Algorithm REF .",
"We managed the rank growth of the approximation $\\tilde{M}^{(i)}$ by means of incorporating Remark REF .",
"Namely, in the first iteration, we let $r = \\rho $ and $\\tilde{M}^{(1)} = {\\bf 0} + \\Delta ^{(0)}$ ; in the subsequent iterations, we let $r = 2\\rho $ and let $\\tilde{M}^{(i+1)}$ be the rank-$\\rho $ truncation of the approximation $\\tilde{M}^{(i)} + \\Delta ^{(i)}$ .",
"For each input matrix with each type of multipliers, we run Algorithm REF 100 times with independent random choices of multipliers $F$ and $H$ ; we terminated each run in three refinement iterations, although the approximations have become quite accurate in just two iterations in our experiments.",
"We recorded the mean relative error ratio of the approximation before and after the truncation for every iteration step in Table REF .",
"With our choice of subalgorithm and multipliers, the approximation rank before truncation could potentially be twice the target rank $\\rho $ in the 2nd and 3rd iterations, and for this reason the relative error before truncation were much less than 1 in our tests.",
"Table: Test resultsfor Algorithm on input matrices: synthetic matrix with fast decaying spectra (Fast Decay),synthetic matrix with slow decaying spectra (Slow Decay), Shaw, Gravity, and discretized single layer potential operator (SLP).The results of our tests with abridged Hadamard multipliers are similar to the results with Gaussian random multipliers and are only slightly worse in few places.",
"This minor deterioration of the output accuracy was a reasonable price for using abridged (very sparse) Hadamard multipliers, with which we only access a small fraction of the input matrix at each iteration step.",
"In the tests with the input matrices of the class Shaw, the output relative error was close to 1, but not quite as close as in case of the other inputs, possibly because of the clusters of the trailing singular values in the spectrum.",
"In our tests with some other inputs that were not “well-mixed\", it was necessary to increase the recursion depth of the abridged Hadamard multipliers in order to bring the relative error ratio close to 1.",
"Appendix"
],
[
"Small families of hard inputs for\nsuperfast computation of their LRA",
"Any superfast algorithm fails to compute a close LRA of the following small families of matrices.",
"Example A.1 Let $\\Delta _{i,j}$ denote an $m\\times n$ matrix of rank 1 filled with 0s except for its $(i,j)$ th entry filled with 1.",
"The $mn$ such matrices $\\lbrace \\Delta _{i,j}\\rbrace _{i,j=1}^{m,n}$ form a family of $\\delta $ -matrices.",
"We also include the $m\\times n$ null matrix $O_{m,n}$ filled with 0s into this family.",
"Now fix any superfast algorithm; it does not access the $(i,j)$ th entry of its input matrices for some pair of $i$ and $j$ .",
"Therefore it outputs the same approximation of the matrices $\\Delta _{i,j}$ and $O_{m,n}$ , with an undetected error at least 1/2.",
"Arrive at the same conclusion by applying the same argument to the set of $mn+1$ small-norm perturbations of the matrices of the above family and to the $mn+1$ sums of the latter matrices with any fixed $m\\times n$ matrix of low rank.",
"Finally, the same argument shows that a posteriori estimation of the output errors of an LRA algorithm applied to the same input families cannot run superfast."
],
[
"Superfast a posteriori error estimation for LRA of a matrix filled with i.i.d. values of a single variable",
"Clearly, the value $|e_{i,j}|$ for every entry $e_{i,j}$ of the error matrix $E=(e_{i,j})_{i,j}=\\tilde{M}-M$ of an LRA of (REF ) is a lower bound on the norm $|||E|||$ of this matrix.",
"This deterministic lower bound on the LRA error norm $|||E|||$ also implies its a posteriori randomized upper bound if the error matrix $E$ is filled with independent identically distributed (i.i.d.)",
"values of a single random variable and has sufficiently many entries, e.g., 100 entries or more.",
"In our randomized a posteriori error estimation below we assume that the error matrix $E$ of an LRA has enough entries, say, 100 or more, and that they are the observed i.i.d.",
"values of a single random variable.",
"These assumptions are realistic, for example, where the deviation of the matrix $W$ from its rank-$\\rho $ approximation is due to the errors of measurement or rounding.",
"In this case randomized a posteriori error estimation can be achieved at a very low cost because the Central Limit Theorem implies that the distribution of the variable is close to Gaussian (see [7]).",
"Fix a pair of integers $q$ and $s$ such that $qs$ is large enough (say, exceeds 100), but $qs=O((m+n)kl )$ and $qs\\ll mn$ ; then apply our tests just to a random $q\\times s$ submatrix of the $m\\times n$ error matrix.",
"Under this policy we compute the error matrix at a dominated arithmetic cost in $O((m+n)kl )$ but still verify correctness with high confidence, by applying the customary rules of hypothesis testing for the variance of a Gaussian variable.",
"Namely, suppose that we have observed the values $g_1,\\dots ,g_K$ of a Gaussian random variable $g$ with a mean value $\\mu $ and a variance $\\sigma ^2$ and that we have computed the observed average value and variance $\\mu _K=\\frac{1}{K}\\sum _{i=1}^K |g_i|~{\\rm and }~\\sigma _K^2=\\frac{1}{K}\\sum _{i=1}^K |g_i-\\mu _K|^2,$ respectively.",
"Then, for a fixed reasonably large $K$ , both ${\\rm Probability}~\\lbrace |\\mu _K-\\mu |\\ge t|\\mu |\\rbrace ~{\\rm and~Probability}\\lbrace |\\sigma _K^2-\\sigma ^2|\\ge t\\sigma ^2\\rbrace $ converge to 0 exponentially fast as $t$ grows to the infinity (see [4])."
],
[
"From $\\rho $ -top SVD to CUR LRA",
"In Section we have readily computed $\\rho $ -top SVD of an LRA of an $m\\times n$ matrix $M$ , and this computation is superfast where $\\rho ^2=o(m+n)$ .",
"Next we complement that algorithm with transformation of the $\\rho $ -top SVD of a matrix $M$ into its rank-$\\rho $ CUR decomposition; the algorithm is also superfast for small $\\rho $ .",
"$\\rho $ -top SVD of a matrix $M$ is SVD of $M_{\\rho }$ , and so every nonsingular $\\rho \\times \\rho $ submatrix $G$ of $M_{\\rho }$ generates its exact CUR decomposition, which is an optimal rank-$\\rho $ approximation of $M$ , but next we are going to stabilize this decomposition numerically by bounding the norm of $G^+$ .",
"Algorithm C.1 [Transition from $\\rho $ -top SVD to CUR LRA.]",
"Input: Five integers $k$ , $l$ , $m$ , $n$ , and $\\rho $ satisfying (REF ) and four matrices $M\\in \\mathbb {R}^{m\\times n}$ , $\\Sigma \\in \\mathbb {R}^{\\rho \\times \\rho }$ (diagonal), $U\\in \\mathbb {R}^{m\\times \\rho }$ , and $V\\in \\mathbb {R}^{n\\times \\rho }$ (both orthogonal) such that $M:=U\\Sigma V^*$ is SVD.",
"Output: Three matricesHere we denote nucleus by $N$ rather than $U$ in order to avoid conflict with the factor $U$ in SVD.",
"$C\\in \\mathbb {R}^{m\\times l}$ , $N\\in \\mathbb {R}^{l\\times k}$ , and $R\\in \\mathbb {R}^{k\\times n}$ such that $C$ and $R$ are submatrices made up of $l$ columns and $k$ rows of $M$ , respectively, and $M=CNR.$ Computations: By applying to the matrices $U$ and $V$ the algorithms of [9] or [20] compute the submatrices $U_{\\mathcal {I},:}\\in \\mathbb {R}^{k\\times \\rho }$ and $V^*_{:,\\mathcal {J}}\\in \\mathbb {R}^{\\rho \\times l}$ , respectively.",
"Output the CUR factors $C=U\\Sigma V^*_{:,\\mathcal {J}}$ and $R=U_{\\mathcal {I},:}\\Sigma V^*$ .",
"Define a CUR generator $G:=U_{\\mathcal {I},:}\\Sigma V^*_{:,\\mathcal {J}}$ and output a nucleus $N:=G^{+}=V_{:,\\mathcal {J}}^{*+}\\Sigma ^{-1} U_{\\mathcal {I},:}^{+}$ .",
"[Prove the latter equation by verifying the Moore – Penrose conditions for the matrix $G^+$ .]",
"Correctness verification.",
"Substitute the expressions for $C$ , $N$ and $R$ and obtain $CNR=(U\\Sigma V^*_{:,\\mathcal {J}})(V_{:,\\mathcal {J}}^{*+}\\Sigma ^{-1} U_{\\mathcal {I},:}^+)(U_{\\mathcal {I},:}\\Sigma V^*)$ .",
"Substitute the equations $V^*_{:,\\mathcal {J}}V^{*+}_{:,\\mathcal {J}}=U_{\\mathcal {I},:}^+U_{\\mathcal {I},:}=I_{\\rho }$ , which hold because $V^*_{:,\\mathcal {J}}\\in \\mathbb {R}^{\\rho \\times l}$ , $U_{\\mathcal {I},:}^+\\in \\mathbb {R}^{k\\times \\rho }$ , and $\\rho \\le \\min \\lbrace k,l\\rbrace $ by assumption, and obtain $CNR=U\\Sigma V^*=M^{\\prime }$ .",
"Cost bounds.",
"The algorithm uses $nk+ml+kl$ memory cells and $O(mk^2+nl^2)$ flops; these cost bounds are dominated at stage 2 and are in $O(mn)$ if $k^2=o(n)$ and if $l^2=o(m)$ .",
"Let us also estimate the spectral norm of the nucleus $||N||_2$ .",
"Notice that $\\operatorname{rank}(V_{:,\\mathcal {J}})=\\operatorname{rank}(\\Sigma )= \\operatorname{rank}(U_{\\mathcal {I},:})=\\rho $ , apply Lemma REF , and obtain $||N||_2\\le || V_{:,\\mathcal {J}}^{*+}||_2 ~||\\Sigma ^{-1}||_2~ ||U_{\\mathcal {I},:}^{+}||_2$ .",
"Recall that $||\\Sigma ^{-1}||_2=||M^+||_2=1/\\sigma _{\\rho }(M)$ .",
"Write $t_{q,s,h}^2:=(q-s)sh^2+1$ , allow any choice of $h>1$ , say, $h=1.1$ , and then recall that $||U_{\\mathcal {I},:}^{+}||_2\\le t_{m,k,h}^a$ , $||(V_{:,\\mathcal {J}}^{*})^{+}||_2\\le t_{n,l,h}^a$ , and consequently $||N||_2\\le t_{m,\\rho ,h}^at_{n,\\rho ,h}^a/\\sigma _{\\rho }(M) $ where $a=1$ if we apply the algorithms of [9] at stage 1 and $a=2$ if we apply those of [20].",
"Acknowledgements: Our work has been supported by NSF Grants CCF–1563942 and CCF–1733834 and PSC CUNY Awards 62797-00-50 and 63677-00-51."
]
] | 1906.04223 |
[
[
"Bayesian Estimation of Economic Simulation Models using Neural Networks"
],
[
"Abstract Recent advances in computing power and the potential to make more realistic assumptions due to increased flexibility have led to the increased prevalence of simulation models in economics.",
"While models of this class, and particularly agent-based models, are able to replicate a number of empirically-observed stylised facts not easily recovered by more traditional alternatives, such models remain notoriously difficult to estimate due to their lack of tractable likelihood functions.",
"While the estimation literature continues to grow, existing attempts have approached the problem primarily from a frequentist perspective, with the Bayesian estimation literature remaining comparatively less developed.",
"For this reason, we introduce a Bayesian estimation protocol that makes use of deep neural networks to construct an approximation to the likelihood, which we then benchmark against a prominent alternative from the existing literature.",
"Overall, we find that our proposed methodology consistently results in more accurate estimates in a variety of settings, including the estimation of financial heterogeneous agent models and the identification of changes in dynamics occurring in models incorporating structural breaks."
],
[
"Recent advances in computing power and the potential to make more realistic assumptions due to increased flexibility have led to the increased prevalence of simulation models in economics.",
"While models of this class, and particularly agent-based models, are able to replicate a number of empirically-observed stylised facts not easily recovered by more traditional alternatives, such models remain notoriously difficult to estimate due to their lack of tractable likelihood functions.",
"While the estimation literature continues to grow, existing attempts have approached the problem primarily from a frequentist perspective, with the Bayesian estimation literature remaining comparatively less developed.",
"For this reason, we introduce a Bayesian estimation protocol that makes use of deep neural networks to construct an approximation to the likelihood, which we then benchmark against a prominent alternative from the existing literature.",
"Overall, we find that our proposed methodology consistently results in more accurate estimates in a variety of settings, including the estimation of financial heterogeneous agent models and the identification of changes in dynamics occurring in models incorporating structural breaks.",
"Keywords: Agent-based modelling, Simulation modelling, Bayesian estimation, Machine learning, Neural networks JEL Classification: C13 $\\cdot $ C52"
],
[
"Introduction",
"Recent years have, to some extent, seen the emergence of a paradigm shift in how economic models are constructed.",
"Traditionally, a need to facilitate mathematical tractability and limited computational resources have led to a dependence on strong assumptionsThese include, but are not limited to, assumptions of perfect rationality and the existence of representative agents., many of which are inconsistent with the heterogeneity and non-linearity that characterise real economic systems [18], [15], [13].",
"Ultimately, the Great Recession of the late 2000s and the perceived failings of traditional approaches, particularly those built on general equilibrium theory, would lead to the birth of a growing community arguing that the adoption of new paradigms harnessing contemporary advances in computing power could lead to richer and more robust insights [15], [13].",
"Perhaps the most prominent examples of this new wave of computational approaches are agent-based models (ABMs), which attempt to model systems by directly simulating the actions of and interactions between their microconstituents [31].",
"In theory, the flexibility offered by simulation should allow for more empirically-motivated assumptions and this, in turn, should result in a more principled approach to the modelling of the economy [8], [29].",
"The extent to which this has been achieved in practice, however, remains open for debate [23].",
"While ABMs initially found success by demonstrating an ability to replicate a wide array of stylised facts not recovered by more traditional approaches [29], [4], their simulation-based nature makes their estimation nontrivial [14].",
"Therefore, while the last decade has seen the emergence of increasingly larger and more realistic macroeconomic models, such as the Eurace [10] and Schumpeter Meeting Keynes [11] models, their acceptance in mainstream policy-making circles remains limited due to these and other challenges.",
"The aforementioned estimation difficulties largely stem from the simulation-based nature of ABMs, which, in all but a few exceptional casesSee, for example, the work of [1], [2] and [3]., renders it impossible to obtain a tractable expression for the likelihood function.",
"As a result, most existing approaches have attempted to circumvent these difficulties by directly comparing model-simulated and empirically-measured data using measures of dissimilarity (or similarity) and searching the parameter space for appropriate values that minimise (or maximise) these metrics [20], [30].",
"The most pervasive of these approaches, which [19] call simulated minimum distance (SMD) methods, is the method of simulated moments (MSM), which constructs an objective function by considering weighted sums of the squared errors between simulated and empirically-measured moments (or summary statistics).",
"Though MSM has been widely applied in a number of different contextsSee [16], [17], [12], [19], [9] and [36] for examples.",
"and has desirable mathematical propertiesThe estimator is both consistent and asymptotically normal [32]., it suffers from a critical weakness.",
"In more detail, the choice of moments or summary statistics is entirely arbitrary and the quality of the associated parameter estimates depends critically on selecting a sufficiently comprehensive set of moments, which has proven to be nontrivial in practice.",
"In response, recent years have seen the development of a new generation of SMD methods that largely eliminate the need to transform data into a set of summary statistics and instead harness its full informational content [20].",
"These new methodologies vary substantially in their sophistication and theoretical underpinnings.",
"Among the simplest of these approaches is attempting to match time series trajectories directly, as suggested by [38].",
"More sophisticated alternatives include information-theoretic approaches [5], [27], simulated maximum likelihood estimation [26], and comparing the causal mechanisms underlying real and simulated data through the use of SVAR regressions [22].",
"In addition to the development of similarity metrics, attempts have also been made to reduce the large computational burden imposed by SMD methods by replacing the costly model simulation process with computationally efficient surrogates [41], [28].",
"Interestingly, the aforementioned approaches are all frequentist in nature, with Bayesian estimation being significantly less prevalentThere is a rather substantial literature on what are called approximate bayesian computation methods that has gained a significant following in biology and ecology [43].",
"Unfortunately, the vast majority of these methods rely on converting data to a set of summary statistics and their appeal for estimating economic ABMs is therefore limited.. As it currently stands, only one study in the literature [20] has focused extensively on the use of Bayesian techniques and recent work by [30] involving sequential Monte Carlo methods includes attempts at Bayesian estimation, though the work as a whole focuses more on a frequentist approach.",
"While the estimation literature has certainly been growing, it still suffers from a number of key weaknesses.",
"Perhaps the most significant of these is a lack of a standard benchmark against which to compare the performance of new methods.",
"For this reason, most new approaches have traditionally only been tested in isolation and comparative exercises have been relatively rare.",
"For this reason, we compared a number of prominent estimation techniques in a previous investigation [35] and found, rather surprisingly, that the Bayesian estimation procedure proposed by [20] consistently outperformed a number of prominent frequentist alternatives in a series of head-to-head tests, despite its relative simplicity.",
"We therefore argued that more interest in Bayesian methods is warranted and suggested that increased emphasis should be placed on their development.",
"In line with this recommendation, we introduce a method for the Bayesian estimation of economic simulation modelsIt is worth noting that while we focus on ABMs, the proposed methodology is applicable to any model capable of simulating time series or panel data.",
"For this reason, the methodology would be equally applicable to competing modelling approaches.",
"that relaxes a number of the assumptions made by the approach of [20] through the use of a neural network-based likelihood approximation.",
"We then benchmark our proposed methodology through a series of computational experiments and finally conclude with discussions related to practical considerations, such as the setting of the method's hyperparameters and the associated computational costs."
],
[
"Estimation and Experimental Procedures",
"In this section, we introduce the reader to a number of the essential elements of our investigation, including a brief discussion of the fundamentals of Bayesian estimation, a description of the approach of [20] (our chosen benchmark), and an introduction to our proposed estimation methodology."
],
[
"Bayesian Estimation of Simulation Models",
"For our purposes, we consider a simulation model to be any mathematical or algorithmic representation of a real world system capable of producing time series (panel) data of the form $\\mathbf {X} ^{sim} (\\mathbf {\\theta }, T, i) = \\left[\\mathbf {x} ^{sim} _{1, i}(\\mathbf {\\theta }), \\mathbf {x} ^{sim} _{2, i}(\\mathbf {\\theta }), \\dots , \\mathbf {x} ^{sim} _{T, i}(\\mathbf {\\theta }) \\right],$ where $\\mathbf {\\theta }$ is a model parameter set in the space of feasible parameter values, $T$ is the length of the simulation, $i$ represents the seed used to initialise the model's random number generators, and $\\mathbf {x} ^{sim} _{t, i}(\\mathbf {\\theta }) \\in \\mathbb {R}^{n}$ for all $t = 1, 2, \\dots , T$ .",
"In general, estimation or calibration procedures aim to determine appropriate values for $\\mathbf {\\theta }$ such that $\\mathbf {X} ^{sim} (\\mathbf {\\theta }, T, i)$ produces dynamics that are as close as possible to those observed in an empirically-measured equivalent, $\\mathbf {X} = \\left[\\mathbf {x}_{1}, \\mathbf {x}_{2}, \\dots , \\mathbf {x}_{T} \\right],$ where $\\mathbf {x}_{t} \\in \\mathbb {R}^{n}$ for all $t = 1, 2, \\dots , T$ .",
"Bayesian estimation attempts to achieve the above by first assuming that the parameter values follow a given distribution, $p(\\mathbf {\\theta })$ , which is chosen to reflect one's prior knowledge or beliefs regarding the parameter values.",
"This is then updated in light of empirically-measured data, yielding a modified distribution, $p(\\mathbf {\\theta } | \\mathbf {X})$ , called the posterior.",
"Bayesian estimation can therefore be framed in terms of Bayes' theorem as follows: $p(\\mathbf {\\theta } | \\mathbf {X}) = \\frac{p(\\mathbf {X} | \\mathbf {\\theta })p(\\mathbf {\\theta })}{p(\\mathbf {X})}.$ Unfortunately, obtaining an analytical expression for the posterior is typically not feasible.",
"Firstly, the normalisation constant, $p(\\mathbf {X})$ , is unknown or determining it is nontrivial.",
"Secondly, the likelihood, $p(\\mathbf {X} | \\mathbf {\\theta })$ , is intractable for most simulation models, particularly large-scale macroeconomic ABMs.",
"Nevertheless, these limitations can be overcome to some extent.",
"[20] provide a method for approximating $p(\\mathbf {X} | \\mathbf {\\theta })$ for a particular value of $\\mathbf {\\theta }$ , which then allows us to evaluate the right-hand side of $p(\\mathbf {\\theta } | \\mathbf {X}) \\propto p(\\mathbf {X} | \\mathbf {\\theta })p(\\mathbf {\\theta }).$ The above may then be used along with Markov chain Monte Carlo (MCMC) methods, such as the Metropolis-Hastings algorithm, to sample the posterior.",
"This is possible since most MCMC techniques only require that we are able to determine the value of a function proportional to the density function of interest rather than the density function itself.",
"It should be apparent, however, that the overall estimation error will depend critically on the method used to approximate the likelihood."
],
[
"The Approach of {{cite:bace85b05bf71af2d92a25db7d496c6fde75ffdb}}",
"As previously stated, [20] provide a method to approximate the likelihood for simulation models, which we now discuss in more detail.",
"In essence, the approach is based on the assumption that, for all $t \\ge \\tilde{T}$ , we reach a statistical equilibrium such that $\\mathbf {x}_{t, i} ^{sim}(\\mathbf {\\theta })$ fluctuates around a stationary level, $\\mathbb {E}[\\mathbf {x}_{t, i} ^{sim}(\\mathbf {\\theta }) | t \\ge \\tilde{T}]$ , which allows us to further assume that $\\mathbf {x}_{\\tilde{T}, i} ^{sim}(\\mathbf {\\theta }), \\mathbf {x}_{\\tilde{T} + 1} ^{sim}(\\mathbf {\\theta }), \\dots , \\mathbf {x}_{T, i} ^{sim}(\\mathbf {\\theta })$ constitutes a random sample from a given distributionThe samples need not all be drawn from a single Monte Carlo replication and may instead be drawn from the statistical equilibria reached by each replication in an ensemble generated using various random seeds.",
"In practice, we simulate an ensemble of $R$ such Monte Carlo replications for each candidate set of $\\mathbf {\\theta }$ values and combine the samples from each replication into a single random sample..",
"It is then possible to determine a density function that describes this distribution, which we denote by $\\tilde{f}(\\mathbf {x} | \\mathbf {\\theta })$ , using kernel density estimation (KDE), finally allowing us to approximate the likelihood of the empirically-sampled dataNote that we have assumed, as in the case of the simulated data, that the empirically-sampled data fluctuates around a stationary level.",
"for a given value of $\\mathbf {\\theta }$ as follows: $p(\\mathbf {X} | \\mathbf {\\theta }) = \\prod _{t = 1} ^{T} \\tilde{f}(\\mathbf {x}_t | \\mathbf {\\theta }).$ It should be apparent that the above results in a simple strategy that is easy to apply in most contexts.",
"It must be noted, however, that this is largely made possible through strong assumptions that seldom hold in practice.",
"In more detail, notice that ordered time series (panel) data is essentially being treated as an i.i.d.",
"random sample, implying that $\\mathbf {x}_{t, i} ^{sim}(\\mathbf {\\theta }) \\perp \\mathbf {x}_{1, i} ^{sim}(\\mathbf {\\theta }), \\dots , \\mathbf {x}_{t - 1, i} ^{sim}(\\mathbf {\\theta })$ for all $t = 2, 3, \\dots , T$ .",
"Unfortunately, such independence assumptions do hold for most simulation models, since $\\mathbf {x}_{t, i} ^{sim}(\\mathbf {\\theta })$ is likely be dependent on at least some of the previously realised values, whether this dependence is explicit or mediated through latent variables.",
"Additionally, such assumptions result in a likelihood function that makes no distinction between $\\mathbf {\\theta }$ values that result in identical unconditional distributions but differing temporal trends.",
"Since many economic simulation models and particularly large-scale macroeconomic ABMs produce datasets that are characterised by seasonality or structural breaks, there is likely to be some impact on the quality of the resultant parameter estimates.",
"Nevertheless, [35] demonstrates that despite the above shortcomings, the method of [20] is able to provide reasonable parameter estimates in many contexts, while also outperforming several more sophisticated frequentist approaches.",
"This warrants further investigation and naturally leads one to ask whether relaxing the required independence assumptions would allow for the construction of a superior Bayesian estimation method."
],
[
"Likelihood Approximation using Neural Networks",
"We now begin our discussion of a relatively simple extension to the likelihood approximation procedure proposed by [20] that is capable of capturing some of the dependence of $\\mathbf {x}_{t, i} ^{sim}(\\mathbf {\\theta })$ on past realised values.",
"As a starting point, we assume that $p \\left(\\mathbf {x}_{t, i} ^{sim} \\big | \\mathbf {x}_{1, i} ^{sim}, \\dots , \\mathbf {x}_{t - 1, i} ^{sim}: \\mathbf {\\theta }\\right) = p \\left(\\mathbf {x}_{t, i} ^{sim} \\big | \\mathbf {x}_{t - L, i} ^{sim}, \\dots , \\mathbf {x}_{t - 1, i} ^{sim}: \\mathbf {\\theta }\\right)$ for all $L < t \\le T$ , implying that $\\mathbf {x}_{t, i} ^{sim}(\\mathbf {\\theta })$ depends only on the past $L$ realised values.",
"Our task, therefore, is the estimation of the above conditional densities, $\\tilde{f}\\left(\\mathbf {x}_{t - L, i} ^{sim}, \\dots , \\mathbf {x}_{t - 1, i} ^{sim}, \\mathbf {x}_{t, i} ^{sim}, \\mathbf {\\phi } \\right) \\simeq p \\left(\\mathbf {x}_{t, i} ^{sim} \\big | \\mathbf {x}_{t - L, i} ^{sim}, \\dots , \\mathbf {x}_{t - 1, i} ^{sim}: \\mathbf {\\theta }\\right),$ for all $L < t \\le T$ , where $\\mathbf {\\phi } = \\mathbf {\\phi }(\\mathbf {\\theta })$ are parameters associated with the density estimation procedure.",
"In our context, we make use of a mixture density network (MDN), a neural network-based approach to conditional density estimation introduced by [6].",
"The aforementioned scheme consists of two primary componentsNote that these discussions are primarily illustrative and serve to briefly describe and motivate our approach.",
"A detailed technical description of its implementation is provided in Appendix ., a mixture of $K$ Gaussian random variables, $ \\tilde{f}\\left(\\mathbf {x}, \\mathbf {y}, \\mathbf {\\phi } \\right) = \\sum _{k = 1} ^{K} \\alpha _k \\left(\\mathbf {x}\\right) \\mathcal {N}\\left(\\mathbf {y} \\big | \\mathbf {\\mu }_k\\left(\\mathbf {x}\\right), \\mathbf {\\Sigma }_k\\left(\\mathbf {x}\\right)\\right),$ where we denote $\\mathbf {x}_{t, i} ^{sim}$ by $\\mathbf {y}$ and $\\mathbf {x}_{t - L, i} ^{sim}, \\dots , \\mathbf {x}_{t - 1, i} ^{sim}$ by $\\mathbf {x}$ , and functions $\\alpha _k$ , $\\mathbf {\\mu }_k$ and $\\mathbf {\\Sigma }_k$ of $\\mathbf {x}$ which determine the mixture parameters.",
"Here, $\\alpha _k$ , $\\mathbf {\\mu }_k$ and $\\mathbf {\\Sigma }_k$ are the outputs of a feedforward neural network taking $\\mathbf {x}$ as input and having weights and biases $\\mathbf {\\phi }(\\mathbf {\\theta })$ , which are determined by training the network on an ensemble of $R$ Monte Carlo replications simulated by the candidate model for parameter set $\\mathbf {\\theta }$ .",
"Using the trained MDN, it is then possible to approximate the likelihood of the empirically-sampled data for a given value of $\\mathbf {\\theta }$ as follows: $p(\\mathbf {X} | \\mathbf {\\theta }) = \\prod _{t = 1} ^{T - L} \\tilde{f}(\\mathbf {x}_{t}, \\dots , \\mathbf {x}_{t + L - 1}, \\mathbf {x}_{t + L}, \\mathbf {\\phi }).$ While alternative density estimation procedures could potentially have been employed, our consideration of MDNs is motivated primarily by their desirable properties.",
"Specifically, MDNs are, in theory, capable of approximating fairly complex conditional distributions.",
"This follows directly from the fact that mixtures of normal random variables are universal density approximators for sufficiently large $K$ [42] and the fact that neural networks are universal function approximators [24], provided they are sufficiently expressive.",
"Therefore, provided that $K$ is sufficiently large and the constructed neural network sufficiently deep (and wide), the above methodology should result in accurate conditional density estimates."
],
[
"Method Comparison and Benchmarking",
"Given that we have now described our proposed estimation methodology, we proceed to discuss our strategy for benchmarking it against the approach of [20], where we follow a similar strategy to that employed in [35].",
"We begin by letting $\\mathbf {X}^{sim}(\\mathbf {\\theta }, T, i)$ be the output of a candidate model, $M$ .",
"Since empirically-observed data is nothing more than a single realisation of the true data-generating process, which may itself be viewed as a model with its own set of parameters, it follows that we may consider $\\mathbf {X} = \\mathbf {X}^{sim}(\\mathbf {\\theta }^{true}, T^{emp}, i^{*})$ as a proxy for real data to which $M$ may be calibrated.",
"In this case, we are essentially estimating a perfectly-specified model using data for which the true parameter values, $\\mathbf {\\theta }^{true}$ , are known.",
"It can be argued that a good estimation method would, in this idealised setting, be able to recover these true values to some extent and that methods which produce estimates closer to $\\mathbf {\\theta }^{true}$ would be considered superior.",
"This leads us to define the following loss function $LS(\\mathbf {\\theta }^{true}, \\hat{\\mathbf {\\theta }}) = || \\mathbf {\\theta }^{true} - \\hat{\\mathbf {\\theta }} ||_{2},$ where $\\hat{\\mathbf {\\theta }}$ is the parameter estimate (posterior mean) produced by a given Bayesian estimation method.",
"In practice, it is important that both $\\hat{\\mathbf {\\theta }}$ and $\\mathbf {\\theta }^{true}$ are normalised to take values in the interval $[0, 1]$ before the loss function value is calculated.",
"This is because even relatively small estimation errors associated with parameters that typically take on larger values will increase the loss function value substantially more than relatively large estimation errors associated with parameters that typically take on smaller values if no normalisation is performed.",
"Therefore, for each free parameter, $\\theta _j \\in [a, b]$ , we set $\\hat{\\theta }_{j} ^{[0, 1]} = \\frac{\\hat{\\theta }_{j} - a}{b - a},$ with an analogous transformation being applied to $\\theta ^{true} _{j}$ .",
"The above allows us to develop a series of benchmarking exercises in which we compare the loss function values associated with our proposed method and that of [20] for a number of different models, free parameter sets, and $\\mathbf {\\theta }^{true}$ valuesWhile the constructed loss function will act as our primary metric, we will also consider a number of other relevant criteria, such as the standard deviation of the obtained posteriors..",
"In all of these comparative exercises, we aim to ensure that the overall conditions of the experiments are consistent throughout, regardless of the method used to approximate the likelihood.",
"Therefore, in all cases, we set the length of the proxy for real data to be $T_{emp} = 1000$ , the number of Monte Carlo replications in the simulated ensembles to be $R = 100$ , the length of each series in the simulated ensembles to be $T_{sim} = 1000$ , and the priors for all free parameters to be uniform over the explored parameter ranges.",
"Additionally, we have also used the same lag length, $L = 3$ , for all estimation attempts involving our neural network-based method.",
"While seemingly arbitrary, this choice has very clear motivations that are discussed in detail in Section REF .",
"Finally, the MCMC algorithm used to sample the posterior and its associated hyperparameters remain unchanged in all experiments.",
"Rather than using a standard random walk Metropolis-Hastings algorithm, we have instead employed the adaptive scheme proposed by [21], which allows for more effective initialisation, faster convergence, and better handling of multimodal posteriorsA complete description of the procedure is presented in Appendix .."
],
[
"Candidate Models",
"With our estimation and benchmarking strategies now described, we introduce the candidate models that we attempt to estimate.",
"Their selection is primarily justified by their ubiquity; each has appeared in a number of calibration and estimation studiesFor example, the [7] model is considered by [38], [28], and [26] and the [17] model is considered by [17] and [30]., leading them to become standard test cases in the field.",
"While computationally-inexpensive to simulate, most are capable of producing nuanced dynamics and thus still prove to be a reasonable challenge for most contemporary estimation approaches.",
"Since our focus here is the benchmarking of the proposed estimation procedure as opposed to estimating the candidate models using empirical data, our discussion will be relatively brief.",
"In empirical investigations, however, it would be necessary to provide some justification that the chosen models were reasonable representations of the considered systems."
],
[
"{{cite:454d1c21374a733ebef8f862509ed8ef5344ad6b}} Model",
"The first model we introduce, and by far the most popular in the existing literature, is the [7] model, an early example of a class of simulation models that attempt to model the trading of assets on an artificial stock market by simulating the interactions of heterogenous traders that follow various trading strategies.",
"We focus on a particular version of the model that can be expressed as a system of coupled equationsThe interested reader should refer to [7] for a detailed discussion of the model's underlying assumptions and the derivation of the above system of equations., $y_{t + 1} &= \\frac{1}{1 + r} \\sum _{h = 1} ^{H} n_{h, t + 1} (g_h y_t + b_h) + \\epsilon _{t + 1} \\text{, } \\epsilon _t \\sim \\mathcal {N}(0, \\sigma ^2), \\\\n_{h, t + 1} &= \\frac{\\exp (\\beta U_{h, t})}{\\sum _{h = 1} ^{H}\\exp (\\beta U_{h, t})}, \\\\U_{h, t} &= (y_t - R y_{t - 1})(g_h y_{t - 2} + b_h - R y_{t - 1}),$ where $y_{t}$ is the asset price at time $t$ (in deviations from the fundamental value $p_{t} ^{*}$ ), $n_{h, t}$ is the fraction of trader agents following strategy $h \\in \\left\\lbrace 1, 2, \\dots , H \\right\\rbrace $ at time $t$ , and $R = 1 + r$ .",
"Each strategy, $h$ , has an associated trend following component, $g_h$ , and bias, $b_h$ , both of which are real-valued parameters.",
"The model also includes positive-valued parameters that affect all trader agents, regardless of the strategy they are currently employing, specifically $\\beta $ , which controls the rate at which agents switch between various strategies, and the prevailing market interest rate, $r$ .",
"Finally, assuming an i.i.d.",
"dividend process, the fundamental value $p_{t} ^{*} = p^{*}$ is constant, allowing us to obtain the asset price at time $t$ , $p_t = y_t + p^{*}.$"
],
[
"Random Walks with Structural Breaks",
"The second model we consider is a random walk capable of replicating simple structural breaks, defined according to $x_{t + 1}= x_{t} + d_{t + 1} + \\epsilon _{t + 1} \\text{, } \\epsilon _t \\sim \\mathcal {N}(0, \\sigma _t ^2),$ where $d_t, \\sigma _t ={\\left\\lbrace \\begin{array}{ll}d_1, \\sigma _1 & t \\le \\tau \\\\d_2, \\sigma _2 & t > \\tau .\\end{array}\\right.",
"}$ Unlike the [7] model, the above is not a representation of a real-world system, but rather an artificially-constructed test example designed to challenge estimation methodologiesThis particular instantiation of the model was first used by [27] to test an information-theoretic criterion called the GSL-div.. Its inclusion is justified on the grounds that, as previously discussed, many large-scale ABMs produce dynamics that are characterised by structural breaks and the fact that it allows us to compare our approach against that of [20] in cases where the considered data demonstrates clear temporal changes in the prevailing dynamics."
],
[
"{{cite:ef647b0b37346da35f4c6f5d98084bef06343ca4}} Model",
"The final model we discuss shares a number of conceptual similarities with the previously described [7] model, being a heterogeneous agent model that simulates the interactions of traders following a number of trading strategies.",
"It is, however, different in a number of key areas, particularly in how the probability of an agent switching from one strategy to another is determined and in its incorporation of only two trader types, chartists and fundamentalists.",
"As in the case of the [7] model, the core elements of the model can be expressed as a system of coupled equations $ p_t &= p_{t - 1} + \\mu \\left(n_{t - 1} ^{f} d_{t - 1} ^{f} + n_{t - 1} ^{c} d_{t - 1} ^{c} \\right), \\\\d_{t} ^{f} &= \\phi (p^{*} - p_t) + \\epsilon _{t} ^{f} \\text{, } \\epsilon _{t} ^{f} \\sim \\mathcal {N}(0, \\sigma _{f} ^{2}),\\\\d_{t} ^{c} &= \\chi (p_{t} - p_{t - 1}) + \\epsilon _{t} ^{c} \\text{, } \\epsilon _{t} ^{c} \\sim \\mathcal {N}(0, \\sigma _{c} ^{2}),\\\\n_{t} ^{f} &= \\frac{1}{1 + \\exp (-\\beta a_{t - 1})},\\\\n_{t} ^{c} &= 1 - n_{t} ^{f}, $ where $p_t$ is the log asset price at time $t$ , $p^{*}$ is the log of the (constant) fundamental value, $n_{t} ^{f}$ and $n_{t} ^{c}$ are the market fractions of fundamentalists and chartists respectively at time $t$ , $d_{t} ^{f}$ and $d_{t} ^{c}$ are the corresponding average demands, and the remaining symbols all correspond to positive-valued parameters.",
"At this point, it is worth pointing out that [17] do not introduce a single model, but rather a family of related formulations built on the same foundation (Eqns.",
"REF -).",
"These models differ in how they define $a_{t}$ , the attractiveness of fundamentalism relative to chartism at the end of period $t$ , and incorporate a number of different mechanisms, including wealth, herding and price misalignment.",
"This makes the consideration of multiple versions of the model worthwhile and we thus consider two of the proposed versions$\\alpha _n$ , $\\alpha _w$ , and $\\alpha _p$ are strictly positive while $\\alpha _0$ may take on any real value.",
": $a_t = \\alpha _n (n_{t} ^ {f} - n_{t} ^{c}) + \\alpha _0 + \\alpha _p (p_t - p^{*}) ^{2},$ referred to as herding, predisposition and misalignment (HPM), and $g_{t} ^{s} &= \\left[\\exp (p_t) - \\exp (p_{t - 1})\\right] d_{t - 2} ^{s} \\text{, } s = \\lbrace f, c\\rbrace , \\\\w_{t} ^{s} &= \\eta w_{t - 1} ^{s} + (1 - \\eta ) g_{t} ^{s}, \\\\a_t &= \\alpha _w (w_{t} ^ {f} - w_{t} ^{c}) + \\alpha _0, $ referred to as wealth and predisposition (WP).",
"As a final remark, we consider $r_t = p_t - p_{t - 1}$ , the log return process, rather than $p_t$ in our estimation attempts."
],
[
"{{cite:454d1c21374a733ebef8f862509ed8ef5344ad6b}} Model",
"We now proceed with the presentation of the results of our comparative experiments, beginning with the [7] modelFrom this point onwards, we use KDE to refer to the method of [20] and MDN to refer to our proposed method in all tables and figures..",
"In these experiments, we consider a market with $H = 4$ trading strategies and focus on estimating $g_2$ , $b_2$ , $g_3$ , and $b_3$ , the trend following and bias components for two of these strategies.",
"For the first free parameter set, we consider $g_2, b_2 \\in [-1, 0]$ and $g_3, b_3 \\in [0, 1]$ , corresponding to a contrarian strategy with a negative bias and a trend following strategy with a positive bias respectively.",
"For the second free parameter set, we instead consider $g_2, b_2, g_3 \\in [0, 1]$ and $b_3 \\in [-1, 0]$ , corresponding to trend following strategies with positive and negatives biases respectively.",
"Referring to Figure REF , we observe that, for the first free parameter set, there is a pronounced difference in performance between our proposed methodology and that of [20].",
"While both approaches perform similarly when estimating the bias components, our proposed procedure results in marginal posteriors for $g_2$ and $g_3$ that not only have means noticeably closer to the true parameter values, but are also significantly narrower and more peaked, with their density concentrated in a smaller region of the parameter space.",
"This can be seen as indicative of reduced estimation uncertainty.",
"Figure: Marginal posterior distributions for free parameter set 1 of the model.Table REF elaborates on these findings and reveals that similar behaviours also emerge in the case of the second free parameter set.",
"Specifically, we find that the posterior means ($\\mathbf {\\mu }_{posterior}$ ) for both methods result in more or less equivalent estimates for $b_2$ and $b_3$ , while the posterior mean for our proposed method appears to result in noticeably superior estimates for $g_2$ and $g_3$ in both cases, ultimately leading to lower loss function values.",
"We also observe that our approach results in reduced posterior standard deviations ($\\mathbf {\\sigma }_{posterior}$ ) consistently for all free parameters, in line with our observation of reduced estimation uncertainty in Figure REF .",
"Table: *In Appendix , where we describe the method used to sample the posteriors, we indicate that we run the procedure multiple times with different initial conditions and combine the obtained samples into a single, larger sample from which we estimate $\\mathbf {\\mu }_{posterior}$ and $\\mathbf {\\sigma }_{posterior}$ .",
"We can, however, estimate the posterior mean for each of these runs individually and determine the standard deviation of $\\mathbf {\\mu }_{posterior}$ across the instantiations of the algorithm, which we call $\\mathbf {\\sigma }_{sample}$ .",
"As shown in Table REF , this standard deviation is generally very small for both methods, suggesting that the posterior mean estimates are generally robustThis is true for all free parameter sets and models considered in this investigation.."
],
[
"Random Walks with Structural Breaks",
"Moving on from the [7] model, we now discuss the estimation of a random walk incorporating a structural break.",
"In these experiments, we consider a fixed structural break location, $\\tau = 700$This induces a degree of asymmetry in the data and results in a more challenging and realistic estimation problem than $\\tau = 500$ ., and determine the extent to which both methods are capable of estimating the pre- and post-break drift, $d_1, d_2 \\in [0, 1]$ , and volatility, $\\sigma _1, \\sigma _2 \\in [0, 10]$ , for differing underlying changes in the dynamics.",
"While the loss function described in Section REF will still be used as our primary metric, we note that since the considered free parameters directly define the dynamics that characterise the different regimes of the data, it would also be worthwhile to assess the extent to which the competing approaches are able to correctly identify the relationships between the parameters and hence the shift in the pre- and post-break dynamics ($\\Delta _d$ and $\\Delta _{\\sigma }$ ).",
"Table: *Before proceeding, however, there are a number of nuances that should be highlighted.",
"Being a random walk, the model clearly produces non-stationary time series and therefore violates a key assumption of the method of [20].",
"For this reason, it is necessary to consider the series of first differences, $x_t - x_{t - 1}$ , rather than $x_t$ itself.",
"While our approach does not make stationarity assumptions, we have none the less considered the series of first differences when applying both methods to make the comparison as fair as possible.",
"It should also be noted that we have assumed the location of the structural break to be unknown or difficult to determine a-priori (as is the case in most practical problems), meaning that we apply both estimation approaches to the full time series data to estimate both the pre- and post-break parameters simultaneously.",
"If, however, the location of the structural break was known, it would be possible to estimate the relevant parameters separately using appropriate subsets of the data, a less challenging undertaking that we do not consider here.",
"Now, referring to Table REF , we see that both our proposed estimation methodology and that of [20] perform similarly well when attempting to estimate the pre- and post-break volatility, with both producing reasonable estimates for the free parameters and both being able to identity the correct shift in the dynamics.",
"Referring to Tables REF and REF , however, we see that more pronounced differences emerge when attempting to estimate the pre- and post-break drift.",
"While this is clearly evident from the fact that the loss function values associated with our proposed methodology are noticeably lower in all cases, a more detailed analysis reveals further distinctions worth mentioning.",
"Table REF , which presents the results for cases involving an increasing drift, reveals that our proposed methodology has correctly identified an increasing trend in both cases and has also correctly identified that the increase in drift for parameter set 4 is three times that of parameter set 3.",
"In contrast to this, the method of [20] incorrectly suggests a decreasing trend in both cases.",
"Table REF , which presents the results for cases involving a decreasing drift, similarly shows that our proposed methodology delivers superior performance when attempting to identify the change in drift.",
"Table: *This change in the relative performances of each method when estimating the drift rather than the volatility is a direct consequence of the relationship between the deterministic and stochastic components of the model.",
"For the selected parameter ranges, the random fluctuations, $\\epsilon _t$ , dominate the evolution of the model, with the drift producing a more subtle effect, particularly after the structural break occurs.",
"For this reason, correctly estimating the pre- and post-break volatility is a far less challenging task than estimating the pre- and post-break drift.",
"Therefore, while both methods perform well when estimating parameters associated with dominant effects like volatility, our method's incorporation of dependence on previously observed values seems to be important when estimating parameters related to more nuanced and less dominant aspects of a model.",
"Table: *"
],
[
"{{cite:ef647b0b37346da35f4c6f5d98084bef06343ca4}} Model",
"As stated in Section REF , the final model we consider has a number of alternate configurations differing in how the attractiveness of fundamentalism relative to chartism, $a_t$ , is determined during each period.",
"For this reason, we consider two of these configurations, HPM and WP, and focus on estimating the parameters associated with the rules governing $a_t$ : $\\alpha _n \\in [0, 2]$ , $\\alpha _0 \\in [-1, 1]$ , $\\alpha _p \\in [0, 20]$ , $\\alpha _w \\in [0, 15000]$ , and $\\eta \\in [0, 1]$ , while also estimating the standard deviation of the noise term appearing in the chartist demand equation, $\\sigma _c \\in [0, 5]$We originally attempted to estimate $\\sigma _f$ as well, but found this to exhibit a degree of collinearity with $\\sigma _c$ ..",
"Referring to Table REF , we see that our proposed estimation methodology appears slightly more effective than that of [20] for the HPM parameter set, producing superior estimates for all but one of the considered free parameters and resulting in a lower loss function value.",
"Nevertheless, the estimates do not differ substantially when comparing the methods.",
"Despite this, we see, in what is a seemingly analogous trend to what was observed in the random walk experiments, that the differences in performance are more pronounced for the WP parameter set.",
"In particular, we see a substantial difference in the loss function values associated with each method, brought about by differences in the quality of estimates produced for $\\eta $ .",
"Table: *As illustrated in Figure REF , the method of [20] produces a wide posterior for $\\eta $ that is dispersed across the entirety of the explored parameter range, which results in a relatively poor estimate.",
"In contrast to this, we see that the proposed methodology fares better, producing a far narrower posterior and a significantly more accurate estimate.",
"While it is nontrivial to identify any definitive causes for the observed behaviours due to the nonlinear nature of heterogeneous agent models, it is worth pointing out that the inclusion of wealth dynamics in the WP version of the model introduces a dependence of $a_t$ on the previous return via Eqns.",
"REF -, which may in turn increase the strength of the relationship between the current and previously observed values in the log return time series.",
"As a final remark, notice that for the vast majority of the free parameters considered, the proposed methodology also results in lower posterior standard deviations, as was the case for the [7] model."
],
[
"Overall Summary",
"In the preceding subsections, we have focused primarily on analysing the results on a case-by-case basis.",
"Here, however, we provide a summative comparison across all of the considered models.",
"This is achieved though the consideration of a number of key performance metrics, presented in Table REF , which compare the approaches at both a global and individual parameter level.",
"Figure: Marginal posterior distributions for the WP parameter set of the model.The first of the aforementioned metrics, and the most important, $LS_{mdn} < LS_{kde}$ , indicates how often the proposed methodology results in lower loss function values, and hence measures its relative ability to recover the true parameter set.",
"We observe that in all cases considered, our methodology results in lower loss function values, which can be seen as indicative of dominance at the global level.",
"Table: Estimation Result Summary Across All ModelsThe second metric, $|\\mu _{mdn} ^{i} - \\theta _{true} ^{i}| < |\\mu _{kde} ^{i} - \\theta _{true} ^{i}|$ , determines how often our proposed methodology produces superior estimates for individual parameters in a free parameter set.",
"In some situations, one might find that the estimates obtained for a subset of the free parameters by the method of [20] are superior, even if the overall estimate for the entire free parameter set is not as good.",
"Nevertheless, we find that in over $80\\%$ of cases, our methodology also results in superior estimates at the level of individual parameters, a comfortable majority.",
"It should also be noted that in virtually all situations where $|\\mu _{mdn} ^{i} - \\theta _{true} ^{i}| > |\\mu _{kde} ^{i} - \\theta _{true} ^{i}|$ , such as some cases of $b_2$ and $b_3$ in the [7] model, and $\\sigma _1$ and $\\sigma _2$ in the random walk model, the differences in the estimates produced by both methods are incredibly small.",
"In contrast to this, a sizeable number of cases where $|\\mu _{mdn} ^{i} - \\theta _{true} ^{i}| < |\\mu _{kde} ^{i} - \\theta _{true} ^{i}|$ , such as $g_2$ and $g_3$ in the [7] model, and $\\eta $ in the [17] model, are characterised by comparatively large differences in the estimates obtained by the competing approaches.",
"This suggests that our proposed methodology also demonstrates a degree of dominance at the level of individual parameters.",
"The final metric, $\\sigma _{mdn} ^{i} < \\sigma _{kde} ^{i}$ , indicates how frequently our proposed methodology results in reduced posterior standard deviations for individual parameters, which occurs in slightly below $80\\%$ of the considered cases, again a comfortable majorityOn closer inspection, it appears that our methodology results in reduced posterior standard deviations more often for parameter sets consisting of more than 2 free parameters, which may hint at the possibility of the uncertainty of estimation increasing less rapidly for our approach than for the method of [20] as the number of free parameters is increased.",
"Ultimately, further investigation would be required to verify this hypothesis.. Based on the evidence presented by the above metrics as a whole, it would appear that our proposed methodology does indeed compare favourably to that of [20], which was itself already shown to dominate a number of other contemporary approaches in the literature by [35].",
"This ultimately validates our method as a worthwhile addition to the growing toolbox of estimation methods for economic simulation models."
],
[
"Choosing the Lag Length",
"As previously stated, we set $L = 3$ in all estimation experiments involving our proposed method.",
"Naturally, one may wonder whether this is an arbitrary choice or if there is a systematic way of choosing $L$ .",
"Similarly, one may also wonder if the obtained results are robust to this choice, even if only to some extent.",
"We now address both issues.",
"Figure: A demonstration of the sensitivity of the conditional density estimates to the choice of lag length for a typical example of the model.When applying the proposed methodology, we observed a phenomenon that appeared to be relatively consistent throughout the experiments.",
"In more detail, we observe that while increasing $L$ initially has a pronounced effect on the estimated conditional densities, there exists some $L^{*} \\ge 0$ such that for $L \\ge L^{*}$ , $p \\left(\\mathbf {x}_{t, i} ^{sim} \\big | \\mathbf {x}_{t - L, i} ^{sim}, \\dots , \\mathbf {x}_{t - 1, i} ^{sim}: \\mathbf {\\theta }\\right) \\simeq p \\left(\\mathbf {x}_{t, i} ^{sim} \\big | \\mathbf {x}_{t - L - 1, i} ^{sim}, \\dots , \\mathbf {x}_{t - 1, i} ^{sim}: \\mathbf {\\theta }\\right),$ or, in other words, the MDN essentially ignores the additional lags.",
"Figure: A demonstration of the sensitivity of the conditional density estimates to the choice of lag length for i.i.d.",
"random samples following a log-normal distribution, LN(0,0.25)LN(0, 0.25).Figure: A demonstration of the sensitivity of the conditional density estimates to the choice of lag length for an AR(2) model, x t+1 =0.45x t +0.45x t-1 +ϵ t x_{t + 1} = 0.45 x_t + 0.45 x_{t - 1} + \\epsilon _t, where ϵ t ∼𝒩(0,1)\\epsilon _t \\sim \\mathcal {N}(0, 1).We illustrate this graphically in Figure REF .",
"Here, we train an MDN on 100 realisations of length 1000 generated using the [7] model initialised using parameter set 1.",
"We then randomly draw an arbitrary sequence of 6 consecutive values from a time series of length 1000, also generated by the [7] model.",
"This then allows us to use the MDN to plot the conditional density functions for differing choices of $L$ , conditioned on the values generated in the previous step, and observe the aforementioned trend.",
"Repeating this exercise on models for which the true lag, $L_{true}$ , is known a-priori (see Figures REF and REF ), we see that $L^{*} = L_{true}$ .",
"This has a number of important implications.",
"Firstly, it implies that plots of the type we have constructed here can be used as a means to systematically inform the choice of $L$ for arbitrary models.",
"Secondly, and perhaps more importantly, it implies that if $L \\ge L_{true}$ , the procedure should demonstrate at least some robustness to the choice of lag, provided that the MDN is sufficiently expressive and sufficiently well-trained.",
"This explains why simply setting $L = 3$ resulted in a high level of estimation performance in our experiments, regardless of the considered model, since the models considered are not characterised by long-range dependenciesThe interested reader should refer to Appendix for additional discussions.."
],
[
"Computational Costs",
"At this point, one may ask whether the proposed estimation routine compares favourably to other contemporary alternatives in terms of computational costs.",
"As stated by [20], the cost of generating simulated data using a candidate model is generally dominant, particularly for large-scale models that may need to be run for several minutes in order to generate a single realisation.",
"It is therefore imperative that any estimation methodology keep the simulated ensemble size, which we call $R$ , to a minimum.",
"As previously stated, we have selected $R = 100$ , which results in a relatively large training set of $R(T_{sim} - L) = 99700$ training examples.",
"This compares favourably to most alternatives in the literature on a number of grounds.",
"Firstly, most studies which have attempted to estimate models of similar complexity make use of ensembles consisting of a far greater number of realisations, typically in excess of $R = 1000$ [5], [27], [30].",
"Secondly, the training set associated with $R = 100$ is already large relative to the complexity of the network architecture we employSee Appendix REF .. To illustrate this point, we repeat the experiments associated with parameter set 1 of the [7] model, changing only the simulated ensemble size, which has been halved to $R = 50$ .",
"We find that even with this drastic decrease in the number of Monte Carlo replications, the proposed methodology still performs well and results in a lower loss function value than was obtained using the method of [20] in the original experiments, with a ratio of $LS_{MDN} / LS_{KDE} = 0.7249$Here $LS_{KDE}$ is determined from the results of the original experiment involving the method of [20] with $R = 100$ , while $LS_{MDN}$ is determined from the results of the supplementary experiment involving our proposed methodology with $R = 50$ ..",
"This provides some evidence that even for greatly reduced ensemble sizes, our approach remains viable, and implies that the complexity of the candidate model and hence the employed neural network would likely need to be increased substantially before any increase in $R$ beyond 100 is required.",
"In addition to concerns related to the size of the simulated ensemble, it is also worthwhile to consider the actual computational costs of the neural network training procedure relative to those associated with the generation of a single model realisation.",
"For this reason, Figure REF demonstrates the total training time required by various neural network configurations, most of which are larger than that of the network employed in this investigation, which typically takes $\\sim 5$ seconds to be completely trained.",
"We find that even for substantially more complex neural networks than those considered in our investigation, the overall training time is still typically less than 40 seconds, which compares favourably to the simulation time of large-scale models, and we additionally find that the increase in computational time is linear for both increases in the lag length and network width.",
"Figure: Training time for various MDN configurations on an ensemble of 100 realisations of length 1000 generated using the model initialised using parameter set 1.",
"The point indicated on both the left and right panels corresponds to the configuration employed in our estimation experiments.Further, it should be noted that GPU parallelisation was not employed when generating the aforementioned computational cost diagrams.",
"Given the significant speedup that could be expected with the use of such hardware, typically in the region of $20\\times $ [34], we find there to be at least some evidence that the time taken to train the neural network will generally be negligible in comparison to the time taken to generate a single model realisation, even for far more sophisticated neural networks and candidate models.",
"This would, however, require further testing that is beyond the scope of this investigation and we thus suggest that the proposed routine be applied to more sophisticated models in future work."
],
[
"Conclusion",
"In the preceding sections, we have introduced a neural network-based protocol for the Bayesian estimation of economic simulation models (with a particular focus on ABMs) and demonstrated its estimation capabilities relative to a leading method in the existing literature.",
"Overall, we find that our method delivers compelling performance in a number of scenarios, including the estimation of heterogeneous agent models typically used to test estimation procedures, and less orthodox examples, such as identifying dynamic shifts in data generated by a random walk model.",
"In all of the cases tested, we find that our proposed methodology produces estimates closer to known ground truth values than the approach proposed by [20] and also find that it typically results in narrower and more sharply peaked posteriors for larger free parameter sets.",
"In addition to our primary findings, we also discuss practical issues related to the applicability of the proposed routine.",
"We demonstrate that the lag length, which can be viewed as our approach's primary hyperparameter, can be systematically chosen and that the overall estimation performance demonstrates at least some robustness to this choice.",
"Further, we provide a number of arguments as to the protocol's computational efficiency relative to a number of prominent alternatives in the literature and therefore suggest that attempts be made to apply it to models of a larger scale in future research."
],
[
"Acknowledgements",
"The author would like to thank J. Doyne Farmer for helpful discussions that greatly aided the process of preparing this manuscript and the UK government for the award of a Commonwealth Scholarship.",
"Responsibility for the conclusions herein lies entirely with the author."
],
[
"Technical Details of the Proposed Estimation Procedure ",
"While we presented an overview of our estimation procedure in Section , the associated discussions were primarily illustrative and omitted several key details.",
"We thus provide a more technical, step-by-step discussion of our approach in this section."
],
[
"Training Set Construction",
"The primary aim of our methodology is the construction of an approximation to the likelihood function for a given set of parameter values, $\\mathbf {\\theta }$ .",
"In order to facilitate this process, we make the simplifying assumption that $\\mathbf {x} ^{sim} _{t, i}(\\mathbf {\\theta })$ depends only on $\\mathbf {x} ^{sim} _{t - L, i}(\\mathbf {\\theta }), \\dots , \\mathbf {x} ^{sim} _{t - 1, i}(\\mathbf {\\theta })$ , for all $L < t \\le T$ .",
"Our problem therefore reduces to the estimation of conditional densities of the form $p \\left(\\mathbf {x}_{t, i} ^{sim} \\big | \\mathbf {x}_{t - L, i} ^{sim}, \\dots , \\mathbf {x}_{t - 1, i} ^{sim}: \\mathbf {\\theta }\\right)$ .",
"In order to estimate the above conditional densities, we will require an appropriate dataset, which is constructed in a number of stages.",
"The first of these stages involves the use of the candidate model to generate an ensemble of $R$ Monte Carlo replications, $\\mathbf {X} ^{sim} (\\mathbf {\\theta }, T ^{sim}, i), i = i_0, i_0 + 1, \\dots , i_0 + R - 1$ , for a given value of $\\mathbf {\\theta }$ .",
"This is then followed by the construction of two ordered sets for each Monte Carlo replication $i$ in the ensemble, $\\begin{split}\\mathbf {X} ^{train}_i(\\mathbf {\\theta }) = \\Big \\lbrace &\\left\\lbrace \\mathbf {x} ^{sim} _{1, i}(\\mathbf {\\theta }), \\dots , \\mathbf {x} ^{sim} _{L, i}(\\mathbf {\\theta })\\right\\rbrace , \\left\\lbrace \\mathbf {x} ^{sim} _{2, i}(\\mathbf {\\theta }), \\dots , \\mathbf {x} ^{sim} _{L + 1, i}(\\mathbf {\\theta })\\right\\rbrace , \\dots , \\\\&\\left\\lbrace \\mathbf {x} ^{sim} _{T - L, i}(\\mathbf {\\theta }), \\dots , \\mathbf {x} ^{sim} _{T - 1, i}(\\mathbf {\\theta })\\right\\rbrace \\Big \\rbrace ,\\end{split}$ and $\\mathbf {Y} ^{train}_i(\\mathbf {\\theta }) = \\left\\lbrace \\mathbf {x} ^{sim} _{L + 1, i}(\\mathbf {\\theta }), \\mathbf {x} ^{sim} _{L + 2, i}(\\mathbf {\\theta }), \\dots , \\mathbf {x} ^{sim} _{T, i}(\\mathbf {\\theta })\\right\\rbrace .$ Finally, the sets $\\mathbf {X} ^{train}_i(\\mathbf {\\theta }), i = i_0, i_0 + 1, \\dots , i_0 + R - 1$ are concatenated, in order, to produce a single, larger ordered set, $\\mathbf {X} ^{train}(\\mathbf {\\theta })$ , with an analogous procedure being applied to $\\mathbf {Y} ^{train}_i(\\mathbf {\\theta })$ to yield $\\mathbf {Y} ^{train}(\\mathbf {\\theta })$ .",
"In essence, $\\mathbf {X} ^{train}(\\mathbf {\\theta })$ consists of rolling windows of length $L$ drawn from the ensemble of Monte Carlo replications, while $\\mathbf {Y} ^{train}(\\mathbf {\\theta })$ consists of the $\\mathbf {x} ^{sim} _{t, i}(\\mathbf {\\theta })$ values that directly follow each window in $\\mathbf {X} ^{train}(\\mathbf {\\theta })$ .",
"Together, they form a training set of size $R(T - L)$ that can be used to approximate the required conditional densities."
],
[
"Neural Network Specification and Training",
"With an appropriate dataset now constructed, we proceed with a more detailed discussion of the MDN itself.",
"As a starting point, let $H$ be a feedforward neural network with input layer $\\mathbf {h}_0$ (taking in windows of length $L$ ), hidden layers $\\mathbf {h}_1, \\mathbf {h}_2, \\dots , \\mathbf {h}_{n - 1}$ , output layer $\\mathbf {h}_{n}$ , and weights and biases $\\mathbf {\\psi }$ .",
"The mixture parameters are then defined as $ \\mathbf {\\alpha } = softmax(\\mathbf {W}_{\\alpha }\\mathbf {h}_{n} + \\mathbf {b}_{\\alpha }),$ $\\mathbf {\\mu }_{k} = \\mathbf {W}_{\\mu _k}\\mathbf {h}_{n} + \\mathbf {b}_{\\mu _k},$ and $ \\mathbf {\\Sigma }_{k} = diag(\\mathbf {\\sigma }_{k} ^2),$ where $diag(\\mathbf {x})$ is a diagonal matrix with diagonal $\\mathbf {x}$ and $ \\log \\mathbf {\\sigma } ^{2} _{k} = \\mathbf {W}_{\\sigma _k}\\mathbf {h}_{n} + \\mathbf {b}_{\\sigma _k}.$ This results in an expanded neural network with weights and biases $\\mathbf {\\phi } = \\left\\lbrace \\mathbf {\\psi }, \\mathbf {W}_{\\alpha }, \\mathbf {b}_{\\alpha }, \\mathbf {W}_{\\mu _k}, \\mathbf {b}_{\\mu _k}, \\mathbf {W}_{\\sigma _k}, \\mathbf {b}_{\\sigma _k}\\right\\rbrace $ that takes windows of length $L$ as input and outputs $\\mathbf {\\alpha }$ , $\\mathbf {\\mu }_{k}$ , and $\\mathbf {\\Sigma }_{k}$ as defined above.",
"At this stage, there are a number of nuances worth highlighting.",
"In Eqn.",
"REF , notice that we make use of the $softmax$ function.",
"This ensures that the mixture weights, $\\mathbf {\\alpha }$ , are strictly positive and sum to one, as required.",
"Additionally, notice that in Eqn.",
"REF we consider a diagonal rather than a full covariance matrixIt should be noted that the universal density approximation properties of Gaussian mixtures still apply for diagonal covariance matrices..",
"If we had not made such an assumption, we would have to ensure that the covariance matrices returned by our neural network were positive definite.",
"Though possible in principle, this would significantly increase the number of network parameters and have a potentially detrimental effect on computational performance [40].",
"Finally, it should be apparent from Eqn.",
"REF that the neural network outputs a vector of log variances rather than the diagonal covariance matrix, allowing us to avoid imposing positivity constraints on the network output.",
"Now, all that remains is the training of our constructed network, which is achieved through the application of maximum likelihood estimation to our training set.",
"Denoting by $\\mathbf {X}_m ^{train}$ the $m$ -th entry in $\\mathbf {X} ^{train}(\\mathbf {\\theta })$ (with $\\mathbf {Y}_m ^{train}$ being similarly defined), maximum likelihood estimation is equivalent to solving $\\arg \\min _{\\mathbf {\\phi }} -\\sum _{m = 1} ^{R (T - L)} \\log \\sum _{k = 1} ^{K} \\alpha _k \\left(\\mathbf {X}_m ^{train}\\right) \\mathcal {N}\\left(\\mathbf {Y}_m ^{train} \\big | \\mathbf {\\mu }_k\\left(\\mathbf {X}_m ^{train}\\right), \\mathbf {\\Sigma }_k\\left(\\mathbf {X}_m ^{train}\\right)\\right)$ using stochastic gradient descent methods."
],
[
"Data Normalisation and Regularisation",
"While the scheme we have just described could be applied as is, it is likely to perform suboptimally in its current form.",
"This is because neural networks, like most machine learning techniques with a large number of free parameters, have a tendency to overfit the training data and thus perform poorly out-of-sample, particularly when the training set is small [33].",
"In practice, this is often addressed using early stopping, a technique that requires a percentage of the data to be kept separate from the training set in order to evaluate out-of-sample performance during each epoch [37].",
"Such a solution is, however, undesirable in our context, since it requires the generation of additional data, an expensive undertaking for large-scale simulation models.",
"Fortunately, [40] present a set of best practices for conditional density estimation using neural networks that provides an alternative solution for overfitting.",
"In particular, a technique called noise regularisation is employed, in which small random perturbations are applied to the data during the training process.",
"It can be shown that this ultimately results in a complexity penalty that favours smoother density estimates that are less prone to overfitting [40].",
"For this reason, we apply Gaussian perturbations to training examples in $\\mathbf {X} ^{train}(\\mathbf {\\theta })$ and $\\mathbf {Y} ^{train}(\\mathbf {\\theta })$ , which we denote by $\\mathbf {\\xi }_{x} \\sim \\mathcal {N}(0, \\eta _x\\mathbf {I}) \\text{ and } \\mathbf {\\xi }_{y} \\sim \\mathcal {N}(0, \\eta _y\\mathbf {I}),$ respectively.",
"It should be apparent that the degree of regularisation depends directly on the magnitudes of the standard deviations $\\eta _x$ and $\\eta _y$ relative to the range of variation in the training dataAs an example, setting $\\eta _x = 0.5$ would result in a substantial amount of regularisation for training examples that take values in $[0, 1]$ , while essentially having no effect for training examples taking values in $[0, 1000]$ ..",
"This implies that $\\eta _x$ and $\\eta _y$ would have to be adjusted for each candidate model in order to result in the same degree of regularisation.",
"[40] therefore propose a data normalisation scheme that ensures the training data exhibits zero mean and unit variance, eliminating the need to retune these hyperparameters for each candidate model.",
"This is achieved through the application of a simple transformation to each training example.",
"Letting $\\hat{\\mathbf {\\mu }}_x$ and $\\hat{\\mathbf {\\sigma }}_x$ be vectors that contain estimates of the mean and standard deviation along each dimension for training examples in $\\mathbf {X} ^{train}(\\mathbf {\\theta })$ , this transformation is given by $ \\tilde{\\mathbf {X}}_m ^{train} = diag(\\hat{\\mathbf {\\sigma }}_x) ^{-1} (\\mathbf {X}_m ^{train} - \\hat{\\mathbf {\\mu }}_x),$ with $\\hat{\\mathbf {\\mu }}_y$ , $\\hat{\\mathbf {\\sigma }}_y$ and $\\tilde{\\mathbf {Y}}_m ^{train}$ being defined analogously.",
"Once the network has been trained on the normalised dataset, we are required to evaluate $\\tilde{f}(\\mathbf {x}, \\mathbf {y}, \\mathbf {\\phi })$ , originally defined in Eqn.",
"REF .",
"This is achieved through a simple procedure.",
"Firstly, the normalisation transform is applied to $\\mathbf {x}$ and $\\mathbf {y}$ using the same $\\hat{\\mathbf {\\mu }}_y$ , $\\hat{\\mathbf {\\sigma }}_y$ , $\\hat{\\mathbf {\\mu }}_x$ and $\\hat{\\mathbf {\\sigma }}_x$ values defined in Eqn.",
"REF , yielding $\\tilde{\\mathbf {x}}$ and $\\tilde{\\mathbf {y}}$ .",
"$\\tilde{\\mathbf {x}}$ is then fed through the trained neural network to yield corresponding mixture parameters, allowing us to evaluate the density at $\\tilde{\\mathbf {y}}$ , which we denote by $\\tilde{g}(\\tilde{\\mathbf {x}}, \\tilde{\\mathbf {y}}, \\tilde{\\mathbf {\\phi }})$ .",
"It should be noted that $\\tilde{g}$ does not directly correspond to $\\tilde{f}$ , since we have made a change of variables and the volume of the probability density is not preserved under the normalisation transform for $\\hat{\\mathbf {\\sigma }}_y \\ne 1$ .",
"[40] do, however, prove that $\\tilde{f}(\\mathbf {x}, \\mathbf {y}, \\mathbf {\\phi }) = \\frac{1}{\\prod _{j = 1} ^{J} \\hat{\\sigma }_y ^{(j)}} \\tilde{g}(\\tilde{\\mathbf {x}}, \\tilde{\\mathbf {y}}, \\tilde{\\mathbf {\\phi }}),$ where $\\hat{\\sigma }_y ^{(j)}$ is the $j$ -th element of $\\hat{\\mathbf {\\sigma }}_y$ , allowing us to easily calculate the required density."
],
[
"Neural Network Architecture",
"In essence, we have defined a general neural network-based approach to simulation model estimation that is independent of the specific network architecture (number of hidden layers, number of neurons, type of activation functions, and so on) used.",
"Nevertheless, for the sake of completeness, we briefly introduce the (relatively simple) architecture employed in our study, which is used consistently throughout unless stated otherwise.",
"For the mixture model itself, we set the number of mixture components to be $K = 16$ , with the associated mixture parameter network consisting of 3 hidden layers, each with 32 neurons and ReLU activations.",
"This was trained using the well-known Adam optimiser [25] over 12 epochsAny improvements in the likelihood for subsequent epochs were generally negligible., with a batch size of 512 and noise regularisation parameters $\\eta _x = \\eta _y = 0.2$ .",
"The above architecture, which performed well for all of the estimation tasks conducted, was, perhaps rather surprisingly, the first architecture we considered and was chosen by hand rather than through an automated optimisation procedure.",
"Attempts to improve performance by increasing the number of hidden layers, neurons, and mixture components seemed to have little effect, suggesting that the proposed network is sufficiently expressive to produce high-quality density estimates for our considered set of problems.",
"We suspect that this will likely hold for other models of similar complexity and therefore make the recommendation that our proposed architecture be used as a baseline for future investigations employing this estimation methodology.",
"For more complex models, however, it may be necessary to construct more expressive networks and in such cases we would suggest that some form of hyperparameter optimisation be carried out.",
"This is beyond the scope of our investigation, however, and we thus leave it to future research."
],
[
"Technical Details of the Employed Sampling Strategy",
"In this section, we briefly discuss the adaptive Metropolis-Hastings algorithm that has been employed in all of the conducted estimation experiments.",
"Our discussion here is mainly illustrative and positioned in the context of our investigation.",
"The interested reader should therefore refer to the original contribution by [21] for theoretical justifications and a more general discussion.",
"In essence, the approach is centred on the idea of maintaining a set of samples, $\\mathbf {\\theta }_s = \\left\\lbrace \\mathbf {\\theta _{s}} ^{(1)}, \\mathbf {\\theta _{s}} ^{(2)}, \\dots , \\mathbf {\\theta _{s}} ^{(N)} \\right\\rbrace , s = 1, 2, \\dots , S$ , that is updated for a desired number of iterations.",
"Initially, the set consists of samples drawn uniformly from the space of feasible parameter values, $\\mathbf {\\Theta }$ , but eventually converges to be distributed according to $p(\\mathbf {\\theta }|\\mathbf {X})$ .",
"This is achieved through the construction of an adaptive proposal distribution that is dependent on the current samples, $\\mathbf {\\theta }_s$ , which can be summarised algorithmically as follows: Sample $\\mathbf {z}$ according to $\\tilde{p}\\left(\\mathbf {z} \\big | \\mathbf {\\theta _{s}} ^{(1)}, \\mathbf {\\theta _{s}} ^{(2)}, \\dots , \\mathbf {\\theta _{s}} ^{(N)} \\right)$ , which is determined by applying KDE to $\\mathbf {\\theta _{s}} ^{(1)}, \\mathbf {\\theta _{s}} ^{(2)}, \\dots , \\mathbf {\\theta _{s}} ^{(N)}$ .",
"Propose the switch of $\\mathbf {z}$ with $\\mathbf {\\theta _{s}} ^{(n)}$ , where $n$ is chosen uniformly from $\\left\\lbrace 1, 2, \\dots , N \\right\\rbrace $ .",
"Accept the switch with probability $\\alpha = \\min \\left\\lbrace 1, \\frac{p\\left(\\mathbf {z} \\big | \\mathbf {X}\\right) \\tilde{p}\\left(\\mathbf {\\theta _{s}} ^{(n)} | \\mathbf {\\theta _{s}} ^{(1)}, \\mathbf {\\theta _{s}} ^{(2)}, \\dots , \\mathbf {\\theta _{s}} ^{(n - 1)}, \\mathbf {z}, \\mathbf {\\theta _{s}} ^{(n + 1)}, \\dots , \\mathbf {\\theta _{s}} ^{(N)}\\right)}{p\\left(\\mathbf {\\theta _{s}} ^{(n)} | \\mathbf {X}\\right) \\tilde{p}\\left(\\mathbf {z} | \\mathbf {\\theta _{s}} ^{(1)}, \\mathbf {\\theta _{s}} ^{(2)}, \\dots , \\mathbf {\\theta _{s}} ^{(N)}\\right)} \\right\\rbrace .$ If accepted, set $\\mathbf {\\theta }_{s + 1} = \\mathbf {\\theta }_{s}$ with $\\mathbf {\\theta _{s}} ^{(n)}$ replaced by $\\mathbf {z}$ , otherwise simply set $\\mathbf {\\theta }_{s + 1} = \\mathbf {\\theta }_{s}$ .",
"Repeating the above for $S$ iterations, we obtain a sequence of sample sets that can be used to compute expectations of the form $\\mathbb {E}\\left[g(\\mathbf {\\theta })\\right] = \\frac{1}{NS} \\sum _{s = 1} ^{S} \\sum _{n = 1} ^{N} g\\left(\\mathbf {\\theta _{s}} ^{(n)}\\right).$ In our investigation, we set $S = 5000$ and $N = 70$ in all cases, with convergence typically observed at some point before $s = 1500$ , leading us to discard the first 1500 sets as part of a burn-in period.",
"When constructing the posterior samples, we repeat this entire sampling process 5 times and collect the obtained sets to form a larger collection of $5 \\times 3500 \\times 70 = 1225000$ samplesNote that since we only update a single sample during each step, the Monte Carlo variance still decreases at the standard rate of $\\frac{1}{\\sqrt{S}}$ ..",
"Ultimately, this has become our MCMC algorithm of choice for two main reasons: The number of iterations required to reach convergence in random walk Metropolis-Hastings algorithms depends significantly on the initialisation of the algorithm.",
"If, for example, the initial candidate parameter set has a particularly low posterior density, it could take a substantial period of time before convergence is observed.",
"Since the algorithm proposed by [21] is initialised using a sample of points from a number of areas of the parameter space, this problem is less pronounced.",
"Most random walk Metropolis-Hastings algorithms require careful tuning of the proposal distribution, usually with the aim of obtaining an acceptance rate of roughly $25\\%$ , in order to ensure a good balance between local exploration of high density areas of the parameter space and global coverage of the parameter space as a whole [39].",
"This can be difficult to achieve in practice, making an adaptive approach that determines the proposal distribution automatically particularly appealing."
],
[
"Robustness Tests",
"In Section REF , we provided evidence that our proposed estimation procedure demonstrates some robustness relative to the choice of lag length, $L$ .",
"Here, we provide a more complete demonstration by repeating all of the previously conducted estimation experiments involving our approach, changing only the lag length, which we have increased to $L = 4$ .",
"Referring to the summary presented in Table REF , we find that the overall performance of the procedure relative to our chosen benchmark is virtually unchangedSince there are a total of 27 individual parameter cases, the percentage shifts correspond to changes in only a single binary relation for both $|\\mu _{mdn} ^{i} - \\theta _{true} ^{i}| < |\\mu _{kde} ^{i} - \\theta _{true} ^{i}|$ and $\\sigma _{mdn} ^{i} < \\sigma _{kde} ^{i}$ ., verifying the robustness of our conclusions.",
"Table: Estimation Result Summary Across All Models for L=4L = 4"
]
] | 1906.04522 |
[
[
"A short proof of two shuffling theorems for tilings and a weighted\n generalization"
],
[
"Abstract Recently, Lai and Rohatgi discovered a shuffling theorem for lozenge tilings of doubly-dented hexagons, which generalized the earlier work of Ciucu.",
"Later, Lai proved an analogous theorem for centrally symmetric tilings, which generalized some other previous work of Ciucu.",
"In this paper, we give a unified proof of these two shuffling theorems, which also covers the weighted case.",
"Unlike the original proofs, our arguments do not use the graphical condensation method but instead rely on a well-known tiling enumeration formula due to Cohn, Larsen, and Propp.",
"Fulmek independently found a similar proof of Lai and Rohatgi's original shuffling theorem.",
"Our proof also gives a combinatorial explanation for Ciucu's recent conjecture relating the total number and the number of centrally symmetric lozenge tilings."
],
[
"Introduction",
"The enumeration of lozenge tilings of a region on a triangular lattice has received much attention during the last three decades.",
"In particular, people have tried to find regions whose number of lozenge tilings is expressed as a simple product formula.",
"In [3], Ciucu defined a structure called a fern, which is an arbitrary string of triangles of alternating orientations that touch at corners and are lined up along a common axis.",
"He considered a hexagon with a fern removed from its center and proved that the ratio of the number of lozenge tilings of two such regions is given by a simple product formula.",
"In [4], Ciucu proved that the ratio of the number of centrally symmetric lozenge tilings of two such regions is also given by a simple product formula.",
"In particular, he showed that the ratio for the centrally symmetric lozenge tilings is equal to the square root of the ratio for the total number of lozenge tilings.",
"From this observation, he conjectured that this square root phenomenon holds in a more general setting.",
"Recently, Lai and Rohatgi [11] found and proved a shuffling theorem for lozenge tilings of doubly-dented hexagons, generalizing the work of Ciucu [3].",
"Later, Lai showed that similar theorems also exist for reflectively symmetric tilings [9] and centrally symmetric tilings [10].",
"In [10], he generalized the work of Ciucu and proved Ciucu's Conjecture in [4].",
"The proofs in [10] and [11] were based on Kuo's graphical condensation method (more precisely, Kuo's original recurrence from [8] for [11] and Ciucu's extension from [4] for [10]).",
"The goal of this paper is to give a unified and shorter proof for these two shuffling theorems, which also covers the weighted case.",
"Unlike the original proofs, our arguments do not use the graphical condensation method but instead rely on a well-known tiling enumeration formula due to Cohn, Larsen, and Propp.",
"Fulmek independently found a similar proof of Lai and Rohatgi's original shuffling theorem in [6].",
"Our proof also gives a combinatorial explanation for Ciucu's recent conjecture relating the total number and the number of centrally symmetric lozenge tilings of regions with removed ferns."
],
[
"A shuffling theorem",
"In this paper, we are dealing with bounded regions on a triangular lattice and their lozenge tilings.",
"A lozenge-shaped tile (or lozenge for short) is a union of two adjacent unit triangles on the triangular lattice, and a lozenge tiling of a region is a collection of lozenges that covers the entire region without overlapping.",
"We will always draw the lattice so that one family of lattice lines is horizontal.",
"There are three kinds of lozenges that we can use.",
"According to their orientation, we call them left-, vertical-, and right-lozenge, respectively (see Figure 2.1).",
"For any region $G$ on a triangular lattice, let $M(G)$ be the number of its lozenge tilings.",
"We now describe the region we are interested in.",
"Figure: Two figures V 3,8,4 ({2,3,5,8,9,11},{3,7})V_{3,8,4}(\\lbrace 2,3,5,8,9,11\\rbrace ,\\lbrace 3,7\\rbrace ) (left) and V 6,5,7 ({3,7,9},{2,3,5,8,11})V_{6,5,7}(\\lbrace 3,7,9\\rbrace ,\\lbrace 2,3,5,8,11\\rbrace ) (right).",
"The figure on the right is obtained from the left figure by flipping 4 of the removed up-pointing unit triangles (indexed by 2,5,82,5,8 and 11), 1 of the removed down-pointing unit triangle (indexed by 7) and moving up the horizontal diagonal 3(=4-1)3 (=4-1) units while preserving the height and width of the hexagon.For non-negative integers $a,b$ , and $c$ , let $V_{a,b,c}$ be the hexagon of side lengths $a$ , $b$ , $c$ , $a+b-c$ , $c$ , and $b$ (clockwise from top).",
"Let $\\ell $ be the horizontal diagonal of $V_{a,b,c}$ .",
"This diagonal has length $a+b$ .",
"Label the unit segments on it from left to right by $1,\\cdots ,a+b$ .",
"For any subsets X and Y of $[a+b]:=\\lbrace 1,2,\\cdots ,a+b\\rbrace $ , let $V_{a,b,c}(X,Y)$ be the region obtained from $V_{a,b,c}$ by removing up-pointing unit triangles whose bases are along the unit segments of $\\ell $ labeled by the elements of $X$ , and down-pointing unit triangles whose bases are along the unit segments of $\\ell $ labeled by the elements of $Y$ (two examples are shown in Figure 2.2).",
"Without loss of generality, we may assume the following two conditions: $(i)$ $c\\le a+b$ , and $(ii)$ $0 \\le b-|X| = c-|Y| \\le |[a+b]\\setminus (X\\cup Y)|$ .",
"Indeed, $(i)$ follows because the bottom side of our hexagon has length $a+b-c$ .",
"All the statements in $(ii)$ follow from the assumption that a tiling exists (which we can clearly assume without loss of generality).",
"Indeed, the first inequality in $(ii)$ follows by encoding lozenge tilings by families of non-intersecting paths of lozenges (if $|X|>b$ , the $|X|$ paths starting at the right sides of removed up-pointing triangles do not have enough room to end on the northeastern side of the hexagon).",
"The equality in $(ii)$ follows from the requirement that $V_{a,b,c}(X,Y)$ contains the same number of up- and down-pointing unit triangles (a necessary condition for the existence of a tiling).",
"The last inequality in $(ii)$ again follows by encoding lozenge tilings by families of non-intersecting paths of lozenges (if $c>|Y|+|[a+b]\\setminus (X\\cup Y)|$ , the $c$ paths starting at the southwestern side of the hexagon would not have enough room to go through $\\ell $ ).",
"Therefore, we assume that conditions $(i)$ and $(ii)$ hold for all regions that we will encounter in this paper.",
"Definition 2.1 For any finite subsets of the integers $X$ , $X^{\\prime }$ , $Y$ , and $Y^{\\prime }$ , we say that a pair $(X^{\\prime }, Y^{\\prime })$ is a shuffling of $(X, Y)$ if the following two conditions hold: $\\begin{aligned}&1) X\\cup Y=X^{\\prime }\\cup Y^{\\prime } \\text{, and}\\\\&2) X\\cap Y=X^{\\prime }\\cap Y^{\\prime }.\\end{aligned}$ In particular, if $X$ and $Y$ record the positions of the removed up- and down-pointing unit triangles along $\\ell $ , and we are allowed to freely flip removed up-pointing unit triangles down and removed down-pointing unit triangles up — with the one restriction that pairs of removed unit triangles that form a vertical-lozenge are preserved —, then the pair $(X^{\\prime },Y^{\\prime })$ recording the new positions of the removed unit triangles is a shuffling of $(X,Y)$ .",
"In the first part of this paper, we give a short proof of the following theorem of Lai and Rohatgi presented in [11].",
"Theorem 2.1 $[$ 11, Theorem 2.1$]$ Let $a, b$ , and $c$ be non-negative integers and $X,Y$ be subsets of $[a+b]$ satisfying conditions $(i)$ and $(ii)$ above.",
"Consider the region $V_{a,b,c}(X,Y)$ .",
"While preserving removed unit triangles that form a vertical-lozenge, freely flip $d$ of the removed up-pointing unit triangles along $\\ell $ down and $u$ of the down-pointing up, so that their new positions are recorded by the sets $X^{\\prime }$ and $Y^{\\prime }$ .",
"Modify the boundary of $V_{a,b,c}(X,Y)$ so that the height and width of the hexagon are preserved, but $\\ell $ is moved up $d-u$ units (see Figure 2.2).",
"This leads to the region $V_{a^{\\prime },b^{\\prime },c^{\\prime }}(X^{\\prime },Y^{\\prime })$ , where $(X^{\\prime },Y^{\\prime })$ is a shuffling of $(X,Y)$ , and $\\bullet $ $a+b=a^{\\prime }+b^{\\prime }$ , $b+c=b^{\\prime }+c^{\\prime }$ $\\bullet $ $a+b\\ge c$ , $a^{\\prime }+b^{\\prime }\\ge c^{\\prime }$ $\\bullet $ $b-x=c-y=b^{\\prime }-x^{\\prime }=c^{\\prime }-y^{\\prime }\\ge 0$ , where $x, x^{\\prime }, y,$ and $y^{\\prime }$ are the cardinalities of $X, X^{\\prime }, Y,$ and $Y^{\\prime }$ , respectively.",
"For these two regions, we have $\\frac{M(V_{a^{\\prime }, b^{\\prime }, c^{\\prime }}(X^{\\prime },Y^{\\prime }))}{M(V_{a, b, c}(X,Y))}=\\frac{H(b)H(c)}{H(b^{\\prime })H(c^{\\prime })}\\frac{\\Delta _{1}(X^{\\prime })\\Delta _{1}(Y^{\\prime })}{\\Delta _{1}(X)\\Delta _{1}(Y)}$ where $\\Delta _{1}(S):=\\displaystyle \\prod _{s, s^{\\prime } \\in S, s<s^{\\prime }}{(s^{\\prime }-s)}$ and $H(n):=\\Delta _{1}([n])=\\displaystyle \\prod _{i=0}^{n-1}i!$ .",
"We point out that given a region $V_{a, b, c}(X,Y)$ , the region $V_{a^{\\prime }, b^{\\prime }, c^{\\prime }}(X^{\\prime },Y^{\\prime })$ is completely determined by the shuffling $(X^{\\prime },Y^{\\prime })$ of $(X,Y)$ .",
"The rest of this paper is organized as follows.",
"In section 3, we give a short proof of the above shuffling theorem.",
"In section 4, we state and prove the weighted generalization.",
"We end the paper by stating and giving a short proof of the shuffling theorem for centrally symmetric tilings."
],
[
"Proof of the shuffling theorem",
"This argument was presented before in the author’s earlier work [2].",
"We present it here in a clearer form.",
"One readily sees that each tiling of $V_{a,b,c}(X,Y)$ must contain precisely $b-|X|$ ($=c-|Y|$ ) vertical-lozenges crossing the horizontal diagonal $\\ell $ .",
"Similarly, for $V_{a^{\\prime },b^{\\prime },c^{\\prime }}(X^{\\prime },Y^{\\prime })$ , the corresponding number is $b^{\\prime }-|X^{\\prime }|$ ($=c^{\\prime }-|Y^{\\prime }|$ ).",
"Note that the lengths of the horizontal diagonals in $V_{a,b,c}(X,Y)$ and $V_{a^{\\prime },b^{\\prime },c^{\\prime }}(X^{\\prime },Y^{\\prime })$ are the same.",
"Furthermore, $b-|X|=b^{\\prime }-|X^{\\prime }|$ .",
"Partition the set of tilings of $V_{a,b,c}(X,Y)$ — and also the set of tilings of $V_{a^{\\prime },b^{\\prime },c^{\\prime }}(X^{\\prime },Y^{\\prime })$ — in classes, according to the positions of the $b-|X|$ ($=b^{\\prime }-|X^{\\prime }|$ ) vertical-lozenges that straddle the diagonal $\\ell $ .",
"The proof will follow from the simple fact that the ratio of the cardinalities of corresponding classes in these two partitions is equal to a concrete simple product, which — crucially — turns out to be the same for all classes of the partitions.",
"This follows from the following result, which is the lozenge tilings interpretation given by Cohn, Larsen, and Propp [5] of a classical result due to Gelfand and Tsetlin [7].",
"Proposition 3.1 For non-negative integers $m$ and $n$ , let $T_{m,n}$ be the trapezoid on the triangular lattice of side lengths $m$ , $n$ , $m+n$ , and $n$ (clockwise from top).",
"Label the unit segments on the bottom from left to right by $1,\\cdots ,m+n$ .",
"For any subset $S=\\lbrace s_1, s_2,\\cdots ,s_n\\rbrace \\subset [m+n]$ , let $T_{m,n}(S)$ be the region obtained from $T_{m,n}$ by removing the up-pointing unit triangles whose bases have labels in $S$ (see Figure 3.1 for an example).",
"Then $M(T_{m,n}(S))=\\frac{\\Delta _{1}(S)}{\\Delta _{1}([n])}=\\frac{\\Delta _{1}(S)}{H(n)}.$ Figure: A region T 8,5 ({1,4,5,9,12})T_{8,5}(\\lbrace 1,4,5,9,12\\rbrace )Proof of Theorem 2.1.",
"The set of lozenge tilings of $V_{a, b, c}(X,Y)$ can be partitioned according to the positions of $b-x$ vertical-lozenges crossing the horizontal diagonal $\\ell $ .",
"Let $Z$ be an index set for positions of the vertical-lozenges.",
"If we fix these vertical-lozenges, the remaining region to be tiled is $V_{a, b, c}(X\\cup Z,Y\\cup Z)$ .",
"Furthermore, since tilings of $V_{a, b, c}(X\\cup Z,Y\\cup Z)$ have no vertical-lozenges along $\\ell $ , tilings of $V_{a, b, c}(X\\cup Z,Y\\cup Z)$ is in bijection with pairs of tilings, one from tilings of $T_{a,b}(X\\cup Z)$ and the other from tilings of $T_{a+b-c,c}(Y\\cup Z)_{(a+b)}$ where $(Y\\cup Z)_{(a+b)}:=\\lbrace a+b+1-y|y\\in Y\\cup Z\\rbrace $ (see Figure 3.2).",
"Hence, $\\begin{aligned}M(V_{a, b, c}(X,Y))&=\\sum _{Z}M(V_{a, b, c}(X\\cup Z,Y\\cup Z))\\\\&=\\sum _{Z}M(T_{a,b}(X\\cup Z)) M(T_{a+b-c,c}(Y\\cup Z)_{(a+b)})\\end{aligned}$ where $Z$ runs over all subsets of $[a+b]\\setminus (X\\cup Y)$ whose cardinality is $b-x$ .",
"By Proposition 3.1, we have $\\begin{aligned}M(T_{a,b}(X\\cup Z))M(T_{a+b-c,c}(Y\\cup Z)_{(a+b)})&=\\frac{\\Delta _{1}(X\\cup Z)}{H(b)}\\frac{\\Delta _{1}((Y\\cup Z)_{(a+b)})}{H(c)}\\\\&=\\frac{\\Delta _{1}(X\\cup Z)}{H(b)}\\frac{\\Delta _{1}(Y\\cup Z)}{H(c)}.\\end{aligned}$ Hence, by (3.2)-(3.3), we have $M(V_{a, b, c}(X,Y))=\\frac{\\displaystyle \\sum _{Z} \\Delta _{1}(X\\cup Z) \\Delta _{1}(Y\\cup Z)}{H(b) H(c)}.$ Figure: A lozenge tiling of V 3,8,4 ({2,3,5,8,9,11},{3,7})V_{3,8,4}(\\lbrace 2,3,5,8,9,11\\rbrace ,\\lbrace 3,7\\rbrace ) with vertical lozenges at positions {1,10}\\lbrace 1,10\\rbrace (left) and corresponding pair of lozenge tilings of two subregions (right).",
"In this paper, pink lozenges represent lozenges crossing the horizontal diagonal.Similarly, $M(V_{a^{\\prime }, b^{\\prime }, c^{\\prime }}(X^{\\prime },Y^{\\prime }))$ can be written as follows: $M(V_{a^{\\prime }, b^{\\prime }, c^{\\prime }}(X^{\\prime },Y^{\\prime }))=\\frac{\\displaystyle \\sum _{Z} \\Delta _{1}(X^{\\prime }\\cup Z) \\Delta _{1}(Y^{\\prime }\\cup Z)}{H(b^{\\prime }) H(c^{\\prime })}$ where the summation is taken over all subsets of $[a^{\\prime }+b^{\\prime }]\\setminus (X^{\\prime }\\cup Y^{\\prime }) (= [a+b]\\setminus (X\\cup Y))$ whose cardinality is $b^{\\prime }-x^{\\prime } (= b-x)$ .",
"Thus, we have $\\frac{V_{a^{\\prime }, b^{\\prime }, c^{\\prime }}(X^{\\prime },Y^{\\prime })}{V_{a, b, c}(X,Y)}=\\frac{H(b) H(c)}{H(b^{\\prime }) H(c^{\\prime })}\\frac{\\displaystyle \\sum _{Z} \\Delta _{1}(X^{\\prime }\\cup Z) \\Delta _{1}(Y^{\\prime }\\cup Z)}{\\displaystyle \\sum _{Z} \\Delta _{1}(X\\cup Z) \\Delta _{1}(Y\\cup Z)}.$ Observe that summations in numerator and denominator are taken over the same sets $Z$ .",
"For any such $Z$ , the ratio of corresponding summands is $\\begin{aligned}\\frac{\\Delta _{1}(X^{\\prime }\\cup Z) \\Delta _{1}(Y^{\\prime }\\cup Z)}{\\Delta _{1}(X\\cup Z) \\Delta _{1}(Y\\cup Z)}&=\\frac{\\Delta _{1}(X^{\\prime }) \\Delta _{2}(X^{\\prime },Z) \\Delta _{1}(Z) \\Delta _{1}(Y^{\\prime }) \\Delta _{2}(Y^{\\prime },Z) \\Delta _{1}(Z)}{\\Delta _{1}(X) \\Delta _{2}(X,Z) \\Delta _{1}(Z) \\Delta _{1}(Y) \\Delta _{2}(Y,Z) \\Delta _{1}(Z)}\\\\&=\\frac{\\Delta _{1}(X^{\\prime }) \\Delta _{1}(Y^{\\prime })}{\\Delta _{1}(X) \\Delta _{1}(Y)}\\end{aligned}$ where $\\Delta _{2}(S,T):=\\displaystyle \\prod _{s \\in S ,t \\in T}{|t-s|}$ for finite disjoint subsets $S, T\\subset \\mathbb {Z}_{+}$ .",
"The right-hand side of (3.7) does not depend on the set $Z$ .",
"Hence, $\\frac{V_{a^{\\prime }, b^{\\prime }, c^{\\prime }}(X^{\\prime },Y^{\\prime })}{V_{a, b, c}(X,Y)}=\\frac{H(b)H(c)}{H(b^{\\prime })H(c^{\\prime })}\\frac{\\Delta _{1}(X^{\\prime })\\Delta _{1}(Y^{\\prime })}{\\Delta _{1}(X)\\Delta _{1}(Y)}.$ This completes the proof.",
"$\\square $"
],
[
"Weighted shuffling theorem",
"For any set $X=\\lbrace x_1,x_2,\\cdots ,x_n\\rbrace \\subset \\mathbb {Z}_{+}$ of positive integers, where elements are written in increasing order, let $\\lambda (X)$ be the partition $(x_n-n,\\cdots ,x_2-2,x_1-1)$ , which may contain 0 as a part.",
"Also, for any finite disjoint subsets $S, T\\subset \\mathbb {Z}_{+}$ , let $\\Delta _{1,q}(S):=\\displaystyle \\prod _{s, s^{\\prime } \\in S, s<s^{\\prime }}{([s^{\\prime }]_q-[s]_q)}$ and $\\Delta _{2,q}(S,T):=\\displaystyle \\prod _{s \\in S, t \\in T, s<t}{([t]_q-[s]_q)}\\cdot \\prod _{s \\in S ,t \\in T, t<s}{([s]_q-[t]_q)}$ , where $[n]_q:=\\frac{1-q^n}{1-q}$ denotes the $q$ -analogue of $n\\in \\mathbb {Z}_{+}$ .",
"Figure: A tiling of V 5,4,2 ({1,3,7},{6})V_{5,4,2}(\\lbrace 1,3,7\\rbrace ,\\lbrace 6\\rbrace ) with weighted lozenges.",
"Weight of this tiling is q 33 q^{33}.For any bounded region $G$ on the lattice, we give weight $q^k$ to each right-lozenge (recall Figure 2.1) whose distance between the bottom side of the lozenge and the top side of $G$ (= highest horizontal line that intersects with the closure of the region $G$ ) is $\\frac{k\\sqrt{3}}{2}$ , and give weight 1 to all vertical- and left-lozenges (see Figure 4.1).",
"When certain weights are given on lozenges and a lozenge tiling of the region is also given, the weight of the tiling is the product of weights of all tiles that the given tiling contains.",
"Also, the tiling generating function of a region $G$ is the sum of weights of tilings of $G$ where the sum is taken over all lozenge tilings of the region $G$ .",
"We will consider the tiling generating function under the weight described above and will denote it by $M(G;q)$ .",
"Note that if we take $q\\rightarrow 1$ , $M(G;q)$ becomes $M(G)$ .",
"The following is the weighted generalization of Theorem 2.1.",
"Lai and Rohatgi already provided the weighted generalization (see Theorem 2.4 of [11].",
"Their statement is stronger than ours — it involves a concept of “barrier”, which gives restriction on available vertical-lozenges that cross the horizontal diagonal —, and the weight they used is different.",
"However, our argument can also provide a proof of their version of weighted generalization).",
"The arguments in Section 3 can be adapted to provide a short proof of it.",
"Theorem 4.1 For the same regions $V_{a, b, c}(X,Y)$ and $V_{a^{\\prime }, b^{\\prime }, c^{\\prime }}(X^{\\prime },Y^{\\prime })$ as in Theorem 2.1, we have $\\frac{M(V_{a^{\\prime }, b^{\\prime }, c^{\\prime }}(X^{\\prime },Y^{\\prime });q)}{M(V_{a, b, c}(X,Y);q)}=q^{\\alpha }\\frac{\\Delta _{1,q}([b])\\Delta _{1,q}([c])}{\\Delta _{1,q}([b^{\\prime }])\\Delta _{1,q}([c^{\\prime }])}\\frac{\\Delta _{1,q}(X^{\\prime })\\Delta _{1,q}(Y^{\\prime })}{\\Delta _{1,q}(X)\\Delta _{1,q}(Y)}$ where $\\alpha :=\\displaystyle \\Bigg [\\sum _{i^{\\prime }\\in X^{\\prime }}i^{\\prime }-(b^{\\prime }+c^{\\prime })\\sum _{j^{\\prime }\\in Y^{\\prime }}j^{\\prime }-\\frac{b^{\\prime }(b^{\\prime }+1)}{2}+(a^{\\prime }+b^{\\prime }+1)(b^{\\prime }+1)c^{\\prime }-\\frac{(b^{\\prime }+1)c^{\\prime }(c^{\\prime }+1)}{2}+\\frac{(a^{\\prime }+b^{\\prime }+1)c^{\\prime }(c^{\\prime }-1)}{2}\\Bigg ]-\\Bigg [\\sum _{i\\in X}i-(b+c)\\sum _{j\\in Y}j-\\frac{b(b+1)}{2}+(a+b+1)(b+1)c-\\frac{(b+1)c(c+1)}{2}+\\frac{(a+b+1)c(c-1)}{2}\\Bigg ]$ .",
"Our strategy is almost the same as that of the unweighted case, except we need the following identity involving Schur function instead of Proposition 3.1.",
"Lemma 4.2 Let $X$ , $X^{\\prime }$ , $Y$ , and $Y^{\\prime }$ be any sets of positive integers whose cardinalities are $x, x^{\\prime }, y$ , and $y^{\\prime }$ , respectively, so that a pair $(X^{\\prime }, Y^{\\prime })$ is a shuffling of $(X, Y)$ .",
"Also, let $Z$ be any finite set of positive integers disjoint from $X\\cup Y(=X^{\\prime }\\cup Y^{\\prime })$ whose cardinality is $z$ .",
"Then we have $\\begin{aligned}&\\frac{s_{\\lambda (X^{\\prime }\\cup Z)}(1,q,\\cdots ,q^{x^{\\prime }+z-1})s_{\\lambda (Y^{\\prime }\\cup Z)}(1,q,\\cdots ,q^{y^{\\prime }+z-1})}{s_{\\lambda (X\\cup Z)}(1,q,\\cdots ,q^{x+z-1})s_{\\lambda (Y\\cup Z)}(1,q,\\cdots ,q^{y+z-1})}\\\\&=\\frac{\\Delta _{1,q}([x+z])\\Delta _{1,q}([y+z])}{\\Delta _{1,q}([x^{\\prime }+z])\\Delta _{1,q}([y^{\\prime }+z])}\\frac{\\Delta _{1,q}(X^{\\prime })\\Delta _{1,q}(Y^{\\prime })}{\\Delta _{1,q}(X)\\Delta _{1,q}(Y)}\\end{aligned}$ where $s_{\\lambda }$ represents a Schur function associated to a partition $\\lambda $ .",
"It can be easily deduced from the following proposition from Stanley [12].",
"Proposition 4.3 $[$ 12, (7.105)$]$ For any set $X=\\lbrace x_1,x_2,\\cdots ,x_n\\rbrace $ of positive integers, where elements are written in increasing order, we have $s_{\\lambda (X)}(1,q,\\cdots ,q^{n-1})=\\frac{\\Delta _{1,q}(X)}{\\Delta _{1,q}([n])}.$ Proof of Lemma 4.2.",
"By Proposition 4.3, for any such set $Z$ , we have $\\begin{aligned}&\\frac{s_{\\lambda (X^{\\prime }\\cup Z)}(1,q,\\cdots ,q^{x^{\\prime }+z-1})s_{\\lambda (Y^{\\prime }\\cup Z)}(1,q,\\cdots ,q^{y^{\\prime }+z-1})}{s_{\\lambda (X\\cup Z)}(1,q,\\cdots ,q^{x+z-1})s_{\\lambda (Y\\cup Z)}(1,q,\\cdots ,q^{y+z-1})}\\\\&=\\frac{\\Delta _{1,q}([x+z])\\Delta _{1,q}([y+z])}{\\Delta _{1,q}([x^{\\prime }+z])\\Delta _{1,q}([y^{\\prime }+z])}\\frac{\\Delta _{1,q}(X^{\\prime }\\cup Z)\\Delta _{1,q}(Y^{\\prime }\\cup Z)}{\\Delta _{1,q}(X\\cup Z)\\Delta _{1,q}(Y\\cup Z)}.\\end{aligned}$ We can simplify terms containing $Z$ on the right hand side of (4.4) as follows: $\\begin{aligned}&\\frac{\\Delta _{1,q}(X^{\\prime }\\cup Z)\\Delta _{1,q}(Y^{\\prime }\\cup Z)}{\\Delta _{1,q}(X\\cup Z)\\Delta _{1,q}(Y\\cup Z)}\\\\&=\\frac{\\Delta _{1,q}(X^{\\prime })\\Delta _{2,q}(X^{\\prime }, Z)\\Delta _{1,q}(Z)\\Delta _{1,q}(Y^{\\prime })\\Delta _{2,q}(Y^{\\prime }, Z)\\Delta _{1,q}(Z)}{\\Delta _{1,q}(X)\\Delta _{2,q}(X, Z)\\Delta _{1,q}(Z)\\Delta _{1,q}(Y)\\Delta _{2,q}(Y, Z)\\Delta _{1,q}(Z)}\\\\&=\\frac{\\Delta _{1,q}(X^{\\prime })\\Delta _{1,q}(Y^{\\prime })}{\\Delta _{1,q}(X)\\Delta _{1,q}(Y)}.\\end{aligned}$ Hence, by (4.4) and (4.5), $\\begin{aligned}&\\frac{s_{\\lambda (X^{\\prime }\\cup Z)}(1,q,\\cdots ,q^{x^{\\prime }+z-1})s_{\\lambda (Y^{\\prime }\\cup Z)}(1,q,\\cdots ,q^{y^{\\prime }+z-1})}{s_{\\lambda (X\\cup Z)}(1,q,\\cdots ,q^{x+z-1})s_{\\lambda (Y\\cup Z)}(1,q,\\cdots ,q^{y+z-1})}\\\\&=\\frac{\\Delta _{1,q}([x+z])\\Delta _{1,q}([y+z])}{\\Delta _{1,q}([x^{\\prime }+z])\\Delta _{1,q}([y^{\\prime }+z])}\\frac{\\Delta _{1,q}(X^{\\prime })\\Delta _{1,q}(Y^{\\prime })}{\\Delta _{1,q}(X)\\Delta _{1,q}(Y)}.\\end{aligned}$ This completes the proof.",
"$\\square $ The above identity will be used in the proof of Theorem 4.1 via the following well-known relation between Schur function and weighted enumeration of lozenge tilings of the trapezoidal region with some dents.",
"For the reference, we state the version of Ayyer and Fischer in [1] (in [1], they stated it in terms of matching generating function of a certain graph, which is equivalent).",
"Theorem 4.4 $[$ 1, Theorem 2.3$]$ Consider the region $T_{m,n}(S)$ that we described in Proposition 3.1.",
"On this region, we give weight $t_k$ to each right-lozenge whose distance between the bottom side of the lozenge and the top side of the region is $\\frac{k\\sqrt{3}}{2}$ , and give weight 1 to all vertical- and left-lozenges.",
"Let $M(T_{m,n}(S);(t_1,t_2,\\cdots ,t_n))$ be the tiling generating function of $T_{m,n}(S)$ under this weight.",
"Then we have $M(T_{m,n}(S);(t_1,t_2,\\cdots ,t_n))=s_{\\lambda (S)}(t_1,t_2,\\cdots ,t_n)$ Recall that Schur functions are symmetric and homogeneous.",
"Theorem 4.4 allows us to convert these properties of Schur functions into the following properties of tiling generating functions: $\\begin{aligned}&M(T_{m,n}(S);(t_1,t_2,\\cdots ,t_n))=M(T_{m,n}(S);(t_{\\sigma (1)},t_{\\sigma (2)},\\cdots ,t_{\\sigma (n)})), \\forall \\sigma \\in S_n \\text{, and}\\\\&M(T_{m,n}(S);(qt_1,qt_2,\\cdots ,qt_n))=q^{|\\lambda (S)|}M(T_{m,n}(S);(t_1,t_2,\\cdots ,t_n)).\\end{aligned}$ Note that for any bounded region $G$ whose height is $\\frac{n\\sqrt{3}}{2}$ , we have $M(G;(q,q^2,\\cdots ,q^n))=M(G;q).$ By using Lemma 4.2 and above properties of tiling generating functions, we can now give a simple proof of Theorem 4.1.",
"Figure: A lozenge tiling of V 3,8,4 ({2,3,5,8,9,11},{3,7})V_{3,8,4}(\\lbrace 2,3,5,8,9,11\\rbrace ,\\lbrace 3,7\\rbrace ) with vertical lozenges at positions {1,10}\\lbrace 1,10\\rbrace (left) and corresponding pair of lozenge tilings of two subregions (right).",
"Note that each lozenge has weight.Proof of Theorem 4.1.",
"By the same partitioning as we did in the proof of Theorem 2.1, we have $M(V_{a,b,c}(X,Y);q)=\\sum _{Z}M(V_{a,b,c}(X\\cup Z,Y\\cup Z);q)$ where $Z$ runs over all subsets of $[a+b]\\setminus (X\\cup Y)$ with cardinality $b-x$ , and $\\begin{aligned}&M(V_{a,b,c}(X\\cup Z,Y\\cup Z);q)\\\\&=M(T_{a,b}(X\\cup Z);q) M(T_{a+b-c,c}(Y\\cup Z)_{(a+b)};(q^{b+c},q^{b+c-1},\\cdots ,q^{b+1}))\\end{aligned}$ (see Figure 4.2).",
"Also, by Theorem 4.4 and the properties of tiling generating functions, $\\begin{aligned}M(T_{a,b}(X\\cup Z);q)&=M(T_{a,b}(X\\cup Z);(q,q^2,\\cdots ,q^{b}))\\\\&=q^{|\\lambda (X\\cup Z)|}M(T_{a,b}(X\\cup Z);(1,q,\\cdots ,q^{b-1}))\\\\&=q^{(\\sum _{i\\in X}i+\\sum _{k\\in Z}k-\\sum _{l=1}^{b}l)}s_{\\lambda (X\\cup Z)}(1,q,\\cdots ,q^{b-1})\\end{aligned}$ and similarly $\\begin{aligned}&M(T_{a+b-c,c}(Y\\cup Z)_{(a+b)};(q^{b+c},q^{b+c-1},\\cdots ,q^{b+1}))\\\\&=M(T_{a+b-c,c}(Y\\cup Z)_{(a+b)};(q^{b+1},q^{b+2},\\cdots q^{b+c}))\\\\&=q^{(b+1)|\\lambda ((Y\\cup Z)_{(a+b)})|}M(T_{a+b-c,c}(Y\\cup Z)_{(a+b)};(1,q,\\cdots q^{c-1}))\\\\&=q^{(b+1)\\lbrace (a+b+1)c-\\sum _{j\\in Y}j-\\sum _{k\\in Z}k-\\sum _{l=1}^{c}l\\rbrace }s_{\\lambda ((Y\\cup Z)_{(a+b)})}(1,q,\\cdots ,q^{c-1}).\\end{aligned}$ By Proposition 4.3, we have $\\begin{aligned}&s_{\\lambda ((Y\\cup Z)_{(a+b)})}(1,q,\\cdots ,q^{c-1})\\\\&=\\frac{\\Delta _{1,q}((Y\\cup Z)_{(a+b)})}{\\Delta _{1,q}([c])}\\\\&=\\frac{1}{\\Delta _{1,q}([c])}\\displaystyle \\prod _{i, j \\in (Y\\cup Z)_{(a+b)}, i<j}([j]_q-[i]_q)\\\\&=\\frac{1}{\\Delta _{1,q}([c])}\\displaystyle \\prod _{i, j \\in Y\\cup Z, j<i}([a+b+1-j]_q-[a+b+1-i]_q)\\\\&=\\frac{1}{\\Delta _{1,q}([c])}\\displaystyle \\prod _{i, j \\in Y\\cup Z, j<i}q^{a+b+1-i-j}([i]_q-[j]_q)\\\\&=\\Bigg (\\prod _{i, j \\in Y\\cup Z, j<i}q^{a+b+1-i-j}\\Bigg ) \\frac{\\Delta _{1,q}(Y\\cup Z)}{\\Delta _{1,q}([c])}\\\\&=q^{\\lbrace (a+b+1)\\frac{c(c-1)}{2}-(c-1)(\\sum _{j\\in Y}j+\\sum _{k\\in Z}k)\\rbrace }s_{\\lambda (Y\\cup Z)}(1,q,\\cdots ,q^{c-1}).\\\\\\end{aligned}$ Thus, by (4.9)-(4.12), $\\begin{aligned}&M(V_{a,b,c}(X\\cup Z,Y\\cup Z);q)\\\\&=q^{\\alpha (Z)} s_{\\lambda (X\\cup Z)}(1,q,\\cdots ,q^{b-1})s_{\\lambda (Y\\cup Z)}(1,q,\\cdots ,q^{c-1})\\end{aligned}$ where $\\alpha (Z):=\\sum _{i\\in X}i-(b+c)\\sum _{j\\in Y}j+(1-b-c)\\sum _{k\\in Z}k-\\frac{b(b+1)}{2}+(a+b+1)(b+1)c-\\frac{(b+1)c(c+1)}{2}+\\frac{(a+b+1)c(c-1)}{2}$ .",
"Similarly, we also have $M(V_{a^{\\prime },b^{\\prime } ,c^{\\prime }}(X^{\\prime },Y^{\\prime });q)=\\sum _{Z}M(V_{a^{\\prime },b^{\\prime },c^{\\prime }}(X^{\\prime }\\cup Z,Y^{\\prime }\\cup Z);q)$ where $Z$ runs over all subsets of $[a^{\\prime }+b^{\\prime }]\\setminus (X^{\\prime }\\cup Y^{\\prime }) (= [a+b]\\setminus (X\\cup Y))$ whose cardinality is $b^{\\prime }-x^{\\prime } (= b-x)$ , and $\\begin{aligned}&M(V_{a^{\\prime },b^{\\prime },c^{\\prime }}(X^{\\prime }\\cup Z,Y^{\\prime }\\cup Z);q)\\\\&=q^{\\alpha ^{\\prime }(Z)} s_{\\lambda (X^{\\prime }\\cup Z)}(1,q,\\cdots ,q^{b-1})s_{\\lambda (Y^{\\prime }\\cup Z)}(1,q,\\cdots ,q^{c-1})\\end{aligned}$ where $\\alpha ^{\\prime }(Z):=\\sum _{i^{\\prime }\\in X^{\\prime }}i^{\\prime }-(b^{\\prime }+c^{\\prime })\\sum _{j^{\\prime }\\in Y^{\\prime }}j^{\\prime }+(1-b^{\\prime }-c^{\\prime })\\sum _{k\\in Z}k-\\frac{b^{\\prime }(b^{\\prime }+1)}{2}+(a^{\\prime }+b^{\\prime }+1)(b^{\\prime }+1)c^{\\prime }-\\frac{(b^{\\prime }+1)c^{\\prime }(c^{\\prime }+1)}{2}+\\frac{(a^{\\prime }+b^{\\prime }+1)c^{\\prime }(c^{\\prime }-1)}{2}$ .",
"One can readily check that summations in (4.8) and (4.14) are taken over the same sets $Z$ .",
"For any such $Z$ , by (4.13), (4.15) and Lemma 4.2, the ratio of corresponding summands is $\\begin{aligned}&\\frac{M(V_{a^{\\prime },b^{\\prime },c^{\\prime }}(X^{\\prime }\\cup Z,Y^{\\prime }\\cup Z);q)}{M(V_{a,b,c}(X\\cup Z,Y\\cup Z);q)}\\\\&=q^{\\alpha ^{\\prime }(Z)-\\alpha (Z)}\\frac{s_{\\lambda (X^{\\prime }\\cup Z)}(1,q,\\cdots ,q^{b^{\\prime }-1})s_{\\lambda (Y^{\\prime }\\cup Z)}(1,q,\\cdots ,q^{c^{\\prime }-1})}{s_{\\lambda (X\\cup Z)}(1,q,\\cdots ,q^{b-1})s_{\\lambda (Y\\cup Z)}(1,q,\\cdots ,q^{c-1})}\\\\&=q^{\\alpha }\\frac{\\Delta _{1,q}([b])\\Delta _{1,q}([c])}{\\Delta _{1,q}([b^{\\prime }])\\Delta _{1,q}([c^{\\prime }])}\\frac{\\Delta _{1,q}(X^{\\prime })\\Delta _{1,q}(Y^{\\prime })}{\\Delta _{1,q}(X)\\Delta _{1,q}(Y)}\\end{aligned}$ where $\\alpha =\\alpha ^{\\prime }(Z)-\\alpha (Z)=[\\sum _{i^{\\prime }\\in X^{\\prime }}i^{\\prime }-(b^{\\prime }+c^{\\prime })\\sum _{j^{\\prime }\\in Y^{\\prime }}j^{\\prime }-\\frac{b^{\\prime }(b^{\\prime }+1)}{2}+(a^{\\prime }+b^{\\prime }+1)(b^{\\prime }+1)c^{\\prime }-\\frac{(b^{\\prime }+1)c^{\\prime }(c^{\\prime }+1)}{2}+\\frac{(a^{\\prime }+b^{\\prime }+1)c^{\\prime }(c^{\\prime }-1)}{2}]-[\\sum _{i\\in X}i-(b+c)\\sum _{j\\in Y}j-\\frac{b(b+1)}{2}+(a+b+1)(b+1)c-\\frac{(b+1)c(c+1)}{2}+\\frac{(a+b+1)c(c-1)}{2}]$ .",
"The expression in the right hand side of (4.16) does not depend on the set $Z$ .",
"Therefore, by (4.8), (4.14), and (4.16), $\\frac{M(V_{a^{\\prime },b^{\\prime },c^{\\prime }}(X^{\\prime },Y^{\\prime });q)}{M(V_{a,b,c}(X,Y);q)}=q^{\\alpha }\\frac{\\Delta _{1,q}([b])\\Delta _{1,q}([c])}{\\Delta _{1,q}([b^{\\prime }])\\Delta _{1,q}([c^{\\prime }])}\\frac{\\Delta _{1,q}(X^{\\prime })\\Delta _{1,q}(Y^{\\prime })}{\\Delta _{1,q}(X)\\Delta _{1,q}(Y)}.$ This completes the proof.",
"$\\square $"
],
[
"Centrally symmetric shuffling theorem",
"A region $G$ is centrally symmetric if it is invariant under rotation by 180$^{\\circ }$ with respect to a certain point (= the center).",
"A lozenge tiling of a centrally symmetric region is centrally symmetric if the tiling is invariant under rotation by 180$^{\\circ }$ with respect to the center.",
"For any centrally symmetric region $G$ , let $M_\\odot (G)$ be the number of its centrally symmetric lozenge tilings.",
"Also, for any positive integer $k$ , we say that two sets $X$ and $Y\\subset [k]$ are k-symmetric if $Y=\\lbrace k+1-x|x\\in X\\rbrace $ (or equivalently $X=\\lbrace k+1-y|y\\in Y\\rbrace $ ) holds, and denote this relation by $Y=X_{(k)}$ (or $X=Y_{(k)}$ ).",
"Obviously, if two sets are $k$ -symmetric to each other, then they have the same cardinality.",
"Note that the region $V_{a,b,c}(X,Y)$ is centrally symmetric if and only if $b=c$ and $Y=X_{(a+b)}$ (two examples are shown in Figure 5.1.)",
"In [10], Lai presented a shuffling theorem for centrally symmetric tilings.",
"By using the simple argument that we have used in the proof of previous theorems, we can also give a short proof of it.",
"Like the weighted shuffling theorem discussed in the previous section, the original version of this theorem involves a concept of “barrier.” However, since the proof is basically the same, we show the simple case when there is no barrier.",
"Figure: Two regions V 5,9,9 ({2,4,6,8,11},{4,7,9,11,13})V_{5,9,9}(\\lbrace 2,4,6,8,11\\rbrace ,\\lbrace 4,7,9,11,13\\rbrace ) (left) and V 5,9,9 ({4,8,9,11,13},{2,4,6,7,11})V_{5,9,9}(\\lbrace 4,8,9,11,13\\rbrace ,\\lbrace 2,4,6,7,11\\rbrace ) (right): Both regions are centrally symmetric.",
"The region on the right is obtained from the left region by flipping two removed up-pointing unit triangles (indexed by 2 and 6) and two removed down-pointing unit triangles (indexed by 9 and 13).Theorem 5.1 $[$ 10, Theorem 1.2$]$ Let $a$ and $b$ be any non-negative integers, and let $X$ be a subset of $[a + b]$ of cardinality $x$ satisfying $|[a+b]\\setminus (X\\cup X_{(a+b)})|\\ge b-x \\ge 0$ .",
"Consider a centrally symmetric region $V_{a, b, b}(X,X_{(a+b)})$ .",
"While preserving removed unit triangles that form a vertical-lozenge, freely flip removed up- and down-pointing unit triangles in $(a+b)$ -symmetric way (which means flipping removed up-pointing triangle whose position is indexed by $i$ and removed down-pointing triangle whose position is indexed by $(a+b+1-i)$ at the same time), so that their positions are recorded by the sets $X^{\\prime }$ and $X^{\\prime }_{(a+b)}$ .",
"This leads to the region $V_{a, b, b}(X^{\\prime },X^{\\prime }_{(a+b)})$ , where $(X^{\\prime }, X^{\\prime }_{(a+b)})$ is a shuffling of $(X, X_{(a+b)})$ .",
"For these two regions, we have $\\frac{M_\\odot (V_{a, b, b}(X^{\\prime },X^{\\prime }_{(a+b)}))}{M_\\odot (V_{a, b, b}(X,X_{(a+b)}))}= \\sqrt{\\frac{M(V_{a, b, b}(X^{\\prime },X^{\\prime }_{(a+b)}))}{M(V_{a, b, b}(X,X_{(a+b)}))}}=\\frac{\\Delta _{1}(X^{\\prime })}{\\Delta _{1}(X)}.$ Note that first equality in (5.1) implies Ciucu's conjecture on centrally symmetric lozenge tilings.",
"To prove this shuffling theorem for centrally symmetric tilings, we need the following simple lemma.",
"Lemma 5.2 Let $k$ be a positive integer, and let $X$ , $X^{\\prime }$ be subsets of $[k]$ such that a pair $(X^{\\prime }, X^{\\prime }_{(k)})$ is a shuffling of $(X, X_{(k)})$ .",
"Also, let $Z$ be any subset of $[k]$ with cardinality $z$ that is disjoint from $X\\cup X_{(k)} (=X^{\\prime }\\cup X^{\\prime }_{(k)})$ and is $k$ -symmetric with itself ($Z=Z_{(k)}$ ).",
"Then we have $\\frac{\\Delta _{1}(X^{\\prime }\\cup Z)}{\\Delta _{1}(X\\cup Z)}=\\frac{\\Delta _{1}(X^{\\prime })}{\\Delta _{1}(X)}$ The proof of Lemma 5.2 is analogous to that of Lemma 4.2.",
"Additionally, we have to use the $k$ -symmetric relations between some sets.",
"Proof of Lemma 5.2.",
"By shuffling condition and $k$ -symmetric relations, we have $\\begin{aligned}\\Delta _{2}(X,Z)=\\sqrt{\\Delta _{2}(X,Z)\\cdot \\Delta _{2}(X_{(k)},Z_{(k)})}&=\\sqrt{\\Delta _{2}(X,Z)\\cdot \\Delta _{2}(X_{(k)},Z)}\\\\&=\\sqrt{\\Delta _{2}(X^{\\prime },Z)\\cdot \\Delta _{2}(X^{\\prime }_{(k)},Z)}\\\\&=\\sqrt{\\Delta _{2}(X^{\\prime },Z)\\cdot \\Delta _{2}(X^{\\prime }_{(k)},Z_{(k)})}\\\\&=\\Delta _{2}(X^{\\prime },Z).\\\\\\end{aligned}$ Then, by (5.3), we have $\\begin{aligned}\\frac{\\Delta _{1}(X^{\\prime }\\cup Z)}{\\Delta _{1}(X\\cup Z)}=\\frac{\\Delta _{1}(X^{\\prime }) \\Delta _{2}(X^{\\prime },Z) \\Delta _{1}(Z)}{\\Delta _{1}(X) \\Delta _{2}(X,Z) \\Delta _{1}(Z)}=\\frac{\\Delta _{1}(X^{\\prime })}{\\Delta _{1}(X)}.\\end{aligned}$ This completes the proof.",
"$\\square $ Figure: A centrally symmetric lozenge tiling of V 5,9,9 ({2,4,6,8,11},{4,7,9,11,13})V_{5,9,9}(\\lbrace 2,4,6,8,11\\rbrace ,\\lbrace 4,7,9,11,13\\rbrace ) with vertical lozenges at positions {1,5,10,14}\\lbrace 1, 5, 10, 14\\rbrace (left) and a corresponding lozenge tiling of its subregion T 5,9 ({1,2,4,5,6,8,10,11,14})T_{5,9}(\\lbrace 1,2,4,5,6,8,10,11,14\\rbrace ) (right).Proof of Theorem 5.1.",
"Again, we partition the set of centrally symmetric lozenge tilings of the region $V_{a,b,b}(X,X_{(a+b)})$ according to $(b-x)$ vertical-lozenges crossing the horizontal diagonal.",
"Let $Z$ be the index set that record the positions of the vertical-lozenges.",
"Then, this set $Z$ should be $(a+b)$ -symmetric with itself ($Z=Z_{(a+b)}$ ).",
"If we fix these vertical-lozenges, the remaining region to be tiled is $V_{a, b, b}(X\\cup Z,X_{(a+b)}\\cup Z_{(a+b)})$ and one can readily see that its centrally symmetric lozenge tilings are in bijection with lozenge tilings of its subregion above the horizontal diagonal, $T_{a,b}(X\\cup Z)$ (see Figure 5.2).",
"Hence, $M_\\odot (V_{a, b, b}(X,X_{(a+b)}))=\\sum _{Z}M(T_{a,b}(X\\cup Z))=\\frac{\\sum _{Z}\\Delta _{1}(X\\cup Z)}{H(b)}$ where the sum is taken over all $(b-x)$ -element sets $Z\\subset [a+b]\\setminus (X\\cup X_{(a+b)})$ such that $Z=Z_{(a+b)}$ holds.",
"By exactly the same argument, $M_\\odot (V_{a, b, b}(X^{\\prime },X^{\\prime }_{(a+b)}))=\\sum _{Z}M(T_{a,b}(X^{\\prime }\\cup Z))=\\frac{\\sum _{Z}\\Delta _{1}(X^{\\prime }\\cup Z)}{H(b)}$ where the sum is taken over all $(b-x)$ -element sets $Z\\subset [a+b]\\setminus (X^{\\prime }\\cup X^{\\prime }_{(a+b)})(=[a+b]\\setminus (X\\cup X_{(a+b)}))$ such that $Z=Z_{(a+b)}$ holds.",
"Note that summations in (5.5) and (5.6) are taken over the same sets $Z$ .",
"For any such $Z$ , Lemma 5.2 says the ratio of the corresponding summands is $\\frac{\\Delta _{1}(X^{\\prime }\\cup Z)}{\\Delta _{1}(X\\cup Z)}=\\frac{\\Delta _{1}(X^{\\prime })}{\\Delta _{1}(X)}.$ This ratio does not depend on the set $Z$ .",
"Thus, by (5.5)-(5.7), we have $\\frac{M_\\odot (V_{a, b, b}(X^{\\prime },X^{\\prime }_{(a+b)}))}{M_\\odot (V_{a, b, b}(X,X_{(a+b)}))}= \\frac{\\sum _{Z}M(T_{a,b}(X^{\\prime }\\cup Z))}{\\sum _{Z}M(T_{a,b}(X\\cup Z))}=\\frac{\\Delta _{1}(X^{\\prime })}{\\Delta _{1}(X)}.$ Remaining part of the theorem is clear from Theorem 2.1 and the two facts $\\Delta _{1}(X)=\\Delta _{1}(X_{(a+b)})$ and $\\Delta _{1}(X^{\\prime })=\\Delta _{1}(X^{\\prime }_{(a+b)})$ , which can be easily deduced from $(a+b)$ -symmetric relations between sets.",
"This completes the proof.",
"$\\square $ It is clear from the proof that the ratio between the number of centrally symmetric tilings of the two regions is equal to the square roots of that of the total number of tilings.",
"This is because centrally symmetric tilings are determined by lozenges above the horizontal line, while both lozenges above and below the horizontal line contribute in the other case."
],
[
"Acknowledgment",
"The author would like to thank his advisor, Professor Mihai Ciucu, for his encouragement and useful discussions.",
"This paper could not have been written without his continued guidance.",
"Also, the author thanks Jeff Taylor for installing software and helpful assistance.",
"David Wilson's program, vaxmacs, was extremely useful when the author made an observation."
]
] | 1906.04533 |
[
[
"Quantum Coherence from Periodic Driving with Laser Pulses and Decay"
],
[
"Abstract Non-equilibrium physics is a particularly fascinating field of current research.",
"Generically, driven systems are gradually heated up so that quantum effects die out.",
"In contrast, we show that a driven central spin model including controlled dissipation in a highly excited state allows us to distill quantum coherent states, indicated by a substantial reduction of entropy.",
"The model is experimentally accessible in quantum dots or molecules with unpaired electrons.",
"The potential of preparing and manipulating coherent states by designed driving potentials is pointed out."
],
[
"Abstract",
"Non-equilibrium physics is a particularly fascinating field of current research.",
"Generically, driven systems are gradually heated up so that quantum effects die out.",
"In contrast, we show that a driven central spin model including controlled dissipation in a highly excited state allows us to distill quantum coherent states, indicated by a substantial reduction of entropy; the key resource is the commensurability between the periodicity of the pump pulses and the internal processes.",
"The model is experimentally accessible in purified quantum dots or molecules with unpaired electrons.",
"The potential of preparing and manipulating coherent states by designed driving potentials is pointed out."
],
[
"Introduction",
"Controlling a quantum mechanical system in a coherent way is one of the long-standing goals in physics.",
"Obviously, coherent control is a major ingredient for handling quantum information.",
"In parallel, non-equilibrium physics of quantum systems is continuing to attract significant interest.",
"A key issue in this field is to manipulate systems in time such that their properties can be tuned and changed at will.",
"Ideally, they display properties qualitatively different from what can be observed in equilibrium systems.",
"These current developments illustrate the interest in understanding the dynamics induced by time-dependent Hamiltonians $H(t)$ .",
"The unitary time evolution operator $U(t_2,t_1)$ induced by $H(t)$ is formally given by $U(t_2,t_1) = {\\cal T}\\exp \\left(-i\\int _{t_1}^{t_2}H(t)dt\\right)$ where ${\\cal T}$ is the time ordering operator.",
"While the explicit calculation of $U(t_2,t_1)$ can be extremely difficult it is obvious that the dynamics induced by a time-dependent Hamiltonian maps quantum states at $t_1$ to quantum states at $t_2$ bijectively and conserves the mutual scalar products.",
"Hence, if initially the system is in a mixed state with high entropy $S>0$ it stays in a mixed state for ever with exactly the same entropy.",
"No coherence can be generated in this way even for a complete and ideal control of $H(t)$ in time.",
"Hence, one has to consider open systems.",
"The standard way to generate a single state is to bring the system of interest into thermal contact with a cold system.",
"Generically, this is an extremely slow process.",
"The targeted quantum states have to be ground states of some given system.",
"Alternatively, optical pumping in general and laser cooling in particular [1] are well established techniques to lower the entropy of microscopic systems using resonant pumping and spontaneous decay.",
"Quite recently, engineered dissipation has been recognized as a means to generate targeted entangled quantum states in small [2], [3], [4] and extended systems [5], [6].",
"Experimentally, entanglement has been shown for two quantum bits [7], [8] and for two trapped mesoscopic cesium clouds [9].",
"In this article, we show that periodic driving can have a quantum system converge to coherent quantum states if an intermediate, highly excited and decaying state is involved.",
"The key aspect is the commensurability of the period of the pump pulses to the time constants of the internal processes, here Larmor precessions.",
"This distinguishes our proposal from established optical pumping protocols.",
"The completely disordered initial mixture can be made almost coherent.",
"The final mixture only has an entropy $S\\approx k_\\text{B}\\ln 2$ corresponding to a mixture of two states.",
"An appealing asset is that once the driving is switched off the Lindbladian decay does not matter anymore and the system is governed by Hamiltonian dynamics only.",
"The focus of the present work is to exemplarily demonstrate the substantial reduction of entropy in a small spin system subject to periodic laser pulses.",
"The choice of system is motivated by experiments on the electronic spin in quantum dots interacting with nuclear spins [10], [11], [12], [13], [14], [15], [16], [17].",
"The model studied is also applicable to the electronic spin in molecular radicals [18] or to molecular magnets, see Refs.",
"[19], [20], [21].",
"In organic molecules the spin bath is given by the nuclear spins of the hydrogen nuclei in organic ligands."
],
[
"Model",
"The model comprises a central, electronic spin $S=1/2$ which is coupled to nuclear spins $H_\\text{spin} = H_\\text{CS} + H_\\text{eZ} + H_\\text{nZ}$ where $H_\\text{eZ}=h S^x$ is the electronic Zeeman term with $h=g\\mu _\\text{B} B$ ($\\hbar $ is set to unity here and henceforth) with the gyromagnetic factor $g$ , the Bohr magneton $\\mu _\\text{B}$ , the external magnetic field $B$ in $x$ -direction and the $x$ -component $S^x$ of the central spin.",
"The nuclear Zeeman term is given by $H_\\text{nZ} = z h \\sum _{i=1}^N I^x_i$ where $z$ is the ratio of the nuclear $g$ -factor multiplied by the nuclear magneton and their electronic counterparts $g_\\text{nuclear}\\mu _\\text{nuclear}/(g\\mu _\\text{B})$ .",
"The operator $I^x_i$ is the $x$ -component of the nuclear spin $i$ .",
"For simplicity we take $I=1/2$ for all nuclear spins.",
"Due to the large nuclear mass, the factor $z$ is of the order of $10^{-3}$ , but in principle other $z$ -values can be studied as well, see also below.",
"In the central spin part $H_\\text{CS}=\\vec{S}\\cdot \\vec{A}$ the so-called Overhauser field $\\vec{A}$ results from the combined effect of all nuclear spins each of which is interacting via the hyperfine coupling $J_i$ with the central spin $\\vec{A} = \\sum _{i=1}^N J_i \\vec{I}_i.$ If the central spin results from an electron the hyperfine coupling is a contact interaction at the location of the nucleus stemming from relativistic corrections to the non-relativistic Schrödinger equation with a Coulomb potential.",
"It is proportional to the probability of the electron to be at the nucleus, i.e., to the modulus squared of the electronic wave function [22], [23].",
"Depending on the positions of the nuclei and on the shape of the wave function various distributions of the $J_i$ are plausible.",
"A Gaussian wave function in one dimension implies a parametrization by a Gaussian while in two dimensions an exponential parametrization is appropriate [24], [25] distribution.",
"We will first use a uniform distribution for simplicity and consider the Gaussian and exponential case afterwards.",
"Besides the spin system there is an important intermediate state given by a single trion state ${|\\mathrm {T}\\rangle }$ consisting of the single fermion providing the central spin bound to an additional exciton.",
"This trion is polarised in $z$ -direction at the very high energy $\\varepsilon $ ($\\approx 1$ eV).",
"The other polarisation exists as well, but using circularly polarised light it is not excited.",
"A Larmor precession of the trion is not considered here for simplicity.",
"Then, the total Hamiltonian reads $H = H_\\text{spin} + \\varepsilon {|\\mathrm {T}\\rangle }{\\langle \\mathrm {T}|}.$ The laser pulse is taken to be very short as in experiment where its duration $\\tau $ is of the order of picoseconds.",
"Hence, we describe its effect by a unitary time evolution operator $\\exp (-i\\tau H_\\text{puls})=U_\\text{puls}$ which excites the ${|\\uparrow \\rangle }$ state of the central spin to the trion state or de-excites it $U_\\text{puls} = c^\\dag + c +{|\\downarrow \\rangle } {\\langle \\downarrow |}.$ where $c:={|\\uparrow \\rangle }{\\langle \\mathrm {T}|}$ and $c^\\dagger :={|\\mathrm {T}\\rangle }{\\langle \\uparrow |}$ .",
"This unitary operator happens to be hermitian as well, but this is not an important feature.",
"One easily verifies $U_\\text{puls}U_\\text{puls}^\\dag =\\mathbb {1}$ .",
"Such pulses are applied in long periodic trains lasting seconds and minutes.",
"The repetition time between two consecutive pulses is ${T_\\mathrm {rep}}$ of the order of 10 ns.",
"The decay of the trion is described by the Lindblad equation for the density matrix $\\rho $ $\\partial _t \\rho (t) = -i[H,\\rho ] - \\gamma (c^\\dag c\\rho + \\rho c^\\dag c- 2c\\rho c^\\dag )$ where the prefactor $\\gamma >0$ of the dissipator term [26] defines the decay rate.",
"The corresponding process with $c$ and $c^\\dag $ swapped needs not be included because its decay rate is smaller by $\\exp (-\\beta \\varepsilon )$ , i.e., it vanishes for all physical purposes.",
"We emphasize that we deal with an open quantum system by virtue of the Lindblad dynamics in (REF ).",
"Since the decay of the trion generically implies the emission of a photon at high energies the preconditions for using Lindblad dynamics are perfectly met [26]."
],
[
"Mathematical Properties of Time Evolution",
"The key observation is that the dynamics from just before the $n$ th pulse at $t=n{T_\\mathrm {rep}}-$ to just before the $n+1$ st pulse at $t=(n+1){T_\\mathrm {rep}}-$ is a linear mapping $M: \\rho (n{T_\\mathrm {rep}}-) \\rightarrow \\rho ((n+1){T_\\mathrm {rep}}-)$ which does not depend on $n$ .",
"Since it is acting on operators one may call it a superoperator.",
"Its matrix form is derived explicitly in Appendix .",
"If no dissipation took place ($\\gamma =0$ ) the mapping $M$ would be unitary.",
"But in presence of the dissipative trion decay it is a general matrix with the following properties: The matrix $M$ has an eigenvalue 1 which may be degenerate.",
"If the dynamics of the system takes place in $n$ separate subspaces without transitions between them the degeneracy is at least $n$ .",
"All eigenoperators to eigenvalues different from 1 are traceless.",
"At least one eigenoperator to eigenvalue 1 has a finite trace.",
"The absolute values of all eigenvalues of $M$ are not larger than 1.",
"If there is a non-real eigenvalue $\\lambda $ with eigenoperator $C$ , the complex conjugate $\\lambda ^*$ is also an eigenvalue with eigenoperator $C^\\dag $ .",
"The eigenoperators to eigenvalues 1 can be scaled to be hermitian.",
"While the above properties can be shown rigorously, see Appendix , for any Lindblad evolution, the following ones are observed numerically in the analysis of the particular model (REF ) under study here: (a) The matrix $M$ is diagonalizable; it does not require a Jordan normal form.",
"(b) For pairwise different couplings $i\\ne j\\Rightarrow J_i\\ne J_j$ the eigenvalue 1 is non-degenerate.",
"(c) The eigenoperators to eigenvalue 1 can be scaled to be hermitian and non-negative.",
"In the generic, non-degenerate case we denote the properly scaled eigenoperator $V_0$ with $\\text{Tr}(V_0)=1$ .",
"(d) No eigenvalue $\\ne 1$ , but with absolute value 1, occurs, i.e., all eigenvalues different from 1 are smaller than 1 in absolute value.",
"(e) Complex eigenvalues and complex eigenoperators do occur.",
"The above properties allow us to understand what happens in experiment upon application of long trains of pulses corresponding to $10^{10}$ and more applications of $M$ .",
"Then it is safe to conclude that all contributions from eigenoperators to eigenvalues smaller than 1 have died out completely.",
"Only the (generically) single eigenoperator $V_0$ to eigenvalue 1 is left such that $\\lim _{n\\rightarrow \\infty } \\rho (n{T_\\mathrm {rep}}-) = V_0.$ The quasi-stationary state after long trains of pulses is given by $V_0$ We use the term `quasi-stationary' state because it is stationary only if we detect it stroboscopically at the time instants $t=n{T_\\mathrm {rep}}-$ ..",
"This observation simplifies the calculation of the long-time limit greatly compared to previous quantum mechanical studies [13], [14], [27], [17].",
"One has to compute the eigenoperator of $M$ to the eigenvalue 1.",
"Below this is performed by diagonalization of $M$ which is a reliable approach, but restricted to small systems $N\\lessapprox 6$ .",
"We stress that no complete diagonalization is required to know $V_0$ because only the eigenoperator to the eigenvalue 1 is needed.",
"Hence we are optimistic that further computational improvements are possible.",
"If, however, the speed of convergence is of interest more information on the spectrum and the eigenoperators of $M$ is needed, see also Sect.",
"."
],
[
"Results on Entropy",
"It is known that in pulsed quantum dots nuclear frequency focusing occurs (NFF) [10], [11], [28] which can be explained by a significant change in the distribution of the Overhauser field [12], [13], [14], [27], [16], [17] which is Gaussian initially.",
"This distribution develops a comb structure with equidistant spikes.",
"The difference $\\Delta A_x$ between consecutive spikes is such that it corresponds to a full additional revolution of the central spin ${T_\\mathrm {rep}}\\Delta A_x=2\\pi $ .",
"A comb-like probability distribution is more structured and contains more information than the initial featureless Gaussian.",
"For instance, the entropy reduction of the Overhauser field distributions computed in Ref.",
"[17], Fig.",
"12, relative to the initial Gaussians is $\\Delta S =-0.202k_\\text{B}$ at $B=0.93$ T and $\\Delta S =-0.018k_\\text{B}$ at $B=3.71$ T. Hence, NFF decreases the entropy, but only slightly for large spin baths.",
"This observation inspires us to ask to which extent continued pulsing can reduce entropy and which characteristics the final state has.",
"Inspired by the laser experiments on quantum dots [10], [11], [28] we choose an (arbitrary) energy unit $J_\\text{Q}$ and thus $1/J_\\text{Q}$ , recalling that we have set $\\hbar =1$, as time unit which can be assumed to be of the order of 1ns.",
"The repetition time ${T_\\mathrm {rep}}$ is set to $4\\pi /J_\\text{Q}$ which is on the one hand close to the experimental values where ${T_\\mathrm {rep}}=13.2\\text{ns}$ and on the other hand makes it easy to recognize resonances, see below.",
"The trion decay rate is set to $2\\gamma =2.5 J_\\text{Q}$ to reflect a trion life time of $\\approx 0.4\\leavevmode {\\color {black}\\text{ns}}$ .",
"The bath size is restricted to $N\\in \\lbrace 1,2,\\ldots ,6\\rbrace $ , but still allows us to draw fundamental conclusions and to describe electronic spins coupled to hydrogen nuclear spins in small molecules [18], [19], [20], [21].",
"The individual couplings $J_i$ are chosen to be distributed according to $J_i = J_\\text{max}(\\sqrt{5}-2)\\left(\\sqrt{5}+2({i-1})/({N-1})\\right),$ which is a uniform distribution between $J_\\text{min}$ and $J_\\text{max}$ with $\\sqrt{5}$ inserted to avoid accidental commensurabilities of the different couplings $J_i$.",
"The value $J_\\text{min}$ results from $J_i$ for $i=1$ .",
"Other parametrizations are motivated by the shape of the electronic wave functions [29], [22], [23].",
"Results for a frequently used exponential parameterization [24] $J_i = J_\\text{max}\\exp (-\\alpha (i-1)/(N-1))$ with $\\alpha \\in \\lbrace 0.5, 1\\rbrace $ and for a Gaussian parametrization, motivated by the electronic wave function in quantum dots [23], $J_i = J_\\text{max}\\exp (-\\alpha [(i-1)/(N-1)]^2).$ are given in the next section and in Appendix .",
"For both parametrizations the minimum value $J_\\text{min}$ occurs for $i=N$ and takes the value $J_\\text{min}=J_\\text{max}\\exp (-\\alpha )$ .",
"Figure: (a) Residual entropy of the limiting density matrix V 0 V_0 obtained afterinfinite number of pulses vs. the applied magnetic field forJ max =0.02J Q J_\\text{max}=0.02J_\\text{Q} and z=1/1000z=1/1000; 1 Tesla corresponds roughly to 50J Q 50J_\\text{Q}.Resonances of the electronic spin occur everyΔh=0.5J Q \\Delta h=0.5J_\\text{Q}; resonances of the nuclear spins occur every Δh=500J Q \\Delta h = 500J_\\text{Q}.The blue dashed line depicts an offset of Δh=±J max /(2z)\\Delta h=\\pm J_\\text{max}/(2z) from the nuclear resonance.",
"(b) Zooms into intervals of the magnetic field where the lowest entropiesare reached.",
"The blue dashed lines depict an offset of Δh=±A max \\Delta h =\\pm A_\\text{max} from theelectronic resonance.Figure REF displays a generic dependence on the external magnetic field $h=g\\mu _\\text{B}B_x$ of the entropy of the limiting density matrix $V_0$ obtained after infinite number of pulses.",
"Two nested resonances of the Larmor precessions are discernible: the central electronic spin resonates for $h{T_\\mathrm {rep}}= 2\\pi n, \\qquad n\\in \\mathbb {Z}$ where $n$ is the number of Larmor revolutions that fit into the interval ${T_\\mathrm {rep}}$ between two pulses.",
"This means that for an increase of the magnetic field from $h$ to $h+\\Delta h$ with $\\Delta h=2\\pi /{T_\\mathrm {rep}}$ the central spin is in the same state before the pulse as it was at $h$ .",
"The other resonance is related to the Larmor precession of the nuclear bath spins which leads to the condition $zh{T_\\mathrm {rep}}=2\\pi n^{\\prime }, \\qquad n^{\\prime }\\in \\mathbb {Z}$ where $n^{\\prime }$ indicates the number of Larmor revolutions of the nuclear spins which fit between two pulses.",
"Upon increasing the magnetic field $h$ , the nuclear spins are in the same state before the next pulse if $h$ is changed to $h+\\Delta h$ with $\\Delta h=2\\pi /(z{T_\\mathrm {rep}})$ .",
"But the two resonance conditions (REF ) and (REF ) for the central spin and for the bath spins apply precisely as given only without coupling between the spins.",
"The coupled system displays important shifts.",
"The nuclear resonance appears to be shifted by $z\\Delta h \\approx \\pm J_\\text{max}/2$ , see right panel of Fig.",
"REF (a).",
"The explanation is that the dynamics of the central spin $S=1/2$ creates an additional magnetic field similar to a Knight shift acting on each nuclear spin of the order of $J_i/2$ which is estimated by $J_\\text{max}/2$ .",
"Further support of the validity of this explanation is given in Appendix .",
"The electronic resonance is shifted by $\\Delta h = \\pm A_\\text{max}$ where $A_\\text{max}$ is the maximum possible value of the Overhauser field given by $A_\\text{max}:=(1/2)\\sum _{i=1}^N J_i$ for maximally polarized bath spins.",
"This is shown in the right panel of Fig.",
"REF (b).",
"Fig.",
"REF shows that the effect of the periodic driving on the entropy strongly depends on the precise value of the magnetic field.",
"The entropy reduction is largest close to the central resonance (REF ) and to the bath resonance (REF ).",
"This requires that both resonances must be approximately commensurate.",
"In addition, the precise position of the maximum entropy reduction depends on the two above shifts, the approximate Knight shift and the shift by the maximum Overhauser field (REF ).",
"We pose the question to which extent the initial entropy of complete disorder $S_\\text{init}=k_\\text{B}(N+1)\\ln 2$ (in the figures and henceforth $k_\\text{B}$ is set to unity) can be reduced by commensurate periodic pumping.",
"The results in Fig.",
"REF clearly show that remarkably low values of entropy can be reached.",
"The residual value of $S\\approx 0.5k_\\text{B}$ in the minima of the right panel of Fig.",
"REF (b) corresponds to a contribution of less than two states ($S=\\ln 2k_\\text{B}\\approx 0.7k_\\text{B}$ ) while initially 16 states were mixed for $N=3$ so that the initial entropy is $S_\\text{init}=4\\ln 2k_\\text{B}\\approx 2.77k_\\text{B}$ .",
"This represents a remarkable distillation of coherence.",
"Figure: (a) Residual entropy of the limiting density matrix V 0 V_0for various bath sizes; other parameters as in Fig.",
".The dashed lines indicate the shifts of the electronicresonance by -A max -A_\\text{max}.",
"(b) Corresponding normalized polarization of the spin bathin the external field direction, i.e.",
"the xx-direction.Hence, we focus on the minima and in particular on the left minimum.",
"We address the question whether the distillation of coherence still works for larger systems.",
"Unfortunately, the numerical analysis cannot be extended easily due to the dramatically increasing dimension $D= 2^{2(N+1)}$ because we are dealing with the Hilbert space of density matrices of the spin bath and the central spin.",
"Yet a trend can be deduced from results up to $N=6$ displayed in Fig.",
"REF (a).",
"The entropy reduction per $N+1$ spins is $-0.58k_\\text{B}$ for $N=3$ , $-0.57k_\\text{B}$ for $N=4$ , $-0.55k_\\text{B}$ for $N=5$ , and $-0.52k_\\text{B}$ for $N=6$ .",
"The reduction is substantial, but slowly decreases with system size.",
"Presently, we cannot know the behavior for $N\\rightarrow \\infty $ .",
"The finite value $\\approx -0.2k_\\text{B}$ found in the semiclassical simulation [16], [17] indicates that the effect persists for large baths.",
"In Appendix , results for the couplings defined in (REF ) or in (REF ) are given which corroborate our finding.",
"The couplings may be rather close to each other, but not equal.",
"It appears favorable that the spread of couplings is not too large.",
"Which state is reached in the minimum of the residual entropy?",
"The decisive clue is provided by the lower panel Fig.",
"REF (b) displaying the polarization of the spin bath.",
"It is normalized such that its saturation value is unity.",
"Clearly, the minimum of the residual entropy coincides with the maximum of the polarization.",
"The latter is close to its saturation value though not quite with a minute decrease for increasing $N$ .",
"This tells us that the limiting density matrix $V_0$ essentially corresponds to the polarized spin bath.",
"The central electronic spin is also almost perfectly polarized (not shown), but in $z$ -direction.",
"These observations clarify the state which can be retrieved by long trains of pulses.",
"Additionally, Fig.",
"REF (b) explains the shift of the electronic resonance.",
"The polarized spin bath renormalizes the external magnetic field by (almost) $\\pm A_\\text{max}$ .",
"To the left of the resonance, it enhances the external field ($+A_\\text{max}$ ) while the external field is effectively reduced ($-A_\\text{max}$ ) to the right of the resonance.",
"Note that an analogous direct explanation for the shift of the nuclear resonance in the right panel of Fig.",
"REF is not valid.",
"The computed polarization of the central spin points in $z$ -direction and thus does not shift the external field."
],
[
"Results on Convergence",
"In order to assess the speed of convergence of the initially disordered density matrix $\\rho _0=\\mathbb {1}/Z$ to the limiting density matrix $V_0$ we proceed as follows.",
"Let us assume that the matrices $v_i$ are the eigen matrices of $M$ and that they are normalized $||v_i||^2:=\\text{Tr}(v_i^\\dag v_i)=1$ .",
"Since the mapping $M$ is not unitary, orthogonality of the eigenmatrices cannot be assumed.",
"Note that the standard normalization generically implies that there is some factor between $V_0$ with $\\text{Tr}(V_0)=1$ and $v_0$ .",
"The initial density matrix $\\rho _0$ can be expanded in the $\\lbrace v_i\\rbrace $ $\\rho _0 = \\sum _{j=0}^{D-1} \\alpha _j v_j.$ After $n$ pulses, the density matrix $\\rho _n$ is given by $\\rho _n = \\sum _{j=0}^{D-1} \\alpha _j \\lambda _j^n v_j$ where $\\lambda _j$ are the corresponding eigenvalues of $M$ and $\\lambda _0=1$ by construction.",
"We aim at $\\rho _0$ being close to $V_0$ within $p_\\text{thresh}$ , i.e., $|| \\rho _n -V_0 ||\\le p_\\text{thresh} ||V_0||$ should hold for an appropriate $n$ .",
"A generic value of the threshold $p_\\text{thresh}$ is $1\\%$ .",
"To this end, the minimum $n$ which fulfills (REF ) has to be estimated.",
"Figure: Number of pulses for a convergence within 1%1\\% (p thresh =0.0p_\\text{thresh}=0.0)are plotted for various bath sizes; couplings given by (),other parameters as in Fig.",
".The corresponding residual entropies and magnetizationsare depicted in Fig.",
".",
"The vertical dashed lines indicatethe estimates () for the entropy minima as before.Such an estimate can be obtained by determining $n_j := 1+\\text{trunc}\\left[\\frac{\\ln (|p_\\text{thresh}\\alpha _0/\\alpha _j|)}{\\ln (|\\lambda _j|)}\\right]$ for $j\\in \\lbrace 1, 2,3, \\ldots ,D-1 \\rbrace $ .",
"The estimate of the required number of pulses is the maximum of these number, i.e., $n_\\text{puls} := \\max _{1\\le j < D} n_j.$ We checked exemplarily that the number determined in this way implies that the convergence condition (REF ) is fulfilled.",
"This is not mathematically rigorous because it could be that there are very many slowly decreasing contributions which add up to a significant deviation from $V_0$ .",
"But generically, this is not the case.",
"In Fig.",
"REF the results are shown for various bath sizes and the parameters for which the data of the previous figures was computed.",
"Since the entropy minima are located at the positions of the vertical dashed lines to good accuracy one can read off the required number of pulses at the intersections of the solid and the dashed lines.",
"Clearly, about $2\\cdot 10^{12}$ pulses are necessary to approach the limiting, relatively pure density matrices $V_0$ .",
"Interestingly, the number of required pulses does not depend much on the bath size, at least for the accessible bath sizes.",
"This is a positive message in view of the scaling towards larger baths in experimental setups.",
"Figure: Number of pulses for a convergence within 1%1\\% (p thresh =0.01p_\\text{thresh}=0.01)for N=5N=5, J max =0.02J Q J_\\text{max}=0.02J_\\text{Q}, and z=10 -3 z=10^{-3} for the exponential parametrizationin () (legend “expo”) and the Gaussian parametrizationin () (legend “gaus”).The corresponding residual entropies and magnetizationsare depicted in Figs.",
"and , respectively.The vertical dashed lines indicatethe estimates for the entropy minima which are shifted from theresonances without interactions according to ().Figure REF depicts the required minimum number of pulses for the two alternative parametrizations of the couplings (REF ) and (REF ).",
"Again, the range is about $3\\cdot 10^{12}$ .",
"Still, there are relevant differences.",
"The value $n_\\text{puls}$ is higher for $\\alpha =1$ ($\\approx 4\\cdot 10^{12}$ ) than for $\\alpha =1/2$ ($\\lessapprox 2\\cdot 10^{12}$ ).",
"This indicates that the mechanism of distilling quantum states by commensurability with periodic external pulses works best if the couplings $J_i$ are similar, i.e., if their spread given by $J_\\text{min}/J_\\text{max}=\\exp (-\\alpha )$ is small.",
"The same qualitative result is obtained for the residual entropy, see Appendix .",
"Note that this argument also explains why the Gaussian parametrized couplings (REF ) require slightly less pulses than the exponential parametrized couplings (REF ).",
"The couplings $J_i$ cumulate at their maximum $J_\\text{max}$ in the Gaussian case so that their variance is slightly smaller than the one of the exponential parametrization.",
"One could have thought that the cumulated couplings $J_i \\approx J_\\text{max}$ in the Gaussian case require longer pulsing in order to achieve a given degree of distillation because mathematically equal couplings $J_i=J_{i^{\\prime }}$ imply degeneracies preventing distillation, see the mathematical properties discussed in Sect.",
".",
"But this appears not to be the case.",
"Figure: Residual entropies (panel a) and number of pulses(panel b) for a convergence within 1%1\\% (p thresh =0.0p_\\text{thresh}=0.0)for N=3N=3, J max =0.1J Q J_\\text{max}=0.1J_\\text{Q}, and z=0.1z=0.1 for the equidistant parametrizationin () (legend “equidist”), the exponential parametrizationin () (legend “expo”) and the Gaussian parametrizationin () (legend “gaus”).The vertical dashed lines indicatethe estimates for the entropy minima which are shifted from theresonances without interactions according to ().The total numbers of pulses is rather high.",
"As many as $2\\cdot 10^{12}$ pulses for a repetition time ${T_\\mathrm {rep}}\\approx 10$ ns imply about six hours of pulsing.",
"This can be achieved in the lab, but the risk that so far neglected decoherence mechanisms spoil the process is real.",
"If, however, the pulses can be applied more frequently, for instance with ${T_\\mathrm {rep}}=1$ ns, the required duration shrinks to about 30 minutes.",
"The question arises why so many pulses are required.",
"While a comprehensive study of this aspect is beyond the scope of the present article, first clue can be given.",
"It suggests itself that the slow dynamics in the bath is responsible for the large number of pulses required for convergence.",
"This idea is corroborated by the results displayed in Fig.",
"REF where a larger maximum coupling and, importantly, a larger $z$ factor is assumed.",
"Recall that the $z$ -factor is the ratio of the Larmor frequency of the bath spins to the Larmor frequency of the central spin.",
"If it is increased, here by a factor of 100, the bath spins precess much quicker.",
"Indeed, the range of the required number of pulses is much lower with $2\\cdot 10^7$ which is five orders of magnitude less than for the previous parameters.",
"The former six hours then become fractions of seconds.",
"Of course, the conventional $g$ -factors of nuclear and electronic spins do not allow for $z=0.1$ .",
"But the central spin model as such, built by a central spin and a bath of spins supplemented by a damped excitation can also be realized in a different physical system.",
"Alternatively, optimized pulses can improve the efficiency of the distillation by periodic driving.",
"One may either consider modulated pulses of finite duration [30] or repeated cycles of several instantaneous pulses applied at optimized time instants [31], [32] or combinations of both schemes [33].",
"Thus, further research is called for.",
"The focus, however, of the present work is to establish the fundamental mechanism built upon periodic driving, dissipation and commensurability."
],
[
"Conclusion",
"Previous work has established dynamic nuclear polarization (DNP), for a review see Ref.",
"[34].",
"But it must be stressed that the mechanism of this conventional DNP is fundamentally different from the one described here.",
"Conventionally, the polarization of an electron is transferred to the nuclear spins, i.e., the polarization of the electrons induces polarization of the nuclei in the same direction.",
"In contrast, in the setup studied here, the electron is polarized in $z$ -direction while the nuclear spins are eventually polarized perpendicularly in $x$ -direction.",
"Hence, the mechanism is fundamentally different: it is NFF stemming essentially from commensurability.",
"This is also the distinguishing feature compared to standard optical pumping.",
"States in the initial mixture which do not allow for a time evolution commensurate with the repetition time ${T_\\mathrm {rep}}$ of the pulses are gradually suppressed while those whose time evolution is commensurate are enhanced.",
"This means that the weight of the former in the density matrix is reduced upon periodic application of the pulses while the weight of the latter is enhanced.",
"Note that the trace of the density matrix is conserved so that the suppression of the weight of some states implies that the weight of other states is increased.",
"The effect of the pulses on other norms of the density matrix is not obvious since the dynamics is not unitary, but dissipative.",
"For particular magnetic fields, there may be only one particular state allowing for a dynamics commensurate with ${T_\\mathrm {rep}}$ .",
"This case leads to the maximum entropy reduction.",
"Such a mechanism can be used also for completely different physical systems, e.g., in ensembles of oscillators.",
"The studied case of coupled spins extends the experimental and theoretical observations of NFF for large spin baths [10], [11], [12], [13], [14], [15], [16], [17] where many values of the polarization of the Overhauser field can lead to commensurate dynamics.",
"Hence, only a partial reduction of entropy occurred.",
"The above established DNP by NFF comprises the potential for a novel experimental technique for state preparation: laser pulses instead of microwave pulses as in standard NMR can be employed to prepare coherent states which can be used for further processing, either to perform certain quantum protocols or for analysis of the systems under study.",
"The combination of optical and radio frequency pulsing appears promising because it enlarges the possibilities of experimental manipulations.",
"Another interesting perspective is to employ the concept of state distillation by commensurability to physical systems other than localized spins, for instance to spin waves in quantum magnets.",
"A first experimental observations of commensurability effects for spin waves in ferromagnets are already carried out [35].",
"Studies on how to enhance the efficiency of the mechanism by optimization of the shape and distribution of the pulses constitute an interesting route for further research.",
"In summary, we showed that dissipative dynamics of a highly excited state is sufficient to modify the dynamics of energetically low-lying spin degrees of freedom away from unitarity.",
"The resulting dynamic map acts like a contraction converging towards a single density matrix upon iterated application.",
"The crucial additional ingredient is commensurability between the external periodic driving and the internal dynamic processes, for instance Larmor precessions.",
"If commensurability is possible a substantial entropy reduction can be induced, almost to a single pure state.",
"This has been explicitly shown for an exemplary small central spin model including electronic and nuclear Zeeman effect.",
"This model served as proof-of-principle model to establish the mechanism of distillation by commensurability.",
"Such a model describes the electronic spin in quantum dots with diluted nuclear spin bath or the spin of unpaired electrons in molecules, hyperfine coupled to nuclear hydrogen spins.",
"We stress that the mechanism of commensurability can also be put to use in other systems with periodic internal processes.",
"The fascinating potential to create and to manipulate coherent quantum states by such approaches deserves further investigation."
],
[
"Acknowledgements",
"The author thanks A. Greilich, J. Schnack, and O. P. Sushkov for useful discussions and the School of Physics of the University of New South Wales for its hospitality."
],
[
"Funding information",
"This work was supported by the Deutsche Forschungsgemeinschaft (DFG) and the Russian Foundation of Basic Research in TRR 160, by the DFG in project no.",
"UH 90-13/1, and by the Heinrich-Hertz Foundation of Northrhine-Westfalia."
],
[
"Derivation of the Linear Mapping",
"The goal is to solve the time evolution of $\\rho (t)$ from just before a pulse until just before the next pulse.",
"Since the pulse leads to a unitary time evolution which is linear $\\rho (n{T_\\mathrm {rep}}-)\\rightarrow \\rho (n{T_\\mathrm {rep}}+)=U_\\text{puls}\\rho (n{T_\\mathrm {rep}}-)U_\\text{puls}^\\dag $ with $U_\\text{puls}$ from (5) and the subsequent Lindblad dynamics defined by the linear differential equation (6) is linear as well the total propagation in time is given by a linear mapping $M: \\rho (n{T_\\mathrm {rep}}-) \\rightarrow \\rho ((n+1){T_\\mathrm {rep}}-)$ .",
"This mapping is derived here by an extension of the approach used in Ref.",
"[17].",
"The total density matrix acts on the Hilbert space given by the direct product of the Hilbert space of the central spin comprising three states (${|\\uparrow \\rangle },{|\\downarrow \\rangle }, {|\\text{T}\\rangle }$ ) and the Hilbert space of the spin bath.",
"We focus on $\\rho _\\text{TT}:={\\langle \\text{T}|}\\rho {|\\text{T}\\rangle }$ which is a $2^N\\times 2^N$ dimensional density matrix for the spin bath alone because the central degree of freedom is traced out.",
"By $\\rho _\\text{S}$ we denote the $d\\times d$ dimensional density matrix of the spin bath and the central spin, i.e., $d=2^{N+1}$ since no trion is present: $\\rho _\\text{S}{|\\text{T}\\rangle }=0$ .",
"The number of entries in the density matrix is $D=d^2$ , i.e., the mapping we are looking for can be represented by a $D\\times D$ matrix.",
"The time interval ${T_\\mathrm {rep}}$ between two consecutive pulses is sufficiently long so that all excited trions have decayed before the next pulse arrives.",
"In numbers, this means $2\\gamma {T_\\mathrm {rep}}\\gg 1$ and implies that $\\rho (n{T_\\mathrm {rep}}-)=\\rho _\\text{S}(n{T_\\mathrm {rep}}-)$ and hence inserting the unitary of the pulse (5) yields $\\rho (n{T_\\mathrm {rep}}+) &= U_\\text{puls}\\rho _\\text{S}(n{T_\\mathrm {rep}}-)U_\\text{puls}^\\dag \\\\\\rho _\\text{TT}(n{T_\\mathrm {rep}}+) &={\\langle \\uparrow |} \\rho _\\text{S}(n{T_\\mathrm {rep}}-) {|\\uparrow \\rangle }\\\\\\rho _\\text{S}(n{T_\\mathrm {rep}}+) &={|\\downarrow \\rangle }{\\langle \\downarrow |} \\rho _\\text{S}(n{T_\\mathrm {rep}}-){|\\downarrow \\rangle }{\\langle \\downarrow |} \\ =\\ S^-S^+ \\rho _\\text{S}(n{T_\\mathrm {rep}}-) S^-S^+$ where we used the standard ladder operators $S^\\pm $ of the central spin to express the projection ${|\\downarrow \\rangle }{\\langle \\downarrow |}$ .",
"The equations () set the initial values for the subsequent Lindbladian dynamics which we derive next.",
"For completeness, we point out that there are also non-diagonal contributions of the type ${\\langle \\text{T}|}\\rho {|\\uparrow \\rangle }$ , but they do not matter for $M$ .",
"Inserting $\\rho _\\text{TT}$ into the Lindblad equation (6) yields $\\partial _t \\rho _\\text{TT}(t) = -i [H_\\text{nZ},\\rho _\\text{TT}(t)] -2\\gamma \\rho _\\text{TT}(t).$ No other parts contribute.",
"The solution of (REF ) reads $\\rho _\\text{TT}(t) = e^{-2\\gamma t} e^{-iH_\\text{nZ}t} \\rho _\\text{TT}(0+)e^{iH_\\text{nZ}t}.$ By the argument $0+$ we denote that the initial density matrix for the Lindbladian dynamics is the one just after the pulse.",
"For $\\rho _\\text{S}$ , the Lindblad equation (6) implies $\\partial _t \\rho _\\text{S}(t) = -i[H_\\text{spin},\\rho _\\text{S}(t)]+2\\gamma {|\\uparrow \\rangle } \\rho _\\text{TT}(t) {\\langle \\uparrow |}.$ Since we know the last term already from its solution in (REF ) we can treat it as given inhomogeneity in the otherwise homogeneous differential equation.",
"With the definition $U_\\text{S}(t):= \\exp (-iH_\\text{spin}t)$ we can write $\\partial _t \\left(U_\\text{S}^\\dag (t) \\rho _\\text{S}(t) U_\\text{S}(t)\\right) = 2\\gamma U_\\text{S}^\\dag (t) {|\\uparrow \\rangle }\\rho _\\text{TT}(t){\\langle \\uparrow |}U_\\text{S}(t).$ Integration leads to the explicit solution $\\rho _\\text{S}(t) = U_\\text{S}(t) \\rho _\\text{S}(0+) U_\\text{S}^\\dag (t)+2\\gamma \\int _0^t U_\\text{S}^\\dag (t-t^{\\prime }) {|\\uparrow \\rangle }\\rho _\\text{TT}(t^{\\prime }){\\langle \\uparrow |}U_\\text{S}(t-t^{\\prime })dt^{\\prime }.$ If we insert (REF ) into the above equation we encounter the expression ${|\\uparrow \\rangle } \\exp (-iH_\\text{nZ} t) = \\exp (-iH_\\text{nZ} t){|\\uparrow \\rangle }\\ =\\ \\exp (-izh I^x_\\text{tot} t) \\exp (izh S^x t) {|\\uparrow \\rangle }.$ where $I^x_\\text{tot} :=S^x+\\sum _{i=1}^NI^x_i$ is the total momentum in $x$ -direction.",
"It is a conserved quantity commuting with $H_\\text{spin}$ so that a joint eigenbasis with eigenvalues $m_\\alpha $ and $E_\\alpha $ exists.",
"We determine such a basis $\\lbrace {|\\alpha \\rangle }\\rbrace $ by diagonalization in the $d$ -dimensional Hilbert space ($d=2^{N+1}$ ) of central spin and spin bath and convert (REF ) in terms of the matrix elements of the involved operators.",
"For brevity, we write $\\rho _{\\alpha \\beta }$ for the matrix elements of $\\rho _\\text{S}$ .",
"$\\rho _{\\alpha \\beta }(t) &= e^{-i(E_\\alpha -E_\\beta )t}\\Big \\lbrace \\rho _{\\alpha \\beta }(0+)\\nonumber \\\\&+2\\gamma \\int _0^t e^{i(E_\\alpha -E_\\beta -zh(m_\\alpha -m_\\beta ))t^{\\prime }}{\\langle \\alpha |} e^{izhS^xt^{\\prime }}{|\\uparrow \\rangle } \\rho _\\text{TT}(0+){\\langle \\uparrow |} e^{izhS^xt^{\\prime }}{|\\beta \\rangle }dt^{\\prime }\\Big \\rbrace .$ Elementary quantum mechanics tells us that $e^{izhS^xt^{\\prime }}{|\\uparrow \\rangle } =\\frac{1}{2}e^{ia}({|\\uparrow \\rangle }+{|\\downarrow \\rangle })+ \\frac{1}{2}e^{-ia}({|\\uparrow \\rangle }-{|\\downarrow \\rangle })$ with $a:=zht^{\\prime }/2$ which we need for the last row of equation (REF ).",
"Replacing $\\rho _\\text{TT}(0+)$ by ${\\langle \\uparrow |} \\rho _\\text{S}(n{T_\\mathrm {rep}}-) {|\\uparrow \\rangle }$ according to (REF ) and inserting (REF ) we obtain ${\\langle \\alpha |} e^{izhS^xt^{\\prime }}{|\\uparrow \\rangle } \\rho _\\text{TT}(0+){\\langle \\uparrow |} e^{izhS^xt^{\\prime }}{|\\beta \\rangle }& ={\\langle \\alpha |} e^{izhS^xt^{\\prime }}{|\\uparrow \\rangle } {\\langle \\uparrow |} \\rho _\\text{S}(0-) {|\\uparrow \\rangle }{\\langle \\uparrow |} e^{izhS^xt^{\\prime }}{|\\beta \\rangle }\\\\& = \\frac{1}{2} \\left( R^{(0)} + e^{izht^{\\prime }} R^{(1)} + e^{-izht^{\\prime }} R^{(-1)}\\right)_{\\alpha \\beta }$ with the three $d\\times d$ matrices $R^{(0)} &:= S^+S^- \\rho _\\text{S}(0-) S^+S^- + S^- \\rho _\\text{S}(0-) S^+\\\\R^{(1)} &:= \\frac{1}{2}(S^++\\mathbb {1}_d)S^- \\rho _\\text{S}(0-) S^+(S^--\\mathbb {1}_d)\\\\R^{(-1)} &:= \\frac{1}{2}(S^+-\\mathbb {1}_d)S^- \\rho _\\text{S}(0-) S^+(S^-+\\mathbb {1}_d).$ In this derivation, we expressed ket-bra combinations by the spin ladder operators according to ${|\\uparrow \\rangle } {\\langle \\uparrow |} = S^+S^- \\qquad {|\\uparrow \\rangle } {\\langle \\downarrow |} = S^+ \\qquad {|\\downarrow \\rangle } {\\langle \\uparrow |} = S^-.$ The final step consists in inserting (REF ) into (REF ) and integrating the exponential time dependence straightforwardly from 0 to ${T_\\mathrm {rep}}$ .",
"Since we assume that $2\\gamma {T_\\mathrm {rep}}\\gg 1$ so that no trions are present once the next pulse arrives the upper integration limit ${T_\\mathrm {rep}}$ can safely and consistently be replaced by $\\infty $ .",
"This makes the expressions $G_{\\alpha \\beta }(\\tau ) := \\frac{\\gamma }{2\\gamma -i[E_\\alpha -E_\\beta +zh(m_\\beta -m_\\alpha +\\tau )]}$ appear where $\\tau \\in \\lbrace -1,0,1\\rbrace $ .",
"Finally, we use () and summarize $\\rho _{\\alpha \\beta }(t) = e^{-i(E_\\alpha -E_\\beta )t}\\Big \\lbrace (S^-S^+ \\rho _\\text{S}(0-) S^-S^+ )_{\\alpha \\beta }+\\sum _{\\tau =-1}^1 G_{\\alpha \\beta }(\\tau ) R^{(\\tau )}_{\\alpha \\beta }\\Big \\rbrace .$ This provides the complete solution for the dynamics of $d\\times d$ matrix $\\rho _\\text{S}$ from just before a pulse ($t=0-$ ) till just before the next pulse for which we set $t={T_\\mathrm {rep}}$ in (REF ).",
"In order to set up the linear mapping $M$ as $D\\times D$ dimensional matrix with $D=d^2$ we denote the matrix elements $M_{\\mu ^{\\prime }\\mu }$ where $\\mu $ is a combined index for the index pair $\\alpha \\beta $ and $\\mu ^{\\prime }$ for $\\alpha ^{\\prime }\\beta ^{\\prime }$ with $\\alpha ,\\beta ,\\alpha ^{\\prime },\\beta ^{\\prime }\\in \\lbrace 1,2\\ldots ,d\\rbrace $ .",
"For brevity, we introduce $P_{\\alpha \\beta } := [(S^++\\mathbb {1}_d)S^-]_{\\alpha \\beta } \\qquad Q_{\\alpha \\beta } := [(S^+-\\mathbb {1}_d)S^-]_{\\alpha \\beta }.$ Then, (REF ) implies $M_{\\mu ^{\\prime }\\mu } &= \\frac{1}{2}e^{-i(E_{\\alpha ^{\\prime }}-E_{\\beta ^{\\prime }}){T_\\mathrm {rep}}}\\Big \\lbrace 2(S^-S^+)_{\\alpha ^{\\prime }\\alpha } (S^-S^+)_{\\beta \\beta ^{\\prime }}\\nonumber \\\\&+2G_{\\alpha ^{\\prime }\\beta ^{\\prime }}(0)\\left[(S^+S^-)_{\\alpha ^{\\prime }\\alpha } (S^+S^-)_{\\beta \\beta ^{\\prime }}+ S^-_{\\alpha ^{\\prime }\\alpha } S^+_{\\beta \\beta ^{\\prime }}\\right]\\nonumber \\\\&+\\left[G_{\\alpha ^{\\prime }\\beta ^{\\prime }}(1) P_{\\alpha ^{\\prime }\\alpha } Q^*_{\\beta ^{\\prime }\\beta }+ G_{\\alpha ^{\\prime }\\beta ^{\\prime }}(-1) Q_{\\alpha ^{\\prime }\\alpha } P^*_{\\beta ^{\\prime }\\beta }\\right]\\Big \\rbrace .$ This concludes the explicit derivation of the matrix elements of $M$ .",
"Note that they are relatively simple in the sense that no sums over matrix indices are required on the right hand side of (REF ).",
"This relative simplicity is achieved because we chose to work in the eigenbasis of $H_\\text{spin}$ .",
"Other choices of basis are possible, but render the explicit respresentation significantly more complicated."
],
[
"Preliminaries",
"Here we state several mathematical properties of the mapping $M$ which hold for any Lindblad dynamics over a given time interval which can be iterated arbitrarily many times.",
"We assume that the underlying Hilbert space is $d$ dimensional so that $M$ acts on the $D=d^2$ dimensional Hilbert space of $d\\times d$ matrices, i.e., $M$ can be seen as $D\\times D$ matrix.",
"We denote the standard scalar product in the space of operators by $(A|B):=\\text{Tr}(A^\\dag B)$ where the trace refers to the $d\\times d$ matrices $A$ and $B$ .",
"Since no state of the physical system vanishes in its temporal evolution $M$ conserves the trace of any density matrix $\\text{Tr}(M\\rho )=\\text{Tr}(\\rho ).$ This implies that $M$ conserves the trace of any operator $C$ .",
"This can be seen by writing $C=(C+C^\\dag )/2+ (C-C^\\dag )/2=R+iG$ where $R$ and $G$ are hermitian operators.",
"They can be diagonalized and split into their positive and their negative part $R=p_1-p_2$ and $G=p_3-p_4$ .",
"Hence, each $p_i$ is a density matrix up to some real, positive scaling and we have $C = p_1-p_2+i(p_3-p_4).$ Then we conclude $\\text{Tr}(MC) &= \\text{Tr}(Mp_1)-\\text{Tr}(Mp_2)+ i(\\text{Tr}(Mp_3)-\\text{Tr}(Mp_4))\\\\& = \\text{Tr}(p_1)-\\text{Tr}(p_2)+i(\\text{Tr}(p_3)-\\text{Tr}(p_4))\\ =\\ \\text{Tr}(C).$"
],
[
"Property 1.",
"The conservation of the trace for any $C$ implies $\\text{Tr}(C)=(\\mathbb {1}_d|C)=(\\mathbb {1}_d|MC)=(M^\\dag \\mathbb {1}_d|C)$ where $\\mathbb {1}_d$ is the $d\\times d$ -dimensional identity matrix and $M^\\dag $ is the $D\\times D$ hermitian conjugate of $M$ .",
"From (REF ) we conclude $M^\\dag \\mathbb {1}_d = \\mathbb {1}_d$ which means that $\\mathbb {1}_d$ is an eigenoperator of $M^\\dag $ with eigenvalue 1.",
"Since the characteristic polynomial of $M$ is the same as the one of $M^\\dag $ up to complex conjugation we immediately see that 1 is also an eigenvalue of $M$ .",
"If the dynamics of the system takes place in $n$ independent subspaces without transitions between them, the $n$ different traces over these subspaces are conserved separately Such a separation occurs in case conserved symmetries split the Hilbert space, for instance the total spin is conserved in the dynamics given by (6) if all couplings are equal.",
"Then, the above argument implies the existence of $n$ different eigenoperators with eigenvalue 1.",
"Hence the degeneracy is (at least) $n$ which proves property 1. in the main text."
],
[
"Properties 2. and 3.",
"As for property 2, we consider an eigenoperator $C$ of $M$ with eigenvalue $\\lambda \\ne 1$ so that $MC=\\lambda C$ .",
"Then $\\text{Tr}(C) = \\text{Tr}(MC) \\ = \\ \\lambda \\text{Tr}(C)$ implies $\\text{Tr}(C)=0$ , i.e., tracelessness as stated.",
"Since all density matrices can be written as linear combinations of eigenoperators there must be at least one eigenoperator with finite trace.",
"In view of property 2., this needs to be an eigenoperator with eigenvalue 1 proving property 3.",
"The latter conclusion holds true if we assume that $M$ cannot be diagonalized, but only has a Jordan normal form.",
"If $d_\\text{J}$ is the dimension of the largest Jordan block, the density matrix $M^{d_\\text{J}-1}\\rho $ will be a linear combination of eigenoperators while still having the trace 1."
],
[
"Property 4.",
"Next, we show that no eigenvalue $\\lambda $ can be larger than 1 in absolute value.",
"The idea of the derivation is that the iterated application of $M$ to the eigenoperator belonging to $|\\lambda |>1$ would make this term grow exponentially $\\propto |\\lambda |^n$ beyond any bound which cannot be true.",
"The formal proof is a bit intricate.",
"First, we state that for any two density matrices $\\rho $ and $\\rho ^{\\prime }$ their scalar product is non-negative $0\\le (\\rho |\\rho ^{\\prime })$ because it can be viewed as expectation value of one of them with respect to the other and both are positive operators.",
"In addition, the Cauchy-Schwarz inequality implies $0 \\le (\\rho |\\rho ^{\\prime }) \\le \\sqrt{(\\rho |\\rho )(\\rho ^{\\prime }|\\rho ^{\\prime })}\\ =\\ \\sqrt{\\text{Tr}(\\rho ^2)\\text{Tr}((\\rho ^{\\prime })^2)} \\ \\le \\ 1.$ Let $C$ be the eigenoperator of $M^\\dag $ belonging to $\\lambda $ ; it may be represented as in (REF ) and scaled such that the maximum of the traces of the $p_i$ is 1.",
"Without loss of generality this is the case for $p_1$ , i.e., $\\text{Tr}(p_1)=1$ .",
"Otherwise, $C$ is simply rescaled: by $C\\rightarrow -C$ to switch $p_2$ to $p_1$ , by $C\\rightarrow -iC$ to switch $p_3$ to $p_1$ , or by $C\\rightarrow iC$ to switch $p_4$ to $p_1$ .",
"On the one hand, we have for any density matrix $\\rho _n$ $|(C|\\rho _n)| \\le |\\Re (C|\\rho _n)| + |\\Im (C|\\rho _n)| \\le 2$ where the last inequality results form (REF ).",
"On the other hand, we set $\\rho _n:= M^np_1$ and obtain $2 &\\ge |(C|\\rho _n)|= |((M^\\dag )^nC|p_1)|= |\\lambda ^*|^n |(C|p_1)|=|\\lambda |^n \\sqrt{(\\Re (C|p_1))^2+ (\\Im (C|p_1))^2}\\\\&\\ge |\\lambda |^n|\\Re (C|p_1)|= |\\lambda |^n(p_1|p_1)$ where we used $(p_1|p_2)=0$ in the last step; this holds because $p_1$ and $p_2$ result from the same diagonalization, but refer to eigenspaces with eigenvalues of different sign.",
"In essence we derived $2 \\ge |\\lambda |^n(p_1|p_1)$ which clearly implies a contradiction for $n \\rightarrow \\infty $ because the right hand side increases to infinity for $|\\lambda |>1$ .",
"Hence there cannot be eigenvalues with modulus larger than 1."
],
[
"Property 5.",
"The matrix $M$ can be represented with respect to a basis of the Krylov space spanned by the operators $\\rho _n:=M^n\\rho _0$ where $\\rho _0$ is an arbitrary initial density matrix which should contain contributions from all eigenspaces of $M$ .",
"For instance, a Gram-Schmidt algorithm applied to the Krylov basis generates an orthonormal basis $\\tilde{\\rho }_n$ .",
"Due to the fact, that all the operators $\\rho _n$ from (REF ) are hermitian density matrices $\\tilde{\\rho }_n = \\tilde{\\rho }_n^\\dag $ , we know that all overlaps $(\\rho _m|\\rho _n)$ are real and hence the constructed orthonormal basis $\\tilde{\\rho }_n$ consists of hermitian operators.",
"Also, all matrix elements $(\\rho _m|M\\rho _n)=(\\rho _m|\\rho _{n+1})$ are real so that the resulting representation $\\tilde{M}$ is a matrix with real coefficients whence $\\tilde{M} c = \\lambda c$ implies $\\tilde{M} c^* = \\lambda ^* c^*$ by complex conjugation.",
"Here $c$ is a vector of complex numbers $c_n$ which define the corresponding eigenoperators by $C = \\sum _{n=1}^D c_n \\tilde{\\rho }_n.$ Thus, $c$ and $c^*$ define $C$ and $C^\\dag $ , respectively."
],
[
"Property 6.",
"In view of the real representation $\\tilde{M}$ of $M$ with respect to an orthonormal basis of hermitian operators derived in the previous paragraph the determination of the eigenoperators with eigenvalue 1 requires the computation of the kernel of $\\tilde{M}-\\mathbb {1}_D$ .",
"This is a linear algebra problem in $\\mathbb {R}^D$ with real solutions which correspond to hermitian operators by means of (REF ).",
"This shows the stated property 6..",
"Figure: (a) Residual entropy as function of the appliedmagnetic field for N=3,J max =0.02N=3, J_\\text{max}=0.02, and z=1/1000z=1/1000 to show the positionat h=2π/(zT rep )h=2\\pi /(z{T_\\mathrm {rep}}) and the shift, dashed line at ≈500J Q J max /(2z)\\approx 500J_\\text{Q}J_\\text{max}/(2z)of the nuclear magnetic resonance.",
"(b) Same as (a) for z=1/500z=1/500.",
"(c) Same as (a) for z=1/250z=1/250."
],
[
"Shift of the Nuclear Resonance",
"In the main text, the shift of the nuclear resonance due to the coupling of the nuclear spins to the central, electronic spin was shown in the right panel of Fig.",
"1(a).",
"The effect can be estimated by $z\\Delta h \\approx \\pm J_\\text{max}/2.$ This relation is highly plausible, but it cannot be derived analytically because no indication for a polarization of the central, electronic spin in $x$ -direction was found.",
"Yet, the numerical data corroborates the validity of (REF ).",
"In Fig.",
"REF , we show that the nuclear resonance without shift occurs for $zh{T_\\mathrm {rep}}=2\\pi n^{\\prime }$ where $n^{\\prime }\\in \\mathbb {Z}$ .",
"But it is obvious that an additional shift occurs which is indeed captured by (REF ).",
"In order to support (REF ) further, we also study various values of $J_\\text{max}$ in Fig.",
"REF .",
"The estimate (REF ) captures the main trend of the data, but it is not completely quantitative because the position of the dashed lines relative to the minimum of the envelope of the resonances varies slightly for different values of $J_\\text{max}$ .",
"Hence, a more quantitative explanation is still called for."
],
[
"Entropy Reduction for Other Distributions of Couplings",
"In the main text, we analyzed a uniform distribution of couplings, see Eq.",
"(8).",
"In order to underline that our results are generic and not linked to a special distribution, we provide additional results for two distributions which are often considered in literature, namely an exponential parameterization as defined in (REF ) and a Gaussian parametrization as defined in (REF ).",
"Figure: Residual entropy as function of the appliedmagnetic field for various bath sizes NN forthe exponentially distributed couplings given by (); panel (a) forα=1\\alpha =1 and panel (b) for α=0.5\\alpha =0.5 and hence smaller ratioJ min /J max J_\\text{min}/J_\\text{max}.The key difference between both parametrizations (REF ) and (REF ) is that due to the quadratic argument in (REF ) the large couplings in this parametrization are very close to each other, in particular for increasing $N$ .",
"Hence, one can study whether this feature is favorable of unfavorable for entropy reduction.",
"Figure: Residual entropy as function of the appliedmagnetic field for various bath sizes NN forthe Gaussian distributed couplings given by (); panel (a) forα=1\\alpha =1 and panel (b) for α=0.5\\alpha =0.5 and hence smaller ratioJ min /J max J_\\text{min}/J_\\text{max}.Additionally, the difference between $\\alpha =0.5$ and $\\alpha =1$ consists in a different spread of the couplings.",
"For $\\alpha =1$ , one has $J_\\text{min}/J_\\text{max}=1/e$ in both parametrizations while one has $J_\\text{min}/J_\\text{max}=1/\\sqrt{e}$ for $\\alpha =0.5$ , i.e., the spread is smaller.",
"Figure REF displays the results for the exponential parametrization (REF ) while Fig.",
"REF depicts the results for the Gaussian parametrization (REF ).",
"Comparing both figures shows that the precise distribution of the couplings does not matter much.",
"Exponential and Gaussian parametrization lead to very similar results.",
"They also strongly ressemble the results shown in Fig.",
"2a in the main text for a uniform distribution of couplings.",
"This is quite remarkable since the Gaussian parametrization leads to couplings which are very close to each other and to the maximum coupling.",
"This effect does not appear to influence the achievable entropy reduction.",
"The ratio $J_\\text{min}/J_\\text{max}$ between the smallest to the largest coupling appears to have an impact.",
"If it is closer to unity, here for $\\alpha =0.5$ , the reduction of entropy works even better than for smaller ratios."
]
] | 1906.04283 |
[
[
"Artificial Noisy MIMO Systems under Correlated Scattering Rayleigh\n Fading -- A Physical Layer Security Approach"
],
[
"Abstract The existing investigations on artificial noise (AN) security systems assumed that only null spaces is used to send AN signals, and all eigen-subchannels should be used to transmit messages.",
"Our previous work proposed an AN scheme that allocates some of eigen-subchannels to transmit AN signals for improving secrecy rates.",
"Nevertheless, our previous work considered only uncorrelated MIMO Rayleigh fading channels.",
"In fact, the correlations among antennas exist in realistic scattering channel environments.",
"In this paper, we extend our previous AN scheme to spatially correlated Rayleigh fading channels at both legitimate receiver- and eavesdropper-sides and derive an exact theoretical expression for the ergodic secrecy rate of the AN scheme, along with an approximate analysis.",
"Both numerical and simulation results show that the proposed AN scheme offers a higher ergodic secrecy rate than the existing schemes, revealing a fact that the correlation among eavesdropper's antennas can potentially improve the secrecy rate of an MIMO system."
],
[
"INTRODUCTION",
"Physical layer security has attracted a lot of attention due to its potential to offer low-cost and high-level security in wireless communications[1], [2], [3], [4], [5].",
"The idea of utilizing artificial noise (AN) or jamming signals, as a physical layer security scheme, was proposed for the first time in Negi and Goel's work [6], [7].",
"Recently, AN schemes have been extended to different channels to safeguard sensitive and confidential data [8], [9], [10], [11].",
"For instance, [8], [9], [10] considered Rayleigh MIMO channels, whereas [11] assumed Rician MIMO channels.",
"The basic idea of the aforementioned AN schemes is that message streams are sent in a multiplex mode via all eigen-subchannels (positive eigenvalue channels) at desired directions, and the AN signals are transmitted to a null space of desired directions, such that they do not interfere desired users but only impair eavesdropped channels.",
"However, these AN schemes use a null space for AN signals only under the condition that the number of transmit antennas is larger than that of receivers [6], [7], [12], [8], [9], [10], [11], [13].",
"In addition, using all eigen-subchannels in an MIMO system for message transmission may degrade secrecy rate if compared to the schemes, which properly allocate some of eigen-subchannels for AN signals.",
"In our previous work [14], we took the number of eigen-subchannels of message streams as a variable that can be leveraged to maximize ergodic secrecy rate and showed that, when the number of transmit antennas is smaller than that of receivers, it is possible to find eigen-subchannels used by AN signals in order not to interfere desired users with the help of AN elimination technique at the desired users.",
"The work in [14] was done based on a Rayleigh fading channel, as Rayleigh fading is a reasonable model for heavily built-up urban environments [12], which has been extended to uncorrelated Rician fading channels via a non-central Wishart matrix in [15].",
"Zheng et al.",
"also used eigen-subchannels for AN signal transmission, but treated the AN as interference signals due to the lack of proper AN elimination techniques [16].",
"It is noted that all of the aforementioned schemes assumed the presence of uncorrelated fading in MIMO channels.",
"Unfortunately, in many real applications, the correlation among antennas may exist due to poor-scattering environments or small spacing between antenna elements [17], [18], [19].",
"It motivates us to design a better AN scheme to suit for correlated fading environments.",
"Recently, Li [20] investigated secure transmissions in an MISO-based system with receiver-side correlation in satellite-terrestrial channels.",
"The effect of double-side correlation in the main and wiretap channels of MIMO systems was studied via Monte Carlo simulations in [21] and [22].",
"All of the above investigations showed that the correlation has its impacts on security performance.",
"However, the effect of receiver-side correlation of MIMO-aided AN systems has not been fully investigated so far, and an exact expression for ergodic secrecy rate of AN schemes is far more useful than Monte Carlo simulation results because it provides us an objective function to disclose the relationship between secrecy rate and channel correlation.",
"This paper focuses on receiver-side correlated fading scenarios at both legitimate receiver- and eavesdropper-sides.",
"As shown in a report on the downlink channel correlation by 3GPP [17], transmitters are located at base stations with enough space to deploy multiple antennas, and the size of a receiver (e.g., a mobile terminal) is usually small.",
"Thus, receiver-side correlation more likely occurs than transmitter-side correlation in downlink channels.",
"In addition, in 5G and beyond systems, the receivers, such as vehicles and unmanned aerial vehicles (UAVs), may move to an appropriate location for secrecy transmission, whose channel correlation parameters may change from time to time based on statistical channel information [23], such as mean angles of arrival (AoA) and receive angle spread (RAS).",
"Some devices with a very small antenna separation distance (such as massive MIMO) will emerge for secure communications in the future.",
"The main contributions of this work can be summarized as follows.",
"We extend the AN scheme [14] to receiver-side correlated MIMO channels, and derive an exact expression for ergodic secrecy rates.",
"To the best of our knowledge, this is the first time to give such an exact expression in terms of spatial correlation parameters (i.e., mean AoA, RAS, and antenna spacing, etc.).",
"A suitable number of eigen-subchannels for messages and AN can be easily identified based on the derived ergodic secrecy rate expression.",
"Then, we simplify the expression and give its approximate analysis.",
"In addition, we derive an exact closed-form expression for marginal probability density function (pdf) of the $k$ th eigenvalue of receiver-side correlated Wishart matrices.",
"The work in [24] required two expressions to formulate this function.",
"We need only one expression as a more generalized form.",
"We identify the properties of the correlated matrices in terms of spatial correlation parameters.",
"The mathematical investigations given in the paper are general, which can also be used for analyzing ergodic secrecy rates of an AN scheme and channel capacities of traditional MIMO systems.",
"The remainder of this paper can be outlined as follows.",
"Section II introduces the system model and AN scheme.",
"Section III aims to derive an exact mathematical expression for ergodic secrecy rates, along with an approximate analysis.",
"Section IV is dedicated for numerical analysis and simulations, followed by the conclusions in Section V. The notations are explained as follows.",
"Bold uppercase letters denote matrices and bold lowercase letters denote column vectors.",
"$\\mathbf {A}^{\\dagger }$ represents the Hermitian transpose of $\\mathbf {A}$ .",
"$\\mathbf {I}_a$ is an identity matrix with its rank $a$ .",
"$\\mathbf {S}_a$ denotes an $(a \\times a)$ square matrix with its order $a$ .",
"E$[\\cdot ]$ denotes the expectation operator.",
"$[\\mathbf {A}]_{i,j}$ gives the $i$ th row and the $j$ th column element of $\\mathbf {A}$ .",
"$[\\mathbf {A}]_{(i\\sim u),(j\\sim v)}$ is a submatrix of $\\mathbf {A}$ , including the $i$ th to the $u$ th rows and the $j$ th to the $v$ th columns of $\\mathbf {A}$ .",
"$\\exp (x)$ denotes an exponential function of $x$ .",
"$\\det [\\mathbf {A}]$ is the determinant of $\\mathbf {A}$ .",
"$\\text{etr}(\\mathbf {X})$ denotes $\\exp [\\text{Tr}(\\mathbf {X})]$ , where $\\text{Tr}(\\mathbf {X})$ is the trace of $\\mathbf {X}$ .",
"$\\otimes $ stands for a Kronecker product.",
"An $[a\\times (b+c)]$ matrix $[\\mathbf {A},\\mathbf {B}]$ denotes a combined matrix between an $(a\\times b)$ matrix $\\mathbf {A}$ and an $(a\\times c)$ matrix $\\mathbf {B}$ .",
"$(\\mathbf {A})^{1/2}$ represents matrix square root operation such that $\\mathbf {A}^{1/2}(\\mathbf {A}^{1/2})^{\\dagger }=\\mathbf {A}$ .",
"$\\binom{x}{y}$ is the combination between $x$ and $y$ such that $\\binom{x}{y}=\\frac{x!}{(x-y)!y!",
"}$ .",
"Figure: Illustration of an artificial noisy MIMO wiretap channel model, where Alice, Bob, and Eve use uniformly linear array antennas, and θ\\theta is AoA between a scattered path and the antenna array."
],
[
"SYSTEM MODEL",
"In this section, we introduce a system model that specifies a spatial correlation channel, as well as the AN scheme."
],
[
"MIMO Wiretap Channel with Spatial Correlation",
"Let us consider an MIMO communication system in the presence of correlated Rayleigh fading at both legitimate receiver- and the eavesdropper-sides.",
"The system consists of a transmitter (Alice) with $t$ transmit antennas, a legitimate receiver (Bob) with $r$ receive antennas, and an eavesdropper (Eve) with $e$ receive antennas, as shown in Fig.",
"REF , where $t>e$ and $r$ is arbitrary.",
"In general, the main channel between Alice and Bob and the wiretap channel between Alice and Eve are defined by receiver-side correlated complex Gaussian matrices $\\mathbf {H}\\in \\mathbb {C}^{r\\times t}$ and $\\mathbf {H}_e\\in \\mathbb {C}^{e\\times t}$ , as given in Definition 1.",
"$\\mathbf {R}_r\\in \\mathbb {C}^{r\\times r}$ and $\\mathbf {R}_e\\in \\mathbb {C}^{e\\times e}$ are the receiver-side correlated channel matrices of Bob and Eve, respectively, as given in Definition 2.",
"Definition 1 (Central complex Gaussian matrix): Each element of a random matrix $\\mathbf {A}\\in \\mathbb {C}^{a\\times b}$ takes a complex value, whose real and imaginary parts follow a normal distribution $\\mathcal {N}(0, 1/2$ ).",
"$\\mathbf {A}$ is defined as a central complex Gaussian matrix with a covariance matrix $\\mathbf {\\Phi }_a\\otimes \\mathbf {\\Psi }_b$ , which is expressed as $\\mathbf {A}\\sim \\mathcal {CN}_{a,b}(\\mathbf {0},\\mathbf {\\Phi }_a\\otimes \\mathbf {\\Psi }_b),$ where $\\mathbf {\\Psi }_b=\\text{E}[\\mathbf {a}_{i,(1\\sim b)}\\mathbf {a}_{i,(1\\sim b)}^{\\dagger }]$ for $i=1, ..., a$ , and $\\mathbf {\\Phi }_a=\\text{E}[\\mathbf {a}_{(1\\sim a),j}\\mathbf {a}_{(1\\sim a),j}^{\\dagger }]$ for $j=1, ..., b$ .",
"The $(a\\times a)$ matrix $\\mathbf {\\Phi }_a$ and $(b\\times b)$ matrix $\\mathbf {\\Psi }_b$ are the Hermitian positive definite matrices.",
"A similar definition of this complex Gaussian matrix can be found in [24], [25].",
"In order to investigate $\\mathbf {H}$ and $\\mathbf {H}_e$ in the model, let us use a Kronecker model to define $\\mathbf {H}=\\mathbf {R}_r^{1/2}\\mathbf {H}_{\\text{Bob}} \\sim \\mathcal {CN}_{r,t}(\\mathbf {0},\\mathbf {R}_r\\otimes \\mathbf {I}_t),$ $\\mathbf {H}_e=\\mathbf {R}_e^{1/2}\\mathbf {H}_{\\text{Eve}} \\sim \\mathcal {CN}_{e,t}(\\mathbf {0},\\mathbf {R}_e\\otimes \\mathbf {I}_t),$ where $\\mathbf {H}_{\\text{Bob}}\\in \\mathbb {C}^{r\\times t}$ and $\\mathbf {H}_{\\text{Eve}}\\in \\mathbb {C}^{e\\times t}$ are complex Gaussian random matrices with independent complex Gaussian elements.",
"Similar to $\\mathbf {A}$ , the real and imaginary parts of each element of $\\mathbf {H}_{\\text{Bob}}$ and $\\mathbf {H}_{\\text{Eve}}$ follow a normal distribution $\\mathcal {N}(0, 1/2$ ).",
"$\\mathbf {H}_{\\text{Bob}}$ and $\\mathbf {H}_{\\text{Eve}}$ can be expressed respectively as $&&\\mathbf {H}_{\\text{Bob}}\\sim \\mathcal {CN}_{r,t}(\\mathbf {0},\\mathbf {I}_r \\otimes \\mathbf {I}_t),\\\\&&\\mathbf {H}_{\\text{Eve}}\\sim \\mathcal {CN}_{e,t}(\\mathbf {0},\\mathbf {I}_e \\otimes \\mathbf {I}_t).$ The correlated matrices $\\mathbf {R}_r$ and $\\mathbf {R}_e$ are the key factors in deriving channel state information (CSI) matrices.",
"From [26], [19], we know that a correlated matrix $\\mathbf {R}_a$ (a generalized version of $\\mathbf {R}_r$ and $\\mathbf {R}_e$ ) is a function of AoA distribution (defined by $\\theta $ ), as given in Definition 2, which is a way to generate a receiver-side correlated matrix.",
"Definition 2 (Receiver-side correlated matrix): Assume that all antennas form a uniformly linear antenna array with $d=d_{\\min }/\\omega $ , where $d$ is the normalized minimum distance, $d_{\\min }$ is the spacing between any two neighbor antennas, and $\\omega $ is the wavelength.",
"Each element of a receiver-side correlated matrix $\\mathbf {R}_a$ , i.e., $[\\mathbf {R}_a]_{u,v}$ is $[\\mathbf {R}_a]_{u,v}&=\\exp \\big \\lbrace -j2\\pi d(u-v)\\cos \\bar{\\theta }\\big \\rbrace \\\\ &\\times \\exp \\big \\lbrace -\\frac{1}{2}\\big [2\\pi d \\delta (u-v)\\sin \\bar{\\theta }\\big ]^2\\big \\rbrace , $ where $u\\in \\lbrace 1,...,a\\rbrace $ and $v\\in \\lbrace 1,...,a\\rbrace $ are the receive antenna index numbers.",
"For Bob, we have $a=r$ , and for Eve, we have $a=e$ .",
"The AoA, i.e., $\\theta $ , follows a Gaussian distribution, where the mean AoA of $\\theta $ is $\\bar{\\theta }$ and the RAS (variance) of $\\theta $ is $\\delta $ .",
"A similar definition of this receiver-side correlated matrix can be found in [19] and [26].",
"Based on the calculations of Eqn.",
"(REF ), we can see that when $u=v$ , $\\big |[\\mathbf {R}_a]_{u,v}\\big |$ equals to one.",
"When $u\\ne v$ , $\\big |[\\mathbf {R}_a]_{u,v}\\big |$ approaches to zero with an increasing $d$ , $\\bar{\\theta }$ , or $\\delta $ .",
"Hence, $\\mathbf {R}_a$ approaches to $\\mathbf {I}_a$ with an increasing $d$ , $\\bar{\\theta }$ , or $\\delta $ .",
"This means that the correlation will be reduced with an increasing $d$ , $\\bar{\\theta }$ , or $\\delta $ .",
"Let us use Theorem 1 to specify the properties of the correlated matrix $\\mathbf {R}_a$ , which is useful for the approximate analysis of ergodic secrecy rates in the next section.",
"Theorem 1: Let $\\mathbf {R}_a(d)$ , $\\mathbf {R}_a(\\bar{\\theta })$ , and $\\mathbf {R}_a(\\delta )$ be the functions of $d$ , $\\bar{\\theta }$ , and $\\delta $ , respectively, as given in Definition 2.",
"The largest eigenvalue of $\\mathbf {R}_a$ is defined as $\\sigma _1(\\mathbf {R}_a)$ , and the determinant of $\\mathbf {R}_a$ is defined as $\\det [\\mathbf {R}_a]$ .",
"Then, we get the conclusions as follows.",
"If $d_1>d_2$ , we have $\\sigma _1\\big [\\mathbf {R}_a(d_1)\\big ]<\\sigma _1\\big [\\mathbf {R}_a(d_2)\\big ]$ and $\\det [\\mathbf {R}_a(d_1)]>\\det [\\mathbf {R}_a(d_2)]$ .",
"If $\\bar{\\theta }_1>\\bar{\\theta }_2$ , we have $\\sigma _1\\big [\\mathbf {R}_a(\\bar{\\theta }_1)\\big ]<\\sigma _1\\big [\\mathbf {R}_a(\\bar{\\theta }_2)\\big ]$ and $\\det [\\mathbf {R}_a(\\bar{\\theta }_1)]>\\det [\\mathbf {R}_a(\\bar{\\theta }_2)]$ .",
"If $\\delta _1>\\delta _2$ , we have $\\sigma _1\\big [\\mathbf {R}_a(\\delta _1)\\big ]<\\sigma _1\\big [\\mathbf {R}_a(\\delta _2)\\big ]$ and $\\det [\\mathbf {R}_a(\\delta _1)]>\\det [\\mathbf {R}_a(\\delta _2)]$ .",
"Proof: See Appendix A.",
"Let us define the mean AoAs at Bob and Eve as $\\bar{\\theta }_{\\text{Bob}}$ and $\\bar{\\theta }_{\\text{Eve}}$ , respectively, define RASs at Bob and Eve as $\\delta _{\\text{Bob}}$ and $\\delta _{\\text{Eve}}$ , respectively, and define the normalized distances at Bob and Eve as $d_{\\text{Bob}}$ and $d_{\\text{Eve}}$ , respectively."
],
[
"Artificial Noise Precoding",
"In this paper, we use the AN scheme as proposed in [14].",
"There are $s_1$ eigen-subchannels for sending confidential messages selected by Alice based on CSI feedback from Bob.",
"$s_1$ is a variable that can be adjusted by Alice.",
"More specifically, Alice performs the eigenvalue decomposition (eig) of $\\mathbf {H}^{\\dagger }\\mathbf {H}$ , which outputs two unitary matrices, i.e., $\\mathbf {U}\\in \\mathbb {C}^{t\\times t}$ and its Hermitian transpose $\\mathbf {U}^{\\dagger }\\in \\mathbb {C}^{t\\times t}$ .",
"The eig process also outputs a diagonal matrix $\\mathbf {\\Lambda }\\in \\mathbb {R}^{t\\times t}$ , which consists of the positive and zero eigenvalues of $\\mathbf {H}^{\\dagger }\\mathbf {H}$ , i.e., $(\\lambda _1,...,\\lambda _t)$ , where the positive eigenvalues are defined as $\\lambda _1>...>\\lambda _n$ , where $n=\\min (t,r)$ .",
"Alice generates a message precoding matrix $\\mathbf {B} \\in \\mathbb {C}^{t \\times s_1}$ , whose columns are the eigenvectors corresponding to the first to the $s_1$ th largest eigenvalues of $\\mathbf {H}^{\\dagger }\\mathbf {H}$ , and an AN precoding matrix $\\mathbf {Z} \\in \\mathbb {C}^{ t\\times s_2}$ $(s_1+s_2=t)$ , whose columns are the eigenvectors of the remaining eigenvalues of $\\mathbf {H}^{\\dagger }\\mathbf {H}$ .",
"Remark 1: (Proved in [14]): We can readily show $[\\mathbf {H}\\mathbf {B}]^{\\dagger }\\mathbf {HZ}=\\mathbf {0}$ , $[\\mathbf {H}\\mathbf {B}]^{\\dagger }\\mathbf {H}_e\\mathbf {Z}\\ne \\mathbf {0}$ , and $[\\mathbf {H}_e\\mathbf {B}]^{\\dagger }\\mathbf {H}_e\\mathbf {Z}\\ne \\mathbf {0}$ .",
"As the CSI is extremely important in this work, we would like to discuss about the CSI at Alice and Eve as follows.",
"CSI at Alice: As mentioned earlier, let us consider a slow-fading environment that Alice knows full CSI of Bob, including $\\mathbf {H}$ and $\\mathbf {R}_r$ , via a unprotected broadcast feedback channel from Bob due to FDD or non-reciprocal TDD systems [27], but knows only $\\mathbf {R}_e$ and the channel distribution information (CDI) of Eve.",
"Alice can get the knowledge of $\\mathbf {R}_e$ and the CDI of Eve, because Eve can be just a normal receiver in the same communication system with Alice, and may exchange messages without security protection.",
"Hence, Alice can obtain $\\mathbf {R}_e$ via historical CSI of $\\mathbf {H}_e$ , i.e., $\\mathbf {R}_e=\\text{E}(\\mathbf {H}_e\\mathbf {H}_e^{\\dagger }/t)$ or statistical AoA information as shown in Definition 2.",
"Otherwise, Alice should assume that there is no correlation at Eve side, i.e., $\\mathbf {R}_e=\\mathbf {I}_e$ , which is the worst assumption because $\\mathbf {R}_e=\\mathbf {I}_e$ will maximize the ergodic wiretap channel capacity among all realizations of $\\mathbf {R}_e$ [18].",
"CSI at Eve: Let us consider a pessimistic scenario that Eve knows the CSI of all channels, includes $\\mathbf {H}$ , $\\mathbf {H}_e$ , $\\mathbf {R}_r$ , and $\\mathbf {R}_e$ .",
"This scenario usually exists in feedback-based CSI estimation.",
"The investigation in [28] provided an example of the leaked CSI, where Alice sends a training signal to Bob and Bob uses feedback channels to inform Alice of CSI, which allows Bob and Alice to obtain accurate knowledge of $\\mathbf {H}$ .",
"However, Eve can obtain $\\mathbf {H}$ due to the broadcasting nature of feedback channels, and Eve can intercept the training signals to get $\\mathbf {H}_e$ .",
"In addition, Eve can obtain $\\mathbf {R}_r=\\text{E}(\\mathbf {H}\\mathbf {H}^{\\dagger }/t)$ and $\\mathbf {R}_e=\\text{E}(\\mathbf {H}_e\\mathbf {H}_e^{\\dagger }/t)$ according to long-term realizations of $\\mathbf {H}$ and $\\mathbf {H}_e$ or statistical AoA information as shown in Definition 2.",
"Based on the precoding of $\\mathbf {B}$ and $\\mathbf {Z}$ , Alice transmits a combined signal $\\mathbf {w}$ via $t$ antennas as $\\mathbf {w}=\\mathbf {Bx}+\\mathbf {Zv}$ , and the received signals at Bob and Eve can be expressed as $&\\mathbf {y}=\\mathbf {HBx}+\\mathbf {H}\\mathbf {Zv}+\\mathbf {n},\\\\& \\mathbf {y}_e=\\mathbf {H}_e\\mathbf {Bx}+\\mathbf {H}_e\\mathbf {Zv}+\\mathbf {n}_e, $ respectively.",
"Here, $\\mathbf {x}$ is a transmit signal of the desired user, and $\\mathbf {v}$ is an AN signal.",
"We follow a convention used in [6], [7], which used Gaussian input alphabets and Gaussian AN, i.e., both $\\mathbf {x}$ and $\\mathbf {v}$ are circularly symmetric complex Gaussian vectors with zero-means and covariance matrices $P/t\\mathbf {I}_{s_1}$ and $P/t\\mathbf {I}_{s_2}$ , respectively, where $P$ is an average transmit power constraint.",
"For analytical simplicity, we distribute total power over all antennas equally as $\\rho =P/t$ .",
"$\\mathbf {n}$ and $\\mathbf {n}_e$ are the additive white Gaussian noise (AWGN) vectors with their covariance matrices $\\mathbf {I}_r$ and $\\mathbf {I}_e$ , respectively.",
"It is obvious that each antenna transmits a combination of message and AN components, but the AN components can be eliminated by the pre-processor at Bob, who eliminates the AN signal $\\mathbf {v}$ by pre-processing ($[\\mathbf {H}\\mathbf {B}]^{\\dagger }\\mathbf {HZ}=\\mathbf {0}$ ), and the received signal $\\mathbf {y}$ is $\\mathbf {\\tilde{y}}=[\\mathbf {H}\\mathbf {B}]^{\\dagger }\\mathbf {y}=\\mathbf {\\Lambda }_{s_1}\\mathbf {x}+\\mathbf {\\tilde{n}},$ where $\\mathbf {\\tilde{n}}=[\\mathbf {H}\\mathbf {B}]^{\\dagger }\\mathbf {n} \\in \\mathbb {C}^{s_1\\times 1}$ is an AWGN vector with its distribution $\\mathcal {CN}(\\mathbf {0},\\mathbf {\\Lambda }_{s_1})$ .",
"$\\mathbf {\\Lambda }_{s_1}\\in \\mathbb {R}^{s_1\\times s_1}$ is a diagonal matrix formed by the first to the $s_1$ th eigenvalues of $\\mathbf {H}^{\\dagger }\\mathbf {H}$ .",
"In the AN elimination process, the channel, where the received signal is left-multiplied by a given matrix $[\\mathbf {H}\\mathbf {B}]^{\\dagger }$ , will not change its capacity if $\\mathbf {B}$ includes all eigenvectors of $\\mathbf {H}^{\\dagger }\\mathbf {H}$ .",
"Since we have $[\\mathbf {H}\\mathbf {B}]^{\\dagger }\\mathbf {H}_e\\mathbf {Z}\\ne \\mathbf {0}$ and $[\\mathbf {H}_e\\mathbf {B}]^{\\dagger }\\mathbf {H}_e\\mathbf {Z}\\ne \\mathbf {0}$ , Eve can not eliminate this AN signal under the condition of $t>e$ , such that the AN signal degrades Eve's channel capacity even if Eve has the knowledge of $\\mathbf {H}$ , $\\mathbf {H}_e$ , $\\mathbf {B}$ , and $\\mathbf {Z}$ .",
"In this way, we can enlarge the capacity difference between the main and wiretap channels."
],
[
"EXACT AND APPROXIMATE ERGODIC SECRECY RATES",
"Next, we derive an exact ergodic secrecy rate expression, as well as perform an approximate analysis to show the impacts of correlated matrices on the ergodic secrecy rates."
],
[
"Exact Expression for Ergodic Secrecy Rate",
"In the proposed scheme, $P$ , $\\mathbf {H}$ , $\\mathbf {R}_r$ , and $\\mathbf {R}_e$ are system parameters.",
"The numbers of message and AN streams, denoted by $s_1$ and $s_2$ , are the variables controlled by us.",
"Then, we can get a real ergodic secrecy rate expression $\\tilde{R}_s$ as $&\\tilde{R}_s(P,\\mathbf {H},\\mathbf {R}_r, \\mathbf {R}_e;s_1,s_2)=\\text{E}_{\\mathbf {H}_e,\\mathbf {H}}[C_m-C_w]^+\\\\ &\\ge \\big [\\text{E}_\\mathbf {H}[C_m]-\\text{E}_{\\mathbf {H}_e,\\mathbf {H}}[C_w]\\big ]^{+}, $ where we have $[x]^+=\\max (x,0)$ , and $C_m=&\\log _2\\det (\\mathbf {I}_r+\\rho \\mathbf {H}_1\\mathbf {H}_1^{\\dagger }), \\\\ C_w=&\\log _2\\text{det}\\bigg (\\mathbf {I}_e+\\frac{\\rho \\mathbf {H}_2\\mathbf {H}_2^{\\dagger }}{\\rho \\mathbf {H}_3\\mathbf {H}_3^{\\dagger }+\\mathbf {I}_e}\\bigg ) \\\\=&\\log _2\\text{det}\\big (\\mathbf {I}_e+\\rho \\mathbf {H}_4\\mathbf {H}_4^{\\dagger }\\big )-\\log _2\\text{det}\\big (\\mathbf {I}_e+\\rho \\mathbf {H}_3\\mathbf {H}_3^{\\dagger }\\big ) .$ Here, we have $\\mathbf {H}_1=\\mathbf {HB}\\in \\mathbb {C}^{r\\times s_1}$ , $\\mathbf {H}_2=\\mathbf {H}_e\\mathbf {B}\\in \\mathbb {C}^{e\\times s_1}$ , $\\mathbf {H}_3=\\mathbf {H}_e\\mathbf {Z}\\in \\mathbb {C}^{e\\times s_2}$ , and $\\mathbf {H}_4=[\\mathbf {H}_2,\\mathbf {H}_3]=\\mathbf {H}_e\\mathbf {U}\\in \\mathbb {C}^{e\\times t}$ .",
"Note that $C_m$ is the main channel capacity that can be achieved by the pre-processor as shown in Eqn.",
"(REF ) [14].",
"The pre-processor can eliminate the interference among antennas and AN-induced interference, so that Bob can decode confidential message streams individually.",
"Assume that Eve sees the Gaussian AN signal and AWGN as a combined AWGN, views $\\mathbf {H}_2$ as its CSI, and then uses the minimum mean squared error (MMSE) with successive interference cancellation (SIC) technique based on $\\mathbf {H}_2$ to achieve a wiretap channel capacity, i.e., $C_w$ .",
"From the conclusions made in [9] and [29], the MMSE with SIC technique is the best choice for Eve without knowledge of Gaussian AN signals.",
"We have an equality in Eqn.",
"() if and only if the secrecy rates are always nonnegative over all channel states.",
"With a large $s_2$ , i.e., more eigen-subchannels are allocated for sending AN signals, $C_m$ is much larger than $C_w$ with a high probabilityThe test results are available in https://github.com/yiliangliu1990/liugit_pub..",
"However, due to the lack of the knowledge of $\\mathbf {H}_e$ , we can not determine if an instantaneous secrecy rate is nonnegative or not, and thus we resort to derive a lower bound of the real ergodic secrecy rate as $R_s(P,\\mathbf {H},\\mathbf {R}_r, \\mathbf {R}_e;s_1,s_2)=[\\text{E}_\\mathbf {H}[C_m]-\\text{E}_{\\mathbf {H}_e,\\mathbf {H}}[C_w]]^{+},$ assuming that both $\\mathbf {H}$ and $\\mathbf {H}_e$ are independent receiver-side correlated complex Gaussian matrices.",
"In order to calculate the ergodic secrecy rate, we need to calculate $\\text{E}_{\\mathbf {H}_e,\\mathbf {H}}[C_w]$ , and we should find out the distributions of random matrices $\\mathbf {H}_2$ , $\\mathbf {H}_3$ , and $\\mathbf {H}_4$ , all of which are the product of a complex Gaussian matrix and an independent unitary matrix.",
"The corresponding results are given in Theorem 2.",
"Theorem 2: Define $\\mathbf {H}_e\\sim \\mathcal {CN}_{e,t}(\\mathbf {0},\\mathbf {R}_e\\otimes \\mathbf {I}_t)$ as a receiver-side correlated central complex Gaussian matrix, and establish an independent $(t\\times f)$ unitary matrix $\\mathbf {F}$ (generalized for $\\mathbf {B}$ and $\\mathbf {Z}$ ).",
"We have $\\mathbf {H}_e\\mathbf {F}\\sim \\mathcal {CN}_{e,f}(\\mathbf {0},\\mathbf {R}_e\\otimes \\mathbf {I}_f),$ where $f\\in \\mathbb {N}$ and $t\\ge f$ .",
"Proof: See Appendix B.",
"From Theorem 2, we know that $\\mathbf {H}_2$ , $\\mathbf {H}_3$ , and $\\mathbf {H}_4$ are complex Gaussian matrices with their distributions as $&\\mathbf {H}_2=\\mathbf {H}_e\\mathbf {B}\\sim \\mathcal {CN}_{e,s_1}(\\mathbf {0},\\mathbf {R}_e\\otimes \\mathbf {I}_{s_1}),\\\\&\\mathbf {H}_3=\\mathbf {H}_e\\mathbf {Z}\\sim \\mathcal {CN}_{e,s_2}(\\mathbf {0},\\mathbf {R}_e\\otimes \\mathbf {I}_{s_2}),\\\\&\\mathbf {H}_4=\\mathbf {H}_e\\mathbf {U}\\sim \\mathcal {CN}_{e,t}(\\mathbf {0},\\mathbf {R}_e\\otimes \\mathbf {I}_t),$ respectively.",
"In order to evaluate the performance of the AN scheme further, we should use the pdf of the $k$ th eigenvalue of complex Wishart matrices to derive a theoretical ergodic secrecy rate expression of Eqn.",
"(REF ).",
"Here, we give the definition of the Wishart matrix, as shown in Definition 3.",
"Definition 3 (Receiver-side correlated central complex Wishart matrix): For $\\mathbf {A}\\sim \\mathcal {CN}_{a,b}(\\mathbf {0},\\mathbf {R}_a\\otimes \\mathbf {I}_b)$ , $m=\\max (a,b)$ , and $n=\\min (a,b)$ , a Hermitian matrix $\\mathbf {W}\\in \\mathbb {C}^{n\\times n}$ is defined as $\\mathbf {W}={\\left\\lbrace \\begin{array}{ll}\\mathbf {A}\\mathbf {A}^{\\dagger },&\\mbox{$b\\ge a$},\\\\\\mathbf {A}^{\\dagger }\\mathbf {A}, &\\mbox{$b<a$},\\end{array}\\right.",
"}$ where $\\mathbf {W}$ is called a receiver-side correlated central Wishart matrix defined as $\\mathbf {W}\\sim W_n(m,\\mathbf {0}_n,\\mathbf {R}_a)$ with $n$ degrees of freedom, and a receiver-side correlated matrix $\\mathbf {R}_a$ has its eigenvalues $\\sigma _i, 1\\le i\\le a$ , where $\\sigma _1>\\sigma _2>...>\\sigma _a$ .",
"The Wishart matrix was investigated first in [30].",
"An arbitrary MIMO channel ($\\mathbf {H}$ , $\\mathbf {H}_3$ , or $\\mathbf {H}_4$ ) can be effectively decomposed into multiple parallel SISO eigen-subchannels.",
"With the help of transmit and receive signal processing as described in Section II, $s_1$ eigen-subchannels are selected for sending messages.",
"Then, we can re-write the ergodic secrecy rate function Eqn.",
"(REF ) as $&R_s(P,\\mathbf {R}_r, \\mathbf {R}_e;s_1,s_2)\\\\ &=\\big [C_{\\mathbf {H}}(\\mathbf {R}_r, \\rho ,s_1)+C_{\\mathbf {H}_3}(\\mathbf {R}_e, \\rho ,n_1)-C_{\\mathbf {H}_4}(\\mathbf {R}_e,\\rho ,e)\\big ]^{+},$ where $C_{\\mathbf {A}}(\\mathbf {R}_a, \\rho ,\\eta )=\\sum _{k=1}^{\\eta }\\int _0^{\\infty }\\log _2(1+\\rho x)f_{\\lambda _k}(x)dx,$ in which we have $\\rho =P/t$ , $\\mathbf {A}\\in \\mathbb {C}^{a\\times b}$ , $\\mathbf {R}_a$ is an $a\\times a$ matrix, $\\lambda _k$ is the $k$ th largest eigenvalue of $\\mathbf {A}\\mathbf {A}^{\\dagger }$ (or $\\mathbf {A}^{\\dagger }\\mathbf {A}$ ), $n_1=\\min (s_2,e)$ , and $f_{\\lambda _k}(x)$ is given in Theorem 3.",
"Note that the ergodic secrecy rate function takes an integral form rather than a closed form because $f_{\\lambda _k}(x)$ is very complicated.",
"Theorem 3: For $k=1,...,n$ , the marginal pdf of the $k$ th largest eigenvalue $\\lambda _k$ of a receiver-side correlated central Wishart matrix $\\mathbf {W}\\sim W_n(m,\\mathbf {0}_n,\\mathbf {R}_a)$ is given by $f_{\\lambda _k}(x)=K^{-1}\\sum _{i=1}^k\\sum _{\\mathbf {\\mu }\\in \\mathcal {P}(i)}\\sum _{j=1}^{n}\\det \\big [\\mathbf {G},\\mathbf {\\Omega }(\\mathbf {\\mu },\\mathbf {\\sigma },i,j;x)\\big ], $ where we have $K=\\prod _{i<j}^n\\sigma _i-\\sigma _j \\prod _{i=1}^n(b-i)!,$ and $\\mathcal {P}(i)$ is a set of all permutations $(\\mu _1,...,\\mu _n)$ of integers $(1,...,n)$ such that $(\\mu _1<\\mu _2<...<\\mu _{i-1})$ and $(\\mu _i<\\mu _{i+1}<...<\\mu _n)$ .",
"The set has $\\binom{n}{i-1}$ permutations of $\\mathbf {\\mu }$ , each of which is a representation of the matrix function $\\mathbf {\\Omega }(\\cdot )$ .",
"Hence, $\\sum _{\\mathbf {\\mu }\\in \\mathcal {P}(i)}$ denotes a summation over these $\\binom{n}{i-1}$ matrices.",
"$\\mathbf {G}$ is an $a\\times (a-n)$ matrix, whose $(i,j)$ th element is $\\sigma _i^{j-1}$ .",
"Note that $\\mathbf {G}$ is a null matrix when $b\\ge a$ .",
"The $a \\times n$ real matrix $\\mathbf {\\Omega }(\\mathbf {\\mu },\\mathbf {\\sigma },i,j;x)$ is defined as $&\\big [\\mathbf {\\Omega }(\\mathbf {\\mu },\\mathbf {\\sigma },i,j;x)\\big ]_{u,\\mu _v}\\\\ &\\!\\!\\!={\\left\\lbrace \\begin{array}{ll}\\sigma _u^{a-n+\\mu _v-1}\\Gamma (b-n+\\mu _v,\\frac{x}{\\sigma _u}),&\\!\\!\\!\\text{$v=1,...,k-1$, $\\mu _v \\ne j$},\\\\-\\sigma _u^{a-b-1}\\exp (-\\frac{x}{\\sigma _u})x^{b-n+\\mu _v-1},&\\!\\!\\!\\text{$v=1,...,k-1$, $\\mu _v = j$},\\\\\\sigma _u^{a-n+\\mu _v-1}\\gamma (b-n+\\mu _v,\\frac{x}{\\sigma _u}),&\\!\\!\\!\\text{$v=k,...,n$, $\\mu _v \\ne j$},\\\\\\sigma _u^{a-b-1}\\exp (-\\frac{x}{\\sigma _u})x^{b-n+\\mu _v-1},&\\!\\!\\!\\text{$v=k,...,n$, $\\mu _v = j$},\\end{array}\\right.",
"}$ for $u=1,...,a$ and $v=1,...,n$ , where $\\Gamma (\\cdot ,\\cdot )$ and $\\gamma (\\cdot ,\\cdot )$ are the upper and lower incomplete Gamma functions [31] defined as $&\\Gamma (\\epsilon ,x)=\\int _x^{\\infty }\\exp (-z)z^{\\epsilon -1}\\text{d}z,\\\\&\\gamma (\\epsilon ,x)=\\int _0^x\\exp (-z)z^{\\epsilon -1}\\text{d}z.$ Proof: See Appendix.",
"C. Note that when $\\mathbf {R}_r=\\mathbf {I}_r$ or $\\mathbf {R}_e=\\mathbf {I}_e$ , $C_\\mathbf {A}(\\mathbf {I}_a,\\rho ,\\eta )$ will be replaced by the equation in [14].",
"Remark 2: We can use Eqn.",
"(REF ), as a theoretical ergodic secrecy rate expression of Eqn.",
"(REF ), to maximize the ergodic secrecy rate via a one-dimensional search, which takes the number of eigen-subchannels of message streams, i.e., $s_1$ , as a search direction.",
"Although the results from the search are not globally optimal and the achieved ergodic secrecy rates are the lower bounds of ergodic secrecy capacities, the search with its complexity $O(n)$ avoids complicated convex optimization processes.",
"The eigen-subchannels of larger eigenvalues should be selected for sending messages because $C_{\\mathbf {H}}(\\mathbf {R}_r, \\rho ,s_1)$ in Eqn.",
"(REF ) is larger when using the eigen-subchannels of larger eigenvalues for a fixed $s_1$ .",
"Meanwhile, which one is selected for AN signals has no effect on the ergodic secrecy rate for a fixed $s_1$ , because $C_{\\mathbf {H}_3}(\\mathbf {R}_e, \\rho ,n_1)$ in Eqn.",
"(REF ) is a constant for a given $s_1$ .",
"In addition, $C_{\\mathbf {H}_4}(\\mathbf {R}_e,\\rho ,e)$ in Eqn.",
"(REF ) is an average value over $\\mathbf {H}_4$ , and is fixed for given $t$ and $e$ , which have nothing to do with $s_1$ .",
"Hence, the optimal method must be that the eigen-subchannels for messages are selected from their large to small corresponding eigenvalues.",
"For example, given $t=4$ , $s_1=2$ , and $s_2=2$ , the maximization is achieved if the 1st and 2nd eigen-subchannels are selected for sending messages, while the 3rd and 4th eigen-subchannels are selected for sending AN signals.",
"In this case, maximization is done over an array with $n$ elements, and the eigen-subchannel allocation is a one-dimensional search problem with its complexity $O(n)$ , where $n=\\min (t,r)$ ."
],
[
"Approximate Ergodic Secrecy Rate",
"The derived ergodic secrecy rate expression in Eqn.",
"(REF ) is not in a closed form.",
"We can simplify the expression to an approximate form, to show the impacts of correlated matrices $\\mathbf {R}_r$ (a function of $d_{\\text{Bob}}, \\bar{\\theta }_{\\text{Bob}}, \\text{and } \\delta _{\\text{Bob}}$ ) and $\\mathbf {R}_e$ (a function of $d_{\\text{Eve}}, \\bar{\\theta }_{\\text{Eve}}, \\text{and } \\delta _{\\text{Eve}}$ ) on the ergodic secrecy rates.",
"Theorem 4: The ergodic secrecy rate, i.e., Eqn.",
"(REF ), can be expressed approximately as $R^{\\text{app}}_s&=[\\chi _1+\\chi _2]^{+}, $ where $&\\chi _1=\\sum _{i=1}^{s_1}\\log _2\\big \\lbrace 1+\\rho \\text{E}[\\lambda _i(\\mathbf {H}\\mathbf {H}^{\\dagger })]\\big \\rbrace , \\\\&\\chi _2=\\log _2\\bigg [\\frac{1+\\sum _{k=1}^{e}\\rho ^k\\prod _{i=0}^{k-1}(m_1-i)\\varrho _k}{1+\\sum _{k=1}^{e}\\rho ^k\\prod _{i=0}^{k-1}(t-i)\\varrho _k} \\bigg ], \\\\&\\varrho _k=\\sum _{\\ell _1<\\ell _2<...<\\ell _k}\\det [\\mathbf {R}_{e,(\\ell _1,...\\ell _k)}], \\\\&\\mathbf {R}_{e,(\\ell _1,...\\ell _k)}=\\left[\\begin{matrix} [\\mathbf {R}_e]_{(\\ell _1,\\ell _1)}&\\ldots &[\\mathbf {R}_e]_{(\\ell _1,\\ell _k)}\\\\[\\mathbf {R}_e]_{(\\ell _2,\\ell _1)}&\\ldots &[\\mathbf {R}_e]_{(\\ell _2,\\ell _k)}\\\\\\vdots & &\\vdots \\\\[\\mathbf {R}_e]_{(\\ell _k,\\ell _1)}&\\ldots &[\\mathbf {R}_e]_{(\\ell _k,\\ell _k)}\\end{matrix}\\right], \\\\&n_1 =\\min (e,d), \\quad m_1=\\max (e,d), \\quad 1\\le k \\le e. $ Here, we denote a subset $(\\ell _1,...,\\ell _k)$ of $(1,2,...,e)$ such that $\\ell _1<\\ell _2<...<\\ell _k$ , which means $\\mathbf {R}_{e,(\\ell _1,...\\ell _n)}=\\mathbf {R}_e$ .",
"Similar matrix structures and more explanations were given in [32] and [33].",
"$\\lambda _1(\\mathbf {H}\\mathbf {H}^{\\dagger })>\\lambda _2(\\mathbf {H}\\mathbf {H}^{\\dagger })>...>\\lambda _n(\\mathbf {H}\\mathbf {H}^{\\dagger })$ are the ordered eigenvalues of $\\mathbf {H}\\mathbf {H}^{\\dagger }$ .",
"Proof: See Appendix.",
"D. Remark 3: (The impact of $d_{\\text{Bob}}, \\bar{\\theta }_{\\text{Bob}}, \\text{and} \\delta _{\\text{Bob}}$ ): $\\mathbf {R}_r$ only affects $\\chi _1$ .",
"When $s_1=1$ , Eqn.",
"(REF ) can be re-written as $\\chi _1=&\\log _2\\big \\lbrace 1+P\\text{E}[\\lambda _1(\\mathbf {H}\\mathbf {H}^{\\dagger }/t)]\\big \\rbrace .$ From [34], when $r/t=c<1$ , $t\\rightarrow +\\infty $ , and $r \\rightarrow +\\infty $ , we can get $\\text{E}[\\lambda _1(\\mathbf {H}\\mathbf {H}^{\\dagger }/t)]&\\rightarrow {\\left\\lbrace \\begin{array}{ll}\\sigma _1(1+\\frac{c}{\\sigma _1-1}),&\\!\\!\\!\\text{$\\sigma _1>1+\\sqrt{c}$},\\\\(1+\\sqrt{c})^2,&\\!\\!\\!\\text{$\\sigma _1\\le 1+\\sqrt{c}$},\\end{array}\\right.",
"}$ where $\\sigma _1$ is the largest eigenvalue of $\\mathbf {R}_r$ , i.e., $\\lambda _1(\\mathbf {R}_r)$ .",
"Based on Theorem 1, we know that $\\sigma _1$ decreases monotonically with increasing $d_{\\text{Bob}}$ , $\\bar{\\theta }_{\\text{Bob}}$ , and $\\delta _{\\text{Bob}}$ , respectively.",
"Thus, $\\text{E}[\\lambda _1(\\mathbf {H}\\mathbf {H}^{\\dagger }/t)]$ decreases monotonically with increasing $d_{\\text{Bob}}$ , $\\bar{\\theta }_{\\text{Bob}}$ , and $\\delta _{\\text{Bob}}$ , respectively, and then keeps constant.",
"We can conclude that, when $s_1=1$ , an ergodic secrecy rate decreases monotonically with increasing $d_{\\text{Bob}}$ , $\\bar{\\theta }_{\\text{Bob}}$ , and $\\delta _{\\text{Bob}}$ , respectively.",
"When $s_1=n=\\min (t,r)$ , based on [32], Eqn.",
"(REF ) can be expressed approximately as $\\chi _1&=\\sum _{i=1}^n \\log _2\\big \\lbrace 1+\\rho \\text{E}[\\lambda _i(\\mathbf {H}\\mathbf {H}^{\\dagger })]\\big \\rbrace , \\\\ &\\simeq n\\log _2\\rho +\\hbar +\\log _2\\det [\\mathbf {R}_r],$ where $\\hbar ={\\left\\lbrace \\begin{array}{ll}\\log _2\\sum _{i=0}^{n-1}(m-i)&\\text{$\\rho \\text{E}[\\lambda _i(\\mathbf {H}\\mathbf {H}^{\\dagger })]\\gg 1$},\\\\\\sum _{i=0}^{n-1}\\psi (m-i)&\\text{$\\rho \\text{E}[\\lambda _i(\\mathbf {H}\\mathbf {H}^{\\dagger })]\\ll 1$},\\end{array}\\right.",
"}$ and $m=\\max (t,r)$ , $n=\\min (t,r)$ , and $\\psi (x)$ is defined as $\\psi (x)=-\\xi +\\sum _{i=1}^{x-1}\\frac{1}{i},$ where $\\xi \\simeq 0.5772156649$ is the Euler's constant.",
"It is obvious that Eqn.",
"(REF ) increases monotonically with an increasing $\\det [\\mathbf {R}_r]$ .",
"Based on Theorem 1, we know that $\\det [\\mathbf {R}_r]$ increases monotonically with increasing $d_{\\text{Bob}}$ , $\\bar{\\theta }_{\\text{Bob}}$ , and $\\delta _{\\text{Bob}}$ , respectively.",
"In conclusion, when $s_1=n$ , an ergodic secrecy rate increases monotonically with increasing $d_{\\text{Bob}}$ , $\\bar{\\theta }_{\\text{Bob}}$ , and $\\delta _{\\text{Bob}}$ , respectively.",
"However, when $n>s_1>1$ , it is very hard to find a simple relationship between an ergodic secrecy rate and correlation parameters, and thus we simulate these scenarios, as given in Section IV.",
"Remark 4: (The impact of $d_{\\text{Eve}}, \\bar{\\theta }_{\\text{Eve}}, \\text{and } \\delta _{\\text{Eve}}$ ): In Eqn.",
"(REF ), $\\mathbf {R}_e$ affects $\\chi _2$ only.",
"Based on [32], we can get $\\det [\\mathbf {I}+\\mathbf {R}_e]=1+\\sum _{k=1}^e\\sum _{\\ell _1<\\ell _2<...<\\ell _k}\\det [\\mathbf {R}_{e,(\\ell _1,...\\ell _k)}].$ Since $\\mathbf {R}_e$ is a Hermitian positive definite matrix, $\\det [\\mathbf {I}+\\mathbf {R}_e]$ increases monotonically with an increasing $\\det [\\mathbf {R}_e]$ .",
"Certainly, we know that $\\sum _{k=1}^e\\sum _{\\ell _1<\\ell _2<...<\\ell _k}\\det [\\mathbf {R}_{e,(\\ell _1,...\\ell _k)}]$ increases monotonically with an increasing $\\det [\\mathbf {R}_e]$ .",
"In addition, we will introduce an auxiliary function $f(\\mathbf {x})$ as $f(\\mathbf {x})=\\frac{1+\\sum _{k=1}^e a_k x_k}{1+\\sum _{k=1}^e b_k x_k},$ where $b_k>a_k$ , $\\forall k$ .",
"$f(\\mathbf {x})$ decreases monotonically with $\\sum _{k=1}^e x_k$ .",
"Hence, $\\chi _2$ decreases monotonically with $\\sum _{k=1}^e\\sum _{\\ell _1<\\ell _2<...<\\ell _k}\\det [\\mathbf {R}_{e,(\\ell _1,...\\ell _k)}]$ as well as $\\det [\\mathbf {R}_e]$ .",
"Therefore, Eqn.",
"() decreases with $\\det [\\mathbf {R}_e]$ .",
"Similarly, $\\det [\\mathbf {R}_e]$ increases monotonically with $d_{\\text{Eve}}$ , $\\bar{\\theta }_{\\text{Eve}}$ , and $\\delta _{\\text{Eve}}$ , respectively.",
"Thus, an ergodic secrecy rate decreases monotonically with increasing $d_{\\text{Eve}}$ , $\\bar{\\theta }_{\\text{Eve}}$ , and $\\delta _{\\text{Eve}}$ , respectively.",
"Table REF shows the impacts of $\\lbrace d_{\\text{Bob}}, \\bar{\\theta }_{\\text{Bob}}, \\text{and } \\delta _{\\text{Bob}}\\rbrace $ , as well as $\\lbrace d_{\\text{Eve}}, \\bar{\\theta }_{\\text{Eve}}, \\text{and } \\delta _{\\text{Eve}}\\rbrace $ on the ergodic secrecy rates.",
"We use $\\uparrow $ and $\\downarrow $ to represent monotonically “increase\" and “decrease\", respectively.",
"For example, “$R^{\\text{app}}_s\\downarrow $ with $\\lbrace d_{\\text{Eve}}, \\bar{\\theta }_{\\text{Eve}}, \\delta _{\\text{Eve}}\\rbrace \\uparrow $ \" means that “the ergodic secrecy rate decreases monotonically with increasing $d_{\\text{Eve}}$ , $\\bar{\\theta }_{\\text{Eve}}$ , and $\\delta _{\\text{Eve}}$ , respectively\".",
"We must point out that “increase\" or “decrease\" will not take place forever, because when the correlation parameters grow to a certain extent, the correlation disappears and ergodic secrecy rates will be constant.",
"Table: Impacts of correlation on ergodic secrecy rates.Figure: Numerical and simulation results of ergodic secrecy rates of a correlated MIMO channel in terms of transmit SNR, where t=6t=6, r=e=4r=e=4, d Bob =d Eve =0.8d_{\\text{Bob}}=d_{\\text{Eve}}=0.8, θ ¯ Bob =θ ¯ Eve =30 ∘ \\bar{\\theta }_{\\text{Bob}}=\\bar{\\theta }_{\\text{Eve}}=30^{\\circ }, and δ Bob =δ Eve =10 ∘ \\delta _{\\text{Bob}}=\\delta _{\\text{Eve}}=10^{\\circ }."
],
[
"NUMERICAL AND SIMULATE RESULTS",
"In this section, numerical and simulation results are given.",
"As shown in the figures below, the theoretical results (theo.)",
"from Eqn.",
"(REF ) are in a good agreement with the Monte Carlo simulations (simu.)",
"of $10^5$ independent runs on Eqn.",
"(REF ).",
"The ergodic secrecy rates of the proposed scheme are compared to the traditional AN schemes [8], [9], [10], [11], which did not consider the correlation and used all eigen-subchannels to transmit messages, i.e., $s_1=n$ .",
"In the proposed scheme, the number of eigen-subchannels for sending messages, i.e., $s_1$ , is a variable.",
"The channel model in the simulations is a receiver-side correlated Rayleigh fading channel.",
"Fig.",
"REF illustrates the impact of transmit SNR on ergodic secrecy rates with different choices of $\\lbrace s_1$ , $s_2\\rbrace $ .",
"As shown in Fig.",
"REF , the achievable ergodic secrecy rates increase almost exponentially with SNR, and $s_1=2$ is the best choice when SNR$<16$ .",
"There exists a crossing point between $s_1=2$ and $s_1=3$ because $s_1=3$ offers a better performance with an increasing SNR, which is consistent with [14].",
"In addition, the black and dashed lines are simulation results without the awareness of the correlated fading that are conformed to the scenarios $\\lbrace s_1=2,s_2=4\\rbrace $ and $\\lbrace s_1=1,s_2=5\\rbrace $The more results are given in https://github.com/yiliangliu1990/liugit_pub..",
"If we do not consider (or do not know) correlation parameters at Eve's sides, the ergodic secrecy rates will be reduced compared to the performance with the knowledge of Eve's correlation parameters, because the ergodic wiretap channel rate will be enlarged if there is on correlation among Eve's antennas.",
"Fig.",
"REF shows the results in high SNR regions, where ergodic secrecy rates grow almost linearly with SNR.",
"We see that $s_1=3$ is the best choice, and the simulations of $s_1=2$ and $s_1=4$ show similar performance.",
"The results indicate that it is better to allocate stronger eigen-subchannels to transmit messages and weaker eigen-subchannels to send AN signals, especially in high SNR regions, which coincides with the results given in the uncorrelated scenarios [14].",
"Figure: Numerical and simulation results of ergodic secrecy rates in a correlated MIMO channel in terms of the number of antennas of Bob, where t=5t=5, e=3e=3, transmit SNR=5 dB, d Bob =d Eve =0.8d_{\\text{Bob}}=d_{\\text{Eve}}=0.8, θ ¯ Bob =θ ¯ Eve =30 ∘ \\bar{\\theta }_{\\text{Bob}}=\\bar{\\theta }_{\\text{Eve}}=30^{\\circ }, and δ Bob =δ Eve =10 ∘ \\delta _{\\text{Bob}}=\\delta _{\\text{Eve}}=10^{\\circ }.Fig.",
"REF shows the relationship between ergodic secrecy rates and the number of antennas of Bob.",
"We can observe that an increasing number of antennas at Bob is beneficial for any $s_1$ and $s_2$ chosen.",
"If Bob has many receive antennas, it can enlarge the channel gains of the message streams because Bob has enough antennas to decode them and gather the received power from all antennas.",
"The previous works in [8], [9], [10], [11] did not consider the scenarios with $t<r$ , and thus we do not compare them here.",
"Figure: Numerical and simulation results of ergodic secrecy rates in a correlated MIMO channel in terms of the normalized minimum distances d Bob d_{\\text{Bob}} and d Eve d_{\\text{Eve}}, where transmit SNR=5 dB, t=6t=6, r=e=4r=e=4, θ ¯ Bob =θ ¯ Eve =30 ∘ \\bar{\\theta }_{\\text{Bob}}=\\bar{\\theta }_{\\text{Eve}}=30^{\\circ }, and δ Bob =δ Eve =10 ∘ \\delta _{\\text{Bob}}=\\delta _{\\text{Eve}}=10^{\\circ }.Figure: Numerical and simulation results of ergodic secrecy rates in a correlated MIMO channel in terms of mean AoA θ ¯ Bob \\bar{\\theta }_{\\text{Bob}} and θ ¯ Eve \\bar{\\theta }_{\\text{Eve}}, where transmit SNR==5 dB, t=6t=6, r=e=4r=e=4, d Bob =d Eve =0.8d_{\\text{Bob}}=d_{\\text{Eve}}=0.8, and δ Bob =δ Eve =10 ∘ \\delta _{\\text{Bob}}=\\delta _{\\text{Eve}}=10^{\\circ }.Figure: Numerical and simulation results of ergodic secrecy rates in a correlated MIMO channel in terms of RAS δ Bob \\delta _{\\text{Bob}} and δ Eve \\delta _{\\text{Eve}}, where transmit SNR==5 dB, t=6t=6, r=e=4r=e=4, d Bob =d Eve =0.8d_{\\text{Bob}}=d_{\\text{Eve}}=0.8, and θ ¯ r =θ ¯ e =30 ∘ \\bar{\\theta }_r=\\bar{\\theta }_e=30^{\\circ }.Fig.",
"REF shows the ergodic secrecy rate simulations in terms of antenna spacing in wavelength, where we set SNR$=5$ dB, $\\bar{\\theta }_{\\text{Bob}}=\\bar{\\theta }_{\\text{Eve}}=30^{\\circ }$ , and $\\delta _{\\text{Bob}}=\\delta _{\\text{Eve}}=10^{\\circ }$ .",
"Assume that $d_{\\text{Eve}}$ is fixed in Fig.",
"REF .",
"When $s_1=3$ and $s_1=4$ , the ergodic secrecy rates grow with the antenna spacing; when $s_1=1$ , the ergodic secrecy rates decrease with the antenna spacing.",
"If $s_1=2$ , we see a peak value of the ergodic secrecy rates, where the rates rise at the beginning, then reduce, and keep constant.",
"Intuitively, the peak occurs due to the fact that the curve of $s_1=2$ is affected by variations of the largest and the second largest eigen-subchannels, where the gain of the second largest eigen-subchannel increases fast at the beginning, and the gain of the largest eigen-subchannel decreases fast in the second half of the simulation diagram.",
"In Fig.",
"REF , we assumed that $d_{\\text{Bob}}$ is fixed, and we can see that the ergodic secrecy rates decrease with the antenna spacing.",
"In particular, the more eigen-subchannels are allocated for messages, the more quickly the ergodic secrecy rates will decrease.",
"Note that in Figs.",
"REF and REF , when the normalized minimum distances are larger than three, the receiver-side correlation almost disappears.",
"Therefore, the curves of ergodic secrecy rates tend to be constant.",
"This phenomenon also appears in traditional MIMO systems[18], [35].",
"Fig.",
"REF examines the ergodic secrecy rates in terms of mean AoA in a correlated MIMO channel, where $d_{\\text{Bob}}=d_{\\text{Eve}}=0.8$ and $\\delta _{\\text{Bob}}=\\delta _{\\text{Eve}}=10^{\\circ }$ .",
"Fig.",
"REF shows the effects of Bob's mean AoA with a fixed $\\bar{\\theta }_{\\text{Eve}}$ , where an increasing mean AoA will reduce the gain of the strongest eigen-subchannel, such that the ergodic secrecy rates will be reduced if we only chose the strongest one, i.e., $s_1=1$ .",
"However, an increasing mean AoA will reduce the correlation at receiver-side, and thus the ergodic secrecy rate will increase if we use most of the eigen-subchannels to transmit messages.",
"As shown in Fig.",
"REF , assuming $\\bar{\\theta }_{\\text{Bob}}$ is fixed, we see that the ergodic secrecy rates will reduce with an increasing AoA of Eve because an increasing AoA of Eve will reduce the receiver-side correlation, which enlarges the wiretap channel capacities but does not affect the main channel capacities at all.",
"RAS has also its impact on the ergodic secrecy rates, which has a similar effect as the mean AoA.",
"As shown in Fig.",
"REF , an increasing RAS of Bob will reduce receiver-side correlation and reduce the gain of the strongest eigen-subchannel.",
"Hence, when $s_1=1$ , the ergodic secrecy rates will decrease with $\\delta _{\\text{Bob}}$ , and when $s_1=2, 3, \\text{and }4$ , the ergodic secrecy rates will increase because a weaker receiver-side correlation enlarges the main channel capacities.",
"As shown in Fig.",
"REF with a fixed $\\delta _{\\text{Bob}}$ , an increasing RAS of Eve reduces the ergodic secrecy rate with an arbitrary number of message streams.",
"The curve of $s_1=4$ grows fast if compared to $s_1$ =1, 2, and 3."
],
[
"CONCLUSIONS AND FUTURE WORKS",
"In this paper, we investigated the ergodic secrecy rate of spatially correlated scattering Rayleigh fading channels in an artificial noisy MIMO system, along with theoretical and approximate ergodic secrecy rate analysis.",
"The suitable number of eigen-subchannels for sending messages and AN signals can be identified via a one-dimensional search based on the derived ergodic secrecy rate expressions.",
"According to the results given in the analyses and simulations, we revealed that the correlation parameters, i.e., mean AoA, RAS, and antenna spacing, have significant influence on ergodic secrecy rates.",
"Nevertheless, a real MIMO channel may be transmitter-side correlated or doubly-correlated at both sides.",
"When considering the transmitter-side correlated or doubly-correlated channels, the derivation of statistical distribution of $\\mathbf {H}_e\\mathbf {F}$ , as shown in Theorem 2, is still an open issue.",
"Hence, in the future, we should establish a new Wishart matrix model first before investigating the impacts of those correlated channels on ergodic secrecy rates."
],
[
"Proof of Theorem 1",
"Lemma 1 (Proved in [36]): For $r\\times r$ matrices $\\mathbf {A}=[a_{ij}]$ and $\\mathbf {B}=[b_{ij}]$ , if $\\mathbf {A}$ and $\\mathbf {B}$ are Hermitian positive (semi-)definite, then $\\sigma _1(\\mathbf {A}\\circ \\mathbf {B})\\le \\max _{1\\le i \\le r}a_{ii}\\sigma _1(\\mathbf {B}),$ where $a_{ii}$ is a diagonal element of $\\mathbf {A}$ , and “$\\circ $ \" denotes the Schur product defined as $\\mathbf {A} \\circ \\mathbf {B}=[a_{ij}b_{ij}]$ .",
"Lemma 2 (Proved in [37]): For two $r\\times r$ matrices $\\mathbf {A}=[a_{ij}]$ and $\\mathbf {B}=[b_{ij}]$ , if $\\mathbf {A}$ and $\\mathbf {B}$ are Hermitian positive (semi-)definite, then $\\prod _{i=1}^r\\sigma _i(\\mathbf {A} \\circ \\mathbf {B}) \\ge \\prod _{i=1}^r\\sigma _i(\\mathbf {B})a_{ii}.$ We begin to prove Theorem 1 as follows.",
"For $d_1>d_2$ , we can build up $\\mathbf {R}_a(d_1)$ via $\\mathbf {R}_a(d_2)$ as $\\mathbf {R}_a(d_1)=\\mathbf {M} \\circ \\mathbf {R}_a(d_2),$ where $\\mathbf {M}$ is a Hermitian matrix whose diagonal elements are all one.",
"If $d_1>d_2$ and $i \\ne j$ , based on Eqn.",
"(REF ), we can find that the modulus value of $[\\mathbf {R}_a(d_1)]_{i,j}$ is smaller, i.e., $|[\\mathbf {R}_a(d_1)]_{i,j}|<|[\\mathbf {R}_a(d_2)]_{i,j}|.$ Thus, $\\mathbf {M}$ is positive (semi-)definite because all diagonal elements of $\\mathbf {M}$ are one, and the modulus of non-diagonal elements is smaller than one.",
"Based on Lemma 1 and Eqn.",
"(REF ), we can get $\\sigma _1[\\mathbf {R}_a(d_1)]\\le \\max _{1\\le i \\le n}m_{ii}\\sigma _1[\\mathbf {R}_a(d_2)],$ and $m_{ii}$ is a diagonal element of $\\mathbf {M}$ such that $m_{ii}=1$ .",
"Hence, $\\sigma _1[\\mathbf {R}_a(d_1)]<\\sigma _1[\\mathbf {R}_a(d_2)]$ .",
"With the same argument, we can show that $\\sigma _1[\\mathbf {R}_a(\\bar{\\theta })]$ and $\\sigma _1[\\mathbf {R}_a(\\delta )]$ have the same property.",
"Based on Lemma 2 and $a_{ii}=1$ , we get $\\det [\\mathbf {R}_a(d_1)]&=\\prod _{i=1}^r\\sigma _i\\big [\\mathbf {M} \\circ \\mathbf {R}_a(d_2)\\big ] \\\\ &\\ge \\prod _{i=1}^r\\sigma _i\\big [\\mathbf {R}_a(d_2)\\big ]m_{ii} =\\det [\\mathbf {R}_a(d_2)].$ Note $\\det [\\mathbf {R}_a(d_1)]\\ne \\det [\\mathbf {R}_a(d_2)]$ , and thus “$>$ \" is held.",
"Similarly, $\\det [\\mathbf {R}_a(\\bar{\\theta })]$ and $\\det [(\\mathbf {R}_a(\\delta )]$ have the same property.",
"$\\blacksquare $"
],
[
"Proof of Theorem 2",
"Lemma 3 (Proved in [38]): If $\\mathbf {H}_e$$\\sim $ $\\mathcal {CN}_{e,t}(\\mathbf {0},\\mathbf {R}_e\\otimes \\mathbf {I}_t)$ , the characteristic function of $\\mathbf {H}_e$ is $\\phi _{\\mathbf {H}_e}(\\mathbf {X})=\\text{E}\\big \\lbrace \\text{etr}[i\\mathbf {H}_e\\mathbf {X}^{\\dagger })]\\big \\rbrace =\\text{etr}\\big (-\\frac{1}{2}\\mathbf {X}^{\\dagger }\\mathbf {R}_e\\mathbf {X}\\mathbf {I}_t\\big ),$ where $i=\\sqrt{-1}$ .",
"Next, we can prove Theorem 2 based on Lemma 3.",
"For a given $(t\\times s)$ unitary matrix $\\mathbf {B}$ , the characteristic function of $\\mathbf {H}_e\\mathbf {B}$ is $\\phi _{\\mathbf {H}_e\\mathbf {B}}(\\mathbf {X})=\\text{E}[\\text{etr}(i\\mathbf {H}_e\\mathbf {B}\\mathbf {X}^{\\dagger })]=\\text{E}[\\text{etr}(i\\mathbf {H}_e\\mathbf {Y}^{\\dagger })],$ where $\\mathbf {Y}^{\\dagger }=\\mathbf {B}\\mathbf {X}^{\\dagger }$ .",
"Viewing $\\mathbf {Y}$ as a variable, from Lemma 1, we get $\\text{E}[\\text{etr}(i\\mathbf {H}_e\\mathbf {Y}^{\\dagger })]&=\\text{etr}\\big (-\\frac{1}{2}\\mathbf {Y}^{\\dagger }\\mathbf {R}_e\\mathbf {Y}\\big )\\\\ &=\\text{etr}\\big (-\\frac{1}{2}\\mathbf {X}^{\\dagger }\\mathbf {R}_e\\mathbf {X}\\mathbf {B}^{\\dagger }\\mathbf {B}\\big ).$ Since $\\mathbf {B}$ is a $(t\\times s)$ unitary matrix, we have $\\mathbf {B}^{\\dagger }\\mathbf {B}=\\mathbf {I}_s$ .",
"Then, Eqn.",
"(REF ) can be written as $\\phi _{\\mathbf {H}_e\\mathbf {B}}(\\mathbf {X})&=\\text{etr}\\big (-\\frac{1}{2}\\mathbf {X}^{\\dagger }\\mathbf {R}_e\\mathbf {X}\\mathbf {B}^{\\dagger }\\mathbf {B}\\big )\\\\ &=\\text{etr}\\big (-\\frac{1}{2}\\mathbf {X}^{\\dagger }\\mathbf {R}_e\\mathbf {X}\\mathbf {I}_s\\big ).$ As Eqn.",
"(REF ) is the characteristic function of a complex Gaussian matrix with its covariance matrix $\\mathbf {R}_e\\otimes \\mathbf {I}_s$ , the proof is completed.",
"$\\blacksquare $"
],
[
"Proof of Theorem 3",
"Let us define the cdf $F_{\\lambda _k}(x)$ as $F_{\\lambda _k}(x)&=P(\\lambda _k\\le x)\\\\ &=P(\\lambda _{k-1}\\le x)+p,$ where $p=P(\\lambda _n<\\dots <\\lambda _k<x<\\lambda _{k-1}<\\dots <\\lambda _1)$ .",
"Let the domain be $D_1=\\lbrace 0<\\lambda _1<\\dots <\\lambda _n<x\\rbrace $ , $D_2=\\lbrace x<\\lambda _1<\\dots <\\lambda _n<\\infty \\rbrace $ , and $D_3=\\lbrace \\lambda _n<\\dots <\\lambda _k<x<\\lambda _{k-1}<\\dots <\\lambda _1\\rbrace $ .",
"Lemma 6 (Proved in [39]): The joint pdf of the ordered eigenvalues $\\lambda _1>\\cdots >\\lambda _n>0$ of a receiver-side correlated central Wishart matrix $\\mathbf {W}\\sim W_n(m,\\mathbf {0}_n,\\mathbf {R}_a)$ is $f_{\\mathbf {\\lambda }}(\\mathbf {\\lambda })=K_0^{-1}\\det \\big [\\mathbf {G}, \\mathbf {E}(\\mathbf {\\lambda })\\big ]\\prod _{i<j}^n(\\lambda _i-\\lambda _j)\\prod _{i=1}^n\\lambda _i^{b-n},$ where $K_0={\\left\\lbrace \\begin{array}{ll}\\prod _{i=1}^a \\sigma _i^{b-n}(b-i)!\\prod _{i<j}^a\\sigma _i-\\sigma _j, &\\text{$b\\ge a$},\\\\\\prod _{i=1}^b(b-i)!\\prod _{i<j}^a\\sigma _i-\\sigma _j, & \\text{$b<a$},\\end{array}\\right.",
"}$ and $\\mathbf {G}$ is a $a\\times (a-n)$ matrix, whose $(i,j)$ th element is $\\sigma _i^{j-1}$ .",
"$\\mathbf {\\sigma }=(\\sigma _1,...\\sigma _a)$ are the eigenvalues of $\\mathbf {R}_a$ , such that $\\sigma _1>...>\\sigma _a>0$ .",
"$\\mathbf {E}(\\mathbf {\\lambda })$ is a $a\\times n$ matrix, whose $(i,j)$ th element is $[\\sigma _i^{a-n-1}\\exp (-\\lambda _{j-a+n}/\\sigma _i)]$ .",
"Integrating Eqn.",
"(REF ) over $D_3$ , we can get the probability $p$ as $p&=K_0^{-1}\\int _{D_3}\\det [\\mathbf {G},\\mathbf {E}(\\mathbf {\\lambda })]\\prod _{i<j}^n(\\lambda _i-\\lambda _j)\\prod _{i=1}^n\\lambda _i^{b-n}d\\lambda _i.$ Performing the Laplace expansion over the first $a-n$ columns of $[\\mathbf {G}, \\mathbf {E}(\\mathbf {\\lambda })]$ , we gave $\\det [\\mathbf {G}, \\mathbf {E}(\\mathbf {\\lambda })]=\\sum _{\\mathbf {\\kappa }\\in \\mathcal {Q}(i)}(-1)^{\\sum _{i=1}^{a-n}(\\kappa _i+i)}\\det [\\mathbf {G}^{\\mathbf {\\kappa }}]\\det [\\mathbf {E}^{\\mathbf {\\kappa }}(\\mathbf {\\lambda })],$ where $\\mathcal {Q}(i)$ is a set of all permutations $(\\kappa _1,...,\\kappa _a)$ of the integers $(1,...,a)$ , such that $(\\kappa _1<\\kappa _2<...<\\kappa _{a-n})$ and $(\\kappa _{a-n+1}<\\kappa _{a-n+2}<...<\\kappa _a)$ .",
"Hence, $\\sum _{\\mathbf {\\kappa }\\in \\mathcal {Q}(i)}$ denotes the summation over two combinations $(\\kappa _1<\\kappa _2<...<\\kappa _{a-n})$ and $(\\kappa _{a-n+1}<\\kappa _{a-n+2}<...<\\kappa _a)$ .",
"$[\\mathbf {E}^{\\mathbf {\\kappa }}(\\mathbf {\\lambda })]$ is a $n\\times n$ matrix, i.e., $[\\mathbf {E}^{\\mathbf {\\kappa }}(\\mathbf {\\lambda })]_{i,j}=\\sigma ^{a-n-1}_{\\kappa _{a-n+i}} \\exp (-\\lambda _j/\\sigma _{\\kappa _{a-n+i}})$ for $i, j=1,...,n$ .",
"$[\\mathbf {G}^{\\mathbf {\\kappa }}]$ is a $(a-n)\\times (a-n)$ Vandermonde matrix, i.e., $[\\mathbf {G}^{\\mathbf {\\kappa }}]_{i,j}=\\sigma _{\\kappa _{i}}^{j-1}$ for $i, j=1,...,a-n$ .",
"When $a=n$ , we set $\\det [\\mathbf {G}^{\\mathbf {\\kappa }}]=1$ .",
"Figure: NO_CAPTIONNext, we prove Eqn.",
"(REF ) for simplifying Eqn.",
"(REF ).",
"In Eqn.",
"(REF ), $\\sum ^{\\sim }_q$ denotes the summation over all permutations $(q_1,\\dots ,q_n)$ of $(1,\\dots ,n)$ , $\\sum ^{\\sim }_\\iota $ is the summation over all permutations $(\\iota _1,\\dots ,\\iota _n)$ of $(1,\\dots ,n)$ , and per($\\iota _1,\\dots ,\\iota _n$ ) is either 0 or 1, corresponding to even or odd value of the permutation $(\\iota _1,\\dots ,\\iota _n)$ .",
"Then, $p$ can be written as $p&=K_0^{-1}\\int _{D_3}\\det [\\mathbf {G},\\mathbf {E}(\\mathbf {\\lambda })]\\prod _{i<j}^n(\\lambda _i-\\lambda _j)\\prod _{i=1}^n\\lambda _i^{b-n}d\\lambda _i \\\\ &=K_0^{-1}\\sum _{\\mathbf {\\kappa }\\in \\mathcal {Q}(i)}(-1)^{\\sum _{i=1}^{a-n}(\\kappa _i+i)}\\det [\\mathbf {G}^{\\mathbf {\\kappa }}] \\prod _{i=1}^n\\sigma _{\\kappa _{a-n+i}}^{a-n-1} \\sum ^{\\sim }_q\\sum ^{\\sim }_\\iota \\\\ &\\times (-1)^{\\text{per}(\\iota _1,\\dots ,\\iota _n)}\\int _{D_3}\\prod _{i=1}^n\\lambda _{q_i}^{\\iota _i-1}\\exp (-\\frac{\\lambda _{q_i}}{\\sigma _{\\kappa _{a-n+i}}})\\prod _{i=1}^n\\lambda _i^{b-n}d\\lambda _{q_i}\\\\ &=K_0^{-1}\\sum _{\\mathbf {\\mu }\\in \\mathcal {P}(k)} \\sum _{\\mathbf {\\kappa }\\in \\mathcal {Q}(i)}(-1)^{\\sum _{i=1}^{a-n}(\\kappa _i+i)}\\det [\\mathbf {G}^{\\mathbf {\\kappa }}]\\prod _{i=1}^n\\sigma _{\\kappa _{a-n+i}}^{a-n-1} \\\\ &\\times \\sum ^{\\sim }_\\iota (-1)^{\\text{per}(\\iota _1,\\dots ,\\iota _n)}I_1(\\mu ,\\iota ,\\kappa )I_2(\\mu ,\\iota ,\\kappa ),$ where $\\sum _q^{\\sim }=\\sum _{\\mathbf {\\mu }\\in \\mathcal {P}(k)}\\sum ^{\\sim }_{q_{\\mu _{\\psi }}}\\sum ^{\\sim }_{q_{\\mu _{\\omega }}}$ , and $\\sum ^{\\sim }_{q_{\\mu _{\\psi }}}$ denotes the summation over the permutations $(q_{\\mu _1},\\dots ,q_{\\mu _{k-1}})$ of $(1,\\dots ,k-1)$ , $\\sum ^{\\sim }_{q_{\\mu _{\\omega }}}$ calculates the summation over the permutations $(q_{\\mu _k},\\dots ,q_{\\mu _n})$ of $(k,\\dots ,n)$ , $\\sum _{\\mathbf {\\mu }\\in \\mathcal {P}(k)}$ is the summation over the combination of sets $(\\mu _1<\\mu _2<\\dots <\\mu _{k-1})$ and $(\\mu _k<\\mu _{k+1}<\\dots <\\mu _n)$ , and $(\\mu _1,\\dots ,\\mu _n)$ is a permutation of $(1,\\dots ,n)$ .",
"From [25], we obtain $I_1(\\mu ,\\iota ,\\kappa )&=\\sum ^{\\sim }_{q_{\\mu _{\\psi }}}\\int _{D_4}\\prod _{i=1}^{k-1}\\lambda _{q_{\\mu _i}}^{b-n+\\iota _{i}-1}\\exp (-\\frac{\\lambda _{q_{\\mu _i}}}{\\sigma _{\\kappa _{a-n+\\mu _i}}})d\\lambda _{q_{\\mu _i}} \\\\&=\\prod _{i=1}^{k-1}\\int _x^{\\infty }\\lambda _{\\mu _i}^{b-n+\\iota _{i}-1}\\exp (-\\frac{\\lambda _{\\mu _i}}{\\sigma _{\\kappa _{a-n+\\mu _i}}})d\\lambda _{\\mu _i} \\\\&=\\prod _{i=1}^{k-1}\\sigma _{\\kappa _{a-n+\\mu _i}}^{b-n+\\iota _i}\\Gamma (b-n+\\iota _i,\\frac{\\lambda _{\\mu _i}}{\\sigma _{\\kappa _{a-n+\\mu _i}}}), \\\\ I_2(\\mu ,\\iota ,\\kappa )&=\\sum ^{\\sim }_{q_{\\mu _{\\omega }}}\\int _{D_5}\\prod _{i=k}^{n}\\lambda _{q_{\\mu _i}}^{b-n+\\iota _{i}-1}\\exp (-\\frac{\\lambda _{q_{\\mu _i}}}{\\sigma _{\\kappa _{a-n+\\mu _i}}})d\\lambda _{q_{\\mu _i}} \\\\&=\\prod _{i=k}^{n}\\int _0^x\\lambda _{\\mu _i}^{\\iota _{i}-1}\\exp (-\\frac{\\lambda _{\\mu _i}}{\\sigma _{\\kappa _{a-n+\\mu _i}}})d\\lambda _{\\mu _i} \\\\&=\\prod _{i=k}^{n}\\sigma _{\\kappa _{a-n+\\mu _i}}^{b-n+\\iota _i}\\gamma (b-n+\\iota _i,\\frac{\\lambda _{\\mu _i}}{\\sigma _{\\kappa _{a-n+\\mu _i}}}),$ where $D_4=\\lbrace x<\\lambda _{k-1}<\\dots <\\lambda _1<\\infty \\rbrace $ and $D_5=\\lbrace 0<\\lambda _n<\\dots <\\lambda _k<x\\rbrace $ .",
"$\\iota _i$ is the $i$ th position after re-ordering $(\\iota _1,\\dots ,\\iota _n)$ , which can be viewed as the column index of the determinant of an $(n\\times n)$ matrix.",
"$\\mu _i$ is the row index of the determinant of the $(n\\times n)$ matrix dependent on $k$ .",
"Hence, $\\sum ^{\\sim }_\\iota (-1)^{\\text{per}(\\iota _1,\\dots ,\\iota _n)}I_1(\\mu ,\\iota ,\\kappa )I_2(\\mu ,\\iota ,\\kappa )$ denotes the determinant of a matrix, each element of which is expressed by $[\\mathbf {\\Theta }(\\mathbf {\\mu },\\mathbf {\\sigma },\\mathbf {\\kappa },k;x)]_{\\mu _i,i}$ .",
"We can re-define the order index numbers of rows and columns of the determinant as $u$ and $\\mu _v$ .",
"Finally, we get $p&=K_0^{-1}\\sum _{\\mathbf {\\mu }\\in \\mathcal {P}(k)} \\sum _{\\mathbf {\\kappa }\\in \\mathcal {Q}(i)}(-1)^{\\sum _{i=1}^{a-n}(\\kappa _i+i)}\\det [\\mathbf {G}^{\\mathbf {\\kappa }}] \\\\ & \\times \\prod _{i=1}^n\\sigma _{\\kappa _{a-n+i}}^{a-n-1}\\det \\big [\\mathbf {\\Theta }(\\mathbf {\\mu },\\mathbf {\\sigma },\\mathbf {\\kappa },k;x)\\big ],$ where $(n\\times n)$ real matrix $\\mathbf {\\Theta }(\\mathbf {\\mu },\\mathbf {\\sigma },\\mathbf {\\kappa },k;x)$ is defined as $&\\big [\\mathbf {\\Theta }(\\mathbf {\\mu },\\mathbf {\\sigma },\\mathbf {\\kappa },k;x)\\big ]_{u,\\mu _v} \\\\ &={\\left\\lbrace \\begin{array}{ll}\\sigma _{\\kappa _{a-n+u}}^{b-n+\\mu _v}\\Gamma (b-n+\\mu _v,\\frac{x}{\\sigma _{\\kappa _{a-n+u}}}),&\\text{$v=1,...,k-1$},\\\\\\sigma _{\\kappa _{a-n+u}}^{b-n+\\mu _v}\\gamma (b-n+\\mu _v,\\frac{x}{\\sigma _{\\kappa _{a-n+u}}}),&\\text{$v=k,...,n$},\\end{array}\\right.",
"}$ for $u,v=1, ..., n$ , where $\\Gamma (\\cdot ,\\cdot )$ and $\\gamma (\\cdot ,\\cdot )$ are the upper and lower incomplete Gamma functions defined in Eqns.",
"(REF ) and ().",
"Since we have $&\\prod _{i=1}^n\\sigma _{\\kappa _{a-n+i}}^{a-n-1}\\det \\big [\\mathbf {\\Theta }(\\mathbf {\\mu },\\mathbf {\\sigma },\\mathbf {\\kappa },k;x)\\big ]\\\\ &= \\prod _{i=1}^n\\sigma _{\\kappa _{a-n+i}}^{b-n}\\det [\\mathbf {\\Psi }(\\mathbf {\\mu },\\mathbf {\\sigma },\\mathbf {\\kappa },k;x)],$ where $(n\\times n)$ real matrix $\\mathbf {\\Psi }(\\mathbf {\\mu },\\mathbf {\\sigma },k,\\kappa ;x)$ is defined as $&[\\mathbf {\\Psi }(\\mathbf {\\mu },\\mathbf {\\sigma },\\mathbf {\\kappa },k;x)]_{u,\\mu _v}\\\\ &={\\left\\lbrace \\begin{array}{ll}\\sigma _{\\kappa _{a-n+u}}^{a-n+\\mu _v-1}\\Gamma (b-n+\\mu _v,\\frac{x}{\\sigma _{\\kappa _{a-n+u}}}),&\\text{$v=1,...,k-1$},\\\\\\sigma _{\\kappa _{a-n+u}}^{a-n+\\mu _v-1}\\gamma (b-n+\\mu _v,\\frac{x}{\\sigma _{\\kappa _{a-n+u}}}),&\\text{$v=k,...,n$},\\end{array}\\right.",
"}$ for $u,v=1, ..., n$ .",
"Substituting Eqn.",
"(REF ) to Eqn.",
"(REF ) and performing the inverse Laplace expansion of Eqn.",
"(REF ), we obtain $p&=K_0^{-1}\\prod _{i=1}^n\\sigma _{i}^{b-n}\\sum _{\\mathbf {\\mu }\\in \\mathcal {P}(k)} \\det [\\mathbf {G},\\mathbf {\\Psi }(\\mathbf {\\mu },\\mathbf {\\sigma },k;x)] \\\\ &=K^{-1}\\sum _{\\mathbf {\\mu }\\in \\mathcal {P}(k)} \\det [\\mathbf {G},\\mathbf {\\Psi }(\\mathbf {\\mu },\\mathbf {\\sigma },k;x)],$ where $K=\\prod _{i<j}^n\\sigma _i-\\sigma _j \\prod _{i=1}^n(b-i)!.$ $F_{\\lambda _k}(x)$ can be expressed by $F_{\\lambda _k}(x)=K^{-1}\\sum _{i=1}^k\\sum _{\\mathbf {\\mu }\\in \\mathcal {P}(i)}\\det [\\mathbf {G},\\mathbf {\\Psi }(\\mathbf {\\mu },\\mathbf {\\sigma },i;x)],$ which is the marginal cdf of the $k$ th largest eigenvalue $\\lambda _k$ of a receiver-side correlated central Wishart matrix $\\mathbf {W}$$\\sim $$W_n(m,\\mathbf {0}_n,\\mathbf {R}_a)$ .",
"The marginal pdf of the $k$ th largest eigenvalue can be easily derived from the derivative of a determinant as shown in [40], which is $f_{\\lambda _k}(x)&=\\frac{d}{dx}\\bigg \\lbrace K^{-1}\\sum _{i=1}^k\\sum _{\\mathbf {\\mu }\\in \\mathcal {P}(i)}\\det \\big [\\mathbf {G},\\mathbf {\\Psi }(\\mathbf {\\mu },\\mathbf {\\sigma },i;x)\\big ]\\bigg \\rbrace \\\\&=K^{-1}\\sum _{i=1}^k\\sum _{\\mathbf {\\mu }\\in \\mathcal {P}(i)}\\sum _{j=1}^{n}\\det \\big [\\mathbf {G},\\mathbf {\\Omega }(\\mathbf {\\mu },\\mathbf {\\sigma },i,j;x)\\big ],$ where $(n\\times n)$ real matrix $\\mathbf {\\Omega }(\\mathbf {\\mu },\\mathbf {\\sigma },i,j;x)$ is defined in Eqn.",
"(REF ).",
"This completes the proof.",
"$\\blacksquare $"
],
[
"Proof of Theorem 4",
"According to Jensen's inequality, we have $C_{\\mathbf {A}}(\\mathbf {R}_a, \\rho ,\\eta )=&\\sum _{i=1}^{\\eta }\\text{E}\\big \\lbrace \\log _2[1+(P/t)\\lambda _i(\\mathbf {A}\\mathbf {A}^{\\dagger })]\\big \\rbrace \\\\ \\le & \\sum _{i=1}^{\\eta }\\log _2\\big \\lbrace 1+(P/t)\\text{E}[\\lambda _i(\\mathbf {A}\\mathbf {A}^{\\dagger })]\\big \\rbrace ,$ where $\\lambda _1(\\mathbf {A}\\mathbf {A}^{\\dagger })>\\lambda _2(\\mathbf {A}\\mathbf {A}^{\\dagger })>\\cdots >\\lambda _n(\\mathbf {A}\\mathbf {A}^{\\dagger })$ are the ordered eigenvalues of $\\mathbf {A}\\mathbf {A}^{\\dagger }$ .",
"Thus, $C_{\\mathbf {H}}(\\mathbf {R}_r, \\rho ,s_1)$ in Eqn.",
"(REF ) can be expressed as $C_{\\mathbf {H}}(\\mathbf {R}_r, \\rho ,s_1)=\\chi _1=\\sum _{i=1}^{s_1}\\log _2\\big \\lbrace 1+\\rho \\text{E}[\\lambda _i(\\mathbf {H}\\mathbf {H}^{\\dagger })]\\big \\rbrace .$ From [32] or [33], we get $ C_{\\mathbf {H}_3}(\\mathbf {R}_e, \\rho ,n_1)=\\log _2\\bigg [1+\\sum _{k=1}^{e}\\rho ^k\\prod _{i=0}^{k-1}(m_1-i)\\varrho _k\\bigg ],$ and $ C_{\\mathbf {H}_4}(\\mathbf {R}_e,\\rho ,e)=\\log _2\\bigg [1+\\sum _{k=1}^{e}\\rho ^k\\prod _{i=0}^{k-1}(t-i)\\varrho _k \\bigg ],$ respectively, where $\\varrho _k$ , $n_1$ , and $m_1$ are defined in Eqns.",
"() and ().",
"We can simplify $C_{\\mathbf {H}_3}(\\mathbf {R}_e, \\rho ,n_1)-C_{\\mathbf {H}_4}(\\mathbf {R}_e,\\rho ,e)$ as $&C_{\\mathbf {H}_3}(\\mathbf {R}_e, \\rho ,n_1)-C_{\\mathbf {H}_4}(\\mathbf {R}_e,\\rho ,e)\\\\ &=\\chi _2= \\log _2\\bigg [\\frac{1+\\sum _{k=1}^{e}\\rho ^k\\prod _{i=0}^{k-1}(m_1-i)\\varrho _k}{1+\\sum _{k=1}^{e}\\rho ^k\\prod _{i=0}^{k-1}(t-i)\\varrho _k} \\bigg ].$ Hence, Eqn.",
"(REF ) can be expressed approximately by $R^{\\text{app}}_s&=[\\chi _1+\\chi _2]^{+}.$ This completes the proof.",
"$\\blacksquare $ [Figure: NO_CAPTIONFigure: NO_CAPTIONFigure: NO_CAPTIONFigure: NO_CAPTION"
]
] | 1906.04289 |
[
[
"Identification of taxon through classification with partial reject\n options"
],
[
"Abstract Identification of taxa can significantly be assisted by statistical classification based on trait measurements in two major ways; either individually or by phylogenetic (clustering) methods.",
"In this paper we present a general Bayesian approach for classifying species individually based on measurements of a mixture of continuous and ordinal traits as well as any type of covariates.",
"It is assumed that the trait vector is derived from a latent variable with a multivariate Gaussian distribution.",
"Decision rules based on supervised learning are presented that estimate model parameters through blockwise Gibbs sampling.",
"These decision regions allow for uncertainty (partial rejection), so that not necessarily one specific category (taxon) is output when new subjects are classified, but rather a set of categories including the most probable taxa.",
"This type of discriminant analysis employs reward functions with a set-valued input argument, so that an optimal Bayes classifier can be defined.",
"We also present a way of safeguarding against outlying new observations, using an analogue of a $p$-value within our Bayesian setting.",
"Our method is illustrated on an original ornithological data set of birds.",
"We also incorporate model selection through cross-validation, examplified on another original data set of birds."
],
[
"Introduction",
"The general field of population ecology relies on correct classification of organisms to taxa, which in turn requires that reliable and detailed data is available.",
"Classification is, in its general form, performed using observations of traits.",
"These can be morphological or genetic, but in either case there is some kind of reference material (or a training data set) to which the registered data is compared and matched.",
"Deciding which taxa to assign a particular subject to is based on which referential material is the most similar.",
"Constructing the referential material is in itself a difficult task, but one we shall not consider in this paper.",
"Without a reference material, which serves as training data, the nature of our problem would correspond to unsupervised learning and to a large extent fall under general model-based clustering [2], [10], [11] of observations taken from a mixture of multivarian Gaussian distributions.",
"The popular R-package mclust's [14] comprehensive take on classification and clustering for Gaussian mixtures, in a frequentist setting, includes many of these methods.",
"To the authors' knowledge, there has been no published interpretable and general solution to the issue of statistically comparing the observed taxonomic traits to the referential material.",
"The widely used work of [15] in bird species identification makes rich use of numerical combinations (often refered to as “wing formulae”) of trait observation.s to classify birds to taxa, but offers no general solution.",
"The method presented in this paper is a generalization of the approaches suggested in [15] and [9].",
"Modern approaches to species classification include the Merlin Bird ID software, by the Cornell Lab of OrnithologySee merlin.allaboutbirds.org.. Merlin Bird ID uses convolutional neural networks for image recognition [8] and gives the user a ranked list of species to choose from.",
"Due to the machine learning approach parameter interpretability is complicated, and not accessible to the user.",
"In this paper we develop a method that makes it possible to conclude what trait measurements signify various taxa, as an alternative to neural networks and other black box methods.",
"It can be summarized as an instance of Bayesian discriminant analysis, where parameters are estimated from training data by means of blockwise Gibbs sampling [4].",
"Since we also allow for uncertainty of the classified taxa, our methods could also be labeled as fuzzy classfication or fuzzy discriminant analysis [1], [12], [5].",
"The primary scenario for which we envision use of our proposed method in as follows: A researcher has a desire to construct a procedure to classify subjects into a, usually small, number of taxa.",
"The procedure should not require any advanced or expensive technology, but rather be viable using only visual observations or simple measurements from everyday tools, such as rulers, callipers and scales.",
"Such a procedure might be widely used in data collection by population ecologists, putting very high requirements on the classification procedure in order to avoid misclassifications.",
"It is assumed that the researcher has access to training data where each subject has a know taxon, and the traits shared by the taxa are observed for the subjects, although not neccessarily perfectly.",
"To achieve the goal, the researcher would apply our proposed methodology and, if feasible, derive a comprehensive list of which trait observations that signify the various taxa, and which are ambiguous.",
"The secondary scenario for using our methodology is that in which an agent desires to classify an organism to a taxon, but is unable to do so without some kind of aid.",
"This could be due to inexperience of the considered taxa, or because the number of traits and the number of different outcomes of the traits are too vast to remember.",
"In such a situation, our method can provide tailored prediction of a taxon given the observed traits.",
"Already at this point we notice some of the requirements on a general statistical methodology for the scenarios that we envision; the data can be a mixture of various types of observations (continuous, integer valued, ordered categorical), there might be partial observations (due to e.g.",
"rounding or grouping) or missing values.",
"We will use the umbrella term obfuscated observations for partial and missing observations.",
"As an additional requirement, we must also be able to provide some kind of cutoff for the risk of misclassifying a subject, to adhere to the importance of correct classification for population ecology purposes.",
"For instance, an observation of a species outside of it's expected area of occurence has much more impact than an observation of a frequently occuring species.",
"Hence one may prefer that the classifier signals for the rare species when the degree of belief in it is much higher than for the other species.",
"Another aspect of population ecology data is that taxonomic trait measurments are usually informative conditional on other information.",
"In other words, the same value of a trait measurement might indicate different taxa for a subject, depending on the particular subject's additional information.",
"We will summarize this additional information in terms of a set of covariates, and construct our classifier conditional on the covariate information for each subject.",
"Moreover, we will allow variances of and covariances between traits to depend on covariates, essentially relaxing any assumption of homoscedasticity.",
"Our paper is organized as follows.",
"In Section , we present the real world application, with original data, which we will employ as we present and analyze our method of classification.",
"The core models, multivariate and multiple regression with Gaussian errors for perfectly observed data, and the corresponding latent Gaussian models for obfuscated data, are presented in Section in general but concise terms, and specified for our type of data.",
"In our classification setting, this core model is strongly inspired by Bayesian Quadratic Discriminant Analysis [3], and the way in which we extend it to encompass various types of observations and obfuscations draws on work by [10] and [11] for Bayesian Item Response Theory models.",
"In both of the aforementioned papers, the inclusion of covariates is mentioned as an extension in the discussion sections.",
"Here we contribute to this extension, assuming that both the location and the scale parameter of the latent Gaussian distribution are allowed to depend on covariates.",
"Section presents canonical maximum aposteriori classification (assigning new observations to categories/taxa) and a fuzzy extension which allows several taxa to be classified, based on a maximizing a general set valued reward function.",
"Moreover, we add an option of rejecting all categories for outlying trait measurement, where outlyingness is quantified in terms of a multivariate $p$ -value.",
"[1] and [5] present rejection options for binary classification, and here we extend it to an arbitrary number of categories.",
"In Section we propose model selection (both for traits and covariates) through cross-validation and apply it on another original data set.",
"Section contains the discussion.",
"We have moved most of the mathematical details to the appendices, and will refer to them accordingly.",
"As a final note, we stress that the presented approach in no way is limited to population ecology data and classification of subjects to taxa.",
"It is applicable more generally for fuzzy classification of traits vectors whose components might be of different types."
],
[
"Bird species classification",
"In order to showcase in detail the use of our method, we will present an example of bird species classification using an original data set collected in part by the first author.",
"A minor subset of this primary data set was used in [17] and [9], articles that apply various statistical classification approaches for identifying taxa, but the data set as a whole is unpublished.",
"We will also make use of a second original, unpublished data set in Section , the details of which we present in that section.",
"Both data sets were collected by the Falsterbo Bird Observatory.",
"The primary data set concerns four morphologically similar warblers in the Acrocephalus genus; Eurasian Reed Warbler (Acrocephalus scirpaceus), Marsh Warbler (Acrocephalus palustris), Blyth's Reed Warbler (Acrocephalus dumetorum) and Paddyfield Warbler (Acrocephalus agricola).",
"They constitute a typical example of species that experts with long experience can classify through visual observation, whereas those with less experience usually need measurements of particular traits to aid their classification.",
"Our material contains measurements of these traits for birds that have been identified by experts.",
"Data on the species was collected from wild birds captured and ringed in Falsterbo, Sweden, and from museum specimens at the museums of natural history in Stockholm, Copenhagen, and Tring.",
"All birds have three traits of main concern, the wing length [15], the notch length of feather P2 [9] and the relative position of the P2 notch to the rest of the wing, referred to as the notch position [15].",
"For those with less experience of bird topography, these are all measurements of different parts of the wing.",
"The only covariate is age, and it is binary, with levels juvenile and adult.",
"Age is included as a covariate due to a suspected change in the trait distribution (both location and scale) between the birds' juvenile and adult plumages.",
"Ideally, wing length and notch length are measured continuously, but in reality this is impossible.",
"Instead, wing length is rounded to integer millimeters, as this is the typical unit that gives consistent measurements.",
"By the same rationale notch length is rounded to half millimeters.",
"Finally, notch position is ordered categorical by definition.",
"It measures where in relation to the other wing feathers the notch is positioned and it is denoted by the feather closest to the notch.",
"Overall we have 54 155 observed birds, and these constitute the referential data for this classification problem.",
"These are distributed as presented in Table REF .",
"The uneven distribution of observations across taxa will be commented on throughout the analysis of the data set.",
"Table: The distribution of the Warbler dataset over measured covariates and traits.",
"Note that the Paddyfield Warbler has few observations overall, but almost all of them are complete, whereas the Reed Warbler has many observations but very few of them are complete."
],
[
"Model formulation",
"We will now present the model for the case where all trait measurements are continuous and perfectly observed and thereafter for the case where the vector of trait measurements contains different types of values, some of which might be missing."
],
[
"Ideal case; no obfuscated trait measurements",
"Suppose we have $N$ different categories, contained in the set $\\mathsf {N}= \\lbrace 1,\\ldots ,N\\rbrace $ , with prior probabilities $\\pi = (\\pi _1,\\ldots ,\\pi _N)$ .",
"With full data we measure $q$ traits and $p$ covariates of each subject.",
"Let $Y_{ijk}$ be the measurement of trait $k$ for subject $j$ in category $i$ , where $1\\le i\\le N$ , $1\\le j\\le n_i$ , $1\\le k\\le q$ and $n_i$ is the number of subjects in category $i$ .",
"We assume that $Y_{ij} = (Y_{ij1},\\ldots ,Y_{ijq}) \\sim \\text{N}\\left(m_{ij}, \\mathbf {\\Sigma }_{ij}\\right)$ are independent random vectors having a multivariate normal distribution, with $m_{ij} = (m_{ij1},\\ldots ,m_{ijq}) \\qquad \\text{and} \\qquad \\mathbf {\\Sigma }_{ij} = \\left(\\Sigma _{ijkl}\\right)_{k,l=1}^q$ being the mean vector and the covariance matrix of subject $j$ of category $i$ .",
"Let also $x_{ij} = \\left(1, x_{ij1},\\ldots ,x_{ijp}\\right) = \\left(x_{ijm}\\right)_{m=0}^p$ be the covariate vector of subject $j$ of category $i$ .",
"Trait vectors and covariate vectors of category $i$ are rows in the matrices $\\mathbf {Y}_i = \\left(Y_{i1}^\\top ,\\ldots ,Y_{in_i}^\\top \\right)^\\top $ and $\\mathbf {X}_i= \\left(x_{i1}^\\top ,\\ldots ,x_{in_i}^\\top \\right)^\\top $ respectively, where $\\top $ refers to matrix transposition.",
"We now proceed by formulating a multivariate and multiple regression model $ \\mathbf {Y}_i = \\mathbf {X}_i\\mathbf {B}_i + \\mathbf {E}_i$ for category $i$ , where $\\mathbf {B}_i=\\left(B_{imk}; m=0,\\ldots ,p; k=1,\\ldots ,q\\right)$ is the regression parameter matrix, whose first row consists of intercepts for the $q$ traits, $m_{ij}$ is the $j^{\\text{th}}$ row of $\\mathbf {X}_i\\mathbf {B}_i$ , and $\\mathbf {E}_i = \\left(E_{i1}^\\top ,\\ldots ,E_{in_i}^\\top \\right)^\\top $ is an error term matrix with independent rows $E_{ij} \\sim \\text{N}(0,\\mathbf {\\mathbf {\\Sigma }}_{ij})$ .",
"For use in the construction of a joint prior, and later the derivation of the marginal posterior distributions of the parameters, the vectorized form of our regression model is needed.",
"Denote the operation of appending columns of a matrix by $\\text{vec}(\\cdot )$ (we will also make use of the inverse operation $\\text{vec}^{-1}(\\cdot )$ on column vectors) and rewrite (REF ) as $ \\mathbf {U}_i = \\text{vec}(\\mathbf {Y}_i) = \\mathbf {Z}_i \\beta _i + \\text{vec}(\\mathbf {E}_i)$ with $\\beta _i = \\text{vec}(\\mathbf {B}_i)$ .",
"Denoting an identity matrix of rank $q$ with $\\mathbf {I}_q$ and using the matrix tensor product $\\otimes $ , $ \\mathbf {Z}_i = \\mathbf {I}_q \\otimes \\mathbf {X}_i = \\begin{pmatrix} \\mathbf {X}_i & 0 & \\cdots & 0 \\\\0 & \\mathbf {X}_i & \\ddots & \\vdots \\\\\\vdots & \\ddots & \\ddots & 0 \\\\0 & \\cdots & 0 & \\mathbf {X}_i \\end{pmatrix}$ is a block-diagonal matrix with $q$ blocks along the diagonal.",
"Now suppose we have $A$ covariance classes $\\alpha = 1,\\ldots ,A$ for category $i$ such that $ \\mathbf {\\Sigma }_{ij} = \\mathbf {\\Sigma }^\\alpha _i \\quad \\text{if } x_{ij} \\in \\mathcal {X}^\\alpha ,$ where $\\mathcal {X}= \\mathcal {X}^1 \\cup \\ldots \\cup \\mathcal {X}^A$ is a disjoint decomposition of the predictor space $\\mathcal {X}$ .",
"Assuming a prior on each of the columns of $\\mathbf {B}_i$ , and letting it be $\\text{N}\\left(\\left(b_{i0k},\\ldots ,b_{ipk}\\right)^\\top = b_{ik}, \\mathbf {\\Sigma }_{\\mathbf {B}_i}\\right)$ for $k=1,\\ldots ,q$ , implies the prior $\\text{N}\\left(\\left(b_{i1}^\\top ,\\ldots ,b_{iq}^\\top \\right)^\\top = \\beta _{i0}, \\mathbf {I}_q \\otimes \\mathbf {\\Sigma }_{\\mathbf {B}_i} = \\mathbf {\\Sigma }_{\\beta _i}\\right)$ on $\\beta _i$ .",
"Further, assuming prior independence and imposing an Inverse-Wishart distribution $\\mathbf {\\Sigma }_i^\\alpha \\sim IW(\\nu _0, \\mathbf {V}_0)$ on the covariance matrices in (REF ) for $\\alpha = 1,\\ldots ,A$ , we get the joint prior $ p(\\beta _i, \\mathbf {\\Sigma }_i^1, \\ldots , \\mathbf {\\Sigma }_i^A) = p(\\beta _i)\\prod _{\\alpha =1}^A p(\\mathbf {\\Sigma }_i^\\alpha )$ for the parameters of category $i$ .",
"Write ${\\cal D}_i=(\\mathbf {X}_i,\\mathbf {Y}_i)$ for the training data of category $i$ and let $(x,Y)$ be a new obervation that we want to classify.",
"The classifier in Section will involve the density $\\omega _i = f(Y;x|{\\cal D}_i) = \\mathbb {E}[f(Y;x,\\theta _i|{\\cal D}_i)]$ of this new observation, in case it belongs to category $i$ , where $f(Y;x;\\theta _i)$ is the density function of the trait vector $Y$ conditional on the covariate vector $x$ and the parameter vector $\\theta _i$ .",
"For a detailed derivation of the collection of model parameters $\\theta _i = \\left(\\mathbf {B}_i, \\mathbf {\\Sigma }^1_i,\\ldots ,\\mathbf {\\Sigma }_i^A\\right)$ and the posterior category weights $(\\omega _1,\\ldots ,\\omega _N)$ we refer to Appendix .",
"The Monte Carlo approximations of $\\theta _i$ and $\\omega _i$ , $i \\in \\mathsf {N}$ are $ \\hat{\\theta }^{\\text{(Bayes)}}_i = \\frac{1}{R_i}\\sum _{r=1}^{R_i} \\theta _{ir}$ and $ \\hat{\\omega }_i = \\frac{1}{R_i}\\sum _{r=1}^{R_i} f(Y;x,\\theta _{ir})$ respectively, where $R_i$ is the number of samples drawn from the posterior distribution of $\\theta _i$ , with $\\theta _{ir}$ the parameter vector obtained through blockwise Gibbs sampling in simulation run $r$ ."
],
[
"General case; all types of obfuscation may occur",
"Overall our setup is the same as in Section REF , but now we suppose there is only partial information about the complete training data set ${\\cal D}= \\lbrace (\\mathbf {X}_i,\\mathbf {Y}_i);\\, i=1,\\ldots ,N\\rbrace $ .",
"Due to some obfuscation, which could be due to rounding, grouping, categorization or lost measurements of some traits, we only know that $Y_{ij} \\in \\mathsf {S}_{ij} = \\mathsf {S}_{ij1}\\times \\cdots \\times \\mathsf {S}_{ijq},$ i.e.",
"the complete trait vector $Y_{ij}$ for subject $j$ of category $i$ is contained in a hyperrectangle $\\mathsf {S}_{ij}$ , whose components are given by $\\lbrace \\mathsf {S}_{ijk}\\rbrace _{k=1}^q$ .",
"These components are sets, ranging in possible size from singletons to infinite intervals of $\\mathbb {R}$ , and they are given by $\\mathsf {S}_{ijk} = {\\left\\lbrace \\begin{array}{ll} Y_{ijk}, & k \\notin \\mathsf {K}_{ij}, \\\\\\left(c_{ijk},d_{ijk}\\right], & k \\in \\mathsf {K}_{ij},\\end{array}\\right.",
"}$ where $\\mathsf {K}_{ij}= \\left\\lbrace k; 1\\le k \\le q; \\, Y_{ijk} \\text{ obfuscated}\\right\\rbrace $ .",
"The obfuscations are of three main types.",
"First, if a trait $Y_{ijk}$ is unobserved, written as $Y_{ijk} = \\texttt {NA}$ , the $k$ :th component of $\\mathsf {S}_{ij}$ is of infinite length; e.g.",
"$c_{ijk} = -\\infty $ , $d_{ijk} = \\infty $ , and we let the interval be open.",
"That is, the interval $\\mathsf {S}_{ijk}$ equals $\\mathbb {R}$ .",
"Secondly, a trait may be obfuscated in such a way that interval limits are observed.",
"Rounding is a typical example of this; consider a measurement of a trait $y_{ijk} \\in \\mathbb {R}^+$ that has been rounded to $z_{ijk} \\in 2\\tau \\cdot \\mathbb {Z}^+$ .",
"We put $c_{ijk} = z_{ijk} - \\tau $ and $d_{ijk} = z_{ijk} + \\tau $ , which constitute the limits of the interval around $z_{ijk}$ .",
"Generally, we assume rounding to the midpoint of an interval.",
"We can always scale so that the interval is of unit length, which would be equivalent to $\\tau = 1/2$ .",
"Lastly we have the case when we observe an ordered categorical random variable $Z_{ijk}$ .",
"We assume there is an underlying normally distributed variable $Y_{ijk}$ , and that each category $z_{ijk}$ corresponds to an interval of possible values of $y_{ijk}$ .",
"Count data can be treated as an instance of this type of obfuscation, i.e.",
"a trait may be measured in counting occurences of something, and we can handle that type of data similarily as ordered categorical data.",
"We will treat all types of obfuscations in the following unified way.",
"Suppose trait $k$ of subject $j$ of category $i$ is imperfectly observed, i.e.",
"$k \\in \\mathsf {K}_{ij}$ .",
"Let $g_{k}$ be the number of categories of this trait, which we number as $0,1,\\ldots ,g_{k}-1$ .",
"The observed category is $z_{ijk} \\in \\left\\lbrace 0,1,\\ldots ,g_{k} - 1\\right\\rbrace $ , where $g_{k} =2$ for binary data and $g_{k} = \\infty $ for count data.",
"The corresponding side of $\\mathsf {S}_{ij}$ is $\\mathsf {S}_{ijk} = {\\left\\lbrace \\begin{array}{ll}\\left(-\\infty , \\frac{1}{2}\\right], & \\text{if } z_{ijk} = 0, \\\\\\left(z_{ijk} - \\frac{1}{2}, z_{ijk} + \\frac{1}{2}\\right], & \\text{if } 1\\le z_{ijk} \\le g_{k}-2, \\\\\\left(g_{k}-\\frac{3}{2}, \\infty \\right), & \\text{if } z_{ijk} = g_{k} - 1.\\end{array}\\right.",
"}$ Here, a useful trick would be to add auxiliary categories, that never were observed, to take the place of $z_{ijk}=0$ and $z_{ijk}= g_{k} -1$ .",
"That ensures all observed intervals are of unit length, although we may let intervals vary in length if there is reason to construct such a model.",
"We also write $Z_{ijk} = z(\\mathsf {S}_{ijk}) = {\\left\\lbrace \\begin{array}{ll}0, & \\text{if } \\mathsf {S}_{ijk} = \\left(-\\infty , \\frac{1}{2}\\right], \\\\\\frac{c_{ijk} + d_{ijk}}{2}, & \\text{if $\\mathsf {S}_{ijk}$ is bounded}, \\\\g_{k} - 1, & \\text{if } \\mathsf {S}_{ijk} = \\left(g_{k}-\\frac{3}{2}, \\infty \\right],\\end{array}\\right.",
"}$ for the center point of a finite or half-open, infinite $\\mathsf {S}_{ijk}$ , whereas $Z_{ijk} =z\\left(\\mathsf {S}_{ijk}\\right) = Y_{ijk}$ when $Y_{ijk} = \\mathsf {S}_{ijk}$ is perfectly observed.",
"We will write the observed training data set as ${\\cal D}^{\\text{obs}}= \\left\\lbrace \\left(x_{ij},\\mathsf {S}_{ij}\\right);\\, i=1,\\ldots ,N,j=1,\\ldots ,n_i\\right\\rbrace .$ Finally, we remark on the importance (or lack thereof) of taking rounding into account.",
"Consider rounding a Gaussian trait $Y_{ijk}$ to $z_{ijk}\\in \\mathbb {Z}$ for some $k$ , $i=1,\\ldots ,N$ and $j=1,\\ldots ,n_i$ .",
"Suppose we have no covariates and that $Y_{ijk} \\sim N(m_{ik},\\sigma _{ik}^2)$ for $j=1,\\ldots ,n_i$ .",
"An unbiased estimator of $m_{ik}$ is the average $\\bar{Y}_{ik}$ , whereas $\\bar{Z}_{ik}$ is a biased estimator of $m_{ik}$ .",
"It is possible to quantify the size of the bias using $\\sigma _k$ and the width of the rouding interval $\\mathsf {w} = \\eta \\sigma _k$ [16].",
"In short, the larger $\\mathsf {w}$ is relative to $\\sigma _k$ , the larger the bias is, as measured by $\\eta $ .",
"Already when $\\sigma =\\mathsf {w} = 1$ , the bias is very small, and hence, unless we need an extremely precise mean estimate, the bias is small compared to the uncertainty of the parameter estimate.",
"Therefore, one might regard rounded values as true values, if the standard deviation of the trait that is rounded, is large enough.",
"For full details on the derivation of exact estimators of the model parameters $\\theta _i$ and the posterior weights $\\omega _i$ and Monte Carlo appproximations thereof, we refer to Appendices and respectively.",
"To summarize, let $(x,\\mathsf {S})$ refer to a new obserservation, for which the trait vector $Y\\in \\mathsf {S}$ is obfuscated for traits $k\\in K$ .",
"For each category $i$ we want to generate the set $ \\left\\lbrace \\theta _{ir}, Y_{ijkr}, \\, 1\\le j\\le n_i, \\, k\\in \\mathsf {K}_{ij}; Y_{kr}, k\\in \\mathsf {K}\\right\\rbrace _{r=1}^{R_i}$ of $R_i$ blockwise Gibbs sampling iterates from the joint density $ p(\\theta _i|{\\cal D}^{\\text{obs}}_i)\\prod _{j=1}^{n_i} f\\left(y_{ij\\mathsf {K}_{ij}} \\mid x_{ij}, \\mathsf {S}_{ij},\\theta _i\\right) f\\left(y_{\\mathsf {K}} \\mid x, Y_{\\mathsf {K}^\\complement };\\theta _i\\right)$ of the parameters $\\theta _i$ , the imperfectly observed training data and the imperfectly observed new data point, where $\\theta _{ir} = \\left(\\beta _{ir}, \\mathbf {\\Sigma }_{ir}^1,\\ldots ,\\mathbf {\\Sigma }_{ir}^A\\right)$ , $y_{ij\\mathsf {K}_{ij}r} = \\left(y_{ijkr}; k\\in \\mathsf {K}_{ij}\\right)$ , and $y_{\\mathsf {K}r}=\\left(y_{kr} ; k\\in \\mathsf {K}\\right)$ refer to the values of these quantities for Monte Carlo iteration $r$ , whereas $Y_{\\mathsf {K}^\\complement } = \\left(Y_k ; k\\notin \\mathsf {K}\\right)$ .",
"In particular, $\\theta _{ir}$ are drawn from the posterior distribution $p(\\theta _i|{\\cal D}^{\\text{obs}}_i)$ of $\\theta _i$ , conditional on observed data ${\\cal D}^{\\text{obs}}_i$ for category $i$ ."
],
[
"Example model fit",
"We now have the tools to fit the model to our primary data set, that on the four species of Acrocephalus genus.",
"As we can tell from the end of Section , all traits are obfuscated in some way (many are missing), there is one covariate influencing the interpretation of the trait values and we have reason to believe there are different covariance classes.",
"Our trait vectors $Y_{ij}$ are attributed to species $i=1,\\ldots ,4$ , where $i=1$ corresponds to Eurasian Reed Warbler, $i=2$ to Marsh Warbler, $i=3$ to Paddyfield Warbler and $i=4$ to Blyth's Reed Warbler, whereas $j = 1,\\ldots ,n_i$ denote individual birds within species.",
"Each $Y_{ij}$ is of length $q=3$ , where $Y_{ij1}$ is the wing length, $Y_{ij2}$ is the notch length and $Y_{ij3}$ is the notch position.",
"All traits are obfuscated, but in different ways: $Y_{ij1}$ is continuous, but rounded to nearest integer on a millimeter scale; $Y_{ij2}$ is continuous, but rounded to nearest half millimeter; whereas $Y_{ij3}$ is ordered categorical.",
"We have one covariate, age, which takes on values juveline or adult, which we code as 0 and 1 respectively.",
"We denote it by $x_1$ , and it determines to which of the $A=2$ covariance classes each observation belongs.",
"We denote the respective covariance matrices with $\\mathbf {\\Sigma }^\\texttt {juv}$ and $\\mathbf {\\Sigma }^\\texttt {ad}$ .",
"The hyperparameter values of the covariance matrix prior are $\\nu _0 = 10$ , whereas $V_0$ has a diagonal of 15 and all other elements equal to 5.",
"Since there is $p=1$ covariate, the matrix used in the construction of the prior on the vectorized regression parameters, $\\mathbf {\\Sigma }_{\\mathbf {B}_i}$ , is a diagonal matrix with diagonal $(4,1)$ for all $i$ .",
"The mean parameter values $\\mathbf {B}_{i0}=\\mathbb {E}(\\mathbf {B}_i)$ of the prior on $\\mathbf {B}_i$ are informative for each $i$ and based on the results in [9], as shown in Table REF .",
"Table: Hyperparameter values for the prior of the regression parameter matrices.",
"These are informed by , except for the first element of the second row of 𝐁 30 \\mathbf {B}_{30} and 𝐁 40 \\mathbf {B}_{40}, which we put to 0.750.75, since we strongly believe that the pattern of slightly longer wings in adult plumages also hold for these two species.Fitting the model where we consider all traits to be obfuscated, we get the Bayes estimates presented in Table REF , along with highest posterior density intervals for each parameter (see Appendix ).",
"Overall, the effect of age, our covariate, is to increase the trait values.",
"However, the increase is different across traits and across species.",
"Table: These are the Bayes estimates 𝐁 ^ imk =𝔼(𝐁 imk |𝒟 i )\\hat{\\mathbf {B}}_{imk}=\\mathbb {E}(\\mathbf {B}_{imk}|{\\cal D}_i) and Σ ^ ikl α =𝔼(Σ ikl α |𝒟 i )\\hat{\\mathbf {\\Sigma }}_{ikl}^\\alpha =\\mathbb {E}(\\mathbf {\\Sigma }_{ikl}^\\alpha |{\\cal D}_i) of all model parameters of the Acrocephalus model, rounded to two decimals, except the notch position trait where we present the categories that the regression parameters estimates fall into.",
"The coding of these categories is explained in Table 2 of ."
],
[
"Classification",
"After having fitted the model, this brings us to the next step.",
"We now have interpretable results on the variation of traits, but ultimately we also want to use our knowledge to classify new birds to species.",
"Throughout this section we will present in detail classification using this kind of model.",
"First, we define more generally the posterior category weights $\\omega _i$ that were introduced in Section REF and then look at canonical classification.",
"Then we introduce set-valued classifiers, including the possibility of classifying to empty sets in order to handle outliers.",
"We also showcase the flexibility of the underlying model for classifying among a subset of categories and end with remarks on the choice of the classifier's two tuning parameters $\\rho $ and $\\tau $ .",
"Let ${\\cal D}^{\\text{new}}= (x,\\mathsf {S})$ denote a new observation with obfuscated traits $\\mathsf {K}$ .",
"We define the posterior weight of category $i$ as $ \\omega _i = \\iint _{\\mathsf {S}} \\!",
"f(Y;x,\\theta _i) \\prod _{k\\in \\mathsf {K}} \\mathrm {d}y_{k} \\, p(\\theta _i \\mid {\\cal D}^{\\text{obs}}_i) \\, \\mathrm {d}\\theta _i,$ where $f$ is the density function of the trait vector $Y=(y_1,\\ldots ,y_q)$ of the new observation, i.e.",
"the multivariate Gaussian density function.",
"As shown in Appendix , the Markov chain in (REF ) can be used to find estimates $\\hat{\\omega }_i$ of these weights.",
"We may then approximate the posterior probability $\\hat{p}_i = \\hat{\\mathbb {P}}(I=i \\mid {\\cal D}^{\\text{new}}, {\\cal D}^{\\text{obs}})$ of ${\\cal D}^{\\text{new}}$ to be of any considered category as $ \\hat{p}_i = \\hat{\\mathbb {P}}(I=i \\mid {\\cal D}^{\\text{new}}, {\\cal D}^{\\text{obs}}) = \\frac{\\pi _i\\hat{\\omega }_i}{\\pi _1\\hat{\\omega }_1 + \\ldots + \\pi _N\\hat{\\omega }_N},$ where $I\\in \\mathsf {N}$ is the true but unknow category of the future observation, with prior distribution $\\mathbb {P}(I=i)=\\pi _i$ .",
"Let $\\mathcal {N}= \\mathcal {P}(\\mathsf {N}) \\setminus \\emptyset $ denote the collection of all non-empty subsets of $\\mathsf {N}$ .",
"Let $\\hat{\\mathrm {I}}\\in \\mathcal {N}$ be a classifier with $\\vert \\hat{\\mathrm {I}}\\vert \\ge 1$ .",
"In order to define $\\hat{\\mathrm {I}}$ we introduce a reward function $\\mathcal {N}\\times \\mathsf {N}\\ni (\\mathcal {I}, \\mathrm {I}) \\mapsto R(\\mathcal {I},\\mathrm {I})$ for all $\\mathcal {I}\\in \\mathcal {N}$ and put $\\hat{\\mathrm {I}}&= \\operatornamewithlimits{arg\\,max}_{\\mathcal {I}} \\mathbb {E}\\left[R(\\mathcal {I},\\mathrm {I}) \\mid {\\cal D}^{\\text{obs}}, {\\cal D}^{\\text{new}}\\right] \\\\&= \\operatornamewithlimits{arg\\,max}_{\\mathcal {I}} \\sum _{i=1}^N R(\\mathcal {I},i)p_i$ as the optimal Bayesian classifier, with the complete training data set ${\\cal D}^{\\text{obs}}$ and $p_i$ defined as in (REF ).",
"So, $\\hat{\\mathrm {I}}$ is the set in $\\mathcal {N}$ that maximizes the expected posterior reward.",
"Each classifier $\\hat{\\mathrm {I}}= \\hat{\\mathrm {I}}({\\cal D}^{\\text{obs}}, {\\cal D}^{\\text{new}})$ , viewed as a function of ${\\cal D}^{\\text{new}}$ , partitions the test data space into decision regions $\\Omega _{\\mathcal {I}} = \\lbrace (x,Y); \\hat{\\mathrm {I}}= \\mathcal {I}\\rbrace $ for all $\\mathcal {I}\\in \\mathcal {N}$ .",
"This gives rise to an indecisive region $\\Lambda = \\bigcup _{|\\mathcal {I}| > 1} \\Omega _{\\mathcal {I}},$ where we cannot distinguish one particular category with acceptable confidence, only eliminate some of the categories with low degree of belief.",
"There is considerable freedom in choosing the reward function $R$ .",
"For instance, [1] introduced a reward function for $N=2$ categories, with $R(\\lbrace i\\rbrace ,\\mathrm {I})=I(i\\in \\mathrm {I})$ for $i=1,2$ and $R(\\lbrace 1,2\\rbrace ,I)=c$ for some constant $0.5\\le c\\le 1$ .",
"In the next two subsections we will consider two $R$ functions, the first of which extends Chow's reward function with $c=0.5$ to arbitrary $N$ ."
],
[
"Classification to one category",
"Let $R(\\mathcal {I}, \\mathrm {I}) = {\\left\\lbrace \\begin{array}{ll} 0; & \\mathrm {I}\\notin \\mathcal {I}\\\\1/|\\mathcal {I}|; & \\mathrm {I}\\in \\mathcal {I}\\end{array}\\right.}",
"$ which has expected posterior reward $\\mathbb {E}\\left[R(\\mathcal {I}, \\mathrm {I}) \\mid {\\cal D}^{\\text{obs}}, {\\cal D}^{\\text{new}}\\right] = \\frac{1}{|\\mathcal {I}|}\\sum _{i\\in \\mathcal {I}} p_i$ and optimal classifier $ \\hat{\\mathrm {I}}= \\lbrace (N)\\rbrace = \\operatornamewithlimits{arg\\,max}_i \\pi _i \\omega _i$ where $p_{(1)} < \\ldots < p_{(N)}$ are the ordered posterior category probabilities.",
"Notice that $\\Lambda = \\emptyset $ , i.e.",
"this reward function leads to a consistent, but potentially overzealous, classifier.",
"To estimate the probability of classifying wrongly using this approach, we simulated a large number of realisations from the predictive posterior distribution of Section REF for each species, under the assumption of a uniform prior distribution over species.",
"Each draw was then attributed to a hyperrectangle in trait space in order to represent obfuscation as described in Section , resulting in half side lengths or $\\tau $ -values $(1/2, 1/4, 1/2)$ for trait $j=1,2,3$ respectively.",
"We then computed which species an observation each hyperrectangle would predict, using the classifier in (REF ).",
"This gives a numerical approximation under current obfuscations of the probability of observing a new bird and classifying it wrongly, when using the fitted model on the Acrocephalus data: $\\begin{matrix}\\hat{\\mathbb {P}}\\left(\\hat{\\mathrm {I}}\\ne \\mathrm {I}\\mid x_1=0 \\right) = 0.0251, &\\hat{\\mathbb {P}}\\left(\\hat{\\mathrm {I}}\\ne \\mathrm {I}\\mid x_1=1 \\right) = 0.0264.\\end{matrix}$ Roughly 1 in 50 birds would be classified erraneously, when birds appear uniformly from the species under consideration.",
"In Section REF we will show how to reduce this error by allowing for classification to sets of species."
],
[
"Classification to at least one category",
"Choosing the reward function $R(\\mathcal {I}, \\mathrm {I}) = I_{\\lbrace \\mathrm {I}\\in \\mathcal {I}\\rbrace } - \\rho \\vert \\left\\lbrace i \\in \\mathcal {I}; i\\ne (N) \\right\\rbrace \\vert p_{(N)},$ we get the expected posterior reward $\\mathbb {E}\\left[ R(\\mathcal {I}, \\mathrm {I}) \\mid {\\cal D}^{\\text{obs}}, {\\cal D}^{\\text{new}}\\right] &= \\sum _{i \\in \\mathcal {I}} p_i - \\rho \\left(\\vert \\mathcal {I}\\vert - I_{\\lbrace (N) \\in \\mathcal {I}\\rbrace }\\right)p_{(N)}$ which is maximized by $ \\hat{\\mathrm {I}}&= \\left\\lbrace i; p_i \\ge \\rho p_{(N)} \\right\\rbrace = \\left\\lbrace i; \\pi _i\\omega _i \\ge \\rho \\pi _{(N)}\\omega _{(N)} \\right\\rbrace .$ Thus we can tune the risk of classifying wrongly by picking $\\rho \\in [0,1]$ adequately, as it specifies an upper bound on the fraction of the largest posterior probability $p_{(N)}$ other posterior probabilities may attain and still be excluded.",
"If we choose $\\rho =0$ , we get the classifier $\\hat{\\mathrm {I}}= \\mathsf {N}$ which means $\\mathbb {P}(\\mathrm {I}\\in \\hat{\\mathrm {I}}) = 1$ for all new observations, but that prediction method does not provide any information at all.",
"The other extreme, choosing $\\rho = 1$ , leads to $\\hat{\\mathrm {I}}= \\lbrace (N)\\rbrace $ , and thus our classifier will be the same as (REF ).",
"In conclusion, our first classifier is a special case of the second.",
"Choosing $\\rho = 0.1$ yields the exclusion critera $\\hat{p}_i < \\hat{p}_{(N)}/10$ .",
"With this value on $\\rho $ , we find that the estimated probability of classifying wrongly rounded to four decimals are $\\begin{matrix}\\hat{\\mathbb {P}}\\left(\\mathrm {I}\\notin \\hat{\\mathrm {I}}\\mid x_1=0 \\right) = 0.0058, &\\hat{\\mathbb {P}}\\left(\\mathrm {I}\\notin \\hat{\\mathrm {I}}\\mid x_1=1 \\right) = 0.0058\\end{matrix}$ and that the probability of not singling out a particular species is $\\begin{matrix}\\hat{\\mathbb {P}}\\left(\\vert \\hat{\\mathrm {I}}\\vert > 1 \\mid x_1=0\\right) = 0.0790, &\\hat{\\mathbb {P}}\\left(\\vert \\hat{\\mathrm {I}}\\vert > 1 \\mid x_1=1\\right) = 0.0819.\\end{matrix}$ This means we have reduced the probability of choosing the wrong species by $76.8\\%$ for juvenile birds and $77.9\\%$ for adult birds, at a price of not singling out a species for about 8% of all observations.",
"Of these cases, only $0.0044 \\%$ will result in a classifier containing three species, meaning that we will be able to exclude half of the potential species for the vast majority of observations."
],
[
"Classification allowing for empty outputs",
"If none of the $N$ categories support test data ${\\cal D}^{\\text{new}}$ we would like to include $\\emptyset $ as a possible output of the classifier $\\hat{\\mathrm {I}}$ , so that $\\hat{\\mathrm {I}}\\ \\in \\mathcal {P}(\\mathsf {N})$ .",
"To this end, we denote the posterior weight of (REF ) as $\\omega _i(x,\\mathsf {S})$ in order to emphasize the dependence on the test data set ${\\cal D}^{\\text{new}}= (x,\\mathsf {S})$ .",
"Then let $ \\bar{\\omega }_i(x,\\mathsf {S}) = \\iint \\!",
"p(\\theta _i \\mid {\\cal D}^{\\text{obs}}_i)p(\\mathsf {S}^\\prime ,x;\\theta _i)\\, \\mathrm {d}\\theta _i \\mathrm {d}\\mathsf {S}^\\prime $ where the outer integral is taken over all $\\mathsf {S}^\\prime $ such that $\\omega _i(x,\\mathsf {S}^\\prime ) \\le \\omega _i(x,\\mathsf {S})$ .",
"We interpret $\\bar{\\omega }_i(x,\\mathsf {S})$ as a $p$ -value of test data $(x,\\mathsf {S})$ for category $i$ , i.e.",
"the probability of observing an obfuscated trait vector $\\mathsf {S}^\\prime $ of category $i$ with covariate vector $x$ , whose posterior weight $\\omega _i(x,\\mathsf {S}^\\prime )$ is at most as large as that of $(x,\\mathsf {S})$ .",
"As such, it is a measure of the degree of outlyingness of ${\\cal D}^{\\text{new}}$ .",
"Then, for a given value of $\\rho $ , we generalize the classifier (REF ) to $ \\hat{\\mathrm {I}}= \\left\\lbrace i; \\pi _i\\omega _i \\ge \\rho \\pi _{(N)}\\omega _{(N)} \\wedge \\pi _i\\bar{\\omega }_i \\ge \\tau \\right\\rbrace $ where $\\bar{\\omega }_i = \\bar{\\omega }_i(x,\\mathsf {S})$ .",
"Notice that (REF ) is as special case of (REF ) with $\\tau = 0$ .",
"Choosing $\\tau = 0.001$ results in $\\begin{matrix}\\hat{\\mathbb {P}}\\left(\\mathrm {I}= \\emptyset \\mid x_1=0 \\right) = 6.62\\cdot 10^{-4}, &\\hat{\\mathbb {P}}\\left(\\mathrm {I}= \\emptyset \\mid x_1=1 \\right) = 6.32\\cdot 10^{-4}.\\end{matrix}$ The probability of not choosing a set containing the correct species is $\\begin{matrix}\\hat{\\mathbb {P}}\\left(\\mathrm {I}\\notin \\hat{\\mathrm {I}}\\mid x_1=0 \\right) = 0.0066, &\\hat{\\mathbb {P}}\\left(\\mathrm {I}\\notin \\hat{\\mathrm {I}}\\mid x_1=1 \\right) = 0.0065,\\end{matrix}$ whereas the probability of not singling out a particular species ($|\\hat{\\mathrm {I}}|>1$ ) or getting an outlier ($|\\hat{\\mathrm {I}}|=0$ ) is $\\begin{matrix}\\hat{\\mathbb {P}}\\left( |\\hat{\\mathrm {I}}|\\ne 1\\right) = 0.0791, &\\hat{\\mathbb {P}}\\left( |\\hat{\\mathrm {I}}|\\ne 1\\right) = 0.0820.\\end{matrix}$ These probabilities are very close to the ones in Section REF , meaning we can hedge the risk of classifying something that might be a new species (not belonging to $\\lbrace 1,\\ldots ,N\\rbrace $ ) entirely at a low cost.",
"The decision regions for these values on $\\rho $ and $\\tau $ are presented graphically in Appendix , where we cover all classification scenarios with missing trait values."
],
[
"Subproblem accessibility",
"Having fitted the model to the whole set of species, one may use the fit for any subproblem, e.g.",
"classifying a bird between two species since the others are, for some reason, ruled out.",
"Taking species 1 and 2, i.e.",
"Eurasian Reed Warbler and Marsh Warbler, we estimate the probability of classifying wrongly and the probability of ending up in the indecisive region $\\Lambda $ analogously with Section REF .",
"Using $\\rho =0.1$ and $\\tau = 0$ , we find that $\\begin{matrix}\\hat{\\mathbb {P}}\\left(\\mathrm {I}\\notin \\hat{\\mathrm {I}}\\mid x_1=0 \\right) = 0.0037, &\\hat{\\mathbb {P}}\\left(\\mathrm {I}\\notin \\hat{\\mathrm {I}}\\mid x_1=1 \\right) = 0.000971\\end{matrix}$ $\\begin{matrix}\\hat{\\mathbb {P}}\\left(\\vert \\hat{\\mathrm {I}}\\vert > 1 \\mid x_1=0\\right) = 0.0495, &\\hat{\\mathbb {P}}\\left(\\vert \\hat{\\mathrm {I}}\\vert > 1 \\mid x_1=1\\right) = 0.0099.\\end{matrix}$"
],
[
"Choosing $\\rho $ and {{formula:5b58d166-54f0-420e-93bf-04fa641ea7f0}}",
"Choosing $\\rho $ is intentionally a subjective matter.",
"In the case of a known cost of misclassifications, and a known cost of having a large indecisive region, one could certainly compute which $\\rho $ to use, in order to get the minimal expected cost.",
"However, many applications are more vague, where the user believes it is worse to misclassify, i.e.",
"predict a category to which the obervation does not belong, than to not be precise, i.e.",
"only rule out categories with sufficiently low degrees of belief, without being able to put a value on the cost.",
"Together with the choice of prior category probabilities $(\\pi _1,\\ldots , \\pi _N)$ a user may completely accomodate the prior beliefs about the classification problem at hand.",
"Indeed, $(\\pi _1,\\ldots , \\pi _N)$ captures the apriori beliefs about how expected an observation from each category is relative to the others and $\\rho $ is intended to represent the user's idea of the cost of misclassification.",
"Finally, $\\tau $ represents how much outlyingness we accept without losing trust in our classifier.",
"A potential risk for misclassification is observing a subject of a category not even considered for classification.",
"To allow for mitigation of this, we introduced $\\tau $ as a cut-off value for the trait distributions.",
"Indeed, the value of $\\tau $ determines how large deviations in trait measurements we accept without suspecting that we actually observe a subject from an unconsidered category.",
"Choosing $\\tau = 0$ allows us to classify any point in the whole trait space, i.e.",
"we believe the model is perfect in the sense that no unconsidered categories will be observed.",
"Finally, we remark that the parameter $\\rho $ could be used as a measure of how managable a classification problem is.",
"For any classification problem, we may compute the lowest $\\rho $ -value for which $\\mathrm {I}\\in \\hat{\\mathrm {I}}$ in at most $\\psi \\,\\%$ of our test cases.",
"A more managable problem would then be one for which the $\\rho $ -value is higher.",
"It can be used as a model selection tool too; for any combination of traits and/or covariates, we compute the lowest $\\rho $ -value for which $\\mathrm {I}\\in \\hat{\\mathrm {I}}$ in at most $\\psi \\,\\%$ of our test cases, and then choose a model where $\\rho $ is acceptably high, and the model is as parsimonious as possible."
],
[
"Model selection using cross-validation",
"In this section we will present an approach to model selection based on $\\kappa $ -fold cross-validation.",
"It can be used to select covariates and/or traits from a larger set, based on predictive performance and parsimonity of the model.",
"We will illustrate it on another original, unpublished data set, on two subspecies of Common chiffchaff (Phylloscopus collybita), the collybita and abietinus subspecies.",
"It contains measurements for birds classified to subspecies visually by an expert at Falsterbo Bird Observatory."
],
[
"Cross-validation for our type of models",
"The idea of $\\kappa $ -fold cross-validation is well established, and used for a very wide range of model families, see e.g.",
"[18].",
"Choosing $\\kappa = n_i$ for category $i$ corresponds to the basic form of cross-validation.",
"This procedure is however computationally expensive, since one has to fit as many models ($=\\sum _{i=1}^N n_i$ ) as there are observations.",
"Since the method under study is already computationally intensive, in particular under widespread obfuscation, large $q$ and large data sets, we recommend using $\\kappa $ -fold cross-validation with $\\kappa $ a bit smaller.",
"In an interesting paper by [6] examines cross-validation in general when choosing between classifiers.",
"The author of this paper concludes that $\\kappa < n_i$ is generally preferred when picking the best classifier using cross-validation.",
"To perform $\\kappa $ -fold cross-validation in general for our class of models, we begin by choosing $\\kappa \\in \\mathbb {Z}^+$ independently of $i$ .",
"Then create fold $l$ for species $i$ by choosing uniformly at random a set $J_{il} \\subset \\lbrace 1,\\ldots ,n_i\\rbrace $ comprising $n_i/\\kappa $ or $n_i/\\kappa +1$ observations of ${\\cal D}_i$ , the training data at hand for category $i$ .",
"Repeat this for all categories until we have left-out test data sets $J_l = \\cup _{i=1}^N J_{il}$ for $l = 1,\\ldots , \\kappa $ .",
"Then for each $l$ proceed to fit models on the observations ${\\cal D}^{\\text{obs}}_{(-l)} = {\\cal D}^{\\text{obs}}\\setminus \\lbrace (x_{ij}, \\mathsf {S}_{ij}); j \\in J_{il}\\rbrace _{i=1}^N$ that were not left out and estimate the posterior category probabilities $\\hat{p}_1,\\ldots ,\\hat{p}_N$ for each observation in $J_l$ .",
"Choosing a reward function $R$ , the corresponding classifier $\\hat{\\rm {I}}$ may be applied to each set of posterior probabilities in order to generate predictions $\\hat{\\mathrm {I}}_{(-l)ij}$ of the category of the left-out observations in $J_l$ .",
"To assess the predictive performance of the $M$ models under consideration, let $w_i >0$ be weights such that $\\sum _{i=1}^N w_i = 1$ , and compute $ R^{\\texttt {cv}}_m = \\sum _{i=1}^N \\frac{w_i}{n_i} \\sum _{l=1}^\\kappa \\sum _{j\\in J_{il}} R(\\hat{\\mathrm {I}}_{(-il)ij},i),$ with a reward function that corresponds to a prespecified value of $\\rho $ and $\\tau $ , for $m=1,\\ldots ,M$ .",
"One could e.g.",
"use the weights $w_i = n_i / \\sum _{a=1}^N n_a$ or $w_i = 1/N$ , depending on whether it is more valuable to be able to predict a category with many observations or not.",
"Based on (REF ), the best classifier is $m^\\star = \\operatornamewithlimits{arg\\,max}_{m} \\left(R^{\\texttt {cv}}_1, \\ldots , R^{\\texttt {cv}}_M\\right).$ When having computed $R^{\\texttt {cv}}_m$ , for $m=1,\\ldots ,M$ , one has the possibility to choose a more parsimonious model that performes almost as well as $m^\\star $ , under the usual tradeoff between simplicity and predictive performance."
],
[
"Choosing traits and covariates to use for classification",
"We will now examplify usage of cross validation with $\\kappa =10$ folds for model selection.",
"At hand we have a data set over two subspecies of Common chiffchaff (Phylloscopus collybita).",
"Each subspecies has $q=9$ continuous traits measured.",
"Although they are measured to half millimeters, we will regard them as perfect observations (${\\cal D}^{\\text{obs}}= {\\cal D}$ ), since the rounding bias is extremely small.",
"Moreover we have two binary covariates (age and season), and we are interested in which combination of covariates and traits that best predicts the subspecies of a new bird.",
"We will consider models with one covariate, both covariates and both covariates including an interaction term, meaning that the number of predictors $p$ ranges from 1 to 3.",
"For each covariate scenario, we fit a model with one trait, for each trait in turn, and then choose the one with the highest $R^{\\texttt {cv}}$ value.",
"Keeping this trait, we then add another trait and repeat the procedure, and keep the two traits that yield the highest $R^{\\texttt {cv}}$ value.",
"The procedure is repeated until we have a model with 8 traits, and for each step, the value of $R^{\\texttt {cv}}$ is stored.",
"The main reason for doing this forward selection-like procedure is to reduce the number of models that are fitted from about $(2^9-1)\\cdot 4 \\cdot 10 = 20440$ (if every combination of traits was tried) to $(9+8+\\ldots +1)\\cdot 4\\cdot 10 = 1800$ .",
"Also, for each included trait, if the value of $R^{cv}$ does not increase, we may choose a more parsimonious model.",
"Figure REF shows a plot of how $R^{\\texttt {cv}}$ changes with the number of included traits for the various covariate combinations.",
"Figure: Each line represents, for a particular covariate combination, the change in average prediction success of the left-out data in the cross validation, as the number of traits increases.",
"That is, we use () as our reward function and w i =1/Nw_i = 1/N for all categories ii in ().",
"No clear differences between the covariate combinations are visible.",
"It is, however, clear that 3 of the covariate scenarios reach their maximum cross-validates prediction accuracy with 7 traits.",
"The set of included traits are not the same for all covariate scenarios."
],
[
"Discussion",
"Throughout this paper, we have defined and analysed a classification problem where classification is aided by two sets of observations for each subject; its trait vector $Y$ and its covariate vector $x$ , where $x$ is informative about the interpretation of $Y$ .",
"Since the trait values are often subject to various types of obfuscation, we set up a unified Bayesian framework for these situations, using a latent multivariate Gaussian distribution, with parameters estimated through supervised learning and a blockwise Gibbs sampling algorithm.",
"To formalize the classification, we introduced reward functions and two tuning parameters $\\rho \\in [0,1]$ and $\\tau \\in [0,1)$ .",
"The choice of $\\rho $ affects the size and location of the indecisive region $\\Lambda $ of our fuzzy discriminant rule.",
"This region is that part of observational space where our classifer does not have sufficient information to rule out all but one category, whereas $\\tau $ puts a limit on how much we allow an observation to deviate from the bulk of the data and still allowing it to be classified by our decision rule.",
"Finally, we present a method of covariate and/or trait selection, through cross-validation, in order to obtain classifers that are both parsimonious and efficient.",
"Overall, there are two main usages of the method presented in this paper.",
"First, one may derive distinguishing characteristics of the categories considered.",
"Secondly, one may use a fitted model to classify new observations with statistical rigour.",
"An example of the usefulness of the first case would be an ornithologist with a set of birds of known taxa, who doesn't know what morphologically separates these taxa.",
"Using this method, she may extract for which trait measurements there is a high probability of certain taxa and thereby create (and write down) an identification key.",
"Further, if there are too many combinations of trait levels to memorize, the Bayesian procedure we have described may perform the classification in an automized way.",
"An adjustment that is conceptually critical but often negligable numerically, is to correct all latent Gaussian distributions by truncating them to positive values for some of the traits.",
"The trait wing length in our first real world example has to be positive by definition, and hence we should adjust our Gaussian distribution accordingly.",
"However, considering the values of the parameter estimates (see Table REF ), it would essentially make no difference to impose such a restriction for this particular data set.",
"In other cases, it could be more important.",
"The reliance on training data with known categories can potentially be relaxed, or at least partially relaxed.",
"To take a step towards unsupervised learning, one would need to add a clustering layer to the model, still allowing for an indecisive region, where classification is ambigous.",
"Such a fuzzy clustering algorithm would transfer ideas in [1], [12], [5] and this paper of having incomplete decisions, to a framework of unsupervised learning [2], [10].",
"A challenge in this context is to incorporate the effect of covariates.",
"Specifying the number of clusters in advance would make the problem more well behaved, but might oftentimes be impossible, since the fundamental problem to solve in practice, would be to determine this number of clusters.",
"Still, this would allow the method to be used in situations where it is not known how to classify observations at all, and thus investigate how many clusters a given data set supports.",
"Modifying the method slightly in order to handle repeated measurements is a straightforward task within our multivariate Gaussian framework.",
"The benefit with repeated measurements of the traits is a better understanding of the magnitude of the meaurement error, when trait vectors of observations are replaced by averaged trait vectors, for all repeated measurements.",
"One could then incldue the number of measurements into the classification method, with direct effects on the size of the indecisive region and the accuracy of the classifier.",
"As mentioned in Section , we assume independence between the regression parameters within different categories.",
"This allows the effect of a covariate to vary between the categories, as opposed to forcing apriori the same effect of a covariate across categories.",
"However, Appendix lists the posterior means of the covariate effects from our real data example of Section , and one may notice that the effect is similar for some traits across categories, and to some extent even across traits.",
"This indicates that there is a general effect of our covariate, and hence we could construct a model that emphasizes such a general effect, by introducing apriori dependencies between the regression parameters.",
"Finally, it is of interest to apply and extend our fuzzy discrminant analysis method in order to analyze data sets where some observations are known to belong to several clusters [7].",
"This requires an extended type of reward function, $R(\\mathcal {I},\\mathrm {I})$ , where not only the classification $\\mathcal {I}$ is allowed to contain more than one category, but also the true set of categories, $\\mathrm {I}$ ."
],
[
"Acknowledgements",
"The authors are grateful for the data on Acrocephalus warblers provided by Falsterbo Bird Observatory, in particular Lennart Karlsson, Björn Malmhagen and Göran Walinder, and for helpful methodological feedback from Felix Wahl and Mathias Lindholm.",
"We thank the Museum of Natural History in Stockholm, Copenhagen and Tring.",
"Helpful comments from an anonymous reviewer are also acknowledged."
],
[
"Model formulation, complete data",
"Appendices - contain further mathematical details about the Bayesian model and classification procedure defined in Sections and respectively of the main article.",
"In order to make the text self-contained, some details of the main article have been repeated.",
"This Appendix contains, in full detail, the derivation of estimators and the posterior category weights for a model using perfectly observed data.",
"Posterior distributions for Bayesian multivariate linear regression with homoscedasticity assumption is readily available in [13], and we extend this to allow for heteroscedasticity.",
"Suppose we have $N$ different categories, contained in the set $\\mathsf {N}= \\lbrace 1,\\ldots ,N\\rbrace $ , with prior probabilities $\\pi = (\\pi _1,\\ldots ,\\pi _N)$ .",
"With full data we measure $q$ traits and $p$ covariates of each subject.",
"Let $Y_{ijk}$ be the measurement of trait $k$ for subject $j$ in category $i$ , where $1\\le i\\le N$ , $1\\le j\\le n_i$ , $1\\le k\\le q$ and $n_i$ is the number of subjects in category $i$ .",
"We assume that $Y_{ij} = (Y_{ij1},\\ldots ,Y_{ijq}) \\sim \\text{N}\\left(m_{ij}, \\mathbf {\\Sigma }_{ij}\\right)$ are independent random vectors having a multivariate normal distribution, with $m_{ij} = (m_{ij1},\\ldots ,m_{ijq}) \\qquad \\text{and} \\qquad \\mathbf {\\Sigma }_{ij} = \\left(\\Sigma _{ijkl}\\right)_{k,l=1}^q$ being the mean vector and the covariance matrix of subject $j$ of category $i$ .",
"Let also $x_{ij} = \\left(1, x_{ij1},\\ldots ,x_{ijp}\\right) = \\left(x_{ijm}\\right)_{m=0}^p$ be the covariate vector of subject $j$ of category $i$ .",
"Trait vectors and covariate vectors of category $i$ are rows in the matrices $\\mathbf {Y}_i = \\left(Y_{i1}^\\top ,\\ldots ,Y_{in_i}^\\top \\right)^\\top $ and $\\mathbf {X}_i= \\left(x_{i1}^\\top ,\\ldots ,x_{in_i}^\\top \\right)^\\top $ respectively.",
"We now proceed by formulating a multivariate and multiple regression model $ \\mathbf {Y}_i = \\mathbf {X}_i\\mathbf {B}_i + \\mathbf {E}_i$ for category $i$ , where $\\mathbf {B}_i=\\left(B_{imk}; m=0,\\ldots ,p; k=1,\\ldots ,q\\right)$ is the regression parameter matrix, whose first row consists of intercepts for the $q$ traits, $m_{ij}$ is the $j^{\\text{th}}$ row of $\\mathbf {X}_i\\mathbf {B}_i$ , and $\\mathbf {E}_i = \\left(E_{i1}^\\top ,\\ldots ,E_{in_i}^\\top \\right)^\\top $ is an error term matrix with independent rows $E_{ij} \\sim \\text{N}(0,\\mathbf {\\mathbf {\\Sigma }}_{ij})$ .",
"For use in the construction of a joint prior, and later the derivation of the marginal posterior distributions of the parameters, the vectorized form of our regression model is needed.",
"Denote the operation of appending columns of a matrix by $\\text{vec}(\\cdot )$ (we will also use the inverse operation $\\text{vec}^{-1}(\\cdot )$ on column vectors) and rewrite (REF ) as $ \\mathbf {U}_i = \\text{vec}(\\mathbf {Y}_i) = \\mathbf {Z}_i \\beta _i + \\text{vec}(\\mathbf {E}_i)$ with $\\beta _i = \\text{vec}(\\mathbf {B}_i)$ .",
"Denoting an identity matrix of rank $q$ with $\\mathbf {I}_q$ and using the matrix tensor product $\\otimes $ , $ \\mathbf {Z}_i = \\mathbf {I}_q \\otimes \\mathbf {X}_i = \\begin{pmatrix} \\mathbf {X}_i & 0 & \\cdots & 0 \\\\0 & \\mathbf {X}_i & \\ddots & \\vdots \\\\\\vdots & \\ddots & \\ddots & 0 \\\\0 & \\cdots & 0 & \\mathbf {X}_i \\end{pmatrix}$ is a block-diagonal matrix with $q$ blocks along the diagonal.",
"Now suppose we have $A$ covariance classes $\\alpha = 1,\\ldots ,A$ for category $i$ such that $ \\mathbf {\\Sigma }_{ij} = \\mathbf {\\Sigma }^\\alpha _i \\quad \\text{if } x_{ij} \\in \\mathcal {X}^\\alpha ,$ where $\\mathcal {X}= \\mathcal {X}^1 \\cup \\ldots \\cup \\mathcal {X}^A$ is a disjoint decomposition of the predictor space $\\mathcal {X}$ .",
"Assuming a prior on each of the columns of $\\mathbf {B}_i$ , and letting it be $\\text{N}\\left(\\left(b_{i0k},\\ldots ,b_{ipk}\\right)^\\top = b_{ik}, \\mathbf {\\Sigma }_{\\mathbf {B}_i}\\right)$ for $k=1,\\ldots ,q$ , implies the prior $\\text{N}\\left(\\left(b_{i1}^\\top ,\\ldots ,b_{iq}^\\top \\right)^\\top = \\beta _{i0}, \\mathbf {I}_q \\otimes \\mathbf {\\Sigma }_{\\mathbf {B}_i} = \\mathbf {\\Sigma }_{\\beta _i}\\right)$ on $\\beta _i$ .",
"Further, assuming prior independence of $\\beta _i, \\mathbf {\\Sigma }_i^1, \\ldots ,\\mathbf {\\Sigma }_i^A$ and imposing an Inverse-Wishart distribution $\\mathbf {\\Sigma }_i^\\alpha \\sim IW(\\nu _0, \\mathbf {V}_0)$ on the covariance matrices in (REF ) for $\\alpha = 1,\\ldots ,A$ , we get the joint prior $ p(\\beta _i, \\mathbf {\\Sigma }_i^1, \\ldots , \\mathbf {\\Sigma }_i^A) = p(\\beta _i)\\prod _{\\alpha =1}^A p(\\mathbf {\\Sigma }_i^\\alpha )$ for the parameters of category $i$ ."
],
[
"Estimation",
"Let $\\theta _i = \\left(\\mathbf {B}_i, \\mathbf {\\Sigma }^1_i,\\ldots ,\\mathbf {\\Sigma }_i^A\\right)$ represent all parameters of category $i$ .",
"In the following, we assume that $\\theta _1, \\ldots , \\theta _N$ are independent random vectors with probability densities $p(\\theta _1),\\ldots ,p(\\theta _N)$ defined in (REF ).",
"Introducing dependencies is of course possible, and may be important for specific problems.",
"This is briefly mentioned in Section of the main paper.",
"From Bayes' Theorem we get an aposteriori density $p(\\theta _i\\mid {\\cal D}_i) &= p(\\theta _i)C({\\cal D}_i)\\prod _{j=1}^{n_i} f\\left(y_{ij}; x_{ij}, \\theta _i\\right)\\\\&= p(\\theta _i)C({\\cal D}_i){\\cal L}(\\theta _i; {\\cal D}_i) \\\\&\\propto p(\\theta _i){\\cal L}(\\theta _i; {\\cal D}_i)$ of $\\theta _i$ given the complete training data set ${\\cal D}_i = \\lbrace (x_{ij}, Y_{ij}); \\, j=1, \\ldots ,n_i\\rbrace = \\lbrace \\mathbf {X}_i, \\mathbf {Y}_i\\rbrace $ for category $i$ .",
"The function ${\\cal L}\\left(\\theta _i; {\\cal D}_i\\right) = p\\left(\\mathbf {Y}_i \\mid \\mathbf {X}_i, \\theta _i\\right)$ is the likelihood.",
"In the last step we removed the normalizing factor $C({\\cal D}_i)=(p({\\cal D}_i))^{-1}$ , since it does not depend on $\\theta _i$ .",
"The Maximum Aposteriori (MAP)-estimator of $\\theta _i$ is $\\theta ^{(\\text{MAP})}_i &= \\operatornamewithlimits{arg\\,max}_{\\theta _i} p(\\theta _i\\mid {\\cal D}_i) \\\\&= \\operatornamewithlimits{arg\\,max}_{\\theta _i} p(\\theta _i){\\cal L}(\\theta _i; {\\cal D}_i),$ whereas the Bayes' estimator of $\\theta _i$ is $\\theta ^{(\\text{Bayes})}_i &= \\mathbb {E}\\left[\\theta _i\\mid {\\cal D}_i\\right] \\\\&= \\int \\!",
"\\theta _i p(\\theta _i\\mid {\\cal D}_i) \\, \\mathrm {d}\\theta _i \\\\&= C({\\cal D}_i)\\int \\!",
"\\theta _i p(\\theta _i){\\cal L}(\\theta _i; {\\cal D}_i) \\, \\mathrm {d}\\theta _i.$ Finally, given a new observation ${\\cal D}^{\\text{new}}= (x,Y)$ , define the posterior probability of the new observation belonging to category $i$ as $ p_i = \\mathbb {P}(\\mathrm {I}=i \\mid {\\cal D}, {\\cal D}^{\\text{new}}) = \\frac{\\pi _i\\omega _i}{\\pi _1\\omega _1 + \\ldots + \\pi _N\\omega _N},$ where $\\omega _i &= \\int \\!",
"f(Y;x,\\theta _i)p(\\theta _i \\mid {\\cal D}_i) \\, \\mathrm {d}\\theta _i \\\\&= C({\\cal D}_i) \\int \\!",
"f(Y;x,\\theta _i)p(\\theta _i) \\mathcal {L}(\\theta _i; {\\cal D}_i) \\, \\mathrm {d}\\theta _i$ are the posterior category weights given ${\\cal D}^{\\text{new}}$ for all categories, before the prior probabilities $\\pi _i$ have been taken into account."
],
[
"Monte Carlo Approximations",
"It is usually difficult to evaluate the normalizing constants $C({\\cal D}_i)$ for high-dimensional data sets, and hence also $\\theta ^{(\\text{Bayes})}_i$ and $\\omega _i$ .",
"However, it is possible to estimate $\\theta ^{(\\text{Bayes})}_i$ and $\\omega _i$ by Monte Carlo simulation, with $ \\hat{\\theta }^{\\text{(Bayes)}}_i = \\frac{1}{R_i}\\sum _{r=1}^{R_i} \\theta _{ir}$ and $ \\hat{\\omega }_i = \\frac{1}{R_i}\\sum _{r=1}^{R_i} f(Y;x,\\theta _{ir})$ respectively, if $\\theta _{i1},\\ldots ,\\theta _{iR_i}$ are $R_i$ replicates drawn from the posterior distribution $p(\\theta _i\\mid {\\cal D}_i)$ , with $\\theta _{ir} = \\left(\\beta _{ir}, \\mathbf {\\Sigma }_{ir}^1,\\ldots ,\\mathbf {\\Sigma }_{ir}^A\\right)$ .",
"We will generate $\\theta _{i1}, \\ldots , \\theta _{iR_i}$ by blockwise Gibbs sampling, and for this we need the conditional posterior distributions of $\\beta _i$ and $\\mathbf {\\Sigma }_i^\\alpha $ for $\\alpha =1,\\ldots ,A$ .",
"To derive those, we need some additional notation.",
"Let $\\mathbf {Z}_i^\\alpha $ , $\\mathbf {X}_i^\\alpha $ , $\\mathbf {Y}_i^\\alpha $ and $\\mathbf {U}_i^\\alpha $ denote the submatrices of $\\mathbf {Z}_i$ , $\\mathbf {X}_i$ , $\\mathbf {Y}_i$ and $\\mathbf {U}_i$ corresponding to covariance class $\\alpha $ , and let $\\mathbf {I}_n$ be the identity matrix of order $n$ .",
"Recall also that $\\mathbf {B}_i = \\text{vec}^{-1}(\\beta _i)$ , meaning that we know $\\mathbf {B}_i$ from $\\beta _i$ , and vice versa.",
"For simplicity of notation we omit index $i$ in the following proposition: Proposition 1 Denote the parameter vector of a Bayesian multivariate multiple regression model with $A$ covariance classes by $\\theta = \\left(\\beta , \\mathbf {\\Sigma }^1,\\ldots ,\\mathbf {\\Sigma }^A\\right)$ , where $\\beta $ is the regression parameter vector and $\\mathbf {\\Sigma }^1,\\ldots ,\\mathbf {\\Sigma }^A$ are the $A$ covariance matrices.",
"Let the prior of $\\theta $ be $p(\\theta ) = p(\\beta )\\prod _{\\alpha =1}^A p(\\mathbf {\\Sigma }^\\alpha )$ , where $\\beta \\sim \\text{N}(\\beta _0,\\mathbf {\\Sigma }_\\beta )$ and $\\mathbf {\\Sigma }^\\alpha \\sim IW(\\nu _0, \\mathbf {V}_0)$ for $\\alpha =1,\\ldots ,A$ .",
"Then the posterior distribution of $\\beta \\mid \\mathbf {U}, \\mathbf {Z}, \\mathbf {\\Sigma }^1,\\ldots ,\\mathbf {\\Sigma }^\\alpha $ is $\\text{N}(\\tilde{\\beta }, \\tilde{\\mathbf {\\Sigma }})$ , where $\\tilde{\\mathbf {\\Sigma }} &= \\left[ \\mathbf {\\Sigma }_{\\beta }^{-1} + \\sum _{\\alpha =1}^A (\\mathbf {\\Sigma }^\\alpha )^{-1} \\otimes \\left(\\mathbf {X}^\\alpha \\right)^\\top \\mathbf {X}^\\alpha \\right]^{-1}$ and $\\tilde{\\beta } &= \\tilde{\\mathbf {\\Sigma }} \\times \\left[\\mathbf {\\Sigma }_{\\beta }^{-1}\\beta _0 + \\sum _{\\alpha =1}^A \\left(\\left(\\mathbf {\\Sigma }^\\alpha \\right)^{-1} \\otimes \\left(\\mathbf {X}^{\\alpha }\\right)^\\top \\right)\\mathbf {U}^{\\alpha }\\right].$ By applying Bayes' theorem $p\\left( \\beta \\mid \\mathbf {U}, \\mathbf {Z}, \\mathbf {\\Sigma }^1, \\ldots \\mathbf {\\Sigma }^A\\right) &\\propto \\exp \\left\\lbrace -\\frac{1}{2} (\\beta - \\beta _0)^\\top \\mathbf {\\Sigma }_{\\beta }^{-1} (\\beta - \\beta _0) \\right\\rbrace \\cdot \\nonumber \\\\&\\cdot \\prod _{\\alpha =1}^A \\exp \\left\\lbrace -\\frac{1}{2} \\left(\\mathbf {U}^\\alpha - \\mathbf {Z}^\\alpha \\beta \\right)^\\top \\left(\\mathbf {\\Sigma }^\\alpha \\otimes \\mathbf {I}_{n^\\alpha } \\right)^{-1} \\left(\\mathbf {U}^\\alpha - \\mathbf {Z}^\\alpha \\beta \\right) \\right\\rbrace \\\\&= \\exp \\left\\lbrace -\\frac{1}{2} \\beta \\mathbf {C}\\beta + \\beta \\mathbf {D}\\right\\rbrace $ where $n^\\alpha $ is the number of observations in covariance class $\\alpha $ , $\\mathbf {C}&= \\mathbf {\\Sigma }_{\\beta }^{-1} + \\sum _{\\alpha =1}^A (\\mathbf {Z}^\\alpha )^\\top (\\mathbf {\\Sigma }^\\alpha \\otimes \\mathbf {I}_{n^\\alpha })^{-1} \\mathbf {Z}^\\alpha \\\\&= \\mathbf {\\Sigma }_{\\beta }^{-1} + \\sum _{\\alpha =1}^A \\left(\\mathbf {Z}^\\alpha (\\mathbf {\\Sigma }^\\alpha \\otimes \\mathbf {I}_{p+1})^{-1}\\right)^\\top \\mathbf {Z}^\\alpha \\\\&= \\mathbf {\\Sigma }_\\beta ^{-1} + \\sum _{\\alpha =1}^A ((\\mathbf {\\Sigma }^\\alpha )^{-1}\\otimes \\mathbf {I}_{p+1}) (\\mathbf {Z}^\\alpha )^\\top \\mathbf {Z}^\\alpha \\\\&= \\mathbf {\\Sigma }_{\\beta }^{-1} + \\sum _{\\alpha =1}^A \\left(\\mathbf {\\Sigma }^\\alpha \\right)^{-1} \\otimes \\left(\\mathbf {X}^{\\alpha }\\right)^\\top \\mathbf {X}^{\\alpha },$ where in the second step of the last equation we used Lemma REF below, and $\\mathbf {D}&= \\mathbf {\\Sigma }_{\\beta }^{-1}\\beta _0 + \\sum _{\\alpha =1}^A (\\mathbf {Z}^\\alpha )^\\top (\\mathbf {\\Sigma }^\\alpha \\otimes \\mathbf {I}_{n^\\alpha } )^{-1} \\mathbf {U}^\\alpha \\\\&= \\mathbf {\\Sigma }_{\\beta }^{-1}\\beta _0 + \\sum _{\\alpha =1}^A \\left(\\left(\\mathbf {\\Sigma }^\\alpha \\right)^{-1} \\otimes \\left(\\mathbf {X}^{\\alpha }\\right)^\\top \\right)\\mathbf {U}^{\\alpha }.$ Consequently, $\\beta \\mid \\mathbf {U}, \\mathbf {Z}, \\mathbf {\\Sigma }^1,\\ldots ,\\mathbf {\\Sigma }^A \\sim N(\\tilde{\\beta }, \\tilde{\\mathbf {\\Sigma }})$ where $\\tilde{\\beta } &= \\mathbf {C}^{-1}\\mathbf {D}\\\\&= \\left[ \\mathbf {\\Sigma }_{\\beta }^{-1} + \\sum _{\\alpha =1}^A (\\mathbf {\\Sigma }^\\alpha )^{-1} \\otimes \\left(\\mathbf {X}^\\alpha \\right)^\\top \\mathbf {X}^\\alpha \\right]^{-1} \\\\&\\times \\left[\\mathbf {\\Sigma }_{\\beta }^{-1}\\beta _0 + \\sum _{\\alpha =1}^A \\left(\\left(\\mathbf {\\Sigma }^\\alpha \\right)^{-1} \\otimes \\left(\\mathbf {X}^{\\alpha }\\right)^\\top \\right)\\mathbf {U}^{\\alpha }\\right]$ and $\\tilde{\\mathbf {\\Sigma }} = \\mathbf {C}^{-1} = \\left[ \\mathbf {\\Sigma }_{\\beta }^{-1} + \\sum _{\\alpha =1}^A (\\mathbf {\\Sigma }^\\alpha )^{-1} \\otimes \\left(\\mathbf {X}^\\alpha \\right)^\\top \\mathbf {X}^\\alpha \\right]^{-1}.$ Notice that $\\tilde{\\beta }$ is a multivariate version of a weighted average of the prior vector $\\beta _0$ and the least squares estimates of $\\beta $ obtained from the $A$ covariance classes.",
"The following lemma was used in the proof of Proposition REF .",
"It admits a considerable gain in computational speed, when calculating the posterior covariance matrix $\\tilde{\\mathbf {\\Sigma }}$ .",
"Lemma 1 Let $\\mathbf {Z}$ be a block-diagonal $nq \\times (p+1)q$ matrix, where there are $q$ blocks $\\mathbf {X}$ , which are $n \\times (p+1)$ -matrices, along the diagonal.",
"Let $\\mathbf {\\Sigma }$ be a symmetric, positive definite $q\\times q$ -matrix.",
"Then it holds that $\\mathbf {Z}^\\top \\left(\\mathbf {\\Sigma }\\otimes \\mathbf {I}_n\\right)^{-1}\\mathbf {Z}= \\left(\\mathbf {Z}\\left(\\mathbf {\\Sigma }\\otimes \\mathbf {I}_{p+1}\\right)^{-1}\\right)^\\top \\mathbf {Z}.$ We prove the lemma by iterated use of the mixed-product property of the tensor product.",
"Since $\\mathbf {Z}= \\left(\\mathbf {I}_q \\otimes \\mathbf {X}\\right)$ , the left hand side becomes $\\left(\\mathbf {I}_q \\otimes \\mathbf {X}\\right)^\\top \\left(\\mathbf {\\Sigma }\\otimes \\mathbf {I}_n\\right)^{-1} \\left(\\mathbf {I}_q \\otimes \\mathbf {X}\\right) &= \\left(\\mathbf {I}_q \\otimes \\mathbf {X}^\\top \\right) \\left(\\mathbf {\\Sigma }^{-1} \\otimes \\mathbf {I}_n\\right) \\left(\\mathbf {I}_q \\otimes \\mathbf {X}\\right) \\\\&= \\left(\\mathbf {I}_q\\mathbf {\\Sigma }^{-1} \\otimes \\mathbf {X}^\\top \\mathbf {I}_n \\right) \\left(\\mathbf {I}_q \\otimes \\mathbf {X}\\right) \\\\&= \\left(\\mathbf {\\Sigma }^{-1} \\otimes \\mathbf {X}^\\top \\right) \\left(\\mathbf {I}_q \\otimes \\mathbf {X}\\right) \\\\&= \\mathbf {\\Sigma }^{-1}\\mathbf {I}_q \\otimes \\mathbf {X}^\\top \\mathbf {X}\\\\&= \\mathbf {\\Sigma }^{-1} \\otimes \\mathbf {X}^\\top \\mathbf {X}\\\\$ and the right hand side becomes $\\left(\\left(\\mathbf {I}_q \\otimes \\mathbf {X}\\right)\\left(\\mathbf {\\Sigma }\\otimes \\mathbf {I}_{p+1}\\right)^{-1}\\right)^\\top \\left(\\mathbf {I}_q \\otimes \\mathbf {X}\\right) &= \\left(\\mathbf {I}_q\\mathbf {\\Sigma }^{-1} \\otimes \\mathbf {X}\\mathbf {I}_{p+1}\\right)^\\top \\left(\\mathbf {I}_q \\otimes \\mathbf {X}\\right) \\\\&= \\left(\\mathbf {\\Sigma }^{-1} \\otimes \\mathbf {X}\\right)^\\top \\left(\\mathbf {I}_q \\otimes \\mathbf {X}\\right) \\\\\\lbrace \\text{by symmetry of $\\mathbf {\\Sigma }$}\\rbrace &= \\left(\\mathbf {\\Sigma }^{-1} \\otimes \\mathbf {X}^\\top \\right) \\left(\\mathbf {I}_q \\otimes \\mathbf {X}\\right) \\\\&= \\mathbf {\\Sigma }^{-1}\\mathbf {I}_q \\otimes \\mathbf {X}^\\top \\mathbf {X}\\\\&= \\mathbf {\\Sigma }^{-1} \\otimes \\mathbf {X}^\\top \\mathbf {X}\\\\$ which proves the lemma already in the third equalities.",
"Using the vector form, we may express the conditional posterior of the regression parameters $\\beta _{i} \\mid \\mathbf {U}_i, \\left\\lbrace \\mathbf {\\Sigma }_{i}^\\alpha \\right\\rbrace _{\\alpha =1}^A \\sim \\text{N}(\\tilde{\\beta },\\tilde{\\mathbf {\\Sigma }})$ where $\\tilde{\\mathbf {\\Sigma }} = \\left[ \\mathbf {\\Sigma }_{\\beta }^{-1} + \\sum _{\\alpha =1}^A \\left(\\mathbf {\\Sigma }_i^\\alpha \\right)^{-1} \\otimes \\left(\\left(\\mathbf {X}_i^\\alpha \\right)^\\top \\mathbf {X}_i^\\alpha \\right) \\right]^{-1},$ and $\\tilde{\\beta } &= \\tilde{\\mathbf {\\Sigma }} \\times \\left[\\mathbf {\\Sigma }_{\\beta }^{-1}\\beta _0 + \\sum _{\\alpha =1}^A \\left(\\left(\\mathbf {\\Sigma }_i^\\alpha \\right)^{-1} \\otimes \\left(\\mathbf {X}_i^\\alpha \\right)^\\top \\right)\\mathbf {U}_i^\\alpha \\right].$ Meanwhile, the conditional posteriors of the covariance matrices are $\\mathbf {\\Sigma }_i^\\alpha \\mid \\mathbf {B}_{i}, \\mathbf {Y}_i^\\alpha , \\mathbf {X}_i^\\alpha &\\sim IW(\\nu _0 + n_i^\\alpha , \\mathbf {V}_0 + \\mathbf {S}^\\alpha _i),$ where $n_i^\\alpha $ denotes the number of observations in the covariance class $\\alpha $ for category $i$ and $\\mathbf {S}_i^\\alpha = \\left(\\mathbf {Y}_i^\\alpha - \\mathbf {X}_i^\\alpha \\mathbf {B}_{i}\\right)^\\top \\left(\\mathbf {Y}_i^\\alpha - \\mathbf {X}_i^\\alpha \\mathbf {B}_{i}\\right) $ Having computed $\\hat{\\omega }_1,\\ldots ,\\hat{\\omega }_N$ , for ${\\cal D}^{\\text{new}}$ , we may compute the Monte Carlo-estimated aposteriori probability of ${\\cal D}^{\\text{new}}$ being in category $i$ as $\\hat{p}_i = \\hat{\\mathbb {P}}(I=i\\mid {\\cal D},{\\cal D}^{\\text{new}}) = \\frac{\\pi _i\\hat{\\omega }_i}{\\pi _1\\hat{\\omega }_1+\\ldots +\\pi _N\\hat{\\omega }_N},$ where ${\\cal D}= {\\cal D}_1 \\cup \\ldots \\cup {\\cal D}_N$ is the complete training data set."
],
[
"Model Formulation, obfuscated data",
"Overall our setup is the same as in Appendix , but we now suppose there is only partial information about the complete training data set ${\\cal D}$ .",
"Due to some obfuscation, which could be due to rounding, grouping, categorization or lost measurements of some traits, we only know that $Y_{ij} \\in \\mathsf {S}_{ij} = \\mathsf {S}_{ij1}\\times \\cdots \\times \\mathsf {S}_{ijq},$ i.e.",
"the complete trait vector $Y_{ij}$ for subject $j$ of category $i$ is contained in a hyperrectangle $\\mathsf {S}_{ij}$ , whose components are given by $\\lbrace \\mathsf {S}_{ijk}\\rbrace _{k=1}^q$ .",
"The components are sets, ranging in possible size from singletons to infinite intervals of $\\mathbb {R}$ , and are given by $\\mathsf {S}_{ijk} = {\\left\\lbrace \\begin{array}{ll} Y_{ijk}, & k \\notin \\mathsf {K}_{ij}, \\\\\\left(c_{ijk},d_{ijk}\\right], & k \\in \\mathsf {K}_{ij},\\end{array}\\right.",
"}$ where $\\mathsf {K}_{ij}= \\left\\lbrace k; 1\\le k \\le q; \\, Y_{ijk} \\text{ obfuscated}\\right\\rbrace $ .",
"As described in the main article, without loss of generality we assume that $z_{ijk}$ , the mid point of $(c_{ijk},d_{ijk}]$ for finite sets, is integer-valued.",
"We can treat all types of obfuscations uniformly in the following way.",
"Suppose trait $k$ of subject $j$ of category $i$ is imperfectly observed, i.e.",
"$k \\in \\mathsf {K}_{ij}$ .",
"Let $g_{k}$ be the number of categories of this trait, which we number as $0,1,\\ldots ,g_{k}-1$ .",
"The observed category is $z_{ijk} \\in \\left\\lbrace 0,1,\\ldots ,g_{k} - 1\\right\\rbrace $ , where $g_{k} =2$ for binary data and $g_{k} = \\infty $ for count data.",
"The corresponding side of $\\mathsf {S}_{ij}$ is $\\mathsf {S}_{ijk} = {\\left\\lbrace \\begin{array}{ll}\\left(-\\infty , \\frac{1}{2}\\right], & \\text{if } z_{ijk} = 0, \\\\\\left(z_{ijk} - \\frac{1}{2}, z_{ijk} + \\frac{1}{2}\\right], & \\text{if } 1\\le z_{ijk} \\le g_{k}-2, \\\\\\left(g_{k}-\\frac{3}{2}, \\infty \\right), & \\text{if } z_{ijk} = g_{k} - 1.\\end{array}\\right.",
"}$ We also write $Z_{ijk} = z(\\mathsf {S}_{ijk}) = {\\left\\lbrace \\begin{array}{ll}0, & \\text{if } \\mathsf {S}_{ijk} = \\left(-\\infty , \\frac{1}{2}\\right], \\\\\\frac{c_{ijk} + d_{ijk}}{2}, & \\text{if $\\mathsf {S}_{ijk}$ is bounded}, \\\\g_{k} - 1, & \\text{if } \\mathsf {S}_{ijk} = \\left(g_{k}-\\frac{3}{2}, \\infty \\right],\\end{array}\\right.",
"}$ for the center point of a finite or half-open, infinite $\\mathsf {S}_{ijk}$ , whereas $z\\left(\\mathsf {S}_{ijk}\\right) = Y_{ijk}$ when $Y_{ijk} = \\mathsf {S}_{ijk}$ is perfectly observed.",
"We may write the actually observed training data set as ${\\cal D}^{\\text{obs}}= \\left\\lbrace \\left(x_{ij},\\mathsf {S}_{ij}\\right);\\, i=1,\\ldots ,N,j=1,\\ldots ,n_i\\right\\rbrace .$"
],
[
"Estimation",
"Using ${\\cal D}^{\\text{obs}}_i = \\lbrace (x_{ij},\\mathsf {S}_{ij}) ; j=1,\\ldots ,n_i\\rbrace $ , the posterior distribution of $\\theta _i$ becomes $ p(\\theta _i \\mid {\\cal D}^{\\text{obs}}_i) &= p(\\theta _i)C({\\cal D}^{\\text{obs}}_i)\\prod _{j=1}^{n_i} p(\\mathsf {S}_{ij};x_{ij},\\theta _{i}) \\nonumber \\\\&= p(\\theta _i) C\\left({\\cal D}^{\\text{obs}}_i \\right) {\\cal L}(\\theta _i ; {\\cal D}^{\\text{obs}}_i),$ where the normalizing factor is $C\\left({\\cal D}^{\\text{obs}}_i \\right) = \\left(\\int \\!",
"p(\\theta _i) {\\cal L}(\\theta _i; {\\cal D}^{\\text{obs}}_i) \\, \\mathrm {d}\\theta _i\\right)^{-1}$ and $ p(\\mathsf {S}_{ij},x_{ij};\\theta _i) = {\\left\\lbrace \\begin{array}{ll} f(Y_{ij};x_{ij},\\theta _i), & \\mathsf {K}_{ij}=\\emptyset , \\\\\\int _{\\mathsf {S}_{ij}} \\!",
"f(y_{ij};x_{ij},\\theta _i) \\prod _{k\\in \\mathsf {K}_{ij}} \\mathrm {d}y_{ijk}, & \\mathsf {K}_{ij}\\ne \\emptyset .\\end{array}\\right.",
"}$ Thus, with perfect observations ($\\mathsf {K}_{ij}=\\emptyset $ ), we evaluate the density of the trait vector $f$ at the observed point $Y_{ij}$ and the model is exactly as specified in Appendix .",
"Otherwise, we construct a $\\left|\\mathsf {K}_{ij}\\right|$ -dimensional integral over $f$ and the contribution to the likelihood is this integral, with the function evaluated exactly at the remaining perfectly observed traits, if any exist.",
"In particular, if all traits are imperfectly observed, the integral is $q$ -dimensional.",
"We may approximate the integral in (REF ) by $\\left|\\mathsf {S}_{ij}\\right| f\\left(z(\\mathsf {S}_{ij}), x_{ij}; \\theta _i\\right) = \\prod _{k\\in \\mathsf {K}_{ij}} \\left|\\mathsf {S}_{ijk}\\right| \\cdot f\\left(z(\\mathsf {S}_{ij1}),\\ldots ,z(\\mathsf {S}_{ijq}), x_{ij}; \\theta _i\\right)$ whenever all $\\left|\\mathsf {S}_{ijk}\\right| < \\infty $ for $k \\in \\mathsf {K}_{ij}$ , which is the case when employing the trick with auxiliary categories, mentioned in Section REF of the main article.",
"Alternatively, since $p(\\mathsf {S}_{ij},x_{ij};\\theta _i)$ potentially contains integrals of a multivariate Gaussian density function and there in general is a lack of a CDF on closed form for this distribution, the integrals in (REF ) need to be solved numerically.",
"However, in the case of $|\\mathsf {K}_{ij}| = 1$ , with $\\mathsf {K}_{ij} = \\lbrace k\\rbrace $ and $\\mathsf {S}_{ijk}=\\left(c_{ijk}, d_{ijk}\\right]$ , the integral is univariate and thusThe notation with subscript $(-k)$ means dropping element $k$ from a vector; dropping row $k$ from a matrix when not being the last index of a matrix; and dropping column $k$ when being the last index.",
"$ p(\\mathsf {S}_{ij},x_{ij};\\theta _i) &= f(Y_{ij(-k)}, x_{ij}; \\theta _i) \\bigg [ \\Phi \\left(\\frac{d_{ijk}-m_{ijk}(y_{ij(-k)})}{\\sigma _{ijk}}\\right) \\\\&- \\Phi \\left(\\frac{c_{ijk}-m_{ijk}(y_{ij(-k)})}{\\sigma _{ijk}}\\right)\\bigg ], \\nonumber $ where $m_{ijk}(y_{ij(-k)})=m_{ijk}+\\mathbf {\\Sigma }_{ijk(-k)}\\mathbf {\\Sigma }_{ij(-k)(-k)}^{-1}(y_{ij(-k)}-m_{ij(-k)})$ is the conditional expectation of $Y_{ijk}$ given that $Y_{ij(-k)} = (Y_{ijk^\\prime };\\, k^\\prime \\ne k) = y_{ij(-k)}$ , $\\sigma _{ijk}=\\sqrt{\\mathbf {\\Sigma }_{ijkk}-\\mathbf {\\Sigma }_{ijk(-k)}\\mathbf {\\Sigma }_{ij(-k)(-k)}^{-1}\\mathbf {\\Sigma }_{ij(-k)k}}$ is the conditional standard deviation of $Y_{ijk}$ given any value of $Y_{ij(-k)}$ , and $\\Phi $ is the CDF of the univariate Gaussian distribution with mean 0 and standard deviation 1.",
"Using ${\\cal D}^{\\text{obs}}_i$ , we find from (REF ) that the estimators $\\theta ^{(\\text{MAP})}_i$ and $\\theta ^{(\\text{Bayes})}_i$ are $\\theta ^{(\\text{MAP})}_i &= \\operatornamewithlimits{arg\\,max}_{\\theta _i} p(\\theta _i \\mid {\\cal D}^{\\text{obs}}_i) \\nonumber \\\\&= \\operatornamewithlimits{arg\\,max}_{\\theta _i} p(\\theta _i) {\\cal L}({\\cal D}^{\\text{obs}}_i ; \\theta _i)$ and $ \\theta ^{(\\text{Bayes})}_i &= \\mathbb {E}\\left[\\theta _i \\mid {\\cal D}^{\\text{obs}}_i\\right] = \\int \\!",
"\\theta _i p(\\theta _i \\mid {\\cal D}^{\\text{obs}}_i) \\, \\mathrm {d}\\theta _i \\nonumber \\\\&= C\\left({\\cal D}^{\\text{obs}}_i\\right) \\int \\!",
"\\theta _i p(\\theta _i) {\\cal L}({\\cal D}^{\\text{obs}}_i ; \\theta _i) \\, \\mathrm {d}\\theta _i$ respectively.",
"Furthermore, redefining ${\\cal D}^{\\text{new}}:= (x,\\mathsf {S})$ for a new observation, where $\\mathsf {S}=\\mathsf {S}_1\\times \\ldots \\times \\mathsf {S}_q$ , and denoting the corresponding set of imperfectly observed traits by $\\mathsf {K}$ , leads to the posterior category weights $\\omega _i &= \\iint _{\\mathsf {S}} \\!",
"f(y;x,\\theta _i) \\prod _{k\\in \\mathsf {K}} \\mathrm {d}y_{k} \\, p(\\theta _i \\mid {\\cal D}^{\\text{obs}}_i) \\, \\mathrm {d}\\theta _i \\nonumber \\\\&= C({\\cal D}^{\\text{obs}}_i) \\iint _{\\mathsf {S}} \\!",
"f(y;x,\\theta _i) \\prod _{k\\in \\mathsf {K}} \\mathrm {d}y_{k} \\, p(\\theta _i) {\\cal L}({\\cal D}^{\\text{obs}}_i ; \\theta _i) \\, \\mathrm {d}\\theta _i $ of this observation."
],
[
"Monte Carlo Approximations",
"The integral over $\\mathsf {S}$ in (REF ) is, as mentioned in conjunction with (REF ), potentially impossible to compute analytically, but could also be well behaved.",
"We can in theory approximate $\\theta ^{(\\text{Bayes})}_i$ in (REF ) as in (REF ) by sampling $\\theta _i$ from $p(\\theta _i \\mid {\\cal D}^{\\text{obs}})$ a total of $R_i$ times.",
"However, this entails a large number of numerical evaluations of integrals, see (REF )-(REF ).",
"Similarly, we may estimate $\\omega _i$ for $1\\le i \\le N$ in (REF ) through $ \\hat{\\omega }_i = \\frac{1}{R_i} \\sum _{r=1}^{R_i} \\int _{\\mathsf {S}} f(y;x,\\theta _{ir}) \\prod _{k\\in \\mathsf {K}} \\mathrm {d}y_{k},$ which in addition to previously presented integrals, involves computation of an integral over $\\mathsf {S}$ .",
"As an alternative way of computing (REF ) and (REF ), we also present an approach where complete data is sampled, based on the obfuscated data, as one step of the Monte Carlo algorithm, whereas the parameters are sampled as another step of the same algorithm.",
"This allows us to estimate all $\\theta ^{(\\text{Bayes})}_i$ and $\\omega _i$ under widespread obfuscation, given that we are able to simulate $\\mathbf {Y}_i$ , $i=1,\\ldots ,N$ .",
"Overall, we want to generate $ \\left\\lbrace \\theta _{ir}, Y_{ijkr}, \\, 1\\le j\\le n_i, \\, k\\in \\mathsf {K}_{ij}; Y_{kr}, k\\in \\mathsf {K}\\right\\rbrace _{r=1}^{R_i}$ from $p(\\theta _i|{\\cal D}^{\\text{obs}}_i)\\prod _{j=1}^{n_i} f\\left(y_{ij\\mathsf {K}_{ij}r} \\mid x_{ij}, \\mathsf {S}_{ij},\\theta _i\\right) f\\left(y_{\\mathsf {K}r}, \\mid x, Y_{\\mathsf {K}^\\complement };\\theta _i\\right)$ where $\\theta _{ir} = \\left(\\beta _{ir}, \\mathbf {\\Sigma }_{ir}^1,\\ldots ,\\mathbf {\\Sigma }_{ir}^A\\right)$ , $y_{ij\\mathsf {K}_{ij}r} = \\left(y_{ijkr}; k\\in \\mathsf {K}_{ij}\\right)$ , $y_{\\mathsf {K}r}=\\left(y_{kr} ; k\\in \\mathsf {K}\\right)$ and $Y_{\\mathsf {K}^\\complement } = \\left(Y_k ; k\\notin \\mathsf {K}\\right)$ .",
"Note that we do not condition on $\\mathsf {S}$ in the conditional density of the unobserved traits for the new observation that we want to classify, as this would introduce a bias in the Monte Carlo estimate of $\\omega _i$ below.",
"The details of the specific Gibbs sampling approach we use are presented in Appendix .",
"Having generated a sample $\\theta _{i1},\\ldots ,\\theta _{iR_i}$ , we may compute the estimated category weights of ${\\cal D}^{\\text{new}}$ as $ \\hat{\\omega }_i = \\frac{1}{R_i} \\sum _{r=1}^{R_i} f\\left(Y_{\\mathsf {K}^\\complement }; x , \\theta _{ir}\\right) I_{\\left\\lbrace Y_{\\mathsf {K}r} \\in _{k\\in \\mathsf {K}} \\mathsf {S}_k \\right\\rbrace },$ where $Y_{\\mathsf {K}^\\complement }$ is as above, and $Y_{\\mathsf {K}r} = \\left(Y_{kr} ; k\\in \\mathsf {K}\\right)$ .",
"For every $\\theta _{ir}$ , one could generate many $Y_{\\mathsf {K}}$ and replace the indicator with an average of the indicators for each sampled $Y_{\\mathsf {K}}$ .",
"A potentially more efficient method would be to define $\\lbrace y_t\\rbrace _{t=1}^T$ , where $y_t= (y_{t1},\\ldots ,y_{tq})$ with $y_{t\\mathsf {K}^\\complement } = Y_{\\mathsf {K}^\\complement }$ and $y_{t\\mathsf {K}} \\in _{k\\in \\mathsf {K}} \\mathsf {S}_k$ , in such a way that $\\lbrace y_{tk}, k \\in \\mathsf {K}\\rbrace $ is a grid approximation of $_{k\\in \\mathsf {K}} \\mathsf {S}_k$ .",
"Then we can estimate $\\omega _i$ through $ \\hat{\\omega }_i = \\frac{\\prod _{k\\in \\mathsf {K}} \\vert \\mathsf {S}_k \\vert }{TR_i} \\sum _{r=1}^{R_i}\\sum _{t=1}^T f\\left(y_t ; x, \\theta _{ir}\\right).$ If we use the trick with auxiliary categories described in Section REF of the main article, we can we can choose $y_t$ uniformly at random on $_{k\\in \\mathsf {K}} \\mathsf {S}_k$ , as long as we do not have any missing observations, since those are represented with an infinite interval.",
"Thus, (REF ) is potentially more effcient than (REF ), but comes at a cost of generality, since (REF ) is applicable to any new observation.",
"Finally, the Monte Carlo-estimated aposteriori probability of ${\\cal D}^{\\text{new}}= (x,\\mathsf {S})$ being of category $i$ is $ \\hat{p}_i = \\hat{\\mathbb {P}}(I=i \\mid {\\cal D}^{\\text{new}}, {\\cal D}^{\\text{obs}}) = \\frac{\\pi _i\\hat{\\omega }_i}{\\pi _1\\hat{\\omega }_1 + \\ldots + \\pi _N\\hat{\\omega }_N}$ and we may apply (REF ) with replacement of $\\omega _i$ by (REF ) for prediction.",
"If (REF ) is inserted into (REF ) we notice that $\\prod _{k\\in \\mathsf {K}} \\vert \\mathsf {S}_k \\vert $ and $T$ cancel out, and in case $R_i = R$ for $i=1,\\ldots ,N$ , also $R$ cancels out."
],
[
"Gibbs sampling details",
"The focus of this appendix is Procedure , in which we describe in detail how to generate a sample of size $R_i$ from the posterior distribution of the parameter vector $\\theta _i$ , using blockwise Gibbs sampling.",
"It describes the general case, i.e.",
"when we have obfuscated trait measurements in ${\\cal D}^{\\text{obs}}$ .",
"For the case with perfectly observed trait measurements, we skip the sampling of $\\mathbf {Y}_i$ and use the observed values instead, otherwise the procedure is the same.",
"Applying the procedure to data from each category $i$ will yield all the samples we need from the posterior distribution in order to perform classification.",
"In Procedure , $TN(\\mu ,\\mathbf {\\Sigma },\\mathsf {S})$ refers to the truncated Gaussian distribution, where $\\mu $ is the mean vector, $\\mathbf {\\Sigma }$ is the covariance matrix and $\\mathsf {S}$ is a hyper rectangle specifying the truncation limits.",
"Simulating from this distribution can be done exactly using rejection sampling, or approximately using an inner Gibbs algorithm.",
"Depending on application, either approach can be preferred, as the tradeoff is exact sampling versus efficiency.",
"Also, more advanced algorithms such as Importance Sampling-techniques can be used in this step.",
"The Monte Carlo approach to sampling the parameters' posterior distribution under obfuscation.",
"${\\cal D}^{\\text{obs}}, \\nu _0, \\mathbf {V}_0, \\beta _{i0} = \\text{vec}(\\mathbf {B}_{i0})$ A sample of size $R_i$ from the posterior distribution of $\\theta _i$ .",
"$\\alpha =1 \\rightarrow A$ draw $\\mathbf {\\Sigma }_{i0}^\\alpha \\sim IW\\left(\\nu _0, \\mathbf {V}_0\\right)$ draw $\\beta _{i0} \\sim \\text{N}\\left(\\beta _{i0}, \\mathbf {\\Sigma }_{\\beta _i}\\right)$ $\\theta _{i0} \\leftarrow \\left(\\beta _{i0}, \\mathbf {\\Sigma }_{i0}^1,\\ldots ,\\mathbf {\\Sigma }_{i0}^A\\right)$ $r=1 \\rightarrow R_i$ $j=1 \\rightarrow n_i$ draw $\\mathbf {Y}_{ij,r-1} \\mid x_{ij}, \\mathsf {S}_{ij}, \\theta _{i,r-1} \\sim TN\\left(\\mathbf {X}_{ij}\\mathbf {B}_{i,r-1},\\mathbf {\\Sigma }_{ij,r-1}, \\mathsf {S}_{ij}\\right)$ $\\mathbf {U}_{i,r-1} \\leftarrow \\text{vec}(\\mathbf {Y}_{i,r-1})$ draw $\\beta _{ir} \\mid \\mathbf {U}_{i(r-1)}, \\left\\lbrace \\mathbf {\\Sigma }_{i(r-1)}^\\alpha \\right\\rbrace _{\\alpha =1}^A \\sim \\text{N}\\left(\\tilde{\\beta }, \\tilde{\\mathbf {\\Sigma }}\\right)$ $\\alpha = 1 \\rightarrow A$ draw $\\mathbf {\\Sigma }_{ir}^\\alpha \\mid \\mathbf {U}_{i(r-1)}^\\alpha , \\mathbf {X}_i^\\alpha , \\beta _{ir} \\sim IW(\\nu _0 + n_i^\\alpha , \\mathbf {V}_0 + \\mathbf {S}_i^\\alpha )$ $\\theta _{ir} \\leftarrow \\left(\\beta _{ir}, \\mathbf {\\Sigma }_{ir}^1,\\ldots ,\\mathbf {\\Sigma }_{ir}^A\\right)$ save $\\theta _{ir}$"
],
[
"Visualized decision regions",
"All of these visualizations are done using the same model fit and the same generated new observations from the posterior predictive distribution as in Section of the main text.",
"As a reminder, we used the values $\\rho =0.1$ and $\\tau = 0.001$ for the tuning parameters of the classifier.",
"Figure: Decision regions when observing all three traits of the Acrocephaluswarblers.",
"Completely transparent blocks represent observations that will be classified as outliers, i.e.",
"not get any species assigned to them.",
"The indecisive regionΛ\\Lambda is less transparent, and colored according to which species there is uncertainty about.",
"The probability of observing an individual that belongs to the indecisive region is 0.08190.0819 for (a) and 0.07900.0790 for (b), when each species is equally likely apriori to occur.",
"Thedecision region of Paddyfield Warbler partially engulfs Blyth's Reed Warblerfor adult birds, reflecting the large uncertainty in the parameter estimatesfor adult Paddyfield Warblers.",
"Notice also that we introduce unnamed categoriesfor notch position, as the predictive posterior distribution requires this.Figure: Decision regions when only observing wing and notch length.Figure: Decision regions when only observing wing length and notch position.Figure: Decision regions when only observing notch length and notch position.Figure: In (a) and (b), decision regions are shown when only wing length is observed;in (c) and (d) decision regions are shown when only notch length isobserved; and in (e) and (f) decision regions are shown when onlynotch position is observed.",
"In all plots, kerneldensity estimates of each aposteriori trait distribution for eachspecies is shown with black lines of different types.",
"The plot highlightsthe larger degree of separation in the traits wing lengthand notch position."
]
] | 1906.04538 |
[
[
"Quantum Random Numbers generated by the Cloud Superconducting Quantum\n Computer"
],
[
"Abstract A cloud quantum computer is similar to a random number generator in that its physical mechanism is inaccessible to its users.",
"In this respect, a cloud quantum computer is a black box.",
"In both devices, its users decide the device condition from the output.",
"A framework to achieve this exists in the field of random number generation in the form of statistical tests for random number generators.",
"In the present study, we generated random numbers on a 20-qubit cloud quantum computer and evaluated the condition and stability of its qubits using statistical tests for random number generators.",
"As a result, we observed that some qubits were more biased than others.",
"Statistical tests for random number generators may provide a simple indicator of qubit condition and stability, enabling users to decide for themselves which qubits inside a cloud quantum computer to use."
],
[
"Introduction",
"Given a coin with an unknown probability distribution, there are two approaches to decide whether the coin is fair [1].",
"The first approach is to examine the coin itself; one expects an evenly shaped coin to yield fair results.",
"The second approach is to actually toss the coin a number of times to see if the output is sound.",
"In this approach, the coin is treated as a black box.",
"A random number generator is similar to a coin in that it is expected to produce unbiased and independent 0s and 1s.",
"Unlike a coin, however, the physical mechanism of a random number generator is often inaccessible to its users.",
"Therefore, users rely on statistical tests to decide the fairness of the device from its output.",
"Random number generators play an important role in cryptography, particularly in the context of key generation.",
"For example, the security of the RSA cryptosystem is based on keys that are determined by random choices of two large prime numbers [2].",
"If the choices of prime numbers are not random, an adversary could predict future keys and hence compromise the security of the system.",
"Randomness in cryptography derives from what is called the seed.",
"The seed is provided by physical random number generators [3], [4].",
"It is required that the physical mechanism of a physical random number generator remains a black box for the seed to be unpredictable.",
"Given that the measurement outcomes are theoretically unpredictable in quantum mechanics, random number generators based on quantum phenomena are a promising source of unpredictability [5], [6], [7].",
"Cloud quantum computers are quantum computers that are accessed online [8], [9], [10], [11], [12], [13].",
"In order to use a cloud quantum computer, users are required to send programs specifying the quantum circuit to be executed and the number of times the circuit should be run [14].",
"When a user's turn arrives, the quantum computer executes the program and returns the results [15].",
"A similarity between random number generators and cloud quantum computers is that its users do not have direct access to the physical mechanism of the device.",
"So, as far as the users are concerned, both random number generators and cloud quantum computers are black boxes.",
"In the field of random number generation, much research has been done on how to characterize the device from its output.",
"This lead to the creation of statistical tests for random number generators.",
"The present study aims to introduce the idea of statistical tests for random number generators to the field of cloud quantum computing.",
"This aim is supported by three points.",
"Firstly, the cloud quantum computer is a black box to its users, which is also the case with random number generators.",
"Secondly, quantum computers become random number generators when given certain programs.",
"Finally, the cloud quantum computer lacks a simple benchmark that would enable its users to decide the condition of the device.",
"The rest of this article is organized as follows.",
"In Section 2, statistical tests for random number generators is generally explained.",
"In Section 3, a group of statistical tests called the NIST SP 800-22 is reviewed.",
"In Section 4, we present the results of the statistical analysis of random number samples obtained from the cloud quantum computer, IBM 20Q Poughkeepsie, and the test results of the eight statistical tests from the NIST SP 800-22.",
"Finally, Section 5 is devoted to the conclusion.",
"In the appendix, a measure of uniformity often employed in the field of cryptography, the min-entropy, is explained."
],
[
"Statistical Tests for Random Number Generators",
"Statistical tests for random number generators are necessary to confirm that a random number generator is suitable for use in encryption processes [17].",
"Random number generators used in this context are required to have unpredictability.",
"This means that given any subset of a sequence produced by the device, no adversary can predict the rest of the sequence, including the output from the past.",
"Statistical tests aim to detect random number generators that produce sequences with a significant bias and/or correlation.",
"When subjected to statistical tests, a random number generator is considered a black box.",
"This means that the only information available is its output.",
"Under the null hypothesis that the generator is unbiased and independent, one expects its output to have certain characteristics.",
"The characteristics of the output are quantified by the test statistic, whose probability distribution is known.",
"From the test statistic, the probability that a true random number generator produces an output with a worse test statistic value is calculated.",
"This probability is called the p-value.",
"If the p-value is below the level of significance $\\alpha $ , the generator fails the test and the null hypothesis that the generator is unbiased and independent is rejected.",
"Since statistical tests for random number generators merely rule out significantly biased and/or correlated generators, these tests do not verify that a device is the ideal random number generator.",
"Nevertheless, a generator that passes the tests is more reliable than a generator that doesn't.",
"This is why statistical tests are usually organized in the form of test suites, so as to be comprehensive.",
"Some well known test suites are the NIST SP 800-22 [18], TestU01 [19], and the Dieharder test.",
"Because statistical tests are designed to check for statistical anomalies under the hypothesis that the generator is unbiased, a biased random number generator would naturally fail the tests.",
"This can be a problem when testing quantum random number generators, as they can be biased and unpredictable at the same time.",
"Given that statistically faulty generators can still be unpredictable, the framework of statistical tests fails to capture the essence of randomness: unpredictability.",
"There have been attempts to assure the presence of unpredictability by exploiting quantum inequalities, but they have not reached the point of replacing statistical tests altogether."
],
[
"NIST SP 800-22",
"The NIST SP 800-22 is a series of statistical tests for cryptographic random number generators provided by the National Institute of Standards and Technology [18].",
"Random number generators for cryptographic purposes are required to have unpredictability, which is not strictly necessary in other applications such as simulation and modeling, but is a crucial element of randomness.",
"The test suite contains 16 tests, each test with a different test statistic to characterize deviations of binary sequences from randomness.",
"The entire testing procedure of the NIST SP 800-22 is divided into 3 steps.",
"The first step is to subject all samples to the 16 tests.",
"For each sample, each test returns the probability that the sample is obtained from an unbiased and independent RNG.",
"This probability, which is called the p-value, is then compared to the level of significance $\\alpha = 0.01$ .",
"If the p-value is under the level of significance, the sample fails the test.",
"The second step involves the proportion of passed samples for each test.",
"Under the level of significance $\\alpha = 0.01$ , 1% of samples obtained from an unbiased and independent RNG is expected to fail each test.",
"If the proportion of passed samples is too high or too low, the RNG fails the test.",
"Finally, p-value uniformity is checked for each test.",
"Suppose one tested 100 binary samples.",
"This yields 100 p-values per test.",
"If the samples are independent, the p-values should be uniformly distributed for all tests.",
"The distribution of p-values is checked via the chi-squared test.",
"In the following sections, 8 tests from the NIST SP 800-22 are explained.",
"The input sequence will be denoted $\\varepsilon = \\varepsilon _1, \\varepsilon _2, \\cdots , \\varepsilon _n$ , and the $i$ th element $\\varepsilon _i$ .",
"Table: The minimum length nn required for each test in order to obtain meaningful results.",
"The tests not employed in the present study are shaded in grey.",
"Note that the tests will be referred to by their test # in Sec.",
"."
],
[
"Frequency Test",
"The frequency test aims to test whether a sequence contains a reasonable proportion of 0s and 1s.",
"If the probability of obtaining the sequence from an independent and unbiased random number generator is lower than 1 %, it follows that the random number generator is not “independent and unbiased\".",
"The minimum sample length required for this test is 100.",
"[colback=white!5!white,colframe=black!5!black,enforce breakable,pad at break*=1mm,title=Test Description] Convert the sequence into $\\pm {1}$ using the formula: $X_i = 2\\varepsilon _i-1$ .",
"Add the elements of $X$ together to obtain $S_n$ .",
"Compute test statistic: $s_{{\\mathrm {obs}}} = |S_n|/\\sqrt{n}$ .",
"Compute p-value $={\\mathrm {erfc}}(s_{{\\mathrm {obs}}} / \\sqrt{2})$ using complementary error function shown as erfc(z) = 2ze-u2du.",
"Compare p-value to 0.01.",
"If p-value $\\ge $ 0.01, then the sequence passes the test.",
"Otherwise, the sequence fails.",
"[colback=white,enforce breakable,pad at break*=1mm] Example: $\\varepsilon = 1001100010$ , length $n = 10$ .",
"$1,0,0,1,1,0,0,0,1,0 \\rightarrow +1,-1,-1,+1,+1,-1,-1,-1,+1,-1$ .",
"$S_{10} = 1-1-1+1+1-1-1-1+1-1 = -2$ .",
"$s_{{\\mathrm {obs}}} = |-2|/\\sqrt{10} \\approx 0.632455$ .",
"P-value $={\\mathrm {erfc}}(s_{\\mathrm {obs}} / \\sqrt{2}) \\approx 0.527089$ .",
"P-value $=0.527089 > 0.01 \\rightarrow $ the sequence passes the test.",
"This test is equivalent to testing the histogram for bias.",
"Because the test only considers the proportion of 1s, sequences such as 0000011111 or 0101010101 would pass the test.",
"Failing this test means that the sample is overall biased."
],
[
"Frequency Test Within a Block",
"Firstly, the sequence is divided into $N$ blocks of size $M$ .",
"The frequency test is then applied to the respective blocks.",
"As a result, one obtains $N$ p-values.",
"The second part of this test aims to check whether the variance of the p-values is by chance or not.",
"This is called the chi-squared ($\\chi ^2$ ) test.",
"For meaningful results, a sample with a length of at least 100 is required.",
"The following is the test description.",
"[colback=white!5!white,colframe=black!5!black,enforce breakable,pad at break*=1mm,title=Test Description] Divide the sequence into $N = \\lfloor \\frac{n}{M} \\rfloor $ non-overlapping blocks of size $M$ .",
"Determine the proportion of 1s in each block using i = j=1M (i-1)M+jM.",
"Compute $\\chi ^2$ statistic $\\chi ^2_{{\\mathrm {obs}}} = 4M\\sum _{i=1}^N \\left(\\pi _i-\\frac{1}{2} \\right)^2$ .",
"Compute p-value $= 1 - {\\mathrm {igamc}}\\left(\\frac{N}{2}, \\frac{\\chi ^2_{{\\mathrm {obs}}}}{2} \\right)$ .",
"Note that ${\\mathrm {igamc}}$ stands for the incomplete gamma function.",
"(z) = 0tz-1 e-t igamc(a,x) 1(a)0x e-t t(a-1)dt Compare p-value to 0.01.",
"If p-value $\\ge $ 0.01, then the sequence passes the test.",
"Otherwise, the sequence fails.",
"[colback=white,enforce breakable,pad at break*=1mm] Example: $\\varepsilon = 1001100010$ , length: $n = 10$ .",
"If $M=3$ , then $N=3$ and the blocks are 100, 110, 001.",
"The final 0 is discarded.",
"$\\pi _1 = 1/3$ , $\\pi _2 = 2/3$ , $\\pi _3 = 1/3$ .",
"$\\chi ^2_{{\\mathrm {obs}}} = 4M\\sum _{i=1}^N(\\pi _i-\\frac{1}{2})^2$ .",
"$\\chi ^2_{{\\mathrm {obs}}} = 4\\times 3 \\times \\left\\lbrace (\\frac{1}{3}-\\frac{1}{2})^2+ (\\frac{2}{3}-\\frac{1}{2})^2+ (\\frac{1}{3}-\\frac{1}{2})^2\\right\\rbrace = 1$ .",
"P-value $=1 - {\\mathrm {igamc}}(\\frac{3}{2}, \\frac{1}{2})=0.801252$ .",
"P-value $=0.801252 > 0.01 \\rightarrow $ the sequence is passes the test.",
"This test divides the sequence into blocks and checks each block for bias.",
"Depending on the block size, samples such as 001100110011 or 101010101010 could pass the test.",
"Failing this test means that certain sections of the sequence are biased."
],
[
"Runs Test",
"The proportion of 0s and 1s does not suffice to identify a random sequence.",
"A run, which is an uninterrupted sequence of identical bits, is also a factor to be taken into account.",
"The runs test determines whether the lengths and oscillation of runs in a sequence is as expected from a random sequence.",
"A minimum sample length of 100 is required for this test.",
"The following is the test description.",
"[colback=white!5!white,colframe=black!5!black,enforce breakable,pad at break*=1mm,title=Test Description] Compute proportion of ones $\\pi = \\left( \\sum _j \\varepsilon _j \\right) / n$ .",
"If the sequence passes frequency test, proceed to next step.",
"Otherwise, the p-value of this test is 0.",
"Compute test statistic $V_n({\\mathrm {obs}}) = \\sum _{k=1}^{n-1}(\\varepsilon _k\\oplus \\varepsilon _{k+1})+1$ , where $\\oplus $ stands for the XOR operation.",
"Compute p-value $= {\\mathrm {erfc}}\\left(\\frac{|V_n({\\mathrm {obs}})-2n\\pi (1-\\pi )|}{2\\sqrt{2n}\\pi (1-\\pi )} \\right)$ .",
"Compare p-value to 0.01.",
"If p-value $\\ge $ 0.01, then the sequence passes the test.",
"Otherwise, the sequence fails.",
"[colback=white,enforce breakable,pad at break*=1mm] Example: $\\varepsilon = 1010110001$ , length $n = 10$ .",
"$\\pi = \\frac{5}{10} = 0.5$ .",
"$|\\pi - 0.5| = 0 < \\frac{2}{\\sqrt{n}} = \\frac{2}{\\sqrt{10}} = 0.63\\rightarrow $ test is applicable.",
"$V_{10}({\\mathrm {obs}}) = (1 + 1 + 1 + 1 + 0 + 1 + 0 + 0+ 1) + 1 = 7$ .",
"P-value $= {\\mathrm {erfc}}\\left(\\frac{|7-2\\times 10\\times 0.5\\times (1-0.5)|}{2\\times \\sqrt{2\\times 10}\\times 0.5 \\times (1-0.5)} \\right) = 0.21$ .",
"P-value $= 0.21\\ge 0.01$ , so sequence passes the test."
],
[
"The Longest Run of Ones Within a Block Test",
"This test determines whether the longest runs of ones $111\\cdots $ within blocks of size M is consistent with what would be expected in a random sequence.",
"The possible values of M for this test are limited to three values, namely, 8, 128 and 10,000, depending on the length of the sequence to be tested.",
"[colback=white!5!white,colframe=black!5!black,enforce breakable,pad at break*=1mm,title=Test Description] Divide the sequence into blocks of size M. The choices of M and N are determined in regard to the length of the sequence.",
"N denotes the number of blocks, and the elements exceeding the number of blocks are discarded.",
"Table: Choices of M for the longest runs of ones within a block test.",
"Classify each block into the following categories regarding $M$ and the length of the longest run in each block.",
"See Table REF .",
"Table: Classifications of each block.",
"Compute $\\chi ^2({\\mathrm {obs}}) = \\sum _{i = 0}^{K}\\frac{(v_i - N\\pi _i)^2}{N\\pi _i}$ .",
"Note that $K$ , $N$ and $\\pi _i$ are determined by $M$ .",
"See Tables REF and REF .",
"Table: Values of KK and NN corresponding to MM.Table: Values of π i \\pi _i corresponding to KK and MM.",
"Compute p-value $= 1 - {\\mathrm {igamc}}\\left(\\frac{K}{2},\\frac{\\chi ^2({\\mathrm {obs}})}{2}\\right)$ .",
"Compare p-value to 0.01.",
"If p-value $\\ge $ 0.01, then the sequence passes the test.",
"Otherwise, the sequence fails.",
"[colback=white,enforce breakable,pad at break*=1mm] Example: $n = 10000$ $M = 128$ and $N = 49$ .",
"The remaining 3728 elements are discarded.",
"The counts for the longest run of ones are $v_0 = 6$ , $v_1 = 10$ , $v_2 = 10$ , $v_3 = 7$ , $v_4 = 7$ , and $v_5 = 9$ .",
"2(obs) = (6 - 490.1174)2490.1174 + (10 - 490.2430)2490.2430 + (10 - 490.2493)2490.2493 + (7 - 490.1752)2490.1752 + (7 - 490.1027)2490.1027 + (9 - 490.1124)2490.1124 = 3.994459.",
"P-value $= 1 - {\\mathrm {igamc}}\\left(\\frac{5}{2},\\frac{3.994459}{2}\\right) = 0.550214$ .",
"P-value $= 0.550214 \\ge 0.01$ , so the sequence passes the test."
],
[
"Discrete Fourier Transform Test",
"This test checks for periodic patterns in the sequence by performing a discrete Fourier transform (DFT).",
"The minimum sample length required for this test is 1000.",
"The following is the test description.",
"[colback=white!5!white,colframe=black!5!black,enforce breakable,pad at break*=1mm,title=Test Description] Convert the sequence $\\varepsilon $ of 0s and 1s into a sequence $X$ of $-1$ s and $+1$ s. Apply a DFT on X: $S = DFT(X)$ .",
"This should yield a sequence of complex variables representing the periodic components of the sequence of bits at different frequencies.",
"Compute $M = {\\mathrm {modulus}}(S^{\\prime }) \\equiv |S^{\\prime }|$ , where $S^{\\prime }$ is the first $\\frac{n}{2}$ elements of $S$ .",
"This produces a sequence of peak heights.",
"Compute $T = \\sqrt{\\left(\\log _e\\frac{1}{0.05}\\right)}$ .",
"This is the 95 % peak height threshold value.",
"95 % of the values obtained by the test should not exceed $T$ for a random sequence.",
"Compute $N({{\\mathrm {ideal}}}) = \\frac{0.95n}{2}$ , which is the expected theoretical number of peaks that are less than $T$ .",
"Compute $N({\\mathrm {obs}})$ , which is the actual number of peaks in $M$ that are less than $T$ .",
"Compute $d = \\frac{N({{\\mathrm {ideal}}})-N({\\mathrm {obs}})}{\\sqrt{n\\cdot 0.95\\cdot 0.05\\cdot \\frac{1}{4}}}$ .",
"Compute p-value $= {\\mathrm {erfc}}\\left(\\frac{|d|}{\\sqrt{2}}\\right)$ .",
"Compare p-value to 0.01.",
"If p-value $\\ge $ 0.01, then the sequence is passes the test.",
"Otherwise, the sequence fails.",
"This test checks for periodic features.",
"Samples with periodic features may look like 0110011001100110 or 010010100101001 among various other possibilities.",
"Failing this test suggests that the sample has periodic patterns.",
"It is noted that the probability distribution of the test statistic $d$ should be rectified as it does not converge to the standard normal distribution [20].",
"[colback=white,enforce breakable,pad at break*=1mm] Example: $\\varepsilon = 1001010011$ , length $n = 10$ .",
"$X = 2\\varepsilon _1 - 1, 2\\varepsilon _2 - 1, \\ldots , 2\\varepsilon _n - 1= 1,-1,-1,1,-1,1,-1,-1,1,1$ .",
"$N({{\\mathrm {ideal}}}) = 4.75$ .",
"$N({\\mathrm {obs}}) = 4$ .",
"$d = \\frac{(4.75-4)}{\\sqrt{10\\cdot 0.95\\cdot 0.05\\cdot \\frac{1}{4}}} = 2.147410$ .",
"P-value $= {\\mathrm {erfc}}\\left(\\frac{|2.147410|}{\\sqrt{2}}\\right) = 0.031761$ .",
"P-value $= 0.031761 \\ge 0.01$ , so the sequence passes the test."
],
[
"Approximate Entropy Test",
"The approximate entropy test compares the frequency of $m$ -bit overlapping patterns with that of $(m+1)$ -bit patterns in the sequence.",
"It checks whether the relation of two frequencies is what is expected from an unbiased and independent RNG.",
"The level of significance is $\\alpha = 0.01$ .",
"This test can be applied to samples with lengths equal to or larger than 64.",
"The test description is below.",
"[colback=white!5!white,colframe=black!5!black,enforce breakable,pad at break*=1mm,title=Test Description] Append the first $m-1$ bits of the sequence to the end of the sequence.",
"Divide the sequence into overlapping blocks with a length of $m$ .",
"There are $2^m$ possible m-bit blocks.",
"Count how many of each possible block there are in the sequence.",
"Compute $\\frac{\\mathrm {count}}{n}\\log _e(\\frac{\\mathrm {count}}{n})$ for each count.",
"Compute the sum of all counts $\\varphi _m$ .",
"Replace $m$ with $m+1$ and repeat steps 1 through 5 to obtain $\\varphi _{m+1}$ .",
"Calculate test statistic $\\mathrm {obs} = 2n(\\log _e(n) - (\\varphi _m - \\varphi _{m+1}))$ .",
"Derive p-value $= 1 - {\\mathrm {igamc}}(2^{(m-1)}, \\mathrm {obs}/2)$ .",
"Compare p-value with level of significance $\\alpha = 0.01$ .",
"If p-value $\\ge 0.01$ , the result is pass.",
"Otherwise, the sequence fails the test.",
"[colback=white,enforce breakable,pad at break*=1mm] Example: $\\varepsilon = 1011010010$ , length $n = 10$ , $m = 3$ .",
"$\\varepsilon = {red}{10}11010010$ $\\rightarrow $ $1011010010{red}{10}$ .",
"101101001010 $\\rightarrow $ $101, 011, 110, 101, 010, 100, 001, 010, 101, 010$ .",
"$\"000\": 0,\"001\": 1,\\\\\"010\": 3,\"011\": 1,\"100\": 1,\"101\": 3,\"110\": 1 ,\"111\": 0$ .",
"$\"000\": 0, \"001\": 0.1\\log _e (0.1), \"010\": 0.3 \\log _e(0.3),\\\\ \"011\": 0.1\\log _e(0.1), \"100\": 0.1\\log _e(0.1),\"101\": 0.3\\log _e(0.3),\\\\ \"110\": 0.1\\log _e(0.1), \"111\": 0$ .",
"$\\varphi _3 = -1.643418$ $\\varphi _{3+1} = -2.025326$ .",
"$\\mathrm {obs} = 2\\times 10\\times (\\log _e(10) - (-1.643418 - (-2.025326))) = 6.224774$ .",
"P-value $= 1 - {\\mathrm {igamc}}(2^{(3-1)}, \\mathrm {6.224774}/2) = 0.622069$ .",
"P-value $= 0.622069 \\ge 0.01$ .",
"The sequence passes the test.",
"The approximate entropy test checks for correlation between the number of $m$ -bit patterns and $(m+1)$ -bit patterns in the sequence.",
"The difference between the number of possible $m$ -bit patterns and the number of possible $(m+1)$ -bit patterns in the sequence is computed, and if this difference is too small or too large, the two patterns are correlated."
],
[
"Cumulative Sums Test",
"The cumulative sums test is basically a random walk test.",
"It checks how far from 0 the sum of the sequence in terms of $\\pm 1$ reaches.",
"For a sequence that contains uniform and independent 0s and 1s, the sum should be close to 0.",
"This test requires a minimum sample length of 100.",
"[colback=white!5!white,colframe=black!5!black,enforce breakable,pad at break*=1mm,title=Test Description] Convert 0 to -1 and 1 to +1.",
"In forward mode, compute the sum of the first $i$ elements of $X$ .",
"In backward mode, compute the sum of the last $i$ elements of $X$ .",
"Find the maximum value $z$ of the sums.",
"Compute the following p-value.",
"$\\Phi $ is the cumulative distribution function for the standard normal distribution.",
"Pvalue = 1 - k = (-nz + 1 )/4(nz - 1)/4 [ ( (4k+1)zn ) - ( (4k-1)zn ) ] +k = (-nz - 3 )/4(nz - 1 )/4 [ ( (4k+3)zn ) - ( (4k+1)zn ) ].",
"Compare p-value to $\\alpha = 0.01$ .",
"If p-value $\\ge 0.01$ , the result is pass.",
"Otherwise, the sequence fails the test.",
"[colback=white,enforce breakable,pad at break*=1mm] Example: $\\varepsilon = 1011010010$ , length $n = 10$ .",
"$\\varepsilon = 1011010010$ $\\rightarrow $ $X = 1, -1, 1, 1, -1, 1, -1, -1, 1, -1$ .",
"Forward mode: $S_1 = 1$ , $S_2 = 1 + (-1) = 0$ , $S_3 = 1 + (-1) + 1 = 2$ , $S_4 = 1 + (-1) + 1 + 1$ , $S_5 = 1 + (-1) + 1 + 1 + (-1) = 1$ , $S_6 = 1 + (-1) + 1 + 1 + (-1) + 1 = 2$ , $S_7 = 1 + (-1) + 1 + 1 + (-1) + 1 + (-1) = 1$ , $S_8 = 1 + (-1) + 1 + 1 + (-1) + 1 + (-1) + 1 = 2$ , $S_9 = 1 + (-1) + 1 + 1 + (-1) + 1 + (-1) + 1 + (-1) = 1$ .",
"In forward mode, the maximum value is $z = 2$ .",
"P-value $= 0.941740$ for both forward and backward.",
"P-value $= 0.941740 \\ge 0.01$ .",
"The sequence passes the test.",
"Once the p-value has been calculated for all tests and samples, the proportion of samples that passed the test is computed for each test.",
"Let us consider a case where 1000 samples were subjected to each of the 15 tests.",
"This results in 1000 p-values per test.",
"For example, if 950 out of 1000 samples passed the frequency test, the proportion of passed samples is $0.95$ .",
"If the proportion of passed samples falls within the following range for all 15 tests, the samples pass the second step of the NIST SP 800-22.",
"The acceptable range of proportion is calculated with (1-)3(1-)m, where $\\alpha $ stands for the level of significance and $m$ the sample size.",
"It is noted that it is controversial whether the coefficient should be 3.",
"A suggestion that the coefficient should be 2.6 exists [21].",
"In the case of the current example, Eq.",
"(REF ) can be calculated using $\\alpha = 0.01$ and $m = 1000$ as (1-0.01)30.01(1-0.01)1000 = 0.99 0.0094.",
"From the fact that $0.95$ is not within the acceptable range, it follows that the samples fail the frequency test.",
"The same process is done with all 16 tests, and unless the samples pass all tests, the result is that the hypothesis that the RNG is unbiased and independent is rejected.",
"The final step of the NIST SP 800-22 is to evaluate the p-value uniformity of each test.",
"In order to perform the chi-squared ($\\chi ^2$ ) test, the p-value is divided into 10 regions: $[k,k+0.1)$ for $k = 0, 1, \\ldots , 9$ .",
"The test statistic is given by 2 = i=110(number of samples in i-th region-sample size/10)2sample size/10.",
"When the number of samples in each region is 2, 8, 10, 13, 17, 17, 13, 10, 8, 2, the test statistic REF is calculated as $\\chi ^2 = 25.200000$ .",
"From $\\chi ^2$ , the p-value is pvalue = igamc( 92, 22 ).",
"Therefore, in the current example where $\\chi ^2 = 25.200000$ , the p-value is $0.002758$ .",
"The level of significance for the p-value uniformity is $\\alpha = 0.0001$ .",
"So when the p-value is $0.002758$ , it follows that the p-value distribution is uniform.",
"The p-value uniformity test requires at least 55 samples.",
"As mentioned before, it is remarked that passing the NIST SP 800-22 does not ensure a sequence to be truly random [22], [23], [24]."
],
[
"Quantum Random Number Generation on the Cloud Quantum Computer",
"According to quantum mechanics, the measurement outcomes of the superposition state $(\\mathinner {|{0}\\rangle }+\\mathinner {|{1}\\rangle })/\\sqrt{2}$ along the computational basis ideally form random number sequences.",
"This means that the resulting sequences are expected to pass the statistical tests for RNGs explained previously.",
"Here, the computational basis, $\\mathinner {|{0}\\rangle }$ and $\\mathinner {|{1}\\rangle }$ , spans the two-dimensional Hilbert space.",
"In a quantum computer, the desired state $(\\mathinner {|{0}\\rangle }+\\mathinner {|{1}\\rangle })/\\sqrt{2}$ is generated from the initial state $\\mathinner {|{0}\\rangle }$ by applying the Hadamard gate to a single quantum bit (qubit).",
"Note that in this process, the initial state is always the same.",
"Unlike classical random number generators and pseudorandom number generators that require random seeds to produce independent sequences, quantum random number generators are capable of producing independent sequences with the same seed.",
"This reduces the risk of the output of a random number generator being predicted from the seed, because all possible outputs come from the same seed.",
"Figure: (a): QRNG quantum circuit using the Hadamard gate.",
"(b): device topology of IBM 20Q Poughkeepsie provided by Qiskit.In the present study, the cloud superconducting quantum computer, IBM 20Q Poughkeepsie, was used.",
"The device was given the circuit in Fig.",
"REF (a) and was repeatedly instructed to execute the circuit 8192 times without interruption from 2019/05/09 11:24:27 GMT.",
"Because the quantum computer has multiple users across the globe, interruption between jobs occur [25].",
"8192 is the maximum number of uninterrupted executions (shots) available.",
"Running the circuit with 8192 shots yields a binary sequence with a length of 8192 per qubit.",
"This process was automatically repeated across calibrations.",
"The device goes through calibration once a day as seen in Table REF .",
"As a result, 579 samples were obtained from the IBM 20Q Poughkeepsie device.",
"Note that each qubit produced 579 samples, each with a length of 8192.",
"The samples were subjected to the eight tests from the NIST SP 800-22, which are: the frequency test, frequency within a block test, runs test, longest runs within a block test, DFT test, approximate entropy test, and the cumulative sums test (forward, backward).",
"The p-value of each test corresponding to the respective samples was computed.",
"For each test, the proportion of passed samples was checked.",
"The acceptable range of the proportion of passed samples for 579 samples under the level of significance $\\alpha = 0.01$ is $> 0.977595$ .",
"Table: The correspondence between calibration start/end time and time of job sent.",
"All dates and times are in GMT.By constantly running the IBM 20Q Poughkeepsie device for five days, we obtained 579 samples for each of the 20 qubits.",
"In theory, these samples should qualify as the output of an ideal random number generator.",
"In random number generation, the output sequences are checked for two properties: bias and patterns.",
"When the sequences show signs of bias or patterns, the device is not in ideal condition.",
"The same logic applies to the cloud quantum computer.",
"We also simulated the same quantum circuit on the simulator with the obtained noise parameters such as the T1 and T2 time, the coherent error, the single-qubit error, and the readout error, all of which are updated.",
"The simulator is referred to as the noisy simulator in the following.",
"The noisy simulator program was also provided by IBM [25].",
"In the present section, the random number output of each qubit inside the IBM 20Q Poughkeepsie device is analyzed.",
"The qubits that are connected by arrows in Fig.",
"REF (b) represent the pairs of qubits on which the controlled NOT gate can operate.",
"The controlled NOT gate is a two-qubit gate.",
"Figure: Min-enrtopy transition of qubit [0]∼\\sim [19].",
"The blue plots are the experimental results and the red plots the noisy simulation results.",
"The figure has been rotated 90 degrees.",
"The horizontal axis ranges from 2019/05/09 11:24 GMT to 2019/05/14 07:54 GMT.The min-entropy, whose definition and properties are seen in the Appendix, was computed for each qubit from the 579 samples.",
"This resulted in 579 min-entropy transition plots for 20 qubits.",
"Figure REF is organized to form the topology of the IBM 20Q Poughkeepsie.",
"The min-entropy takes values from 0 to 1 depending on the highest probability of the probability distribution.",
"When the probability distribution is uniform, the min-entropy is 1.",
"Figure REF shows how each qubit has a unique tendency for min-entropy.",
"Qubit [17], for example, shows a sudden drop in min-entropy at around 60 hours.",
"This does not occur in simulation.",
"A sudden drop in min-entropy suggests that the measurement results can vary depending on when the cloud quantum computer executes a circuit.",
"Overall, the noisy simulator tends to have a higher min-entropy compared to the actual device.",
"According to Ref.",
"[25], the readout error that IBM provides does not reflect the asymmetry between the error output 1 on the state $\\mathinner {|{0}\\rangle }$ and the error output 0 on the state $\\mathinner {|{1}\\rangle }$ .",
"The discrepancy between the min-entropy of the actual device and the simulator suggests that readout asymmetry exists.",
"Figure: The proportion of 1s of qubit[0]∼\\sim [19].",
"The acceptable range under the level of significance α=0.01\\alpha = 0.01 is between the two dotted lines.",
"The blue bars are the experimental results and the red plots the noisy simulation results.Next, the samples were checked for bias.",
"Each qubit produced 579 samples with a length of 8192, which form 4,743,168-bit sequences when chronologically connected.",
"Figure REF demonstrates the proportion of 1s in the entire sequence output by each qubit.",
"Under the level of significance $\\alpha = 0.01$ , the proportion of 1s of a 4,743,168-bit sequence should fall between the red lines.",
"The result is that none of the qubits produced acceptable proportions of 1s as seen in Fig.",
"REF .",
"Furthermore, Fig.",
"REF shows that the actual device failed to pass the eight statistical tests, which indicates that the current quantum computing device does not have the statistical properties of a uniform random number generator.",
"Figure: The proportion of passed samples for each test.",
"The test names corresponding to the test # can be found in Table .",
"The acceptable range provided by the NIST is above the red line marking the proportion 0.9775950.977595.",
"The blue plots are the experimental results and the red plots the noisy simulation results.",
"The figure has been rotated 90 degrees.The problem with histograms as seen in Fig.",
"REF is that they fail to detect certain anomalies.",
"For example, a sequence consisting of all 0s for the former half and all 1s for the latter half yields a perfect histogram.",
"However, such a sequence is clearly not random.",
"To compensate for this flaw, we focused on the transition of the number of 1s in the sequence.",
"Ideally, the number of 1s in a random number sequence should always be roughly half of the sequence length.",
"The difference between the ideal number of 1s and the observed number of 1s for the 4,743,168-bit sequence of each qubit is examined in Fig.",
"REF .",
"Note that here, too, the figures are aligned topologically.",
"Figure REF shows the stability of each qubit in terms of the proportion of 1s in its output; a linear plot suggests that the qubit is being stably operated.",
"While qubit[7] is more biased than qubit[17] overall, the line representing qubit[7] shows more stability than that of qubit[17].",
"Furthermore, the noisy simulator does not capture the trend of the qubits.",
"Therefore, the discrepancy between the output of the actual device and the noisy simulator may not only be a result of readout asymmetry, but also time-varying parameters.",
"Figure: The difference between the ideal and observed increase in the number of 1s of qubit [0]∼\\sim [19].",
"The blue plots are the experimental results and the red plots the noisy simulation results.",
"The figure has been rotated 90 degrees."
],
[
"Conclusion",
"We characterized the qubits in a cloud quantum computer by using statistical tests for random number generators to provide a potential indicator of the device's condition.",
"The IBM 20Q Poughkeepsie device was repeatedly run for a period of five days, and 579 samples with a length of 8192 were obtained for each of the 20 qubits.",
"For comparison, the noise parameters obtained in experiment were used to run the noisy simulator.",
"Samples from both the actual device and the simulator were statistically analyzed for bias and patterns.",
"To evaluate the uniformity of each sample, the min-entropy was computed.",
"The transition of min-entropy showed that the qubits have unique characteristics.",
"We identified a sudden drop of min-entropy in qubit [17].",
"The histogram of the proportion of 1s in the 4,743,168-bit sequences produced by each qubit revealed that, overall, none of the qubits produced acceptable proportions of 1s.",
"However, we evaluated each qubit's stability from the time-series data of the proportion of 1s, and found that qubits [0] and [12] were relatively stable.",
"Finally, eight tests from the NIST SP 800-22 were applied to the 529 samples of the 20 qubits.",
"None of the qubits cleared the standards of the test suite.",
"However, the test results showed that qubits [0] and [12] were the closest to the ideal in terms of the proportion of passed samples for each test.",
"As is the case with random number generators, a cloud quantum computer is a black box to its users.",
"Therefore, users are required to decide for themselves when to use a cloud quantum computer and which qubits to choose.",
"Statistical tests for random number generators are a potential candidate for a simple indicator of qubit condition and stability inside a cloud quantum computer.",
"The authors thank Hidetoshi Okutomi, Atsushi Iwasaki, Shumpei Uno and Rudy Raymond for valuable discussions.",
"This work is partially supported by JSPS KAKENHI (Grant Nos.",
"17K05082 and 19H05156).",
"The results presented in this paper were obtained in part using an IBM Q quantum computing system as part of the IBM Q Network.",
"The views expressed are those of the authors and do not reflect the official policy or position of IBM or the IBM Q team."
],
[
"Appendix: Min-entropy",
"tocsectionAppendix Among various entropy measures for uniformity, the min-entropy is often used in the context of cryptography.",
"The min-entropy for a random variable $X$ is defined as follows: H(X) = -2( x {0, 1}Pr[X = x] ).",
"On the other hand, Shannon's entropy, which is also a measure for uniformity, is defined as follows: Hsh(X) = -x {0, 1} Pr[X = x]2 Pr[X = x].",
"Both measures (REF ) and (REF ) take values ranging from 0 to 1 for a random variable on $\\lbrace 0, 1\\rbrace $ .",
"The reason why the min-entropy is more appropriate in the context of cryptography is that it is more sensitive than Shannon's entropy.",
"This is apparent from Fig.",
"REF .",
"Figure REF compares the min-entropy and Shannon's entropy corresponding to the probability of $X$ yielding 1.",
"The min-entropy provides a clearer distinction of probability distributions close to uniform than Shannon's entropy.",
"Figure: Relation between Shannon's entropy and min-entropy.The min-entropy also indicates the probability that an adversary with knowledge of the probability distribution of $X$ predicts the outcome of $X$ correctly [16].",
"Here, the adversary predicts the value that appears with the highest probability.",
"For this reason, the min-entropy considers the maximum probability of $X$ ."
]
] | 1906.04410 |
[
[
"A method for identifying stability regimes using roots of a\n reduced-order polynomial"
],
[
"Abstract For dispersive Hamiltonian partial differential equations of order 2N+1, N integer, there are two criteria to analyse to examine the stability of small-amplitude, periodic travelling wave solutions to high-frequency perturbations.",
"The first necessary condition for instability is given via the dispersion relation.",
"The second criterion for instability is the signature of the eigenvalues of the spectral stability problem given by the sign of the Hamiltonian.",
"In this work, we show how to combine these two conditions for instability into a polynomial of degree N. If the polynomial contains no real roots, then the travelling wave solutions are stable.",
"We present the method for deriving the polynomial and analyse its roots using Sturm's theory via an example."
],
[
"Introduction",
"Partial differential equations (PDEs) are used in a wide variety of applications to describe physical phenomena where this physical relevance imposes the requirement that the solutions to the PDEs are real.",
"Moreover, if the description is of a closed system, there is usually an associated conservation of energy and the equations used are Hamiltonian.",
"As more methodology for solving PDEs is developed , the natural question to ask is then how realistic are these solutions are and how likely we are to observe them in nature.",
"Thus, analysing their stability also becomes important , , .",
"The purpose of this work is to present a simplified method for stability analysis, illustrated by an explicit example.",
"We focus on high-frequency instabilities arising from spectral analysis of a perturbation of periodic travelling waves and restrict our focus to stability of solutions of dispersive Hamiltonian equations.",
"We show how working with the dispersion relation, we can methodically construct a parameter regime where there is only spectral stability with respect to particular perturbations and in the regions where we expect instability, we show what types of instabilities can arise.",
"In recent work , a method for establishing the presence of high-frequency instabilities of travelling wave solutions for both scalar PDEs as well as for systems of equations was described.",
"In this method, there are two important conditions to consider: collisions of eigenvalues of the spectral stability problem and the signature of these eigenvalues.",
"Furthermore, it was shown that in order for the solutions to become unstable, the system had to admit waves travelling in different directions (bi-directional waves).",
"In the follow-up work , the authors showed that a different way to meet the instability criteria, was for equations to contain what is referred to as a generalised resonance.",
"An equation contains a resonance if there is a certain set of parameters for which travelling wave solutions are predominantly composed of at least two distinct frequencies which can travel at the same speed.",
"Physically, this implies that there are at least two different forces that can influence the travelling waves that are of the same order of magnitude.",
"For example, if we are considering water waves, then these waves are in a resonant regime if surface tension and gravity are competing forces of the same order of magnitude.",
"The result is that the travelling wave profiles contain two different prominent modes, otherwise referred to as Wilton ripples , .",
"If we restrict ourselves to scalar, dispersive and Hamiltonian PDEs where the solution $u$ depends on one spatial and one time variable, i.e.",
"$u = u(x,t)$ with a period $L$ and up to $2N+1$ derivatives, then it has been shown that all we need is a polynomial dispersion relation $\\omega (k)$ of order $2N+1$ to describe both of the necessary conditions for instability.",
"In , it was shown that the two necessary conditions for instability can be collapsed into one criterion on the roots of a polynomial of order $N$ to be in an interval $I$ defined in Section .",
"This greatly simplifies the analysis, leading to closed-form results for stability regions of specific PDEs.",
"This work presents a method for the single criteria for instability of periodic travelling wave solutions to a dispersive, Hamiltonian PDE using an example with three competing terms.",
"The formulation and underlying theory is described in Section .",
"Working with the dispersion relation, we show the general methodology for the stability analysis in Section .",
"Section explicitly shows how to implement the method via an example, demonstrating how to construct the coefficients systematically and use Sturm's theory to analyse the roots of the reduced polynomial.",
"In Section , figures of the stability and instability regions are shown and we conclude in Section ."
],
[
"Summary of Stability Theory",
"Consider a scalar Hamiltonian PDE of the form $u_t = \\partial _x \\frac{\\delta H}{\\delta u},$ where the function $u = u(x,t)$ describes a periodic travelling wave, with $H$ the Hamiltonian and $\\frac{\\delta H}{\\delta u}$ a variational derivative.",
"More specifically $u(x,t)$ is a solution of $u_t = \\sum _{n=1}^{N} C_{2n+1} \\frac{\\partial ^{2n+1}u}{\\partial x^{2n+1}} + f(u,u_x,...,u_{(2N)x})_x,$ where $N$ is positive integer and $\\frac{\\partial ^{2n+1}u}{\\partial x^{2n+1}}$ are $2n+1$ (odd) derivatives up to order $2N+1$ with the nonlinearity $f$ that can depend on $u$ as well as its derivatives up to order $2N$ (denoted as $u_{(2N)x}$ ), keeping the overall system dispersive.",
"For ease, we consider the equation with real coefficients $C_{2n+1}$ .",
"We obtain the dispersion relation $\\omega (k)$ if we let $u(x,t) \\sim e^{ikx - i\\omega t}$ with $k$ a Fourier mode, and substitute into (REF ) to obtain $\\omega (k) = \\sum _{n=1}^N (-1)^{(n+1)}C_{2n+1}k^{2n+1}.$ Furthermore, if we restrict the space of solutions $u(x,t)$ to periodic, travelling waves moving at speed $V$ such that $u(x,t)\\rightarrow u^{(0)}(x-Vt)$ , then we can write (REF ) in the travelling frame of reference and consider the steady-state equation $Vu_x + \\sum _{n=1}^{N} C_{2n+1} \\frac{\\partial ^{2n+1}u}{\\partial x^{2n+1}} + f(u,u_x,...,u_{(2N)x})_x = 0,$ and setting $x\\rightarrow x-Vt$ from now on.",
"Despite restricting the space of solutions to travelling waves $u^{(0)}(x)$ , we can still gather information about the time dependence by perturbing about this steady-state with a small perturbation governed by $\\delta $ , i.e.",
"$u(x,t) & = u^{(0)}(x) + \\delta \\bar{u}^{(1)}(x,t) \\nonumber \\\\& = u^{(0)}(x) + \\delta e^{\\lambda t} u^{(1)}(x).$ We have made an assumption about the time dependence of the perturbation by introducing $\\lambda \\in \\mathbb {C}$ .",
"Recall that $u^{(0)}(x)$ is periodic of period $L$ (for convenience, $L=2\\pi $ ) .",
"We allow the perturbations to be of any period, but bounded in space using the Fourier-Floquet expansion $u^{(1)}(x) = e^{i\\mu x}\\sum _{m=-M}^{M} b_{m} e^{imx},$ with $\\mu \\in \\mathbb {R}$ the Floquet parameter governing the period of the perturbation and a Fourier mode $m \\in \\mathbb {Z}$ .",
"We note that this perturbation can grow exponentially in time if Re$(\\lambda )>0$ , where $\\lambda = \\lambda (\\mu +m)$ depends on the Fourier-Floquet modes $m$ and $\\mu $ .",
"For solutions with $|u^{(0)}(x)| = O(\\epsilon )$ with $\\epsilon \\rightarrow 0$ , $\\lambda (\\mu +m) = i(m+\\mu )V -i\\omega (m+\\mu ),$ if we consider $O(\\delta )$ term when substituting (REF ) and (REF ) into (REF ), staying in the travelling frame of reference.",
"For ease of notation, we introduce the dispersion relation $\\Omega $ in the travelling frame of reference as $\\Omega (m+\\mu ) = \\omega (m+\\mu ) - (m+\\mu )V$ with $\\lambda (\\mu + m) = -i\\Omega (m+\\mu )$ .",
"Since $\\lambda $ is purely imaginary when we consider the linear regime, the perturbation will not grow exponentially in time and thus $u^{(0)}(x)$ is spectrally stable.",
"However, as the nonlinearity is increased with increasing $\\epsilon $ , the eigenvalues which depend continuously on the amplitude of the solution will change and may develop some non-zero real part.",
"Since the equation is Hamiltonian, they will do so symmetrically in the complex plane to conserve the energy, keeping the solution real.",
"The possible configurations of the symmetries in eigenvalues are shown in Figure REF .",
"In order to leave the imaginary axis and develop instability, the eigenvalues first have to collide in order to maintain the symmetry of the equation.",
"In Figure REF , even if eigenvalues move and collide, they do not necessarily leave the imaginary axis as shown in the left panel.",
"This implies a necessary condition for instability is collisions of eigenvalues for different modes $m$ and $n$ in a perturbation given by $\\lambda (\\mu + m) = \\lambda (\\mu + n).$ Also in the linear regime (considering the $O(\\delta )$ term when substituting (REF ) into (REF ) with $|u^{(0)}(x)| \\rightarrow 0$ ), we can explicitly write the Hamiltonian of the system as $H_{\\text{lin}} =\\int _0^{L} \\frac{1}{2}\\left(\\sum _{n=1}^N (-1)^{n} C_{2n+1} (u^{(1)}_{nx})^2 + V (u^{(1)})^2\\right) dx,$ with $0 = \\partial _x \\frac{\\delta H_{\\text{lin}}}{\\delta u^{(1)}}.$ An unstable solution has to conserve energy given by (REF ).",
"This implies that for a collision of eigenvalues arising from two different modes, for every mode that is contributing positively to the Hamiltonian, there needs to be a negatively contributing mode as well.",
"This contribution of eigenvalues to the Hamiltonian (known as their signature) is simply given by the sign of the Hamiltonian.",
"The signature is derived from (REF ) by substituting $u^{(1)} \\sim e^{i(\\mu + m)x}$ to obtain $\\text{sign}(H_{\\text{lin}}) = \\text{sign}\\left(\\sum _{m=1}^N(-1)^mC_{2m+1}(i(\\mu + m))^{2m} + V \\right).$ Using the definition of the dispersion relation in the moving frame and dividing by $i$ , we can write the sign of the Hamiltonian as $\\text{sign}(H_{\\text{lin}}) = \\text{sign}\\left(\\frac{\\Omega (\\mu + m)}{\\mu + m}\\right),$ where we have used (REF ) and the definition of the dispersion relation incorporating the travelling frame of reference.",
"With more algebra described in , , we can introduce $s$ which will govern if two colliding eigenvalues for modes $m$ and $n$ will have opposing signature as $s = (\\mu +m)(\\mu +n) < 0.$ To reduce the number of unknowns in (REF ), we set $(\\mu +m) \\rightarrow \\mu $ therefore letting $n \\rightarrow (n-m)$ , shifting the focus instead on the difference in Fourier modes of the perturbation.",
"This implies that if we wish to consider when the periodic travelling wave solutions are unstable to perturbations of the form shown in (REF ), then we need examine the collision condition $\\lambda (\\mu )=\\lambda (\\mu +n)$ as well as the corresponding combination of signatures of colliding eigenvalues given by $s = \\mu (\\mu +n)$ In the following sections, we show this can be further simplified to one condition using a reduced order polynomial of degree $N$ and examine where the polynomial has real roots thereby meeting the necessary conditions for instability.",
"Figure: Three different configurations of the smallest number of eigenvalues λ\\lambda of the spectral stability problem of a Hamiltonian system, showing the symmetry about the real and imaginary axes.",
"On the left (in blue), is the stable regime.",
"The centre and right panel are the unstable regimes (in red)."
],
[
"General Methodology",
"In general, if we are given a polynomial with $p(\\mu )=\\mu ^N$ with $N$ odd (for example one term in a dispersion relation), then a collision of eigenvalues is of the form $p(\\mu +n)-p(\\mu ) = 0.$ Setting $s = \\mu (\\mu +n)$ , we can equivalently write the collision condition as a reduced-order polynomial $q(s,n)$ of order $\\frac{N-1}{2}$ that is indirectly dependent on the Floquet parameter $\\mu $ as $q(s,n) = \\sum _{i=0}^{\\frac{N-1}{2}}a_{i,N-2i}s^in^{N-2i}.$ The coefficients can be computed recursively as $a_{i,j} ={\\left\\lbrace \\begin{array}{ll}{N \\atopwithdelims ()j} \\ &\\text{for} \\ i=0,j=2,...,N, \\\\a_{i-1,j+1} - a_{i,j+1} \\ &\\text{for} \\ i=1,...,\\frac{N-1}{2}, j=1,...,N-2i,\\\\0 \\ &\\text{otherwise}.\\end{array}\\right.",
"}$ Rewriting the collision condition as a signature condition is always possible as shown by Kollar et al.",
"in .",
"In the following section we will focus on the simplicity of constructing this polynomial for the signature.",
"The main consequence of being able to rewrite the polynomial of lower order, is that it simplifies the equation and the number of roots we have to consider.",
"From (REF ), we can solve for the Floquet parameter as $\\mu = \\frac{1}{2}\\left(-n \\pm \\sqrt{n^2+4 s} \\right).$ To satisfy both the collision condition and signature condition for instability while maintaining that perturbations are bounded in space, we need the roots of (REF ) to be real and for the signatures to remain opposite, i.e.",
"$-\\frac{n^2}{4} < s < 0.$ Checking that the roots of a polynomial are within a certain interval $I$ , in this case given by (REF ), becomes a relatively straightforward procedure and is in some respect easier than computing exact roots.",
"This can be done using Sturm's theory , via a sequence of polynomials (sometimes known as a Sturm chain).",
"Given a polynomial $g(x) = g_0(x)$ of degree $N$ with real coefficients, a sequence of polynomials of decreasing order is constructed by using the following criteria $g_1(x) & = \\frac{\\partial }{\\partial x} g_0(x)\\ \\text{and} \\\\g_n(x) & = -\\left( g_{n-2}(x)-g_{n-1}(x)\\frac{g_{n-2}(x)}{g_{n-1}(x)} \\right) = -\\text{Rem}(g_{n-2}(x),g_{n-1}(x))$ where $\\frac{g_{n-2}(x)}{g_{n-1}(x)}$ is a polynomial quotient and Rem($g_{n-2}(x),g_{n-1}(x)$ ) is the remainder.",
"The sequence terminates at $n=N$ when the last term is a constant and therefore independent of $x$ .",
"If we are interested in how many real roots $r_n$ occur in the interval $I = (a_i,a_f)$ , where $a_i$ and $a_f$ are not themselves roots, then we need to examine the difference in the number of sign changes of the polynomials evaluated at the endpoints of the interval (as shown in (REF ), in this case $a_i = -n^2/4$ and $a_f = 0$ ).",
"To obtain the number of real roots in the interval, we subtract the number of sign changes at $a_f$ from the number of sign changes at $a_i$ .",
"To summarise, in order to analyse spectral stability of periodic travelling waves of (REF ) to high-frequency instabilities of the form given by (REF ), we must Write the dispersion relation $\\omega $ given by the general form in (REF ).",
"Compute the travelling wave speed $V$ for a non-trivial solution.",
"Solve for the polynomial that governs the collision condition of the form (REF ).",
"Reduce the order of the polynomial by substituting $s=\\mu (\\mu +n)$ .",
"Generate the Sturm sequence of polynomials using (REF ).",
"Compute the number of roots in $I$ by examining the number of sign changes in the Sturm sequence of polynomials at each end point and noting the difference.",
"If the result is that we have no real roots contained in $I$ , then the periodic travelling waves are spectrally stable to high-frequency perturbations.",
"In order to show how this method works, we proceed with an example."
],
[
"Example",
"In this section we examine an equation of the form $u_t + \\alpha u_{3x} + \\beta u_{5x} + \\gamma u_{7x} + f(u)_x = 0,$ where $\\alpha $ , $\\beta $ and $\\gamma $ are real coefficients and the subscripts represent the number of derivatives of $u(x,t)$ and go through the process outlined in Section 3 to compute the regions of stability, referring to step number in parentheses.",
"In this section, we will keep these as variables however in practice, they are defined by the scaling in the partial differential equation that is being considered.",
"We begin by introducing a travelling frame of reference, moving with speed $V$ and considering a steady-state solution $\\alpha u_{3x} + \\beta u_{5x} + \\gamma u_{7x} + f(u)_x + V u_x = 0.$ The dispersion relation (step 1 in the process) of this equation is given by $\\omega = -\\alpha k^3 + \\beta k^5 - \\gamma k^7.$ Linearizing about a small amplitude solution with $u^{(0)}=\\epsilon e^{ikx}$ (where $f(u^{(0)}_x) \\approx 0$ ), we obtain $\\alpha (ik)^3 + \\beta (ik)^5 + \\gamma (ik)^7 + V(ik) = 0,$ or $- \\alpha k^2 + \\beta k^4 - \\gamma k^6 + V = 0.$ If we assume the solution we are linearising about is $2 \\pi $ periodic, we can show it is symmetric and without loss of generality we can set $k=1$ .",
"This gives $V_0 = \\alpha - \\beta + \\gamma $ (completing step 2) as a bifurcation point from which we can compute non-trivial solutions $u^{(0)}(x)$ travelling at speed $V_0$ .",
"We will sub in for $V=V_0$ in the equations from now on.",
"The polynomial in terms of $(\\mu ,n)$ (step 3) for the collision condition is given by $p(\\mu ,n) = \\gamma (\\mu +n)^7 - \\beta (\\mu +n)^5 + \\alpha (\\mu +n)^3 - \\gamma \\mu ^9 + \\beta \\mu ^5 -\\alpha \\mu ^3 - (\\alpha - \\beta + \\gamma )n.$ The above can be simplified if we set $s = \\mu (\\mu + n)$ .",
"In order to do this, we first note that we can use binomial theorem gives us the polynomial expansion $(\\mu +n)^N = \\sum _{k=0}^{N}{N \\atopwithdelims ()k}\\mu ^{N-k}n^{k}.$ Table: Acknowledgements"
]
] | 1906.04275 |
[
[
"When random walkers help solving intriguing integrals"
],
[
"Abstract We revisit a family of integrals that delude intuition, and that recently appeared in mathematical literature in connection with computer algebra package verification.",
"We show that the remarkable properties displayed by these integrals become transparent when formulated in the language of random walks.",
"In turn, the random walk view naturally leads to a plethora of nontrivial generalizations, that are worked out.",
"Related complex identities are also derived, without the need of explicit calculation.",
"The crux of our treatment lies in a causality argument where a message that travels at finite speed signals the existence of a boundary."
],
[
"When random walkers help solving intriguing integrals Satya N. Majumdar and Emmanuel Trizac LPTMS, CNRS, Univ.",
"Paris-Sud, Université Paris-Saclay, 91405 Orsay, France We revisit a family of integrals that delude intuition, and that recently appeared in mathematical literature in connection with computer algebra package verification.",
"We show that the remarkable properties displayed by these integrals become transparent when formulated in the language of random walks.",
"In turn, the random walk view naturally leads to a plethora of nontrivial generalizations, that are worked out.",
"Related complex identities are also derived, without the need of explicit calculation.",
"The crux of our treatment lies in a causality argument where a message that travels at finite speed signals the existence of a boundary.",
"Introduction.",
"While intuitions and experimentations are both crucial in mathematical works, inductive thinking may be spectacularly misguided in some cases.",
"A celebrated illustration of the dangers of pattern extrapolation is provided by the question of circle division by chords [1]: Consider $n$ points on the circumference of a circle and join every pair of points by a chord such that at any point inside the circle at most two chords can intersect.",
"How many regions $S_n$ gets the circle divided into?",
"By simple drawing, one sees that $S_1=1$ (by convention), $S_2=2$ , $S_3=4$ , $S_4=8$ , $S_5=16$ .",
"At this point one may naively guess that for general $n$ , $S_n=2^{n-1}$ .",
"Wrong !",
"It turns out that $S_6=31$ .",
"Indeed, the correct answer is $S_n= {n\\atopwithdelims ()4}+{n\\atopwithdelims ()2}+1$ , which happens to coincide with the sequence $2^{n-1}$ up to $n=5$ , but starts differing from it for $n=6$ onwards !",
"Our interest goes here to a lesser known such problem, and the surprising behavior of integrals of the type $I_N &=& \\int _{-\\infty }^{\\infty } \\prod _{n=1}^N \\hbox{sinc}\\left(\\frac{k}{2n-1}\\right) \\, dk\\\\J_N &=& \\int _{-\\infty }^{\\infty } \\cos (k) \\, \\prod _{n=1}^N \\hbox{sinc}\\left(\\frac{k}{2n-1}\\right) \\, dk$ where $\\hbox{sinc}(x)=\\sin (x)/x$ denotes the cardinal sine function [2], [3].",
"We do not dwell on the prevalence of $\\hbox{sinc}$ function in mathematics (geometry, spectral analysis...) and physics (signal processing, optics...), see e.g.",
"[4].",
"It was shown that $I_1=I_2=I_3=I_4=I_5=I_6 = I_7= \\pi $ , whereas $I_N < \\pi $ , for all $N\\ge 8$ [5].",
"In the latter situation, the difference $\\pi -I_N$ is minute, less than $10^{-10}$ for $N=8$ , which was first realized numerically, and attributed to a bug in the software [5].",
"A related phenomenon was observed for the $J$ -family: $J_N = \\pi /2$ for $N\\le 56$ , but $J_N < \\pi /2$ for all $N\\ge 57$ [6].",
"A theorem shown in [5] rationalizes this matter of fact: it states that provided $\\sum _{n=2}^N |a_n| \\, < \\, |a_1|$ , $\\frac{1}{2\\pi }\\, \\int _{-\\infty }^\\infty \\prod _{n=1}^N \\hbox{sinc}(a_n\\, k) \\, dk \\, =\\, \\frac{1}{2|a_1|} .$ Without loss of generality, one can choose the coefficients $a_n$ to be positive real quantities, and $a_1$ can then be taken as the largest of them.",
"Given that $\\sum _{n=2}^7 1/(2n-1)<1$ while $\\sum _{n=2}^8 1/(2n-1)>1$ , this explains the behavior of the $I$ -family, for which $a_1=1$ .",
"The $J$ -family falls under the same argument [7].",
"When the equality in (REF ) breaks, the explicit integrals could be computed.",
"The corresponding values, related to the volume of hypercubes, cut by parallel hyperplanes, is immaterial for our purposes [5].",
"Our goal is rather to provide a transparent understanding of the statement (REF ).",
"To this end, we will show that the language of random walks, and physical intuition not only provide a natural framework to understand this change of behavior, but also leads to relevant and interesting generalizations, thereby offering an explicit and effort-free calculation of complex multidimensional integrals [8].",
"At the heart of our approach lies a causality argument, formulated in terms of a message that signals the existence of a boundary.",
"Random walkers in a finite or infinite “world”.",
"We start by considering a random walk making $N$ steps on a line, $x_{N} = \\sum _{n=1}^N \\eta _n$ starting at $x_0=0$ , where $\\eta _n$ is uniformly distributed in $[-a_n,a_n]$ and the $\\eta _n$ 's are independent random variables.",
"The probability density function (pdf) of each $\\eta _n$ is thus a rectangle function, with simple characteristic function (Fourier-Transform) $\\langle e^{i k \\eta _n} \\rangle \\,=\\, \\hbox{sinc}(a_n\\, k)$ .",
"The characteristic function of $x_N$ , sum of independent increments, thus reads $\\Big \\langle e^{i k x_N} \\Big \\rangle \\, =\\, \\prod _{n=1}^N \\Big \\langle e^{i k \\eta _n}\\Big \\rangle \\,\\, =\\,\\, \\prod _{n=1}^N \\hbox{sinc}(a_n\\, k)$ which allows us to write its pdf as the inverse Fourier transform $&&p_N(x_N) = \\int _{-\\infty }^\\infty \\, \\frac{dk}{2\\pi } \\, \\prod _{n=1}^N \\hbox{sinc}(a_n\\, k) \\, e^{-i k x_N} \\\\\\Longrightarrow ~~ &&\\frac{1}{2\\pi }\\, \\int _{-\\infty }^\\infty \\prod _{n=1}^N \\hbox{sinc}(a_n\\, k) \\, dk \\, = p_N(0).$ The $I_N$ integral under scrutiny is thus isomorphic to $p_N(0)$ , i.e., the probability density of the random walk to be back at the origin, while starting from the origin ($x_0=0$ ).",
"To proceed further, it is useful to reinterpret $p_N(0)$ as follows.",
"Consider a large (infinite actually) number of independent random walkers, all starting at $x_0=0$ .",
"Then $p_N(0)$ is just the fraction of walkers at $x=0$ after $N$ steps, i.e., the density of this gas of independent particles at the origin after $N$ steps.",
"After step 1, their density is uniform in $[-a_1,a_1]$ so that $p_1(0)=1/(2 a_1)$ (incidentally meaning that $I_1=\\pi $ ).",
"A second step is then made, with amplitude $a_2<a_1$ .",
"Because the jump $a_2$ is finite and $a_2<a_1$ , it is clear that all the walkers that were, after step 1, in the range $[-(a_1-a_2), a_1-a_2]$ will not leave their step-1 domain $[-a_1,a_1]$ following the second step.",
"Only walkers near the two edges, e.g., those in the range $[a_1-a_2,a_1]$ or $[-a_1, -a_1+a_2]$ may leave the step-1 domain after the second jump.",
"Hence, the walkers in $[-(a_1-a_2), a_1-a_2]$ do not `see' the edges of the step-1 domain–for them, it is as if the system was infinite with uniform density $1/(2a_1)$ .",
"In such an “infinite world”, the gain and loss contribution balancing those walkers leaving the origin and those reaching it after the second step do cancel: $p_1(0)=p_2(0)$ .",
"This is illustrated in Fig.",
"REF where one can appreciate that the flatness of the density near the origin is preserved, although in a range that diminishes with the number $N$ of steps performed.",
"The argument does not depend on the left/right symmetry of the random steps [9].",
"In other words, we invoke causality and the boundedness of the steps to state that the only possibility for $p_N(0)$ to be affected by a new step is when walkers having started from the edges at $\\pm a_1$ do reach the origin.",
"Those `messengers' carry the information that the “world” is not infinite, which in turn impinges on $p_N(0)$ .",
"If $\\sum _{n=2}^N a_n < a_1$ , the distance traveled by the messengers is not sufficient to reach the origin, and $p_N(0)=1/{2a_1}$ is $N$ -independent.",
"We therefore recover statement (REF ) [10].",
"Besides, while our random walk argument directly applies to $I_N$ integrals, it also is relevant for the $J_N$ -type, as explained in the supplemental material [11].",
"It is nevertheless necessary here to supplement the analysis with a new property, the left-right symmetry of the random steps.",
"The key feature becomes the preservation of the edge density $p_N(a_1)$ under performing random steps, while it pertained to the preservation of $p_N(0)$ when treating $I_N$ integrals [11].",
"Figure: Probability density function of the random walk x N x_N afterN=1N=1, N=2N=2 and N=3N=3 steps, as indicated.",
"Here, the amplitudes of the steps area 1 =1a_1=1, a 2 =1/3a_2=1/3, a 3 =1/5a_3=1/5, in line with the definition of II integrals in Eq.().",
"The density at the origin is invariant(equal to 1/2), which means that I 1 =I 2 =I 3 =πI_1=I_2=I_3=\\pi .",
"For respectivelyN=1,2N=1,2 and 3, the extension of the flatregion near the origin is 2a 1 2a_1, 2(a 1 -a 2 )2(a_1-a_2), 2(a 1 -a 2 -a 3 )2(a_1-a_2-a_3), due to the progressionof the walkers arriving from the boundaries (so-called “messengers” in the main text).Note that the density at x=1x=1 is preserved as well, and half that at the origin (leaving aside theN=1N=1 case).The random walk reformulation provides us with an immediate generalization.",
"Consider, for instance, the case where the first step is of Pearson's type [12] (i.e.",
"of a fixed amplitude $a_1$ , ending at $\\pm a_1$ ), while the subsequent steps are again uniform as before with $\\eta _n\\in [-a_n, a_n]$ for $n\\ge 2$ , then $p_N(0)=0$ for $\\sum _{n=2}^N a_n< a_1$ .",
"In this case, the characteristic function after $N$ steps is given by: $\\langle e^{ik\\, x_n}\\rangle = \\cos (k\\, a_1)\\, \\prod _{n=2}^N \\hbox{sinc}(a_n k)$ .",
"Then our causality argument tells us that $p_N(0)=0$ for $\\sum _{n=2}^N a_n< a_1$ .",
"Evidently, the origin remains void of walkers, until the messengers arrive at $x=0$ .",
"Provided that $\\sum _{n=2}^N a_n \\, < \\, a_1$ , this implies [13]: $\\int _{-\\infty }^\\infty \\cos (a_1 k)\\prod _{n=2}^N \\hbox{sinc}(a_n\\, k) \\, dk \\, =\\,0 .$ This identity can be recovered by invoking a different random walk sharing with the previous one steps $2,3\\ldots n$ , but not the first step [11].",
"One dimensional generalizations.",
"A natural extension of the above results consists in considering that all steps except the first are arbitrary, but of finite range.",
"The corresponding pdfs are therefore of finite support.",
"The argument now involves the associated characteristic functions, that we denote $\\widehat{\\cal F}_n(k)$ .",
"These $\\widehat{\\cal F}_n(k)$ are defined as the Fourier transforms of pdfs ${\\cal F}_n(x)$ that have a finite support, taken for convenience to be unity.",
"Provided $\\sum _{n=2}^N a_n \\, < \\, a_1$ , one can write: $\\frac{1}{2\\pi }\\, \\int _{-\\infty }^\\infty \\hbox{sinc}(a_1 k)\\prod _{n=2}^N \\widehat{\\cal F}_n(a_n\\, k) \\, dk \\, =\\, \\frac{1}{2 a_1} .$ Under the same condition and taking once more advantage of causality, we obtain [13] $\\int _{-\\infty }^\\infty \\cos (a_1 k) \\prod _{n=2}^N \\widehat{\\cal F}_n(a_n\\, k) \\, dk \\, =\\, 0$ where again all amplitudes $a_n$ are considered positive.",
"Some particular cases have been addressed in earlier studies [5], [14], but we stress that many more are subsumed under Eqs.",
"(REF ) and (REF ).",
"The task amounts to establishing a catalogue of eligible $\\widehat{\\cal F}_n(k)$ .",
"It is not our purpose here, and we simply mention some emblematic such functions: the Bessel functions $J_0(k)$ , $J_1(k)/k$ and more generally $J_\\nu (k)/k^\\nu $ for $\\nu >-1/2$ , $(1-\\cos k)/k^2$ , $(k-\\sin k)/k^3$ and a number of hyper-geometric functions.",
"To generate candidates, advantage can be taken from the study of hyper-uniform systems, that feature potentials of bounded Fourier Transform, see e.g.",
"[15].",
"Beyond dimension one.",
"A second natural extention of previous considerations consist in considering $d$ dimensional random walks, with $d>1$ .",
"A straightforward calculation shows that the counterparts of the one dimensional $\\hbox{sinc}(ka)$ and $\\cos (ka)$ functions are given as follows.",
"For a one-step walk with jump $\\mathbf {\\eta }$ , chosen respectively (i) uniformly within a $d$ -dimensional sphere of radius $a$ and (ii) uniformly on the surface of the same sphere (Pearson's type jump), the associated characteristic functions of the jumps are $\\langle e^{i\\, {\\mathbf {k}} \\cdot {\\mathbf {\\eta }} }\\rangle ={\\left\\lbrace \\begin{array}{ll}& \\frac{(2\\pi )^{d/2}}{V_d}\\,\\frac{J_{d/2}(ka)}{(ka)^{d/2}}\\quad \\quad \\,\\, {\\rm (i)} \\\\&\\\\&\\frac{(2\\pi )^{d/2}}{S_d}\\,\\frac{J_{d/2-1}(ka)}{(ka)^{d/2-1}} \\quad \\quad {\\rm (ii)}\\end{array}\\right.",
"}$ where $V_d= \\pi ^{d/2}/{\\Gamma (d/2+1)}$ and $S_d=d\\, V_d$ are respectively the volume and surface of a $d$ -dimensional unit sphere.",
"For $d=1$ , using $J_{1/2}(z)= \\sqrt{2/{\\pi \\, z}}\\, \\sin (z)$ and $J_{-1/2}(z)= \\sqrt{2/{\\pi z}}\\, \\cos (z)$ , one recovers respectively $\\hbox{sinc}(ka)$ and $\\cos (ka)$ .",
"A new catalogue of functions $\\widehat{\\cal F}_n^{(d)}({\\mathbf {k}})$ can then be established, such that their $d$ -dimensional Fourier Transform is of bounded support (several interesting candidates can also be found in [15]; a rather generic one being the hyper-geometric function $_1F_2\\left( \\frac{d+m}{2};\\frac{d}{2};1+\\frac{m+d}{2},-k^2 \\right)$ , where $m$ is some arbitrary parameter).",
"Knowing the eligible building blocks $\\widehat{\\cal F}_n^{(d)}({\\mathbf {k}})$ , one can write upon setting $k=|{\\mathbf {k}}|$ $\\int _{\\mathbb {R}^d} \\frac{J_{d/2}(a_1 k)}{k^{d/2}} \\,\\prod _{n=2}^N \\widehat{\\cal F}_n^{(d)}(a_n\\, {\\mathbf {k}}) \\, d^d {\\mathbf {k}}\\, =\\left(\\frac{2\\pi }{a_1}\\right)^{d/2},$ provided $\\sum _{n=2}^N a_n \\, < \\, a_1$ .",
"A nontrivial identity follows from Eq.",
"(REF ) by considering a special case.",
"Choose the $n$ -th step uniformly from a $d_n$ -dimensional sphere of radius $b_n$ (for $n\\ge 1$ ).",
"Consequently, using $\\widehat{\\cal F}_n^{(d_n)}(a_n\\, {\\mathbf {k}})\\propto J_{d_n/2}(b_n k)/k^{d_n/2}$ in Eq.",
"(REF ), we get upon setting $d_n=2 \\mu _n$ the following identity $\\int _0^\\infty && dk \\, k^{\\mu _1-1} \\, J_{\\mu _1}(b_1\\,k) \\,\\prod _{j=2}^N\\frac{J_{\\mu _j}(b_j k)}{k^{\\mu _j}} \\nonumber \\\\&&= \\frac{2^{\\mu _1-1-\\sum _{j=2}^N \\mu _j}}{{b^{\\mu _1}_1}} \\,\\frac{\\Gamma (\\mu _1)}{\\prod _{j=2}^N \\Gamma (1+\\mu _j)} \\,\\prod _{j=2}^N b_j^{\\mu _j}$ provided $\\sum _{j=2}^N b_j \\,<\\, b_1$ [16].",
"Not surprisingly, with a Pearson first step that depopulates the origin and for $\\sum _{n=2}^N a_n \\, < \\, a_1$ : $\\int _{\\mathbb {R}^d} \\frac{J_{d/2-1}(a_1 k)}{k^{d/2-1}} \\,\\prod _{n=2}^N \\widehat{\\cal F}_n^{(d)}(a_n\\, {\\mathbf {k}}) \\, d^d {\\mathbf {k}}\\, = \\, 0 .$ Mixing dimensions, we note that the steps $n\\ge 2$ can be of any type provided the associated Fourier Transform is bounded: in (REF ) and (REF ), the building blocks $\\widehat{\\cal F}_n^{(d)}({\\mathbf {k}})$ can be some $\\widehat{\\cal F}_n^{(d^{\\prime })}({\\mathbf {k}})$ , borrowed from a lower dimensional catalogue with $d^{\\prime }<d$ .",
"In doing so, we generate a wealth of complex integrals.",
"Some of the simplest are known [17], [18], for instance $\\int _0^\\infty k^{\\nu -\\mu +1}\\, J_{\\nu } (a_1 k) \\, J_\\mu {(a_2 k)} \\, \\cos (a_3 k)\\,\\hbox{sinc}(a_4 k) \\, = 0 \\,$ for $a_2+a_3+a_4<a_1$ , which follows from (REF ) with $\\nu =d/2-1$ , $\\widehat{\\cal F}_{n=2}^{(d)}({\\mathbf {k}})=J_\\mu (k)/k^\\mu $ and that appears under section 6.711.2 in [17], when $a_4=0$ [19].",
"Yet, infinitely many other identities that are subsumed in (REF ) or (REF ) are complex and seemingly unknown.",
"It is worth stressing that in some cases the random walk reformulation may not immediately lead to an explicit result, however it may nevertheless offer a direct means of calculation.",
"As an example, we find the following nontrivial identity $\\int _{\\mathbb {R}^2} && d k_x \\, dk_y\\, \\cos (k_x) \\cos (k_y) \\, J_0(b k)\\, J_0(a k) \\nonumber \\\\&& = \\frac{4}{\\sqrt{\\left(\\left(a+b\\right)^2-2\\right)\\,\\left(2-\\left(a-b\\right)^2\\right)}}$ with $k=\\sqrt{k_x^2+k_y^2}$ .",
"The above result holds for $\\sqrt{2} \\in [|b-a|,b+a]$ (and say $a>0$ , $b>0$ ); otherwise, the integral vanishes.",
"The proof is provided in the supplemental material [11].",
"We outline here the main steps.",
"Consider a 2-d random walk, starting at the origin ${\\mathbf {0}}$ and making 4 successive steps: a first jump $\\pm 1$ along $x$ -direction, a second jump $\\pm 1$ along $y$ -direction, then a Pearson jump on the circle of radius $b$ with a final fourth step distributed as the third, but with a radius $a$ and say $a<b$ , see Fig.",
"REF .",
"Then the lhs of Eq.",
"(REF ), using the results from line (ii) of Eq.",
"(REF ) ($d=1$ for the first two steps and $d=2$ for the last two steps), is precisely $4\\pi ^2\\, p_4(\\mathbf {0})$ where $p_4(\\mathbf {0})$ denotes the density at the origin ${\\mathbf {0}}$ after 4 steps.",
"This density can, in turn be computed by elementary means, see the Suppl.",
"Mat.",
"[11], leading to the rhs of Eq.",
"(REF ).",
"If $\\sqrt{2} \\notin [b-a,b+a]$ , the random walk which is at a distance $\\sqrt{2}$ from the origin after step 2 cannot be back to be origin after step four, exploring an annulus with inner radius $b-a$ and outer radius $b+a$ (see Fig.",
"REF ).",
"Equally tractable is the case where the fourth step is not Pearson but uniform within the disc of radius $a$ (for the 4-th step we use (i) of Eq.",
"(REF ) with $d=2$ ), leading to $\\int _{\\mathbb {R}^2} && d k_x \\, dk_y\\, \\cos (k_x) \\cos (k_y) \\,J_0(b k) \\, \\frac{J_1(a k)}{k} \\nonumber \\\\&& =\\frac{2}{a}\\, \\arccos \\left(\\frac{2+b^2-a^2}{2\\sqrt{2}\\,b}\\right)\\, .$ The lhs can be made more complex without sacrificing the possibility of an explicit calculation of $p_4(\\mathbf {0})$ .",
"Figure: Sketch of the random walk geometry considered.",
"After step 2, the walker is on one of four cornersof a square, chosen to be (1,1)(1,1) on the graph.",
"The third step is uniform onthe large circle with radius bb,while the fourth is uniform on a smaller circle with radius a.",
"The walker can then end up, non uniformly,at any point within the shaded annulus of inner radius b-ab-a and outer radius b+ab+a.Since the origin lies in that region, the integral considered in non-vanishing.When sums and integrals coincide.",
"We now turn to a distinct problem, that bears a similarity with the previous ones, after suitable reformulation.",
"There was some interest recently in identities of the form [20], [21] $\\int _{-\\infty }^\\infty \\prod _{n=1}^N \\hbox{sinc}(a_n k) \\, dk = \\sum _{k=-\\infty }^\\infty \\prod _{n=1}^N \\hbox{sinc}(a_n k)$ which hold provided $\\sum _{n=1}^N a_n < 2\\pi $ [22].",
"The latter condition can be compared to that applying to (REF ), $\\sum _{n=2}^N a_n < a_1$ , that can be rewritten as $\\sum _{n=1}^N a_n < 2 a_1$ .",
"The difference between the two criteria, where the same quantity if bounded either by $2\\pi $ or by $2 a_1$ , indicates that the identity (REF ) cannot be reduced to any of the previous arguments.",
"Yet, the random walk reformulation also is insightful to show, and understand, relation (REF ).",
"The idea is to compare two population of random walkers, one on the infinite line (case F, for “flat”), and the other on the unit circle (case C).",
"Both population, starting from the origin, undergo the same random jumps.",
"Provided that the front-runners (the random walkers having travelled the greater distance from the origin) did not travel round the circle in case C, moving on a flat line or on a finite circle is immaterial.",
"The corresponding condition reads $\\sum _{n=1}^N a_n < 2\\pi $ .",
"When this inequality is fulfilled, the probability density of the walkers at the origin is thus the same in cases C and F. Expressing the pdf as either a Fourier series for case C or a Fourier transform for case F, we then get $\\int _{-\\infty }^\\infty \\prod _{n=1}^N \\widehat{\\cal F}_n(a_n k) \\, dk = \\sum _{k=-\\infty }^\\infty \\prod _{n=1}^N \\widehat{\\cal F}_n(a_n k) ,$ which includes (REF ) and many other cognate relations [23].",
"As above, the $\\widehat{\\cal F}_n$ refer to arbitrary functions, the Fourier transform of which are bounded with unit support.",
"Loosely speaking, Eq.",
"(REF ) can thus be viewed as the “flat world equation”.",
"Conclusion and discussion.",
"We have proposed a random walk interpretation of a curious phenomenon, exhibited by integrals of type (REF )-().",
"The underlying physical image is that of an ensemble of random walkers starting from the origin, and performing a first step so as to populate uniformly the interval $[-a_1,a_1]$ .",
"The walkers then undergo a series of $N$ smaller steps with respective amplitudes $a_n$ .",
"If the maximal span of these steps cannot lead walkers from the edge (i.e.",
"at $x=\\pm a_1$ ) back to the origin $x=0$ , then the walkers near $x=0$ have a fixed density (given by the lhs in Eq.",
"(REF )), specified by the first step and thus equal to $1/(2 a_1)$ .",
"In pictorial terms, the walkers near the origin cannot know they live in a finite world, unless the messengers starting from the confines at $\\pm a_1$ reach them.",
"This may never happen if $\\sum _{n=2}^\\infty a_n \\, < \\, a_1$ , in which case an equality like (REF ) will hold at all orders $N$ [24].",
"It is interesting here to note that the model of random walks with shrinking steps directly applies to physico-chemical problems such as line broadening for single molecule spectroscopy in disordered media [25], [26].",
"The random walk reformulation naturally leads to non-trivial extensions, since it is irrelevant that the steps $n=2,3,$ etc.",
"be uniformly distributed, provided the first one (labeled $n=1$ ) has the desired property (uniform to lead to (REF ) or Pearson to lead to (REF )) and that the subsequent steps ($n=1, 2$ ...) are bounded.",
"Generalizations in higher dimensions appear of particular interest, and provide calculation-free results that would otherwise require considerable effort and ingenuity.",
"While the mathematical problem at stake deceives intuition, we have shown that physical arguments may take over.",
"Physics' insight takes the form of a causality rule, where a message travels from a boundary.",
"Applied to random walkers undergoing jumps of bounded amplitude, this yields a clear account for the change of behavior of a class of multidimensional integrals (of the types (REF ) and (REF )).",
"The random walk picture also allows for simple calculation of complex multidimensional integrals (such as in (REF )) and (REF )).",
"Besides, related probabilistic ideas can be generalized to compute other classes of integrals that are otherwise hard to obtain.",
"Assume for instance that a 1d walker starts with a Pearson jump of amplitude $\\pm a_1$ , so that $\\langle |x_1| \\rangle =a_1$ .",
"The boundedness of subsequent steps and the causality rule mean that $\\langle |x_N| \\rangle $ remain at $a_1$ , as long as $\\sum _{n=2}^N a_n \\le a_1$ , i.e.",
"as long as the message starting from $x=a_1$ does not hit $x=0$ .",
"This implies $\\int _0^\\infty \\frac{1}{k^2} \\, \\left(1-\\cos {(k a_1)} \\prod _{n=2}^N \\widehat{\\cal F}_n(a_n k) \\right) \\, dk \\,=\\,\\frac{\\pi }{2} \\, a_1$ after a straightforward re-expression of $\\langle |x_N| \\rangle $ presented in [11], section III.",
"It thus appears that the curious phenomenon at work for Borwein integrals is much more general and applies to a much broader class of complex integrals and discrete sums."
]
] | 1906.04545 |
[
[
"FASTER Recurrent Networks for Efficient Video Classification"
],
[
"Abstract Typical video classification methods often divide a video into short clips, do inference on each clip independently, then aggregate the clip-level predictions to generate the video-level results.",
"However, processing visually similar clips independently ignores the temporal structure of the video sequence, and increases the computational cost at inference time.",
"In this paper, we propose a novel framework named FASTER, i.e., Feature Aggregation for Spatio-TEmporal Redundancy.",
"FASTER aims to leverage the redundancy between neighboring clips and reduce the computational cost by learning to aggregate the predictions from models of different complexities.",
"The FASTER framework can integrate high quality representations from expensive models to capture subtle motion information and lightweight representations from cheap models to cover scene changes in the video.",
"A new recurrent network (i.e., FAST-GRU) is designed to aggregate the mixture of different representations.",
"Compared with existing approaches, FASTER can reduce the FLOPs by over 10x?",
"while maintaining the state-of-the-art accuracy across popular datasets, such as Kinetics, UCF-101 and HMDB-51."
],
[
"Introduction",
"Video classification has made tremendous progress since the popularity of deep learning.",
"Though the accuracy of Convolutional Neural Networks (CNNs) [25] on standard video datasets continues to improve, their computational cost is also soaring.",
"For example, on the popular Kinetics dataset [20], the pioneering C3D [35] reported a top-1 accuracy of $63.4\\%$ with a single-clip FLOPs of 38.5G.",
"The recent I3D [5] model improved the accuracy to an impressive $72.1\\%$ while its FLOPs also increased to 108G.",
"Non-local networks [41] achieved the state-of-the-art accuracy of $77.7\\%$ on Kinetics with FLOPs as high as 359G.",
"Moreover, this accuracy is achieved by sampling 30 clips from each video, which increases the computational cost by a factor of 30 at test time, as shown in Fig.",
"REF (a).",
"Though modern architectures have achieved incredible accuracy for action recognition, their high computational costs prohibit them from being widely used for real-world applications such as video surveillance, or being deployed on hardware with limited computation power, like mobile devices.",
"Figure: (a) Typical video classification framework processes each clip repeatedly with an expensive model; (b) The FASTER framework exploits the combination of expensive and cheap networks for better accuracy/FLOPs trade-offs.Fig.",
"REF (a) illustrates the typical framework for video classification.",
"Multiple clips are sampled from a video, and fed to a computationally expensive network to generate predictions for each clip.",
"Clip-level predictions are then aggregated, often by taking the average, to form the final video-level results.",
"To reduce the computational cost, recent efforts have focused on designing more efficient architectures [16] at the clip-level.",
"However, less attention has been paid to the efficiency of the overall framework, including how to aggregate the clip-level predictions over time, since they have a strong temporal structure.",
"In this work we address the problem of reducing the computational cost of video classification by focusing on the temporal aggregation stage.",
"In particular, we leverage the observation that video data has a strong temporal structure and is highly redundant in time.",
"We argue that it is computationally inefficient to process many video clips close in time with an expensive video model.",
"As adjacent clips are visually similar, computational cost can be saved by processing most of clips with a lightweight network, shown in Fig.",
"REF (b).",
"A Recurrent Neural Network (RNN) is learned to aggregate the representations from both the expensive and lightweight models.",
"We summarize the contributions of our paper in the following.",
"First, we propose a novel framework for efficient video classification that we call FASTER for Feature Aggregation for Spatio-TEmporal Redundancy (Fig.",
"REF (b)).",
"FASTER explicitly leverages the redundancy of video sequences to reduce FLOPs.",
"Instead of processing every clip with an expensive video model, FASTER uses a combination of an expensive model that captures the details of the action, and a lightweight model which captures scene changes over time, avoids redundant computation, and provides a global coverage of the entire video at a low cost.",
"We show that up to 75% of the clips can be processed with a much cheaper model without losing classification accuracy, maintaining the state-of-the-art accuracy with 10$\\times $ less FLOPs.",
"Second, we design a new RNN architecture, namely FAST-GRU, dedicated to the problem of learning to aggregate the representations from different clip models.",
"As shown in our experiments, the prevailing average pooling fails to encode predictions from different distributions.",
"On the contrary, FAST-GRU is capable of learning integration of representations from multiple models for longer periods of time than popular RNNs, such as GRU [8] and LSTM [15].",
"Compared with typical GRU, FAST-GRU keeps the spatio-temporal resolution of the feature maps unchanged, whereas GRU collapses the resolution with global average pooling.",
"FAST-GRU also has a channel reduction mechanism for feature compression to reduce the number of parameters and introduce more non-linearity.",
"As a result, the FASTER framework achieves significantly better accuracy/FLOPs trade-offs for video classification."
],
[
"Related Work",
"Before the deep learning era, hand-crafted features [37], [38], [24], [21], [9] were widely used for action recognition.",
"In this section, we cover related works that apply deep learning for efficient video classification.",
"paragraph4 .5em plus1ex minus.2ex-.5emClip-level Classification.",
"Many efforts in video classification focus on designing models to classify short clips.",
"3D convolutions [3], [18], [19], [35] are widely adopted for learning spatio-temporal information jointly.",
"While these methods do improve the accuracy, they also dramatically increases computational cost due to expensive 3D convolutions.",
"Recent progress includes factorizing 3D convolution into 2D spatial convolution and 1D temporal convolution [33], [28], [36], [43], [45], and designing new building blocks to learn global information on top of 3D convolution [41], [7].",
"Another line of research designs two-stream networks [29], [44], [40], [11], [5], [43] that use both RGB and optical flow as input.",
"Though two-stream networks tend to improve the accuracy, accurate optical flow algorithms are often computationally expensive.",
"paragraph4 .5em plus1ex minus.2ex-.5emVideo-level Aggregation.",
"To aggregate the clip-level predictions, average pooling is the most popular approach and works remarkably well in practice.",
"Besides average pooling, [19] proposed convolutional temporal fusion networks for aggregation.",
"NetVLAD [2] also has been used in [12] for feature pooling on videos.",
"[44] learned a global video representation by max-pooling over the last convolutional layer.",
"[40] proposed the TSN network that divides the video into segments, which are averaged for classification.",
"RNNs are also used for sequential learning in video classification [32], [44], [26].",
"In contrast with existing work, FAST-GRU aggregates the representations from different models, and aims to leverage spatio-temporal redundancy for efficient video classification.",
"paragraph4 .5em plus1ex minus.2ex-.5emTowards Efficient Video Classification.",
"[46] proposed the ECO model for online video understanding, where a 3D network is used to aggregate frame-level features.",
"Recent work [1], [42], [22], [4] tried to reduce the computational cost by either reducing the number of frames that need to be processed, or sampling discriminative clips to avoid redundant computation.",
"The FASTER framework is orthogonal to these approaches.",
"Instead of applying sophisticated sampling strategies, we focus on designing a general framework with a new architecture that can learn to aggregate representations from models of different complexities."
],
[
"Learning to Aggregate",
"As shown in Fig.",
"REF (b), FASTER is a general framework to aggregate expensive and lightweight representations from different clips.",
"In this section ,we investigate how to learn the temporal structure of clips, aggregate diverse representations to reduce computational cost.",
"We first formulate the problem of learning to aggregate, and introduce the FAST-GRU architecture for aggregation.",
"We then describe other aggregation operators used for comparison and the backbones for extracting the expensive and lightweight representations."
],
[
"Problem Formulation",
"Given a sequence of $N$ clips from a video, we denote their feature representations as $\\mathbf {x}_{t}$ , where $t \\in [0 \\dots N-1]$ .",
"The problem of learning to aggregate can be formulated as: $\\begin{aligned}\\mathbf {o}_{t} = f(\\mathbf {o}_{t-1}, \\mathbf {x}_{t}), t \\in [1 \\dots N-1],\\end{aligned}$ where $\\mathbf {o}_{t-1}$ encodes the historical information before the current clip $\\mathbf {x}_{t}$ , and $\\mathbf {o}_{0}$ equals to $\\mathbf {x}_{0}$ .",
"Note that Eq.",
"REF is exactly a recurrent neural network.",
"All previous features $\\mathbf {x}_{0}$ , $\\mathbf {x}_{1}$ , ..., $\\mathbf {x}_{t}$ are recursively encoded into $\\mathbf {o}_{t}$ .",
"The FASTER framework is suitable for online applications, which requires to provide classification results at any time.",
"Note also FASTER does not make any assumption about the form of $\\mathbf {x}_{t}$ .",
"In this paper, we use the feature map from the last convolutional layer of the clip models, which is a tensor of shape $l \\times h \\times w \\times c$ , as shown in Fig.",
"REF .",
"$l$ is the temporal length of $\\mathbf {x}_{t}$ , whereas $ h \\times w$ is the spatial resolution and $c$ is the number of channels.",
"At any time $t$ , $\\mathbf {x}_{t}$ can come from either the expensive or lightweight model.",
"Once all the features are aggregated, we apply global average pooling to $\\mathbf {o}_{N-1} $ , then feed it to a fully-connected layer, which is trained with SoftMax loss to generate classification scores."
],
[
"FAST-GRU",
"To implement the aggregation function in Eq.",
"REF , we propose the FAST-GRU architecture shown in Fig.",
"REF (b).",
"While FAST-GRU is based on GRU, we highlight their differences in the following.",
"paragraph4 .5em plus1ex minus.2ex-.5em3D Convolutional RNN.",
"RNNs are usually used to model a sequence of inputs represented as 1D feature vectors.",
"These feature vectors can be extracted by applying global average pooling on convolutional feature maps.",
"While these representations are very compact, they do not contain any spatio-temporal information due to pooling.",
"The FASTER framework requires the RNN to handle representations generated by different models.",
"Thus keeping the spatio-temporal resolution of the feature maps may lead to better performance for aggregation.",
"We empirically show that incorporating spatio-temporal information is beneficial for long sequence modeling, which has been largely ignored in recent video pooling methods based on RNNs.",
"Instead of applying a fully-connected layer to the pooled 1D features, we use a 3D $1\\times 1\\times 1$ convolution and keep the original resolution of the feature maps in state transitions, as shown in Fig.",
"REF (b).",
"The matrix multiplication in gating functions is also replaced with 3D $1\\times 1\\times 1$ convolutions, which enable feature gating in each spatio-temporal location.",
"Figure: (a) The “Concat\" baseline simply concatenates 𝐱 t \\mathbf {x}_{t} with 𝐨 t-1 \\mathbf {o}_{t-1};(b) Unlike typical GRU collapses the resolution of 𝐱 t \\mathbf {x}_{t} with global average pooling, FAST-GRU keeps the resolution of 𝐱 t \\mathbf {x}_{t} to learn spatio-temporal information.We also introduce bottleneck structures (in dash rectangles) in the gate functions to reduce the number of parameters and increase non-linearity.paragraph4 .5em plus1ex minus.2ex-.5emBottleneck Gating.",
"In RNNs, the importance of the current input $\\mathbf {x}_{t}$ is decided by a subnetwork conditioned on the historical information encoded in $\\mathbf {o}_{t-1}$ .",
"The subnetwork learns to combine the state $\\mathbf {o}_{t-1}$ and input $\\mathbf {x}_{t}$ with different gating mechanisms.",
"Different gates are designed for different functionality for sequential modeling in RNNs.",
"Here we first briefly describe the original GRU architecture, then introduce the bottleneck gating mechanism we proposed.",
"In GRU, the subnetwork first concatenates $\\mathbf {x}_t$ and $\\mathbf {o}_{t-1}$ , then project them to a vector of the same dimension, , the number of channels $c$ .",
"There are two gates in GRU, , the read gate $\\mathbf {r}$ and update gate $\\mathbf {z}$ , which are defined as $\\begin{aligned}\\mathbf {r}_t &=\\sigma (\\mathbf {G}_{rx}\\mathbf {x}_t + {\\mathbf {G}_{ro}\\mathbf {o}_{t-1}}), \\\\\\mathbf {z}_t &=\\sigma (\\textstyle {\\mathbf {G}_{zx}\\mathbf {x}_t + {\\mathbf {G}_{zo}\\mathbf {o}_{t-1}}}), \\\\\\end{aligned}$ where $\\mathbf {G}_*$ are $c\\times c$ matrices, and $\\sigma $ is the sigmoid function.",
"After the calculation of $\\mathbf {r}_t$ and $\\mathbf {z}_t$ , The output of GRU is $\\begin{aligned}\\bar{\\mathbf {o}_t} &= \\textstyle {\\tanh (\\mathbf {V}_{\\bar{o}} \\mathbf {x}_t + {\\mathbf {V}_{\\bar{o}} (\\mathbf {r}_t \\odot \\mathbf {o}_{t-1})})}, \\\\\\mathbf {o}_t &= \\textstyle {(1 - \\mathbf {z}_t) \\odot \\mathbf {o}_{t-1} + \\mathbf {z}_t \\odot \\bar{\\mathbf {o}_{t}}}, \\end{aligned}$ where $\\mathbf {V}_*$ are $c\\times c$ matrices, and $\\odot $ represents element-wise multiplication.",
"Both $\\mathbf {G}_*$ and $\\mathbf {V}_*$ are learnable parameters.",
"Inspired by ResNet [14], we introduce additional bottleneck structures to increases the expressiveness of the subnetwork in FAST-GRU, shown in the dash rectangles of Fig.",
"REF (b).",
"In the bottleneck layer, $\\mathbf {x}_t$ and $\\mathbf {o}_{t-1}$ are projected to a lower dimensional feature space after concatenation.",
"This compresses the number of channels to reduce parameters and introduces more non-linearity.",
"The compressed features are then recovered to the original dimension $c$ with another projection.",
"In practice, channel compression is implemented by $1\\times 1\\times 1$ convolution to reduce the dimension by a factor of $r$ .",
"After that, there is an ReLU followed by $1\\times 1\\times 1$ convolution to recover the dimensionality.",
"The read and update gates of FAST-GRU are defined as: $\\begin{aligned}\\mathbf {r^{\\prime }}_t &= \\text{ReLU}(\\mathbf {U}_{rx} * \\mathbf {x}_t + {\\mathbf {U}_{ro} * \\mathbf {o}_{t-1}}), \\\\\\mathbf {z^{\\prime }}_t &= \\text{ReLU}(\\textstyle {\\mathbf {U}_{zx} * \\mathbf {x}_t + {\\mathbf {U}_{zo} * \\mathbf {o}_{t-1}}}), \\\\\\mathbf {r}_t &= \\textstyle {\\sigma (\\mathbf {W}_{r^{\\prime }} * \\mathbf {r^{\\prime }})}, \\\\\\mathbf {z}_t &= \\textstyle {\\sigma (\\mathbf {W}_{z^{\\prime }} * \\mathbf {z^{\\prime }})}, \\\\\\end{aligned}$ where $\\mathbf {U}_*$ are $1\\times 1\\times 1$ convolutions to compress the number of channels by a factor of $r$ , and $\\mathbf {W}_*$ are $1\\times 1\\times 1$ convolutions to recover the dimensionality.",
"Our gates are generated by taking the jointly compressed features $\\mathbf {r}_t^{\\prime }$ and $\\mathbf {z}_t^{\\prime }$ , which enables more powerful gating for feature aggregation.",
"We empirically set $r$ to be 4, which gives good results in our experiments."
],
[
"Other Aggregation Operator Variants",
"Besides FAST-GRU, we also consider other baselines for the learning-to-aggregate problem for a comprehensive comparison.",
"We detail each baseline in the following.",
"Average pooling is the most popular approach for aggregation.",
"Despite its simplicity, it performs remarkably well in practice.",
"In the context of our FASTER framework, we just average the classification scores from each clip regardless of whether the scores are generated by the expensive or lightweight model.",
"The concat baseline is illustrated in Fig.",
"REF , where $\\mathbf {x}_{t}$ and $\\mathbf {o}_{t-1}$ are concatenated together, and then projected to a joint feature space.",
"It is defined as: $\\begin{aligned}f(\\mathbf {o}_{t-1}, \\mathbf {x}_{t}) = \\text{ReLU}(\\text{BN}(\\mathbf {W}\\mathbf {o}_{t-1} + \\mathbf {U}\\mathbf {x}_{t})),\\end{aligned}$ where $\\mathbf {W}$ and $\\mathbf {U}$ are learnable parameters.",
"Batch normalization [17] and ReLU are also applied in the concat baseline to further improve the performance.",
"Popular RNNs such as LSTM [15] and GRU [8] are the most related baselines.",
"These are the go-to methods for tasks that involve sequential modeling [34], such as language or speech.",
"For the LSTM baseline, we use the variant that consists of three gates and an additional cell between time steps [13].",
"LSTM has a forget gate that can reset the history for the current input $\\mathbf {x}_{t}$ .",
"For video classification, this could be useful in the case of different camera shots or unrelated actions.",
"The GRU baseline we used has two gates, , the read and update gate as we described in the previous section.",
"The state transition function uses the update gate to perform a weighted average of the historical state $\\mathbf {o}_{t-1}$ and the current input $\\mathbf {x}_{t}$ ."
],
[
"Clip-level Backbones",
"FASTER is a general framework and does not make any assumption about the underlying clip-level backbones.",
"We can potentially choose any popular networks, such as I3D [5], R(2+1)D [36] or non-local network [41].",
"We implement FASTER based on R(2+1)D because it is one of the state-of-the-art methods, and its Github repositoryhttps://github.com/facebookresearch/VMZ provides a family of networks that can be used as the expensive and lightweight models.",
"In particular, we choose R(2+1)D with 50 layers as the expensive model, and R2D with 26 layers as the lightweight model, as detailed in Table REF .",
"Comparing with the original R(2+1)D, we make two changes to further improve its performance.",
"First, we replace convolutional blocks with bottleneck blocks, which have been widely used in the family of ResNet architectures and shown to both reduce computational cost and improve accuracy.",
"Second, we insert a max-pooling layer after $\\mathbf {conv}_1$ , which enables R(2+1)D to support a spatial resolution of $224\\times 224$ without significantly increasing its FLOPs.",
"For the lightweight R2D model, bottleneck layers are used in the same way as R(2+1)D. To reduce the FLOPs of R2D, a temporal stride of 8 is used in $\\mathbf {conv}_1$ , which effectively reduces the temporal length of the clip by a factor of 8.",
"Unlike R(2+1)D, R2D only has 26 layers to further reduce the computational cost.",
"To integrate R(2+1)D-50 and R2D-26 in FASTER, we simply use the two backbones as feature extractors.",
"In Table REF , the output from res$_5$ generates a feature map of size $\\frac{L}{8} \\times 7 \\times 7 \\times 2048$ , which is used as input $\\mathbf {x}_{t}$ to the RNNs.",
"Table: Clip-level backbones for extracting the expensive and lightweight representations.",
"The FLOPs of R(2+1)D-50 is about 10×\\times of R2D-26."
],
[
"Experimental Setups",
"In this section, we describe the experimental setups, , the datasets, training and test protocols for both the clip-level backbones and FASTER framework.",
"paragraph4 .5em plus1ex minus.2ex-.5emDatasets.",
"We choose the Kinetics [20] dataset as the major testbed for FASTER.",
"Kinetics is among the most popular datasets for video classification.",
"To simplify, all reported results on Kinetics are trained from scratch, without pre-training on other datasets (, Sports1M or ImageNet).",
"Kinetics has 400 action classes and about 240K training videos.",
"We report top-1 accuracy on the validation set as labels on the testing set is not public available.",
"We also report results on UCF-101 [31] and HMDB-51 [23].",
"These datasets are much smaller, thus we use Kinetics for pre-training and report mean accuracy on three testing splits.",
"Table: Accuracy of clip-level backbones on Kinetics.The significant performance gap between the two backbones offers a great opportunity to explore better accuracy/FLOPs trade-offs when combining them.Due to limited GPU memory, we train the FASTER framework in a two-stage process.",
"First, two clip-level backbones are trained by themselves separately.",
"In the second stage, two backbones are fixed and we train different RNNs using the feature maps extracted by the backbones.",
"paragraph4 .5em plus1ex minus.2ex-.5emSetups for clip-level backbones.",
"We mostly follow the procedure in [36] to train the clip-level backbones except two changes.",
"First, we scale the input video whose shorter side is randomly sampled in [256, 320] pixels, following [41], [30].",
"Second, we adopt the cosine learning rate schedule [27].",
"During training, we randomly sample $L$ consecutive frames from a given video.",
"For testing, we uniformly sample $N$ clips to cover the whole video, and average pool the classification scores of each clip, as shown in Table REF .",
"We fix the total number of frames processed to be 256, , $N \\times L = 256$ .",
"As the average length of Kinetics videos is about 10 seconds in 30 FPS, 256 frames give us enough coverage over the whole video.",
"We only use RGB frames as input as computing optical flow can be very expensive.",
"paragraph4 .5em plus1ex minus.2ex-.5emSetups for learning to aggregate.",
"To train different RNNs for learning to aggregate, we randomly sample $N$ clips, and each clip has $L$ consecutive frames.",
"As mentioned above, the clip-level backbones are fixed in the second stage due to limited GPU memory.",
"The initial state $\\mathbf {o}_0$ is set to be the features from the first clip $\\mathbf {x}_{0}$ .",
"The training procedure is similar to the first stage except we train much fewer epochs.",
"For testing, the $N$ clips are uniformly sampled to cover the whole video.",
"Table: Comparison of different architectures for aggregation on Kinetics.",
"The clip length LL is 8.",
"For all the methods, only the first clip is processed by the expensive model, and the remaining clips are processed by the lightweight model."
],
[
"Experimental Results",
"In this section, we demonstrate the advantage of FASTER and discuss accuracy/FLOPs trade-offs in different settings on Kinetics.",
"We also compare FASTER with the state of the art on Kinetics, UCF-101, and HMDB-51.",
"paragraph4 .5em plus1ex minus.2ex-.5emAccuracy of the backbone architectures.",
"Table.",
"REF summarizes the top-1 accuracy of the two clip-level backbones on Kinetics.",
"We consider the clip length $L$ to be 8, 16 or 32 frames, and sample 32, 16 or 8 clips, respectively.",
"This is to ensure that we always process the same number of frames under different settings.",
"Note that the GFLOPs of R(2+1)D-50 is about 10$\\times $ of R2D-26, and the accuracy of R(2+1)D-50 is about $7\\%$ better than R2D-26.",
"The significant performance gap between the two backbones gives us the opportunity to explore different trade-offs in the following experiments.",
"For the same backbone, longer clip length $L$ often leads to better accuracy.",
"But we can not train R(2+1)D-50 with $L=64$ frames due to GPU memory limit.",
"paragraph4 .5em plus1ex minus.2ex-.5emFAST-GRU .",
"other RNN variants.",
"We compare the accuracy of different architectures for the learning-to-aggregate problem across a wide range of number of clips ($N=\\lbrace 2, 4, 8, 16, 32\\rbrace $ ).",
"The first clip is always processed with the expensive model and the following $N-1$ clips are processed with the cheap model.",
"We use clip length $L=8$ frames.",
"This allows us to observe the behavior of different architectures in shorter and longer sequences.",
"The results are shown in Table REF .",
"For “Avg.",
"pool”, we simply average the classification scores of all the clips.",
"The performance of “Avg.",
"pool” saturates as more cheap clips are added.",
"Less accurate predictions dominate the final score, since the accuracy of 16 clips (, 64.0%) is quite similar to the accuracy of 32 clips (, 63.8%).",
"This suggests that the expensive representations from the first clip are not fully utilized.",
"Figure: FASTER typical video classification frameworks with average pooling.",
"We show accuracy GFLOPs of different methods as a function of the number of clips.",
"We compare the results of FAST-GRU from Table with average pooling of all expensive (Avg.",
"pool (expensive)) and lightweight models (Avg.",
"pool (cheap)).“Concat” outperforms “Avg.",
"pool” by 2.1% when $N=8$ , showing the benefits of learning temporal structure.",
"However, as $N$ increases, the performance of “Concat” decreases, and by $N=32$ , the accuracy is 8.1% lower than $N=8$ .",
"This shows that learning to aggregate over a longer time period is difficult.",
"Interestingly, “Concat”, and all recurrent networks have similar performances when the number of clips is small ($N=2,4$ ), showing that the advantage of RNNs is not evident for short sequences.",
"As the number of clips increases ($N\\ge 16$ ), recurrent networks show their strength in modeling long sequences and perform better than the simple “Concat” baseline.",
"Of all the methods, FAST-GRU achieves the best performance for long sequences.",
"For example, FAST-GRU outperforms LSTM by 2.5% and GRU by 1.4% when $N=32$ .",
"The results show that our proposed FAST-GRU is beneficial for long sequence learning, even when sequence samples come from different distributions.",
"paragraph4 .5em plus1ex minus.2ex-.5emFASTER .",
"average pooling.",
"How well does the FASTER framework do with respect to the prevailing average pooling?",
"Recall that our main focus is not only to increase accuracy, but also to reduce computational cost.",
"For this, we compare the results of FAST-GRU from Table REF with the traditional framework (in Fig.",
"REF (a)), where all clips are cheap (or all clips are expensive) and aggregated with average pooling.",
"These can be considered as the lower and upper bound of the FASTER framework.",
"We plot the GFLOPs .",
"accuracy comparisons in Fig.",
"REF .",
"We use clip length $L=8$ frames, and consider the number of clips $N=\\lbrace 2, 4, 8, 16, 32\\rbrace $ , like Table REF .",
"When $N=4$ , it takes 119 GFLOPs for “Avg.",
"pool (expensive)” to achieve 67.1%.",
"FASTER can achieve comparable results with only half the GFLOPs.",
"Comparing with “Avg.",
"pool (cheap)” when $N=16$ , FASTER is over 4% better with similar amount of GFLOPs.",
"Note that there is still an accuracy gap between FASTER and “Avg.",
"pool (expensive)”.",
"In the next experiment, we introduce more expensive clips and study the optimal ratio between the expensive and cheap clips, achieving superior results.",
"Table: Optimal ratio between the expensive and cheap clips.For a fixed number of total frames, we vary the number of clips NN and clip length LL, and measure accuracy and GFLOPs on Kinetics.",
"Empty cells correspond to combinations that are not feasible.",
"“E” denotes the expensive model, while “C” denotes the cheap model.GFLOPs combine the costs of the cheap, expensive and FAST-GRU models together.paragraph4 .5em plus1ex minus.2ex-.5emBetter accuracy/FLOPs trade-offs.",
"There are three parameters in our framework that affect accuracy and computational cost: the clip length ($L$ ), the number of clips ($N$ ), and the proportion of expensive and lightweight clips used as input pattern (#E :#C).",
"We now provide an empirical study of these parameters to achieve the best trade-off.",
"Since we are interested in the behavior of FASTER, we “fix\" the input data by keeping the number of frames constant, which spans the entire video.",
"Thus, $L\\times N=256$ for all settings.",
"Given $N$ clips, we experiment with input patterns of $1:x$ .",
"It is the ratio between the number of expensive clips and cheap clips, $x \\in \\lbrace 1, 3, 7, 15, 31\\rbrace $ .",
"For example, when $x=3$ and $L=16$ , there will be 4 expensive clips and 12 cheap clips.",
"Additionally, we evaluate the case where all the inputs are expensive clips ($x=0$ ).",
"Results are shown in Table REF .",
"As expected, a higher ratio of expensive models leads to a higher accuracy.",
"For example, when $L=8$ , there is a large gap of 1.8% when $x$ goes from 31 to 15, with a modest increase of $25.6$ GFLOPs.",
"The gap is less obvious for higher ratios of expensive models.",
"A remarkable observation is that FASTER achieves 70.1% when $x=3$ , which is the same accuracy as the average pooling baseline with all expensive clips from Table REF .",
"But FASTER only consumes 35% of its GFLOPs.",
"Even more, for $x=1$ , the proposed method outperforms the average pooling baseline in Table REF , at 57% of the cost.",
"This shows the benefit of learning temporal aggregation of hybrid inputs.",
"We also observe that for a fixed cost, it is often beneficial to use longer clip length $L$ .",
"For example, for a fixed proportion of cheap models $x=3$ , the accuracy of 32-frames$\\times $ 8-clips is higher than 16-frames$\\times $ 16-clips, and that in turn is considerably higher than 8-frames$\\times $ 32-clips.",
"In other words, it is better to have fewer clips with more frames.",
"However, for higher ratios of cheap clips (, $x=7$ ), this pattern does not hold.",
"Table: Comparisons of runtime between different methods.",
"The results of FASTER is reported in the 32-frames×\\times 8-clips setting.Table: Compare different architectures as the expensive clip-level backbone.",
"Our FASTER framework can generalize to different architectures, and R3D even performs slightly better than R(2+1)D.We choose the parameter setting $L=16$ and $x=7$ as the best inexpensive configuration, and call this setting FASTER16 (, $L=16, x=7$ ).",
"It indicates that it is possible to achieve better performance with shorter clip length $L$ , which reflects the importance of learning a good aggregation function.",
"When $L=32$ and $x=1$ , the proposed framework achieves 74.6% accuracy with 550.4 GFLOPs.",
"We denote this setting as FASTER32 (, $L=32, x=1$ ) and it is the best model when the cost is around $\\sim $ 550 GFLOPs .",
"FASTER also works when all input features come from the same network.",
"When the inputs are all expensive features, FASTER outperforms the average pooling baseline from Table REF in all settings, with a negligible increase of FLOPs from FAST-GRU.",
"paragraph4 .5em plus1ex minus.2ex-.5emRuntime FLOPs.",
"We measure the runtime speed of different methods on a TITAN X GPU with Intel i7 CPU.",
"We sum up the runtime over 100 Kinetics videos.",
"The results are listed in Table REF , which confirms that the theoretical FLOPs are consistent with the runtime for our experiments.",
"The reduction of FLOPs is roughly proportional to the decrease of runtime.",
"paragraph4 .5em plus1ex minus.2ex-.5emGeneralize to different backbones.",
"Our FASTER framework does not make any assumption about the clip-level backbones.",
"To demonstrate that FASTER can generalize to different backbones, we replace the expensive model, , R(2+1)D-50, with R3D-50 [36] and keep everything else unchanged.",
"In Table REF , R3D even works slightly better than R(2+1)D. Comparing FASTER32 with the average pooling baseline, using R3D leads to a larger improvement of 0.9%.",
"Table: Comparisons to the state-of-the-art methods on Kinetics.The inputs for all the methods are RGB frames.",
"For “GFLOPs ×\\times clips”, we report the cost of a single clip and the number of clips used.",
"“N/A”' indicates the number of clips are not reported in the paper.Table: Comparisons on UCF-101 and HMDB-51.",
"“I” denotes pre-training on ImageNet, while “K” denotes pre-training on Kinetics.Figure: Log-scale GFLOPs accuracy comparisons on Kinetics.",
"In this chart, optimal methods are closer to the top-left corner.",
"The two proposed variants of FASTER are the closest to the corner.paragraph4 .5em plus1ex minus.2ex-.5emComparison to state-of-the-art We now compare the FASTER framework to state-of-the-art methods across three most popular video datasets, , Kinetics, UCF-101 and HMDB-51.",
"We only include methods that use RGB frames as inputs.",
"Results on Kinetics are shown in Table REF .",
"These same numbers are also visualized in terms of accuracy GFLOPs in Fig.",
"REF .",
"Our proposed FASTER framework achieves the best accuracy/GFLOPs trade-offs.",
"This is specially impressive since FASTER is not pre-trained on ImageNet, and the clip-level backbones used are not the most cost effective ones.",
"In this comparison, we use FASTER32 with R3D as its backbone.",
"FASTER32 outperforms R(2+1)D-34 by 3.3% with only 4% of its GFLOPs.",
"FASTER16 outperforms I3D trained from scratch by 3.3% (71.7% 68.4%), using only half of I3D's cost (432 GFLOPs).",
"Even though we do not leverage pre-training, our framework outperforms all existing pre-trained models except NL-I3D.",
"However, FASTER32 only uses 5% of the GFLOPs of NL-I3D.",
"Our FASTER32 model outperforms $A^2$ -Net by 0.7%, while the cost is reduced by over 50%.",
"The results validate the effectiveness of our FASTER framework, showing it is promising to leverage the combination of expensive and lightweight models for better accuracy/FLOPs trade-off.",
"Competitive results are also achieved on UCF-101 and HMDB-51 (Table REF ).",
"It shows learning the aggregation function on larger datasets can facilitate the generalization on smaller datasets.",
"The improvements are less profound as the performance on small datasets tends to be saturated."
],
[
"Conclusion",
"In this paper, we propose a novel framework called FASTER for efficient video classification.",
"To exploit the spatio-temporal redundancy of the video sequence, we combine the representations from both expensive and lightweight models.",
"We propose a recurrent unit called FAST-GRU to learn to aggregate mixed representations.",
"Experimental results show that our FASTER framework has significantly better accuracy/FLOPs trade-offs, achieving the state-of-the-art accuracy with $10\\times $ less FLOPs."
]
] | 1906.04226 |
[
[
"Fast-forward approach to adiabatic quantum dynamics of regular spin\n clusters: nature of geometry-dependent driving interactions"
],
[
"Abstract The fast forward scheme of adiabatic quantum dynamics is applied to finite regular spin clusters with various geometries and the nature of driving interactions is elucidated.",
"The fast forward is the quasi-adiabatic dynamics guaranteed by regularization terms added to the reference Hamiltonian, followed by a rescaling of time with use of a large scaling factor.",
"With help of the regularization terms consisting of pair-wise and 3-body interactions, we apply the proposed formula (Phys.",
"Rev.A 96, 052106(2017)) to regular triangle and open linear chain for N = 3 spin systems, and to triangular pyramid, square, primary star graph and open linear chain for N = 4 spin systems.",
"The geometry-induced symmetry greatly decreases the rank of coefficient matrix of the linear algebraic equation for regularization terms.",
"Choosing a transverse Ising Hamiltonian as a reference, we find: (1) for N = 3 spin clusters, the driving interaction consists of only the geometry-dependent pairwise interactions and there is no need for the 3-body interaction; (2) for N = 4 spin clusters, the geometry-dependent pair-wise interactions again constitute major part of the driving interaction, whereas the universal 3-body interaction free from the geometry is necessary but plays a subsidiary role.",
"Our scheme predicts the practical driving interaction in accelerating the adiabatic quantum dynamics of structured regular spin clusters."
],
[
"INTRODUCTION",
"Effectively manipulating and optimizing the dynamics of given systems constitutes one of big experimental and theoretical subjects in the current technology.",
"In particular, it is a challenging theme to find suitable driving fields for tailoring a quantum system to rapidly generate a target state from a given initial state.",
"In designing quantum computers, the acceleration of adiabatic quantum dynamics is desirable because the coherence of systems is degraded by their interaction with the environment.",
"Since naive numerical trial-and-error methods are time- and resource-consuming, we must deeply understand relevant quantum dynamics to find useful schemes for such accelerations.",
"In this context, various researches on the way to the shortcut to adiabaticity (STA) have been developed, which include invariant-based inverse engineering [1], [2], [3], transitionless counter-diabatic (CD) driving [4], [5], [6], fast-forward approach [7], [8], [9], and variational methods to generate approximate CD protocols[10], [11], [12].",
"The fast-forward theory proposed by Masuda and Nakamura [7] was originally concerned with acceleration of general reference quantum dynamics.",
"This theory was developed to accelerate the adiabatic quantum dynamics by introducing the large time-scaling factor in the quasi-adiabatic dynamics guaranteed by regularization terms added to the reference Hamiltonian [8], [9], and was then used to enhance the quantum tunneling power [13] and to construct the non-equilibrium equation of state under a rapid piston [14].",
"The relation between the fast-forward approach and other methods was rigorously investigated in [15].",
"Recently, we proposed a fast forward scheme of adiabatic spin dynamics [16].",
"Confining to a single and two spin systems there, we showed the acceleration of Landau-Zener transition and that of a generation of entangled states, as can be shown in other methods [4], [5], [6], [17], [10], [18], [19] .",
"The fast forward scheme of adiabatic quantum dynamics has advantages as addressed by [20], [21]: (1) No need of writing the driving interaction in the spectral representation with use of full spectral properties of given spin systems.",
"No necessity of worrying about the divergence of the driving interaction due to the level crossing; (2) A great flexibility in choosing the regularization Hamiltonian which leads to the driving interaction.",
"Namely, users can specify the regularization Hamiltonian by themselves so as to satisfy the core equation (see Eq.",
"(REF ) of this paper).",
"The latter advantage will play an important role when we shall investigate spin clusters of various geometries.",
"However, no technical guide was so far presented in solving the core equation for unknown regularization terms.",
"Within a framework of the transitionless CD driving [4], [5], [6], on the other hand, there exist intensive works on a linear chain of many quantum spins described by the Ising model in a transverse field [22], [23], [24], [25] and the related model [26], which showed the complicated non-local multi-body CD terms that are hard to achieve in experiment.",
"While a variational method to generate approximate local CD protocols[11], [12] is being cultivated, it is timely to sharpen the fast-forward approach by showing a guiding principle to manage spin clusters with various geometries on the basis of the proposed formula in [16].",
"In this paper the fast forward scheme of adiabatic dynamics is applied to regular spin clusters of various geometries with number of spins $N$ up to 4, i.e., regular triangle and open linear chain for $N=3$ spins, and triangular pyramid, square, primary star graph and open linear chain for $N=4$ spins.",
"(Note: the geometry is irrelevant for systems with $N=1$ and 2 spins.)",
"Choosing the Hamiltonian for a transverse Ising model as a reference, we shall reveal the nature of driving interactions.",
"In Section , a brief summary is given on the fast forward scheme of adiabatic quantum spin dynamics.",
"In Section we propose a candidate regularization Hamiltonian consisting of geometry-dependent pair-wise interactions and a universal 3-body interaction, and describe a method of solving the linear algebraic equation for regularization terms.",
"Sections and are devoted to the analysis of spin clusters of various geometries with $N=3$ and $N=4$ , respectively.",
"Summary and discussions are given in Section .",
"Appendix gives matrices for some regularization Hamiltonians."
],
[
"Fast-forward scheme of adiabatic spin dynamics",
"For self-containedness, we shall sketch the fast forward scheme of adiabatic spin dynamics[16].",
"Our strategy is as follows: (i) A given original (reference) Hamiltonian $H_{0}$ is assumed to change adiabatically and to generate a stationary state $\\Psi _{0}$ , which is an eigenstate of the time-independent Schrödinger equation with the instantaneous Hamiltonian.",
"Then $H_{0}$ is regularized so that $\\Psi _{0}$ should satisfy the time-dependent Schrödinger equation (TDSE); (ii) Taking $\\Psi _{0}$ as a reference state, we shall rescale time in TDSE with use of the scaling factor $\\alpha \\left(t\\right)$ , where the mean value $\\bar{\\alpha }$ of the infinitely-large time scaling factor $\\alpha (t)$ will be chosen to compensate the infinitesimally-small growth rate $\\epsilon $ of the quasi-adiabatic parameter and to satisfy $\\bar{\\alpha } \\times \\epsilon =finite$ .",
"Consider the Hamiltonian for spin systems to be characterized by a slowly time-changing parameter $R(t)$ such as the exchange interaction, magnetic field, etc.",
"Then we can study the eigenvalue problem for the time-independent Schrödinger equation : $H_0(R)\\textbf {C}^{(n)}(R)=E_n(R)\\textbf {C}^{(n)}(R)$ with $\\textbf {C}^{(n)}(R)=\\begin{pmatrix}C^{(n)}_1(R) \\\\ \\vdots \\\\C^{(n)}_N(R)\\end{pmatrix},$ where $R\\equiv R(t)= R_0 + \\epsilon t$ is the adiabatically-changing parameter with $\\epsilon \\ll 1$ .",
"In Eq.",
"(REF ), $n$ stands for the quantum number for each eigenvalue and eigenstate.",
"Let us assume $\\Psi ^{(n)}_0(R(t)) =\\textbf {C}^{(n)}(R(t))e^{-\\frac{i}{\\hbar }\\int _{0}^{t}E_n(R(t^{\\prime }))dt^{\\prime }} e^{i\\xi _n(R(t))},$ to be a quasi-adiabatic state, i.e., adiabatically evolving state, where $\\xi _n$ is the adiabatic phase: $\\xi _n(R(t))= i \\int _{0}^{t} dt^{\\prime } \\textbf {C}^{(n)\\dagger }\\partial _t\\textbf {C}^{(n)} = i \\epsilon \\int _{0}^{t} dt^{\\prime } \\textbf {C}^{(n)\\dagger }\\partial _R\\textbf {C}^{(n)}.",
"\\nonumber \\\\$ $\\Psi ^{(n)}_0(R(t))$ in Eq.",
"(REF ) is not a solution of TDSE.",
"To make it to satisfy the TDSE, we must regularize the Hamiltonian as $H_0^{reg}(R(t)) = H_0(R(t)) + \\epsilon \\mathcal {\\tilde{H}}_n(R(t)).$ Then TDSE becomes $i \\hbar \\frac{\\partial }{\\partial t} \\Psi ^{(n)}_0(R(t))=(H_0+\\epsilon \\mathcal {\\tilde{H}}_n)\\Psi ^{(n)}_0(R(t)).$ Here $\\mathcal {\\tilde{H}}_n$ is the $n$ -th state-dependent regularization term.",
"Substituting $\\Psi ^{(n)}_0(R(t))$ in Eq.",
"(REF ) into the above TDSE, we see the eigenvalue problem in Eq.",
"(REF ) in order of $O(\\epsilon ^0)$ , and the algebraic equation for $\\mathcal {\\tilde{H}}_n$ , $\\mathcal {\\tilde{H}}_n\\textbf {C}^{(n)}(R)=i \\hbar \\partial _R\\textbf {C}^{(n)}(R) -i\\hbar ( \\textbf {C}^{(n)\\dagger }\\partial _R\\textbf {C}^{(n)})\\textbf {C}^{(n)}(R),$ in order of $O(\\epsilon ^1)$ .",
"Equation (REF ) is the core of the present study.",
"The state in Eq.",
"(REF ) and TDSE in Eq.",
"(REF ) are working on a very slow time scale.",
"We shall innovate them so that they can work on a laboratory time scale.",
"With time $t$ rescaled by the advanced time $\\Lambda (t)$ , the fast-forward state is introduced as $\\Psi ^{(n)}_{FF}(t) &\\equiv & \\Psi ^{(n)}_0(R(\\Lambda (t))) \\nonumber \\\\&=& \\textbf {C}^{(n)}(R(\\Lambda (t)))e^{-\\frac{i}{\\hbar }\\int _{0}^{t}E_n(R(\\Lambda (t^{\\prime })))dt^{\\prime }} e^{i\\xi _n(R(\\Lambda (t)))}, \\nonumber \\\\$ where $\\Lambda (t)$ is defined by $\\Lambda (t) = \\int _{0}^{t} \\alpha (t^{\\prime })dt^{\\prime },$ with the standard time $t$ .",
"$\\alpha (t)$ is an arbitrary magnification time-scale factor which satisfies $\\alpha (0)$ = 1, $\\alpha (t) > 1(0 < t < T_{FF})$ and $\\alpha (t)$ = $1 (t\\ge T_{FF})$ .",
"For a long final time $T$ in the original adiabatic dynamics, we can consider the fast forward dynamics with a new time variable which reproduces the target state $\\Psi ^{(n)}_0(R(T))$ in a shorter final time $T_{FF}$ defined by $T = \\int _{0}^{T_{FF}}\\alpha (t)dt.$ The simplest expression for $\\alpha (t)$ in the fast-forward range ($0\\le t \\le T_{FF} $ ) is given by [8] as : $\\alpha (t) = \\bar{\\alpha }-(\\bar{\\alpha }-1) \\cos (\\frac{2 \\pi }{T_{FF}}t),$ where $\\bar{\\alpha }$ is the mean value of $\\alpha (t)$ and is given by $\\bar{\\alpha } = T/T_{FF}$ .",
"Then by taking the time derivative of $\\Psi ^{(n)}_{FF}$ in Eq.",
"(REF ) and using the equalities $\\partial _t \\textbf {C}^{(n)}(R(\\Lambda (t))) =\\alpha \\epsilon \\partial _R \\textbf {C}^{(n)}$ and $\\partial _t\\xi _n(R(\\Lambda (t)))$ = $i \\textbf {C}^{(n)\\dagger }\\partial _t\\textbf {C}^{(n)}$ = $i \\alpha \\epsilon \\textbf {C}^{(n)\\dagger } \\partial _R \\textbf {C}^{(n)}$ , we have $i \\hbar \\dot{\\Psi }^{(n)}_{FF} &=& \\Big [ i \\hbar \\alpha \\epsilon \\left(\\partial _R\\textbf {C}^{(n)}-(\\textbf {C}^{(n)\\dagger }\\partial _R\\textbf {C}^{(n)})\\textbf {C}^{(n)}\\right)+ E \\textbf {C}^{(n)}\\Big ]\\nonumber \\\\&\\times & e^{-\\frac{i}{\\hbar }\\int _{0}^{t}E_n(R(\\Lambda (t^{\\prime })))dt^{\\prime }} e^{i\\xi _n(R(\\Lambda (t)))}.$ The first and second terms in the angular bracket on the r.h.s are replaced by $\\alpha \\epsilon \\mathcal {\\tilde{H}}_n\\textbf {C}^{(n)}(R(\\Lambda (t)))$ and $H_0\\textbf {C}^{(n)}(R(\\Lambda (t)))$ , respectively, by using Eqs.",
"(REF ) and (REF ).",
"Using the definition of $\\Psi ^{(n)}_{FF}(t)$ and taking the asymptotic limit $\\bar{\\alpha } \\rightarrow \\infty $ and $\\epsilon \\rightarrow 0$ under the constraint $\\bar{\\alpha } \\cdot \\epsilon \\equiv \\bar{v}= finite$ , we obtain $i \\hbar \\frac{\\partial \\Psi ^{(n)}_{FF}}{\\partial t} &=& \\left(H_0(R(\\Lambda (t)))+v(t) \\mathcal {\\tilde{H}}_n(R(\\Lambda (t))) \\right) \\Psi ^{(n)}_{FF}\\nonumber \\\\&\\equiv & H^{(n)}_{FF} \\Psi ^{(n)}_{FF}.$ Here $v(t)$ is a velocity function available from $\\alpha (t)$ in the asymptotic limit: $v(t) = \\lim _{\\epsilon \\rightarrow 0, \\bar{\\alpha } \\rightarrow \\infty } \\epsilon \\alpha (t) = \\bar{v}\\left(1-\\cos \\frac{2 \\pi }{T_{FF}}t\\right).$ Consequently, for $0 \\le t \\le T_{FF}$ , $R(\\Lambda (t))&=&R_0+\\lim _{\\epsilon \\rightarrow 0, \\bar{\\alpha }\\rightarrow \\infty }\\varepsilon \\Lambda (t)=R_{0}+\\int ^{t}_{0}v(t^{\\prime })dt^{\\prime }\\nonumber \\\\&=& R_{0}+\\bar{v}\\left[t-\\frac{T_{FF}}{2\\pi }\\sin \\left(\\frac{2\\pi }{T_{FF}}t\\right)\\right].$ $H^{(n)}_{FF}$ is the fast-forward Hamiltonian and $\\mathcal {\\tilde{H}}_n$ is the regularization term obtained from Eq.",
"(REF ) to generate the fast-forward scheme in spin system.",
"Eqs.",
"(REF ) and (REF ) work on a laboratory time scale.",
"There is a relationship between our formula for $\\mathcal {\\tilde{H}}_n$ in Eq.",
"(REF ) and Demirplak-Rice-Berry (DRB)'s formula [4], [5], [6] for the CD term $\\mathcal {H}$ .",
"If there is a $n$ -independent regularization term $\\mathcal {\\tilde{H}}$ among $\\lbrace \\mathcal {\\tilde{H}}_n\\rbrace $ , we can define $\\mathcal {H} \\equiv v(t) \\mathcal {\\tilde{H}}$ with use of $v(t) = \\frac{\\partial R(\\Lambda (t))}{\\partial t}$ .",
"Then Eq.",
"(REF ) gives a solution $\\mathcal {H} $ which agrees with DRB's formula for the CD term (See the proof in [16]).",
"It should be noted, however, that the above correspondence works well only in the case that we can find $n$ -independent regulariztion terms $\\mathcal {\\tilde{H}}$ among $\\lbrace \\mathcal {\\tilde{H}}_n\\rbrace $ .",
"Using the above notion, one may call $v(t) \\mathcal {\\tilde{H}}_n$ as a state-dependent CD term.",
"Hereafter we shall be concerned with the fast forward of adiabatic dynamics of one of the adiabatic states (i.e., the ground state) and therefore the suffix $n$ in $ \\mathcal {\\tilde{H}}_n$ will be suppressed.",
"Figure: (a) Regular triangle; (b) Open linear 3 spin chain.",
"Solid lines stand for the original exchange interactions.",
"Dashed and dotted lines mean the pair-wise regularization interactions.",
"Each line species denotes the geometrically-identical regularization interactions.Figure: (a) Triangular pyramid; (b) Square; (c) Primary star graph; (d) Open linear 4 spin chain.",
"Solid lines stand for the original exchange interactions.",
"Dashed, dotted, dotted dashed and double-dotted dashed lines mean the pair-wise regularization interactions.",
"Each line species denotes the geometrically-identical regularization interactions."
],
[
"Fast-forward driving interactions for spin clusters of various geometries",
"To begin with, let us explain the method of solving the linear algebraic equation for unknown regularization terms in Eq.",
"(REF ).",
"Then in the succeeding Sections, we shall treat regular spin clusters of various geometries with $N$ up to 4, i.e., regular triangle and open linear chain for $N=3$ spins (see Fig.REF ), and triangular pyramid, square, primary star graph and open linear chain for $N=4$ spins (see Fig.REF ).",
"Our scheme is free from obtaining all eigenvectors for a given adiabatic Hamiltonian.",
"As shown in the core equation in Eq.",
"(REF ), we need only information of a single eigenstate, typically of the ground state.",
"As an original (reference) model, we choose the transverse Ising mode, whose Hamiltonian for $N$ spin systems is written as $H_0 = J(R(t)) \\sum _{(i,j) \\in N.N.",
"}\\sigma _i^z \\sigma _j^z- \\frac{1}{2}B_x(R(t))\\sum _{i=1}^{N}\\sigma _i^x,$ where $J(R(t))=R(t)=R_0+\\epsilon t$ and $B_x(R(t))=B_0-R(t)$ with $\\epsilon \\ll 1$ are adiabatically-changing exchange interaction and transverse magnetic field, respectively.",
"$(i,j) \\in N.N.$ means nearest-neighbouring pairs.",
"Using the spin configuration bases, the dimension of Hilbert space is $2^N$ .",
"Energy matrix corresponding to the Hamiltonian in Eq.",
"(REF ) is real symmetric, which makes the eigenstates real, and the ground state is expressed by the real components $\\lbrace C_k: k=1, \\cdots , 2^N\\rbrace $ .",
"This, in combination with the fact that the length of the corresponding eigenvector is constant and equal to 1, leads to the conclusion that the adiabatic phase $\\xi _n$ in Eq.",
"(REF ) is zero in all spin clusters in the present work.",
"Further, because of the geometrical symmetry of spin clusters in Figs.",
"REF and REF , some of the components $C_k$ s are degenerate which reduce the number of independent equations in the core equation in Eq.",
"(REF ).",
"As for the unknown regularization term ($\\mathcal {\\tilde{H}}$ ) in Eq.",
"(REF ) , we must impose a form which makes its matrix elements pure imaginary because the right-hand side of Eq.",
"(REF ) is now pure imaginary.",
"Among several possibilities, we assume the regularization term consisting of pair-wise interactions described by $\\tilde{W}_{ij}^{yz} = \\tilde{W}_{ij}^{yz}(\\epsilon t)$ and 3-body interactions $\\tilde{Q}_{ijk}^{xyz} = \\tilde{Q}_{ijk}^{xyz}(\\epsilon t)$ .",
"Other possible contributions such as a single-particle energy due to $y$ -component of the magnetic field ($\\tilde{B}_y$ ), pair-wise interaction $\\tilde{W}_{ij}^{xy}$ and 3-body interaction $\\tilde{Q}_{ijk}^{xxy}$ lead to incompatible algebraic equations in Eq.",
"(REF ), and should be excluded.",
"The candidate for regularization Hamiltonian $\\mathcal {\\tilde{H}}$ then takes the following form : $\\mathcal {\\tilde{H}} &=& \\sum _{(i,j) \\in all}\\tilde{W}_{ij}^{yz} (\\sigma _i^y \\sigma _j^z + \\sigma _i^z \\sigma _j^y) \\nonumber \\\\&+& \\sum _{(i,j,k) \\in all}\\tilde{Q}_{ijk}^{xyz} (\\sigma _i^x \\sigma _j^y + \\sigma _i^y \\sigma _j^x)\\cdot \\sigma _k^z,$ where $(i,j) \\in all$ and $(i,j,k) \\in all$ mean all possible combinations (not permutations), and are not limited to nearest neighbours.",
"The 3-body interaction here is not brought as a result of the truncation of long-range and multi-body counter-diabatic interactions, but is introduced in advance to make the core equation solvable.",
"Since regular spin clusters have geometric symmetry, some of the interactions ($\\tilde{W}_{ij}^{yz}$ ) are degenerate as shown in Figs.",
"REF and REF , and the reduced number of independent interactions should be equal to the number of independent equations in Eq.",
"(REF ).",
"In the present paper, the 3-body interaction will play a subsidiary role.",
"Below we shall solve the regularization terms and obtain the fast-forward Hamiltonian for spin clusters of various geometries."
],
[
"Regular triangle and open linear 3 spins",
"In this Section we investigate a regular triangle and open linear 3 spins in Fig.",
"REF .",
"We use the spin configuration bases as $\\left|1\\right\\rangle =\\left|\\uparrow \\uparrow \\uparrow \\right\\rangle $ , $\\left|2\\right\\rangle =\\left|\\uparrow \\uparrow \\downarrow \\right\\rangle $ , $\\left|3\\right\\rangle =\\left|\\uparrow \\downarrow \\uparrow \\right\\rangle $ , $\\left|4\\right\\rangle =\\left|\\downarrow \\uparrow \\uparrow \\right\\rangle $ , $\\left|5\\right\\rangle =\\left|\\uparrow \\downarrow \\downarrow \\right\\rangle $ , $\\left|6\\right\\rangle =\\left|\\downarrow \\uparrow \\downarrow \\right\\rangle $ , $\\left|7\\right\\rangle =\\left|\\downarrow \\downarrow \\uparrow \\right\\rangle $ and $\\left|8\\right\\rangle =\\left|\\downarrow \\downarrow \\downarrow \\right\\rangle $ ."
],
[
"Regular triangle",
"In the case of the regular triangle, the eigenvalue for the ground state is $E_0 = -\\sqrt{B_x^2+2 B_x J+4 J^2}-\\frac{B_x}{2}+J$ .",
"We have confirmed in Fig.",
"REF (a) that all eight eigenvalues show no mutual energy crossing in the fast-forward time range where we choose $J(R(\\Lambda (t)))\\equiv R(\\Lambda (t))$ and $B_x(R(\\Lambda (t)))\\equiv B_0 - R(\\Lambda (t))$ with $R(\\Lambda (t))$ defined in Eq.",
"(REF ).",
"Figure: The time dependence in the case of the regular triangle in the fast-forward time range where we choose J=R(Λ(t))J=R(\\Lambda (t)) and B x =B 0 -R(Λ(t))B_x = B_0-R(\\Lambda (t)) withR(Λ(t))R(\\Lambda (t)) defined in Eq.().",
"B 0 =10B_0 =10 and v ¯=100\\bar{v}= 100.T FF =0.1T_{FF}=0.1 and R 0 =0R_0=0.",
"(a) All eight eigenvalues.",
"From the bottom, the 2nd and 4th lines are each doubly degenerate;(b) Regularization term v(t)W ˜v(t)\\tilde{W};(c) Probability amplitudes for the solution Ψ FF (t)\\Psi _{FF}(t) of TDSE, |C 2 FF | 2 =|C 3 FF | 2 =|C 4 FF | 2 =|C 5 FF | 2 =|C 6 FF | 2 =|C 7 FF | 2 |C_2^{FF}|^2=|C_3^{FF}|^2=|C_4^{FF}|^2=|C_5^{FF}|^2=|C_6^{FF}|^2=|C_7^{FF}|^2 (solid line)and |C 1 FF | 2 =|C 8 FF | 2 |C_1^{FF}|^2=|C_8^{FF}|^2 (dashed line).The components of the eigenvector for the ground state are : $ C_1 = V_1 \\zeta $ , $C_2 =V_2 \\zeta $ , $C_3 = V_3 \\zeta $ , $C_4 = V_4 \\zeta $ , $C_5 = V_5 \\zeta $ , $C_6 =V_6 \\zeta $ , $C_7 = V_7 \\zeta $ , $C_8 = V_8 \\zeta $ , where $V_1 = V_8 = 1$ , $V_2 = V_3 = V_4 =V_5 = V_6 =V_7 = \\frac{2\\sqrt{B_x^2+2 B_x J+4 J^2}+B_x +4J}{3 B_x}$ , and $\\zeta =\\frac{1}{\\sqrt{2 +6 V_2^2}}$ .",
"Here we see the symmetry: $C_1=C_8$ , $C_2 =C_3=C_4=C_5=C_6=C_7$ .",
"From $R$ -derivative of the normalization ($\\sum _{j=1}^8C_j^2=2C_1^2+6C_2^2 = 1$ ), we see $C_1 \\frac{\\partial C_1}{\\partial R}+ 3 C_2\\frac{\\partial C_2}{\\partial R} =0 ,$ and then the adiabatic phase $\\xi = 0$ .",
"As for the regularization Hamiltonian for the regular triangle, we can proceed without having recourse to the 3-body interaction.",
"Three $\\tilde{W}_{ij}^{yz}$ s should be identical due to the triangular symmetry in Fig.",
"REF (a).",
"Therefore the unknown pairwise interaction is only one: $\\tilde{W}\\equiv \\tilde{W}_{ij}^{yz}$ , independent of the pairs $(i,j)$ .",
"By using the spin configuration bases as above, the regularization Hamiltonian in Eq.",
"(REF ) is characterized by the matrix elements: $\\mathcal {\\tilde{H}}_{1j}=-\\mathcal {\\tilde{H}}_{j1}=-2i\\tilde{W}$ with $j=2,3,4$ , $\\mathcal {\\tilde{H}}_{8j}=-\\mathcal {\\tilde{H}}_{j8}=-2i\\tilde{W}$ with $j=5,6,7$ and all other elements $=0$ .",
"The explicit expression for $\\mathcal {\\tilde{H}}$ will help us to solve Eq.",
"(REF ).",
"Due to the symmetry of $\\lbrace C_j\\rbrace $ , the number of independent equations are only two in Eq.",
"(REF ) : $-6\\tilde{W}C_2&=&\\hbar \\frac{\\partial C_1}{\\partial R}, \\nonumber \\\\2\\tilde{W}C_1 &=&\\hbar \\frac{\\partial C_2}{\\partial R}.$ Noting the normalization-assisted relation in Eq.",
"(REF ), one of the above two equations becomes trivial, and Eq.",
"(REF ) has the solution: $\\tilde{W}&=&\\hbar \\frac{\\partial _R C_2}{2C_1}=\\hbar (C_1 \\partial _RC_2 - C_2 \\partial _RC_1) \\nonumber \\\\&=&\\frac{B_x \\frac{\\partial J}{\\partial R}-J\\frac{\\partial B_x}{\\partial R}}{4 (B_x^2+2 B_x J+4 J^2)}.$ The second equality above is due to the normalization condition and Eq.",
"(REF ).",
"Including the regularization term followed by rescaling of time, the fast forward Hamiltonian is written as $H_{FF}= H_0(R(\\Lambda (t)))+v(t) \\tilde{\\mathcal {H}} (R(\\Lambda (t)))$ with $H_0$ = $J(R(\\Lambda (t))) (\\sigma _1^z \\sigma _2^z+ \\sigma _2^z \\sigma _3^z+ \\sigma _3^z \\sigma _1^z)- \\frac{1}{2}(\\sigma _1^x+ \\sigma _2^x+\\sigma _3^x)B_x(R(\\Lambda (t)))$ , and $v\\mathcal {\\tilde{H}}$ = $v(t) \\tilde{W}(R(\\Lambda (t)))\\big [ (\\sigma _1^y \\sigma _2^z+ \\sigma _1^z \\sigma _2^y)+(\\sigma _2^y \\sigma _3^z+ \\sigma _2^z \\sigma _3^y)+(\\sigma _3^y \\sigma _1^z+ \\sigma _3^z \\sigma _1^y)\\big ]$ .",
"The fast forward Hamiltonian guarantees the fast forward of the adiabatic dynamics of the ground state wave function.",
"Figures REF (b) and REF (c) show the time dependence of the regularization term and that of the wave function, respectively.",
"The wave function starts from the ground state with $J=0$ , i.e., $C_1 = C_2 = C_3 = C_4 = C_5= C_6= C_7= C_8= \\frac{1}{2\\sqrt{2}}$ .",
"The initial state is a linear combination of $\\left|\\uparrow \\uparrow \\uparrow \\right\\rangle $ , $\\left|\\uparrow \\uparrow \\downarrow \\right\\rangle $ , $\\left|\\uparrow \\downarrow \\uparrow \\right\\rangle $ , $\\left|\\downarrow \\uparrow \\uparrow \\right\\rangle $ , $\\left|\\uparrow \\downarrow \\downarrow \\right\\rangle $ , $\\left|\\downarrow \\uparrow \\downarrow \\right\\rangle $ , $\\left|\\downarrow \\downarrow \\uparrow \\right\\rangle $ and $\\left|\\downarrow \\downarrow \\downarrow \\right\\rangle $ states.",
"As $J$ is increased from 0 and $B_x$ is decreased, the system rapidly changes to the final state, a linear combination of reduced bases $\\left|\\uparrow \\uparrow \\downarrow \\right\\rangle $ , $\\left|\\uparrow \\downarrow \\uparrow \\right\\rangle $ , $\\left|\\downarrow \\uparrow \\uparrow \\right\\rangle $ , $\\left|\\uparrow \\downarrow \\downarrow \\right\\rangle $ , $\\left|\\downarrow \\uparrow \\downarrow \\right\\rangle $ , and $\\left|\\downarrow \\downarrow \\uparrow \\right\\rangle $ .",
"In Fig.",
"REF (c) the solution $\\Psi _{FF}(t)$ of TDSE in Eq.",
"(REF ) has reproduced the time-rescaled ground state wave function, which means the perfect fidelity of $\\Psi _{FF}(t)$ during the fast-forward time range $0 \\le t \\le T_{FF}$ ."
],
[
"Open linear 3 spin chain",
"In a similar way we can obtain the regularization term and fast-forward Hamiltonian in the case of open linear 3 spin chain.",
"In this case the eigenvalue for the ground state is $E_0 = -\\frac{1}{6}\\big (B_x+(\\beta +\\bar{\\beta }) -\\sqrt{3}i(\\beta -\\bar{\\beta })\\big )$ , where $\\beta =\\big (18J^2B_x-8B_x^3+6Ji\\sqrt{48J^4+39B_x^2J^2+24B_x^4} \\big )^{1/3}$ .",
"We have confirmed in Fig.",
"REF (a) that all eight eigenvalues show no mutual energy crossing in the fast-forward time range where we choose $J(R(\\Lambda (t)))\\equiv R(\\Lambda (t))$ and $B_x(R(\\Lambda (t)))\\equiv B_0 - R(\\Lambda (t))$ with $R(\\Lambda (t))$ defined in Eq.",
"(REF ).",
"The components of the eigenvector for the ground state are : $ C_1 =C_8= V_1 \\zeta $ , $C_2 =C_4 =C_5 = C_7 = V_2 \\zeta $ , $C_3=C_6= V_3 \\zeta $ , where $V_1 = {\\frac{3{B_x}^{2}-8JB_x-4B_x E_0 -4E_0^2-8 E_0 J}{4JB_x}}$ , $V_2 =-\\frac{1}{2}V_1 - \\frac{2J+E_0}{B_x}$ , $V_3=1$ , and $\\zeta =\\frac{1}{\\sqrt{2V_1^2+4V_2^2+2}}$ .",
"Here we see the symmetry: $C_1=C_8$ , $C_2 =C_4 =C_5 = C_7$ and $C_3=C_6$ .",
"From $R$ -derivative of the normalization ($\\sum _{j=1}^8C_j^2=2C_1^2+4C_2^2+2C_3^2 = 1$ ), we see $C_1 \\frac{\\partial C_1}{\\partial R} + 2 C_2\\frac{\\partial C_2}{\\partial R}+ C_3 \\frac{\\partial C_3}{\\partial R} = 0,$ and then the adiabatic phase $\\xi = 0$ .",
"Figure: The same time dependence as in Fig.",
", but in the case of the open linear 3 spin chain.",
"(a) All eight eigenvalues;(b) Regularization terms v(t)W ˜ 1 v(t)\\tilde{W}_{1} (dashed line) and v(t)W ˜ 2 v(t)\\tilde{W}_{2} (dotted line) ;(c) Probability amplitudes for the solution Ψ FF (t)\\Psi _{FF}(t) of TDSE, |C 3 FF | 2 =|C 6 FF | 2 |C_3^{FF}|^2=|C_6^{FF}|^2 (solid line), |C 1 FF | 2 =|C 8 FF | 2 |C_1^{FF}|^2=|C_8^{FF}|^2 (dashed line) and |C 2 FF | 2 =|C 4 FF | 2 =|C 5 FF | 2 =|C 7 FF | 2 |C_2^{FF}|^2=|C_4^{FF}|^2=|C_5^{FF}|^2=|C_7^{FF}|^2 (dotted line).The regularization Hamiltonian for the linear 3 spin system can also be available without using the 3-body interaction.",
"Because of the geometric symmetry seen in Fig.",
"REF (b), $\\mathcal {\\tilde{H}}$ is then characterized by two independent pairwise interactions: $\\tilde{W}_{1} \\equiv \\tilde{W}_{12}^{yz}=\\tilde{W}_{23}^{yz}$ and $\\tilde{W}_{2} \\equiv \\tilde{W}_{31}^{yz}$ .",
"$\\tilde{W}_{1}$ and $\\tilde{W}_{2}$ correspond to the nearest-neighboring (N.N.)",
"and 2nd N.N.",
"interactions, respectively.",
"With use of the spin configuration bases, the matrix form for $\\mathcal {\\tilde{H}}$ in Eq.",
"(REF ) is given by $\\mathcal {\\tilde{H}} =i \\begin{pmatrix}0 & -\\tilde{W}_1-\\tilde{W}_2& -2\\tilde{W}_1 & -\\tilde{W}_1-\\tilde{W}_2 & 0 & 0 & 0 & 0 \\\\\\tilde{W}_1+\\tilde{W}_2& 0 & 0& 0& 0 & -\\tilde{W}_1+\\tilde{W}_2 & 0 & 0 \\\\2\\tilde{W}_1 & 0 & 0 & 0 & \\tilde{W}_1-\\tilde{W}_2 & 0 &\\tilde{W}_1-\\tilde{W}_2& 0 \\\\\\tilde{W}_1+\\tilde{W}_2& 0 & 0& 0& 0 & -\\tilde{W}_1+\\tilde{W}_2 & 0 & 0 \\\\0 & 0 & -\\tilde{W}_1+\\tilde{W}_2 & 0 & 0 & 0& 0 & \\tilde{W}_1+\\tilde{W}_2\\\\0 & \\tilde{W}_1-\\tilde{W}_2 & 0 & \\tilde{W}_1-\\tilde{W}_2 & 0 & 0 & 0 & 2\\tilde{W}_1\\\\0 & 0 & -\\tilde{W}_1+\\tilde{W}_2 & 0 & 0 & 0& 0 & \\tilde{W}_1+\\tilde{W}_2\\\\0 & 0 & 0 & 0 & -\\tilde{W}_1-\\tilde{W}_2 & -2\\tilde{W}_1& -\\tilde{W}_1-\\tilde{W}_2 & 0\\end{pmatrix}.$ Due to the symmetry of $\\lbrace C_j\\rbrace $ , the number of independent equations in Eq.",
"(REF ) are three: $ -2(\\tilde{W}_{1}+\\tilde{W}_{2})C_2-2 \\tilde{W}_{1}C_3&=& \\hbar \\frac{\\partial C_1}{\\partial R}\\nonumber \\\\(\\tilde{W}_{1}+\\tilde{W}_{2})C_1+( -\\tilde{W}_{1}+\\tilde{W}_{2})C_3&=& \\hbar \\frac{\\partial C_2}{\\partial R}\\nonumber \\\\2\\tilde{W}_{1}C_1+2(\\tilde{W}_{1}-\\tilde{W}_{2})C_2&=& \\hbar \\frac{\\partial C_3}{\\partial R}.\\nonumber \\\\$ By using Eq.",
"(REF ), the 3rd line (for example) of the above equation proves trivial.",
"Then Eq.",
"(REF ), whose coefficient matrix has the rank 2, gives the solution: $\\tilde{W}_{1}&=&-\\frac{\\hbar }{2}\\frac{\\partial (C_1-C_3)}{\\partial R}/(C_1+2C_2+C_3),\\nonumber \\\\\\tilde{W}_{2}&=&-\\frac{\\hbar }{2}\\frac{\\partial (C_1-2C_2+C_3)}{\\partial R}/(C_1+2C_2+C_3).\\nonumber \\\\$ Including the regularization terms followed by rescaling of time, the fast forward Hamiltonian are written as $H_{FF}= H_0(R(\\Lambda (t)))+v(t) \\tilde{\\mathcal {H}} (R(\\Lambda (t)))$ with $H_0$ = $J(R(\\Lambda (t))) (\\sigma _1^z \\sigma _2^z+ \\sigma _2^z \\sigma _3^z)- \\frac{1}{2}(\\sigma _1^x+ \\sigma _2^x+\\sigma _3^x)B_x(R(\\Lambda (t)))$ , and $v\\mathcal {\\tilde{H}}$ = $v(t) \\tilde{W}_{1}(R(\\Lambda (t)))\\big [ (\\sigma _1^y \\sigma _2^z+ \\sigma _1^z \\sigma _2^y)+(\\sigma _2^y \\sigma _3^z+ \\sigma _2^z \\sigma _3^y)\\big ]+v(t)\\tilde{W}_{2}(R(\\Lambda (t)))(\\sigma _1^y \\sigma _3^z+ \\sigma _1^z \\sigma _3^y)$ .",
"The fast forward Hamiltonian guarantees the fast forward of the adiabatic dynamics of the ground state wave function.",
"Figures REF (b) and REF (c) show the time dependence of the regularization terms and that of the wave function, respectively.",
"The wave function starts from the ground state with $J=0$ , i.e., $C_j = \\frac{1}{2\\sqrt{2}}$ for $j=1,\\cdots , 8$ .",
"As $J$ is increased from 0 and $B_x$ is decreased, the system rapidly changes to the final state, i.e., a linear combination of reduced bases.",
"In Fig.",
"REF (c) the solution $\\Psi _{FF}(t)$ of TDSE in Eq.",
"(REF ) has exactly reproduced the time-rescaled ground state wave function.",
"In case of $N=3$ spin systems, we have obtained the regularization terms and the fast-forward Hamiltonian without having recourse to the 3-body interaction.",
"Of course, we can see regularization terms which include the 3-body interaction: For a regular triangle we can have an extra solution consisting of only the 3-body interaction ($\\tilde{Q}$ ), and for the open linear 3 spin system there can be solutions where $\\tilde{Q}\\ne 0$ and one of $\\tilde{W_1}$ and $\\tilde{W_2}$ is non-vanishing.",
"But these extra solutions are less interesting from the viewpoint of searching for simpler controls.",
"In the case of $N=4$ spin systems in next Section, however, we cannot proceed without the 3-body interaction, although it will play only a subsidiary role."
],
[
"triangular pyramid, square, star graph and open linear 4 spin chain",
"Now we shall investigate regular spin clusters with $N=4$ spins, namely, a triangular pyramid, square, star graph and open linear 4 spin chain in Fig.",
"REF .",
"Their original (reference) and regularization Hamiltonians are already given by Eq.",
"(REF ) and Eq.",
"(REF ), respectively, where we put $N=4$ .",
"By using the spin configuration bases, $\\left|1\\right\\rangle =\\left|\\uparrow \\uparrow \\uparrow \\uparrow \\right\\rangle $ , $\\left|2\\right\\rangle =\\left|\\uparrow \\uparrow \\uparrow \\downarrow \\right\\rangle $ , $\\left|3\\right\\rangle =\\left|\\uparrow \\uparrow \\downarrow \\uparrow \\right\\rangle $ , $\\left|4\\right\\rangle =\\left|\\uparrow \\downarrow \\uparrow \\uparrow \\right\\rangle $ , $\\left|5\\right\\rangle =\\left|\\downarrow \\uparrow \\uparrow \\uparrow \\right\\rangle $ $\\left|6\\right\\rangle =\\left|\\uparrow \\uparrow \\downarrow \\downarrow \\right\\rangle $ , $\\left|7\\right\\rangle =\\left|\\uparrow \\downarrow \\downarrow \\uparrow \\right\\rangle $ , $\\left|8\\right\\rangle =\\left|\\downarrow \\downarrow \\uparrow \\uparrow \\right\\rangle $ , $\\left|9\\right\\rangle =\\left|\\downarrow \\uparrow \\uparrow \\downarrow \\right\\rangle $ , $\\left|10\\right\\rangle =\\left|\\uparrow \\downarrow \\uparrow \\downarrow \\right\\rangle $ , $\\left|11\\right\\rangle =\\left|\\downarrow \\uparrow \\downarrow \\uparrow \\right\\rangle $ , $\\left|12\\right\\rangle =\\left|\\downarrow \\downarrow \\downarrow \\uparrow \\right\\rangle $ , $\\left|13\\right\\rangle =\\left|\\downarrow \\downarrow \\uparrow \\downarrow \\right\\rangle $ , $\\left|14\\right\\rangle =\\left|\\downarrow \\uparrow \\downarrow \\downarrow \\right\\rangle $ , $\\left|15\\right\\rangle =\\left|\\uparrow \\downarrow \\downarrow \\downarrow \\right\\rangle $ , and $\\left|16\\right\\rangle =\\left|\\downarrow \\downarrow \\downarrow \\downarrow \\right\\rangle $ , the matrix form for original Hamiltonian $H_0$ in Eq.",
"(REF ) can be constructed."
],
[
"Triangular pyramid",
"The eigenvalue of the ground state is $E_0= \\frac{1}{3} (-(\\beta +\\bar{\\beta })+4J+\\sqrt{3}i(\\beta -\\bar{\\beta }) )$ , where $\\beta =(35\\,{J}^{3}-18{B_x}^{2}J+3i\\sqrt{108{J}^{6}+309{B_x}^{2}{J}^{4}+3{B_x}^{4}{J}^{2}+3{B_x}^{6}})^{1/3}.$ For all regular clusters with $N=4$ spins in Fig.",
"REF , as is the case of the previous Section, we have numerically confirmed that there is no level crossing between the ground and 1st excited states in the fast-forward time range.",
"So figures of 16 eigenvalues will be suppressed in this Section.",
"The components of the eigenvector of the ground state are: $C_1=C_{16}=V_1\\zeta $ , $C_2=C_3=C_4=C_5=C_{12}=C_{13}=C_{14}=C_{15}=V_2\\zeta $ , and $C_6=C_7=C_8=C_9=C_{10}=C_{11}=V_6\\zeta $ .",
"Here $\\zeta =(2+8V_2^2+6V_6^2)^{-1/2}$ , $V_1=1$ , $V_2=\\frac{(\\beta +\\bar{\\beta })+14J-\\sqrt{3}i(\\beta -\\bar{\\beta })}{6B_x}$ , and $V_6=-{\\frac{2\\left(\\beta ^2+\\bar{\\beta }^2\\right) -10J \\left(\\beta +\\bar{\\beta }\\right)-\\left(48 J^2+15B_x^2\\right)+i\\sqrt{3}(2\\left(\\beta ^2-\\bar{\\beta }^2\\right) +10J \\left(\\beta -\\bar{\\beta }\\right))}{27{B_x}^{2}}}$ , where the equality $|\\beta |^2=13J^2+3B_x^2$ is used.",
"From $R$ -derivative of the normalization ($2C_1^2+8C_2^2+6C_6^2=1$ ), we see $C_1 \\frac{\\partial C_1}{\\partial R} + 4 C_2\\frac{\\partial C_2}{\\partial R}+ 3C_6 \\frac{\\partial C_6}{\\partial R} = 0 .$ If we suppress the 3-body interaction, the regularization Hamiltonian consists of only one pairwise interaction $\\tilde{W} \\equiv \\tilde{W}_{ij}^{yz}$ , due to the high symmetry of the triangular pyramid in Fig.REF (a).",
"The corresponding matrix for the regularization term can be written as $\\mathcal {\\tilde{H}} =i \\left(\\begin{array}{cccccccccccccccc}0 & -3\\tilde{W} & -3\\tilde{W} & -3\\tilde{W} & -3\\tilde{W} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\3\\tilde{W} & 0 & 0 & 0 & 0 & -\\tilde{W} & 0 & 0 & -\\tilde{W} & -\\tilde{W} & 0 & 0 & 0 & 0 & 0 & 0 \\\\3\\tilde{W} & 0 & 0 & 0 & 0 & -\\tilde{W} & -\\tilde{W} & 0 & 0 & 0 & -\\tilde{W} & 0 & 0 & 0 & 0 & 0 \\\\3\\tilde{W} & 0 & 0 & 0 & 0 & 0 & -\\tilde{W} & -\\tilde{W} & 0 & -\\tilde{W} & 0 & 0 & 0 & 0 & 0 & 0 \\\\3\\tilde{W} & 0 & 0 & 0 & 0 & 0 & 0 & -\\tilde{W} & -\\tilde{W} & 0 & -\\tilde{W} & 0 & 0 & 0 & 0 & 0 \\\\0 & \\tilde{W} & \\tilde{W} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\tilde{W} & \\tilde{W} & 0 \\\\0 & 0 & \\tilde{W} & \\tilde{W} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\tilde{W} & 0 & 0 & \\tilde{W} & 0 \\\\0 & 0 & 0 & \\tilde{W} & \\tilde{W} & 0 & 0 & 0 & 0 & 0 & 0 & \\tilde{W} & \\tilde{W} & 0 & 0 & 0 \\\\0 & \\tilde{W} & 0 & 0 & \\tilde{W} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\tilde{W} & \\tilde{W} & 0 & 0 \\\\0 & \\tilde{W} & 0 & \\tilde{W} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\tilde{W} & 0 & \\tilde{W} & 0 \\\\0 & 0 & \\tilde{W} & 0 & \\tilde{W} & 0 & 0 & 0 & 0 & 0 & 0 & \\tilde{W} & 0 & \\tilde{W} & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & -\\tilde{W} & -\\tilde{W} & 0 & 0 & -\\tilde{W} & 0 & 0 & 0 & 0 & 3\\tilde{W} \\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & -\\tilde{W} & -\\tilde{W} & -\\tilde{W} & 0 & 0 & 0 & 0 & 0 & 3\\tilde{W} \\\\0 & 0 & 0 & 0 & 0 & -\\tilde{W} & 0 & 0 & -\\tilde{W} & 0 & -\\tilde{W} & 0 & 0 & 0 & 0 & 3\\tilde{W} \\\\0 & 0 & 0 & 0 & 0 & -\\tilde{W} & -\\tilde{W} & 0 & 0 & -\\tilde{W} & 0 & 0 & 0 & 0 & 0 & 3\\tilde{W} \\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -3\\tilde{W} & -3\\tilde{W} & -3\\tilde{W} & -3\\tilde{W} & 0\\end{array}\\right) .$ Due to the symmetry of $\\lbrace C_j\\rbrace $ , the number of independent equations in Eq.",
"(REF ) are three: $ -12 \\tilde{W}C_2&=& \\hbar \\frac{\\partial C_1}{\\partial R}\\nonumber \\\\3\\tilde{W}C_1- 3\\tilde{W}C_6&=& \\hbar \\frac{\\partial C_2}{\\partial R}\\nonumber \\\\4\\tilde{W}C_2&=& \\hbar \\frac{\\partial C_6}{\\partial R}.$ While one of the above equations is trivial due to Eq.",
"(REF ), we need one more unknown variable to make meaningful the algebraic equations in Eq.",
"(REF ).",
"Here we evaluate the contribution of the 3-body interaction.",
"The geometrical symmetry allows a universal 3-body interaction $\\tilde{Q} \\equiv \\tilde{Q}_{ijk}^{xyz}$ , independent of all possible 3-body configurations $(i,j,k)$ .",
"The inclusion of the 3-body interaction improves some matrix elements of $\\mathcal {\\tilde{H}}$ in Eq.",
"(REF ) as follows: $\\mathcal {\\tilde{H}}_{1,j}&=&\\mathcal {\\tilde{H}}_{16,j}=-4i\\tilde{Q} \\quad \\rm {for} \\quad \\textit {j}=6,\\cdots ,11, \\nonumber \\\\\\mathcal {\\tilde{H}}_{i,1}&=&\\mathcal {\\tilde{H}}_{i,16}=4i\\tilde{Q} \\quad \\rm {for} \\quad \\textit {i}=6, \\cdots ,11.",
"\\nonumber \\\\$ After the above improvements, the algebraic equations in Eq.",
"(REF ) are revised as: $ -12 \\tilde{W}C_2-24\\tilde{Q}C_6&=& \\hbar \\frac{\\partial C_1}{\\partial R}\\nonumber \\\\3\\tilde{W}C_1- 3\\tilde{W}C_6&=& \\hbar \\frac{\\partial C_2}{\\partial R}\\nonumber \\\\8\\tilde{Q}C_1+4\\tilde{W}C_2&=& \\hbar \\frac{\\partial C_6}{\\partial R},$ where one of the above lines is again trivial because of Eq.",
"(REF ).",
"Equation (REF ), whose coefficient matrix has the rank 2, gives the solution: $\\tilde{W}&=&\\frac{\\hbar \\partial _R C_2}{3(C_1-C_6)}, \\nonumber \\\\\\tilde{Q}&=&\\frac{\\hbar \\partial _R (C_1+3C_6)}{24(C_1-C_6)}.$ The fast-forward Hamiltonian is given by $H_{FF}&=& J(R(\\Lambda (t))) \\sum _{(i,j) \\in N.N.",
"}\\sigma _i^z \\sigma _j^z- \\frac{1}{2}B_x(R(\\Lambda (t)))\\sum _{i=1}^{4}\\sigma _i^x, \\nonumber \\\\&+&v(t) \\tilde{\\mathcal {H}} (R(\\Lambda (t)))$ with $v\\mathcal {\\tilde{H}}&=& \\sum _{(i,j) \\in all}v(t) \\tilde{W}(R(\\Lambda (t))) (\\sigma _i^y \\sigma _j^z + \\sigma _i^z \\sigma _j^y) \\nonumber \\\\&+& \\sum _{(i,j,k) \\in all}v(t)\\tilde{Q}(R(\\Lambda (t))) (\\sigma _i^x \\sigma _j^y + \\sigma _i^y \\sigma _j^x)\\cdot \\sigma _k^z.\\nonumber \\\\$ In the triangular pyramid, $\\sum _{(i,j) \\in N.N.",
"}$ is equivalent to $\\sum _{(i,j) \\in all}$ .",
"The fast forward Hamiltonian guarantees the fast forward of the adiabatic dynamics of the ground state wave function.",
"Figures REF (a) and REF (a) show the time dependence of regularization terms and that of the wave function, respectively.",
"The wave function starts from the ground state with $J=0$ , i.e., $C_j=\\frac{1}{4}$ for $j=1,\\cdots , 16$ .",
"In Fig.",
"REF (a) the solution $\\Psi _{FF}(t)$ of TDSE in Eq.",
"(REF ) has exactly reproduced the time-rescaled ground state wave function during the fast-forward time range $0 \\le t \\le T_{FF}$ .",
"Figure: The time dependence of regularization terms multiplied by v(t)v(t) in the fast-forward time range where we choose J=R(Λ(t))J=R(\\Lambda (t)) and B x =B 0 -R(Λ(t))B_x = B_0-R(\\Lambda (t)) withR(Λ(t))R(\\Lambda (t)) defined in Eq.().",
"B 0 =10B_0 =10 and v ¯=100\\bar{v}= 100.T FF =0.1T_{FF}=0.1 and R 0 =0R_0=0.",
"(a) Triangular pyramid.",
"v(t)W ˜v(t)\\tilde{W} (dashed line) and v(t)Q ˜v(t)\\tilde{Q} (solid line); (b) Square.",
"v(t)W ˜ 1 v(t)\\tilde{W}_{1} (dashed line), v(t)W ˜ 2 v(t)\\tilde{W}_{2} (dotted line) and v(t)Q ˜v(t)\\tilde{Q} (solid line) ; (c) Primary star graph.",
"v(t)W ˜ 1 v(t)\\tilde{W}_{1} (dashed line), v(t)W ˜ 2 v(t)\\tilde{W}_{2} (dotted line) and v(t)Q ˜v(t)\\tilde{Q} (solid line) ; (d) Open linear 4 spins.",
"v(t)W ˜ 1 v(t)\\tilde{W}_{1} (dashed line), v(t)W ˜ 2 v(t)\\tilde{W}_{2} (dotted line), v(t)W ˜ 3 v(t)\\tilde{W}_{3} (dotted dashed line), v(t)W ˜ 4 v(t)\\tilde{W}_{4} (double-dotted dashed line) and v(t)Q ˜v(t)\\tilde{Q} (solid line).Figure: Time dependence of probability amplitudes |C j FF | 2 |C_j^{FF}|^2 with j=1∼16j=1\\sim 16 for the solution Ψ FF (t)\\Psi _{FF}(t) of TDSE in the fast-forward time range, where we choose J=R(Λ(t))J=R(\\Lambda (t)) and B x =B 0 -R(Λ(t))B_x = B_0-R(\\Lambda (t)) with R(Λ(t))R(\\Lambda (t)) defined in Eq.().",
"B 0 =10B_0 =10 and v ¯=100\\bar{v}= 100.T FF =0.1T_{FF}=0.1 and R 0 =0R_0=0.",
"(a) Triangular pyramid.",
"j=1,16j=1,16 (dashed line), j=2∼5 j=2 \\sim 5 and 12∼1512\\sim 15 (dotted line), and j=6∼11j=6 \\sim 11 (solid line); (b) Square.",
"j=1,16j=1,16 (dotted dashed line), j=2∼5 j=2\\sim 5 and 12∼1512\\sim 15 (dotted line), j=6∼9j=6\\sim 9 (dashed line), and j=10,11j=10,11 (solid line); (c) Primary star graph.",
"j=1,16j=1,16 (dashed line), j=2,3,5,12,13,15 j=2,3,5,12,13,15 (dotted line) , j=4,14j=4,14 (solid line), and j=6∼11j=6 \\sim 11 (dotted dashed line); (d) Open linear 4 spins.j=1,16j=1,16 (lower solid line), j=2,5,12,15 j=2,5,12,15 (dotted dashed line), j=3,4,13,14j=3,4,13,14 (dashed line), j=6,8j=6,8 (dotted line), j=7,9j=7,9 (double-dotted dashed line), and j=10,11j=10,11(upper solid line)."
],
[
"Square",
"The eigenvalue of the ground state is $E_0=-\\beta _1$ , where $\\beta _1=\\sqrt{8{J}^{2}+2{B_x}^{2}+2\\beta _2}$ with $ \\beta _2=\\sqrt{16{J}^{4}+{B_x}^{4}}$ .",
"The components of the eigenvector of the ground state are: $C_1=C_{16}=V_1\\zeta $ , $C_2=C_3=C_4=C_5=C_{12}=C_{13}=C_{14}=C_{15}=V_2\\zeta $ , $C_6=C_7=C_8=C_9=V_6\\zeta $ , and $C_{10}=C_{11}=V_{10}\\zeta $ with $\\zeta =(8+2V_1^2+4V_6^2+2V_{10}^2)^{-1/2}$ .",
"Here $V_1=\\frac{(\\beta _1-4J)(4J^2-B_x^2+\\beta _2)}{8J^2B_x}$ , $V_2=1$ , $V_6=\\frac{(\\beta _1^2-4\\beta _2)\\beta _1}{16J^2B_x}$ , and $V_{10}=\\frac{(\\beta _1+4J)(4J^2-B_x^2+\\beta _2)}{8J^2B_x}$ .",
"From $R$ -derivative of the normalization ($2C_1^2+8C_2^2+4C_6^2+2C_{10}^2=1$ ), we see $C_1 \\frac{\\partial C_1}{\\partial R} + 4C_2\\frac{\\partial C_2}{\\partial R}+ 2C_6 \\frac{\\partial C_6}{\\partial R}+ C_{10} \\frac{\\partial C_{10}}{\\partial R} = 0$ The geometric symmetry of the square spin system in Fig.REF (b) allows two candidates as regularization terms, which are $\\tilde{W}_{12}=\\tilde{W}_{23}=\\tilde{W}_{34}=\\tilde{W}_{41}= \\tilde{W}_1$ and $\\tilde{W}_{31}=\\tilde{W}_{42}= \\tilde{W}_2$ .",
"$\\tilde{W}_1$ and $\\tilde{W}_2$ correspond to N.N.",
"and the second N.N.",
"interactions, respectively.",
"The regularization matrix $\\mathcal {\\tilde{H}}$ is given in Eq.",
"(REF ).",
"To add one more unknown variable, we include a contribution of the universal 3-body interaction $\\tilde{Q}\\equiv \\tilde{Q}_{ijk}^{xyz}$ .",
"This inclusion requires the same improvement of some matrix elements of $\\mathcal {\\tilde{H}}$ as in Eq.",
"(REF ).",
"Due to the symmetry of $\\lbrace C_j\\rbrace $ , the number of independent algebraic equations are four: $ (-8\\tilde{W}_1- 4\\tilde{W}_2)C_2-16\\tilde{Q}C_6-8\\tilde{Q}C_{10}&=& \\hbar \\frac{\\partial C_1}{\\partial R}\\nonumber \\\\(2\\tilde{W}_1+\\tilde{W}_2)C_1+( -2\\tilde{W}_2)C_6+ (-2\\tilde{W}_1+\\tilde{W}_2)C_{10}&=& \\hbar \\frac{\\partial C_2}{\\partial R}\\nonumber \\\\8\\tilde{Q}C_1+4\\tilde{W}_2C_2&=& \\hbar \\frac{\\partial C_6}{\\partial R}.\\nonumber \\\\8\\tilde{Q}C_1+(8\\tilde{W}_1-4 \\tilde{W}_2)C_2&=& \\hbar \\frac{\\partial C_{10}}{\\partial R}.\\nonumber \\\\$ Because of Eq.",
"(REF ), one of the above equations is trivial.",
"Ignoring the second line for example, Eq.",
"(REF ), whose coefficient matrix has the rank 3, gives the solution: $\\tilde{W}_1&=&-\\frac{\\hbar }{8(3C_1-C_{10}-2C_6)C_2}\\times ( C_1\\partial _RC_1-4C_2\\partial _RC_2 \\nonumber \\\\&+&(C_1+C_{10})\\partial _RC_6-(C_1-2C_6)\\partial _RC_{10} ), \\nonumber \\\\\\tilde{W}_2&=&-\\hbar \\frac{C_1\\partial _RC_1-(C_1-2C_6-C_{10})\\partial _RC_6+C_1\\partial _RC_{10}}{4(3C_1-C_{10}-2C_6)C_2}, \\nonumber \\\\\\tilde{Q}&=&\\frac{\\hbar \\partial _R(C_1+2C_6+C_{10})}{8(3C_1-C_{10}-2C_6)C_2}.$ The fast-forward Hamiltonian is given by Eq.",
"(REF ), where $v(t)\\tilde{\\mathcal {H}} (R(\\Lambda (t)))$ is now replaced by: $v\\mathcal {\\tilde{H}}&=& \\sum _{(i,j)=(1,2),(2,3),(3,4),(4,1)}v(t) \\tilde{W}_1(R(\\Lambda (t))) (\\sigma _i^y \\sigma _j^z + \\sigma _i^z \\sigma _j^y) \\nonumber \\\\&+&\\sum _{(i,j)=(3,1),(4,2)}v(t) \\tilde{W}_2(R(\\Lambda (t))) (\\sigma _i^y \\sigma _j^z + \\sigma _i^z \\sigma _j^y) \\nonumber \\\\&+& \\sum _{(i,j,k) \\in all}v(t)\\tilde{Q}(R(\\Lambda (t))) (\\sigma _i^x \\sigma _j^y + \\sigma _i^y \\sigma _j^x)\\cdot \\sigma _k^z.\\nonumber \\\\$ Figures REF (b) and REF (b) show the time dependence of regularization terms and that of wave function, respectively.",
"The wave function starts from the ground state with $J=0$ , i.e., $C_j=\\frac{1}{4}$ for $j=1,\\cdots , 16$ .",
"In Fig.",
"REF (b) the solution $\\Psi _{FF}(t)$ of TDSE in Eq.",
"(REF ) has exactly reproduced the time-rescaled ground state wave function."
],
[
"Primary star graph",
"The eigenvalue of the ground state is $E_0=-\\beta $ , where $\\beta =\\sqrt{2{B_x}^{2}+5{J}^{2}+2\\sqrt{{B_x}^{4}+{B_x}^{2}{J}^{2}+4{J}^{4}}}$ .",
"The components of the eigenvector of the ground state are: $C_1=C_{16}=V_1\\zeta $ , $C_2=C_3=C_5=C_{12}=C_{13}=C_{15}=V_2\\zeta $ , $C_4=C_{14}=V_4\\zeta $ , and $C_6=C_7=C_8=C_9=C_{10}=C_{11}=V_{6}\\zeta $ with $\\zeta =(6+2V_1^2+6V_2^2+2V_{4}^2)^{-1/2}$ .",
"Here $V_1=\\frac{-J(7{B_x}^{2}+3J^2)+\\beta (4{B_x}^2+3\\beta J-{\\beta }^{2}+J^2)}{5J{B_x}^2}$ , $V_2=\\frac{-2J(9J^2-4{B_x}^{2})+\\beta (4{B_x}^2-2\\beta J-{\\beta }^{2}+21J^2)}{30J^2 B_x}$ , $V_4=\\frac{-2J(J^2+4{B_x}^{2})-\\beta (4{B_x}^2-2\\beta J-{\\beta }^{2}+J^2)}{10J^2 B_x}$ , and $V_6=1$ .",
"From $R$ -derivative of the normalization ($2C_1^2+6C_2^2+2C_4^2+ 6C_6^2=1$ ), we see $C_1 \\frac{\\partial C_1}{\\partial R} + 3 C_2\\frac{\\partial C_2}{\\partial R}+ C_{4} \\frac{\\partial C_4}{\\partial R}+ 3C_6 \\frac{\\partial C_6}{\\partial R} = 0.$ The geometric symmetry of the primary star-graph spin system in Fig.REF (c) allows two candidates as regularization terms, which are $\\tilde{W}_{12}=\\tilde{W}_{23}=\\tilde{W}_{24}= \\tilde{W}_1$ and $\\tilde{W}_{14}=\\tilde{W}_{13}=\\tilde{W}_{34}= \\tilde{W}_2$ .",
"$\\tilde{W}_1$ and $\\tilde{W}_2$ correspond to N.N.",
"and the 2nd N.N.",
"interactions, respectively.",
"The matrix for regularization term $\\mathcal {\\tilde{H}}$ can be written in Eq.",
"(REF ).",
"To add one more unknown variable, we include a contribution of the universal 3-body interaction $\\tilde{Q}\\equiv \\tilde{Q}_{ijk}^{xyz}$ .",
"This inclusion requires the same improvement of some matrix elements of $\\mathcal {\\tilde{H}}$ as in Eq.",
"(REF ).",
"One might have an idea to include two species of 3-body interactions with one among N.N.s and another among the 2nd N.N.s.",
"But this idea results in incompatible equations in Eq.",
"(REF ) and cannot be acceptable.",
"Due to the symmetry of $\\lbrace C_j\\rbrace $ , the number of independent equations are four: $ (-6\\tilde{W}_2-3\\tilde{W}_1)C_2+( -3 \\tilde{W}_1)C_4-24\\tilde{Q}C_6&=& \\hbar \\frac{\\partial C_1}{\\partial R}\\nonumber \\\\(2\\tilde{W}_2+\\tilde{W}_1)C_1+( -3\\tilde{W}_1)C_6&=& \\hbar \\frac{\\partial C_2}{\\partial R}\\nonumber \\\\(3\\tilde{W}_1) C_1+(3\\tilde{W}_1-6 \\tilde{W}_2)C_6&=& \\hbar \\frac{\\partial C_4}{\\partial R}.\\nonumber \\\\8\\tilde{Q}C_1+(3\\tilde{W}_1)C_2+(-\\tilde{W}_1+2\\tilde{W}_2)C_4&=& \\hbar \\frac{\\partial C_{6}}{\\partial R}.\\nonumber \\\\$ Because of Eq.",
"(REF ), one of the above 4 equations becomes trivial.",
"Ignoring the first line for example, Eq.",
"(REF ), whose coefficient matrix has the rank 3, gives the solution: $\\tilde{W}_1&=&\\hbar \\frac{C_1\\partial _RC_4+3C_6\\partial _RC_2}{3(C_1-C_{6})(C_1+3C_6)}, \\nonumber \\\\\\tilde{W}_2&=&\\hbar \\frac{3(C_1+C_6)\\partial _RC_2-(C_1-3C_6)\\partial _RC_{4}}{6(C_1-C_{6})(C_1+3C_6)}, \\nonumber \\\\\\tilde{Q}&=&\\frac{\\hbar }{24C_1(C_1-C_{6})(C_1+3C_6)} \\nonumber \\\\& \\times & \\bigl ( 3(C_1^2+2C_1C_6-3C_6^2)\\partial _RC_6 \\nonumber \\\\&-&3(3C_2C_6+C_1C_4)\\partial _RC_2 \\nonumber \\\\& -&(3C_1C_2-2C_1C_4+3C_4C_6)\\partial _RC_4\\bigr ).\\nonumber \\\\$ The fast-forward Hamiltonian is given by Eq.",
"(REF ), where $v(t)\\tilde{\\mathcal {H}} (R(\\Lambda (t)))$ is replaced by: $v\\mathcal {\\tilde{H}}&=&\\sum _{(i,j)=(1,2),(2,3),(2,4)}v(t) \\tilde{W}_1(R(\\Lambda (t))) (\\sigma _i^y \\sigma _j^z + \\sigma _i^z \\sigma _j^y) \\nonumber \\\\&+&\\sum _{(i,j)=(1,4),(1,3),(3,4)}v(t) \\tilde{W}_2(R(\\Lambda (t))) (\\sigma _i^y \\sigma _j^z + \\sigma _i^z \\sigma _j^y) \\nonumber \\\\&+& \\sum _{(i,j,k) \\in all}v(t)\\tilde{Q}(R(\\Lambda (t))) (\\sigma _i^x \\sigma _j^y + \\sigma _i^y \\sigma _j^x)\\cdot \\sigma _k^z.\\nonumber \\\\$ Figures REF (c) and REF (c) show the time dependence of regularization terms and that of wave function, respectively.",
"The wave function starts from the ground state with $J=0$ , i.e., $C_j=\\frac{1}{4}$ for $j=1,\\cdots , 16$ .",
"In Fig.",
"REF (c) the solution $\\Psi _{FF}(t)$ of TDSE in Eq.",
"(REF ) has exactly reproduced the time-rescaled ground state wave function."
],
[
"Open linear 4 spin chain",
"The eigenvalue of the ground state is $E_0=-\\frac{\\beta _2}{\\sqrt{3}}$ , where $\\beta _2=\\sqrt{2(\\beta _1+\\bar{\\beta _1})+11J^2+4B_x^2}$ with $\\beta _1=\\bigl (64{J}^{6}+15{J}^{4}{B_x}^{2}+21{B_x}^{4}{J}^{2}+8{B_x}^{6}+3\\sqrt{3}J^2B_x i\\sqrt{128{J}^{6}+93{J}^{4}{B_x}^{2}+51{B_x}^{4}{J}^{2}+25{B_x}^{6} }\\bigr )^{1/3}$ and $|\\beta _1|^2=4{B_x}^{4}+7{B_x}^{2}J^2+16J^4$ .",
"The components of the eigenvector of the ground state are: $C_1=C_{16}=V_1\\zeta $ , $C_2=C_5=C_{12}=C_{15}=V_2\\zeta $ , $C_3=C_{4}=C_{13}=C_{14}=V_3\\zeta $ , $C_6=C_8=V_{6}\\zeta $ , $C_7=C_9=V_{7}\\zeta $ , and $C_{10}=C_{11}=V_{10}\\zeta $ with $\\zeta =(2+4V_2^2+4V_3^2+2V_6^2+2V_7^2+2V_{10}^2)^{-1/2}$ .",
"Here $V_1=1$ , $V_2=-\\frac{\\sqrt{3}{J}^{2}\\beta _2\\left(180{B_x}^{2}+144{J}^{2}\\right)-\\sqrt{3}\\beta _2^3 \\left(12{B_x}^{2}+33{J}^{2}\\right)+\\sqrt{3} \\beta _2^5-162{J}^{5}}{ 162 {J}^{4}B_x}$ , $V_3=\\frac{\\sqrt{3} {J}^{2} \\beta _2\\left(180{B_x}^{2} +198J^2\\right)- \\sqrt{3} \\beta _2^3\\left( 12 {B_x}^{2}+33{J}^{2} \\right)+\\sqrt{3} \\beta _2^5+324{J}^{5}}{ 162 {J}^{4}B_x}$ , $V_6=\\frac{-\\sqrt{3} {J}^{2}\\beta _2(144 {B_x}^{2}+81{J}^{2})-J\\beta _2^2(36{B_x}^{2}+90 {J}^{2})+\\sqrt{3} \\beta _2^3(12{B_x}^{2}+30{J}^{2})+3J \\beta _2^4-\\sqrt{3}\\beta _2^5+243{J}^{5}+ 648{B_x}^{2}{J}^{3}}{216 {B_x}^{2}{J}^{3}}$ , $V_7=-\\frac{\\sqrt{3} {J}^{2}\\beta _2 \\left(144 {B_x}^{2}+81{J}^{2} \\right)-J \\beta _2^2 \\left(36{B_x}^{2}+90{J}^{2}\\right)-\\sqrt{3}\\beta _2^3\\left(12{B_x}^{2}+30J^2\\right)+3J \\beta _2^4+\\sqrt{3} \\beta _2^5+243{J}^{5}+324{J}^{3} B_x^2 }{108{B_x}^{2}{J}^{3} }$ , and $V_{10}= -\\frac{\\sqrt{3} J^2 \\beta _2\\left(144{B_x}^{2}-9{J}^{2}\\right)+ J \\beta _2^2\\left(108{B_x}^{2}+54 {J}^{2}\\right)+6\\sqrt{3}\\beta _2^3\\left(2{B_x}^{2}+J^2 \\right)-9J \\beta _2^4-\\sqrt{3} \\beta _2^5+648{B_x}^{2}{J}^{3}-81{J}^{5}}{648{B_x}^{2}{J}^{3} }$ .",
"Since $\\beta _2$ is real, all components of the ground state are also real.",
"From $R$ -derivative of the normalization ($2C_1^2+4C_2^2+4C_3^2+ 2C_6^2+2C_7^2+2C_{10}^2=1$ ), we see $C_1 \\frac{\\partial C_1}{\\partial R} + 2 C_2\\frac{\\partial C_2}{\\partial R}+2 C_{3} \\frac{\\partial C_3}{\\partial R}+ C_6 \\frac{\\partial C_6}{\\partial R}+ C_7 \\frac{\\partial C_7}{\\partial R}+C_{10} \\frac{\\partial C_{10}}{\\partial R} = 0$ In case of the open linear 4 spin system in Fig.REF (d), the symmetry consideration allows 4 regularization terms which consist of $\\tilde{W}_{12}=\\tilde{W}_{34}=\\tilde{W}_1, \\tilde{W}_{23}=\\tilde{W}_2$ , $\\tilde{W}_{13}=\\tilde{W}_{24}=\\tilde{W}_3$ , and $\\tilde{W}_{14}= \\tilde{W}_4$ .",
"The regularization Hamiltonian $\\mathcal {\\tilde{H}}$ is given in Eq.",
"(REF ).",
"To add one more unknown variable, we include a contribution of the universal 3-body interaction $\\tilde{Q}\\equiv \\tilde{Q}_{ijk}^{xyz}$ .",
"This inclusion requires the same improvement of some matrix elements of $\\mathcal {\\tilde{H}}$ as in Eq.",
"(REF ).",
"The idea to include plural species of 3-body interactions results in incompatible equations in Eq.",
"(REF ) and can not be employed.",
"Due to the symmetry of $\\lbrace C_j\\rbrace $ , the number of independent equations are six: $ (-2\\tilde{W}_1-2\\tilde{W}_4-2\\tilde{W}_3)C_2+( -2 \\tilde{W}_2-2\\tilde{W}_1-2\\tilde{W}_3)C_3-8\\tilde{Q}C_6-8\\tilde{Q}C_7-8\\tilde{Q}C_{10}&=& \\hbar \\frac{\\partial C_1}{\\partial R}\\nonumber \\\\(\\tilde{W}_1+\\tilde{W}_4+\\tilde{W}_3)C_1+(-\\tilde{W}_2+\\tilde{W}_1-\\tilde{W}_3)C_6+(-\\tilde{W}_1+\\tilde{W}_4-\\tilde{W}_3)C_7+(-\\tilde{W}_1-\\tilde{W}_2+\\tilde{W}_3)C_{10}&=& \\hbar \\frac{\\partial C_2}{\\partial R}\\nonumber \\\\(\\tilde{W}_1+\\tilde{W}_2+\\tilde{W}_3)C_1+(\\tilde{W}_1-\\tilde{W}_4-\\tilde{W}_3)C_6+(-\\tilde{W}_1+\\tilde{W}_2-\\tilde{W}_3)C_7+(-\\tilde{W}_1-\\tilde{W}_4+\\tilde{W}_3)C_{10}&=& \\hbar \\frac{\\partial C_3}{\\partial R}\\nonumber \\\\8\\tilde{Q}C_1+(2\\tilde{W}_2-2\\tilde{W}_1+2\\tilde{W}_3)C_2+( -2 \\tilde{W}_1+2\\tilde{W}_4+2\\tilde{W}_3)C_3&=& \\hbar \\frac{\\partial C_6}{\\partial R}\\nonumber \\\\8\\tilde{Q}C_1+(2\\tilde{W}_1-2\\tilde{W}_2+2\\tilde{W}_3)C_3+(2 \\tilde{W}_1-2\\tilde{W}_4+2\\tilde{W}_3)C_2&=& \\hbar \\frac{\\partial C_7}{\\partial R}\\nonumber \\\\8\\tilde{Q}C_1+(2\\tilde{W}_1+2\\tilde{W}_2-2\\tilde{W}_3)C_2+(2 \\tilde{W}_1+2\\tilde{W}_4-2\\tilde{W}_3)C_3&=& \\hbar \\frac{\\partial C_{10}}{\\partial R} .$ The constraint in Eq.",
"(REF ) renders one of the above 6 equations trivial, and Eq.",
"(REF ), whose coefficient matrix has the rank 5, gives the following solution: $\\tilde{W}_1&=&\\hbar \\frac{\\partial _R (C_7 + C_{10})}{4(C_2+C_3)} +\\kappa , \\nonumber \\\\\\tilde{W}_2&=&-\\gamma _1+\\gamma _2, \\nonumber \\\\\\tilde{W}_3&=&\\hbar \\frac{\\partial _R (C_6 + C_7)}{4(C_2+C_3)}+\\kappa ,\\nonumber \\\\\\tilde{W}_4&=&\\gamma _1+\\gamma _2, \\nonumber \\\\\\tilde{Q}&=&\\frac{\\hbar }{8( 3C_1-C_6-C_7- C_{10}) (C_1+C_6+C_7+C_{10})}\\nonumber \\\\&\\times & \\bigl (4(C_2-C_3)\\partial _R (C_2-C_3)\\nonumber \\\\&+&(C_1+C_6+C_7+C_{10})\\partial _R (C_1 + C_6+ C_7 + C_{10})\\bigr ),\\nonumber \\\\$ with $\\kappa &\\equiv &\\frac{\\hbar }{2(C_2+C_3) (-3C_1+C_6+C_7+C_{10})} \\nonumber \\\\&\\times &\\bigl (C_1\\partial _R C_1+(C_2 - C_3)\\partial _R(C_2-C_3) \\nonumber \\\\&+&C_1\\partial _R(C_6+C_7+C_{10})\\bigr ), \\nonumber \\\\\\gamma _1&\\equiv &\\hbar \\frac{\\partial _R (C_2-C_3)}{2(C_1+C_6+C_7+C_{10})}, \\nonumber \\\\\\gamma _2&\\equiv &\\hbar \\frac{\\partial _R (C_6+C_{10})}{4(C_2+C_3)} +\\kappa .\\nonumber \\\\$ The fast-forward Hamiltonian is given by Eq.",
"(REF ), where $v(t)\\tilde{\\mathcal {H}} (R(\\Lambda (t)))$ is replaced by: $v\\mathcal {\\tilde{H}}&=&\\sum _{(i,j)=(1,2),(3,4)}v(t) \\tilde{W}_1(R(\\Lambda (t))) (\\sigma _i^y \\sigma _j^z + \\sigma _i^z \\sigma _j^y) \\nonumber \\\\&+&\\sum _{(i,j)=(2,3)}v(t) \\tilde{W}_2(R(\\Lambda (t))) (\\sigma _i^y \\sigma _j^z + \\sigma _i^z \\sigma _j^y) \\nonumber \\\\&+&\\sum _{(i,j)=(1,3),(2,4)}v(t) \\tilde{W}_3(R(\\Lambda (t))) (\\sigma _i^y \\sigma _j^z + \\sigma _i^z \\sigma _j^y) \\nonumber \\\\&+&\\sum _{(i,j)=(1,4)}v(t) \\tilde{W}_4(R(\\Lambda (t))) (\\sigma _i^y \\sigma _j^z + \\sigma _i^z \\sigma _j^y) \\nonumber \\\\&+& \\sum _{(i,j,k) \\in all}v(t)\\tilde{Q}(R(\\Lambda (t))) (\\sigma _i^x \\sigma _j^y + \\sigma _i^y \\sigma _j^x)\\cdot \\sigma _k^z.\\nonumber \\\\$ Figures REF (d) and REF (d) show the time dependence of regularization terms and that of wave function, respectively.",
"The wave function starts from the ground state with $J=0$ , i.e., $C_j=\\frac{1}{4}$ for $j=1,\\cdots , 16$ .",
"In Fig.",
"REF (d) the solution $\\Psi _{FF}(t)$ of TDSE in Eq.",
"(REF ) has reproduced the time-rescaled ground state wave function, which means the perfect fidelity during the fast-forward time range $0 \\le t \\le T_{FF}$ .",
"In this Section, the number of independent equations to determine the pair-wise interactions ($\\tilde{W}_i$ ) is varied depending on the symmetry of clusters.",
"To make Eq.",
"(REF ) solvable, however, these equations always require only one extra unknown 3-body interaction, whose contribution to $\\mathcal {\\tilde{H}}$ is commonly given in Eq.",
"(REF ) for all spin clusters with $N=4$ spins.",
"Therefore the 3-body interaction ($\\tilde{Q}$ ) here is geometry-independent and played a subsidiary role."
],
[
"Summary and discussions",
"The fast forward is the quasi-adiabatic dynamics guaranteed by regularization terms added to the reference Hamiltonian, followed by a rescaling of time with use of a large scaling factor.",
"Assuming the regularization terms consisting of pair-wise and 3-body interactions, we applied the core formula in Eq.",
"(REF ) to regular spin clusters with various geometries, e.g., regular triangle and open linear chain for $N=3$ spin systems, and triangular pyramid, square, primary star graph and open linear chain for $N=4$ spin systems.",
"The geometry-induced symmetry greatly decreases the rank of coefficient matrix of the linear algebraic equation for regularization terms, namely, the rank is determined by the geometric symmetry of the regular spin cluster.",
"Choosing a transverse Ising Hamiltonian as a reference, we find: (1) for $N=3$ spin clusters, the driving interaction consists of only the geometry-dependent pair-wise interactions and there is no need for the 3-body interaction.",
"The regular triangle and open linear 3 spins require, respectively, one and 2 species of the pair-wise driving interactions; (2) for $N=4$ spin clusters, the main part of the driving interaction again consists of pair-wise interactions.",
"The triangular pyramid and open linear 4 spins require, respectively, one and 4 species of the pair-wise driving interactions.",
"On the other hand, two species of the pair-wise driving interactions are necessary for the square and primary star graph.",
"For $N=4$ spin clusters, besides these geometry-dependent pair-wise interactions, we need a common geometry-independent 3-body interaction just to make the core equation in Eq.",
"(REF ) solvable.",
"The 3-body interaction here plays a subsidiary role.",
"The geometric symmetry of regular spin clusters determines the number of independent species of pair-wise driving interactions, and the clusters with the highest symmetry have only one species of pair-wise driving interaction.",
"Our fast-forward scheme provides a flexible method in designing the practical driving interaction in accelerating the adiabatic quantum dynamics of structured regular spin clusters.",
"The scheme may also be useful in our inventing a variational method for treating much bigger regular clusters.",
"We are grateful to S. Masuda for valuable discussions in the early stage of the present work.",
"The work is supported by Hibah Disertasi Doktor Kemenristekdikti 2018.",
"The work of B.E.G is supported by PUPT Ristekdikti-ITB 2017-2018."
],
[
"Square",
"The matrix for regularization term can be written as $\\mathcal {\\tilde{H}} =i \\left(\\begin{array}{cccccccccccccccc}0 & -A_1 & -A_1 & -A_1 & -A_1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\A_1 & 0 & 0 & 0 & 0 & -A_2 & 0 & 0 & -A_2 & -A_3 & 0 & 0 & 0 & 0 & 0 & 0 \\\\A_1 & 0 & 0 & 0 & 0 & -A_2 & -A_2 & 0 & 0 & 0 & -A_3 & 0 & 0 & 0 & 0 & 0 \\\\A_1 & 0 & 0 & 0 & 0 & 0 & -A_2 & -A_2 & 0 & -A_3 & 0 & 0 & 0 & 0 & 0 & 0 \\\\A_1 & 0 & 0 & 0 & 0 & 0 & 0 & -A_2 & -A_2 & 0 & -A_3 & 0 & 0 & 0 & 0 & 0 \\\\0 & A_2 & A_2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & A_2 & A_2 & 0 \\\\0 & 0 & A_2 & A_2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & A_2 & 0 & 0 & A_2 & 0 \\\\0 & 0 & 0 & A_2 & A_2 & 0 & 0 & 0 & 0 & 0 & 0 & A_2 & A_2 & 0 & 0 & 0 \\\\0 & A_2 & 0 & 0 & A_2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & A_2 & A_2 & 0 & 0 \\\\0 & A_3 & 0 & A_3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & A_3 & 0 & A_3 & 0 \\\\0 & 0 & A_3 & 0 & A_3 & 0 & 0 & 0 & 0 & 0 & 0 &A_3 & 0 & A_3 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & -A_2 & -A_2 & 0 & 0 & -A_3 & 0 & 0 & 0 & 0 & A_1 \\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & -A_2 & -A_2 & -A_3 & 0 & 0 & 0 & 0 & 0 & A_1 \\\\0 & 0 & 0 & 0 & 0 & -A_2 & 0 & 0 & -A_2 & 0 & -A_3 & 0 & 0 & 0 & 0 & A_1 \\\\0 & 0 & 0 & 0 & 0 & -A_2 & -A_2 & 0 & 0 & -A_3 & 0 & 0 & 0 & 0 & 0 & A_1\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -A_1 & -A_1 & -A_1 & -A_1 & 0\\end{array}\\right)$ where $A_1$ = $2\\tilde{W}_1 + \\tilde{W}_2$ , $A_2$ = $\\tilde{W}_2$ , $A_3$ = $2\\tilde{W}_1-\\tilde{W}_2$ ."
],
[
"Primary star graph",
"The matrix for regularization term can be written as $\\mathcal {\\tilde{H}} =i \\left(\\begin{array}{cccccccccccccccc}0 & -A_1 &-A_1 & -A_2 & -A_1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\A_1 & 0 & 0 & 0 & 0 & -A_3 & 0 & 0 & -A_3 & -A_3 & 0 & 0 & 0 & 0 & 0 & 0 \\\\A_1 & 0 & 0 & 0 & 0 & -A_3 & -A_3& 0 & 0 & 0 & -A_3 & 0 & 0 & 0 & 0 & 0 \\\\A_2 & 0 & 0 & 0 & 0 & 0 & A_4 & A_4 & 0 & A_4 & 0 & 0 & 0 & 0 & 0 & 0 \\\\A_1 & 0 & 0 & 0 & 0 & 0 & 0 & -A_3 & -A_3 & 0 & -A_3 & 0 & 0 & 0 & 0 & 0 \\\\0 & A_3 & A_3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -A_4 & A_3 & 0 \\\\0 & 0 & A_3 & -A_4 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & A_3 & 0 & 0 & A_3 & 0 \\\\0 & 0 & 0 & -A_4 & A_3 & 0 & 0 & 0 & 0 & 0 & 0 & A_3 & A_3 & 0 & 0 & 0 \\\\0 & A_3 & 0 & 0 & A_3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &A_3 & -A_4 & 0 & 0 \\\\0 & A_3 & 0 & -A_4 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & A_3 & 0 & A_3 & 0 \\\\0 & 0 & A_3& 0 & A_3& 0 & 0 & 0 & 0 & 0 & 0 & A_3 & 0 & -A_4 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & -A_3 & -A_3 & 0 & 0 & -A_3 & 0 & 0 & 0 & 0 & A_1 \\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & -A_3 & -A_3 & -A_3 & 0 & 0 & 0 & 0 & 0 & A_1 \\\\0 & 0 & 0 & 0 & 0 & A_4 & 0 & 0 & A_4 & 0 & A_4 & 0 & 0 & 0 & 0 & A_2 \\\\0 & 0 & 0 & 0 & 0 & -A_3 & -A_3 & 0 & 0 & -A_3 & 0 & 0 & 0 & 0 & 0 & A_1 \\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -A_1 & -A_1& -A_2 & -A_1 & 0\\end{array}\\right)$ where $A_1$ = $2\\tilde{W}_2+\\tilde{W}_1$ , $A_2$ = $3\\tilde{W}_1$ , $A_3$ = $\\tilde{W}_1$ , $A_4$ = $\\tilde{W}_1-2\\tilde{W}_2$ ."
],
[
"Open linear 4 spins",
"The matrix for regularization term can be written as $\\mathcal {\\tilde{H}} =i \\left(\\begin{array}{cccccccccccccccc}0 & -A_1 & -A_2 & -A_2 & -A_1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\A_1 & 0 & 0 & 0 & 0 & -A_3& 0 & 0 & -A_4 & -A_5 & 0 & 0 & 0 & 0 & 0 & 0 \\\\A_2 & 0 & 0 & 0 & 0 & -A_6 & -A_7 & 0 & 0 & 0 & -A_8 & 0 & 0 & 0 & 0 & 0 \\\\A_2 & 0 & 0 & 0 & 0 & 0 & -A_7 & -A_6 & 0 & -A_8 & 0 & 0 & 0 & 0 & 0 & 0 \\\\A_1 & 0 & 0 & 0 & 0 & 0 & 0 & -A_3 & -A_4 & 0 & -A_5 & 0 & 0 & 0 & 0 & 0 \\\\0 & A_3 & A_6 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & A_6 & A_3 & 0 \\\\0 & 0 & A_7 & A_7 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & A_4 & 0 & 0 & A_4 & 0 \\\\0 & 0 & 0 & A_6 & A_3 & 0 & 0 & 0 & 0 & 0 & 0 & A_3 & A_6 & 0 & 0 & 0 \\\\0 & A_4 & 0 & 0 & A_4 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & A_7 & A_7 & 0 & 0 \\\\0 & A_5 & 0 & A_8 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & A_8 & 0 & A_5 & 0 \\\\0 & 0 & A_8 & 0 & A_5 & 0 & 0 & 0 & 0 & 0 & 0 & A_5 & 0 & A_8 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & -A_4 & -A_3 & 0 & 0 & -A_5& 0 & 0 & 0 & 0 & A_1 \\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & -A_6 & -A_7 & -A_8 & 0 & 0 & 0 & 0 & 0 & A_2 \\\\0 & 0 & 0 & 0 & 0 & -A_6 & 0 & 0 & -A_7 & 0 & -A_8 & 0 & 0 & 0 & 0 & A_2 \\\\0 & 0 & 0 & 0 & 0 & -A_3 & -A_4 & 0 & 0 & -A_5 & 0 & 0 & 0 & 0 & 0 & A_1 \\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -A_1 & -A_2 & -A_2 & -A_1 & 0\\end{array}\\right)$ where $A_1$ = $\\tilde{W}_1+\\tilde{W}_3+\\tilde{W}_4$ , $A_2$ = $\\tilde{W}_1+\\tilde{W}_2+\\tilde{W}_3$ , $A_3$ = $\\tilde{W}_2-\\tilde{W}_1+\\tilde{W}_3$ , $A_4$ = $\\tilde{W}_1-\\tilde{W}_4+\\tilde{W}_3$ , $A_5$ = $\\tilde{W}_1+\\tilde{W}_2-\\tilde{W}_3$ , $A_6$ = $-\\tilde{W}_1+\\tilde{W}_4+\\tilde{W}_3$ , $A_7$ = $\\tilde{W}_1-\\tilde{W}_2+\\tilde{W}_3$ , $A_8$ = $\\tilde{W}_1+\\tilde{W}_4-\\tilde{W}_3$ ."
]
] | 1906.04433 |
[
[
"Using Hoare logic in a process algebra setting"
],
[
"Abstract This paper concerns the relation between process algebra and Hoare logic.",
"We investigate the question whether and how a Hoare logic can be used for reasoning about how data change in the course of a process when reasoning equationally about that process.",
"We introduce an extension of ACP (Algebra of Communicating Processes) with features that are relevant to processes in which data are involved, present a Hoare logic for the processes considered in this process algebra, and discuss the use of this Hoare logic as a complement to pure equational reasoning with the equational axioms of the process algebra."
],
[
"Introduction",
" (Algebra of Communicating Processes) and its extensions provide a setting for equational reasoning about processes of some kind.",
"The processes about which reasoning is in demand are often processes in which data are involved.",
"It is quite common for such a process that the data that are involved change in the course of the process and that the process proceeds at certain stages in a way that depends on the changing data.",
"This means that reasoning about a process often involves reasoning about how data change in the course of that process.",
"The question arises whether and how a Hoare logic can be used for the second kind of reasoning when reasoning equationally about a process.",
"After all, processes of the kind described above are reminiscent of the processes that arise from the execution of imperative programs.",
"This paper is concerned with the above-mentioned question.",
"We investigate it using an extension of [10] with features that are relevant to processes in which data are involved and a Hoare logic of asserted processes based on this extension of .",
"The extension concerned is called .",
"Its additional features include assignment actions to deal with data that change in the course of a process and guarded commands to deal with processes that proceed at certain stages in a way that depends on certain data.",
"In the Hoare logic concerned, an asserted process is a formula of the form ${\\phi }{p}{\\psi }$ , where $p$ is a term of that denotes a process and $\\phi $ and $\\psi $ are terms of that denote conditions.",
"We define what it means that an asserted process is true in such a way that ${\\phi }{p}{\\psi }$ is true iff a set of equations that represents this judgment is derivable from the axioms of .",
"Such a definition is a prerequisite for an affirmative answer to the question whether and how a Hoare logic can be used for reasoning about how data change in the course of a process when reasoning equationally about that process.",
"The set of equations that represents the judgment expresses that a certain equivalence relation holds between processes determined by the asserted process.",
"The equivalence relation concerned may be a useful equivalence relation when reasoning about processes in which data are involved.",
"However, it is not a congruence relation, i.e.",
"it is not preserved by all contexts.",
"This complicates pure equational reasoning considerably.",
"The presented Hoare logic can be considered to be a means to get partially round the complications concerned.",
"This paper is organized as follows.",
"We begin with presenting , an extension of with the empty process constant $$ and the binary iteration operator $$ , and , an extension of with features that are relevant to processes in which data are involved (Sections and ).",
"We also present a structural operational semantics of , define a notion of bisimulation equivalence based on this semantics, and show that the axioms of are sound with respect to this bisimulation equivalence (Section ).",
"After that, we present a Hoare logic of asserted processes based on , define what it means that an asserted process is true, and show that the axioms and rules of this Hoare logic are sound with respect to this meaning (Section ).",
"Following this, we go further into the connection of the presented Hoare logic with by way of the equivalence relation referred to in the previous paragraph (Section ).",
"We also go into the use of the presented Hoare logic as a complement to pure equational reasoning with the axioms of by means of examples (Section ).",
"Finally, we discuss related work and make some concluding remarks (Sections and )."
],
[
" with the Empty Process and Iteration",
"In this section, we present , [10] extended with the empty process constant $$ as in [7] and the binary iteration operator ${} {}$ as in [8].",
"In , it is assumed that a fixed but arbitrary finite set $$ of basic actions, with $,\\notin $ , and a fixed but arbitrary commutative and associative communication function ${}{({}) ({})}{({})}$ , such that $(,a) = $ for all $a \\in {}$ , have been given.",
"Basic actions are taken as atomic processes.",
"The function $$ is regarded to give the result of synchronously performing any two basic actions for which this is possible, and to be $$ otherwise.",
"Henceforth, we write $$ for ${}$ .",
"The algebraic theory has one sort: the sort $$ of processes.",
"We make this sort explicit to anticipate the need for many-sortedness later on.",
"The algebraic theory has the following constants and operators to build terms of sort $$ : the inaction constant ${}{}$ ; the empty process constant ${}{}$ ; for each $a \\in $ , the basic action constant ${a}{}$ ; the binary alternative composition operator ${}{}{}$ ; the binary sequential composition operator ${}{}{}$ ; the binary iteration operator ${}{}{}$ ; the binary parallel composition operator ${}{}{}$ ; the binary left merge operator ${}{}{}$ ; the binary communication merge operator ${}{}{}$ ; for each $H \\subseteq $ , the unary encapsulation operator ${{H}}{}{}$ .",
"We assume that there is a countably infinite set of variables of sort $$ , which contains $x$ , $y$ and $z$ .",
"Terms are built as usual.",
"We use infix notation for the binary operators.",
"The following precedence conventions are used to reduce the need for parentheses: the operator ${} {}$ binds stronger than all other binary operators and the operator ${} {}$ binds weaker than all other binary operators.",
"The constants and operators of are the constants and operators of [7] and additionally the iteration operator ${} $ .",
"Let $p$ and $q$ be closed terms, $a \\in $ , and $H \\subseteq $ .As usual, a term in which no variables occur is called a closed term.",
"Then the constants and operators of can be explained as follows: the constant $$ denotes the process that is not capable of doing anything, not even terminating successfully; the constant $$ denotes the process that is only capable of terminating successfully; the constant $a$ denotes the process that is only capable of first performing action $a$ and next terminating successfully; a closed term of the form $p q$ denotes the process that behaves either as the process denoted by $p$ or as the process denoted by $q$ , but not both; a closed term of the form $p q$ denotes the process that first behaves as the process denoted by $p$ and on successful termination of that process next behaves as the process denoted by $q$ ; a closed term of the form $p q$ denotes the process that behaves either as the process denoted by $q$ or as the process that first behaves as the process denoted by $p$ and on successful termination of that process next behaves as $p q$ again; a closed term of the form $p q$ denotes the process that behaves as the processes denoted by $p$ and $q$ taking place in parallel, by which we understand that, each time an action is performed, either a next action of one of the two processes is performed or a next action of the former process and a next action of the latter process are performed synchronously; a closed term of the form $p q$ denotes the process that behaves the same as the process denoted by $p q$ , except that it starts with performing an action of the process denoted by $p$ ; a closed term of the form $p q$ denotes the process that behaves the same as the process denoted by $p q$ , except that it starts with performing an action of the process denoted by $p$ and an action of the process denoted by $q$ synchronously; a closed term of the form ${H}(p)$ denotes the process that behaves the same as the process denoted by $p$ , except that actions from $H$ are blocked.",
"The axioms of are the equations given in Table REF .",
"Table: Axioms ofIn these equations, $a$ and $b$ stand for arbitrary constants of that differ from $$ and $H$ stands for an arbitrary subset of $$ .",
"So, CM3, CM7, and D0–D4 are actually axiom schemas.",
"Axioms A1–A9, CM1T, CM2T, CM3, CM4, CM5T, CM6T, CM7–CM9, and D0–D4 are the axioms of (cf. [7]).",
"Axioms BKS1 and RSP* have been taken from [9].",
"The iteration operator originates from [8], where it is called the binary Kleene star operator.",
"The unary counterpart of this operator can be defined by the equation $x = x $ .",
"From this defining equation, it follows, using RSP*, that $x = x {x } $ and also that $x y = {x } y$ .",
"Among the equations derivable from the axioms of are the equations concerning the iteration operator given in Table REF .",
"Table: Derivable equations for iterationIn the axiom system of given in [8], the axioms for the iteration operator are BKS1–BKS4 instead of BKS1 and RSP*.",
"There exist equations derivable from the axioms of that are not derivable from the axioms of with BKS1 and RSP* replaced by BKS1–BKS5.",
"For example, the equation $a = (a a) $ is derivable with BKS1 and RSP*, but not with BKS1–BKS5 (cf. [28]).",
"Moreover, we do not see how Theorem of this paper can be proved if RSP* is replaced by BKS2–BKS5 (see the remark following the proof of the theorem)."
],
[
"Data Enriched ",
"In this section, we present , data enriched .",
"This extension of has been inspired by [12].",
"It extends with features that are relevant to processes in which data are involved, such as guarded commands (to deal with processes that only take place if some data-dependent condition holds), data parameterized actions (to deal with process interactions with data transfer), and assignment actions (to deal with data that change in the course of a process).",
"In , it is assumed that the following has been given with respect to data: a (single- or many-sorted) signature $_$ that includes a sort $$ of data and constants and/or operators with result sort $$ ; a minimal algebra $$ of the signature $_$ .",
"Moreover, it is assumed that a countably infinite set $$ of flexible variables has been given.",
"A flexible variable is a variable whose value may change in the course of a process.The term flexible variable is used for this kind of variables in e.g.",
"[27], [20].",
"Flexible variables are found under the name program variables in imperative programming.",
"We write $$ for the set of all closed terms over the signature $_$ that are of sort $$ .",
"An evaluation map is a function $\\sigma $ from $$ to $$ where, for all $v \\in $ , $\\sigma (v) = v$ if $\\sigma (v) \\in $ .",
"Let $\\sigma $ be an evaluation map and let $V$ be a finite subset of $$ .",
"Then $\\sigma $ is a $V$ -evaluation map if, for all $v \\in $ , $\\sigma (v) \\in $ iff $v \\in V$ .",
"Evaluation maps are intended to provide the data values assigned to flexible variables of sort $$ when a term of sort $$ is evaluated.",
"However, in order to fit better in an algebraic setting, they provide closed terms over the signature $_$ that denote those data values instead.",
"The requirement that $$ is a minimal algebra guarantees that each data value can be represented by a closed term.",
"The possibility to map flexible variables to themselves may be used for partial evaluation, i.e.",
"evaluation where some flexible variables are not evaluated.",
"The algebraic theory has three sorts: the sort $$ of processes, the sort $$ of conditions, and the sort $$ of data.",
"has the constants and operators from $_$ and in addition the following constants to build terms of sort $$ : for each $v \\in $ , the flexible variable constant ${v}{}$ .",
"has the following constants and operators to build terms of sort $$ : the binary equality operator ${}{}{}$ ; the truth constant ${}{}$ ; the falsity constant ${}{}$ ; the unary negation operator ${}{}{}$ ; the binary conjunction operator ${}{}{}$ ; the binary disjunction operator ${}{}{}$ ; the binary implication operator ${}{}{}$ ; the unary variable-binding universal quantification operator ${\\forall }{}{}$ that binds a variable of sort $$ ; the unary variable-binding existential quantification operator ${\\exists }{}{}$ that binds a variable of sort $$ .",
"has the constants and operators of and in addition the following operators to build terms of sort $$ : the binary guarded command operator ${}{}{}$ ; for each $n \\in $ , for each $a \\in $ , the $n$ -ary data parameterized action operator ${a}{\\underbrace{\\cdots }_{n\\; \\mathrm {times}}}{}$ ; for each $v \\in $ , a unary assignment action operator ${{v}}{}{}$ ; for each evaluation map $\\sigma $ , a unary evaluation operator ${{\\sigma }}{}{}$ .",
"We assume that there are countably infinite sets of variables of sort $$ and $$ and that the sets of variables of sort $$ , $$ , and $$ are mutually disjoint and disjoint from $$ .",
"The formation rules for terms are the usual ones for the many-sorted case (see e.g.",
"[26], [29]) and in addition the following rule: if $O$ is a variable-binding operator ${O}{S_1 \\ldots S_n}{S}$ that binds a variable of sort $S^{\\prime }$ , $t_1,\\ldots ,t_n$ are terms of sorts $S_1,\\ldots ,S_n$ , respectively, and $X$ is a variable of sort $S^{\\prime }$ , then $O X (t_1,\\ldots ,t_n)$ is a term of sort $S$ (cf. [25]).",
"We use the same notational conventions as before.",
"We also use infix notation for the additional binary operators.",
"Moreover, we use the notation ${v}{e}$ , where $v \\in $ and $e$ is a term of sort $$ , for the term ${v}(e)$ .",
"We use the notation $\\phi \\psi $ , where $\\phi $ and $\\psi $ are terms of sort $$ , for the term $(\\phi \\psi ) (\\psi \\phi )$ .",
"Moreover, we use the notation $\\Phi $ , where $\\Phi = {\\phi _1,\\ldots ,\\phi _n}$ and $\\phi _1, \\ldots , \\phi _n$ are terms of sort $$ , for the term $\\phi _1 \\ldots \\phi _n$ .",
"We write $$ for the set of all closed terms of sort $$ , $$ for the set of all closed terms of sort $$ , and $$ for the set of all closed terms of sort $$ .",
"Each term from $$ can be taken as a formula of a first-order language with equality of $$ by taking the flexible variable constants as additional variables of sort $$ .",
"We implicitly take the flexible variable constants as additional variables of sort $$ wherever the context asks for a formula.",
"In this way, each term from $$ can be interpreted as a formula in $$ .",
"The axioms of (given below) include an equation $\\phi = \\psi $ for each two terms $\\phi $ and $\\psi $ from $$ for which the formula $\\phi \\psi $ holds in $$ .",
"Let $p$ be a term from $$ , $\\phi $ be a term from $$ , and $e_1,\\ldots ,e_n$ and $e$ be terms from $$ .",
"Then the additional operators can be explained as follows: the term $\\phi p$ denotes the process that behaves as the process denoted by $p$ under condition $\\phi $ ; the term $a(e_1,\\ldots ,e_n)$ denotes the process that is only capable of first performing action $a(e_1,\\ldots ,e_n)$ and next terminating successfully; the term ${v}{e}$ denotes the process that is only capable of first performing action ${v}{e}$ , whose intended effect is the assignment of the result of evaluating $e$ to flexible variable $v$ , and next terminating successfully; the term ${\\sigma }(p)$ denotes the process that behaves the same as the process denoted by $p$ except that each subterm of $p$ that belongs to $$ is evaluated using the evaluation map $\\sigma $ updated according to the assignment actions that have taken place at the point where the subterm is encountered.",
"Evaluation operators are a variant of state operators (see e.g. [3]).",
"The guarded command operator is often used to construct terms that are reminiscent of control flow statements of imperative programming languages.",
"For example, terms of the form $\\phi t (\\, \\phi ) t^{\\prime }$ are reminiscent of if-then-else statements and terms of the form $(\\phi t) ((\\, \\phi ) )$ are reminiscent of while-do statements.",
"The following term contains a subterm of the latter form ($i, j, q, r \\in $ ): q0 ri (((r j) qq + 1 rr - j) (( r j) )).",
"This term is reminiscent of a program that computes the quotient and remainder of dividing two integers by repeated subtraction.",
"That is, the final values of $q$ and $r$ are the quotient and remainder of dividing the initial value of $i$ by the initial value of $j$ .",
"An evaluation operator can be used to show that this is the case for given initial values of $i$ and $j$ .",
"For example, consider the case where the initial values of $i$ and $j$ are 11 and 3, respectively.",
"Let $\\sigma $ be an evaluation map such that $\\sigma (i) = 11$ and $\\sigma (j) = 3$ .",
"Then the following equation can be derived from the axioms of given below: (q0 ri (((r j) qq + 1 rr - j) (( r j) ))) = q0 r11 q1 r8 q2 r5 q3 r2.",
"This equation shows that in the case where the initial values of $i$ and $j$ are 11 and 3 the final values of $q$ and $r$ are 3 and 2 (which are the quotient and remainder of dividing 11 by 3).",
"An evaluation map $\\sigma $ can be extended homomorphically from flexible variables to terms of sort $$ and terms of sort $$ .",
"These extensions are denoted by $\\sigma $ as well.",
"We write $\\sigma {e}{v}$ for the evaluation map $\\sigma ^{\\prime }$ defined by $\\sigma ^{\\prime }(v^{\\prime }) = \\sigma (v^{\\prime })$ if $v^{\\prime } \\lnot \\equiv v$ and $\\sigma ^{\\prime }(v) = e$ .",
"The axioms of are the axioms of and in addition the equations given in Table REF .",
"Table: Axioms ofIn these equations, $\\phi $ and $\\psi $ stand for arbitrary terms from $$ , $e$ , $e_1,e_2,\\ldots $ , and $e^{\\prime }$ , $e^{\\prime }_1,e^{\\prime }_2,\\ldots $ stand for arbitrary terms from $$ , $v$ stands for an arbitrary flexible variable from $$ , $\\sigma $ stands for an arbitrary evaluation map, $a$ and $b$ stand for arbitrary constants of that differ from $$ , $c$ stands for an arbitrary constant of that differ from $$ and $$ , and $H$ stands for an arbitrary subset of $$ .",
"Axioms GC1–GC11 have been taken from [4] (using a different numbering), but with the axioms with occurrences of conditional expressions of the form ${p}{\\phi }{q}$ replaced by simpler axioms.",
"Axioms CM3D, CM7Da, CM7Db, D1D, and D2D have been inspired by [12].",
"The set $$ of actions of is inductively defined by the following rules: if $a \\in $ , then $a \\in $ ; if $a \\in $ and $e_1,\\dots ,e_n \\in $ , then $a(e_1,\\dots ,e_n) \\in $ ; if $v \\in $ and $e \\in $ , then ${v}{e} \\in $ .",
"The elements of $$ are the processes that are considered to be atomic.",
"The set $$ of head normal forms of is inductively defined by the following rules: $\\in $ ; if $\\phi \\in $ , then $\\phi \\in $ ; if $\\phi \\in $ , $\\alpha \\in $ , and $p \\in $ , then $\\phi a p \\in $ ; if $p,p^{\\prime } \\in $ , then $p p^{\\prime } \\in $ .",
"The following lemma about head normal forms is used in later sections.",
"For all terms $p \\in $ , there exists a term $q \\in $ such that $p = q$ is derivable from the axioms of .",
"This is straightforwardly proved by induction on the structure of $p$ .",
"The cases where $p$ is of the form $$ , $$ or $\\alpha $ ($\\alpha \\in $ ) are trivial.",
"The case where $p$ is of the form $p_1 p_2$ follows immediately from the induction hypothesis.",
"The case where $p$ is of the form $p_1 p_2$ follows immediately from the case that $p$ is of the form $p_1 p_2$ and the case that $p$ is of the form $p_1 p_2$ .",
"Each of the other cases follow immediately from the induction hypothesis and a claim that is easily proved by structural induction.",
"In the case where $p$ is of the form $p_1 p_2$ , each of the cases to be considered in the inductive proof demands an additional proof by structural induction.",
"$\\Box $ Some earlier extensions of include Hoare's ternary counterpart of the binary guarded command operator (see e.g. [4]).",
"This operator can be defined by the equation ${x}{u}{y} = u x (\\, u) y$ .",
"From this defining equation, it follows that $u x = {x}{u}{}$ .",
"In [15], a unary counterpart of the binary guarded command operator is used.",
"This operator can be defined by the equation ${u} = u $ .",
"From this defining equation, it follows that $u x = {u} x$ and also that ${} = $ and ${} = $ .",
"In [15], the processes denoted by closed terms of the form ${\\phi }$ are called guards."
],
[
"Structural Operational Semantics and Bisimulation Equivalence",
"In this section, we present a structural operational semantics of , define a notion of bisimulation equivalence based on this semantics, and show that the axioms of are sound with respect to this bisimulation equivalence.",
"We write $$ for the set of all terms $\\phi \\in $ for which ${}{\\phi }$ .",
"As formulas of a first-order language with equality of $$ , the terms from $$ are the formulas that are satisfiable in $$ .",
"We start with the presentation of the structural operational semantics of .",
"The following transition relations on $$ are used: for each $\\phi \\in $ , a unary relation ${{\\phi }}$ ; for each $\\ell \\in $ , a binary relation ${{\\ell }}$ .",
"We write ${p}{\\phi }$ instead of $p \\in {{\\phi }}$ and ${p}{{\\phi }{\\alpha }}{q}$ instead of ${p,q} \\in {{{\\phi ,\\alpha }}}$ .",
"The relations ${{\\phi }}$ and ${{\\ell }}$ can be explained as follows: ${p}{\\phi }$ : $p$ is capable of terminating successfully under condition $\\phi $ ; ${p}{{\\phi }{\\alpha }}{q}$ : $p$ is capable of performing action $\\alpha $ under condition $\\phi $ and then proceeding as $q$ .",
"The structural operational semantics of is described by the transition rules given in Table REF .",
"Table: Transition rules forIn this table, $a$ , $b$ , and $c$ stand for arbitrary basic actions from $$ , $v$ stands for an arbitrary flexible variable from $$ , $e$ and $e_1,e_2,\\ldots $ stand for arbitrary terms from $$ , $\\phi $ and $\\psi $ stand for arbitrary terms from $$ , $\\alpha $ stands for an arbitrary term from $$ , $H$ stands for arbitrary subset of $$ , and $\\sigma $ stands for an arbitrary evaluation map.",
"Two process are considered equal if they can simulate each other.",
"In order to make this precise, we will define the notion of bisimulation equivalence on the set $$ below.",
"In the definition concerned, we need an equivalence relation on the set $$ .",
"Two actions $\\alpha ,\\alpha ^{\\prime } \\in $ are data equivalent, written $\\alpha \\simeq \\alpha ^{\\prime }$ , iff one of the following holds: there exists an $a \\in $ such that $\\alpha = a$ and $\\alpha ^{\\prime } = a$ ; there exist an $a \\in $ and $e_1,\\dots ,e_n,e^{\\prime }_1,\\dots ,e^{\\prime }_n \\in $ such that ${}{{e_1 = e^{\\prime }_1 \\linebreak [2] \\ldots e_n = e^{\\prime }_n}}$ , $\\alpha = a(e_1,\\dots ,e_n)$ , and $\\alpha ^{\\prime } = a(e^{\\prime }_1,\\dots ,e^{\\prime }_n)$ ; there exist a $v \\in $ and $e,e^{\\prime } \\in $ such that ${}{{e = e^{\\prime }}}$ , $\\alpha = {v}{e}$ , and $\\alpha ^{\\prime } = {v}{e^{\\prime }}$ .",
"We write $[\\alpha ]$ , where $\\alpha \\in $ , for the equivalence class of $\\alpha $ with respect to $\\simeq $ .",
"A bisimulation is a binary relation $R$ on $$ such that, for all terms $p,q \\in $ with $(p,q) \\in R$ , the following conditions hold: if ${p}{{\\phi }{\\alpha }}{p^{\\prime }}$ , then there exists a finite set $\\Psi \\subseteq $ such that ${}{\\phi \\Psi }$ and, for all $\\psi \\in \\Psi $ , there exist an $\\alpha ^{\\prime } \\in [\\alpha ]$ and a $q^{\\prime } \\in $ such that ${q}{{\\psi }{\\alpha ^{\\prime }}}{q^{\\prime }}$ and $(p^{\\prime },q^{\\prime }) \\in R$ ; if ${q}{{\\phi }{\\alpha }}{q^{\\prime }}$ , then there exists a finite set $\\Psi \\subseteq $ such that ${}{\\phi \\Psi }$ and, for all $\\psi \\in \\Psi $ , there exist an $\\alpha ^{\\prime } \\in [\\alpha ]$ and a $p^{\\prime } \\in $ such that ${p}{{\\psi }{\\alpha ^{\\prime }}}{p^{\\prime }}$ and $(p^{\\prime },q^{\\prime }) \\in R$ ; if ${p}{\\phi }$ , then there exists a finite set $\\Psi \\subseteq $ such that ${}{\\phi \\Psi }$ and, for all $\\psi \\in \\Psi $ , ${q}{\\psi }$ ; if ${q}{\\phi }$ , then there exists a finite set $\\Psi \\subseteq $ such that ${}{\\phi \\Psi }$ and, for all $\\psi \\in \\Psi $ , ${p}{\\psi }$ .",
"Two terms $p,q \\in $ are bisimulation equivalent, written $p q$ , if there exists a bisimulation $R$ such that $(p,q) \\in R$ .",
"Let $R$ be a bisimulation such that $(p,q) \\in R$ .",
"Then we say that $R$ is a bisimulation witnessing $p q$ .",
"The above definition of a bisimulation deviates from the standard definition because a transition on one side may be simulated by a set of transitions on the other side.",
"For example, the transition ${(\\phi _1 \\phi _2) a b}{{\\phi _1 \\phi _2}{a}}{b}$ is simulated by the set of transitions consisting of ${\\phi _1 a b}{{\\phi _1}{a}}{b}$ and ${\\phi _2 a b}{{\\phi _2}{a}}{b}$ .",
"A bisimulation as defined above is called a splitting bisimulation in [11].",
"Bisimulation equivalence is a congruence with respect to the operators of of which the result sort and at least one argument sort is $$ .",
"[Congruence] For all terms $p,q,p^{\\prime },q^{\\prime } \\in $ and all terms $\\phi \\in $ , $p p^{\\prime }$ and $q q^{\\prime }$ only if $p q p^{\\prime } q^{\\prime }$ , $p q p^{\\prime } q^{\\prime }$ , $p q p^{\\prime } q^{\\prime }$ , $\\phi p \\phi p^{\\prime }$ , $p q p^{\\prime } q^{\\prime }$ , $p q p^{\\prime } q^{\\prime }$ , $p q p^{\\prime } q^{\\prime }$ , ${H}(p) {H}(p^{\\prime })$ , and ${\\sigma }(p) {\\sigma }(p^{\\prime })$ .",
"We can reformulate the transition rules such that: bisimulation equivalence based on the reformulated transition rules according to the standard definition of bisimulation equivalence coincides with bisimulation equivalence based on the original transition rules according to the definition of bisimulation equivalence given above; the reformulated transition rules make up a transition system specification in path format.",
"The reformulation is similar to the one for the transition rules for BPAps outlined in [5].",
"The proposition follows now immediately from the well-known result that bisimulation equivalence according to the standard definition of bisimulation equivalence is a congruence if the transition rules concerned make up a transition system specification in path format (see e.g. [6]).",
"$\\Box $ The underlying idea of the reformulation referred to above is that we replace each transition ${p}{{\\phi }{\\alpha }}{p^{\\prime }}$ by a transition ${p}{{\\nu }{[\\alpha ]}}{p^{\\prime }}$ for each valuation of variables $\\nu $ such that ${}{\\phi }\\,[\\nu ]$ , and likewise ${p}{\\phi }$ .",
"Thus, in a bisimulation, a transition on one side must be simulated by a single transition on the other side.",
"We did not present the reformulated structural operational semantics in this paper because it is, in our opinion, intuitively less appealing.",
"The axioms of are sound with respect to $$ for equations between terms from $$ .",
"[Soundness] For all terms $p,q \\in $ , $p = q$ is derivable from the axioms of only if $p q$ .",
"Because ${}$ is a congruence, it is sufficient to prove the theorem for all substitution instances of each axiom of .",
"We will loosely say that a relation contains all closed substitution instances of an equation if it contains all pairs $(p,q)$ such that $p = q$ is a closed substitution instance of the equation.",
"For each axiom, we can construct a bisimulation $R$ witnessing $p q$ for all closed substitution instances $p = q$ of the axiom as follows: in the case of A1–A6, A8, A9, BKS1, CM3, CM4, CM7–CM9, D1, D3, D4, GC1, GC4–GC11, V1–V5, CM3D, CM7Da, D1D, CM3A, and D1A, we take the relation $R$ that consists of all closed substitution instances of the axiom concerned and the equation $x = x$ ; in the case of A7, CM2T, CM5T, CM6T, D0, D2, GC2, GC3, V0, CM7Db–CM7Dd, D2D, CM5A, and CM6A, we take the relation $R$ that consists of all closed substitution instances of the axiom concerned; in the case of CM1T, we take the relation $R$ that consists of all closed substitution instances of CM1T, the equation $x y = y x$ , and the equation $x = x$ ; in the case of RSP*, we take the relation $R$ that consists of all closed substitution instances $r = p q$ of the consequent of RSP* for which $r p r q$ and all closed substitution instances of the equation $x = x$ .",
"$\\Box $ We have not been able to prove the completeness of the axioms of with respect to $$ for equations between terms from $$ .",
"Such a proof would give an affirmative answer to an open question about the axiomatization of the iteration operator already posed in 1984 by Milner [23].",
"Until now, all attempts to answer this question have failed (see [14])."
],
[
"A Hoare Logic of Asserted Processes",
"In this section, we present , a Hoare logic of asserted processes based on , define what it means that an asserted process is true, and show that the axioms and rules of this logic are sound with respect to this meaning.",
"We write $$ for the set of all closed terms of sort $$ in which the evaluation operators ${\\sigma }$ and the auxiliary operators $$ and $$ do not occur and we write $$ for the set of all terms of sort $$ in which variables of sort $$ do not occur.",
"Clearly, $\\subset $ and $\\subset $ .",
"An asserted process is a formula of the form ${\\phi }{p}{\\psi }$ , where $p \\in $ and $\\phi ,\\psi \\in $ .",
"Here, $\\phi $ is called the pre-condition of the asserted process and $\\psi $ is called the post-condition of the asserted process.",
"The intuitive meaning of an asserted process ${\\phi }{p}{\\psi }$ is as follows: if $\\phi $ holds at the start of $p$ and $p$ eventually terminates successfully, then $\\psi $ holds at the successful termination of $p$ .",
"The conditions $\\phi $ and $\\psi $ concern the data values assigned to flexible variables at the start and at successful termination, respectively.",
"Therefore, in general, one or more flexible variables occur in $\\phi $ and $\\psi $ .",
"Unlike in $p$ , (logical) variables of sort $$ may also occur in $\\phi $ and $\\psi $ .",
"This allows of referring in $\\psi $ to the data values assigned to flexible variables at the start, like in ${v = u}{{v}{v+1}}{v = u + 1}$ .",
"Below, we use the notion of equivalence under $V$ -evaluation to make the intuitive meaning of asserted processes more precise.",
"We write $(p)$ , where $p \\in $ , for the set of all $v \\in $ that occur in $p$ and likewise $(\\phi )$ , where $\\phi \\in $ , for the set of all $v \\in $ that occur in $\\phi $ .",
"We write $(p)$ , where $p \\in $ , for the set of all $v \\in (p)$ that occur in subterms of $p$ that are of the form ${v}{e}$ .",
"Moreover, we write $_V$ , where $V$ is a finite subset of $$ , for the set ${p \\in (p) \\subseteq V}$ .",
"Let $V$ be a finite subset of $$ and let $p,q \\in _V$ .",
"Then $p$ and $q$ are equivalent under $V\\!$ -evaluation, written $p {V} q$ , if, for all $V$ -evaluation maps $\\sigma $ , ${\\sigma }(p) = {\\sigma }(q)$ is derivable from the axioms of .",
"Notice that ${V}$ , where $V$ be a finite subset of $$ , is an equivalence relation indeed.",
"Notice further that, for all $p,q \\in _W$ , $W \\subset V$ and $p {W} q$ only if $p {V} q$ .",
"Let ${\\phi }{p}{\\psi }$ be an asserted process and let $V = (\\phi ) (p) (\\psi )$ .",
"Then ${\\phi }{p}{\\psi }$ is true if, for all closed substitution instances ${\\phi ^{\\prime }}{p}{\\psi ^{\\prime }}$ of ${\\phi }{p}{\\psi }$ , $\\phi ^{\\prime } p {V} (\\phi ^{\\prime } p) (\\psi ^{\\prime } )$ .",
"To justify the claim that the definition given above reflects the intuitive meaning given earlier, we mention that $\\phi ^{\\prime } p {V} (\\phi ^{\\prime } p) (\\psi ^{\\prime } )$ only if, for all $V$ -evaluation maps $\\sigma $ , there exists a $V$ -evaluation map $\\sigma ^{\\prime }$ such that ${\\sigma }(\\phi ^{\\prime } p) {\\sigma }(\\phi ^{\\prime } p) {\\sigma ^{\\prime }}(\\psi ^{\\prime } )$ .",
"Notice that, using the unary guard operator mentioned in Section , we can write ${\\phi ^{\\prime }} p {V}{\\phi ^{\\prime }} p {\\psi ^{\\prime }}$ instead of $\\phi ^{\\prime } p {V} (\\phi ^{\\prime } p) (\\psi ^{\\prime } )$ .",
"Below, we will present the axioms and rules of .",
"In addition to axioms and rules that concern a particular constant or operator of , there is a rule concerning auxiliary flexible variables and a rule for precondition strengthening and/or postcondition weakening.",
"We use some special terminology and notations with respect to auxiliary variables.",
"Let $p \\in $ , and let $A \\subseteq (p)$ .",
"Then $A$ is a set of auxiliary variables of $p$ if each flexible variable in $A$ occurs in $p$ only in subterms of the form ${v}{e}$ with $v \\in A$ .",
"We write $(p)$ , where $p \\in $ , for the set of all sets of auxiliary variables of $p$ .",
"Moreover, we write $p_A$ , where $p \\in $ and $A \\in (p)$ , for $p$ with all occurrences of subterms of the form ${v}{e}$ with $v \\in A$ replaced by $$ .",
"The axioms and rules of are given in Table REF .",
"Table: Axioms and rules ofIn this table, $p$ and $q$ stand for arbitrary terms from $$ , $\\phi $ , $\\psi $ , $\\chi $ , $\\phi ^{\\prime }$ , and $\\psi ^{\\prime }$ stand for arbitrary terms from $$ , $a$ stands for an arbitrary basic action from $$ , $v$ stands for an arbitrary flexible variable from $$ , and $e$ and $e_1,e_2,\\ldots $ stand for arbitrary terms from $$ .",
"The parallel composition rule may only be applied if the premises are disjoint.",
"Premises ${\\phi }{p}{\\psi }$ and ${\\phi ^{\\prime }}{q}{\\psi ^{\\prime }}$ are disjoint if $(p) (q) = \\emptyset $ , $(p) (\\phi ^{\\prime }) = \\emptyset $ , and $(p) (\\psi ^{\\prime }) = \\emptyset $ ; $(q) (p) = \\emptyset $ , $(q) (\\phi ) = \\emptyset $ , and $(q) (\\psi ) = \\emptyset $ .",
"In the consequence rule, the first premise and the last premise are not asserted processes.",
"They assert that $\\phi \\phi ^{\\prime } = $ and $\\psi ^{\\prime } \\psi = $ are derivable from the axioms of .",
"Before we move on to the soundness of the axioms and rules of , we consider two congruence related properties of the equivalences ${V}$ that are relevant to the soundness proof.",
"[Congruence] For all finite $V \\subseteq $ , for all terms $p,q,p^{\\prime },q^{\\prime } \\in _V$ , $p {V} p^{\\prime }$ and $q {V} q^{\\prime }$ only if $p q {V} p^{\\prime } q^{\\prime }$ , $p q {V} p^{\\prime } q^{\\prime }$ , and $p q {V} p^{\\prime } q^{\\prime }$ .",
"Moreover, for all finite $V \\subseteq $ , for all terms $p,p^{\\prime } \\in _V$ and all terms $\\phi \\in $ with $(\\phi ) \\subseteq V$ , $p {V} p^{\\prime }$ only if $\\phi p {V} \\phi p^{\\prime }$ and ${H}(p) {V} {H}(p^{\\prime })$ .",
"Assume $p {V} p^{\\prime }$ and $q {V} q^{\\prime }$ .",
"Then $p q {V} p^{\\prime } q^{\\prime }$ follows immediately and $p q {V} p^{\\prime } q^{\\prime }$ and $p q {V} p^{\\prime } q^{\\prime }$ follow easily by induction on the number of proper subprocesses of $p$ , where use is made of Lemma .",
"Assume $p {V} p^{\\prime }$ .",
"Then $\\phi p {V} \\phi p^{\\prime }$ follows immediately and ${H}(p) {V} {H}(p^{\\prime })$ follows easily by induction on the number of proper subprocesses of $p$ , where use is made of Lemma .",
"$\\Box $ [Limited Congruence] For all finite $V \\subseteq $ , for all terms $p,q,p^{\\prime },q^{\\prime } \\in _V$ with $(p) (q) = \\emptyset $ , $(q) (p) = \\emptyset $ , $(p^{\\prime }) (q^{\\prime }) = \\emptyset $ , and $(q^{\\prime }) (p^{\\prime }) = \\emptyset $ , $p {V} p^{\\prime }$ and $q {V} q^{\\prime }$ only if $p q {V} p^{\\prime } q^{\\prime }$ .",
"Assume $(p) (q) = \\emptyset $ and $(q) (p) = \\emptyset $ , $(p^{\\prime }) \\linebreak [2] (q^{\\prime }) = \\emptyset $ and $(q^{\\prime }) (p^{\\prime }) = \\emptyset $ , $p {V} p^{\\prime }$ and $q {V} q^{\\prime }$ .",
"Then $p q {V} p^{\\prime } q^{\\prime }$ follows easily by induction on the number of proper subprocesses of $p$ , where use is made of Lemma .",
"$\\Box $ [Soundness] For all terms $p \\in $ , for all terms $\\phi ,\\psi \\in $ , the asserted process ${\\phi }{p}{\\psi }$ is derivable from the axioms and rules of only if ${\\phi }{p}{\\psi }$ is true.",
"We will assume that $\\phi ,\\psi \\in $ .",
"We can do so without loss of generality because, by the definition of the truth of asserted processes, it is sufficient to consider arbitrary closed substitution instances of $\\phi $ and $\\psi $ if $\\phi ,\\psi \\notin $ .",
"We will prove the theorem by proving that each of the axioms is true and each of the rules is such that only true conclusions can be drawn from true premises.",
"The theorem then follows by induction on the length of the proof.",
"The proofs for the axioms and the consequence rule are trivial.",
"Theorems and facilitate the proofs for the other rules.",
"By these theorems, the proofs for the alternative composition rule, the sequential composition rule, and the guarded command rule are also trivial and the proofs for the parallel composition rule, the encapsulation rule, and the auxiliary variables rule are straightforward proofs by induction on the number of proper subprocesses, in which use is made of Lemma .",
"The parallel composition rule is proved simultaneously with similar rules for the left merge operator and the communication merge operator.",
"The proof for the iteration rule goes in a less straightforward way.",
"In case of the iteration rule, we assume that (1) for all $V$ -evaluation maps $\\sigma $ , ${\\sigma }(\\phi p) ={\\sigma }((\\phi p) (\\phi ))$ is derivable; (2) for all $V$ -evaluation maps $\\sigma $ , ${\\sigma }(\\phi q) ={\\sigma }((\\phi q) (\\psi ))$ is derivable; and we prove that (3) for all $V$ -evaluation maps $\\sigma $ , ${\\sigma }(\\phi (p q)) ={\\sigma }((\\phi (p q)) (\\psi ))$ is derivable; where $V = (\\phi ) (p q) (\\psi )$ .",
"We do so by induction on the number of proper subprocesses of ${\\sigma }(\\phi (p q))$ .",
"The basis step is trivial.",
"The inductive step is proved in the following way.",
"It follows easily from assumption (1), making use of BKS1, that (4) for all $V$ -evaluation maps $\\sigma $ , for some evaluation map $\\sigma ^{\\prime }$ , ${\\sigma }(\\phi (p q)) ={\\sigma }(\\phi p) {\\sigma ^{\\prime }}(\\phi (p q)) {\\sigma }(\\phi q)$ is derivable.",
"We distinguish two cases: $\\sigma \\ne \\sigma ^{\\prime }$ and $\\sigma = \\sigma ^{\\prime }$ .",
"In the case where $\\sigma \\ne \\sigma ^{\\prime }$ , (3) follows easily from (4), the induction hypothesis, and assumption (2), making use of BKS1.",
"In the case where $\\sigma = \\sigma ^{\\prime }$ , it follows immediately from (4), making use of RSP*, that (5) for all $V$ -evaluation maps $\\sigma $ , ${\\sigma }(\\phi (p q)) ={\\sigma }((\\phi p) (\\phi q))$ is derivable; and (3) follows easily from (5) and assumption (2), making use of BKS1.",
"$\\Box $ In the proof of Theorem , RSP* is used in the part concerning the iteration rule.",
"We do not see how that part of the proof can be done if RSP* is replaced by BKS2–BKS5.",
"The following is a corollary of the definition of the truth of asserted processes and Theorem .",
"For all terms $p, p^{\\prime } \\in $ , for all terms $\\phi ,\\psi \\in $ , the asserted process ${\\phi }{p}{\\psi }$ is derivable from the axioms and rules of and $p = p^{\\prime }$ is derivable from the axioms of only if ${\\phi }{p^{\\prime }}{\\psi }$ is true.",
"If it is possible at all, equational reasoning with the axioms of a process algebra about how data change in the course of a process is often rather cumbersome.",
"In many cases, but not all, reasoning with the axioms and rules of a Hoare logic is much more convenient.",
"We have not strived for a Hoare logic that covers the cases where it does not simplify reasoning.",
"Actually, the axioms and rules of are not complete (in the sense of Cook [13]).",
"The side condition of the parallel composition rule precludes completeness.",
"We have, for example, that the asserted process ${i = 0}{{i}{i+1} {i}{i+1} {i}{0}}{i = 0 i = 1 i = 2}$ is true, but this cannot be derived by means of the axioms and rules of alone because a premise of the form ${\\phi }{{i}{i+1} {i}{i+1}}{\\psi }$ and a premise of the form ${\\phi ^{\\prime }}{{i}{0}}{\\psi ^{\\prime }}$ are never disjoint.",
"We could have replaced the disjointness side condition by an interference-freedom side condition to cover cases such as the example given above and perhaps this would lead to completeness.",
"However, unless the disjointness side condition would suffice, fulfillment of the interference-freedom side condition generally needs a sophisticated proof.",
"These interference-freedom proofs partly outweigh the advantage of using a Hoare logic for reasoning about how data change in the course of a process.",
"As will be shown by means of an example in Section , equational reasoning with the axioms of offers an alternative without interference-freedom proofs.",
"That is why we have chosen for the parallel composition rule with the disjointness side condition."
],
[
"On the Connection between the Hoare Logic and ",
"In this section, we go into the connection of with by way of the equivalence relations ${V}$ .",
"Let ${\\phi }{p}{\\psi }$ be an asserted process, and let $V = (\\phi ) (p) (\\psi )$ .",
"Suppose that ${\\phi }{p}{\\psi }$ has been derived from the axioms and rules of .",
"Then, by Theorem , ${\\phi }{p}{\\psi }$ is true.",
"This means that, for all closed substitution instances ${\\phi ^{\\prime }}{p}{\\psi ^{\\prime }}$ of ${\\phi }{p}{\\psi }$ , $\\phi ^{\\prime } p {V} (\\phi ^{\\prime } p) (\\psi ^{\\prime } )$ .",
"In other words, for all closed substitution instances ${\\phi ^{\\prime }}{p}{\\psi ^{\\prime }}$ of ${\\phi }{p}{\\psi }$ , for all $V$ -evaluation maps $\\sigma $ , ${\\sigma }(\\phi ^{\\prime } p) ={\\sigma }(\\phi ^{\\prime } p) {\\sigma ^{\\prime }}(\\psi ^{\\prime } )$ is derivable from the axioms of .",
"Thus, the derivation of ${\\phi }{p}{\\psi }$ from the axioms and rules of has made a collection of equations available that can be considered to be derived by equational reasoning from the axioms of .",
"Let us have a closer look at the equivalence relation ${V}$ on $_V$ .",
"Clearly, this equivalence relation is useful when reasoning about processes in which data are involved.",
"However, it is plain from the proof of Theorem that ${V}$ is not a congruence relation on $_V$ .",
"This complicates the use of equational reasoning to derive, among other things, the collection of equations referred to above considerably.",
"The presented Hoare logic can be considered to be a means to get partially round the complications concerned.",
"Dissociated from its connection with , ${V}$ remains an interesting equivalence relation on $_V$ when it comes to reasoning about processes in which data is involved.",
"Therefore, we mention below a result on this equivalence relation which is a corollary of results from Section used to prove the soundness of .",
"The fact that ${V}$ is not a congruence relation on $_V$ , and consequently that ${V}$ is not preserved by all contexts, makes this corollary to the point.",
"In order to formulate the corollary, we first define a set of contexts, using $\\Box $ as a placeholder.",
"For each finite $V \\subseteq $ , the set ${V}$ of sequential evaluation supporting contexts for $V$ is the set $_{W \\subseteq V} {V,W}$ , where the sets ${V,W}$ , for finite $V,W \\subseteq $ with $W \\subseteq V$ , are defined by simultaneous induction as follows: $\\Box \\in {V,W}$ ; if $p \\in $ , $C \\in {V,W}$ , $(p) \\subseteq V$ , and $(p) \\subseteq W$ , then $p C,\\; C p,\\linebreak [2]p C,\\; C p,\\; p C,\\; C p \\in {V,W}$ ; if $\\phi \\in $ and $C \\in {V,W}$ , $(\\phi ) \\subseteq V$ , then $\\phi C \\in {V,W}$ ; if $p \\in $ , $C \\in {V,W}$ , $(p) V = \\emptyset $ , and $(p) W = \\emptyset $ , then $p C,\\;C p \\in {V (p),W (p)}$ ; if $H \\subseteq $ and $C \\in {V,W}$ , then ${H}(C) \\in {V,W}$ .",
"We write $C[p]$ , where $C \\in {V}$ and $p \\in $ , for $C$ with the occurrence of $\\Box $ replaced by $p$ .",
"The following is a corollary of Theorems and .",
"Let $V$ be a finite subset of $$ .",
"Then, for all $p,p^{\\prime } \\in _V$ , for all $C \\in {V}$ , $p {V} p^{\\prime }$ only if $C[p] {V} C[p^{\\prime }]$ .",
"Of course, Corollary can be applied to results from using .",
"Let ${\\phi }{p}{\\psi }$ be an asserted process, let $V = (\\phi ) (p) (\\psi )$ , and let $C \\in {V}$ .",
"Suppose that ${\\phi }{p}{\\psi }$ has been derived from the axioms and rules of .",
"Then, for all closed substitution instances ${\\phi ^{\\prime }}{p}{\\psi ^{\\prime }}$ of ${\\phi }{p}{\\psi }$ , we have that $C[\\phi ^{\\prime } p] {V} C[(\\phi ^{\\prime } p) (\\psi ^{\\prime } )]$ ."
],
[
"On the Role of the Hoare Logic for ",
"Process algebras focus on the main role of a reactive system, namely maintaining some ongoing interaction with its environment.",
"Hoare logics focus on the main role of a transformational system, namely producing, without interruption by its environment, outputs from inputs.The terms reactive system and transformational system were coined in [16].",
"However, actual systems are often reactive systems composed of reactive components and transformational components.",
"provides a setting for equational reasoning about the behaviour of such systems, but it does not offer by itself the possibility to reason in Hoare-logic style about the behaviour of the transformational components.",
"Below, we will take the behaviour of a very simple transformational component and reason about how it changes data both in Hoare-logic style with the axioms and rules of and equationally with the axioms of .",
"We assume that $$ is the group of integers.",
"We also assume that $i$ and $j$ are flexible variables from $$ and $n$ and $n^{\\prime }$ are variables of sort $$ .",
"Moreover, we use $e - e^{\\prime }$ as an abbreviation of $e + (-e^{\\prime })$ .",
"The behaviour of the very simple transformational component concerned is described by the closed term ${i}{i + j} {j}{i - j} {i}{i - j}$ .",
"We begin with showing by means of that this behaviour swaps the values of $i$ and $j$ .",
"We derive i = n j = n'ii + ji = n + n' j = n' using the assignment axiom and the consequence rule.",
"Similarly, we derive i = n + n' j = n'ji - j i = n + n' j = n and i = n + n' j = nii - ji = n' j = n. From these three asserted processes, we derive i = n j = n' ii + j ji - j ii - j i = n' j = n using the sequential composition rule twice.",
"We continue with showing the same by means of .",
"This means that we have to derive from the axioms of , for all $e,e^{\\prime } \\in $ , for all ${i,j}$ -evaluation maps $\\sigma $ : (*) [c]@l@ ((i = e j = e') ii + j ji - j ii - j) = ((i = e j = e') = ( ii + j ji - j ii - j (i = e' j = e) ).",
"We derive ((i = e j = e') ii + j) = (i = e j = e') i(i + j) using axioms V3 and V5; and ((i = e j = e') ii + j (i = e + e' j = e') ) = (i = e j = e') = i(i + j) (e + e')i(i = e + e' j = e') using axioms V0, V3, and V5.",
"We can derive the following equation for all ${i,j}$ -evaluation maps $\\sigma $ : (**) [c]@l@ (i = e j = e') i(i + j) = (i = e j = e') = i(i + j) (e + e')i(i = e + e' j = e') .",
"In the case $\\sigma (i) = e$ and $\\sigma (j) = e^{\\prime }$ , we derive $\\sigma {\\sigma (e + e^{\\prime })}{i}(i = e + e^{\\prime } j = e^{\\prime }) = $ using IMP2.",
"From this, we derive equation (**) using axioms GC1 and A8.",
"In the case $\\sigma (i) \\ne e$ or $\\sigma (j) \\ne e^{\\prime }$ , we derive $\\sigma {\\sigma (e + e^{\\prime })}{i}(i = e + e^{\\prime } j = e^{\\prime }) = $ using IMP2.",
"From this, we derive equation (**) using axiom GC2.",
"Hence, we have for all ${i,j}$ -evaluation maps $\\sigma $ : ((i = e j = e') ii + j) = ((i = e j = e') ii + j (i = e + e' j = e') ).",
"Similarly, we find for all ${i,j}$ -evaluation maps $\\sigma $ : ((i = e + e' j = e') ji - j) = ((i = e + e' j = e') ji - j (i = e + e' j = e) ) and ((i = e + e' j = e) ii - j) = ((i = e + e' j = e) ii - j (i = e' j = e) ).",
"From the last three equations, we derive equation (*) using axioms A5, A9, GC5, V3, and V5.",
"By this we have finally shown by means of that the values of $i$ and $j$ are swapped by the process described by ${i}{i + j} {j}{i - j} {i}{i - j}$ .",
"In this case, it is clear that Hoare-logic style reasoning with the axioms and rules of is much more convenient than equational reasoning with the axioms of .",
"Because a single application of a rule of cannot be justified by a single application of an axiom of , we expect that this also holds for virtually all other cases of reasoning about how the behaviour of a transformational system changes data.",
"Now, we turn our attention to the rather restrictive side condition of the parallel composition rule of .",
"As mentioned before at the end of Section , we have that the asserted process i = 0ii+1 ii+1 i0 i = 0 i = 1 i = 2 is true, but this cannot be derived by means of the axioms and rules of alone because a premise of the form ${\\phi }{{i}{i+1} {i}{i+1}}{\\psi }$ and a premise of the form ${\\phi ^{\\prime }}{{i}{0}}{\\psi ^{\\prime }}$ are never disjoint.",
"However, we can derive the following equation from the axioms of : ii+1 ii+1 i0 = ii+1 (ii+1 i0 i0 ii+1) = i0 ii+1 ii+1.",
"By Corollary , it is sound to replace in the above asserted process the left-hand side of this equation by the right-hand side of this equation.",
"This yields the asserted process {i = 0} {ii+1 (ii+1 i0 i0 ii+1) i0 ii+1 ii+1} {i = 0 i = 1 i = 2}, which can be derived using the assignment axiom, the alternative composition rule, the sequential composition rule, and the consequence rule of several times.",
"If the disjointness side condition of the parallel composition rule of is replaced by an interference-freedom side condition, like in [24], then the original asserted process becomes derivable using the axioms and rules of the Hoare logic alone (see e.g. [1]).",
"The interference-freedom proof involved needs proof outlines (see [24]) for ${i = 0}{{i}{i+1} {i}{i+1}}{}$ and ${}{{i}{0}}{i = 0 i = 1 i = 2}$ .",
"In this very simple case, the interference-freedom proof already amounts to seven interference-freedom checks.",
"However, for two processes in which $k$ and $k^{\\prime }$ assignment actions occur, the number of interference-freedom checks is at least $2 k k^{\\prime } + k + k^{\\prime }$ .",
"Therefore, we expect that interference-freedom proofs partly outweigh the advantage of using a Hoare logic."
],
[
"Related Work",
"The approach to the formal verification of programs that is now known as Hoare logic was proposed in [17].",
"The illustration of this approach was at the time confined to the very simple deterministic sequential programs that are mostly referred to as while programs (cf. [1]).",
"The axioms, the sequential composition rule, the iteration rule, the guarded command rule, and the consequence rule from our Hoare logic savour strongly of the common rules for while programs.",
"The alternative composition rule is the or rule due to [21], the parallel composition rule was proposed in [18], and the auxiliary variables rule was first introduced in [24].",
"The parallel composition rules proposed in [2], [22], [24] are more complicated than our parallel composition rule.",
"In the case of [2], [22], the intention was to provide a Hoare logic for the first design of CSP [19].",
"In that design, one program may force another program to assign a data value sent by the former program to a program variable used by the latter program.",
"This feature complicates the parallel composition rule considerably.",
"Moreover, incorporating this feature in an ACP-like process algebra would lead to the situation that, in equational reasoning, certain axioms may not be applied in contexts of parallel processes (like in [15], see below).",
"Because our concern is in the use of a Hoare logic as a complement to pure equational reasoning, we have not considered incorporating this feature.",
"In the case of [24], the rule is more complicated because, in the parallel programs covered, program variables may be shared variables, i.e.",
"program variables that are assigned to in one program may be used in another program.",
"Our process algebra also covers shared variables.",
"However, covering shared variables in our Hoare logic as well would mean that the simple disjointness proof required by our parallel composition rule has to be replaced a sophisticated interference-freedom proof.",
"We believe that this would diminish the usefulness of our Hoare logic as a complement to equational reasoning considerably.",
"Therefore, we have not considered covering shared variables in the parallel composition rule.",
"In [15], an extension of ACP with the empty process constant and the unary counterpart of the binary guarded command operator is presented, the truth of an asserted sequential process is defined in terms of the transition relations from the given structural operational semantics of the presented extension of ACP, and it is shown that an asserted sequential process ${\\phi }{p}{\\psi }$ is true according to that definition iff ${\\phi } p ^{\\prime } {\\phi } p {\\psi }$ , where $^{\\prime }$ is bisimulation equivalence as defined in [15] for sequential processes.",
"Moreover, a Hoare logic of sequential asserted processes is presented and its soundness is shown.",
"However, [15] does not go into the use of that Hoare logic as a complement to pure equational reasoning from the equational axioms.",
"Regarding the bisimulation equivalence $^{\\prime }$ defined in [15] for sequential processes, we can mention that, if the data-states are evaluation maps, $p ^{\\prime } q$ iff ${\\sigma }(p) {\\sigma }(q)$ for all $V$ -evaluation maps $\\sigma $ , where $V = (p) (q)$ .",
"Due to the possibility of interference between parallel processes, a different bisimulation equivalence $^{\\prime \\prime }$ , finer than $^{\\prime }$ , is needed in [15] for parallel processes.",
"As a consequence, in equational reasoning, certain axioms may not be applied in contexts of parallel processes.",
"Moreover, $$ together with the operators ${\\sigma }$ allows of dealing with local data-states, whereas the combination of $^{\\prime }$ and $^{\\prime \\prime }$ does not allow of dealing with local data-states."
],
[
"Concluding Remarks",
"We have taken an extension of with features that are relevant to processes in which data are involved, devised a Hoare logic of asserted processes based on this extension of , and gone into the use of this Hoare logic as a complement to pure equational reasoning from the axioms of the extension of .",
"We have defined what it means that an asserted process is true in terms of an equivalence relation (${V}$ ) that had been found to be central to relating the extension of and the Hoare logic.",
"That this equivalence relation is not a congruence relation with respect to parallel composition is related to the fact that in the extension of presented in [15] certain axioms may not be applied in contexts of parallel processes.",
"In this paper, we build on earlier work on ACP.",
"The axioms of have been taken from [7], the axioms for the iteration operator have been taken from [9], and the axioms for the guarded command operator have been taken from [4].",
"The evaluation operators have been inspired by [11] and the data parameterized action operator has been inspired by [12]."
]
] | 1906.04491 |
[
[
"The effective magnetic field decay of radio pulsars: insights from the\n statistical properties of their spin frequency's second derivatives"
],
[
"Abstract We present a new method to investigate the effective magnetic field decay of isolated neutron stars, from the analysis of the long-term timing data of a large sample of radio pulsars \\citep{2010MNRAS.402.1027H}.",
"There are some differences between the distributions of frequency's second derivatives of the pulsar spins with different effective field decay timescales.",
"Kolmogorov-Smirnov tests are performed to reexamine the consistency of distributions of the simulated and reported data for a series values of decay timescales.",
"\\textbf{We show that the timescale of the effective field decay exceeds $\\sim 5~\\rm {Myr}$ for pulsars with spin-down age $\\tau_{\\rm C}< 10^{7}~{\\rm yr}$ or $\\sim 100~\\rm {Myr}$ for pulsars with $10^{7}<\\tau_{\\rm C}< 10^{9}~{\\rm yr}$ in the sample.",
"The result does not depend on any specific theories of the field evolution, the inclination decay or the variation in the moment of inertia.",
"It is also found that the extent of the closed line region of the magnetic field is close to the light cylinder $r_{\\rm lc}$, i.e., the corotating radius $r_{\\rm c}\\approx r_{\\rm lc}$ is a good approximation for the observed pulsar population."
],
[
"Introduction",
"The magnetic field is probably one of the most important physical quantities affecting the evolution and the observational behaviours of radio pulsars.",
"The field strength determines the loss rate of rotational energy, the luminosity of pulses, and thus the spin evolution and observability of a pulsar.",
"The primary method used to determine the magnetic field is by measuring each pulsar's spin parameters, which have actually provided a surprising amount of information on the nature of the pulsed radio sources.",
"As such, the knowledge of the spin parameters is particularly valuable in elucidating whether magnetic field decay occurs in isolated neutron stars.",
"Many impressive studies have been done on this issue during the past few decades [43], [22], [39], [40], [59], [42], [2], [25], [37], [24], [7], [60], [61], [15], [10], [32], [33].",
"Unfortunately, the conclusions of pulsar population investigations have often been conflicting (See e.g.",
"[23], [55], [38] for reviews).",
"The lack of conclusive evidence on magnetic field decay is mainly due to the fact that the true age of a pulsar is unavailable, and the characteristic (spin-down) age $\\tau _{\\rm C}$ is normally significantly different from its true age [67].",
"This makes the evolution of the magnetic field remaining as one of the most important unresolved issues of the physics of neutron stars.",
"[29] studied the timing noise in the residuals of 366 pulsars that had been regularly observed for 10 to 36 years, and showed that the magnitude of the frequency's second derivatives of the pulsar spins, i.e.",
"$|\\ddot{\\nu }|$ , is much larger than that caused by magnetic braking of the neutron star, and the numbers of negative and positive $\\ddot{\\nu }$ are almost equal in the sample.",
"It had also been noticed that the distributions between the positive and negative signs in $\\ddot{\\nu }-\\tau _{\\rm C}$ diagram show an approximate symmetry.",
"In this paper we present a new method to investigate the magnetic field decay of radio pulsars, from the analysis of the long-term timing data.",
"We find that the decay timescale can be constrained by measuring the difference between the distributions of $\\ddot{\\nu }$ with different decay timescales in $\\ddot{\\nu }-\\tau _{\\rm C}$ diagram.",
"This method can effectively avoid the “true age problem\".",
"The method and its validity are described in section 2, the revealed restrictions on the timescale of effective magnetic field decay are shown in section 3, and the magnetospheric effects are also tested in the section.",
"The physical implications of the decay timescales are discussed in section 4, and the results are summarized and discussed in section 5.",
"A basic model for a pulsar's spin-down is the magnetic dipole radiation model [44], [43], [22], [56].",
"The standard dipole (SD) radiation model assumes that the pure magnetic dipole radiation in vacuum as the braking mechanism, i.e.",
"$\\dot{\\nu }=-K \\nu ^3,$ where $\\nu $ and $\\dot{\\nu }$ are the spin frequency and its first derivative, respectively.",
"The parameter $K=8\\pi ^2R^6 B^2\\sin ^2 \\theta /3c^3I$ is a constant, $B=3.2\\times 10^{19}\\sqrt{-\\dot{\\nu }/\\nu ^3}/\\sin \\theta $ is the effective dipole magnetic field at equator, $R~(\\simeq 10^6~{\\rm cm})$ and $I~(\\simeq 10^{45}~{\\rm g~cm^2})$ are the radius and moment of inertia, respectively.",
"[66] improved the model by incorporating the effect of a longitudinal current outflow (relativistic particle wind) powered by a unipolar generator into the rotation energy-loss rate.",
"This effect was confirmed by a few intermittent pulsars, whose rotation slows down faster when the pulsar is on than when it is off [35].",
"Further, [57] developed a numerical method for evolving time dependent force-free MHD equations and applied it to solving a dynamic pulsar magnetosphere.",
"Similarly, the dynamic magnetosphere (DM) model also included both the dipole radiation and the unipolar generator mechanisms to contribute the total braking torque of an oblique pulsar, and they found a formula that gives a very good fit to the oblique spin-down for all inclinations.",
"The formula have the same form with Eq.",
"(REF ) but with a different parameter $K$ , $K=\\frac{4\\pi ^2 B^2 R^6}{c^3I}(1+\\sin ^2 \\theta ).$ The inferred effective magnetic field at the magnetic equator is then $B=2.6\\times 10^{19}\\sqrt{-\\dot{\\nu }/\\nu ^3}(1+\\sin ^2 \\theta )^{-1/2}$ , which can be up to $1.7$ times smaller than the estimate from the SD formula."
],
[
"The Frequency's Second Derivatives and The Revised Spin-down Model",
"Both the SD and the DM model predict that the frequency second derivative of a pulsar spin $\\ddot{\\nu }_{\\rm SD}=\\ddot{\\nu }_{\\rm DM}=3\\dot{\\nu }^2/\\nu >0$ .",
"However, it is widely known that the observed $\\ddot{\\nu }$ for the majority of pulsars cannot be explained by these model with a constant field strength $B$ .",
"Particularly, the recent large-sample analysis showed [29], [68], [69] that $|\\ddot{\\nu }|\\gg \\ddot{\\nu }_{\\rm SD}$ , $\\ddot{\\nu }^{+}(\\tau _{\\rm C})\\approx -\\ddot{\\nu }^{-}(\\tau _{\\rm C})$ and $N(\\ddot{\\nu }^{+})\\approx N(\\ddot{\\nu }^{-})$ , where $N$ indicates the total number, the superscripts `+' and `-' indicate positive and negative signs of $\\ddot{\\nu }$ and $\\tau _{\\rm C}\\equiv -\\nu /2\\dot{\\nu }$ is the characteristic age of a pulsar.",
"All the pulsars in the sample are shown in the $|\\ddot{\\nu }|-\\tau _{\\rm c}$ diagram in panel (1) of Fig.",
"REF , in which 193 pulsars have $\\ddot{\\nu }>0$ and the remainder 173 pulsars have $\\ddot{\\nu }<0$ .",
"Following [3], we re-formulate the braking law of a pulsar as $\\dot{\\nu }=-K(t)\\nu ^3$ , which assumes that the SD or DM model is responsible for the instantaneous spin-down of a pulsar, but $K(t)$ is time-dependent.",
"Generically and without depending-upon any specific model for the time-dependence, $K(t)$ can be decomposed into a long-term monotonic term plus a short-term perturbation to the monotonic term.",
"Again assuming $R$ and $I$ are constants, the decomposition is equivalent to $B(t)=B_{\\rm M}(t)+B_{\\rm O}(t)$ , where $B_{\\rm M}(t)$ is the long-term monotonic component and $B_{\\rm O}(t)$ is the short-term perturbation around $B_{\\rm M}$ .",
"The quasi-periodic oscillation structures, which is widespread in pulsar timing behaviours [28], [29], can be phenomenologically described by $B_{\\rm O}(t)$ [70], [65].",
"One possible source of the perturbation may be their magnetospheric activities, which are known to influence their timing behaviours significantly [41].",
"Then after some simple algebra, we get $\\ddot{\\nu }&=&3\\dot{\\nu }^2/\\nu +2\\dot{\\nu }\\dot{B}_{\\rm M}/B_{\\rm M}+2\\dot{\\nu }\\dot{B}_{\\rm O}/B_{\\rm M}\\\\ \\nonumber &=&\\ddot{\\nu }_{\\rm SD}+\\ddot{\\nu }_{\\rm M}+\\ddot{\\nu }_{\\rm O}$ where $\\ddot{\\nu }_{\\rm M}=2\\dot{\\nu }\\dot{B}_{\\rm M}/B_{\\rm M}$ and $\\ddot{\\nu }_{\\rm O}=2\\dot{\\nu }\\dot{B}_{\\rm O}/B_{\\rm M}$ .",
"For relatively old pulsars without significant glitch activities, $\\dot{\\nu }<0$ and $\\dot{B}_{\\rm M}\\leqslant 0$ , therefore $\\ddot{\\nu }_{\\rm M}\\geqslant 0$ .",
"Given the case of a quasi-periodic oscillation, $\\ddot{\\nu }_{\\rm O}$ has a positive or a negative value with almost equal chances [68], [69].",
"Observationally, since generically and statistically $\\ddot{\\nu }^{+}(\\tau _{\\rm C})\\approx -\\ddot{\\nu }^{-}(\\tau _{\\rm C})$ and $N(\\ddot{\\nu }^{+})\\approx N(\\ddot{\\nu }^{-})$ for large number of pulsars, clearly $\\ddot{\\nu }_{\\rm O}$ in Eq.",
"(REF ) dominates the observed statistical properties of $\\ddot{\\nu }$ .",
"On the other hand, $\\ddot{\\nu }_{\\rm SD}>0$ and $\\ddot{\\nu }_{\\rm M}>0$ will cause some differences on the distributions of $\\ddot{\\nu }$ with different decay timescales, and some asymmetry between the observed $\\ddot{\\nu }^{+}$ and $\\ddot{\\nu }^{-}$ .",
"These effects might in turn provide clues of long-term magnetic field decay of pulsars, since $\\ddot{\\nu }_{\\rm SD}$ can be calculated from the observed $\\nu $ and $\\dot{\\nu }$ ."
],
[
"Monte-Carlo Simulations",
"We assume that the magnetic fields of pulsars in the sample have a typical decay timescale.",
"Define the timescale as $\\tau _{B}\\equiv -B_{\\rm M}/\\dot{B}_{\\rm M}$ , we have $\\ddot{\\nu }_{\\rm M}=-2\\dot{\\nu }/\\tau _{\\rm B}$ and $\\ddot{\\nu }=\\ddot{\\nu }_{\\rm O}+\\ddot{\\nu }_{\\rm SD}-2\\dot{\\nu }/\\tau _{\\rm B}.$ We can simulate the distributions of $\\ddot{\\nu }$ with different $\\tau _{B}$ , as described below.",
"Our strategy is then to search for the typical $\\tau _{B}$ that can maximize the p-value of the Kolmogorov-Smirnov test against the hypothesis that the reported distribution is the same as the simulated distribution in $|\\ddot{\\nu }|-\\tau _{\\rm C}$ diagram for pulsars in the sample.",
"We construct a phenomenological model for the dipole magnetic field evolution of pulsars with a long-term decay modulated by short-term oscillations, $B(t)=B_d(t)(1+\\sum k\\sin (\\phi +2\\pi \\frac{t}{T})),$ where $t$ is the pulsar's age, and $k$ , $\\phi $ , $T$ are the amplitude, phase and period of the oscillation,respectively.",
"$B_d(t)=B_0 \\exp (-t/\\tau _{\\rm B})$ , in which $B_0$ is the field strength at the age $t_0$ .",
"Substituting Equation (REF ) into Equation (REF ), we get the differential equation describing the the spin frequency evolution of a pulsar as follows $\\dot{\\nu }=-A B(t)^2 \\nu ^{3},$ in which $A=\\frac{8\\pi ^2R^6\\sin \\theta ^2}{3c^3I}$ is a constant, $R~(\\simeq 10^6~{\\rm cm})$ , $I~(\\simeq 10^{45}~{\\rm g~cm^2})$ , and $\\theta ~(\\simeq \\pi /2)$ is the radius, moment of inertia, and angle of magnetic inclination of the neutron star, respectively.",
"The constant $A$ and $B(t)$ are actually inseparable in Equation (REF ), which means that the decay timescales of the effective magnetic fields can be attributed not only to the magnetic field evolution, by also the changes of the inclination angle, or the small changes in the moment of inertia.",
"In order to model the $\\ddot{\\nu }$ distributions in $\\tau _{\\rm C}- |\\ddot{\\nu }|$ diagram, we first obtain $\\nu (t)$ by integrating the pulsar spin-down law described as Equation (REF ), and the phase $\\Phi (t)=\\int _{t_0}^{t}\\nu (t^{\\prime }){\\rm d}t^{\\prime }.$ Then, these observable quantities, $\\nu $ , $\\dot{\\nu }$ and $\\ddot{\\nu }$ can be obtained by fitting the phases to the third order of its Taylor expansion over a time span $T_{\\rm s}$ , $\\Phi (t_i) =\\Phi _0 + \\nu (t_i-t_0) + \\frac{1}{2}\\dot{\\nu }(t_i-t_0)^2+ \\frac{1}{6}\\ddot{\\nu }(t_i-t_0)^3.$ We thus get $\\nu $ , $\\dot{\\nu }$ and $\\ddot{\\nu }$ from fitting to Equation (REF ), with a certain time interval of phases $\\Delta T_{\\rm int}=10^6~{\\rm s}$ .",
"We assume that $k=10^{-4.6}\\sim 10^{-1.9}$ and $T$ follows a uniform random distribution in the range from $0.1\\sim 10 ~{\\rm yr}$ , and the reasons will be shown in the next subsection.",
"It is also assumed that the sample of the phase $\\phi $ of the field oscillations uniformly distributed in the range from 0 to $2\\pi $ .",
"Drawing randomly a data set $\\lbrace \\nu , \\dot{\\nu }, T_{\\rm s}\\rbrace $ from the reported sample space (i.e.",
"from Table 1 of [29]), and calculating a corresponding start time $t_0$ , we can obtain a rotation phase set $\\lbrace \\Phi (t_i)\\rbrace $ using Equation (REF ).",
"Then the values of $\\nu $ , $\\dot{\\nu }$ and $\\ddot{\\nu }$ can be obtained by fitting $\\lbrace \\Phi (t_i)\\rbrace $ to Equation (REF ).",
"Hence one has each $\\lbrace \\tau _{\\rm c},|\\ddot{\\nu }| \\rbrace $ .",
"Repeat this procedure for $N$ times, we will have $N$ ($=283$ ) data points in the $|\\ddot{\\nu }|$ -$\\tau _{\\rm c}$ diagram.",
"We exclude all the pulsars which have glitch records from the sample, since a small variation in the moment of inertial due to glitches may have an impact on the overall spin-down evolution, for instance, a change on the amount of superfluid content of the star may impact the braking index (i.e.",
"[27]), and actually the timing noise of younger pulsars can be mainly attributed to glitch recovery [29].",
"Meanwhile, millisecond pulsars ($\\tau _{\\rm C}\\gtrsim 10^{9}~{\\rm yr}$ or $\\nu >100 ~{\\rm s^{-1}}$ ) are also excluded from the sample, since the characteristic age of millisecond pulsars is highly deceptive, and the millisecond period in these systems reflects the recycling spin-up mechanism rather than secular spin-down evolution.",
"As examples, we show the simulated $\\ddot{\\nu }$ distribution with $\\tau _{\\rm B}=10^{6}~{\\rm yr}$ in left panel of Fig.",
"REF , the simulated distribution without field decay ($\\tau _{\\rm B}=10^{15}~{\\rm yr}$ is taken, which is longer than the age of the universe) in the right panel of Fig.",
"REF , and the reported $\\ddot{\\nu }$ distribution in both panels.",
"In the left panel, one can see that the simulated data are much more sparse in the lower part than the reported data, especially inside the triangular area surrounded by dashed lines.",
"In the right panel, the two distributions are completely consistent.",
"Figure: |ν ¨|-τ C |\\ddot{\\nu }|-\\tau _{\\rm C} distributions.",
"The published data from Hobbs et al.",
"(2010) are shown in both panels.",
"The simulated distribution with τ B =10 6 \\tau _{B}=10^6 yr is shown in the left panel.",
"The distribution without magnetic decay (with τ B =10 15 \\tau _{B}=10^{15} yr) is shown in the right panel.Some of the simulated pulsars in the bottom right part of the sample may have turned off as a radio pulsar and have crossed the death line in the $P-\\dot{P}$ diagram.",
"In most models, the period $P$ at turnoff depends upon the structure and the magnitude of the neutron star's surface magnetic field [4], [71].",
"Assuming a multipole magnetic field configuration, i.e.",
"the polar cap area is similar to that of the pure dipole field, but with very curved field lines at the surface, and the radius of the curvature $r_{\\rm c}\\sim R=10^6~{\\rm cm}$ , and this field configuration will be discussed in section .",
"The theoretical death line of the pulsar is then [4], $4 \\log B_{\\rm p}-6.5 \\log P=45.7.$ After taking into account this observational effect, about 30 simulated pulsars have been excluded from the simulated sample in the right panel of Fig.",
"REF .",
"The analysis on the scatter of $\\ddot{\\nu }$ versus $\\tau _{\\rm C}$ in Fig.",
"REF is potentially misleading, since the two quantities may not be entirely independent.",
"We carry out a similar analysis as Lyne et al.",
"(1975) to assure the reader that inherent correlations could not be found by plotting random pairings of pulsars, i.e.",
"the value of $\\nu $ , $\\dot{\\nu }$ , and $\\ddot{\\nu }$ are randomly taken from different pulsars in the sample.",
"In this case, there is no clear correlations between $\\ddot{\\nu }$ and $\\tau _{\\rm C}$ ."
],
[
"Kolmogorov-Smirnov Tests",
"We perform two-dimensional Kolmogorov-Smirnov (2DKS) test to reexamine the consistency of distributions of the simulated and reported $\\ddot{\\nu }$ for a series values of $\\tau _{B}$ .",
"The 2DKS packagehttp://www.downloadplex.com/Scripts/Matlab/Development-Tools/two-sample-two-diensional-kolmogorov-smirnov-test432625.html [45] is adopted for the test.",
"Our strategy is then to search for a typical $\\tau _{B}$ that can maximize the p-value of the 2DKS test against the hypothesis that the two distributions are consistent for the pulsars in the sample.",
"We let $\\tau _{\\rm B}$ vary from $10^5$ to $10^{8}$ yr.",
"The returned p-values are shown with solid lines in the upper panel of Fig.",
"REF .",
"The p-value $0.1$ is considered as the threshold level with probability $95\\%$ .",
"From the panel, one can see that the decay timescale can be well constrained with p-values larger than $\\sim 0.1$ .",
"It is shown apparently that the decay timescale $\\tau _{B}\\gtrsim 5\\times 10^{6}~\\rm {yr}$ .",
"In the bottom panel, we show the $N(\\ddot{\\nu }^{+})$ and $N(\\ddot{\\nu }^{-})$ as functions of $\\tau _{\\rm B}$ , giving a constraint $\\tau _{\\rm B}\\gtrsim 10^{5}$ yr for $95\\%$ probability, which is much loose than the constraint from 2DKS tests.",
"It should be noticed that the 2DKS test has only an approximate and stochastic p-value in each simulation, thus very intensive tests (with $\\log \\Delta \\tau _{B}=0.01$ ) were performed, as shown in the upper panel.",
"We also checked the validity of 2DKS tests with one-dimensional Kolmogorov-Smirnov (1DKS) test.",
"A very similar result is obtained with 1DKS but $\\tau _{B}\\gtrsim 10^{6} \\rm {yr}$ .",
"We also performed the 2DKS test only for young pulsars with $\\tau _{\\rm C}< 10^{7}~{\\rm yr}$ , the returned values also give $\\tau _{B}\\gtrsim 5\\times 10^{6}~\\rm {yr}$ , as shown in the upper panel of Fig.",
"REF .",
"However, for the sample of middle-age pulsars with $10^{7}<\\tau _{\\rm C}< 10^{9}~{\\rm yr}$ , the returned values give $\\tau _{B}\\gtrsim 10^{8}~\\rm {yr}$ , as shown in the bottom panel of Fig.",
"REF .",
"In addition, 23 millisecond recycled pulsars are excluded from our samples, and the number is too small to be tested independently with 2DKS.",
"Figure: Upper panel: the p-values of 2DKS for |ν ¨|-τ|\\ddot{\\nu }|-\\tau distributions from simulated data.",
"The ranges of p-values larger than 0.10.1 is identified by the transverse line.",
"The boundary of p-value ≳0.1\\gtrsim 0.1 indicates τ B ≳5×10 6 yr \\tau _{B}\\gtrsim 5\\times 10^{6}~\\rm {yr}.",
"Bottom panel: the pulsar number ratios of the positive and negative ν ¨\\ddot{\\nu } against τ B \\tau _{\\rm B} are represented by red and black lines, respectively.",
"σ\\sigma is the standard deviation of Poisson distribution.Figure: Upper panel: the p-values of 2DKS for |ν ¨|-τ|\\ddot{\\nu }|-\\tau distributions from young pulsars with τ C <10 7 yr \\tau _{\\rm C}< 10^{7}~{\\rm yr}.",
"Bottom panel: the p-values of 2DKS for |ν ¨|-τ|\\ddot{\\nu }|-\\tau distributions from young pulsars with 10 7 <τ C <10 9 yr 10^{7}<\\tau _{\\rm C}< 10^{9}~{\\rm yr}.It is very important to explore the parameter space of the simulations, especially regarding the dependence on the oscillation period $T$ and magnitude $k$ .",
"However, it is found that the method cannot place effective restrictions on $T$ .",
"For instance, there is no significant difference between the returned p-values for $T$ distributing uniformly from 5 to $25~{\\rm yr}$ or from $0.1$ to $100~{\\rm yr}$ .",
"There are extreme examples of magnetars with torque variations which could be interpreted as a change in the spin-down magnetic field by a generous fraction within months (i.e.",
"[1] in 1E 1048.1-5937).",
"Thus, for a wider coverage the latter ($0.1\\sim 100~{\\rm yr}$ ) is chosen for all the simulations in this paper.",
"For the magnitude $k$ , the returned p-values for the lower limit and the upper limit ($k=10^{-k_{2}}\\sim 10^{- k_{1}}$ ) of the power index are shown in panel (a) of Fig.",
"REF .",
"It can easily be seen that $3.9 \\lesssim k_{1} \\lesssim 4.6$ and $1.9\\lesssim k_{2} \\lesssim 2.5$ .",
"Therefore, $k=10^{-4.6}\\sim 10^{-1.9}$ is taken for the wider coverage.",
"As an example, the young pulsar PSR ${\\rm B1828 - 11}$ shows correlated shape and spin-down changes [58], and the observed $0.7\\%$ variation in $\\dot{P}$ implies a fractional change of similar magnitude in the oscillation magnitude $k\\simeq 3.5 \\times 10^{-3}$ , which falls well within the limits."
],
[
"The magnetospheric effects",
"Aside from the two prevailing models (SD and DM model), [5] proposed a spin-down formula that takes into account the magnetospheric particle acceleration gaps and the misalignment of magnetic and rotation axes, as well as the mechanism of the magnetic field reconnection around the equatorial extent $r_{\\rm c}$ of the closed-line region.",
"This formula can be simply expressed as $\\dot{\\Omega }=\\frac{B^2 R^6 \\Omega }{4cIr_{\\rm c}^2}[\\sin ^2\\theta +(1-\\frac{\\Omega _{\\rm death}}{\\Omega })\\cos ^2\\theta ],$ where the corotating region follows the light cylinder as $r_{\\rm c}=r_{\\rm lc}(\\Omega /\\Omega _0)^\\alpha $ , $\\Omega _0$ is the value of the angular velocity $\\Omega $ at pulsar birth, and the parameter $\\alpha $ ($0<\\alpha <1$ ) depends on the efficiency of the reconnection around $r_{\\rm c}$ .",
"If the reconnection is very efficient, $r_{\\rm c}\\approx r_{\\rm lc}$ , i.e.",
"$\\alpha =0$ .",
"However, if the reconnection is very inefficient, then the closed-line region cannot grow, thus $r_{\\rm c}\\approx {\\rm const}.$ [5].",
"$\\Omega _{\\rm death}$ ($=2\\pi /P_{\\rm death}$ ) describes a pulsar is “death\", i.e., the cessation of pulsar emission.",
"$P_{\\rm death}$ can be written as $\\begin{split}P_{\\rm death} =& 8.1^{1/(2-\\alpha )}~{\\rm s}~(\\frac{B}{10^{12}~{\\rm G}})^{1/(2-\\alpha )}\\\\& \\times (\\frac{V_{\\rm gap}}{10^{12}~{\\rm V}})^{-1/(2-\\alpha )} (\\frac{P_0}{1~\\rm s})^{-1/(2-\\alpha )},\\end{split}$ in which $V_{\\rm gap}$ is the gap potential.",
"Figure: The p-values of 2DKS for |ν ¨|-τ|\\ddot{\\nu }|-\\tau distributions from simulated data.",
"The ranges of p-values larger than 0.10.1 is identified by the transverse line.",
"Panel (a): the p-values for various values of the upper limit k 1 k_{1} and the lower limit k 2 k_{2} of -logk-\\log k; Panel (b): the p-values for various values of 〈log(B/G)〉\\langle \\log (B/{\\rm G})\\rangle ; Panel (c): the p-values for various values of σ logB \\sigma _{\\log B}; Panel (d): the p-values for various values of α\\alpha .We perform Monte Carlo simulations to confront the model with observations.",
"The main procedures of the simulations are the same as in the previous subsections.",
"We assume a lognormal distribution of polar magnetic fields with mean value $\\langle \\log (B/{\\rm G})\\rangle $ and standard derivation $\\sigma _{\\log B}$ .",
"Following [5], $V_{\\rm gap}=10^{13}~{\\rm V}$ is taken, and the initial period $P_0$ is uniformly distributed between $10~{\\rm ms}$ and $0.2~{\\rm s}$ .",
"We assume the distribution of inclination angle $\\theta $ is also uniform from 0 to $\\pi /2 $ .",
"The returned p-values for $\\langle \\log (B/{\\rm G})\\rangle $ , $\\sigma _{\\log B}$ and $\\alpha $ are shown in the panels (b), (c) and (d) of Fig.",
"REF , respectively.",
"The results show that $12.05\\lesssim \\langle \\log (B/{\\rm G})\\rangle \\lesssim 12.35$ , $\\sigma _{\\log B}\\lesssim 0.50 $ , and $\\alpha \\lesssim 0.11$ .",
"The result of no floor for $\\sigma _{\\log B}$ implies that the distribution width of $\\ddot{\\nu }$ is determined by the oscillation magnitude $k$ , rather than by the distribution width of the magnetic field $B$ .",
"The parameter $\\alpha \\lesssim 0.11$ means that our method cannot prove or rule out the spin-down law, but suggests that the reconnection of the north-south poloidal magnetic field around $r_{\\rm lc}$ is very efficient, and the extent of the closed line region is close to the light cylinder.",
"We thus propose that $\\alpha \\sim 0$ or $r_{\\rm c}\\approx r_{\\rm lc}$ , is a good approximation for the observed pulsar population."
],
[
"Physical Implications",
"The time-dependent behaviors of $B(t)$ , and thus the decay timescales of the effective magnetic fields, can be attributed not only to the magnetic field evolution, by also the changes of the inclination angle, or the small changes in the moment of inertia.",
"However, since these tests only prescribe lower limits on the evolution timescales, the results are valid for all the three mechanisms.",
"Magnetic field are crucial for neutron stars's activities.",
"Understanding the long-term evolution of neutron stars' magnetic fields might be key to unifying the observational diversity of isolated neutron stars [64].",
"The magnetic field in slow-rotating ultra-magnetized neutron stars, so-called magnetars (AXPs and SGRs), is believed to be decay on timescales of $10^3-10^5$ years [62], since their rotational energy is not sufficient to power the observed emission.",
"The isolated X-ray pulsars with spin periods longer than $12~\\rm s$ are still rarely observed.",
"However, they are not subject to physical limits to the emission mechanism nor observational biases against longer periods.",
"This puzzle could be well understood if their magnetic field is dissipated by one or even two orders of magnitude for $1~{\\rm Myr}$ , which is probably due to a highly resistive layer in the innermost part of the crust of neutron stars [50].",
"For normal radio pulsars, some population synthesis studies suggest that $\\tau _{\\rm B}$ must be longer than $10~{\\rm Myr}$ [26], [53].",
"However, there are also some other studies claimed short decay timescales, i.e.",
"$0.1\\lesssim \\tau _{\\rm B}\\lesssim 10~{\\rm Myr}$ [40], [42], [16], [51], [21], [31].",
"The present method imposes a piecewise restriction on the decay timescale, i.e.",
"$\\tau _{\\rm B}\\gtrsim 5~{\\rm Myr}$ for young pulsars ($\\tau _{\\rm C}< 10^{7}~{\\rm yr}$ ), and $\\tau _{\\rm B}\\gtrsim 100~{\\rm Myr}$ for middle-age pulsars ($10^{7}<\\tau _{\\rm C}< 10^{9}~{\\rm yr}$ ), and may contribute to our understanding of actual mechanisms of the field decay and magnetic configurations in neutron stars.",
"Three avenues for the magnetic field decay in isolated neutron stars have been intensively studied, i.e.",
"Ohmic decay, ambipolar diffusion, and Hall drift [14], [63], [11], [54], [30], [8], [47], [48], [49], [34], [12], [17].",
"Depending on the strength of the magnetic fields, each of these processes may dominate the evolution.",
"Ohmic decay occurs in both the fluid core and solid crust.",
"It is inversely proportional to the electric conductivity and independent of the field strength.",
"The Hall drift is non-dissipative and thus cannot be a direct cause of magnetic field decay.",
"However, it can enhance the rate of ohmic dissipation, since only electrons are mobile in the solid crust, and their Hall angle is large.",
"This causes that the evolution of magnetic fields resembles that of vorticity, and then the fields undergo a turbulent cascade terminated by ohmic dissipation at small scales [14], [8].",
"Compared with the Hall drift, the timescale of the ambipolar diffusion is much longer for normal pulsars, however, it may be very important for magnetars [62].",
"For a typical density and conductivity profile in the crustal region [47], [18], the Ohmic timescale is $\\tau _{\\rm Ohm}\\sim \\frac{4\\pi \\sigma L^2}{c^2}=13.5(\\frac{\\sigma }{3\\times 10^{24}~{s^{-1}}})(\\frac{L}{\\rm km})^2~{\\rm Myr},$ where $\\sigma $ is the electric conductivity, and $L$ is the characteristic length scale of magnetic field in the crust.",
"For the Hall timescale, one reads, $\\tau _{\\rm Hall}\\sim \\frac{4\\pi eL^2 n_{\\rm e}}{cB}=\\frac{16.8}{B_{13}}(\\frac{n_{\\rm e}}{{2.5\\times 10^{36}~{\\rm cm}^{-3}}})(\\frac{L}{\\rm km})^2~{\\rm Myr},$ in which $B_{13}\\equiv B/(10^{13}~\\rm G)$ .",
"The combined effect, i.e.",
"Hall cascade, could cause a fast field evolution on a timescale of the order of $10~{\\rm Myr}$ [20].",
"All these contradictory facts can be well understood by the natural assumption that the magnetic field is maintained by two current systems.",
"The large scale dipolar field which is responsible for the pulsar spin down are supported by long living currents in the superconducting core.",
"Currents in the crust support the small scale multipolar fields which decay on timescale that are comparable to the pulsar spin-down ages [47].",
"The two current systems and the corresponding field configurations are particularly demonstrated in the burst activities of a low dipole magnetic field magnetar, SGR 0418+5729, which is expected to harbor a sufficiently intense internal toroidal component [52].",
"The present result for middle-age pulsars, i.e.",
"$\\tau _{B}\\gtrsim 100~\\rm {Myr}$ , suggests that the dipole component that anchored in the crust are relatively low, and thus its decay has no observable influence on the spin frequency's second derivatives of pulsars in the sample.",
"In addition, the core-anchored field could be expelled and subsequently dissipated in the crust, and our result also implys that the timescale exceeds $5~\\rm {Myr}$ .",
"This may be helpful to understand the poorly known physics at the crust-core boundary.",
"Our results are also suitable for changes of the inclination angle, which could be either due to rotation-magnetic axis alignment or three-dimensional magnetic field evolution [46], [19].",
"Using polarization data for a large number of isolated pulsars, [60] found that the magnetic beam axis align with the spin axis on a timescale of $\\sim 10~{\\rm Myr}$ .",
"With new data, [72] found a shorter alignment timescale of $\\sim 1~{\\rm Myr}$ .",
"Theoretically, the electromagnetic torque which brakes the rotation of a pulsar also tends to align the magnetic axis with the rotation axis [9], [13].",
"The electromagnetic alignment timescale is related to the spin-down age as [36], $\\tau _{\\rm A}\\equiv \\frac{\\sin {\\theta }}{\\frac{d}{dt}\\sin {\\theta }} =2\\frac{\\sin ^2{\\theta }}{\\cos ^2{\\theta }}\\tau _{\\rm c}.$ For young or middle-age pulsars, our results imply the alignment timescale is most likely longer than $\\sim 5~{\\rm Myr}$ or $\\sim 100~{\\rm Myr}$ , which is roughly consistent with the relation.",
"A small variation in the moment of inertia may have an impact on the overall spin-down evolution, for instance, a decrease in the effective moment of inertia due to an increase on the amount of superfluid content as the star cools through neutrino emission may impact the braking index [27].",
"However, most of the stars are typically young and glitching pulsars, which have been excluded from our sample.",
"Our results imply that the populations without glitch record shows no long-term variation in the moment of inertia with timescale short than $5~{\\rm Myr}$ ."
],
[
"Summary and Discussion",
"The perturbation from the long-term dipole magnetic field decay will produce some differences on the distributions for the second derivatives of pulsars' spin frequency with different decay timescales.",
"This in turn provides a new method to investigate the magnetic field decay of radio pulsars, which does not depend on any specific theories of field evolution or inclination decay.",
"We made use of the published large-sample timing data of radio pulsars to find evidence of their magnetic field decay with 2DKS tests.",
"The method impose a piecewise restriction on the decay timescale, i.e.",
"$\\tau _{\\rm B}\\gtrsim 5~{\\rm Myr}$ for young pulsars with $\\tau _{\\rm C}< 10^{7}~{\\rm yr}$ , and $\\tau _{\\rm B}\\gtrsim 100~{\\rm Myr}$ for middle-age pulsars with $10^{7}<\\tau _{\\rm C}< 10^{9}~{\\rm yr}$ .",
"It is also proposed that the corotating radius $r_{\\rm c}\\approx r_{\\rm lc}$ is a good approximation for the observed pulsar population.",
"Though pulsars with major glitches have been excluded from the data, tiny glitch activities and other types of timing irregularities may still have some influences on the observed $\\ddot{\\nu }$ , which may cause a small deviation.",
"We expect to gain much deeper understanding of pulsars from future larger sample of radio pulsars with higher precision data on $\\ddot{\\nu }$ , to be brought by China's soon-to-be operating Five-hundred-meter Aperture Spherical radio-Telescope (FAST) and the future Square Kilometer Array (SKA)."
],
[
"Acknowledgments",
"We thank J. Y. Liao for discussions.",
"We thank the anonymous referee for comments and suggestions that led to a significant improvement in this manuscript.",
"This work is supported by National Natural Science Foundation of China under grant Nos.",
"11603009, 11803009, 11373036,and 11133002, by the National Program on Key Research and Development Project under grant Nos.",
"2016YFA0400802, by the Key Research Program of Frontier Sciences, CAS, Grant No.",
"QYZDY-SSW-SLH008, and by the Natural Science Foundation of Fujian Province under grant Nos.",
"2016J05013 and 2018J05006."
]
] | 1906.04470 |
[
[
"Fermion Number 1/2 of Sphalerons and Spectral Mirror Symmetry"
],
[
"Abstract We present a rederivation of the baryon and lepton numbers $\\frac{1}{2}$ of the $SU(2)_L$ S sphaleron of the standard electroweak theory based on spectral mirror symmetry.",
"We explore the properties of a fermionic Hamiltonian under discrete transformations along a noncontractible loop of field configurations that passes through the sphaleron and whose endpoints are the vacuum.",
"As is well known, CP transformation is not a symmetry of the system anywhere on the loop, except at the endpoints.",
"By augmenting CP with a chirality transformation, we observe that the Dirac Hamiltonian is odd under the new transformation precisely at the sphaleron, and this ensures the mirror symmetry of the spectrum, including the continua.",
"As a consistency check, we show that the fermionic zero mode presented by Ringwald in the sphaleron background is invariant under the new transformation.",
"The spectral mirror symmetry which we establish here, together with the presence of the zero mode, are the two necessary conditions whence the fermion number $\\frac{1}{2}$ of the sphaleron can be inferred using the reasoning presented by Jackiw and Rebbi or, equivalently, using the spectral deficiency $\\frac{1}{2}$ of the Dirac sea.",
"The relevance of this analysis to other solutions is also discussed."
],
[
"Introduction",
"In their seminal paper on the subject of charge fractionalization, Jackiw and Rebbi studied the Dirac equation in classical bosonic backgrounds for a number of field theories [1].",
"Their key discovery was the existence of states with half-integer fermion numbers in theories where all the fundamental fields have integer fermion numbers.",
"As was pointed out in [1], [2], [3], [4], in order for a bosonic configuration to have half-integer fermion numbers, the following two conditions must be simultaneously satisfied: The existence of a normalizable fermionic zero mode precisely at the bosonic configuration; Mirror symmetry of the entire fermionic spectrum, consisting of the bound and continuum states, at the bosonic configuration, or equivalently, the fermion number conjugation invariance of the system.",
"Since then, charge fractionalization has been widely studied and has found many applications in different areas, such as particle physics [2], [3], [4], [5], [6], [7], [8], condensed matter physics [9], [10], [11], polymer physics [12], [13], [14], quantum wires [15] and topological insulators [16], [17].",
"One class of solutions that can be found in certain field theories are sphalerons, which are saddle-point solutions in field configuration space [18], [19].",
"An important member of this class of solutions is the `S' sphaleron [20] of the standard electroweak theory.",
"Its importance is due to the role that it is believed to play in the early Universe, including the generation of the matter-antimatter asymmetry of the Universe [21], [22], [23].",
"Following the discovery of this solution in hadronic models [24], [25], it was rediscovered [18] in $SU(2)_L$ theory and its properties and implications for cosmology were detailed in [21].",
"In 1974 Dashen et.",
"al.",
"not only constructed a sphaleron solution as an extended model of hadrons, but also presented a framework for calculating the bound state energies of fermions which couple to the $SU(2)$ gauge field component of the sphalerons [24].",
"The coupling of the fermions to the Higgs component was represented as an explicit fermionic mass term.",
"Based on their work, the author of [26] showed that in the classical $SU(2)$ gauge field of the sphaleron, a fermion has a single bound-state solution with zero energy.",
"In the standard electroweak theory, a single normalizable zero energy solution of the Dirac equation in the sphaleron background was shown to exist for massless fermions in [27].",
"Shortly after, this result was extended by Ringwald to the case of massive fermions [28].",
"Later on, in the level-crossing picture for the $SU(2)_L$ theory, the change in the bound state energy of fermions coupled to the bosonic fields of the noncontractible loop (NCL) was numerically determined [29].",
"There, it was shown that as one traverses the NCL passing through the sphaleron, a single negative eigenvalue of the Dirac Hamiltonian arises from the Dirac sea, crosses the zero energy level precisely at the sphaleron and enters the positive energy continuum as one returns to the vacuum configuration.",
"The numerical study of [29] thus reconfirmed the existence of a zero energy bound state in the sphaleron background.",
"It is well known that the baryon and lepton numbers of the S sphaleron are $\\frac{1}{2}$ [21].",
"This can be seen by using the chiral anomaly and integrating the temporal component of the Chern-Simons current over one-point compactified 3-space and obtaining the resultant Chern-Simons charge for the sphaleron configuration, which is half-integer due to $SO(3)$ and reflection symmetries in the bosonic sector [19].",
"However, the use of the one-point compactification scheme corresponds to a gauge that breaks the reflection symmetry of the bosonic fields of the NCL about the sphaleron [21], [19].",
"On the other hand, the spectral flow for the Dirac Hamiltonian along the NCL and symmetries of the fermionic sector are independent of the gauge, and constitute the core of this paper.",
"The spectra of fermions coupled to the $SU(2)_L$ gauge-Higgs fields of the NCL passing through the S sphaleron have been studied in great detail over the past three decades by various authors.",
"In doing so, many of the symmetries of the spectra have been pointed out and explored [21], [30], [29], [31], [32], [19].",
"However, a symmetry that seems to have not been fully elucidated hitherto in the literature is a certain conjugation symmetry precisely at the sphaleron.",
"This manifests itself as the mirror symmetry of the fermionic spectrum about $E=0$ .",
"Following the path of the bound state as the NCL is traversed, this symmetry can be obviously seen to exist for the bound state precisely at the sphaleron, where it crosses $E=0$ .",
"Now, the question is whether the entire fermionic spectrum, including the continuum states, has mirror symmetry at the sphaleron.",
"As we shall show, this is indeed the case, and this has an important implication which brings us to the subject of this paper.",
"The main goal of this paper is to present a rederivation of the half-integer fermion numbers of $SU(2)_L$ S sphalerons by adopting an approach that is based on discrete symmetries.",
"To do this, we explicitly construct the transformation operator, which consists of the chiral and CP transformations, under which the Dirac Hamiltonian at the sphaleron is odd.",
"Hence we show that the entire spectrum of the Dirac Hamiltonian has mirror symmetry in the presence of the sphaleron.",
"We then use the results presented by Jackiw and Rebbi [1] to argue that the presence of the zero mode mandates half-integer fermion numbers for the sphaleron.",
"We should mention that the relation between the results of [1] and sphalerons had been hinted at in works such as [21], [33], [27], [20].",
"However, as mentioned before, a necessary condition to make such a connection is the mirror symmetry of the entire spectrum, which we establish here.",
"Furthermore, whereas some works have based their arguments on a fermionic zero mode in the limit of vanishing fermion mass [33], [27], in this work we have used the Ansatz given by [28], which is an extension to massive fermions within the standard electroweak theory.",
"In this case, the Higgs component of the sphaleron plays a nontrivial and essential role, which is beyond an explicit mass term.",
"The outline of this paper is as follows.",
"In Section , we briefly review the bosonic sector of the standard electroweak theory and the sphaleron Ansatz of $SU(2)_L$ Yang-Mills-Higgs theory in the limit of vanishing weak mixing angle.",
"In Section , we analyze the behavior of the Dirac Hamiltonian operator under a CP transformation for all configurations along the NCL.",
"Then, we augment CP to arrive at a suitable choice of conjugation operator which reveals the spectral mirror symmetry at the sphaleron.",
"Then, in Section , we perform the symmetry transformation on the zero mode given by Ringwald [28] in the sphaleron background.",
"In Section , we summarize our results and present an outlook."
],
[
"$SU(2)_L$ Sphaleron",
"Consider the bosonic sector of the well-established electroweak Lagrangian $\\mathcal {L}= -\\frac{1}{4}G_{\\mu \\nu }^a G^{\\mu \\nu } _a -\\frac{1}{4}F_{\\mu \\nu }F^{\\mu \\nu } + \\left(D_{\\mu } \\Phi )^{\\dagger }(D^{\\mu }\\Phi \\right) - \\lambda \\left(\\Phi ^{\\dagger } \\Phi - \\eta ^2 \\right)^2,$ where the $U(1)$ field strength tensor is given by $F_{\\mu \\nu } = \\partial _{\\mu }A_{\\nu } - \\partial _{\\nu }A_{\\mu },$ the $SU(2)$ field strength tensor is given by $G_{\\mu \\nu }^a = \\partial _{\\mu }B_{\\nu }^a - \\partial _{\\nu }B_{\\mu }^a + g\\epsilon ^{abc}B_{\\mu }^b B_{\\nu }^c,$ and the covariant derivative of the Higgs field is $D_{\\mu }\\Phi = \\left(\\partial _{\\mu } - ig\\frac{\\tau ^a}{2}B_{\\mu }^a -ig^{\\prime }YA_{\\mu }\\right)\\Phi .$ In the limit of vanishing mixing angle, the $U(1)$ field decouples and this allows for a spherically symmetric Ansatz for the gauge and Higgs fields of the NCL.",
"To this end, consider the following map: $U: S^1 \\wedge S^2 \\sim S^3 \\rightarrow SU(2),\\;\\;\\;\\;\\;\\;\\left(\\mu ,\\theta ,\\phi \\right)\\mapsto U\\left(\\mu ,\\theta ,\\phi \\right),$ where $\\wedge $ is the smash productFor a definition, see [20].",
"and $\\mu $ is the loop parameter.",
"A suitable representation is [18], [20] $U\\left(\\mu ,\\theta ,\\phi \\right)=-iy^1 \\tau _1 -iy^2 \\tau _2 -iy^3\\tau _3 + y^4 \\mathbb {1},$ where $\\begin{pmatrix}y^1\\\\y^2\\\\y^3\\\\y^4\\\\\\end{pmatrix}=\\begin{pmatrix}-\\sin \\mu \\sin \\theta \\sin \\phi \\\\-\\sin \\mu \\sin \\theta \\cos \\phi \\\\\\sin \\mu \\cos \\mu \\left(\\cos \\theta -1\\right)\\\\\\cos ^2\\mu + \\sin ^2\\mu \\cos \\theta \\\\\\end{pmatrix},$ and $\\tau ^i$ are chosen to be the usual Pauli matrices while $\\tau ^i /2$ are the generators in weak isospace.",
"Using the above map, the AnsatzIt can be shown that, even when the Ansatz is not manifestly spherically symmetric, it can always be transformed to one that is , .",
"for the static gauge and Higgs fields of the $SU(2)_L$ sphaleron barrier becomes [18] $\\begin{split}B\\left(\\mu ,r,\\theta ,\\phi \\right)&=-\\frac{f\\left(r\\right)}{g} dU\\left(\\mu ,\\theta ,\\phi \\right)U^{-1}\\left(\\mu ,\\theta ,\\phi \\right),\\\\\\Phi \\left(\\mu ,r,\\theta ,\\phi \\right)&= \\eta h\\left(r\\right)U\\left(\\mu ,\\theta ,\\phi \\right)\\begin{pmatrix}0\\\\1\\\\\\end{pmatrix}+\\eta \\left[1-h\\left(r\\right)\\right]\\begin{pmatrix}0\\\\e^{-i\\mu }\\cos \\mu \\end{pmatrix},\\end{split}$ where the radial functions have the following boundary conditions: $\\begin{split}\\lim _{r\\rightarrow 0}\\frac{f\\left(r\\right)}{r}=0,\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\lim _{r\\rightarrow \\infty }f\\left(r\\right)=1,\\\\\\lim _{r\\rightarrow 0}h\\left(r\\right)=0,\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\lim _{r\\rightarrow \\infty }h\\left(r\\right)=1.\\end{split}$ The field $B $ is an $SU(2)$ -valued one-form, $B\\left(\\mu ,r,\\theta ,\\phi \\right)=B_rdr+B_{\\theta }d\\theta +B_{\\phi }d\\phi = B_idx^i,$ for which we impose the radial gauge condition $B_r=0$ [18].",
"We assume that in the radial gauge there exists a limiting field $\\Phi ^{\\infty }\\left(\\theta ,\\phi \\right)\\equiv \\lim _{r\\rightarrow \\infty }\\Phi \\left(r,\\theta ,\\phi \\right)\\;\\;,$ such that $\\left|\\Phi ^{\\infty } \\right|=1$ and $\\Phi ^{\\infty }\\left(\\theta =0\\right)=\\begin{pmatrix}0\\\\1\\\\\\end{pmatrix}.$"
],
[
"CP Transformation along NCL",
"In this section we study the behavior of the Dirac Hamiltonian under discrete transformations including C and P in a sphaleron background.",
"When the weak mixing angle goes to zero, we perform our analysis for arbitrary loop parameter $\\mu $ .",
"For the $SU(2)_L\\times U(1)_Y$ sphaleron, only the sphaleron Ansatz has been constructed and not the full barrier.",
"This restricts our analysis to the sphaleron when $\\theta _w=0$ .",
"Nevertheless, this strategy can be readily extended to the full barrier once it is constructed.",
"Consider the Dirac Hamiltonian operator of the standard electroweak theory at $\\theta _w=0$ [20] $\\hat{\\mathcal {H}}= -i\\gamma ^0 \\gamma ^j D_j + k\\gamma ^0\\left(\\Phi _M^{\\dagger }P_L + \\Phi _M P_R\\right),$ where the matrix $\\Phi _M$ contains the scalar fields of the Higgs doublet and its charge-conjugated doublet and is given by $\\Phi _M =\\begin{pmatrix}\\begin{array}{cc}\\phi _2^* & \\phi _1\\\\-\\phi _1^*& \\phi _2\\\\\\end{array}\\end{pmatrix},$ and the projection operators are defined as $P_L=\\frac{1}{2}\\left(1-\\gamma ^5\\right),\\;\\;\\;\\;\\;P_R=\\frac{1}{2}\\left(1+\\gamma ^5\\right).$ We now use the Ansatz given in Eq.",
"(REF ) to construct $\\Phi _M$ shown in Eq.",
"(REF ) and insert it into Eq.",
"(REF ) to obtain the expression for $\\hat{\\mathcal {H}}$ along the NCL.",
"The final expression for $\\hat{\\mathcal {H}}$ is shown in the appendix.",
"We use the following choice of Weyl representation for our gamma matrices $\\gamma ^0 =\\begin{pmatrix}\\begin{array}{cc}0 & I_2 \\\\I_2 & 0 \\\\\\end{array}\\end{pmatrix},\\;\\;\\;\\gamma ^i =\\begin{pmatrix}\\begin{array}{cc}0 & \\sigma ^i \\\\-\\sigma ^i & 0\\\\\\end{array}\\end{pmatrix},\\;\\;\\;\\gamma ^5 =\\begin{pmatrix}\\begin{array}{cc}-I_2 & 0 \\\\0 & I_2 \\\\\\end{array}\\end{pmatrix}.$ In this representation, charge conjugation acts non-trivially on scalars and spinors, both of which transform in the fundamental representation of $SU(2)$ , as $\\Phi \\left(\\vec{x},t\\right) \\xrightarrow{} i\\tau ^2 \\Phi ^*\\left(\\vec{x},t\\right) ,\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\; \\Psi \\left(\\vec{x},t\\right) \\xrightarrow{} i\\tau ^2 \\gamma ^2 \\Psi ^*\\left(\\vec{x},t\\right),$ while under the combined transformation of C and P, $\\Phi \\left(\\vec{x},t\\right) \\xrightarrow{} i\\tau ^2 \\Phi ^*\\left(-\\vec{x},t\\right) ,\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\; \\Psi \\left(\\vec{x},t\\right) \\xrightarrow{} i\\tau ^2 \\gamma ^2 \\gamma ^0 \\Psi ^*\\left(-\\vec{x},t\\right).$ Therefore, the charge-conjugated, parity-inverted Hamiltonian becomes $\\hat{\\mathcal {H}}^{CP}\\left(\\vec{x},t,\\mu \\right) = \\gamma ^2\\gamma ^0\\begin{pmatrix}\\begin{array}{cc}-\\hat{\\mathcal {H}}_{22}^* & \\hat{\\mathcal {H}}_{21}^*\\\\\\hat{\\mathcal {H}}_{12}^* & -\\hat{\\mathcal {H}}_{11}^*\\\\\\end{array}\\end{pmatrix}_{(-\\vec{x},t,\\mu )}\\gamma ^0\\gamma ^2.$ After inserting the explicit expressions for the matrix elements of $\\hat{\\mathcal {H}}$ given in Eq.",
"() into Eq.",
"(REF ), we observe that nowhere along the NCL is $\\hat{\\mathcal {H}}$ odd under CP, except at the trivial vacuum ($\\mu =0,\\pi $ ).",
"However, at $\\mu =\\frac{\\pi }{2}$ , there are many cancellations and the even part reduces to $\\begin{split}\\hat{\\mathcal {H}}^{CP}&\\left(\\vec{x},t,\\mu =\\frac{\\pi }{2}\\right) + \\;\\hat{\\mathcal {H}}\\left(\\vec{x},t,\\mu =\\frac{\\pi }{2}\\right) \\\\&= 2k\\eta h(r)\\gamma ^0\\begin{pmatrix}\\begin{array}{cc}\\cos \\theta \\left(P_L + P_R \\right) &-\\sin \\theta e^{i\\phi } \\left(P_L - P_R \\right)\\\\\\sin \\theta e^{-i\\phi } \\left(P_L - P_R \\right) &\\cos \\theta \\left(P_L + P_R \\right) \\\\\\end{array}\\end{pmatrix}.\\end{split}$ This reflects the fact that the pseudoscalar sphaleron configuration breaks the CP invariance of the one-generation electroweak theory that we have been considering.",
"We now define a new conjugation transformation, $\\widetilde{CP}$ , which consists of CP and is augmented by an additional operation as follows $\\widetilde{CP} \\equiv CP \\mathcal {X},$ where $\\mathcal {X}= e^{-i\\mu \\gamma ^5}$ .",
"By repeating the calculation leading to Eq.",
"(REF ) for the new operation, Eq.",
"(REF ), we see that $\\hat{\\mathcal {H}}^{\\widetilde{CP}}\\left(\\vec{x},t,\\mu =\\frac{\\pi }{2}\\right) = - \\;\\hat{\\mathcal {H}}\\left(\\vec{x},t,\\mu =\\frac{\\pi }{2}\\right).$ The existence of a transformation under which $\\hat{\\mathcal {H}}$ is odd ascertains the spectral mirror symmetry.",
"That is, under such a transformation every eigenstate of $\\hat{\\mathcal {H}}$ with energy $E$ is transformed into one with energy $-E$ , the only exception being a zero energy bound state which must then be invariant under such a transformation.",
"From the viewpoint of spectral deficiency [7], this implies that as the NCL is traversed, spectral deficiency $\\mathcal {D}$ in the positive continuum caused by the bound state starts to replenish, while deficiency starts to build up in the Dirac sea.",
"At $\\mu = \\frac{\\pi }{2}$ the fermionic bound state is at $E=0$ , and the spectral deficiencies in both continua are [7] $\\mathcal {D} = \\frac{\\mu }{\\pi }.$ Therefore, at $\\mu = \\frac{\\pi }{2}$ , the quantum field theoretic expectation value of the fermion number operator is [4] $\\left| \\left\\langle N \\right\\rangle \\right| = \\frac{1}{2}.$ This number can be interpreted as the fermion number of the bosonic configuration.",
"In the next section, we check the invariance of the zero mode presented by Ringwald (which is the one that is relevant to our model) [28] under $\\widetilde{CP}$ .",
"The fermion numbers $\\frac{1}{2}$ of the sphaleron then follow immediately from the reasoning presented by Jackiw and Rebbi or, equivalently, from the spectral deficiency $\\frac{1}{2}$ of the Dirac sea.",
"It is worth mentioning that at the trivial vacuum ($\\mu =0,\\pi $ ), $\\hat{\\mathcal {H}}$ is odd under CP, showing that the spectrum has mirror symmetry there.",
"However, there are no bound states in the trivial vacuum."
],
[
"The Zero Mode",
"Recall that in the original analysis of Jackiw and Rebbi, the fermionic zero mode in the soliton background was fermion number self-conjugate [1].",
"Thus, an important consistency check on our symmetry transformation would be to operate it on the fermionic zero mode that was given by Ringwald at the electroweak S sphaleron [28].",
"To this end, consider the zero-energy solution of the Dirac equation in the sphaleron background.",
"The Ansatz for the left-handed isodoublet of the zero mode is given by [28] $\\Psi _{0,L}^{i\\alpha }\\left(\\vec{x},t\\right) = \\epsilon ^{i\\alpha } z(r),$ where $i=1,2$ is the weak isospin index, $\\alpha =1,2$ is the spinor index and $\\epsilon ^{ij}$ is the Levi-Civita symbol ($\\epsilon ^{12}= +1$ ).",
"The functional form of $z(r)$ is obtained by solving the radial part of the Dirac equation.",
"Depending on whether the fermions are massive or massless, $z(r)$ will take on a different form [28].",
"For a single generation of left-handed quarks, denoted by $\\Psi _{0,L}^{\\alpha } =\\begin{pmatrix}u_{0,L}^{\\alpha } \\\\d_{0,L}^{\\alpha }\\\\\\end{pmatrix},$ this implies that $u_{0,L}\\left(\\vec{x},t\\right) = z(r)\\begin{pmatrix}0\\\\1\\end{pmatrix}\\equiv z(r){\\downarrow },\\;\\;\\;\\;\\;d_{0,L}\\left(\\vec{x},t\\right) = z(r) \\begin{pmatrix}-1\\\\0\\\\\\end{pmatrix}\\equiv -z(r){\\uparrow }.$ Thus, Eq.",
"(REF ) can also be written as $\\Psi _{0,L}\\left(\\vec{x},t\\right) = z(r)\\begin{pmatrix}{\\downarrow } \\\\-{\\uparrow } \\\\\\end{pmatrix}.$ A CP transformation on Eq.",
"(REF ) or, equivalently, Eq.",
"(REF ) yields $\\Psi _{0,L}^{CP}\\left(\\vec{x},t\\right) = i\\gamma ^5 \\Psi _{0,L}\\left(\\vec{x},t\\right),$ which shows that, as expected, the zero mode is not CP-invariant.",
"By noting that we are performing the symmetry transformation at the sphaleron ($\\mu = \\frac{\\pi }{2}$ ), implementing the additional factor of $-i\\gamma ^5$ required by a $\\widetilde{CP}$ transformation, we obtain $\\Psi _{0,L}^{\\widetilde{CP}}\\left(\\vec{x},t\\right) = \\Psi _{0,L}\\left(\\vec{x},t\\right).$ Thus, we observe that the fermionic zero mode of Ringwald in the sphaleron background is $\\widetilde{CP}$ -invariant."
],
[
"Summary and Discussion",
"In this paper, we have studied the behavior of fermions under discrete transformations in a sphaleron background.",
"For the fields of the NCL passing through the S sphaleron, it is well known that the system is not CP-invariant except at the vacua.",
"However, we have constructed a new transformation, denoted by $\\widetilde{CP}$ , by augmenting a CP transformation with an additional operation that acts nontrivially in the Yukawa sector and has the following important property.",
"We see that for field configurations along the NCL, the Dirac Hamiltonian is odd under $\\widetilde{CP}$ precisely at the S sphaleron sitting at the top of the barrier that begins and ends at the trivial vacuum.",
"This ensures that the spectrum has mirror symmetry.",
"That is, for every positive energy eigenstate there is a corresponding negative energy one and the zero mode, if any, is self-conjugate.",
"As an important consistency check, by performing the symmetry transformation on the fermionic zero mode given by Ringwald [28] in the sphaleron background, we observe that the zero mode is indeed $\\widetilde{CP}$ -invariant.",
"This is closely analogous to the analyses of [1], [4].",
"There, fermion number conjugation symmetry of the spectrum including the zero mode in the background of the classical solution was an important condition that led to the derivation of the half-integer fermion numbers of the background bosonic fields.",
"In the analyses of [1], [4], fermion number conjugation was charge conjugation.",
"Our transformation operator is $\\widetilde{CP}$ which reveals the spectral mirror symmetry at the sphaleron.",
"In this configuration, the spectral deficiency in the Dirac sea is exactly $\\frac{1}{2}$ and one associates this to the fermion number of the background field which is the sphaleron.",
"Overall, this analysis offers a number of other potential advantages.",
"At a basic level, it can provide a useful consistency check for the numerous fractionally-charged sphaleron Ansätze that have been discovered so far , , , , , , , , and helps place constraints on their functional forms.",
"An example of this can be seen in the Ansatz for the axially symmetric sphaleron, where the arbitrary functions acquire a z-dependence .",
"Furthermore, the analyses of [1], [4] required C-invariance, while the present analysis led to $\\widetilde{CP}$ -invariance.",
"It may be that other solutions require other novel symmetry transformations for the fermionic sector to correctly explain their fractional charges.",
"An important issue that our study has not addressed is what happens when one considers three generations of fermions, where CP symmetry is violated through the CKM and PMNS mixing matrices in the background of the even-parity Higgs field vacuum.",
"Finally, from a more practical perspective, one should bear in mind that sphalerons currently play an omnipresent role in physics and show up in many field theories, such as gravitation, electroweak theory and quantum chromodynamics.",
"Thus, it seems worthy to delve even deeper into their structure to see if new symmetries emerge.",
"It may be that studying these symmetries paves the way for a more systematic understanding of the topological properties of unstable solutions in gauge field theories and their physical applications.",
"Acknowledgments: M. M. would like to thank N. Manton and J. Kunz for useful discussions and N. Dadkhah for useful comments on the manuscript.",
"We would like to thank the research council of the Shahid Beheshti University for financial support."
],
[
"Dirac Hamiltonian along NCL",
"In this section we give the explicit functional form of the components of Eq.",
"(REF ).",
"As an $SU(2)$ -valued $2\\times 2$ matrix, $\\hat{\\mathcal {H}}$ is $\\hat{\\mathcal {H}}=\\begin{pmatrix}\\begin{array}{cc}\\hat{\\mathcal {H}}_{11}& \\hat{\\mathcal {H}}_{12}\\\\\\hat{\\mathcal {H}}_{21}& \\hat{\\mathcal {H}}_{22}\\\\\\end{array}\\end{pmatrix}.$ In the background of the gauge and Higgs fields of the NCL, Eq.",
"(REF ), the components of $\\hat{\\mathcal {H}}$ are $\\begin{split}\\hat{\\mathcal {H}}_{11} = &-i\\gamma ^0\\gamma ^j\\partial _j + f(r)r\\gamma ^0\\gamma ^3P_L\\sin \\mu \\cos \\mu \\sin ^2\\theta \\\\ &- f(r)r\\gamma ^0\\gamma ^1P_L\\sin \\mu \\sin \\theta \\left(\\cos \\mu \\cos \\theta \\cos \\phi - \\sin \\mu \\sin ^2\\theta \\sin \\phi \\right) \\\\ &- f(r)r\\gamma ^0\\gamma ^2P_L\\sin \\mu \\sin \\theta \\left(\\cos \\mu \\cos \\theta \\sin \\phi + \\sin \\mu \\sin ^2\\theta \\cos \\phi \\right) \\\\& + k\\eta h(r)\\gamma ^0\\left[e^{-i\\mu } \\left( \\frac{\\cos \\mu }{h(r)} + i\\sin \\mu \\cos \\theta \\right)P_L + e^{i\\mu }\\left(\\frac{\\cos \\mu }{h(r)} - i\\sin \\mu \\cos \\theta \\right) P_R \\right],\\end{split}\\\\[2ex]\\begin{split}\\hat{\\mathcal {H}}_{12} = &\\;if(r)r\\gamma ^0\\gamma ^1P_Le^{i(\\mu +\\phi )}\\sin \\mu \\\\&\\times \\left[\\cos \\theta \\cos \\phi \\left(\\cos \\mu \\cos \\theta -i\\sin \\mu \\right)-i\\sin ^2\\theta \\sin \\phi \\left(\\cos \\mu -i\\sin \\mu \\cos \\theta \\right)\\right]\\\\ +&\\; if(r)r\\gamma ^0\\gamma ^2P_Le^{i(\\mu +\\phi )}\\sin \\mu \\\\&\\times \\left[\\cos \\theta \\sin \\phi \\left(\\cos \\mu \\cos \\theta -i\\sin \\mu \\right)+i\\sin ^2\\theta \\cos \\phi \\left(\\cos \\mu -i\\sin \\mu \\cos \\theta \\right)\\right]\\\\-&\\;if(r)r\\gamma ^0\\gamma ^3P_Le^{i(\\mu +\\phi )}\\sin \\mu \\sin \\theta \\left(\\cos \\mu \\cos \\theta -i\\sin \\mu \\right)\\\\-&\\;k\\eta h(r)\\gamma ^0\\sin \\mu \\sin \\theta e^{i\\phi }\\left(P_L - P_R\\right),\\end{split}$ $\\begin{split}\\hat{\\mathcal {H}}_{21} = &\\;-if(r)r\\gamma ^0\\gamma ^1P_Le^{-i(\\mu +\\phi )}\\sin \\mu \\\\&\\times \\left[\\cos \\theta \\cos \\phi \\left(\\cos \\mu \\cos \\theta +i\\sin \\mu \\right)+i\\sin ^2\\theta \\sin \\phi \\left(\\cos \\mu +i\\sin \\mu \\cos \\theta \\right)\\right]\\\\ -&\\; if(r)r\\gamma ^0\\gamma ^2P_Le^{-i(\\mu +\\phi )}\\sin \\mu \\\\&\\times \\left[\\cos \\theta \\sin \\phi \\left(\\cos \\mu \\cos \\theta +i\\sin \\mu \\right)-i\\sin ^2\\theta \\cos \\phi \\left(\\cos \\mu +i\\sin \\mu \\cos \\theta \\right)\\right]\\\\+&\\;if(r)r\\gamma ^0\\gamma ^3P_Le^{-i(\\mu +\\phi )}\\sin \\mu \\sin \\theta \\left(\\cos \\mu \\cos \\theta +i\\sin \\mu \\right)\\\\+&\\;k\\eta h(r)\\gamma ^0\\sin \\mu \\sin \\theta e^{-i\\phi }\\left(P_L - P_R\\right),\\end{split}\\\\[2ex]\\begin{split}\\hat{\\mathcal {H}}_{22} = &-i\\gamma ^0\\gamma ^j\\partial _j - f(r)r\\gamma ^0\\gamma ^3P_L\\sin \\mu \\cos \\mu \\sin ^2\\theta \\\\ &+ f(r)r\\gamma ^0\\gamma ^1P_L\\sin \\mu \\sin \\theta \\left(\\cos \\mu \\cos \\theta \\cos \\phi - \\sin \\mu \\sin ^2\\theta \\sin \\phi \\right) \\\\ &+ f(r)r\\gamma ^0\\gamma ^2P_L\\sin \\mu \\sin \\theta \\left(\\cos \\mu \\cos \\theta \\sin \\phi + \\sin \\mu \\sin ^2\\theta \\cos \\phi \\right) \\\\& + k\\eta h(r)\\gamma ^0\\left[e^{+i\\mu } \\left( \\frac{\\cos \\mu }{h(r)} - i\\sin \\mu \\cos \\theta \\right)P_L + e^{-i\\mu }\\left(\\frac{\\cos \\mu }{h(r)} + i\\sin \\mu \\cos \\theta \\right) P_R \\right].\\end{split}$"
]
] | 1906.04427 |
[
[
"Gravitational-wave detection rates for compact binaries formed in\n isolation: LIGO/Virgo O3 and beyond"
],
[
"Abstract Using simulations performed with the population synthesis code MOBSE, we compute the merger rate densities and detection rates of compact binary mergers formed in isolation for second- and third-generation gravitational-wave detectors.",
"We estimate how rates are affected by uncertainties on key stellar-physics parameters, namely common envelope evolution and natal kicks.",
"We estimate how future upgrades will increase the size of the available catalog of merger events, and we discuss features of the merger rate density that will become accessible with third-generation detectors."
],
[
"Introduction",
"The detection of gravitational waves (GWs) from 10 binary black holes (BBHs) and a binary neutron star (BNS) in the first two LIGO/Virgo observing runs [1], and the subsequent detections of numerous compact binary candidates in the third observing run, naturally lead to the question: how do these binaries form, and what is the physics that drives their evolution?",
"Advanced LIGO (AdLIGO) is expected to reach design sensitivity in the near future, the so-called A$+$ upgrade to current detectors was already approved for funding, and further upgrades (A$++$ and Voyager) are expected in the near future [2], [3], [4], [5], [6].",
"The GW community is also planning future, “third-generation” (3G) facilities, such as the Einstein Telescope (ET) [7], [8] and Cosmic Explorer (CE) [6], which will extend the observable horizon to the very early Universe.",
"As GW detectors improve and the number of detections grows, we will gather information about the environments in which compact binaries form, and constrain the physical parameters that drive their evolution.",
"Future GW detectors will measure compact binary parameters (such as masses and spins) within few per cent accuracy [9], reconstructing fine details of distribution of these observables.",
"They will observe sources up to redshifts as large as $z\\sim 10^2$ [10], allowing us to study how the merger rate density evolves with redshift, and ultimately to constrain astrophysical models [11], [12], [13].",
"The large number of detections that comes with increased sensitivity will also reduce statistical errors on the parameters that describe compact binary populations to few per cent with $\\sim 10^3$ observations [14].",
"Compact-object binaries could form either in the field [15], [16] or through dynamical interactions in young [17], [18], [19], nuclear [20], [21] or globular clusters [22], [23].",
"In this paper we present updated detection rates, and a roadmap of our prospects for constraining the astrophysics of compact binaries in the near future.",
"We study how detection rates for binaries formed in isolation (“field binaries”) will evolve with future improvements of GW detectors, with the goal to understand if and when characteristic features of the astrophysical populations will become visible.",
"The plan of the paper is as follows.",
"In Sec.",
"we present our astrophysical populations based on the MOBSE population-synthesis code [24], [25].",
"In Sec.",
"we investigate how uncertainties in binary evolution affect the evolution of the merger rate density, and what new generation of detectors can tell us about this evolution.",
"In Sec.",
"we compute detection rates for each of the six models we consider and for different detector sensitivities.",
"In Sec.",
"we summarize our findings and out line directions for future work.",
"Appendix gives details on how detection rates are computed from the MOBSE simulations.",
"Throughout the paper we use the standard cosmological parameters determined by the Planck Collaboration [26].",
"We assume that a source is detected if the single-detector signal-to-noise ratio (SNR) $\\rho $ is such that $\\rho >8$ ."
],
[
"Astrophysical populations",
"We use simulations performed with the population-synthesis code MOBSE [25].",
"MOBSE is an upgrade of the BSE code [16] which includes up-to-date prescriptions for the evolution of massive stars.",
"The treatment of stellar winds accounts for the stellar metallicity and luminosity dependence of the mass loss.",
"Compact objects are produced via different channels, including core-collapse, electron-capture and (pulsational) pair instability supernovae (SNe).",
"In our simulations, the primary star's mass $m_{\\mathrm {1}}$ is distributed according to the Kroupa mass function [27] $\\mathcal {F}(m_1) \\propto m_1^{-2.3} \\qquad {\\rm with} \\;\\; m_1 \\in [5-150]M_\\odot \\,,$ while the mass ratio $q=m_2/m_1$ scales like [28] $\\mathcal {F}(q) \\propto q^{-0.1} \\qquad {\\rm with} \\;\\; q \\in [0.1-1]\\,.$ The orbital period $P$ is drawn from $\\mathcal {F}({\\mathcal {P}}) \\propto {\\mathcal {P}}^{-0.55} \\quad {\\rm with} \\;\\; {\\mathcal {P}} = \\mathrm {log_{10}}\\left(\\frac{P}{\\mathrm {day}}\\right) \\in [0.15-5.5]$ and the eccentricity $e$ follows the distribution [28] $\\mathcal {F}(e) \\propto e^{-0.42} \\qquad \\mathrm {with}\\;\\; 0\\le e < 1\\,.$ Among the many physical processes involved in the formation of compact binaries that can merge within a Hubble time, the so called common-envelope phase is believed to be critical [29], [30].",
"When a star in a binary system overfills its Roche lobe, it starts transferring mass, and eventually forms a common envelope that engulfs the companion.",
"The common envelope does not corotate with the stars or their cores, and this leads to a drag force.",
"As a result, the stars spiral in and transfer their orbital energy to the envelope.",
"The system will survive only if the energy transferred is sufficient to eject the envelope [31], [32], [33].",
"The efficiency of this mechanism constitutes a main uncertainty in compact-binary formation modelling.",
"Another important source of uncertainty are natal kicks.",
"If a compact object forms from a supernova explosion, it is expected to receive a birth kick because of asymmetric mass ejection.",
"A non-zero kick (the so-called Blaauw kick [34]) is expected even in the unlikely case where mass loss is symmetric, but the compact object is part of a binary system.",
"This natal kick can disrupt the binary or substantially modify its orbit.",
"Kicks set the fraction of stellar binaries which are unbound by the SN explosion and, consequently, play a major role in determining GW detection rates [15], [24], [35].",
"Table: Catalog of MOBSE models considered in this study.As described by [36] and summarized in Table REF , we consider six representative populations of merging binaries, aiming at bracketing the uncertainties in the physics of both common envelope and natal kicks.",
"These two parameters might be the first to be constrained with GW data (see e.g.",
"[37], [14]).",
"The common envelope phase is treated using the so-called $\\alpha \\lambda $ formalism [38], [32], where $\\alpha $ quantifies the efficiency of energy transfer to the envelope and $\\lambda $ represents the binding energy of the envelope.",
"In this work we consider $\\alpha $ as a free parameter, while $\\lambda $ depends on the stellar type [39] and it is computed by using the prescriptions derived in Ref. [40].",
"Kicks are extracted from a Maxwellian distribution with root-mean-square speed (rms) $\\sigma _{\\rm CCSN}$ for core-collapse SNe that produce neutron stars.Neutron stars can also form through electron-capture SNe, which are less energetic, faster and do not develop large asymmetries.",
"This is generally expected to lead to small kicks, and therefore we assume $\\sigma _{\\rm ECSN}=15 \\,{\\rm km/s}$ [41].",
"For black holes, we reduce the kick velocity $v_{\\rm BH}$ by taking into account fallback: $v_{\\rm BH} = (1 - f_{\\rm fb})v_{\\rm NS}$ , where $v_{\\rm NS}$ is the natal kick for neutron stars and $f_{\\rm fb}$ parametrizes the amount of fallback on the proto-compact object [42].",
"Models CC15 produce natal kicks $\\le {}100$ km s$^{-1}$ , and therefore they are in tension with the proper motions of the fastest single Galactic neutron stars [43].",
"These models were chosen because they give a local merger rate density of binary neutron stars consistent with the one inferred from GW170817 [44], without requiring exotic assumptions about common envelope.",
"MOBSE predicts the NS masses from $1.1$ to $2 M_\\odot $ where light (heavy) NSs are preferred during BNS (NSBH) mergers.",
"On the other hand, NSBH mergers favor low BH masses ($<15 M_\\odot $ ) while BBH mergers could have BHs as heavy as $45 M_\\odot $ with most binaries having mass ratios close to unity [36]."
],
[
"Merger rate densities",
"The merger rate density ${\\mathcal {R}}(z_m)$ as a function of merger redshift $z_m$ tracks the distribution of merging binaries across cosmic time, and it depends on two factors: (i) the rate of binary formation at a given redshift $z_f$ , and (ii) the distribution of time delays $t_{\\rm delay}$ between the formation of the parent stars in the binary and the merger of their compact object remnants.",
"In turn, binary formation at $z_f$ depends on the star formation rate and the metallicity, both of which evolve over time.",
"The time delay distribution is sensitive to the physics that drives binary evolution (see e.g.",
"[46], [22], [47]).",
"In Fig.",
"REF we plot the evolution of the merger rate density for the six MOBSE models considered in this study.",
"The low-redshift behavior is often parametrized as a power law: ${\\mathcal {R}}(z)\\approx {\\mathcal {R}}_0 (1+z)^{\\lambda _0}$ [13], [11], where ${\\mathcal {R}}_0$ is the local merger rate density and $\\lambda _0$ is a model-dependent parameter that describes its evolution.",
"The parameter $\\lambda _0$ can be used to infer astrophysical information.",
"The star formation rate is well approximated by $\\lambda _0\\simeq 2.4$ for $0.1<z<1$ [11].",
"Therefore, an observed $\\lambda _0<2.4$ would imply that mergers peaked before the peak of star formation, which is only possible if compact-object binary formation is high at low metallicities and if the time delays are short enough [11].",
"Current detectors can only investigate the evolution of the merger rate at low redshift, but in the near future we will be able to trace the redshift evolution of the merger rate density.",
"Figure REF shows that the BNS rate density follows quite closely the star formation rate, with a peak at slightly lower redshift (because of the short but finite time delays).",
"Current observations favor models with low kicks and large $\\alpha $ : as shown by the red shaded region in the top panel of Fig.",
"REF , only low-kick models with $\\alpha =3$ and $\\alpha =5$ can explain the high local merger rates resulting from the detection of GW170817 [48], [36].",
"Most BNS formation models have weak dependence on metallicity.",
"Quite interestingly, some of them show a bimodal distribution, with a dip at $z_m\\approx 5.6$ and a second peak at $z_m\\approx 9$ .",
"Indeed, the efficiency in forming merging BNS has a minimum at intermediate metallicity $Z \\sim 0.1 Z_{\\odot }$ (see e.g.",
"Fig.",
"14 of [36]).",
"Stars at intermediate metallicities tend to develop larger radii, and this leads to the formation of wide BNS systems that either do not merge in a Hubble time, or are easily disrupted by a SN explosion (because of their large orbital separation).",
"However, not all models that show a dip in the merger efficiency lead to a bimodal merger rate density.",
"Since most detectors are not sensitive to binaries from such large redshifts, 3G detectors are needed to observe this behavior in the early Universe.",
"By contrast, BBH production is very efficient at low metallicities because of the impact of metallicity on stellar radii and evolutionary stages.",
"At solar metallicity massive stars become Wolf-Rayet stars quite rapidly, after leaving the giant branch, because of stellar wind efficiency.",
"Wolf-Rayet stars have small radii ($1-2~R_\\odot $ ); thus, it is highly unlikely that such stars enter common envelope.",
"Without common envelope, the binary star evolves into a BBH with a large orbital separation, which will not be able to merge within a Hubble time.",
"In contrast, metal-poor massive stars can retain a large fraction of their hydrogen envelope and avoid the Wolf-Rayet stage, increasing the probability of undergoing mass transfer and entering common envelope.",
"The rate density peaks at $z\\gtrsim 2$ , earlier than the peak of star formation, and the merger rate density at small redshifts is not as steep as the star formation rate (i.e., it has $\\lambda _0<2.4$ ).",
"We should soon be able to verify this trend with current detectors."
],
[
"Detection rates",
"To study how detection rates will benefit from detector improvements, here we will consider noise power spectral densities for the AdLIGO design sensitivity noise [2]; planned upgrades to existing LIGO detectors (A+, A++ and Voyager [4], [3], [5]); and 3G detectors, including CE [6] and the Einstein Telescope (more specifically, ET-B [7]).",
"We approximate the detector noise for the O2 and O3 runs by rescaling the AdLIGO noise curve in such a way that the resulting BNS range is 90 Mpc [1] and 140 Mpc [49], respectively.",
"In Fig.",
"REF we plot the distribution of signal-to-noise ratios (SNRs) for these detectors using the low-kick model with $\\alpha =5$ .",
"Most of the binaries with very large SNRs come from local Universe, so their distribution scales like $1/ \\rho {}^4$ [50].In the local Universe, the total number of binaries within luminosity distance $D_*$ is $N(D<D_*)\\propto D_*^3$ , or equivalently $N(\\rho >\\rho _*)\\ \\propto \\rho _*^{-3}$ , so the SNR probability distribution scales like $N(\\rho _*)=\\frac{dN(\\rho >\\rho _*)}{d\\rho _*}\\propto \\rho _*^{-4}$ .",
"Since CE (and, for BBHs, also ET) will see past the peak of the merger rate density (cf.",
"Fig.",
"REF ), the maximum detection redshift is not controlled by the detector capabilities, but by the physics that governs the merger rate density ${\\mathcal {R}}(z_m)$ .",
"Figure REF shows the detection rates, $R_{\\rm det}$ for different astrophysical models and different detectors, comparing them with the intrinsic merger rate in the Universe that would correspond to an ideal, noiseless detector (see Appendix for details of the detection-rate calculations).",
"According to our models, AdLIGO at design sensitivity could see $220-360$ BBH, up to 9 NSBH and 9 BNS mergers per year.",
"Upgrading AdLIGO detectors to a configuration like A+ would increase the detection rates by a factor of 3.",
"With 3G detectors, BBH rates would increase by up to 2–3 orders of magnitude, while NSBH and BNS detection rates would increase by up to 3–4 orders of magnitude.",
"CE would see at least $92\\%$ of all BBH mergers in the Universe, compared to the $0.06$ –$0.24\\%$ seen by AdLIGO at design sensitivity.",
"Current-generation detectors like AdLIGO have low BNS and NSBH detection rates, detecting only $10^{-5}$ ($\\sim 10^{-4}$ ) of all BNS (NSBH) mergers in the Universe.",
"By contrast, CE will see more than $50\\%$ ($\\sim 75\\%$ ) of all BNS (NSBH) mergers.",
"It is also clear from Fig.",
"REF that $\\alpha $ and $\\sigma _{\\rm CCSN}$ can affect detection rates of all compact binary systems by up to an order of magnitude.",
"In particular, BBH and BNS rates are affected in different ways by the common-envelope efficiency parameter $\\alpha $ : lower values of $\\alpha $ yield smaller rates for BNSs and larger rates for BBHs.",
"This can be understood as follows.",
"BBHs form from massive stars that can develop very large radii during their evolution, and therefore enter the common envelope phase with a wide orbital separation.",
"If $\\alpha >1$ , the envelope will be ejected easily while the binary is still widely separated, and the outcome will be a wide binary that is unlikely to merge in a Hubble time [36].",
"In contrast, BNSs form from smaller stars, and the orbital separation at the beginning of the common envelope phase is smaller.",
"Therefore high values of $\\alpha $ lead to the formation of a close binary that can merge in a Hubble time, while small values of $\\alpha $ cause a premature merger of the system.",
"Low kicks (CC$15\\alpha 1$ , CC$15\\alpha 3$ , CC$15\\alpha 5$ ) lead to higher detections rates for BNS and NSBH mergers, because strong kicks are efficient at disrupting these binaries.",
"On the other hand, most BBH progenitors undergo direct collapse in the models presented here: nearly all of the star's mass falls back onto the compact object, and kicks are suppressed.",
"For this reason, BBH detection rates are nearly insensitive to natal kicks.BBH merger rates are found to strongly depend on SN kicks if fallback is suppressed [24], [51], [35].",
"Local NSBH merger rates for low-kick models are larger than high-kick models by a factor of 3–10.",
"If we assume low (high) SN kicks, the NSBH merger rate increases (decreases) with $\\alpha $ .",
"This is because large SN kicks tend to unbind the binary.",
"If the natal kick is high, a small value of $\\alpha {}$ increases the probability that the system merges, because if $\\alpha $ is small the system's semi-major axis shrinks considerably during CE, after the first supernova.",
"Thus, if the kick is high a small value of $\\alpha {}$ increases the NSBH merger rate.",
"In contrast, if the kick is low, a small value of $\\alpha {}$ might trigger the premature merger of the binary, before the second compact object has formed.",
"Thus, if the kick is low, the highest NSBH merger rate is achieved for a rather large value of $\\alpha {}$ , as already explained in [48].",
"We list minimum and maximum rates across all models in Table REF ."
],
[
"Conclusions",
"We studied the detection rates and redshift evolution of BNS, NSBH and BBH merger rate densities.",
"The redshift distribution of the merger rates contains important clues about the physics that drives the evolution of these compact objects (see also the companion papers [24], [48], [52]).",
"The merger rate history of compact-object binaries is obtained by convolving their formation history with the time-delay distribution.",
"The formation rate depends on both star formation rate and metallicity.",
"The formation of BNSs depends only mildly on metallicity, and therefore their formation across cosmic time follows quite closely the star formation rate (but it is shifted to slightly lower redshifts, because of finite delay times).",
"Therefore for BNSs we expect $\\lambda _0\\gtrsim 2.4$ , i.e.",
"the merger rate peak occurs after, but very close to the peak of star formation.",
"Current detectors have small BNS horizons, so they will mainly see binaries that formed in the local Universe, where metallicity is high, but 3G detectors should allow us to observe large-redshift BNSs and to verify this prediction.",
"In contrast, BBH production (and, marginally, NSBH production) is very efficient at low metallicities.",
"Most BBHs form at $z \\gtrsim 2$ , before the peak of star formation, and their merger rate density evolves slowly compared to BNSs: most BBHs and NSBHs formed before the peak of star formation, yielding $\\lambda _0<2.4$ .",
"Only CE (and, in the case of BBHs, ET) will allow us to see beyond the merger rate peak of compact object binaries.",
"We also investigated how these rates are affected by common-envelope efficiency and natal kicks, considering both second- and third-generation detectors.",
"We found that a lower common envelope efficiency leads to smaller BNS detection rates, and larger BBH detection rates.",
"This is because lower efficiency causes a longer inspiral of the stellar cores, leading to BNS progenitors that merge prematurely, before they can collapse into a neutron star.",
"By contrast, BBH progenitors are much larger, and their orbits are wider compared to BNS progenitors.",
"Natal kick assumptions affects only BNS and NSBH mergers in our models: high kicks can more easily disrupt binaries and usually lead to lower detection rates.",
"On the other hand, BBH kicks are suppressed because of the large amount of material that falls back onto the compact object after the supernova explosion.",
"In Fig.",
"REF we plot the growth of the GW catalog size as detectors improve, based on the rate calculations of Fig.",
"REF .",
"We assume 1 year of observations for O3, which started in 2019.",
"The observing run O4 for AdLIGO at design sensitivity is expected to start in 2021, and it should last for $\\sim 2$ years, followed by 1 year of commissioning period for upgrades to A+ (which is currently targeted to be operational by 2024 [53]).",
"We assume the operational time for A+ to be 6 years [54], with further upgrades to “A++” in 2027.",
"By the beginning of the 2030s, when new detectors – Voyager in the existing LIGO facilities, and CE/ET in separate facilities – may start operations, we could have a GW catalog of up to $10^4$ events.",
"In Fig.",
"REF we assume a 5-year observation period before Voyager is superseded by CE.",
"As the detectors improve, the rapid growth of the GW catalog should allow us to place stringent constraints on the population parameters that influence the final stages of the lives of massive stars.",
"Figure: Growth of catalog size as detectors improve for models in agreement with current observations.",
"The timeline for different detectors and their upgrades is estimated following Refs.",
", , .",
"We assume an optimistic duty cycle of 100%, which is compatible with expectations for future observations with multiple detectors."
],
[
"Acknowledgments",
"MM and YB acknowledge financial support by the European Research Council for the ERC Consolidator grant DEMOBLACK, under contract no.",
"770017.",
"EB and VB are supported by NSF Grant No.",
"PHY-1841464, NSF Grant No.",
"AST-1841358, NSF-XSEDE Grant No.",
"PHY-090003, and NASA ATP Grant No.",
"17-ATP17-0225.",
"This work has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No.",
"690904.",
"The authors would like to acknowledge networking support by the COST Action GWverse CA16104.",
"Computational work was performed on the University of Birmingham's BlueBEAR cluster and at the Maryland Advanced Research Computing Center (MARCC)."
],
[
"Detection rate calculations ",
"The detection rate is given by [56], [57] $R_{\\rm det} = \\int _0^{t_0} p_{\\rm det}{\\mathcal {R}} (z_m) \\frac{dV_c}{dt_m} \\frac{dt_m}{dt_{\\rm det}} dt_m ,$ where $t_0$ is the age of universe and $p_{\\rm det}$ is the probability of detecting a given binary, defined in Eq.",
"(REF ) below.",
"The factor ${dt_m}/{dt_{\\rm det}} = {1}/{(1+z_m)}$ accounts for the different clock rates at merger and at the detector.",
"The source-frame merger rate density at redshift $z_m$ is ${\\mathcal {R}} (z_m) &\\equiv & \\frac{dN}{dV_c dt_m} = \\int _0^{t_m} {\\rm sfr}(z_f)\\frac{dN}{dt_m dM_f} dt_f,$ where the star-formation rate is ${\\rm sfr}(z_f)\\equiv \\frac{dM_f}{dV_c dt_f} $ .",
"The second term in the integrand accounts for the number of binaries per unit star-forming mass that form at $t_f$ and merge at $t_m$ .",
"Here, we have marginalized over the distribution of component masses and time delays.",
"We can rewrite Eq.",
"(REF ) (after switching the order of the integrals over $t_f$ and $t_m$ ) as $R_{\\rm det} &=& \\int _0^{t_0} {\\rm sfr}(z_f)\\frac{d\\quad }{dM_f}\\left( \\int _{t_f}^{t_0}\\frac{dN}{dt_m} \\frac{p_{\\rm det}(z_m)}{1+z_m} \\frac{dV_c}{dt_m} dt_m \\right) dt_f,\\nonumber \\\\&=& \\int _0^{t_0} {\\rm sfr}(z_f)\\frac{d\\quad }{dM_f}\\left( \\sum \\frac{p_{\\rm det}(z_m)}{1+z_m} \\frac{dV_c}{dz_m} \\frac{dz_m}{dt_m} \\right) dt_f.$ In the second line above, we converted the integral over a distribution to a Monte-Carlo sum, $\\int \\frac{dN}{dt_m} f(t_m) dt_m\\rightarrow \\sum _{i} f(t^{i}_{m})\\;.$ In practice, the term in parentheses is evaluated by Monte Carlo integrations, where the samples $t^{i}_{m}$ are generated from the distribution ${dN}/{dt_m}$ .",
"The comoving volume element $dV_{\\rm c}/dz$ is given by $\\frac{dV_c}{dz}(z) =4 \\pi \\frac{c}{H_0} \\frac{D_{\\rm c}^2 }{ E(z)},$ where $E(z)$ is the function that describes the evolution of Hubble parameter, i.e.",
"$H(z)=H_0 E(z)$ , and $D_{\\rm c}$ is comoving distance [58].",
"The factor of $4\\pi $ takes into account the angular integration over the sky.",
"In practice, at a given metallicity $Z_f$ , MOBSE starts with a given total mass $M_{\\rm sim}$ and outputs a distribution of binaries.",
"For each set of free parameters in Table REF , we have 12 simulations of $10^7$ binaries each, with metallicities $Z= 0.01$ –$1\\,Z_\\odot $ .",
"We simulate a set of compact-object binaries formed at different times $t_f$ inside bins of $\\Delta t_f=10{\\rm ~Myr}$ .",
"At the time of formation $t_f$ , we assume that the metallicity is given by $\\log {\\frac{Z(z_f)}{Z_{\\odot }}}={\\left\\lbrace \\begin{array}{ll}-0.19\\ z_f, & z_f \\le 1.5\\\\-0.22\\ z_f, & z_f > 1.5\\,,\\end{array}\\right.",
"}$ i.e.",
"we follow the metallicity evolution of Ref.",
"[59], but we rescale it so that $Z(0)=Z_\\odot $ .",
"Each formation time bin is assigned one the 12 metallicities according to Eq.",
"(REF ).",
"However, since the MOBSE simulation started with total binary mass, $M_{\\rm sim}$ , we need to rescale this mass according to the star formation in that particular time bin.",
"We have adopted the following fit for star formation rate [60]: ${\\rm sfr}(z) = \\frac{0.015(1+z)^{2.7}}{1 + [(1+z)/2.9]^{5.6}} \\, M_{\\odot } {\\rm Mpc}^{-3} .$ These binaries are then evolved in time until they merge at $t_m$ .",
"This produces a catalog of binaries that form at $t_f$ and merge at $z_m$ .",
"The integral in Eq.",
"(REF ) can be now be written as $R_{\\rm det}=\\sum _i ( s_i(t_f)\\Delta t_f ) {\\frac{p_{\\rm det}}{ 1+z_m}} {\\frac{dV_{\\rm c}}{dz_m}} {\\frac{dz_m}{dt_m}} \\,,$ where all terms except the first are evaluated at the merger redshift $z_m$ .",
"The first term is the number density of binaries formed at redshift $z_f$ , $s_i(z_f)\\Delta t_f = f_{\\rm bin} f_{\\rm IMF} \\frac{ {\\rm sfr}(z_f)}{M_{\\rm sim}(Z_f)} \\Delta t_f \\,.$ The factors $f_{\\rm bin}=0.5$ and $f_{\\rm IMF}=0.285$ take into account the fact that MOBSE only simulates binaries with primary mass larger than $5 M_{\\odot }$ .",
"Finally, a binary is assumed to be detected if it has the signal-to-noise ratio (SNR) $\\rho =\\rho _0 w>8$ , where $\\rho _0$ is the SNR assuming that the binary is optimally oriented and located in the sky, while $0\\le w \\le 1$ is the projection factor that depends on the binary's sky position and orientation.",
"The optimal SNR is calculated as $\\rho ^2_0 = 4\\int _0^\\infty \\frac{\\tilde{h}^*(f) \\tilde{h}(f)}{S_h(f)}df\\,,$ where $h(f)$ is the frequency-domain GW signal and $S_h(f)$ is the detector noise power spectral density [61], [62].",
"The horizon $z_h$ is the farthest redshift for which a binary with component masses $m_1$ and $m_2$ can be detected, i.e.",
"$\\rho _0(m_1,m_2,z_h)=8$ .",
"The quantity $\\rho _0$ determines the probability of detecting a binary that lies within the detector's horizon (i.e.",
"$\\rho _0>8$ , or equivalently $z<z_h$ ): $p_{\\rm det}=\\int _{8/\\rho _0}^1 p(\\omega ) d\\omega $ where $p(w)$ is the probability distribution function of $\\omega $ [63].",
"Detection rates only depend on $p_{\\rm det}$ , hence $\\rho _0$ .",
"We calculate the signal-to-noise ratio of BBH mergers using the waveform approximant IMRPhenomD, while for NSBH and BNS mergers we use TaylorF2.",
"Since MOBSE does not have any prescriptions to evolve the spins, we assume black holes and neutron stars to be non-spinning.",
"Spins are expected to impact detection rates within a factor 1.5 [35], which should be added to the error budget of our estimates.",
"Note that in Fig REF , where we looked at the distribution of $\\rho = \\rho _0 w$ , we sample $p(\\omega )$ for each binary in the catalogs mentioned above and assign the SNR accordingly."
]
] | 1906.04197 |
[
[
"Revisiting the 16 Cygni planet host at unprecedented precision and\n exploring automated tools for precise abundances"
],
[
"Abstract The binary system 16 Cygni is key in studies of the planet-star chemical composition connection, as only one of the stars is known to host a planet.",
"This allows us to better assess the possible influence of planet interactions on the chemical composition of stars that are born from the same cloud and thus, should have a similar abundance pattern.",
"In our previous work, we found clear abundance differences for elements with Z$\\leq30$ between both components of this system, and a trend of these abundances as a function of the condensation temperature (T$_{c}$), which suggests a spectral chemical signature related to planet formation.",
"In this work we show that our previous findings are still consistent even if we include more species, like the volatile N and neutron capture elements (Z $>$ 30).",
"We report a slope with T$_{c}$ of $1.56 \\pm 0.24 \\times 10^{-5}$ dex K$^{-1}$, that is good agreement with both our previous work and recent results by Nissen and collaborators.",
"We also performed some tests using ARES and iSpec to automatic measure the equivalent width and found T$_c$ slopes in reasonable agreement with our results as well.",
"In addition, we determine abundances for Li and Be by spectral synthesis, finding that 16 Cyg A is richer not only in Li but also in Be, when compared to its companion.",
"This may be evidence of planet engulfment, indicating that the T$_{c}$ trend found in this binary system may be a chemical signature of planet accretion in the A component, rather than a imprint of the giant planet rocky core formation on 16 Cyg B."
],
[
"Introduction",
"The most accepted hypothesis of star formation is the nebular collapse, when stars are formed from gravitationally unstable molecular clouds.",
"Therefore, it is expected that stars that are born from the same interstellar material cloud should have the same abundance pattern during the Main Sequence, with exception of the light elements Li and Be, that can be destroyed in regions deeper than the convective zone, in solar type stars.",
"Thereby, differences in the chemical content of stars born from the same natal cloud, may suggest that extra processes, that are not necessary connected to stellar evolution, may have influenced its photospheric chemical composition.",
"In particular, planet formation or planet engulfment may imprint important chemical signatures in the host star.",
"Such phenomenon is expected to leave subtle signs on the stellar abundance pattern, of the order of 0.01 dex [15], that can only be detected with a high-precision analysis.",
"This can only be achieved with the differential method [57], that requires a comparison between the target star with a similar star of known parameters, that will serve as the standard for the abundance calculations.",
"So, the sample should be restricted to objects that are very similar among themselves, like in the case of solar twinsMore recently, solar twins have been defined as stars with effective temperature within 100K; log g and [Fe/H] within 0.1 dex from the Sun [69]..",
"Following this premise, [47] analyzed the abundances of 11 solar twins, achieving a high-precision abundance determination, with uncertainties of $\\sim $ 0.01 dex, and found not only a depletion of refractory elements, when compared to the average of the sample, but also a trend with condensation temperature (T$_{c}$ ).",
"The authors suggested that the correlation of the refractory elements abundances with condensation temperature is probably due to rocky planet formation.",
"This hypothesis has been corroborated by [15], who showed that the depleted material in the Sun's convective zone is comparable to the mass of terrestrial planets of our Solar System (see also [23]).",
"However, other hypothesis have been proposed to explain the abundance trend, like the stellar environment in which the star was formed [60], although according to the recent theoretical estimates by [28], the mechanism is hardly significant; dust segregation in the protostellar disc [22]; the influence of the stellar age [1]; and the planet engulfment scenario [83].",
"In this context, twin stars in binary systems are extremely important, because the effects connected to the stellar environment of its formation and to the Galaxy chemical evolution, would be canceled out in a comparative analysis between both components.",
"Thus, investigating wide binaries can bring more light into the subject of planets interactions (or other astrophysical event) influencing photospheric abundance of their host stars.",
"Some authors have already reported a T$_{c}$ trend on binary stars.",
"[91] found an abundance trend on the WASP94 system, where both stars are planet hosts.",
"The planet-hosting binaries XO-2N/XO-2S [70], [11], [90], HD 133131A/B [92] and HAT-P-4 [74], also show chemical anomalies most likely due to planets, and the binary $\\zeta ^{1,2}$ Ret, where one of the stars hosts a debris disk, also shows a trend with condensation temperature [73], [2].",
"Albeit no differences have been found in the HAT-P-1 [33], HD80606/HD80607 [72], [43] and HD20782/HD20781 [42] and HD 106515 [75], the evidences are inconclusive in the latter three due to the high abundance errors.",
"Indeed, a more precise abundance analysis of the pair HD80606/HD80607 by [36], shows small but detectable abundance differences between the binary components.",
"A binary system of twin stars with large abundance differences, is HD 240429/HD 240430 [59], for which no planets are known yet.",
"Furthermore, it was found that Kepler-10, a star hosting a rocky planet, is deficient in refractory elements when compared to stars with similar stellar parameters and from the same stellar population [34].",
"For 16 Cygni, a binary pair of solar twins, where the B component hosts a giant planet [16], [32] clearly detected that 16 Cyg A is more metal rich $\\Delta $ [Fe/H] = +0.025$\\pm $ 0.009 dex than its companion.",
"Later, [67] expanded the analysis to 23 chemical elements, showing abundance differences in all of them by about +0.04 dex and finding a T${_C}$ trend similar to [47], when the binary stars are compared to the Sun.",
"This was confirmed in our previous work [93], where we show that 16 Cyg A is $0.047 \\pm 0.005$ dex metal richer than B and also finding a T$_{c}$ slope of $+1.99 \\pm 0.79 \\times 10^{-5}$ dex K$^{-1}$ for the refractory elements, as reported in [67]).",
"This result was then associated with the rocky core formation of the gas giant 16 Cyg Bb.",
"Recently, [56] also found a $\\Delta $ [Fe/H](A-B)= $+0.031 \\pm 0.010$ dex and a T$_{c}$ slope of $+0.98 \\pm 0.35 \\times 10^{-5}$ dex K$^{-1}$ .",
"In contrast, there are studies that challenge the metallicity difference and the T$_c$ trend between the two components of this system.",
"[78] find a T$_{c}$ trend for both stars relative to the Sun but, however, do not find any significant abundance differences between the pair, in agreement with [18] and [88].",
"Also, [2], analyzing the case of $\\zeta ^{1,2}$ Ret, argues that the T$_c$ slope trend could be due to nonphysical factors and related to the quality of spectra employed, which is expected as high precision abundances can only be obtained in spectra of adequate quality.",
"In this context, the initial motivation for this work is to assess if, by revisiting this binary system now with better data with higher resolving power, higher S/N and broader spectral coverage, our previous results (obtained with lower resolving power) would still be consistent; in addition to provide improvements in the precision of the abundance determination, we include the analysis of elements that were not available before.",
"We also challenge our results by employing automated tools to derive stellar parameters and T$_c$ while using the same methodology in all of the cases.",
"On the following sections, we will present the differential abundances of 34 elements and also the abundances of Li and Be, through spectral synthesis, which may present a possible evidence of planetary engulfment on 16 Cyg A.",
"The observations of 16 Cyg A and B were carried out with the High Dispersion Spectrograph [58] on the 8.2m Subaru Telescope of the National Astronomical Observatory of Japan (NAOJ), located at the Mauna Kea summit, in June 2015.",
"Besides the 16 Cyg binary system, we also observed the asteroid Vesta, which was used as an initial reference for our differential analysis.",
"For the optical, we obtained an S/N ratio of $\\sim $ 750 at 600nm and $\\sim $ 1000 at 670nm (Li region), on the highest resolution possible (R$\\sim $ 160 000) using the 0.2\" slit.",
"The UV observations with HDS were made using the 0.4\" slit, which provides a R = 90 000 that results in an S/N$\\sim $ 350 per pixel at 340 nm, corresponding to the NH region, and S/N$\\sim $ 200 at 310 nm (Be region).",
"This gave us the opportunity to analyze volatile elements like nitrogen and neutron-capture elements in the UV with S/N $>$ 300.",
"The stars from the binary system and the Sun (Vesta) were both observed using the same instrumental setup to minimize errors in a differential analysis, which requires comparisons between the spectra of all sample stars for the continuum placement and comparison of the line profiles, to achieve consistent equivalent width (EW) measurements.",
"The extraction of the orders and wavelength calibration were performed immediately after the observations by Subaru staff, with routines available at the observatory.",
"The continuum normalization and Doppler correction were performed using standard routines with IRAF."
],
[
"Stellar parameters",
"Our method to determine stellar parameters and elemental abundances follows the approach described on previous papers [67], [69], [47], [48], [94], [85], by imposing differential excitation and ionization equilibrium for Fe I and Fe II lines (Figure REF ).",
"Since the 16 Cygni system is a pair of solar twins, they have similar physical characteristics to the Sun, and thus we initially used the Sun as a reference for our analysis.",
"The abundance determination was performed by using the line-by-line differential method, employing the EW manually measured by fitting Gaussian profiles with the IRAF splot task and deblending when necessary.",
"Very special care was taken for the continuum placement during the measurements, always comparing and overplotting the spectral lines region for the sample, focusing on a consistent determination.",
"With the measured EW, we first determined the Fe I and Fe II abundances to differentially obtain the stellar parameters.",
"For this we employed the 2014 version of the LTE code MOOG [80] with the MARCS grid of 1D-LTE model atmospheres [27].",
"It is important to highlight that the choice of a particular atmospheric model has a minor impact on the determination of stellar parameters and chemical abundances in a strictly differential analysis, as long the stars that are being studied are similar to the star of reference [94].",
"Figure: Excitation and ionization equilibrium of Fe abundances (manually measured) using the Sun as the standard star for 16 Cyg A.",
"Crosses represent Fe I and filled circles Fe II.To make the analysis more efficient, we employed the Python q${\\rm ^2}$ codehttps://github.com/astroChasqui/q2 [69], which operates in a semi-automatic mode, by calling MOOG routines to determine the elemental abundances and perform a line-by-line differential analysis using these results.",
"This code also performs corrections of hyperfine structure (HFS) and the determination of uncertainties.",
"In this work, we take into account the HFS for V, Mn, Co, Cu, Y, Ag, La and Pr using the line list from [48].",
"The errors are computed considering both observational (due to uncertainties in the measurements, represented by the standard error) and systematic uncertainties (from the stellar parameters and their inter-dependences), as described in in [70].",
"Observational and systematic errors are added in quadrature.",
"Table REF shows the stellar parameters obtained for 16 Cyg A and B using the Sun (T$_{eff}$ = 5777 K, log g = 4.44 dex, [Fe/H] = 0.0 dex) as reference.",
"Note that these results are practically the same within the errors as the ones found in [93], which are T$_{eff}$ = 5830 $\\pm $ 7 K, log g = 4.30 $\\pm $ 0.02 and [Fe/H] = 0.101 $\\pm $ 0.008 dex for 16 Cyg A, and T$_{eff}$ = 5751 $\\pm $ 7 K, log g = 4.35 $\\pm $ 0.02 and [Fe/H] = 0.054 $\\pm $ 0.008 dex for 16 Cyg B.",
"The final difference in metallicity between the components of this binary system is remarkably similar to the one reported in [93], that is $\\Delta $ [Fe/H] = 0.047 $\\pm $ 0.005 dex, while we find in this work $\\Delta $ [Fe/H] = 0.040 $\\pm $ 0.006 dex.",
"This confirms with a significance of $\\sim 7\\sigma $ that 16 Cyg A is indeed more metal rich when compared to 16 Cyg B, in agreement with [67] and the earlier work by [32], as well as the recent work by [56]."
],
[
"Trigonometric surface gravity",
"New parallaxes for the binary stars of 16 Cygni have been measured by the Gaia mission DR 2 [21].",
"The new values are $47.2771 \\pm 0.0327$ mas and $47.2754 \\pm 0.0245$ mas for 16 Cyg A and B, respectively.",
"Adopting the magnitudes from the General Catalogue of Photometric Data [50] with V(A) = $5.959 \\pm 0.009$ and V(B) = $6.228 \\pm 0.019$ , we determined the absolute magnitudes $M_{A} = 4.332 \\pm 0.012$ and $M_{B} = 4.599 \\pm 0.026$ .",
"Using this information with the values of the T$_{eff}$ , metallicity and mass, we estimate the trigonometric surface gravity for the pair of stars.",
"For 16 Cyg A we found log g($A_{T}$ ) = 4.293 $\\pm $ 0.005 dex and for 16 Cyg A log g($B_{T}$ ) = 4.364 $\\pm $ 0.006.",
"Notice that, while the surface gravity for 16 Cyg B has a good agreement with the one found through the ionization equilibrium of Fe lines (Table REF ), for 16 Cyg A the trigonometric value is $\\sim $ 0.02 dex lower, albeit they agree within 1.5 $\\sigma $ .",
"Comparing with the results of [67], both the trigonometric and Fe-lines-based surface gravity are in agreement for the A component, while our surface gravities for B are somewhat higher in both cases."
],
[
"Age, mass and radius",
"The age and mass of the binary stars were determined using customized Yonsei-Yale isochrones [98], as described in [68], [69].",
"This method provides good relative ages, due to the high precision achieved for the atmospheric parameters.",
"We estimate the ages and masses with probability distribution functions, through the comparison of atmospheric parameters position of the star with the values predicted by the isochrones.",
"Initially, the calculations were based on the [Fe/H], T$_{\\rm eff}$ and log $g$ , and later we replaced the gravity by the parallax values and magnitudes, to obtain the isochronal ages using the absolute magnitudes.",
"The results are shown in Table REF .",
"Our masses and radii shows a very good agreement when compared to the asteroseismology determinations of M$_{A} = 1.08 \\pm 0.02 M_{\\odot }$ , M$_{B}= 1.04 \\pm 0.02 M_{\\odot }$ , R$_{A}= 1.229 \\pm 0.008 R_{\\odot }$ and R$_{B}= 1.116 \\pm 0.006 R_{\\odot }$ , as reported by [51].",
"Table: Stellar parameters for the 16 Cygni binary system using EW measured manuallyThe inferred isochronal ages of A and B based on log $g$ , are 6.0 $\\pm $ 0.3 Gyr and 6.7 $\\pm $ 0.4 Gyr, respectively.",
"This shows that both components of the system have roughly the same age (within error bars).",
"We have also estimated the ages of 16 Cyg A and B using the correlation of [Y/Mg] and [Y/Al] as a function of stellar age.",
"The abundance clock [Y/Mg] for solar-type stars was first suggested by [79] and the correlation between [Y/Mg] or [Y/Al] and stellar age was quantified for solar twins by [55], [94] and [85] Notice that the correlation between [Y/Mg] and age is only valid for solar-metallicity stars [20].",
"The derived [Y/Mg] ages are A= 6.2 Gyr and B= 6.3 Gyr using the relation of [94].",
"Similar ages are found using the [85] relations, A= 6.0 $\\pm 1.0$ Gyr and B= 6.1 $\\pm 1.0$ Gyr.",
"These results are consistent with the values calculated using the isochronal method, while the [Y/Al] ages [85] give A =6.6 $\\pm 1.0$ Gyr and B= 6.8 $\\pm 1.0$ Gyr.",
"The values of age, mass and radius are in agreement with asteroseismic with values around 7 Gyrs [71], [10], with 16 Cyg A being slightly more massive and with bigger radius than its companion."
],
[
"Activity",
"The chromospheric activity is an important constrain on stellar ages [39].",
"In order to measure the activity differences between 16 Cyg A and B, we defined an instrumental activity index based on H$\\alpha $ line which is a well-known chromospheric indicator of late-type stars [61], [41], [54]: $\\mathcal {H} = \\frac{F_{\\rm H\\alpha }}{(F_{\\rm B}+F_{\\rm V})},$ where $F_{\\rm H\\alpha }$ is the flux integrated around the H$\\alpha $ line ($\\Delta \\lambda $ = 6562.78 $\\pm $ 0.3 Å).",
"We chose this narrow spectral interval to minimize the effective temperature effects that might be present along the H$\\alpha $ wingsSmall residual photospheric effects are still expected to be affecting our index measurements, however, this residual feature should have negligible impact on our results since we are not interested in absolute activity scale determination for a wide range of effective temperatures.. $F_{\\rm B}$ and $F_{\\rm V}$ are the fluxes integrated around 0.3 Å continuum windows, centered at 6500.375 and 6625.550 Å, respectively.",
"In table REF , we show the estimated $\\mathcal {H}$ for 16 Cyg AB and the Sun.",
"The uncertainties were estimated by quadratic error propagation of equation REF , assuming Poisson error distribution.",
"Table: Activity indexes for 16 Cyg A, B, and the Sun.",
"The last row is the mean activity level of 16 Cyg AB.Accordingly to $\\mathcal {H}$ , none of 16 Cyg components show unexpected level of chromospheric activity for a typical 6-7 Gyr-old star [45].",
"Furthermore, 16 Cyg A and B seem to be chromospherically quiet stars ($\\mathcal {H}$ = 0.188 $\\pm $ 0.001) and slightly more inactive than the Sun ($\\mathcal {H}$ = 0.1909 $\\pm $ 0.0019), indicating a chromospheric age older than 4-5 Gyr.",
"This result is in line with Ca II H & K multi-epoch observations of [30] who found $\\log (R^\\prime _{\\rm HK})$ $\\approx $ -5.05 dex for this system, in good agreement with the mean activity level of $\\log (R^\\prime _{\\rm HK})$ = -5.03 $\\pm $ 0.1 dex derived for 49 solar-type stars from the 6-7 Gyr old open cluster NGC 188 [38].",
"We inspected the chromospheric signature of other classical indicators along the spectral coverage of our observations such as Ca II H & K [45], [38], H$\\beta $ [54] and Ca II infrared triplet [37].",
"All of them show the same behavior found by H$\\alpha $ lines, 16 Cyg A and B are chromospherically older than the Sun (age $>$ 4-5 Gyr) and the activity differences between the components are negligible.",
"In summary, the different activity indicators reinforce the age results from isochrones and seismology."
],
[
"Abundance analysis",
"We present high-precision abundances for the light elements C, N, O, Na, Mg, Al, Si, S, K, Ca, Sc, Ti, V, Cr, Mn, Co, Ni, Cu and Zn; and the heavy elements Sr, Y, Zr, Ba, Ru, Rh, Pd, Ag, La, Ce, Nd, Sm, Eu, Gd, and Dy.",
"The abundances of these elements were differentially determined using initially the Sun as our standard star and then using 16 Cyg B as the reference to obtain the $\\Delta [$ X/H$]_{(A-B)}$ .",
"The calculations were performed with the same method as described for the iron lines (see also [94]).",
"Taking into account only the elements with Z $\\le 30$ , there is a clear chemical trend as a function of the condensation temperature (T$_{c}$ ) in the pattern of both stars relative to the Sun (Figure REF ), in agreement with [67], [78] and [93].",
"In addition to results based on atomic lines, abundances for the volatiles elements C, N, and O were also determined using the molecules CH, NH and OH (red triangles in Figures REF and REF ).",
"There is a very good agreement between the C and O abundances based on high excitation atomic lines and low excitation molecular lines, while for N we only present the abundance based on NH.",
"The excellent agreement between atomic and molecular-based differential abundances reinforces the reliability of our adopted atmospheric parameters."
],
[
"Abundance vs. condensation temperature trend",
"A possible indication of rocky planet formation (or planet engulfment) can be found in the distribution of the differential elemental abundances as a function of condensation temperature.",
"Refractory elements have high condensation temperature (T$_{C} \\gtrsim $ 900 K), easily forming dust, being thus an important component of rocky bodies.",
"Terrestrial planets (or the core of giant planets) may influence the surface abundance of its host star in two ways: ${\\it i)}$ the accretion of rocky material (planetary engulfment) depleted of hydrogen that enrich the stellar atmosphere in refractories [83], [49], [63]; ${\\it ii)}$ imprisonment of refractory rich material into rocky objects (i.e, planetesimals, rocky planets, core of giant planets), that deplete the material accreted by the star during its formation [47], [67], [93].",
"In the case of planet engulfment, the thermohaline mixing should dilute the overabundance in a few milion year [89], however, the thermohaline mixing could not be as effective and still leave some enhancement on the outer layers of the star, that can only be detected with a precision of $\\sim 0.01$ dex.",
"An important point to highlight is that the signature of planet formation or planet engulfment is directly connected to the size of the convective zone during the event.",
"If a solar-mass protostar would go through a fully convective phase that would last longer than the lifetime of the protoplanetary disk [29], any event of planetary formation that occur during such phase would be masked by a significant dilution with the stellar material enclosed into the convective zone, which would homogenize the chemical content throughout the star [83].",
"In contrast to the classic steady accretion, there is the scenario of episodic material accretion onto the star (with observational evidence reported by [35]).",
"Models that include episodic accretion can reach the stabilization of the convective zone earlier than 10 Myr with initial mass of $10 M_{Jup}$ and accretion rate bursts of $5 \\times 10^{-4} M_{\\odot }$ yr$^{-1}$ , reaching a final mass of 1 $M_{\\odot }$ [6].",
"Although is an extreme of their models, it is important to highlight that due to the effects of episodic accretion, the higher the mass of the accretion rate bursts for a given initial mass (or lower the initial mass for a given accretion rate) the greater the impact on the internal structure, reaching the necessary central temperature for the development of the radiative core ($\\sim 2 - 3 \\times 10^{6}$ K) earlier than what is predicted by the model of a non accreting star [6].",
"This effect makes plausible the assumption that the formation of rocky bodies can chemically alter the surface abundance pattern of its parent star.",
"Following this premise, [47] suggested that the depletion of refractory elements in the Sun, when compared to a sample of 11 solar twins (without information regarding planets), is due to the formation of terrestrial planets in the solar system (see also further work by [65], [66]).",
"However, in the literature, there are different suggestions for the Sun's abundance trend with condensation temperature.",
"[1] proposed that the trend with condensation temperature is an effect of the chemical evolution of the Galaxy or depends on the star's birthplace.",
"Investigating the influence of age on solar twins, [55] found a strong correlation of $\\alpha $ and s-process elements abundances with stellar age, findings that were confirmed by [94] and [85].",
"According to [60], if the star is formed in a dense stellar environment, the gas of the proto-stellar disk could have its dust cleansed before its birth by radiation of hot stars in the cluster, but recent theoretical estimates by [28], suggest that the mechanism is not significant.",
"[22] associate this effect to the gas-dust segregation in the protoplanetary disk.",
"[44] find differences in the T$_{c}$ -slopes of refractory elements between stars with and without known planets, but this effect depends on the evolutionary stage, since it has been detected on main-sequence and subgiant stars, while no trend is found in their sample of giants.",
"The authors also suggest that there is a correlation of both the mass and age, with T$_{c}$ .",
"In this context, the investigation of abundance peculiarities in binary stars with and without planets is essential, because in a binary system there is no effect due to the chemical evolution of the Galaxy and other external factors, because it would equally affect both stars and thus be minimized in a differential analysis.",
"In this sense, the 16 Cygni system is a very interesting case, where both components are solar twins with the same age from asteroseismology [71].",
"On top of that, 16 Cyg B has a detected giant planet with a minimal mass of 1.5 Jupiter mass [16] while 16 Cyg A has no planet detected up to now, being thus a key target to study the effect of planets on the chemical composition of stars.",
"However, the abundance pattern of 16 Cyg A relative to B is still a controversy.",
"A few authors suggest that there is no difference on the metallicity of the pair [18], [78], [88], while most found abundance differences of about 0.05 dex [56], [2], [52], [93], [67], [32], [25]."
],
[
"16 Cygni",
"A linear fit was performed with orthogonal distance regression (ODS) using the individual abundance errors for each element, excluding K due to its uncertain non-LTE effects.",
"It was necessary to assume a minimum threshold for the abundances uncertainties because some species were returning very small error bars (0.001 dex), heavily impacting the abundance vs. condensation temperature slope, because some species does not have many lines.",
"In order to address this issue, we adopt a minimum abundance error of 0.009 dex, which is the average error of all species analyzed.",
"We obtain the slopes $3.99 \\pm 0.58\\times 10^{-5}$ dex K$^{-1}$ and $2.78 \\pm 0.57\\times 10^{-5}$ dex K$^{-1}$ for 16 Cyg A $-$ Sun, and 16 Cyg B $-$ Sun, vs condensation temperature, respectively.",
"In contrast to our past work [93], we do not break the linear fit into two distinct curves for the volatiles and refractory elements, as a simple linear fit represents well the trend with T$_{c}$ .",
"We include nitrogen from NH, and for the abundances of C and O we assumed the average between the molecular and atomic abundances.",
"Figure: Elemental abundances of 16 Cyg A (upper panel) and B (lower panel) based on our manually measured solar abundances as a function of condensation temperature for light elements (Z≤30\\le 30).",
"Solid linesrepresent the linear fits with a slope of 3.99±0.58×10 -5 3.99 \\pm 0.58\\times 10^{-5} for the A component and 2.78±0.57×10 -5 2.78 \\pm 0.57\\times 10^{-5} for 16 Cyg B.Red triangles correspond to the molecule-based abundances of C, N, and O.In Figure REF we plot the abundances of the heavy elements (Z $>$ 30).",
"In this case, the abundances do not clearly follow the same trend as in the previous case, with slopes $-0.16 \\pm 3.99 \\times 10^{-5}$ dex K$^{-1}$ and $-0.05 \\pm 2.97 \\times 10^{-5}$ dex K$^{-1}$ for 16 Cyg A $-$ Sun and 16 Cyg B $-$ Sun, repectively, with a minimum uncertanty threshold of 0.02 dex in both cases.",
"However, due to the large errors in the [X/H] and the small range in T$_{c}$ it is not possible to claim if there is indeed a trend by considering only the heavy elements, but we stress that, within the uncertainties, the slope is not actually different from those of Z $\\le $ 30.",
"Although no T${c}$ trend is detected, there is a difference $\\Delta $ (A-B) = 0.043 $\\pm $ 0.075 dex regarding the abundances of these heavy elements, that somewhat follow the difference of Fe between the stars of the pair, however, due to the high uncertainty we cannot conclude if this discrepancy is real.",
"Figure: As Figure but for the heavy elements (Z >> 30).",
"There is no clear trend with condensation temperature.The abundances of 16 Cyg A relative to 16 Cyg B were also determined and are presented in Figure REF .",
"There is an evident trend between the (A-B) abundances and T$_{c}$ .",
"The slope of the linear fit (without including the n-capture elements) is $1.56 \\pm 0.24 \\times 10^{-5}$ dex K$^{-1}$ (with a threshold of 0.005 dex).",
"This result agrees with Tucci Maia et al.",
"(2014; slope = $1.88 \\pm 0.79 \\times 10^{-5}$ dex K$^{-1}$ ) within error bars, showing once again the consistency and robustness of our analysis.",
"If we include the heavy elements on the fit, we find a slope of $1.38 \\pm 0.41 \\times 10^{-5}$ (with a threshold of 0.010 dex).",
"Although the abundance of potassium presented in Table REF has been corrected for non-LTE effects using the grid by [87], we did not use it for the linear fit as the non-LTE grid is too sparse for a precise correction.",
"Our slope is also in good agreement with the recent result by [56], $+ 0.98 \\pm 0.35 \\times 10^{-5}$ dex K$^{-1}$ , , based on high-resolution high-S/N HARPS-N spectra.",
"Figure: Differential abundances (manually measured) of (A - B) as function of T c _{c} for elements with Z ≤\\le 30 (left panel) and adding also the neutron-capture elements (right panel).",
"The red triangles correspond to the molecule-based abundances of C, N and O.",
"The slope found is 1.56±0.24×10 -5 1.56 \\pm 0.24 \\times 10^{-5} dex K -1 ^{-1}, based on the fit to the elements with Z ≤\\le 30.Table: Elemental abundances of 16 Cygni system relative to the Sun and to 16 Cyg B."
],
[
"Automated codes",
"We conducted tests utilizing iSpec version 2016 [12] and ARES v2 [82] to automatically measure the EWs of 16 Cyg A, B and the Sun, in order to differentially determine the stellar parameters.",
"The aim of these tests is to evaluate if these codes, when applied to high-resolution data and following our methodology (same as in Section 2.2), could return a similar result to what we find by \"hand\".",
"Our motivations for this is to assess if our procedure could be automatized and applied to a bigger sample of stars and still retrieve stellar parameters with the same precision as ours, and to also find out if the chemical composition differences that we found between the 16 Cygni components is consistent by applying different methods of EW meassurement.",
"As discussed earlier, the differential method minimize most of the error sources while, for a solar twin sample, the uncertainty is almost entirely related to the EW.",
"One big concern in a differential analysis is the continuum normalization to achieve a consistent continuum placement for all stars being analyzed.",
"In this test, the spectra is the same as our manual analysis, which was previously normalized, and the EW measured following the same line list as ours.",
"In this way, any discrepancy in the values would be due to how each code interprets and places the continuum, and how the fit is performed.",
"Table: Atmospheric parameters for 16 Cygni determined with automated EW measurements for R= 160 000 and 81000 spectra.In addition to that, we use another set of spectra with lower resolving power (R $\\sim $ 81 000, from [93]), in order to evaluate if by using the same tools but providing different resolution spectra, the results could be somewhat discrepant.",
"In Table REF we present the determined stellar parameters obtained using the EW measurements obtained with the automated codes.",
"Comparing the results, we find that all the codes return stellar parameters in good agreement to ours, based on the \"manual\" measurements.",
"In overall, as we go to lower resolution, the uncertainty gets higher, as expected, because it can lead to more blends around the lines measured and thus a more contaminated value for the EW, a phenomenon that also happens with manual measurements as well, as we can see from the stellar parameters from our previous work [93].",
"Table: Slopes of abundances versus condensation temperature, for the elements with Z ≤\\le 30 for the EWs measurements from iSpec and ARES for the R ∼\\sim 160 000 and 81 000 spectra.We also used iSpec and ARES to determine the elemental abundance of our sample for the elements with Z $\\le $ 30, with the same method and spectra described in the previous section.",
"The trends with condensation temperature are presented in Table REF with its respective abundance thresholds.",
"In both resolution sets iSpec and ARES returns a T$_{c}$ slope that is in agreement with our value within error bars.",
"The codes confirm not only that 16 Cyg A is more metal rich than B, but also the existence of T$_c$ trend even though we use spectra with almost half the resolving power (but still high-resolution spectra) as the other.",
"However, iSpec shows a higher significance on its values."
],
[
"Li and Be",
"Lithium and beryllium abundances were determined by performing spectral synthesis calculations, using a method similar to the outlined in [95].",
"For lithium, we used the Li-7 doublet at 670.7 nm and, for beryllium, we used the doublet resonance lines of Be II at 313.0420 nm and 313.1065 nm.",
"The line list for the Li synthesis is from [48], while for Be we used a modified list of [3], as described in [95].",
"For the spectral synthesis, we used the synth driver of the 2014 version of the 1D LTE code MOOG [80].",
"We adopted A(Be) = 1.38 dex as the standard solar Be abundance from [5].",
"The model atmospheres were interpolated from the MARCS grid [27] using the stellar parameters previously obtained.",
"The abundances of Li were corrected for non-LTE effects using the online grids of the INSPECT projecthttp://inspect-stars.com/.",
"Beryllium lines are insensitive to non-LTE effects in the solar type stars, according to [4].",
"To determine the macroturbulence line broadening, we first analyzed the line profiles of the Fe I 602.7050 nm, 609.3644 nm, 615.1618 nm, 616.5360 nm, 670.5102 nm and Ni I 676.7772 lines in the Sun; the synthesis also included a rotational broadening of v sin i = 1.9 km.s$^{-1}$ [13] and the instrumental broadening.",
"The macroturbulent velocity found for the Sun is V$_{macro}$ = 3.6 km s$^{-1}$ .",
"For 16 Cygni, we estimate the macroturbulence following the relation of [76], which takes into account the dependence with effective temperature and log g. With the macroturbulence fixed, v $\\sin i$ was estimated for 16 Cyg A and B by fitting the profiles of the six lines mentioned above, also including the instrumental broadening.",
"Table REF shows the abundances of Li and Be with their estimated macroturbulence and v $\\sin i$ .",
"Table: Abundances of Li and Be for the binary 16 Cyg using spectral synthesisIn Figure REF we show the synthetic spectra of 16 Cyg A and B plotted against the observed spectra.",
"Figure: Comparison between the observed (blue dots) and synthetic (red solid line) spectra of 16 Cyg A (top) and 16 Cyg B (bottom).Comparing our results with previous works in the literature, we found that Li and Be on the binary system 16 Cygni is hardly a consensus, but for lithium there is a qualitative agreement in the A component being more abundant in lithium than the B component.",
"What is important to highlight here is that we found a higher Be abundance on the A component when compared to B, in contrast to the results of [18] and [24], while [88] does not find any significant Be variation between components, maybe because of the different parameters (including broadening parameters) and different spectra (resolving power, S/N, normalization).",
"Table: Comparison of Li and Be abundances.Lithium and beryllium are elements that are destroyed at different temperatures ($2.5\\times 10^{6}$ K and $3.5\\times 10^{6}$ K, respectively) and therefore at different depths in the stellar interiors.",
"According to standard stellar evolution models, these temperatures are only achieved below the base of the convective zone.",
"However, the solar photospheric Li abundance is approximately 150 times lower than the meteoritic value, indicating that extra mixing processes are acting in solar-type stars and need to be taken into account.",
"In solar-type stars, it is known that Li has a strong correlation with age and surface rotation [9], [14], [8], [19], which suggests internal depletion of lithium.",
"However, it is a more challenging task to do the same analysis for Be due to difficulties related to its detection utilizing instruments from the ground, with only two accessible lines of Be II being near the atmospheric cutoff in the UV, at 313 nm, in a heavily populated region of the spectrum.",
"In [95], we determined the abundance of Be in a sample of 8 solar twins through a “differential” spectral synthesis, where the line list was calibrated to match the observed solar spectrum, which was observed with the same setup as the other stars.",
"We found that the Be content of solar twins is barely depleted, if at all, during the Main Sequence ($\\sim 0.04$ dex in a time span of 8 Gyrs).",
"Thus, in a probable scenario of a planet being engulfed by its host star, if this event happens after the stabilization of its convective zone, one could expect an enhancement of Li and Be, in a similar way as for refractory elements.",
"If we analyze our result with this hypothesis in mind, the overabundance of lithium and beryllium on 16 Cyg A relative to B (in addition to the enhancement of refractory elements) could indicate an accretion of mass.",
"In fact, [77] suggests that the Li abundance can be used as a signal of pollution enrichment of the outer layers of solar-type stars, if stellar ages are well known.",
"The majority of previous studies agree that both components of the binary system have the same stellar ages.",
"However, as seen in Table REF , we found that 16 Cyg A is 0.70 dex more rich in Li than 16 Cyg B, in accord with the results of [88], [31], [25], and [67].",
"Furthermore, on the lithium-age trend of [14] and [53], 16 Cyg B shows a normal Li abundance for a solar twin of its age, while 16 Cyg A has a Li abundance above the curve, thus, when compared to a sample of solar twins, the A component shows an anomalous abundance of lithium.",
"On top of that, 16 Cyg A also seems to have a higher $v \\sin i$ velocity (Table REF ), which may indicate momentum transferred by mass accretion.",
"[25] also suggests that the odd lithium abundance of 16 Cyg A may be due to planet accretion of a 1-2 Jupiter mass planet.",
"This would increase the abundance not only of Li but of Fe as well.",
"This is reinforced by [46] who propose that the separation between the two stars [64] is sufficiently small to permit planet-planet interactions to cause orbit instabilities on the binary pair.",
"Furthermore, the high eccentricity [97] of 16 Cyg Bb could also be evidence of the interaction between the stars.",
"Similar results were found by [32], who find a difference of 0.025 $\\pm $ 0.009 dex in [Fe/H] between the 16 Cygni pair (the A component being more metal rich), suggesting a self-pollution scenario.",
"[26] also investigated abundance differences in six main-sequence binaries with separations on the order of hundreds AU (enough to permit orbit instabilities on possible exoplanets) with components with almost the same mass, using the differential abundance technique (errors of the order of 0.01 dex).",
"Four of these systems did not show any chemical differences between components, while the two remaining binary systems (HD 219542 and HD 200466) display a clear metallicity difference, being the primary stars more rich in iron (and in the most analyzed elements) than the secondary.",
"The authors also support the idea that the difference in chemical composition of those binary stars is due to infall of rocky material.",
"By taking into account the hypothesis of planet accretion pollution, one could expect that the Be abundance would also be enriched on the outer layers of the star, in a similar way as Li.",
"As discussed earlier, according to [95] beryllium is not depleted in a very effective way (if it is at all) on solar twins stars during the Main Sequence, making it also a good proxy for planet accretion after the stabilization of the convective zone.",
"Comparing the pair of stars, we found that 16 Cyg A has 0.07 $\\pm $ 0.03 dex more beryllium than 16 Cyg B, in line with the planet engulfment hypothesis.",
"Following the procedure outlined in [23], we estimate that, if we add 2.5 - 3.0 earth masses of Earth-like material into the convective zone of 16 Cyg B, we would alter the content of Be in about 0.07 dex, thus canceling the abundance difference between the stars.",
"This estimate is close to the one derived by [93], who derived that the addition of 1.5 earth mass of a material with a mixture of the composition of the Earth and CM chondrites is necessary to reproduce the refractory elements abundances as function of the condensation temperature pattern on 16 Cyg B.",
"However, [93] assumed that this abundance pattern is a spectral signature of the 16 Cyg Bb rocky core formation and, now considering also the abundances of Li and Be, it may be a signature of planet accretion rather than planet formation.",
"In contrast, [89] discuss that engulfment of rocky planets can induce instabilities on a stellar surface, by the dilution of a metal-rich material in young Main Sequence stars, which creates an unstable $\\mu $ -gradient at the bottom of the convective zone, activating fingering (thermohaline) convection.",
"This would be responsible for the depletion of the abundances that enriched the convective zone, thus quenching any signature of accretion.",
"However, the authors also discuss that the mixing process would not completely erase the enhanced abundances, meaning that the engulfment event would still be detected in the high precision abundances domain.",
"In this scenario, [17] argues that, during the early periods on the Main Sequence, 16 Cyg B was able to accrete rocky material from its planetary disk, whereas no accretion may have developed around 16 Cyg A due to the presence of a red dwarf (16 Cyg C) orbiting at 73 AU around the A component [96], [62].",
"The models of [17], that take into account the mixing by fingering convection, could reproduce the observed difference of lithium on the binary pair by adding 0.66 Earth mass to 16 Cyg B.",
"However, in those same models, Be does not show any depletion with the addition of 0.66 Earth mass, with the destruction starting to be more effective with the accretion of higher masses.",
"Notice that among the two dozen chemical elements showing abundance differences between 16 Cyg A and B, the model of [17] can only explain the difference in lithium.",
"We find this scenario very unlikely because the lithium content of 16 Cyg B seems to be normal for its age when compared to other solar twins, while 16 Cyg A, on the other hand, is the one that displays an enhanced Li content for a solar twin with $\\sim $ 7 Gyr [14], [53].",
"Another explanation for the discrepancy in Li abundances could be different initial rotation rates [16].",
"However, notice that although young solar-type stars of a given mass may have different rotation rates [40], they all seem to converge to the same rotation period at an age of about 0.2 Gyr [7], which is much earlier than the age of 16 Cyg.",
"Furthermore, the companion 16 Cyg C at 73 AU may be too far as to have any significant impact on 16 Cyg A."
],
[
"Conclusions",
"We present a detailed study of elemental abundances on the 16 Cygni solar twin binary system using higher quality data (R = 160 000, S/N = 1000 at 670nm).",
"We confirm the difference of 0.04 dex in [Fe/H] between 16 Cyg A and B.",
"We also confirm the positive trend of differential abundances (A - B) as a function of condensation temperature, in very good agreement with our previous work [93], which was obtained with spectra of lower resolving power and signal to noise ratio on a different instrument.",
"There is also good agreement with the slope obtained independently by [56], and also using a different spectrograph (HARPS-N).",
"We also find the same result by employing the ARES and iSpec codes to measure the EWs.",
"This shows that our differential analysis method is consistent and a powerful tool to unveil physical characteristics that can only be seen with high precision abundance determinations, and that the T$_c$ trend is a physical phenomenon, being thus unlikely to be related to some instrumental effect, as we show that high-quality spectra obtained with different spectrographs (Espadons at CFHT, HDS at Subaru, HARPS-N at the Telescopio Nationale Galileo) give essentially the same results (within error bars).",
"We also determine the abundance of Li and Be through a “differential” spectral synthesis analysis, using the solar spectrum (obtained with the same instrumental configuration) to calibrate the line list that was used to perform the calculations.",
"We found that 16 Cyg A exhibits an overabundance of not only Li (as reported by previous studies) but of Be as well, relative to 16 Cyg B.",
"This discrepancy is compatible with a 2.5 - 3.0 Earth masses of earth like material if we assume a convective zone similar to Sun for both stars.",
"Interestingly, the amount of rocky material needed to explain the Li and Be overabundances is also compatible with the trend of the (A-B) abundances vs. condensation temperature, reinforcing thus the hypothesis of planet engulfment, although a similar opposite trend in (B-A) could be attributed to the effect of the rocky core in 16 Cyg B [93].",
"However, the overabundant Li content in 16 Cyg A, above what is expected for its age, suggest that the signature that we are observed is due to a planet engulfment event.",
"MTM acknowledges support by financial support of Joint committee ESO Chile and CNPq (312956/2016-9).",
"JM, LS and DLO thanks FAPESP (2014/15706-9, 2016/20667-8, and 2018/04055-8) and CNPq (Bolsa de produtividade).",
"PJ acknowledges FONDECYT Iniciación 11170174 grant for financial support."
]
] | 1906.04195 |
[
[
"Schur ring and Codes for $S$-subgroups over $\\Z_{2}^{n}$"
],
[
"Abstract In this paper the relationship between $S$-subgroups in $\\Z_{2}^{n}$ and binary codes is shown.",
"If the codes used are both $P(T)$-codes and $G$-codes, then the $S$-subgroup is free.",
"The codes constructed are cyclic, decimated or symmetric and the $S$-subgroups obtained are free under the action the cyclic permutation subgroup, invariants under the action the decimated permutation subgroup and symmetric under the action of symmetric permutation subgroup, respectively.",
"Also it is shows that there is no codes generating whole $\\Z_{2}^{n}$ in any $\\G_{n}(a)$-complete $S$-set of the $S$-ring $\\mathfrak{S}(\\Z_{2}^{n},S_{n})$."
],
[
"Introduction",
"Let $G$ be a finite group with identity element $e$ and $G]$ the group algebra of all formal sums $\\sum _{g\\in G}a_{g}g$ , $a_{g}\\in , $ gG$.",
"For $ TG$, the element $ gTg$ willbe denoted by $T$.",
"Such an element is also called a $simple quantity$.The transpose of $T = gGagg$ is defined as $T = gGag(g-1)$.",
"Let $ {T0,T1,...,Tr}$ be a partition of $ G$ and let $ S$ be the subspaceof $ G]$ spanned by $T1,T2,...,Tr$.",
"We say that $ S$ isa $Schur ring$ ($ S$-ring, for short) over $ G$ if:$ $T_{0} = \\lbrace e\\rbrace $ , for each $i$ , there is a $j$ such that $\\overline{T_{i}}^{\\top } = \\overline{T_{j}}$ , for each $i$ and $j$ , we have $\\overline{T_{i}}\\cdot \\overline{T_{j}} = \\sum _{k=1}^{r}\\lambda _{i,j,k}\\overline{T_{k}}$ , for constants $\\lambda _{i,j,k}\\in .$ The numbers $\\lambda _{i,j,k}$ are the structure constants of $S$ with respect to the linear base $\\lbrace \\overline{T_{0}},\\overline{T_{1}},...,\\overline{T_{r}}\\rbrace $ .",
"The sets $T_{i}$ are called the basic sets of the $S$ -ring $S$ .",
"Any union of them is called an $S$ -sets.",
"Thus, $X\\subseteq G$ is an $S$ -set if and only if $\\overline{X}\\in S$ .",
"The set of all $S$ -set is closed with respect to taking inverse and product.",
"Any subgroup of $G$ that is an $S$ -set, is called an $S$ -subgroup of $G$ or $S$ -group.",
"A partition $\\lbrace T_{0},...,T_{r}\\rbrace $ of $G$ is called Schur partition or $S$ -partition if $T_{0}=\\lbrace e\\rbrace $ and if for each $i$ there is some $j$ such that $T_{i}^{-1}=\\lbrace g^{-1}:g\\in T_{i}\\rbrace =T_{j}$ .",
"It is known that there is a 1-1 correspondence between $S$ -rings over $G$ and $S$ -partitions of $G$ .",
"By using this correspondence, in this paper we will refer to an $S$ -ring by mean of its $S$ -partition.",
"The concept of $S$ -ring was iniciated by I. Schur in their classical paper [1] which was published in 1933.",
"Later, the theory of $S$ -ring was developed for Wielandt [2].",
"But the main objective of theory was purely group theoretical concept, especially in problem concerning the permutations groups.",
"In the 80s and 90s, the theory received a notable impulse by the study of $S$ -ring over cyclic groups and their applications to the graph theory [3],[4],[5],[6].",
"With the papers [8],[9] and [11] was initiated the research of $S$ -ring over the group $\\mathbb {Z}_{2}^{n}$ and was shown the relationship between this and Hadamard matrices, perfect binary sequences and periodic compatible binary sequences.",
"In this paper we will show the relationship between $S$ -ring over $\\mathbb {Z}_{2}^{n}$ and binary codes.",
"In particular we will use codes for to construct $S$ -subgroups over $\\mathbb {Z}_{2}^{n}$ .",
"This point of view will be shown as an alternative to the cocyclic matrices and to the difference sets used for to research hadamard matrices, since with the schur rings and its generator codes will be possible to understand the structure of the special binary sequences above.",
"In this paper a code $\\mathcal {X}_{n}$ generating whole $\\mathbb {Z}_{2}^{n}$ is found.",
"Then the others codes for $S$ -subgroups are constructed by using $\\mathcal {X}_{n}$ as a base.",
"A code $\\mathcal {X}_{n}^{\\prime }$ will be called $G$ -codes if there exists a permutation subgroup $G$ in $Aut(\\mathbb {Z}_{2}^{n})$ such that $G\\mathcal {X}_{n}^{\\prime }=\\mathcal {X}_{n}^{\\prime }$ .",
"Other types of codes studied are the $P(T)$ -codes, closely related to the partition $P(T)$ of the subset $T$ of $N=\\lbrace 0,1,...,n-1\\rbrace $ .",
"In fact, the $S$ -subgroups constructed are both $P(T)$ -codes and $G$ -codes.",
"This codes generating free $S$ -subgroups in $\\mathbb {Z}_{2}^{n}$ .",
"This paper is organized as follows.",
"In section 2 basic concepts of theory of codes are shown.",
"In section 3 properties of $P(T)$ -codes are stablished.",
"Also is shown that a $P(T)$ -code generates a free subgroup in $\\mathbb {Z}_{2}^{n}$ and that a $G$ -code generates an $S$ -subgroup in the same group.",
"In section 4 is shown the connection between $G$ -codes and $\\mathcal {G}_{n}(a)$ -complete $S$ -set.",
"$S$ -subgroups and its generator codes in $S$ -rings induced by the permutation subgroups $C_{n}$ , $\\Delta _{n}$ , $H_{n}C_{n}$ , $\\Delta _{n}C_{n}$ and $H_{n}\\Delta _{n}C_{n}$ of $Aut(\\mathbb {Z}_{2}^{n})$ are studied in the sections 5 to 10.",
"Some terminology of the Theory of Codes In [11] the following terminology related to the theory of codes can be found An arbitrary set $\\mathcal {A}$ will be called an alphabet and its elements are called letters.",
"A finite sequence of letters written in the form $s_{1}s_{2}\\cdots s_{n}$ , $n\\ge 0$ , with every $s_{i}$ in $\\mathcal {A}$ , is called a word.",
"Any subsequence of consecutive letters of a word is a subword.",
"When $n=0$ the word is the empty word and denoted by $1_{\\mathcal {A}}$ .",
"Given a word $w=s_{1}s_{2}\\cdots s_{n}$ , the number $n$ is called the lenght of $w$ and is denoted $l(w)$ .",
"Then the empty word $1_{\\mathcal {A}}$ has lenght 0, i.e., $l(1_{\\mathcal {A}})=0$ .",
"Let $\\mathcal {A}^{*}$ denote all finite words defined on $\\mathcal {A}$ and let $\\mathcal {A}^{+}$ denote all finite nonempty words on $\\mathcal {A}$ .",
"$\\mathcal {A}^{*}$ is equipped with an associative binary operation obtained by concatenating two sequences: $s_{1}s_{2}\\cdots s_{n}\\cdot t_{1}t_{2}\\cdots t_{m}=s_{1}s_{2}\\cdots s_{n}t_{1}t_{2}\\cdots t_{m}.$ The empty word is the unit element with respect to this operation and consequently the sets $\\mathcal {A}^{*}$ and $\\mathcal {A}^{+}$ are a monoid and a semigroup, respectivelly.",
"A factorization of a word $s\\in \\mathcal {A}^{*}$ is a sequence $\\lbrace s_{1},s_{2},...,s_{n}\\rbrace $ of $n\\ge 0$ words in $\\Sigma ^{*}$ such that $s=s_{1}s_{2}\\cdots s_{n}$ .",
"For a subset $X$ of $\\mathcal {A}^{*}$ , we denote by $X^{*}$ the submonoid generated by $X$ , $X^{*}=\\lbrace x_{1}x_{2}\\cdots x_{n}\\vert \\ n\\ge 0,\\ x_{i}\\in X\\rbrace .$ Similarly, we denote by $X^{+}$ the subsemigroup generated by $X$ , $X^{^{+}}=\\lbrace x_{1}x_{2}\\cdots x_{n}\\vert \\ n\\ge 1,\\ x_{i}\\in X\\rbrace .$ By definition, each word $s$ in $X^{*}$ admits a least one factorization $x_{1}x_{2}\\cdots x_{n}$ with all $x_{i}$ in $X$ .",
"Such a factorization is called an $X$ -factorization.",
"A monoid $M$ is called free if it has a subset $B$ such that: $M=B^{*}$ , and For all $n,m\\ge 1$ and $x_{1},\\cdots x_{n}$ , $y_{1},\\cdots y_{m}\\in B$ we have $x_{1}\\cdots x_{n}=y_{1}\\cdots y_{m}\\ \\Rightarrow \\ n=m\\ \\textsl {and}\\ x_{i}=y_{i}\\ \\textsl {for}\\ i=1,2,...,n.$ From condition 1., $B$ is a generating set of $M$ and condition 2. say us that each element in $M$ has an unique representation as a product of elements of $B$ .",
"The set $B$ satisfying 1. y 2. is called a base of $M$ .",
"Let $M$ be a monoid and $B$ its generating set.",
"We say that $B$ is minimal generating set if no proper subset of $B$ is a generating set.",
"A element $x$ of $M$ is called indecomposable or atomic if it cannot be expressed in the form $x=yz$ with $y,z\\ne 1$ .",
"Now, by reformulating the condition (REF ) we obtain the following definition.",
"A subset $\\mathcal {X}\\subseteq \\mathcal {A}^{*}$ is a code if it satisfies the following condition: For all $n,m\\ge 1$ and $x_{1},...,x_{n}$ , $y_{1},...,y_{m}\\in \\mathcal {X}$ $x_{1}\\cdots x_{n}=y_{1}\\cdots y_{m}\\ \\Rightarrow \\ n=m\\ \\textsl {and}\\ x_{i}=y_{i}\\ \\textsl {for}\\ i=1,2,...,n.$ Note that the empty word $1_{\\mathcal {A}}$ is never in a code.",
"The following theorem shows the equivalence between codes and free generating set Theorem 1 Let $\\mathcal {X}\\subseteq \\mathcal {A}^{*}$ .",
"Then the following conditions are equivalent: $\\mathcal {X}$ is a code, $\\mathcal {X}$ is a free generating set, or a base, of the monoid $\\mathcal {X}^{*}$ , $\\mathcal {X}^{*}$ is free and $\\mathcal {X}$ is its minimal generating set.",
"On the other hand, let $G$ be a group with $\\mathcal {A}\\subseteq G$ .",
"Elements of $\\mathcal {A}^{*}$ represent elements of $G$ closed under concatenation and inversion.",
"The empty word represents $1_{G}$ , the unity of $G$ .",
"Then $\\mathcal {A}^{*}$ is a subgroup of $G$ .",
"A word $w=s_{1}s_{2}\\cdots s_{n}$ in $\\mathcal {A}^{*}$ is called reduced if $w$ contains no subwords $xx^{-1}$ or $x^{-1}x$ for $x\\in \\mathcal {A}$ .",
"A group $G$ is called a free group if there exists a generating set $X$ of $G$ such that every non-empty reduced group word in $X$ defines a non-trivial element of $G$ .",
"Let $G$ be a free group on $X$ .",
"Then the cardinality of $X$ is called the rank of $G$ .",
"Schur rings and codes for $S$ -subgroups over $\\mathbb {Z}_{2}^{n}$ In this paper denote by $\\mathbb {Z}_{2}$ the cyclic group of order 2 with elements $+$ and $-$ (where + and $-$ mean 1 and $-1$ respectively).",
"Let $\\mathbb {Z}_{2}^{n}=\\overset{n}{\\overbrace{\\mathbb {Z}_{2}\\times \\cdots \\times \\mathbb {Z}_{2}}}$ .",
"Then all $X\\in \\mathbb {Z}_{2}^{n}$ are sequences of $+$ and $-$ and will be called $\\mathbb {Z}_{2}$ -sequences or binary sequences.",
"All binary sequence in $\\mathbb {Z}_{2}^{n}$ is of the form $(x_{0},x_{1},...,x_{n-1})$ .",
"Let $\\textbf {1}$ denote the sequence $(1,1,...,1)$ .",
"As $X^{2}=\\textbf {1}$ for all $X$ in $\\mathbb {Z}_{2}^{n}$ , then all reduced word in $\\mathbb {Z}_{2}^{n}$ contains no the subword $XX$ .",
"Now we will find a code generating whole $\\mathbb {Z}_{2}^{n}$ .",
"We define the following subset of $\\mathbb {Z}_{2}^{n}$ $\\mathcal {X}_{n}=\\left\\lbrace \\begin{array}{c}X_{0}=-+\\cdots ++\\\\X_{1}=+-\\cdots ++\\\\\\vdots \\\\X_{n-2}=++\\cdots -+\\\\X_{n-1}=++\\cdots +-\\end{array}\\right\\rbrace $ where each $-$ is in the $i$ -th position.",
"In the following theorem we shall show that $\\mathcal {X}_{n}$ is a base for all $\\mathbb {Z}_{2}^{n}$ Theorem 2 $\\mathcal {X}_{n}$ is a code for $\\mathbb {Z}_{2}^{n}$ .",
"As $\\vert \\mathcal {X}_{n}\\vert =n$ , then all word on $\\mathcal {X}_{n}$ has the form $w=X_{0}^{\\epsilon _{0}}X_{1}^{\\epsilon _{1}}\\cdots X_{n-1}^{\\epsilon _{n-1}},$ with $\\epsilon _{i}=0,1$ .",
"Thus, the number of words on $\\mathcal {X}_{n}$ of lenght $k$ is $\\binom{n}{k}$ , with $k$ ranging in $[1,n-1]$ .",
"As the empty word corresponds to $\\textbf {1}$ and as $X_{0}X_{1}\\cdots X_{n-1}=-\\textbf {1}$ , the number total of words constructed with the codewords $X_{i}$ is $2^{n}$ .",
"Hence there exist a 1-1 correspondence between all words on $\\mathcal {X}_{n}$ and all binary sequences in $\\mathbb {Z}_{2}^{n}$ .",
"Consequently $\\mathcal {X}_{n}^{*}=\\mathbb {Z}_{2}^{n}$ as we announce.",
"Let $Aut(\\mathbb {Z}_{2}^{n})$ denote the automorphism group of $\\mathbb {Z}_{2}^{n}$ and take $G$ by a subgroup of $Aut(\\mathbb {Z}_{2}^{n})$ .",
"We shall denote with $\\mathfrak {S}(\\mathbb {Z}_{2}^{n},G)$ an $S$ -partition of $\\mathbb {Z}_{2}^{n}$ under the action of $G$ .",
"As $\\mathcal {X}_{n}$ generates whole $\\mathbb {Z}_{2}^{n}$ , we wish to find codes on $\\mathcal {X}_{n}$ , this is, with codewords factorizable on $\\mathcal {X}_{n}$ , for $S$ -subgroups of $\\mathfrak {S}(\\mathbb {Z}_{2}^{n},G)$ .",
"We start with the following definition Definition 1 Let $T=\\lbrace i_{1},i_{2},...,i_{r}\\rbrace $ be a subset of $N=\\lbrace 0,1,...,n-1\\rbrace $ and let $P(T)=\\lbrace T_{1},...,T_{s}\\rbrace $ denote a partition on $T$ .",
"A code $\\mathcal {X}_{n}^{\\prime }$ on $\\mathcal {X}_{n}$ is a $P(T)$ -code if $\\mathcal {X}_{n}^{\\prime }=\\lbrace Y_{T_{1}},...,Y_{T_{s}}\\rbrace $ , where $Y_{T_{j}}=X_{j_{1}}\\cdots X_{j_{k}}$ and $T_{j}=\\lbrace j_{1},...,j_{k}\\rbrace $ .",
"The map $T_{j}\\mapsto Y_{T_{j}}$ establishes an 1-1 correspondence between the blocks $T_{j}$ of $T$ and the codewords $Y_{T_{j}}$ of $\\mathcal {X}_{n}^{\\prime }$ .",
"Then it is easily inferred that Theorem 3 Let $\\mathcal {X}_{n}^{\\prime }$ be a $P(T)$ -code.",
"Then $\\vert \\mathcal {X}_{n}^{\\prime *}\\vert =2^{\\vert \\mathcal {X}_{n}^{\\prime }\\vert }.$ We will call to $\\mathcal {X}_{n}^{\\prime *}$ a $P(T)$ -free group.",
"Let $P(T)=\\lbrace T_{1},...,T_{s}\\rbrace $ be a partition of some subset $T$ of $N$ .",
"Then all word on $\\mathcal {X}_{n}^{\\prime }$ is irreducible.",
"Hence the number of words on $\\mathcal {X}_{n}^{\\prime }$ of lenght $k$ is $\\binom{\\vert \\mathcal {X}^{\\prime }\\vert }{k}$ and $\\vert \\mathcal {X}_{n}^{\\prime *}\\vert =1+\\sum _{k=1}^{\\vert \\mathcal {X}_{n}^{\\prime }\\vert }\\binom{\\vert \\mathcal {X}_{n}^{\\prime }\\vert }{k}=2^{\\vert \\mathcal {X}_{n}^{\\prime }\\vert }$ Corollary 1 $\\mathcal {X}_{n}$ in (REF ) is the only $P(T)$ -code generating whole $\\mathbb {Z}_{2}^{n}$ .",
"$\\mathcal {X}_{n}$ is a $P(T)$ -code with $P(T)=\\lbrace \\lbrace 0\\rbrace ,\\lbrace 1\\rbrace ,...,\\lbrace n-1\\rbrace \\rbrace $ .",
"The statement is followed from here.",
"Now, we will find the number of $P(T)$ -free subgroups in $\\mathbb {Z}_{2}^{n}$ Theorem 4 The number of $P(T)$ -free subgroup in $\\mathbb {Z}_{2}^{n}$ is $B_{\\vert T\\vert +1}$ where $B_{\\vert T\\vert }$ are the Bell numbers.",
"By the correspondence is clear that the number of $P(T)$ -free subgroups for any subset $T$ is $B_{\\vert T\\vert }$ .",
"As $\\binom{n}{\\vert T\\vert }$ indicates the number of $\\vert T\\vert $ -element subsets of an $n$ -element set, then $\\binom{n}{\\vert T\\vert }B_{\\vert T\\vert }$ indicates the number of $P(T)$ -free subgroups with fixed size.",
"Assuming that $B_{0}$ is the number of empty words we obtain $\\sum _{\\vert T\\vert =0}^{n}\\binom{n}{\\vert T\\vert }B_{\\vert T\\vert }=B_{\\vert T\\vert +1}$ .",
"For example, the following are all $P(T)$ -free subgroup of $\\mathbb {Z}_{2}^{3}$ $&&\\lbrace X_{0},X_{1},X_{2}\\rbrace ^{*},\\lbrace X_{0}X_{1},X_{2}\\rbrace ^{*},\\lbrace X_{0}X_{2},X_{1}\\rbrace ^{*},\\lbrace X_{1}X_{2},X_{0}\\rbrace ^{*}\\\\&&\\lbrace X_{0}X_{1}X_{2}\\rbrace ^{*}\\\\&& \\lbrace X_{0},X_{1}\\rbrace ^{*},\\lbrace X_{0}X_{1}\\rbrace ^{*},\\lbrace X_{0},X_{2}\\rbrace ^{*},\\lbrace X_{0}X_{2}\\rbrace ^{*},\\lbrace X_{1},X_{2}\\rbrace ^{*},\\lbrace X_{1}X_{2}\\rbrace ^{*}\\\\&&\\lbrace X_{0}\\rbrace ^{*},\\lbrace X_{1}\\rbrace ^{*},\\lbrace X_{2}\\rbrace ^{*},\\\\&&\\lbrace 1\\rbrace ^{*}$ The reason for deal with $P(T)$ -code will be showed now.",
"Let $\\mathcal {X}_{7}=\\left\\lbrace \\begin{array}{c}X_{0}=-++++++\\\\X_{1}=+-+++++\\\\X_{2}=++-++++\\\\X_{3}=+++-+++\\\\X_{4}=++++-++\\\\X_{5}=+++++-+\\\\X_{6}=++++++-\\\\\\end{array}\\right\\rbrace $ a code for $\\mathbb {Z}_{2}^{7}$ and define the set $\\mathcal {X}_{7}^{\\prime }=\\lbrace X_{3}X_{5}X_{6},X_{2}X_{4}X_{5},X_{1}X_{3}X_{4},X_{0}X_{2}X_{3},X_{6}X_{1}X_{2},X_{5}X_{0}X_{1},X_{4}X_{6}X_{0}\\rbrace .$ on $\\mathcal {X}_{7}$ .",
"It is easy to show that $\\mathcal {X}_{7}^{\\prime }$ is not a code.",
"For example, $X_{0}X_{2}X_{5}X_{6}$ has at least two factorization in $\\mathcal {X}^{\\prime }$ , namely $X_{0}X_{2}X_{3}\\cdot X_{3}X_{5}X_{6}$ and $X_{2}X_{4}X_{5}\\cdot X_{4}X_{6}X_{0}$ .",
"Also, $\\vert \\mathcal {X}_{7}^{\\prime *}\\vert =16$ and not 128 as desirable.",
"Hence $\\mathcal {X}_{7}^{\\prime *}$ is not free group.",
"However, from theorem REF a $P(T)$ -code always generates a free group.",
"The following theorem say us as to obtain new $P(T)$ -free subgroup from old.",
"Theorem 5 Let $\\mathcal {X}_{i}$ be $P(T_{i})$ -codes, $1\\le i\\le r$ , in $\\mathbb {Z}_{2}^{n}$ such that $T_{i}\\cap T_{j}=\\emptyset $ , $i\\ne j$ , and $T_{i}\\subset N$ .",
"Then $\\left(\\bigcup _{i=1}^{r}\\mathcal {X}_{i}\\right)^{*}=\\prod _{i=1}^{r}\\mathcal {X}_{i}^{*}.$ and $\\vert \\prod _{i=1}^{r}\\mathcal {X}_{i}^{*}\\vert =2^{\\sum _{i=1}^{r}\\vert \\mathcal {X}_{i}\\vert }$ Follows by induction on number of $T_{i}$ -codes $\\mathcal {X}_{i}$ .",
"Now we will obtain a necessary condition for the existence of an $S$ -subgroup Theorem 6 Let $G$ be a permutation automorphic subgroup of $Aut(\\mathbb {Z}_{2}^{n})$ acting on some set $\\mathcal {X}$ in $\\mathbb {Z}_{2}^{n}$ .",
"Then $\\mathcal {X}^{*}$ is an $S$ -subgroup in $\\mathfrak {S}(\\mathbb {Z}_{2}^{n},G)$ .",
"We take a word $Y_{i_{1}}Y_{i_{2}}\\cdots Y_{i_{r}}$ in $\\mathcal {X}^{\\prime *}$ and a $g$ in $G$ .",
"Then $g(Y_{i_{1}}Y_{i_{2}}\\cdots Y_{i_{r}})&=&g(Y_{i_{1}})g(Y_{i_{2}})\\cdots g(Y_{i_{r}})\\\\&=&Y_{j_{1}}Y_{j_{2}}\\cdots Y_{j_{r}}.$ As $g$ is arbritary, then $g(Y_{i_{1}}Y_{i_{2}}\\cdots Y_{i_{r}})$ is in $\\mathcal {X}^{\\prime *}$ for all $g$ in $G$ .",
"Hence $G$ defines a partition on $\\mathcal {X}^{\\prime *}$ and $\\mathcal {X}^{\\prime *}$ is an $S$ -subgroup of $\\mathfrak {S}(\\mathbb {Z}_{2}^{n},G)$ .",
"Not all $S$ -subgroup is free.",
"$\\mathcal {X}_{7}^{\\prime *}$ in (REF ) is an $S$ -subgroup of $\\mathfrak {S}(\\mathbb {Z}_{2}^{7},C_{7})$ , where $C_{7}=\\left\\langle C\\right\\rangle $ is the cyclic permutation automorphic subgroup of $Aut(\\mathbb {Z}_{2}^{7})$ of order 7 with $C$ the cyclic permutation acting on all component of some $Y$ in $\\mathbb {Z}_{2}^{7}$ .",
"But $\\mathcal {X}_{7}^{\\prime *}$ is not a free subgroup of $\\mathbb {Z}_{2}^{7}$ .",
"Again let $G$ be a permutation automorphic subgroup of $Aut(\\mathbb {Z}_{2}^{n})$ and let $Y_{G}$ denote the orbit of some $Y$ in $\\mathbb {Z}_{2}^{n}$ under the action of $G$ .",
"From previous theorem $Y_{G}^{*}$ is an $S$ -subgroup.",
"We will called to $Y_{G}^{*}$ a basic $S$ -subgroup of $\\mathfrak {S}(\\mathbb {Z}_{2}^{n},G)$ .",
"If for a code $\\mathcal {X}$ it is true that $\\mathcal {X}=Y_{G}$ for some $Y$ in $\\mathbb {Z}_{2}^{n}$ , then we will say that $\\mathcal {X}$ is a $G$ -code.",
"In the following sections we construct $S$ -subgroups by using $G$ -codes.",
"Schur ring $\\mathfrak {S}(\\mathbb {Z}_{2}^{n},S_{n})$ Let $\\omega (X)$ denote the Hamming weight of $X\\in \\mathbb {Z}_{2}^{n}$ .",
"Thus, $\\omega (X)$ is the number of $+$ in any $\\mathbb {Z}_{2}-$ sequences $X$ of $\\mathbb {Z}_{2}^{n}$ .",
"Now let $\\mathcal {G}_{n}(k)$ be the subset of $\\mathbb {Z}_{2}^{n}$ such that $\\omega (X)=k$ for all $X\\in \\mathcal {G}_{n}(k)$ , where $0\\le k\\le n$ .",
"We let $T_{i}=\\mathcal {G}_{n}(n-i)$ .",
"It is straightforward to prove that the partition $\\mathfrak {S}(\\mathbb {Z}_{2}^{n},S_{n})=\\lbrace \\mathcal {G}_{n}(0),...,\\mathcal {G}_{n}(n)\\rbrace $ induces an $S$ -partition over $\\mathbb {Z}_{2}^{n}$ , where $S_{n}\\le Aut(\\mathbb {Z}_{2}^{n})$ is the permutation group on $n$ objects.",
"From [7] it is know that the constant structure $\\lambda _{i,j,k}$ is equal to $\\lambda _{i,j,k}={\\left\\lbrace \\begin{array}{ll}0&\\mbox{if } i+j-k\\ \\mbox{is an odd number}\\\\\\binom{k}{(j-i+k)/2}\\binom{n-k}{(j+i-k)/2} &\\mbox{if } i+j-k\\ \\mbox{is an even number}\\end{array}\\right.",
"}$ From (REF ) follows that $\\mathcal {G}_{n}(a)\\mathcal {G}_{n}(b)={\\left\\lbrace \\begin{array}{ll}\\bigcup \\limits _{i=0}^{a}\\mathcal {G}_{n}(n-a-b+2i), & 0\\le a\\le \\left[\\dfrac{n}{2}\\right], a\\le b\\le n-a,\\\\\\bigcup \\limits _{i=0}^{n-a}\\mathcal {G}_{n}(a+b-n+2i), & \\left[\\dfrac{n}{2}\\right]+1\\le a\\le n, n-a\\le b\\le a.\\end{array}\\right.",
"}$ From (REF ) we know that $\\mathcal {X}_{n}=\\mathcal {G}_{n}(n-1)$ .",
"Then $\\mathcal {X}_{n}$ is an $S_{n}$ -code for $\\mathfrak {S}(\\mathbb {Z}_{2}^{n},S_{n})$ .",
"In the following corollary we found another $S_{n}$ -code for $\\mathfrak {S}(\\mathbb {Z}_{2}^{n},S_{n})$ Corollary 2 $\\mathcal {G}_{n}(1)$ is an $S_{n}$ -code for $\\mathfrak {S}(\\mathbb {Z}_{2}^{n},S_{n})$ .",
"It is enough to take into account that $-\\mathcal {G}_{n}(n-1)=\\mathcal {G}_{n}(1)$ .",
"We prefer to use the $S_{n}$ -code $\\mathcal {G}_{n}(n-1)$ and not $\\mathcal {G}_{n}(1)$ because in $\\mathcal {G}_{n}(n-1)$ the positions of the negative components are easily obtained.",
"Indeed, $+++-\\cdot +-++=+-+-$ in $\\mathcal {G}_{4}(3)$ but in $\\mathcal {G}_{4}(1)$ we have $---+\\cdot -+--=+-+-$ .",
"Next we will see that a $G$ -code is contained in any $S$ -set of $\\mathfrak {S}(\\mathbb {Z}_{2}^{n},S_{n})$ Proposition 1 A $G$ -code $\\mathcal {X}$ is contained in $\\mathcal {G}_{n}(a)$ for some $a$ ranging in $[1,n-1]$ .",
"Take $X$ in $\\mathcal {X}$ .",
"It is easy to note that $\\omega (gX)=\\omega (X)$ for all $g\\in G$ .",
"Hence $\\mathcal {X}\\subseteq \\mathcal {G}_{n}(a)$ for some $a$ in $[1,n-1]$ .",
"On the other hand, it is follows directly from (REF ) that $\\lambda _{i,j,2k+1}=0$ if $i+j$ is even and $\\lambda _{i,j,2k}=0$ if $i+j$ is odd.",
"The union of all basic sets $\\mathcal {G}_{n}(2a)$ in $\\mathfrak {S}(\\mathbb {Z}_{2}^{n},S_{n})$ will be denoted by $\\mathcal {E}_{n}$ and the union of all basic sets $\\mathcal {G}_{n}(2a+1)$ in $\\mathfrak {S}(\\mathbb {Z}_{2}^{n},S_{n})$ will be denoted $\\mathcal {O}_{n}$ .",
"The sets $\\mathcal {E}_{2n}$ and $\\mathcal {O}_{2n+1}$ are subgroups of order $2^{2n-1}$ and $2^{2n}$ , respectively.",
"Then $\\mathfrak {S}(\\mathcal {E}_{2n},S_{n})=\\lbrace \\mathcal {G}_{2n}(0),\\mathcal {G}_{2n}(2),...,\\mathcal {G}_{2n}(2n)\\rbrace $ and $\\mathfrak {S}(\\mathcal {O}_{2n+1},S_{n})=\\lbrace \\mathcal {G}_{2n+1}(1),\\mathcal {G}_{2n+1}(3),...,\\mathcal {G}_{2n+1}(2n+1)\\rbrace $ are $S$ -subgroups of $\\mathfrak {S}(\\mathbb {Z}_{2}^{2n},S_{2n})$ and $\\mathfrak {S}(\\mathbb {Z}_{2}^{2n+1},S_{2n+1})$ , respectively.",
"From (REF ), $\\mathcal {G}_{2n}(n)^{2}=\\bigcup _{i=0}^{n}\\mathcal {G}_{2n}(2i)=\\mathcal {E}_{2n}$ and $\\mathcal {G}_{2n+1}(n)^{2}=\\bigcup _{i=0}^{n}\\mathcal {G}_{2n+1}(2i+1)=\\mathcal {O}_{2n+1}$ .",
"Therefore, neither $\\mathcal {G}_{4n}(2n)$ nor $\\mathcal {G}_{4n+3}(2n+1)$ contains some code $\\mathcal {X}$ generating whole $\\mathbb {Z}_{2}^{4n}$ and $\\mathbb {Z}_{2}^{2n+1}$ , respectively.",
"This remark is generalized below.",
"From [7] is obtained the following definition Definition 2 Take $\\mathcal {G}_{n}(a)$ in $\\mathfrak {S}(\\mathbb {Z}_{2}^{n},S_{n})$ .",
"Let $\\mathfrak {S}^{\\prime }\\subset \\mathfrak {S}(\\mathbb {Z}_{2}^{n},S_{n})$ be a set of basic sets.",
"We will call $\\mathfrak {S}^{\\prime }$ a $\\mathcal {G}_{n}(a)$ -complete $S$ -set if it holds $\\mathcal {G}_{n}(i)\\mathcal {G}_{n}(j)\\supset \\mathcal {G}_{n}(a)$ for all $\\mathcal {G}_{n}(i),\\mathcal {G}_{n}(j)\\in \\mathfrak {S}^{\\prime }$ , There is no $\\mathcal {G}_{n}(b)\\in \\mathfrak {S}(\\mathbb {Z}_{2}^{n},S_{n})$ such that $\\mathcal {G}_{n}(b)^{2}\\supset \\mathcal {G}_{n}(a)$ and $\\mathcal {G}_{n}(b)\\mathcal {G}_{n}(k)\\supset \\mathcal {G}_{n}(a)$ for all $\\mathcal {G}_{n}(k)\\in \\mathfrak {S}^{\\prime }$ .",
"A important result obtained is that there is no $\\mathcal {G}_{n}(a)$ -complete for all $n$ and all $a$ Theorem 7 There is no $\\mathcal {G}_{2n}(2a+1)$ -complete $S$ -sets in $\\mathfrak {S}(\\mathbb {Z}_{2}^{2n},S_{n})$ .",
"There is no $\\mathcal {G}_{2n+1}(2a)$ -complete $S$ -sets in $\\mathfrak {S}(\\mathbb {Z}_{2}^{2n+1},S_{n})$ .",
"In the following theorem is shown the relationship between codes generating whole $\\mathbb {Z}_{2}^{n}$ and non $\\mathcal {G}_{n}(a)$ -complete $S$ -sets in $\\mathfrak {S}(\\mathbb {Z}_{2}^{n},S_{n})$ Theorem 8 There is no a code $\\mathcal {X}$ generating whole $\\mathbb {Z}_{2}^{n}$ in a $\\mathcal {G}_{n}(a)$ -complete $S$ -set.",
"Let $\\mathfrak {S}^{\\prime }$ denote a $\\mathcal {G}_{2n}(2a)$ -complete $S$ -set.",
"From (REF ) $\\mathcal {G}_{2n}(2b)^{2}=\\bigcup _{i=0}^{2b}\\mathcal {G}_{2n}(2n-4b+2i)$ for all $\\mathcal {G}_{2n}(2b)$ in $\\mathfrak {S}^{\\prime }$ .",
"Then all powers of $\\mathcal {G}_{2n}(2b)$ will contain basic sets $\\mathcal {G}_{2n}(2k)$ only.",
"Therefore the basic sets in a $\\mathcal {G}_{2n}(2a)$ -complete can generate the $S$ -subgroup $\\mathcal {E}_{2n}$ at the most.",
"With a similar argument is shown for basic sets in $\\mathcal {G}_{2n+1}(2a+1)$ -complete $S$ -sets.",
"We finish this section showing some basic sets of $\\mathfrak {S}(\\mathbb {Z}_{2}^{n},S_{n})$ that can to contain $G$ -codes generating all $\\mathbb {Z}_{2}^{n}$ .",
"Proposition 2 $_{}$ $\\mathcal {G}_{4n}(2n-1)^{3}=\\mathcal {O}_{4n}$ , $\\mathcal {G}_{4n}(2n-1)^{4}=\\mathcal {E}_{4n}$ .",
"$\\mathcal {G}_{4n+2}(2n-1)^{3}=\\mathcal {O}_{4n+2}$ , $\\mathcal {G}_{4n}(2n-1)^{4}=\\mathcal {E}_{4n+2}$ .",
"$\\mathcal {G}_{4n+1}(2n-2)^{3}=\\mathcal {E}_{4n+1}$ , $\\mathcal {G}_{4n+1}(2n-2)^{4}=\\mathcal {O}_{4n+1}$ .",
"$\\mathcal {G}_{4n+3}(2n)^{3}=\\mathcal {E}_{4n+3}$ , $\\mathcal {G}_{4n+3}(2n)^{4}=\\mathcal {O}_{4n+3}$ .",
"By using (REF ) we have $\\mathcal {G}_{4n}(2n-1)^{2}&=&\\mathcal {G}_{4n}(2)\\cup \\cdots \\cup \\mathcal {G}_{4n}(4n)\\\\&\\supset &\\mathcal {G}_{4n}(2n)\\nonumber .$ As $\\mathcal {G}_{4n}(2n-1)\\mathcal {G}_{4n}(2n)=\\mathcal {G}_{4n}(1)\\cup \\cdots \\cup \\mathcal {G}_{4n}(4n-1)$ then $\\mathcal {G}_{4n}(2n-1)^{3}=\\mathcal {O}_{4n}$ .",
"From $\\mathcal {G}_{4n}(2n-1)\\mathcal {G}_{4n}(2n+1)\\supset \\mathcal {G}_{4n}(0)$ and from (REF ) is followed that $\\mathcal {G}_{4n}(2n-1)^{4}=\\mathcal {E}_{4n}$ .",
"As $S_{n}$ induces a $S$ -partition on $\\mathbb {Z}_{2}^{n}$ is straightforward to prove that $G$ induces a $S$ -partition on $\\mathbb {Z}_{2}^{n}$ for all $G\\le S_{n}\\le Aut(\\mathbb {Z}_{2}^{n})$ .",
"In the following sections we will construct $S$ -subgroups by using $G$ -codes in $S$ -ring $\\mathfrak {S}(\\mathbb {Z}_{2}^{n},G)$ .",
"Schur ring $\\mathfrak {S}(\\mathbb {Z}_{2}^{n},C_{n})$ Let $C$ denote the cyclic permutation on the components $+$ and $-$ of $X$ in $\\mathbb {Z}_{2}^{n}$ such that $C(X)=C\\left( x_{0},x_{1},...,x_{n-2},x_{n-1}\\right) =\\left(x_{1},x_{2},x_{3},...,x_{0}\\right),$ that is, $C(x_{i})=x_{(i+1) mod n}$ .",
"The permutation $C$ is a generator of cyclic group $C_{n}=\\left\\langle C\\right\\rangle $ of order $n$ .",
"Let $X_{C}=Orb_{C_{n}}X=\\lbrace C^{i}(X):C^{i}\\in C_{n} \\rbrace $ .",
"Therefore, $C_{n}$ defines a partition in equivalent class on $\\mathbb {Z}_{2}^{n}$ which is an $S$ -partition and this we shall denote by $\\mathbb {Z}_{2C}^{n}=\\mathfrak {S}(\\mathbb {Z}_{2}^{n},C_{n})$ .",
"It is worth mentioning that this Schur ring corresponds to the orbit Schur ring induced by the cyclic permutation automorphic subgroup $C_{n}\\le S_n\\le Aut(\\mathbb {Z}_2^n)$ .",
"On the other hand, let $X=\\lbrace x_{i}\\rbrace $ and $Y=\\lbrace y_{i}\\rbrace $ be two complex-valued sequences of period $n$ .",
"The periodic correlation of $X$ and $Y$ at shift $k$ is the product defined by: $\\mathsf {P}_{X,Y}(k)=\\sum \\limits _{i=0}^{n-1}x_{i}\\overline{y}_{i+k},\\ k=0,1,...,n-1,$ where $\\overline{a}$ denotes the complex conjugation of $a$ and $i+k$ is calculated modulo $n$ .",
"If $Y=X$ , the correlation $\\mathsf {P}_{X,Y}(k)$ is denoted by $\\mathsf {P}_{X}(k)$ and is the autocorrelation of $X$ .",
"Obviously, $\\mathsf {P}_{X}(k)&=&\\overline{\\mathsf {P}_{X}(n-k)},\\\\\\mathsf {P}_{-X}(k)&=&\\mathsf {P}_{X}(k),\\\\\\mathsf {P}_{C^{i}X}(k)&=&\\mathsf {P}_{X}(k),$ for all $0\\le i\\le n-1$ and for all $X$ in $\\mathbb {Z}_{2}^{n}$ .",
"If $X$ is a $\\mathbb {Z}_{2}$ -sequence of length $n$ , $\\mathsf {P}_{X}(k)= 2\\omega \\left\\lbrace Y_{k}\\right\\rbrace -n$ , where $ Y_{k}=XC^{k}X $ .",
"Also by (REF ), if $X\\in \\mathcal {G}_{n}(a)$ , then $\\mathsf {P}_{X}(k)=n-4a+4i_{k},$ for some $0\\le i_{k}\\le a$ and $n-\\mathsf {P}_{X}(k)$ is divisible by 4 for all $k$ .",
"We know from theorem REF that $\\mathcal {G}_{n}(n-1)$ is a code for $\\mathfrak {S}(\\mathbb {Z}_{2}^{n},S_{n})$ for all $n$ .",
"As $\\mathcal {G}_{n}(n-1)=\\lbrace X,CX,C^{2}X,...,C^{n-1}\\rbrace =\\mathcal {X}_{C}$ , then $\\mathcal {X}_{C}$ is a $C_{n}$ -code for $\\mathfrak {S}(\\mathbb {Z}_{2}^{n},C_{n})$ .",
"Then for to obtain information from each basic set $Y_{C}$ in $\\mathfrak {S}(\\mathbb {Z}_{2}^{n},C_{n})$ we must do it through of its $\\mathcal {X}_{C}$ -factorization with the $C_{n}$ -code $\\mathcal {X}_{C}$ .",
"The advantage of using this code lies in its simplicity, since each $C^{i}X$ has exactly a $-$ as its component and thereby it is possible to know exactly the Hamming weight of each word writing with this basis.",
"All word $Y$ in $\\mathcal {G}_{n}(a)$ has the form $C^{i_{1}}XC^{i_{2}}X\\cdots C^{i_{r}}X$ with length $\\vert Y\\vert =r$ and with $a=n-r$ .",
"Then, every basic set in $\\mathcal {G}_{nC}(a)$ has form $Y_{C}=\\bigcup _{k=0}^{n-1}C^{i_{1}+k}XC^{i_{2}+k}X\\cdots C^{i_{r}+k}X,$ and in this way if $Z=C^{j_{1}}XC^{j_{2}}X\\cdots C^{j_{s}}X$ , we have $Y_{C}Z_{C}&=&\\bigcup _{k=0}^{n-1}(YC^{k}Z)_{C}\\\\&=&\\bigcup _{k=0}^{n-1}(C^{i_{1}}X\\cdots C^{i_{r}}XC^{j_{1}+k}X\\cdots C^{j_{s}+k}X)_{C}.$ Each word $YC^{k}Z=C^{i_{1}}X\\cdots C^{i_{r}}XC^{j_{1}+k}X\\cdots C^{j_{s}+k}X$ can be reduced if exist two equal letters.",
"Thereupon $YC^{k}Z$ decreases its length an even number.",
"Therefore $YC^{k}Z$ belong to $\\mathcal {G}_{n}(b)$ with $b=n-(r+s)+2w$ where $2w$ is the number of canceled letters.",
"If both $Y$ and $Z$ belongs to $\\mathcal {G}_{n}(a)$ , then $b=n-2r+2w$ and $\\mathsf {P}_{Y,Z}(k)=n-4r+4w_{k}$ .",
"Next we will obtain the algebraic version of (REF ), () and () Proposition 3 Let $Y$ denote the binary sequence $C^{i_{1}}XC^{i_{2}}X\\cdots C^{i_{r}}X$ .",
"If $YC^{k}Y\\in \\mathcal {G}_{n}(a)$ , then $YC^{n-k}Y$ and $(C^{j}X)C^{k}(C^{j}X)$ are in $\\mathcal {G}_{n}(a)$ too.",
"$(-Y)C^{k}(-Y)\\in \\mathcal {G}_{n}(a)$ .",
"1.",
"It is clear that $YC^{k}Y=C^{i_{1}}XC^{i_{2}}X\\cdots C^{i_{r}}XC^{i_{1}+k}XC^{i_{2}+k}X\\cdots C^{i_{r}+k}X\\in \\mathcal {G}_{n}(a)$ with $a=n-2r+2w$ , where $2w$ are the number of canceled letters.",
"We wish to show that the cancellation numbers of $YC^{k}Y$ and $YC^{n-k}Y$ coincide.",
"Suppose that $i_{j}=i_{1}+k$ for some $j$ and some $k$ .",
"Then this implies that $n-k+i_{j}=i_{1}$ reduced module $n$ .",
"Therefore $YC^{n-k}Y=C^{i_{1}}X\\cdots C^{i_{r}}XC^{n-k+i_{1}}X\\cdots C^{n-k+i_{r}}X$ has the same number of cancellations as $YC^{k}Y$ .",
"Equally is proved for $(C^{j}X)C^{k}(C^{j}X)$ .",
"2.",
"As $C^{k}(-Y)=-C^{k}Y$ , then $(-Y)C^{k}(-Y)=YC^{k}Y\\in \\mathcal {G}_{n}(a)$ .",
"Now we will show other advantage of to use the $C_{n}$ -code $\\mathcal {X}_{C}$ Proposition 4 For all $n\\le 2$ we have $\\overline{\\mathcal {G}_{n}(n-2)}^{2}=n+2\\overline{\\mathcal {G}_{n}(n-2)}+(n-3)\\overline{\\mathcal {G}_{n}(n-4)}.$ All word $Y$ in $\\mathcal {G}_{n}(n-2)$ has the form $C^{i}XC^{j}X$ with $i<j$ .",
"Then $YC^{k}Y=C^{i}XC^{j}XC^{i+k}XC^{j+k}X$ and there exist a $k$ such that either $i+k=j$ or $j+k=i$ and for all the remaining values of $k$ we have that $C^{i}XC^{j}XC^{i+k}XC^{j+k}X$ is a reduced word.",
"As $YC^{k}Y$ and $YC^{n-k}Y$ are in $\\mathcal {G}_{n}(a)$ for some $a$ , then $Y_{C}^{2}$ contains 2 words in $\\mathcal {G}_{n}(n-2)$ , $n-3$ words in $\\mathcal {G}_{n}(n-4)$ and the trivial word in $\\mathcal {G}_{n}(n)$ .",
"On the other hand, let $F_{d}(\\mathbb {Z}_{2}^{n})=\\bigcup _{\\vert X\\vert =d}X.$ Clearly $d$ divides to $n$ and the $X\\in F_{d}(\\mathbb {Z}_{2}^{n})$ have the form $X=(Y,Y,...,Y)$ , with $Y\\in \\mathbb {Z}_{2}^{d}$ .",
"Then $F_{d}(\\mathbb {Z}_{2C}^{n})=\\bigcup _{\\vert X_{C}\\vert =d}X_{C}$ is an $S$ -set of $\\mathbb {Z}_{2C}^{n}$ , for each $d\\vert n$ .",
"When $d=n$ , we will to say that $C_{n}$ acts freely on $X_{C}$ and we denote $F_{n}(\\mathbb {Z}_{2C}^{n})$ as $F(\\mathbb {Z}_{2C}^{n})$ .",
"When $d<n$ , we will to say that $C_{n}$ don't act freely on $X_{C}$ and let $\\widehat{F}(\\mathbb {Z}_{2C}^{n})$ denote the set of the $X_{C}$ which are not frees under the action of $C_{n}$ , namely $\\widehat{F}(\\mathbb {Z}_{2C}^{n})=\\bigcup _{d\\mid n,d<n}F_{d}(\\mathbb {Z}_{2C}^{n}).$ Therefore, $\\mathbb {Z}_{2C}^{n}&=&F(\\mathbb {Z}_{2C}^{n})\\cup \\widehat{F}(\\mathbb {Z}_{2C}^{n})\\nonumber \\\\&=&\\bigcup _{d\\mid n}F_{d}(\\mathbb {Z}_{2C}^{n}).$ The set $F_{d}(\\mathbb {Z}_{2}^{n})$ is constructed with codewords in $\\mathcal {X}_{F,d}=\\lbrace A_{0,d}X,A_{1,d}X,\\cdots ,A_{d-1,d}X\\rbrace $ where $A_{i,d}X=C^{i}XC^{i+d}X\\cdots C^{i+kd}X$ for $i=0,1,2,...,d-1$ with $k=\\frac{n}{d}-1$ , $X\\in \\mathcal {G}_{n}(n-1)$ and $\\lbrace i,i+d,i+2d,\\cdots ,i+kd\\rbrace $ is an arithmetic progression.",
"Then $\\mathcal {X}_{F,d}$ is a $P(T)$ -code with $P(T)=\\lbrace \\lbrace i,i+d,i+2d,...,i+kd\\rbrace :\\ i=0,1,...,d-1\\rbrace $ and any word in $\\mathcal {X}_{F,d}^{*}$ has the form $Y=A_{0,d}^{\\epsilon _{0}}XA_{1,d}^{\\epsilon _{1}}X\\cdots A_{d-1,d}^{\\epsilon _{d-1}}X$ $\\epsilon _{i}=0,1$ .",
"It is clear that $C^{i}A_{j,d}X=A_{i+j,d}X$ .",
"Hence $\\mathcal {X}_{F,d}$ is a $C_{n}$ -code.",
"If $d=1$ , then $k=n-1$ , $i=0$ and $\\mathcal {X}_{F,1}=\\lbrace A_{0,1}X\\rbrace =\\lbrace XCXC^{2}X\\cdots C^{n-1}X\\rbrace $ is a code with a codeword.",
"As $XCXC^{2}X\\cdots C^{n-1}X=-\\textbf {1}$ , then $\\mathcal {X}_{F,1}^{*}=\\lbrace \\textbf {1},-\\textbf {1}\\rbrace $ for all $n$ .",
"If $d=n$ , then $k=0$ , $i=0,1,\\cdots ,n-1$ and $\\mathcal {X}_{F,n}=\\lbrace A_{0,n}X,A_{1,n}X,\\cdots ,A_{n-1,n}X\\rbrace =\\lbrace X,CX,C^{2}X,\\cdots ,C^{n-1}X\\rbrace $ is an code with exactly $n$ codewords and $\\mathcal {X}_{F,d}=\\mathcal {X}_{C}$ .",
"Let $\\vert \\mathcal {X}_{F,d}\\vert $ be the rank of $\\mathcal {X}_{F,d}$ .",
"We then note that $1<\\vert \\mathcal {X}_{F,d}\\vert <n$ for $1<d<n$ .",
"Now we will see the relationship between free subgroup $\\mathcal {X}_{F,d}^{*}$ and the sets $F_{d}(\\mathbb {Z}_{2}^{n})$ Theorem 9 $\\mathcal {X}_{F,d}^{*}$ is a subgroup of $\\mathbb {Z}_{2}^{n}$ of order $2^{d}$ with $\\mathcal {X}_{F,d}^{*}=\\bigcup _{r\\vert d}F_{d}(\\mathbb {Z}_{2}^{n})$ .",
"We will donote this subgroup with $\\mathbb {G}_{d}(n)$ .",
"As $\\mathcal {X}_{F,d}$ is a $P(T)$ -code, then $\\mathbb {G}_{d}(n)$ is a subgroup of $\\mathbb {Z}_{2}^{n}$ of order $2^{d}$ for all divisor $d$ of $n$ .",
"Then we only will show that $\\mathbb {G}_{d}(n)$ has the desired structure.",
"If $d$ is a prime divisor of $n$ , then all words of $\\mathbb {G}_{d}(n)$ are in $F_{d}(\\mathbb {Z}_{2}^{n})$ , except for $\\textbf {1}$ and $-\\textbf {1}$ .",
"Hence $\\mathbb {G}_{d}(n)=F_{1}(\\mathbb {Z}_{2}^{n})\\cup F_{d}(\\mathbb {Z}_{2}^{n})$ with $-\\textbf {1}&=&\\prod _{i=0}^{d-1}A_{i,d}X\\\\\\textbf {1}&=&(A_{i,d}X)^{2}\\ for\\ all\\ i.$ Suppose that $d$ is no prime.",
"Then $A_{i,d}XA_{i+r,d}X\\cdots A_{i+(\\frac{d}{r}-1)r,d}X$ is contained in $F_{r}(\\mathbb {Z}_{2}^{n})$ , $r\\vert d$ and $i=0,1,...,r-1$ .",
"Therefore $\\mathbb {G}_{d}(n)=\\bigcup _{r\\vert d}F_{r}(\\mathbb {Z}_{2}^{n})$ .",
"Corollary 3 $\\mathbb {G}_{dC}(n)$ is an $S$ -subgroup of $\\mathfrak {S}(\\mathbb {Z}_{2}^{n},C_{n})$ .",
"The group $\\left\\langle C_{n}\\right\\rangle $ defines a partition on each $F_{r}(\\mathbb {Z}_{2}^{n})$ in $\\mathbb {G}_{d}(n)$ and hence we obtain the desired statement.",
"Example 1 The subgroup $\\mathbb {G}_{3}(9)$ of $\\mathbb {Z}_{2}^{9}$ is given by $\\mathbb {G}_{3}(9)&=&F_{1}(\\mathbb {Z}_{2}^{9})\\cup F_{3}(\\mathbb {Z}_{2}^{9})\\\\&=&\\lbrace -\\textbf {1},\\textbf {1}\\rbrace \\cup \\lbrace XC^{3}XC^{6}X,\\ CXC^{4}XC^{7}X,\\ C^{2}XC^{5}XC^{8}X,\\\\&&XCXC^{3}XC^{4}XC^{6}XC^{7}X,\\ XC^{2}XC^{3}XC^{5}XC^{6}XC^{8}X,\\\\&&CXC^{2}XC^{4}XC^{5}XC^{7}XC^{8}X\\rbrace \\\\&=&\\lbrace -\\textbf {1},\\textbf {1},-++-++-++,++-++-++-,\\\\&&+-++-++-+,-+--+--+-,--+--+--+,\\\\&&+--+--+--\\rbrace .$ And the $S$ -subgroup $\\mathbb {G}_{3C}(9)$ of $\\mathfrak {S}(\\mathbb {Z}_{2}^{9},C_{9})$ is given by $\\mathbb {G}_{3C}(9)&=&\\lbrace \\textbf {1},-\\textbf {1},(-++-++-++)_{C},(-+--+--+-)_{C}\\rbrace .$ Theorem 10 $\\mathbb {G}_{d}(n)\\subseteq \\bigcup _{a=0}^{d}\\mathcal {G}_{n}\\left(\\frac{na}{d}\\right)$ with equality only for $d=1,n$ .",
"Follows from $F_{d}(\\mathbb {Z}_{2}^{n})\\subseteq \\mathcal {G}_{n}\\left(\\frac{n}{d}\\right)\\cup \\mathcal {G}_{n}\\left(n-\\frac{n}{d}\\right)$ .",
"Now we show the lattice of $S$ -subgroups $\\mathbb {G}_{dC}(60)$ of $\\mathfrak {S}(\\mathbb {Z}_{2}^{60},C_{60})$ ordered by inclusion.",
"maxmed) at (1,3) $\\mathbb {G}_{30C}(60)$ ; a) at (-3,2) $\\mathbb {G}_{15C}(60)$ ; b) at (1,0) $\\mathbb {G}_{10C}(60)$ ; c) at (4,2) $\\mathbb {G}_{6C}(60)$ ; d) at (-3,-1) $\\mathbb {G}_{5C}(60)$ ; e) at (0,1) $\\mathbb {G}_{3C}(60)$ ; f) at (4,-1) $\\mathbb {G}_{2C}(60)$ ; min) at (0,-2) $\\mathbb {G}_{1C}(60)=\\lbrace \\textbf {1},-\\textbf {1}\\rbrace $ ; max) at (5,4) $\\mathbb {G}_{60C}(60)=\\mathbb {Z}_{2C}^{60}$ ; g) at (5,1) $\\mathbb {G}_{20C}(60)$ ; h) at (8,3) $\\mathbb {G}_{12C}(60)$ ; i) at (8,0) $\\mathbb {G}_{4C}(60)$ ; (min) – (d) – (a) – (maxmed) – (b) – (f) – (i) – (h) – (max) (e) – (min) – (f) – (c) – (maxmed) – (max) (d) – (b) – (g) – (max) (c) – (h) (i) – (g); [preaction=draw=white, -,line width=6pt] (a)– (e) – (c) – (maxmed) (c) – (h) (f) – (c) (e) – (min); We finish this section we provide other proof to the Theorems 6 and 7 in [7].",
"We start with the lemma Lemma 1 $XC^{n}X\\in F_{n}(\\mathbb {Z}_{2}^{2n})$ for all $X$ in $\\mathcal {X}_{C}=\\mathcal {G}_{2n}(2n-1)$ Clearly $XC^{n}X$ is in $F_{n}(\\mathbb {Z}_{2}^{2n})$ when $X=-+++\\cdots +++$ .",
"As $(C^{i}X)C^{n}(C^{i}X)=C^{i}(XC^{n}X)$ for any other codeword $C^{i}X$ in the code $\\mathcal {X}_{C}$ , then $C^{i}(XC^{n}X)\\in F_{n}(\\mathbb {Z}_{2}^{2n})$ for all $1\\le i\\le 2n-1$ , since $\\left\\langle C_{n}\\right\\rangle $ defines a partition on $F_{n}(\\mathbb {Z}_{2}^{2n})$ .",
"Theorem 11 If $Y_{C}\\in F(\\mathbb {Z}_{2C}^{2n})$ , then $Y_{C}^{2}\\setminus \\lbrace \\textbf {1}\\rbrace \\notin F(\\mathbb {Z}_{2C}^{2n})$ .",
"Take $Y=C^{i_{1}}X\\cdots C^{i_{r}}X$ in $F(\\mathbb {Z}_{2}^{2n})$ .",
"Then $YC^{n}Y&=&C^{i_{1}}X\\cdots C^{i_{r}}XC^{i_{1}+n}X\\cdots C^{i_{r}+n}X\\\\&=&C^{i_{1}}(XC^{n}X)\\cdots C^{i_{r}}(XC^{n}X).$ From the above lemma $XC^{n}X$ is in $F_{n}(\\mathbb {Z}_{2}^{2n})$ for all $X$ in $\\mathcal {X}_{C}$ .",
"Then $YC^{n}Y\\in F_{n}(\\mathbb {Z}_{2}^{2n})$ for all $Y$ in $F(\\mathbb {Z}_{2}^{n})$ and $Y_{C}^{2}\\setminus \\lbrace \\textbf {1}\\rbrace \\notin F(\\mathbb {Z}_{2C}^{2n})$ as we promised to show.",
"Lemma 2 $XC^{k}X\\notin F_{d}(\\mathbb {Z}_{2}^{2n+1})$ for no $X$ in $\\mathcal {X}_{C}=\\mathcal {G}_{2n+1}(2n)$ and for no $k$ ranging in $[1,2n]$ , $d<2n+1$ .",
"Take $X=-++\\cdots ++$ in $\\mathcal {X}_{C}$ .",
"Then $XC^{k}X=-\\overset{2n-k}{\\overbrace{+\\cdots +}}-\\overset{k-1}{\\overbrace{+\\cdots +}}$ .",
"If we want $k-1=2n-k$ , then $k=\\frac{2n+1}{2}$ , which is not possible.",
"Hence $XC^{k}X$ is no contained in $F_{d}(\\mathbb {Z}_{2}^{2n+1})$ , $d<2n+1$ .",
"As $C^{i}(XC^{k}X)=(C^{i}X)C^{k}(C^{i}X)$ , it is followed the statement.",
"Theorem 12 If $X_{C}\\in F(\\mathbb {Z}_{2C}^{2n+1})$ , then $X_{C}^{2}\\setminus \\lbrace \\textbf {1}\\rbrace \\in F(\\mathbb {Z}_{2C}^{2n+1})$ .",
"Let $Y=C^{i_{1}}XC^{i_{2}}X\\cdots C^{i_{r}}X$ such that all the $i_{j}$ are not in arithmetic progression.",
"If $YC^{k}Y=C^{i_{1}}(XC^{k}X)C^{i_{2}}(XC^{k}X)\\cdots C^{i_{r}}(XC^{k}X)$ is contained in some $F_{d}(\\mathbb {Z}_{2}^{2n+1})$ , $d\\vert (2n+1)$ , $d<2n+1$ , then $XC^{k}X$ must be contained in $F_{d}(\\mathbb {Z}_{2}^{2n+1})$ , but is not possible by the previous lemma.",
"Schur ring $\\mathfrak {S}(\\mathbb {Z}_{2}^{n},\\Delta _{n})$ Let $\\delta _{a}\\in S_{n}$ act on $X\\in \\mathbb {Z}_{2}^{n}$ by decimation, that is, $\\delta _{a}(x_{i})=x_{ai(\\mod {n})}$ for all $x_{i}$ in $X$ , $(a,n)=1$ and let $\\Delta _{n}$ denote the set of this $\\delta _{a}$ .",
"The set $\\Delta _{n}$ is a group of order $\\phi (n)$ isomorphic to $\\mathbb {Z}_{n}^{*}$ , the group the units of $\\mathbb {Z}_{n}$ , where $\\phi $ is called the Euler totient function.",
"Clearly $\\mathfrak {S}(\\mathbb {Z}_{2}^{n},\\Delta _{n})$ is an $S$ -partition of $\\mathbb {Z}_{2}^{n}$ .",
"In this section, we will construct $\\Delta _{n}$ -codes for $S$ -subgroups of $\\mathfrak {S}(\\mathbb {Z}_{2}^{n},\\Delta _{n})$ .",
"We will use the commutation relation $C^{i}\\delta _{a}=\\delta _{a}C^{ia}$ for to prove all of results in this section.",
"We begin for show that $\\mathcal {G}_{n}(n-1)$ is partitioned in three equivalence class Proposition 5 $\\Delta _{n}\\mathcal {G}_{n}(n-1)=\\lbrace X\\rbrace \\cup \\mathcal {X}_{\\mathbb {Z}_{n}^{*}}\\cup \\mathcal {X}_{\\mathbb {Z}_{n}\\setminus \\mathbb {Z}_{n}^{*}}$ where $X=-++\\cdots ++$ and $\\mathcal {X}_{\\mathbb {Z}_{n}^{*}}&=&\\lbrace C^{a_{1}}X,C^{a_{2}}X,...,C^{a_{\\phi (n)}}X:\\ (a_{i},n)=1\\rbrace \\\\\\mathcal {X}_{\\mathbb {Z}_{n}\\setminus \\mathbb {Z}_{n}^{*}}&=&\\lbrace C^{d_{1}}X,C^{d_{2}}X,...,C^{d_{r}}X:\\ (d_{i},n)\\ne 1\\rbrace $ $r=n-\\phi (n)-1$ .",
"It is very easy to see that $X=-++\\cdots ++$ is fixed under the action of $\\Delta _{n}$ .",
"Also, as $\\delta _{a}C^{i}X=C^{a^{-1}i}\\delta _{a}X=C^{a^{-1}i}X$ , then $\\Delta _{n}\\mathcal {X}_{\\mathbb {Z}_{n}^{*}}=\\mathcal {X}_{\\mathbb {Z}_{n}^{*}}$ and $\\Delta _{n}\\mathcal {X}_{\\mathbb {Z}_{n}\\setminus \\mathbb {Z}_{n}^{*}}=\\mathcal {X}_{\\mathbb {Z}_{n}\\setminus \\mathbb {Z}_{n}^{*}}$ .",
"As each $C^{i}X$ in $\\mathcal {X}_{\\mathbb {Z}_{n}^{*}}$ or in $\\mathcal {X}_{\\mathbb {Z}_{n}\\setminus \\mathbb {Z}_{n}^{*}}$ is atomic, then $\\mathcal {X}_{\\mathbb {Z}_{n}^{*}}$ and $\\mathcal {X}_{\\mathbb {Z}_{n}\\setminus \\mathbb {Z}_{n}^{*}}$ are $\\Delta _{n}$ -code and hence $\\mathcal {X}_{\\mathbb {Z}_{n}^{*}}^{*}$ and $\\mathcal {X}_{\\mathbb {Z}_{n}\\setminus \\mathbb {Z}_{n}^{*}}^{*}$ are $S$ -subgroups in $\\mathfrak {S}(\\mathbb {Z}_{2}^{n},\\Delta _{n})$ with $\\vert \\mathcal {X}_{\\mathbb {Z}_{n}^{*}}^{*}\\vert =2^{\\phi (n)}$ and $\\vert \\mathcal {X}_{\\mathbb {Z}_{n}\\setminus \\mathbb {Z}_{n}^{*}}^{*}\\vert =2^{n-\\phi (n)-1}$ .",
"On the other hand, let $(\\mathsf {P}_{Y}(0),\\mathsf {P}_{Y}(1),...,\\mathsf {P}_{Y}(n-1))$ denote the autocorrelation vector of $Y$ in $\\mathbb {Z}_{2}^{n}$ and let $\\mathfrak {A}(\\mathbb {Z}_{2}^{n})$ denote the set of all this.",
"Let $X_{1}+X_{2}+\\cdots +X_{n}=a$ denote the plane in $\\mathbb {Z}^{n}$ in the indeterminates $X_{i}$ , $i=1,2,...,n$ and let $\\theta :\\mathbb {Z}_{2}^{n}\\rightarrow \\mathfrak {A}(\\mathbb {Z}_{2}^{n})$ be the map defined by $\\theta (Y)=(\\mathsf {P}_{Y}(0),\\mathsf {P}_{Y}(1),\\dots ,\\mathsf {P}_{Y}(n-1))$ .",
"The decimation group $\\Delta _{n}$ do not alter the set of values which $\\mathsf {P}_{X}(k)$ takes on, but merely the order in which they appear, i.e., if $Y=\\delta _{a}X$ then $\\mathsf {P}_{Y}(k)=\\mathsf {P}_{X}(ka)$ .",
"Therefore, we have the commutative diagram ${\\mathbb {Z}_{2}^{n} [d]^{\\theta } [r]^{\\delta _{r}} & \\mathbb {Z}_{2}^{n} [d]^{\\theta }\\\\\\mathfrak {A}(\\mathbb {Z}_{2}^{n}) [r]^{\\delta _{r}} & \\mathfrak {A}(\\mathbb {Z}_{2}^{n})}$ and $\\theta \\circ \\delta _{r} = \\delta _{r}\\circ \\theta .$ Let $Y\\in \\mathbb {Z}_{2}^{n}$ such that $\\theta (Y)=(n,d,d,...,d)$ .",
"Such a binary sequence is known as binary sequence with 2-levels autocorrelation value and are important by its applications on telecommunication.",
"We want to construct a $\\Delta _{n}$ -code for some $S$ -subgroup $H$ of $\\mathfrak {S}(\\mathbb {Z}_{2}^{n},\\Delta _{n})$ containing such $Y$ .",
"From (REF ) is followed that $\\theta (Y)=\\delta _{a}\\theta (Y)=\\theta (\\delta _{a}Y)$ , for all $\\delta _{a}\\in \\Delta _{n}$ .",
"Hence $Y$ and $\\delta _{a}Y$ have the same autocorrelation vector.",
"For $Y$ fullfilling $\\delta _{a}Y=Y$ for some $\\delta _{a}$ in $\\Delta _{n}$ we have the following definition Definition 3 Let $a$ be a unit in $\\mathbb {Z}_{n}^{*}$ .",
"A word $Y$ in $\\mathbb {Z}_{2}^{n}$ is $\\delta _{a}$ -invariant if $\\delta _{a}Y=Y$ .",
"Denote by $\\mathbb {I}_{n}(a)$ the set of these $Y$ .",
"If $Y$ is in $\\mathbb {I}_{n}(a)$ , then $\\delta _{r}Y$ is in $\\mathbb {I}_{n}(a)$ , too.",
"Also $\\delta _{a}(YZ)=\\delta _{a}Y\\delta _{a}Z=YZ$ for all $Y,Z$ in $\\mathbb {I}_{n}(a)$ .",
"Then $\\mathbb {I}_{n}(a)$ is an $S$ -subgroup of $\\mathfrak {S}(\\mathbb {Z}_{2}^{n},\\Delta _{n})$ .",
"Now, we shall see that all factorization of words in $\\mathbb {I}_{n}(a)$ is relationated with cyclotomic coset of $a$ module $n$ .",
"First, we have the following definition Definition 4 Let $a$ relative prime to $n$ .",
"The cyclotomic coset of $a$ module $n$ is defined by $\\mathsf {C}_{s}=\\lbrace s,sa,sa^{2},\\cdots ,sa^{t-1}\\rbrace .$ where $sa^{t}\\equiv s\\mod {n}$ .",
"A subset $\\lbrace s_{1},s_{2},\\dots ,s_{r}\\rbrace $ of $\\mathbb {Z}_{n}$ is called complete set of representatives of cyclotomic coset of $a$ modulo $n$ if $\\mathsf {C}_{i_{1}}$ ,$\\mathsf {C}_{i_{2}}$ ,..., $\\mathsf {C}_{i_{r}}$ are distinct and are a partition of $\\mathbb {Z}_{n}$ .",
"Take $Y=C^{i_{1}}XC^{i_{2}}X\\cdots C^{i_{r}}X$ in $\\mathbb {I}_{n}(a)$ with $X=-++\\cdots ++$ .",
"We want $\\delta _{a}Y=Y$ .",
"Then $\\delta _{a}Y&=&\\delta _{a}C^{i_{1}}X\\delta _{a}C^{i_{2}}X\\cdots \\delta _{a}C^{i_{r}}X\\\\&=&C^{i_{1}a^{-1}}\\delta _{a}XC^{i_{2}a^{-1}}\\delta _{a}X\\cdots C^{i_{r}a^{-1}}\\delta _{a}X\\\\&=&C^{i_{1}a^{-1}}XC^{i_{2}a^{-1}}X\\cdots C^{i_{r}a^{-1}}X$ since $\\delta _{a}X=X$ .",
"As must be $\\delta _{a}Y=Y$ , then $i_{k}=a^{-1}i_{j}$ or $i_{j}=ai_{k}$ for $1\\le k,j\\le r$ .",
"Let $\\mathsf {C}_{s}X$ denote the word $C^{s}XC^{sa}X\\cdots C^{sa^{t_{s}-1}}X$ .",
"Then all $Y$ in $\\mathbb {I}_{n}(a)$ has the form $Y=\\mathsf {C}_{s_{1}}^{\\epsilon _{1}}X\\mathsf {C}_{s_{2}}^{\\epsilon _{r}}X\\cdots \\mathsf {C}_{s_{r}}^{\\epsilon }X$ , with $\\epsilon _{i}=0,1$ .",
"As $\\mathbb {I}_{n}(a)$ is an $S$ -subgroup in $\\mathfrak {S}(\\mathbb {Z}_{2}^{n},\\Delta _{n})$ , $\\delta _{r}\\mathsf {C}_{s_{i}}X=\\mathsf {C}_{s_{j}}X$ and $\\mathcal {X}_{\\mathbb {I}(a)}=\\lbrace X,\\mathsf {C}_{s_{1}}X,\\ \\mathsf {C}_{s_{2}}X,...,\\ \\mathsf {C}_{s_{r}}X\\rbrace $ is a $\\Delta _{n}$ -code for $\\mathbb {I}_{n}(a)$ .",
"Also $\\mathcal {X}_{\\mathbb {I}(a)}$ is a $P(T)$ -code with $P(T)=\\lbrace \\lbrace 0\\rbrace ,\\mathsf {C}_{s_{1}},\\mathsf {C}_{s_{2}},...,\\mathsf {C}_{s_{r}}\\rbrace $ and $\\lbrace s_{1},s_{2},...,s_{r}\\rbrace $ a complete set of representatives.",
"Hence $\\mathcal {X}_{\\mathbb {I}(a)}^{*}$ has order $2^{r+1}$ , where $r$ is the number of cyclotomic cosets of $a$ module $n$ In the table 1, binary sequences with 2-level autocorrelation values with their respective $\\delta _{a}$ -invariants $S$ -subgroups are shown Table: Binary sequences with 2-level autocorrelation valuesOn the other hand, we have the following theorem Theorem 13 If $\\left\\langle b\\right\\rangle $ is a subgroup of $\\left\\langle a\\right\\rangle $ , then $\\mathbb {I}_{n}(a)\\le \\mathbb {I}_{n}(b)$ .",
"Let $\\mathsf {C}_{1}^{a}$ and $\\mathsf {C}_{1}^{b}$ denote the classes $\\lbrace 1,a,a^{2},...,a^{t-1}\\rbrace $ and $\\lbrace 1,b,b^{2},...,b^{s-1}\\rbrace $ .",
"By hypothesis $\\left\\langle b\\right\\rangle \\le \\left\\langle a\\right\\rangle $ , then $\\mathsf {C}_{1}^{b}\\subseteq \\mathsf {C}_{1}^{a}$ .",
"Hence there exists $y_{i}$ in $\\left\\langle a\\right\\rangle $ such that $\\mathsf {C}_{1}^{a}=\\mathsf {C}_{1}^{b}\\cup y_{1}\\mathsf {C}_{1}^{b}\\cup \\cdots \\cup y_{k}\\mathsf {C}_{1}^{b},$ and $k=[\\left\\langle a\\right\\rangle :\\left\\langle b\\right\\rangle ]$ .",
"Then is follows that $\\mathsf {C}_{s}^{a}=\\mathsf {C}_{s}^{b}\\cup y_{1}\\mathsf {C}_{s}^{b}\\cup \\cdots \\cup y_{k}\\mathsf {C}_{s}^{b}.$ Therefore $\\vert \\mathcal {X}_{\\mathbb {I}_{n}(a)}\\vert \\le \\vert \\mathcal {X}_{\\mathbb {I}_{n}(b)}\\vert $ and $\\mathbb {I}_{n}(a)\\le \\mathbb {I}_{n}(b)$ .",
"We finish this section constructing some $\\delta _{a}$ -invariants $S$ -subgroups Proposition 6 $_{}$ $\\mathbb {I}_{2n+1}(2n)=\\lbrace X,\\ \\mathsf {C}_{q}X:\\ q\\in \\lbrace 1,2,...,n\\rbrace \\rbrace ^{*}$ , $\\mathsf {C}_{q}=\\lbrace q,2n+1-q\\rbrace $ $\\mathbb {I}_{2n}(2n-1)=\\lbrace X,\\ C^{n}X,\\ \\mathsf {C}_{q}X:\\ q\\in \\lbrace 1,2,...,n-1\\rbrace \\rbrace ^{*}$ , $\\mathsf {C}_{q}=\\lbrace q,2n-q\\rbrace $ .",
"We proof 1.",
"The proof of 2 it is analogous.",
"We note that $(2n)^{2}=(2n+1-1)^{2}=(2n+1)^{2}-2(2n+1)+1\\equiv 1\\mod {(}2n+1),$ then $\\mathsf {C}_{1}=\\lbrace 1,2n\\rbrace $ .",
"As $2n+1\\nmid 2n-1$ and $q<2n+1$ , then $2nq\\lnot \\equiv q\\mod {(}2n+1)$ and the $\\mathsf {C}_{q}=\\lbrace q,2nq\\rbrace $ are cyclotomic cosets of $2n$ module $2n+1$ .",
"Finally, it is easy to note that $2nq$ is congruent to $2n+1-q$ module $2n+1$ , $2nq-2n-1+q=(2n+1)(q-1)\\equiv 0\\mod {(}2n+1).$ Proposition 7 Let $2p+1$ be an prime number with $p$ an odd prime number.",
"The $S$ -subgroups invariants in $\\mathbb {Z}_{2}^{2p+1}$ are $\\mathbb {I}_{2p+1}(x)$ , $\\mathbb {I}_{2p+1}(y)$ and $\\mathbb {I}_{2p+1}(2p)$ , where $x$ is a primitive root module $2p+1$ and $y$ is not neither primitive root module $2p+1$ nor $2p$ .",
"Let $P=\\lbrace x_{1},x_{2},...,x_{t}\\rbrace $ denote the set of primitive roots module $2p+1$ .",
"Then $\\mathbb {I}_{2p+1}(x_{1})=\\mathbb {I}_{2p+1}(x_{2})=\\cdots =\\mathbb {I}_{2p+1}(x_{t}).$ As $\\vert \\left\\langle x_{i}\\right\\rangle \\vert =2p$ for any $x_{i}\\in P$ , then $\\left\\langle x_{i}\\right\\rangle $ has exactly a subgroup of order 2 and a subgroup of order $p$ .",
"Therefore there exist $S$ -subgroups invariants $\\mathbb {I}_{2p+1}(y)$ and $\\mathbb {I}_{2p+1}(2p)$ where $y\\in P^{c}\\setminus \\lbrace 2p\\rbrace $ , with $P^{c}$ the complement of $P$ in $\\mathbb {Z}_{2p+1}^{*}$ .",
"Schur ring $\\mathfrak {S}(\\mathbb {Z}_{2}^{n},H_{n})$ We note by $RY$ the reversed sequence $RY = (y_{n-1},...,y_{1},y_{0})$ and let $H_{n}$ denote the permutation automorphic subgroup $H_{n}=\\lbrace 1,R\\rbrace \\le S_{n}\\le Aut(Z_{2}^{n})$ .",
"Hence $H_{n}$ defines a partition on $\\mathbb {Z}_{2}^{n}$ and $\\mathfrak {S}(\\mathbb {Z}_{2}^{n},H_{n})$ is a schur ring.",
"Definition 5 Let $Y\\in \\mathbb {Z}_{2}^{n}$ .",
"We shall call $Y$ symmetric if $RY=Y$ and otherwise we say it is non symmetric.",
"We make $Sym(\\mathbb {Z}_{2}^{n})$ the set of all $Y$ symmetric and $\\widehat{Sym}(\\mathbb {Z}_{2}^{n})$ the set of all $Y$ nonsymmetric.",
"Take $Y\\in \\mathbb {Z}_{2}^{n}$ such that $Y$ is of the form $C^{i_{1}}XC^{i_{2}}X\\cdots C^{i_{r}}X$ .",
"We want to understand the structure of the words in $Sym(\\mathbb {Z}_{2}^{n})$ .",
"As it must be fulfilled that $RY=Y$ , then taking $X=+\\cdots +-+\\cdots +$ in $\\mathcal {G}_{n}(n-1)$ with $n$ an odd number we have $RY&=&RC^{i_{1}}XRC^{i_{2}}X\\cdots RC^{i_{r}}X\\\\&=&C^{-i_{1}}RXC^{-i_{2}}RX\\cdots C^{-i_{r}}RX\\\\&=&C^{n-i_{1}}XC^{n-i_{2}}X\\cdots C^{n-i_{r}}X$ where we have used that $RX=X$ .",
"Hence if $Y$ is symmetric, then must be $n-i_{j}=i_{k}$ for $j\\ne k$ ranging in $[1,r]$ .",
"Thereby $Y$ has the form $Y_{0}^{\\epsilon _{0}}Y_{1}^{\\epsilon _{1}}\\cdots Y_{(n-1)/2}^{\\epsilon _{(n-1)/2}}$ for $\\epsilon _{i}=0,1$ and $Y_{0}=X$ and $Y_{i}=C^{i}XC^{n-i}X$ .",
"As $C^{i}XC^{n-i}X$ is in $Sym(\\mathbb {Z}_{2}^{n})$ for all $i$ , then $\\mathcal {X}_{Sym^{O}}=\\lbrace X,\\ CXC^{n-1}X,\\ C^{2}XC^{n-2}X,...,\\ C^{(n-1)/2}XC^{(n+1)/2}X\\rbrace $ is a $H_{n}$ -code for $Sym(\\mathbb {Z}_{2}^{n})$ .",
"For the case $n$ an even number it is easily followed that $\\mathcal {X}_{Sym^{E}}=\\lbrace XC^{n-1}X,\\ CXC^{n-2}X,\\ C^{2}XC^{n-3}X,...,\\ C^{(n-2)/2}XC^{n/2}X\\rbrace $ is a $H_{n}$ -code for $Sym(\\mathbb {Z}_{2}^{n})$ , where $X=+++\\cdots ++-$ .",
"Also it is clear that $\\mathcal {X}_{Sym^{E}}$ and $\\mathcal {X}_{Sym^{O}}$ are $P(T)$ -codes, therefore $Sym(\\mathbb {Z}_{2}^{n})$ is an free $S$ -subgroup in $\\mathfrak {S}(\\mathbb {Z}_{2}^{n},H_{n})$ and $Sym(\\mathbb {Z}_{2}^{2n+1})$ and $Sym(\\mathbb {Z}_{2}^{2n})$ have order $2^{n+1}$ and $2^{n}$ , respectively.",
"Finally, the relationship between the symmetric subgroup $Sym(\\mathbb {Z}_{2}^{^{2n+1}})$ and the $\\delta _{2n}$ -invariant $S$ -subgroup $\\mathbb {I}_{2n+1}(2n)$ is shown Theorem 14 $Sym(\\mathbb {Z}_{2}^{^{2n+1}})=\\mathbb {I}_{2n+1}(2n)$ .",
"By proposition REF the codewords in $\\mathcal {X}_{\\mathbb {I}_{2n+1}(2n)}$ are $X$ and $C^{q}XC^{2n+1-q}X$ , $1\\le q\\le n$ , with $X=+\\cdots +-+\\cdots +$ .",
"We want to show that all of codewords in $\\mathcal {X}_{\\mathbb {I}_{2n+1}(2n)}$ is symmetric.",
"For this we note that $RX=X$ and $R(C^{q}XC^{2n+1-q}X)&=&RC^{q}XRC^{2n+1-q}X\\\\&=&C^{2n+1-q}RXC^{q}RX\\\\&=&C^{2n+1-q}XC^{q}X.$ Schur ring $\\mathfrak {S}(\\mathbb {Z}_{2}^{n},H_{n}C_{n})$ In this section we will use the commutation relation $RC^{i}=C^{n-i}R$ to show that the $S$ -subgroups $\\mathbb {G}_{d}(n)$ and $Sym(\\mathbb {Z}_{2}^{n})$ are $S$ -subgroups in $\\mathfrak {S}(\\mathbb {Z}_{2}^{n},H_{n}C_{n})$ .",
"Theorem 15 $\\mathbb {G}_{dC}(n)$ is an $S$ -subgroup in $\\mathfrak {S}(\\mathbb {Z}_{2}^{n},H_{n}C_{n})$ By having in mind the commutation relation (REF ), we can to show that $\\mathcal {X}_{F,d}$ is an $H_{n}$ -code.",
"In this way we have $RA_{i,d}X&=&R(C^{i}XC^{i+d}X\\cdots C^{i+n-d}X)\\\\&=&RC^{i}XRC^{i+d}X\\cdots RC^{i+n-d}X\\\\&=&C^{n-i}RXC^{n-i-d}RX\\cdots C^{d-i}RX$ As $X=-++\\cdots ++$ , then $RX=CX$ .",
"Hence $RA_{i,d}X&=&C^{n-i}CXC^{n-i-d}CX\\cdots C^{d-i}CX\\\\&=&C(C^{n-i}XC^{n-i-d}X\\cdots C^{d-i}X)$ Finally, reordering and rewriting $RA_{i,d}X&=&C(C^{d-i}X\\cdots C^{d-i+(n-2d)}XC^{d-i+(n-d)}X)\\\\&=&CA_{d-i,d}X\\\\&=&A_{d-i+1,d}X.$ Definition 6 A $S$ -set $Y_{C}$ in $\\mathbb {Z}_{2C}^{n}$ is symmetric if $R\\cdot Y_{C}=Y_{C}$ , where $R\\cdot Y_{C}$ means the action of $R$ on the elements of $Y_{C}$ .",
"The set of all symmetric $S$ -sets will be denoted by $Sym(\\mathbb {Z}_{2C}^{n})$ and the set of all non-symmetric $S$ -sets will be denoted by $\\widehat{Sym}(\\mathbb {Z}_{2C}^{n})$ .",
"It is easy to note that $R\\cdot Y_{C}=(RY)_{C}$ .",
"Therefore $Y_{C}$ is a symmetric $S$ -set if and only if contains some $C^{i}Y$ symmetric.",
"Theorem 16 $Sym(\\mathbb {Z}_{2C}^{n})$ is an $S$ -subgroup of $\\mathfrak {S}(\\mathbb {Z}_{2}^{n},H_{n}C_{n})$ .",
"Take $Y_{C},Z_{C}$ in $Sym(\\mathbb {Z}_{2C}^{n})$ and suppose that $Y,Z\\in Sym(\\mathbb {Z}_{2}^{n})$ .",
"As $R(YC^{k}Z)_{C}=(RYRC^{k}Z)_{C}=(YC^{n-k}Z)_{C},$ then $R(Y_{C}Z_{C})=R(Y_{C})R(Z_{C})=Y_{C}Z_{C}$ .",
"Schur ring $\\mathfrak {S}(\\mathbb {Z}_{2}^{n},\\Delta _{n}C_{n})$ In this section we will use the commutation relation $\\delta _{a}C^{i}=C^{ia^{-1}}\\delta _{a}$ to show that the $S$ -subgroups $\\mathbb {G}_{d}(n)$ and $\\mathbb {I}_{n}(a)$ are $S$ -subgroups in $\\mathfrak {S}(\\mathbb {Z}_{2}^{n},\\Delta _{n}C_{n})$ .",
"Theorem 17 $\\mathbb {G}_{dC}(n)$ is an $S$ -subgroup in $\\mathfrak {S}(\\mathbb {Z}_{2}^{n},\\Delta _{n}C_{n})$ .",
"We want to show that $\\mathcal {X}_{F,d}$ is a $\\Delta _{n}$ -code.",
"Take $A_{i,d}X$ in $\\mathcal {X}_{F,d}$ .",
"Then if $k=\\frac{n}{d}-1$ we have $\\delta _{a}A_{i,d}X&=&\\delta _{a}(C^{i}XC^{i+d}X\\cdots C^{i+kd}X)\\\\&=&\\delta _{a}C^{i}X\\delta _{a}C^{i+d}X\\cdots \\delta _{a}C^{i+kd}X)\\\\&=&C^{ia^{-1}}\\delta _{a}XC^{(i+d)a^{-1}}\\delta _{a}X\\cdots C^{(i+kd)a^{-1}}\\delta _{a}X\\\\&=&C^{ia^{-1}}XC^{(i+d)a^{-1}}X\\cdots C^{(i+kd)a^{-1}}X.$ Define the map $\\vartheta :\\mathcal {X}_{F,d}\\rightarrow \\lbrace ni/d:\\ i=1,2,...,d-1\\rbrace $ by $\\vartheta (A_{i,d}X)&=&\\vartheta (C^{i}XC^{i+d}X\\cdots C^{i+kd}X)\\\\&=&\\sum _{j=0}^{k}(i+jd)\\\\&=&i(k+1)+\\frac{k(k+1)}{2}d\\\\&=&\\frac{ni}{d}+\\left(\\frac{n}{2d}-\\frac{1}{2}\\right)n\\\\&\\equiv &\\frac{ni}{d}\\mod {n}$ Then $\\vartheta $ is a biyection.",
"As $\\vartheta (\\delta _{a}A_{i,d}X)\\equiv \\frac{a^{-1}ni}{d}\\mod {n}$ and $\\vartheta (C^{l}\\delta _{a}A_{i,d}X)\\equiv (a^{-1}+l)\\frac{ni}{d}\\mod {n}$ , is followed that $\\delta _{a}A_{i,d}X\\in \\mathcal {X}_{F,d}$ and therefore $\\mathcal {X}_{F,d}$ is a $\\Delta _{n}$ -code.",
"Definition 7 Let $a$ be a unit in $\\mathbb {Z}_{n}^{*}$ .",
"A basic set $Y_{C}$ in $\\mathbb {Z}_{2C}^{n}$ is $\\delta _{a}$ -invariant if $\\delta _{a}\\cdot Y_{C}=Y_{C}$ .",
"Denote by $\\mathbb {I}_{nC}(a)$ the set of these $Y$ .",
"By having in mind the basic set $Y_{C}$ , we can note that $\\delta _{a}\\cdot Y_{C}=(\\delta _{a}Y)_{C}$ .",
"Hence $Y_{C}$ is $\\delta _{a}$ -invariant in $\\mathbb {Z}_{2C}^{n}$ if and only if contains some $C^{i}Y$ $\\delta _{a}$ -invariant in $\\mathbb {Z}_{2}^{n}$ .",
"Theorem 18 $\\mathbb {I}_{nC}(a)$ is an $S$ -subgroup in $\\mathfrak {S}(\\mathbb {Z}_{2}^{n},\\Delta _{n}C_{n})$ .",
"Take $Y_{C},Z_{C}$ in $\\mathbb {I}_{nC}(a)$ and suppose that $Y,Z\\in \\mathbb {I}_{n}(a)$ .",
"As $\\delta _{a}(YC^{k}Z)_{C}=(\\delta _{a}Y\\delta _{a}C^{k}Z)_{C}=(YC^{ka^{-1}}Z)_{C}$ , then $\\delta _{a}(Y_{C}Z_{C})=\\delta _{a}(Y_{C})\\delta _{a}(Z_{C})=Y_{C}Z_{C}$ .",
"Schur ring $\\mathfrak {S}(\\mathbb {Z}_{2}^{n},H_{n}\\Delta _{n}C_{n})$ Finally we show that the $S$ -subgroups $\\mathbb {G}_{d}(n)$ , $Sym(\\mathbb {Z}_{2}^{n})$ and $\\mathbb {I}_{n}(a)$ are $S$ -subgroups in $\\mathfrak {S}(\\mathbb {Z}_{2}^{n},H_{n}\\Delta _{n}C_{n})$ .",
"Let $Y_{C}$ be any basic set in $\\mathbb {Z}_{2C}^{n}$ .",
"It is a very easy to notice that $Y_{C}=(C^{k}Y)_{C}$ for all $k$ .",
"We will use this fact for to prove the following lemma Lemma 3 $_{}$ $\\delta _{a}Y_{iC}=Y_{a^{-1}iC}$ for $Y_{i}=C^{i}XC^{2n+1-i}X\\in \\mathcal {X}_{Sym^{O}}$ .",
"$\\delta _{a}Y_{iC}=Y_{\\left(a^{-1}i+\\frac{a^{-1}-1}{2}\\right)C}$ for $Y_{i}=C^{i}XC^{2n-1-i}X\\in \\mathcal {X}_{Sym^{E}}$ .",
"$1.$ Take $\\delta _{a}$ in the group $\\Delta _{n}$ .",
"It is clear that $\\delta _{a}X=C^{k_{a}}X$ for some $k_{a}$ depending on $a$ , where $X=+\\cdots +-+\\cdots +$ is the word in $\\mathcal {X}_{C}$ used to construct all codewords in $\\mathcal {X}_{Sym^{O}}$ .",
"Then $\\delta _{a}Y_{iC}&=&(\\delta _{a}C^{i}X\\delta _{a}C^{2n+1-i}X)_{C}\\\\&=&(C^{a^{-1}i}\\delta _{a}XC^{a^{-1}(2n+1-i)}\\delta _{a}X)_{C}\\\\&=&(C^{k_{a}}(C^{a^{-1}i}XC^{2n+1-a^{-1}i}X))_{C}\\\\&=&(C^{a^{-1}i}XC^{2n+1-a^{-1}i}X)_{C}\\\\&=&Y_{a^{-1}iC}.$ $2.$ Equally, $\\delta _{a}X=C^{k_{a}}X$ for some $k_{a}$ depending on $a$ , where $X=++\\cdots ++-$ in $\\mathcal {X}_{C}$ is used to construct all codewords in $\\mathcal {X}_{Sym^{E}}$ .",
"Then $\\delta _{a}Y_{iC}&=&(\\delta _{a}C^{i}X\\delta _{a}C^{2n-1-i}X)_{C}\\\\&=&(C^{a^{-1}i}\\delta _{a}XC^{a^{-1}(2n-1-i)}\\delta _{a}X)_{C}\\\\&=&(C^{k_{a}}(C^{a^{-1}i}XC^{2n-a^{-1}(i+1)}X))_{C}\\\\&=&\\left(C^{\\frac{a^{-1}-1}{2}}(C^{a^{-1}i}XC^{2n-a^{-1}(i+1)}X)\\right)_{C}\\\\&=&\\left(C^{a^{-1}i+\\frac{a^{-1}-1}{2}}XC^{2n-a^{-1}(i+1)+\\frac{a^{-1}-1}{2}}X\\right)_{C}\\\\&=&Y_{\\left(a^{-1}i+\\frac{a^{-1}-1}{2}\\right)C}.$ We will use this lemma for to show that the symmetric binary sequences form an $S$ -subgroup in $\\mathfrak {S}(\\mathbb {Z}_{2}^{n},H_{n}\\Delta _{n}C_{n})$ Theorem 19 $Sym(\\mathbb {Z}_{2C}^{n})$ is an $S$ -subgroup in $\\mathfrak {S}(\\mathbb {Z}_{2}^{n},H_{n}\\Delta _{n}C_{n})$ .",
"Clearly $Sym(\\mathbb {Z}_{2C}^{n})$ is an $S$ -subgroup of $\\mathfrak {S}(\\mathbb {Z}_{2}^{n},H_{n}C_{n})$ .",
"From the previous lemma is followed that $\\Delta _{n}$ defines a partition on $\\mathcal {X}_{Sym^{E}}$ and $\\mathcal {X}_{Sym^{O}}$ .",
"Hence they are $\\Delta _{n}$ -codes and $Sym(\\mathbb {Z}_{2C}^{n})$ is an $S$ -subgroup of $\\mathfrak {S}(\\mathbb {Z}_{2}^{n},H_{n}\\Delta _{n}C_{n})$ .",
"Theorem 20 $\\mathbb {G}_{dC}(n)$ is an $S$ -subgroup in $\\mathfrak {S}(\\mathbb {Z}_{2}^{n},H_{n}\\Delta _{n}C_{n})$ .",
"Follows from theorems REF and REF .",
"Theorem 21 $\\mathbb {I}_{nC}(a)$ is an $S$ -subgroup in $\\mathfrak {S}(\\mathbb {Z}_{2}^{n},H_{n}\\Delta _{n}C_{n})$ .",
"From theorem REF the case $a=n-1$ is excluded.",
"In the previous section already was proved that $\\mathbb {I}_{nC}(a)$ is an $S$ -subgroup in $\\mathfrak {S}(\\mathbb {Z}_{2}^{n},\\Delta _{n}C_{n})$ .",
"Now, we wish to show that $H_{n}$ defines a partition on $\\mathbb {I}_{nC}(a)$ by using (REF ).",
"Take the codeword $\\mathsf {C}_{s}X$ in $\\mathcal {X}_{\\mathbb {I}_{n}(a)}$ .",
"We have then $R(\\mathsf {C}_{s}X)_{C}&=&R(C^{s}XC^{sa}X\\cdots C^{sa^{t_{s}-1}}X)_{C}\\\\&=&(RC^{s}XRC^{sa}X\\cdots RC^{sa^{t_{s}-1}}X)_{C}\\\\&=&(C^{n-s}CXC^{n-sa}CX\\cdots C^{n-sa^{t_{s}-1}}CX)_{C}\\\\&=&(C(C^{n-s}XC^{n-sa}X\\cdots C^{n-sa^{t_{s}-1}}X))_{C}\\\\&=&(C^{n-s}XC^{(n-s)a}X\\cdots C^{(n-s)a^{t_{s}-1}}X))_{C}\\\\&=&(\\mathsf {C}_{n-s}X)_{C}.$ E-mail address, [email protected]"
],
[
"Some terminology of the Theory of Codes",
"In [11] the following terminology related to the theory of codes can be found An arbitrary set $\\mathcal {A}$ will be called an alphabet and its elements are called letters.",
"A finite sequence of letters written in the form $s_{1}s_{2}\\cdots s_{n}$ , $n\\ge 0$ , with every $s_{i}$ in $\\mathcal {A}$ , is called a word.",
"Any subsequence of consecutive letters of a word is a subword.",
"When $n=0$ the word is the empty word and denoted by $1_{\\mathcal {A}}$ .",
"Given a word $w=s_{1}s_{2}\\cdots s_{n}$ , the number $n$ is called the lenght of $w$ and is denoted $l(w)$ .",
"Then the empty word $1_{\\mathcal {A}}$ has lenght 0, i.e., $l(1_{\\mathcal {A}})=0$ .",
"Let $\\mathcal {A}^{*}$ denote all finite words defined on $\\mathcal {A}$ and let $\\mathcal {A}^{+}$ denote all finite nonempty words on $\\mathcal {A}$ .",
"$\\mathcal {A}^{*}$ is equipped with an associative binary operation obtained by concatenating two sequences: $s_{1}s_{2}\\cdots s_{n}\\cdot t_{1}t_{2}\\cdots t_{m}=s_{1}s_{2}\\cdots s_{n}t_{1}t_{2}\\cdots t_{m}.$ The empty word is the unit element with respect to this operation and consequently the sets $\\mathcal {A}^{*}$ and $\\mathcal {A}^{+}$ are a monoid and a semigroup, respectivelly.",
"A factorization of a word $s\\in \\mathcal {A}^{*}$ is a sequence $\\lbrace s_{1},s_{2},...,s_{n}\\rbrace $ of $n\\ge 0$ words in $\\Sigma ^{*}$ such that $s=s_{1}s_{2}\\cdots s_{n}$ .",
"For a subset $X$ of $\\mathcal {A}^{*}$ , we denote by $X^{*}$ the submonoid generated by $X$ , $X^{*}=\\lbrace x_{1}x_{2}\\cdots x_{n}\\vert \\ n\\ge 0,\\ x_{i}\\in X\\rbrace .$ Similarly, we denote by $X^{+}$ the subsemigroup generated by $X$ , $X^{^{+}}=\\lbrace x_{1}x_{2}\\cdots x_{n}\\vert \\ n\\ge 1,\\ x_{i}\\in X\\rbrace .$ By definition, each word $s$ in $X^{*}$ admits a least one factorization $x_{1}x_{2}\\cdots x_{n}$ with all $x_{i}$ in $X$ .",
"Such a factorization is called an $X$ -factorization.",
"A monoid $M$ is called free if it has a subset $B$ such that: $M=B^{*}$ , and For all $n,m\\ge 1$ and $x_{1},\\cdots x_{n}$ , $y_{1},\\cdots y_{m}\\in B$ we have $x_{1}\\cdots x_{n}=y_{1}\\cdots y_{m}\\ \\Rightarrow \\ n=m\\ \\textsl {and}\\ x_{i}=y_{i}\\ \\textsl {for}\\ i=1,2,...,n.$ From condition 1., $B$ is a generating set of $M$ and condition 2. say us that each element in $M$ has an unique representation as a product of elements of $B$ .",
"The set $B$ satisfying 1. y 2. is called a base of $M$ .",
"Let $M$ be a monoid and $B$ its generating set.",
"We say that $B$ is minimal generating set if no proper subset of $B$ is a generating set.",
"A element $x$ of $M$ is called indecomposable or atomic if it cannot be expressed in the form $x=yz$ with $y,z\\ne 1$ .",
"Now, by reformulating the condition (REF ) we obtain the following definition.",
"A subset $\\mathcal {X}\\subseteq \\mathcal {A}^{*}$ is a code if it satisfies the following condition: For all $n,m\\ge 1$ and $x_{1},...,x_{n}$ , $y_{1},...,y_{m}\\in \\mathcal {X}$ $x_{1}\\cdots x_{n}=y_{1}\\cdots y_{m}\\ \\Rightarrow \\ n=m\\ \\textsl {and}\\ x_{i}=y_{i}\\ \\textsl {for}\\ i=1,2,...,n.$ Note that the empty word $1_{\\mathcal {A}}$ is never in a code.",
"The following theorem shows the equivalence between codes and free generating set Theorem 1 Let $\\mathcal {X}\\subseteq \\mathcal {A}^{*}$ .",
"Then the following conditions are equivalent: $\\mathcal {X}$ is a code, $\\mathcal {X}$ is a free generating set, or a base, of the monoid $\\mathcal {X}^{*}$ , $\\mathcal {X}^{*}$ is free and $\\mathcal {X}$ is its minimal generating set.",
"On the other hand, let $G$ be a group with $\\mathcal {A}\\subseteq G$ .",
"Elements of $\\mathcal {A}^{*}$ represent elements of $G$ closed under concatenation and inversion.",
"The empty word represents $1_{G}$ , the unity of $G$ .",
"Then $\\mathcal {A}^{*}$ is a subgroup of $G$ .",
"A word $w=s_{1}s_{2}\\cdots s_{n}$ in $\\mathcal {A}^{*}$ is called reduced if $w$ contains no subwords $xx^{-1}$ or $x^{-1}x$ for $x\\in \\mathcal {A}$ .",
"A group $G$ is called a free group if there exists a generating set $X$ of $G$ such that every non-empty reduced group word in $X$ defines a non-trivial element of $G$ .",
"Let $G$ be a free group on $X$ .",
"Then the cardinality of $X$ is called the rank of $G$ ."
],
[
"Schur rings and codes for $S$ -subgroups over {{formula:e143b868-1680-4d7c-8d17-44208d880b84}}",
"In this paper denote by $\\mathbb {Z}_{2}$ the cyclic group of order 2 with elements $+$ and $-$ (where + and $-$ mean 1 and $-1$ respectively).",
"Let $\\mathbb {Z}_{2}^{n}=\\overset{n}{\\overbrace{\\mathbb {Z}_{2}\\times \\cdots \\times \\mathbb {Z}_{2}}}$ .",
"Then all $X\\in \\mathbb {Z}_{2}^{n}$ are sequences of $+$ and $-$ and will be called $\\mathbb {Z}_{2}$ -sequences or binary sequences.",
"All binary sequence in $\\mathbb {Z}_{2}^{n}$ is of the form $(x_{0},x_{1},...,x_{n-1})$ .",
"Let $\\textbf {1}$ denote the sequence $(1,1,...,1)$ .",
"As $X^{2}=\\textbf {1}$ for all $X$ in $\\mathbb {Z}_{2}^{n}$ , then all reduced word in $\\mathbb {Z}_{2}^{n}$ contains no the subword $XX$ .",
"Now we will find a code generating whole $\\mathbb {Z}_{2}^{n}$ .",
"We define the following subset of $\\mathbb {Z}_{2}^{n}$ $\\mathcal {X}_{n}=\\left\\lbrace \\begin{array}{c}X_{0}=-+\\cdots ++\\\\X_{1}=+-\\cdots ++\\\\\\vdots \\\\X_{n-2}=++\\cdots -+\\\\X_{n-1}=++\\cdots +-\\end{array}\\right\\rbrace $ where each $-$ is in the $i$ -th position.",
"In the following theorem we shall show that $\\mathcal {X}_{n}$ is a base for all $\\mathbb {Z}_{2}^{n}$ Theorem 2 $\\mathcal {X}_{n}$ is a code for $\\mathbb {Z}_{2}^{n}$ .",
"As $\\vert \\mathcal {X}_{n}\\vert =n$ , then all word on $\\mathcal {X}_{n}$ has the form $w=X_{0}^{\\epsilon _{0}}X_{1}^{\\epsilon _{1}}\\cdots X_{n-1}^{\\epsilon _{n-1}},$ with $\\epsilon _{i}=0,1$ .",
"Thus, the number of words on $\\mathcal {X}_{n}$ of lenght $k$ is $\\binom{n}{k}$ , with $k$ ranging in $[1,n-1]$ .",
"As the empty word corresponds to $\\textbf {1}$ and as $X_{0}X_{1}\\cdots X_{n-1}=-\\textbf {1}$ , the number total of words constructed with the codewords $X_{i}$ is $2^{n}$ .",
"Hence there exist a 1-1 correspondence between all words on $\\mathcal {X}_{n}$ and all binary sequences in $\\mathbb {Z}_{2}^{n}$ .",
"Consequently $\\mathcal {X}_{n}^{*}=\\mathbb {Z}_{2}^{n}$ as we announce.",
"Let $Aut(\\mathbb {Z}_{2}^{n})$ denote the automorphism group of $\\mathbb {Z}_{2}^{n}$ and take $G$ by a subgroup of $Aut(\\mathbb {Z}_{2}^{n})$ .",
"We shall denote with $\\mathfrak {S}(\\mathbb {Z}_{2}^{n},G)$ an $S$ -partition of $\\mathbb {Z}_{2}^{n}$ under the action of $G$ .",
"As $\\mathcal {X}_{n}$ generates whole $\\mathbb {Z}_{2}^{n}$ , we wish to find codes on $\\mathcal {X}_{n}$ , this is, with codewords factorizable on $\\mathcal {X}_{n}$ , for $S$ -subgroups of $\\mathfrak {S}(\\mathbb {Z}_{2}^{n},G)$ .",
"We start with the following definition Definition 1 Let $T=\\lbrace i_{1},i_{2},...,i_{r}\\rbrace $ be a subset of $N=\\lbrace 0,1,...,n-1\\rbrace $ and let $P(T)=\\lbrace T_{1},...,T_{s}\\rbrace $ denote a partition on $T$ .",
"A code $\\mathcal {X}_{n}^{\\prime }$ on $\\mathcal {X}_{n}$ is a $P(T)$ -code if $\\mathcal {X}_{n}^{\\prime }=\\lbrace Y_{T_{1}},...,Y_{T_{s}}\\rbrace $ , where $Y_{T_{j}}=X_{j_{1}}\\cdots X_{j_{k}}$ and $T_{j}=\\lbrace j_{1},...,j_{k}\\rbrace $ .",
"The map $T_{j}\\mapsto Y_{T_{j}}$ establishes an 1-1 correspondence between the blocks $T_{j}$ of $T$ and the codewords $Y_{T_{j}}$ of $\\mathcal {X}_{n}^{\\prime }$ .",
"Then it is easily inferred that Theorem 3 Let $\\mathcal {X}_{n}^{\\prime }$ be a $P(T)$ -code.",
"Then $\\vert \\mathcal {X}_{n}^{\\prime *}\\vert =2^{\\vert \\mathcal {X}_{n}^{\\prime }\\vert }.$ We will call to $\\mathcal {X}_{n}^{\\prime *}$ a $P(T)$ -free group.",
"Let $P(T)=\\lbrace T_{1},...,T_{s}\\rbrace $ be a partition of some subset $T$ of $N$ .",
"Then all word on $\\mathcal {X}_{n}^{\\prime }$ is irreducible.",
"Hence the number of words on $\\mathcal {X}_{n}^{\\prime }$ of lenght $k$ is $\\binom{\\vert \\mathcal {X}^{\\prime }\\vert }{k}$ and $\\vert \\mathcal {X}_{n}^{\\prime *}\\vert =1+\\sum _{k=1}^{\\vert \\mathcal {X}_{n}^{\\prime }\\vert }\\binom{\\vert \\mathcal {X}_{n}^{\\prime }\\vert }{k}=2^{\\vert \\mathcal {X}_{n}^{\\prime }\\vert }$ Corollary 1 $\\mathcal {X}_{n}$ in (REF ) is the only $P(T)$ -code generating whole $\\mathbb {Z}_{2}^{n}$ .",
"$\\mathcal {X}_{n}$ is a $P(T)$ -code with $P(T)=\\lbrace \\lbrace 0\\rbrace ,\\lbrace 1\\rbrace ,...,\\lbrace n-1\\rbrace \\rbrace $ .",
"The statement is followed from here.",
"Now, we will find the number of $P(T)$ -free subgroups in $\\mathbb {Z}_{2}^{n}$ Theorem 4 The number of $P(T)$ -free subgroup in $\\mathbb {Z}_{2}^{n}$ is $B_{\\vert T\\vert +1}$ where $B_{\\vert T\\vert }$ are the Bell numbers.",
"By the correspondence is clear that the number of $P(T)$ -free subgroups for any subset $T$ is $B_{\\vert T\\vert }$ .",
"As $\\binom{n}{\\vert T\\vert }$ indicates the number of $\\vert T\\vert $ -element subsets of an $n$ -element set, then $\\binom{n}{\\vert T\\vert }B_{\\vert T\\vert }$ indicates the number of $P(T)$ -free subgroups with fixed size.",
"Assuming that $B_{0}$ is the number of empty words we obtain $\\sum _{\\vert T\\vert =0}^{n}\\binom{n}{\\vert T\\vert }B_{\\vert T\\vert }=B_{\\vert T\\vert +1}$ .",
"For example, the following are all $P(T)$ -free subgroup of $\\mathbb {Z}_{2}^{3}$ $&&\\lbrace X_{0},X_{1},X_{2}\\rbrace ^{*},\\lbrace X_{0}X_{1},X_{2}\\rbrace ^{*},\\lbrace X_{0}X_{2},X_{1}\\rbrace ^{*},\\lbrace X_{1}X_{2},X_{0}\\rbrace ^{*}\\\\&&\\lbrace X_{0}X_{1}X_{2}\\rbrace ^{*}\\\\&& \\lbrace X_{0},X_{1}\\rbrace ^{*},\\lbrace X_{0}X_{1}\\rbrace ^{*},\\lbrace X_{0},X_{2}\\rbrace ^{*},\\lbrace X_{0}X_{2}\\rbrace ^{*},\\lbrace X_{1},X_{2}\\rbrace ^{*},\\lbrace X_{1}X_{2}\\rbrace ^{*}\\\\&&\\lbrace X_{0}\\rbrace ^{*},\\lbrace X_{1}\\rbrace ^{*},\\lbrace X_{2}\\rbrace ^{*},\\\\&&\\lbrace 1\\rbrace ^{*}$ The reason for deal with $P(T)$ -code will be showed now.",
"Let $\\mathcal {X}_{7}=\\left\\lbrace \\begin{array}{c}X_{0}=-++++++\\\\X_{1}=+-+++++\\\\X_{2}=++-++++\\\\X_{3}=+++-+++\\\\X_{4}=++++-++\\\\X_{5}=+++++-+\\\\X_{6}=++++++-\\\\\\end{array}\\right\\rbrace $ a code for $\\mathbb {Z}_{2}^{7}$ and define the set $\\mathcal {X}_{7}^{\\prime }=\\lbrace X_{3}X_{5}X_{6},X_{2}X_{4}X_{5},X_{1}X_{3}X_{4},X_{0}X_{2}X_{3},X_{6}X_{1}X_{2},X_{5}X_{0}X_{1},X_{4}X_{6}X_{0}\\rbrace .$ on $\\mathcal {X}_{7}$ .",
"It is easy to show that $\\mathcal {X}_{7}^{\\prime }$ is not a code.",
"For example, $X_{0}X_{2}X_{5}X_{6}$ has at least two factorization in $\\mathcal {X}^{\\prime }$ , namely $X_{0}X_{2}X_{3}\\cdot X_{3}X_{5}X_{6}$ and $X_{2}X_{4}X_{5}\\cdot X_{4}X_{6}X_{0}$ .",
"Also, $\\vert \\mathcal {X}_{7}^{\\prime *}\\vert =16$ and not 128 as desirable.",
"Hence $\\mathcal {X}_{7}^{\\prime *}$ is not free group.",
"However, from theorem REF a $P(T)$ -code always generates a free group.",
"The following theorem say us as to obtain new $P(T)$ -free subgroup from old.",
"Theorem 5 Let $\\mathcal {X}_{i}$ be $P(T_{i})$ -codes, $1\\le i\\le r$ , in $\\mathbb {Z}_{2}^{n}$ such that $T_{i}\\cap T_{j}=\\emptyset $ , $i\\ne j$ , and $T_{i}\\subset N$ .",
"Then $\\left(\\bigcup _{i=1}^{r}\\mathcal {X}_{i}\\right)^{*}=\\prod _{i=1}^{r}\\mathcal {X}_{i}^{*}.$ and $\\vert \\prod _{i=1}^{r}\\mathcal {X}_{i}^{*}\\vert =2^{\\sum _{i=1}^{r}\\vert \\mathcal {X}_{i}\\vert }$ Follows by induction on number of $T_{i}$ -codes $\\mathcal {X}_{i}$ .",
"Now we will obtain a necessary condition for the existence of an $S$ -subgroup Theorem 6 Let $G$ be a permutation automorphic subgroup of $Aut(\\mathbb {Z}_{2}^{n})$ acting on some set $\\mathcal {X}$ in $\\mathbb {Z}_{2}^{n}$ .",
"Then $\\mathcal {X}^{*}$ is an $S$ -subgroup in $\\mathfrak {S}(\\mathbb {Z}_{2}^{n},G)$ .",
"We take a word $Y_{i_{1}}Y_{i_{2}}\\cdots Y_{i_{r}}$ in $\\mathcal {X}^{\\prime *}$ and a $g$ in $G$ .",
"Then $g(Y_{i_{1}}Y_{i_{2}}\\cdots Y_{i_{r}})&=&g(Y_{i_{1}})g(Y_{i_{2}})\\cdots g(Y_{i_{r}})\\\\&=&Y_{j_{1}}Y_{j_{2}}\\cdots Y_{j_{r}}.$ As $g$ is arbritary, then $g(Y_{i_{1}}Y_{i_{2}}\\cdots Y_{i_{r}})$ is in $\\mathcal {X}^{\\prime *}$ for all $g$ in $G$ .",
"Hence $G$ defines a partition on $\\mathcal {X}^{\\prime *}$ and $\\mathcal {X}^{\\prime *}$ is an $S$ -subgroup of $\\mathfrak {S}(\\mathbb {Z}_{2}^{n},G)$ .",
"Not all $S$ -subgroup is free.",
"$\\mathcal {X}_{7}^{\\prime *}$ in (REF ) is an $S$ -subgroup of $\\mathfrak {S}(\\mathbb {Z}_{2}^{7},C_{7})$ , where $C_{7}=\\left\\langle C\\right\\rangle $ is the cyclic permutation automorphic subgroup of $Aut(\\mathbb {Z}_{2}^{7})$ of order 7 with $C$ the cyclic permutation acting on all component of some $Y$ in $\\mathbb {Z}_{2}^{7}$ .",
"But $\\mathcal {X}_{7}^{\\prime *}$ is not a free subgroup of $\\mathbb {Z}_{2}^{7}$ .",
"Again let $G$ be a permutation automorphic subgroup of $Aut(\\mathbb {Z}_{2}^{n})$ and let $Y_{G}$ denote the orbit of some $Y$ in $\\mathbb {Z}_{2}^{n}$ under the action of $G$ .",
"From previous theorem $Y_{G}^{*}$ is an $S$ -subgroup.",
"We will called to $Y_{G}^{*}$ a basic $S$ -subgroup of $\\mathfrak {S}(\\mathbb {Z}_{2}^{n},G)$ .",
"If for a code $\\mathcal {X}$ it is true that $\\mathcal {X}=Y_{G}$ for some $Y$ in $\\mathbb {Z}_{2}^{n}$ , then we will say that $\\mathcal {X}$ is a $G$ -code.",
"In the following sections we construct $S$ -subgroups by using $G$ -codes."
],
[
"Schur ring $\\mathfrak {S}(\\mathbb {Z}_{2}^{n},S_{n})$",
"Let $\\omega (X)$ denote the Hamming weight of $X\\in \\mathbb {Z}_{2}^{n}$ .",
"Thus, $\\omega (X)$ is the number of $+$ in any $\\mathbb {Z}_{2}-$ sequences $X$ of $\\mathbb {Z}_{2}^{n}$ .",
"Now let $\\mathcal {G}_{n}(k)$ be the subset of $\\mathbb {Z}_{2}^{n}$ such that $\\omega (X)=k$ for all $X\\in \\mathcal {G}_{n}(k)$ , where $0\\le k\\le n$ .",
"We let $T_{i}=\\mathcal {G}_{n}(n-i)$ .",
"It is straightforward to prove that the partition $\\mathfrak {S}(\\mathbb {Z}_{2}^{n},S_{n})=\\lbrace \\mathcal {G}_{n}(0),...,\\mathcal {G}_{n}(n)\\rbrace $ induces an $S$ -partition over $\\mathbb {Z}_{2}^{n}$ , where $S_{n}\\le Aut(\\mathbb {Z}_{2}^{n})$ is the permutation group on $n$ objects.",
"From [7] it is know that the constant structure $\\lambda _{i,j,k}$ is equal to $\\lambda _{i,j,k}={\\left\\lbrace \\begin{array}{ll}0&\\mbox{if } i+j-k\\ \\mbox{is an odd number}\\\\\\binom{k}{(j-i+k)/2}\\binom{n-k}{(j+i-k)/2} &\\mbox{if } i+j-k\\ \\mbox{is an even number}\\end{array}\\right.",
"}$ From (REF ) follows that $\\mathcal {G}_{n}(a)\\mathcal {G}_{n}(b)={\\left\\lbrace \\begin{array}{ll}\\bigcup \\limits _{i=0}^{a}\\mathcal {G}_{n}(n-a-b+2i), & 0\\le a\\le \\left[\\dfrac{n}{2}\\right], a\\le b\\le n-a,\\\\\\bigcup \\limits _{i=0}^{n-a}\\mathcal {G}_{n}(a+b-n+2i), & \\left[\\dfrac{n}{2}\\right]+1\\le a\\le n, n-a\\le b\\le a.\\end{array}\\right.",
"}$ From (REF ) we know that $\\mathcal {X}_{n}=\\mathcal {G}_{n}(n-1)$ .",
"Then $\\mathcal {X}_{n}$ is an $S_{n}$ -code for $\\mathfrak {S}(\\mathbb {Z}_{2}^{n},S_{n})$ .",
"In the following corollary we found another $S_{n}$ -code for $\\mathfrak {S}(\\mathbb {Z}_{2}^{n},S_{n})$ Corollary 2 $\\mathcal {G}_{n}(1)$ is an $S_{n}$ -code for $\\mathfrak {S}(\\mathbb {Z}_{2}^{n},S_{n})$ .",
"It is enough to take into account that $-\\mathcal {G}_{n}(n-1)=\\mathcal {G}_{n}(1)$ .",
"We prefer to use the $S_{n}$ -code $\\mathcal {G}_{n}(n-1)$ and not $\\mathcal {G}_{n}(1)$ because in $\\mathcal {G}_{n}(n-1)$ the positions of the negative components are easily obtained.",
"Indeed, $+++-\\cdot +-++=+-+-$ in $\\mathcal {G}_{4}(3)$ but in $\\mathcal {G}_{4}(1)$ we have $---+\\cdot -+--=+-+-$ .",
"Next we will see that a $G$ -code is contained in any $S$ -set of $\\mathfrak {S}(\\mathbb {Z}_{2}^{n},S_{n})$ Proposition 1 A $G$ -code $\\mathcal {X}$ is contained in $\\mathcal {G}_{n}(a)$ for some $a$ ranging in $[1,n-1]$ .",
"Take $X$ in $\\mathcal {X}$ .",
"It is easy to note that $\\omega (gX)=\\omega (X)$ for all $g\\in G$ .",
"Hence $\\mathcal {X}\\subseteq \\mathcal {G}_{n}(a)$ for some $a$ in $[1,n-1]$ .",
"On the other hand, it is follows directly from (REF ) that $\\lambda _{i,j,2k+1}=0$ if $i+j$ is even and $\\lambda _{i,j,2k}=0$ if $i+j$ is odd.",
"The union of all basic sets $\\mathcal {G}_{n}(2a)$ in $\\mathfrak {S}(\\mathbb {Z}_{2}^{n},S_{n})$ will be denoted by $\\mathcal {E}_{n}$ and the union of all basic sets $\\mathcal {G}_{n}(2a+1)$ in $\\mathfrak {S}(\\mathbb {Z}_{2}^{n},S_{n})$ will be denoted $\\mathcal {O}_{n}$ .",
"The sets $\\mathcal {E}_{2n}$ and $\\mathcal {O}_{2n+1}$ are subgroups of order $2^{2n-1}$ and $2^{2n}$ , respectively.",
"Then $\\mathfrak {S}(\\mathcal {E}_{2n},S_{n})=\\lbrace \\mathcal {G}_{2n}(0),\\mathcal {G}_{2n}(2),...,\\mathcal {G}_{2n}(2n)\\rbrace $ and $\\mathfrak {S}(\\mathcal {O}_{2n+1},S_{n})=\\lbrace \\mathcal {G}_{2n+1}(1),\\mathcal {G}_{2n+1}(3),...,\\mathcal {G}_{2n+1}(2n+1)\\rbrace $ are $S$ -subgroups of $\\mathfrak {S}(\\mathbb {Z}_{2}^{2n},S_{2n})$ and $\\mathfrak {S}(\\mathbb {Z}_{2}^{2n+1},S_{2n+1})$ , respectively.",
"From (REF ), $\\mathcal {G}_{2n}(n)^{2}=\\bigcup _{i=0}^{n}\\mathcal {G}_{2n}(2i)=\\mathcal {E}_{2n}$ and $\\mathcal {G}_{2n+1}(n)^{2}=\\bigcup _{i=0}^{n}\\mathcal {G}_{2n+1}(2i+1)=\\mathcal {O}_{2n+1}$ .",
"Therefore, neither $\\mathcal {G}_{4n}(2n)$ nor $\\mathcal {G}_{4n+3}(2n+1)$ contains some code $\\mathcal {X}$ generating whole $\\mathbb {Z}_{2}^{4n}$ and $\\mathbb {Z}_{2}^{2n+1}$ , respectively.",
"This remark is generalized below.",
"From [7] is obtained the following definition Definition 2 Take $\\mathcal {G}_{n}(a)$ in $\\mathfrak {S}(\\mathbb {Z}_{2}^{n},S_{n})$ .",
"Let $\\mathfrak {S}^{\\prime }\\subset \\mathfrak {S}(\\mathbb {Z}_{2}^{n},S_{n})$ be a set of basic sets.",
"We will call $\\mathfrak {S}^{\\prime }$ a $\\mathcal {G}_{n}(a)$ -complete $S$ -set if it holds $\\mathcal {G}_{n}(i)\\mathcal {G}_{n}(j)\\supset \\mathcal {G}_{n}(a)$ for all $\\mathcal {G}_{n}(i),\\mathcal {G}_{n}(j)\\in \\mathfrak {S}^{\\prime }$ , There is no $\\mathcal {G}_{n}(b)\\in \\mathfrak {S}(\\mathbb {Z}_{2}^{n},S_{n})$ such that $\\mathcal {G}_{n}(b)^{2}\\supset \\mathcal {G}_{n}(a)$ and $\\mathcal {G}_{n}(b)\\mathcal {G}_{n}(k)\\supset \\mathcal {G}_{n}(a)$ for all $\\mathcal {G}_{n}(k)\\in \\mathfrak {S}^{\\prime }$ .",
"A important result obtained is that there is no $\\mathcal {G}_{n}(a)$ -complete for all $n$ and all $a$ Theorem 7 There is no $\\mathcal {G}_{2n}(2a+1)$ -complete $S$ -sets in $\\mathfrak {S}(\\mathbb {Z}_{2}^{2n},S_{n})$ .",
"There is no $\\mathcal {G}_{2n+1}(2a)$ -complete $S$ -sets in $\\mathfrak {S}(\\mathbb {Z}_{2}^{2n+1},S_{n})$ .",
"In the following theorem is shown the relationship between codes generating whole $\\mathbb {Z}_{2}^{n}$ and non $\\mathcal {G}_{n}(a)$ -complete $S$ -sets in $\\mathfrak {S}(\\mathbb {Z}_{2}^{n},S_{n})$ Theorem 8 There is no a code $\\mathcal {X}$ generating whole $\\mathbb {Z}_{2}^{n}$ in a $\\mathcal {G}_{n}(a)$ -complete $S$ -set.",
"Let $\\mathfrak {S}^{\\prime }$ denote a $\\mathcal {G}_{2n}(2a)$ -complete $S$ -set.",
"From (REF ) $\\mathcal {G}_{2n}(2b)^{2}=\\bigcup _{i=0}^{2b}\\mathcal {G}_{2n}(2n-4b+2i)$ for all $\\mathcal {G}_{2n}(2b)$ in $\\mathfrak {S}^{\\prime }$ .",
"Then all powers of $\\mathcal {G}_{2n}(2b)$ will contain basic sets $\\mathcal {G}_{2n}(2k)$ only.",
"Therefore the basic sets in a $\\mathcal {G}_{2n}(2a)$ -complete can generate the $S$ -subgroup $\\mathcal {E}_{2n}$ at the most.",
"With a similar argument is shown for basic sets in $\\mathcal {G}_{2n+1}(2a+1)$ -complete $S$ -sets.",
"We finish this section showing some basic sets of $\\mathfrak {S}(\\mathbb {Z}_{2}^{n},S_{n})$ that can to contain $G$ -codes generating all $\\mathbb {Z}_{2}^{n}$ .",
"Proposition 2 $_{}$ $\\mathcal {G}_{4n}(2n-1)^{3}=\\mathcal {O}_{4n}$ , $\\mathcal {G}_{4n}(2n-1)^{4}=\\mathcal {E}_{4n}$ .",
"$\\mathcal {G}_{4n+2}(2n-1)^{3}=\\mathcal {O}_{4n+2}$ , $\\mathcal {G}_{4n}(2n-1)^{4}=\\mathcal {E}_{4n+2}$ .",
"$\\mathcal {G}_{4n+1}(2n-2)^{3}=\\mathcal {E}_{4n+1}$ , $\\mathcal {G}_{4n+1}(2n-2)^{4}=\\mathcal {O}_{4n+1}$ .",
"$\\mathcal {G}_{4n+3}(2n)^{3}=\\mathcal {E}_{4n+3}$ , $\\mathcal {G}_{4n+3}(2n)^{4}=\\mathcal {O}_{4n+3}$ .",
"By using (REF ) we have $\\mathcal {G}_{4n}(2n-1)^{2}&=&\\mathcal {G}_{4n}(2)\\cup \\cdots \\cup \\mathcal {G}_{4n}(4n)\\\\&\\supset &\\mathcal {G}_{4n}(2n)\\nonumber .$ As $\\mathcal {G}_{4n}(2n-1)\\mathcal {G}_{4n}(2n)=\\mathcal {G}_{4n}(1)\\cup \\cdots \\cup \\mathcal {G}_{4n}(4n-1)$ then $\\mathcal {G}_{4n}(2n-1)^{3}=\\mathcal {O}_{4n}$ .",
"From $\\mathcal {G}_{4n}(2n-1)\\mathcal {G}_{4n}(2n+1)\\supset \\mathcal {G}_{4n}(0)$ and from (REF ) is followed that $\\mathcal {G}_{4n}(2n-1)^{4}=\\mathcal {E}_{4n}$ .",
"As $S_{n}$ induces a $S$ -partition on $\\mathbb {Z}_{2}^{n}$ is straightforward to prove that $G$ induces a $S$ -partition on $\\mathbb {Z}_{2}^{n}$ for all $G\\le S_{n}\\le Aut(\\mathbb {Z}_{2}^{n})$ .",
"In the following sections we will construct $S$ -subgroups by using $G$ -codes in $S$ -ring $\\mathfrak {S}(\\mathbb {Z}_{2}^{n},G)$ ."
],
[
"Schur ring $\\mathfrak {S}(\\mathbb {Z}_{2}^{n},C_{n})$",
"Let $C$ denote the cyclic permutation on the components $+$ and $-$ of $X$ in $\\mathbb {Z}_{2}^{n}$ such that $C(X)=C\\left( x_{0},x_{1},...,x_{n-2},x_{n-1}\\right) =\\left(x_{1},x_{2},x_{3},...,x_{0}\\right),$ that is, $C(x_{i})=x_{(i+1) mod n}$ .",
"The permutation $C$ is a generator of cyclic group $C_{n}=\\left\\langle C\\right\\rangle $ of order $n$ .",
"Let $X_{C}=Orb_{C_{n}}X=\\lbrace C^{i}(X):C^{i}\\in C_{n} \\rbrace $ .",
"Therefore, $C_{n}$ defines a partition in equivalent class on $\\mathbb {Z}_{2}^{n}$ which is an $S$ -partition and this we shall denote by $\\mathbb {Z}_{2C}^{n}=\\mathfrak {S}(\\mathbb {Z}_{2}^{n},C_{n})$ .",
"It is worth mentioning that this Schur ring corresponds to the orbit Schur ring induced by the cyclic permutation automorphic subgroup $C_{n}\\le S_n\\le Aut(\\mathbb {Z}_2^n)$ .",
"On the other hand, let $X=\\lbrace x_{i}\\rbrace $ and $Y=\\lbrace y_{i}\\rbrace $ be two complex-valued sequences of period $n$ .",
"The periodic correlation of $X$ and $Y$ at shift $k$ is the product defined by: $\\mathsf {P}_{X,Y}(k)=\\sum \\limits _{i=0}^{n-1}x_{i}\\overline{y}_{i+k},\\ k=0,1,...,n-1,$ where $\\overline{a}$ denotes the complex conjugation of $a$ and $i+k$ is calculated modulo $n$ .",
"If $Y=X$ , the correlation $\\mathsf {P}_{X,Y}(k)$ is denoted by $\\mathsf {P}_{X}(k)$ and is the autocorrelation of $X$ .",
"Obviously, $\\mathsf {P}_{X}(k)&=&\\overline{\\mathsf {P}_{X}(n-k)},\\\\\\mathsf {P}_{-X}(k)&=&\\mathsf {P}_{X}(k),\\\\\\mathsf {P}_{C^{i}X}(k)&=&\\mathsf {P}_{X}(k),$ for all $0\\le i\\le n-1$ and for all $X$ in $\\mathbb {Z}_{2}^{n}$ .",
"If $X$ is a $\\mathbb {Z}_{2}$ -sequence of length $n$ , $\\mathsf {P}_{X}(k)= 2\\omega \\left\\lbrace Y_{k}\\right\\rbrace -n$ , where $ Y_{k}=XC^{k}X $ .",
"Also by (REF ), if $X\\in \\mathcal {G}_{n}(a)$ , then $\\mathsf {P}_{X}(k)=n-4a+4i_{k},$ for some $0\\le i_{k}\\le a$ and $n-\\mathsf {P}_{X}(k)$ is divisible by 4 for all $k$ .",
"We know from theorem REF that $\\mathcal {G}_{n}(n-1)$ is a code for $\\mathfrak {S}(\\mathbb {Z}_{2}^{n},S_{n})$ for all $n$ .",
"As $\\mathcal {G}_{n}(n-1)=\\lbrace X,CX,C^{2}X,...,C^{n-1}\\rbrace =\\mathcal {X}_{C}$ , then $\\mathcal {X}_{C}$ is a $C_{n}$ -code for $\\mathfrak {S}(\\mathbb {Z}_{2}^{n},C_{n})$ .",
"Then for to obtain information from each basic set $Y_{C}$ in $\\mathfrak {S}(\\mathbb {Z}_{2}^{n},C_{n})$ we must do it through of its $\\mathcal {X}_{C}$ -factorization with the $C_{n}$ -code $\\mathcal {X}_{C}$ .",
"The advantage of using this code lies in its simplicity, since each $C^{i}X$ has exactly a $-$ as its component and thereby it is possible to know exactly the Hamming weight of each word writing with this basis.",
"All word $Y$ in $\\mathcal {G}_{n}(a)$ has the form $C^{i_{1}}XC^{i_{2}}X\\cdots C^{i_{r}}X$ with length $\\vert Y\\vert =r$ and with $a=n-r$ .",
"Then, every basic set in $\\mathcal {G}_{nC}(a)$ has form $Y_{C}=\\bigcup _{k=0}^{n-1}C^{i_{1}+k}XC^{i_{2}+k}X\\cdots C^{i_{r}+k}X,$ and in this way if $Z=C^{j_{1}}XC^{j_{2}}X\\cdots C^{j_{s}}X$ , we have $Y_{C}Z_{C}&=&\\bigcup _{k=0}^{n-1}(YC^{k}Z)_{C}\\\\&=&\\bigcup _{k=0}^{n-1}(C^{i_{1}}X\\cdots C^{i_{r}}XC^{j_{1}+k}X\\cdots C^{j_{s}+k}X)_{C}.$ Each word $YC^{k}Z=C^{i_{1}}X\\cdots C^{i_{r}}XC^{j_{1}+k}X\\cdots C^{j_{s}+k}X$ can be reduced if exist two equal letters.",
"Thereupon $YC^{k}Z$ decreases its length an even number.",
"Therefore $YC^{k}Z$ belong to $\\mathcal {G}_{n}(b)$ with $b=n-(r+s)+2w$ where $2w$ is the number of canceled letters.",
"If both $Y$ and $Z$ belongs to $\\mathcal {G}_{n}(a)$ , then $b=n-2r+2w$ and $\\mathsf {P}_{Y,Z}(k)=n-4r+4w_{k}$ .",
"Next we will obtain the algebraic version of (REF ), () and () Proposition 3 Let $Y$ denote the binary sequence $C^{i_{1}}XC^{i_{2}}X\\cdots C^{i_{r}}X$ .",
"If $YC^{k}Y\\in \\mathcal {G}_{n}(a)$ , then $YC^{n-k}Y$ and $(C^{j}X)C^{k}(C^{j}X)$ are in $\\mathcal {G}_{n}(a)$ too.",
"$(-Y)C^{k}(-Y)\\in \\mathcal {G}_{n}(a)$ .",
"1.",
"It is clear that $YC^{k}Y=C^{i_{1}}XC^{i_{2}}X\\cdots C^{i_{r}}XC^{i_{1}+k}XC^{i_{2}+k}X\\cdots C^{i_{r}+k}X\\in \\mathcal {G}_{n}(a)$ with $a=n-2r+2w$ , where $2w$ are the number of canceled letters.",
"We wish to show that the cancellation numbers of $YC^{k}Y$ and $YC^{n-k}Y$ coincide.",
"Suppose that $i_{j}=i_{1}+k$ for some $j$ and some $k$ .",
"Then this implies that $n-k+i_{j}=i_{1}$ reduced module $n$ .",
"Therefore $YC^{n-k}Y=C^{i_{1}}X\\cdots C^{i_{r}}XC^{n-k+i_{1}}X\\cdots C^{n-k+i_{r}}X$ has the same number of cancellations as $YC^{k}Y$ .",
"Equally is proved for $(C^{j}X)C^{k}(C^{j}X)$ .",
"2.",
"As $C^{k}(-Y)=-C^{k}Y$ , then $(-Y)C^{k}(-Y)=YC^{k}Y\\in \\mathcal {G}_{n}(a)$ .",
"Now we will show other advantage of to use the $C_{n}$ -code $\\mathcal {X}_{C}$ Proposition 4 For all $n\\le 2$ we have $\\overline{\\mathcal {G}_{n}(n-2)}^{2}=n+2\\overline{\\mathcal {G}_{n}(n-2)}+(n-3)\\overline{\\mathcal {G}_{n}(n-4)}.$ All word $Y$ in $\\mathcal {G}_{n}(n-2)$ has the form $C^{i}XC^{j}X$ with $i<j$ .",
"Then $YC^{k}Y=C^{i}XC^{j}XC^{i+k}XC^{j+k}X$ and there exist a $k$ such that either $i+k=j$ or $j+k=i$ and for all the remaining values of $k$ we have that $C^{i}XC^{j}XC^{i+k}XC^{j+k}X$ is a reduced word.",
"As $YC^{k}Y$ and $YC^{n-k}Y$ are in $\\mathcal {G}_{n}(a)$ for some $a$ , then $Y_{C}^{2}$ contains 2 words in $\\mathcal {G}_{n}(n-2)$ , $n-3$ words in $\\mathcal {G}_{n}(n-4)$ and the trivial word in $\\mathcal {G}_{n}(n)$ .",
"On the other hand, let $F_{d}(\\mathbb {Z}_{2}^{n})=\\bigcup _{\\vert X\\vert =d}X.$ Clearly $d$ divides to $n$ and the $X\\in F_{d}(\\mathbb {Z}_{2}^{n})$ have the form $X=(Y,Y,...,Y)$ , with $Y\\in \\mathbb {Z}_{2}^{d}$ .",
"Then $F_{d}(\\mathbb {Z}_{2C}^{n})=\\bigcup _{\\vert X_{C}\\vert =d}X_{C}$ is an $S$ -set of $\\mathbb {Z}_{2C}^{n}$ , for each $d\\vert n$ .",
"When $d=n$ , we will to say that $C_{n}$ acts freely on $X_{C}$ and we denote $F_{n}(\\mathbb {Z}_{2C}^{n})$ as $F(\\mathbb {Z}_{2C}^{n})$ .",
"When $d<n$ , we will to say that $C_{n}$ don't act freely on $X_{C}$ and let $\\widehat{F}(\\mathbb {Z}_{2C}^{n})$ denote the set of the $X_{C}$ which are not frees under the action of $C_{n}$ , namely $\\widehat{F}(\\mathbb {Z}_{2C}^{n})=\\bigcup _{d\\mid n,d<n}F_{d}(\\mathbb {Z}_{2C}^{n}).$ Therefore, $\\mathbb {Z}_{2C}^{n}&=&F(\\mathbb {Z}_{2C}^{n})\\cup \\widehat{F}(\\mathbb {Z}_{2C}^{n})\\nonumber \\\\&=&\\bigcup _{d\\mid n}F_{d}(\\mathbb {Z}_{2C}^{n}).$ The set $F_{d}(\\mathbb {Z}_{2}^{n})$ is constructed with codewords in $\\mathcal {X}_{F,d}=\\lbrace A_{0,d}X,A_{1,d}X,\\cdots ,A_{d-1,d}X\\rbrace $ where $A_{i,d}X=C^{i}XC^{i+d}X\\cdots C^{i+kd}X$ for $i=0,1,2,...,d-1$ with $k=\\frac{n}{d}-1$ , $X\\in \\mathcal {G}_{n}(n-1)$ and $\\lbrace i,i+d,i+2d,\\cdots ,i+kd\\rbrace $ is an arithmetic progression.",
"Then $\\mathcal {X}_{F,d}$ is a $P(T)$ -code with $P(T)=\\lbrace \\lbrace i,i+d,i+2d,...,i+kd\\rbrace :\\ i=0,1,...,d-1\\rbrace $ and any word in $\\mathcal {X}_{F,d}^{*}$ has the form $Y=A_{0,d}^{\\epsilon _{0}}XA_{1,d}^{\\epsilon _{1}}X\\cdots A_{d-1,d}^{\\epsilon _{d-1}}X$ $\\epsilon _{i}=0,1$ .",
"It is clear that $C^{i}A_{j,d}X=A_{i+j,d}X$ .",
"Hence $\\mathcal {X}_{F,d}$ is a $C_{n}$ -code.",
"If $d=1$ , then $k=n-1$ , $i=0$ and $\\mathcal {X}_{F,1}=\\lbrace A_{0,1}X\\rbrace =\\lbrace XCXC^{2}X\\cdots C^{n-1}X\\rbrace $ is a code with a codeword.",
"As $XCXC^{2}X\\cdots C^{n-1}X=-\\textbf {1}$ , then $\\mathcal {X}_{F,1}^{*}=\\lbrace \\textbf {1},-\\textbf {1}\\rbrace $ for all $n$ .",
"If $d=n$ , then $k=0$ , $i=0,1,\\cdots ,n-1$ and $\\mathcal {X}_{F,n}=\\lbrace A_{0,n}X,A_{1,n}X,\\cdots ,A_{n-1,n}X\\rbrace =\\lbrace X,CX,C^{2}X,\\cdots ,C^{n-1}X\\rbrace $ is an code with exactly $n$ codewords and $\\mathcal {X}_{F,d}=\\mathcal {X}_{C}$ .",
"Let $\\vert \\mathcal {X}_{F,d}\\vert $ be the rank of $\\mathcal {X}_{F,d}$ .",
"We then note that $1<\\vert \\mathcal {X}_{F,d}\\vert <n$ for $1<d<n$ .",
"Now we will see the relationship between free subgroup $\\mathcal {X}_{F,d}^{*}$ and the sets $F_{d}(\\mathbb {Z}_{2}^{n})$ Theorem 9 $\\mathcal {X}_{F,d}^{*}$ is a subgroup of $\\mathbb {Z}_{2}^{n}$ of order $2^{d}$ with $\\mathcal {X}_{F,d}^{*}=\\bigcup _{r\\vert d}F_{d}(\\mathbb {Z}_{2}^{n})$ .",
"We will donote this subgroup with $\\mathbb {G}_{d}(n)$ .",
"As $\\mathcal {X}_{F,d}$ is a $P(T)$ -code, then $\\mathbb {G}_{d}(n)$ is a subgroup of $\\mathbb {Z}_{2}^{n}$ of order $2^{d}$ for all divisor $d$ of $n$ .",
"Then we only will show that $\\mathbb {G}_{d}(n)$ has the desired structure.",
"If $d$ is a prime divisor of $n$ , then all words of $\\mathbb {G}_{d}(n)$ are in $F_{d}(\\mathbb {Z}_{2}^{n})$ , except for $\\textbf {1}$ and $-\\textbf {1}$ .",
"Hence $\\mathbb {G}_{d}(n)=F_{1}(\\mathbb {Z}_{2}^{n})\\cup F_{d}(\\mathbb {Z}_{2}^{n})$ with $-\\textbf {1}&=&\\prod _{i=0}^{d-1}A_{i,d}X\\\\\\textbf {1}&=&(A_{i,d}X)^{2}\\ for\\ all\\ i.$ Suppose that $d$ is no prime.",
"Then $A_{i,d}XA_{i+r,d}X\\cdots A_{i+(\\frac{d}{r}-1)r,d}X$ is contained in $F_{r}(\\mathbb {Z}_{2}^{n})$ , $r\\vert d$ and $i=0,1,...,r-1$ .",
"Therefore $\\mathbb {G}_{d}(n)=\\bigcup _{r\\vert d}F_{r}(\\mathbb {Z}_{2}^{n})$ .",
"Corollary 3 $\\mathbb {G}_{dC}(n)$ is an $S$ -subgroup of $\\mathfrak {S}(\\mathbb {Z}_{2}^{n},C_{n})$ .",
"The group $\\left\\langle C_{n}\\right\\rangle $ defines a partition on each $F_{r}(\\mathbb {Z}_{2}^{n})$ in $\\mathbb {G}_{d}(n)$ and hence we obtain the desired statement.",
"Example 1 The subgroup $\\mathbb {G}_{3}(9)$ of $\\mathbb {Z}_{2}^{9}$ is given by $\\mathbb {G}_{3}(9)&=&F_{1}(\\mathbb {Z}_{2}^{9})\\cup F_{3}(\\mathbb {Z}_{2}^{9})\\\\&=&\\lbrace -\\textbf {1},\\textbf {1}\\rbrace \\cup \\lbrace XC^{3}XC^{6}X,\\ CXC^{4}XC^{7}X,\\ C^{2}XC^{5}XC^{8}X,\\\\&&XCXC^{3}XC^{4}XC^{6}XC^{7}X,\\ XC^{2}XC^{3}XC^{5}XC^{6}XC^{8}X,\\\\&&CXC^{2}XC^{4}XC^{5}XC^{7}XC^{8}X\\rbrace \\\\&=&\\lbrace -\\textbf {1},\\textbf {1},-++-++-++,++-++-++-,\\\\&&+-++-++-+,-+--+--+-,--+--+--+,\\\\&&+--+--+--\\rbrace .$ And the $S$ -subgroup $\\mathbb {G}_{3C}(9)$ of $\\mathfrak {S}(\\mathbb {Z}_{2}^{9},C_{9})$ is given by $\\mathbb {G}_{3C}(9)&=&\\lbrace \\textbf {1},-\\textbf {1},(-++-++-++)_{C},(-+--+--+-)_{C}\\rbrace .$ Theorem 10 $\\mathbb {G}_{d}(n)\\subseteq \\bigcup _{a=0}^{d}\\mathcal {G}_{n}\\left(\\frac{na}{d}\\right)$ with equality only for $d=1,n$ .",
"Follows from $F_{d}(\\mathbb {Z}_{2}^{n})\\subseteq \\mathcal {G}_{n}\\left(\\frac{n}{d}\\right)\\cup \\mathcal {G}_{n}\\left(n-\\frac{n}{d}\\right)$ .",
"Now we show the lattice of $S$ -subgroups $\\mathbb {G}_{dC}(60)$ of $\\mathfrak {S}(\\mathbb {Z}_{2}^{60},C_{60})$ ordered by inclusion.",
"maxmed) at (1,3) $\\mathbb {G}_{30C}(60)$ ; a) at (-3,2) $\\mathbb {G}_{15C}(60)$ ; b) at (1,0) $\\mathbb {G}_{10C}(60)$ ; c) at (4,2) $\\mathbb {G}_{6C}(60)$ ; d) at (-3,-1) $\\mathbb {G}_{5C}(60)$ ; e) at (0,1) $\\mathbb {G}_{3C}(60)$ ; f) at (4,-1) $\\mathbb {G}_{2C}(60)$ ; min) at (0,-2) $\\mathbb {G}_{1C}(60)=\\lbrace \\textbf {1},-\\textbf {1}\\rbrace $ ; max) at (5,4) $\\mathbb {G}_{60C}(60)=\\mathbb {Z}_{2C}^{60}$ ; g) at (5,1) $\\mathbb {G}_{20C}(60)$ ; h) at (8,3) $\\mathbb {G}_{12C}(60)$ ; i) at (8,0) $\\mathbb {G}_{4C}(60)$ ; (min) – (d) – (a) – (maxmed) – (b) – (f) – (i) – (h) – (max) (e) – (min) – (f) – (c) – (maxmed) – (max) (d) – (b) – (g) – (max) (c) – (h) (i) – (g); [preaction=draw=white, -,line width=6pt] (a)– (e) – (c) – (maxmed) (c) – (h) (f) – (c) (e) – (min); We finish this section we provide other proof to the Theorems 6 and 7 in [7].",
"We start with the lemma Lemma 1 $XC^{n}X\\in F_{n}(\\mathbb {Z}_{2}^{2n})$ for all $X$ in $\\mathcal {X}_{C}=\\mathcal {G}_{2n}(2n-1)$ Clearly $XC^{n}X$ is in $F_{n}(\\mathbb {Z}_{2}^{2n})$ when $X=-+++\\cdots +++$ .",
"As $(C^{i}X)C^{n}(C^{i}X)=C^{i}(XC^{n}X)$ for any other codeword $C^{i}X$ in the code $\\mathcal {X}_{C}$ , then $C^{i}(XC^{n}X)\\in F_{n}(\\mathbb {Z}_{2}^{2n})$ for all $1\\le i\\le 2n-1$ , since $\\left\\langle C_{n}\\right\\rangle $ defines a partition on $F_{n}(\\mathbb {Z}_{2}^{2n})$ .",
"Theorem 11 If $Y_{C}\\in F(\\mathbb {Z}_{2C}^{2n})$ , then $Y_{C}^{2}\\setminus \\lbrace \\textbf {1}\\rbrace \\notin F(\\mathbb {Z}_{2C}^{2n})$ .",
"Take $Y=C^{i_{1}}X\\cdots C^{i_{r}}X$ in $F(\\mathbb {Z}_{2}^{2n})$ .",
"Then $YC^{n}Y&=&C^{i_{1}}X\\cdots C^{i_{r}}XC^{i_{1}+n}X\\cdots C^{i_{r}+n}X\\\\&=&C^{i_{1}}(XC^{n}X)\\cdots C^{i_{r}}(XC^{n}X).$ From the above lemma $XC^{n}X$ is in $F_{n}(\\mathbb {Z}_{2}^{2n})$ for all $X$ in $\\mathcal {X}_{C}$ .",
"Then $YC^{n}Y\\in F_{n}(\\mathbb {Z}_{2}^{2n})$ for all $Y$ in $F(\\mathbb {Z}_{2}^{n})$ and $Y_{C}^{2}\\setminus \\lbrace \\textbf {1}\\rbrace \\notin F(\\mathbb {Z}_{2C}^{2n})$ as we promised to show.",
"Lemma 2 $XC^{k}X\\notin F_{d}(\\mathbb {Z}_{2}^{2n+1})$ for no $X$ in $\\mathcal {X}_{C}=\\mathcal {G}_{2n+1}(2n)$ and for no $k$ ranging in $[1,2n]$ , $d<2n+1$ .",
"Take $X=-++\\cdots ++$ in $\\mathcal {X}_{C}$ .",
"Then $XC^{k}X=-\\overset{2n-k}{\\overbrace{+\\cdots +}}-\\overset{k-1}{\\overbrace{+\\cdots +}}$ .",
"If we want $k-1=2n-k$ , then $k=\\frac{2n+1}{2}$ , which is not possible.",
"Hence $XC^{k}X$ is no contained in $F_{d}(\\mathbb {Z}_{2}^{2n+1})$ , $d<2n+1$ .",
"As $C^{i}(XC^{k}X)=(C^{i}X)C^{k}(C^{i}X)$ , it is followed the statement.",
"Theorem 12 If $X_{C}\\in F(\\mathbb {Z}_{2C}^{2n+1})$ , then $X_{C}^{2}\\setminus \\lbrace \\textbf {1}\\rbrace \\in F(\\mathbb {Z}_{2C}^{2n+1})$ .",
"Let $Y=C^{i_{1}}XC^{i_{2}}X\\cdots C^{i_{r}}X$ such that all the $i_{j}$ are not in arithmetic progression.",
"If $YC^{k}Y=C^{i_{1}}(XC^{k}X)C^{i_{2}}(XC^{k}X)\\cdots C^{i_{r}}(XC^{k}X)$ is contained in some $F_{d}(\\mathbb {Z}_{2}^{2n+1})$ , $d\\vert (2n+1)$ , $d<2n+1$ , then $XC^{k}X$ must be contained in $F_{d}(\\mathbb {Z}_{2}^{2n+1})$ , but is not possible by the previous lemma."
],
[
"Schur ring $\\mathfrak {S}(\\mathbb {Z}_{2}^{n},\\Delta _{n})$",
"Let $\\delta _{a}\\in S_{n}$ act on $X\\in \\mathbb {Z}_{2}^{n}$ by decimation, that is, $\\delta _{a}(x_{i})=x_{ai(\\mod {n})}$ for all $x_{i}$ in $X$ , $(a,n)=1$ and let $\\Delta _{n}$ denote the set of this $\\delta _{a}$ .",
"The set $\\Delta _{n}$ is a group of order $\\phi (n)$ isomorphic to $\\mathbb {Z}_{n}^{*}$ , the group the units of $\\mathbb {Z}_{n}$ , where $\\phi $ is called the Euler totient function.",
"Clearly $\\mathfrak {S}(\\mathbb {Z}_{2}^{n},\\Delta _{n})$ is an $S$ -partition of $\\mathbb {Z}_{2}^{n}$ .",
"In this section, we will construct $\\Delta _{n}$ -codes for $S$ -subgroups of $\\mathfrak {S}(\\mathbb {Z}_{2}^{n},\\Delta _{n})$ .",
"We will use the commutation relation $C^{i}\\delta _{a}=\\delta _{a}C^{ia}$ for to prove all of results in this section.",
"We begin for show that $\\mathcal {G}_{n}(n-1)$ is partitioned in three equivalence class Proposition 5 $\\Delta _{n}\\mathcal {G}_{n}(n-1)=\\lbrace X\\rbrace \\cup \\mathcal {X}_{\\mathbb {Z}_{n}^{*}}\\cup \\mathcal {X}_{\\mathbb {Z}_{n}\\setminus \\mathbb {Z}_{n}^{*}}$ where $X=-++\\cdots ++$ and $\\mathcal {X}_{\\mathbb {Z}_{n}^{*}}&=&\\lbrace C^{a_{1}}X,C^{a_{2}}X,...,C^{a_{\\phi (n)}}X:\\ (a_{i},n)=1\\rbrace \\\\\\mathcal {X}_{\\mathbb {Z}_{n}\\setminus \\mathbb {Z}_{n}^{*}}&=&\\lbrace C^{d_{1}}X,C^{d_{2}}X,...,C^{d_{r}}X:\\ (d_{i},n)\\ne 1\\rbrace $ $r=n-\\phi (n)-1$ .",
"It is very easy to see that $X=-++\\cdots ++$ is fixed under the action of $\\Delta _{n}$ .",
"Also, as $\\delta _{a}C^{i}X=C^{a^{-1}i}\\delta _{a}X=C^{a^{-1}i}X$ , then $\\Delta _{n}\\mathcal {X}_{\\mathbb {Z}_{n}^{*}}=\\mathcal {X}_{\\mathbb {Z}_{n}^{*}}$ and $\\Delta _{n}\\mathcal {X}_{\\mathbb {Z}_{n}\\setminus \\mathbb {Z}_{n}^{*}}=\\mathcal {X}_{\\mathbb {Z}_{n}\\setminus \\mathbb {Z}_{n}^{*}}$ .",
"As each $C^{i}X$ in $\\mathcal {X}_{\\mathbb {Z}_{n}^{*}}$ or in $\\mathcal {X}_{\\mathbb {Z}_{n}\\setminus \\mathbb {Z}_{n}^{*}}$ is atomic, then $\\mathcal {X}_{\\mathbb {Z}_{n}^{*}}$ and $\\mathcal {X}_{\\mathbb {Z}_{n}\\setminus \\mathbb {Z}_{n}^{*}}$ are $\\Delta _{n}$ -code and hence $\\mathcal {X}_{\\mathbb {Z}_{n}^{*}}^{*}$ and $\\mathcal {X}_{\\mathbb {Z}_{n}\\setminus \\mathbb {Z}_{n}^{*}}^{*}$ are $S$ -subgroups in $\\mathfrak {S}(\\mathbb {Z}_{2}^{n},\\Delta _{n})$ with $\\vert \\mathcal {X}_{\\mathbb {Z}_{n}^{*}}^{*}\\vert =2^{\\phi (n)}$ and $\\vert \\mathcal {X}_{\\mathbb {Z}_{n}\\setminus \\mathbb {Z}_{n}^{*}}^{*}\\vert =2^{n-\\phi (n)-1}$ .",
"On the other hand, let $(\\mathsf {P}_{Y}(0),\\mathsf {P}_{Y}(1),...,\\mathsf {P}_{Y}(n-1))$ denote the autocorrelation vector of $Y$ in $\\mathbb {Z}_{2}^{n}$ and let $\\mathfrak {A}(\\mathbb {Z}_{2}^{n})$ denote the set of all this.",
"Let $X_{1}+X_{2}+\\cdots +X_{n}=a$ denote the plane in $\\mathbb {Z}^{n}$ in the indeterminates $X_{i}$ , $i=1,2,...,n$ and let $\\theta :\\mathbb {Z}_{2}^{n}\\rightarrow \\mathfrak {A}(\\mathbb {Z}_{2}^{n})$ be the map defined by $\\theta (Y)=(\\mathsf {P}_{Y}(0),\\mathsf {P}_{Y}(1),\\dots ,\\mathsf {P}_{Y}(n-1))$ .",
"The decimation group $\\Delta _{n}$ do not alter the set of values which $\\mathsf {P}_{X}(k)$ takes on, but merely the order in which they appear, i.e., if $Y=\\delta _{a}X$ then $\\mathsf {P}_{Y}(k)=\\mathsf {P}_{X}(ka)$ .",
"Therefore, we have the commutative diagram ${\\mathbb {Z}_{2}^{n} [d]^{\\theta } [r]^{\\delta _{r}} & \\mathbb {Z}_{2}^{n} [d]^{\\theta }\\\\\\mathfrak {A}(\\mathbb {Z}_{2}^{n}) [r]^{\\delta _{r}} & \\mathfrak {A}(\\mathbb {Z}_{2}^{n})}$ and $\\theta \\circ \\delta _{r} = \\delta _{r}\\circ \\theta .$ Let $Y\\in \\mathbb {Z}_{2}^{n}$ such that $\\theta (Y)=(n,d,d,...,d)$ .",
"Such a binary sequence is known as binary sequence with 2-levels autocorrelation value and are important by its applications on telecommunication.",
"We want to construct a $\\Delta _{n}$ -code for some $S$ -subgroup $H$ of $\\mathfrak {S}(\\mathbb {Z}_{2}^{n},\\Delta _{n})$ containing such $Y$ .",
"From (REF ) is followed that $\\theta (Y)=\\delta _{a}\\theta (Y)=\\theta (\\delta _{a}Y)$ , for all $\\delta _{a}\\in \\Delta _{n}$ .",
"Hence $Y$ and $\\delta _{a}Y$ have the same autocorrelation vector.",
"For $Y$ fullfilling $\\delta _{a}Y=Y$ for some $\\delta _{a}$ in $\\Delta _{n}$ we have the following definition Definition 3 Let $a$ be a unit in $\\mathbb {Z}_{n}^{*}$ .",
"A word $Y$ in $\\mathbb {Z}_{2}^{n}$ is $\\delta _{a}$ -invariant if $\\delta _{a}Y=Y$ .",
"Denote by $\\mathbb {I}_{n}(a)$ the set of these $Y$ .",
"If $Y$ is in $\\mathbb {I}_{n}(a)$ , then $\\delta _{r}Y$ is in $\\mathbb {I}_{n}(a)$ , too.",
"Also $\\delta _{a}(YZ)=\\delta _{a}Y\\delta _{a}Z=YZ$ for all $Y,Z$ in $\\mathbb {I}_{n}(a)$ .",
"Then $\\mathbb {I}_{n}(a)$ is an $S$ -subgroup of $\\mathfrak {S}(\\mathbb {Z}_{2}^{n},\\Delta _{n})$ .",
"Now, we shall see that all factorization of words in $\\mathbb {I}_{n}(a)$ is relationated with cyclotomic coset of $a$ module $n$ .",
"First, we have the following definition Definition 4 Let $a$ relative prime to $n$ .",
"The cyclotomic coset of $a$ module $n$ is defined by $\\mathsf {C}_{s}=\\lbrace s,sa,sa^{2},\\cdots ,sa^{t-1}\\rbrace .$ where $sa^{t}\\equiv s\\mod {n}$ .",
"A subset $\\lbrace s_{1},s_{2},\\dots ,s_{r}\\rbrace $ of $\\mathbb {Z}_{n}$ is called complete set of representatives of cyclotomic coset of $a$ modulo $n$ if $\\mathsf {C}_{i_{1}}$ ,$\\mathsf {C}_{i_{2}}$ ,..., $\\mathsf {C}_{i_{r}}$ are distinct and are a partition of $\\mathbb {Z}_{n}$ .",
"Take $Y=C^{i_{1}}XC^{i_{2}}X\\cdots C^{i_{r}}X$ in $\\mathbb {I}_{n}(a)$ with $X=-++\\cdots ++$ .",
"We want $\\delta _{a}Y=Y$ .",
"Then $\\delta _{a}Y&=&\\delta _{a}C^{i_{1}}X\\delta _{a}C^{i_{2}}X\\cdots \\delta _{a}C^{i_{r}}X\\\\&=&C^{i_{1}a^{-1}}\\delta _{a}XC^{i_{2}a^{-1}}\\delta _{a}X\\cdots C^{i_{r}a^{-1}}\\delta _{a}X\\\\&=&C^{i_{1}a^{-1}}XC^{i_{2}a^{-1}}X\\cdots C^{i_{r}a^{-1}}X$ since $\\delta _{a}X=X$ .",
"As must be $\\delta _{a}Y=Y$ , then $i_{k}=a^{-1}i_{j}$ or $i_{j}=ai_{k}$ for $1\\le k,j\\le r$ .",
"Let $\\mathsf {C}_{s}X$ denote the word $C^{s}XC^{sa}X\\cdots C^{sa^{t_{s}-1}}X$ .",
"Then all $Y$ in $\\mathbb {I}_{n}(a)$ has the form $Y=\\mathsf {C}_{s_{1}}^{\\epsilon _{1}}X\\mathsf {C}_{s_{2}}^{\\epsilon _{r}}X\\cdots \\mathsf {C}_{s_{r}}^{\\epsilon }X$ , with $\\epsilon _{i}=0,1$ .",
"As $\\mathbb {I}_{n}(a)$ is an $S$ -subgroup in $\\mathfrak {S}(\\mathbb {Z}_{2}^{n},\\Delta _{n})$ , $\\delta _{r}\\mathsf {C}_{s_{i}}X=\\mathsf {C}_{s_{j}}X$ and $\\mathcal {X}_{\\mathbb {I}(a)}=\\lbrace X,\\mathsf {C}_{s_{1}}X,\\ \\mathsf {C}_{s_{2}}X,...,\\ \\mathsf {C}_{s_{r}}X\\rbrace $ is a $\\Delta _{n}$ -code for $\\mathbb {I}_{n}(a)$ .",
"Also $\\mathcal {X}_{\\mathbb {I}(a)}$ is a $P(T)$ -code with $P(T)=\\lbrace \\lbrace 0\\rbrace ,\\mathsf {C}_{s_{1}},\\mathsf {C}_{s_{2}},...,\\mathsf {C}_{s_{r}}\\rbrace $ and $\\lbrace s_{1},s_{2},...,s_{r}\\rbrace $ a complete set of representatives.",
"Hence $\\mathcal {X}_{\\mathbb {I}(a)}^{*}$ has order $2^{r+1}$ , where $r$ is the number of cyclotomic cosets of $a$ module $n$ In the table 1, binary sequences with 2-level autocorrelation values with their respective $\\delta _{a}$ -invariants $S$ -subgroups are shown Table: Binary sequences with 2-level autocorrelation valuesOn the other hand, we have the following theorem Theorem 13 If $\\left\\langle b\\right\\rangle $ is a subgroup of $\\left\\langle a\\right\\rangle $ , then $\\mathbb {I}_{n}(a)\\le \\mathbb {I}_{n}(b)$ .",
"Let $\\mathsf {C}_{1}^{a}$ and $\\mathsf {C}_{1}^{b}$ denote the classes $\\lbrace 1,a,a^{2},...,a^{t-1}\\rbrace $ and $\\lbrace 1,b,b^{2},...,b^{s-1}\\rbrace $ .",
"By hypothesis $\\left\\langle b\\right\\rangle \\le \\left\\langle a\\right\\rangle $ , then $\\mathsf {C}_{1}^{b}\\subseteq \\mathsf {C}_{1}^{a}$ .",
"Hence there exists $y_{i}$ in $\\left\\langle a\\right\\rangle $ such that $\\mathsf {C}_{1}^{a}=\\mathsf {C}_{1}^{b}\\cup y_{1}\\mathsf {C}_{1}^{b}\\cup \\cdots \\cup y_{k}\\mathsf {C}_{1}^{b},$ and $k=[\\left\\langle a\\right\\rangle :\\left\\langle b\\right\\rangle ]$ .",
"Then is follows that $\\mathsf {C}_{s}^{a}=\\mathsf {C}_{s}^{b}\\cup y_{1}\\mathsf {C}_{s}^{b}\\cup \\cdots \\cup y_{k}\\mathsf {C}_{s}^{b}.$ Therefore $\\vert \\mathcal {X}_{\\mathbb {I}_{n}(a)}\\vert \\le \\vert \\mathcal {X}_{\\mathbb {I}_{n}(b)}\\vert $ and $\\mathbb {I}_{n}(a)\\le \\mathbb {I}_{n}(b)$ .",
"We finish this section constructing some $\\delta _{a}$ -invariants $S$ -subgroups Proposition 6 $_{}$ $\\mathbb {I}_{2n+1}(2n)=\\lbrace X,\\ \\mathsf {C}_{q}X:\\ q\\in \\lbrace 1,2,...,n\\rbrace \\rbrace ^{*}$ , $\\mathsf {C}_{q}=\\lbrace q,2n+1-q\\rbrace $ $\\mathbb {I}_{2n}(2n-1)=\\lbrace X,\\ C^{n}X,\\ \\mathsf {C}_{q}X:\\ q\\in \\lbrace 1,2,...,n-1\\rbrace \\rbrace ^{*}$ , $\\mathsf {C}_{q}=\\lbrace q,2n-q\\rbrace $ .",
"We proof 1.",
"The proof of 2 it is analogous.",
"We note that $(2n)^{2}=(2n+1-1)^{2}=(2n+1)^{2}-2(2n+1)+1\\equiv 1\\mod {(}2n+1),$ then $\\mathsf {C}_{1}=\\lbrace 1,2n\\rbrace $ .",
"As $2n+1\\nmid 2n-1$ and $q<2n+1$ , then $2nq\\lnot \\equiv q\\mod {(}2n+1)$ and the $\\mathsf {C}_{q}=\\lbrace q,2nq\\rbrace $ are cyclotomic cosets of $2n$ module $2n+1$ .",
"Finally, it is easy to note that $2nq$ is congruent to $2n+1-q$ module $2n+1$ , $2nq-2n-1+q=(2n+1)(q-1)\\equiv 0\\mod {(}2n+1).$ Proposition 7 Let $2p+1$ be an prime number with $p$ an odd prime number.",
"The $S$ -subgroups invariants in $\\mathbb {Z}_{2}^{2p+1}$ are $\\mathbb {I}_{2p+1}(x)$ , $\\mathbb {I}_{2p+1}(y)$ and $\\mathbb {I}_{2p+1}(2p)$ , where $x$ is a primitive root module $2p+1$ and $y$ is not neither primitive root module $2p+1$ nor $2p$ .",
"Let $P=\\lbrace x_{1},x_{2},...,x_{t}\\rbrace $ denote the set of primitive roots module $2p+1$ .",
"Then $\\mathbb {I}_{2p+1}(x_{1})=\\mathbb {I}_{2p+1}(x_{2})=\\cdots =\\mathbb {I}_{2p+1}(x_{t}).$ As $\\vert \\left\\langle x_{i}\\right\\rangle \\vert =2p$ for any $x_{i}\\in P$ , then $\\left\\langle x_{i}\\right\\rangle $ has exactly a subgroup of order 2 and a subgroup of order $p$ .",
"Therefore there exist $S$ -subgroups invariants $\\mathbb {I}_{2p+1}(y)$ and $\\mathbb {I}_{2p+1}(2p)$ where $y\\in P^{c}\\setminus \\lbrace 2p\\rbrace $ , with $P^{c}$ the complement of $P$ in $\\mathbb {Z}_{2p+1}^{*}$ ."
],
[
"Schur ring $\\mathfrak {S}(\\mathbb {Z}_{2}^{n},H_{n})$",
"We note by $RY$ the reversed sequence $RY = (y_{n-1},...,y_{1},y_{0})$ and let $H_{n}$ denote the permutation automorphic subgroup $H_{n}=\\lbrace 1,R\\rbrace \\le S_{n}\\le Aut(Z_{2}^{n})$ .",
"Hence $H_{n}$ defines a partition on $\\mathbb {Z}_{2}^{n}$ and $\\mathfrak {S}(\\mathbb {Z}_{2}^{n},H_{n})$ is a schur ring.",
"Definition 5 Let $Y\\in \\mathbb {Z}_{2}^{n}$ .",
"We shall call $Y$ symmetric if $RY=Y$ and otherwise we say it is non symmetric.",
"We make $Sym(\\mathbb {Z}_{2}^{n})$ the set of all $Y$ symmetric and $\\widehat{Sym}(\\mathbb {Z}_{2}^{n})$ the set of all $Y$ nonsymmetric.",
"Take $Y\\in \\mathbb {Z}_{2}^{n}$ such that $Y$ is of the form $C^{i_{1}}XC^{i_{2}}X\\cdots C^{i_{r}}X$ .",
"We want to understand the structure of the words in $Sym(\\mathbb {Z}_{2}^{n})$ .",
"As it must be fulfilled that $RY=Y$ , then taking $X=+\\cdots +-+\\cdots +$ in $\\mathcal {G}_{n}(n-1)$ with $n$ an odd number we have $RY&=&RC^{i_{1}}XRC^{i_{2}}X\\cdots RC^{i_{r}}X\\\\&=&C^{-i_{1}}RXC^{-i_{2}}RX\\cdots C^{-i_{r}}RX\\\\&=&C^{n-i_{1}}XC^{n-i_{2}}X\\cdots C^{n-i_{r}}X$ where we have used that $RX=X$ .",
"Hence if $Y$ is symmetric, then must be $n-i_{j}=i_{k}$ for $j\\ne k$ ranging in $[1,r]$ .",
"Thereby $Y$ has the form $Y_{0}^{\\epsilon _{0}}Y_{1}^{\\epsilon _{1}}\\cdots Y_{(n-1)/2}^{\\epsilon _{(n-1)/2}}$ for $\\epsilon _{i}=0,1$ and $Y_{0}=X$ and $Y_{i}=C^{i}XC^{n-i}X$ .",
"As $C^{i}XC^{n-i}X$ is in $Sym(\\mathbb {Z}_{2}^{n})$ for all $i$ , then $\\mathcal {X}_{Sym^{O}}=\\lbrace X,\\ CXC^{n-1}X,\\ C^{2}XC^{n-2}X,...,\\ C^{(n-1)/2}XC^{(n+1)/2}X\\rbrace $ is a $H_{n}$ -code for $Sym(\\mathbb {Z}_{2}^{n})$ .",
"For the case $n$ an even number it is easily followed that $\\mathcal {X}_{Sym^{E}}=\\lbrace XC^{n-1}X,\\ CXC^{n-2}X,\\ C^{2}XC^{n-3}X,...,\\ C^{(n-2)/2}XC^{n/2}X\\rbrace $ is a $H_{n}$ -code for $Sym(\\mathbb {Z}_{2}^{n})$ , where $X=+++\\cdots ++-$ .",
"Also it is clear that $\\mathcal {X}_{Sym^{E}}$ and $\\mathcal {X}_{Sym^{O}}$ are $P(T)$ -codes, therefore $Sym(\\mathbb {Z}_{2}^{n})$ is an free $S$ -subgroup in $\\mathfrak {S}(\\mathbb {Z}_{2}^{n},H_{n})$ and $Sym(\\mathbb {Z}_{2}^{2n+1})$ and $Sym(\\mathbb {Z}_{2}^{2n})$ have order $2^{n+1}$ and $2^{n}$ , respectively.",
"Finally, the relationship between the symmetric subgroup $Sym(\\mathbb {Z}_{2}^{^{2n+1}})$ and the $\\delta _{2n}$ -invariant $S$ -subgroup $\\mathbb {I}_{2n+1}(2n)$ is shown Theorem 14 $Sym(\\mathbb {Z}_{2}^{^{2n+1}})=\\mathbb {I}_{2n+1}(2n)$ .",
"By proposition REF the codewords in $\\mathcal {X}_{\\mathbb {I}_{2n+1}(2n)}$ are $X$ and $C^{q}XC^{2n+1-q}X$ , $1\\le q\\le n$ , with $X=+\\cdots +-+\\cdots +$ .",
"We want to show that all of codewords in $\\mathcal {X}_{\\mathbb {I}_{2n+1}(2n)}$ is symmetric.",
"For this we note that $RX=X$ and $R(C^{q}XC^{2n+1-q}X)&=&RC^{q}XRC^{2n+1-q}X\\\\&=&C^{2n+1-q}RXC^{q}RX\\\\&=&C^{2n+1-q}XC^{q}X.$"
],
[
"Schur ring $\\mathfrak {S}(\\mathbb {Z}_{2}^{n},H_{n}C_{n})$",
"In this section we will use the commutation relation $RC^{i}=C^{n-i}R$ to show that the $S$ -subgroups $\\mathbb {G}_{d}(n)$ and $Sym(\\mathbb {Z}_{2}^{n})$ are $S$ -subgroups in $\\mathfrak {S}(\\mathbb {Z}_{2}^{n},H_{n}C_{n})$ .",
"Theorem 15 $\\mathbb {G}_{dC}(n)$ is an $S$ -subgroup in $\\mathfrak {S}(\\mathbb {Z}_{2}^{n},H_{n}C_{n})$ By having in mind the commutation relation (REF ), we can to show that $\\mathcal {X}_{F,d}$ is an $H_{n}$ -code.",
"In this way we have $RA_{i,d}X&=&R(C^{i}XC^{i+d}X\\cdots C^{i+n-d}X)\\\\&=&RC^{i}XRC^{i+d}X\\cdots RC^{i+n-d}X\\\\&=&C^{n-i}RXC^{n-i-d}RX\\cdots C^{d-i}RX$ As $X=-++\\cdots ++$ , then $RX=CX$ .",
"Hence $RA_{i,d}X&=&C^{n-i}CXC^{n-i-d}CX\\cdots C^{d-i}CX\\\\&=&C(C^{n-i}XC^{n-i-d}X\\cdots C^{d-i}X)$ Finally, reordering and rewriting $RA_{i,d}X&=&C(C^{d-i}X\\cdots C^{d-i+(n-2d)}XC^{d-i+(n-d)}X)\\\\&=&CA_{d-i,d}X\\\\&=&A_{d-i+1,d}X.$ Definition 6 A $S$ -set $Y_{C}$ in $\\mathbb {Z}_{2C}^{n}$ is symmetric if $R\\cdot Y_{C}=Y_{C}$ , where $R\\cdot Y_{C}$ means the action of $R$ on the elements of $Y_{C}$ .",
"The set of all symmetric $S$ -sets will be denoted by $Sym(\\mathbb {Z}_{2C}^{n})$ and the set of all non-symmetric $S$ -sets will be denoted by $\\widehat{Sym}(\\mathbb {Z}_{2C}^{n})$ .",
"It is easy to note that $R\\cdot Y_{C}=(RY)_{C}$ .",
"Therefore $Y_{C}$ is a symmetric $S$ -set if and only if contains some $C^{i}Y$ symmetric.",
"Theorem 16 $Sym(\\mathbb {Z}_{2C}^{n})$ is an $S$ -subgroup of $\\mathfrak {S}(\\mathbb {Z}_{2}^{n},H_{n}C_{n})$ .",
"Take $Y_{C},Z_{C}$ in $Sym(\\mathbb {Z}_{2C}^{n})$ and suppose that $Y,Z\\in Sym(\\mathbb {Z}_{2}^{n})$ .",
"As $R(YC^{k}Z)_{C}=(RYRC^{k}Z)_{C}=(YC^{n-k}Z)_{C},$ then $R(Y_{C}Z_{C})=R(Y_{C})R(Z_{C})=Y_{C}Z_{C}$ ."
],
[
"Schur ring $\\mathfrak {S}(\\mathbb {Z}_{2}^{n},\\Delta _{n}C_{n})$",
"In this section we will use the commutation relation $\\delta _{a}C^{i}=C^{ia^{-1}}\\delta _{a}$ to show that the $S$ -subgroups $\\mathbb {G}_{d}(n)$ and $\\mathbb {I}_{n}(a)$ are $S$ -subgroups in $\\mathfrak {S}(\\mathbb {Z}_{2}^{n},\\Delta _{n}C_{n})$ .",
"Theorem 17 $\\mathbb {G}_{dC}(n)$ is an $S$ -subgroup in $\\mathfrak {S}(\\mathbb {Z}_{2}^{n},\\Delta _{n}C_{n})$ .",
"We want to show that $\\mathcal {X}_{F,d}$ is a $\\Delta _{n}$ -code.",
"Take $A_{i,d}X$ in $\\mathcal {X}_{F,d}$ .",
"Then if $k=\\frac{n}{d}-1$ we have $\\delta _{a}A_{i,d}X&=&\\delta _{a}(C^{i}XC^{i+d}X\\cdots C^{i+kd}X)\\\\&=&\\delta _{a}C^{i}X\\delta _{a}C^{i+d}X\\cdots \\delta _{a}C^{i+kd}X)\\\\&=&C^{ia^{-1}}\\delta _{a}XC^{(i+d)a^{-1}}\\delta _{a}X\\cdots C^{(i+kd)a^{-1}}\\delta _{a}X\\\\&=&C^{ia^{-1}}XC^{(i+d)a^{-1}}X\\cdots C^{(i+kd)a^{-1}}X.$ Define the map $\\vartheta :\\mathcal {X}_{F,d}\\rightarrow \\lbrace ni/d:\\ i=1,2,...,d-1\\rbrace $ by $\\vartheta (A_{i,d}X)&=&\\vartheta (C^{i}XC^{i+d}X\\cdots C^{i+kd}X)\\\\&=&\\sum _{j=0}^{k}(i+jd)\\\\&=&i(k+1)+\\frac{k(k+1)}{2}d\\\\&=&\\frac{ni}{d}+\\left(\\frac{n}{2d}-\\frac{1}{2}\\right)n\\\\&\\equiv &\\frac{ni}{d}\\mod {n}$ Then $\\vartheta $ is a biyection.",
"As $\\vartheta (\\delta _{a}A_{i,d}X)\\equiv \\frac{a^{-1}ni}{d}\\mod {n}$ and $\\vartheta (C^{l}\\delta _{a}A_{i,d}X)\\equiv (a^{-1}+l)\\frac{ni}{d}\\mod {n}$ , is followed that $\\delta _{a}A_{i,d}X\\in \\mathcal {X}_{F,d}$ and therefore $\\mathcal {X}_{F,d}$ is a $\\Delta _{n}$ -code.",
"Definition 7 Let $a$ be a unit in $\\mathbb {Z}_{n}^{*}$ .",
"A basic set $Y_{C}$ in $\\mathbb {Z}_{2C}^{n}$ is $\\delta _{a}$ -invariant if $\\delta _{a}\\cdot Y_{C}=Y_{C}$ .",
"Denote by $\\mathbb {I}_{nC}(a)$ the set of these $Y$ .",
"By having in mind the basic set $Y_{C}$ , we can note that $\\delta _{a}\\cdot Y_{C}=(\\delta _{a}Y)_{C}$ .",
"Hence $Y_{C}$ is $\\delta _{a}$ -invariant in $\\mathbb {Z}_{2C}^{n}$ if and only if contains some $C^{i}Y$ $\\delta _{a}$ -invariant in $\\mathbb {Z}_{2}^{n}$ .",
"Theorem 18 $\\mathbb {I}_{nC}(a)$ is an $S$ -subgroup in $\\mathfrak {S}(\\mathbb {Z}_{2}^{n},\\Delta _{n}C_{n})$ .",
"Take $Y_{C},Z_{C}$ in $\\mathbb {I}_{nC}(a)$ and suppose that $Y,Z\\in \\mathbb {I}_{n}(a)$ .",
"As $\\delta _{a}(YC^{k}Z)_{C}=(\\delta _{a}Y\\delta _{a}C^{k}Z)_{C}=(YC^{ka^{-1}}Z)_{C}$ , then $\\delta _{a}(Y_{C}Z_{C})=\\delta _{a}(Y_{C})\\delta _{a}(Z_{C})=Y_{C}Z_{C}$ ."
],
[
"Schur ring $\\mathfrak {S}(\\mathbb {Z}_{2}^{n},H_{n}\\Delta _{n}C_{n})$",
"Finally we show that the $S$ -subgroups $\\mathbb {G}_{d}(n)$ , $Sym(\\mathbb {Z}_{2}^{n})$ and $\\mathbb {I}_{n}(a)$ are $S$ -subgroups in $\\mathfrak {S}(\\mathbb {Z}_{2}^{n},H_{n}\\Delta _{n}C_{n})$ .",
"Let $Y_{C}$ be any basic set in $\\mathbb {Z}_{2C}^{n}$ .",
"It is a very easy to notice that $Y_{C}=(C^{k}Y)_{C}$ for all $k$ .",
"We will use this fact for to prove the following lemma Lemma 3 $_{}$ $\\delta _{a}Y_{iC}=Y_{a^{-1}iC}$ for $Y_{i}=C^{i}XC^{2n+1-i}X\\in \\mathcal {X}_{Sym^{O}}$ .",
"$\\delta _{a}Y_{iC}=Y_{\\left(a^{-1}i+\\frac{a^{-1}-1}{2}\\right)C}$ for $Y_{i}=C^{i}XC^{2n-1-i}X\\in \\mathcal {X}_{Sym^{E}}$ .",
"$1.$ Take $\\delta _{a}$ in the group $\\Delta _{n}$ .",
"It is clear that $\\delta _{a}X=C^{k_{a}}X$ for some $k_{a}$ depending on $a$ , where $X=+\\cdots +-+\\cdots +$ is the word in $\\mathcal {X}_{C}$ used to construct all codewords in $\\mathcal {X}_{Sym^{O}}$ .",
"Then $\\delta _{a}Y_{iC}&=&(\\delta _{a}C^{i}X\\delta _{a}C^{2n+1-i}X)_{C}\\\\&=&(C^{a^{-1}i}\\delta _{a}XC^{a^{-1}(2n+1-i)}\\delta _{a}X)_{C}\\\\&=&(C^{k_{a}}(C^{a^{-1}i}XC^{2n+1-a^{-1}i}X))_{C}\\\\&=&(C^{a^{-1}i}XC^{2n+1-a^{-1}i}X)_{C}\\\\&=&Y_{a^{-1}iC}.$ $2.$ Equally, $\\delta _{a}X=C^{k_{a}}X$ for some $k_{a}$ depending on $a$ , where $X=++\\cdots ++-$ in $\\mathcal {X}_{C}$ is used to construct all codewords in $\\mathcal {X}_{Sym^{E}}$ .",
"Then $\\delta _{a}Y_{iC}&=&(\\delta _{a}C^{i}X\\delta _{a}C^{2n-1-i}X)_{C}\\\\&=&(C^{a^{-1}i}\\delta _{a}XC^{a^{-1}(2n-1-i)}\\delta _{a}X)_{C}\\\\&=&(C^{k_{a}}(C^{a^{-1}i}XC^{2n-a^{-1}(i+1)}X))_{C}\\\\&=&\\left(C^{\\frac{a^{-1}-1}{2}}(C^{a^{-1}i}XC^{2n-a^{-1}(i+1)}X)\\right)_{C}\\\\&=&\\left(C^{a^{-1}i+\\frac{a^{-1}-1}{2}}XC^{2n-a^{-1}(i+1)+\\frac{a^{-1}-1}{2}}X\\right)_{C}\\\\&=&Y_{\\left(a^{-1}i+\\frac{a^{-1}-1}{2}\\right)C}.$ We will use this lemma for to show that the symmetric binary sequences form an $S$ -subgroup in $\\mathfrak {S}(\\mathbb {Z}_{2}^{n},H_{n}\\Delta _{n}C_{n})$ Theorem 19 $Sym(\\mathbb {Z}_{2C}^{n})$ is an $S$ -subgroup in $\\mathfrak {S}(\\mathbb {Z}_{2}^{n},H_{n}\\Delta _{n}C_{n})$ .",
"Clearly $Sym(\\mathbb {Z}_{2C}^{n})$ is an $S$ -subgroup of $\\mathfrak {S}(\\mathbb {Z}_{2}^{n},H_{n}C_{n})$ .",
"From the previous lemma is followed that $\\Delta _{n}$ defines a partition on $\\mathcal {X}_{Sym^{E}}$ and $\\mathcal {X}_{Sym^{O}}$ .",
"Hence they are $\\Delta _{n}$ -codes and $Sym(\\mathbb {Z}_{2C}^{n})$ is an $S$ -subgroup of $\\mathfrak {S}(\\mathbb {Z}_{2}^{n},H_{n}\\Delta _{n}C_{n})$ .",
"Theorem 20 $\\mathbb {G}_{dC}(n)$ is an $S$ -subgroup in $\\mathfrak {S}(\\mathbb {Z}_{2}^{n},H_{n}\\Delta _{n}C_{n})$ .",
"Follows from theorems REF and REF .",
"Theorem 21 $\\mathbb {I}_{nC}(a)$ is an $S$ -subgroup in $\\mathfrak {S}(\\mathbb {Z}_{2}^{n},H_{n}\\Delta _{n}C_{n})$ .",
"From theorem REF the case $a=n-1$ is excluded.",
"In the previous section already was proved that $\\mathbb {I}_{nC}(a)$ is an $S$ -subgroup in $\\mathfrak {S}(\\mathbb {Z}_{2}^{n},\\Delta _{n}C_{n})$ .",
"Now, we wish to show that $H_{n}$ defines a partition on $\\mathbb {I}_{nC}(a)$ by using (REF ).",
"Take the codeword $\\mathsf {C}_{s}X$ in $\\mathcal {X}_{\\mathbb {I}_{n}(a)}$ .",
"We have then $R(\\mathsf {C}_{s}X)_{C}&=&R(C^{s}XC^{sa}X\\cdots C^{sa^{t_{s}-1}}X)_{C}\\\\&=&(RC^{s}XRC^{sa}X\\cdots RC^{sa^{t_{s}-1}}X)_{C}\\\\&=&(C^{n-s}CXC^{n-sa}CX\\cdots C^{n-sa^{t_{s}-1}}CX)_{C}\\\\&=&(C(C^{n-s}XC^{n-sa}X\\cdots C^{n-sa^{t_{s}-1}}X))_{C}\\\\&=&(C^{n-s}XC^{(n-s)a}X\\cdots C^{(n-s)a^{t_{s}-1}}X))_{C}\\\\&=&(\\mathsf {C}_{n-s}X)_{C}.$ E-mail address, [email protected]"
]
] | 1906.04250 |
[
[
"The study of unclassified B[e] stars and candidates in the Galaxy and\n Magellanic Clouds"
],
[
"Abstract We investigated 12 unclassified B[e] stars or candidates, 8 from the Galaxy, 2 from the Large Magellanic Cloud (LMC) and 2 from the Small Magellanic Cloud (SMC).",
"Based on the analysis of high-resolution spectroscopic (FEROS) and photometric data, we confirmed the presence of the B[e] phenomenon for all objects of our sample, except for one (IRAS 07455-3143).",
"We derived their effective temperature, spectral type, luminosity class, interstellar extinction and, using the distances from Gaia DR2, we obtained their bolometric magnitude, luminosity and radius.",
"Modeling of the forbidden lines present in the FEROS spectra revealed information about the kinematics and geometry of the circumstellar medium of these objects.",
"In addition, we analyzed the light curves of four stars, finding their most probable periods.",
"The evolutionary stage of 11 stars of our sample is suggested from their position on the HR diagram, taking into account evolutionary tracks of stars with solar, LMC and SMC metallicities.",
"As results, we identified B and B[e] supergiants, B[e] stars probably at the main sequence or close to its end, post-AGB and HAeB[e] candidates, and A[e] stars in the main sequence or in the pre-main sequence.",
"However, our most remarkable results are the identification of the third A[e] supergiant (ARDB\\,54, the first one in the LMC), and of an \"LBV impostor\" in the SMC (LHA 115-N82)."
],
[
"Introduction",
"The nomenclature “B[e] stars” was first used by to designate B-type stars that present forbidden emission lines in the optical spectrum.",
"Later, suggested the expression “stars with the B[e] phenomenon” to describe these objects.",
"This phenomenon was revised by , who associated it to the presence in the optical spectrum of B-type stars with: (i) intense Balmer emission lines, and (ii) permitted and forbidden emission lines of neutral and singly ionized metals, such as O i and Fe ii.",
"In addition, these stars also present strong excess in the near-IR and mid-IR, due to circumstellar (CS) dust.",
"However, these spectral characteristics are associated to the circumstellar medium and not to the object itself.",
"noted a great heterogeneity among these objects, suggesting the existence of four classes of stars with the B[e] phenomenon, based on their evolutionary stage: pre-main sequence intermediate-mass stars, or Herbig Ae/B[e] or simply HAeB[e]; massive supergiant stars, or B[e] supergiants or sgB[e]; compact planetary nebulae, or cPNB[e]; and symbiotic stars, or SymB[e].",
"Thus, an important question that needs to be answered is how such different objects can have similar spectroscopic features.",
"A possible answer is linked to the presence of a complex circumstellar environment, composed of a disk, as confirmed by polarimetric and interferometric measurements , or by rings .",
"The effect of binarity cannot be discarded either.",
"On the other hand, there is a large number of objects whose evolutionary stage is still unknown or poorly known, due to the absence of reliable stellar parameters, also including distance and interstellar extinction.",
"This group of objects is usually called as simply unclassified B[e] stars or unclB[e] .",
"proposed a new group of stars associated to the B[e] phenomenon, called as FS CMa stars, which is mainly formed by unclB[e] objects that would be close to or still on the main sequence in binary systems with mass exchange.",
"Table: Acknowledgements"
]
] | 1906.04268 |
[
[
"Translation hyperovals and $\\mathbb{F}_2$-linear sets of pseudoregulus\n type"
],
[
"Abstract In this paper, we study translation hyperovals in PG$(2,q^k)$.",
"The main result of this paper characterises the point sets defined by translation hyperovals in the Andr\\'e/Bruck-Bose representation.",
"We show that the affine point sets of translation hyperovals in PG$(2,q^k)$ are precisely those that have a scattered $\\mathbb{F}_2$-linear set of pseudoregulus type in PG$(2k-1,q)$ as set of directions.",
"This correspondence is used to generalise the results of Barwick and Jackson who provided a characterisation for translation hyperovals in PG$(2,q^2)$."
],
[
"Introduction",
"Let $\\operatorname{PG}(n,q)$ denote the $n$ -dimensional projective space over the finite field $\\mathbb {F}_q$ with $q$ elements.",
"A $k$ -arc in $\\operatorname{PG}(2,q)$ is a set of $k$ points such that no three of them are collinear.",
"A hyperoval in $\\operatorname{PG}(2,q)$ is a $(q+2)$ -arc.",
"Hyperovals only exist when $q$ is even.",
"A translation hyperoval is a hyperoval $H$ such that there exists a bisecant $\\ell $ of $H$ such that the group of elations with axis $\\ell $ acts transitively on the points of $H$ not on $\\ell $ .",
"It is well-known (see e.g.",
"[7]) that every translation hyperoval in $\\operatorname{PG}(2,q)$ is $\\mathrm {PGL}$ -equivalent to a point set $\\lbrace (1,t,t^{2^i})|t\\in \\mathbb {F}_{q}\\rbrace \\cup \\lbrace (0,1,0),(0,0,1)\\rbrace ,$ where $q=2^h$ and $\\gcd (i,h)=1$ .",
"In [4], Barwick and Jackson provided a chararacterisation of translation hyperovals in $\\operatorname{PG}(2,q^2)$ : they considered a set $\\mathcal {C}$ of points in $\\operatorname{PG}(4,q)$ , $q$ even, with certain combinatorial properties with respect to the planes of $\\operatorname{PG}(4,q)$ (see Section for details).",
"They proved that the set $\\mathcal {C}^{\\prime }$ of directions determined by the points of $\\mathcal {C}$ has the property that every line intersects $\\mathcal {C}^{\\prime }$ in $0,1,3$ or $q-1$ points.",
"They then used this to construct a Desarguesian line spread $\\mathcal {S}$ in $\\operatorname{PG}(3,q)$ , such that in the corresponding André/Bruck-Bose plane $\\mathcal {P}(\\mathcal {S})\\cong \\operatorname{PG}(2,q^2)$ , the points corresponding to $\\mathcal {C}$ form a translation hyperoval.",
"This extended the work done in [3], where the same authors gave a similar characterisation of André/Bruck-Bose representation of conics for $q$ odd.",
"In this paper, we will generalise the combinatorial characterisation provided by Barwick and Jackson for translation hyperovals.",
"In order to do this, we prove our main theorem, linking translation hyperovals with $\\mathbb {F}_2$ -linear sets of pseudoregulus type: Theorem 1.1 Let $\\mathcal {Q}$ be a set of $q^k$ affine points in $\\operatorname{PG}(2k,q)$ , $q=2^h$ , $h\\ge 4$ , $k\\ge 2$ , determining a set $D$ of $q^k-1$ directions in the hyperplane at infinity $H_\\infty =\\operatorname{PG}(2k-1,q)$ .",
"Suppose that every line has 0, 1, 3 or $q-1$ points in common with the point set $D$ .",
"Then (1) $D$ is an $\\mathbb {F}_2$ -linear set of pseudoregulus type.",
"(2) There exists a Desarguesian spread $\\mathcal {S}$ in $H_\\infty $ such that, in the Bruck-Bose plane $\\mathcal {P}(\\mathcal {S})\\cong \\operatorname{PG}(2,q^k)$ , with $H_\\infty $ corresponding to the line $l_\\infty $ , the points of $\\mathcal {Q}$ together with 2 extra points on $\\ell _\\infty $ , form a translation hyperoval in $\\operatorname{PG}(2,q^k)$ .",
"Vice versa, via the André/Bruck-Bose construction, the set of affine points of a translation hyperoval in $\\operatorname{PG}(2,q^k)$ , $q> 4, k\\ge 2$ , corresponds to a set $\\mathcal {Q}$ of $q^k$ affine points in $\\operatorname{PG}(2k,q)$ whose set of determined directions $D$ is an $\\mathbb {F}_2$ -linear set of pseudoregulus type.",
"Consequently, every line meets $D$ in $0,1,3$ or $q-1$ points.",
"This paper is organised as follows.",
"In Section we give the necessary definitions and background, in section , we provide a proof of Theorem REF .",
"Finally, we use this result in Section to generalise the result of Barwick-Jackson [4]."
],
[
"Linear sets",
"Linear sets are a central object in finite geometry and have been studied intensively, mainly due to the connection with other objects such as semifield planes, blocking sets, and more recently, MRD codes.",
"(see e.g.",
"[9], [10], [14]).",
"Let $V$ be an $r$ -dimensional vector space over $\\mathbb {F}_{q^n}$ , let $\\Omega $ be the projective space $\\operatorname{PG}(V) = \\operatorname{PG}(r-1, q^n)$ .",
"A set $T$ is said to be an $\\mathbb {F}_q$ -linear set of $\\Omega $ of rank $t$ if it is defined by the non-zero vectors of an $\\mathbb {F}_q$ -vector subspace $U$ of $V$ of dimension $t$ , i.e.",
"$T = L_U = \\lbrace \\langle u \\rangle _{\\mathbb {F}_{q^n}}| u \\in U \\setminus \\lbrace 0\\rbrace \\rbrace .$ The points of $\\operatorname{PG}(r-1,q^n)$ correspond to 1-dimensional subspaces of $\\mathbb {F}_{q^n}^{r}$ , and hence to $n$ -dimensional subspaces of $\\mathbb {F}_q^{rn}$ .",
"In this way, the point set of $\\operatorname{PG}(r-1,q^n)$ corresponds to a set $\\mathcal {D}$ of $(n-1)$ -dimensional subspaces of $\\operatorname{PG}(rn-1,q)$ , which partitions the point set of $\\operatorname{PG}(rn-1,q)$ .",
"The set $\\mathcal {D}$ is called a Desarguesian spread, and we have a one-to-one correspondence between the points of $\\operatorname{PG}(r-1,q^n)$ and the elements of $\\mathcal {D}$ .",
"Using coordinates, we see that a point $P=(x_0,x_1,\\dots x_{r-1})_{q^n} \\in \\operatorname{PG}(r-1,q^n)$ corresponds to the set $\\lbrace (\\alpha x_0,\\alpha x_1, \\dots , \\alpha x_{r-1})_q| \\alpha \\in \\mathbb {F}_{q^n} \\rbrace $ in $\\operatorname{PG}(rn-1,q)$ .",
"Note that we have used $r$ coordinates from $\\mathbb {F}_{q^n}$ , defined up to $\\mathbb {F}_q$ -scalar multiple to define points of $\\operatorname{PG}(rn-1,q)$ , and the set $\\lbrace (\\alpha x_0,\\alpha x_1, \\dots , \\alpha x_{r-1})_q| \\alpha \\in \\mathbb {F}_{q^n} \\rbrace $ consists of $\\frac{q^n-1}{q-1}$ different points forming an $(n-1)$ -dimensional space.",
"Hence, we find that $\\mathcal {D}$ is given by the set of $(n-1)$ -spaces $\\lbrace (\\alpha x_0,\\alpha x_1, \\dots , \\alpha x_{r-1})_q| \\alpha \\in \\mathbb {F}_{q^n} \\rbrace \\; \\mathrm {for\\ all\\ }(x_0,x_1,\\ldots ,x_{r-1})\\in \\mathbb {F}_{q^n}^r.$ Note that these coordinates for points in $\\operatorname{PG}(rn-1,q)$ can be transformed into the usual coordinates consisting of $rn$ elements of $\\mathbb {F}_q$ by representing the elements of $\\mathbb {F}_{q^n}$ as the $n$ coordinates with respect to a fixed basis of $\\mathbb {F}_{q^n}$ over $\\mathbb {F}_q$ .",
"We also have a more geometric perspective on the notion of a linear set; namely, an $\\mathbb {F}_q$ -linear set is a set $T$ of points of $\\operatorname{PG}(r-1,q^n)$ for which there exists a subspace $\\pi $ in $\\operatorname{PG}(rn-1,q)$ such that the points of $T$ correspond to the elements of $\\mathcal {D}$ that have a non-empty intersection with $\\pi $ .",
"For more on this approach to linear sets, we refer to [10].",
"If the subspace $\\pi $ intersects each spread element in at most a point, then $\\pi $ is called scattered with respect to $\\mathcal {D}$ and the associated linear set is called a scattered linear set.",
"Note that if $\\pi $ is $(n-1)$ -dimensional and scattered, then the associated $\\mathbb {F}_q$ -linear set has rank $n$ and has exactly $\\frac{q^n-1}{q-1}$ points, and conversely.",
"In this paper, we will make use of the following bound on the rank of a scattered linear set.",
"Result 2.1 ([5]) The rank of a scattered $\\mathbb {F}_q$ -linear set in $\\operatorname{PG}(r-1,q^n)$ is at most $ rn/2$ .",
"A maximum scattered linear set is a scattered $\\mathbb {F}_q$ -linear set in $\\operatorname{PG}(r-1,q^n)$ with rank $rn/2$ .",
"In this article we work with maximum scattered linear sets to which a geometric structure, called pseudoregulus, can be associated.",
"For more information, we refer to [12] and [13].",
"Definition 2.2 Let $S$ be a scattered $\\mathbb {F}_q$ -linear set of $\\operatorname{PG}(2k-1,q^n)$ of rank $kn$ , where $n, k\\ge 2$ .",
"We say that $S$ is of pseudoregulus type if there exist $m=\\frac{q^{nk}-1}{q^n-1}$ pairwise disjoint lines of $\\operatorname{PG}(2k-1, q^n)$ , say $s_1, s_2, \\dots , s_m$ , such that $| S\\cap s_i|=\\frac{q^n-1}{q-1}\\quad \\forall i=1,\\dots ,m,$ there exist exactly two $(k-1)$ -dimensional subspaces $T_1$ and $T_2$ of $\\operatorname{PG}(2k-1,q^n)$ disjoint from $S$ such that $T_j\\cap s_i \\ne \\emptyset $ for each $i = 1,\\dots , m$ and $j = 1, 2$ .",
"The set of lines $s_i$ , $i=1,\\dots m$ is called the pseudoregulus of $\\operatorname{PG}(2k-1,q^n)$ associated with the linear set $S$ and we refer to $T_1$ and $T_2$ as transversal spaces to this pseudoregulus.",
"Since a maximum scattered linear set spans the whole space, we see that the transversal spaces are disjoint.",
"Throughout this paper we need the following result of [13] on pseudoreguli .",
"Applied to $\\mathbb {F}_2$ -linear sets, this gives us the following result.",
"Result 2.3 ([13]) Each $\\mathbb {F}_2$ -linear set of $\\operatorname{PG}(2k-1, q)$ , $q$ even, of pseudoregulus type, is of the form $L_{\\rho ,f}$ with $L_{\\rho ,f} = \\lbrace ( u, \\rho f(u))_q| u\\in U_0 \\rbrace ,$ with $\\rho \\in \\mathbb {F}_{q}^*$ , $U_0,U_\\infty $ the $k$ -dimensional vector spaces corresponding to the transversal spaces $T_0,T_\\infty $ and with $f:U_0\\rightarrow U_\\infty $ an invertible semilinear map with companion automorphism $\\sigma \\in Aut(\\mathbb {F}_q)$ , $Fix(\\sigma )=\\lbrace 0,1\\rbrace $ .",
"Note that in the previous result, $\\operatorname{PG}(2k-1,q)$ is identified with $\\operatorname{PG}(V)$ , $V= U_0 \\oplus U_\\infty $ and a point, corresponding to a vector $v=v_1+v_2\\in U_0 \\oplus U_\\infty $ , has coordinates $(v_1,v_2)_q$ ."
],
[
"The Barlotti-Cofman and André/Bruck-Bose constructions",
"In this paper, we will switch between three different representations of a projective plane $\\operatorname{PG}(2,q^k)$ , $q=2^h$ .",
"Using the André/Bruck-Bose correspondence, we can, on one hand, model this plane as a subset of points and $k$ -spaces in $\\operatorname{PG}(2k,q)$ , determined by a $(k-1)$ -spread at infinity.",
"On the other hand, we can see it as a subset of points and $hk$ -spaces of $\\operatorname{PG}(2hk,2)$ determined by a $(hk-1)$ -spread at infinity.",
"We can switch between the $\\operatorname{PG}(2k,q)$ -setting an the $\\operatorname{PG}(2hk,2)$ -setting by the Barlotti-Cofman correspondence, which is a natural generalization of the André/Bruck-Bose correspondence.",
"The Barlotti-Cofman representation of the projective space $\\operatorname{PG}(2k,2^h)$ in $\\operatorname{PG}(2hk,2)$ is defined as follows (see [2]).",
"Let $\\mathcal {S}^{\\prime }$ be a Desarguesian $(h-1)$ -spread in $\\operatorname{PG}(2hk-1, 2)$ .",
"Embed $\\operatorname{PG}(2hk-1, 2)$ as a hyperplane $\\tilde{H_\\infty }$ in $\\operatorname{PG}(2hk, 2)$ .",
"Consider the following incidence structure $\\mathcal {P}(\\mathcal {S}) = (\\mathcal {P},\\mathcal {L} )$ , where incidence is natural: The set $\\mathcal {P}$ of points consists of the $2^{2hk}$ affine points $P_i$ in $\\operatorname{PG}(2hk,2)$ (i.e.",
"the points not in $\\tilde{H_\\infty }$ ) together with elements of the $(h-1)$ -spread $\\mathcal {S}^{\\prime }$ in $\\tilde{H_\\infty }$ The set $\\mathcal {L}$ of lines consist of the following two sets of subspaces in $\\operatorname{PG}(2hk,2)$ .",
"The set of $h$ -spaces spanned by an element of $\\mathcal {S}^{\\prime }$ and an affine point of $\\operatorname{PG}(2hk,2)$ .",
"The set of $(2h-1)$ -spaces in $\\tilde{H_\\infty }$ spanned by two different elements of $\\mathcal {S}^{\\prime }$ .",
"This incidence structure $(\\mathcal {P},\\mathcal {L} )$ is isomorphic to $\\operatorname{PG}(2k,2^h)$ .",
"We use the notation $P_i$ for the affine point of $\\operatorname{PG}(2k,2^h)$ (i.e.",
"a point not contained in $H_\\infty $ ) which corresponds to the affine point $\\tilde{P_i}\\in \\operatorname{PG}(2hk,2)$ .",
"A point, say $R_i$ in $H_\\infty $ , corresponds to the element $\\mathcal {S}^{\\prime }(R_i)$ of the $(h-1)$ -spread $\\mathcal {S}^{\\prime }$ in $\\tilde{H_\\infty }$ .",
"As already mentioned above, during this paper we will work in the following three projective spaces: The $2k$ -dimensional projective space $\\Pi _q=\\operatorname{PG}(2k,q)$ , $q=2^h, h>2$ , with the $(2k-1)$ -space at infinity called $H_\\infty $ .",
"The projective plane $\\Pi _{q^k}=\\operatorname{PG}(2,q^k)$ , $q=2^h$ with line at infinity called $\\ell _\\infty $ .",
"Given a Desarguesian $(k-1)$ -spread $\\mathcal {S}$ in $H_\\infty $ in $\\Pi _q$ , the plane $\\Pi _{q^k}$ is obtained by the André-Bruck-Bose construction using $\\mathcal {S}$ .",
"The $2hk$ -dimensional projective space $\\Pi _2=\\operatorname{PG}(2hk,2)$ , with the $(2hk-1)$ -space $\\tilde{H_\\infty }$ at infinity.",
"Note that the Barlotti-Cofman representation of $\\Pi _q$ defines a Desarguesian $(h-1)$ -spread $\\mathcal {S}^{\\prime }$ in $\\tilde{H_\\infty }$ .",
"Moreover, if $\\mathcal {S}$ is the $(k-1)$ -spread in $H_\\infty $ in $\\Pi _q$ such that $\\Pi _{q^ k}$ is the corresponding projective plane, the André-Bruck-Bose representation of $\\Pi _{q^k}$ in $\\Pi _2$ gives rise to a Desarguesian $(hk-1)$ -spread $\\tilde{\\mathcal {S}}$ in $\\tilde{H_\\infty }$ , such that $\\mathcal {S}^{\\prime }$ is a subspread of $\\tilde{\\mathcal {S}}$ ."
],
[
"The proof of the main theorem",
"Consider $\\Pi _q=\\operatorname{PG}(2k,q)$ and the hyperplane $H_\\infty $ of $\\operatorname{PG}(2k,q)$ .",
"Recall that a point of $\\operatorname{PG}(2k,q)$ is called affine if it is not contained in $H_\\infty $ .",
"Likewise, a line is called affine if it is not contained in $H_\\infty $ .",
"Let $P_1,P_2$ be affine points, then the point $P_1P_2\\cap H_\\infty $ is the direction determined by the line $P_1P_2$ .",
"If $\\mathcal {Q}$ is a set of affine points, then the directions determined by $\\mathcal {Q}$ are all points of $H_\\infty $ that appear as the direction of a line $P_iP_j$ for some $P_i,P_j\\in \\mathcal {Q}$ .",
"From now on, we consider a set $\\mathcal {Q}$ satisfying the conditions of Theorem REF : $\\mathcal {Q}$ is a set of $q^k$ affine points in $\\operatorname{PG}(2k,q)$ , $q=2^h$ , $h\\ge 4$ , $k \\ge 2$ ; $D$ , the set of directions determined by $\\mathcal {Q}$ at the hyperplane at infinity $H_\\infty $ has size $q^k-1$ ; Every line has 0, 1, 3 or $q-1$ points in common with the point set $D$ ."
],
[
"The $(q-1)$ -secants to {{formula:824f1829-06a2-4e3c-9246-2c2a3663df00}} are disjoint",
"Definition 3.1 A 0-point in $H_\\infty $ is a point $P\\notin D$ such that $P$ is contained in at least one $(q-1)$ -secant to $D$ .",
"From Proposition REF , it will follow that a 0-point is contained in precisely one $(q-1)$ -secant to $D$ .",
"We first start with two lemmas.",
"Lemma 3.2 No three points of $\\mathcal {Q}$ are collinear.",
"Let $l$ be an affine line in $\\operatorname{PG}(2k,q)$ containing $3\\le t\\le q$ points of $\\mathcal {Q}$ , and let $P^{\\prime }=l\\cap H_\\infty $ .",
"A point $P_i\\in \\mathcal {Q}\\setminus l$ determines a plane $\\alpha _i=\\langle P_i,l\\rangle $ such that the line $l_i= \\alpha _i\\cap H_\\infty $ is a $(q-1)$ -secant: the lines through $P_i$ and a point of $l\\cap \\mathcal {Q}$ determine $t\\ge 3$ directions of $D$ on the line $l_i$ , different from the point $P^{\\prime } \\in D$ .",
"So $l$ contains more than three points of $D$ , showing that $l_i$ is a $(q-1)$ -secant.",
"Furthermore, the plane $\\alpha _i$ contains at most $q$ affine points of $\\mathcal {Q}$ , as every affine line in $\\alpha $ through a 0-point of $l_i$ contains at most one element of $\\mathcal {Q}$ .",
"This implies that each of the $q^k-t\\ge q^k-q$ points of $\\mathcal {Q}\\setminus l$ define a plane $\\alpha $ , with $\\alpha \\cap H_\\infty $ a $(q-1)$ -secant, and so that $\\alpha $ contains at most $q-t\\le q-3$ points of $\\mathcal {Q}\\setminus l$ .",
"This shows that the number of such planes $\\alpha _i$ through $l$ , and hence the number of $(q-1)$ -secants through $P^{\\prime }$ , is at least $\\frac{q^k-q}{q-3}$ .",
"This gives that there are at least $1+\\frac{q^k-q}{q-3}(q-2)>q^k-1$ points of $D$ , a contradiction.",
"Lemma 3.3 Let $\\gamma $ be a plane in $\\operatorname{PG}(2k,q)$ containing 4 points $P_1,P_2,P_3$ and $P_4$ of $\\mathcal {Q}$ , such that $P_1P_2 \\cap P_3P_4\\notin \\mathcal {Q}\\cup D$ .",
"Then $\\gamma $ meets $H_\\infty $ in a $(q-1)$ -secant to $D$ .",
"By Lemma REF , no three points of $P_1, P_2,P_3,P_4$ are collinear.",
"Since $P_1P_2 \\cap P_3P_4\\notin D$ , we see that $P_1P_2$ and $P_3P_4$ define two different directions in $H_\\infty $ .",
"The four points $P_1,P_2,P_3$ and $P_4$ determine at least 4 directions on the line $\\gamma \\cap H_\\infty $ .",
"The statement follows since a line contains $0,1,3$ or $q-1$ points of $D$ .",
"Proposition 3.4 Every two $(q-1)$ -secants to $D$ are disjoint.",
"Consider a point $P_0\\in \\mathcal {Q}$ .",
"Then, by Lemma REF , all points of $D$ are defined by the lines $P_0 P_i$ with $P_i\\in \\mathcal {Q}\\setminus \\lbrace P_0\\rbrace $ .",
"Let $P_i^{\\prime }$ denote the direction of the line $P_0 P_i$ , that is, the point $P_0P_i\\cap H_\\infty $ .",
"We see that a line through a point $P_i^{\\prime }\\in D$ contains 0 or 2 points of $\\mathcal {Q}$ .",
"Let $l_\\alpha $ and $l_\\beta $ be two lines, both containing $q-1$ points of $D$ , with $P^{\\prime }=l_\\alpha \\cap l_\\beta $ .",
"Let $\\alpha =\\langle P_0, l_\\alpha \\rangle $ and $\\beta =\\langle P_0, l_\\beta \\rangle $ and let $\\lbrace P_{1 \\alpha },P_{2 \\alpha }\\rbrace $ and $\\lbrace P_{1 \\beta },P_{2 \\beta }\\rbrace $ be the 0-points in $l_\\alpha $ and $l_\\beta $ .",
"Note that $P^{\\prime }$ may be amongst these points.",
"It follows from the argument above that there are precisely $q$ points in $\\alpha \\cap \\mathcal {Q}$ and that the affine points of $\\mathcal {Q}$ in $\\alpha $ together with the two points $P_{1 \\alpha },P_{2 \\alpha }$ form a hyperoval $H_\\alpha $ .",
"Similarly, we find a hyperoval $H_\\beta $ in $\\beta $ .",
"We first suppose that $P^{\\prime }\\in D$ .",
"This implies that there is a point $P\\ne P_0$ of $\\mathcal {Q}$ on the line $P_0 P^{\\prime }$ .",
"Note that $P_0$ and $P$ are contained in $H_\\alpha \\cap H_\\beta $ .",
"Consider a point $R\\in l_\\alpha $ , different from $P^{\\prime }, P_{1 \\alpha },P_{2 \\alpha }$ .",
"Then $R\\in D$ and through $R$ , there are $\\frac{q}{2}$ bisecants to $H_\\alpha \\ne l_\\alpha $ .",
"One of these bisecants contains $P$ and another one contains $P_0$ .",
"Since $q>8$ , there exists a bisecant to $H_\\alpha $ through $R$ which intersects the line $P_0 P$ in a point $R_0 \\notin \\lbrace P_0,P,P^{\\prime }\\rbrace $ .",
"Through $R_0$ , there are $\\frac{q}{2}-2$ bisecants $r_i$ to $H_\\beta $ , different from the lines $R_0 P$ , $R_0P_{1\\beta }$ and $R_0P_{2\\beta }$ .",
"Let $r_i\\cap l_\\beta =R_i, i=1,\\dots ,\\frac{q}{2}-2$ .",
"A plane $\\langle R, r_i \\rangle $ contains two lines, $r_i$ and $m=R R_0$ , both containing two points of $\\mathcal {Q}$ and $r_i\\cap m=R_0\\notin \\mathcal {Q}$ .",
"Hence, by Lemma REF we find that every line $R R_i$ is a $(q-1)$ -secant to $D$ .",
"So there are $\\frac{q}{2}-2$ $(q-1)$ -secants of the form $RR_i$ , and the total number of 0-points on these lines is $2(\\frac{q}{2}-2)=q-4$ .",
"Let $\\Omega $ be the set of these 0-points.",
"We call a $(\\le 3)$ -secant in $\\langle l_\\alpha , l_\\beta \\rangle $ a line with at most 3 points of $D$ .",
"A line through $P^{\\prime }$ in $\\langle l_\\alpha ,l_\\beta \\rangle $ intersects all lines $RR_i$ .",
"The $q-4$ points of $\\Omega $ lie on the $q-1$ lines through $P^{\\prime }$ different from $l_\\alpha $ and $l_\\beta $ .",
"Since every line $RR_i$ contains precisely two 0-points, we find that for $q>8$ there are at most 3 $(\\le 3)$ -secants through $P^{\\prime }$ : if there are at least four $(\\le 3)$ -secants through $P^{\\prime }$ in $\\langle l_\\alpha , l_\\beta \\rangle $ , then there are at least $\\frac{q}{2}-2-2$ 0-points of $\\Omega $ on each of these lines, as we supposed that $P^{\\prime }\\in D$ .",
"This implies that there would be at least $4(\\frac{q}{2}-4)>q-4$ 0-points in $\\Omega $ , which gives a contradiction for $q\\ge 16$ .",
"Now we distinguish different cases depending on the number of $(\\le 3)$ -secants through $P^{\\prime }$ .",
"In each of the cases we will show that there exists at least two $(\\le 3)$ -secants $l_1,l_2$ in $\\langle l_\\alpha ,l_\\beta \\rangle $ , and a point $X\\notin D$ not on these lines.",
"This leads to a contradiction since there are at least $q+1-7$ lines through $X$ , both intersecting $l_1$ and $l_2$ in a point not in $D$ , and not through $l_1\\cap l_2$ .",
"These lines contain at least 3 points not in $D$ so they have to be $(\\le 3)$ -secants.",
"But this implies that there are at least $1+(q-6)(q-3)=q^2-9q+19$ points in $\\langle l_\\alpha ,l_\\beta \\rangle $ , not contained in $D$ .",
"On the other hand, there are at most three $(\\le 3)$ -secants through $P^{\\prime }$ and the other lines through $P^{\\prime }$ contain two 0-points.",
"This implies that there are at most $3q+ 2(q-2)=5q-4<q^2-9q+19$ points in $\\langle l_\\alpha ,l_\\beta \\rangle $ , not contained in $D$ .",
"This gives a contradiction for $q\\ge 16$ .",
"It remains to show that in every case there exists at least two $(\\le 3)$ -secants and a point $X\\notin D$ , not on these lines.",
"Suppose first that there are two or three $(\\le 3)$ -secants through $P^{\\prime }$ .",
"These lines are different from $l_\\alpha $ , so they do not contain the point $P_{1\\alpha }$ .",
"Then $X=P_{1 \\alpha }\\notin D$ is a point not on the $(\\le 3)$ -secants.",
"Suppose there is an unique $(\\le 3)$ -secant $l$ through $P^{\\prime }$ .",
"Then every other line through $P^{\\prime }$ contains two 0-points.",
"Suppose first that there exists a 0-point $P_1$ so that $P_{1\\alpha } P_1 \\cap l \\notin D$ .",
"Then $l^{\\prime }=P_{1\\alpha } P_1$ contains 3 points not in $D$ , so $l^{\\prime }$ is a $(\\le 3)$ -secant.",
"Note that $P_1\\ne P_{2\\alpha }$ as otherwise $P_{1\\alpha } P_1 \\cap l=l_\\alpha \\cap l=P^{\\prime } \\in D $ .",
"Hence $X=P_{2\\alpha }\\notin D$ is not contained in $l\\cup l^{\\prime }$ .",
"If there is no point $P_1$ so that $P_{1 \\alpha } P_1 \\cap l \\notin D$ , then all $2q-4$ 0-points on the $(q-1)$ -secants through $P^{\\prime }$ , different from $l_\\alpha , l_\\beta $ , lie on at most 2 lines $P_{1\\alpha }P_1$ and $P_{1\\alpha }P_2$ , with $P_1,P_2\\in D\\cap l\\setminus \\lbrace P^{\\prime }\\rbrace $ .",
"But then $P_{1\\alpha }P_1$ and $P_{1\\alpha }P_2$ are $(\\le 3)$ -secants.",
"Note that these lines are different from $l_\\alpha $ , and so, they do not contain $P_{2\\alpha }$ .",
"Hence we may take $X=P_{2\\alpha }$ .",
"Suppose all lines through $P^{\\prime }$ are $(q-1)$ -secants with $\\Gamma $ the corresponding set of $2q+2$ 0-points.",
"Let $G\\in \\Gamma $ and consider the $q+1$ lines through $G$ in $\\langle l_\\alpha , l_\\beta \\rangle $ .",
"The $2q+1$ other points of $\\Gamma $ lie on these lines and since every line contains 2 or at least $q-2$ points not in $D$ , we find that through $G$ there is at least one $(\\le 3)$ -secant $l_1$ .",
"Consider now a point $G^{\\prime }\\in \\Gamma \\setminus l_1$ .",
"Through this point there is also a $(\\le 3)$ -secant $l_2$ .",
"The lines $l_1\\cup l_2$ contain at most $2q+1$ points of $\\Gamma $ , so there is at least one 0-point $X$ not contained in these two lines.",
"This shows that two $(q-1)$ -secants cannot meet in a point $P^{\\prime }$ of $D$ .",
"Suppose now that $P^{\\prime }\\notin D$ .",
"As above, we find for a given point $R\\in D\\cap l_\\alpha $ , at least $\\frac{q}{2}-2$ $(q-1)$ -secants $R R_i$ , different from $l_\\alpha $ .",
"But by the previous part, we know that there are no two $(q-1)$ -secants through a point $R \\in D$ .",
"As $\\frac{q}{2}-2\\ge 2$ , we find a contradiction.",
"We now deduce a corollary that will be useful later.",
"Corollary 3.5 A $(q-1)$ -secant and a 3-secant to $D$ in $H_\\infty $ cannot have a 0-point in common.",
"Let $l_\\alpha $ be a 3-secant to $D$ , $l_\\beta $ be a $(q-1)$ -secant to $D$ , and $P^{\\prime }=l_\\alpha \\cap l_\\beta $ be a 0-point.",
"Pick $P_0 \\in \\mathcal {Q}$ and let $\\alpha =\\langle P_0, l_\\alpha \\rangle $ and $\\beta =\\langle P_0, l_\\beta \\rangle $ .",
"The points of $Q\\cup D$ in $\\alpha $ form a Fano plane: let $P^{\\prime }_i$ , $i=1,2,3$ , be the three points of $D$ on the line $l_\\alpha $ and let $P_i$ , $i=1,2,3$ be the corresponding affine points of $\\mathcal {Q}$ so that $P_0P_i\\cap l_\\alpha =P_i^{\\prime }$ .",
"Since there are only three directions $P_1^{\\prime },P_2^{\\prime },P_3^{\\prime }$ of $D$ in $\\alpha $ , we find that $\\lbrace P_1,P_3,P_2^{\\prime }\\rbrace $ ,$\\lbrace P_1,P_2,P_3^{\\prime }\\rbrace $ and $\\lbrace P_2,P_3,P_1^{\\prime }\\rbrace $ are triples of collinear points.",
"Since also $\\lbrace P_1^{\\prime },P^{\\prime }_2,P^{\\prime }_3\\rbrace $ and $\\lbrace P_0,P_i,P^{\\prime }_i\\rbrace $ , $i=1,2,3$ are triples of collinear points, we find that the points $\\lbrace P_0,P_1,P_2,P_3,P^{\\prime }_1,P^{\\prime }_2,P^{\\prime }_3\\rbrace $ define a Fano plane $\\operatorname{PG}(2,2)$ .",
"Let $R _0$ be the point $P_1^{\\prime }P_2\\cap P^{\\prime }P_0$ .",
"Note that $R_0\\notin \\mathcal {Q}$ .",
"As the points of $\\mathcal {Q}$ in $\\beta $ form a $q$ -arc, we know that there are at least two lines $R_0 R_1$ and $R_0 R_2$ in $\\beta $ , with $R_1,R_2 \\in l_\\beta \\cap D$ , such that both lines contain 2 points of $\\mathcal {Q}$ .",
"By Lemma REF we see that the lines $P_1^{\\prime }R_1$ and $P_1^{\\prime }R_2$ are both $(q-1)$ -secants through $P_1^{\\prime }$ .",
"This gives a contradiction by Proposition REF ."
],
[
"The set $D$ of directions in {{formula:8fe3f599-46be-4744-a88b-75081cb474e5}} is a linear set",
"Recall that we use the notation $\\tilde{P}$ for the affine point in $\\Pi _2$ , corresponding to the affine point $P\\in \\Pi _q$ .",
"Let $\\mathcal {S}^{\\prime }$ be the $(h-1)$ -spread in the hyperplane $\\tilde{H_\\infty }$ of $\\operatorname{PG}(2hk,2)$ corresponding to the points of the hyperplane $H_\\infty $ of $\\Pi _q$ .",
"We use the notation $\\mathcal {S}^{\\prime }(P^{\\prime })$ for the element of $\\mathcal {S}^{\\prime }$ corresponding to the point $P^{\\prime }\\in H_\\infty $ .",
"We will now show that $D$ is an $\\mathbb {F}_2$ -linear set in $H_\\infty $ by showing that its points correspond to spread elements in $\\tilde{H_\\infty }$ intersecting some fixed $(hk-1)$ -subspace of $\\tilde{H_\\infty }$ .",
"Let ${Q}=\\mathcal {Q}\\cup D$ , $\\tilde{{Q}}=\\tilde{\\mathcal {Q}} \\cup \\tilde{D}$ , with $\\tilde{\\mathcal {Q}}$ the union of the points $\\tilde{P},$ with $P\\in Q$ , and $\\tilde{D}$ the directions in $\\tilde{H_\\infty }$ determined by the points of $\\tilde{\\mathcal {Q}}$ .",
"Lemma 3.6 Let $P_0, P_1, P_2 \\in \\mathcal {Q}$ and $P^{\\prime }_i=P_0P_i \\cap H_\\infty $ , $i=1,2$ .",
"If $P_1^{\\prime }P_2^{\\prime }$ is a 3-secant to $D$ , then the plane in $\\operatorname{PG}(2hk,2)$ spanned by $\\tilde{P_0}$ , $\\tilde{P_1}$ and $\\tilde{P_2}$ is contained in $\\tilde{{Q}}$ .",
"Since $P_1^{\\prime }P_2^{\\prime }$ is not a $(q-1)$ -secant, we know that there is a unique point $P_3^{\\prime }\\ne P^{\\prime }_1,P^{\\prime }_2$ in $P_1^{\\prime }P_2^{\\prime }\\cap D$ , and a point $P_3 \\in \\mathcal {Q}$ such that $P^{\\prime }_3 \\in P_0P_3$ .",
"Let $\\alpha $ be the plane spanned by the points $P_0, P_1$ and $P_2$ .",
"As $\\alpha \\cap D=\\lbrace P_1^{\\prime },P_2^{\\prime },P_3^{\\prime }\\rbrace $ , we find that $\\lbrace P_1,P_3,P_2^{\\prime }\\rbrace $ ,$\\lbrace P_1,P_2,P_3^{\\prime }\\rbrace $ and $\\lbrace P_2,P_3,P_1^{\\prime }\\rbrace $ are triples of collinear points.",
"As in the proof of REF , we find that these points define a Fano plane $\\operatorname{PG}(2,2)$ .",
"We claim that the corresponding points $\\tilde{P_0}$ , $\\tilde{P_1}$ , $\\tilde{P_2}$ and $\\tilde{P_3}$ lie in a plane in $\\operatorname{PG}(2hk,2)$ .",
"Suppose these points are not contained in a plane in $\\operatorname{PG}(2hk,2)$ , then they span a 3-space $\\beta $ .",
"Since $P_1^{\\prime }=P_0P_1\\cap P_2P_3$ , $\\tilde{P_0}\\tilde{P_1}$ meets $\\mathcal {S}^{\\prime }(P_1^{\\prime })$ in a point, say $A_1$ .",
"Similarly, $\\tilde{P_2}\\tilde{P_3}$ meets $\\mathcal {S}^{\\prime }(P_1^{\\prime })$ in a point, say $B_1$ .",
"Since $\\tilde{P_0},\\tilde{P_1},\\tilde{P_2},\\tilde{P_3}$ span a 3-space, $A_1\\ne B_1$ .",
"Similarly, the points $A_2=\\tilde{P_0}\\tilde{P_2}\\cap \\mathcal {S}^{\\prime }(P_2^{\\prime })$ and $B_2=\\tilde{P_1}\\tilde{P_3}\\cap \\mathcal {S}^{\\prime }(P_2^{\\prime })$ are different and span the line $A_2 B_2$ .",
"But now $A_1B_1\\in \\mathcal {S}^{\\prime }(\\tilde{P_1^{\\prime }})$ and $A_2B_2\\in \\mathcal {S}^{\\prime }(\\tilde{P_2^{\\prime }})$ are two lines in the plane $\\beta \\cap \\tilde{H_\\infty }$ , so they intersect, a contradiction since the spread elements $\\mathcal {S}^{\\prime }(P_1^{\\prime })$ and $\\mathcal {S}^{\\prime }(P_2^{\\prime })$ are disjoint.",
"Theorem 3.7 The set $D$ is an $\\mathbb {F}_2$ -linear set.",
"We will show that the set $\\tilde{{Q}}$ of points in $\\operatorname{PG}(2hk,2)$ forms a subspace.",
"By Lemma REF , we have the following property: if $\\tilde{P_0}$ , $\\tilde{P_1}$ and $\\tilde{P_2}$ are three points in $\\tilde{\\mathcal {Q}}$ such that the line at infinity of the plane spanned by these points corresponds to a 3-secant in $\\Pi _q$ , then we know that all points of $\\langle \\tilde{P_0},\\tilde{P_1},\\tilde{P_2}\\rangle $ are included in $\\tilde{{Q}}$ .",
"Consider now a point $\\tilde{P_0}\\in \\tilde{\\mathcal {Q}}$ and a point $\\tilde{P_1} \\in \\tilde{\\mathcal {D}}$ .",
"Let $P_1$ be the point corresponding to the spread element through $\\tilde{P_1}$ (i.e.",
"$P_1$ is the unique point such that $\\tilde{P_1}$ is contained in $\\mathcal {S}^{\\prime }(P_1)$ ).",
"By Proposition REF we can take two 3-secants to $D$ , say $L_\\alpha $ and $L_\\beta $ through $P_1$ in $\\Pi _q$ .",
"Let $l_\\alpha $ and $l_\\beta $ denote the unique line through $\\tilde{P_1}$ such that the spread elements intersecting $l_\\alpha $ and $l_\\beta $ correspond precisely the the points of $D$ on $L_\\alpha \\cup L_\\beta $ .",
"Let $\\alpha =\\langle \\tilde{P_0}, l_\\alpha \\rangle $ and $\\beta =\\langle \\tilde{P_0}, l_\\beta \\rangle $ .",
"Let $\\lbrace \\tilde{P_1}, \\tilde{P_2}, \\tilde{P_3} \\rbrace = \\alpha \\cap \\tilde{H_\\infty }$ and let $\\lbrace \\tilde{P_1}, \\tilde{P_4}, \\tilde{P_5}\\rbrace = \\beta \\cap \\tilde{H_\\infty }$ .",
"Consider an affine point $\\tilde{P}$ in $ \\gamma =\\langle \\alpha ,\\beta \\rangle $ , $\\tilde{P}\\notin \\alpha \\cup \\beta $ .",
"We want to show that $\\tilde{P}$ lies in $\\tilde{\\mathcal {Q}}$ .",
"Let $\\tilde{P^{\\prime }}$ be the point at infinity of the line $\\tilde{P_0} \\tilde{P}$ .",
"W.l.o.g.",
"we suppose that $\\tilde{P^{\\prime }}= \\tilde{P_2}\\tilde{P_5}\\cap \\tilde{P_3}\\tilde{P_4}$ .",
"Let $P_2,P_3,P_4,P_5$ be the points in $H_\\infty $ corresponding to the spreadelements of $\\mathcal {S}^{\\prime }$ through $\\tilde{P_2}, \\tilde{P_3}, \\tilde{P_4}, \\tilde{P_5}$ .",
"We know that $P_2P_5$ and $P_3P_4$ cannot both be $(q-1)$ -secants by Proposition REF .",
"So suppose that $P_2P_5$ is a 3-secant in $\\operatorname{PG}(2k-1,q)$ .",
"By Lemma REF , we know that all points of the plane $\\langle \\tilde{P_2}\\tilde{P_5}, \\tilde{P_0}\\rangle $ lie in $\\tilde{{Q}}$ .",
"This proves that $\\tilde{P}\\in \\tilde{\\mathcal {Q}}$ , and so that $\\gamma \\setminus \\tilde{H_\\infty } \\subset \\tilde{\\mathcal {Q}}$ .",
"As $\\tilde{\\mathcal {D}}$ is the set of directions determined by $\\tilde{\\mathcal {Q}}$ , we also find that $\\gamma \\cap \\tilde{H_\\infty } \\in \\tilde{\\mathcal {D}}$ .",
"We conclude that all points of a 3-space through a point $\\tilde{P_0}$ of $\\tilde{\\mathcal {Q}}$ , whose point set at infinity corresponds to two intersecting 3-secants at infinity, are contained in $\\tilde{{Q}}$ .",
"Now suppose that there is a $t$ -space $\\beta $ , with $\\beta \\subset \\tilde{{Q}}$ .",
"By the previous part of this proof, we may assume that the points in $H_\\infty $ , corresponding to the spread elements intersecting $\\beta \\cap \\tilde{H_\\infty }$ , are not all contained in a single $(q-1)$ -secant.",
"If $t=hk$ , then our proof is finished, so assume that $t<hk$ .",
"This implies that there exists a point $\\tilde{G}\\in \\tilde{\\mathcal {Q}}\\setminus \\beta $ .",
"Let $G$ be the corresponding point in $\\mathcal {Q}$ in $\\operatorname{PG}(2k,q)$ , and let $\\gamma =\\langle \\beta ,\\tilde{G}\\rangle $ .",
"We show that every point $\\tilde{X}$ in $\\gamma \\setminus \\beta $ is a point of $\\tilde{{Q}}$ .",
"Suppose first that $\\tilde{X}$ is a point at infinity of $\\gamma \\setminus \\beta $ , then the line $\\tilde{X}\\tilde{G}$ contains an affine point $\\tilde{Y}$ of $\\beta $ , as $\\beta $ is a hyperplane of $\\gamma $ .",
"But since $\\tilde{G}$ and $\\tilde{Y}$ are points of $\\tilde{\\mathcal {Q}}$ , we find that $\\tilde{X}\\in \\tilde{D}\\subset \\tilde{{Q}}$ .",
"Suppose now that $\\tilde{X}$ is an affine point in $\\gamma \\setminus \\beta $ , and let $X$ be the corresponding point in $\\operatorname{PG}(2k,q)$ .",
"As the field size in $\\operatorname{PG}(2hk,2)$ is 2, the line $\\tilde{X}\\tilde{G}$ contains 1 extra point $\\tilde{Y}$ .",
"This point has to lie in $\\beta $ and in the hyperplane at infinity, so $\\tilde{Y}\\in \\beta \\cap \\tilde{H_\\infty }$ .",
"Let $l_1$ be a line through $\\tilde{Y}$ in $\\beta $ corresponding to a 3-secant, which exists since we have seen that not all points corresponding to points of $\\beta \\cap H_\\infty $ are contained in one single $(q-1)$ -secant.",
"The plane spanned by $\\tilde{G}$ and $l_1$ is contained in $\\tilde{{Q}}$ by Lemma REF , and hence, since $X$ lies on the line $\\tilde{Y}\\tilde{G}$ which is contained in this plane, $X\\in \\tilde{{Q}}$ .",
"This implies that $\\gamma \\subseteq {Q}$ .",
"We can repeat this argument until we find that $\\tilde{{Q}}$ is a $hk$ -space in $\\operatorname{PG}(2hk,2)$ .",
"Note that $D$ is a scattered linear set since $|D|=q^k-1=2^{hk}-1=|\\operatorname{PG}(hk-1,2)|$ .",
"As $D$ has rank $hk$ , we find that $D$ is maximum scattered.",
"Remark 3.8 In Lemma REF , we showed that the $(q-1)$ -secants to $D$ were disjoint.",
"In Theorem REF , we have used this to show that $D$ is a maximum scattered $\\mathbb {F}_2$ -linear set.",
"The fact that $(q-1)$ -secants to a maximum scattered $\\mathbb {F}_2$ -linear set are disjoint, is well-known (see e.g.",
"[13])."
],
[
"The set $D$ is an {{formula:1a97cd8c-803c-45b2-877b-9b63d674b483}} -linear set of pseudoregulus type",
"The proof that $D$ is of pseudoregulus type, is based on some ideas of [12].",
"Lemma 3.9 There are $\\frac{q^k-1}{q-1}$ pairwise disjoint $(q-1)$ -secants to $D$ in $\\operatorname{PG}(2k-1,q), q>4$ .",
"Let $K$ be the $(hk-1)$ -dimensional subspace in $\\operatorname{PG}(2hk-1,2)$ defining the $\\mathbb {F}_2$ -linear set $D$ and let $\\mathcal {S}^{\\prime }$ be the $(h-1)$ -spread that corresponds to the point set of $\\operatorname{PG}(2k-1,q)$ .",
"For every $hk$ -space $Y$ through $K$ in $\\operatorname{PG}(2hk-1,2)$ , we find at least one element of $\\mathcal {S}^{\\prime }$ that intersects $Y$ in a line since $D$ is maximum scattered.",
"Every line $l$ , through a point of $K$ , such that $l$ lies in an element of $\\mathcal {S}^{\\prime }$ , defines a $hk$ -space through $K$ , and the number of $hk$ -spaces through $K$ is $2^{hk}-1$ .",
"This implies that there are on average $2^{h-1}-1>2$ lines contained in different spread elements of $\\mathcal {S}^{\\prime }$ in a $hk$ -space through $K$ in $\\operatorname{PG}(2hk-1,2)$ .",
"Take a $hk$ -space $Y$ through $K$ with at least two lines contained in spread elements, and let $S_1$ and $S_2$ be two elements of $\\mathcal {S}^{\\prime }$ that intersect $Y$ in the lines $y_1$ and $y_2$ respectively.",
"The $(2h-1)$ -space $\\langle S_1,S_2\\rangle $ intersects $K$ in at least a plane, as $y_1$ and $y_2$ span a 3-space.",
"But this implies that the line $l$ in $\\operatorname{PG}(2k-1,q)$ , corresponding with $\\langle S_1,S_2\\rangle $ contains at least 7 points of $D$ .",
"This implies that $l$ is a $(q-1)$ -secant of $D$ , and that $\\langle S_1,S_2\\rangle $ intersects $K$ in a $(h-1)$ -space $\\alpha $ as a $(h-1)$ -space contains $2^h-1=q-1$ points.",
"Consider now the $h$ -space $\\beta =Y\\cap \\langle S_1,S_2 \\rangle $ through $\\alpha $ .",
"Since all of the $2^h+1$ $(h-1)$ -spaces of $\\mathcal {S}^{\\prime }$ in $\\langle S_1,S_2 \\rangle $ intersect $\\beta $ in a point or a line, we find that there are precisely $2^{h-1}-1$ elements of $\\mathcal {S}^{\\prime }$ , meeting $\\beta $ , and so $Y$ , in a line.",
"Hence, this proves that a $hk$ -space $Y$ through $K$ , containing at least 2 lines $y_1,y_2$ in $S_1,S_2$ respectively, contains at least $2^{h-1}-1$ lines $y_i$ in different spread elements of $\\mathcal {S}^{\\prime }$ .",
"Now we prove, by contradiction, that $Y$ cannot contain more lines $y_i$ contained in a spread element.",
"Suppose $Y$ contains another line $y_0 \\subset S_0$ with $S_0 \\in \\mathcal {S}^{\\prime }$ , then $y_0 \\notin \\langle S_1, S_2 \\rangle $ .",
"Repeating the previous argument for $y_1$ and $y_2$ shows that there are two $(2h-1)$ -spaces $\\langle S_1, S_2 \\rangle $ and $\\langle S_0, S_1 \\rangle $ , both meeting $K$ in a $(h-1)$ -space and so, there are two $(q-1)$ -secants through $P_1\\in H_\\infty $ , the point corresponding to the spread element $S_1$ .",
"This gives a contradiction by Proposition REF .",
"Since the average number of lines contained in a spreadelement in a $hk$ -space through $K$ is $2^{h-1}-1>2$ , we find that every $hk$ -space through $K$ contains exactly $2^{h-1}-1$ lines contained in a spreadelement.",
"In particular, every line $y_i \\subset S_i$ , with $S_i \\in \\mathcal {S}^{\\prime }$ and $y_i$ through a point of $K$ , defines a $hk$ -space though $K$ , and so a $(q-1)$ -secant.",
"So we find that every point in $D$ is contained in at least one $(q-1)$ -secant.",
"As we already proved that two $(q-1)$ -secants are disjoint (see Lemma REF ), we find $\\frac{q^k-1}{q-1}$ pairwise disjoint $(q-1)$ -secants in $\\operatorname{PG}(2k-1,q)$ .",
"We will first show that the linear set is of pseudoregulus type when $k=2$ .",
"To prove this, we begin with a lemma.",
"Lemma 3.10 Assume that $k=2$ .",
"Let $l$ be a line in $H_\\infty $ through two 0-points, not on the same $(q-1)$ -secant, then $l$ contains no points of $D$ .",
"Let $l_1$ and $l_2$ be two $(q-1)$ -secants in $\\operatorname{PG}(3,q)$ and let $l$ be a line through a 0-point of $l_1$ and through a 0-point of $l_2$ .",
"Recall that $l_1$ and $l_2$ are disjoint by Lemma REF .",
"Every two points $(A,B)$ , $A\\in l_1$ , $B\\in l_2$ , define a third point in $D$ on the line $AB$ .",
"Hence we find, since $|D|=q^2-1$ , that every point $P\\in D\\setminus \\lbrace l_1,l_2\\rbrace $ is uniquely defined as a third point on a line, defined by two points $A$ and $B$ of $D$ in $l_1$ and $l_2$ respectively.",
"Now suppose that $l$ contains a point $X\\in D$ , then $X$ lies on a unique line $l^{\\prime }$ , intersecting $l_1$ and $l_2$ in precisely one point.",
"But then $l_1$ and $l_2$ lie in a plane spanned by $l$ and $l^{\\prime }$ , a contradiction since $l_1$ and $l_2$ are disjoint by Lemma REF .",
"Proposition 3.11 Assume that $k=2$ .",
"The $(q-1)$ -secants to $D$ in $\\operatorname{PG}(3,q)$ form a pseudoregulus.",
"By Lemma REF it is sufficient to prove that there exist 2 lines in $\\operatorname{PG}(3,q)$ that have a point in common with all $(q-1)$ -secants to $D$ .",
"Consider three $(q-1)$ -secants $l_1$ , $l_2$ and $l_3$ and let $P_i,Q_i\\in l_i$ , $i=1,2,3$ be the corresponding 0-points.",
"Let $l_0$ be the unique line through $P_1$ that intersects $l_2$ and $l_3$ both in a point, say $R_2=l_0\\cap l_2$ and $R_3=l_0\\cap l_3$ respectively.",
"By Corollary $\\ref {q-1en3disjunct}$ , $R_2$ and $R_3$ cannot both belong to $\\mathcal {Q}$ , so suppose $R_2$ is a 0-point of $l_2$ (w.l.o.g.",
"$R_2=P_2$ ).",
"We see that $l_0=P_1P_2$ is a line through two 0-points, so $R_3$ is also a 0-point by Corollary REF , w.l.o.g.",
"$R_3=P_3$ .",
"By the same argument, we see that $Q_1,Q_2$ and $Q_3$ are contained in a line, say $l_\\infty $ .",
"Now we want to show that every other $(q-1)$ -secant has a 0-point in common with both $l_0$ and $l_\\infty $ .",
"Consider an $(q-1)$ -secant $l_4$ , different from $l_1,l_2,l_3$ , with 0-points $P_4$ and $Q_4$ .",
"Consider now again the unique line $m$ through $P_4$ that intersects $l_1$ and $l_2$ in a point.",
"By the previous arguments $m$ has to contain a 0-point of $l_1$ and a 0-point of $l_2$ , so $m=l_0$ , $m=l_\\infty $ , $m=P_1Q_2$ or $m=Q_1P_2$ .",
"We will show that only the first two possibilities can occur, which then proves that every other 0-point lies on $l_0$ or $l_\\infty $ .",
"Suppose to the contrary that $m=P_1Q_2P_4$ (the case $m=Q_1P_2P_4$ is completely analogous).",
"Then the unique line through $Q_4$ , meeting $l_1$ and $l_2$ is the line $Q_1P_2$ .",
"Consider now the unique line $m^{\\prime }$ through $P_4$ meeting $l_2$ and $l_3$ in a point.",
"As we supposed that $m\\ne l_0$ and $m\\ne l_\\infty $ , we see that $P_4$ cannot lie on these lines, so $m^{\\prime }$ contains the points $P_4,P_2,Q_3$ or the points $P_4,Q_2,P_3$ .",
"In the former case both lines $l_0$ and $l_\\infty $ are contained in the plane spanned by $m^{\\prime }=P_4Q_3P_2$ and $m=P_1Q_2P_4$ .",
"This implies that the disjoint lines $l_1$ and $l_2$ are contained in this plane, a contradiction.",
"If $m^{\\prime }=P_4P_3Q_2$ , then $m$ and $m^{\\prime }$ both contain $P_4$ and $Q_2$ but intersect $l_0$ in different points, a contradiction.",
"We conclude that $P_4$ , and analoguously $P_4^{\\prime }$ , is contained in the line $l_0$ or $l_\\infty $ .",
"Using the previous proposition, we will prove that for all $k$ , the $\\mathbb {F}_2$ -linear set $D$ in $\\operatorname{PG}(2k-1,q)$ is of pseudoregulus type.",
"Theorem 3.12 The $(q-1)$ -secants to $D$ in $\\operatorname{PG}(2k-1,q)$ form a pseudoregulus.",
"By Lemma REF it is sufficient to prove that there exist two $(k-1)$ -spaces in $\\operatorname{PG}(2k-1,q)$ that both have a point in common with all $(q-1)$ -secants to $D$ .",
"Consider a $(q-1)$ -secant $l_0$ , and let $P_0$ and $P_0^{\\prime }$ be the 0-points on $l_0$ .",
"Let $l_i$ be a $(q-1)$ -secant, different from $l_0$ .",
"The lines $l_0$ and $l_i$ span a 3-space $\\gamma $ and since $D$ is a scattered $\\mathbb {F}_2$ -linear set, $\\gamma \\cap D$ is also a scattered $\\mathbb {F}_2$ -linear set.",
"Since $\\gamma $ contains $2(q-1)$ points of $D$ on the lines $l_i,l_j$ and $(q-1)^2$ points of $D$ defined in a unique way as a third point on the line $A_1 A_2$ , with $A_1 \\in l_0$ , $A_2\\in l_i$ , we have that $|D\\cap \\gamma |=q^2-1$ , and hence it is a maximum scattered linear set.",
"By Theorem REF , we find that $\\gamma \\cap D$ is of pseudoregulus type.",
"This means that it has transversal lines, say $m_i$ and $m_i^{\\prime }$ , where $P_0$ lies on $m_i$ and $P_0^{\\prime }$ lies on $m_i^{\\prime }$ .",
"This holds for every $(q-1)$ -secant $l_i$ .",
"Since there are exactly $\\frac{q^k-1}{q-1}$ $(q-1)$ -secants to $D$ , which are mutually disjoint, there are exactly $2\\frac{q^k-1}{q-1}$ 0-points.",
"We have proven that a 0-point lies on $\\frac{q^{k-1}-1}{q-1}$ lines full of 0-points (call such lines 0-lines) and on $\\frac{q^k-1}{q-1}$ lines containing exactly 1 other 0-point.",
"Let $A$ and $A^{\\prime }$ be the set of all points on the lines $m_i$ and $m_i^{\\prime }$ respectively.",
"Then we will to show that $A\\cup A^{\\prime }$ is the union of two disjoint $(k-1)$ -spaces.",
"Consider a line containing two 0-points $P_1,P_2$ , with $l_1$ and $l_2$ the $(q-1)$ -secants through $P_1,P_2$ .",
"Then, as seen before, the intersection of the 3-space spanned by $l_1$ and $l_2$ with $D$ is a linear set of pseudoregulus type, and hence the line $P_1P_2$ contains 2 or $q+1$ 0-points.",
"This shows that every line in $\\operatorname{PG}(2k-1,q)$ intersects $A \\cup A^{\\prime }$ in $0,1,2$ or $q+1$ points.",
"This in turn implies that a plane with three 0-lines only contains 0-points.",
"Consider now a point $P_3$ on a 0-line through $P_0$ , and consider a 0-line $m\\ne P_0P_3$ through $P_3$ .",
"If $m$ contains a point $P_4 \\ne P_3$ such that $P_4P_0$ is a 0-line through $P_0$ , then we see that the plane $\\langle P_0,m \\rangle $ only contains 0-points.",
"In the other case, $M$ contains at least two 0-points on 0-lines through $P_0^{\\prime }$ .",
"In this case, all the points in the plane $\\langle P_0^{\\prime }, m \\rangle $ are 0-points, and hence the line $P_1P_0^{\\prime }$ is a 0-line, a contradiction.",
"So we find that every 0-line through a 0-point of $A$ is contained in $A$ .",
"Since every point of $A$ lies on $\\frac{q^{k-1}-1}{q-1}$ 0-lines, and $A$ contains $\\frac{q^k-1}{q-1}$ 0-points, we find that every 2 points of $A$ are contained in a 0-line of $A$ .",
"The same argument works for the set $A^{\\prime }$ .",
"This shows that $A$ forms a subspace and likewise $A^{\\prime }$ forms a subspace.",
"Since $|A|=|A^{\\prime }|=\\frac{q^k-1}{q-1}$ , these subspaces are $(k-1)$ -dimensional."
],
[
"There exists a suitable Desarguesian $(k-1)$ -spread {{formula:5d299a62-d518-48c3-80a5-84f442bd5791}} in {{formula:32a97683-3610-493b-8c79-c869c3ed8c0b}}",
"Consider the scattered linear set $D\\subset H_\\infty $ of pseudoregulus type.",
"Let $T_0$ and $T_\\infty $ be the transversal $(k-1)$ -spaces to the pseudoregulus defined by $D$ found in Theorem REF .",
"Now we want to show that there exists a Desarguesian $(k-1)$ -spread $\\mathcal {S}$ in $\\operatorname{PG}(2k-1,q)$ such that $T_0,T_\\infty \\in \\mathcal {S}$ and such that every other $(k-1)$ -space of $\\mathcal {S}$ has precisely one point in common with $D$ .",
"Lemma 3.13 There exists a Desarguesian $(k-1)$ -spread $\\mathcal {S}$ in $\\operatorname{PG}(2k-1,q)$ , such that $T_0,T_\\infty \\in \\mathcal {S}$ and such that every other element of $\\mathcal {S}$ has precisely one point in common with $D$ .",
"We prove this lemma using the representation of Result REF .",
"By [13] we find that the linear sets $L_{\\rho ,f}$ and $L_{\\rho ^{\\prime },g}$ are equivalent if and only if $\\sigma _f=\\sigma _g^{\\pm 1}$ , where $\\sigma _f$ and $\\sigma _g$ are the automorphisms associated with $f$ and $g$ respectively.",
"Hence, up to equivalence, we may suppose that $\\rho =1$ and $f:\\mathbb {F}_{q^k} \\rightarrow \\mathbb {F}_{q^k}: t\\rightarrow t^{2^i}$ , $\\gcd (i,hk)=1$ .",
"Considering $U_0,U_\\infty $ as $\\mathbb {F}_{q^k}$ , it follows that $D$ is equivalent to the set of points $P_u$ with $P_u:=(u,u^{2^i})_q,u\\in \\mathbb {F}_{q^k}^*.$ The transversal spaces $T_0$ and $T_\\infty $ are the point sets $T_0=\\lbrace (u,0)|u\\in \\mathbb {F}_{q^k}^*\\rbrace $ and $T_\\infty =\\lbrace (0,u)|u\\in \\mathbb {F}_{q^k}^*\\rbrace $ .",
"Consider now the set $\\mathcal {S}_0$ of $(k-1)$ -spaces $T_u$ , $u\\in \\mathbb {F}_{q^k}^*$ with $T_u:=\\lbrace (\\alpha u, \\alpha u^{2^i})_{q}|\\alpha \\in \\mathbb {F}_{q^k}^*\\rbrace .$ We will show that the set $\\mathcal {S}=\\mathcal {S}_0\\cup \\lbrace T_0,T_\\infty \\rbrace $ is a $(k-1)$ -spread of $\\operatorname{PG}(2k-1,q)$ .",
"Suppose that $P=T_{u_1}\\cap T_{u_2}$ , for some $u_1, u_2\\notin \\lbrace 0, \\infty \\rbrace $ , then there exists elements $\\alpha _1,\\alpha _2 \\in \\mathbb {F}_{q^k}^*, \\mu \\in \\mathbb {F}_{q}^*$ such that $\\left\\lbrace \\begin{array}{ll}\\alpha _1 u_1&=\\mu \\alpha _2 u_2 \\\\\\alpha _1 u_1^{2^i}&=\\mu \\alpha _2 u_2^{2^i} \\end{array}\\right.$ With $\\mu \\in \\mathbb {F}_q^*$ .",
"This implies that $u_1^{2^i-1}=u_2^{2^i-1}$ or $\\left(\\frac{u_1}{u_2}\\right)^{2^i}=\\frac{u_1}{u_2}$ .",
"Hence $\\frac{u_1}{u_2}\\in \\mathbb {F}_{2^i}\\cap \\mathbb {F}_{2^{hk}}$ which is $\\mathbb {F}_2$ since $\\gcd (i,hk)=1$ .",
"Since $u_1,u_2\\in \\mathbb {F}_{q^k}^*$ , this implies that $u_1=u_2$ , and that $T_{u_1}=T_{u_2}$ .",
"In particular, we see that $T_u \\ne T_{u^{\\prime }}$ for $u\\ne u^{\\prime } \\in \\mathbb {F}_{q^k}^*$ .",
"Since $T_0$ and $T_\\infty $ are distinct from $T_u$ for all $u\\in \\mathbb {F}_{q^k}^*$ , we obtain that $|\\mathcal {S}|=q^k+1$ .",
"We will now show that $T_u\\cap T_0=\\emptyset $ for all $u\\in \\mathbb {F}_{q^k}^*$ .",
"If $P=T_u\\cap T_0$ , $u\\notin \\lbrace 0, \\infty \\rbrace $ for some $u\\in \\mathbb {F}_{q^k}^*$ then $P=(u^{\\prime },0)_q$ with $u^{\\prime }\\in \\mathbb {F}_{q^k}^*$ and $\\left\\lbrace \\begin{array}{ll}\\alpha u&=\\mu u^{\\prime } \\\\\\alpha u^{2^i}&=0 \\end{array}\\right.$ for some $\\mu \\in \\mathbb {F}_q^*$ and $\\alpha \\in \\mathbb {F}_{q^k}^*$ .",
"The second equality gives a contradiction since $u\\ne 0 \\ne \\alpha $ .",
"Hence $T_u \\cap T_0 = \\emptyset $ .",
"It follows from a similar argument that $T_u \\cap T_\\infty =\\emptyset $ .",
"This shows that $\\mathcal {S}$ is a spread which is Desarguesian as seen in Subsection REF .",
"Remark 3.14 In [13] a geometric construction of the Desarguesian spread, found in Lemma REF , using indicator sets, is given."
],
[
"The point set $\\mathcal {Q}$ defines a translation hyperoval in the André/Bruck-Bose plane {{formula:fe441269-42d1-4751-97d9-6854ca55cc19}}",
"The spread $\\mathcal {S}$ found in Lemma REF defines a projective plane $\\mathcal {P}(\\mathcal {S})=\\Pi _{q^k}\\cong \\operatorname{PG}(2,q^k)$ by the André/Bruck-Bose construction.",
"The transversal $(k-1)$ -spaces $T_0, T_\\infty \\in \\mathcal {S}$ to the pseudoregulus associated with $D$ correspond to points $P_0, P_\\infty $ contained in the line $\\ell _\\infty $ at infinity of $\\operatorname{PG}(2,q^k)$ .",
"Theorem 3.15 The set $\\mathcal {Q}$ , together with $T_0$ and $T_\\infty $ , defines a translation hyperoval in $\\Pi _{q^k}\\cong \\operatorname{PG}(2,q^k)$ .",
"Let $\\mathcal {A}$ be the set of points in $\\Pi _{q^k}$ corresponding to the point set $\\mathcal {Q}$ of $\\Pi _q$ .",
"Recall that $T_0$ corresponds to a point $P_0$ and $T_\\infty $ to a point $P_\\infty $ , contained in the line $\\ell _\\infty $ of $\\Pi _{q^k}$ .",
"We first show that every line in $\\operatorname{PG}(2,q^k)$ contains at most 2 points of the set $\\mathcal {H}=\\mathcal {A}\\cup P_0\\cup P_\\infty $ .",
"The line $\\ell _\\infty $ at infinity only contains the points $P_0$ and $P_\\infty $ .",
"Consider a line $l\\ne \\ell _\\infty $ through $P_0$ in $\\operatorname{PG}(2,q^k)$ .",
"This line corresponds to a $k$ -space through $T_0$ in $\\operatorname{PG}(2k,q)$ .",
"As $P_0\\in l\\cap \\mathcal {H}$ , we have to show that this $k$ -space contains at most one affine point of $\\mathcal {Q}$ .",
"If this space would contain 2 (or more) affine points $X_1,X_2\\in \\mathcal {Q}$ , then they would define a direction of $D$ at infinity in $T_0$ .",
"But this is impossible as $T_0$ has no points of $D$ , see Corollary REF .",
"This argument also works for the lines through $P_\\infty $ , different from $\\ell _\\infty $ .",
"Consider a line $l$ through a point $P_i$ , $i\\notin \\lbrace 0,\\infty \\rbrace $ at infinity.",
"This point $P_i$ corresponds to an element $T_i\\in \\mathcal {S}$ that intersects the pseudoregulus $D$ in a unique point $X_i$ .",
"The line $l$ corresponds to a $k$ -space $\\gamma $ in $\\operatorname{PG}(2k,q)$ through $T_i$ .",
"Suppose that $\\gamma $ contains at least 3 points from $\\mathcal {Q}$ , say $X,Y,Z$ .",
"By Lemma REF these points are not collinear, hence they determine at least two different points of $D$ which are contained in $T_i$ , a contradiction.",
"This proves that $\\gamma $ contains at most two points of $\\mathcal {Q}$ , which implies that the line $l$ contains at most two points of $\\mathcal {A}$ .",
"Since $\\mathcal {H}$ has size $q^k+2$ , it follows that $\\mathcal {H}$ is a hyperoval.",
"Finally consider the group $G$ of elations in $\\operatorname{PG}(2hk,2)$ with axis the hyperplane at infinity $\\tilde{H}_\\infty $ .",
"Since the points of $\\tilde{{Q}}$ form a subspace, we see that $G$ acts transitively on the points of $\\tilde{\\mathcal {Q}}$ .",
"Every element of $G$ induces an element of the group $G^{\\prime }$ of elations in $\\operatorname{PG}(2,q^k)$ with axis the line $P_0P_\\infty $ .",
"Hence, $G^{\\prime }$ acts transitively on the points of $\\mathcal {A}$ in $\\operatorname{PG}(2,q^k)$ .",
"This shows that $\\mathcal {H}$ is a translation hyperoval."
],
[
"Every translation hyperoval defines a linear set of pseudoregulus type",
"In this section, we show that the vice versa part of Theorem REF holds.",
"Proposition 3.16 Via the André/Bruck-Bose construction, the set of affine points of a translation hyperoval in $\\operatorname{PG}(2,q^k)$ , $q=2^h$ , where $h,k\\ge 2$ corresponds to a set $\\mathcal {Q}$ of $q^k$ affine points in $\\operatorname{PG}(2k,q)$ whose set of determined directions $D$ is an $\\mathbb {F}_2$ -linear set of pseudoregulus type.",
"Consider a translation hyperoval $H\\in \\operatorname{PG}(2,q^k)$ .",
"Without loss of generality we may suppose that $H=\\lbrace (1,t,t^{2^i})_{q^k} | t\\in \\mathbb {F}_{q^k}\\rbrace \\cup \\lbrace (0,1,0)_{q^k},(0,0,1)_{q^k}\\rbrace $ with $\\gcd (i,hk)=1$ .",
"The set of affine points of $H$ corresponds to the set of points $H^{\\prime }=\\lbrace (1,t,t^{2^i})_{q} \\in \\mathbb {F}_q \\oplus \\mathbb {F}_{q^k} \\oplus \\mathbb {F}_{q^k} | t\\in \\mathbb {F}_{q^k}\\rbrace $ in $\\operatorname{PG}(2k,q)$ (for more information about the use of these coordinates for $H$ and $H^{\\prime }$ , see [15]).",
"The determined directions in the hyperplane at infinity $H_\\infty : X_0=0$ have coordinates $(0,t_1-t_2,t_1^{2^i}-t_2^{2^i})_q$ where $t_1,t_2 \\in \\mathbb {F}_{q^k}$ .",
"So the set $D=\\lbrace (0,u,u^{2^i} ) _q| u\\in \\mathbb {F}_{q^k} \\rbrace $ is precisely the set of directions determined by the points of $H$ .",
"By Result REF we find that this set of directions $D$ is an $\\mathbb {F}_2$ -linear set of pseudoregulus type in the hyperplane $H_\\infty $ .",
"We will now show that every line in $\\operatorname{PG}(2k-1,q)$ intersects the points of the linear set $D$ in $0,1,3$ or $q-1$ points.",
"Proposition 3.17 Let $D$ be the set of points of an $\\mathbb {F}_2$ -linear set of pseudoregulus type in $\\operatorname{PG}(2k-1,q)$ , $q=2^h$ , $h>2$ , $k\\ge 2$ .",
"Then every line of $\\operatorname{PG}(2k-1,q)$ meets $D$ in $0,1,3$ or $q-1$ points.",
"We use the representation of Result REF for the points of $D$ .",
"Let $R_1=(u_1,f(u_1))_q$ and $R_2=(u_2,f(u_2))_q$ , $u_1,u_2 \\in U_0$ , be two points of $D$ not on the same line of the pseudoregulus, so the vectors $\\langle u_1 \\rangle $ and $\\langle u_2 \\rangle $ in $V(k,q)$ are not an $\\mathbb {F}_q$ -multiple (in short $\\langle u_1 \\rangle _q \\ne \\langle u_2 \\rangle _q$ ).",
"Recall that $f$ is a invertible semilinear map with automorphism $\\sigma \\in Aut(\\mathbb {F}_q)$ , $Fix(\\sigma )=\\lbrace 0,1\\rbrace $ .",
"A third point $R_3=(u_3,f(u_3))_q\\in D$ is contained in $R_1R_2$ if and only if there are $\\mu ,\\lambda \\in \\mathbb {F}_q$ such that $\\left\\lbrace \\begin{array}{ll}u_1+\\lambda u_2&=\\mu u_3 \\\\f(u_1)+\\lambda f(u_2) &= \\mu f(u_3)\\end{array}\\right.\\Leftrightarrow \\left\\lbrace \\begin{array}{ll}f(u_1)+\\lambda ^\\sigma f(u_2)&=\\mu ^\\sigma f(u_3) \\\\f(u_1)+\\lambda f(u_2) &= \\mu f(u_3)\\end{array} \\right.\\\\\\Leftrightarrow \\left\\lbrace \\begin{array}{ll}u_1+\\lambda u_2&=\\mu u_3 \\\\(\\lambda ^\\sigma - \\lambda ) f(u_2) &= f((\\mu -\\mu ^{\\sigma ^{-1}})u_3)\\end{array} \\right.\\Leftrightarrow \\left\\lbrace \\begin{array}{ll}u_1+\\lambda u_2&=\\mu u_3 \\\\(\\lambda ^\\sigma - \\lambda )^{\\sigma ^{-1}} u_2 &= (\\mu -\\mu ^{\\sigma ^{-1}})u_3\\end{array} \\right.$ As $R_2$ and $R_3$ lie on different $(q-1)$ -secants to $D$ , we have that $\\langle u_2 \\rangle _q \\ne \\langle u_3 \\rangle _q$ .",
"It follows that $\\lambda ^\\sigma - \\lambda =\\mu -\\mu ^{\\sigma ^{-1}}=0$ , so $\\lambda ,\\mu \\in Fix(\\sigma )=\\lbrace 0,1\\rbrace $ .",
"We find that there is only one solution of this system, such that $R_1\\ne R_3$ (i.e.",
"$\\langle u_1 \\rangle _q \\ne \\langle u_3 \\rangle _q$ ), namely when $\\lambda =\\mu =1$ .",
"Hence, given two points $R_1,R_2$ in $D$ , there is an unique point $R_3\\in D\\cap R_1R_2$ , different from $R_1$ and $R_2$ ."
],
[
"The generalisation of a characterisation of Barwick and Jackson",
"Using Theorem REF , we are now able to generalise the following result of Barwick-Jackson which concerns translation hyperovals in $\\operatorname{PG}(2,q^2)$ ([4]).",
"Result 4.1 [4] Consider $\\operatorname{PG}(4,q)$ , $q$ even, $q>2$ , with the hyperplane at infinity denoted by $\\Sigma _\\infty $ .",
"Let $\\mathcal {C}$ be a set of $q^2$ affine points, called $\\mathcal {C}$ -points and consider a set of planes called $\\mathcal {C}$ -planes which satisfies the following: (A1) Each $\\mathcal {C}$ -plane meets $\\mathcal {C}$ in a $q$ -arc.",
"(A2) Any two distinct $\\mathcal {C}$ -points lie in a unique $\\mathcal {C}$ -plane.",
"(A3) The affine points that are not in $\\mathcal {C}$ lie on exactly one $\\mathcal {C}$ -plane.",
"(A4) Every plane which meets $\\mathcal {C}$ in at least 3 points either meets $\\mathcal {C}$ in 4 points or is a $\\mathcal {C}$ -plane.",
"Then there exists a Desarguesian spread $\\mathcal {S}$ in $\\Sigma _\\infty $ such that in the Bruck-Bose plane $\\mathcal {P}(\\mathcal {S})\\cong \\operatorname{PG}(2,q^2)$ , the $\\mathcal {C}$ -points, together with 2 extra points on $\\ell _\\infty $ form a translation hyperoval in $\\operatorname{PG}(2,q^2)$ .",
"Remark 4.2 At two different points, the proofs of [4] are inherently linked to the fact that they are dealing with hyperovals in $\\operatorname{PG}(2,q^2)$ .",
"In [4] the authors show the existence of a design which is isomorphic to an affine plane, of which they later need to use the parallel classes.",
"In [4], they use the Klein correspondence to represent lines in $\\operatorname{PG}(3,q)$ in $\\operatorname{PG}(5,q)$ .",
"Both techniques cannot be extended in a straightforward way to $q^k$ , $k>2$ .",
"The following Proposition shows that a set of $\\mathcal {C}$ -planes as defined by Barwick and Jackson in [4] (using $\\operatorname{PG}(2k,q)$ instead of $\\operatorname{PG}(4,q)$ ) satisfies the conditions of Theorem REF .",
"Proposition 4.3 Consider $\\operatorname{PG}(2k,q)$ , $q$ even, $q>2$ , with the hyperplane at infinity denoted by $\\Sigma _\\infty $ .",
"Let $\\mathcal {C}$ be a set of $q^k$ affine points, called $\\mathcal {C}$ -points and consider a set of planes called $\\mathcal {C}$ -planes which satisfies the following: (A1) Each $\\mathcal {C}$ -plane meets $\\mathcal {C}$ in a $q$ -arc.",
"(A2) Any two distinct $\\mathcal {C}$ -points lie in a unique $\\mathcal {C}$ -plane.",
"(A3) The affine points that are not in $\\mathcal {C}$ lie on exactly one $\\mathcal {C}$ -plane.",
"(A4) Every plane which meets $\\mathcal {C}$ in at least 3 points either meets $\\mathcal {C}$ in 4 points or is a $\\mathcal {C}$ -plane.",
"Then $\\mathcal {C}$ determines a set of $q^k-1$ directions $D$ in $\\Sigma _\\infty $ such that every line of $\\Sigma _\\infty $ meets $D$ in $0,1,3$ or $q-1$ points.",
"As before, we call the points that are not contained in $\\Sigma _\\infty $ affine points.",
"Note that all $\\mathcal {C}$ -points are affine.",
"Since every two $\\mathcal {C}$ -points lie on a $\\mathcal {C}$ -plane which meets $\\mathcal {C}$ in a $q$ -arc, we have that no three $\\mathcal {C}$ -points are collinear.",
"Let $P_0$ be a $\\mathcal {C}$ -point and let $D_0$ be the set of points of the form $P_0P_i\\cap \\Sigma _\\infty $ , where $P_i\\ne P_0$ is a point of $\\mathcal {C}$ .",
"We first show that every line meets $D_0$ in $0,1,3$ or $q-1$ points.",
"Let $M$ be a line of $\\Sigma _\\infty $ containing 2 points of $D_0$ , say $R_1^{\\prime }=P_0R_1\\cap \\Sigma _\\infty $ , $R_2^{\\prime }=P_0R_2\\cap \\Sigma _\\infty $ , where $R_1,R_2\\in \\mathcal {C}$ .",
"Then $\\langle M,P_0\\rangle $ contains at least 3 points of $\\mathcal {C}$ , and hence, by (A4), either it is a $\\mathcal {C}$ -plane or it contains exactly 4 points of $\\mathcal {C}$ .",
"If $\\langle M,P_0\\rangle $ is a $\\mathcal {C}$ -plane, it contains $q$ points of $\\mathcal {C}$ forming a $q$ -arc, and hence, $M$ contains $q-1$ points of $D_0$ .",
"Now suppose that $\\langle M,P_0\\rangle $ contains exactly 4 $\\mathcal {C}$ -points, then $M$ contains 3 points of $D_0$ .",
"Now let $P_1\\ne P_0$ be a point of $\\mathcal {C}$ and let $D_1$ be the set of points of the form $P_1P_i\\cap \\Sigma _\\infty $ , where $P_i\\ne P_1$ is a point of $\\mathcal {C}$ .",
"We claim that $D_0=D_1$ .",
"Let $P_1^{\\prime }=P_0P_1\\cap \\Sigma _\\infty $ .",
"We see that $P_1^{\\prime }\\in D_0\\cap D_1$ .",
"Consider a point $P_2^{\\prime }\\ne P_1^{\\prime }$ in $D_0$ , then $P_0P_2\\cap \\Sigma _\\infty =P_2^{\\prime }$ for some $P_2\\in \\mathcal {C}$ .",
"Consider the plane $\\pi =\\langle P_0,P_1,P_2\\rangle $ .",
"Suppose first that $\\pi $ is not a $\\mathcal {C}$ -plane, then, by (A4), $\\pi $ contains exactly one extra point, say $P_3$ of $\\mathcal {C}$ .",
"The lines $P_0P_1$ and $P_2P_3$ lie in $\\pi $ and hence, meet in a point $Q$ .",
"By $(A2)$ , there is a $\\mathcal {C}$ -plane $\\mu $ through $P_0P_1$ , and likewise, there is a $\\mathcal {C}$ -plane $\\mu ^{\\prime }$ through $P_2P_3$ .",
"Since $\\pi $ is not a $\\mathcal {C}$ -plane, $\\mu $ and $\\mu ^{\\prime }$ are two distinct $\\mathcal {C}$ -planes through $Q$ .",
"By (A3) his implies that $Q$ is a point of $\\Sigma _\\infty $ .",
"Likewise, $P_0P_2\\cap P_1P_3$ and $P_0P_3\\cap P_1P_2$ are points of $\\Sigma _\\infty $ .",
"It follows that $D_0\\cap \\pi =D_1\\cap \\pi $ .",
"This argument shows that for all points $R\\ne P_1^{\\prime }\\in D_0$ such that $\\langle P_0,P_1,R\\rangle $ is not a $\\mathcal {C}$ -plane, we have that $R\\in D_1$ .",
"Now $P_0P_1$ lies on a unique $\\mathcal {C}$ -plane, say $\\nu $ .",
"Let $\\nu \\cap \\Sigma _\\infty =L$ , then we have shown that $\\langle P_0,P_1,R\\rangle $ is not a $\\mathcal {C}$ -plane as long as $R\\in \\Sigma _\\infty $ is not on $L$ .",
"We conclude that $D_0\\setminus L=D_1\\setminus L$ .",
"Now assume that $D_0\\ne D_1$ and let $X$ be a point in $D_1$ which is not contained in $D_0$ .",
"Then $X\\in L$ and $P_1X$ contains a point $Y\\ne P_1\\in \\mathcal {C}$ .",
"Consider a point $P_4^{\\prime }\\in D_1$ , not on $L$ , then $P_1P_4^{\\prime }$ contains a point $P_4\\ne P_1$ of $\\mathcal {C}$ .",
"Since $P_4^{\\prime }\\in D_1\\setminus L$ , $P_4^{\\prime }\\in D_0$ so the line $P_4^{\\prime }P_0$ contains a point $P_5\\ne P_1$ of $\\mathcal {C}$ .",
"The plane $\\langle P_1,P_4^{\\prime },X\\rangle $ is not a $\\mathcal {C}$ -plane since otherwise, the points $P_1$ and $Y$ of $\\mathcal {C}$ would lie in two different $\\mathcal {C}$ -planes.",
"This implies that $\\langle P_1,P_4,X\\rangle $ which contains the $\\mathcal {C}$ -points $P_1,P_4,Y$ contains exactly one extra point of $\\mathcal {C}$ , say $P_6$ .",
"Denote $P_1P_6\\cap \\Sigma _\\infty $ by $P^{\\prime }_6$ .",
"We see that there are exactly 3 points of $D_1$ on the line $P_4^{\\prime }X$ , namely $P_4^{\\prime },X$ and $P_6^{\\prime }$ .",
"Now $P^{\\prime }_6$ is a point of $D_1$ , not on $L$ , so $P^{\\prime }_6\\in D_0$ .",
"Hence, there is a point $S\\ne P_0\\in \\mathcal {C}$ on the line $P_0P^{\\prime }_6$ .",
"If $\\langle P_4^{\\prime },P^{\\prime }_6,P_0\\rangle $ is not a $\\mathcal {C}$ -plane, then, since it contains $P_0,P_5,S$ of $\\mathcal {C}$ it contains precisely 3 points of $D_0$ at infinity.",
"These are the points $P_4^{\\prime },P_6^{\\prime }$ and one other point, say $T$ , which needs to be different from $X$ by our assumption that $X\\notin D_0$ .",
"That implies that $T$ is not on $L$ , and hence, $T\\in D_1$ .",
"This is a contradiction since we have seen that the only points of $D_1$ on $P_4^{\\prime }X$ are $P_4^{\\prime },X$ and $P_6^{\\prime }$ .",
"Now if $\\langle P_4^{\\prime },P_6,P_0\\rangle $ is a $\\mathcal {C}$ -plane, we find $q-1$ points of $D_0$ on $P_4^{\\prime }X$ , all of them are not on $L$ .",
"Hence, we find $q-1$ points of $D_1$ on $P_4^{\\prime }X$ , not on $L$ .",
"This is again a contradiction since $P_4^{\\prime }X$ has only the points $P_4^{\\prime }$ and $P_6^{\\prime }$ of $D_1$ not on $L$ .",
"This proves our claim that $D_0=D_1$ .",
"Since $P_1$ was chosen arbitrarily, different from $P_0$ , and $D_0=D_1$ , we find that the set $D$ of directions determined by $\\mathcal {C}$ is precisely the set $D_0$ .",
"The statement now follows from the fact that a line meets $D_0$ in $0,1,3$ or $q-1$ points.",
"Proposition REF shows that the set $\\mathcal {C}$ satisfies the criteria of Theorem REF .",
"Hence, we find the following generalisation of Result REF .",
"Theorem 4.4 Consider $\\operatorname{PG}(2k,q)$ , $q$ even, $q>2$ , with the hyperplane at infinity denoted by $\\Sigma _\\infty $ .",
"Let $\\mathcal {C}$ be a set of $q^k$ affine points, called $\\mathcal {C}$ -points and consider a set of planes called $\\mathcal {C}$ -planes which satisfies the following: (A1) Each $\\mathcal {C}$ -plane meets $\\mathcal {C}$ in a $q$ -arc.",
"(A2) Any two distinct $\\mathcal {C}$ -points lie in a unique $\\mathcal {C}$ -plane.",
"(A3) The affine points that are not in $\\mathcal {C}$ lie on exactly one $\\mathcal {C}$ -plane.",
"(A4) Every plane which meets $\\mathcal {C}$ in at least 3 points either meets $\\mathcal {C}$ in 4 points or is a $\\mathcal {C}$ -plane.",
"Then there exists a Desarguesian spread $\\mathcal {S}$ in $\\Sigma _\\infty $ such that in the Bruck-Bose plane $\\mathcal {P}(\\mathcal {S})\\cong \\operatorname{PG}(2,q^k)$ , the $\\mathcal {C}$ -points, together with 2 extra points on $\\ell _\\infty $ form a translation hyperoval in $\\operatorname{PG}(2,q^k)$ ."
]
] | 1906.04537 |
[
[
"Stochastic Neural Network with Kronecker Flow"
],
[
"Abstract Recent advances in variational inference enable the modelling of highly structured joint distributions, but are limited in their capacity to scale to the high-dimensional setting of stochastic neural networks.",
"This limitation motivates a need for scalable parameterizations of the noise generation process, in a manner that adequately captures the dependencies among the various parameters.",
"In this work, we address this need and present the Kronecker Flow, a generalization of the Kronecker product to invertible mappings designed for stochastic neural networks.",
"We apply our method to variational Bayesian neural networks on predictive tasks, PAC-Bayes generalization bound estimation, and approximate Thompson sampling in contextual bandits.",
"In all setups, our methods prove to be competitive with existing methods and better than the baselines."
],
[
"Introduction",
"Work done while Chin-Wei was an intern at Element AI and Ahmed at Facebook Research.",
"Stochastic neural networks (SNN) are a central tool in many subfields of machine learning, including (1) Bayesian deep learning [33], [6], [19], [14], (2) exploration in reinforcement learning [23], [39], [43], and (3) statistical learning theory such as PAC-Bayesian learning [34], [29], [12].",
"Perturbations of the network parameters induce a distribution over the model, and this intrinsic uncertainty is the subject of great interest to machine learning practitioners and theoreticians alike.",
"For example, deep Bayesian models are often used to adequately measure uncertainty, and determine whether the model itself is inherently familiar with the unseen data.",
"This is especially important in the context of autonomous vehicles, where decisions must be made to meet specific safety standards [35].",
"Conversely, the lack of confidence can be leveraged to efficiently guide exploration in reinforcement learning, via randomizing the approximate value function [2], [49] or maximizing intrinsic rewards [20].",
"Furthermore, a considerable proportion of statistical learning theory is devoted to understanding what implies generalization, or what constitutes an appropriate measure of complexity [3], [1], [38].",
"PAC-Bayesian learning theory [34] specifically explores the generalization property of a randomized prediction rule, and has been recently studied in the context of stochastic neural networks [12].",
"In this particular study, the working hypothesis is that good generalization can be guaranteed on the premise that stochastic gradient descent [45] finds a solution that obtains certain structural properties, such as flatness.",
"For computational reasons, considerable effort has been devoted to modelling uncertainty through the injection of independent noise to the network parameters [18], [6], [26].",
"However, noise independence largely restricts the expressivity of the noise distribution and thus the resulting uncertainty measures are ill-calibrated [36], [50].",
"Attempts have been made to correlate parameters of a neural network, including [32], [28], [40], for example, by adapting expressive non-linear invertible transformations developed in the variational inference literature [42], [25], [21], or via implicit methods [17].",
"However, these methods are limited due to their inability to scale well.",
"[32], for instance, resort to a specific multiplicative noise sampled from a lower dimensional space and have to use an auxiliary method to bound the entropy.",
"[28], on the other hand, give up on injecting noise on the entire set of parameters and model the distribution of the scale and shift parameter of the pre-activations.",
"In attempts to address some of the challenges articulated above and efficiently model the joint distribution of a network's parameters, we take inspiration from the Kronecker product, which we notice can be thought of as left-transforming a matrix via a linear map, and then right-transforming it using another linear map, thus providing us an efficient way to correlate the weight parameters.",
"We propose the Kronecker Flow, an invertible transformation-based method that generalizes the Kronecker product to its nonlinear counterparts.",
"Our contributions are as follows.",
"We extend the idea of Kronecker product to more general invertible mappings to induce non-linear dependencies, and apply this trick to parameterizing deep stochastic neural networks.",
"We apply our method to predictive tasks and show that our methods work better on larger architectures compared to existing methods.",
"We are the first to apply flow-based methods to tighten the PAC-Bayes bound.",
"We show that the KL divergence in the PAC-Bayes bound can be estimated with high probability, and demonstrate the generalization gap can be further reduced and explained by leveraging the structure in the parameter space.",
"Our methods prove to be competitive over other methods in approximate Thompson sampling in contextual bandit problems."
],
[
"Background",
"Stochastic neural networks with parameter perturbation normally follow the stochastic process: $\\Theta \\sim q_\\phi (\\Theta )$ , $y|x\\sim p(y|x, \\Theta )=f_\\Theta (x)$ , where $\\Theta $ is the parameters of the neural network $f$ , which outputs the prediction probability vector for classification or the predicted values for regression.",
"We let $D=\\lbrace (x_i,y_i):{i\\in [m]}\\rbrace $ be the training set of size $m$ We use the notation $[n]$ to compactly describe the set of integers $\\lbrace 1,2,\\dots ,n\\rbrace $ ., $H$ be the differential entropy $H[q]=- \\mathbb {E}_q[\\log q]$ , $\\beta >0$ be the coefficient controlling the amount of noise injected into the model and the degree of regularization, $l(y,\\bar{y})$ be the loss function and $\\hat{R}_{D}(\\Theta )=\\frac{1}{m}\\sum _{i=1}^{m} l(y_i, f_\\Theta (x_i))$ be the empirical risk."
],
[
"Variational Bayesian neural networks",
"Variational Bayesian neural networks are a type of stochastic neural network.",
"Bayesian inference updates our prior belief $p(\\Theta )$ over the model parameters according to the Bayes rule $p(\\Theta |D)\\propto p(D|\\Theta )p(\\Theta )$ , by incorporating information from the training set through the likelihood function $p(D|\\Theta )$ .",
"Variational inference is a computational realization of Bayesian inference, which casts inference as an optimization problem, where one maximizes the variational lower bound (also known as the evidence lower bound, or the ELBO) on the log marginal likelihood: $\\log p(D) \\ge \\mathbb {E}_{q_\\phi }[ \\log p(D|\\Theta ) + \\log p(\\Theta )] + H(q_\\phi (\\Theta )),$ where $q_\\phi $ is the variational approximate posterior and $p(D|\\Theta )$ can be decomposed into $\\prod _{i=1}^{m} p(y_i|x_i, \\Theta )$ due to conditional independence assumption.",
"The optimal $q$ is the true posterior, i.e.",
"$q^*(\\Theta )=\\frac{p(D|\\Theta )p(\\Theta )}{p(D)}$ .",
"In our case, we use $\\Theta $ to parameterize a neural network.",
"Prediction can be carried out via the (approximate) predictive posterior $p(y|x,D) & = \\mathbb {E}_{\\Theta \\sim p(\\Theta |D)} [p(y|x,\\Theta )] \\\\& \\approx \\mathbb {E}_{\\Theta \\sim q_\\phi (\\Theta )}[p(y|x,\\Theta )] \\\\& \\approx \\frac{1}{K}\\sum _{k=1}^K p(y|x,\\Theta _k)$ for $\\lbrace \\Theta _k\\rbrace _{k\\in [K]}$ drawn i.i.d.",
"from $q_\\phi (\\Theta )$ , where we use the variational distribution $q$ to approximate $p(\\Theta |D)$ and a Monte Carlo estimate to estimate the integral.",
"The prior distribution can be used to encode some form of inductive bias, such as one that is in favor of parameter values closer to some $\\Theta _0$ chosen a priori.",
"We choose the prior to be an isotropic Gaussian, centered at the random initialization $\\Theta _0$ , i.e., $p(\\Theta ) = {\\mathcal {N}}(\\Theta ;\\Theta _0, \\lambda {\\mathbf {I}})$ .",
"The entropy term ensures the variational posterior does not collapse to a point estimate.",
"Both of them can be thought of as some form of regularizer, so we attach a coefficient $\\beta $ in front of them as a hyperparameter Like the $\\lambda $ parameter in [53]."
],
[
"PAC-Bayes generalization bound",
"Another use case of stochastic neural networks is to understand generalization, via PAC-Bayes bounds.",
"The aim is to bound a divergence between the empirical risk, $\\hat{{\\mathcal {L}}}[q]=\\mathbb {E}_q[\\hat{R}_D(\\Theta )]$ , and the risk measured on the true distribution $\\mathcal {D}$ , ${\\mathcal {L}}[q]=\\mathbb {E}_q[\\mathbb {E}_{\\mathcal {D}}[l(y,f_\\Theta (x))]]$ .",
"While this quantity is unbounded in the general case, assuming a bounded loss function $l$ (e.g.",
": zero-one loss), we can obtain a probabilistic bound that holds with probability $1-\\delta $ over the choice of $D$ , for $\\delta >0$ .",
"More specifically, with probability $1-\\delta $ , $\\Delta (\\hat{{\\mathcal {L}}}[q], {\\mathcal {L}}[q])\\le \\Omega (D_{\\mathrm {KL}}(q||p),m,\\delta )$ , where $\\Omega $ is a measure of complexity that scales proportionally with the Kullback–Leibler (KL) divergence and $\\Delta $ a measure of divergence (e.g.",
": square distance or convex functions [16]).",
"For instance, [12] minimize the following bound originally due to [34] and then tightened by [29]: Theorem 1 Let $l$ be the zero-one loss.",
"For any $\\delta >0$ and data distribution ${\\mathcal {D}}$ , and any distribution $p$ on the space of $\\Theta $ , with probability at least $1-\\delta $ over the choice of a training set $D\\sim {\\mathcal {D}}^m$ , for all distributions $q$ on the space of $\\Theta $ , $D_{\\mathrm {KL}}(\\hat{{\\mathcal {L}}}[q]||{\\mathcal {L}}[q])\\le \\frac{D_{\\mathrm {KL}}(q||p)+\\log \\frac{m}{\\delta }}{m-1},$ where the KL on the LHS is between two Bernoulli distributions, defined by the probability of performing an error.",
"We refer to the above bound as the McAllester bound.",
"The KL divergence on the RHS of the bound, also known as the information gain, tells us to what extent the posterior $q$ is dependent on the training data.",
"The sharper and more confident $q$ is, and the farther away it is from the prior $p$ , the larger the KL will be, which in turn is reflected by the larger bound on the generalization gap.",
"This is consistent with traditional notion of bias-variance trade-off.",
"Alternatively, we consider the following bound due to [9]: Theorem 2 With the $l$ , $\\delta $ , $\\mathcal {D}$ , and $p$ as defined in Theorem REF , and with a fixed $\\beta >1/2$ , the following bound holds with probability over $1-\\delta $ : ${\\mathcal {L}}[q] &\\le \\frac{1}{1-\\frac{1}{2\\beta }} \\left(\\hat{{\\mathcal {L}}}[q] + \\frac{\\beta }{m} \\left( D_{\\mathrm {KL}}(q||p) + \\ln \\frac{1}{\\delta } \\right) \\right).$ We refer to this bound as the Catoni bound.",
"We notice the linear relationship (which is also noticed by [15]) between the empirical risk and the KL divergence.",
"This allows us to make use of the linearity of expectation to perform change of variable (see the next section).",
"We also note that the optimal $\\beta $ in Equation REF is always larger than 1, so the PAC-Bayes bound is actually more conservative than Bayesian inference in this sense."
],
[
"Normalizing flows",
"Minimization of Equation REF and REF requires (i) computing the gradient with respect to the parameter of the (PAC-)Bayesian posterior $\\phi $ , and (ii) computing the entropy of $q$ .",
"One approach to do this is via change of variable under an invertible mapping.",
"Let $\\mathbf {\\epsilon }\\sim q_0$ be a random variable in $\\mathbb {R}^d$ , and ${\\mathbf {g}}_\\phi :\\mathbb {R}^d\\rightarrow \\mathbb {R}^d$ be a bijection parameterized by $\\phi $ .",
"Let $\\Theta ={\\mathbf {g}}_\\phi (\\mathbf {\\epsilon })$ and $q_\\phi $ be its density.",
"Then we can rewrite the loss function as Since the weighting coefficient $\\beta $ can be absorbed into the loss function $l$ , we neglect it for simplicity now.",
"$\\mathbb {E}_{\\Theta }[\\hat{R}_{D}(\\Theta ) + \\log q_\\phi (\\Theta )]& =\\mathbb {E}_{\\mathbf {\\epsilon }}[\\hat{R}_{D}({\\mathbf {g}}_\\phi (\\mathbf {\\epsilon })) + \\log q_0(\\mathbf {\\epsilon }) \\\\& - \\log \\left|\\det \\frac{\\partial {\\mathbf {g}}_\\phi (\\mathbf {\\epsilon })}{\\partial \\mathbf {\\epsilon }}\\right|],$ where we apply the change of variable (see Appendix for the detailed derivation).",
"The log-determinant (logdet) term ensures that we obtain a valid probability density function after $g_\\phi $ is applied, which can be a sequence of invertible mappings itself, hence referred to as the normalizing flow [42].",
"This way, the random variable and the parameters are decoupled, so that we can differentiate the integrand to have an unbiased estimate of the gradient (fixing some $\\mathbf {\\epsilon }\\sim q_0$ ).",
"We let $q_0$ be the standard normal."
],
[
"Kronecker Flows",
"We consider maximizing the ELBO and minimizing the Catoni bound via normalizing flow-based SNNs.",
"Conventionally, mean-field approximation using factorized distributions (such as multivariate Gaussian with diagonal covariance) has been well explored in the variational inference (VI) literature [6].",
"We are interested in better capturing the structure in the parameter space as restricted VI methods are known to exhibit overconfidence [36], [50].",
"However, the parameters of a neural network are usually very high dimensional (on the order of millions), requiring a novel way to parameterize the joint distribution over the parameters.",
"In its general form, neural networks can be represented by a collection of tensors i.e.",
"$\\Theta =\\lbrace {\\mathbf {W}}_l:{l\\in [L]}\\rbrace $ .",
"While our method below can easily be generalized to high-dimensional tensors (such as for convolutional kernels), to simplify notation, we describe the matrix form."
],
[
"Linear Kronecker Flow",
"The matrix-variate normal ($\\mathcal {MN}$ ) distribution generalizes the multivariate normal distribution to matrix-valued random variables, such as weight matrices of a neural network [31].",
"Matrix normal is a multivariate normal distribution whose covariance matrix is a Kronecker product ($\\otimes $ ), which allows us to model the correlation among the parameters to some degree.",
"More concretely, assume ${\\mathbf {E}}_{ij}\\overset{\\textnormal {i.i.d.",
"}}{\\sim } {\\mathcal {N}}(0,1)$ is an $n\\times p$ random Gaussian matrix, and ${\\mathbf {A}}\\in \\mathbb {R}^{n\\times n}$ , ${\\mathbf {B}}\\in \\mathbb {R}^{p\\times p}$ and ${\\mathbf {M}}\\in \\mathbb {R}^{n\\times p}$ are real-valued matrices.",
"Then ${\\mathbf {M}}+ {\\mathbf {A}}{\\mathbf {E}}{\\mathbf {B}}$ has a matrix normal distribution, as $\\textnormal {vec}({\\mathbf {M}}+ {\\mathbf {A}}{\\mathbf {E}}{\\mathbf {B}})\\sim {\\mathcal {N}}(\\textnormal {vec}({\\mathbf {M}}), {\\mathbf {B}}^\\top {\\mathbf {B}}\\otimes {\\mathbf {A}}{\\mathbf {A}}^\\top ),$ where $\\textnormal {vec}$ is the vectorization of a matrix that concatenates all the columns.",
"This allows us to represent the covariance matrix in a more compact manner ($n^2p^2/2$ parameters versus $n^2/2+p^2/2$ parameters for Kronecker product)."
],
[
"Limitation of the Kronecker product.",
"The Kronecker product covariance matrix is not a strict generalization of diagonal covariance matrix.",
"To observe this, let ${\\mathbf {U}}=\\mathop {\\mathrm {diag}}\\nolimits ({\\mathbf {u}})$ , ${\\mathbf {V}}=\\mathop {\\mathrm {diag}}\\nolimits ({\\mathbf {v}})$ (this is the case of [31]), and ${\\mathbf {S}}=\\mathop {\\mathrm {diag}}\\nolimits ({\\mathbf {s}})$ , where ${\\mathbf {u}}\\in \\mathbb {R}^n_{>0}$ , ${\\mathbf {v}}\\in \\mathbb {R}^p_{>0}$ , and ${\\mathbf {s}}\\in \\mathbb {R}^{np}_{>0}$ .",
"Then ${\\mathbf {U}}\\otimes {\\mathbf {V}}$ is also a diagonal matrix of size $np\\times np$ .",
"Equating ${\\mathbf {U}}\\otimes {\\mathbf {V}}={\\mathbf {S}}$ to solve for ${\\mathbf {u}}$ and ${\\mathbf {v}}$ will result in $np$ nonlinear equations with $n+p$ variables, which can be over-determined for $n,p>2$ .",
"For example, let $n=2,p=3$ , and ${\\mathbf {s}}=[1,\\epsilon ,\\epsilon ,1,1,1]$ for some $\\epsilon >0$ .",
"Then the nonlinear system below does not have a solution: ${\\mathbf {U}}\\otimes {\\mathbf {V}}= {\\mathbf {S}}\\quad \\Longleftrightarrow & \\quad {\\mathbf {u}}_1{\\mathbf {v}}_1\\overset{(a)}{=}1 \\quad {\\mathbf {u}}_1{\\mathbf {v}}_2\\overset{(b)}{=}\\epsilon \\quad {\\mathbf {u}}_1{\\mathbf {v}}_3\\overset{(c)}{=}\\epsilon \\quad \\\\& {\\mathbf {u}}_2{\\mathbf {v}}_1\\overset{(d)}{=}1 \\quad {\\mathbf {u}}_2{\\mathbf {v}}_2\\overset{(e)}{=}1 \\quad {\\mathbf {u}}_2{\\mathbf {v}}_3\\overset{(f)}{=}1$ To see this, dividing $(a)$ by $(b)$ and dividing $(d)$ by $(e)$ yield ${\\mathbf {v}}_1={\\mathbf {v}}_2/\\epsilon $ and ${\\mathbf {v}}_1={\\mathbf {v}}_2$ , respectively, which doesn't have a solution if $\\epsilon \\ne 1$ .",
"This is because the Kronecker product is essentially parameter sharing, which can heavily restrict the matrix it can represent.",
"To remedy the above limitation, we can further decouple the reparameterization of the parameter matrix into two parts: (1) one that models the marginal variance and (2) one that models correlations.",
"Assume ${\\mathbf {S}}\\in \\mathbb {R}^{n\\times p}_{>0}$ is a positive-valued matrix, and let ${\\mathbf {W}}:={\\mathbf {M}}+ {\\mathbf {A}}({\\mathbf {E}}\\circ {\\mathbf {S}}){\\mathbf {B}}$ .",
"Then $\\textnormal {vec}({\\mathbf {W}})$ is a Gaussian distribution with the following property, which is useful in calculating the KL divergence: Property 1 Let ${\\mathbf {W}}$ be given as above, with $\\mu =\\mathbb {E}[\\textnormal {vec}({\\mathbf {W}})]$ and $\\Sigma =\\mathrm {Var}(\\textnormal {vec}({\\mathbf {W}}))$ .",
"Then $\\mu =\\textnormal {vec}({\\mathbf {M}})$ , and $\\Sigma =({\\mathbf {B}}^\\top \\otimes {\\mathbf {A}}) \\mathop {\\mathrm {diag}}\\nolimits (\\textnormal {vec}({\\mathbf {S}}^2)) ({\\mathbf {B}}\\otimes {\\mathbf {A}}^\\top )$ $\\det (\\Sigma )=\\det ({\\mathbf {A}})^{2p}\\det ({\\mathbf {B}})^{2n}\\prod _{ij}{\\mathbf {S}}_{ij}^2$ $\\textnormal {Tr}(\\Sigma )=\\sum _{ij}\\left({\\mathbf {A}}^2{\\mathbf {S}}^2{\\mathbf {B}}^2\\right)_{ij}$ See Appendix for the derivation and interpretation of the property.",
"Naive implemetations of this can be inefficient and numerically unstable, as the entropy term involves computing the log-determinant of ${\\mathbf {A}}$ and ${\\mathbf {B}}$ , requiring the standard automatic differentiation libraries to resort to singular value decomposition when the matrix is near-singular.",
"Thus, we choose to parameterize ${\\mathbf {A}}$ and ${\\mathbf {B}}$ as lower triangular matricesThis is achieved by masking.",
"with ones on the diagonal, leaving the uncertainty to be modeled by ${\\mathbf {S}}$ .",
"This means $\\det (\\Sigma )=\\prod _{ij}{\\mathbf {S}}_{ij}^2$ .",
"Figure: Random 3D Gaussian tensor"
],
[
"Simulation.",
"To validate the limited expressiveness of kronecker product, we randomly initialize a target density $p$ to be a multivariate Gaussian with mean zero, and covariance being the square of a random standard Gaussian matrix.",
"We choose the dimensionality $d$ of the Gaussian such that it can be decomposed into a product of integers, and parameterize $q$ using independent Gaussian (dubbed Diag), the Kronecker product with diagonal $A$ and $B$ (K-Diag), and the Kronecker product with elementwise scaling (K-Linear).",
"We minimize $D_{\\mathrm {KL}}(q||p)$ ; see Figure REF for the results.",
"We also conduct the same experiment with 3D tensors (instead of matrices).",
"We see that K-Diag consistently underperforms when compared to Diag, which indicates parameter sharing does restrict the family of distributions it can represent, and K-Linear is consistently better as it captures some correlation."
],
[
"Nonlinear Kronecker Flow",
"In this section, we generalize the Kronecker product to more general non-linear mappings.",
"In Appendix , we make a connection to non-decreasing triangle maps [52] that are general enough to model any probability distributions.",
"First, notice that left-multiplying ${\\mathbf {E}}$ by ${\\mathbf {A}}$ amounts to introducing linear correlation among the $n$ rows of ${\\mathbf {E}}$ , applied to each of the $p$ columns.",
"Likewise, right-multiplying ${\\mathbf {E}}$ by ${\\mathbf {B}}$ amounts to correlating column entries of each row of ${\\mathbf {E}}$ .",
"Inspired by this, we consider applying an invertible mapping to each row of the random weight matrix, and another invertible mapping to each column of the matrix.",
"We call this the Kronecker Flow To differentiate this from K-Linear from the previous section, we refer to using non-linear ${\\mathbf {g}}$ as K-Nonlinear..",
"Specifically, let ${\\mathbf {g}}_A:\\mathbb {R}^{n}\\rightarrow \\mathbb {R}^{n}$ and ${\\mathbf {g}}_B:\\mathbb {R}^{p}\\rightarrow \\mathbb {R}^{p}$ be invertible mappings.",
"We define the matrix-matrix function ${\\mathbf {G}}:\\mathbb {R}^{n\\times p}\\rightarrow \\mathbb {R}^{n\\times p}$ as ${\\mathbf {G}}_B({\\mathbf {G}}_A({\\mathbf {E}}^\\top )^\\top )$ , with the following batch-operations (for $i\\in [n]$ and $j\\in [p]$ ): ${\\mathbf {G}}_A({\\mathbf {E}}^\\top )_{j:} := {\\mathbf {g}}_A({\\mathbf {E}}_{:j}) \\qquad \\qquad {\\mathbf {G}}_B({\\mathbf {E}})_{i:} := {\\mathbf {g}}_B({\\mathbf {E}}_{i:}) $ It is easy to verify that ${\\mathbf {G}}$ is invertible.",
"Due to the partial dependency of ${\\mathbf {G}}_A$ and ${\\mathbf {G}}_B$ , the Jacobians of the vectorized forms (after proper permutation) are block-diagonal, so we have $\\det &\\frac{\\partial \\textnormal {vec}({\\mathbf {G}}({\\mathbf {E}}))}{\\partial \\textnormal {vec}({\\mathbf {E}})} \\\\&= \\prod _{j\\in [p]}\\det \\frac{\\partial {\\mathbf {g}}_A({\\mathbf {E}}_{:j})}{\\partial {\\mathbf {E}}_{:j}}\\cdot \\prod _{i\\in [n]}\\det \\frac{\\partial {\\mathbf {g}}_B({\\mathbf {G}}_A({\\mathbf {E}}^\\top )_{:i})}{\\partial {\\mathbf {G}}_A({\\mathbf {E}}^\\top )_{:i}}.$ In practice, we use the volume preserving version of RealNVP [11] and inverse autoregressive flow (IAF) [25] to parameterize ${\\mathbf {g}}_A$ and ${\\mathbf {g}}_B$ for our experiments We also experimented with Block NAF by [10], but did not include it in the manuscript: its performance was similar to IAF, but it is much slower to sample from (we adapted the open implementation from [10])..",
"The K-Linear from the previous section can be thought of as using a linear map as ${\\mathbf {g}}_A$ and ${\\mathbf {g}}_B$ ."
],
[
"Concentration of empirical KL with normalizing flows",
"In their study, [12] use independent Gaussian for $q$ to minimize the McAllester bound, so they can compute the KL between Gaussians analytically.",
"This is no longer feasible when we use more flexible families for $q$ , such as normalizing flows.",
"Moreover, a Monte Carlo estimate might result in underestimating the bound after inverting the KL between Bernoullis on the LHS of Equation REF (which is a concave function; see Appendix A of [41] for an illustration).",
"This necessitates a high probability bound on the concentration of the empirical estimate.",
"In Section REF , we have established $D_{\\mathrm {KL}}(q||p)$ can be written in the following form $\\leavevmode \\xbox {resizebox}{\\XMLaddatt {width}{427.0pt}\\mathbb {E}_{\\mathbf {\\epsilon }} \\left[\\log {\\mathcal {N}}(\\mathbf {\\epsilon }; \\mathbf {0},{\\mathbf {I}}) - \\log \\left|\\det \\frac{\\partial {\\mathbf {g}}_\\phi (\\mathbf {\\epsilon })}{\\partial \\mathbf {\\epsilon }}\\right|-\\log {\\mathcal {N}}({\\mathbf {g}}_\\phi (\\mathbf {\\epsilon }); \\mathbf {0}, {\\mathbf {I}}) \\right],}$ where both $q_0$ and $p$ are standard Gaussian (the mean and variance can be absorbed into the invertible mapping ${\\mathbf {g}}$ if this is not the case).",
"The first term in the KL can be computed analytically.",
"The second term usually can be almost surely bounded (e.g.",
"using Block neural autoregressive flows) so that we can use Hoeffding-type concentration or it can simply be made zero using e.g.",
"volume preserving flows.",
"The challenge now lies in the third term, which has a quadratic form $\\frac{1}{2}||{\\mathbf {g}}(\\mathbf {\\epsilon })||^2$ , neglecting the normalizing constant.",
"Now assume ${\\mathbf {g}}$ is a $L_0$ -Lipschitz The following flows are all Lipschitz (with Lipschitz activation functions): volume preserving version of [11], [25], [5], [4], [10], etc.. Let $g(\\mathbf {\\epsilon })=\\frac{1}{\\sqrt{2}}||{\\mathbf {g}}(\\mathbf {\\epsilon })||$ .",
"Then $g$ is $L_0/\\sqrt{2}$ -Lipschitz: $\\big |g(\\mathbf {\\epsilon }_1)-g(\\mathbf {\\epsilon }_2)\\big |= &\\frac{1}{\\sqrt{2}}\\big |||{\\mathbf {g}}(\\mathbf {\\epsilon }_1)|| - ||{\\mathbf {g}}(\\mathbf {\\epsilon }_2)||\\big | \\\\& \\le \\frac{1}{\\sqrt{2}}||{\\mathbf {g}}(\\mathbf {\\epsilon }_1) - {\\mathbf {g}}(\\mathbf {\\epsilon }_2)||\\le \\frac{L_0}{\\sqrt{2}}||\\mathbf {\\epsilon }_1-\\mathbf {\\epsilon }_2||.$ This is key in deriving a tail bound on $g^2$ , as Lipschitz functions of canonical Gaussian random variables are sub-Gaussians, meaning they have a tail that decays faster than a Gaussian random variable.",
"The following theorem provides a concentration bound for the empirical average of $g^2$ similar to that of a Chi-square random variable, as $g^2$ (square of a sub-Gaussian) is sub-exponential.",
"Theorem 3 Let $g$ be defined as above with a Lipschitz constant $L=L_0/\\sqrt{2}$ .",
"Let $\\bar{g}^2=\\frac{1}{K}\\sum _{k=1}^K g_k^2$ .",
"Then the following concentration bound holds ${\\mathbb {P}}(\\bar{g}^2-\\mathbb {E}[g^2]>\\epsilon )\\le \\exp \\left(-\\frac{K\\epsilon ^2}{2(4C^2+C\\epsilon )}\\right),$ where $C=\\left(6L^2+\\frac{L}{\\sqrt{\\log 2}}(\\sqrt{d}+||{\\mathbf {g}}^{-1}(\\mathbf {0})||)\\right)^2$ .",
"Note that in practice the empirical KL that we use is inversely scaled by the size of the training set $m$ (see Equation REF ), so the Lipschitz constant can be made small in practice to dominate the dimensionality."
],
[
"Experiments",
"We evaluate our proposed method in the context of two prediction tasks (Section REF ), PAC-Bayes bound minimization (Section REF ) and contextual bandit (Section REF ).",
"For the two prediction tasks, we use the MNIST handwritten digit dataset [30] and CIFAR-10 [27].",
"See Appendix for a detailed description.",
"Table: Test error with LeNet (%) on MNIST and the first 5 classes of CIFAR-10.",
"First 3 columns are from .",
"K-Diag on CIFAR-5 diverged, so we did not include the result.Table: Test error with modified version of VGG16 (%) on CIFAR10.First 4 columns are from .R means regular training and D means training with data augmentation.Table: PAC-Bayes bound estimation: We minimize the Pinsker bound (an upper bound on the McAllester bound) and the Catani bound using different flows, and estimate the McAllester bound at inference time using Newton's method.Table: Cumulative regret incurred by different algorithms on the bandit benchmarks described in .",
"Values reported are the mean over 3 independent trials with standard error of the mean, normalized with respect to the performance of the uniform policy."
],
[
"Classification",
"One benefit of Bayesian neural networks compared to the regular ones is that the trade-off between the prior and the likelihood is a form of regularization.",
"In this section, we evaluate the generalization performance of our method applied to Bayesian neural networks.",
"We consider two architectures: LeNet-5 [30] and a modified version VGG-16 [46] proposed by [53].",
"We first compare to the multiplicative normalizing flow (MNFG) proposed by [32], applying our method to LeNet-5 (see Table REF ).",
"Our Diag matches the performance of their FFG (fully factorized Gaussian).",
"K-Diag outperforms Diag in this case, perhaps due to the smaller number of parameters which makes it easier to optimize.",
"K-Nonlinear yields the best generalization error in this case.",
"On the CIFAR-5 experiment (we take the first 5 classes of CIFAR-10), our methods are on par with MNFG.",
"Second, we compare with the noisy K-FAC proposed by [53], applying our methods to the larger architecture VGG-16 (see Table REF ).",
"Noisy K-FAC applies an approximate natural gradient method.",
"Despite this advantage, our methods (K-Linear and K-Nonlinear) have simiar prediction accuracy in the regular setup.",
"We also include the results of data augmentation with horizontal flip and random crop where K-Nonlinear outperforms all the other methods."
],
[
"PAC Bayes bound minimization",
"For the PAC-Bayes bound estimation, we minimize Equation REF .",
"We follow the recipe of [12].",
"We upper bound the zero-one loss by cross-entropy divided by $\\log |{\\mathcal {Y}}|$ (where $|{\\mathcal {Y}}|$ is the number of classes) to make the upper bound tight.",
"We set the prior to be ${\\mathcal {N}}(\\Theta _0,\\lambda {\\mathbf {I}})$ , where $\\Theta _0$ is the initial value of the parameters, and apply a union bound to tune the prior variance $\\lambda $ .",
"We also tune the $\\beta $ coefficient as a parameter during training We are allowed to do so since we treat Equation REF as an optimization objective, rather than report it as a bound.",
"We report the McAllester bound, which holds for any $q$ , even if it depends on $\\beta $ ., and report the McAllester bound for comparison (since it is the tightest).",
"For more details, see [12] for reference.",
"We test with a multi-layer perceptron with 1 or 2 hidden layers with 600 neurons and LeNet-5, evaluated on the MNIST dataset (see Table REF ).",
"For further clarification, we follow the steps of [12] by minimizing the McAllester bound, using Pinsker's inequality to bound the inverse of the Bernoulli KL (which we call the Pinsker bound).",
"Since this bound has a square root in the complexity term, we can only use the Gaussian family with an analytic form of the KL.",
"The result we have is slightly looser than [12] since we have a 10-class problem and they deal with a binary version of MNIST.",
"We see that the bound can indeed be improved by capturing the correlation among the parameters.",
"We then compare to minimizing the Catoni bound, which is slightly looser since the linear relationship between the empirical risk and the KL term penalizes the latter more when the KL is larger.",
"However, by modelling the non-linear dependencies, K-Nonlinear clearly outperforms the other methods (even compared to the ones minimizing the Pinsker bound).",
"This indicates there exists a considerable amount of structure in the parameter space that may explain the gap between the test error and the generalization bound.",
"We also notice that, despite the linear relationship, the Catoni bound focuses more on the complexity term than the ELBO.",
"For example, the empirical risks of LeNet-5 in Table REF are much higher compared to the test loss of Table REF .",
"The reasons are: (1) the optimal $\\beta $ in Equation REF is larger than 1 (depending on the relative value of the KL), and (2) to properly upper bound the zero-one loss, we scale down the cross-entropy loss by $\\log |{\\mathcal {Y}}|$ during optimization.",
"This means a learning algorithm based on a tight PAC-Bayes risk bound cannot overfit by much; see [13] for a recent demonstration.",
"A smaller value of $\\beta $ could bring down the risk and empirical risk, resulting in a looser bound but usually better test performance.",
"This trade-off between test set performance and the tightness of the bound is a general issue for generalization bounds.",
"One reason for the interest in PAC-Bayes bounds is that their optimization leads to training algorithms with generalization guarantees.",
"The bounds, however, are considerably looser than held-out estimates.",
"A recent work by [44] shows PAC-Bayes bound can potentially be made tight; however, we could not reproduce the results and the best bound using Gaussian prior was around 40% in their setting.",
"Our work produces much tighter bounds by building flexible families of distributions on neural network weight matrices."
],
[
"Contextual bandit",
"Uncertainty modeling lies at the heart of the exploration-exploitation dilemma in sequential decision-making.",
"In order to maximize its collected cumulative rewards, an agent should trade off exploring different actions and gaining more knowledge about the reward estimate vs. exploiting the current estimate and allocating resources to the actions that are likely rewarding.",
"Thompson sampling (TS) [48] is one the popular approaches that deals with the latter trade-off by maintaining posterior distribution over reward models and randomizing actions on the basis of their probability of being optimal.",
"In this section, we investigate the effectiveness of our proposed method for performing an approximate Thompson sampling in the particular setting of contextual bandits.",
"In the latter setting, at each time $t = 1 \\ldots T$ , the agent sees a $d$ -dimensional context $X_t$ , selects one of the $k$ available actions, $a_t$ , and earns a reward $r_t$ generated by the environment.",
"The agent aims to minimize its cumulative regret defined as $R = \\mathbb {E}[\\sum _{t=1}^T r^{\\star }_t - r_t]$ where $r^\\star _t$ is the highest expected reward given the context $X_t$ and the expectation is over the randomness of both environment and the agent's choice of actions.",
"We compare different methods on a range of real-world bandit problems introduced by [43].",
"We train the models every 50 time steps for 200 iterations using a batch-size of 512.",
"We ran each experiment with 3 different random seeds and we report the means and standard errors of cumulative regret normalized with respect to the uniform baseline in the table REF .",
"We include the functional variational Bayesian neural networks (fBNN), recently introduced by [47] as a baseline, and we use their open sourced implementation of fBNN in the bandit setting.",
"From table REF , we see that across the 6 bandit problems, our proposed method (K-Linear and K-Nonlinear) provides competitive and consistent results.",
"They outperform other baselines in 4 problems out of 6."
],
[
"Conclusion",
"In this work, we present the Kronecker Flow, a flow-based method to induce complex distribution inspired by the Kronecker product.",
"Our methods scale to larger architectures such as VGG-16 since it takes advantage of the shape of the parameters.",
"We demonstrate our methods work better than vanilla Kronecker product with diagonal matrices on multiple setups, including classification and approximate Thompson sampling in contexual bandit, and prove to be competitive with existing methods in the Bayesian neural network literature.",
"We are also the first to apply flow-based methods to obtain a tighter numerical generalization bound.",
"Our work shows that the dependencies among network parameters constitute a non-negligible portion of the gap between risk and PAC-Bayes generalization bound."
],
[
"Acknowledgement",
"CWH would like to thank Kris Sankaran for pointing to the TIS inequality for Gaussian concentration, which is a key component in deriving the tail bound on Lipschitz flows.",
"Law of the unconscious statistician Let $(\\Omega , {\\mathcal {F}}, {\\mathbb {P}})$ be our probability space.",
"Let $\\mathbf {\\epsilon }\\in \\mathbb {R}^d$ be a random variable following the (Lebesgue) density $q_0(\\mathbf {\\epsilon })=\\frac{\\mathrm {d}\\mathbf {\\epsilon }_*{\\mathbb {P}}}{\\mathrm {d}\\mu }$ and $\\mathbf {\\epsilon }_*{\\mathbb {P}}$ being its pushforward measure, and write $\\Theta ={\\mathbf {g}}_\\phi (\\mathbf {\\epsilon })\\in \\mathbb {R}^d$ with $q_\\phi =\\frac{\\mathrm {d}({\\mathbf {g}}_\\phi \\circ \\mathbf {\\epsilon })_*{\\mathbb {P}}}{\\mathrm {d}\\mu }$ being its density and $({\\mathbf {g}}_\\phi \\circ \\mathbf {\\epsilon })_*{\\mathbb {P}}$ being its pushforward measure, and $A=\\hat{R}_D(\\Theta )+\\log q_\\phi (\\Theta )\\in \\mathbb {R}$ .",
"Then $\\mathbb {E}[A] &= \\int _{\\mathbb {R}^d} A \\,\\,\\mathrm {d}({\\mathbf {g}}_\\phi \\circ \\mathbf {\\epsilon })_*{\\mathbb {P}}&&= \\int _{\\mathbb {R}^d} \\left(\\hat{R}_D(\\Theta ) + \\log q_\\phi (\\Theta ) \\right) q_\\phi (\\Theta ) \\,\\mathrm {d}\\Theta \\\\&= \\int _{\\mathbb {R}^d} A\\circ \\Theta \\,\\,\\mathrm {d}\\mathbf {\\epsilon }_*{\\mathbb {P}}&&= \\int _{\\mathbb {R}^d} \\left(\\hat{R}_D({\\mathbf {g}}_\\phi (\\mathbf {\\epsilon })) + \\log q_\\phi ({\\mathbf {g}}_\\phi (\\mathbf {\\epsilon })) \\right) q_0(\\mathbf {\\epsilon }) \\,\\mathrm {d}\\mathbf {\\epsilon }\\\\&{}&&=\\int _{\\mathbb {R}^d} \\left(\\hat{R}_{D}({\\mathbf {g}}_\\phi (\\mathbf {\\epsilon })) + \\log q_0(\\mathbf {\\epsilon }) - \\log \\left|\\det \\frac{\\partial {\\mathbf {g}}_\\phi (\\mathbf {\\epsilon })}{\\partial \\mathbf {\\epsilon }}\\right|\\right)q_0(\\mathbf {\\epsilon })\\,\\mathrm {d}\\mathbf {\\epsilon }$ Derivation and interpretation of Property REF We first derive Property REF algebraically, and give an interpretation that can be genralized to higher dimensional tensor operation.",
"Recall that we have the following givens: Assume ${\\mathbf {E}}_{ij}\\overset{\\textnormal {i.i.d.",
"}}{\\sim } {\\mathcal {N}}(0,1)$ is a $n\\times p$ random Gaussian matrix.",
"Assume ${\\mathbf {A}}\\in \\mathbb {R}^{n\\times n}$ and ${\\mathbf {B}}\\in \\mathbb {R}^{p\\times p}$ .",
"${\\mathbf {S}}\\in \\mathbb {R}^{n\\times p}_{>0}$ .",
"${\\mathbf {M}}\\in \\mathbb {R}^{n\\times p}$ .",
"If we rescale ${\\mathbf {E}}$ elementwise by ${\\mathbf {S}}$ before inducing the column-wise and row-wise correlation, we have: (superscript is Hadamard power) $\\Sigma :=\\mathrm {Var}(\\textnormal {vec}({\\mathbf {M}}+ {\\mathbf {A}}({\\mathbf {E}}\\circ {\\mathbf {S}}){\\mathbf {B}}))&=\\mathrm {Var}( ({\\mathbf {B}}^\\top \\otimes {\\mathbf {A}})\\textnormal {vec}({\\mathbf {E}}\\circ {\\mathbf {S}}))\\\\&=({\\mathbf {B}}^\\top \\otimes {\\mathbf {A}}) \\mathop {\\mathrm {diag}}\\nolimits (\\textnormal {vec}({\\mathbf {S}}^2)) ({\\mathbf {B}}^\\top \\otimes {\\mathbf {A}})^\\top \\\\&=({\\mathbf {B}}^\\top \\otimes {\\mathbf {A}}) \\mathop {\\mathrm {diag}}\\nolimits (\\textnormal {vec}({\\mathbf {S}}^2)) ({\\mathbf {B}}\\otimes {\\mathbf {A}}^\\top )$ If ${\\mathbf {S}}$ is a matrix of ones, the RHS equals $({\\mathbf {B}}^\\top \\otimes {\\mathbf {A}}) ({\\mathbf {B}}\\otimes {\\mathbf {A}}^\\top )= ({\\mathbf {B}}^\\top {\\mathbf {B}})\\otimes ({\\mathbf {A}}{\\mathbf {A}}^\\top )$ , which is the covariance of the matrix normal.",
"Generally, ${\\mathbf {S}}$ might not be a matrix of ones.",
"But we can still compute the determinant and trace of the covariance matrix (useful in computing KL): $\\det (\\Sigma )=\\det ({\\mathbf {A}})^{2p}\\det ({\\mathbf {B}})^{2n}\\prod _{ij}{\\mathbf {S}}_{ij}^2$ $\\textnormal {Tr}(\\Sigma )&=\\sum _i \\Sigma _{ii} \\\\&=\\sum _i \\sum _{j} ({\\mathbf {B}}^\\top \\otimes {\\mathbf {A}})_{ij}\\textnormal {vec}({\\mathbf {S}}^2)_j ({\\mathbf {B}}\\otimes {\\mathbf {A}}^\\top )_{ji}\\\\&=\\sum _i \\sum _{j} ({\\mathbf {B}}^{2\\top }\\otimes {\\mathbf {A}}^2)_{ij}\\textnormal {vec}({\\mathbf {S}}^2)_j\\\\&=\\sum _i\\left( ({\\mathbf {B}}^{2\\top }\\otimes {\\mathbf {A}}^2)\\textnormal {vec}({\\mathbf {S}}^2) \\right)_i\\\\&=\\sum _{ij}\\left({\\mathbf {A}}^2{\\mathbf {S}}^2{\\mathbf {B}}^2\\right)_{ij}$ Interpretation of the determinant and trace.",
"The determinant measures the change in volume due to the linear map.",
"Since each operation (elementwise multiplication with ${\\mathbf {S}}$ , left-multiplication with ${\\mathbf {A}}$ , and right-multiplication with ${\\mathbf {B}}$ ) is an invertible map, the determinant of the composition is a product of determinants.",
"After elementwise multiplication with ${\\mathbf {S}}$ (hence $\\prod {\\mathbf {S}}_{ij}$ )The power 2 comes from the fact that we are looking at the determinant of the covariance.",
"Direct computation of the likelihood involves $\\frac{1}{2}\\log \\det (\\Sigma )$ , which is equivalent to the log-determinant of the invertible map., we apply the same linear map ${\\mathbf {A}}$ to the columns of $({\\mathbf {E}}\\circ {\\mathbf {S}})$ ; in vector form, this corresponds to left-multiplication with a block diagonal of $p$ ${\\mathbf {A}}$ 's, hence $\\det ({\\mathbf {A}})^{p}$ .",
"The same reasoning explains $\\det ({\\mathbf {B}})^n$ .",
"The trace of the covariance can be written as $\\textnormal {Tr}(\\Sigma )=\\sum _{ij}\\mathrm {Var}({\\mathbf {W}}_{ij})$ , i.e.",
"the sum of marginal variances.",
"Each of the ${\\mathbf {W}}_{ij}$ is a linear combination of the entries of ${\\mathbf {E}}$ , which have unit variance and are uncorrelated, so by the additive property of variance of sum of uncorrelated random variables and the quadratic scaling property of variance, $\\mathrm {Var}({\\mathbf {W}}_{ij})=A_{i:}^2{\\mathbf {S}}^2{\\mathbf {B}}_{:j}^2$ .",
"Connection to triangular maps.",
"Much of the recent work on normalizing flows has been dedicated to inverse autoregressive transformations [25], [21], [37], [10], [24], as they are general enough to induce any density function [22], [7], [52].",
"When such transformations are used for ${\\mathbf {g}}_A$ and ${\\mathbf {g}}_B$ , the overall transformation ${\\mathbf {G}}$ is also a triangle map, since ${\\mathbf {G}}({\\mathbf {E}})_{ij}$ depends on ${\\mathbf {E}}_{i^{\\prime }j^{\\prime }}$ for $i^{\\prime }\\le i$ and $j^{\\prime }\\le j$ .",
"Such a function has some “blind spots” similar to the ones discovered by [51].",
"One avenue for improvement is to design a transformation that increase the connectivity.",
"Another avenue for improvement is to condition each (row-wise or column-wise) transformation on a learnable embedding of the row/column, such that each row/column is transformed by a slightly different function than another We try this idea in the preliminary stage of the project, but find it harder to optimize.",
"This is potentially due to the extra parameters that have to be learned.. Tail bound of empirical KL We begin with some preliminaries and lemmas in Section REF , and prove the main result in Section REF .",
"Basic tail bounds and Bernstein inequality The tools developed in this section is to translate the coefficients (such as variance) of sub-Gaussian random variables and sub-exponential random variables.",
"We start with the definition of sub-Gaussians: Definition 1 We write $X\\sim \\textnormal {sub}{\\mathcal {N}}(L^2)$ if $X$ is a random variable satisfying ${\\mathbb {P}}(|X|>t)\\le 2\\exp \\left(-\\frac{t^2}{2L^2}\\right)$ We write $\\Gamma (\\cdot )$ as the Gamma function: $\\Gamma (z)=\\int _0^\\infty e^{-u}u^{z-1}\\mathrm {d}u$ .",
"Note that for positive integers $z$ , $\\Gamma (z)=(z-1)!$ .",
"The following lemma gives an upper-bound on the moments of a sub-Gaussian.",
"Lemma 4 For $X\\sim \\textnormal {sub}{\\mathcal {N}}(L^2)$ , for any integer $p\\ge 1$ , $\\mathbb {E}[|X|^p]\\le (2L^2)^{p/2}p\\Gamma (p/2)$ .",
"Since $|X|^p$ is non-negative, similar to Lemma REF , we have $\\mathbb {E}[|X|^p]&= \\int _{0}^\\infty {\\mathbb {P}}(|X|^p\\ge s)\\mathrm {d}s=\\int _0^\\infty {\\mathbb {P}}(|X|\\ge t)pt^{p-1} \\mathrm {d}t \\\\&\\le 2p\\int _0^\\infty e^{-{t^2}/{2L^2}}t^{p-1} \\mathrm {d}t=\\le p(2L^2)^{p/2}\\int _0^\\infty e^{-u}u^{p/2-1}\\mathrm {d}u = p(2L^2)^{p/2}\\Gamma (p/2)$ where we let $s=t^p$ and $u=t^2/2L^2$ .",
"The following definition is the main tool for translating the coefficients.",
"Definition 2 Let $X$ be a random variable.",
"For integer $k\\ge 1$ , define the $\\psi _k$ -Orlicz norm as $||X||_{\\psi _k}:=\\inf \\lbrace t>0:\\mathbb {E}[\\exp (|X|^k/t^k)]\\le 2\\rbrace $ i.e, the smallest constant $t>0$ for which the super-exponential moment of $X^k/t^k$ is bounded by 2.",
"The Orlicz norm is infinity if there's no finite $t$ for which $\\mathbb {E}[\\exp (|X|^k/t^k)]$ exists.",
"It is easy to verify that the Orlicz norm is indeed a norm.",
"We call $||\\cdot ||_{\\psi _2}$ the sub-Gaussian norm, and $||\\cdot ||_{\\psi _1}$ the sub-exponential norm.",
"Note that $||X^2||_{\\psi _1} = ||X||_{\\psi _2}^2$ .",
"The following lemma upper bounds the sub-Gaussian norm by its variance.",
"Lemma 5 If $X\\sim \\textnormal {sub}{\\mathcal {N}}(L^2)$ , $||X||_{\\psi _2}\\le 6L^2$ .",
"By power series expansion of the exponential function, $\\mathbb {E}[e^{cX^2}]=1+\\sum _{p=1}^\\infty \\frac{c^p\\mathbb {E}[X^{2p}]}{p!",
"}\\le 1+\\sum _{p=1}^\\infty \\frac{c^p}{p!}",
"2(2L^2)^pp!=1+2\\sum _{p=1}^\\infty (2cL^2)^p$ where we used Lemma REF for the inequality.",
"The RHS converges and is equal to 2 if $c=1/6L^2$ .",
"Thus, $||X||_{\\psi _2}\\le 6L^2$ .",
"The following lemma gives an upper bound on the moments of sub-exponential random variables.",
"Lemma 6 If for some $C>0$ , $\\mathbb {E}[\\exp (|X|/C)]\\le 2$ , then $\\mathbb {E}[|X|^p]\\le 2C^pp!$ .",
"By Markov's inequality, ${\\mathbb {P}}(|X|>t)\\le \\frac{\\mathbb {E}[\\exp (|X|/C)]}{\\exp (t/C)}\\le 2e^{-t/C}$ For $p\\in {\\mathbb {Z}}^+$ , since $|X|^p$ is non-negative, $\\mathbb {E}[|X|^p] &= \\int _{0}^\\infty {\\mathbb {P}}(|X|^p\\ge s)ds= \\int _{0}^\\infty {\\mathbb {P}}(|X|\\ge t)pt^{p-1}dt \\\\&\\le 2p\\int _{0}^\\infty e^{-t/C}t^{p-1}dt =2pC^p\\int _{0}^\\infty e^{-u}u^{p-1} du=2pC^p\\Gamma (p)=2C^pp!$ where we let $s=t^p$ and $u=t/C$ .",
"Finally, we derive a concentration bound for sub-exponential random variables.",
"Theorem 7 (Bernstein's inequality for sub-exponential random variables) Let $(X_i)_{i\\in [n]}$ be independent real-valued random variables satisfying $\\mathbb {E}[\\exp (|X|/C)]\\le 2$ for some $C>0$ , with mean $\\mu _X=\\mathbb {E}[X]$ , and let $\\bar{X}=\\frac{1}{n}\\sum _{i=1}^nX_i$ .",
"Then, for any $\\epsilon >0$ , the following concentration bound holds: ${\\mathbb {P}}(\\bar{X}-\\mu _X > \\epsilon ) \\le \\exp \\left(-\\frac{n\\epsilon ^2}{2(4C^2+C\\epsilon )}\\right)$ Let $\\nu =4nC^2$ and $c=C$ .",
"Then by Lemma REF , $\\sum _{i=1}^n\\mathbb {E}[X_i^2]\\le n\\cdot 4C^2=\\nu $ and for integers $p>2$ : $\\sum _{i=1}^n \\mathbb {E}[|X_i|^p] \\le 2nC^pp!",
"= \\nu C^{p-2}p!/2 = \\nu c^{p-2}p!/2$ .",
"Then by Corollary 2.11 of [8], we have ${\\mathbb {P}}(\\bar{X}-\\mu _X > \\epsilon )={\\mathbb {P}}\\left(\\sum _{i=1}^n(X_i-\\mu _X)>n\\epsilon \\right)\\le \\exp \\left(-\\frac{n\\epsilon ^2}{2(4C^2+C\\epsilon )}\\right)$ Proof of Theorem REF Since $\\bar{g}(\\mathbf {\\epsilon }):=g(\\mathbf {\\epsilon })-\\mathbb {E}[g(\\mathbf {\\epsilon })]$ is $L$ -Lipschitz, according to Theorem 5.5 and 5.6 of [8], $\\bar{g}\\sim \\textnormal {sub}{\\mathcal {N}}(L^2)$ .",
"And we have that $||g^2||_{\\psi _1}=||g||_{\\psi _2}^2=||\\bar{g}+\\mathbb {E}[g]||_{\\psi _2}^2\\le \\left(||\\bar{g}||_{\\psi _2} + \\frac{\\mathbb {E}[g]}{\\sqrt{\\log 2}}\\right)^2$ due to triangle inequality of the norm.",
"Now since $g$ is $L$ -Lipschitz, its expectation can be bounded by $\\mathbb {E}[g]=\\mathbb {E}\\left[\\frac{1}{\\sqrt{2}}||{\\mathbf {g}}(\\mathbf {\\epsilon })-\\mathbf {0}||\\right]\\le L\\mathbb {E}\\left[||\\mathbf {\\epsilon }-{\\mathbf {g}}^{-1}(\\mathbf {0})||\\right]\\le L(\\mathbb {E}\\left[||\\mathbf {\\epsilon }||\\right]+||{\\mathbf {g}}^{-1}(\\mathbf {0})||) $ Since $\\mathbf {\\epsilon }$ is standard-normally distributed, $||\\mathbf {\\epsilon }||$ follows the chi distribution with $d$ degrees of freedom, which has an expectation that can be upper-bounded using Gautschi's inequality (using Wendel's version of the upper bound): $\\mathbb {E}[||\\mathbf {\\epsilon }||]=\\sqrt{2}\\frac{\\Gamma ((d+1)/2)}{\\Gamma (d/2)}\\le \\sqrt{2}\\left(\\frac{d}{2}\\right)^{1/2}=\\sqrt{d}$ Combining the above and using Lemma REF , we have $||g^2||_{\\psi _1}\\le \\left(6L^2+\\frac{L}{\\sqrt{\\log 2}}(\\sqrt{d}+||{\\mathbf {g}}^{-1}(\\mathbf {0})||)\\right)^2$ Setting $C$ to be the RHS and applying Theorem REF yield the desired result.",
"Experimental Details For the predictive tasks (Section REF ), we use a cosine annealing schedule for the learning rate, scaling down to 0.01 of the initial learning rate, and pretrain a deterministic network for 10 epochs using the Adam optimizer with a learning rate of 0.001, to initialize the mean of the Gaussian $q_0$ , and train $q$ for 200 epochs.",
"LeNet-5 MNIST.",
"We use a linear annealing schedule of the $\\beta $ coefficient (from 0 back to 1) for 50,000 iterations.",
"We use the Adam optimizer with a learning rate of 0.0005.",
"The result we get for K-Linear uses polyak averaging with exponential decay coefficient 0.995.",
"We use the volume preserving version of RealNVP for the K-Nonlinear.",
"We use the standard Gaussian prior for $p$ .",
"LeNet-5 CIFAR-5 We use the same architecture as [32] (192 convolutional kernels and 1,000 hidden units for the fully connected layers).",
"We use a linear annealing schedule of the $\\beta $ coefficient (from 0 back to 1) for 20,000 iterations for Diag, and no annealing for K-Linear and K-Nonlinear.",
"We use the Adam optimizer with a learning rate of 0.0003, 0.0003, 0.0005 for Diag, K-Linear and K-Nonlinear, respectively.",
"We use the volume preserving version of RealNVP for K-Nonlinear.",
"We use the standard Gaussian prior for $p$ .",
"VGG-16 CIFAR-10 We use the modified version of VGG-16 proposed by [53].",
"We use a learning rate of 0.0005 for all experiments but K-Nonlinear in the regular setup (where we use 0.001).",
"We use the isotropic Gaussian prior with variance being 0.1, and set $\\beta $ to be [0.5, 0.1, 0.1, 0.5] in the regular setup and [0.5, 0.1, 0.1, 0.1] in the data augmented setup for Diag, K-Diag, K-Linear, and K-Nonlinear, respectively.",
"We use the volume preserving version of RealNVP for the K-Nonlinear.",
"PAC-Bayes MLP We follow the same steps as [12], except we did not discretize the prior variance after tuning.",
"In practice this does not affect the bound much.",
"We also did not initialize the mean of $q_0$ in our setup using SGD for our experiments.",
"We train the stochastic network for 300 epochs, with a learning rate of 0.002.",
"The bound holds with probability at least 0.965 over the choice of prior and the training set.",
"The $b$ and $c$ coefficients in [12] are set as 100 and 0.1.",
"We use the volume preserving version of IAF for the K-Nonlinear.",
"PAC-Bayes LeNet-5 The same setup as PAC-Bayes MLP, except with polyak averaging with coefficient 0.995.",
"We use the volume preserving version of IAF for the K-Nonlinear.",
"Bandit Benchmark All the models share the same architechture: one hidden layer with 50 units.",
"We use the volume preserving version of RealNVP for K-NonLinear.",
"We train models every 50 time steps for 200 training iterations using a batch-size of 512.",
"Table: Description of bandit problem: number of actions and number of contexts used for experiments.",
"Comparing to benchmark, we restrict ourself to 50000 contexts for Covertype instead of 150000 contexts.Table: Additional results: Cumulative regret incurred by different algorithms on the bandit benchmarks described in .",
"Values reported are the mean over 5 independent trials with standard error of the mean, normalized with respect to the performance of the uniform policy.",
"We use the same hyperparameters for different algorithms without any finetuning: learning rate = 0.0001 and 100 training epochs.Figure: Posterior predictive (with 20 samples) on rotated MNIST digit 3 and 5.",
"Top left: Diag; top right: K-Diag; bottom left: K-Linear; bottom right: K-Nonlinear."
],
[
"Law of the unconscious statistician",
"Let $(\\Omega , {\\mathcal {F}}, {\\mathbb {P}})$ be our probability space.",
"Let $\\mathbf {\\epsilon }\\in \\mathbb {R}^d$ be a random variable following the (Lebesgue) density $q_0(\\mathbf {\\epsilon })=\\frac{\\mathrm {d}\\mathbf {\\epsilon }_*{\\mathbb {P}}}{\\mathrm {d}\\mu }$ and $\\mathbf {\\epsilon }_*{\\mathbb {P}}$ being its pushforward measure, and write $\\Theta ={\\mathbf {g}}_\\phi (\\mathbf {\\epsilon })\\in \\mathbb {R}^d$ with $q_\\phi =\\frac{\\mathrm {d}({\\mathbf {g}}_\\phi \\circ \\mathbf {\\epsilon })_*{\\mathbb {P}}}{\\mathrm {d}\\mu }$ being its density and $({\\mathbf {g}}_\\phi \\circ \\mathbf {\\epsilon })_*{\\mathbb {P}}$ being its pushforward measure, and $A=\\hat{R}_D(\\Theta )+\\log q_\\phi (\\Theta )\\in \\mathbb {R}$ .",
"Then $\\mathbb {E}[A] &= \\int _{\\mathbb {R}^d} A \\,\\,\\mathrm {d}({\\mathbf {g}}_\\phi \\circ \\mathbf {\\epsilon })_*{\\mathbb {P}}&&= \\int _{\\mathbb {R}^d} \\left(\\hat{R}_D(\\Theta ) + \\log q_\\phi (\\Theta ) \\right) q_\\phi (\\Theta ) \\,\\mathrm {d}\\Theta \\\\&= \\int _{\\mathbb {R}^d} A\\circ \\Theta \\,\\,\\mathrm {d}\\mathbf {\\epsilon }_*{\\mathbb {P}}&&= \\int _{\\mathbb {R}^d} \\left(\\hat{R}_D({\\mathbf {g}}_\\phi (\\mathbf {\\epsilon })) + \\log q_\\phi ({\\mathbf {g}}_\\phi (\\mathbf {\\epsilon })) \\right) q_0(\\mathbf {\\epsilon }) \\,\\mathrm {d}\\mathbf {\\epsilon }\\\\&{}&&=\\int _{\\mathbb {R}^d} \\left(\\hat{R}_{D}({\\mathbf {g}}_\\phi (\\mathbf {\\epsilon })) + \\log q_0(\\mathbf {\\epsilon }) - \\log \\left|\\det \\frac{\\partial {\\mathbf {g}}_\\phi (\\mathbf {\\epsilon })}{\\partial \\mathbf {\\epsilon }}\\right|\\right)q_0(\\mathbf {\\epsilon })\\,\\mathrm {d}\\mathbf {\\epsilon }$"
],
[
"Derivation and interpretation of Property ",
"We first derive Property REF algebraically, and give an interpretation that can be genralized to higher dimensional tensor operation.",
"Recall that we have the following givens: Assume ${\\mathbf {E}}_{ij}\\overset{\\textnormal {i.i.d.",
"}}{\\sim } {\\mathcal {N}}(0,1)$ is a $n\\times p$ random Gaussian matrix.",
"Assume ${\\mathbf {A}}\\in \\mathbb {R}^{n\\times n}$ and ${\\mathbf {B}}\\in \\mathbb {R}^{p\\times p}$ .",
"${\\mathbf {S}}\\in \\mathbb {R}^{n\\times p}_{>0}$ .",
"${\\mathbf {M}}\\in \\mathbb {R}^{n\\times p}$ .",
"If we rescale ${\\mathbf {E}}$ elementwise by ${\\mathbf {S}}$ before inducing the column-wise and row-wise correlation, we have: (superscript is Hadamard power) $\\Sigma :=\\mathrm {Var}(\\textnormal {vec}({\\mathbf {M}}+ {\\mathbf {A}}({\\mathbf {E}}\\circ {\\mathbf {S}}){\\mathbf {B}}))&=\\mathrm {Var}( ({\\mathbf {B}}^\\top \\otimes {\\mathbf {A}})\\textnormal {vec}({\\mathbf {E}}\\circ {\\mathbf {S}}))\\\\&=({\\mathbf {B}}^\\top \\otimes {\\mathbf {A}}) \\mathop {\\mathrm {diag}}\\nolimits (\\textnormal {vec}({\\mathbf {S}}^2)) ({\\mathbf {B}}^\\top \\otimes {\\mathbf {A}})^\\top \\\\&=({\\mathbf {B}}^\\top \\otimes {\\mathbf {A}}) \\mathop {\\mathrm {diag}}\\nolimits (\\textnormal {vec}({\\mathbf {S}}^2)) ({\\mathbf {B}}\\otimes {\\mathbf {A}}^\\top )$ If ${\\mathbf {S}}$ is a matrix of ones, the RHS equals $({\\mathbf {B}}^\\top \\otimes {\\mathbf {A}}) ({\\mathbf {B}}\\otimes {\\mathbf {A}}^\\top )= ({\\mathbf {B}}^\\top {\\mathbf {B}})\\otimes ({\\mathbf {A}}{\\mathbf {A}}^\\top )$ , which is the covariance of the matrix normal.",
"Generally, ${\\mathbf {S}}$ might not be a matrix of ones.",
"But we can still compute the determinant and trace of the covariance matrix (useful in computing KL): $\\det (\\Sigma )=\\det ({\\mathbf {A}})^{2p}\\det ({\\mathbf {B}})^{2n}\\prod _{ij}{\\mathbf {S}}_{ij}^2$ $\\textnormal {Tr}(\\Sigma )&=\\sum _i \\Sigma _{ii} \\\\&=\\sum _i \\sum _{j} ({\\mathbf {B}}^\\top \\otimes {\\mathbf {A}})_{ij}\\textnormal {vec}({\\mathbf {S}}^2)_j ({\\mathbf {B}}\\otimes {\\mathbf {A}}^\\top )_{ji}\\\\&=\\sum _i \\sum _{j} ({\\mathbf {B}}^{2\\top }\\otimes {\\mathbf {A}}^2)_{ij}\\textnormal {vec}({\\mathbf {S}}^2)_j\\\\&=\\sum _i\\left( ({\\mathbf {B}}^{2\\top }\\otimes {\\mathbf {A}}^2)\\textnormal {vec}({\\mathbf {S}}^2) \\right)_i\\\\&=\\sum _{ij}\\left({\\mathbf {A}}^2{\\mathbf {S}}^2{\\mathbf {B}}^2\\right)_{ij}$"
],
[
"Interpretation of the determinant and trace.",
"The determinant measures the change in volume due to the linear map.",
"Since each operation (elementwise multiplication with ${\\mathbf {S}}$ , left-multiplication with ${\\mathbf {A}}$ , and right-multiplication with ${\\mathbf {B}}$ ) is an invertible map, the determinant of the composition is a product of determinants.",
"After elementwise multiplication with ${\\mathbf {S}}$ (hence $\\prod {\\mathbf {S}}_{ij}$ )The power 2 comes from the fact that we are looking at the determinant of the covariance.",
"Direct computation of the likelihood involves $\\frac{1}{2}\\log \\det (\\Sigma )$ , which is equivalent to the log-determinant of the invertible map., we apply the same linear map ${\\mathbf {A}}$ to the columns of $({\\mathbf {E}}\\circ {\\mathbf {S}})$ ; in vector form, this corresponds to left-multiplication with a block diagonal of $p$ ${\\mathbf {A}}$ 's, hence $\\det ({\\mathbf {A}})^{p}$ .",
"The same reasoning explains $\\det ({\\mathbf {B}})^n$ .",
"The trace of the covariance can be written as $\\textnormal {Tr}(\\Sigma )=\\sum _{ij}\\mathrm {Var}({\\mathbf {W}}_{ij})$ , i.e.",
"the sum of marginal variances.",
"Each of the ${\\mathbf {W}}_{ij}$ is a linear combination of the entries of ${\\mathbf {E}}$ , which have unit variance and are uncorrelated, so by the additive property of variance of sum of uncorrelated random variables and the quadratic scaling property of variance, $\\mathrm {Var}({\\mathbf {W}}_{ij})=A_{i:}^2{\\mathbf {S}}^2{\\mathbf {B}}_{:j}^2$ ."
],
[
"Connection to triangular maps.",
"Much of the recent work on normalizing flows has been dedicated to inverse autoregressive transformations [25], [21], [37], [10], [24], as they are general enough to induce any density function [22], [7], [52].",
"When such transformations are used for ${\\mathbf {g}}_A$ and ${\\mathbf {g}}_B$ , the overall transformation ${\\mathbf {G}}$ is also a triangle map, since ${\\mathbf {G}}({\\mathbf {E}})_{ij}$ depends on ${\\mathbf {E}}_{i^{\\prime }j^{\\prime }}$ for $i^{\\prime }\\le i$ and $j^{\\prime }\\le j$ .",
"Such a function has some “blind spots” similar to the ones discovered by [51].",
"One avenue for improvement is to design a transformation that increase the connectivity.",
"Another avenue for improvement is to condition each (row-wise or column-wise) transformation on a learnable embedding of the row/column, such that each row/column is transformed by a slightly different function than another We try this idea in the preliminary stage of the project, but find it harder to optimize.",
"This is potentially due to the extra parameters that have to be learned.."
],
[
"Tail bound of empirical KL",
"We begin with some preliminaries and lemmas in Section REF , and prove the main result in Section REF ."
],
[
"Basic tail bounds and Bernstein inequality",
"The tools developed in this section is to translate the coefficients (such as variance) of sub-Gaussian random variables and sub-exponential random variables.",
"We start with the definition of sub-Gaussians: Definition 1 We write $X\\sim \\textnormal {sub}{\\mathcal {N}}(L^2)$ if $X$ is a random variable satisfying ${\\mathbb {P}}(|X|>t)\\le 2\\exp \\left(-\\frac{t^2}{2L^2}\\right)$ We write $\\Gamma (\\cdot )$ as the Gamma function: $\\Gamma (z)=\\int _0^\\infty e^{-u}u^{z-1}\\mathrm {d}u$ .",
"Note that for positive integers $z$ , $\\Gamma (z)=(z-1)!$ .",
"The following lemma gives an upper-bound on the moments of a sub-Gaussian.",
"Lemma 4 For $X\\sim \\textnormal {sub}{\\mathcal {N}}(L^2)$ , for any integer $p\\ge 1$ , $\\mathbb {E}[|X|^p]\\le (2L^2)^{p/2}p\\Gamma (p/2)$ .",
"Since $|X|^p$ is non-negative, similar to Lemma REF , we have $\\mathbb {E}[|X|^p]&= \\int _{0}^\\infty {\\mathbb {P}}(|X|^p\\ge s)\\mathrm {d}s=\\int _0^\\infty {\\mathbb {P}}(|X|\\ge t)pt^{p-1} \\mathrm {d}t \\\\&\\le 2p\\int _0^\\infty e^{-{t^2}/{2L^2}}t^{p-1} \\mathrm {d}t=\\le p(2L^2)^{p/2}\\int _0^\\infty e^{-u}u^{p/2-1}\\mathrm {d}u = p(2L^2)^{p/2}\\Gamma (p/2)$ where we let $s=t^p$ and $u=t^2/2L^2$ .",
"The following definition is the main tool for translating the coefficients.",
"Definition 2 Let $X$ be a random variable.",
"For integer $k\\ge 1$ , define the $\\psi _k$ -Orlicz norm as $||X||_{\\psi _k}:=\\inf \\lbrace t>0:\\mathbb {E}[\\exp (|X|^k/t^k)]\\le 2\\rbrace $ i.e, the smallest constant $t>0$ for which the super-exponential moment of $X^k/t^k$ is bounded by 2.",
"The Orlicz norm is infinity if there's no finite $t$ for which $\\mathbb {E}[\\exp (|X|^k/t^k)]$ exists.",
"It is easy to verify that the Orlicz norm is indeed a norm.",
"We call $||\\cdot ||_{\\psi _2}$ the sub-Gaussian norm, and $||\\cdot ||_{\\psi _1}$ the sub-exponential norm.",
"Note that $||X^2||_{\\psi _1} = ||X||_{\\psi _2}^2$ .",
"The following lemma upper bounds the sub-Gaussian norm by its variance.",
"Lemma 5 If $X\\sim \\textnormal {sub}{\\mathcal {N}}(L^2)$ , $||X||_{\\psi _2}\\le 6L^2$ .",
"By power series expansion of the exponential function, $\\mathbb {E}[e^{cX^2}]=1+\\sum _{p=1}^\\infty \\frac{c^p\\mathbb {E}[X^{2p}]}{p!",
"}\\le 1+\\sum _{p=1}^\\infty \\frac{c^p}{p!}",
"2(2L^2)^pp!=1+2\\sum _{p=1}^\\infty (2cL^2)^p$ where we used Lemma REF for the inequality.",
"The RHS converges and is equal to 2 if $c=1/6L^2$ .",
"Thus, $||X||_{\\psi _2}\\le 6L^2$ .",
"The following lemma gives an upper bound on the moments of sub-exponential random variables.",
"Lemma 6 If for some $C>0$ , $\\mathbb {E}[\\exp (|X|/C)]\\le 2$ , then $\\mathbb {E}[|X|^p]\\le 2C^pp!$ .",
"By Markov's inequality, ${\\mathbb {P}}(|X|>t)\\le \\frac{\\mathbb {E}[\\exp (|X|/C)]}{\\exp (t/C)}\\le 2e^{-t/C}$ For $p\\in {\\mathbb {Z}}^+$ , since $|X|^p$ is non-negative, $\\mathbb {E}[|X|^p] &= \\int _{0}^\\infty {\\mathbb {P}}(|X|^p\\ge s)ds= \\int _{0}^\\infty {\\mathbb {P}}(|X|\\ge t)pt^{p-1}dt \\\\&\\le 2p\\int _{0}^\\infty e^{-t/C}t^{p-1}dt =2pC^p\\int _{0}^\\infty e^{-u}u^{p-1} du=2pC^p\\Gamma (p)=2C^pp!$ where we let $s=t^p$ and $u=t/C$ .",
"Finally, we derive a concentration bound for sub-exponential random variables.",
"Theorem 7 (Bernstein's inequality for sub-exponential random variables) Let $(X_i)_{i\\in [n]}$ be independent real-valued random variables satisfying $\\mathbb {E}[\\exp (|X|/C)]\\le 2$ for some $C>0$ , with mean $\\mu _X=\\mathbb {E}[X]$ , and let $\\bar{X}=\\frac{1}{n}\\sum _{i=1}^nX_i$ .",
"Then, for any $\\epsilon >0$ , the following concentration bound holds: ${\\mathbb {P}}(\\bar{X}-\\mu _X > \\epsilon ) \\le \\exp \\left(-\\frac{n\\epsilon ^2}{2(4C^2+C\\epsilon )}\\right)$ Let $\\nu =4nC^2$ and $c=C$ .",
"Then by Lemma REF , $\\sum _{i=1}^n\\mathbb {E}[X_i^2]\\le n\\cdot 4C^2=\\nu $ and for integers $p>2$ : $\\sum _{i=1}^n \\mathbb {E}[|X_i|^p] \\le 2nC^pp!",
"= \\nu C^{p-2}p!/2 = \\nu c^{p-2}p!/2$ .",
"Then by Corollary 2.11 of [8], we have ${\\mathbb {P}}(\\bar{X}-\\mu _X > \\epsilon )={\\mathbb {P}}\\left(\\sum _{i=1}^n(X_i-\\mu _X)>n\\epsilon \\right)\\le \\exp \\left(-\\frac{n\\epsilon ^2}{2(4C^2+C\\epsilon )}\\right)$"
],
[
"Proof of Theorem ",
"Since $\\bar{g}(\\mathbf {\\epsilon }):=g(\\mathbf {\\epsilon })-\\mathbb {E}[g(\\mathbf {\\epsilon })]$ is $L$ -Lipschitz, according to Theorem 5.5 and 5.6 of [8], $\\bar{g}\\sim \\textnormal {sub}{\\mathcal {N}}(L^2)$ .",
"And we have that $||g^2||_{\\psi _1}=||g||_{\\psi _2}^2=||\\bar{g}+\\mathbb {E}[g]||_{\\psi _2}^2\\le \\left(||\\bar{g}||_{\\psi _2} + \\frac{\\mathbb {E}[g]}{\\sqrt{\\log 2}}\\right)^2$ due to triangle inequality of the norm.",
"Now since $g$ is $L$ -Lipschitz, its expectation can be bounded by $\\mathbb {E}[g]=\\mathbb {E}\\left[\\frac{1}{\\sqrt{2}}||{\\mathbf {g}}(\\mathbf {\\epsilon })-\\mathbf {0}||\\right]\\le L\\mathbb {E}\\left[||\\mathbf {\\epsilon }-{\\mathbf {g}}^{-1}(\\mathbf {0})||\\right]\\le L(\\mathbb {E}\\left[||\\mathbf {\\epsilon }||\\right]+||{\\mathbf {g}}^{-1}(\\mathbf {0})||) $ Since $\\mathbf {\\epsilon }$ is standard-normally distributed, $||\\mathbf {\\epsilon }||$ follows the chi distribution with $d$ degrees of freedom, which has an expectation that can be upper-bounded using Gautschi's inequality (using Wendel's version of the upper bound): $\\mathbb {E}[||\\mathbf {\\epsilon }||]=\\sqrt{2}\\frac{\\Gamma ((d+1)/2)}{\\Gamma (d/2)}\\le \\sqrt{2}\\left(\\frac{d}{2}\\right)^{1/2}=\\sqrt{d}$ Combining the above and using Lemma REF , we have $||g^2||_{\\psi _1}\\le \\left(6L^2+\\frac{L}{\\sqrt{\\log 2}}(\\sqrt{d}+||{\\mathbf {g}}^{-1}(\\mathbf {0})||)\\right)^2$ Setting $C$ to be the RHS and applying Theorem REF yield the desired result."
],
[
"Experimental Details",
"For the predictive tasks (Section REF ), we use a cosine annealing schedule for the learning rate, scaling down to 0.01 of the initial learning rate, and pretrain a deterministic network for 10 epochs using the Adam optimizer with a learning rate of 0.001, to initialize the mean of the Gaussian $q_0$ , and train $q$ for 200 epochs."
],
[
"LeNet-5 MNIST.",
"We use a linear annealing schedule of the $\\beta $ coefficient (from 0 back to 1) for 50,000 iterations.",
"We use the Adam optimizer with a learning rate of 0.0005.",
"The result we get for K-Linear uses polyak averaging with exponential decay coefficient 0.995.",
"We use the volume preserving version of RealNVP for the K-Nonlinear.",
"We use the standard Gaussian prior for $p$ ."
],
[
"LeNet-5 CIFAR-5",
"We use the same architecture as [32] (192 convolutional kernels and 1,000 hidden units for the fully connected layers).",
"We use a linear annealing schedule of the $\\beta $ coefficient (from 0 back to 1) for 20,000 iterations for Diag, and no annealing for K-Linear and K-Nonlinear.",
"We use the Adam optimizer with a learning rate of 0.0003, 0.0003, 0.0005 for Diag, K-Linear and K-Nonlinear, respectively.",
"We use the volume preserving version of RealNVP for K-Nonlinear.",
"We use the standard Gaussian prior for $p$ ."
],
[
"VGG-16 CIFAR-10",
"We use the modified version of VGG-16 proposed by [53].",
"We use a learning rate of 0.0005 for all experiments but K-Nonlinear in the regular setup (where we use 0.001).",
"We use the isotropic Gaussian prior with variance being 0.1, and set $\\beta $ to be [0.5, 0.1, 0.1, 0.5] in the regular setup and [0.5, 0.1, 0.1, 0.1] in the data augmented setup for Diag, K-Diag, K-Linear, and K-Nonlinear, respectively.",
"We use the volume preserving version of RealNVP for the K-Nonlinear."
],
[
"PAC-Bayes MLP",
"We follow the same steps as [12], except we did not discretize the prior variance after tuning.",
"In practice this does not affect the bound much.",
"We also did not initialize the mean of $q_0$ in our setup using SGD for our experiments.",
"We train the stochastic network for 300 epochs, with a learning rate of 0.002.",
"The bound holds with probability at least 0.965 over the choice of prior and the training set.",
"The $b$ and $c$ coefficients in [12] are set as 100 and 0.1.",
"We use the volume preserving version of IAF for the K-Nonlinear."
],
[
"PAC-Bayes LeNet-5",
"The same setup as PAC-Bayes MLP, except with polyak averaging with coefficient 0.995.",
"We use the volume preserving version of IAF for the K-Nonlinear."
],
[
"Bandit Benchmark",
"All the models share the same architechture: one hidden layer with 50 units.",
"We use the volume preserving version of RealNVP for K-NonLinear.",
"We train models every 50 time steps for 200 training iterations using a batch-size of 512.",
"Table: Description of bandit problem: number of actions and number of contexts used for experiments.",
"Comparing to benchmark, we restrict ourself to 50000 contexts for Covertype instead of 150000 contexts.Table: Additional results: Cumulative regret incurred by different algorithms on the bandit benchmarks described in .",
"Values reported are the mean over 5 independent trials with standard error of the mean, normalized with respect to the performance of the uniform policy.",
"We use the same hyperparameters for different algorithms without any finetuning: learning rate = 0.0001 and 100 training epochs."
]
] | 1906.04282 |
[
[
"Leptogenesis after superconformal subcritical hybrid inflation"
],
[
"Abstract We consider an extended version of superconformal subcritical hybrid inflation model by introducing three right-handed neutrinos that have the Majorana mass terms.",
"In the model one of the right-handed sneutrinos plays a role of the inflaton field, and it decays to reheat the universe after inflation.",
"The vacuum expectation value for the waterfall field gives an unconventional pattern of the light neutrino mass matrix, and the neutrino Yukawa couplings that determine the reheating temperature are not constrained by the neutrino oscillation data.",
"Consequently thermal leptogenesis or sneutrino leptogenesis is realized."
],
[
"Introduction",
"Inflation paradigm is strongly supported by the observations of the cosmic microwave background radiation (CMB).",
"Slow-roll scalar field in the early universe is a promising candidate for inflation, and many types of inflation models have been proposed so far.",
"In a theoretical point of view, it would be tempting to ask what the underlying physics or symmetry of the inflaton field is.",
"Supersymmetry (SUSY) might be one of the answers.",
"It protects the flatness of the inflaton direction, which is suitable for inflation.",
"Recently supersymmetric D-term hybrid inflation has been revisited in various point of view.",
"Under shift symmetric Kähler potential [1], subcritical hybrid inflation was found, where inflation continues for subcritical point value of the inflaton field [2], [3].",
"On the other hand, it was shown in Refs.",
"[4], [5] that Starobinsky model [6] emerges in the framework of superconformal supergravity [7], [8], [9], [10].",
"It turned out in the following study that this framework has another new regime of inflation.",
"It was shown that a general class of superconformal $\\alpha $ -attractor model [11], [12] appears in the subcritical regime of inflation, which we call superconformal subcritical hybrid inflation [13].",
"In addition, the energy scale of inflation should coincide with the grand unification scale to be consistent with the Planck observation data, which is the feature found in the subcritical hybrid inflation [2], [3].",
"Namely, the superconformal subcritical hybrid inflation has both features of the superconformal $\\alpha $ -attractor models and the subcritical hybrid inflation.",
"The shift symmetry and superconformality are crucial for them.",
"In this paper we will study the thermal history after the end of superconformal subcritical hybrid inflation.",
"(See Refs.",
"[14], [15] that study the phenomenology of Pati-Salam version of subcritical hybrid inflation.",
"Recently Ref.",
"[16] comprehensively studies the D-term hybrid inflation, including reheating, leptogenesis, and the SUSY breaking mechanism.)",
"For the purpose, we introduce three right-handed neutrinos that interact with the minimal supersymmetric standard model (MSSM) sector.",
"In fermionic sector, the mass matrix for the light neutrinos is given by the seesaw mechanism [17], but it has an unconventional structure.",
"In bosonic sector, on the other hand, it will be shown that one of sneutrinos can play a role of the inflaton field.",
"In addition, baryon asymmetry that is sufficient amount to explain the observed value is generated via leptogenesis [18].",
"This paper is organized as follows.",
"In the next section the model that we consider is described.",
"Then Sec.",
"shows the conditions required for the superconformal subcritical hybrid inflation in this model.",
"Mass matrices of the heavy and light (s)neutrinos, including parametrization of the neutrino Yukawa couplings, are given in Sec.",
", then we discuss the reheating and leptogenesis after inflation in Sec. .",
"Sec.",
"is dedicated to conclusion."
],
[
"The model",
"We consider a model described by the superpotential $W=W_{\\rm MSSM}+W_{\\rm neu}\\,,$ where $W_{\\rm MSSM}$ is the superpotential of the MSSM sector and $W_{\\rm neu}=\\frac{1}{2}M_{ij}N^c_iN^c_j+y_{\\nu \\,ij}N^c_i L_jH_u+\\lambda _i N_i^c S_+ S_-\\,.$ Here $N_i^c$ , $L_i=(\\nu _{Li},l_{Li})^T$ , and $H_u=(H_u^+,H_u^0)^T$ are the chiral superfields of the right-handed neutrinos, left-handed leptons, and up-type Higgs, respectively, and $L_jH_u=\\nu _{Lj}H_u^0-l_{Lj}H_u^+$ .",
"$S_{\\pm }$ are the local U(1) fields with charge $\\pm q$ $(q>0)$ , one of which plays a role of the waterfall field.1We write the superpartners with tilde for the MSSM fields and right-handed neutrinos.",
"For $S_\\pm $ , the same symbols are used for scalar fields while fermionic parts are expressed with tilde.",
"In the current and the next sections, we adopt the unit in which the reduced Planck mass $M_{pl}\\simeq 2.4\\times 10^{18}\\,{\\rm GeV}$ is taken to be unity unless otherwise mentioned.",
"Indices are summed over $i,j=1$ – 3.",
"(We will use the same contraction in the following discussion unless otherwise mentioned.)",
"$M_{ij}$ terms are explicit superconformal breaking terms that are added by phenomenological purpose.",
"While we do not argue their origin, $M_{ij} \\ll 1$ are expected.",
"The Kähler potential, on the other hand, is given by ${\\cal K}=-3\\log \\Omega ^{-2}\\,,$ where $\\Omega ^{-2}=1-\\frac{1}{3}\\left(\\sum _{\\rm MSSM} |I_{\\rm MSSM}|^2+\\sum _i |N_i^c|^2+|S_+|^2+|S_-|^2\\right)-\\sum _i\\frac{\\chi _i}{6}\\left(N_i^{c\\,2}+\\bar{N}_i^{c\\,2}\\right)\\,.$ Here $I_{\\rm MSSM}$ are chiral superfields in the MSSM sector.",
"The last term is superconformal breaking term that is considered in Refs.",
"[4], [10].",
"With the superpotential and Kähler potential, the scalar potential is given by $V_{\\rm tot}=V_F+V_D,$ where $V_F$ and $V_D$ are F- and D-terms, respectively, and given by [4] $&V_F=\\Omega ^4\\left[\\delta ^{\\alpha \\bar{\\alpha }}W_\\alpha W_{\\bar{\\alpha }}+\\frac{1}{\\Delta }|\\delta ^{\\alpha \\bar{\\alpha }}W_\\alpha \\Phi _{\\bar{\\alpha }}-3W|^2\\right]\\,,\\\\&V_D=\\frac{1}{2}g^2\\left[q\\Omega ^2(|S_+|^2-|S_-|^2)-\\xi \\right]^2\\,.$ Here $\\Phi \\equiv -3\\Omega ^{-2}$ and $\\Delta \\equiv \\Phi -\\delta ^{\\alpha \\bar{\\alpha }}\\Phi _\\alpha \\Phi _{\\bar{\\alpha }}$ have been additionally introduced.",
"Subscript in $W$ and $\\Phi $ stands for the field derivative, e.g., $W_\\alpha \\equiv \\partial W/\\partial z^\\alpha $ where $z^\\alpha $ is a chiral superfield.",
"In the D-term, we have introduced the Fayet-Iliopoulos (FI) term $\\xi \\,(>0)$ associated with the U(1).2The origin of the FI term in canonical superconformal supergravity model [10] is discussed in Ref. [4].",
"See also Ref.",
"[16] for recent development.",
"Due to the FI term, $S_+$ has a vacuum expectation value (VEV) at the global minimum, which is obtained as $\\langle S_+\\rangle =\\sqrt{\\xi /q(1+\\tilde{\\xi })}$ with $\\tilde{\\xi }\\equiv \\xi /3q$ .",
"As in Ref.",
"[13], we take $\\chi _i \\le 0$ without loss of generality.",
"In the present model, we impose the following condition: $\\chi _3\\simeq -1,~~ \\chi _{1},\\,\\chi _2\\simeq 0\\,.$ This distinguishes $N^c_3$ from $N^c_{1}$ and $N^c_{2}$ .",
"$\\phi \\equiv \\sqrt{2}{\\rm Re}\\,\\tilde{N}^c_3 $ has an approximate shift symmetry that is explicitly broken by $\\lambda _3\\,(\\ll 1)$ .",
"Then, $\\phi $ is expected to be the inflaton as studied in Ref. [13].",
"In the inflation model, $ s \\equiv \\sqrt{2}|S_+|$ plays a role of the waterfall field.",
"$N^c_1$ and $N^c_2$ , on the other hand, have no such symmetry.",
"Instead there is a freedom to choose any basis for $N^c_1$ and $N^c_2$ with a redefinition of $M_{ij}$ and $\\lambda _i$ due to $\\chi _{1,2}\\simeq 0$ .",
"For simple notation, hereafter we omit subscripts of $\\chi _3$ and $\\lambda _3$ and introduce $m_\\phi $ , which will be identified as the inflaton mass in Sec.",
"REF (with another assumption in Sec.",
"REF ), $\\chi \\equiv \\chi _3, &~~ \\lambda \\equiv \\lambda _3\\,, \\\\m_\\phi \\equiv &\\, \\lambda \\langle S_+ \\rangle \\,.$ It is sometimes convenient to use $\\delta \\chi $ ($0<\\delta \\chi <1$ ) defined by $\\delta \\chi /(1+\\chi ) = -qg^2\\xi /3\\lambda ^2$ .",
"$0<\\delta \\chi <1$ guarantees $\\Omega (\\phi ,s)^2>0$ and $\\phi _{c,0}^2>0$ , which will be defined in Eqs.",
"(REF ) and (REF ), respectively.",
"Then inflation that is consistent with the observations of the CMB is realized in the parameter space [13], $\\lambda \\simeq (0.5\\, {\\rm }{\\rm }\\,1)\\times 10^{-3}\\,, &~~\\xi ^{1/2}\\simeq (3\\, {\\rm }{\\rm }\\,1)\\times 10^{16}\\,{\\rm GeV}\\,,\\nonumber \\\\m_{\\phi } \\simeq (1\\,{\\rm }{\\rm }\\,2)\\times &10^{13}\\,{\\rm GeV}\\,,$ for $q=g=1$ , $\\delta \\chi =0.9$ and the number of $e$ -folds $N_e=55$ – 60, which we take in the later numerical study.3We will estimate the number of $e$ -folds in Sec.",
"REF to confirm this.",
"In the parameter space, e.g., $\\chi \\simeq -1.16$ for $\\lambda =10^{-3}$ , $\\xi ^{1/2}=10^{16}\\,{\\rm GeV}$ , and $\\delta \\chi =0.9$ .",
"The mass terms in the superpotential, however, have a possibility to alter the inflationary path.",
"In the next section, we will derive the conditions in order not to affect the inflationary dynamics."
],
[
"Inflation",
"We define several variables that are used in the following analysis.",
"During inflation, the other fields except for the inflaton and waterfall field are irrelevant.",
"Thus it is convenient to define following potentials, $& V(\\phi ,s)\\equiv V_{\\rm tot}|_{\\sqrt{2}{\\rm Re}\\,\\tilde{N}_3^c=\\phi ,\\,\\sqrt{2}|S_+|=s,\\,{\\rm the\\,others}=0}\\,,\\\\& \\Omega (\\phi ,s)\\equiv \\Omega |_{\\sqrt{2}{\\rm Re}\\,\\tilde{N}_3^c=\\phi ,\\,\\sqrt{2}|S_+|=s,\\,{\\rm the\\,others}=0}\\,.$ Then the critical point value $\\phi _c$ is defined as a field value below which the waterfall field becomes tachyonic.",
"It receives ${\\cal O}(M_{ij}^2)$ corrections as $\\phi _c^2=\\phi _{c,0}^2\\left[1-\\frac{2\\Delta M^2(\\phi _{c,0})}{3\\lambda ^2}\\right]+{\\cal O}(M_{i3}^4)\\,,$ where $\\phi _{c,0}^2&=\\frac{6qg^2\\xi }{3\\lambda ^2+(1+\\chi )qg^2\\xi }\\,,\\\\\\Delta M^2(\\phi ) &= \\sum _{i=1}^3|M_{i3}|^2-\\frac{(1-2\\chi )^2}{24}\\frac{\\phi ^2}{1+\\phi ^2\\chi (1+\\chi )/6}|M_{33}|^2\\,.$ For example, $\\phi _{c,0}\\simeq 18$ or in terms of canonically-normalized field $\\hat{\\phi }_{c,0}\\simeq 11$ defined in Eq.",
"(REF ) for $\\lambda =10^{-3}$ , $\\xi ^{1/2}=10^{16}\\,{\\rm GeV}$ and $\\delta \\chi =0.9$ .",
"This perturbative expansion is valid when4We have checked that ${\\cal O}(M^4_{i3})$ term is irrelevant when Eq.",
"() is satisfied, thus we ignore it in the following discussion.",
"$|M_{13}|^2+|M_{23}|^2 &\\ll 3\\lambda ^2/2\\sim (10^{15}\\,{\\rm GeV})^2\\,, \\\\|M_{33}|^2 &\\ll 2\\lambda ^4/qg^2\\xi \\sim (10^{14}\\,{\\rm GeV})^2\\,.$ It will be checked in this section that the above conditions are satisfied in this inflation model.",
"Finally it is useful to define $\\Psi \\equiv \\frac{\\Omega (\\phi ,0)\\phi }{\\Omega (\\phi _{c,0},0)\\phi _{c,0}}=\\frac{\\Omega (\\phi ,0)\\phi }{\\sqrt{2qg^2\\xi /\\lambda ^2}}\\,,$ when potential is expressed in terms of canonically-normalized inflaton field."
],
[
"Pre-critical regime",
"Let us begin with the regime where the inflaton is approaching down to the critical point value.",
"Since the waterfall field is stabilized at the origin in this regime, the relevant Lagrangian is given as ${\\cal L}_{\\rm pre}=\\frac{f(\\phi ,0)}{2}(\\partial _\\mu \\phi )^2-V_{\\rm pre}(\\phi )\\,,$ where5It is noted that the term proportional to $\\Delta M^2$ in Eq.",
"(REF ) is equivalent to $\\Omega ^4(\\phi ,0)\\phi ^2\\Delta M^2(\\phi )/2$ .",
"$f(\\phi ,s)&=\\Omega ^2(\\phi ,s)\\left[1+\\Omega ^2(\\phi ,s)\\frac{(1+\\chi )^2}{6}\\phi ^2\\right]\\,, \\\\V_{\\rm pre}(\\phi )&=\\frac{1}{2}g^2\\xi ^2\\left[1+2\\Psi ^2 \\frac{\\Omega ^2(\\phi ,0)\\Delta M^2(\\phi )}{\\lambda ^2\\xi /q}\\right]\\,.$ It is seen that $\\Delta M^2$ term gives a gradient to the inflaton field, which should not invade the slow-roll conditions.",
"To see the impact of $\\Delta M^2$ term, it is instructive to change dynamical variable $\\phi $ to canonically-normalized field $\\hat{\\phi }$ .",
"Since $\\chi \\simeq -1$ , it is a good approximation that $f(\\phi ,0)\\simeq \\Omega ^2(\\phi ,0)$ .",
"Then $d\\phi /d\\hat{\\phi }=f(\\phi )^{-1/2}\\simeq \\Omega ^{-1}(\\phi ,0)$ can be solved easily to obtain, $&\\phi \\simeq \\beta ^{-1/2}\\sinh \\beta ^{1/2}\\hat{\\phi }\\,, \\\\&\\Psi \\simeq \\delta \\chi ^{-1/2}\\tanh \\beta ^{1/2}\\hat{\\phi }\\,, \\\\&\\Omega ^{-1}(\\phi ,0) \\simeq \\cosh \\beta ^{1/2}\\hat{\\phi }\\,,$ where $\\beta =-(1+\\chi )/6=\\lambda ^2\\delta \\chi /2qg^2\\xi $ .",
"Then the potential in terms of $\\hat{\\phi }$ is given as $V_{\\rm pre} &\\simeq \\frac{1}{2}g^2\\xi ^2\\biggr [1+\\frac{\\tanh ^2\\beta ^{1/2}\\hat{\\phi }}{\\cosh ^2\\beta ^{1/2}\\hat{\\phi } }\\frac{2q}{\\lambda ^2\\xi \\delta \\chi }\\biggr \\lbrace \\sum _{i=1}^3|M_{i3}|^2- \\frac{3}{8\\beta }|M_{33}|^2\\tanh ^{2}\\beta ^{1/2}\\hat{\\phi }\\biggl \\rbrace \\biggl ]\\,.$ On the other hand, it was shown in Ref.",
"[4] that there is one-loop corrections to the tree-level potential.",
"In terms of the canonically-normalized field, it is given by $V_{1l}\\simeq \\frac{1}{2}g^2\\xi ^2\\times \\frac{q^2g^2}{8\\pi ^2}\\log \\left[\\delta \\chi ^{-1}\\tanh ^2\\beta ^{1/2}\\hat{\\phi }\\right]\\,.$ Therefore, in order not to affect the inflationary trajectory, it is sufficient that the terms proportional to $|M_{i3}|^2$ are subdominant compared to the one-loop potential.",
"In the parameter space given in Eq.",
"(REF ), $\\sqrt{\\beta }\\hat{\\phi _c}\\simeq \\mathrm {arcsinh} \\sqrt{\\beta }\\phi _c \\sim {\\cal O}(1)$ .",
"Then, the conditions are given as $|M_{13}|^2+|M_{23}|^2 &\\lesssim \\frac{qg^2\\delta \\chi \\lambda ^2\\xi }{16\\pi ^2}\\lesssim (1\\times 10^{12}\\,{\\rm GeV})^2\\,,\\\\|M_{33}|^2 &\\lesssim \\frac{\\delta \\chi ^2\\lambda ^4}{12\\pi ^2}\\lesssim (2\\times 10^{11}\\,{\\rm GeV})^2\\,.$ It is easy to check that the slow-roll conditions are satisfied under the constraints.",
"Since the constraints are more stringent than Eqs.",
"(REF ) and (), it has been confirmed that the perturbative expansion to obtain Eq.",
"(REF ) is valid."
],
[
"Subcritical regime",
"In the previous subsection, we have seen that the slow-roll conditions are satisfied before reaching to the critical point value.",
"After the inflaton field becomes subcritical point value, the tachyonic growth of the waterfall field occurs.",
"It is expected that the inflation continues in the subcritical regime when $M_{i3}\\rightarrow 0$ .",
"In this subsection, we will derive the conditions under which the inflaton and waterfall field dynamics are not affected with non-zero $M_{i3}$ .",
"As seen in the previous section, the perturbative expression for $\\phi _c$ is valid under the conditions given in Eqs.",
"(REF ) and ().",
"Then, the tachyonic growth of the waterfall field is not affected by the additional gradient in the inflaton direction due to $|M_{i3}|^2$ terms since $\\phi _c\\simeq \\phi _{c,0}$ .",
"Consequently, the dynamics of the waterfall field around the critical point is the same as one discussed in Ref. [13].",
"The tachyonic growth is qualitatively the same as the subcritical hybrid inflation [2], [3].",
"Namely, due to the tachyonic growth, the waterfall field relaxes to the local minimum value $s_{\\rm min}$ just after a few Hubble-unit time.",
"(See, for example, Figure 1. of Ref. [2].)",
"In the present model, $s_{\\rm min}$ is found to be $s_{\\rm min}^2&=\\frac{2\\xi }{q(1+\\tilde{\\xi })}\\frac{\\Omega ^{-2}(\\phi ,0)}{1+\\tilde{\\xi }\\Psi ^2/(1+\\tilde{\\xi })}\\left[1-\\Psi ^2\\left\\lbrace 1+\\frac{2\\Omega ^2(\\phi ,0)\\Delta M^2(\\phi )}{3\\lambda ^2} \\right\\rbrace \\right]\\,.$ Then the potential in the subcritical regime of the inflaton field is effectively given by $V(\\phi ,s_{\\rm min})$ and the dynamics reduces to single field inflation that is described by the Lagrangian, ${\\cal L}=\\frac{f(\\phi ,s_{\\rm min})}{2}(\\partial _\\mu \\phi )^2- V_{\\rm inf}(\\phi )\\,,$ where $V_{\\rm inf}(\\phi )&=g^2\\xi ^2\\frac{(1+\\tilde{\\xi })\\Psi ^2}{1+2\\tilde{\\xi }\\Psi ^2}\\biggl [1-\\frac{\\Psi ^2}{2(1+\\tilde{\\xi })}+\\frac{\\lbrace 1+\\tilde{\\xi }\\Psi ^2/(1+\\tilde{\\xi })\\rbrace ^2}{(1+2\\tilde{\\xi }\\Psi ^2)/(1+\\tilde{\\xi })}\\frac{\\Omega ^2(\\phi ,0)\\Delta M^2(\\phi )}{\\lambda ^2\\xi /q}\\biggr ]\\nonumber \\\\ &+{\\cal O}(M_{i3}^4) \\,.$ Recall that $\\tilde{\\xi }\\,,\\xi \\ll 1$ in the parameters in Eq.",
"(REF ).",
"Then it is clear that the additional term proportional to $\\Delta M^2$ is the same as in Eq.",
"(REF ).",
"Note that $s_{\\rm min}$ is negligible in $\\Omega (\\phi ,s_{\\rm min})$ .",
"Then, the canonically-normalized inflaton field is given by Eqs.",
"(REF )–().",
"Consequently, $V_{\\rm inf}$ is given by $V_{\\rm inf} &\\simeq g^2\\xi ^2 \\delta \\chi ^{-1} \\tanh ^2\\beta ^{1/2} \\hat{\\phi }\\biggr [1-\\frac{\\delta \\chi ^{-1}}{2} \\tanh ^2\\beta ^{1/2}\\hat{\\phi } \\nonumber \\\\&+\\frac{\\cosh ^{-2}\\beta ^{1/2}\\hat{\\phi }}{\\lambda ^2\\xi /q}\\biggr \\lbrace \\sum _{i=1}^3|M_{i3}|^2-\\frac{3}{8\\beta }|M_{33}|^2\\tanh ^{2}\\beta ^{1/2}\\hat{\\phi }\\biggl \\rbrace \\biggl ]\\,.$ Therefore, if Eqs.",
"(REF ) and () are satisfied, then the dynamics in the subcritical regime reduces to one in Ref. [13].",
"In addition, it is clear that $m_\\phi $ defined in Eq.",
"(REF ) is indeed the inflaton mass since $V_{\\rm inf}\\simeq \\frac{1}{2}m_\\phi ^2 \\phi ^2$ around $\\phi \\,(\\simeq \\hat{\\phi })\\sim 0$ .",
"To summarize the present and previous subsections, the inflaton-waterfall field dynamics is unchanged when $|M_{i3}|^2=|M_{3i}|^2 \\lesssim (2\\times 10^{11}\\,{\\rm GeV})^2\\,,$ for $i=1$ –3 are satisfied."
],
[
"Stability of inflationary trajectory",
"It was pointed out in Ref.",
"[19] that $\\tilde{L}_iH_u$ may become tachyonic in sneutrino inflation.",
"In order to find out the stability condition, let us derive the mass matrix in $\\tilde{L}_i$ and $H_u$ basis.",
"From $V_{\\rm tot}$ , it is obtained by ${\\cal L}_{\\rm mass}^{\\tilde{L}H_u}=&-\\frac{\\Omega (\\phi ,s)^4 \\phi ^2}{2} |y_{\\nu 3i} \\tilde{L}_i|^2-\\frac{\\Omega (\\phi ,s)^4 \\phi ^2}{2} |y_{\\nu 3i}|^2|H_u|^2\\nonumber \\\\&+\\left[\\frac{M_{33}^* \\Omega (\\phi ,s)^6\\phi ^3}{4\\sqrt{2}}y_{\\nu 3i} \\tilde{L}_iH_u+{\\rm h.c.}\\right]\\,,$ On the other hand, the kinetic terms of $\\tilde{L}_i$ and $H_u$ are given by ${\\cal L}_{\\rm kin}^{\\tilde{L}H_u}=\\Omega (\\phi ,s)^2\\left[|\\partial _\\mu \\tilde{L}_i|^2+|\\partial _\\mu H_u|^2\\right]\\,.$ Therefore, using canonically-normalized fields, $\\hat{\\tilde{L}}_i\\equiv \\Omega (\\phi ,s)\\tilde{L}_i$ and $\\hat{H}_u\\equiv \\Omega (\\phi ,s)H_u$ , the mass terms are rewritten as ${\\cal L}_{\\rm mass}^{\\tilde{L}H_u}=&-\\frac{\\Omega (\\phi ,s)^2 \\phi ^2}{2} |y_{\\nu 3i} \\hat{\\tilde{L}}_i|^2-\\frac{\\Omega (\\phi ,s)^2 \\phi ^2}{2} |y_{\\nu 3i}|^2|\\hat{H}_u|^2\\nonumber \\\\&+\\left[\\frac{M_{33}^* \\Omega (\\phi ,s)^4\\phi ^3}{4\\sqrt{2}}y_{\\nu 3i} \\hat{\\tilde{L}}_i\\hat{H}_u+{\\rm h.c.}\\right]\\nonumber \\\\=&-\\frac{\\Omega (\\phi ,s)^2 \\phi ^2}{2}(y_\\nu y_\\nu ^\\dagger )_{33}|\\hat{\\tilde{L}}_3^\\prime |^2-\\frac{\\Omega (\\phi ,s)^2 \\phi ^2}{2}(y_\\nu y_\\nu ^\\dagger )_{33} |\\hat{H}_u|^2\\nonumber \\\\&+\\left[\\frac{M_{33}^* \\Omega (\\phi ,s)^4\\phi ^3}{4\\sqrt{2}}(y_\\nu y_\\nu ^\\dagger )_{33}^{1/2}\\hat{\\tilde{L}}_3^\\prime \\hat{H}_u+{\\rm h.c.}\\right]\\,,$ where we have defined $(y_\\nu y_\\nu ^\\dagger )_{33}^{1/2}\\hat{\\tilde{L}}_3^\\prime \\equiv y_{\\nu 3i}\\hat{\\tilde{L}}_i$ in the second line following Ref. [19].",
"Then, the stability condition is given by $|M_{33}|<2\\sqrt{2}(y_\\nu y_\\nu ^\\dagger )_{33}^{1/2}\\frac{\\Omega (\\phi ,s)^{-2}}{\\phi }\\,.$ Using Eqs.",
"(REF ) and (), it turns out that $|M_{33}|&<4\\sqrt{2}(y_\\nu y_\\nu ^\\dagger )_{33}^{1/2}\\sqrt{\\beta }\\nonumber \\\\&\\simeq 1.4 \\times 10^{14}\\,{\\rm GeV}\\left(\\frac{(y_\\nu y_\\nu ^\\dagger )_{33}}{10^{-7}}\\frac{\\beta }{10^{-3}}\\right)^{1/2}\\,.$ This upper bound is weaker than (REF ) in most of the parameter space, which will be seen later."
],
[
"Neutrino mass",
"In this section, we derive the mass matrices for the heavy and light neutrinos.",
"Around the global minimum, $\\Omega \\simeq 1$ since $\\xi \\ll 1$ .",
"Consequently, all the fields are canonical.",
"Thus, the mass terms are derived similarly in global SUSY model."
],
[
"Mass matrix",
"The superpotential (REF ) gives the Majorana masses for the light neutrinos.",
"To see how the masses are generated, we write down the mass terms for fermionic part of $N_i^c$ , $\\nu _{Li}$ and $\\tilde{S}_-$ , ${\\cal L}_{\\nu }^{\\rm mass}=-\\frac{1}{2}(\\bar{\\psi }{\\cal M}P_L\\psi +{\\rm h.c.})\\,,$ where $\\psi &=(N^c_1,N^c_2,N^c_3,\\tilde{S}_-,\\nu _{L1},\\nu _{L2},\\nu _{L3})^T\\,,\\\\{\\cal M}&=\\left(\\begin{array}{cc}\\tilde{M} & \\tilde{m}_\\nu \\\\\\tilde{m}_\\nu ^T & {\\bf 0}\\end{array}\\right)\\,.$ $\\tilde{M}$ and $\\tilde{m}_\\nu $ are $4\\times 4$ and $4\\times 3$ matrices, respectively, and given by $\\tilde{M}&=\\left(\\begin{array}{cccc}&&&\\lambda _1\\langle S_+\\rangle \\\\&\\mbox{\\smash{\\Large M}}&&\\lambda _2\\langle S_+\\rangle \\\\&&& m_\\phi \\\\\\lambda _1\\langle S_+\\rangle &\\lambda _2\\langle S_+\\rangle &m_\\phi & 0\\end{array}\\right)\\,, \\\\\\tilde{m}_\\nu &=\\left(\\begin{array}{ccc}&& \\\\&\\mbox{\\smash{\\Large m_\\nu }}&\\\\&& \\\\0&0&0\\end{array}\\right)\\,.$ Here $m_{\\nu \\,ij}=y_{\\nu \\,ij}\\langle H_u^0\\rangle $ with $\\langle H_u^0\\rangle $ being the VEV of the up-type neutral Higgs.",
"Then mass matrix $M_{\\nu }$ for the light neutrinos are obtained by the seesaw mechanism [17], $M_{\\nu } =-\\tilde{m}_\\nu ^T \\tilde{M}^{-1}\\tilde{m}_\\nu \\,.$ An important consequence of the mass matrix is that the one of three light neutrinos is massless.",
"This is because the rank of $M_\\nu $ is two.",
"Using this mass matrix, it is possible to constrain the parameters by the observed neutrino masses.",
"In the later discussion we assume $\\lambda _1\\,,~\\lambda _2\\,\\ll \\lambda \\,.$ Here, recall that there is a freedom to choose a basis for $N^c_1$ and $N^c_2$ .",
"Then, $M_{12}$ can be rotated away.",
"As a result, $M_{\\nu }$ is given in the following simple expression, $M_{\\nu \\, ij} = -\\langle H^0_u\\rangle ^2\\sum _{k=1}^2\\frac{y_{\\nu ki}y_{\\nu kj}}{M_{kk}}+{\\cal O}\\left(\\frac{\\lambda _{1,2}}{\\lambda }\\frac{m_{\\nu \\, il}m_{\\nu \\, jm}}{M_{kk}}\\right)\\,.$ Here in the second term of right-hand side, $k=1$ or 2 and $i$ , $j$ , $l$ , $m$ can be 1–3, and we have ignored $M_{i3}$ based on the discussion of the previous section.",
"The leading order term, on the other hand, is independent of $\\lambda _{1,2}$ , $M_{i3}$ , $m_\\phi $ and $y_{\\nu 3i}$ ($i=1$ –3).",
"Therefore, they are not constrained by the neutrino oscillation data.",
"This fact is important in the estimation of the reheating temperature, which we will see later.",
"To ensure the non-zero $\\lambda _{1,2}$ does not affect our later analysis at percent level, we implicitly assume $\\lambda _{1,2}<0.01\\lambda $ .",
"Before further discussing the light neutrino mass matrix, let us note that the mass matrix $\\tilde{M}$ corresponds to the mass matrix in the superpotential around the global minimum, $W_{\\rm neu}=\\frac{1}{2} (N^c_1,N^c_2,N^c_3,S_-)\\tilde{M}\\left(\\begin{array}{c}N^c_1 \\\\N^c_2 \\\\N^c_3 \\\\S_-\\end{array}\\right)+\\cdots \\,.$ Recall that we have the requirement (REF ) for successful inflation.",
"Therefore, $\\tilde{M}$ should be almost block-diagonal as $\\tilde{M}&\\simeq \\left(\\begin{array}{ccc:c}M_1&&&0 \\\\&M_2&&0\\\\&& M_3& m_\\phi \\\\0 &0&m_\\phi & 0\\end{array}\\right)\\,,$ where $M_i>0$ .",
"Here we have left $M_3$ for later discussion.",
"In the following analysis we use Eq.",
"(REF ) for $\\tilde{M}$ .",
"$\\tilde{M}$ is further diagonalized by a unitary matrix $U_{\\tilde{M}}$ as, $D_{\\tilde{M}}\\simeq {\\rm diag}(M_1,M_2,m_\\phi , m_\\phi )\\simeq U^T_{\\tilde{M}} \\tilde{M} U_{\\tilde{M}}\\,,$ where $U_{\\tilde{M}}=\\left(\\begin{array}{cc:c}1 & & \\\\& 1 & \\\\& &{\\scriptsize \\frac{1}{\\sqrt{2}}\\bigl (\\begin{array}{cc}1 & i\\\\1 & -i\\end{array}\\bigr )}\\end{array}\\right)\\,.$ Here we have omitted ${\\cal O}(M_3/m_\\phi )$ corrections since they are irrelevant in the later analysis.",
"Non-zero $\\lambda _{1,2}$ corrections enter as ${\\cal O}(\\lambda _{1,2}/\\lambda )$ and ${\\cal O}((m_\\phi /M_{1,2})\\lambda _{1,2}/\\lambda )$ for $m_\\phi \\gtrsim M_{1,2}$ and $m_\\phi \\lesssim M_{1,2}$ , respectively.",
"The corrections due to non-zero $\\lambda _{1,2}$ are similar in the scalar sector since the mass matrix in $(\\tilde{N}_1,\\tilde{N}_2,\\tilde{N}_3,S_-)$ basis is given by, $M_{\\rm scalar}^2=\\tilde{M}^\\dagger \\tilde{M}\\,.$ Therefore the assumption (REF ) assures that the dynamics of the inflaton and the waterfall field discussed in the previous section, as well as subsequent reheating and leptogenesis, is not affected.",
"If it is not satisfied, then the inflationary trajectory would be altered so that we need further analysis to find the paramter space that is consistent with the CMB observations.",
"Accordingly, the subsequent reheating and leptogenesis scenario change.",
"We leave the detailed analysis as an interesting future work."
],
[
"Parametrization of neutrino Yukawa couplings",
"Now let us discuss $M_\\nu $ .",
"It can be diagonalized by a unitary matrix $U_\\nu $ as $D_{M_\\nu }={\\rm diag}(m_1,m_2,m_3)=U_\\nu ^T M_\\nu U_\\nu \\,.$ As noted above, one of the three light neutrino masses is zero.",
"We follow the standard convention that $m_3>m_2>m_1(=0)$ for the normal hierarchy (NH) case and $m_2>m_1>m_3(=0)$ for the inverted hierarchy (IH) case and use the values given in Ref.",
"[20], which are listed in Table REF .",
"Before discussing the parametrization of the neutrino Yukawa couplings, it is instructive to count the number of parameters.",
"The situation is the same as one discussed Refs.",
"[19], [21] since the mass matrix of the light neutrinos (REF ) is similar.",
"Since one neutrino is massless, there are 7 parameters in low energy, i.e., 2 neutrino masses $+$ 3 real mixing angles $+$ 2 phases.",
"On the other hand, $M_\\nu $ includes $y_{\\nu ki}$ and $M_k$ where $k=1,2$ and $i=1,2,3$ , which means 12 real parameters (neutrino Yukawa couplings) $+$ 2 real parameters (right-handed neutrino masses).",
"However, 3 phases can be absorbed by lepton doublets and 2 real parameters are unphysical since $M_\\nu $ is unchanged by the rescalings $y_{\\nu ki}\\rightarrow \\gamma _k y_{\\nu ki}$ and $M_k \\rightarrow \\gamma ^2_kM_k$ with $\\gamma _k$ being real constants.",
"Therefore, we have 9 independent parameters in $M_\\nu $ to determine 7 parameters in the light neutrino sector.",
"As it will be seen below, however, the parametrization of the Yukawa couplings is different, especially for $y_{\\nu 3i}$ that are important parameters for the estimation of the reheating temperature.",
"Let us discuss the NH case first.",
"We define $4\\times 3$ matrix $R$ in the similar manner in Refs.",
"[22], [23], $iR=D_{\\tilde{M}}^{-1/2}U_{\\tilde{M}}^T\\tilde{m}_\\nu U_\\nu D_{M_\\nu }^{-1/2}\\,,$ which satisfies $R^TR={\\rm diag}(0,1,1)\\,.$ Here $D_{\\tilde{M}}^{-1/2}$ and $D_{M_\\nu }^{-1/2}$ are matrices that satisfy $(D_{\\tilde{M}}^{-1/2})^2=D_{\\tilde{M}}^{-1}$ and $(D_{M_\\nu }^{-1/2})^2={\\rm diag}(0,m_2^{-1},m_3^{-1})$ , respectively.",
"It is found that $R$ is more restrictive than Eq.",
"(REF ).",
"Namely, $r^Tr=rr^T={\\bf 1}\\,,\\\\R_{42}/R_{32}=R_{43}/R_{33}=i\\,, \\\\R_{l1}=0~~~~(l=14)\\,,$ where $r\\equiv \\left(\\begin{array}{cc}R_{12} & R_{13} \\\\R_{22} & R_{23}\\end{array}\\right)\\,.$ Using the relations, the neutrino Yukawa couplings can be expressed in terms of $R$ .",
"For later discussion, it is useful to give following quantities: $(y_\\nu y_\\nu ^\\dagger )_{ii}&=M_i\\sum _{j=2}^3 |R_{ij}|^2 m_j/\\langle H_u^0 \\rangle ^2~~~~~~~~~(i=1,2)\\,,\\\\(y_\\nu y_\\nu ^\\dagger )_{33}&=2m_\\phi \\sum _{j=2}^3|R_{3j}|^2 m_j/\\langle H_u^0 \\rangle ^2\\,,\\\\{\\rm Im}\\left[(y_\\nu y_\\nu ^\\dagger )_{21}^2\\right]\\frac{M_1}{M_2}&=-\\frac{M_1^2}{\\langle H_u^0\\rangle ^4}{\\rm Im}\\left[\\sum _{j=2}^3R_{1j}^2m_j^2\\right]\\,, \\\\\\sum _{i=1}^2{\\rm Im}\\left[(y_\\nu y_\\nu ^\\dagger )_{i3}^2\\right]\\frac{M_3}{M_i}&=-\\frac{2M_3 m_\\phi }{\\langle H_u^0\\rangle ^4}{\\rm Im}\\left[\\sum _{j=2}^3R_{3j}^2m_j^2\\right]\\,.$ It is seen that $(y_\\nu y_\\nu ^\\dagger )_{11}$ and $(y_\\nu y_\\nu ^\\dagger )_{22}$ are constrained by the neutrino oscillation data, meanwhile $(y_\\nu y_\\nu ^\\dagger )_{33}$ is basically a free parameter since $R_{3j}$ is not constrained.",
"This is consistent with the fact that $M_\\nu $ is independent of $y_{\\nu 3i}$ .",
"The discussion is quite similar in the IH case.",
"The definition of $R$ is the same form as in Eq.",
"(REF ), but satisfies $R^TR={\\rm diag}(1,1,0)$ with $(D_{M_\\nu }^{-1/2})^2={\\rm diag}(m_1^{-1},m_2^{-1},0)$ .",
"Then, defining $r$ as $r\\equiv \\left(\\begin{array}{cc}R_{11} & R_{12} \\\\R_{21} & R_{22}\\end{array}\\right)\\,,$ we get $r^Tr=rr^T={\\bf 1}\\,,\\\\R_{41}/R_{31}=R_{42}/R_{32}=i\\,, \\\\R_{l3}=0~~~~(l=14)\\,,$ and the neutrino Yukawa couplings are given by, $(y_\\nu y_\\nu ^\\dagger )_{ii}&=M_i\\sum _{j=1}^2 |R_{ij}|^2 m_j/\\langle H_u^0 \\rangle ^2~~~~~~~~~(i=1,2)\\,,\\\\(y_\\nu y_\\nu ^\\dagger )_{33}&=2m_\\phi \\sum _{j=1}^2|R_{3j}|^2 m_j/\\langle H_u^0 \\rangle ^2\\,,\\\\{\\rm Im}\\left[(y_\\nu y_\\nu ^\\dagger )_{21}^2\\right]\\frac{M_1}{M_2}&=-\\frac{M_1^2}{\\langle H_u^0\\rangle ^4}{\\rm Im}\\left[\\sum _{j=1}^2R_{1j}^2m_j^2\\right]\\,, \\\\\\sum _{i=1}^{2}{\\rm Im}\\left[(y_\\nu y_\\nu ^\\dagger )_{i3}^2\\right]\\frac{M_3}{M_i}&=-\\frac{2M_3 m_\\phi }{\\langle H_u^0\\rangle ^4}{\\rm Im}\\left[\\sum _{j=1}^2R_{3j}^2m_j^2\\right]\\,.$"
],
[
"Post inflationary regime",
"After the end of inflation, the inflaton oscillates around the global minimum and decays eventually.",
"Due to the decay the universe is reheated and thermal plasma is created.",
"In this section, we estimate the reheating temperature and discuss how the lepton number asymmetry is generated.",
"As in the previous section, we take $\\Omega \\simeq 1$ .",
"After the universe is reheated, gravitinos are produced in various ways.",
"We discuss the gravitino problem at the end of this section."
],
[
"Reheating",
"The reheating temperature $T_R$ due to the inflaton decay is estimated by, $T_R\\simeq (90/\\pi ^2g_*(T_R))^{1/4}\\sqrt{\\Gamma _\\phi M_{pl}}\\,,$ where $g_*(T)$ is the effective degree of freedom of radiation fields at temperature $T$ and $\\Gamma _\\phi $ is the decay rate of the inflaton.",
"This expression is valid when the neutrino Yukawa couplings that are responsible for the decay is sufficiently small to satisfy $T_R\\lesssim m_\\phi $ [24], [25], which is the situation we focus on.6Of course, it is possible to consider a higher reheating temperature than the inflaton mass.",
"Such a case is discussed in Ref. [19].",
"We will comment on the impact of such high reheating temperature on leptogenesis in the next subsection.",
"The inflaton decays as $\\phi \\rightarrow L\\tilde{H}_u$ , $\\bar{L}\\bar{\\tilde{H}}_u$ , $\\tilde{L}H_u$ , $\\tilde{L}^*H_u^*$ .7In general, the inflaton decays to gravitino pair or gravitino and right-handed neutrino.",
"We will discuss those processes in Sec.",
"REF .",
"Here flavor indices and SU(2) doublet components are summed implicitly.",
"Then the decay rates for the modes are given by $&\\Gamma _{\\phi \\rightarrow L\\tilde{H}_u}=\\Gamma _{\\phi \\rightarrow \\bar{L}\\bar{\\tilde{H}}_u}=\\frac{(y_{\\nu }y_\\nu ^\\dagger )_{33}}{16\\pi }m_\\phi \\,, \\\\& \\Gamma _{\\phi \\rightarrow \\tilde{L}H_u}=\\Gamma _{\\phi \\rightarrow \\tilde{L}^*H_u^*}=\\frac{(y_{\\nu }y_\\nu ^\\dagger )_{33}}{16\\pi }\\frac{M_3^2}{m_\\phi }\\,.$ Since $\\Gamma _{\\phi \\rightarrow \\tilde{L}H_u}$ (=$\\Gamma _{\\phi \\rightarrow \\tilde{L}^*H_u^*}$ ) is suppressed by $(M_3/m_\\phi )^2$ , the total decay rate is given by $\\Gamma _\\phi \\simeq \\Gamma _{\\phi \\rightarrow L\\tilde{H}_u}+\\Gamma _{\\phi \\rightarrow \\bar{L}\\bar{\\tilde{H}}_u}=\\frac{(y_{\\nu }y_\\nu ^\\dagger )_{33}}{8\\pi }m_\\phi \\,.$ Then the reheating temperature is estimated as $T_R\\simeq 1.4\\times 10^{10}\\,{\\rm GeV}\\left(\\frac{m_\\phi }{10^{13}\\,{\\rm GeV}}\\right)^{1/2}\\left(\\frac{(y_{\\nu }y_\\nu ^\\dagger )_{33}}{10^{-9}}\\right)^{1/2}\\left(\\frac{g_{*}(T_R)}{228.75}\\right)^{-1/4}\\,.$ Recall that $(y_\\nu y_\\nu ^\\dagger )_{33}$ is not constrained by the neutrino observations.",
"As a consequence, it is possible to consider a wide range of values for the reheating temperature, which is suitable for leptogenesis.",
"To end this subsection, we derive the number of $e$ -folds before the end of inflation.",
"In this model, the inflaton oscillates after the end of inflation and eventually decays to reheat the universe.",
"Therefore, it is given by $N_e\\simeq &\\, 55+\\log \\left(\\frac{L}{{\\rm Gpc}}\\right)+\\frac{1}{3}\\log \\left(\\frac{T_R}{10^{10}\\,{\\rm GeV}}\\right)\\nonumber \\\\&+\\log \\left(\\frac{H_e}{10^{13}\\,{\\rm GeV}}\\right)-\\frac{2}{3}\\log \\left(\\frac{H_{\\rm end}}{10^{13}\\,{\\rm GeV}}\\right)\\,,$ where $L$ is the present cosmological scale, and $H_e$ and $H_{\\rm end}$ are the Hubble scale corresponding to $N_e$ and at the end of inflation, respectively.",
"It is now clear that the parameters given in Eq.",
"(REF ) is consistent with $N_e=55$ – 60."
],
[
"Leptogenesis",
"Now we discuss the lepton number asymmetry.",
"The lepton number is generated via leptogenesis [18] (see, for example, Refs.",
"[26], [27] for review).",
"In the following numerical study, we discuss following representative cases: $({\\rm I}).~& M_1,M_2 < m_\\phi \\,,\\\\({\\rm II}).~& M_1,M_2 > m_\\phi \\,.$ In both cases, $M_3$ should satisfy Eq.",
"(REF ).",
"Let us consider case (I) first.",
"For simplicity, we consider the Majorana masses are further hierarchical, i.e., $M_1\\ll M_2$ .8$M_3$ is irrelevant for leptogenesis if Eq.",
"(REF ) is satisfied.",
"Similar situation has been studied intensively in the literature [28], [29].",
"Even though the lepton number is generated by the inflaton decay, it is possibly washed out when the reheating temperature is comparable or higher than $M_{1}$ .9It would be possible that $\\tilde{N}_i$ ($i=1,2$ ) have initial amplitude of the Hubble parameter during inflation $H_{\\rm inf}\\sim g\\xi /\\sqrt{6}M_{pl}$ .",
"Then $\\tilde{N}_i$ start to oscillate when $H\\sim M_i$ , and eventually decays.",
"However, the effect of coherent oscillation of $\\tilde{N}_i$ is negligible since the energy density ratio of $\\tilde{N}_i$ to radiation at the decay is estimated as less than $\\xi ^2/18M_{pl}^4\\sim 10^{-9}$ .",
"To see this more explicitly, it is convenient to introduce the effective neutrino mass [30] and equilibrium neutrino mass [31], $&\\tilde{m}_1=\\frac{(m_\\nu m_\\nu ^\\dagger )_{11}}{M_1}\\,,\\\\&m_*=\\frac{4\\pi ^2\\sqrt{g_*(M_1)} \\langle H_u^0\\rangle ^2 }{3\\sqrt{10} M_{pl}}\\simeq 3.9 \\times 10^{-4}\\,{\\rm eV}\\left(\\frac{\\langle H_u^0\\rangle }{v/2}\\right)^2\\,,$ where $v\\simeq 246.7~{\\rm GeV}$ .",
"If $\\tilde{m}_1/m_*$ is larger than unity, then it is the strong washout regime and the lepton number generated at the reheating is washed out.",
"$\\tilde{m}_1$ is estimated by using Eqs.",
"(REF ) and (REF ), $\\tilde{m}_1=\\left\\lbrace \\begin{array}{ll}\\sum _{j=2}^3 |R_{1j}|^2 m_j &({\\rm NH}) \\\\\\sum _{j=1}^2 |R_{1j}|^2 m_j &({\\rm IH})\\end{array}\\right.\\,.$ Using the neutrino mass data and Eqs.",
"(REF ) and (REF ), it is straightforward to find that $\\tilde{m}_1$ has a lower bound, $\\tilde{m}_1 \\ge \\left\\lbrace \\begin{array}{ll}m_2 \\simeq 8.6 \\times 10^{-3}\\,{\\rm eV} &({\\rm NH}) \\\\m_1 \\simeq 4.9\\times 10^{-2}\\,{\\rm eV} &({\\rm IH})\\end{array}\\right.\\,.$ Therefore, it is the strong washout regime in either case.",
"Figure: Allowed region for M 1 M_1 as function of effective neutrinomass m ˜ 1 \\tilde{m}_1 defined in Eq.",
"() for the normalhierarchy (NH) and inverted hierarchy (IH) cases.",
"Lower bound onM 1 M_1 is obtained from η B max ≥η B obs \\eta _B^{\\rm max}\\ge \\eta _B^{\\rm obs}while upper bound is ().",
"Lower bound onm ˜ 1 \\tilde{m}_1 is given by Eq.",
"().Although the primordial lepton number is washed out, the lepton number is regenerated by the decay of the lightest right-handed (s)neutrino, i.e., $N_1$ and $\\tilde{N}_1$ in the present case.",
"Then the lepton number, strictly speaking lepton number minus baryon number, is converted to baryon number via the sphaleron effect.",
"This scenario works if $T_R\\gtrsim M_1$ [28], [29], which is always possible as confirmed in the previous subsection.",
"Then the resultant baryon number becomes independent of $T_R$ .",
"In our study, we adopt the analytic expressions in Ref.",
"[29] for the calculation of the baryon number.",
"Note that although the results there are given in non-supersymmetric model, the results in supersymmetric model do not change much both quantitatively and qualitatively [32], [33], [27].",
"In our study we adopt the discussion given in Ref. [27].",
"Then the baryon number is determined by $\\eta _B\\equiv \\frac{n_B}{n_\\gamma }=\\frac{3\\sqrt{2}}{4}\\frac{a_{\\rm sph}}{f}\\epsilon _1 \\kappa _f\\simeq 2.7\\times 10^{-10} \\left(\\frac{\\epsilon _1}{10^{-6}}\\right)\\left(\\frac{\\kappa _f}{2\\times 10^{-2}}\\right)\\,,$ where $n_B$ and $n_\\gamma $ are number densities of baryon and photon at present, respectively, $a_{\\rm sph}=28/79$ , $f=2387/86$ , and a factor of $\\sqrt{2}$ counts the supersymmetric effect.",
"The efficiency factor $\\kappa _f$ is given by [29] $\\kappa _f=(2\\pm 1)\\times 10^{-2}\\left(\\frac{0.01~{\\rm eV}}{\\tilde{m}_1}\\right)^{1.1\\pm 0.1}\\,.$ Finally, referring Ref.",
"[34], the asymmetric parameter $\\epsilon _1$ in our model is given by $\\epsilon _1=-\\frac{3}{16\\pi }\\frac{{\\rm Im}\\,\\left[(y_\\nu y_\\nu ^\\dagger )_{21}^2\\right] }{(y_\\nu y_\\nu ^\\dagger )_{11}}\\frac{M_1}{M_2}=\\frac{3}{16\\pi }\\frac{M_1}{\\langle H^0_u \\rangle ^2}m_{\\rm eff}\\,,$ where we have used Eqs.",
"(REF ), (), (REF ), and () to obtain $m_{\\rm eff} =\\left\\lbrace \\begin{array}{ll}\\frac{{\\rm Im}\\sum _{j=2}^3R_{1j}^2m_j^2}{\\sum _{j=2}^3|R_{1j}|^2m_j}&({\\rm NH}) \\\\[4mm]\\frac{{\\rm Im}\\sum _{j=1}^2R_{1j}^2m_j^2}{\\sum _{j=1}^2|R_{1j}|^2m_j}&({\\rm IH})\\end{array}\\right.\\,.$ It turns out that the maximum value of $m_{\\rm eff}$ , denoted as $m_{\\rm eff}^{\\rm max}$ , is $m_{\\rm eff}^{\\rm max} =\\left\\lbrace \\begin{array}{ll}m_3 - m_2 \\simeq 4.2\\times 10^{-2}\\,{\\rm eV}&({\\rm NH}) \\\\m_2 - m_1\\simeq 7.4 \\times 10^{-4}\\,{\\rm eV}&({\\rm IH})\\end{array}\\right.\\,.$ Therefore, parametrizing $m_{\\rm eff}$ as $m_{\\rm eff}=m_{\\rm eff}^{\\rm max} \\sin \\delta $ , the asymmetric parameter is given by $\\epsilon _1\\simeq \\left\\lbrace \\begin{array}{ll}8.2 \\times 10^{-7}&({\\rm NH}) \\\\1.5\\times 10^{-8}&({\\rm IH})\\end{array}\\right\\rbrace \\times \\left(\\frac{M_1}{10^{10}\\,{\\rm GeV}}\\right)\\left(\\frac{\\langle H^0_u \\rangle }{v/2}\\right)^{-2}\\left(\\frac{\\sin \\delta }{0.5}\\right)\\,.$ Using Eqs.",
"(REF ), (REF ) and (REF ), the lower limit on $M_1$ to explain the present baryon number is determined from $\\eta _B^{\\rm max} \\ge \\eta _B^{\\rm obs}$ where $\\sin \\delta =1$ and $\\eta _B^{\\rm obs}$ is given by [35] $\\eta _B^{\\rm obs}=(6.12\\pm 0.03)\\times 10^{-10}\\,.$ In Fig.",
"REF , allowed regions are depicted for the NH and IH cases.",
"Here we consider so-called high-scale SUSY and take $\\langle H^0_u \\rangle =v/2$ to get 125 GeV Higgs mass [36], [37].",
"In the plot upper bound on $M_1$ is given by (REF ), i.e., $M_1<m_\\phi =10^{13}\\,$ GeV, and the lower bound on $\\tilde{m}_1$ is from Eq.",
"(REF ).10Strictly speaking, the equality should be excluded since baryon number is zero.",
"The theoretical uncertainties in Eq.",
"(REF ) are taken into account.",
"It is found that the present baryon number can be explained in a wide range of parameter space for the NH case.",
"For the IH case, on the other hand, it seems that the parameter space for a successful leptogenesis is relatively limited.",
"The lowest value required for $M_1$ turns out to be $M_1 \\gtrsim \\left\\lbrace \\begin{array}{ll}5.8 \\times 10^{9}\\,{\\rm GeV}&({\\rm NH}) \\\\2.1\\times 10^{12}\\,{\\rm GeV}&({\\rm IH})\\end{array}\\right.",
"\\,.$ The lower limit is near the upper bound in the IH case.",
"Here recall that the upper bound on $M_1$ is just a theoretical one.",
"When $M_1\\sim m_\\phi $ , $T_R$ should be comparable to $m_\\phi $ , which is possible as discussed in Refs.",
"[19], [24], [25].",
"In such a case, sneutrino inflation and leptogenesis can be another source for lepton asymmetry, which will be discussed below in detail.",
"Therefore, the upper bound merely indicates the parameter space for simple thermal leptogenesis to work.",
"Let us move on to case (II).",
"Since they are much heavier than the inflaton, $N_{1,2}$ and $\\tilde{N}_{1,2}$ are never thermalized after the reheating.",
"For $N_3$ and $\\tilde{N_3}$ , on the other hand, it depends on the effective neutrino mass that is defined by $\\tilde{m}_3=\\frac{(m_\\nu m_\\nu ^\\dagger )_{33}}{m_\\phi }\\simeq 1.5\\times 10^{-9}\\,{\\rm eV}\\left(\\frac{(y_{\\nu }y_{\\nu }^\\dagger )_{33}}{10^{-9}}\\right)\\left(\\frac{10^{13}\\,{\\rm GeV}}{m_\\phi }\\right)\\left(\\frac{\\langle H_u^0\\rangle }{v/2}\\right)^2\\,.$ Then, from Eq.",
"(REF ), one obtains $\\frac{T_R}{m_\\phi }\\simeq 0.71 \\times \\left(\\frac{\\tilde{m}_3}{m_*}\\right)^{1/2}\\,.$ Here $m_*$ is defined similarly to Eq.",
"() but replacing $g_*(M_1)$ by $g_*(m_\\phi )$ , and we have taken $g_*(T_R)\\simeq g_*(m_\\phi )$ .",
"As explained in Sec.",
"REF , the expression Eq.",
"(REF ) is valid for $T_R/m_\\phi \\lesssim 1$ that is satisfied for $\\tilde{m}_3<m_*$ .",
"Such a case corresponds to the weak washout regime.",
"In that regime, the $N_3$ and $\\tilde{N_3}$ are not thermalized, and the lepton number produced by the inflaton decay can be the source of the present baryon number.",
"This situation is similar to sneutrino inflation and leptogenesis [38], [39], [40], [41], [42], [43], [44], [45], [46], [19], [21].",
"(See also Refs.",
"[47], [48], [49] for leptogenesis via Afflec-Dine mechanism [50].)",
"Figure: Allowed region for M 3 M_3 as function of m ˜ 3 \\tilde{m}_3defined in Eq. ().",
"Results are the same for the NH andIH cases.",
"Lower bound on M 3 M_3 comes from η B max ≥η B obs \\eta _B^{\\rm max}\\ge \\eta _B^{\\rm obs} meanwhile upper one is given byEqs.",
"() and ().",
"Theformer is not to affect the inflationary trajectory and the latter(dashed orange) is from the stability of the scalarpotential.",
"Vertical lines (dot-dashed purple) are contours ofT R T_R.Let us suppose $\\tilde{m}_3\\ll m_*$ , i.e., $T_R/m_\\phi \\ll 1$ .",
"Then baryon number is given by $\\eta _B=\\frac{3}{4}\\frac{T_R}{m_\\phi }a_{\\rm sph}^{\\rm MSSM}d\\epsilon _\\phi \\,,$ where $a_{\\rm sph}^{\\rm MSSM}=8/23$ and $d=(s/n_\\gamma )_0=43\\pi ^4/495\\zeta (3)$ is the present value of entropy density to and photon density ratio.",
"$\\epsilon _\\phi $ is obtained by an explicit calculation as $\\epsilon _\\phi =-\\frac{3}{4\\pi }\\sum _{i=1}^2\\frac{{\\rm Im}\\,\\left[(y_\\nu y_\\nu ^\\dagger )_{i3}^2\\right] }{(y_\\nu y_\\nu ^\\dagger )_{33}}\\frac{M_3}{M_i}=\\frac{3}{4\\pi }\\frac{M_3}{\\langle H_u^0 \\rangle ^2}m^{\\prime }_{\\rm eff}\\,,$ where $m^{\\prime }_{\\rm eff} =\\left\\lbrace \\begin{array}{ll}\\frac{{\\rm Im}\\sum _{j=2}^3R_{3j}^2m_j^2}{\\sum _{j=2}^3|R_{3j}|^2m_j}&({\\rm NH}) \\\\[4mm]\\frac{{\\rm Im}\\sum _{j=1}^2R_{3j}^2m_j^2}{\\sum _{j=1}^2|R_{3j}|^2m_j}&({\\rm IH})\\end{array}\\right.\\,.$ In the second step, we have used Eqs.",
"(), (), (), and ().",
"It should be noted that $\\epsilon _\\phi $ is independent of the inflaton mass, but it depends on $M_3$ .",
"Even though $R_{3j}$ are not constrained, it has been found that $m^{\\prime }_{\\rm eff}$ is bounded from above.",
"The maximum value turns out to be $m_{\\rm eff}^{\\prime \\,\\rm max} =\\left\\lbrace \\begin{array}{ll}m_3 \\simeq 5.0\\times 10^{-2}\\,{\\rm eV}&({\\rm NH}) \\\\m_2\\simeq 5.0 \\times 10^{-2}\\,{\\rm eV}&({\\rm IH})\\end{array}\\right.\\,.$ Therefore, parametrizing the effective neutrino mass as $m^{\\prime }_{\\rm eff}=m_{\\rm eff}^{\\prime \\,{\\rm max}}\\sin \\delta ^{\\prime }$ the asymmetric parameter is given by $\\epsilon _\\phi \\simeq 3.9\\times 10^{-9}\\times \\left(\\frac{M_3}{10^{7}\\,{\\rm GeV}}\\right)\\left(\\frac{\\langle H^0_u \\rangle }{v/2}\\right)^{-2}\\left(\\frac{\\sin \\delta ^{\\prime }}{0.5}\\right)\\,.$ for both the NH and IH cases.",
"Using the equations, we get $\\eta _B^{\\rm max}\\simeq 1.6\\times 10^{-10}\\left(\\frac{M_3}{10^7\\,{\\rm GeV}}\\right)\\left(\\frac{\\tilde{m}_3}{10^{-7}\\,{\\rm eV}}\\right)^{1/2}\\left(\\frac{\\langle H_u^0 \\rangle }{v/2}\\right)^{-3}\\,.$ Since $\\eta _B$ is independent of $m_\\phi $ , the requirement $\\eta _B^{\\rm max}\\ge \\eta _B^{\\rm obs}$ gives a lower bound on $M_3$ , which is plotted in Fig.",
"REF .",
"Here $\\langle H^0_u\\rangle =v/2$ is taken as in Fig.",
"REF .",
"Upper bound $M_3<2\\times 10^{11}\\,{\\rm GeV}$ is from Eq.",
"(REF ).",
"Another upper bound on $M_3$ from the stability of the inflationary trajectory, Eq.",
"(REF ), is also shown.",
"We quit plotting for region $\\tilde{m}_3>10^{-5}\\,{\\rm eV}$ because $\\tilde{m}_3\\ll m_*$ is no longer valid.",
"In the plot, contours of $T_R$ are depicted.",
"It is found that the leptogenesis is successful in a wide parameter space for both the NH and IH cases.",
"Lower bound on $M_3$ behaves similarly to region C in Fig.",
"1 of Ref.",
"[42] by reading $M_1$ and $\\tilde{m}_1$ as $M_3$ and $\\tilde{m}_3$ , respectively, i.e., the lower bound is proportional to $1/\\sqrt{\\tilde{m}_3}$ .",
"Quantitatively, the lower bound in our model is relaxed by roughly a factor of 4 compared to the result in the reference.",
"This can be understood as follows; first, the decay rate of the inflaton in our model is different from one in the literature (see Eq.",
"(REF )), leading to a $1/\\sqrt{2}$ suppression in $T_R/m_\\phi $ ; as another consequence, the expression (REF ) (with $m_{\\rm eff}^{\\prime }=m_{\\rm eff}^{\\prime {\\rm max}}$ ) is enhanced by a factor of two compared to $|\\epsilon _1^{\\rm max}|$ in the reference; finally $\\tan \\beta =\\infty $ is taken in the work meanwhile $\\tan \\beta =1$ in our model.",
"Thus, in total, a factor of 4 enhancement is obtained in $\\eta _B$ .",
"In the case where $\\tilde{m}_3$ gets much larger than $m_*$ , the situation reduces to case (I).",
"Namely, the reheating temperature is so high that both $\\tilde{N}_3$ and $N_3$ are thermalized and thermal leptogenesis takes place.",
"Resultant allowed region is the same as the NH case of Fig.",
"REF , by replacing $M_1$ and $\\tilde{m}_1$ by $M_3$ and $\\tilde{m}_3$ , respectively, but there is no lower bound on $\\tilde{m}_3$ meanwhile there is the upper bound on $M_3$ .",
"In the intermediate case, $\\tilde{m}_3\\sim m_*$ , on the other hand, the Boltzmann equations should be solved numerically to get the lepton number, which is already done in Ref. [42].",
"The result corresponds to region B in Fig.",
"1 of the reference.",
"Strictly speaking, the effective dissipation rate should be used instead of the decay rate of the inflaton [25] in the Boltzmann equations.",
"As shown in the reference, the reheating process is so efficient when the dissipation rate is taken into account that the reheating temperature can exceed the mass of the inflaton mass and consequently $N_3$ and $\\tilde{N}_3$ are easily thermalized.",
"Once they are thermalized, the thermal leptogenesis takes place, where the resultant baryon number becomes independent of the reheating temperature.",
"Eventually the situation reduces to the case (I).",
"Such qualitative behavior can be confirmed by numerical study, which is left for the future work.",
"Crucial difference from sneutrino leptogenesis [38], [39], [40], [41], [42] is that although $M_3$ and $\\tilde{m}_3$ , i.e., $(y_\\nu y_\\nu ^\\dagger )_{33}$ , are important parameters to determine baryon number, they are sequestered from other physical quantities, such as the heavy right-handed (s)neutrino masses or the light neutrino mass matrix.",
"Therefore, there is no consequence in other low energy experiments.",
"This is a feature of case (II)."
],
[
"Gravitino problem",
"In the framework of supergravity, a fair amount of gravitino $\\psi _\\mu $ can be produced in various ways in the thermal history of the universe.",
"Since the interactions of gravitino with the MSSM particles are Planck-suppressed, gravitino is long-lived and its decay can spoil the successful big-bang nucleosynthesis if it is unstable.",
"Although this problem can be avoided when gravitino is enough heavy to have the lifetime much shorter than 1 sec, gravitino decay produces the lightest superparticle (LSP).",
"Then the LSP produced by the decay may overclose the universe if the R-parity is conserved.",
"There are three types of production mechanism of gravitino in the model we consider; (i) the inflaton decay; (ii) thermal scattering from the thermal bath [51], [52], [53], [54]; (iii) decay of superparticles in the thermal bath [55], [56].",
"In general, process (i) includes gravitino pair production.",
"The decay width of the mode, however, depends on the inflaton VEV [57], [58], [59], [60], [61], [62].",
"In our case, therefore, this process can be ignored since the inflaton does not have a VEV.",
"On the other hand, the inflaton can decay to gravitino and right-handed neutrino.",
"The decay width is given by $\\Gamma _{\\phi \\rightarrow \\psi _{\\mu }N_3}=\\frac{\\beta _f^{3}m_\\phi ^5}{48\\pi M_{pl}^2m_{3/2}^2}\\left[1-\\frac{(m_f+m_{3/2})^2}{m_\\phi ^2}\\right]\\,,$ where $\\beta _f^{2}=1-2(m_f^2+m_{3/2}^2)/m_\\phi ^2+(m_f^2-m_{3/2}^2)^2/m_\\phi ^4\\,.$ Here $m_f$ and $m_{3/2}$ are masses of $N_3$ and $\\psi _\\mu $ , respectively.",
"The mass difference between $\\phi $ and $N_3$ is expected to be given by the soft SUSY breaking mass scale for scalar superpartners, $|m_\\phi -m_f|\\sim \\tilde{m}$ .",
"Let us assume this decay happens by taking $m_\\phi =m_f+\\tilde{m}$ where $\\tilde{m}=k m_{3/2}$ ($k>1$ ).",
"In the limit $km_{3/2}/m_\\phi \\ll 1$ , we obtain $\\Gamma _{\\phi \\rightarrow \\psi _{\\mu }N_3}\\simeq \\frac{Cm_\\phi m_{3/2}^2}{3\\pi M^2_{pl}}\\,,$ where $C=(k-1)(k^2-1)^{3/2}$ .",
"Then the branching fraction of this mode is ${\\rm Br}_{\\phi \\rightarrow \\psi _{\\mu }N_3} =\\frac{\\Gamma _{\\phi \\rightarrow \\psi _{\\mu }N_3}}{\\Gamma _{\\phi }}\\simeq 4.5\\times 10^{-16}C\\left(\\frac{m_{3/2}}{10^{5}\\,{\\rm GeV}}\\right)^2\\left(\\frac{10^{-11}}{(y_\\nu y_\\nu ^\\dagger )_{33}}\\right)\\,.$ Therefore, it is sure that the inflaton decay reheats the universe in wide range of gravitino mass region.",
"On the other hand, the resultant gravitino abundance produced by the decay is estimated as $\\Omega _{3/2}^{\\rm inf}h^2\\sim 1.3\\times 10^{-6} C\\left(\\frac{m_{3/2}}{10^{5}\\,{\\rm GeV}}\\right)^3\\left(\\frac{10^{13}\\,{\\rm GeV}}{m_{\\phi }}\\right)^{1/2}\\left(\\frac{10^{-11}}{(y_\\nu y_\\nu ^\\dagger )_{33}}\\right)^{1/2}\\,,$ where $h$ is the scale factor of Hubble expansion rate.",
"Gravitino abundance via process (ii) is most effective at high temperature, thus it is proportional to $T_R$ , meanwhile in process (iii) gravitino is dominantly produced when the temperature is around the mass of decaying particle.",
"Adopting the expression given in Ref.",
"[63], the abundances via processes (ii) and (iii) are given by $&\\Omega _{3/2}^{\\rm TH}h^2\\sim 4\\left(\\frac{T_R}{10^{9}\\,{\\rm GeV}}\\right)\\left(\\frac{m_{3/2}}{10^{5}\\,{\\rm GeV}}\\right)\\,,\\\\&\\Omega _{3/2}^{\\rm FI}h^2\\sim 5\\times 10^{-4}k^3\\left(\\frac{m_{3/2}}{10^5\\,{\\rm GeV}}\\right)^2\\,.$ Here the contribution of the longitudinal mode of gravitino is suppressed in $\\Omega _{3/2}^{\\rm TH}$ by considering gluino is lighter than gravitino.",
"In $\\Omega _{3/2}^{\\rm FI}$ , we have assumed that all scalar leptons and quarks in the MSSM (whose mass scale is $\\tilde{m}$ ) are heavier than gravitino and that they are thermalized.",
"And as in the discussion of the inflaton decay, $\\tilde{m}=km_{3/2}$ has been taken.11We have checked that the contribution from $\\tilde{N}_1$ is negligible even if $\\tilde{N}_1$ is thermalized, i.e., in case (I).",
"Then the relic abundance of the LSP due to gravitino decay is estimated as $\\Omega _{\\rm LSP}^{\\rm nonth}h^2=\\frac{m_{\\rm LSP}}{m_{3/2}}\\Omega _{3/2}h^2\\,,$ where $m_{\\rm LSP}$ is the LSP mass and $\\Omega _{3/2}=\\Omega _{3/2}^{\\rm inf}+\\Omega _{3/2}^{\\rm TH}+\\Omega _{3/2}^{\\rm FI}$ is the total gravitino abundance.",
"$\\Omega _{\\rm LSP}^{\\rm nonth}h^2$ should not exceed the observed dark matter abundance $\\Omega _{\\rm DM}h^2\\simeq 0.12$ [35], which gives a constraint on gravitino mass.",
"Figure: Relic density of the LSP as function of gravitino mass.“inf” (dotted green), “TH” (dot-dashed red), and “FI”(dashed blue) are the contributions from process (i), (ii), and(iii), respectively.",
"Ω DM h 2 ≃0.12\\Omega _{\\rm DM}h^2\\simeq 0.12 (solid brown) isindicated as a reference.",
"Right region from vertical line (dottedviolet) indicates T 3/2 >m LSP T_{3/2}>m_{\\rm LSP}, and shaded region isexcluded.",
"m LSP =1 TeV m_{\\rm LSP}=1\\,{\\rm TeV}, m φ =10 13 GeV m_\\phi =10^{13}\\,{\\rm GeV},(y ν y ν † ) 33 =10 -11 (y_\\nu y_\\nu ^\\dagger )_{33}=10^{-11}, and k=m ˜/m 3/2 =2k=\\tilde{m}/m_{3/2}=2(left), 10 (right) are taken.Fig.",
"REF shows the resultant $\\Omega _{\\rm LSP}^{\\rm nonth}$ from the each contribution.",
"Here $m_{\\rm LSP}=1\\,{\\rm TeV}$ is taken by considering 1 TeV Higgsino or Wino dark matter with a mass of 2.7 – 3 TeV [64].",
"$m_\\phi =10^{13}\\,{\\rm GeV}$ , $(y_\\nu y_\\nu ^\\dagger )_{33}=10^{-11}$ , and $k=2$ (left), 10 (right) are taken to determine the contributions from processes (i) and (iii).",
"It is seen that in lower gravitino mass region, the dominant contribution to $\\Omega _{\\rm LSP}^{\\rm nonth}$ is from process (ii).",
"In order for the contribution not to exceed the dark matter abundance, $T_R\\lesssim 10^9\\,{\\rm GeV}$ is required, which is well-known result.",
"This gives a stringent constraint on the parameter space for successful leptogenesis shown in Figs.",
"REF and REF if gravitino is not heavy enough.",
"Namely, $\\tilde{m}_1\\sim 10^{-2}\\,{\\rm eV}$ is the allowed region in case (I) meanwhile $10^{-13}\\,{\\rm eV}\\lesssim \\tilde{m}_3\\lesssim 10^{-11}\\,{\\rm eV}$ is allowed in case (II).",
"On the other hand, the contributions from processes (i) and (iii) depend on the parameters, especially $k=\\tilde{m}/m_{3/2}$ (and $m_{3/2}$ ).",
"In order for $\\Omega _{\\rm LSP}^{\\rm nonth}$ not to exceed the dark matter abundance, the upper bound on gravitino mass is obtained depending on the mass spectrum of squarks and sleptons, e.g., $m_{3/2}\\lesssim 10^8$ ($10^{6}$ ) GeV for $\\tilde{m}\\sim m_{3/2}$ $(10m_{3/2})$ .",
"Such gravitino mass is preferred in minimal or mini split supersymmetry [65], [66], pure gravity mediation [67], [68], and spread supersymmetry [63], [69].",
"On the other hand, there is also an allowed region in higher gravitino mass region.",
"This is because gravitino decays before thermal freeze-out of the LSP in that region.",
"The allowed region can be estimated by imposing gravitino decay temperature $T_{3/2}$ larger than the LSP mass.",
"The gravitino decay temperature is defined by $T_{3/2}\\simeq (90/\\pi ^2g_*(T_{3/2}))^{1/4}\\sqrt{\\Gamma _{3/2} M_{pl}}\\,,$ where $\\Gamma _{3/2}$ is the decay rate of gravitino.",
"Then $m_{3/2}\\gtrsim 3\\times 10^{8}\\,{\\rm GeV}$ is obtained from $T_{3/2}>m_{\\rm LSP}$ for $m_{\\rm LSP}=1\\,{\\rm TeV}$ (see e.g., Ref. [70]).",
"Such high gravitino mass can be considered high-scale SUSY [71], intermediate scale supersymmetry [72], and unified inflation model [16].",
"However, gravitino cannot be too heavy because ${\\rm Br}_{\\phi \\rightarrow \\psi _{\\mu }N_3}$ should be less than unity for the reheating.",
"Taking ${\\rm Br}_{\\phi \\rightarrow \\psi _{\\mu }N_3} \\lesssim 0.1$ , for example, upper bound on gravitino mass is obtained as $m_{3/2}\\lesssim 5\\times 10^{11}\\,{\\rm GeV}$ ($2\\times 10^{10}\\,{\\rm GeV}$ ) for $k=2$ (10).",
"Another option is the R-parity violation.",
"Under the R-parity violation, the LSP decays to the standard-model particles.",
"Then, the LSP does not contribute to the matter abundance of the universe so that there is no constraint on $m_{3/2}$ ."
],
[
"Conclusion",
"Superconformal subcritical hybrid inflation is one of attractive inflation models that are consistent with the observed cosmological parameters by the Planck satellite.",
"In this paper we have studied the cosmology of an extended version of the model.",
"In the model three right-handed neutrinos are introduced.",
"The superpotential consists of one in the supersymmetric seesaw model and the interaction terms of the right-handed neutrinos with the additional matter fields, one of which plays the role of the waterfall field.",
"In the Kähler potential, on the other hand, it is possible for the sneutrinos to have shift symmetry by introducing explicit superconformal breaking terms of ${\\cal O}(1)$ .",
"Due to the shift symmetry, one of the sneutrinos becomes the inflaton field similarly in superconformal subcritical hybrid inflation.",
"Although the mass terms of the sneutrinos can affect the trajectory of the inflaton, it has turned out that the effect is restrictive and viable inflation is realized.",
"After inflation, the inflaton field decays to Higgses and sleptons or Higgsinos and leptons to reheat the universe.",
"Light neutrino masses are given by the seesaw mechanism.",
"However, the mass matrix is different from the conventional one.",
"It turns out that one of the neutrinos is massless.",
"Assuming that suppressed couplings of the other right-handed neutrinos to the waterfall field, it has been found that the neutrino Yukawa couplings that couples the inflaton to the MSSM sector are not constrained by the neutrino oscillation data.",
"Consequently, the reheating temperature is a free parameter, which is suitable for leptogenesis.",
"We have considered two representative cases; (I) the other right-handed (s)neutrinos are lighter than the inflaton; (II) the other right-handed (s)neutrinos are heavier than the inflaton.",
"In case (I), thermal leptogenesis is possible if the reheating temperature is larger than $\\sim 10^{9}\\,{\\rm GeV}$ .",
"It has been found leptogenesis is successful in a wide range of parameter space in the normal hierarchy case while the parameter space for leptogenesis is relatively limited in the inverted hierarchy case.",
"In case (II), on the other hand, sneutrino leptogenesis takes place if the reheating temperature is larger than $\\sim 10^{8}\\,{\\rm GeV}$ .",
"It has turned out that in both the normal and inverted hierarchy cases successful leptogenesis is realized in wide range of parameter space."
],
[
"Acknowledgments",
"We are grateful to Wilfried Buchmüller, Pasquale Di Bari, Oleg Lebedev, and Fuminobu Takahashi for valuable discussions.",
"This work was supported by JSPS KAKENHI Grant Numbers JP17H05402, JP17K14278, JP17H02875 and JP18H05542 (KI)."
]
] | 1906.04530 |
[
[
"Retrospective Motion Correction of MR Images using Prior-Assisted Deep\n Learning"
],
[
"Abstract In MRI, motion artefacts are among the most common types of artefacts.",
"They can degrade images and render them unusable for accurate diagnosis.",
"Traditional methods, such as prospective or retrospective motion correction, have been proposed to avoid or alleviate motion artefacts.",
"Recently, several other methods based on deep learning approaches have been proposed to solve this problem.",
"This work proposes to enhance the performance of existing deep learning models by the inclusion of additional information present as image priors.",
"The proposed approach has shown promising results and will be further investigated for clinical validity."
],
[
"Introduction",
"One of the most common causes of artefacts in MR imaging is patient motion , , .",
"Physiological motion, such as cardiac or respiratory movement, can be controlled by gating or by specific sequence design .",
"The focus of this work is on correcting for less predictable voluntary or involuntary movement of the patient, often called, bulk motion , which e.g.",
"in Parkinsonism and leads to ghosting and blurring .",
"If the level of artefacts is too high, the scan must be repeated.",
"Even a moderate level may lead to an incomplete or inaccurate diagnosis [1].",
"The most common strategies for handling motion artefacts can be divided into three main groups: motion prevention, artefact reduction, and motion correction.",
"Motion prevention spans from training the subject, foam restraints, feed and wrap (for babies), sedation till breath-hold.",
"Artefact reduction can be achieved with faster imaging, triggering and gating, phase reordering, and similar techniques.",
"Finally, there are motion correction techniques such as retrospective motion correction (RMC) and prospective motion correction (PMC) .",
"In practice, RMC modifies the MR image data during the reconstruction process, while PMC performs a real-time adaptive update of the data acquisition with the possibility of using different tracking modalities .",
"Recently, an increasing number of attempts using deep learning techniques have shown that it is possible to remove or reduce motion artefacts , , , , , , .",
"It has also been observed that supplying additional prior knowledge, such as image priors, can help improve the performance of deep learning based reconstructions , .",
"This work aims to remove motion artefacts from corrupted brain images by improving existing deep learning models, by exploiting available non-corrupted image priors."
],
[
"Data Preparation",
"T1, T2, and PD images of 100 subjects (for each - training, testing, and validation) from the publicly available IXI DatasetDataset available at: https://brain-development.org/ixi-dataset/ were used in this study.",
"T2-weighted images were artificially corrupted with motion using a modified version of the RandomMotion transformation of TorchIO (v0.17.45).",
"The initial phase of the experiments were performed with 10 simulated movements with a rotation in the range of -1.75 to +1.75 degrees without any translation.",
"In this modified version of the RandomMotion function, in-plane motion corruption was randomly performed in X or Y direction."
],
[
"Image Priors",
"Supplying additional images as prior knowledge along with the corrupted image can help improve the performance of deep learning models , .",
"In this research, experiments were performed using two different types of image priors - similar slices from different subjects of the same MRI contrast and different MRI contrasts of the same subject."
],
[
"Similar Slices:",
"During the motion correction, ten similar (same slice position) slices of the same MRI contrast were randomly chosen from different subjects and supplied as prior along with the motion corrupted image.",
"The motivation behind this type of prior is that while performing motion correction on certain image, images of the same contrast but of different subjects, which are not corrupted by motion, can readily be available.",
"In these experiments, only T2-weighted images from the IXI Dataset were used."
],
[
"Different Contrasts of the Same Subject:",
"Multiple contrasts of the same subject are usually acquired during clinical routine.",
"If one of those various contrasts is corrupted by motion, then that image can be corrected by using the other contrasts of the same subject as prior.",
"All three available contrasts of the IXI Dataset were co-registered against the T2-weighted images.",
"T2-weighted images were artificially corrupted; T1 and PD images were used as priors during the correction process."
],
[
"Network Architectures",
"UNet and a modified version of the ResNet were used as the baselines for this work.",
"The baseline networks were modified such that they can receive priors.",
"Two different methods of supplying priors were tested."
],
[
"Multi-Channel Network:",
"Each motion corrupted image and the priors were concatenated on the channel dimension and were supplied to the network as multi-channel input.",
"The baselines were receiving only one channel image as input, where for the multi-channel approach, the models received $1+n\\_{prior}$ channel images as input."
],
[
"Dual-Branch Network:",
"For this approach, modified versions of the baselines were created by adding an additional branch to the baselines for the priors.",
"The main branch was supplied with the motion corrupted image and the priors were supplied to the auxiliary branch.",
"For the UNet, the auxiliary branch was identical to the contraction path and the latent space of the UNet except for the number of input channels.",
"The skip-connections were supplied only from the main branch of the network, and no skip-connections were supplied from the auxiliary branch.",
"For the ResNet, the auxiliary branch was identical to the downsampling blocks of the ResNet up to the residual blocks, with the only difference being the number of input channels.",
"For both network models, the main branch and the auxiliary branch generated two different latent space representations.",
"These latent representations were combined and forwarded for generating the final output.",
"Two different methods were considered for combining the latent spaces - by simple addition or by concatenation and convolution with a kernel size of one - to generate the final combined latent space.",
"This combined latent space was forwarded to the expansion path of the UNet.",
"In the case of ResNet, this latent representation was forwarded to the residual blocks for further operation."
],
[
"Results and Discussions",
"Figure REF shows the performance of the different methods based on the SSIM values and Figure REF shows a representative example result.",
"Between the two different types of priors, it was observed that supplying ten similar slices of the same contrast but of different subjects did not improve the motion correction.",
"Nonetheless, supplying different contrasts of the same subject improved the motion correction significantly for most of the tests carried out.",
"ResNet's performance improved for both types of prior supply methods - multi-channel and dual-branch.",
"For UNet however, only the multi-channel approach has shown significant improvement.",
"Figure: Plots showing the performance of the various methods, based on SSIMFigure: One example slice to show the motion correction performance of the various methods"
],
[
"Conclusion and Future Work",
"The initial experiments presented here have shown promising results for supplying other contrasts as prior for deep learning based motion correction.",
"For ResNet, the multi-channel and the dual-network approach both have shown promising results.",
"However, for UNet only the multi-channel approach has shown improvements over the baseline.",
"The reason for the failure could be attributed to the lack of skip connections from the auxiliary branch, but the skip connections will be making it similar to the multi-channel approach.",
"Further investigations will be performed to determine the cause of the failure, and also other strategies for supplying the priors to the UNet.",
"Further experiments will be performed to check the performance of these approaches when the corrupted volumes and the prior volumes are not co-registered.",
"Supplying ten similar slices as prior did not show any improvement for any of the networks.",
"Further investigations will be performed to determine the reasons and possible solutions.",
"Furthermore, the strategies will be validated with clinical data.",
"It will be assessed whether pathologies that can only be seen in a corrupted contrast, are preserved after performing the correction."
],
[
"Broader Impact",
"The current stage of the work has been carried out on simulated motion artefacts only and using a publicly available dataset and has been carried out in compliance with ethical standards.",
"To the best of the authors' knowledge, there are not evident direct or indirect effects on societal aspects.",
"It is well known that low-quality images invalidate the clinical diagnosis.",
"In addition, the repetition of scans involves the use of energy, money and human resources.",
"Therefore, researchers and clinicians working in the field of MRI may benefit from this work.",
"As this research is in its preliminary stage, it has still to be investigated whether the usage of priors exclude pathologies from the corrected data.",
"This work was partially conducted within the context of the International Graduate School MEMoRIAL at Otto von Guericke University (OVGU) Magdeburg, Germany, kindly supported by the European Structural and Investment Funds (ESF) under the programme \"Sachsen-Anhalt WISSENSCHAFT Internationalisierung\" (project no.",
"ZS/2016/08/80646).",
"This work was also partially conducted within the context of the Initial Training Network program, HiMR, funded by the FP7 Marie Curie Actions of the European Commission, grant number FP7-PEOPLE-2012-ITN-316716, and supported by the NIH grant number 1R01-DA021146, and by the federal state of Saxony-Anhalt under grant number 'I 88'."
]
] | 2011.14134 |
[
[
"Non-uniform dependence on initial data for the 2D viscous shallow water\n equations"
],
[
"Abstract The failure of uniform dependence on the data is an interesting property of classical solution for a hyperbolic system.",
"In this paper, we consider the solution map of the Cauchy problem to the 2D viscous shallow water equations which is a hyperbolic-parabolic system.",
"We prove that the solution map of this problem is not uniformly continuous in Sobolev spaces $H^s\\times H^{s}$ for $s>2$."
],
[
"Introduction",
"The 2D viscous shallow water equations are given by the following system, which have systematically introduced in [2], [3] $\\left\\lbrace \\begin{array}{ll}\\partial _t\\rho +\\rho u)=0,\\\\\\partial _t(\\rho u)+\\rho u\\otimes u)-\\mu \\rho \\nabla u)+\\rho \\nabla \\rho =0,\\\\\\rho |_{t=0}=\\rho _0,\\;u|_{t=0}=u_0.\\end{array}\\right.$ where $\\rho (x,t)$ is the height of fluid surface, $u(x,t)$ is the horizontal velocity field and $\\mu >0$ is the viscous coefficient.",
"We suppose that the initial data $\\rho _0(x)$ is a small perturbation of some positive constant $\\overline{\\rho }_0$ .",
"Classical solutions and well-posedness of the initial (boundary) value problem for the shallow water equations (REF ) have been studied extensively.",
"By using Lagrangian coordinates and Hölder space estimates, Bui [1] obtained the local existence and uniqueness of classical solutions to the Cauchy-Dirichlet problem for (REF ) with initial data in $C^{2+\\alpha }$ .",
"With the help of the energy method of Matsumura and Nishida [33], Kloeden [26] and Sundbye [34] independently showed the global existence and uniqueness of classical solutions to the Cauchy-Dirichlet problem for (REF ).",
"Subsequently, Sundbye [35] also proved the existence and uniqueness of classical solutions to the Cauchy problem for (REF ) using the method of [33].",
"Due to the strong nonlinearity of system (REF ), the problem of existence of solutions for large initial data is difficult.",
"By applying the Littlewood-Paley decomposition theory for Sobolev spaces to obtain a losing energy estimate in $H^s$ for any $s > 2$ , Wang–Xu [36] obtained local solutions for any initial data and global solutions to (REF ) for small initial data $(u_0, \\rho _0-\\bar{\\rho }_0) \\in H^s\\times H^s$ with $s>2$ .",
"Liu–Yin [30], [31], [32] improved the result of [36] in the Sobolev spaces with low regularity and inhomogeneous Besov spaces.",
"Chen–Miao–Zhang [6] obtained the local well-posedness of system (REF ) for general initial data in critical $L^p$ type Besov spaces with $1\\le p<4$ by developing a new method which relies on the smoothing properties of the heat equations.",
"Recently, Li–Hong–Zhu [28] proved that the system (REF ) is ill-posed in the critical Besov spaces with $p > 4$ .",
"The continuous dependence is particularly important when PDEs are used to model phenomena in the natural world since measurements are always associated with errors.",
"One of the first results of this type was proved by Kato [27] who showed that the solution operator for the (inviscid) Burgers equation is not Hölder continuous in the $H^s(\\mathbb {T})$ -norm $(s > 3/2)$ for any Hölder exponent.",
"After the phenomenon of non-uniform continuity for some dispersive equations was studied by Kenig et al.",
"[24], many results with regard to the non-uniform dependence on the initial data have been obtained for other nonlinear PDEs including the Euler equations [18], the Camassa-Holm equation [19], [20], [29], the Benjamin-Ono equation [25], the compressible gas dynamics [21], [23], the Hunter-Saxton equation [22] and so on.",
"Nevertheless we notice that almost the above system mentioned is hyperbolic.",
"As stated in [23], the exhibition of nonuniform behavior in a hyperbolic system related to the incompressible system indicates that the nonuniform dependence is hyperbolic in nature.",
"From the PDE's point of view, classical solution is uniform dependence on the data for a parabolic system.",
"Naturally, one may wonder if uniform dependence on the data of solution for a coupled system which is hyperbolic-parabolic can persist or not.",
"In this paper, we focus on the 2D viscous shallow water equations $\\left\\lbrace \\begin{array}{ll}\\partial _t\\rho +\\rho u)=0,\\\\\\partial _t(\\rho u)+\\rho u\\otimes u)-\\mu 2\\rho \\mathbb {D}u)+\\rho \\nabla \\rho =0,\\\\\\rho |_{t=0}=\\rho _0,\\;u|_{t=0}=u_0.\\end{array}\\right.$ where the strain tensor $\\mathbb {D}u =\\frac{1}{2}(\\nabla u+\\nabla ^{\\mathsf {T}} u)$ is the symmetric part of the velocity gradient.",
"For the sake of convenience, we take $\\bar{\\rho }_0 = 1$ and denote $\\varrho =\\rho -1$ , we can reformulate the system (REF ) equivalently as follows $\\left\\lbrace \\begin{array}{ll}\\partial _t\\varrho +u̥+u\\cdot \\nabla \\varrho =-\\varrho u̥,\\\\\\partial _tu-{\\mu } \\Delta u-{\\mu } \\nabla u̥+u\\cdot \\nabla u+\\nabla \\varrho =2\\nabla (\\ln (1+\\varrho ))\\cdot \\mathbb {D}u,\\\\\\varrho |_{t=0}=\\varrho _0,\\;u|_{t=0}=u_0.\\end{array}\\right.$ Roughly speaking, the system (REF ) can be regarded as a special case of compressible Navier-Stokes equations (see Refs.",
"[6], [7], [8], [9], [10], [11], [12], [14], [15], [16], [17] and the references therein).",
"We emphasize that the equations (REF ) form a quasi-linear hyperbolic-parabolic system and possess strong nonlinear terms.",
"Although the system (REF ) is partially parabolic, owing to the first equation of (REF ) which is of hyperbolic type, this creates the possibility of the non-uniformity for the solution map.",
"In this paper, we prove the failure of uniform dependence on the data for (REF ).",
"To the best of our knowledge, it allows us to give a first kind of answer to the problem of the non-uniform dependence on the data for the 2D viscous shallow water equations.",
"Our main result of this paper is stated: Theorem 1.1 (Nonuniform dependence on initial data) Let $s>2$ .",
"The data-to-solution map $(u_0,\\varrho _0)\\mapsto \\big (u(t),\\varrho (t)\\big )$ of the Cauchy problem (REF ) is not uniformly continuous from any bounded subset in $H^{s}\\times H^{s}$ into $\\mathcal {C}([0,T];H^{s}\\times H^{s})$ , namely, there exists two sequences of solutions $\\big (u_{1,n}(t),\\varrho _{1,n}(t)\\big )$ and $\\big (u_{2,n}(t),\\varrho _{2,n}(t)\\big )$ such that (C.1) $\\Vert u_{1,n},\\varrho _{1,n}\\Vert _{H^{s}}+\\Vert u_{2,n},\\varrho _{2,n}\\Vert _{H^{s}}\\lesssim 1;$ (C.2) $\\lim \\limits _{n\\rightarrow \\infty }\\big (\\Vert {u}_{2,n}(0)-u_{1,n}(0)\\Vert _{H^{s}}+\\Vert \\varrho _{2,n}(0)-\\varrho _{1,n}(0)\\Vert _{H^{s}}\\big )=0;$ (C.3) $\\liminf \\limits _{n\\rightarrow \\infty }\\big (\\Vert u_{2,n}(t)-u_{1,n}(t)\\Vert _{H^{s}}+\\Vert \\varrho _{2,n}(t)-\\varrho _{1,n}(t)\\Vert _{H^{s}}\\big )\\gtrsim t$ for any $t\\in [0,T_0]$ with small time $T_0$ .",
"Remark 1.1 It should be mentioned that most non-uniformity results for solutions to other nonlinear PDEs were obtained by the Himonas-Misiołek construction in [18].",
"The method we used in proving Theorem REF is motivated by [29] and is completely different from that [18].",
"Remark 1.2 We note that the local well-posedness result holds for $s>1$ .",
"However, the restriction on $s>2$ is essential in the present paper.",
"Outline of the proof to Theorem REF There are two key points in proving Theorem REF : On one hand, we construct one sequence of initial data $(u_{1,n}(0),\\varrho _{1,n}(0))=(0,f_n)$ , which leads to the solutions $(u_{1,n}(t),\\varrho _{1,n}(t))$ to (REF ) and $(u^{\\rm {ap}}_{1,n},\\varrho ^{\\rm {ap}}_{1,n})$ to the linearized system (REF ) respectively.",
"Based on the special choice of $f_n$ , we can prove the distance between the real solution $(u_{1,n}(t),\\varrho _{1,n}(t))$ and the approximate solution $(u^{\\rm {ap}}_{1,n},\\varrho ^{\\rm {ap}}_{1,n})$ will tends to zero in $H^s$ as $n\\rightarrow \\infty $ .",
"See Propositions REF .",
"On the other hand, we construct another sequence of initial data $(u_{2,n}(0),\\varrho _{2,n}(0))=(g_n,f_n)$ , which leads to the solutions $(u_{2,n}(t),\\varrho _{2,n}(t))$ to (REF ) and $(u^{\\rm {ap}}_{2,n},\\varrho ^{\\rm {ap}}_{2,n})$ to the linearized system (REF ) respectively.",
"Next, we aim to show that the mentioned solution $(u^{\\rm {ap}}_{2,n},\\varrho ^{\\rm {ap}}_{2,n})$ cannot approximate the real solution $(u_{2,n}(t),\\varrho _{2,n}(t))$ .",
"For more details, see Propositions REF .",
"Combining the precious steps, we can conclude that the distance of two solution maps at the initial time is converging to zero, while at any later time it is bounded below by a positive constant, which means the solution maps are not uniformly continuous.",
"The structure of the paper In Section we recall some notations and known results which will be used in the sequel.",
"Also, we investigate the spectrum properties of the linearized system corresponding to (REF ), which will play a crucial role in the construction of approximate solutions.",
"In Section we prove Theorem REF based on the well-posedness result and some key error estimates.",
"In Section we present some details in the computations."
],
[
"Notations",
"Firstly, we introduce some notations which shall be used in this paper.",
"The notation $A\\lesssim B$ (resp., $A \\gtrsim B$ ) means that there exists a harmless positive constant $c$ such that $A \\le cB$ (resp., $A \\ge cB$ ).",
"$A\\approx B$ means $A\\lesssim B$ and $A\\gtrsim B$ .",
"Given a Banach space $X$ , we denote its norm by $\\Vert \\cdot \\Vert _{X}$ .",
"We shall use the simplified notation $\\Vert f,\\cdots ,g\\Vert _X=\\Vert f\\Vert _X+\\cdots +\\Vert g\\Vert _X$ if there is no confusion.",
"We shall denote by $\\langle f,g\\rangle $ the $L^2$ inner product of $f$ and $g$ .",
"For all $f\\in \\mathcal {S}^{\\prime }$ , the Fourier transform $\\mathcal {F}f$ (also denoted by $\\widehat{f}$ ) is defined by $\\mathcal {F}f(\\xi )=\\widehat{f}(\\xi )=\\int _{\\mathbb {R}^2}e^{-ix\\xi }f(x)\\mathrm {d}x \\quad \\text{for any}\\; \\xi \\in \\mathbb {R}^2.$ We denote $\\Lambda =(-\\Delta )^{\\frac{1}{2}}$ .",
"For $s\\in \\mathbb {R}$ , the operator $\\Lambda ^s$ is defined by $\\widehat{\\Lambda ^s f}(\\xi )=|\\xi |^{s}\\widehat{f}(\\xi ).$ The homogeneous and nonhomogeneous Sobolev space are defined by $&\\Vert f\\Vert ^2_{\\dot{H}^s}=\\Vert \\Lambda ^sf\\Vert ^2_{L^2}=\\int _{\\mathbb {R}^2}|\\xi |^{2s}|\\hat{f}(\\xi )|^2\\mathrm {d}\\xi ,\\\\&\\Vert f\\Vert ^2_{H^s}=\\int _{\\mathbb {R}^2}(1+|\\xi |^2)^s|\\hat{f}(\\xi )|^2\\mathrm {d}\\xi .$ Then, for $s>0$ , we have $\\Vert f\\Vert ^2_{H^s}\\approx \\Vert f\\Vert ^2_{L^2}+\\Vert f\\Vert ^2_{\\dot{H}^s}.$"
],
[
"Useful Tools",
"Next, we review some useful tools involving the commutator and product estimates.",
"Lemma 2.1 (See [13]) Let $s>1$ and $\\nabla f,g\\in H^s(\\mathbb {R}^2)$ .",
"Then we have $\\Vert [\\Lambda ^s,f]\\cdot \\nabla g\\Vert _{L^2}\\le C\\Vert \\nabla f\\Vert _{H^s}\\Vert g\\Vert _{H^s}.$ Lemma 2.2 (See [27]) Let $s>0$ and $f,g\\in {\\rm {Lip}}\\cap H^s(\\mathbb {R}^2)$ and $g\\in L^\\infty \\cap H^{s-1}(\\mathbb {R}^2)$ .",
"Then we have $\\Vert [\\Lambda ^s,f]g\\Vert _{L^2}\\le C\\big (\\Vert \\nabla f\\Vert _{L^\\infty }\\Vert g\\Vert _{H^{s-1}}+\\Vert f\\Vert _{H^s}\\Vert g\\Vert _{L^\\infty }\\big ).$ Lemma 2.3 (See [4]) Let $s>2$ and $f\\in H^{s-2}(\\mathbb {R}^2),g\\in H^{s-1}(\\mathbb {R}^2)$ .",
"Then we have $\\Vert fg\\Vert _{H^{s-2}}\\le C\\Vert f\\Vert _{H^{s-2}}\\Vert g\\Vert _{H^{s-1}}.$ Lemma 2.4 (See [4]) Let $s>0$ and $f,g\\in L^\\infty \\cap H^s(\\mathbb {R}^2)$ .",
"Then we have $\\Vert fg\\Vert _{H^s}\\le C\\big (\\Vert f\\Vert _{L^\\infty }\\Vert g\\Vert _{H^s}+\\Vert f\\Vert _{H^s}\\Vert g\\Vert _{L^\\infty }\\big ).$ Moreover, for $s>1$ , we have the algebra estimate $\\Vert fg\\Vert _{H^s}\\le C\\Vert f\\Vert _{H^s}\\Vert g\\Vert _{H^s}.$ Lemma 2.5 (See [4]) Let $s>0$ and $f$ be a smooth function such that $f(0)=0.$ If $u \\in H^{s}\\left(\\mathbb {R}^{2}\\right) \\cap L^{\\infty }\\left(\\mathbb {R}^{2}\\right)$ then there exists a function $C$ depending only on $s$ and $f$ such that $\\Vert f(u)\\Vert _{H^{s}} \\le C\\big (\\Vert u\\Vert _{L^{\\infty }}\\big )\\Vert u\\Vert _{H^{s}}.$ Lemma 2.6 (See [4]) Let $s>1$ and $f$ be a smooth function such that $f^{\\prime }(0)=0 .$ If $u, v \\in H^{s}\\left(\\mathbb {R}^{2}\\right) \\cap L^{\\infty }\\left(\\mathbb {R}^{2}\\right)$ , then there exists a function $C$ depending only on $s$ and $f$ such that $\\Vert f(u)-f(v)\\Vert _{H^{s}} \\le C\\big (\\Vert u\\Vert _{L^{\\infty }},\\Vert v\\Vert _{L^{\\infty }}\\big )\\Vert u-v\\Vert _{H^{s}}\\big (\\Vert u\\Vert _{H^{s}}+\\Vert v\\Vert _{H^{s}}\\big ).$"
],
[
"The Linearized System",
"In this subsection, borrowing the idea from [7], we investigate the spectrum properties of the linearized system: $\\left\\lbrace \\begin{array}{ll}\\partial _t\\varrho +u̥=0,\\\\\\partial _tu-{\\mu } \\Delta u-{\\mu } \\nabla u̥+\\nabla \\varrho =0,\\\\\\varrho |_{t=0}=\\varrho _0,\\;u|_{t=0}=u_0.\\end{array}\\right.$ Denote $d=\\Lambda ^{-1}u̥\\quad \\text{and}\\quad c=\\Lambda ^{-1}\\mathrm {curl} u,$ then we deduce from (REF ) $\\left\\lbrace \\begin{array}{ll}\\partial _t\\varrho +\\Lambda d=0,\\\\\\partial _td-2{\\mu } \\Delta d-\\Lambda \\varrho =0,\\\\\\partial _tc-{\\mu } \\Delta c=0\\end{array}\\right.$ Let $\\mathbf {A}=\\left(\\begin{array}{cc}0 & -|\\xi | \\\\|\\xi | & -2\\mu |\\xi |^2 \\\\\\end{array}\\right),$ by taking the Fourier transform of $(\\ref {l1})_1$ and $(\\ref {l1})_2$ , we find that $\\partial _t\\left(\\begin{array}{c}\\widehat{\\varrho }\\\\\\widehat{d} \\\\\\end{array}\\right)=\\mathbf {A}\\left(\\begin{array}{c}\\widehat{\\varrho }\\\\\\widehat{d} \\\\\\end{array}\\right).$ Straightforward calculations give the eigenvalues of the matric $\\mathbf {A}$ as follows $\\lambda _{\\pm }=-\\mu |\\xi |^2\\pm \\sqrt{\\mu ^2|\\xi |^4-|\\xi |^2}$ and $\\mathbf {T}^{-1}\\mathbf {A}\\mathbf {T}=\\left(\\begin{array}{cc}\\lambda _{+} & 0 \\\\0 & \\lambda _{-} \\\\\\end{array}\\right),$ where the matrixes $\\mathbf {T}$ and $\\mathbf {T}^{-1}$ are given by $\\mathbf {T}=\\left(\\begin{array}{cc}\\lambda _{-} & \\lambda _{+} \\\\-|\\xi | & -|\\xi | \\\\\\end{array}\\right)\\quad \\text{and}\\quad \\mathbf {T}^{-1}=\\left(\\begin{array}{cc}-\\frac{1}{\\lambda _{+}-\\lambda _{-}} & -\\frac{\\lambda _{+}}{|\\xi |(\\lambda _{+}-\\lambda _{-})} \\\\\\frac{1}{\\lambda _{+}-\\lambda _{-}} & \\frac{\\lambda _{-}}{|\\xi |(\\lambda _{+}-\\lambda _{-})} \\\\\\end{array}\\right).$ Applying $\\mathbf {T}^{-1}$ to (REF ) and using Duhamel's principle, we get $\\mathbf {T}^{-1}\\left(\\begin{array}{c}\\widehat{\\varrho }\\\\\\widehat{d} \\\\\\end{array}\\right)=\\left(\\begin{array}{cc}e^{\\lambda _+t}&0\\\\0&e^{\\lambda _-t}\\\\\\end{array}\\right)\\mathbf {T}^{-1}\\left(\\begin{array}{c}\\widehat{\\varrho }_0\\\\\\widehat{d}_0 \\\\\\end{array}\\right),$ which implies that $&\\widehat{\\varrho }(t,\\xi )=\\frac{\\lambda _+e^{\\lambda _-t}-\\lambda _-e^{\\lambda _+t}}{\\lambda _+-\\lambda _-}\\widehat{\\varrho }_0-\\frac{e^{\\lambda _+t}-e^{\\lambda _-t}}{\\lambda _+-\\lambda _-}|\\xi |\\widehat{d}_0,\\\\&\\widehat{d}(t,\\xi )=\\frac{e^{\\lambda _+t}-e^{\\lambda _-t}}{\\lambda _+-\\lambda _-}|\\xi |\\widehat{\\varrho }_0+\\frac{\\lambda _+e^{\\lambda _+t}-\\lambda _-e^{\\lambda _-t}}{\\lambda _+-\\lambda _-}\\widehat{d}_0.$ From $(\\ref {l1})_3$ , we also have $&\\widehat{c}(t,\\xi )=e^{-\\mu t|\\xi |^2}\\widehat{c}_0.$ Due to the vector identity $\\Delta u=\\nabla u̥+\\nabla ^{\\bot }\\mathrm {curl}u=\\Lambda (\\nabla d+\\nabla ^{\\bot } c),$ which implies $u=-\\Lambda ^{-1}\\nabla d-\\Lambda ^{-1}\\nabla ^{\\bot } c,$ then we deduce from () and (REF ) $\\widehat{u}(t,\\xi )&=-\\frac{e^{\\lambda _+t}-e^{\\lambda _-t}}{\\lambda _+-\\lambda _-}\\mathcal {F}(\\nabla \\varrho _0)-\\frac{\\lambda _+e^{\\lambda _+t}-\\lambda _-e^{\\lambda _-t}}{\\lambda _+-\\lambda _-}\\mathcal {F}\\big (\\Lambda ^{-2}\\nabla u̥_0\\big )\\nonumber \\\\&\\quad -e^{-\\mu |\\xi |^2t }\\mathcal {F}(\\Lambda ^{-2}\\nabla ^{\\bot }\\mathrm {curl} u_0).$"
],
[
"Proof of Theorem ",
"This section is devoted to proving Theorem REF .",
"We begin with the well-posedness result for (REF ).",
"For the details of the proof, we refer to [30], [31], [36]."
],
[
"Global Well-posedness ",
"Lemma 3.1 Assume that $s>2$ .",
"For any initial data $(u_0,\\varrho _0)$ which belongs to $B_R=\\big \\lbrace (\\psi ,\\phi )\\in H^s\\times H^s: \\Vert \\psi \\Vert _{H^s}+\\Vert \\phi \\Vert _{H^s}\\le c\\big \\rbrace \\quad \\text{for small}\\;c>0,$ then System (REF ) has a unique global solution $\\big (u(t),\\varrho (t)\\big )\\in \\mathcal {C}([0,+\\infty );H^s\\times H^s)$ .",
"Moreover, we have the solution size estimate $\\Vert u(t)\\Vert _{H^{\\sigma }}+\\Vert \\varrho (t)\\Vert _{H^{\\sigma }}\\le C\\Vert u_0,\\varrho _0\\Vert _{H^{\\sigma }}\\quad \\text{for all}\\; \\sigma \\ge s.$"
],
[
"Approximate Solutions",
"Let $\\widehat{\\phi }\\in \\mathcal {C}^\\infty _0(\\mathbb {R})$ be an even, real-valued and non-negative function on $\\mathbb {R}$ and satisfy ()= 1,if $|\\xi |\\le \\frac{1}{4}$ , 0,if $|\\xi |\\ge \\frac{1}{2}$ .",
"Let $(u^{\\rm {ap}}_{i,n},\\varrho ^{\\rm {ap}}_{i,n})$ ($i=1,2$ ) be the solution of the following equation $\\left\\lbrace \\begin{array}{ll}\\partial _t\\varrho ^{\\rm {ap}}_{i,n}+u̥^{\\rm {ap}}_{i,n}=0,\\\\\\partial _tu^{\\rm {ap}}_{i,n}-\\mu \\Delta u^{\\rm {ap}}_{i,n}-\\mu \\nabla u̥^{\\rm {ap}}_{i,n}+\\nabla \\varrho ^{\\rm {ap}}_{i,n}=0,\\\\\\end{array}\\right.$ supplemented with the following initial condition, respectively, $(u^{\\rm {ap}}_{1,n},\\varrho ^{\\rm {ap}}_{1,n})|_{t=0}=(0,f_n)\\quad \\text{and}\\quad (u^{\\rm {ap}}_{2,n},\\varrho ^{\\rm {ap}}_{2,n})|_{t=0}=(g_n,f_n),$ where $f_n=2^{-ns}\\phi (x_1)\\sin (2^nx_1)\\phi (x_2)\\quad \\text{and}\\quad g_n=2^{-n}\\nabla \\big (\\phi (x_1)\\phi (x_2)\\big ),\\quad n\\gg 1.$ The following Lemma will play an important role in the proof of Theorem REF .",
"Lemma 3.2 Let $\\sigma \\in \\mathbb {R}$ .",
"Assume that $(u^{\\rm {ap}}_{i,n},\\varrho ^{\\rm {ap}}_{i,n})$ with $i=1,2$ solves System (REF )–(REF ).",
"Then there exists a constant $C=C(\\sigma )$ such that the following statement holds $&\\Vert \\varrho ^{\\rm {ap}}_{1,n},u^{\\rm {ap}}_{1,n}\\Vert ^2_{H^\\sigma }+\\mu \\int ^t_0\\Vert \\nabla u^{\\rm {ap}}_{1,n},u̥^{\\rm {ap}}_{1,n}\\Vert ^2_{H^\\sigma }\\mathrm {d}\\tau \\le C2^{2n(\\sigma -s)},\\\\&\\Vert \\varrho ^{\\rm {ap}}_{2,n},u^{\\rm {ap}}_{2,n}\\Vert ^2_{H^\\sigma }+\\mu \\int ^t_0\\Vert \\nabla u^{\\rm {ap}}_{2,n},u̥^{\\rm {ap}}_{2,n}\\Vert ^2_{H^\\sigma }\\mathrm {d}\\tau \\le C2^{2n \\max \\lbrace \\sigma -s,-1\\rbrace }.$ Proof.",
"Standard energy method directly gives us that $\\frac{\\mathrm {d}}{\\mathrm {d}t}\\Vert \\varrho ^{\\rm {ap}}_{i,n},u^{\\rm {ap}}_{i,n}\\Vert ^2_{H^\\sigma }+\\mu \\Vert \\nabla u^{\\rm {ap}}_{i,n}\\Vert ^2_{H^\\sigma }+\\mu \\Vert u̥^{\\rm {ap}}_{i,n}\\Vert ^2_{H^\\sigma }=0,$ which implies $&\\Vert \\varrho ^{\\rm {ap}}_{1,n},u^{\\rm {ap}}_{1,n}\\Vert ^2_{H^\\sigma }+\\mu \\int ^t_0\\Vert \\nabla u^{\\rm {ap}}_{1,n},u̥^{\\rm {ap}}_{1,n}\\Vert ^2_{H^\\sigma }\\mathrm {d}\\tau =\\Vert f_n\\Vert ^2_{H^\\sigma },\\\\&\\Vert \\varrho ^{\\rm {ap}}_{2,n},u^{\\rm {ap}}_{2,n}\\Vert ^2_{H^\\sigma }+\\mu \\int ^t_0\\Vert \\nabla u^{\\rm {ap}}_{2,n},u̥^{\\rm {ap}}_{2,n}\\Vert ^2_{H^\\sigma }\\mathrm {d}\\tau =\\Vert f_n,g_n\\Vert ^2_{H^\\sigma }.$ Obviously, we have $&\\Vert g_n\\Vert _{H^\\sigma }\\lesssim 2^{-n}\\Vert \\phi (x_1)\\Vert _{L^2(\\mathbb {R})}\\Vert \\phi (x_2)\\Vert _{L^2(\\mathbb {R})}\\lesssim 2^{-n},\\nonumber \\\\&\\Vert f_n\\Vert _{H^\\sigma }\\lesssim 2^{-ns}\\Vert \\widetilde{f_n}(x_1,x_2)\\Vert _{H^\\sigma (\\mathbb {R}^2)},$ where $\\widetilde{f_n}(x_1,x_2)=\\phi (x_1)\\sin (2^nx_1)\\phi (x_2)$ .",
"Easy computations give that $\\mathcal {F}\\big (\\widetilde{f_n}\\big )(\\xi _1,\\xi _2)=\\frac{i}{2}\\Big [\\hat{\\phi }(\\xi _1+2^n)-\\hat{\\phi }(\\xi _1-2^n)\\Big ]\\hat{\\phi }(\\xi _2),$ which implies $\\mathrm {supp} \\ \\mathcal {F}\\big (\\widetilde{f_n}\\big )\\subset \\mathcal {C}_n:=\\Big \\lbrace \\xi \\in \\mathbb {R}^2: \\ 2^n-1\\le |\\xi |\\le 2^n+1\\Big \\rbrace ,$ By the definition of Sobolev space, we get $\\Vert \\widetilde{f_n}(x_1,x_2)\\Vert _{H^\\sigma (\\mathbb {R}^2)}&\\lesssim \\int _{\\mathbb {R}^2}(1+|\\xi |^2)^{\\sigma /2}|\\mathcal {F}\\big (\\widetilde{f_n}\\big )|^2\\mathrm {d}\\xi \\nonumber \\\\&\\lesssim ~2^{n\\sigma }\\int _{\\mathcal {C}_n }|\\mathcal {F}\\big (\\widetilde{f_n}\\big )|^2\\mathrm {d}\\xi \\nonumber \\\\&\\lesssim 2^{n\\sigma }.$ Inserting (REF ) into (REF ) enables us to finish the proof of Lemma REF .",
"Before proceeding on, we give two data-to-solution maps for the Cauchy problem (REF ) $&\\big (u_{1,n}(0)=0,\\varrho _{1,n}(0)=f_n\\big )\\mapsto \\big (u_{1,n},\\varrho _{1,n}\\big ),\\\\&\\big ({u}_{2,n}(0)=g_n,{\\varrho }_{2,n}(0)=f_n\\big )\\mapsto \\big ({u}_{2,n},{\\varrho }_{2,n}\\big ),$ where $f_{n}$ and $g_n$ are defined in Section REF .",
"Let $\\varepsilon _s=\\frac{1}{2}(s-2)$ and $s^{\\prime }=s-\\varepsilon _s>2$ .",
"For the initial data with $H^{s^{\\prime }}$ norm, we have $\\Vert u_{1,n}(0),u_{2,n}(0)\\Vert _{H^{s^{\\prime }}}+\\Vert \\varrho _{1,n}(0),\\varrho _{2,n}(0)\\Vert _{H^{s^{\\prime }}}\\le 2^{-n\\min \\lbrace \\varepsilon _s,1\\rbrace },$ which tends to 0 when $n$ tends to infinity.",
"Therefore, by Lemma REF , we have the solutions size estimate which will be used implicitly in the sequel $&\\Vert u_{1,n},\\varrho _{1,n}\\Vert _{H^\\gamma }+\\Vert u_{2,n},\\varrho _{2,n}\\Vert _{H^{\\gamma }}\\lesssim 2^{-n\\min \\lbrace (s-\\gamma ),1\\rbrace }\\quad \\text{for all}\\; \\gamma \\ge s^{\\prime }.$ In particular, this implies that the condition $(\\bf {C.1})$ in Theorem REF holds."
],
[
"Error Estimates",
"Letting $\\varrho _{1,n}^{\\rm {er}}=\\varrho _{1,n}-\\varrho _{1,n}^{\\rm {ap}}$ and $u_{1,n}^{\\rm {er}}=u_{1,n}-u_{1,n}^{\\rm {ap}}$ , we can find that $(\\varrho _{1,n}^{\\rm {er}},u_{1,n}^{\\rm {er}})$ satisfies $\\left\\lbrace \\begin{array}{ll}\\partial _t\\varrho _{1,n}^{\\rm {er}}+u̥_{1,n}^{\\rm {er}}+u_{1,n}\\cdot \\nabla \\varrho _{1,n}^{\\rm {er}}=\\mathbf {F_1},\\\\\\partial _tu_{1,n}^{\\rm {er}}-{\\mu } \\Delta u_{1,n}^{\\rm {er}}-{\\mu } \\nabla u̥_{1,n}^{\\rm {er}}+\\nabla \\varrho _{1,n}^{\\rm {er}}+u_{1,n}\\cdot \\nabla u_{1,n}^{\\rm {er}}=\\mathbf {F_2}+\\mathbf {F_3},\\\\\\varrho _{1,n}^{\\rm {er}}|_{t=0}=0,\\; u_{1,n}^{\\rm {er}}|_{t=0}=0, \\end{array}\\right.$ where $&\\mathbf {F_1}:=-u_{1,n}^{\\rm {er}}\\cdot \\nabla \\varrho ^{\\rm {ap}}_{1,n}-\\varrho _{1,n}u̥_{1,n}^{\\rm {er}}-\\varrho _{1,n}^{\\rm {er}}u̥_{1,n}^{\\rm {ap}}-\\varrho ^{\\rm {ap}}_{1,n}u_{1,n}^{\\rm {ap}}),\\\\&\\mathbf {F_2}:=-u_{1,n}^{\\rm {er}}\\cdot \\nabla u^{\\rm {ap}}_{1,n}- u^{\\rm {ap}}_{1,n}\\cdot \\nabla u^{\\rm {ap}}_{1,n},\\\\&\\mathbf {F_3}:=2\\nabla (\\ln (1+\\varrho _{1,n}))\\cdot \\mathbb {D}u_{1,n}.$ The following proposition implies that the error of approximate solution and actual solution will tends to zero in $H^s$ as $n\\rightarrow \\infty $ .",
"Proposition 3.1 Under the assumptions of Theorem REF , then we have for $t\\le 1$ $\\Vert u_{1,n}-u_{1,n}^{\\rm {ap}}\\Vert _{H^s}+\\Vert \\varrho _{1,n}-\\varrho _{1,n}^{\\rm {ap}}\\Vert _{H^s}\\le C2^{-\\frac{n}{2}\\min \\lbrace \\varepsilon _s,1\\rbrace }.$ Proof.",
"Taking the $L^2$ inner product of $(\\ref {u1})_1$ and $(\\ref {u1})_2$ with $(\\varrho _{1,n}^{\\rm {er}},u_{1,n}^{\\rm {er}})$ yields $&\\frac{1}{2}\\frac{\\mathrm {d}}{\\mathrm {d}t}\\Vert \\varrho _{1,n}^{\\rm {er}},u_{1,n}^{\\rm {er}}\\Vert ^2_{L^2}+\\mu \\Vert \\nabla u_{1,n}^{\\rm {er}},u̥_{1,n}^{\\rm {er}}\\Vert ^2_{L^{2}}\\nonumber \\\\=&~\\frac{1}{2} \\big <u̥_{1,n},|\\varrho _{1,n}^{\\rm {er}}|^2+|u_{1,n}^{\\rm {er}}|^2\\big >+ \\big <\\mathbf {F_1}, \\varrho _{1,n}^{\\rm {er}}\\big >+\\big <\\mathbf {F_2}+\\mathbf {F_3}, u_{1,n}^{\\rm {er}}\\big >\\nonumber \\\\\\lesssim &~ \\Vert u_{1,n}\\Vert _{H^{s}}\\Vert \\varrho _{1,n}^{\\rm {er}},u_{1,n}^{\\rm {er}}\\Vert ^2_{L^{2}}+\\Vert \\mathbf {F_1}\\Vert _{L^{2}}\\Vert \\varrho _{1,n}^{\\rm {er}}\\Vert _{L^2}+\\Vert \\mathbf {F_2},\\mathbf {F_3},u_{1,n}^{\\rm {er}}\\Vert ^2_{L^{2}}.$ Applying the operators $\\Lambda ^{s-1}\\varrho _{1,n}^{\\rm {er}}\\Lambda ^{s-1}$ and $\\Lambda ^{s-1}u_{1,n}^{\\rm {er}}\\Lambda ^{s-1}$ to $(\\ref {u1})_1$ and $(\\ref {u1})_2$ , respectively, then integrating the resulting over $\\mathbb {R}^2$ , we obtain $&\\frac{1}{2}\\frac{\\mathrm {d}}{\\mathrm {d}t}\\Vert \\varrho _{1,n}^{\\rm {er}},u_{1,n}^{\\rm {er}}\\Vert ^2_{{\\dot{H}}^{s-1}}+\\mu \\Vert \\nabla u_{1,n}^{\\rm {er}},u̥_{1,n}^{\\rm {er}}\\Vert ^2_{{\\dot{H}}^{s-1}}\\nonumber \\\\=&~\\frac{1}{2} \\big <u̥_{1,n},|\\Lambda ^{s-1}\\varrho _{1,n}^{\\rm {er}}|^2+|\\Lambda ^{s-1}u_{1,n}^{\\rm {er}}|^2\\big >\\nonumber \\\\&-\\big <[\\Lambda ^{s-1},u_{1,n}]\\cdot \\nabla \\varrho _{1,n}^{\\rm {er}},\\Lambda ^{s-1}\\varrho _{1,n}^{\\rm {er}}\\big >-\\big <[\\Lambda ^{s-1},u_{1,n}]\\cdot \\nabla u_{1,n}^{\\rm {er}},\\Lambda ^{s-1}u_{1,n}^{\\rm {er}}\\big >\\nonumber \\\\&+ \\big <\\Lambda ^{s-1}\\mathbf {F_1}, \\Lambda ^{s-1}\\varrho _{1,n}^{\\rm {er}}\\big >+\\big < \\Lambda ^{s-2}\\mathbf {F_2}, \\Lambda ^{s}u_{1,n}^{\\rm {er}}\\big >+\\big < \\Lambda ^{s-2}\\mathbf {F_3}, \\Lambda ^{s}u_{1,n}^{\\rm {er}}\\big >\\nonumber \\\\\\lesssim &~ (1+\\Vert u_{1,n}\\Vert _{H^{s}})\\Vert \\varrho _{1,n}^{\\rm {er}},u_{1,n}^{\\rm {er}}\\Vert ^2_{{{H}}^{s-1}}+\\Vert \\mathbf {F_1}\\Vert _{{\\dot{H}}^{s-1}}\\Vert \\varrho _{1,n}^{\\rm {er}}\\Vert _{{\\dot{H}}^{s-1}}\\nonumber \\\\&+\\Vert \\mathbf {F_2},\\mathbf {F_3}\\Vert ^2_{{\\dot{H}}^{s-2}}+\\varepsilon \\Vert \\nabla u_{1,n}^{\\rm {er}}\\Vert ^2_{{\\dot{H}}^{s-1}},$ where we have used the commutator estimate from Lemma REF .",
"Combing (REF ) and (REF ), we get $\\frac{\\mathrm {d}}{\\mathrm {d}t}\\Vert \\varrho _{1,n}^{\\rm {er}},u_{1,n}^{\\rm {er}}\\Vert ^2_{{{H}}^{s-1}}&+\\mu \\Vert \\nabla u_{1,n}^{\\rm {er}},u̥_{1,n}^{\\rm {er}}\\Vert ^2_{{{H}}^{s-1}}\\lesssim \\Vert \\varrho _{1,n}^{\\rm {er}},u_{1,n}^{\\rm {er}}\\Vert ^2_{{{H}}^{s-1}}\\nonumber \\\\&\\quad +\\Vert \\mathbf {F_1}\\Vert _{{{H}}^{s-1}}\\Vert \\varrho _{1,n}^{\\rm {er}}\\Vert _{{{H}}^{s-1}}+\\Vert \\mathbf {F_2},\\mathbf {F_3}\\Vert ^2_{{{H}}^{s-2}}+\\varepsilon \\Vert \\nabla u_{1,n}^{\\rm {er}}\\Vert ^2_{{\\dot{H}}^{s-1}}.$ Estimate of $\\mathbf {F_1}$ .",
"Notice that $H^{s-1}(\\mathbb {R}^2)$ with $s>2$ is a Banach algebra, we have $\\Vert u_{1,n}^{\\rm {er}}\\cdot \\nabla \\varrho ^{\\rm {ap}}_{1,n}\\Vert _{H^{s-1}}&\\lesssim \\Vert u_{1,n}^{\\rm {er}}\\Vert _{H^{s-1}}\\Vert \\varrho ^{\\rm {ap}}_{1,n}\\Vert _{H^s}\\lesssim \\Vert u_{1,n}^{\\rm {er}}\\Vert _{H^{s-1}},\\\\\\Vert \\varrho _{1,n}u̥_{1,n}^{\\rm {er}}\\Vert _{H^{s-1}}&\\lesssim \\Vert \\varrho _{1,n}\\Vert _{H^{s-1}}\\Vert u̥_{1,n}^{\\rm {er}}\\Vert _{H^{s-1}}\\lesssim \\Vert u̥_{1,n}^{\\rm {er}}\\Vert _{H^{s-1}},\\\\\\Vert \\varrho _{1,n}^{\\rm {er}}u̥_{1,n}^{\\rm {ap}}\\Vert _{H^{s-1}}&\\lesssim \\Vert \\varrho _{1,n}^{\\rm {er}}\\Vert _{H^{s-1}}\\Vert u^{\\rm {ap}}_{1,n}\\Vert _{H^s}\\lesssim \\Vert \\varrho _{1,n}^{\\rm {er}}\\Vert _{H^{s-1}},\\\\\\Vert \\varrho ^{\\rm {ap}}_{1,n}u_{1,n}^{\\rm {ap}})\\Vert _{H^{s-1}}&\\lesssim \\Vert \\varrho ^{\\rm {ap}}_{1,n}u_{1,n}^{\\rm {ap}}\\Vert _{H^{s}}\\lesssim \\Vert \\varrho ^{\\rm {ap}}_{1,n}\\Vert _{L^\\infty }\\Vert u_{1,n}^{\\rm {ap}}\\Vert _{H^{s}}+\\Vert \\varrho ^{\\rm {ap}}_{1,n}\\Vert _{H^{s}}\\Vert u_{1,n}^{\\rm {ap}}\\Vert _{L^\\infty }\\\\&\\lesssim \\Vert \\varrho ^{\\rm {ap}}_{1,n}\\Vert _{H^{s-1-\\varepsilon _s}}\\Vert u_{1,n}^{\\rm {ap}}\\Vert _{H^{s}}+\\Vert \\varrho ^{\\rm {ap}}_{1,n}\\Vert _{H^{s}}\\Vert u_{1,n}^{\\rm {ap}}\\Vert _{H^{s-1-\\varepsilon _s}}\\lesssim 2^{-(1+\\varepsilon _s)n},$ which implies $\\Vert \\mathbf {F_1}\\Vert _{H^{s-1}}&\\lesssim \\Vert u_{1,n}^{\\rm {er}},\\varrho _{1,n}^{\\rm {er}}\\Vert _{H^{s-1}}+\\Vert u̥_{1,n}^{\\rm {er}}\\Vert _{H^{s-1}}+2^{-(1+\\varepsilon _s)n}.$ Estimate of $\\mathbf {F_2}$ .",
"Using Lemma REF yields $\\Vert u_{1,n}^{\\rm {er}}\\cdot \\nabla u^{\\rm {ap}}_{1,n}\\Vert _{H^{s-2}}&\\lesssim \\Vert u_{1,n}^{\\rm {er}}\\Vert _{H^{s-1}}\\Vert u^{\\rm {ap}}_{1,n}\\Vert _{H^{s-1}}\\lesssim \\Vert u_{1,n}^{\\rm {er}}\\Vert _{H^{s-1}},\\\\\\Vert u^{\\rm {ap}}_{1,n}\\cdot \\nabla u^{\\rm {ap}}_{1,n}\\Vert _{H^{s-2}}&\\lesssim \\Vert u^{\\rm {ap}}_{1,n}\\Vert ^2_{H^{s-1}}\\lesssim 2^{-2n},$ which implies $\\Vert \\mathbf {F_2}\\Vert _{H^{s-2}}&\\lesssim \\Vert u_{1,n}^{\\rm {er}}\\Vert _{H^{s-1}}+2^{-2n}.$ Estimate of $\\mathbf {F_3}$ .",
"We can decompose the term $\\nabla (\\ln (1+\\varrho _{1,n}))\\cdot \\mathbb {D}u_{1,n}$ as $\\nabla (\\ln (1+\\varrho _{1,n}))\\cdot \\mathbb {D}u_{1,n}&=\\nabla (\\ln (1+\\varrho _{1,n}))\\cdot \\mathbb {D}u_{1,n}^{\\rm {er}}+\\nabla (\\ln (1+\\varrho _{1,n})-\\ln (1+\\varrho ^{\\rm {ap}}_{1,n}))\\cdot \\mathbb {D}u^{\\rm {ap}}_{1,n}\\\\&\\quad +\\nabla (\\ln (1+\\varrho ^{\\rm {ap}}_{1,n}))\\cdot \\mathbb {D}u^{\\rm {ap}}_{1,n}\\\\&=\\mathbf {F_{3,1}}+\\mathbf {F_{3,2}}+\\mathbf {F_{3,3}}.$ For the first two terms, by Lemma REF and Lemmas REF –REF , we have $&\\Vert \\mathbf {F_{3,1}}\\Vert _{H^{s-2}}\\lesssim \\Vert \\mathbb {D}u_{1,n}^{\\rm {er}}\\Vert _{H^{s-2}}\\Vert \\varrho _{1,n}\\Vert _{H^s}\\lesssim \\Vert u_{1,n}^{\\rm {er}}\\Vert _{H^{s-1}},\\\\&\\Vert \\mathbf {F_{3,2}}\\Vert _{H^{s-2}}\\lesssim \\Vert \\varrho _{1,n}^{\\rm {er}}\\Vert _{H^{s-1}}\\Vert u^{\\rm {ap}}_{1,n}\\Vert _{H^s}\\lesssim \\Vert \\varrho _{1,n}^{\\rm {er}}\\Vert _{H^{s-1}}.$ For the third term, by Lemmas REF and REF , we have $\\Vert \\mathbf {F_{3,3}}\\Vert _{H^{s-2}}&\\lesssim \\Vert \\nabla (\\ln (1+\\varrho ^{\\rm {ap}}_{1,n}))\\Vert _{L^\\infty }\\Vert u_{1,n}^{\\rm {ap}}\\Vert _{H^{s-1}}+\\Vert \\nabla (\\ln (1+\\varrho ^{\\rm {ap}}_{1,n}))\\Vert _{H^{s-2}}\\Vert \\nabla u_{1,n}^{\\rm {ap}}\\Vert _{L^\\infty }\\\\&\\lesssim \\Vert \\varrho ^{\\rm {ap}}_{1,n}\\Vert _{H^{s-\\varepsilon _s}}\\Vert u_{1,n}^{\\rm {ap}}\\Vert _{H^{s-1}}+\\Vert \\varrho ^{\\rm {ap}}_{1,n}\\Vert _{H^{s-1}}\\Vert u_{1,n}^{\\rm {ap}}\\Vert _{H^{s-\\varepsilon _s}}\\lesssim 2^{-(1+\\varepsilon _s)n},$ which implies $\\Vert \\mathbf {F_3}\\Vert _{H^{s-2}}&\\lesssim \\Vert u_{1,n}^{\\rm {er}},\\varrho _{1,n}^{\\rm {er}}\\Vert _{H^{s-1}}+2^{-(1+\\varepsilon _s)n}.$ Putting the above estimates (REF )–(REF ) together with (REF ), we can obtain $\\frac{\\mathrm {d}}{\\mathrm {d}t}\\Vert \\varrho _{1,n}^{\\rm {er}},u_{1,n}^{\\rm {er}}\\Vert ^2_{{{H}}^{s-1}}+\\mu \\Vert \\nabla u_{1,n}^{\\rm {er}},u̥_{1,n}^{\\rm {er}}\\Vert ^2_{{{H}}^{s-1}}\\lesssim &~ \\Vert \\varrho _{1,n}^{\\rm {er}},u_{1,n}^{\\rm {er}}\\Vert ^2_{H^{s-1}}+2^{-2n(1+\\min \\lbrace \\varepsilon _s,1\\rbrace )}\\\\&+\\varepsilon \\Vert \\nabla u_{1,n}^{\\rm {er}},u̥_{1,n}^{\\rm {er}}\\Vert ^2_{{{H}}^{s-1}}.$ Absorbing the $\\varepsilon $ -term and using Gronwall's inequality yield $\\Vert \\varrho _{1,n}^{\\rm {er}},u_{1,n}^{\\rm {er}}\\Vert _{{{H}}^{s-1}}\\le C2^{-n(1+\\min \\lbrace \\varepsilon _s,1\\rbrace )}.$ An interpolation argument leads to $\\Vert \\varrho _{1,n}^{\\rm {er}},u_{1,n}^{\\rm {er}}\\Vert _{{{H}}^{s}}\\lesssim \\Vert \\varrho _{1,n}^{\\rm {er}},u_{1,n}^{\\rm {er}}\\Vert _{{{H}}^{s-1}}^{\\frac{1}{2}}\\Vert \\varrho _{1,n}^{\\rm {er}},u_{1,n}^{\\rm {er}}\\Vert _{{{H}}^{s+1}}^{\\frac{1}{2}}\\lesssim 2^{-\\frac{n}{2}\\min \\lbrace \\varepsilon _s,1\\rbrace }.$ Thus, we have finished the proof of Proposition REF .",
"Denoting $V^{\\rm {ap}}_{n}=-u_{2,n}^{\\rm {ap}}\\cdot \\nabla \\varrho _{2,n}^{\\rm {ap}}$ and introducing the errors $\\varrho _{2,n}^{\\rm {er}}=\\varrho _{2,n}-\\varrho _{2,n}^{\\rm {ap}}-tV^{\\rm {ap}}_{n}\\quad \\text{and}\\quad u_{2,n}^{\\rm {er}}=u_{2,n}-u_{2,n}^{\\rm {ap}},$ we can find that $(\\varrho _{2,n}^{\\rm {er}},u_{2,n}^{\\rm {er}})$ satisfies $\\left\\lbrace \\begin{array}{ll}\\partial _t\\varrho _{2,n}^{\\rm {er}}+u̥_{2,n}^{\\rm {er}}+u_{2,n}\\cdot \\nabla \\varrho _{2,n}^{\\rm {er}}=\\mathbf {G_1}-t\\mathbf {G_4}-t\\partial _tV^{\\rm {ap}}_{n},\\\\\\partial _tu_{2,n}^{\\rm {er}}-{\\mu } \\Delta u_{2,n}^{\\rm {er}}-{\\mu } \\nabla u̥_{2,n}^{\\rm {er}}+\\nabla \\varrho _{2,n}^{\\rm {er}}+u_{2,n}\\cdot \\nabla u_{2,n}^{\\rm {er}}=\\mathbf {G_2}+\\mathbf {G_3}-t\\nabla V^{\\rm {ap}}_{n}, \\\\\\varrho _{2,n}^{\\rm {er}}|_{t=0}=0,\\; u_{2,n}^{\\rm {er}}|_{t=0}=0, \\end{array}\\right.$ where $&\\mathbf {G_1}:=-u_{2,n}^{\\rm {er}}\\cdot \\nabla \\varrho ^{\\rm {ap}}_{2,n}-\\varrho _{2,n}u̥_{2,n}^{\\rm {er}}-\\varrho _{2,n}^{\\rm {er}}u̥_{2,n}^{\\rm {ap}}-\\varrho ^{\\rm {ap}}_{2,n}u̥_{2,n}^{\\rm {ap}},\\\\&\\mathbf {G_2}:=-u_{2,n}^{\\rm {er}}\\cdot \\nabla u^{\\rm {ap}}_{2,n}- u^{\\rm {ap}}_{2,n}\\cdot \\nabla u^{\\rm {ap}}_{2,n},\\\\&\\mathbf {G_3}:=2\\nabla (\\ln (1+\\varrho _{2,n}))\\cdot \\mathbb {D}u_{2,n},\\\\&\\mathbf {G_4}:=u_{2,n}^{\\rm {er}}\\cdot \\nabla V^{\\rm {ap}}_{n}+V^{\\rm {ap}}_{n} u_{2,n}^{\\rm {ap}}).$ Proposition 3.2 Under the assumptions of Theorem REF , then we have for $t\\le 1$ $\\Vert u_{2,n}-u_{2,n}^{\\rm {ap}}\\Vert _{H^s}+\\Vert \\varrho _{2,n}-\\varrho _{2,n}^{\\rm {ap}}-tV^{\\rm {ap}}_{n}\\Vert _{H^s}\\le C2^{-n\\min \\lbrace \\varepsilon _s,1\\rbrace }+Ct^{\\frac{3}{2}}.$ Proof.",
"Following the procedure in (REF ) and (REF ), we obtain $&\\frac{\\mathrm {d}}{\\mathrm {d}t}\\Vert \\varrho _{2,n}^{\\rm {er}},u_{2,n}^{\\rm {er}}\\Vert ^2_{{{H}}^{s}}+\\mu \\Vert \\nabla u_{2,n}^{\\rm {er}},u̥_{2,n}^{\\rm {er}}\\Vert ^2_{{{H}}^{s}}\\nonumber \\\\\\lesssim &~ (1+\\Vert u_{2,n}\\Vert _{H^{s}})\\Vert \\varrho _{2,n}^{\\rm {er}},u_{2,n}^{\\rm {er}}\\Vert ^2_{{{H}}^{s}}\\nonumber \\\\&+\\big (\\Vert \\mathbf {G_1}\\Vert _{{{H}}^{s}}+\\Vert t\\partial _tV^{\\rm {ap}}_{n}\\Vert _{{{H}}^{s}}+t\\Vert \\mathbf {G_4}\\Vert _{{{H}}^{s}}\\big )\\Vert \\varrho _{2,n}^{\\rm {er}}\\Vert _{{{H}}^{s}}\\nonumber \\\\&+\\Vert \\mathbf {G_2},\\mathbf {G_3},t\\nabla V^{\\rm {ap}}_{n}\\Vert ^2_{{{H}}^{s-1}}\\nonumber \\\\\\lesssim &~ \\Vert \\varrho _{2,n}^{\\rm {er}},u_{2,n}^{\\rm {er}}\\Vert ^2_{{{H}}^{s}}+\\Vert \\mathbf {G_1}\\Vert _{{{H}}^{s}}\\Vert \\varrho _{2,n}^{\\rm {er}}\\Vert _{{{H}}^{s}}+\\Vert \\mathbf {G_2},\\mathbf {G_3}\\Vert ^2_{{{H}}^{s-1}}+\\Vert t\\partial _tV^{\\rm {ap}}_{n}\\Vert ^2_{{{H}}^{s}}\\nonumber \\\\&+t^2\\big (\\Vert \\mathbf {G_4}\\Vert ^2_{{{H}}^{s}}+\\Vert V^{\\rm {ap}}_{n}\\Vert ^2_{{{H}}^{s}}\\big ).$ We should mentioned that the commutator estimate from Lemma REF was used here.",
"It is not hard to deduce that for $k\\in \\lbrace -1,0,1\\rbrace $ $\\Vert V^{\\rm {ap}}_{n}\\Vert _{H^{s+k}}&\\lesssim \\Vert u_{2,n}^{\\rm {ap}}\\Vert _{L^\\infty }\\Vert \\nabla \\varrho _{2,n}^{\\rm {ap}}\\Vert _{H^{s+k}}+\\Vert u_{2,n}^{\\rm {ap}}\\Vert _{H^{s+k}}\\Vert \\nabla \\varrho _{2,n}^{\\rm {ap}}\\Vert _{L^\\infty }\\\\&\\lesssim \\Vert u_{2,n}^{\\rm {ap}}\\Vert _{H^{s-1}}\\Vert \\nabla \\varrho _{2,n}^{\\rm {ap}}\\Vert _{H^{s+k}}+\\Vert u_{2,n}^{\\rm {ap}}\\Vert _{H^{s+k}}\\Vert \\nabla \\varrho _{2,n}^{\\rm {ap}}\\Vert _{H^{s-1}}\\\\&\\lesssim 2^{kn}.$ Estimate of $\\mathbf {G_1}$ .",
"Using Lemma REF yields $\\Vert u_{2,n}^{\\rm {er}}\\cdot \\nabla \\varrho ^{\\rm {ap}}_{2,n}\\Vert _{H^{s}}&\\lesssim \\Vert u_{2,n}^{\\rm {er}}\\Vert _{L^\\infty }\\Vert \\varrho ^{\\rm {ap}}_{2,n}\\Vert _{H^{s+1}}+\\Vert u_{2,n}^{\\rm {er}}\\Vert _{H^s}\\Vert \\nabla \\varrho ^{\\rm {ap}}_{2,n}\\Vert _{L^\\infty }\\\\&\\lesssim 2^n\\Vert u_{2,n}^{\\rm {er}}\\Vert _{H^{s-1}}+\\Vert u_{2,n}^{\\rm {er}}\\Vert _{H^{s}},\\\\\\Vert \\varrho _{2,n}u̥_{2,n}^{\\rm {er}}\\Vert _{H^{s}}&\\lesssim \\Vert u̥_{2,n}^{\\rm {er}}\\Vert _{H^{s}},\\\\\\Vert \\varrho _{2,n}^{\\rm {er}}u̥_{2,n}^{\\rm {ap}}\\Vert _{H^{s}}&\\lesssim 2^n\\Vert \\varrho _{2,n}^{\\rm {er}}\\Vert _{H^{s-1}}+\\Vert \\varrho _{2,n}^{\\rm {er}}\\Vert _{H^{s}},\\\\\\Vert \\varrho ^{\\rm {ap}}_{2,n}u̥_{2,n}^{\\rm {ap}}\\Vert _{H^{s}}&\\lesssim \\Vert \\varrho _{2,n}^{\\rm {ap}}\\Vert _{H^{s-1}}\\Vert u̥^{\\rm {ap}}_{2,n}\\Vert _{H^{s}}+\\Vert \\varrho _{2,n}^{\\rm {ap}}\\Vert _{H^s}\\Vert u^{\\rm {ap}}_{2,n}\\Vert _{H^{s-\\varepsilon _s}}\\\\&\\lesssim 2^{-n}\\Vert u̥^{\\rm {ap}}_{2,n}\\Vert _{H^{s}}+2^{-n\\min \\lbrace \\varepsilon _s,1\\rbrace },$ which implies that $\\Vert \\mathbf {G_{1}}\\Vert _{H^{s}}&\\lesssim 2^n\\Vert u_{2,n}^{\\rm {er}},\\varrho _{2,n}^{\\rm {er}}\\Vert _{H^{s-1}}+\\Vert u̥_{2,n}^{\\rm {er}}\\Vert _{H^{s}}+\\Vert u_{2,n}^{\\rm {er}},\\varrho _{2,n}^{\\rm {er}}\\Vert _{H^{s}}\\nonumber \\\\&\\quad +2^{-n}\\Vert u̥^{\\rm {ap}}_{2,n}\\Vert _{H^{s}}+2^{-n\\min \\lbrace \\varepsilon _s,1\\rbrace }.$ Estimate of $\\mathbf {G_2}$ .",
"Notice that $H^{s-1}(\\mathbb {R}^2)$ with $s>2$ is a Banach algebra, we have $\\Vert u_{2,n}^{\\rm {er}}\\cdot \\nabla u^{\\rm {ap}}_{2,n}\\Vert _{H^{s-1}}&\\lesssim \\Vert u_{2,n}^{\\rm {er}}\\Vert _{H^{s-1}}\\Vert u_{2,n}^{\\rm {ap}}\\Vert _{H^{s}}\\lesssim \\Vert u_{2,n}^{\\rm {er}}\\Vert _{H^{s}},\\\\\\Vert u^{\\rm {ap}}_{2,n}\\cdot \\nabla u^{\\rm {ap}}_{2,n}\\Vert _{H^{s-1}}&\\lesssim \\Vert u^{\\rm {ap}}_{2,n}\\Vert _{H^{s-1}}\\Vert u_{2,n}^{\\rm {ap}}\\Vert _{H^{s}}\\lesssim 2^{-n},$ which implies that $\\Vert \\mathbf {G_{2}}\\Vert _{H^{s-1}}&\\lesssim \\Vert u_{2,n}^{\\rm {er}}\\Vert _{H^{s}}+2^{-n}.$ Estimate of $\\mathbf {G_3}$ .",
"For the term $\\nabla (\\ln (1+\\varrho _{2,n}))\\cdot \\mathbb {D}u_{2,n}$ , we can decompose it as $\\nabla (\\ln (1+\\varrho _{2,n}))\\cdot \\mathbb {D}u_{2,n}&=\\nabla (\\ln (1+\\varrho _{2,n}))\\cdot \\mathbb {D}u_{2,n}^{\\rm {er}}\\\\&\\quad +\\nabla (\\ln (1+\\varrho _{2,n})-\\ln (1+\\varrho ^{\\rm {ap}}_{2,n}+tV^{\\rm {ap}}_{n}))\\cdot \\mathbb {D}u^{\\rm {ap}}_{2,n}\\\\&\\quad +\\nabla (\\ln (1+\\varrho ^{\\rm {ap}}_{2,n}+tV^{\\rm {ap}}_{n}))\\cdot \\mathbb {D}u^{\\rm {ap}}_{2,n}\\\\&=\\mathbf {G_{3,1}}+\\mathbf {G_{3,2}}+\\mathbf {G_{3,3}}.$ By Lemmas REF and REF , it is easy to deduce that $&\\Vert \\mathbf {G_{3,1}}\\Vert _{H^{s-1}}\\lesssim \\Vert u_{2,n}^{\\rm {er}}\\Vert _{H^{s}},\\\\&\\Vert \\mathbf {G_{3,2}}\\Vert _{H^{s-1}}\\lesssim \\Vert \\varrho _{2,n}^{\\rm {er}}\\Vert _{H^{s}},\\\\&\\Vert \\mathbf {G_{3,3}}\\Vert _{H^{s-1}}\\lesssim 2^{-n\\min \\lbrace \\varepsilon _s,1\\rbrace },$ which gives that $\\Vert \\mathbf {G_{3}}\\Vert _{H^{s-1}}&\\lesssim \\Vert u_{2,n}^{\\rm {er}},\\varrho _{2,n}^{\\rm {er}}\\Vert _{H^{s}}+2^{-n\\min \\lbrace \\varepsilon _s,1\\rbrace }.$ Estimate of $\\mathbf {G_4}$ .",
"By Lemma REF again, we have $\\Vert u_{2,n}^{\\rm {er}}\\cdot \\nabla V^{\\rm {ap}}_{n}\\Vert _{H^{s}}&\\lesssim \\Vert u_{2,n}^{\\rm {er}}\\Vert _{L^\\infty }\\Vert V^{\\rm {ap}}_{n}\\Vert _{H^{s+1}}+\\Vert u_{2,n}^{\\rm {er}}\\Vert _{H^s}\\Vert \\nabla V^{\\rm {ap}}_{n}\\Vert _{L^\\infty }\\lesssim 2^n\\Vert u_{2,n}^{\\rm {er}}\\Vert _{H^{s-1}}+\\Vert u_{2,n}^{\\rm {er}}\\Vert _{H^{s}},\\\\\\Vert V^{\\rm {ap}}_{2,n}u_{2,n}^{\\rm {ap}})\\Vert _{H^{s}}&\\lesssim \\Vert V^{\\rm {ap}}_{2,n}u̥_{2,n}^{\\rm {ap}}\\Vert _{H^{s}}+ \\Vert u_{2,n}^{\\rm {ap}}\\cdot \\nabla V^{\\rm {ap}}_{2,n}\\Vert _{H^{s}}\\lesssim 1$ and $\\Vert u_{2,n}^{\\rm {er}}\\cdot \\nabla V^{\\rm {ap}}_{n}\\Vert _{H^{s-1}}&\\lesssim \\Vert u_{2,n}^{\\rm {er}}\\Vert _{L^\\infty }\\Vert V^{\\rm {ap}}_{n}\\Vert _{H^{s}}+\\Vert u_{2,n}^{\\rm {er}}\\Vert _{H^{s-1}}\\Vert \\nabla V^{\\rm {ap}}_{n}\\Vert _{L^\\infty }\\lesssim \\Vert u_{2,n}^{\\rm {er}}\\Vert _{H^{s-1}},\\\\\\Vert V^{\\rm {ap}}_{2,n}u_{2,n}^{\\rm {ap}})\\Vert _{H^{s-1}}&\\lesssim \\Vert V^{\\rm {ap}}_{2,n}u_{2,n}^{\\rm {ap}}\\Vert _{H^{s}}\\lesssim 2^{-n},$ from which, we obtain $\\Vert \\mathbf {G_{4}}\\Vert _{H^{s}}&\\lesssim 2^n\\Vert u_{2,n}^{\\rm {er}}\\Vert _{H^{s-1}}+\\Vert u_{2,n}^{\\rm {er}}\\Vert _{H^{s}}+1\\quad \\text{and}\\quad \\Vert \\mathbf {G_{4}}\\Vert _{H^{s-1}}\\lesssim \\Vert u_{2,n}^{\\rm {er}}\\Vert _{H^{s-1}}+2^{-n}.$ Next, we need to deal with the involved term $\\Vert t\\partial _tV^{\\rm {ap}}_{n}\\Vert _{{{H}}^{s}}$ .",
"Estimate of $\\Vert t\\partial _tV^{\\rm {ap}}_{n}\\Vert _{{{H}}^{s}}$ .",
"Direct calculation gives that $&-\\partial _tV_{n}^{\\rm {ap}}=\\partial _tu_{2,n}^{\\rm {ap}}\\cdot \\nabla \\varrho _{2,n}^{\\rm {ap}}+u_{2,n}^{\\rm {ap}}\\cdot \\nabla \\partial _t\\varrho _{2,n}^{\\rm {ap}},$ using the following estimates whose proof are relegated to A.1–A.2 in Section $&\\Vert t\\partial _tu_{2,n}^{\\rm {ap}}\\Vert _{H^{s+k}}\\le C2^{(k-1)n}+Ct2^{-n},\\\\&\\Vert \\partial _t\\varrho _{2,n}^{\\rm {ap}}\\Vert _{H^{s+k}}\\le C2^{kn},$ then we can estimate $\\Vert t\\partial _tV^{\\rm {ap}}_{n}\\Vert _{{{H}}^{s}}$ as $\\Vert t\\partial _tV^{\\rm {ap}}_{n}\\Vert _{H^{s+k}}&\\lesssim \\Vert t\\partial _tu_{2,n}^{\\rm {ap}}\\Vert _{L^\\infty }\\Vert \\nabla \\varrho _{2,n}^{\\rm {ap}}\\Vert _{H^{s+k}}+\\Vert t\\partial _tu_{2,n}^{\\rm {ap}}\\Vert _{H^{s+k}}\\Vert \\nabla \\varrho _{2,n}^{\\rm {ap}}\\Vert _{L^\\infty }\\\\&\\quad +t\\Vert u_{2,n}^{\\rm {ap}}\\Vert _{L^\\infty }\\Vert \\nabla \\partial _t\\varrho _{2,n}^{\\rm {ap}}\\Vert _{H^{s+k}}+t\\Vert u_{2,n}^{\\rm {ap}}\\Vert _{H^{s+k}}\\Vert \\nabla \\partial _t\\varrho _{2,n}^{\\rm {ap}}\\Vert _{L^\\infty }\\\\&\\lesssim \\Vert t\\partial _tu_{2,n}^{\\rm {ap}}\\Vert _{H^{s-1}}\\Vert \\varrho _{2,n}^{\\rm {ap}}\\Vert _{H^{s+k+1}}+\\Vert t\\partial _tu_{2,n}^{\\rm {ap}}\\Vert _{H^{s+k}}\\Vert \\varrho _{2,n}^{\\rm {ap}}\\Vert _{H^{s}}\\\\&\\quad +t\\Vert u_{2,n}^{\\rm {ap}}\\Vert _{H^{s-1}}\\Vert \\partial _t\\varrho _{2,n}^{\\rm {ap}}\\Vert _{H^{s+k+1}}+t\\Vert u_{2,n}^{\\rm {ap}}\\Vert _{H^{s+k}}\\Vert \\partial _t\\varrho _{2,n}^{\\rm {ap}}\\Vert _{H^{s}}\\\\&\\lesssim t2^{kn}+2^{(k-1)n}.$ Putting the above estimates into (REF ) yields $\\frac{\\mathrm {d}}{\\mathrm {d}t}\\Vert \\varrho _{2,n}^{\\rm {er}},u_{2,n}^{\\rm {er}}\\Vert ^2_{{{H}}^{s}}\\lesssim &~\\Vert \\varrho _{2,n}^{\\rm {er}},u_{2,n}^{\\rm {er}}\\Vert ^2_{{{H}}^{s}}+2^{2n}\\Vert \\varrho _{2,n}^{\\rm {er}},u_{2,n}^{\\rm {er}}\\Vert ^2_{H^{s-1}}+2^{-2n}\\Vert u̥_{2,n}^{\\rm {ap}}\\Vert ^2_{H^{s}}\\nonumber \\\\&+2^{-2n\\min \\lbrace \\varepsilon _s,1\\rbrace }+t^2.$ Finally, to close the above, we have to estimate $\\Vert \\varrho _{2,n}^{\\rm {er}},u_{2,n}^{\\rm {er}}\\Vert ^2_{H^{s-1}}$ .",
"Following the procedure in (REF ) and (REF ) once again, we obtain $&\\frac{\\mathrm {d}}{\\mathrm {d}t}\\Vert \\varrho _{2,n}^{\\rm {er}},u_{2,n}^{\\rm {er}}\\Vert ^2_{{{H}}^{s-1}}+\\mu \\Vert \\nabla u_{2,n}^{\\rm {er}},u̥_{2,n}^{\\rm {er}}\\Vert ^2_{{{H}}^{s-1}}\\nonumber \\\\\\lesssim &~ (1+\\Vert u_{2,n}\\Vert _{H^{s}})\\Vert \\varrho _{2,n}^{\\rm {er}},u_{2,n}^{\\rm {er}}\\Vert ^2_{{{H}}^{s-1}}\\nonumber \\\\&+\\big (\\Vert \\mathbf {G_1}\\Vert _{{{H}}^{s-1}}+\\Vert t\\partial _tV^{\\rm {ap}}_{n}\\Vert _{{{H}}^{s-1}}\\big )\\Vert \\varrho _{2,n}^{\\rm {er}}\\Vert _{H^{s-1}}+\\Vert \\mathbf {G_2},\\mathbf {G_3}\\Vert ^2_{{{H}}^{s-2}}\\nonumber \\\\&+t^2\\big (\\Vert \\mathbf {G_4}\\Vert ^2_{H^{s-1}}+\\Vert V^{\\rm {ap}}_{n}\\Vert ^2_{{{H}}^{s-1}}\\big )\\nonumber \\\\\\lesssim &~\\Vert \\varrho _{2,n}^{\\rm {er}},u_{2,n}^{\\rm {er}}\\Vert ^2_{{{H}}^{s-1}}+\\Vert \\mathbf {G_1}\\Vert _{{{H}}^{s-1}}\\Vert \\varrho _{2,n}^{\\rm {er}}\\Vert _{H^{s-1}}+\\Vert \\mathbf {G_2},\\mathbf {G_3}\\Vert ^2_{{{H}}^{s-2}}+\\Vert t\\partial _tV^{\\rm {ap}}_{n}\\Vert ^2_{{{H}}^{s-1}}\\nonumber \\\\&+t^2\\big (\\Vert \\mathbf {G_4}\\Vert ^2_{H^{s-1}}+\\Vert V^{\\rm {ap}}_{n}\\Vert ^2_{{{H}}^{s-1}}\\big ).$ It should be mentioned that the first three terms of $\\mathbf {G_1}$ can be done as that of $\\mathbf {F_1}$ .",
"The only difference is the forth term of $\\mathbf {G_1}$ .",
"In fact, we estimate it as $\\Vert \\varrho ^{\\rm {ap}}_{2,n}u̥_{2,n}^{\\rm {ap}}\\Vert _{{{H}}^{s-1}}\\lesssim \\Vert \\varrho ^{\\rm {ap}}_{2,n}\\Vert _{{{H}}^{s-1}}\\Vert u̥_{2,n}^{\\rm {ap}}\\Vert _{{{H}}^{s-1}}\\lesssim 2^{-n}\\Vert u̥_{2,n}^{\\rm {ap}}\\Vert _{{{H}}^{s-1}}.$ Then we have $\\Vert \\mathbf {G_1}\\Vert _{H^{s-1}}&\\lesssim \\Vert u_{1,n}^{\\rm {er}},\\varrho _{1,n}^{\\rm {er}}\\Vert _{H^{s-1}}+\\Vert u̥_{1,n}^{\\rm {er}}\\Vert _{H^{s-1}}+2^{-n}\\Vert u̥_{2,n}^{\\rm {ap}}\\Vert _{{{H}}^{s-1}}.$ The estimates of $\\mathbf {G_2}$ and $\\mathbf {G_3}$ in $H^{s-2}$ can be done as that of $\\mathbf {F_2}$ and $\\mathbf {F_3}$ , respectively.",
"We have $\\Vert \\mathbf {G_2},\\mathbf {G_3}\\Vert _{H^{s-2}}&\\lesssim \\Vert u_{1,n}^{\\rm {er}},\\varrho _{1,n}^{\\rm {er}}\\Vert _{H^{s-1}}+2^{-(1+\\varepsilon _s)n}+2^{-2n}.$ Combining the above estimates with (REF ), we can obtain $&\\frac{\\mathrm {d}}{\\mathrm {d}t}\\Vert \\varrho _{2,n}^{\\rm {er}},u_{2,n}^{\\rm {er}}\\Vert ^2_{{{H}}^{s-1}}\\lesssim \\Vert \\varrho _{2,n}^{\\rm {er}},u_{2,n}^{\\rm {er}}\\Vert ^2_{{{H}}^{s-1}}+2^{-2n}\\Vert u̥_{2,n}^{\\rm {ap}}\\Vert ^2_{{{H}}^{s-1}}+2^{-2n(1+\\min \\lbrace \\varepsilon _s,1\\rbrace )}+2^{-2n}t^2,$ which follows from Gronwall's inequality and Lemma REF that $\\Vert \\varrho _{2,n}^{\\rm {er}},u_{2,n}^{\\rm {er}}\\Vert ^2_{{{H}}^{s-1}}\\lesssim 2^{-2n(1+\\min \\lbrace \\varepsilon _s,1\\rbrace )}+2^{-2n}t^3.$ Plugging (REF ) into (REF ) yields $\\frac{\\mathrm {d}}{\\mathrm {d}t}\\Vert \\varrho _{2,n}^{\\rm {er}},u_{2,n}^{\\rm {er}}\\Vert ^2_{{{H}}^{s}}\\lesssim &~\\Vert \\varrho _{2,n}^{\\rm {er}},u_{2,n}^{\\rm {er}}\\Vert ^2_{{{H}}^{s}}+2^{-2n}\\Vert u̥_{2,n}^{\\rm {ap}}\\Vert ^2_{H^{s}}+2^{-2n\\min \\lbrace \\varepsilon _s,1\\rbrace }+t^2.$ Using Gronwall's inequality and Lemma REF yields $\\Vert \\varrho _{2,n}^{\\rm {er}},u_{2,n}^{\\rm {er}}\\Vert ^2_{{{H}}^{s}}\\lesssim &~2^{-2n\\min \\lbrace \\varepsilon _s,1\\rbrace }+t^3.$ Thus, we have finished the proof of Proposition REF ."
],
[
"Non-uniform Continuous Dependence",
"With Propositions REF –REF in hand, we can prove Theorem REF .",
"Behavior at time $t=0$ .",
"Obviously, we have $\\Vert {u}_{2,n}(0)-u_{1,n}(0)\\Vert _{H^{s}}+\\Vert \\varrho _{2,n}(0)-\\varrho _{1,n}(0)\\Vert _{H^{s}}=\\Vert g_n\\Vert _{H^{s}}\\le C2^{-n},$ which means that the condition $(\\bf {C.2})$ in Theorem REF holds.",
"Behavior at time $t>0$ .",
"Notice that $&u_{2,n}-u_{1,n}=u^{\\rm {er}}_{2,n}-u^{\\rm {er}}_{1,n}+u^{\\rm {ap}}_{2,n}-u^{\\rm {ap}}_{1,n},\\nonumber \\\\&\\varrho _{2,n}-\\varrho _{1,n}=t{V}_n^{\\rm {ap}}+\\varrho ^{\\rm {er}}_{2,n}-\\varrho ^{\\rm {er}}_{1,n}+\\varrho ^{\\rm {ap}}_{2,n}-\\varrho ^{\\rm {ap}}_{1,n},$ by the triangle inequality and Propositions REF and REF , we deduce that for $t\\le 1$ $&\\Vert \\varrho _{2,n}(t)-\\varrho _{1,n}(t)\\Vert _{H^{s}}+\\Vert u_{2,n}(t)-u_{1,n}(t)\\Vert _{H^{s}}\\nonumber \\\\\\ge &~ t\\Vert V^{\\rm {ap}}_{n}\\Vert _{H^s}-\\big (\\Vert u^{\\rm {ap}}_{2,n}-u^{\\rm {ap}}_{1,n}\\Vert _{H^s}+\\Vert \\varrho ^{\\rm {ap}}_{2,n}-\\varrho ^{\\rm {ap}}_{1,n}\\Vert _{H^s}\\big )-\\big (\\Vert u^{\\rm {er}}_{1,n},\\varrho ^{\\rm {er}}_{1,n}\\Vert _{H^s}+\\Vert u^{\\rm {er}}_{2,n},\\varrho ^{\\rm {er}}_{2,n}\\Vert _{H^s}\\big )\\nonumber \\\\\\gtrsim &~ t\\Vert V^{\\rm {ap}}_{n}\\Vert _{H^s}-2^{-n}-2^{-\\frac{n}{2}\\min \\lbrace \\varepsilon _s,1\\rbrace }-2^{-n\\min \\lbrace \\varepsilon _s,1\\rbrace }-t^{\\frac{3}{2}},$ where we have used $\\Vert u^{\\rm {ap}}_{2,n}-u^{\\rm {ap}}_{1,n}\\Vert ^2_{H^s}+\\Vert \\varrho ^{\\rm {ap}}_{2,n}-\\varrho ^{\\rm {ap}}_{1,n}\\Vert ^2_{H^s}\\lesssim \\Vert g_n\\Vert ^2_{H^s}\\lesssim 2^{-2n}.$ Notice that $-V^{\\rm {ap}}_{n}=u_{2,n}^{\\rm {ap}}\\cdot \\nabla \\varrho _{2,n}^{\\rm {ap}}$ , we decompose it as $-V^{\\rm {ap}}_{n}=(u_{2,n}^{\\rm {ap}})_1\\partial _1 \\varrho _{2,n}^{\\rm {ap}}+(u_{2,n}^{\\rm {ap}})_2\\partial _2 \\varrho _{2,n}^{\\rm {ap}}=(g_n)_1\\partial _1f_n+E,$ where $E:=(u_{2,n}^{\\rm {ap}}-g_n)_1\\partial _1\\varrho _{2,n}^{\\rm {ap}}+(g_n)_1\\partial _1(\\varrho _{2,n}^{\\rm {ap}}-f_n)+(u_{2,n}^{\\rm {ap}})_2\\partial _2\\varrho _{2,n}^{\\rm {ap}}.$ Then we have for $t\\le 1$ $\\Vert E\\Vert _{H^{s}}\\le Ct+C2^{-n}.$ For more details of proof, see A.3 in Section .",
"Furthermore, (REF ) reduces to $\\Vert \\varrho _{2,n}(t)-\\varrho _{1,n}(t)\\Vert _{H^{s}}+\\Vert u_{2,n}(t)-u_{1,n}(t)\\Vert _{H^{s}}\\gtrsim t\\Vert (g_n)_1\\partial _1f_n\\Vert _{H^s}-2^{-\\frac{n}{2}\\min \\lbrace \\varepsilon _s,1\\rbrace }-t^{\\frac{3}{2}},$ combining the following estimate whose proof is postponed to A.4 in Section $\\liminf _{n\\rightarrow \\infty }\\Vert (g_n)_1\\partial _1f_n\\Vert _{H^s}\\gtrsim \\Vert \\phi ^{\\prime }\\phi \\Vert ^2_{L^2}\\Vert \\phi ^2\\Vert _{L^2},$ we get from (REF ) that $\\liminf _{n\\rightarrow \\infty }\\big (\\Vert \\varrho _{2,n}(t)-\\varrho _{1,n}(t)\\Vert _{H^{s}}+\\Vert u_{2,n}(t)-u_{1,n}(t)\\Vert _{H^{s}}\\big )\\gtrsim t\\quad \\text{for} \\ t \\ \\text{small enough},$ which is nothing but the condition $(\\bf {C.3})$ in Theorem REF .",
"Thus, we complete the proof of Theorem REF ."
],
[
"Appendix",
"For the sake of convenience, here we present more details in the computations.",
"A.1 Proof of (REF ).",
"When $|\\xi |\\rightarrow \\infty $ , we have $\\lambda _-(\\xi )\\sim -2\\mu |\\xi |^2 \\quad \\text{and}\\quad \\lambda _+(\\xi )\\sim -\\frac{1}{2\\mu }.$ From (REF ), then $\\mathcal {F}\\big (u_{2,n}^{\\rm {ap}}\\big )(t,\\xi )&=-\\frac{e^{\\lambda _+t}-e^{\\lambda _-t}}{\\lambda _+-\\lambda _-}\\mathcal {F}\\big (\\nabla f_n\\big )-\\frac{\\lambda _+e^{\\lambda _+t}-\\lambda _-e^{\\lambda _-t}}{\\lambda _+-\\lambda _-}\\mathcal {F}\\big (\\Lambda ^{-2}\\nabla g̥_n\\big )$ which gives $&\\mathcal {F}\\big (t\\partial _tu_{2,n}^{\\rm {ap}}\\big )=-\\frac{t\\lambda _+e^{\\lambda _+t}-t\\lambda _-e^{\\lambda _-t}}{\\lambda _+-\\lambda _-}\\mathcal {F}\\big (\\nabla f_n\\big )-t\\frac{\\lambda _{+}^2e^{\\lambda _+t}-\\lambda _{-}^2e^{\\lambda _-t}}{\\lambda _+-\\lambda _-}\\mathcal {F}\\big (\\Lambda ^{-2}\\nabla g̥_n\\big ).$ Using the simple facts $&|t\\lambda _{\\pm }e^{\\lambda _{\\pm }t}|\\le 1\\quad \\text{if }\\;\\xi \\in \\mathcal {C}_n \\\\&\\Big |\\frac{\\lambda _{+}^2e^{\\lambda _+t}-\\lambda _{-}^2e^{\\lambda _-t}}{\\lambda _+-\\lambda _-}\\Big |\\lesssim |\\lambda _++\\lambda _-|+t|\\lambda _-|^2\\lesssim 1\\quad \\text{if }\\;\\xi \\in \\mathcal {B}(0,1),$ then we have $\\big |\\mathcal {F}\\big (t\\partial _tu_{2,n}^{\\rm {ap}}\\big )\\big |&\\le \\Big |\\frac{t\\lambda _+e^{\\lambda _+t}-t\\lambda _-e^{\\lambda _-t}}{\\lambda _+-\\lambda _-}\\mathcal {F}\\big (\\nabla f_n\\big )\\Big |+t\\Big |\\frac{\\lambda _{+}^2e^{\\lambda _+t}-\\lambda _{-}^2e^{\\lambda _-t}}{\\lambda _+-\\lambda _-}\\mathcal {F}\\big (\\Lambda ^{-2}\\nabla g̥_n\\big )\\Big |\\\\&\\le \\frac{2}{|\\lambda _+-\\lambda _-|}\\big |\\mathcal {F}\\big (\\nabla f_n\\big )\\big |+t\\big |\\mathcal {F}\\big (\\Lambda ^{-2}\\nabla g̥_n\\big )\\big |\\\\&\\approx \\big |\\mathcal {F}\\big (\\Lambda ^{-2}\\nabla f_n\\big )\\big |+t\\big |\\mathcal {F}\\big (\\Lambda ^{-2}\\nabla g̥_n\\big )\\big |,$ which in turn gives $\\Vert t\\partial _tu_{2,n}^{\\rm {ap}}\\Vert _{H^{s+k}}&=\\Big [\\int _{\\mathbb {R}^2}(1+|\\xi |^2)^{\\frac{s+k}{2}}\\big |\\mathcal {F}\\big (t\\partial _tu_{2,n}^{\\rm {ap}}\\big )\\big |^2\\mathrm {d}\\xi \\Big ]^{\\frac{1}{2}}\\\\&\\lesssim \\Big [\\int _{\\mathcal {C}_n }(1+|\\xi |^2)^{\\frac{s+k}{2}}\\big |\\mathcal {F}\\big (\\Lambda ^{-2}\\nabla f_n\\big )\\big |^2\\mathrm {d}\\xi \\Big ]^{\\frac{1}{2}}+t\\Big [\\int _{\\mathcal {B}(0,1)}\\big |\\mathcal {F}\\big (\\Lambda ^{-2}\\nabla g̥_n\\big )\\big |^2\\mathrm {d}\\xi \\Big ]^{\\frac{1}{2}}\\\\&\\lesssim \\Vert f_n\\Vert _{H^{s+k-1}}+t\\Vert g_n\\Vert _{L^{2}}.$ A.2 Proof of ().",
"From (REF ), then $\\mathcal {F}\\big (\\partial _t\\varrho _{2,n}^{\\rm {ap}}\\big )&=-\\frac{\\lambda _+\\lambda _-(e^{\\lambda _+t}-e^{\\lambda _-t})}{\\lambda _+-\\lambda _-}\\mathcal {F}\\big (f_n\\big )-\\frac{\\lambda _{+}e^{\\lambda _+t}-\\lambda _{-}e^{\\lambda _-t}}{\\lambda _+-\\lambda _-}\\mathcal {F}\\big ( g̥_n\\big )\\\\&=-\\frac{e^{\\lambda _+t}-e^{\\lambda _-t}}{\\sqrt{\\mu ^2-|\\xi |^{-2}}}\\mathcal {F}\\big (f_n\\big )-\\frac{\\lambda _{+}e^{\\lambda _+t}-\\lambda _{-}e^{\\lambda _-t}}{\\lambda _+-\\lambda _-}\\mathcal {F}\\big ( g̥_n\\big ),$ similarly, we have $\\Vert \\partial _t\\varrho _{2,n}^{\\rm {ap}}\\Vert _{H^{s+k}}&=\\Big [\\int _{\\mathbb {R}^2}(1+|\\xi |^2)^{\\frac{s+k}{2}}\\big |\\mathcal {F}\\big (\\partial _t\\varrho _{2,n}^{\\rm {ap}}\\big )\\big |^2\\mathrm {d}\\xi \\Big ]^{\\frac{1}{2}}\\\\&\\lesssim \\Big [\\int _{\\mathcal {C}_n }(1+|\\xi |^2)^{\\frac{s+k}{2}}\\big |\\widehat{f_n}(\\xi )\\big |^2\\mathrm {d}\\xi \\Big ]^{\\frac{1}{2}}+\\Big [\\int _{\\mathcal {B}(0,1)}(1+|\\xi |^2)^{\\frac{s+k}{2}}\\big |\\mathcal {F}\\big ( g̥_n\\big )\\big |^2\\mathrm {d}\\xi \\Big ]^{\\frac{1}{2}}\\\\&\\lesssim \\Vert f_n\\Vert _{H^{s+k}}+\\Vert g_n\\Vert _{L^{2}}.$ A.3 Proof of (REF ).",
"Notice that $&\\widehat{\\varrho ^{\\rm {ap}}_{2,n}}-\\widehat{f_n}=\\frac{\\lambda _+(e^{\\lambda _-t}-e^{\\lambda _+t})}{\\lambda _+-\\lambda _-}\\widehat{f_n}+(e^{\\lambda _+t}-1)\\widehat{f_n}-\\frac{e^{\\lambda _+t}-e^{\\lambda _-t}}{\\lambda _+-\\lambda _-}\\mathcal {F}(g̥_n),\\\\&\\widehat{u^{\\rm {ap}}_{2,n}}-\\widehat{g_n}=-\\frac{e^{\\lambda _+t}-e^{\\lambda _-t}}{\\lambda _+-\\lambda _-}\\widehat{\\nabla f_n}+\\Big (e^{\\lambda _+t}-1+\\frac{\\lambda _-(e^{\\lambda _+t}-e^{\\lambda _-t})}{\\lambda _+-\\lambda _-}\\Big )\\widehat{g_n},$ then by (REF ) $\\Vert \\varrho ^{\\rm {ap}}_{2,n}-f_n\\Vert _{H^{s+1}}&\\lesssim t\\Big [\\int _{\\mathcal {C}_n }(1+|\\xi |^2)^{\\frac{s+1}{2}}|\\widehat{f_n}|^2\\mathrm {d}\\xi +\\int _{\\mathcal {B}(0,1)}(1+|\\xi |^2)^{\\frac{s+1}{2}}\\big |\\mathcal {F}\\big (g̥_n\\big )\\big |^2\\mathrm {d}\\xi \\Big ]^{\\frac{1}{2}}\\\\&\\lesssim t\\big (\\Vert f_n\\Vert _{H^{s+1}}+\\Vert g_n\\Vert _{L^{2}}\\big )\\lesssim t2^{n}$ and $\\Vert u^{\\rm {ap}}_{2,n}-g_n\\Vert _{H^{s+k}}&\\lesssim \\Big [\\int _{\\mathcal {C}_n }(1+|\\xi |^2)^{\\frac{s+k}{2}}\\big |\\widehat{\\Lambda ^{-1}f_n}\\big |^2\\mathrm {d}\\xi +t^2\\int _{\\mathcal {B}(0,1)}(1+|\\xi |^2)^{\\frac{s+k}{2}}\\big |\\mathcal {F}\\big ( g_n\\big )\\big |^2\\mathrm {d}\\xi \\Big ]^{\\frac{1}{2}}\\\\&\\lesssim \\Vert f_n\\Vert _{H^{s+k-1}}+t\\Vert g_n\\Vert _{L^{2}}\\lesssim 2^{(k-1)n}+t2^{-n}.$ By Lemma REF , we deduce $&\\Vert (u_{2,n}^{\\rm {ap}}-g_n)_1\\partial _1\\varrho _{2,n}^{\\rm {ap}}\\Vert _{H^{s}}\\lesssim \\Vert u_{2,n}^{\\rm {ap}}-g_n\\Vert _{H^{s-1}}\\Vert \\varrho _{2,n}^{\\rm {ap}}\\Vert _{H^{s+1}}+\\Vert u_{2,n}^{\\rm {ap}}-g_n\\Vert _{H^{s}}\\Vert \\varrho _{2,n}^{\\rm {ap}}\\Vert _{H^{s}}\\lesssim t+2^{-n},\\\\&\\Vert (g_n)_1\\partial _1(\\varrho _{2,n}^{\\rm {ap}}-f_n)\\Vert _{H^{s}}\\lesssim \\Vert g_n\\Vert _{H^{s}}\\Vert \\varrho _{2,n}^{\\rm {ap}}-f_n\\Vert _{H^{s+1}}\\lesssim t,\\\\&\\Vert (u_{2,n}^{\\rm {ap}})_2\\partial _2\\varrho _{2,n}^{\\rm {ap}}\\Vert _{H^{s}}\\lesssim \\Vert u_{2,n}^{\\rm {ap}}\\Vert _{H^{s-1}}\\Vert \\partial _2\\varrho _{2,n}^{\\rm {ap}}\\Vert _{H^{s}}+\\Vert u_{2,n}^{\\rm {ap}}\\Vert _{H^{s}}\\Vert \\partial _2\\varrho _{2,n}^{\\rm {ap}}\\Vert _{H^{s-1}}\\lesssim 2^{-n},$ which implies (REF ).",
"A.4 Proof of (REF ).",
"Notice that the support condition of $\\widehat{\\phi }$ , we deduce that $2^{-n}\\Vert \\partial _1\\big (\\phi (x_1)\\phi (x_2)\\big )\\partial _1f_n\\Vert _{H^s}\\gtrsim &~\\Vert \\phi ^{\\prime }(x_1)\\phi (x_1)\\phi ^2(x_2)\\cos (2^nx_1)\\Vert _{L^2}\\nonumber \\\\&-2^{-n}\\Vert (\\phi ^{\\prime }(x_1)\\phi (x_2))^2\\sin (2^nx_1)\\Vert _{L^2}\\nonumber \\\\\\gtrsim &~\\Vert \\phi ^2(x_2)\\Vert _{L^2}\\Vert \\phi ^{\\prime }(x_1)\\phi (x_1)\\cos (2^nx_1)\\Vert _{L^2}\\nonumber \\\\&-2^{-n}\\Vert (\\phi ^{\\prime }(x_1)\\phi (x_2))^2\\Vert _{L^2}.$ Using the simple formula $2\\cos ^2(2^nx_1)=1-\\cos (2^{n+1}x_1),$ we have $\\Vert \\phi ^{\\prime }(x_1)\\phi (x_1)\\cos (2^nx_1)\\Vert ^2_{L^2}=\\frac{1}{2}\\Vert \\phi ^{\\prime }(x_1)\\phi (x_1)\\Vert ^2_{L^2}-\\frac{1}{2}\\int _{\\mathbb {R}}|\\phi ^{\\prime }(x_1)\\phi (x_1)|^2\\cos (2^{n+1}x_1)\\mathrm {d}x_1,$ which follows from Riemann-Lebesgue's Theorem that $\\lim _{n\\rightarrow \\infty }\\Vert \\phi ^{\\prime }(x_1)\\phi (x_1)\\cos (2^nx_1)\\Vert ^2_{L^2}=\\frac{1}{2}\\Vert \\phi ^{\\prime }(x_1)\\phi (x_1)\\Vert ^2_{L^2}.$ Combining (REF ) and (REF ) yields the desired (REF )."
],
[
"Acknowledgments",
"J. Li is supported by the National Natural Science Foundation of China (Grant No.11801090).",
"Y. Yu is supported by the Natural Science Foundation of Anhui Province (No.1908085QA05).",
"W. Zhu is partially supported by the National Natural Science Foundation of China (Grant No.11901092) and Natural Science Foundation of Guangdong Province (No.2017A030310634)."
]
] | 2011.14125 |
[
[
"i3DMM: Deep Implicit 3D Morphable Model of Human Heads"
],
[
"Abstract We present the first deep implicit 3D morphable model (i3DMM) of full heads.",
"Unlike earlier morphable face models it not only captures identity-specific geometry, texture, and expressions of the frontal face, but also models the entire head, including hair.",
"We collect a new dataset consisting of 64 people with different expressions and hairstyles to train i3DMM.",
"Our approach has the following favorable properties: (i) It is the first full head morphable model that includes hair.",
"(ii) In contrast to mesh-based models it can be trained on merely rigidly aligned scans, without requiring difficult non-rigid registration.",
"(iii) We design a novel architecture to decouple the shape model into an implicit reference shape and a deformation of this reference shape.",
"With that, dense correspondences between shapes can be learned implicitly.",
"(iv) This architecture allows us to semantically disentangle the geometry and color components, as color is learned in the reference space.",
"Geometry is further disentangled as identity, expressions, and hairstyle, while color is disentangled as identity and hairstyle components.",
"We show the merits of i3DMM using ablation studies, comparisons to state-of-the-art models, and applications such as semantic head editing and texture transfer.",
"We will make our model publicly available."
],
[
"Introduction",
"3D morphable models (3DMMs) are parametric models of geometry and appearance of human faces, with widespread use in applications such as image and video editing, face recognition and cognitive science [19].",
"These models are trained using 3D scans of humans, e.g., laser scans [4], depth sensor-based scans [9], or photometric multi-view scans [28].",
"As 3DMMs usually learn a deformation space of a fixed template mesh, the training shapes commonly need to be brought into dense surface correspondence with each other.",
"Computing such correspondence for the face region alone is already hard and often requires a challenging non-convex optimizaton problem [19]; computing dense correspondence for the rest of the head and the hair is close to impossible.",
"Therefore, as well as due to the lack of large 3D scan datasets with hair, most 3DMMs only capture the face region.",
"While some recent approaches aim to model the complete head [15], [28], [41], [40], they do not capture the hair region, and the appearance in the head region (if captured) is very limited.",
"In this work, we present i3DMM, the first implicit 3D morphable model which captures the full head region, including hair deformations and appearance.",
"We capture a new dataset of full photogrammetric head scans of 64 people for training.",
"Each subject performs several expressions and natural hair is captured.",
"Since such scans are noisy, especially in the hair region, our training algorithm uses an adaptive sampling strategy based on the quality of reconstructions in different regions of the head, allowing us to effectively handle noise without smoothing out details.",
"In contrast to existing 3DMMs, which use a mesh-representation with a fixed template, we implicitly represent surfaces using signed distance functions.",
"This allows us to capture large deformations easily, which is particularly convenient for the hair region.",
"The implicit representation also allows us to avoid computing dense correspondence between training scans.",
"Our method is inspired by recent works on deep implicit surface modeling [36], [31], [32], which demonstrate the advantages of using an implicit representation compared to voxel grids, point clouds or meshes.",
"Our main technical innovation compared to these works is that we use a novel neural network architecture which decouples the learning process, separating it into learning a reference shape, learning geometry deformations with respect to this reference, and learning a color network.",
"This allows us to automatically compute dense correspondences between any two shapes with only sparse supervision on salient face landmarks.",
"This decoupling is also essential for disentangling the learned space into semantically meaningful parametric components, an important feature of 3DMMs.",
"Most classical models disentangle identity geometry, expression geometry, and appearance of the face.",
"This is sufficient for several applications in face editing [55], [53], [50], but does not allow the explicit control of head parts other than the facial region.",
"Our model allows for unprecedented control over the different semantic modes of full heads.",
"To this end, we learn to disentangle the geometry and color, where geometry is further separated into identity, expression, and hairstyle components, and color is further separated into identity and hairstyle components.",
"In summary, our main contributions are: A method for learning full head 3D morphable models directly from rigidly aligned real-world noisy 3D scans without dense ground truth correspondences.",
"A novel network architecture which can compute dense correspondences between 3D shapes represented using implicit functions.",
"This network is trained only with sparse supervision.",
"A training method for disentangling the color and geometry components, and the identity, expression, and hairstyle components of the geometry; and the identity and hairstyle components of the color.",
"We compare our model to several state-of-the-art 3DMMs by fitting to 3D scans.",
"We also demonstrate the quality of our learned dense correspondences by showing texture transfer between scans, and show that i3DMM benefits applications such as semantic head editing, and one-shot computation of segmentation masks and landmarks on 3D scans."
],
[
"Related Work",
"Most approaches for face and head modeling are mesh-based.",
"We summarize them here and refer the reader to Egger [19] for a more comprehensive report.",
"As our model is based on implicit representations, we also discuss methods which learn implicit shape and color.",
"Face and Head Morphable Models.",
"The work of Blanz and Vetter [5] showed the possibility of representing faces using a 3D Morphable Model (3DMM).",
"The model is learned by transforming the shape and texture of examples into low-dimensional vector representations using principal components analysis (PCA).",
"Further improvements were proposed [22], [38], [8], [6] by using better registration and higher quality scans.",
"Multilinear models have also been proposed to model identity-dependent expressions [10], [20].",
"Li [29] presented FLAME, a head model that combines a linear shape space with an articulated jaw, neck and eyeballs, pose-dependent corrective expression blendshapes, and global expressions.",
"The model is learned from a large 3D dataset, including data from D3DFACS [12].",
"Ranjan [42] learned a head model using an autoencoder based on spectral graph convolutions [17].",
"The LYHM head model [16], [14] uses a hierarchical parts-based template morphing framework to process the head shape, and uses optical flow to refine the texture.",
"The model is built using $1{,}200$ different identities.",
"Ploumpis [40], [39] combined the LYHM and LSFM [7] models as the Universal Head model (UHM), with a focus on face geometry.",
"Note that all existing full head models only model the cranium geometry without hair.",
"Implicit Representation.",
"Implicit representations have a long history in computer vision for inferring 3D shape [27], temporally evolving shape [13], [26] and textured shape [49].",
"Park [36] presented DeepSDF to represent a class of shapes using signed distance fields with an autodecoder.",
"Chen [11] and Mescheder [31] learned a generative model of a class of shapes by classifying a point in space as inside or outside a shape.",
"Recent approaches [56], [18], [21] have proposed to represent shapes using a collection of local implicit patches for higher quality.",
"Oechsle [35] extended OccupancyNets [31] to represent texture along with the geometry using monocular inputs.",
"Niemeyer [34] presented an approach for handling time-varying shapes by learning to predict motion vectors for each point in 3D space using OccupancyNets.",
"PIFu [45], [46] allows for 3D reconstruction of humans from monocular inputs.",
"Pixel-aligned implicit functions estimate a continuous field that determines whether a pixel is inside or outside the surface of the human subject, as well as the color on the surface.",
"Several recent methods [48], [32], [51] have presented ways to achieve higher-quality implicit representations by using periodic functions as activations.",
"Saito [44] presented an approach for 3D hair modeling.",
"Their technique learns a manifold for 3D hairstyles represented as implicit surfaces using a volumetric variational autoencoder."
],
[
"Method",
"In this section, we describe how we obtain our deep implicit 3D morphable model (i3DMM).",
"Our overall processing pipeline is illustrated in Fig.",
"REF .",
"In the following we will describe our data acquisition, the training data preparation, and the neural network architecture.",
"Figure: Overview of the captured expressions (red box: test set; blue box: hairstyles) for each subject.Figure: Multi-view scanning system with 135 cameras around the person (green), and 2 cameras (blue) on top.",
"Right: front view image, and reconstruction."
],
[
"Data Acquisition",
"For training, we have scanned 64 subjects (46 male, 18 female; 22 Caucasian, 9 Asian, 25 Indian Sub-continent, 3 Hispanic, 3 Middle Eastern, 2 undisclosed; with an age ranging from 19 to 69 with average 26).",
"For each subject we have recorded 10 facial expressions (a subset of which are chosen from paGAN [33]), including neutral expressions with 3-4 different “hairstyles” (open hair, tied hair for subjects with long hair, and two different caps), as shown in Fig.",
"REF .",
"Our data was acquired with the Treedyshttps://www.treedys.com/ multi-view scanning system, that comprises of 137 calibrated cameras, see Fig.",
"REF .",
"The total capture time per person amounts to about 20 minutes.",
"Photogrammetry-based 3D reconstruction on the multi-view data was applied to obtain the final textured meshes (see Fig.",
"REF ), where each mesh comprises of around $50{,}000$ vertices."
],
[
"Training Data",
"Given our recorded textured mesh data, we apply several preprocessing steps to make the data is suitable for head model learning.",
"This includes a semi-automated landmark annotation, cropping of the head area to discard the shoulders and parts of the upper body, rigidly aligning the meshes, and closing the hole at the neck to make the mesh watertight.",
"In the following we elaborate on these steps.",
"Landmark Annotation.",
"First, we apply the automated face landmark detector from dlib [24] on the front-facing image obtained from the multi-view camera acquisition setup, which produces a total of 66 face landmarks.",
"We select a subset of 8 landmarks, i.e.",
"the four corner points of the eyes, the tip of the nose, the two corner points of the mouth, and the chin, and then transfer these 2D landmarks to the 3D mesh.",
"We manually correct the inaccurately detected landmarks, and additionally annotate 8 landmarks for the corners of ears (top, bottom, left, and right) directly in 3D.",
"Head Cropping.",
"In order to remove the shoulders and parts of the upper body, we crop the head meshes based on the 3D landmarks.",
"To this end, we first compute the vector $\\mathbf {v}$ from the centroid of the four eye cornerpoints to the tip of the nose.",
"Then, for $\\mathbf {p}$ being the chin landmark in 3D space, we define a virtual plane that goes through the point $\\mathbf {p} + \\frac{1}{2} \\mathbf {v}$ and has the normal $\\mathbf {v}$ .",
"We only keep the side of the plane that contains the facial landmarks.",
"Rigid Alignment.",
"Based on the 3D facial landmarks, we perform a rigid alignment of all head scans in order to ensure that they are oriented consistently in 3D space.",
"We use a numerically stable implementation [3] of the transformation synchronization method [2], which solves the generalized Procrustes problem in an initialization-free and unbiased way.",
"To this end, we first rigidly align all pairs of heads based on Singular Value Decomposition (SVD), i.e.",
"we solve all pairwise Procrustes problems, and subsequently use transformation synchronization to establish cycle-consistency in the set of pairwise transformations.",
"Hole Closing.",
"Since our full head model is based on a signed distance representation, we require that the meshes are watertight.",
"After rigid alignment, we assume that the longitudinal axis of the head aligns with the y-axis.",
"With that, we use a flat patch to close the hole at the neck by extruding the respective boundary vertices to coincide with the smallest y-coordinate.",
"Finally, we scale all meshes with the same value such that all of them fit in the unit cube."
],
[
"Training",
"We learn a vector-valued function $f_{\\theta }(\\mathbf {x},\\mathbf {z})$ , where $\\theta $ are the neural network weights, $\\mathbf {x} \\in \\mathbb {R}^3$ is the query point, and $\\mathbf {z}$ is a code vector that encodes the head instance.",
"The output $f_{\\theta }(\\mathbf {x},\\mathbf {z}) = ({s}, \\mathbf {c})$ includes a scalar ${s}$ that represents the signed distance to the surface, as well as the color $\\mathbf {c} \\in \\mathbb {R}^3$ at the closest surface point from $\\mathbf {x}$ .",
"The shape boundary is represented as the zero level-set of the signed distance function (SDF), while the interior parts of the shapes have a negative signed distance value, and the exterior parts a positive value.",
"We use an autodecoder network architecture [36], where the weights of the network $\\theta $ and the input latent codes $\\mathbf {z}$ for all shapes are learned jointly.",
"Mesh Sampling.",
"We require $(\\mathbf {x},{s}, \\mathbf {c})$ -triplets (query point, signed distance value, color) for training.",
"We use a combination of two strategies for sampling these triplets.",
"First, we sample points on the mesh surface based on uniform random sampling.",
"However, in order to also account for higher accuracy in high-detail facial regions, i.e.",
"the eyes, nose and mouth, we additionally sample more points in these areas.",
"We take the center of each eye, the tip of the nose, and the center of the mouth, and place a sphere around them that covers the respective region of interest.",
"Then, we sample points on the mesh surface that lie within each sphere.",
"Eventually we mix the uniformly sampled points with the landmark-based sampled points in a 1 to 3 ratio.",
"After sampling, the points are perturbed by a uniform 3D Gaussian with standard deviation $0.005$ times the length of the bounding box, such that not only surface points but also points in the interior and exterior are used, see [36].",
"The color value for each sample is obtained by finding the closest point on the mesh surface, and then looking up its color in the texture map.",
"Latent Codes and Disentanglement.",
"As mentioned earlier, latent codes for each object (head scan) are also learned during training.",
"Existing approaches use a single object latent code to describe the shape.",
"In contrast, we design several separate latent spaces for our objects in order to learn a semantically disentangled model.",
"We use two separate latent vector spaces for geometry and color, $\\mathbf {z}_\\text{geo}$ and $\\mathbf {z}_\\text{col}$ , respectively.",
"The geometry space includes three code vectors for identity, expression, and hairstyle, and the color space includes two code vectors for identity and hairstyle.",
"Thus, $\\mathbf {z}_\\text{geo} = (\\mathbf {z}_\\text{geoId}, \\mathbf {z}_\\text{geoEx}, \\mathbf {z}_\\text{geoH})$ and $\\mathbf {z}_\\text{col} = (\\mathbf {z}_\\text{colId}, \\mathbf {z}_\\text{colH})$ .",
"During training, the number of different identity code vectors is equal to the number of training identities, 58.",
"The number of different expression vectors is fixed to 10 (cf.",
"Fig.",
"REF for the training expressions) and hairstyle to 4 (short, long, cap1 or cap2) for geometry, and to 3 (nocap, cap1 or cap2) for color.",
"For all scans of the $i$ -th subject we use the same latent variables for the geometry identity and color identity, $\\mathbf {z}_\\text{geoId}^i$ and $\\mathbf {z}_\\text{colId}^i$ .",
"Similarly, for each expression and hairstyle, the same variables $\\mathbf {z}_\\text{geoEx}, \\mathbf {z}_\\text{geoH}$ and $\\mathbf {z}_\\text{colH}$ are used across all identities.",
"By doing so, we are able to learn disentangled latent variables without imposing any explicit constraints.",
"At test time, we can control each latent space individually, leading to semantically meaningful editing.",
"Network Architecture.",
"Figure: Overview of our network architecture.",
"We learn weights of three network components, a Shape Deformation component, a Reference Shape component, and a Color component.",
"Moreover, the latent codes for each object are also optimized for.",
"The input of the network is a 3D query point, and the output is a signed distance value along with the corresponding color.Our network comprises three components, the Reference Shape Network (RefNet), the Shape Deformation Network (DeformNet) and the Color Network (ColorNet).",
"All networks are composed of fully-connected layers with a ReLU non-linearity after every layer, except the output layer.",
"The inputs to all our networks are encoded using sinusoidal positional encoding [32].",
"Our Reference Shape Network encodes a single reference shape, such that all individual head shapes can be obtained by deforming this shape.",
"The output at a query point $\\mathbf {x}$ is the signed distance value $f_\\theta ^r(\\mathbf {x}) \\in \\mathbb {R}$ .",
"Note that for the reference shape there is no latent code input since only a single reference shape is learned.",
"This can be seen analogously to the mean (or neutral) shape used in classical 3DMMs [5].",
"We use 3 fully connected layers for this network, where each hidden layer has dimensionality 512.",
"The Shape Deformation Network deforms the reference shape to represent the shape of an individual head.",
"The network takes the geometry latent code $\\mathbf {z}^i_\\text{geo}$ for an object $i$ , and a query point $\\mathbf {x}$ as input, and produces the output, $f_\\theta ^s(\\mathbf {x}, \\mathbf {z}^i_\\text{geo}) = \\delta \\in \\mathbb {R}^3\\,,$ which is a displacement vector that deforms the query point to the reference shape.",
"Thus, the signed distance value at any point $\\mathbf {x}$ of the object $i$ with geometry latent $\\mathbf {z}^i_\\text{geo}$ is ${s}(\\mathbf {x}, \\mathbf {z}^i_\\text{geo}) = f_\\theta ^r(\\mathbf {x} + \\delta )\\,,$ where ${s}(\\cdot )$ is the scalar signed distance value, and $\\delta $ is the deformation from Eq.",
"(REF ).",
"This formulation allows us to compute dense correspondences between any input head scan and the reference head shape.",
"The separation between a reference shape and deformations with respect to this reference shape is also common in classic morphable models [19].",
"However, these models require dense correspondences before training, which is not necessary in our approach.",
"We use 8 fully connected layers for this network, where each hidden layer has dimensionality 1024.",
"The introduction of the reference shape makes it possible to disentangle the geometry and color components.",
"The Color Network learns the color of the query point in the reference space.",
"Given a query point $\\mathbf {x}$ , deformation $\\delta $ from Eq.",
"(REF ), and color latent vector $\\mathbf {z}^i_\\text{col}$ for the object $i$ , the output is represented as $f_\\theta ^c(\\mathbf {x}+\\delta , \\mathbf {z}^i_\\text{col}) \\in \\mathbb {R}^3$ , which is the color at point $\\mathbf {x}$ .",
"Note that without this separation of a reference space, the ColorNet would also have to take into account information about object geometry, thus not being able to disentangle shape and color.",
"Note that the latent code for ColorNet, for a given identity and hairstyle, does not change with expressions.",
"DeformNet finds the right colors and geometry for different expressions by achieving dense correspondences.",
"We use 9 fully connected layers for this network, where each hidden layer has dimensionality 1024.",
"Loss Functions.",
"We define the loss function to train our network as $\\mathcal {L}_\\theta (\\mathbf {x}, \\mathbf {z_\\text{geo}}, \\mathbf {z_\\text{col}}) & = \\sum _{i=1}^K(\\mathcal {L}_\\theta ^\\text{geo}(\\mathbf {x}, \\mathbf {z}^i_\\text{geo}) +\\mathcal {L}_\\theta ^\\text{def}(\\mathbf {x}, \\mathbf {z}^i_\\text{geo}) \\nonumber \\\\ &+\\mathcal {L}_\\theta ^\\text{col}(\\mathbf {x}, \\mathbf {z}^i_\\text{col}) + \\mathcal {L}^\\text{reg}(\\mathbf {z}^i_\\text{geo}, \\mathbf {z}^i_\\text{col}) ) \\nonumber \\\\ &+ \\sum _{i\\ne j}\\mathcal {L}^\\text{lm}_\\theta (\\mathbf {z}^i_\\text{geo},\\mathbf {z}^j_\\text{geo})\\,,$ where, $i,\\,j = \\lbrace 1,\\dots ,K\\rbrace $ , $K$ is the number of scans in the batch, $\\mathbf {z}_\\text{geo} = \\lbrace \\mathbf {z}^1_\\text{geo},\\dots ,\\mathbf {z}^K_\\text{geo}\\rbrace $ , and $\\mathbf {z}_\\text{col} = \\lbrace \\mathbf {z}^1_\\text{col},\\dots ,\\mathbf {z}^K_\\text{col}\\rbrace $ .",
"The latent vectors of head $i$ are represented as $\\mathbf {z}_\\text{geo}^i$ and $\\mathbf {z}_\\text{col}^i$ .",
"Here, $\\mathcal {L}_\\theta ^\\text{geo}(\\cdot )$ enforces good geometry reconstructions, $\\mathcal {L}_\\theta ^\\text{def}(\\cdot )$ regularizes the deformation field, $\\mathcal {L}_\\theta ^\\text{col}(\\cdot )$ is used to train the ColorNet, $\\mathcal {L}^\\text{reg}(\\cdot )$ is a regularizer on the latent vectors, and $\\mathcal {L}^\\text{lm}_\\theta (\\cdot )$ is a sparse pairwise landmark supervision loss.",
"For the geometry term that we impose upon the signed distance values, we use the $\\ell _1$ -loss $\\mathcal {L}_\\theta ^\\text{geo}(\\mathbf {x}, \\mathbf {z}^i_\\text{geo}) = w_g \\big \\Vert \\text{cl}(s(\\mathbf {x}, \\mathbf {z}^i_\\text{geo}), t) - \\text{cl}(s_\\text{gt}(\\mathbf {x}), t) \\big \\Vert _1\\,{,}$ where $s_\\text{gt}(\\mathbf {x})$ is the ground truth signed distance value at $\\mathbf {x}$ , and $s(.",
")$ is from Eq.",
"(REF ).",
"These values are (symmetrically) clamped at $t$ =$0.1$ , for which we define $\\text{cl}(x, t) := \\min (t, \\max (x, -t))$ .",
"We use a similar $\\ell _1$ -loss for the color component, i.e.",
"$\\mathcal {L}_\\theta ^\\text{col}(\\mathbf {x}, \\mathbf {z}^i_\\text{col}) = w_c \\big \\Vert f_\\theta ^c(\\mathbf {x}+\\delta , \\mathbf {z}^i_\\text{col}) - c_\\text{gt}(\\mathbf {x}) \\big \\Vert _1\\,{,}$ where $c_\\text{gt} (\\mathbf {x})$ is the ground truth color at $\\mathbf {x}$ .",
"We also enforce that the 16 landmarks $\\lbrace \\mathbf {x}^{\\ell }_i\\rbrace $ , as described in Sec.",
"REF , of each shape $i$ , deform to the same points in the reference space using the pairwise loss $ \\mathcal {L}^\\text{lm}_\\theta (\\mathbf {z}^i_\\text{geo},\\mathbf {z}^j_\\text{geo}) = w_{lm} \\sum _{\\ell =1}^{16} \\big \\Vert (\\mathbf {x}^\\ell _i + \\delta ^\\ell _i) - (\\mathbf {x}^\\ell _j + \\delta ^\\ell _j) \\big \\Vert _2\\,,$ where, $\\delta ^\\ell _i = f_\\theta ^s(\\mathbf {x}^\\ell _i, \\mathbf {z}^i_\\text{geo})$ as in Eq.",
"(REF ).",
"For scans with the ears covered by hair, we do not have any ear annotations.",
"We would like to compress the ear region in the reference shape to a single point in the reconstruction for these shapes.",
"Thus, we additionally optimize for one point for each ear.",
"We enforce pairwise constraints between the learnable ear points and the annotated ear points for the other shapes in the batch using Eq.",
"(REF ).",
"Further, to ensure regularized deformations to the reference shape, we impose a loss on the amount of deformation.",
"We use $\\mathcal {L}_\\theta ^\\text{def}(\\mathbf {x}, \\mathbf {z}^i_\\text{geo}) = w_s || f_\\theta ^s(\\mathbf {x}, \\mathbf {z}^i_\\text{geo}) ||_2$ .",
"Finally, we use an $\\ell _2$ -regularizer on the latent vectors assuming a Gaussian prior distribution, $\\mathcal {L}^\\text{reg}(\\mathbf {z}^i_\\text{geo}, \\mathbf {z}^i_\\text{col}) =w_r ( ||\\mathbf {z}^i_\\text{geo}||_2 + ||\\mathbf {z}^i_\\text{col}||_2)$ .",
"Optimization.",
"Given $N$ batches with $K$ objects per batch, we optimize for the network weights and the latent vectors by solving the optimization problem $\\operatornamewithlimits{argmin}_{\\theta , \\lbrace \\mathbf {z}^b_\\text{geo}, \\mathbf {z}^b_\\text{col}\\rbrace _{b=1}^N}~~ \\sum _{b=1}^N\\sum _{\\mathbf {x}} \\mathcal {L}_\\theta (\\mathbf {x}, \\mathbf {z}^b_\\text{geo}, \\mathbf {z}^b_\\text{col})\\,,$ where, the inner sum takes into account all sampled points, as explained above, and we abuse the notation $\\mathbf {z}^b_\\text{geo}, \\mathbf {z}^b_\\text{col}$ to now represent latent codes of all the scans in batch $b$ ."
],
[
"Experiments",
"In this section, we present an experimental evaluation of our head model.",
"We demonstrate that the model can be used for reconstructing scan data.",
"We present an ablation study to carefully analyze our design choices, and compare i3DMM to state-of-the-art head models.",
"We also show dense correspondence results and applications of our model.",
"Before we present these results, we provide additional information on the neural network training and testing.",
"Training Details.",
"We train our networks using PyTorch [37], where we use the Adam [25] solver with mini-batches of size 64.",
"We train for 1000 epochs with a learning rate of $0.0005$ , which decays by a factor of 2 every 250 epochs.",
"We initialize RefNet by pretraining it using only one mouth-open (top row, third from left in Fig.",
"REF ) training scan.",
"Our network takes about 2 days to train on 2 NVIDIA RTX8000 GPUs.",
"Test Data.",
"We collect a separate test set consisting of scans of 6 identities, which are not part of the training data.",
"We capture each identity in 5 novel expressions that are not part of the training expressions, see Fig.",
"REF .",
"Each scan is preprocessed analogously to the training scans."
],
[
"Reconstruction",
"For a given test scan, we fit our learned model to it.",
"This is done by optimizing for the latent vector that can best reproduce the scan, i.e.",
"by finding the latent variables that minimize the problem $\\operatornamewithlimits{argmin}_{\\mathbf {z}_\\text{geo}^i, \\mathbf {z}_\\text{col}^i}~~ \\sum _{\\mathbf {x}} &\\, (\\mathcal {L}_\\theta ^\\text{geo}(\\mathbf {x}, \\mathbf {z}^i_\\text{geo}) +\\mathcal {L}_\\theta ^\\text{def}(\\mathbf {x}, \\mathbf {z}^i_\\text{geo}) \\nonumber \\\\ + &\\mathcal {L}_\\theta ^\\text{col}(\\mathbf {x}, \\mathbf {z}^i_\\text{col}) + \\mathcal {L}^\\text{reg}(\\mathbf {z}^i_\\text{geo}, \\mathbf {z}^i_\\text{col}))\\,,$ where, $\\mathbf {z}_\\text{geo}^i, \\mathbf {z}_\\text{col}^i$ is the latent code for test scan $i$ .",
"This equation is similar to Eq.",
"(REF ), with the difference that the network weights are fixed here and the pairwise landmark supervision loss in Eq.",
"(REF ) is not enforced.",
"We use Adam [25] with a step size of $0.0005$ to solve this problem.",
"We show several reconstruction results on the test data in Fig.",
"REF .",
"We can generalize to unseen identities and expressions, even though our training data only consists of 58 people with 10 expressions.",
"Note that while we can generally preserve the detailed face region of the scans, we also smooth the scans in the noisy hair area."
],
[
"Correspondences",
"As mentioned in Sec.",
"REF , due to the particular design of our network, where the reference shape and the deformations are separated, our approach also establishes dense correspondences between shapes, with extremely sparse landmark supervision.",
"We demonstrate these correspondences in Fig.",
"REF , where the color is transferred from one scan to the other.",
"The correspondences are also used in the applications of segmentation and landmark transfer in Sec.",
"REF .",
"Our model can reliably find correspondences across different subjects, expressions and hairstyles, including long and short hair.",
"Please refer to the supplementary material for more evaluations.",
"Figure: Reconstruction quality of i3DMM on test data.",
"The top part shows texture and geometry of ground truth scans.",
"The middle part shows texture and geometry of i3DMM fits.",
"The last part shows texture transfer from the scan on right to the scan on left of the image.",
"In column 3, noise in ground truth does not transfer to the i3DMM fit, showing the robustness of our method."
],
[
"Ablative Analysis",
"We design several experiments to evaluate the most important design choices for our approach: landmark-based sampling described in Sec.",
"REF , sparse pairwise landmarks loss in Eq.",
"(REF ), and jointly training ColorNet, DeformNet, and RefNet.",
"We evaluate the qualitative impact of these choices by excluding them one at a time while training our model, see Fig.",
"REF .",
"Without landmarks-based sampling, the model focuses more on the noisy hair region and ignores the details in the face region (first column in Fig.",
"REF ).",
"Without landmark supervision, the network creates small (fake) ear regions in the hair for the long-haired scans (second column in Fig.",
"REF ).",
"It also leads to poor texture transfer around the ear.",
"Finally, for evaluating the importance of joint training of the color and geometry networks, we first train RefNet and DeformNet.",
"After convergence, we train ColorNet.",
"The correspondences in this case are learned only from geometry reconstructions, which leads to artifacts as shown in the third column of Fig.",
"REF .",
"Figure: Ablation study.",
"Top and middle row show reconstructions of two different test scans.",
"Last row shows texture transfer from the scans in the top row to those in the middle row."
],
[
"Comparisons to Existing Models",
"We compare our model with the full head FLAME [29], face only FLAME, and Basel Face Model (BFM) [38] which only models the frontal face.",
"We fit each model to the test set by optimizing for their model parameters.",
"Please refer to the supplemental for more details on the fitting.",
"We show qualitative results in Fig.",
"REF .",
"FLAME only models the cranium without hair, thus fitting to the full head region results in incorrect head shapes.",
"We also evaluate FLAME by only using the the face region for fitting.",
"This leads to higher quality results.",
"We obtain higher quality geometry both for the face and head regions.",
"In addition, we can also reconstruct the color of the head, while FLAME can only model the geometry.",
"We combine the BFM [38] identity geometry and appearance models with the expression model used in Tewari [52], and use this to fit to our test scans.",
"The expression model is a combination of two blendshape models [1], [9].",
"Since BFM also includes color, we jointly optimize to minimize the geometry as well as color alignment errors.",
"As can be seen in Fig.",
"REF (middle), this model can fit to the expression as well as color of the scans.",
"However, it is limited to only the face region, while i3DMM can reconstruct the full head.",
"In addition, our reconstructions are more personalized, with higher quality nose geometry and face colors.",
"We show the quantitative evaluations of the different models in terms of the symmetric Chamfer distance and F-score metrics in Table REF .",
"Chamfer distance is the mean distance of points from one shape to their closest points in another.",
"We compute the symmetric Chamfer distance as the mean of Chamfer distance from the ground truth to the fits and the Chamfer distance in the opposite direction.",
"F-score is (100x) the harmonic mean of these two distances after applying a threshold.",
"We use a similar metric for color.",
"First, the mean error in color at points in the ground truth and the color at their nearest points in the fits is computed, along with the mean error in the opposite direction Finally, the color error is reported as their average.",
"Please refer to the supplemental for details on the metrics used.",
"We sample $150{,}000$ points for computing the metrics.",
"We achieve similar geometric quality as FLAME and BFM in the face region, but significantly outperform FLAME for the full head reconstructions, see Table REF .",
"Moreover, in terms of color, our reconstructions are more realistic than BFM and also outperform it quantitatively.",
"Table: Quantitative comparison of head models.",
"Symmetric Chamfer distance is reported here.",
"F-score is computed with a threshold of 0.010.01."
],
[
"Application: Semantic Head Editing",
"Due to the semantic disentanglement in our model, we can selectively change semantic components of a head.",
"In Fig.",
"REF , we edit one feature of a test scan while keeping the other features fixed.",
"To obtain the edited latent codes, we first find the principle components of variation in each latent-subspace by running PCA on the training latent spaces.",
"We move along the principal components to pick the latent codes to edit the identity, and we pick others from the trained latent codes.",
"Unlike existing models, we can also edit hairstyles and add caps while keeping other components fixed.",
"We also model the mouth interior.",
"These features allow for semantic head editing applications much beyond the capabilities of the existing methods."
],
[
"Application: One-shot Annotation Transfer",
"As i3DMM can predict dense correspondences among head scans, it also enables us to transfer annotations across different reconstructions.",
"As our model has low reconstruction errors, it can also be used to annotate real-world 3D scans.",
"We show this application in Fig.",
"REF .",
"We first transfer the annotations from one manually annotated scan to the reference shape.",
"We can then transfer them to different i3DMM fits, and also to the real-world scans using nearest neighbors from the fits to the scans.",
"As i3DMM can be used to even annotate the hair region, it can be very useful to curate scan datasets.",
"Please refer to the supplemental for more results.",
"Figure: Application: One-shot annotations on real-world scans.",
"Coarsely drawn annotations (top row: semantic segmentation, bottom row: landmarks, not used during training) on one scan (left) can be automatically transferred to other real-world head scans (right)."
],
[
"Discussion and Future Work",
"Although i3DMM disentangles and models an unprecedented amount of variety in head features, and enables reconstructions with very high quality (Fig.",
"REF ), it still has some limitations.",
"Although i3DMM models varying hairstyle shapes, it is only a coarse approximation of the physical nature of hair comprising of individual hair strands.",
"Further, we model hairstyles that cover the ear by trying to collapse the ear to a single point (Eq.",
"(REF )).",
"Ideally, the layered geometry of ears behind the hair should be correctly represented.",
"Finally, there is room for extending i3DMM for more fine-grained control over hairstyles.",
"One challenge is that our scans are noisy in the hair region.",
"This is due to the photometric multi-view reconstruction techniques used, which struggle with hair due to their complex physical interaction with light."
],
[
"Conclusion",
"We have presented the first unified 3D morphable model of geometry and appearance of full heads, which includes different identities, expressions and hairstyles.",
"The head geometry and color is represented using implicit function parameterized with neural networks, learned from 610 multi-view scans.",
"By explicitly separating the network into reference shape, shape deformation and color components, our model can disentangle the different semantic components.",
"It also allows us to obtain dense correspondences across different scans represented using our model, enabling several applications.",
"We will make our code publicly available to the community."
],
[
"Acknowledgements",
"This work has been supported by the ERC Consolidator Grant 4DReply (770784), and by the BMBF-funded Munich Center for Machine Learning.",
"We thank Garvita Tiwari, Navami Kairanda, and Neng Qian for their support in capturing the dataset.",
"We thank all the subjects in the dataset for lending their data for research.",
"Figure: NO_CAPTIONSupplementary Model Visualization Our implicit 3D morphable model models the identity, expression, and hairstyle components of the geometry; as well as the identity and hairstyle components of the color of the head with independent parameteric controls.",
"We visualize these components in Fig.",
"REF .",
"We perform PCA on the identity geometry and color spaces in order to compute the principal components.",
"The expression component is visualized by moving along the directions of the training expressions, since they are semantically well defined.",
"As we model hairstyles that include caps, we show the joint space of geometry and color for hairstyles.",
"Note that the hairstyle geometry can only take four discrete values – short, long, cap1, or cap2.",
"Any variation within these categories is modelled by the identity-geometry component.",
"Similarly, hairstyle color can only take the values – nocap, cap1, or cap2.",
"The color of hair without any cap is determined by the identity-color component.",
"Experiments Here, we provide more details on the evaluations in the main paper, and include further evaluations.",
"Sampling i3DMM One of the important features of a 3D Morphable Model is the ability to randomly sample shapes in the parametric space.",
"This has been used for generating synthetic data for training CNNs [23], [43], [47].",
"We achieve sampling in i3DMM by performing principal component analysis (PCA) over the training latent codes for color-identity, geometry-identity, and expressions.",
"We weigh the singular values using a Gaussian random variable $\\mathcal {N}(0,0.1)$ for color-identity, and geometry-identity, and $\\mathcal {N}(0,0.25)$ for expressions.",
"Since the latent codes for hair shapes and colors can take very limited numbers of values and are very well defined semantically, we sample these from their training values.",
"We show several results in the supplemental video.",
"As can be seen, our model is biased towards generating male heads.",
"This is likely due to our gender-biased training dataset with 46 males and 18 females.",
"While this does not lead to any clear loss of quality when fitting to female test scans, see Fig.",
"6 in the main paper, it might lead to biased quality of results in other problems, for eg., if random samples from the model was used for training another network.",
"Comparisons As mentioned in Sec.",
"4.4 of the main paper, we compare our model to two existing models, BFM [38] and FLAME [29].",
"We show more qualitative model fitting results in Fig.",
"REF .",
"Next, we provide more details on the fitting algorithm used.",
"Fitting: We paint the face region of each model's template mesh to create a mask as shown in Fig.",
"REF .",
"We use these masked regions of the models for fitting the models to head scans.",
"To initialize, we mark 8 landmarks (eye corners, nose, lip corners, and chin) on the template mesh and rigidly align the template to each ground truth scan using Procrustes algorithm.",
"We allow for translation, rotation, and scaling.",
"We use the rigid alignment as initialization and optimize for the parameters of each model using a modified iterative closest point (ICP) algorithm which also updates the model parameters.",
"The fitting algorithm maintains the initial scale and translation but optimizes for rotation.",
"In each optimization step, we first compute the correspondences as the closest points from the masked region in the model, shown in Fig.",
"REF , to the scan data.",
"We compute the loss as shown in Eq.",
"(REF ) and update the model parameters along with Euler angles for global rotation.",
"We run the following optimization program up to convergence to fit the models to our scans: $ \\operatornamewithlimits{argmin}_{\\theta ,K,\\alpha ,\\beta ,\\gamma }& \\,\\, \\sum _{i=1}^N \\,\\,(\\big \\Vert sR(\\alpha ,\\beta ,\\gamma )x_i(\\theta ) + t - x_i \\big \\Vert _2 \\nonumber \\\\ &\\quad \\quad + \\big \\Vert Kc_i(\\theta ) - c_i \\big \\Vert _2 ) \\nonumber \\\\ &+ w_l \\sum _{j=1}^L \\big \\Vert sR(\\alpha ,\\beta ,\\gamma )l_j(\\theta ) + t - l_j\\big \\Vert _2 \\,,$ where, $x_i(\\theta ) \\in \\mathbb {R}^3$ is a vertex $i$ in the masked region of the model (containing $N$ vertices), $x_i$ is the point on the scan data corresponding to $x_i(\\theta )$ ; $R(\\alpha ,\\beta ,\\gamma ) \\in \\mathbb {R}^{3 \\times 3}$ is the global rotation matrix computed using the Euler angles, $\\alpha ,\\beta ,\\text{and } \\gamma $ ($\\in \\mathbb {R}$ ); $s \\in \\mathbb {R} $ , $t \\in \\mathbb {R}^3$ are the global scale and translation computed during initialization; $c_i(\\theta ) \\in \\mathbb {R}^3$ , $c_i \\in \\mathbb {R}^3$ are the colors at the vertices $x_i(\\theta )$ and $x_i$ respectively; $l_j \\in \\mathbb {R}^3$ and $l_j(\\theta ) \\in \\mathbb {R}^3$ are the $L(=8)$ ground truth and model landmarks respectively, as described earlier; and $K \\in R^{3\\times 3}$ is a diagonal matrix.",
"We set $w_l=0.1$ during the fitting process.",
"Note that the color loss is only enforced for BFM, as FLAME does not model colors.",
"Further, as the color intensities of BFM and our scans differ, we globally scale the color values using channel-specific scalars arranged as a diagonal matrix $K$ which we optimize for along with the model parameters.",
"Evaluation details: We describe the evaluation metrics in Sec.",
"4.4 of the main paper.",
"Here, we present details about the masks used to evaluate these metrics.",
"Face region: We manually paint face masks on the ground truth scans to obtain the ground truth masks.",
"We exclude the mouth interior of the ground truth scans.",
"We copy this mask to the i3DMM fits.",
"We do that by annotating a vertex in i3DMM reconstruction if the nearest point from that vertex on the ground truth scan is in the masked region.",
"We show the face masks used to fit BFM and FLAME to ground truth scans in Fig.",
"REF .",
"We obtain the symmetric metrics presented in Table.",
"1 of the main paper for the face region in the following way.",
"In one direction, we compute the errors from masked region of ground truth to the closest points on the (unmasked) models fit to the scan.",
"In the other direction, we compute the errors from the masked region of the models to the (unmasked) ground truth scan.",
"We compute errors between the masked regions of one mesh to unmasked regions of other mesh to avoid large error metrics due to annotation mistakes during manual mask painting.",
"Full Head: We only fit to FLAME full head model as BFM does not model the entire head.",
"We remove the neck region from FLAME as shown in Fig.",
"REF as the ground truth head scans do not have neck regions.",
"We also remove the vertices used to close the neck from ground truth as FLAME has a hole in the mesh at the neck.",
"We compute the metrics as we do for the face region between these two full head meshes.",
"We report the full head metrics for our model in the entire head region, including the closed hole at the neck mesh.",
"Ablative Analysis In Fig.",
"REF , we show additional qualitative results for the ablative analysis.",
"We also show the quantitative results for full head i3DMM fit in comparison to i3DMM variants.",
"We compare the four models that evaluate our design choices in Table REF .",
"The error metrics are computed for the face region using manually annotated face masks as described in Sec.",
"REF .",
"We only evaluate the face region, as the ground truth for hair is noisy, and small quantitative differences are not very indicative of degradation in quality.",
"Although the geometric reconstruction accuracy is marginally better without the landmark supervision loss, as compared to i3DMM, the color reconstruction accuracy of i3DMM is higher.",
"Also, as mentioned in the main paper, Sec.",
"4.3, texture transfer results around the ear regions with landmark supervision loss are worse compared to i3DMM.",
"Table: Quantitative results for ablation study.",
"The columns, from left to right, show results obtained with uniform sampling for SDF instead of landmark-based sampling, without sparse pairwise landmark supervision loss, independently training for representing geometry and color, final model (i3DMM), and ground truth.",
"Correspondence Evaluation We quantitatively evaluate the correspondences predicted by i3DMM by using the FLAME and BFM fits as ground truth correspondences.",
"To this end, we first find the closest points from the vertices of the (masked) model fits to the i3DMM reconstructions for different scans.",
"We will call these correspondences ground truth annotations here.",
"We use a KD tree algorithm for efficiency.",
"The masked face region contains 26370 vertices for BFM, and 1873 vertices for FLAME.",
"We also transfer the annotations for one i3DMM reconstruction, to all the other reconstructions.",
"This process is same as that described in annotation transfer application (see Sec.",
"4.5 of main paper).",
"We compute the correspondence error as the average of error between the transferred and the ground truth annotations.",
"Note that we transfer annotations from one i3DMM fit to every other i3DMM fit.",
"Therefore, we compute a symmetric error metric.",
"The resulting distribution of error is shown in Fig.",
"REF (evaluating with BFM as ground truth is plotted in red, while FLAME is plotted in blue).",
"The mean and median of errors for BFM is $5.08$ mm and $3.02$ mm respectively.",
"The mean and median of errors for FLAME is $2.36$ mm and $1.83$ mm respectively.",
"Note that, this error does not only capture the error in i3DMM's correspondence predictions but also the error in registrations of FLAME and BFM fits, see Table.",
"1 in the main paper.",
"Figure: Distribution of errors in correspondences predicted by i3DMM computed using FLAME fits (face) as ground truth (blue), and using BFM fits (face) as ground truth (red).",
"Applications Full Head Completion Figure: Completing face scans (left) with different hair styles (short hairstyle (middle-left), long hairstyle (middle-right), and cap (right)) using i3DMM as prior.We use the i3DMM prior to complete face scans with different hairstyles as shown in Fig.",
"REF .",
"To obtain the face meshes from our head meshes, we delete all the vertices that are outside a sphere around the the tip of the nose.",
"We learn the latent vector for the given test scan using the SDF samples of the face mesh, as described in Sec.",
"4.1 of the main paper.",
"Additionally, we semantically control the hair style of the completed scan by adding a regularizer that enforces the learned $\\mathbf {z}_\\text{geoH}$ and $\\mathbf {z}_\\text{colH}$ to be close to the hairstyle latent vectors learned during training.",
"It can be inferred from the results in Fig.",
"REF that our model learns a good prior distribution, generating plausible heads for the given faces.",
"i3DMM offers user-guided control for head completion and can be used to turn existing face-only 3DMMs into full head 3DMMs.",
"Further, our method can also be used as a prior distribution for applications such as monocular 3D reconstruction [54].",
"Figure: Additional annotation transfer results.",
"Top: segmentation transfer front view.",
"Middle: segmentation transfer side view.",
"Bottom: landmark transfer.",
"Left column shows i3DMM reconstructions with manual annotations.",
"Right part shows annotations transferred to head scans using i3DMM.",
"Annotation Transfer We show more results for annotation transfer described in Sec.",
"4.5 of the main paper, in Fig.",
"REF .",
"Figure: Additional comparison results between i3DMM (full head) fits, BFM (face) fit, and FLAME (full head and face) fits.Figure: Additional ablation results.",
"From left to right, i3DMM without landmark-based sampling, i3DMM without landmark supervision, i3DMM with independent color and geometry training, i3DMM, and ground truth results are shown.",
"Visualization Details The output of the i3DMM is a signed distance field.",
"Generally, marching cubes algorithm is used to reconstruct the surface from a SDF.",
"However, based on the resolution used, marching cubes algorithm introduces unpleasant surface artifacts.",
"To avoid these artifacts, we used a sphere tracer to render our results.",
"We used the Blinn-Phong reflection model to shade our geometry results.",
"We apply a gamma correction with $\\gamma =0.65$ for the color renders.",
"We use Redner [30] for rendering meshes."
],
[
"Model Visualization",
"Our implicit 3D morphable model models the identity, expression, and hairstyle components of the geometry; as well as the identity and hairstyle components of the color of the head with independent parameteric controls.",
"We visualize these components in Fig.",
"REF .",
"We perform PCA on the identity geometry and color spaces in order to compute the principal components.",
"The expression component is visualized by moving along the directions of the training expressions, since they are semantically well defined.",
"As we model hairstyles that include caps, we show the joint space of geometry and color for hairstyles.",
"Note that the hairstyle geometry can only take four discrete values – short, long, cap1, or cap2.",
"Any variation within these categories is modelled by the identity-geometry component.",
"Similarly, hairstyle color can only take the values – nocap, cap1, or cap2.",
"The color of hair without any cap is determined by the identity-color component."
],
[
"Experiments",
"Here, we provide more details on the evaluations in the main paper, and include further evaluations."
],
[
"Sampling i3DMM",
"One of the important features of a 3D Morphable Model is the ability to randomly sample shapes in the parametric space.",
"This has been used for generating synthetic data for training CNNs [23], [43], [47].",
"We achieve sampling in i3DMM by performing principal component analysis (PCA) over the training latent codes for color-identity, geometry-identity, and expressions.",
"We weigh the singular values using a Gaussian random variable $\\mathcal {N}(0,0.1)$ for color-identity, and geometry-identity, and $\\mathcal {N}(0,0.25)$ for expressions.",
"Since the latent codes for hair shapes and colors can take very limited numbers of values and are very well defined semantically, we sample these from their training values.",
"We show several results in the supplemental video.",
"As can be seen, our model is biased towards generating male heads.",
"This is likely due to our gender-biased training dataset with 46 males and 18 females.",
"While this does not lead to any clear loss of quality when fitting to female test scans, see Fig.",
"6 in the main paper, it might lead to biased quality of results in other problems, for eg., if random samples from the model was used for training another network."
],
[
"Comparisons",
"As mentioned in Sec.",
"4.4 of the main paper, we compare our model to two existing models, BFM [38] and FLAME [29].",
"We show more qualitative model fitting results in Fig.",
"REF .",
"Next, we provide more details on the fitting algorithm used.",
"Fitting: We paint the face region of each model's template mesh to create a mask as shown in Fig.",
"REF .",
"We use these masked regions of the models for fitting the models to head scans.",
"To initialize, we mark 8 landmarks (eye corners, nose, lip corners, and chin) on the template mesh and rigidly align the template to each ground truth scan using Procrustes algorithm.",
"We allow for translation, rotation, and scaling.",
"We use the rigid alignment as initialization and optimize for the parameters of each model using a modified iterative closest point (ICP) algorithm which also updates the model parameters.",
"The fitting algorithm maintains the initial scale and translation but optimizes for rotation.",
"In each optimization step, we first compute the correspondences as the closest points from the masked region in the model, shown in Fig.",
"REF , to the scan data.",
"We compute the loss as shown in Eq.",
"(REF ) and update the model parameters along with Euler angles for global rotation.",
"We run the following optimization program up to convergence to fit the models to our scans: $ \\operatornamewithlimits{argmin}_{\\theta ,K,\\alpha ,\\beta ,\\gamma }& \\,\\, \\sum _{i=1}^N \\,\\,(\\big \\Vert sR(\\alpha ,\\beta ,\\gamma )x_i(\\theta ) + t - x_i \\big \\Vert _2 \\nonumber \\\\ &\\quad \\quad + \\big \\Vert Kc_i(\\theta ) - c_i \\big \\Vert _2 ) \\nonumber \\\\ &+ w_l \\sum _{j=1}^L \\big \\Vert sR(\\alpha ,\\beta ,\\gamma )l_j(\\theta ) + t - l_j\\big \\Vert _2 \\,,$ where, $x_i(\\theta ) \\in \\mathbb {R}^3$ is a vertex $i$ in the masked region of the model (containing $N$ vertices), $x_i$ is the point on the scan data corresponding to $x_i(\\theta )$ ; $R(\\alpha ,\\beta ,\\gamma ) \\in \\mathbb {R}^{3 \\times 3}$ is the global rotation matrix computed using the Euler angles, $\\alpha ,\\beta ,\\text{and } \\gamma $ ($\\in \\mathbb {R}$ ); $s \\in \\mathbb {R} $ , $t \\in \\mathbb {R}^3$ are the global scale and translation computed during initialization; $c_i(\\theta ) \\in \\mathbb {R}^3$ , $c_i \\in \\mathbb {R}^3$ are the colors at the vertices $x_i(\\theta )$ and $x_i$ respectively; $l_j \\in \\mathbb {R}^3$ and $l_j(\\theta ) \\in \\mathbb {R}^3$ are the $L(=8)$ ground truth and model landmarks respectively, as described earlier; and $K \\in R^{3\\times 3}$ is a diagonal matrix.",
"We set $w_l=0.1$ during the fitting process.",
"Note that the color loss is only enforced for BFM, as FLAME does not model colors.",
"Further, as the color intensities of BFM and our scans differ, we globally scale the color values using channel-specific scalars arranged as a diagonal matrix $K$ which we optimize for along with the model parameters.",
"Evaluation details: We describe the evaluation metrics in Sec.",
"4.4 of the main paper.",
"Here, we present details about the masks used to evaluate these metrics.",
"Face region: We manually paint face masks on the ground truth scans to obtain the ground truth masks.",
"We exclude the mouth interior of the ground truth scans.",
"We copy this mask to the i3DMM fits.",
"We do that by annotating a vertex in i3DMM reconstruction if the nearest point from that vertex on the ground truth scan is in the masked region.",
"We show the face masks used to fit BFM and FLAME to ground truth scans in Fig.",
"REF .",
"We obtain the symmetric metrics presented in Table.",
"1 of the main paper for the face region in the following way.",
"In one direction, we compute the errors from masked region of ground truth to the closest points on the (unmasked) models fit to the scan.",
"In the other direction, we compute the errors from the masked region of the models to the (unmasked) ground truth scan.",
"We compute errors between the masked regions of one mesh to unmasked regions of other mesh to avoid large error metrics due to annotation mistakes during manual mask painting.",
"Full Head: We only fit to FLAME full head model as BFM does not model the entire head.",
"We remove the neck region from FLAME as shown in Fig.",
"REF as the ground truth head scans do not have neck regions.",
"We also remove the vertices used to close the neck from ground truth as FLAME has a hole in the mesh at the neck.",
"We compute the metrics as we do for the face region between these two full head meshes.",
"We report the full head metrics for our model in the entire head region, including the closed hole at the neck mesh."
],
[
"Ablative Analysis",
"In Fig.",
"REF , we show additional qualitative results for the ablative analysis.",
"We also show the quantitative results for full head i3DMM fit in comparison to i3DMM variants.",
"We compare the four models that evaluate our design choices in Table REF .",
"The error metrics are computed for the face region using manually annotated face masks as described in Sec.",
"REF .",
"We only evaluate the face region, as the ground truth for hair is noisy, and small quantitative differences are not very indicative of degradation in quality.",
"Although the geometric reconstruction accuracy is marginally better without the landmark supervision loss, as compared to i3DMM, the color reconstruction accuracy of i3DMM is higher.",
"Also, as mentioned in the main paper, Sec.",
"4.3, texture transfer results around the ear regions with landmark supervision loss are worse compared to i3DMM.",
"Table: Quantitative results for ablation study.",
"The columns, from left to right, show results obtained with uniform sampling for SDF instead of landmark-based sampling, without sparse pairwise landmark supervision loss, independently training for representing geometry and color, final model (i3DMM), and ground truth."
],
[
"Correspondence Evaluation",
"We quantitatively evaluate the correspondences predicted by i3DMM by using the FLAME and BFM fits as ground truth correspondences.",
"To this end, we first find the closest points from the vertices of the (masked) model fits to the i3DMM reconstructions for different scans.",
"We will call these correspondences ground truth annotations here.",
"We use a KD tree algorithm for efficiency.",
"The masked face region contains 26370 vertices for BFM, and 1873 vertices for FLAME.",
"We also transfer the annotations for one i3DMM reconstruction, to all the other reconstructions.",
"This process is same as that described in annotation transfer application (see Sec.",
"4.5 of main paper).",
"We compute the correspondence error as the average of error between the transferred and the ground truth annotations.",
"Note that we transfer annotations from one i3DMM fit to every other i3DMM fit.",
"Therefore, we compute a symmetric error metric.",
"The resulting distribution of error is shown in Fig.",
"REF (evaluating with BFM as ground truth is plotted in red, while FLAME is plotted in blue).",
"The mean and median of errors for BFM is $5.08$ mm and $3.02$ mm respectively.",
"The mean and median of errors for FLAME is $2.36$ mm and $1.83$ mm respectively.",
"Note that, this error does not only capture the error in i3DMM's correspondence predictions but also the error in registrations of FLAME and BFM fits, see Table.",
"1 in the main paper.",
"Figure: Distribution of errors in correspondences predicted by i3DMM computed using FLAME fits (face) as ground truth (blue), and using BFM fits (face) as ground truth (red)."
],
[
"Full Head Completion",
"We use the i3DMM prior to complete face scans with different hairstyles as shown in Fig.",
"REF .",
"To obtain the face meshes from our head meshes, we delete all the vertices that are outside a sphere around the the tip of the nose.",
"We learn the latent vector for the given test scan using the SDF samples of the face mesh, as described in Sec.",
"4.1 of the main paper.",
"Additionally, we semantically control the hair style of the completed scan by adding a regularizer that enforces the learned $\\mathbf {z}_\\text{geoH}$ and $\\mathbf {z}_\\text{colH}$ to be close to the hairstyle latent vectors learned during training.",
"It can be inferred from the results in Fig.",
"REF that our model learns a good prior distribution, generating plausible heads for the given faces.",
"i3DMM offers user-guided control for head completion and can be used to turn existing face-only 3DMMs into full head 3DMMs.",
"Further, our method can also be used as a prior distribution for applications such as monocular 3D reconstruction [54].",
"Figure: Additional annotation transfer results.",
"Top: segmentation transfer front view.",
"Middle: segmentation transfer side view.",
"Bottom: landmark transfer.",
"Left column shows i3DMM reconstructions with manual annotations.",
"Right part shows annotations transferred to head scans using i3DMM."
],
[
"Annotation Transfer",
"We show more results for annotation transfer described in Sec.",
"4.5 of the main paper, in Fig.",
"REF .",
"Figure: Additional comparison results between i3DMM (full head) fits, BFM (face) fit, and FLAME (full head and face) fits.Figure: Additional ablation results.",
"From left to right, i3DMM without landmark-based sampling, i3DMM without landmark supervision, i3DMM with independent color and geometry training, i3DMM, and ground truth results are shown."
],
[
"Visualization Details",
"The output of the i3DMM is a signed distance field.",
"Generally, marching cubes algorithm is used to reconstruct the surface from a SDF.",
"However, based on the resolution used, marching cubes algorithm introduces unpleasant surface artifacts.",
"To avoid these artifacts, we used a sphere tracer to render our results.",
"We used the Blinn-Phong reflection model to shade our geometry results.",
"We apply a gamma correction with $\\gamma =0.65$ for the color renders.",
"We use Redner [30] for rendering meshes."
]
] | 2011.14143 |
[
[
"EdgeBERT: Sentence-Level Energy Optimizations for Latency-Aware\n Multi-Task NLP Inference"
],
[
"Abstract Transformer-based language models such as BERT provide significant accuracy improvement for a multitude of natural language processing (NLP) tasks.",
"However, their hefty computational and memory demands make them challenging to deploy to resource-constrained edge platforms with strict latency requirements.",
"We present EdgeBERT, an in-depth algorithm-hardware co-design for latency-aware energy optimization for multi-task NLP.",
"EdgeBERT employs entropy-based early exit predication in order to perform dynamic voltage-frequency scaling (DVFS), at a sentence granularity, for minimal energy consumption while adhering to a prescribed target latency.",
"Computation and memory footprint overheads are further alleviated by employing a calibrated combination of adaptive attention span, selective network pruning, and floating-point quantization.",
"Furthermore, in order to maximize the synergistic benefits of these algorithms in always-on and intermediate edge computing settings, we specialize a 12nm scalable hardware accelerator system, integrating a fast-switching low-dropout voltage regulator (LDO), an all-digital phase-locked loop (ADPLL), as well as, high-density embedded non-volatile memories (eNVMs) wherein the sparse floating-point bit encodings of the shared multi-task parameters are carefully stored.",
"Altogether, latency-aware multi-task NLP inference acceleration on the EdgeBERT hardware system generates up to 7x, 2.5x, and 53x lower energy compared to the conventional inference without early stopping, the latency-unbounded early exit approach, and CUDA adaptations on an Nvidia Jetson Tegra X2 mobile GPU, respectively."
],
[
"Introduction",
"Transformer-based networks trained with large multi-domain datasets have unlocked a series of breakthroughs in natural language learning and representation.",
"A major catalyst of this success is the Bidirectional Encoder Representations from Transformers technique, or BERT [16], which substantially advanced nuance and context understanding.",
"Its pre-training strategy, which consists of learning intentionally hidden sections of text, have proven beneficial for several downstream natural language processing (NLP) tasks.",
"BERT has sparked leading-edge performance in NLP leaderboards [78], [58], and it is now applied at a global scale in web search engines [52] with marked improvements in the quality of query results.",
"Figure: (a) Conventional BERT inference, (b) Conventional latency-unbounded BERT inference with early exit.",
"(c) Proposed latency-bounded inference.",
"The entropy result from the first layer is used to auto-adjust the accelerator supply voltage and clock frequency for energy-optimal operation while meeting an application end-to-end latency target.Advances in NLP models are also fueling the growth of intelligent virtual assistants, which leverage NLP to implement interactive voice interfaces.",
"Currently, these applications are offloaded from the edge device to the cloud.",
"However, they are naturally better suited to deployment on edge devices, where personal data can be kept private and the round trip latency to the cloud is removed.",
"However, the impressive performance of BERT comes with a heavy compute and memory cost, which makes on-device inference prohibitive.",
"Most significantly, the BERT base model consumes a staggering 432 MB of memory in native 32-bit floating-point (FP32).",
"Therefore, the goal of deploying BERT on edge/mobile devices is challenging and requires tight co-design of the BERT model optimizations with dedicated hardware acceleration and memory system design.",
"The constraints on mobile can be quite different to the datacenter scenario, where BERT has been mainly deployed to date.",
"Firstly, since we are dealing with user input, we need to meet real time throughput requirements to prevent a noticeable lag to the user.",
"Secondly, energy consumption is a critical concern on mobile devices, both for the model inference and also the associated data movement cost.",
"A number of prior works have been proposed to reduce BERT storage and computation overheads [21].",
"In fact, most of the compression techniques (weight pruning [49], distillation [63], quantization [89], [68]) originally proposed for convolutional and recurrent neural networks (CNNs, RNNs) have been independently applied to Transformer-based DNNs.",
"In this work, we present EdgeBERT, a principled latency-driven approach to accelerate NLP workloads with minimal energy consumption via early exit prediction, dynamic voltage-frequency scaling (DFVS), and non-volatile memory bitmask encoding of the shared word embeddings.",
"In conventional BERT inference (Fig.",
"REF a), the final classification result is generated by the last Transformer layer.",
"Early exit mechanisms [87], [90], [73], [65] (Fig.",
"REF (b)) have been proposed to reduce the average energy and latency.",
"The early exit entropy, which is a probabilistic measure of the classification confidence, is evaluated at the output of each computed Transformer layer and the inference exits when the entropy value falls below a pre-defined threshold.",
"While this approach can appreciably reduce computation and energy costs, the achieved latency can vary drastically from one input sentence to another, potentially violating the strict real time latency constraint of the application.",
"In contrast, EdgeBERT uses this upper-bound latency and the target entropy as optimization constraints, and then dynamically auto-adjusts the accelerator supply voltage and clock frequency to minimize energy consumption (Fig.",
"REF (c)), while meeting the real time throughput requirement.",
"Since energy scales quadratically with $V_{DD}$ and linearly with the number of computation cycles, our DVFS algorithm finds the lowest possible frequency/voltage, while also minimizing the total number of FLOPs via adaptive attention span predication.",
"While the benefits of early exit and attention predications can be reaped on commodity GPUs, we unlock additional energy savings by co-designing the hardware datapaths.",
"Specifically, we exploit these algorithmic optimizations in the EdgeBERT accelerator system, which integrates a fast-switching low-dropout (LDO) voltage regulator and an all-digital phase-locked loop (ADPLL) for DVFS adjustments.",
"The EdgeBERT accelerator uses bit-mask encoding for compressed sparse computations, while optimizing key operations (entropy assessment, layer normalization, softmax and attention masking) for numerical stability and energy efficiency.",
"Furthermore, edge/IoT devices operate intermittently which motivates powering down as much as possible.",
"The model's weights, typically stored in on-chip SRAMs, either have to be reloaded from DRAM each wake up cycle or the on-chip SRAMs storing the weights must be kept on, wasting leakage power [39].",
"Embedded non-volatile memories (eNVMs), which have shown considerable progress in recent years, offer great promise, if used judiciously, to eliminate the power penalty associated with intermittent operation.",
"For this purpose, we perform monte-carlo fault injection simulations to identify robust and viable eNVM structures for storing the shared NLP multi-task parameters with bitmask encoding.",
"Our resulting eNVM configuration significantly alleviates the energy and latency costs associated with multi-task intermediate computing by as much as 66,000$\\times $ and 50$\\times $ , respectively.",
"Figure: Comparison between (a) BERT, and (b) ALBERT base models.",
"ALBERT uses a smaller embedding size and its Transformer encoder layers share the same parameters.Altogether, EdgeBERT generates on average up to 7$\\times $ , and 2.5$\\times $ per-sentence energy savings compared to the conventional BERT inference, and latency-unaware early exit approaches, respectively.",
"This paper therefore makes the following contributions: We propose EdgeBERT, a novel algorithm-hardware co-design approach to enable latency-bound NLP workloads on resource-constrained embedded devices.",
"Recognizing that BERT word embeddings are shared across NLP tasks, we significantly alleviate off-chip communication costs by identifying viable and robust multi-level eNVM structures for storing the multi-task word embeddings.",
"Leveraging the insights from this broad analysis, we propose and design a 12nm accelerator that integrates a fast-switching LDO, an ADPLL, and a compressed sparse hardware accelerator that efficiently computes the DVFS, entropy, and adaptive attention span predication algorithms and other key Transformer operations using specialized datapaths.",
"We evaluate the energy consumption of latency-bound inference on four NLP tasks, and find that the EdgeBERT hardware accelerator system generates up to 7$\\times $ , 2.5$\\times $ , and 53$\\times $ lower energy compared to the unoptimized baseline inference without early exit, the conventional latency-unaware early exit approach, and CUDA adaptations on an Nvidia Jetson Tegra X2 mobile GPU respectively.",
"The General Language Understanding Evaluation (GLUE) benchmark is the most widely used tool to evaluate NLP performance.",
"It consists of nine English sentence understanding tasks covering three categories: Single-Sentence, Similarity and Paraphrase, and Inference [78].",
"This collection of datasets is specifically designed to favor models that can adapt to a variety of NLP tasks.",
"To validate the robustness and generalization performance of the EdgeBERT methodology, we conduct our evaluation on the four GLUE tasks with the largest corpora, which cover all three GLUE categories: SST-2 (Single-Sentence), QQP (Similarity and Paraphrase), and QNLI and MNLI (Inference)."
],
[
"Variations of BERT",
"Since the advent of BERT with 110M parameters, a number of variants were proposed to alleviate its memory consumption or to further improve its prediction metrics.",
"RoBERTa [44] generalizes better on several GLUE tasks by training on significantly more data, and for a longer amount of time, but remains as computationally intensive as BERT.",
"DistilBERT [63] and MobileBERT [70] leverage knowledge distillation to reduce BERT size by 1.7$\\times $ and 4.3$\\times $ , respectively, with iso-accuracy.",
"SqueezeBERT [29] substitutes several operations in the Transformer encoder with 1D grouped convolutions achieving 4$\\times $ speedup while being 2$\\times $ smaller.",
"Q8BERT [89] employs a symmetric linear quantization scheme for quantizing both weights and activations into 8-bit integers.",
"In contrast, in this work we leverage the higher dynamic range of floating-point encodings for greater quantization resilience.",
"ALBERT [37] yields the smallest footprint to date for a compressed BERT variant with only 12M parameters, with competitive accuracy on the GLUE benchmarks.",
"Fig.",
"REF summarizes the key differences between the ALBERT model and the base BERT model.",
"While each of BERT's twelve encoder layers have a unique set of weights, ALBERT's encoder layers instead share and reuse the same parameters – resulting in significant compression.",
"The encoder block in both models has the same architecture as the legacy Transformer network [75], but with twelve parallel self-attention heads.",
"Moreover, ALBERT employs a smaller embedding size (128 vs. 768) thanks to factorization in the embedding layer.",
"In this work, we adopt the ALBERT variant as an efficient baseline.",
"This work further pursues strategies to reduce latency and storage requirements to suit embedded hardware platform constraints."
],
[
"Alleviating Transformer memory and computation costs",
"An accelerator's energy consumption can be abstracted as: $Energy\\propto \\alpha C V_{DD}^2 N_{cycles}$ where $\\alpha $ , $C$ , $V_{DD}$ and $N_{cycles}$ are the switching activity factor, the effective wire and device capacitance, the supply voltage, and the required number of clock cycles to complete the inference, respectively.",
"While the DVFS algorithm (Sec.",
"REF ) lowers the energy quadratically by bringing $V_{DD}$ down to the lowest optimal voltage, in this section, we explore avenues to further reduce the energy by minimizing $\\alpha $ , $C$ , and $N_{cycles}$ .",
"For this purpose, we carefully incorporate into the multi-task ALBERT inference: 1) adaptive attention span predication and early exit which reduce $N_{cycles}$ ; 2) network pruning, which ultimately reduces $\\alpha $ ; and 3) floating-point quantization helping decrease $C$ , altogether with minimal accuracy degradation.",
"While briefly describing these optimizations individually in this section, we provide a reasoned methodology for applying them to the ALBERT model, as shown in Fig.",
"REF ."
],
[
"Entropy-based Early Exit",
"The motivation behind early exit (EE) is to match linguistically complex sentences with larger (or deeper) models and simple sentences with smaller (or shallower) models [87], [13].",
"This is typically done by adding a lightweight classifier at the output of the Transformer layer so that a given input can exit inference earlier or later in the stack, depending on its structural and contextual complexity.",
"The classifier computes and compares the entropy of an output distribution with a preset “confidence\" threshold, $E_{T}$ , in order to assess whether the prediction should exit or continue inference in the next Transformer encoder layer.",
"The entropy metric quantifies the amount of uncertainty in the data.",
"Smaller entropy values at a Transformer layer output implies greater confidence in the correctness of the classification result.",
"The entropy H on sample x is estimated as: $\\leavevmode \\xbox {resizebox}{\\XMLaddatt {width}{0.0pt}H(x) = - \\sum p(x)\\log p(x)= \\ln (\\sum \\limits _{k=1}^{n} e^{x_k}) -\\frac{\\sum \\limits _{k=1}^{n} x_k e^{x_k}}{\\sum \\limits _{k=1}^{n} e^{x_k}}}$ The early exit condition is met when $H(x)$ $<$ $E_{T}$ .",
"Therefore, the larger $E_{T}$ becomes, the earlier the sample will exit (i.e.",
"$N_{cycles}$ becomes smaller) with potentially lower accuracy.",
"In this work, we modify the conventional EE inference approach by predicting the early exit layer from the output of the first Transformer layer in order to run the rest of the network computation in an energy-optimal and latency-bounded manner (Sec.",
")."
],
[
"Adaptive Attention Span",
"The attention mechanism [8] is a powerful technique that allows neural networks to emphasize the most relevant tokens of information when making predictions.",
"The base ALBERT model contains up to twelve parallel attention heads – each learning their own saliency weights on the full length of the encoder input.",
"However, depending on the complexity of the task, many heads can be redundant and can be safely removed without impacting accuracy [51].",
"Furthermore, the cost of computing the attention mechanism scales quadratically with the sequence length.",
"Therefore, there is potentially a meaningful amount of computations and energy to be saved in optimizing the inspection reach of every attention head.",
"In the quest to avoid needless attention computations in ALBERT, a learnable parameter z is introduced in the datapath of each self-attention head in order to find its own optimal attention span [69].",
"The parameter z is mapped to a masking function with a [0, 1] output range, as shown in Fig.",
"REF .",
"The masked span is then applied on the attention weights in order to re-modulate their saliencies.",
"The optimal span is automatically learned during the fine-tuning process by adding back the average loss from the reduced span to the training cross-entropy loss.",
"The maximum sentence length for fine-tuning the GLUE tasks is 128.",
"As a result, shorter sentences are typically zero-padded to 128 during the tokenization pre-processing.",
"Table REF shows the final attention span learned by each self-attention head when fine-tuning with the adaptive attention span technique.",
"Strikingly, the twelve parallel self-attention heads in ALBERT do not need to inspect their inputs at maximum span.",
"In fact, more than half of the attention heads, 8 for MNLI and QQP and 7 for SST-2 and QNLI, can be completely turned off with minimal accuracy loss.",
"This amounts to a 1.22$\\times $ and 1.18$\\times $ reduction, respectively, in the total number of FLOPS (which linearly correlates with $N_{cycles}$ ) required for single-batch inference.",
"Table: Learned spans of every attention head in ALBERT.Baseline Acc: MNLI=85.16, QQP=90.76, SST-2=92.20, QNLI=89.48The twelve attention spans, learned during fine-tuning, are written to registers in the EdgeBERT accelerator in the form of a 128-wide vector – in order to predicate on the inference computation of the multi-head attention.",
"Notably, all the computations inside any of the twelve attention head units can be effectively skipped in case its associated attention span mask is 100% null.",
"The EdgeBERT accelerator takes advantage of this observation in a proactive manner during inference in the custom hardware (Sec.",
"REF )."
],
[
"Network Pruning",
"The EdgeBERT hardware accelerator (Sec. )",
"executes sparse computations and saves energy by gating MACs whenever input operands are null.",
"Therefore, the extent to which we can prune the ALBERT model, without appreciable accuracy loss, determines the overall accelerator energy efficiency.",
"In this work, we consider both movement pruning [64] and the well-known magnitude pruning [25] methods.",
"Movement pruning is a first-order pruning technique that is applied during model fine-tuning which eliminates weights that are dynamically shrinking towards 0 (i.e., according to the movement of the values).",
"In some cases, magnitude pruning may be a sub-optimal method to use during transfer learning, as pre-trained weights closer to zero may have a high chance of being eliminated regardless of the fine-tuning requirement.",
"We observe that movement pruning particularly outperforms magnitude-based pruning in high sparsity regimes, as each individual remaining weight becomes more important to learn the task at hand.",
"Therefore, choosing between the two pruning techniques would depend on the per-task tolerance to increasing sparsity levels.",
"We note that magnitude pruning is always applied to the ALBERT embedding layer in order to enforce uniformity in the data during multi-domain on-chip acceleration – as using movement pruning on the embedding layer would make its weights unique for each NLP domain, thereby forgoing opportunities for data reuse in hardware."
],
[
"Floating-Point Quantization",
"DNN algorithmic resilience allows for parameters to be represented in lower bit precision without accuracy loss.",
"Fixed-point or integer quantization techniques, commonly adopted in CNN models, suffer from limited range and may be inadequate for NLP models, whose weights can be more than an order of magnitude larger [72].",
"This phenomenon is owed to layer normalization [7], which is commonly adopted in NLP models and has invariance properties that do not reparameterize the network – unlike batch normalization [30], which produces a weight normalization side effect in CNNs.",
"In this work, we employ floating-point based quantization, which provides 2$\\times $ higher dynamic range compared to integer datatypes [32].",
"Both weights and activations are quantized across ALBERT layers to 8-bit precision.",
"We also performed a search on the optimal exponent bit width to satisfy the dynamic range requirements of the ALBERT model.",
"Setting the floating-point exponent space to 4 bits within the 8-bit word size, with the exponent being scaled at a per-layer granularity, provided the best accuracy performance across NLP tasks.",
"In contrast to task-specific encoder weights, word embedding parameters are deliberately fixed during fine-tuning and reused across different NLP tasks.",
"We seek to avoid the energy and latency costs of reloading the word embeddings from off-chip memory for different tasks by storing these shared parameters in embedded non-volatile memories (eNVMs).",
"eNVM storage also enables energy-efficient intermittent computing because the embedding weights will be retained if and when the system-on-chip powers off between inferences.",
"However, despite their compelling storage density and read characteristics, eNVMs exhibit two main drawbacks: potentially high write cost (in terms of energy and latency) and decreased reliability, particularly in multi-level cell (MLC) configurations [15].",
"Fortunately, the word embeddings are acting as read-only parameters on-chip, which makes them highly suitable for eNVM storage, but previous work highlights the need to study the impacts of faulty, highly-dense ReRAM storage on DNN task accuracy [56].",
"On the other hand, encoder weights need to be updated when switching across different NLP tasks.",
"To prevent the energy and latency degradation that would follow from updating the encoder weight values in eNVMs, we map the natural partition of shared and task-specific parameters to eNVMs and SRAMs, respectively [17]."
],
[
"eNVM Modeling Methodology",
"This work specifically considers dense, energy-efficient Resistive RAM (ReRAM) arrays [10], [43] as an on-chip storage solution for shared embedding parameters.",
"We selected ReRAMs for their relative maturity and demonstrated read characteristics.",
"However, we note that there is a larger design space of opportunities to be explored with other emerging MLC-capable NVM technologies such as PCM [14], but is beyond the scope of this work.",
"We evaluate the robustness of storing the 8-bit quantized word embeddings in eNVM storage.",
"In order to quantify the trade-offs between storage density and task accuracy, we use cell characteristics of 28nm ReRAM programmed with varying number of bits per cell [15], and evaluate 100 fault injection trials per storage configuration to identify robust eNVM storage solutions.",
"We leverage and extend Ares [59], which is an existing open-source fault injection framework for quantifying the resilience of DNNs.",
"After pruning, we store non-zero compressed embedding weights using a bitmask-style sparse encoding.",
"Previous work demonstrates that DNN weight bitmask values are vulnerable to MLC faults, so the bitmask is protectively stored in lower-risk SLC devices, while we experiment with MLC storage for the non-zero data values [56]."
],
[
"Optimal eNVM Configuration",
"Table REF uncovers exceptional resilience to storing word embeddings in MLC ReRAM.",
"Across many fault injection trials, we observe that MLC2 (ReRAM programmed at 2 bits-per-cell) does not degrade accuracy across multiple tasks, while MLC3 exhibits potentially catastrophic degradation in minimum accuracy and an appreciable decline in average accuracy for the QNLI task, highlighted in bold.",
"Based on this observation, the EdgeBERT accelerator system leverages MLC2 ReRAMs for word embedding storage (Sec.).",
"[!t] linenosize= $E_T$ := target entropy input sentence $i = 0$ to $n$ encoder layer l = 1 to 12 $z_l = f(x;\\theta | VDD_{nom}, Freq_{max})$ $entropy(z_l) < E_T$ exit inference Conventional early exit inference"
],
[
"EdgeBERT's Latency-Aware Inference",
"The conventional BERT inference (Algorithm REF ) with early exit (EE) can significantly reduce BERT inference latency.",
"To further reduce the energy consumption for NLP inference, a latency-aware inference scheme leveraging the EE predictor and dynamic voltage and frequency scaling (DVFS) is proposed to minimize end-to-end per-sentence energy consumption while satisfying the real-time latency target.",
"[!t] linenosize= $T$ := per-sentence latency target, $E_T$ := entropy target $N_{cycles} := \\text{number of clock cycles to compute the Transformer encoder}$ input sentence $i = 1$ to $n$ encoder layer l = 1 $z_l = f(x;\\theta | VDD_{nom}, Freq_{max})$ $entropy(z_l) < E_T$ exit inference $L_{predict} = LUT(entropy(z_1), E_T)$ $VDD_{opt}, Freq_{opt} = DVFS(L_{predict},T)$ encoder layer $l = 2$ to $L_{predict}$ $z_l = f(x;\\theta | VDD_{opt}, Freq_{opt})$ $entropy(z_l) < E_T$ exit inference exit inference EdgeBERT latency-aware inference.",
"Computations exit at the predicted exit layer or earlier."
],
[
"Methodology",
"DVFS is a widely used technique to dynamically scale down the voltage and frequency for less computationally intensive workloads.",
"In the past, DVFS has been widely deployed in commercial CPUs [74], [28] and GPUs [50].",
"However, these schemes typically adjust the voltage and frequency at a coarse granularity at workload-level.",
"In the era of AI, DVFS has started to be explored for DNN accelerators [38].",
"For example, a recent state-of-the-art AI chip has reported per-layer DVFS to save energy [3].",
"In this work, we explore a fine-grained sentence-level DVFS to reduce the energy consumption for NLP inference while meeting the latency target.",
"The proposed early exit -based latency-aware inference methodology is illustrated in Algorithm .",
"The inference of a sentence starts at nominal voltage and maximum frequency, and the entropy value is calculated at the output of the first Transformer encoder layer.",
"The entropy result is then sent to a trained classifier (EE predictor) to predict which following encoder layer should early exit (e.g.",
"early exit at encoder layer 6 before the final encoder layer 12).",
"Based on the predicted early exit layer, the voltage and frequency is scaled down to proper energy-optimal setting for the rest of encoder layers (e.g.",
"encoder layer 2 to 6) while meeting the latency target for each sentence.",
"This scheme produces a quadratic reduction in the accelerator power consumption.",
"In our work, the EE predictor is a ReLU-activated five-layer perceptron neural network with 64 cells in each of the hidden layers.",
"It takes the entropy of encoder layer 1 as input and forecasts the early exit Transformer layer which has an entropy below the desired threshold.",
"The neural network architecture of the EE predictor was empirically searched with the goal of minimizing the difference between the predicted and the true entropy-based exit layer.",
"For this purpose, we constructed parallel training and test datasets containing the entropy values at the output of the 12 Transformer layers during evaluation on the GLUE benchmarks.",
"The EE predictor is distilled as a lookup table (LUT) leading to negligible one-time (per-sentence) computational overhead.",
"Furthermore, implementing the EE predictor as a LUT simplifies its hardware operation.",
"As the neural network based LUT is error-prone, it may predict a higher exit layer than necessary.",
"Therefore, during the inference, the entropy is checked after each encoder layer for early stopping until the predicted layer.",
"If the computed entropy becomes lower than the exit threshold before the predicted encoder layer, the inference will terminate at that early exit condition point.",
"In case the inference reaches the predicted layer, termination occurs even if the entropy at that layer is still higher than the exit threshold in order to not violate timing constraints.",
"When assessing the impacts of using entropy prediction instead of traditional EE methods, we set a fixed accuracy degradation threshold of 1%, 2%, or 5% (relative to the inference accuracy of the full ALBERT model) and increased the entropy threshold until the accuracy dropped to the desired threshold.",
"This allowed us to compare energy savings between entropy prediction and conventional EE for a fixed accuracy target.",
"For the same accuracy threshold, the entropy threshold for entropy prediction was lower than the entropy threshold for conventional EE, leading to a slightly later average exit layer during inference.",
"However, entropy prediction allows for DVFS since the maximum exit layer is known after the first layer, whereas with the conventional EE appproach, the maximum exit layer is always the final encoder layer.",
"EdgeBERT latency-aware inference therefore achieves greater energy savings than the conventional EE approach by facilitating DVFS (Sec.",
"REF ).",
"Figure: EdgeBERT training and evaluation procedure."
],
[
"On-chip DVFS system",
"To realize fast per-sentence DVFS, the on-chip DVFS system is developed and integrated within EdgeBERT.",
"The DVFS system includes a DVFS controller, an on-chip synthesizable linear voltage regulator (LDO), and an all-digital PLL (ADPLL).",
"Compared with the conventional workload-level DVFS [74], the proposed scheme adjusts voltage and frequency at a finer-grained sentence-level granularity.",
"Based on the predicted early exit layer from the EE predictor, the required run cycles, $N_{cycles}$ , for the rest of the encoder layers before early exit can be known.",
"And, knowing the frontend elapsed time $T_{elapsed}$ up to the EE predictor within the per-sentence latency target $T$ , the optimal running frequency can be calculated as follows: $Freq_{opt} = N_{cycles} / (T - T_{elapsed}) $ Meanwhile, the corresponding energy-optimal supply voltage, $VDD_{opt}$ , is selected by the DVFS controller to achieve the lowest operational voltage value at $Freq_{opt}$ .",
"In the EdgeBERT accelerator system, this is done via indexing the look-up table containing the ADPLL frequency/voltage sweep coordinates.",
"The DVFS is performed for each real-time sentence inference due to its fast response time; the implementation details are shown in Sec.",
"REF ."
],
[
"Algorithmic Synergy",
"In order to quantify the different tradeoffs, and evaluate the synergistic impact on the model accuracy from the memory and latency optimizations, the eNVM modeling, and the EE predictor, we implemented the training and evaluation procedures illustrated in Fig.",
"REF on the base of HuggingFace’s Transformers infrastructure [85]."
],
[
"Training and Evaluation Procedure",
"The training methodology consists of two phases.",
"In the first phase, the model is pruned during fine-tuning: magnitude pruning is applied to the embedding layer and either movement or magnitude pruning is applied to the Transformer encoder layer.",
"An additional loss term comes from knowledge distillation using the base ALBERT model fine-tuned on the target task as a teacher.",
"The embeddings and the encoder layer are subject to separate pruning schedules.",
"At the same time, the attention heads learn their optimal spans.",
"In the second training phase, we freeze the model's parameters prior to fine-tuning the early exit highway off-ramps.",
"At evaluation time, 8-bit floating-point quantization is applied on all the weights and activations.",
"The quantized embedding weights are modeled according to a 2-bit per cell multi-level (MLC2) ReRAM NVM configuration.",
"The learned attention span mask is element-wise multiplied with the attention weights to re-modulate their saliencies.",
"Entropy prediction is then deployed along with early exit during inference according to Algorithm ."
],
[
"Impact on Model Accuracy, Computation, and Storage",
"Using the multi-step procedure illustrated in Fig.",
"REF , we amalgamate into ALBERT the various memory and latency reduction techniques at training and evaluation times.",
"Table REF summarizes the generated benefits of the synergistic inference with the following main observations: EdgeBERT latency-aware inference provides comparable average exit layer for the same accuracy threshold as the conventional EE approach, while allowing the DVFS algorithm to reduce the frequency and voltage in accordance with the predicted exit layer.",
"The EdgeBERT approach requires a lower entropy threshold than the conventional EE approach for the same accuracy target; this demonstrates that the we must predict conservatively due to the classification error introduced by the neural network-based entropy predictor.",
"Across the four corpora, a uniform 40% density in the embedding layer is achieved, establishing a compact memory baseline of 1.73MB to be stored in eNVMs."
],
[
"Required Computations in ALBERT",
"The Transformer encoder is the backbone of ALBERT/BERT, consuming more than 95% of inference computations.",
"Fig.",
"REF summarizes the computations required in this unit.",
"Assuming a sentence length of 128, the transformer encoder requires 1.9GFLOPs to compute matrix multiplications, layer normalizations, element-wise operations (add, mult.",
"), and softmax.",
"The attention span mask learned during fine-tuning is element-wise multiplied with the softmax output.",
"Notably, all the computations inside any of the twelve attention blackhead units can be effectively skipped in case its associated attention span mask is 100% null.",
"The EdgeBERT accelerator reaps this benefit by enforcing adaptive attention span masking during fine-tuning."
],
[
"The EdgeBERT Accelerator System",
"In order to maximize the benefits of the latency and memory reduction techniques during latency-aware inference, we designed a scalable accelerator system that exploits these algorithms for compute and energy efficiency blackwith the following key highlights: blackSpecialized datapath support for (i) early exit assessment, (ii) softmax and attention span masking, and (iii) layer normalization.",
"We notably reformulate their mathematical definitions in order to avoid numerical instability, and where possible, hardware components with long cyclic behaviors such as divisions.",
"blackNon-volatile and high density storage of the shared multi-task parameters substantially improves the accelerator's energy and area efficiency (Sec.",
"REF ).",
"On-demand DVFS aided by the integration of a fast-locking ADPLL and a fast-switching LDO regulator.",
"blackCompressed sparse execution via bitmask encoding.",
"The EdgeBERT hardware accelerator, illustrated in Fig.",
"REF , consists of a processing unit (PU), a special function unit (SFU), a LDO and ADPLL for latency-bounded DVFS.",
"The communication between the PU and SFU occurs via a custom-built bi-directional streaming channel.",
"An AXI splitter arbitrates the CPU-controlled flow of instructions and data bound for the PU and SFU AXI-slave partitions.",
"The multi-task embedding pruned weights and corresponding bitmask are stored in a 2MB ReRAM NVM buffer in order to avoid reloading them when powered on.",
"Specifically, the bitmask embedding values are stored in a single-level cell (SLC) ReRAM configuration while the nonzero embedding parameters are kept in a 2-bit per cell (MLC2) ReRAM structure, according to the learnings from the NVM studies (Sec.",
")."
],
[
"Processing Unit",
"The processing unit (PU) is designed to execute matrix-matrix multiplications in linear layers and attention heads of ALBERT.",
"In the PU datapath in Fig.",
"REF , $n$ defines the number of parallel floating-point vector MACs (VMAC) and the vector size of each VMAC.",
"So, there are $n^2$ MAC units in total.",
"The PU datapath takes two $n*n$ matrices as input and computes $n*n*n$ MAC operations in $n$ clock cycles.",
"We use 8-bit floating point as the input and weight data type as no accuracy degradation was observed, and 32-bit fixed-point during accumulation.",
"The PU accumulator sums activation matrices and quantizes the final matrix back to 8-bit floating-point.",
"To exploit sparsity in both input and weight matrices, we (1) adopt bit-mask encoding and decoding for compressing and decompressing the sparse matrix, and (2) implement skipping logic in the datapath.",
"Bit-masks are binary tags to indicate zero and non-zero entries of a matrix so that only non-zero entries are stored in the decoder SRAM scratchpads.",
"For every cycle during decoding, a size $n$ vector is fetched and decoded.",
"The decoder first reads a $n$ -bit mask from the single-banked mask buffer to figure out what bank in the $n$ -banked input can be neglected, and inserts zero values back to the original zero entries.",
"The encoder also takes a similar approach.",
"It creates a bit mask vector and removes zero entries from the data vector before sending the compressed mask and data vector to one of the PU decoder blocks.",
"To save energy, the PU datapath skips the computation of a VMAC product-sum if one of the operand vectors contains only zero values.",
"Although the cycle-behavior of the datapath is not affected by the sparsity of inputs due to the fixed scheduling of data accesses and computations, skipping VMAC operations saves up to 1.65$\\times $ in energy consumption (Sec.",
"REF )."
],
[
"Special Function Unit",
"The special function unit (SFU) contains specialized datapaths that compute the EE assessment, DVFS control, element-wise addition, layer normalization, and softmax, all of which get invoked during the latency-aware EdgeBERT inference.",
"The SFU also integrates a 32KB auxiliary buffer to house the EE and DVFS LUTs, the layer normalization parameters, and the multi-head attention span masks learned during the fine-tuning process.",
"blackAll the computations in the SFU are in 16-bit fixed-point format."
],
[
"Computing the Multi-Head Attention",
"While the linear layers for the attention query, key and value tensors are computed in the PU, the proceeding softmax operation is optimized in the SFU softmax unit.",
"First, prior to computing an attention head, the SFU controller inspects its associated attention span mask in the auxiliary buffer.",
"In case the attention span mask for an attention head is null, the SFU controller proactively cancels and skips entirely the sequence of computations required for that head, and directly writes zero in the corresponding logical memory for its context vector stored in one of the PU decoder blocks.",
"[!t] attention matrix $A$ , and mask $A_M$ of size ($T*T$ ) masked softmax output matrix $A_O$ $T := \\text{number of tokens}$ ; $n := \\text{tile size}$ ; $i = 0$ to $T-1$ // Step 1: compute max value $max = -\\infty $ $j = 0$ to $T-1$ $vec <= load(A_{[i][n*j:n*j+n-1]})$ $max < max(vec)$ $max = max(vec)$ // Step 2: compute log-exponential-sum $sum_{exp} = 0$ $j = 0$ to $T-1$ $vec <= load(A_{[i][n*j:n*j+n-1]})$ $sum_{exp} += sum(exp(vec - max))$ $logsum_{exp} = ln(sum_{exp})$ // Step 3: Get softmax and modulate with attn span mask $j = 0$ to $T-1$ $vec<= load(A_{[i][n*j:n*j+n-1]})$ $mask <= load(A_{M[i][n*j:n*j+n-1]})$ $vec = exp(vec - max - logsum_{exp})$ $vec = vec * mask$ $store(vec) => A_{O[i][n*j:n*j+n-1]}$ Computing Softmax and Attention Span Masking In case the attention span mask for a head contains non-zero elements, the softmax unit takes advantage of the LogSumExp [19] and Max [48] tricks to vectorize the computation of the softmax function $SM()$ as: $ \\leavevmode \\xbox {resizebox}{\\XMLaddatt {width}{0.0pt}SM(A_{k})= exp[A_{k} - MAX_{k}(A)- ln (\\sum ^{K}_{k=1}exp(A_{k}-MAX_{k}(A)))]}$ By doing so, the hardware prevents numerical instability stemming from exponential overflow, and avoids the computationally intensive division operation from the original softmax function.",
"Upon completing the softmax operation, the softmax unit then performs element-wise multiplication between the resulting attention scores and the attention span mask as described in Algorithm REF ."
],
[
"Performing Early Exit Assessment",
"The EE assessment unit computes the numerically-stable version of the entropy function from equation REF as follows: $\\leavevmode \\xbox {resizebox}{\\XMLaddatt {width}{0.0pt}H(x_k) = \\ln (\\sum \\limits _{k=1}^{n} e^{x_k - MAX_{k}(x) }) - MAX_{k}(x) -\\frac{\\sum \\limits _{k=1}^{n} x_k e^{x_k - MAX_{k}(x)}}{\\sum \\limits _{k=1}^{n} e^{x_k - MAX_{k}(x) }}}$ The EE assessment unit then compares the result with the register value for the entropy threshold.",
"If the EE condition is met, the unit then triggers the accelerator's interrupt (IRQ).",
"Otherwise, the SFU controller initiates the computation of the next Transformer encoder.",
"In the case of latency-aware inference in intermittent mode, the EE assessment unit also indexes the EE predictor LUT stored in the auxiliary buffer in order to acquire the predicted exit layer value, which is then passed on to the DVFS controller."
],
[
"DVFS System",
"During each sentence inference, the DVFS FSM algorithm keeps track of the EE predictor result and manages the operating voltage and frequency accordingly.",
"Based on the predicted early exit layer, the DVFS controller indexes the logical memory for the $V/F$ LUT table in the auxiliary buffer and extracts the lowest corresponding supply voltage value, $VDD_{opt}$ .",
"At the same time, the DVFS controller simultaneously updates the ADPLL and LDO configuration registers with settings for $Freq_{opt}$ and $VDD_{opt}$ , respectively.",
"Table: Performance specs of LDO and ADPLLFigure: Spice simulations of LDO dynamic voltage adjustments.",
"The LDO stabilizes voltage transitions within 100ns.The synthesizable LDO is implemented using standard power header cells [9], and evenly distributed across the EdgeBERT accelerator.",
"The LDO is able to scale the accelerator voltage from 0.5V to 0.8V with a 25mV step.",
"With careful power header selection and layout resistance optimization, the LDO can achieve nearly linear scaled power efficiency and a fast response time of 3.8ns/50mV.",
"The ADPLL is also implemented using all-synthesizable approach with the PLL architecture from the FASoC open-source SoC design framework [4].",
"Following a frequency update request, the all-digital PLL can relock the frequency in a fast speed with low power consumption.",
"The 12nm performance specs of the LDO and ADPLL are shown in Table REF .",
"Fig.",
"REF show the spice-level simulation of the DVFS for a consecutive sequence of sentence inference.",
"For each sentence, the entropy is calculated after the computation of Encoder 1 and sent to the EE predictor to forecast the early exit layer.",
"Based on the predicted early exit encoder and latency requirement for the sentence, the DVFS controller select the lowest voltage level and proper frequency to meet the latency requirement $T_{target}$ .",
"Therefore, the remaining encoder stages will compute at a lower voltage level to save energy.",
"For example, the sentence 1 of Fig.",
"REF , the early exit layer is predicted as 8.",
"Therefore, the rest Encoders (i.e encoder 2-8) in sentence 1 are computed under a lower voltage 0.7V.",
"After the inference of the first sentence, the voltage level ramps back to nominal 0.8V for the computation of layer 1 in the following sentence.",
"As on-chip integrated LDO is used, the transition and settling time is optimized to be within 100ns, which is negligible considering the 50ms latency target.",
"The computation of the next sentence starts once the voltage transition is settled.",
"During idle times, EdgeBERT stays at standby 0.50V to save leakage energy.",
"Figure: Average latency (top row) and energy (Bottom row) per sentence as the PU MAC vector size scales at max frequency (1GHz) and nominal voltage (0.8V), highlighting impact of adaptive attention span (AAS), and sparsity in weights and activations (Sparse) on the EdgeBERT accelerator and TX2 mGPU.",
"MAC size of 16 yields the most energy efficient design.Figure: Average DVFS-driven supply voltage (top row) and clock frequency (middle row), as well as, generated energy expenditures (bottom row) of the EdgeBERT accelerator system with n=16n=16 during latency-aware inference (LAI), and latency-aware inference further improved with adaptive attention span and sparse execution (LAS+AAS+Sparse).",
"Different latency targets of 50ms (T=50), 75ms (T=75), and 100ms (T=100) are used for LAI executions.",
"Results are compared with the baseline 12-layer inference (Base) and the conventional early exit inference (EE)."
],
[
"Design and Verification Methodology",
"The EdgeBERT accelerator is designed in synthesizable SystemC with the aid of hardware components from the MatchLib [34] and HLSLibs [27] open-source libraries.",
"Verilog RTL is auto-generated by the Catapult high-level synthesis (HLS) tool [1] using a commercial 12nm process node.",
"HLS constraints are uniformly set with the goal to achieve maximum throughput on the pipelined design.",
"During the bottom-up HLS phase, the decoder and auxiliary buffers are mapped to synthesized memories from a foundry memory compiler, while the rest of the registers are mapped to D-latches.",
"The energy, performance, and area results are reported on the post-HLS Verilog netlists by the Catapult tool at the 0.8V/25c/typical corner.",
"The 28nm ReRAM cells are characterized in NVSIM [18] and its read latency, energy, and area are back-annotated into the accelerator results after scaling to a 12nm F$^2$ cell definition in order to match the process node used in the rest of the system.",
"To quantify the benefits of non-volatility (Sec.",
"REF ), we quantify the alternative cost of loading embeddings from off-chip using DRAMsim3 [40] to extract cycle-accurate LPDDR4 DRAM energy and latency metrics.",
"GPU results are obtained from CUDA implementations on an Nvidia TX2 mobible GPU (mGPU), whose small form-factor SoC targets embedded edge/IoT applications [2].",
"We start by measuring the energy-performance trade-offs of the EdgeBERT accelerator by scaling its PU MAC vector size.",
"Simultaneously, we further quantify the benefit of bitmask encoding and the predicating logic of the adaptive attention span mechanism by using the attained optimization results (i.e.",
"embedding and encoder sparsity percentage, and attention span) reported in Table REF in which the accuracy drop was at 1%-pt of the baseline.",
"Adaptive adaptive span is also applied to the mGPU platform in order to quantify and compare the extent of these benefits.",
"Fig.",
"REF shows that the per-sentence processing latency decreases by roughly 3.5$\\times $ as the vector size doubles.",
"Across the four tasks, the energy-optimal accelerator design is obtained with a MAC vector size, $n$ , of 16.",
"This is because the increase in the datapath power consumption with $n=32$ starts to subdue throughput gains.",
"The predication/skipping mechanism of adaptive attention span reduces the accelerator processing time and energy consumption by up to 1.2$\\times $ and 1.1$\\times $ , respectively.",
"Compressed sparse execution in the PU datapath amounts to an additional 1.4–1.7$\\times $ energy savings with QQP receiving the benefit the most.",
"The EdgeBERT accelerator starts to outperform the mGPU processing time with $n=16$ .",
"This energy-optimal design generates up 53$\\times $ lower energy compared to the mGPU when all the optimizations are factored in.",
"Fig.",
"REF breaks down the latency, energy, area and power contributions inside the placed-and-routed, energy-optimal ($n$ =16) EdgeBERT accelerator system which occupies 1.4mm$^2$ while consuming an average power of 86mW."
],
[
"DVFS-based Latency-Aware Inference",
"Fig.",
"REF shows the DVFS-controlled supply voltage and clock frequency, and the energy savings of the latency-aware inference (LAI) on the energy-optimal accelerator design (i.e.",
"with MAC vector size $n=16$ ) using latency targets between 50ms and 100ms (common latency thresholds for real-time human perception [57])).",
"The results show that EdgeBERT optimized LAI achieves up to 7$\\times $ , and 2.5$\\times $ per-inference energy savings compared to the conventional inference (Base), and latency-unbounded early exit (EE) approaches, respectively, as seen in the SST-2 case.",
"As AAS further cuts the number of computation cycles, we observe further relaxation of the supply voltage and clock frequency.",
"At some latency targets (e.g., 75ms and 100ms in QQP and SST-2), further energy savings are not possible as V/F scaling bottoms out.",
"blackTo underscore the different contributions to energy savings, at 75ms latency target for example in the case of MNLI, early exit prediction, adaptive attention span, DVFS, sparse execution, and eNVMs account for 21%, 12%, 23%, 39%, and 5%, respectively, of the total accelerator energy reduction.",
"For stricter latency targets (e.g.",
"$<$ 20ms), the proposed DFVS-based scheme can be used by scaling up to even higher MAC vector sizes (i.e.",
"$n \\ge 32$ )."
],
[
"Benefits of NVM Embeddings Storage",
"BERT word embeddings are a natural fit for non-volatile storage, given that in EdgeBERT, we freeze them during fine-tuning and reuses them during inference By virtue of this scheme, we have established a compact 1.73MB baseline wherein the bitmask of the word embeddings is stored in a SLC ReRAM while the nonzero parameters are stored in a 2-bit per cell (MLC2) ReRAM buffer.",
"Fig.",
"REF illustrates the immense gains of leveraging this eNVM configuration during single-batch inference after SoC power-on.",
"In EdgeBERT, ALBERT embeddings would only need to be read from the integrated ReRAM buffers due to being statically pre-loaded.",
"The conventional operation dictates reading the embedding weights from off-chip DRAM, then writing them to dedicated on-chip volatile SRAM memories so they can be reused for future token identifications.",
"The EdgeBERT approach enforces a latency and energy advantage that is, respectively, 50$\\times $ and 66,000$\\times $ greater than the overhead costs in the conventional operation.",
"The non-volatility of this embedded storage means that these benefits can further scale with the frequency of power cycles.",
"Figure: (a) Breakdown of latency and energy consumption in PU and SFU datapaths, and (b) 12nm physical layout, and area and power (@ 0.8V/1GHz) breakdown of the energy-optimal EdgeBERT accelerator (MAC size=16).Figure: Costs of reading all embedding weights after system power-on.",
"Storing embeddings in ReRAMs gives EdgeBERT significant energy and latency advantages compared to the conventional approach requiring DRAM read followed by SRAM write/read."
],
[
"Related Work",
"Over the last decade, there has been extensive research on the design of high-performance and energy-efficient DNN hardware accelerators [24], [60], [11], [82], [83], [12], [47], [84], [5], [6], [26], [42], [31], [33], [54], [53], [62], [61], [66], [36], [35], [41], [45], [22], [86], [67], [71].",
"As these accelerators are increasingly deployed at all computing scales, there is additional interest in the hardware community to automatically generate designs [77], [76], [80], [81].",
"However, most of these works focus on CNN and RNN [20] computations, and not as much scrutiny has been given to accelerating Transformer-based networks with self-attention mechanisms.",
"Recent work in accelerating Transformer-based NLP includes $A^{3}$ [23], which proposed a hardware architecture that reduces the number of computations in attention mechanisms via approximate and iterative candidate search.",
"blackHowever, the $A^{3}$ scheme fetches the full and uncompressed data from DRAM before dynamically reducing computations in the hardware.",
"In contrast, EdgeBERT learns the optimal attention search radius during the finetuning process and then leverages its very sparse mask to avoid unnecessary matrix multiplications.",
"Therefore, our approach substantially eliminates DRAM accesses as the computation and memory optimizations are pre-learned before hardware acceleration.",
"GOBO [88] blackfocuses on BERT quantization only via 3-bit clustering on the majority of BERT weights while storing the outlier weights and activations in full FP32 precision.",
"blackAlthough this scheme significantly reduces DRAM accesses, it requires a mixed-precision computational datapath and a non-uniform memory storage.",
"In contrast, EdgeBERT adopts uniform 8-bit data storage in SRAM and eNVMs memories.",
"Lu et al.",
"[46] proposes a dense systolic array accelerator for the Transformer's multi-head attention and feed-forward layers and optimizes Transformers' computations via matrix partitioning schemes.",
"The EdgeBERT accelerator executes compressed sparse inference for higher energy efficiency.",
"OPTIMUS [55] looks to holistically accelerate Transformers with compressed sparse matrix multiplications and by skipping redundant decoding computations.",
"FlexASR [71] accelerates attention-based RNNs in a specialized attention datapath and only saves energy by gating the MAC when decoder RNN inputs are null.",
"SpAtten [79] accelerates Transformer-based models via progressive cascade token and attention head pruning.",
"The importance of each attention head is determined during the computation via a top-k ranking system.",
"In contrast, EdgeBERT opts to learn the important attention heads during the fine-tuning process by activating adaptive attention spanning.",
"The optimized and sparse attention spans are then used by the EdgeBERT accelerator to predicate the NLP computation.",
"blackFinally, all the aforementioned NLP accelerators stores the embedding weights in traditional volatile SRAM memories.",
"By contrast, this work recognizes that embedding weights do not change across NLP tasks.",
"Therefore, EdgeBERT statically stores the word embeddings in high density eNVMs, generating substantial energy and latency benefits (Sec.",
"REF ).",
"Fig.",
"REF qualitatively contrasts some of the prior work with EdgeBERT.",
"Figure: blackComparison of EdgeBERT with prior work accelerating Transformer-based NLP models."
],
[
"Conclusion",
"As newer Transformer-based pre-trained models continue to generate impressive breakthroughs in language modeling, they characteristically exhibit complexities that levy hefty latency, memory, and energy taxes on resource-constrained edge platforms.",
"EdgeBERT provides an in-depth and principled latency-driven methodology to alleviate these computational challenges in both the algorithm and hardware architecture layers.",
"EdgeBERT adopts first-layer early exit prediction in order to perform dynamic voltage-frequency scaling (DVFS), at a sentence granularity, for minimal energy consumption while adhering to a prescribed target latency.",
"Latency and memory footprint overheads are further alleviated by employing a balanced combination of adaptive attention span, selective network pruning, floating-point quantization.",
"We further exploit and optimize the structure of eNVMs in order to store the shared multi-task parameters, granting EdgeBERT significant performance and energy savings from system power-on.",
"Sentence-level, latency-aware inference on the EdgeBERT accelerator notably consumes 7$\\times $ and 2.5$\\times $ lower energy than the conventional full-model inference, and the latency-unbounded early exit approach, respectively."
],
[
"Acknowledgement",
"This work was supported in part by the Center for Applications Driving Architectures (ADA), one of six centers of JUMP, a Semiconductor Research Corporation (SRC) program co-sponsored by DARPA; DARPA’s DSSoC program; NSF Awards 1704834 and 1718160; Intel Corp.; and Arm Inc."
]
] | 2011.14203 |
[
[
"Learning from Incomplete Features by Simultaneous Training of Neural\n Networks and Sparse Coding"
],
[
"Abstract In this paper, the problem of training a classifier on a dataset with incomplete features is addressed.",
"We assume that different subsets of features (random or structured) are available at each data instance.",
"This situation typically occurs in the applications when not all the features are collected for every data sample.",
"A new supervised learning method is developed to train a general classifier, such as a logistic regression or a deep neural network, using only a subset of features per sample, while assuming sparse representations of data vectors on an unknown dictionary.",
"Sufficient conditions are identified, such that, if it is possible to train a classifier on incomplete observations so that their reconstructions are well separated by a hyperplane, then the same classifier also correctly separates the original (unobserved) data samples.",
"Extensive simulation results on synthetic and well-known datasets are presented that validate our theoretical findings and demonstrate the effectiveness of the proposed method compared to traditional data imputation approaches and one state-of-the-art algorithm."
],
[
"Introduction",
"Learning methods from limited or imperfect data has attracted great attention in the literature recently.",
"Datasets with limited, weak, noisy labels or incomplete features represent an important and still open problem.",
"In this paper, we address the problem of training a classifier on a dataset with incomplete features, which arises in many machine learning applications where sometimes the measurements are incomplete, noisy or affected by artifacts.",
"Examples of this situation include: recommendation systems built upon the information gathered by different users where not all the users have fully completed their forms; medical datasets where typically not all tests can be performed on every patient; or a self-driving vehicle or robot where objects in the view field can be partially occluded.",
"Handling correctly the incomplete-features problem is a classical challenge in machine learning.",
"Skipping missing features by setting them to zero values damages the classification accuracy [1].",
"Most previous studies addressed this problem by using an imputation approach, which consists of performing data completion followed by training the classifier with those reconstructions (referred here as the sequential method).",
"However, this strategy cannot ensure the statistical consistence of the classifier, as data completion is usually fully unsupervised or label information is partially or inefficiently exploited.",
"In this work, a new supervised learning method is developed to train a general classifier, such as a logistic regression or a deep neural network, using only a subset of features per sample, while assuming sparse representations of data vectors on an unknown dictionary.",
"The proposed method simultaneously learns the classifier, the dictionary and the corresponding sparse representation of each input data sample.",
"In this way, we combine the approximation power and simplicity of sparse coding with the extraordinary ability of neural networks (NNs) to model complex decision functions (classifiers) with the goal to successfully train a classifier based on incomplete features.",
"We analyze the limitations of the sequential approach (section REF ), i.e.",
"imputation followed by training, and introduce the simultaneous classification and coding approach in section .",
"Our method consists of incorporating a sparse data representation model into a single cost function that is optimized for training the classifier and, at the same time, finding the best representation of the observed data.",
"A learning algorithm is presented in section REF to train a classifier on incomplete features and sufficient conditions under which such a classifier performs as good as the ideal classifier, i.e.",
"the one that can be obtained from complete observations, is identified (section REF ).",
"Extensive experimental results are presented in section , using synthetic and well known benchmark datasets that validate our theoretical findings and illustrate the effectiveness of the proposed method.",
"Practical sequential methods based on statistical imputation, such as computing the “mean”, “regression” and “multiple” imputation techniques are common practice [2].",
"Remarkably, it was shown that imputing with a constant, e.g.",
"the mean, is Bayes-risk consistent only when missing features are not informative [3].",
"More elaborated completion methods were also explored, such as $K$ -nearest neighbor estimators, multilayer or recurrent NNs and others, see [4] and references therein.",
"However, sequential methods do not fully exploit label information.",
"Data labels can provide valuable information about missing features that could potentially improve the classifier learning process.",
"Recent advances on probabilistic generative models have allowed for a formulation of supervised learning with incomplete features as a statistical inference problem arriving at algorithms that significantly outperformed sequential methods.",
"In the seminal work [5], a framework for maximum likelihood density estimation based on mixtures models was proposed and successfully applied to small incomplete-features problems.",
"In particular, a Gaussian Mixture Model (GMM) was fitted to incomplete-data through an Expectation-Maximization (EM) algorithm.",
"Building upon this generative model strategy, some approaches have considered integrating out the missing values based on a simple logistic regression function [6], [7].",
"Other versions of this approach proposed an explicit simultaneous learning of the model and the decision function [8], [9].",
"While probabilistic generative models provided a nice and elegant approach to the incomplete-features problem showing good results on small datasets, they are not suitable for many modern machine learning applications because: (1) despite some acceleration techniques were explored, e.g.",
"[10], [11], those algorithms are computationally expensive becoming prohibitive for moderate to large datasets; (2) GMM is impractical for modeling high-dimensional datasets because the number of parameters to achieve good approximations becomes unmanageable; and (3) they do not consider complex classification functions as the ones provided by deep NN architectures.",
"Recently, some approaches based on the low-rank property of the features data matrix were investigated and algorithms for data completion were proposed incorporating the label information [12], [13], [14].",
"Since the rank estimation of a matrix is a computationally expensive task, usually based on the Singular Value Decomposition (SVD), the obtained algorithms are prohibitive to solve modern machine learning problems with large datasets.",
"Additionally, as in the case of the probabilistic generative models, none of these methods considered complex classification functions.",
"To overcome this drawback, more recently, a framework based on various NN architectures such as autoencoders, multilayer perceptrons and Radial Basis Function Networks (RBFNs), was proposed for handling missing input data by setting a probabilistic model, e.g.",
"a GMM, for every missing feature, which is trained together with the NN weights [15].",
"This method combined the great capability of NNs to approximate complex decision functions with the nice formulation of the GMM to model missing data.",
"However, it inherited the drawbacks of GMMs, i.e.",
"they are not well suited to higher-dimensional datasets.",
"On the other hand, during the last few years in the signal processing community, there has been a rapid development of theory and algorithms for sparse coding approximations which, by exploiting the redundancy of natural signals, are able to provide simple and accurate models of complex data distributions, see [16], [17], [18], [19], [20] and references therein.",
"Sparse coding is nearly ubiquitous in Nature, for example, it is found in the way that neurons encode sensory information [21], [22].",
"Sparse representations of data showed to be useful also in classification problems.",
"In [23], a Linear Discriminant Analysis (LDA) classifier was trained on corrupted data providing a robust classification method.",
"In [24], algorithms for learning discriminative sparse models, instead of purely reconstructive ones, were proposed based on simple linear and bilinear classifiers.",
"Similar methods were also studied by either using class-specific dictionaries [25], [26] or using a single one for all classes [27].",
"However, these proposed methods neither were applied to the incomplete-features problem nor considered deep NN classifiers."
],
[
"Problem formulation",
"We assume a supervised learning scenario with vector samples and labels $\\lbrace {\\mathbf {x}}_i, y_i\\rbrace $ , $i=1,2,\\dots , I$ , ${\\mathbf {x}}_i \\in {R}^N$ and $y_i \\in \\lbrace 0,1,\\dots , C-1\\rbrace $ ($C$ classes).",
"However, we are constrained to observe only subsets of features and their labels: $\\lbrace {\\mathbf {x}}^o_i, y_i\\rbrace $ , ${\\mathbf {x}}^o_i \\in {R}^{M_i}$ with $M_i < N$ .",
"Unobserved (missing) features are denoted by ${\\mathbf {x}}^m_i \\in {R}^{N-M_i}$ .",
"We consider arbitrary patterns of missing features, which are allowed to be different for each data instance $i$ .",
"The set of indices of missing features at sample $i$ is denoted by ${\\cal {M}}_i$ , i.e.",
"${\\mathbf {x}}^m_i = {\\mathbf {x}}_i({\\cal {M}}_i)$ and ${\\mathbf {x}}^o_i = {\\mathbf {x}}_i\\left({\\overline{\\mathcal {M}}}_i\\right)$ .",
"We define the set of all $K$ -sparse vectors $\\Sigma _K^P = \\lbrace {\\mathbf {s}}\\in {R}^P \\text{ s.t. }",
"\\Vert {\\mathbf {s}}\\Vert _0 \\le K\\rbrace $ (containing at most $K$ non-zero entries) and assume that data vectors ${\\mathbf {x}}_i$ admit $K$ -sparse representations over an unknown dictionary $\\mathbf {D}\\in {R}^{N\\times P}$ ($P \\ge N$ ): ${\\mathbf {x}}_i = \\mathbf {D}{\\mathbf {s}}_i, \\mbox{ with } {\\mathbf {s}}_i \\in \\Sigma _K^{P}.$ The columns of a dictionary are called “atoms” because every data vector can be written as a linear combination of at most $K$ elementary components.",
"Sometimes dictionaries are orthogonal such as the ones derived from the Discrete Cosine or Wavelet [19] transforms.",
"However, overcomplete ($P \\ge N$ ) nonorthogonal dictionaries have demonstrated to play an important role in image processing tasks such as denoising, inpainting, etc [17], [28].",
"By partitioning $\\mathbf {D}$ according to the pattern of missing features at sample $i$ , we obtain $\\mathbf {D}^o_i =\\mathbf {D}\\left({\\overline{\\mathcal {M}}}_i,:\\right) \\in {R}^{M_i\\times P}$ and $\\mathbf {D}^m_i =\\mathbf {D}({\\cal {M}}_i,:)\\in {R}^{{(N-M_i)}\\times P}$ , which according to equation (REF ) implies: ${\\mathbf {x}}^o_i = \\mathbf {D}^o_i {\\mathbf {s}}_i, \\mbox{ and } {\\mathbf {x}}^m_i = \\mathbf {D}^m_i {\\mathbf {s}}_i.$ Let us assume that a perfect classifier, e.g.",
"a logistic regression or deep NN, that assigns probability $p_{\\Theta }(\\hat{y} | {\\mathbf {x}})$ to predicted label $\\hat{y}$ given data ${\\mathbf {x}}$ can be trained on the complete dataset $\\lbrace {\\mathbf {x}}_i, y_i\\rbrace $ , such that, in a two-classes scenario ($C=2$ ), $p_{\\Theta }(\\hat{y} = y_i | {\\mathbf {x}}_i) > p_{\\Theta }(\\hat{y} \\ne y_i | {\\mathbf {x}}_i)$ , $\\forall i =1,2,\\dots , I$ , where $\\Theta $ is the set of trained parameters.",
"Our goal is to develop a method to obtain an estimate $\\hat{\\Theta }$ of parameters using only the incomplete dataset $\\lbrace {\\mathbf {x}}^o_i, y_i\\rbrace $ and to identify conditions under which such a classifier is compatible with the ideal one."
],
[
"Why training after imputation is difficult?",
"If the $K$ -sparse representations of the observations ${\\mathbf {x}}^o_i$ were unique, then ${\\mathbf {x}}_i$ can be perfectly reconstructed from the incomplete observations and the classifier can be successfully trained using these reconstructions.",
"In the particular case where the dictionary is known in advance, there exist conditions on the sampling patterns based on the coherence, spark or RIP (Restricted Isometry Property) of matrix $\\mathbf {D}^o_i$ that can guarantee uniqueness [20].",
"However, these conditions are difficult to meet in practice and determining RIP/Spark properties are NP-hard in general [29].",
"Moreover, in the general case where the dictionary $\\mathbf {D}$ is unknown and needs to be learned from data, it is even more difficult to obtain well separated reconstructions which certainly leads to suboptimal or wrong classifiers.",
"Next, we provide some intuition about the limitation of the sequential approach through a toy example.",
"Let us consider the classification of hand-written digit images belonging to two classes: “3s” and “8s” and assume that they admit 2-sparse representations over a dictionary.",
"Fig.",
"REF (a-b) shows the representations of two example vectors ${\\mathbf {x}}_i$ and ${\\mathbf {x}}_j$ belonging to classes “3” and “8”, respectively.",
"If only the right halves of the images are observed and no label information is provided, we are clearly faced with a problem because our observed samples from two different classes are identical, i.e.",
"${\\mathbf {x}}^o_i = {\\mathbf {x}}^o_j$ .",
"It is obvious that at least two possible 2-sparse representations for the observed data exist as illustrated in Fig.",
"REF (c).",
"When the sparse solution is not unique, we may end up reconstructing wrong vectors that could not be even well separated as illustrated in Fig.",
"REF (d-e).",
"In general, sequential methods using only the information of observed features are prone to fail because the non-uniqueness of solutions can make the training of a good classifier an impossible task.",
"However, we could solve this problem by incorporating the labelling information from the very beginning as it is proposed in the following section."
],
[
"Simultaneous learning and coding approach",
"We propose to train the classifier and find the proper representation, not only as sparse as possible but also providing the best separation of classes.",
"We want to combine the training of the classifier together with the learning of a dictionary and optimal sparse representations such that the reconstructed data vectors are compatible with observations and well separated.",
"To do that we propose to minimize the following global cost function: $J(\\Theta , \\mathbf {D}, {\\mathbf {s}}_i) = \\nonumber \\\\\\textstyle \\underbrace{\\frac{1}{I}\\sum _{i=1}^I\\big \\lbrace J_0(\\Theta , \\hat{{\\mathbf {x}}}_i, y_i) + \\lambda _{1} J_1(\\mathbf {D},{\\mathbf {s}}_i)\\big \\rbrace }_{F(\\Theta , \\mathbf {D}, {\\mathbf {s}}_i)} +\\underbrace{\\frac{1}{I}\\sum _{i=1}^I\\big \\lbrace \\lambda _{2}J_2({\\mathbf {s}}_i)\\big \\rbrace }_{G({\\mathbf {s}}_i)},$ with respect to $\\Theta $ , $\\mathbf {D}$ and ${\\mathbf {s}}_i$ ($i=1,2,\\dots , I$ ), where $\\Theta $ contains the classifier parameters, i.e.",
"the vector of weights in a deep NN classifier architecture; $\\mathbf {D}\\in {R}^{N\\times P}$ ($P \\ge N$ ) is a dictionary and ${\\mathbf {s}}_i \\in \\Sigma _K^{P}$ are the representation coefficients such that the reconstructed data vectors are $\\hat{{\\mathbf {x}}}_i = \\mathbf {D}{\\mathbf {s}}_i$ .",
"$J_0(\\Theta , \\hat{{\\mathbf {x}}}_i, y_i)$ is a measure of the classification error for the reconstructed sample vector $\\hat{{\\mathbf {x}}}_i$ .",
"Typically, we use the crossentropy measure, i.e.",
"$J_0(\\Theta , \\hat{{\\mathbf {x}}}_i, y_i) = -\\log [p_{\\Theta }(y_i | \\hat{{\\mathbf {x}}}_i)]$ , where $p_{\\Theta }(y_i | \\hat{{\\mathbf {x}}}_i)$ is the probability assigned by the classifier to sample $\\hat{{\\mathbf {x}}}_i$ as belonging to class $y_i$ .",
"$J_1(\\mathbf {D},{\\mathbf {s}}_i)$ is a measure of the approximation error of the reconstruction when it is restricted to observed features, which is defined as follows: $J_1(\\mathbf {D},{\\mathbf {s}}_i)=\\frac{M_i}{N}\\Vert {\\mathbf {m}}_i \\odot ({\\mathbf {x}}_i - \\mathbf {D}{\\mathbf {s}}_i)\\Vert ^2$ , where $\\odot $ stands for the entry-wise product, ${\\mathbf {m}}_i \\in {R}^N$ is the observation mask for sample $i$ , i.e.",
"$m_i(n) = 0$ (1) if data entry ${\\mathbf {x}}_i(n)$ is missing (available); and $J_2({\\mathbf {s}}_i) = \\frac{1}{N}\\Vert {\\mathbf {s}}_i\\Vert _1$ is proportional to the $\\ell _1$ -norm whose minimization promotes the sparsity of the representation since $\\ell _1$ -norm is a convenient proxy for $\\ell _0$ -norm [30].",
"Finally, the hyper-parameters $\\lambda _1$ and $\\lambda _2$ allow us to give more or less importance to the representation accuracy and its sparsity, with respect to the classification error.",
"Intuitively, minimizing equation (REF ) favors solutions that not only have sparse representations compatible with observed features, but also providing reconstructions that are best separated in the given classes.",
"Figure: Toy example: (a) 4 out PP dictionary elements 𝐝 i \\mathbf {d}_i (atoms).",
"(b) Digits “3” and “8” can be represented by combining only two atoms in the dictionary (2-sparse representations).",
"(c) A left-half occluded digit “3” or “8” admits more than one 2-sparse representation (sum of 𝐝 1 o \\mathbf {d}^o_1 and 𝐝 2 o \\mathbf {d}^o_2, or 𝐝 3 o \\mathbf {d}^o_3 and 𝐝 4 o \\mathbf {d}^o_4).",
"(d) Linearly separable samples from two classes (A and B) having two features: 𝐱=[x 1 ,x 2 ]∈R 2 \\mathbf {x} = [x_1, x_2]\\in {R}^2 where incomplete observations are taken by observing only one feature.",
"Note that 𝐱 A \\mathbf {x}_A and 𝐱 B \\mathbf {x}_B belong to different classes but their observations are identical.",
"(e) Without using label information, the sequential method could lead to wrong reconstructions of data vectors, i.e.",
"𝐱 ^ A ≠𝐱 A \\hat{\\mathbf {x}}_A\\ne \\mathbf {x}_A and 𝐱 ^ B ≠𝐱 B \\hat{\\mathbf {x}}_B\\ne \\mathbf {x}_B making the set of reconstructed vectors not linearly separable."
],
[
"A sparsity-promoting sub-gradient optimization algorithm",
"To minimize the cost function in equation (REF ) we propose to alternate between the optimization over ${\\mathbf {s}}_i$ ($i=1,2,\\dots , I$ ) and $\\lbrace \\Theta ,\\mathbf {D}\\rbrace $ using the training dataset (incomplete).",
"For fixed $\\lbrace \\Theta ,\\mathbf {D}\\rbrace $ , the optimization with respect to ${\\mathbf {s}}_i$ is a non-smooth separable minimization sub-problem, which was extensively studied in the literature [31], [32].",
"In this sub-problem, the objective function is written as the sum of $F(\\Theta , \\mathbf {D}, {\\mathbf {s}}_i)$ and a non-smooth separable function $G({\\mathbf {s}}_i)$ , for which highly specialized, efficient and provable convergent solvers, namely the Coordinate Gradient Descent (CGD), already exists.",
"However, the following key differences in our setting makes it not suitable for the CGD approach: first, our function $F(\\Theta , \\mathbf {D}, {\\mathbf {s}}_i)$ involves evaluation of a multi-layer NN classifier, which can be non-smooth due to involved activation functions like ReLU or others; second, and more importantly, the computation of its second derivatives (Hessian) becomes prohibitive.",
"Therefore, we choose a simpler and standard first order (stochastic sub-gradient based) search of local minima with back-propagation.",
"We take the strategy similar to the heuristics used in [33].",
"To update ${\\mathbf {s}}_i$ , we need to subtract $\\sigma _{{\\mathbf {s}}} \\frac{\\partial J}{\\partial {\\mathbf {s}}_i}(j)$ from each coordinate $j$ provided that we do not cross zero in the process in order to avoid escaping from a region where $G({\\mathbf {s}}_i)$ is differentiable.",
"In such a case, we let the new value of ${\\mathbf {s}}_i(j)$ be exactly zero.",
"More specifically, we define $\\Delta _i(j)= -\\sigma _{{\\mathbf {s}}} \\frac{\\partial J}{\\partial {\\mathbf {s}}_i}(j)$ and, if ${\\mathbf {s}}_i(j)[{\\mathbf {s}}_i(j)+\\Delta _i(j)] < 0$ (zero crossing condition), we re-define $\\Delta _i(j)=-{\\mathbf {s}}_i(j)$ ; finally we update ${\\mathbf {s}}_i \\leftarrow {\\mathbf {s}}_i + \\Delta _i$ .",
"It is noted that, once a coefficient ${\\mathbf {s}}_i(j)$ reaches zero at a coordinate $j$ , it becomes fixed, in other words, sparsity of solution ${\\mathbf {s}}_i$ is monotonically increasing with iterations.",
"When ${\\mathbf {s}}_i$ is fixed, our problem is reduced to minimize $F(\\Theta , \\mathbf {D}, {\\mathbf {s}}_i)$ with respect to $\\Theta $ and $\\mathbf {D}$ , which is easily done by standard first order (stochastic gradient based) search of local minima.",
"The algorithm proposed for the training phase is presented as Algorithm REF .",
"In addition, for the testing phase, if the test dataset is incomplete, we need to find first the sparsest representation for the given observations, compute the reconstructions $\\hat{{\\mathbf {x}}}_i = \\mathbf {D}{\\mathbf {s}}_i$ and then apply the previously learned classifier to them as presented in Supp.",
"material, Algorithm REF .",
": Simultaneous classification and coding [1] $\\lbrace {\\mathbf {x}}^o_i, y_i\\rbrace $ , $i=1,2,\\dots , I$ , hyper-parameters $\\lambda _1$ and $\\lambda _2$ , $N_{iter}$ and update rates $\\sigma _{\\Theta }$ , $\\sigma _{\\mathbf {D}}$ and $\\sigma _{{\\mathbf {s}}}$ Weights $\\Theta $ and reconstructions $\\hat{{\\mathbf {x}}}_i= \\mathbf {D}{\\mathbf {s}}_i, \\forall i$ Randomly initialize $\\Theta , \\mathbf {D}, {\\mathbf {s}}_i, \\forall i$ $n\\le N_{iter}$ Fix ${\\mathbf {s}}_i$ , update $\\Theta $ and $\\mathbf {D}$ : $\\Theta = \\Theta - \\sigma _{\\Theta } \\frac{\\partial J}{\\partial \\Theta }$ $\\mathbf {D}= \\mathbf {D}- \\sigma _{\\mathbf {D}} \\frac{\\partial J}{\\partial \\mathbf {D}}$ Normalize columns of matrix $\\mathbf {D}$ Fix $\\Theta $ and $\\mathbf {D}$ , update ${\\mathbf {s}}_i$ , $\\forall i$ : $\\Delta _i= -\\sigma _{{\\mathbf {s}}} \\frac{\\partial J}{\\partial {\\mathbf {s}}_i}$ , $\\forall i$ ${\\mathbf {s}}_i(j)[{\\mathbf {s}}_i(j)+\\Delta _i(j)] < 0$ $\\Delta _i(j)=-{\\mathbf {s}}_i(j), \\forall (i, j)$ ; ${\\mathbf {s}}_i = {\\mathbf {s}}_i + \\Delta _i$ , $\\forall i$ return $\\Theta , \\mathbf {D}, {\\mathbf {s}}_i, \\hat{{\\mathbf {x}}}_i = \\mathbf {D}{\\mathbf {s}}_i, \\forall i$"
],
[
"Theoretical analysis",
"Here, we investigate about conditions under which a perfect classifier of the complete data can be obtained from incomplete data samples."
],
[
"Logistic regression",
"Let us first consider a logistic regression classifier [34] where the set of parameters $\\Theta = \\lbrace {\\mathbf {w}}, b\\rbrace $ are a vector ${\\mathbf {w}}\\in {R}^N$ and a scalar $b$ (bias).",
"A perfect classifier exists if there is a hyperplane that separates both classes, i.e., for each data vector ${\\mathbf {x}}_i$ : $f({\\mathbf {x}}_i) = \\langle {\\mathbf {w}}, {\\mathbf {x}}_i \\rangle + b > 0$ if $y_i=1$ , and $f({\\mathbf {x}}_i) \\le 0$ if $y_i=0$ .",
"We consider data samples admitting a $K$ -sparse representations ${\\mathbf {x}}_i = \\mathbf {D}{\\mathbf {s}}_i$ with dictionary $\\mathbf {D}\\in {R}^{N\\times P}$ having unit-norm columns.",
"We also assume an arbitrary pattern of missing features $\\mathcal {M}_i$ such that, data samples and dictionary are partitioned as $\\lbrace {\\mathbf {x}}^m_i,{\\mathbf {x}}^o_i \\rbrace $ and $\\lbrace \\mathbf {D}^m_i,\\mathbf {D}^o_i \\rbrace $ , respectively.",
"The following lemma identifies a sufficient condition under which, if we are able to train a classifier on incomplete observations such that the reconstructed data points are well separated by a hyperplane, then the same classifier correctly separates the original (unobserved) data vectors.",
"Lemma 2.1 (Sufficient condition type I) Suppose that we have obtained an alternative dictionary $\\mathbf {D}^{\\prime } \\ne \\mathbf {D}\\in {R}^{N\\times P}$ such that, for the incomplete observations ${\\mathbf {x}}^o_i \\in {R}^{M_i}$ , the $K$ -sparse representation solutions are non-unique, i.e.",
"$\\exists {\\mathbf {s}}_i, {\\mathbf {s}}_i^{\\prime } \\in \\Sigma _K^P$ such that ${\\mathbf {x}}^o_i = \\mathbf {D}^o_i{\\mathbf {s}}_i = \\mathbf {D}^{\\prime o}_i{\\mathbf {s}}_i^{\\prime }$ , where ${\\mathbf {s}}_i \\in {R}^P$ are the vectors of coefficients of the true data and ${\\mathbf {s}}_i^{\\prime }$ provides reconstructions $\\hat{{\\mathbf {x}}}_i = \\mathbf {D}^{\\prime }{\\mathbf {s}}_i^{\\prime }$ .",
"If a perfect classifier $\\lbrace {\\mathbf {w}}, b\\rbrace $ of the reconstructions $\\hat{{\\mathbf {x}}}_i$ exists s.t.",
"$|f(\\hat{{\\mathbf {x}}}_i)| > \\epsilon _i > 0$ and $\\epsilon _i > |\\langle {\\mathbf {w}}^m_i , {\\mathbf {e}}^m_i\\rangle |$ with ${\\mathbf {e}}^m_i = {\\mathbf {x}}^m_i - \\hat{{\\mathbf {x}}}^m_i$ , then the full data vectors ${\\mathbf {x}}_i$ are also perfectly separated with this classifier, in other words: $f({\\mathbf {x}}_i) = \\langle {\\mathbf {w}}_i, {\\mathbf {x}}_{i} \\rangle + b > 0$ ($\\le 0$ ) if $y_i =1$ ($y_i=0$ ).",
"By using the missing/observed partition and omitting the sample index $i$ , we can write: $f({\\mathbf {x}}) = \\langle {\\mathbf {w}}, {\\mathbf {x}}\\rangle + b = \\langle {\\mathbf {w}}^{o}, {\\mathbf {x}}^o \\rangle + \\langle {\\mathbf {w}}^{m}, {\\mathbf {x}}^m \\rangle + b$ .",
"If we add and subtract the term $\\langle {\\mathbf {w}}^{m}, \\hat{{\\mathbf {x}}}^m \\rangle $ on the right left hand, arrange terms and use the fact that $\\hat{{\\mathbf {x}}}^o = \\mathbf {D}^{\\prime o}{\\mathbf {s}}^{\\prime } = {\\mathbf {x}}^o$ , we get: $f({\\mathbf {x}}) = f(\\hat{{\\mathbf {x}}}) + \\langle {\\mathbf {w}}^{m}, {\\mathbf {e}}^m \\rangle .$ Since we assumed that $f(\\hat{{\\mathbf {x}}}) > \\epsilon >0$ (for $y_i = 1$ ) and $|\\langle {\\mathbf {w}}^{m}, {\\mathbf {e}}^m \\rangle | < \\epsilon $ , it implies that $f({\\mathbf {x}}) > 0$ .",
"Basically, condition (REF ) means requiring that reconstruction vector $\\hat{{\\mathbf {x}}}$ has a distance $\\epsilon $ to the separating hyperplane larger than the absolute dot product between ${\\mathbf {w}}$ and the residual $ {\\mathbf {e}}= {\\mathbf {x}}- \\hat{{\\mathbf {x}}}$ , which of course is true when the reconstruction is accurate, i.e.",
"${\\mathbf {x}}\\approx \\hat{{\\mathbf {x}}}$ .",
"However, in practice, reconstructions are not accurate so we are interested in conditions under which Lemma REF can still holds.",
"Below, we derive a more restrictive but useful sufficient condition: Proposition 2.1 (Sufficient condition type II) Under the same hypothesis of Lemma REF , the following condition is enough to guarantee a proper classifier trained on incomplete data: $\\epsilon > |\\langle {\\mathbf {w}}^{m}, {\\mathbf {x}}^m\\rangle | + |\\langle {\\mathbf {w}}^{m}, \\hat{{\\mathbf {x}}}^m \\rangle |.$ By using the fact that $|\\langle {\\mathbf {w}}^{m}, {\\mathbf {e}}^m \\rangle | = |\\langle {\\mathbf {w}}^{m}, {\\mathbf {x}}^m - \\hat{{\\mathbf {x}}}^m \\rangle | \\le |\\langle {\\mathbf {w}}^{m}, {\\mathbf {x}}^m \\rangle | + |\\langle {\\mathbf {w}}^{m}, \\hat{{\\mathbf {x}}}^m \\rangle |$ , and applying Lemma REF the proof is completed.",
"We highlight that, in our experiments, we were able to verify that Sufficient Condition type II is met in practice (see section , Fig.",
"REF ).",
"In Supp.",
"material section REF , we derive an additional sufficient condition based on the Restricted Isometry Property (RIP) of the dictionary $\\mathbf {D}$ and sparsity level $K$ , showing that sufficient condition (REF ) is easier to hold for datasets admitting highly sparse representations on dictionaries as close to orthogonal ones as possible."
],
[
"Multilayer-perceptron",
"Lemma REF can be straightforwardly generalized to multilayer-perceptron NNs where, if a softmax function is used at the output of the last layer then, as before, the prediction is based on the sign of the linear function: $\\small f({\\mathbf {x}}) = \\langle {\\mathbf {w}}, {\\mathbf {x}}^{(L)}\\rangle + b, \\mbox{ with } {\\mathbf {x}}^{(l)} = h\\left( \\mathbf {W}^T_l {\\mathbf {x}}^{(l-1)} + \\mathbf {B}_l\\right),$ $l=1,2,\\dots ,L$ , where $L+1$ is the total number of layers, $N_l$ is the number of neurons in layer $l$ , ${\\mathbf {w}}\\in {R}^{N_{L+1}}$ contains the weights in the last layer, $h(\\cdot )$ is an activation function, e.g.",
"ReLU, $\\mathbf {W}_l\\in {R}^{N_{l-1}\\times N_l}$ and $\\mathbf {B}_l \\in {R}^N_l$ contain the weights and biases associated to neurons at layer $l$ ; and ${\\mathbf {x}}^{(0)} = {\\mathbf {x}}$ is the input data vector.",
"In this case, the first layer matrix $\\mathbf {W}_1 \\in {R}^{N\\times N_1}$ can be partitioned into submatrices $\\mathbf {W}^o_{1i} \\in {R}^{M \\times N_1}$ and $\\mathbf {W}^m_{1i} \\in {R}^{(N-M) \\times N_1}$ according to the observed and missing input features, respectively.",
"Proposition 2.2 () Under the same conditions of Lemma REF , if a NN-classifier $\\lbrace {\\mathbf {W}_l,\\mathbf {B}_l} (l=1,2,\\dots L),{\\mathbf {w}}, b, h=ReLU\\rbrace $ of the reconstruction $\\hat{{\\mathbf {x}}}_i$ exists such that $\\epsilon _i > A \\max _j |\\langle \\mathbf {W}^m_{1i}(:,j) , {\\mathbf {e}}^m_i\\rangle | ,$ where $A = \\Vert {\\mathbf {w}}\\Vert \\prod _{l=2}^L \\Vert \\mathbf {W}_l \\Vert _2$ and ${\\mathbf {e}}^m_i = {\\mathbf {x}}^m_i - \\hat{{\\mathbf {x}}}^m_i$ , then the full data vector ${\\mathbf {x}}_i$ is also perfectly separated, in other words: $f({\\mathbf {x}}_i) > 0$ ($< 0$ ) if $y_i =1$ ($y_i=0$ ).",
"In the proof of Lemma REF , we were interested in finding a bound of the output error when the input ${\\mathbf {x}}$ of a classifier is perturbed, i.e.",
"we found conditions such that $|f({\\mathbf {x}}) - f({\\mathbf {x}}+ \\mathbf {\\delta })| < \\epsilon $ .",
"By generalizing the classifier to the case of a multilayer perceptron we can derive the proof as follows: Given a perturbation $\\mathbf {\\delta }^{(l-1)}\\in {R}^{N_l}$ at the input of layer $l-1$ , i.e.",
"$\\hat{{\\mathbf {x}}}^{(l-1)} = {\\mathbf {x}}^{(l-1)} + \\mathbf {\\delta }^{(l-1)}$ , it is propagated to the output of layer $l$ .",
"By writing the error at the output we obtain: $\\scriptstyle \\mathbf {\\delta }^{(l)} = h\\left( \\mathbf {W}_l^T {\\mathbf {x}}^{(l-1)} + \\mathbf {B}_l + \\mathbf {W}_l^T \\mathbf {\\delta }^{(l-1)} \\right) - h\\left( \\mathbf {W}_l^T {\\mathbf {x}}^{(l-1)} + \\mathbf {B}_l \\right),$ and, by using the sub-additivity of ReLU function $h(\\cdot )$ , i.e.",
"$h(a+b) \\le h(a) + h(b)$ , we derive the following entry-wise inequality: $\\small \\mathbf {\\delta }^{(l)} \\le h \\left( \\mathbf {W}_l^T \\mathbf {\\delta }^{(l-1)} \\right),$ and, by considering the property of ReLU activation function $\\Vert h({\\mathbf {x}})\\Vert \\le \\Vert {\\mathbf {x}}\\Vert $ , it turns out: $\\small \\Vert \\mathbf {\\delta }^{(l)}\\Vert \\le \\Vert \\mathbf {W}_l^T \\mathbf {\\delta }^{(l-1)} \\Vert ,$ Since the last layer of the NN is a linear classifier as in the case of Lemma REF , we can ask that $ \\langle {\\mathbf {w}}, \\mathbf {\\delta }^{(L)}\\rangle < \\epsilon $ .",
"Thus, by recursively using equation (REF ), we write $ \\langle {\\mathbf {w}}, \\mathbf {\\delta }^{(L)}\\rangle \\le \\Vert {\\mathbf {w}}\\Vert \\Vert \\mathbf {\\delta }^{(L)}\\Vert \\le \\Vert {\\mathbf {w}}\\Vert \\Vert \\mathbf {W}_L\\Vert _2 \\Vert \\mathbf {W}_{l-1}\\Vert _2 \\cdots \\Vert \\mathbf {W}_2\\Vert _2 \\Vert \\mathbf {\\delta }^{(1)} \\Vert .$ By defining $A = \\Vert {\\mathbf {w}}\\Vert \\prod _{l=2}^L\\Vert \\mathbf {W}_l\\Vert _2$ , evaluating equation (REF ) with $l=1$ and taking into account that perturbation at the input of first layer is $\\mathbf {\\delta }^{(0)} = {\\mathbf {e}}$ with ${\\mathbf {e}}^o = \\mathbf {0}$ , we arrive at: $\\small \\langle {\\mathbf {w}}, \\mathbf {\\delta }^{(L)}\\rangle \\le A \\Vert \\mathbf {W}_1^{mT} {\\mathbf {e}}^m\\Vert \\le A \\max _j |\\langle \\mathbf {W}^m_{1}(:,j) , {\\mathbf {e}}^m \\rangle | < \\epsilon ,$ which completes the proof.",
"It is interesting to note that $A=1$ is attained when unit-norm filters (columns of $\\mathbf {W}_l$ ) are orthogonal, which can be imposed by using orthogonality regularization [35]."
],
[
"Experimental results",
"We implemented all the algorithms in Pytorch 1.0.0 on a single GPU.",
"Implementation details are reported in Supplemental material, sections REF and REF .",
"The code is available at https://github.com/ccaiafa/SimultCodClass.",
"Synthetic datasets: We synthetically generated $I=11,000$ ($10,000$ training $+$ $1,000$ test) $K$ -sparse data vectors ${\\mathbf {x}}_i \\in {R}^{100}$ using a dictionary $\\mathbf {D}\\in {R}^{100 \\times 200}$ obtained from a Gaussian distribution with normalized atoms, i.e.",
"$\\Vert \\mathbf {D}(:,j)\\Vert =1, \\forall j$ .",
"A random hyperplane $\\lbrace {\\mathbf {w}}, b\\rbrace $ with ${\\mathbf {w}}\\in {R}^N$ , $b\\in {R}$ was randomly chosen dividing data vectors into two classes according to the sign of the expression $\\langle {\\mathbf {w}}, {\\mathbf {x}}_i \\rangle +b$ , which defined the label $y_i$ .",
"We also controlled the degree of separation between classes by discarding all data vectors with distances to the hyperplane lower than a pre-specified threshold, i.e.",
"$|\\langle {\\mathbf {w}}, {\\mathbf {x}}_i \\rangle +b| < d$ .",
"We used $n=10$ repetitions of each experiment with different masks and input data in order to compute statistics.",
"We applied our simultaneous method (Simult.)",
"with hyperparameters $\\lambda _1$ and $\\lambda _2$ in the cost function (REF ) tuned via cross-validation to train a logistic regression classifier on incomplete datasets with randomly distributed missing features.",
"Then, we computed the classification accuracy on the complete test dataset and compared the results against the following standard sequential methods: Sequential Sparsity based (Seq.",
"Sp.",
"): reconstructions are obtained by finding the sparsest representation compatible with the observations solving a LASSO problem.",
"We used Algorithm REF as shown in the Supp.",
"material; Zero Fill (ZF): missing features are filled with zeros, which is equivalent to ignore unknown values; Mean Unsupervised (MU): missing features are filled with the mean computed on the available values; Mean Supervised (MS): as in the previous case but the mean is computed on the same class vectors only; K-Nearest Neighbor (KNN): as in the previous case but the mean is computed on the K-nearest neighbors of the same class vectors only.",
"To compare the performance of classifiers, we computed the mean accuracy $\\pm $ standard error of the mean (s.e.m.",
"), with $n=10$ , on complete test datasets using all the methods for two levels of separation between classes ($d=0.0, 0.2$ ), two levels of sparsity ($K=4,32$ ) and missing features in the training dataset ranging from $25\\%$ to $95\\%$ as shown in Fig.",
"REF .",
"Our results show that the simultaneous algorithm clearly outperforms all the sequential methods.",
"A t-test was performed to evaluate the statistical significance with $p < 0.05$ of the difference between our algorithm and MS.",
"It is interesting to note that, when classes has some degree of separation ($d=0.2$ ), using the simple MS method, can give good results but not better than our algorithm.",
"Figure: Experimental results on synthetic dataset with random masks using our algorithm (red) and compared to various sequential methods.",
"Test accuracy (mean ±\\pm s.e.m with n=10n=10) is shown as a function of the percentage of missing features for separation of classes d=0.0,0.2d=0.0, 0.2 and levels of sparsity K=4,32K=4,32.",
"Statistical significance for the difference between Simult.",
"and MS is shown (p<0.05p < 0.05).In the second experiment, we generated $I =10,000$ $K$ -sparse data vectors ${\\mathbf {x}}_i \\in {R}^{100}$ using $\\mathbf {D}\\in {R}^{100 \\times 100}$ and we evaluated the sufficient condition of equation (REF ) on $n=10$ repetitions of the experiment with $95\\%$ missing features and separation $d=0.0$ .",
"Fig.",
"REF clearly shows that the sufficient condition is mostly met in practice, especially for highly sparse representations of input data (small $K$ ).",
"This means that in practice it is not necessary to accurately reconstruct the input vectors, it is enough to capture the intrinsic characteristics of the classes such that the distances of reconstructions to the separating hyperplane satisfy the sufficient condition (REF ).",
"Figure: Verification of the sufficient condition () for various levels of sparsity KK: 2D-histogram of ϵ\\epsilon versus g=|〈𝐰 m ,𝐱 m 〉|+|〈𝐰 m ,𝐱 ^ m 〉|g = |\\langle {\\mathbf {w}}^{m}, {\\mathbf {x}}^m\\rangle | + |\\langle {\\mathbf {w}}^{m}, \\hat{{\\mathbf {x}}}^m \\rangle |.",
"Mean + s.e.m (n=10n=10) percentage of correctly classified data samples are shown for ϵ>g\\epsilon > g and ϵ<g\\epsilon < g.Benchmark datasets: We also considered three popular computer vision datasets: MNIST [36] and Fashion [37] consisting of 70,000 images (60,000 train + 10,000 test) each; and CIFAR10 [38] having 60,000 images (50,000 train + 10,000 test).",
"MNIST/Fashion datasets contains $28\\times 28$ gray scale images while CIFAR10 dataset is built upon $32\\times 32\\times 3$ color images of different objects.",
"The corresponding data sample size is $N=28\\times 28 = 784$ for MNIST/Fashion and $N=32\\times 32\\times 3 = 3,072$ for CIFAR10.",
"We considered a dictionary of size $784\\times 784$ (MNIST/Fashion) and $1,024\\times 1,024$ (CIFAR10) and applied our simultaneous algorithm to learn the classifier on incomplete data using uniform random missing masks with several levels of missing data (25%, 50% and 75%) and 50% for random partial occlusions with MNIST/Fashion.",
"We used a logistic regression classifier (single layer NN) and a 4-layer convolutional neural network [39] (CNN4) for the MNIST/Fashion dataset using batch normalization (BN) [40] in the Fashion dataset.",
"For CIFAR10 dataset, an 18-layer residual neural network, Resnet-18 [41] was implemented.",
"We did not use any data augmentation strategy.",
"The hyper-parameters $\\lambda _1$ and $\\lambda _2$ in cost function (REF ) were adjusted by cross-validation through a grid-search, as shown in Supp.",
"material (Table REF and Fig.",
"REF ).",
"We compared our proposed algorithm with the following standard sequential methods: ZF, MS, KNN-10, KNN-20, KNN-50 and KNN-100; and against the recently proposed method from [15], referred here as NN-GMM, which uses the same NN classifier as in our method and models missing features through GMMhttps://github.com/lstruski/Processing-of-missing-data-by-neural-networks.",
"We trained the classifiers on incomplete data with random masks and tested them on complete data for MNIST and CIFAR10 datasets.",
"The obtained mean Test Accuracy $\\pm $ s.e.m ($n=10$ ) are reported in Table REF .",
"It is noted that NN-GMM provided good results with MNIST dataset compared to sequential methods, however, our simultaneous method outperformed all the methods.",
"Interestingly, NN-GMM performed worst than any other method with CIFAR10 dataset.",
"It seems that NN-GMM is not robust to large amount of missing data because, when we reduced the missing entries to $10\\%$ , the test accuracy sensibly increased to $52.57\\%$ .",
"Additionally, our method showed to have little variability (small s.e.m) compared to the second best method (NN-GMM for MNIST and ZF for CIFAR10).",
"Table: Test accuracy (mean ±\\pm s.e.m with n=10n=10) of various methods trained on incomplete data and tested on complete ones for MNIST and CIFAR10.In Table REF , test accuracies obtained when the learned model is applied to incomplete and complete test data, are shown.",
"The right-most column shows the baseline results obtained by training the model on complete datasets using a CNN4 [39] and a Resnet-18 [41], whose implementations can be found at https://github.com/pytorch/examples/tree/master/mnist and https://github.com/kuangliu/pytorch-cifar.",
"It is interesting to note that for the logistic regression classifier, we obtained better results when training with incomplete data rather than using complete data.",
"Also, it is highlighted that training on incomplete data with 50% or fewer random missing features, provides similar test accuracy as training on complete data for MNIST dataset.",
"This could be explained by noting two facts: (1) random missing features is similar to applying dropout, with the exception that missing data do not change during training; and (2) our model has more parameters (Dictionary + sparse coefficients + Linear layer) compared to the baseline logistic regression classifier.",
"To provide a deeper understanding of this effect, we ran the baseline with Dropout at the input and we obtained: 91.95%, 91.97% and 92.01% for $p=0.0$ , $0.1$ and $0.25$ , respectively, which shows that the improvement we obtained with our method is not solely caused by a dropout alike behavior.",
"Table: Test Accuracies obtained with our method on MNIST, Fashion and CIFAR10 datasets training with incomplete data and testing on incomplete/complete data.",
"Baseline results obtained by training the models on complete data are shown for reference in the right-most column.In Fig.",
"REF , we present some randomly selected visual examples comparing the original images in the MNIST/Fashion test dataset, their observations using random masks and partial occlusions, and the reconstructions using the dictionary learned from the incomplete training data.",
"It is clear that, despite the reconstructions may be not very similar to the original images (see “5” digit example), they clearly own the properties of the class to which they belong to.",
"Additional examples are provided in Supp.",
"material (Figs.",
"REF and REF ).",
"Figure: Original (top), observed (middle) and reconstructed (bottom) MNIST and Fashion test images."
],
[
"Discussion",
"It is well known that sparse coding has the ability to accurately model complex distributions of data, such as natural signals (images, audio, EEG, etc).",
"In this work, we demonstrated that assuming a sparse representation for input data allows for the successful training of a general NN when incomplete data is given outperforming traditional sequential approaches and other start-of-the-art methods.",
"It is highlighted that our method can be used with potentially any deep NN architecture, thus relying on their extraordinary capability to accommodate complex decision boundaries as usually needed in modern machine learning.",
"Our method overcomes well known issues of previous approaches: (1) compared to imputation methods, our algorithm successfully incorporates the labelling information into the modeling of missing features; (2) sparse coding allows for a simple way to train dictionaries through linear methods such as stochastic gradient descent with back-propagation compared to the very expensive EM estimators for GMM used in probabilistic generative models, or SVD based algorithms for matrix rank minimization in matrix completion; (3) sparse coding can be more accurate modeling missing values in natural signals compared to GMM, especially for high dimensional data where GMM may require a huge number of parameters making it computationally prohibitive.",
"We analyzed the limitations of the classical imputation approach and demonstrated through experiments with synthetical and real-world datasets that our simultaneous algorithm always outperforms them for various cases such as LASSO, zero-filling, supervised/unsupervised mean and KNN based methods as well as the state-of-the-art method based on NNs and GGM recently proposed in [15].",
"Nevertheless, our experimental results on synthetic and real-world dataset showed that, even though we only constrained dictionaries to have unit-norm columns but not enforcing any other kind of constraint like maximum coherence, the obtained results seem to be satisfactory enough.",
"However, further analysis on the required properties of dictionaries could provide deeper insights and alternative ways to improve the algorithm, which we aim to address in a future work.",
"While current simple sub-gradient based optimization approach provided satisfactory results in terms of performance, it is remarked that observed convergence is slow requiring a thousand of iterations sometimes.",
"We believe, it could be improved by trying to incorporate some second-order derivatives information for computing the updates.",
"Although, full Hessian computation becomes prohibitive with multi-layer NNs a diagonal approximation approach could be explored.",
"Also, a rigorous convergence analysis in the line of the analysis in [31], [32] and taking special properties of multi-layer NN classifier functions can be conducted in a future work.",
"Finally, we provided theoretical insights of the problem by providing sufficient conditions under which, if it is possible to train a classifier on incomplete observations so that its reconstructions are well separated by a hyperplane, then the same classifier also correctly separates the original (unobserved) data samples.",
"Acknowledgments: We are thankful for the RAIDEN computing system and its support team at RIKEN AIP, Tokyo.",
"This work was supported by the JSPS KAKENHI (Grant No.",
"20H04249, 20H04208)."
],
[
"Additional pseudocodes",
"Here, additional pseudocode of the algorithms discussed in the paper are provided.",
"Once the classifier is trained by using Algorithm REF , we are able to apply it to incomplete test data by using Algorithm REF , where for fixed $\\Theta $ and $\\mathbf {D}$ , we need to find the corresponding sparse coefficients ${\\mathbf {s}}_i$ , compute the full data vector estimations and, finally, apply the classifier.",
"A sparsity-based sequential method is presented in Algorithm REF (sequential approach), which consists on learning first the optimal dictionary $\\mathbf {D}$ and sparse coefficients ${\\mathbf {s}}_i$ compatible with the incomplete observations (dictionary learning and coding phase), followed by the training phase, where the classifier weights are tuned in order to minimize the classification error of the reconstructed input data vectors $\\hat{{\\mathbf {x}}}_i = \\mathbf {D}{\\mathbf {s}}_i$ .",
"It is noted that for the imputation stage (lines 2-12) other and more specialized dictionary learning algorithms with missing data can be applied, such as the ones proposed in [42] for high-dimensional data or [43] for color image data.",
": Testing on incomplete data [1] Incomplete data vectors $\\lbrace {\\mathbf {x}}^o_i\\rbrace $ , $i=1,2,\\dots , I$ , classifier parameters $\\Theta $ , dictionary $\\mathbf {D}$ , hyper-parameters $\\lambda _1$ and $\\lambda _2$ , number of iterations $N_{iter}$ and update rate $\\sigma _{{\\mathbf {s}}}$ ${\\hat{y}_i}$ and reconstructions $\\hat{{\\mathbf {x}}}_i= \\mathbf {D}{\\mathbf {s}}_i, \\forall i$ Sparse coding stage: for fixed dictionary $\\mathbf {D}$ find sparse representations of observations ${\\mathbf {x}}^o_i$ Initialize ${\\mathbf {s}}_i, \\forall i$ randomly $n\\le N_{iter}$ $\\Delta _i= -\\sigma _{{\\mathbf {s}}} \\big [\\lambda _1 \\frac{\\partial J_1}{\\partial {\\mathbf {s}}_i} + \\lambda _2 \\frac{\\partial J_2}{\\partial {\\mathbf {s}}_i}\\big ]$ , $\\forall i$ ${\\mathbf {s}}_i(j)[{\\mathbf {s}}_i(j)+\\Delta _i(j)] < 0$ $\\Delta _i(j)=-{\\mathbf {s}}_i(j)$ ; avoid zero crossing ${\\mathbf {s}}_i = {\\mathbf {s}}_i + \\Delta _i$ , $\\forall i$ $ \\hat{{\\mathbf {x}}}_i = \\mathbf {D}{\\mathbf {s}}_i, \\forall i$ ; Compute reconstructions Classification stage: apply classifier to reconstructions $\\hat{{\\mathbf {x}}}_i$ $\\hat{y}_i = \\operatornamewithlimits{arg\\,max}_y (p^{y}_{\\Theta }(\\hat{{\\mathbf {x}}}_i))$ return $\\Theta , \\hat{y}_i, {\\mathbf {s}}_i, \\hat{{\\mathbf {x}}}_i, \\forall i$ : Sequential sparsity based approach [1] Incomplete data vectors and their labels $\\lbrace {\\mathbf {x}}^o_i, y_i\\rbrace $ , $i=1,2,\\dots , I$ , hyper-parameters $\\lambda _1$ and $\\lambda _2$ , number of iterations $N_{iter}$ and update rate $\\sigma _{\\Theta }$ , $\\sigma _{\\mathbf {D}}$ and $\\sigma _{{\\mathbf {s}}}$ Classifier weights $\\Theta $ and reconstructions $\\hat{{\\mathbf {x}}}_i= \\mathbf {D}{\\mathbf {s}}_i, \\forall i$ Randomly initialize $\\mathbf {D}, {\\mathbf {s}}_i, \\forall i$ Imputation stage: learning of $\\mathbf {D}$ and ${\\mathbf {s}}_i$ $n\\le N_{iter}$ $\\mathbf {D}= \\mathbf {D}- \\sigma _{\\mathbf {D}} \\frac{\\partial J_1}{\\partial \\mathbf {D}}$ Normalize columns of matrix $\\mathbf {D}$ $\\Delta _i= -\\sigma _{{\\mathbf {s}}} \\big [\\lambda _1 \\frac{\\partial J_1}{\\partial {\\mathbf {s}}_i} + \\lambda _2 \\frac{\\partial J_2}{\\partial {\\mathbf {s}}_i}\\big ]$ , $\\forall i$ ${\\mathbf {s}}_i(j)[{\\mathbf {s}}_i(j)+\\Delta _i(j)] < 0$ $\\Delta _i(j)=-{\\mathbf {s}}_i(j)$ ; avoid zero crossing ${\\mathbf {s}}_i = {\\mathbf {s}}_i + \\Delta _i$ , $\\forall i$ $ \\hat{{\\mathbf {x}}}_i = \\mathbf {D}{\\mathbf {s}}_i, \\forall i$ ; Compute reconstructions Training stage: update $\\Theta $ $n\\le N_{}$ $\\Theta = \\Theta - \\sigma _{\\Theta } \\frac{\\partial J_0}{\\partial \\Theta }$ ; return $\\Theta , \\mathbf {D}, {\\mathbf {s}}_i, \\hat{{\\mathbf {x}}}_i = \\mathbf {D}{\\mathbf {s}}_i, \\forall i$"
],
[
"A condition based on RIP and sparsity",
"The Restricted Isometry Property (RIP): An overcomplete dictionary $\\mathbf {D}$ satisfies the RIP of order $K$ if there exists $\\delta _K\\in [0,1)$ s.t.",
"$(1-\\delta _K) \\Vert {\\mathbf {s}}\\Vert ^2_2 \\le \\Vert \\mathbf {D}{\\mathbf {s}}\\Vert ^2_2 \\le (1+\\delta _K) \\Vert {\\mathbf {s}}\\Vert ^2_2,$ holds for all ${\\mathbf {s}}\\in \\Sigma _K^P$ .",
"RIP was introduced in [30] and characterizes matrices which are nearly orthonormal when operating on sparse vectors.",
"In the following theorem, we show that, by imposing conditions on the sparsity level of the representation and the RIP constant of a sub-matrix of the dictionary, we can guarantee to meet the sufficient condition (REF ).",
"Theorem 5.1 Given a dataset $\\lbrace {\\mathbf {x}}_i, y_i\\rbrace $ , $i =1,2,\\dots , I$ with normalized data vectors ($\\Vert {\\mathbf {x}}_i\\Vert \\le 1$ ) admitting a $K$ -sparse representation over a dictionary $\\mathbf {D}\\in {R}^{N\\times P}$ with unit-norm columns, whose sub-matrices $\\mathbf {D}^m_i$ satisfy the RIP of order $K$ with constant $\\delta ^i_K$ , and suppose that, we have obtained an alternative dictionary $\\mathbf {D}^{\\prime }\\in {R}^{N\\times P}$ , whose sub-matrices $\\mathbf {D}^{\\prime m}_i$ also satisfy the RIP of order $K$ with constant $\\delta ^i_K$ such that, for the incomplete observation ${\\mathbf {x}}^o_i \\in {R}^{M_i}$ , the $K$ -sparse representation solution is non-unique, i.e.",
"$\\exists {\\mathbf {s}}_i, {\\mathbf {s}}_i^{\\prime } \\in \\Sigma _K^P$ such that ${\\mathbf {x}}^o_i = \\mathbf {D}^o_i{\\mathbf {s}}_i = \\mathbf {D}^{\\prime o}_i{\\mathbf {s}}_i^{\\prime }$ , where ${\\mathbf {s}}_i \\in {R}^P$ is the vector of coefficients of the true data, i.e.",
"${\\mathbf {x}}_i = \\mathbf {D}{\\mathbf {s}}_i$ and ${\\mathbf {s}}_i^{\\prime }$ provides a plausible reconstruction through $\\hat{{\\mathbf {x}}}_i = \\mathbf {D}^{\\prime }{\\mathbf {s}}_i^{\\prime }$ with $\\Vert \\hat{{\\mathbf {x}}}_i\\Vert \\le 1$ .",
"If a perfect classifier $\\lbrace {\\mathbf {w}}, b\\rbrace $ of the reconstruction $\\hat{{\\mathbf {x}}}_i$ exists such that $|f(\\hat{{\\mathbf {x}}})| = |\\langle {\\mathbf {w}}, \\hat{{\\mathbf {x}}} \\rangle + b |> \\epsilon _i >0$ and $\\epsilon _i > 2 \\Vert {\\mathbf {w}}^m_i\\Vert _1\\sqrt{\\frac{K}{1-\\delta ^i_K}},$ then the full data vector ${\\mathbf {x}}_i$ is also perfectly separated with this classifier, in other words: $f({\\mathbf {x}}_i) = \\langle {\\mathbf {w}}_i, {\\mathbf {x}}_{i} \\rangle + b > 0$ ($\\le 0$ ) if $y_i =1$ ($y_i=0$ ).",
"Let us prove that the sufficient condition (REF ) is met under the hypothesis of Theorem REF .",
"Taking into account that ${\\mathbf {x}}_i^m = \\mathbf {D}_i^m {\\mathbf {s}}_i$ , we can write $\\small | \\langle {\\mathbf {w}}^{m}, \\mathbf {D}_i^m{\\mathbf {s}}_i \\rangle | & = \\Big |\\sum _{j=1}^{N-M_i} {\\mathbf {w}}^m(j) \\sum _{n=1}^N\\mathbf {D}_i^m(j,n) {\\mathbf {s}}_i(n) \\Big | \\nonumber \\\\& \\le \\sum _{j=1}^{N-M_i} |{\\mathbf {w}}^m(j)| \\sum _{n=1}^N |\\mathbf {D}_i^m(j,n)| |{\\mathbf {s}}_i(n)|.$ Since we assumed normalized vectors $\\Vert {\\mathbf {x}}_i\\Vert \\le 1$ , by applying the left-hand side of the RIP we obtain: $\\Vert {\\mathbf {s}}_i\\Vert \\le 1/\\sqrt{1-\\delta ^i_K}$ , and, taking into account that $\\Vert {\\mathbf {s}}_i\\Vert _1 \\le \\sqrt{K}\\Vert {\\mathbf {s}}_i\\Vert $ and $ |\\mathbf {D}_i^m(j,n)| \\le 1$ (columns of $\\mathbf {D}$ are unit-norm), we obtain: $\\small | \\langle {\\mathbf {w}}^{m}, \\mathbf {D}_i^m{\\mathbf {s}}_i \\rangle | \\le \\sqrt{\\frac{K}{1-\\delta ^i_K}} \\sum _{j=1}^{N-M_i} |{\\mathbf {w}}^m(j)| = \\sqrt{\\frac{K}{1-\\delta ^i_K}}\\Vert {\\mathbf {w}}^m_i\\Vert _1.$ Similarly, using $\\hat{{\\mathbf {x}}_i}^m = \\mathbf {D}_i^{\\prime m} {\\mathbf {s}}_i^{\\prime }$ , we can obtain that $| \\langle {\\mathbf {w}}^{m}, \\mathbf {D}_i^{\\prime m}{\\mathbf {s}}_i^{\\prime } \\rangle | \\le \\sqrt{\\frac{K}{1-\\delta ^i_K}}\\Vert {\\mathbf {w}}^m_i\\Vert _1.$ Putting equations (REF ) and (REF ) together with equation (REF ) complete the proof of the sufficient condition (REF )."
],
[
"Implementation",
"We implemented all the algorithms in Pytorch 1.0.0 on a single GPU.",
"The code is available athttps://github.com/ccaiafa/SimultCodClass.. Initializations of dictionary $\\mathbf {D}$ and coefficients ${\\mathbf {s}}_i$ were made at random.",
"However, we think some improvements in convergence could be achieved by using some dedicated dictionaries such as the case of Wavelet or Cosine Transform matrices.",
"To update NN weights ($\\Theta $ ), we used standard Stochastic Gradient Descent (SGD) with learning rate $\\sigma _{\\Theta }$ and momentum $m$ , while for updating dictionary $\\mathbf {D}$ and vector coefficients ${\\mathbf {s}}_i$ , we used fixed update rate $\\sigma = \\sigma _{\\mathbf {D}} = \\sigma _{{\\mathbf {s}}}$ .",
"It is noted that we used different update rates for training and testing stages.",
"In Table REF , we report the settings used for experiments for MNIST and CIFAR10 datasets, which includes Number of iterations $N_{iter}$ , batch size $bs$ , learning rate $\\sigma _{\\Theta }$ , momentum $m$ , update rate $\\sigma $ (training and test)."
],
[
"Hyperparameter tuning",
"In Table REF we present the results of the grid search for hyper-parameter tuning on MNIST and CIFAR10 datasets.",
"We fit our model to the training dataset for a range of values of parameters $\\lambda _1$ and $\\lambda _2$ and apply it to a validation data set.",
"Figure REF shows the validation accuracy obtained with different classifiers and levels of missing entries for MNIST dataset.",
"Table: Experimental settings for MNIST and CIFAR10 datasets: Number of iterations N iter N_{iter}, batch size bsbs, learning rate σ Θ \\sigma _{\\Theta }, momentum mm, update rate σ\\sigma (training and test)"
],
[
"Additional visual results",
"To visually evaluate our results, additional randomly selected examples of original (complete) images of the test dataset in MNIST and Fashion, together with their given incomplete observations and obtained reconstructions, are shown in Figure REF and Figure REF Table: Hyper-parameter tuning: crossvalidated hyperparameters λ 1 \\lambda _1 and λ 2 \\lambda _2 obtained for MNIST and CIFAR10 datasets with the classifiers used in our experiments.Figure: Test accuracy in the grid search for hyper-parameter tuning in MNIST dataset: λ 1 \\lambda _1 and λ 2 \\lambda _2 were tuned by cross-validation for various levels of missing entries: 25%, 50% and 75%.Figure: Reconstructions of incomplete test MNIST dataset images by applying our simultaneous classification and coding algorithm with the CNN4 architecture.Figure: Reconstructions of incomplete test Fashion dataset images by applying our simultaneous classification and coding algorithm with the CNN4 architecture with Batch Normalization (BN)."
]
] | 2011.14047 |
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